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Virginia Tech | 3.5 Proof-of-Concept Testing
Several laboratory-scale tests were conducted to evaluate the potential of the HydroFloat
separator for upgrading several types of mineral samples. These samples included mineral
sands, phosphate matrix, carbonaceous slag and coal. Conventional flotation tests, whose results
were used as baseline data in some investigations, were conducted using a laboratory flotation
cell (Denver Model D-12). Aerated teeter-bed investigations were conducted with a laboratory-
scale HydroFloat cell operated in two different modes, i.e., batch and continuous. The test unit
was fabricated from Plexiglas with an open area of approximately 50 cm2. This test unit is the
same device described previously in Chapter 2 (Figure 2.4) except that a static in-line mixer was
added to the elutriation water line. Using compressed air and a glycol frothing agent in
conjunction with the static mixer allowed for the creation of small bubbles which were dispersed
throughout the teeter-bed via the elutriation piping network.
3.5.1 Phosphate Recovery
3.5.1.1 Testing of a North Florida Phosphate Matrix
A sample of a run-of-mine north Florida phosphate matrix (5 mm x 65 mesh) was scalped
at 3 mm to remove the extreme oversize and debris material. The feed ore was classified to
remove the minus 35 mesh fines and conditioned with a fatty acid-diesel fuel mixture at a dosage
of approximately 0.50 kg/t (active fatty acid). In all tests, ammonium hydroxide was used for pH
control and a polyglycol frother was added to stabilize the bubble suspension. Batch HydroFloat
tests were conducted and compared against conventional flotation results.
Figure 3.11 compares the results of the conventional flotation tests with those obtained
using the HydroFloat cell operated in the batch mode. In this series of tests, the HydroFloat cell
133 |
Virginia Tech | achieved a BPL recovery of nearly 95% compared to less than 79% for the conventional cell.
This represents an increase in BPL recovery of more than 20%. Although the conventional cell
was floated to exhaustion, higher recoveries were not possible since many of the coarser particles
remained unfloatable. Furthermore, the recovery improvement was achieved while maintaining
a slighter higher concentrate grade (67.0% versus 65.7% BPL). The insol content of the
HydroFloat concentrate was also lower (7.2% versus 7.7% insols). In fact, the test data show
that the batch HydroFloat cell produced concentrates with a lower insol content over the entire
range of BPL recoveries. The unexpected improvement in the quality of the concentrate
produced by the HydroFloat cell has been attributed to the incremental recovery of very high-
grade coarse particles that could not be floated by the conventional flotation technique.
BPL
Recovery
(%)
Grade
(% BPL)
Insol (%) Conventional
HydroFloat
0 20 40 60 80 100
Figure 3.11 - Comparison of separation data for batch test units for a phosphate matrix.
134 |
Virginia Tech | 3.5.1.2 Testing of a Central Florida Phosphate Matrix
Considering the favorable results of the northern Florida phosphate sample, another
sample of run-of-mine central Florida phosphate matrix (5 mm x 65 mesh) was investigated. It
was deemed necessary in order to prove that these were not ore-specific results. Again, the feed
was scalped at 3 mm to remove the extreme oversize and debris material. In this second
phosphate investigation, two samples were tested. One sample was classified to remove the
minus 35 mesh fines while the other was classified to remove the minus 28 mesh fines. Each
sample was conditioned with a fatty acid-diesel fuel mixture at a dosage of approximately 0.50
kg/t (active fatty acid). For all tests, ammonium hydroxide was used for pH control and a
polyglycol frother was added to stabilize the bubble suspension. For each feed sample, several
tests were conducted in an effort to produce a grade and recovery curve.
Figure 3.13 compares the results of how each of the feeds responded to HydroFloat
testing. Both the plus 35 and plus 28 mesh phosphate matrix responded extremely well to the
aerated hindered-bed separator. It can be seen that the high BPL recoveries (95%) could be
maintained at extremely low insol grades (5-10%). It can also be seen that the plus 28 mesh feed
responded slightly better than the plus 35 mesh material as indicated by the higher separation
curve. This is most likely due to the elutriation water misplacing some of the fine silica particles
that were present in the finer feed sample. A lower water rate may have improved this result.
The optimum results for these tests are presented in Figure 3.14. The BPL content of the
plus 28 and plus 35 mesh product were 69.5 and 68.5%, respectively. These results were
produced at a BPL recovery of 93%. At these high recoveries, the HydroFloat was able to
maintain low insol grades of 5.0% and 6.3% for the coarse and fine feed material, respectively.
136 |
Virginia Tech | 3.5.2 Coal/Carbon Recovery
3.5.2.1 Testing of Anthracite Slag
An industrial slag sample (nominally 6 mm x 200 mesh) was screened at 6.35 mm to
remove oversize tramp material prior to testing. The feed sample contained » 12% fixed carbon
in the form of anthracite coal and » 30% Fe O and » 26% TiO . This sample was ideally suited
2 3 2
for HydroFloat treatment due to the inherent hydrophobicity of the low density component
(anthracite). The objective of these tests was to recover the remaining fixed carbon at a product
quality greater than 80%. Only seven tests were conducted on this sample due to the small
amount of available material. Four tests were conducted utilizing the lab-scale HydroFloat with
full teeter-bed aeration. Three were conducted with the HydroFloat operating as a traditional
hindered-bed separator (i.e., no teeter-bed air was employed).
Figure 3.15 shows the product grade and recovery plot for the plus 28 mesh fraction of
the feed material. Without aeration, the hindered-bed separator was able to make a separation,
although a product quality of over 71% fixed carbon could not be achieved. In contrast, a quality
of over 80% could be achieved when the teeter-bed was aerated. Aeration allowed the product
recovery to be increased by an overage of 10-15%, while simultaneously improving the fixed
carbon content of the product.
138 |
Virginia Tech | 100
80
60
40
20
0
0 20 40 60 80 100
Fixed Carbon (%)
139
)%(
yrevoceR
nobraC
dexiF With Air
Without Air
Figure 3.15 - Testing of anthracite slag using the HydroFloat separator (6.35 mm x 200 mesh).
3.5.2.2 Testing of Central Appalachian Coal
A sample of run-of-mine coal from central Appalachia was used to evaluate the
effectiveness of the HydroFloat separator in treating 2 mm x 0.15 mm coal from an existing
spiral circuit. The feed coal was classified to remove the minus 100 mesh fines and conditioned
with approximately 0.25 kg/t of diesel fuel to enhance particle hydrophobicity. In the first series
of comparison tests, the HydroFloat separator was operated without the addition of air. The
separation performance achieved in this mode of operation would be identical to that obtained
using a traditional hindered-bed separator. In the second series of tests, the HydroFloat was
operated with air bubbles added to the teeter-bed. In this case, approximately 0.1 kg/t of
polyglycol frother was injected into the teeter water to improve air dispersion and minimize
bubble coalescence. |
Virginia Tech | Figure 3.16 shows the recovery-ash curves comparing the performance of the HydroFloat
and hindered-bed separators. For convenience, the data have been reported for both the coarse
(plus 50 mesh) and fine (minus 50 mesh) size fractions. As expected, both devices achieved
good recoveries (>90%) of the minus 50 mesh material. The HydroFloat separator also produced
good recoveries of the plus 50 mesh material. Combustible recoveries in the range of 87-97%
were readily attainable over a wide range of operating conditions. In contrast, the hindered-bed
separator was not able to achieve recoveries greater than about 75% for the plus 50 mesh
material. Attempts were made to improve the recovery of the plus 50 mesh particles by
increasing the flow rate of the fluidization water or by raising the level of the teeter-bed.
However, these attempts generally produced unacceptably high ash products due to (i) short-
circuiting of mineral matter into the product launder and (ii) excessive turbulence within the
teeter-bed. Since more of the feed mass resided in the plus 50 mesh fraction (approximately
60%), the overall performance of the HydroFloat was far superior to that of the hindered-bed
separator in treating the overall 2 x 0.15 mm sample. As shown in Figure 3.17, the recoveries
obtained for the overall feed with the addition of air were approximately 20 percentage points
higher than those obtained without air injection.
140 |
Virginia Tech | 3.5.2.3 Testing of Australian Coal
A sample of froth flotation feed from an Australian coal plant was obtained for testing in
the HydroFloat separator. This sample was considered very coarse for traditional flotation
processes, with over 25% of the coal greater than 0.6 mm (28 mesh). Traditionally, coal
flotation suffers from low recoveries above 28 mesh. It was expected that the HydroFloat could
recover the coarse feed material that was currently being lost in the froth flotation circuit. The
feed to the unit was conditioned with diesel (0.25 kg/ton) and a glycol frother was used for
bubble generation.
Table 3.1 provides a summary of the size-by-size recoveries and qualities obtained using
the HydroFloat separator. Special attention was paid to the effect of particle size since the teeter
water could easily propel fine ash particles (i.e., clay and mineral matter) into the concentrate.
As shown, the preliminary data indicate the plus 28 mesh material could be cleaned to an
acceptable ash content with combustible recoveries well above 90%. However, the ash content of
the finer size fractions (below 65 mesh) increased sharply due to the carry-over of fine mineral
matter. To overcome this problem, a second series of tests were conducted with the lab-scale
HydroFloat in which the minus 65 mesh material was discarded. In this case, the HydroFloat
was capable of combustible recoveries greater than 90% when air was added to the teeter-bed.
In comparison, an average recovery of only 83% was achieved without the addition of air.
However, the separator was capable of producing a higher quality (lower ash) product when no
air was added. The higher ash values obtained using the HydroFloat can be attributed to the
increased recovery of coarse high density middlings that report to the product stream when the
teeter-bed is aerated.
142 |
Virginia Tech | Table 3.1 - Size-by-size HydroFloat results obtained using an Australian coal.
Sample Yield (%) Recovery (%) Product
Mesh Per Size Cumulative Per Size Cumulative Ash (%)
Class Class
+16 85.0 0.5 94.4 0.8 8.4
16 x 20 80.4 8.0 93.1 12.0 13.8
20 x 28 75.8 28.2 91.7 41.6 16.5
28 x 65 61.4 61.6 86.3 83.9 21.2
-65 63.3 70.3 81.4 90.2 25.8
3.5.2.4 Testing of Heavy Mineral Sands
Two samples of heavy mineral sands were tested using the HydroFloat separator. The
first sample contained unwanted carbonaceous matter as well as an undesirably high pyrite
content (average 0.78% sulfur). To process this sample, the feed was first treated with sodium
isopropyl Xanthate to make the pyrite hydrophobic and then passed through the HydroFloat unit.
As shown in Figure 3.18, the HydroFloat separator achieved sulfur and carbon rejections of up to
80% and 55%, respectively. These rejections were maintained at a high product yield of
approximately 97%. The second sample also contained an unacceptably high carbon content
(average 0.92% fixed carbon). However, no Xanthate was added for this sample since the sulfur
content was already within product specifications. Figure 3.19 shows that the HydroFloat
separator was also effective in treating this sample. More than 81% of the carbon was removed
from the feed material at a product yield of nearly 95%. The lower sulfur rejections reflect the
low feed sulfur content (0.05%) of this particular sample.
143 |
Virginia Tech | 3.6 Pilot-Scale Testing
After the successful completion of the laboratory-scale proof-of-concept test work,
sufficient data was obtained to justify pilot-scale testing of the HydroFloat separator. Pilot-scale
work was carried out in two specific areas, namely, coal and phosphate. Two pilot-scale
HydroFloat separators were fabricated to complete these investigations.
3.6.1 Northern Florida Phosphate Matrix
Phosphate testing was conducted at a northern Florida phosphate operation. To this end,
a 0.60 m square by 2.0 m tall test unit was fabricated and installed at the processing facility. A
photograph of the HydroFloat unit is provided in Figure 3.20. A flowsheet for the 10-15 tph test
circuit is provided in Figure 3.21. Circuit feed was supplied from a port located in the feed line
to a bank of plant dewatering cyclones used to prepare feed for existing conventional flotation
cells. The feed slurry was passed through a hydroclassification circuit to produce a plus 0.6 mm
underflow product. This material flowed by gravity into a bank of conditioning tanks where
flotation and pH reagents were added. The operating conditions are summarized in Table 3.2.
The product from the conditioners was then directed to the HydroFloat cell.
Table 3.2 - Parameters for in-plant pilot test program.
Parameter CrossFlow HydroFloat
Feed Rate (tph/sqft) 1-7 2-4
Feed Solids Density (%) 15-50 50-70
Water Addition Rate (gpm) 40-90 40-80
Aeration Rate (scfm) n/a 2-5
Fatty Acid Dosage (lbs/ton) n/a 1-3
145 |
Virginia Tech | Table 3.3 provides a comparison of test data from the pilot-scale HydroFloat unit with
that typically achieved by the existing full-scale conventional flotation circuit currently in
operation at the phosphate plant. At present, the plant typically operates with a BPL recovery of
approximately 35% for the plus 16 mesh feed and approximately 60% for the 16 x 35 mesh feed.
In comparison, the HydroFloat unit achieved a BPL recovery of more than 60% for the plus 16
mesh feed and nearly 85% for the 16 x 35 mesh feed. This represents an increase in recovery of
more than 40%. It is also interesting to note that the plus 16 mesh fraction had a very high BPL
content (72.8% BPL) and very low insol content (4.0%). The combined (i.e., plus 35 mesh)
concentrate from the HydroFloat cell represented at a total recovery of more than 80% with a
BPL grade of 56.8%. This result compares very favorably with the existing plant recoveries of
80-85% normally achieved for the finer 35 x 150 mesh feed.
Table 3.3 - Comparison of typical plant data and pilot-scale HydroFloat test results.
Particle Plant Cells HydroFloat HydroFloat
Size Recovery Recovery Grade
(mesh) (%) (%) (% BPL)
+ 16 » 35% 61.4% 72.8%
16 x 35 » 60% 84.7% 54.6%
Total » 50% 80.5% 56.8%
147 |
Virginia Tech | The HydroFloat was able to process and recover more coal without significantly
increasing the ash content when compared to traditional hindered-bed separations (without air).
Data presented in Figure 3.26 shows that for the plus 28 mesh fraction of coal, ash content
increased by only a few percent. The ash contents of the 28x35 mesh size fractions were nearly
identical. It must also be noted that the HydroFloat produced a significantly lower ash product
when comparing the 35x65 mesh size fraction.
The HydroFloat was tested using feed diverted from a distributor, which was supplying
feed to a bank of spirals. This bank of spirals was part of a rougher-cleaner spiral circuit with
partial middlings, comparable to the spiral circuit advocated in Chapter 1 of this dissertation.
Considering that the HydroFloat was treating the same material as the spiral circuit, a
comparison was warranted. Data from a concurrent coal company directed, full-plant sampling
endeavor were utilized in this effort. Figure 3.27 shows this data. It can be understood from this
figure that the HydroFloat, operating without air, achieved recoveries far below those obtained
by the existing spiral circuit. The coal spiral circuit also generally produced a higher quality
(lower ash) product. However, when the HydroFloat operated with air, combustible recovery
increased by nearly five percent. At optimum operating efficiency, the HydroFloat was able to
slightly improve upon the product quality (ash content) produced by the coal spiral circuit.
152 |
Virginia Tech | 3.7 Conclusions
1. A new separator, known as the HydroFloat unit, has been developed to overcome some of
the shortcomings associated with traditional flotation machines in recovering coarse
particles. The novel characteristic of this separator is the formation of a hindered “teeter”
bed of fluidized solids into which small air bubbles are introduced. The bubbles attach to
hydrophobic particles and create light bubble-particle aggregates that can be separated
from hydrophilic particles based on the principle of differential density. Benefits of this
new separator include enhanced bubble-particle contacting, better control of particle
residence time, lower axial mixing/cell turbulence, and reduced air consumption.
2. Results from simulations conducted with a population balance model show that a
decrease in the apparent specific gravity of one feed component can greatly increase the
recovery of that component. If the density of a feed component can be altered through
the attachment of air bubbles, the density ratio (r /r ) decreases, resulting in large
1 2
improvements in coarse particle recovery.
3. Laboratory tests were conducted with both batch and continuous HydroFloat cells in
order to evaluate the potential of this new technology for upgrading mineral samples
from various sources (e.g., phosphate matrix, coal, anthracite slag, mineral sands, etc.).
The test data indicate that the HydroFloat cell is capable of increasing coarse particle
recoveries by 20% over conventional flotation. Furthermore, the concentrate grades were
also improved in some cases due to a reduction in coarse particle misplacement.
154 |
Virginia Tech | 4. In light of promising laboratory data, a pilot-scale HydroFloat was installed at an
industrial phosphate beneficiation plant. Test data obtained with this unit to date suggest
that the BPL recovery of the plus 16 mesh feed can be nearly doubled (i.e., increased
from approximately 35% to more than 65%) through the application of this new
technology. In addition, the data suggest that the HydroFloat cell may be used to
increase the BPL recovery of the 16 x 35 mesh material from about 60% to nearly 85%.
As a result, the combined recovery of plus 35 mesh product from the HydroFloat cell
compares very favorably with existing plant recoveries of 80-85% normally achieved for
the 16 x 150 mesh feed.
5. A pilot-scale HydroFloat cell was also installed at an Appalachian coal preparation
facility. Data obtained from these in-plant tests indicate increases of coarse coal (+28
mesh) recoveries of nearly 40% over traditional hindered-bed separators. Overall
combustible recovery was increased by nearly 20%. Product quality and combustible
recovery were consistent or better than that produced by an existing coal spiral circuit.
155 |
Virginia Tech | Florida Institute of Phosphate Research, Publications No. 02-090-121, “Understanding the
Basics of Anionic Conditioning in Phosphate Flotation,” Prepared by Jacobs Engineering
Group, Inc., May 1995.
Florida Institute of Phosphate Research, Publication No. 04-045-103, “An Optical Sensor for
On-Line Analysis of Phosphate Minerals”, Prepared by Virginia Polytechnic Institute &
State University, April 1994.
Florida Institute of Phosphate Research, Publication No. 02-080-109, “On-Line Analysis of
Phosphate Rock Slurry by Prompt Neutron Activation Technique”, Prepared by University
of Florida, July 1994.
Florida Institute of Phosphate Research, Publication No. 02-094-108, “Evaluation of Dolomite
Separation Techniques”, Prepared by Hassan El-Shall, Global Marketing and Consulting,
October 1994.
Florida Institute of Phosphate Research, Publication No. 02-067-099, “Enhanced Recovery of
Coarse Particles During Phosphate Flotation”, Prepared by University of Florida, June 1992.
Florida Institute of Phosphate Research, Publication No. 02-070-098, “Development of Novel
Flotation-Elutriation Method for Coarse Phosphate Beneficiation”, Prepared by Laval
University, June 1992.
Florida Institute of Phosphate Research, Publication No. 04-032-091, “An Investigation of
Potential for Improved Efficiencies in Phosphate Rougher Flotation Through On-Line BPL
Measurements”, prepared by Harrison R. Cooper Systems, Inc., March 1991.
Florida Institute of Phosphate Research, Publication No. 02-063-071, “Anionic Flotation of
Florida Phosphate”, prepared by Zellar-Williams, Inc., February 1989.
Furey, J. and Mankosa, M.J., “Column Flotation in the Phosphate Industry”, Presented to Tenth
Annual Regional Phosphate Conference, Lakeland, Florida, October 1995.
158 |
Virginia Tech | CHAPTER 4
General Summary
Mathematical analysis tools, including linear circuit analysis and population balance
modeling, have been utilized to analyze and evaluate some water-based processes. The data
collected from these investigations were used to make modifications and/or improvements in
coal spiral circuitry, hydraulic classification and hindered-bed separation. These improvements
resulted in increased separation efficiency and unit capacity. A better understanding of two
novel pieces of mineral processing equipment (i.e., the CrossFlow and HydroFloat separators)
were also a result of these analyses. In review, several points of summary can be identified from
this work.
Improving Spiral Performance Using Circuit Analysis
1. Linear circuit analysis, a theoretical tool for comparing the relative effectiveness of various
configurations of unit operations, was successfully applied to coal spiral circuitry. Early
studies identified several coal spiral circuits that had the potential to improve separation
efficiency. One circuit in particular, a rougher-cleaner configuration with partial middlings
recycle, was capable of improving separation efficiency (Ep) approximately 1.22 times that
of more traditional coal spiral circuits. A reduced circuit SG and reasonable circulating
50
load were also a benefit of this modified circuit when compared to other preferred unit
configurations.
163 |
Virginia Tech | 2. Based on circuit analysis fundamentals, an alternative method was derived for determining
the partition expressions for any given spiral circuit. This method allows for the prediction
of efficiency (Ep), circuit cut-point or any other partition based result (i.e., SG or SG )
25 75
independent of washability data, provided a proper partition function is used.
3. In-plant testing of a full-scale, two-stage spiral circuit allowed for the comparison of several
alternative circuit configurations. Data obtained during the in-plant circuit evaluations
indicated that for an equivalent number of spirals, rougher-cleaner circuits operated in series
are far superior than parallel circuits for reducing circuit cut-point.
4. In-plant test data also indicated that although the circuit SG for rougher-cleaner coal spiral
50
circuits operated with and without a middlings recycle are very similar (i.e., » 1.65 SG), the
separation efficiency increased when a middling recycle stream was utilized. In fact,
separation efficiency was approximately 1.25 times higher for the circuit incorporating a
middlings recycle stream.
5. Using the in-plant test data, regression equations were developed that were used to simulate
alternative spiral configurations. Although a rougher spiral separation could be adequately
simulated using an expression developed by Reid (1971), the increased loading of near-
gravity material often found on a cleaner bank of spirals necessitated the development of an
alternative expression. This expression (i.e., the modified Reid equation) accurately
simulates an asymmetrical coal spiral partition in which a large amount of high gravity
material is misplaced to the clean coal launder.
164 |
Virginia Tech | 6. Simulations demonstrated that the rougher-cleaner coal spiral circuit incorporating a
middlings recycle was capable of a separation efficiency (Ep) 1.20 times that of a rougher-
cleaner spiral circuit without middlings recycle. This ratio compares favorably with both
the theoretical ratio of 1.22 predicted by circuit analysis and the ratio of 1.25 demonstrated
by actual in-plant performance.
7. The simulations further demonstrated that at a constant ash, the rougher-cleaner spiral
circuit with middlings recycle is capable of increasing circuit yield by as much as 3.86%.
This relatively small increase in yield translates to a growth in revenue of nearly $255,000
for the coal preparation facility at which the on-site tests were conducted. Furthermore, this
increase does not reflect the additional salable coal yielded by the coarse coal dense medium
circuits, which can now operate at higher effective gravities as a result of the lower overall
cut-point of the spiral circuit.
Improving Performance of Hindered-Bed Separators
1. Data from comparative in-plant and laboratory studies show that the feed presentation
system of the CrossFlow separator offers several advantages. Results from tests conducted
with phosphate matrix, limestone aggregate and heavy mineral consistently showed an
increase in capacity and separation efficiency when compared to traditional hindered-bed
separators.
165 |
Virginia Tech | 2. A population balance model was developed to study and understand the operation of the
newly developed CrossFlow hindered-bed separator. This model was based on general
hindered-settling equations for transitional flow regimes. Model input data include feed
rate, feed percent solids (by mass), feed size distribution (up to 9 size fractions), density of
up to two feed components, fluidization water rate, and underflow discharge rate. Output
results included overflow and underflow partition data, size distributions, component
recovery, and classification efficiency in terms of Ep or Imperfection.
3. Validation test work indicated a good correlation existed between the laboratory and model
simulation results. Reliable consistency was found for separation cut-point (d ) and
50
efficiency (i.e., Ep or I). During the validation test work, a correlation between target cut-
point (d ) and the maximum concentration by volume of solids (f ) was confirmed. This
50 max
linear relationship appears to vary with material, feed size distribution, and consequently,
separation cut-point (d ).
50
4. Data produced from simulations using the population balance model indicate that the cross-
flow feed presentation system has several advantages when compared to those used in
conventional hindered-bed separators. These benefits include increased operational stability
and a unit capacity of up to 6 tph/ft2 (71.2 tph/m2). Simulation data also show that the
CrossFlow can maintain an acceptable and less varied efficiency over a wide range of
operating conditions, including low feed percent solids (i.e., approaching 24% by mass).
166 |
Virginia Tech | 5. Laboratory solid and liquid tracer studies of the CrossFlow separator suggest that excess
feed water and solids that should report to the overflow launder are quickly off-loaded by
the cross-flowing action of the feed presentation system. This occurs without disturbing the
volume of material within the separation chamber. In contrast, traditional hindered-bed
separators employing downcomer technology inefficiently use separation chamber volume
to manage excess feed water and segregate overflow material prior to discharge.
Improving Coarse Particle Recovery in Hindered-Bed Separators
1. A hindered-settling population balance model was developed and utilized to identify an
approach to overcome the inherent disadvantages often found in conventional teeter-bed
separators when used as density separation devices. A feed consisting of up to two density
components and nine size fractions could be employed with this model. Output results
include component partition, recovery and rejection data.
2. Results from the modeling investigations suggest that any alteration of apparent density of
any one feed component can greatly effect the recovery of that component. Data showed
that an increase in the recovery of coarse, low density material could be realized if the
apparent specific gravity of that component could be modified (i.e., lowered). In fact,
further simulation indicated that recovery could increase by up to 60%.
3. To this end, the HydroFloat separator was developed based on flotation fundamentals. This
device uses an aerated teeter-bed through which bubbles can rise and attach to hydrophobic
particles. The attachment of air bubbles sufficiently reduces the apparent density of the
167 |
Virginia Tech | hydrophobic particles. These low density bubble-particle aggregates are then separated
from the hydrophilic particles based on the principle of differential density.
4. Data from laboratory proof-of-concept testing indicate that the HydroFloat cell was
successful in upgrading various types of minerals, including phosphate matrix, coal,
anthracite slag, and mineral sands. Data further indicate the HydroFloat cell is capable of
increasing the recovery of coarse (2mm x 50 mesh) particles over that traditionally found in
either froth flotation or conventional hindered-bed separations. Coarse particle
misplacement was also reduced resulting in improved product concentrate grades.
5. In-plant testing was conducted at a north Florida phosphate beneficiation plant. Coarse
phosphate recovery increased substantially using the HydroFloat cell when compared to
existing conventional froth flotation cells. BPL recovery nearly doubled for the +16 mesh
size fraction, and an increase of 25% was also achieved for the 16 x 35 mesh size fraction.
Product quality also improved due to the increased recovery of substantially higher grade,
coarse phosphate.
6. Further in-plant testing of the HydroFloat cell was conducted at an Appalachian coal
processing facility. Increases of up to 40% in coarse coal recovery were realized using the
HydroFloat cell when compared to a traditional hindered-bed separator. As a result, an
increase of 20% in combustible recovery was also achieved. A concurrent in-plant survey
showed that the HydroFloat cell was capable of achieving a product quality and combustible
recovery equivalent to that of an existing coal spiral circuit.
168 |
Virginia Tech | CHAPTER 5
Recommendations for Future Work
Several suggestions for follow-up efforts are offered here. These recommendations
address all aspects of this investigation as seen below.
Improving Spiral Performance Using Circuit Analysis
1. It is recommended that a plant-wide study be conducted in order to quantify the
improvements seen throughout the coal preparation facility as a direct result of the reduced
cut-point in the spiral circuitry. As the specific gravity cut-point of the spiral circuit becomes
lower, the cut-points of other circuits in the plant should increase slightly to compensate for
the decrease in product ash content. These higher cut-points should translate to increased
revenue since these circuits generally treat coarser material at higher tonnages and
efficiencies.
2. It is suggested that several other preferred circuit configurations be tested in plant. Several
configurations as indicated by linear circuit analysis had relative efficiencies better than the
rougher-cleaner configuration that incorporated middlings recycle (See Table 1.1). These
were discounted due to the impracticality and added cost associated with an increased
circulating load in the spiral circuit. Because the improved spiral circuit efficiency will
impact the entire plant, these alternative preferred circuits with high circulating loads may be
feasible when accounting for total improvement in plant performance.
169 |
Virginia Tech | Improving Performance of Hindered-Bed Separators
1. As stated in Chapter 2, the CrossFlow population balance model was capable of predicting
classification cut-point with a high degree of accuracy. However, the model had some
difficulty predicting efficiency, predominantly showing slightly lower efficiencies than what
is seen in actuality. This is most likely caused by a discretization error found in the feed
section of the population balance model. Currently, the feed section of the model is
composed of 25 zones, in a 5 by 5 configuration. It is recommended that this feed section be
expanded to contain a higher number of zones to minimize the discretization error. In fact, a
continuous model would prove to be most useful.
2. It is also recommended that more in-plant test work be completed which directly compares
the CrossFlow to other conventional hindered-bed separators under identical conditions.
This test work would help to quantify the advantages offered by the CrossFlow separator.
Improving Coarse Particle Recovery in Hindered-Bed Separators
1. It is suggested that an advanced, full-scale, in-plant evaluation of the HydroFloat be
undertaken. A beneficiation plant will prove to be the only location where a high rate of
constant feed can be maintained for test purposes. The effect of treating coarse (1mm x 50
mesh) material with the HydroFloat can also be quantified with respect to any performance
changes in the conventional flotation cells.
2. The development of a hybrid, hindered-bed/flotation-based population balance model for the
HydroFloat separator is highly recommended. Although the separation of components in this
170 |
Virginia Tech | Flow Balances
Using the overall CrossFlow zone schematic shown in Figure A1, the following overall
volumetric flow balances can be written:
Q +Q =Q +Q
F W U L
O5
Q =Q - Q
Z W U
where Q , Q , Q , and Q are the feed, elutriation, underflow, and overflow volumetric flows,
F W U LO5
respectively. The feed, elutriation and underflow flow rates are known at time t = 0.
Using the enlarged feed zone schematic seen in Figure A2, the following flow balance can be
written:
Q =Q +Q
L F X
O5 1
where Q is the vertical upwards flow rate exiting zone A and entering the feed zone sections.
X1
The fluidized-bed evenly distributes all vertical flows evenly across the cross-sectional area of
the separator, except for the first five vertical zones (O , B , C , D , and E ). In these zones, it
1 1 1 1 1
can be shown that the falling action of feed solids and associated liquid prevents little or no
upward flow from entering. This assumption is valid as the cross-sectional area of these zones is
minimized. Using this information, the following flow balances can be equated:
(cid:229) (cid:229)
Q = Q +Q + Q - Q
X X L D D
2 1 B1 C2- 4 B2- 4
(cid:229) (cid:229)
Q =Q +Q + Q - Q
X X L D D
3 2 C1 D2- 4 C2- 4
(cid:229) (cid:229)
Q = Q +Q + Q - Q
X X L D D
4 3 D1 E2- 4 D2- 4
(cid:229) (cid:229)
Q = Q +Q + Q - Q
X X L D D
5 4 E1 O2- 4 E2- 4
where Q is the is the downward volumetric flow induced by the settling action of solids in to
D
and out of each respective zone.
173 |
Virginia Tech | The horizontal flows that are found between each zone of the feed section can be determined
from simple flow balances around each zone. Completing these flow balances yield the
following equations:
Q =Q - Q
L F D
O1 O1
Q = 0.25*Q +Q - Q
L X L D
O2 5 O1 O2
Q =0.25*Q +Q - Q
L X L D
O3 5 O2 O3
Q = 0.25*Q +Q - Q
L X L D
O4 5 O3 O4
Q =0.25*Q +Q - Q
L X L D
O5 5 O4 O5
Q =Q - Q
L D D
E1 O1 E1
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
E2 4 O2 E1 5 E2
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
E3 4 O3 E2 5 E3
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
E4 4 O4 E3 5 E4
Q =Q - Q
L D D
D1 E1 D1
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
D2 3 E2 D1 4 D2
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
D3 3 E3 D2 4 D3
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
D4 3 E3 D3 4 D4
Q =Q - Q
L D D
C1 D1 C1
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
C2 2 D2 C1 3 C2
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
C3 2 D3 C2 3 C3
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
C4 2 D4 C3 3 C4
Q =Q - Q
L D D
B1 C1 B1
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
B2 1 C2 B1 2 B2
Q = 0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
B3 1 C3 B2 2 B3
Q =0.25*Q +Q +Q - 0.25*Q - Q
L X D L X D
B4 1 C4 B3 2 B4
176 |
Virginia Tech | Mass Balances
Utilizing the law of mass conservation, mass balance equations were written for each zone in the
dynamic population balance model (mass in = mass out). For each zone and for each size class,
all flows were balanced and multiplied by the concentration of solids (C ) present in the zone
zone
from which the flow emanated. This was completed for increments of time as small as 1/10000th
of one second. For each time increment (iteration), a new concentration for each zone could be
calculated and mathematically added to the concentration from the previous iteration. The term
U A represents the volume of settling solids in a particular zone as defined by the hindered-
p
settling equations presented in the body of this work.
Using Figure A2, the following equations can be written:
C =C +(Q C - C (Q +Q +U A))D t
O O F F O L D P
1NEW 1 1 CO1 CO1
C = C +(Q C +0.25*Q C - C (Q +Q +U A))D t
O O L O X E O L D P
2NEW 2 CO1 1 5 2 2 CO2 CO2
C = C +(Q C +0.25*Q C - C (Q +Q +U A))D t
O O L O X E O L D P
3NEW 3 CO2 2 5 3 3 CO3 CO3
C = C +(Q C +0.25*Q C - C (Q +Q +U A))D t
O O L O X E O L D P
4NEW 4 CO3 3 5 4 4 CO4 CO4
C = C +(Q C +0.25*Q C - C (Q +Q +U A))D t
O O L O X E O L D P
5NEW 5 CO4 4 5 5 5 CO5 CO5
C = C +(Q C - C (Q +Q +U A))D t
E E D O E L D P
1NEW 1 O1 1 1 E1 E1
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
E E L E X D D O E X L D P
2NEW 2 E1 1 4 2 O2 2 2 5 E2 E2
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
E E L E X D D O E X L D P
3NEW 3 E2 2 4 3 O3 3 3 5 E3 E3
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
E E L E X D D O E X L D P
4NEW 4 E3 3 4 4 O4 4 4 5 E4 E4
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
E E L E X D D O E X L D P
5NEW 5 E4 4 4 5 O5 5 5 5 E5 E5
C = C +(Q C - C (Q +Q +U A))D t
D D D E D L D P
1NEW 1 E1 1 1 D1 D1
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
D D L D X C D E D X L D P
2NEW 2 D1 1 3 2 E2 2 2 4 D2 D2
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
D D L D X C D E D X L D P
3NEW 3 D2 2 3 3 E3 3 3 4 D3 D3
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
D D L D X C D E D X L D P
4NEW 4 D3 3 3 4 E4 4 4 4 D4 D4
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
D D L D X C D E D X L D P
5NEW 5 D4 4 3 5 E5 5 5 4 D5 D5
C = C +(Q C - C (Q +Q +U A))D t
C C D D C L D P
1NEW 1 D1 1 1 C1 C1
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
C C L C X B D D C X L D P
2NEW 2 C1 1 2 2 D2 2 2 3 C2 C2
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
C C L C X B D D C X L D P
3NEW 3 C2 2 2 3 D3 3 3 3 C3 C3
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
C C L C X B D D C X L D P
4NEW 4 C3 3 2 4 D4 4 4 3 C4 C4
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
C C L C X B D D C X L D P
5NEW 5 C4 4 2 5 D5 5 5 3 C5 C5
177 |
Virginia Tech | C = C +(Q C - C (Q +Q +U A))D t
B B D C B L D P
1NEW 1 B1 1 1 B1 B1
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
B B L B X A D C B X L D P
2NEW 2 B1 1 1 C2 2 2 2 B2 B2
C =C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
B B L B X A D C B X L D P
3NEW 3 B2 2 1 C3 3 3 2 B3 B3
C = C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
B B L B X A D C B X L D P
4NEW 4 B3 3 1 C4 4 4 2 B4 B4
C = C +(Q C +0.25*Q C +Q C - C (0.25*Q +Q +Q +U A))D t
B B L B X A D C B X L D P
5NEW 5 B4 4 1 C5 5 5 2 B5 B5
C = C + (Q C +U A *C + Q C - C (Q + U A))D t
A A F F P B Z G A X P
New 1
Using Figure A3, the following equations can be written:
C =C +(Q C +U AC - C (Q +U A))D t
G G Z H P A G Z P
New
C =C +(Q C +U AC - C (Q +U A))D t
H H Z I P G H Z P
New
C =C +(Q C +U AC - C (Q +U A))D t
I I Z J P H I Z P
New
C = C +(Q C +U AC - C (Q +U A))D t
J J Z K P I J Z P
New
C = C +(Q C +U AC - C (Q +U A))D t
K K Z L P J K Z P
New
C =C +(U AC - C (Q +Q +U A))D t
L L P K L Z U P
New
C = C +(C (Q +U A)- C Q )D t
U U P U P U U
New
These equations were solved continuously until the change in concentration for each and every
zone in the population balance model was equivalent to zero or C is equal to C .
ZONE(new) ZONE
Steady-state conditions are realized when the change in concentration for the entire separator is
equal to zero.
178 |
Virginia Tech | VITA
Jaisen Nathaniel Kohmuench, son of William C. and Carolyn A. Kohmuench was born in
Teaneck, NJ, on the 23rd day of May, 1973. He graduated from Hunterdon Central Regional
High School in the spring of 1991. The following fall, he was granted admission to Virginia
Polytechnic Institute and State University (Virginia Tech), where he went on to gain a Bachelor
of Science degree in Mining and Minerals Engineering. During his time as an undergraduate, he
was highly involved in the Burkhart Mining Society, the student chapter of the Society for
Mining, Metallurgy and Exploration (SME). It was during the ‘94-‘95 school year that he served
as the vice president of this organization. He also successfully passed the EIT exam.
After graduating in the spring of 1995, he remained at Virginia Tech to pursue a Master
of Science degree in Mining and Minerals Engineering with an emphasis in minerals processing.
He completed his degree in the fall semester of 1997 and immediately enrolled in the Ph.D.
program. He is currently a student member of SME. As a doctoral candidate, he conducted
several presentations at professional meetings in addition to authoring (or co-authoring) over 11
works, including two full peer-reviewed journal publications. While a student, he earned the
Graduate Student of the Year Award for both the 1996 and 1999 school years.
Upon completion of his doctoral dissertation, he and his wife, Kathryn, will relocate to
Erie, Pennsylvania. Here, having already accepted a position, Jaisen will begin his professional
career as a process engineer for Eriez Magnetics.
Jaisen N. Kohmuench
188 |
Virginia Tech | Surface and Hydrodynamic Forces in Wetting Film
Lei Pan
ABSTRACT
The process of froth flotation relies on using air bubbles to collect desired mineral
particles dispersed in aqueous media on the surface, while leaving undesirous mineral particles
behind. For a particle to be collected on the surface of a bubble, the thin liquid films (or wetting
films) of water formed in between must rupture. According to the Frumkin-Derjaguin isotherm,
it is necessary that wetting films can rupture when the disjoining pressures are negative.
However, the negative disjoining pressures are difficult to measure due to the instability and
short lifetimes of the films.
In the present work, two new methods of determining negative disjoining pressures have
been developed. One is to use the modified thin film pressure balance (TFPB) technique, and the
other is to directly determine the interaction forces using the force apparatus for deformable
surfaces (FADS) developed in the present work. The former is designed to obtain spatiotemporal
profiles of unstable wetting films by recording the optical interference patterns. The kinetic
information derived from the spatiotemporal profiles were then used to determine the disjoining
pressures using an analytical expression derived in the present work on the basis of the Reynolds
lubrication theory. The technique has been used to study the effects of surface hydrophobicity,
electrolyte (Al3+ ions) concentration, and bubble size on the stability of wetting films. Further,
the geometric mean combining rule has been tested to see if the disjoining pressures of the
wetting films can be predicted from the disjoining pressures of the colloid films formed between
two hydrophobic surfaces and the disjoining pressures of the foam films formed between two air
bubbles.
The FADS is capable of directly measuring the interaction forces between air bubble and
solid surface, and simultaneously monitoring the bubble deformation. The results were analyzed
using the Reynolds lubrication theory and the extended DLVO theory to determine both the
hydrodynamic and disjoining pressures. The FADS was used to study the effects of surface
hydrophobicity and approach speeds. The results show that hydrophobic force is the major
driving force for the bubble-particle interactions occurring in flotation. |
Virginia Tech | Acknowledgement
First and foremost, I would like to give my utmost appreciation to my advisor, Dr. Roe-Hoan
Yoon, for his support, guidance, and inspiration throughout the course of this study. His
consistent enthusiasm on research and enormous contribution to mineral engineering deeply
impressed me and will continue to influence me in the future. His wisdom and extraordinary
patience on me has made this work possible. He is my mentor for both career and life for my
lifetime.
I would also like to acknowledge my committee members, Dr. Gerald H. Luttrell and Dr.
Gregory T. Adel for their invaluable advices. I appreciate the opportunities to learn the leading
coal research from them. I would also like to thank Dr. Alan R. Esker for his constructive
comments, which helped to improve this work. Special appreciation goes to Dr. Sunghwan Jung
for his help in interfacial fluid mechanics throughout the entire course of this study.
My appreciations are extended to Dr. Anbo Wang and Dr. Bo Dong at Center for Photonics
Technology for their kind help and accommodation for me to learn the fiber interferometry
technique in their lab. I would also like to thank Dr. Rickey Davis and his students for their help
on measuring the zeta potentials.
I would like to thank all the staff and peer graduate students in our Mining department.
Special appreciation is given to Dr. Jialin Wang, Dr. Zuoli Li, Dr. Aaron Noble, and Dr. Jinming
Zhang for the friendship and invaluable discussions. To Chris Hull, Carol Trutt, and Gwen Davis,
I appreciate their care over the time of my study at Virginia Tech. I would also like to express
my gratitude to Donald Leber and Jim Waddell for their technical assistant, and Kirsten Titland
for editing the grammar of this dissertation.
Finally, I sincerely appreciate my family for their unconditional love for all these years. I
wouldn’t have received this great learning opportunity without their financial and mental support.
iii |
Virginia Tech | Chapter 1. Introduction
1.1 General
A wetting film is formed when a vapor condenses on a surface, a liquid spreads on a surface,
or when a gas bubble is pressed against a surface in a liquid. Thus, wetting films are ubiquitous
in our daily lives and play vital roles in a variety of industrial and medical applications, e.g.,
flotation [1-3], enhanced oil recovery [4-6], surface cleaning [7, 8], heat transfer [9, 10],
microfluidics [11, 12], and optoelectronics [13, 14].
In the minerals industry, flotation has been widely used to separate valuable minerals from
valueless gangue minerals for more than a century. The process may appear simple, but is
capable of separating a variety of minerals by control of the surface chemistries of the minerals
involved. In a flotation cell, air bubbles collide with mineral particles, during the course of which
wetting films are formed between the bubbles and particles. Initially, the water in the film drains
due to the surface tension pressure (or Laplace pressure) created by the curvature change. The
film will continue to drain until the excess pressure becomes equal to the disjoining pressure ()
created by the surface forces acting between the bubble and the mineral surfaces. In most cases,
the disjoining pressure is repulsive, because both the electrical double-layer and van der Waals
forces are repulsive. Therefore, the disjoining pressure provides a resistance to film thinning.
When the Laplace pressure is equal to the disjoining pressure, the wetting film is stable and the
film drainage stops at an equilibrium thickness.
Thermodynamically, disjoining pressure is defined as the change in excess Gibbs free energy
(G) per unit area of a flat film with film thickness (h) [15, 16],
(h) G/h (1.1)
p,T,
s
at a constant pressure (p), temperature (T), and chemical potentials of the solutes (μ). According
s
to eq. (1.1), the disjoining pressure becomes positive when G/h < 0, which is the case when
both the double-layer and van der Waals forces are repulsive. For a wetting film to rupture,
1 |
Virginia Tech | should be negative, which is the case when G/h > 0. This condition is realized when the
surface forces in the thin liquid film (TLF) of water in the wetting film are net negative.
According to the classical DLVO theory [17, 18], the stability of a TLF confined between two
surfaces, e.g., between two colloidal particles, is determined by two surface forces, namely,
electrical double-layer force and the van der Waals force. Depending on the relative magnitudes
of the two forces, the disjoining pressure of film can be positive or negative. The film (or the
colloidal suspension) is stable when > 0 and unstable when < 0. Likewise, the stability of
the TLF confined between an air bubble and a mineral surface should be determined by the
disjoining pressure in the wetting film. A problem with this logic is that in most flotation systems,
> 0 according to the DLVO theory, as both the double-layer and the van der Waals forces are
repulsive. In 1969, Laskowski and Kitchener [19] recognized this problem and suggested that for
the wetting film to rupture, the disjoining pressure must be negative and that the negative
pressure may be due to a long-range hydrophobic effect. In 1972, Blake and Kitchener [20]
conducted a series of disjoining pressure measurement in the wetting films of water formed on
the surface of methylated silica. The experimental results showed that > 0 but the wetting film
ruptures at h = 64 nm, which led the authors to suggest that the film rupture observed at long
separation distance was due to the hydrophobic force.
In 1982, Pashley and Israelachvili [21] reported the measurement of the hydrophobic force in
the TLFs of water confined between mica surfaces hydrophobized by the cetylammonium
bromide (CTAB). Since then, numerous other investigators confirmed the presence of the
hydrophobic force in the TLFs confined between hydrophobic surfaces using the surface force
apparatus (SFA) and atomic force microscope (AFM) [22-31]. Eriksson, et al. [25] suggested
that the hydrophobic force may be due to the structuring of the water in the TLFs. The
structuring has something to do with the structure of the solid. Many other investigators had a
difficulty with the suggestion that the structure of a solid can influence the structure of water as
far as h = 80 nm or more [32-35]. As an alternative explanation, Attard and others showed
evidences that the long-range attractive hydrophobic forces may be due to the nanobubbles
preexisting on hydrophobic surfaces [36-41].
More recently, Yoon and his coworkers [42] conducted surface force measurements at several
different temperatures to show that hydrophobic interactions entail entropy decrease, which is
2 |
Virginia Tech | contrary to the hydrophobic interactions at molecular scale. As is well known, the self-assembly
of surfactants and lipids entails entropy increase [43]. The authors attributed the entropy decrease
associated with the macroscopic hydrophobic interaction to the formation of H-bonded structures
of water in the TLFs confined between hydrophobic surfaces. The formation of H-bonded
structures, which is also referred to as partial clathrates, may be a way to minimize the free
energy of the water molecules that cannot be H-bonded to the confining surfaces. In this regard,
the hydrophobic force may be considered a thermodynamic force. More recently, Yoon and his
coworkers [44] reported the measurement of long-range attractive forces in the TLFs of ethanol
and methanol confined between hydrophobic surfaces. It appears, therefore, that the hydrophobic
force is a solvophobic force arising from the structuring of H-bonding liquids.
It is the main objective of the present work to explore the possibility that hydrophobic forces
are present in the wetting films of water confined between air bubbles and hydrophobic surfaces.
The first measurement of disjoining pressures were reported by Derjaguin and his co-workers in
1939 [45] using the bubble-against-plate technique. The initial measurements were conducted on
hydrophilic surfaces such as mica and silica. The disjoining pressures were positive and, hence,
the TLFs were stable, which made it possible to do the measurement. The experimental
technique involves the measurement of the excess pressure (p) in the film while monitoring the
film thicknesses (h) using an optical interference technique. However, it is difficult to do the
measurement when the wetting films are unstable. Typically, the lifetimes of the wetting films
formed hydrophobic surfaces are fractions of a second. In the present work, the optical
interference patterns (Newton rings) generated during the film thinning process is recorded on a
high-speed camera; and the results were used to reconstruct the spatial and temporal film profiles
and calculate the disjoining pressure using a model developed in the present work on the basis of
the Reynolds lubrication theory. The results show that the disjoining pressure is negative in the
wetting films of water formed on hydrophobic surfaces.
Many investigators [46-50] reported the measurement of the surfaces forces involved in
bubble-particle interactions. Typically, they used AFM and SFA for this purpose. A challenge
with this approach has been that it is difficult to determine the film thicknesses accurately as
bubbles deform during the measurement. It is, therefore, another objective of the present work to
develop a new instrument that can directly measure not only the forces involved in bubble-
particle interactions but also the distances between the bubble and the surface. The new
3 |
Virginia Tech | instrument and the numerical methods developed for the analysis of the experimental data allows
one to visualize the dynamic process in the form of nano-scale thickness profiles, while
monitoring the hydrodynamic, surface tension, and surface forces. The results show that the
hydrophobic force is the major driving force for flotation. The new instrument developed in the
present work can also be used to study the interactions between biological membranes.
1.2 Literature Review
The presence of the disjoining pressure () in a thin liquid film (TLFs) has been known since
1939 when its first measurement was reported by Derjaguin and Kussakov [45]. It is now well
recognized that disjoining pressures are present in the TLFs of liquids confined between two
colloidal particles, liquid droplets, or gas bubbles, and its magnitude and sign determines the
stability of the TLFs. In general, a film is stable when > 0 and is unstable when <0. The
classical DLVO theory has been used to predict the disjoining pressures of in the TLFs formed
between colloidal particles and between bubbles in foams.
Attempts have also been made to measure or predict the disjoining pressures in the wetting
films formed between air bubbles and mineral particles [51]. When the disjoining pressure is
positive, the measurement is easy. However, the measurement is difficult when the disjoining
pressure is negative. Laskowski and Kitchener [19] was the first to recognize that the disjoining
pressure must be negative for a contact angle to form or for bubble-particle attachment to occur.
When < 0, however, the film is so unstable that no one has been able to directly measure the
negative disjoining pressures for more than 70 years, nor has anyone developed an analytical
expression for the negative disjoining pressures in wetting films. What follows below is the
review of the experimental techniques that has been developed to date for the measurement
disjoining pressures in wetting films.
The techniques that have been develop for the determination of the disjoining pressure
isotherms in wetting films can be classified into two categories: static and dynamic methods. The
static methods are based upon measuring the film thickness at equilibrium in a well-controlled
environment, when the disjoining pressure in a wetting film is balanced by the pressures across
the film. The dynamic methods involve monitoring of the temporal changes in film thickness, or
directly measuring the interaction force between an air bubble and a solid surface across a thin
4 |
Virginia Tech | layer of liquid. Both methods have been intensively used to study the wetting films of various
liquids on solid surfaces. Each method has unique features and advantages over the others in
view of thickness range, nature, and magnitude. Both techniques have been used interchangeably
over the years.
1.2.1 Static measurement of determining the disjoining pressure isotherm
The first study of the disjoining pressure in a wetting film was carried out by Derjaguin and
Kussakov in 1939 [52]. In their experimental set-up, gas bubbles with sizes from 0.1 mm to 1
mm in radius were slowly pressed against a horizontal glass plate in aqueous solutions. Given
sufficient time, a liquid film of water between the air bubble and the solid surface was stabilized
and reached an equilibrium. The thickness of the thin wetting film between an air bubble and a
solid surface was monitored using the interferometry technique. A thick film, known as β-film,
was formed in equilibrium with a bulk liquid surrounding the air bubble. At equilibrium, the
disjoining pressure in the wetting film is balanced by the Laplace pressure (ΔP) inside the air
bubble. Thus,
P 2/R (1.2)
where σ is the surface tension of the liquid and R is the radius of the bubble. Given an inverse
linear relationship between the disjoining pressure and the bubble size, the disjoining pressure
was determined by measuring the thicknesses of the equilibrium wetting films using the bubbles
with varying sizes. This work was the first experimental evidence supporting the DLVO theory.
In 1968, Read and Kitchener [53] also took advantage of eq. (1.2) while developing a refined
experimental cell for studying the wetting film formed on polished silica plates. Air bubble was
generated in-situ by the electrolysis of a platinum wire and manually placed under a small
inverted silica cup. The thin liquid film was created afterwards by slowly lowering the air bubble
towards a silica plate until an equilibrium film was formed. The intensity of light reflected from
the wetting film was determined by means of a cadmium sulfide photoconductive cell. The
thickness of the thin liquid film was determined from the intensity of the reflected light in
accordance to the Raleigh equations,
5 |
Virginia Tech | I 2 2 2cos
r
I 12 2 2cos
0
n n / n n (1.3)
1 1
n n / n n
2 2
4ndcos /
where n , n and n are the refractive indices for solid, water and vapor, respectively, φ is the
1 2
angle of refraction in the film, d the film thickness and λ is the wavelength of the light. From a
normalized light intensity (I/I ), equilibrium film thicknesses obtained in an aqueous solutions of
r o
electrolytes were compared with the theoretical values predicted using the DLVO theory.
Later, Blake and Kitchener [20] further modified the experimental set-up for an extended
study of a wetting film on a hydrophobic methylated silica surface. Unlike the previous work
done by Read and Kitchener [53], the interference patterns were directly observed under an
inverted microscope by means of a 35 mm camera. The film thickness was determined from a
photo of the ‘Newton rings’ by scanning with a microdensitometer. In their experimental set-up,
an air bubble of nitrogen was formed at the bottom of a capillary tube. The position of the air
bubble was controlled by means of a micrometer. Blake and Kitchener showed variances in the
thickness of the wetting films formed on hydrophobic surfaces in aqueous solutions containing
different concentrations of electrolytes. Similar approach was also applied to study the
equilibrium wetting films of alkanes on alumina surfaces [54].
Aronson and Princen [55, 56] followed the approach developed by Blake and Kitchener, and
studied the wetting films formed on the silica surfaces of surfactant solutions. The sizes of the air
bubbles varied from 0.75 mm to 6 mm with the internal hydrostatic pressures ranging from 2000
to 200 dyne/cm2. The equilibrium film thicknesses were determined at varying hydrostatic
pressures in the thin liquid films by changing the sizes of the air bubbles. The disjoining pressure
isotherm was plotted in the hydrostatic pressures as a function of the film thicknesses.
The methods developed by both Derjaguin’s group and Kitchener’s group were based upon
varying the sizes of the air bubbles. Knowing the sizes of the air bubbles, the disjoining pressure
isotherm was plotted using the Laplace pressure as a function of the equilibrium film thickness,
i.e., disjoining pressure vs. film thickness. In order to visualize the equilibrium wetting films, the
6 |
Virginia Tech | sizes of the air bubbles practically used ranged from 1 mm up to 10 mm with the hydrostatic
pressures inside the air bubbles up to 144 Pa. Thin liquid pressure balance (TFPB) technique, on
the other hand, allows a measurement of the disjoining pressure isotherm at Π above 1000 Pa.
The design of TFPB technique was originally published by Mysels and Jones [57] for a study of
the free film, i.e., a thin liquid film of water between two bubbles. In the TFPB technique, a
meniscus of the liquid film between two air bubbles was formed in a porous porcelain disc by
sucking the liquid out of the cell through a capillary tube. The cell was enclosed in a pressure
chamber sealed by an O-ring. By varying the pressure inside the cell, one can obtain the
equilibrium film thicknesses at the disjoining pressure measuring up to 104 Pa. The film
thickness in Mysels and Jones’s work was determined from an optical set-up by means of a
photomultiplier tube.
The TFPB technique was later adapted for the disjoining pressure measurements in wetting
films [58-60]. Klitzing and his co-workers [59, 60] used the TFPB technique to study the wetting
films formed on silicon surfaces. In their experimental set-up, the substrate was sitting beneath
the porous cell. By sucking the liquid out of a capillary inside the porous cell, a meniscus was
formed between a solid surface and the vapor phase with the liquid wetting the wall of the
capillary hole. The film holder was enclosed in a sealed cell by regulating the pressure using a
piston pump. The film thickness was monitored from the top using the interferometry technique
by means of a photomultiplier tube. The morphologies of the solid surfaces across thin liquid
films were simultaneously monitored by a video microscope.
The monochromatic interferometry technique has proven to be a powerful and feasible tool
for determining the film thickness of the wetting film above 30 nm. However, use of the
monochromatic interferometry technique was rather limited when determining the film thickness
below 30 nm. Instead, the ellipsometry technique allows an accurate measurement of the film
thickness below 30 nm [61-66]. Ellipsometry characterized the reflected light beams from a thin
liquid film or an assembly of thin liquid films at varying incident angles [67, 68]. When the
thicknesses of the thin liquid films vary, the reflectivity and phase of p-polarized and s-polarized
light changes depending upon the reflective indices of the layers that the incident light beams
passed through. Reflectivity and phase changes are related to the Frensel reflection coefficients
r and r for p- and s-polarized light as r /r = tan φ eiΔ. By scanning the polarization of the
p s p s
reflected light beams at varying incident angles, the thickness of the thin liquid film can be
7 |
Virginia Tech | determined from the ellipsometrical angle ((cid:2038)) and the phase difference (∆) upon the reflection.
Ellipsometry can also be operated in an imaging mode using a CCD camera as the detector [39].
It provides detailed, real-time information on 2D profiles of the film thicknesses and refractive
indices across an entire field of view.
Eliseeva et al. [61, 62] combined both ellipsometry and TFPB techniques to determine the
disjoining pressure in wetting films at varying hydrostatic pressures. The thickness of the wetting
film was determined by means of the ellipsometry technique. Fukuzawa et al. [69] used an
ellipsometry microscope to study the dewetting process of the lubricant films on magnetic plates.
A lubricant film with a thickness less than 10 nm was obtained. The surface energy vs. the
thickness was used to reconstruct the disjoining pressure isotherm in a lubricant film.
The methodologies discussed above were valid for the measurement of the disjoining
pressures in the stable wetting films with Π > 0 and dΠ/dh >0. It was typically observed in
wetting films of aqueous solutions formed on the hydrophilic surfaces, such as mica, silica, or
alkanes on alumina and metal surfaces. As the surfaces became hydrophobic, the wetting film
became less stable, leading to a formation of an α-film. An α-film was formed when a continuous
β-film was ruptured at a critical point or by an adsorption of the volatile liquid on surfaces. In an
α-film, the disjoining pressure is equivalent to zero at constant chemical potential and pressure.
In general, the thickness of α-film is less than 10 nm varying with the pattern of the disjoining
pressure isotherm in a wetting film.
Therefore, the measurement of the disjoining pressure in an unstable wetting film at a
thickness below 30 nm was not possible by varying the size of the air bubbles. Shishin et al. [63,
64] developed a modified TFPB technique to determine the disjoining pressure in a liquid film of
tetradecane on a solid surface. The film thickness was obtained by means of the ellipsometry
technique. In their approach, the disjoining pressure was controlled by changing the hydrostatic
pressure of the liquid. From a balance of the disjoining pressure with the hydrostatic pressure,
the disjoining pressure is given by,
gH (1.4)
where ρ is the density of the liquid, g is the gravitational acceleration, and H is the difference in
liquid level between the liquid/vapor interface in a thin film and at bulk liquid.
8 |
Virginia Tech | Alternatively, the disjoining pressure of absorbed vapor on the solid surface was determined
by controlling the chemical potential (μ) of the thin liquid film. Here, the disjoining pressure was
represented in terms of the chemical potential of molecules in the liquid phase ( = ·V ) and
m
the vapor phase ( = RT ln(P/P )) from eq. (1.2) at constant pressure, namely
s
RT /V ln P/P (1.5)
m s
where V is the molar volume of the liquid, T is the temperature, R is molar gas constant and p/p
m s
is the relative vapor pressure. By controlling the relative vapor pressure of molecules in a liquid
film, the disjoining pressure in an adsorbed liquid film could be determined. As calculated using
eq. (1.5), the disjoining pressure can be varied by changing the ratio of p/p up to saturation.
s
Therefore, it is possible to study the property of the adsorbed vapor film over the entire region of
positive Π.
Bangham et al. [70, 71] might be the first to apply the vapor adsorption method to study the
properties of the adsorbed vapor film on a solid surface. They visualized an adsorbed film of the
volatile liquid on a mica surface. Later, Derjaguin et al. [72, 73] investigated the adsorption of
the liquid vapor onto a solid surface by measuring the film thickness ellipsometrically. The
relative vapor pressure (p/p) was controlled by changing the temperatures of the solid surface
s
and the volatile liquid separately in a closed chamber. Simultaneously, the thickness of the liquid
film on the solid surface was monitored in-situ using the ellipsometry technique. Following a
similar approach by Derjaguin et al., Hall studied the adsorption of the water vapor on a quartz
surface by measuring the thickness of an adsorbed water film [65]. The closed chamber was
degassed by a vacuum and refilled with helium repeatedly. The measurements were carried out
at an atmospheric pressure in a helium environment. Hu and Adason [66] determined the
adsorption isotherm (or disjoining pressure isotherm) ellipsometrically for a variety of volatile
liquids, including water, bromobenzene, methane, methanol, and ethanol on a polished
hydrophobic surface.
Instead of using the ellipsometry technique to obtain the thicknesses of the liquid films,
Garbatski and Folman [74] determined the thickness of an absorbed liquid film of water and
isopropanol on a glass plate by measuring the changes in capacity of a quartz crystal at varying
relative vapor pressures. The electrical method was based using a crystal controlled oscillator, in
which the capacity of the quartz crystal was measured at the anode circuit.
9 |
Virginia Tech | Slutsky and his co-workers [75-78] used a quartz crystal microbalance technique to determine
the mass of the vapor molecules absorbed on the solid surface. The thickness of the adsorbed
film was obtained by converting the changes in the mass into the thickness. The adsorption
isotherms of the vapor molecules at varying relative vapor pressures were obtained on both
quartz surfaces and metal surfaces. The quartz crystal functioned as a microbalance by
measuring a shift in the resonance frequency of the crystal when the gas molecules were
absorbed on the quartz surface. Sabisky and Anderson used the acoustic interferometry technique
to measure the thickness of helium films on surfaces of fluoride at 1.38K [79].
1.2.2 Dynamic methods of determining the disjoining pressure isotherm
Above is a review of the static methods that have been used to determine the disjoining
pressure isotherm ((h)) in a thin liquid film between the vapor phase and the solid surface. The
static methods can be used for determining positive disjoining pressure isotherms in both stable
(∂Π/∂h > 0) and unstable wetting films (∂Π/∂h < 0). A positive disjoining pressure was obtained
by balancing the pressure in liquid with the pressure in vapor. In a stable wetting film with a
continuous liquid phase, typically known as a β-film, the disjoining pressure isotherm could be
obtained when the pressure in the thin liquid film was in equilibrium with the bulk liquid. In an
unstable wetting film, the rupture of the β-film led to a formation of an α-film, i.e., a liquid film
of the adsorbed gas molecules in equilibrium with the vapor phase [80]. The disjoining pressure
isotherm was obtained from the adsorption isotherm of the gas molecules on a solid surface.
As discussed above, the static methods only allow one to obtain a positive disjoining pressure
in a wetting film. The negative pressure could not be determined directly, but only from a
theoretical extrapolation of the DLVO theory. On the other hand, the dynamic methods show an
ability of determining both a positive and a negative disjoining pressure in a wetting film. In the
1960s, Scheludko and his co-workers measured the disjoining pressure in a thin liquid film by
monitoring the film drainage of a liquid film between two bubbles. It was designed to study
symmetric free film, i.e., a foam film of a liquid between two vapor/liquid meniscuses [81-83].
With a modification of the liquid cell, the dynamic method was also adapted to study the
drainage of a wetting film [84].
10 |
Virginia Tech | The dynamic method developed by Scheludko et al. [84] was based upon the Reynolds theory
of a thin liquid film between two plane-parallel interfaces. The Reynolds equation yields an
expression for the rate of film drainage by the pressure difference in the thin film relative to the
adjacent bulk liquid [85]. The Reynolds approximation is given by
d 1/h2 4P
(1.6)
dt 3r2
0
where h is the film thickness, η is the viscosity of the liquid, r is the radius of the flat film, t is
0
the time and ΔP is the pressure difference in a thin film relative to the pressure in a bulk liquid.
Equation (1.6) was derived assuming that two adjacent interfaces were parallel with zero slip
velocity at the interfaces. In general, a zero slip velocity is referred to as the non-slip boundary
condition. The pressure difference was given by subtracting the Laplace pressure from the
disjoining pressure; namely,
P P 2/R (1.7)
c
where P is the Laplace pressure due to the changes in curvature in the flat film compared to the
c
meniscus neighboring the flat film. In eq. (1.7), P is equal to 2σ/R, where R is the radius of the
c
meniscus at the edge. As shown in eq. (1.6), the disjoining pressure can be determined when the
rate of the film drainage (dh/dt) is known.
Experimentally, the film thickness was determined using the microinterferometric methods by
monitoring the interference of the light beams reflected from two adjacent interfaces of a wetting
film. Instead of recording one photo of the interference fringe at equilibrium in the static method,
continuous photos of the interference fringes were captured and analyzed to obtain the film
thickness by means of either a photometer or a CCD camera [84, 86-92].
Platikanov [86] experimentally observed the dimpled film by monitoring the temporal
changes in the film thickness between an air bubble and a glass plate in an aqueous solution. He
captured a series of interference patterns when an air bubble was approaching a silica surface
separated by a thin liquid film. The kinetics of the film drainage was compared with the Frankel-
Mysels’ model [93]. It was found that a positive disjoining pressure was present in a wetting film
of water formed on a glass surface. In a thin liquid film of benzene on a mercury surface,
Scheludko and Platikanov [84] showed a negative disjoining pressure acting in a wetting film
11 |
Virginia Tech | due to the van der Waals attractions. Later in 1972, Schulze [87] measured a negative disjoining
pressure in a wetting film of octane on a quartz crystal surface. A negative disjoining pressure
due to the van der Waals attraction was obtained with a maximum attraction at h = 33 nm.
Scheludko's method of determining the disjoining pressure in a wetting film was achieved by
comparing the theoretical prediction with the experimental data. The disjoining pressure was
determined by simulating the kinetics of the film drainage with the theoretical prediction based
upon the Reynolds approximation. The approximation assumes that the drainage occurs between
two plane-parallel surfaces with no thickness variances at the interfaces. It works for the smaller
bubbles with less hydrodynamic resistance, while the theory overestimates the drainage rate
when the deformation occurs due to a larger hydrodynamic resistance. In order to simulate the
drainage process of the thin liquid film between a deformable air bubble and a rigid surface,
many fluid models have been proposed and derived under these dynamic conditions [94-97],
with a number of models considering the effect of the van der Waals force [98-100] and
electrostatic double layers force [101].
Alternatively, Nakamura and Uchida [102] studied the wetting films by monitoring the
interference fringes of a wetting film when an air bubble was freely rising towards a glass plate.
Under the buoyancy of the air bubble, a film confined between a vapor/water interface and a
solid/water interface was drained spontaneously until an equilibrium film was reached. The
kinetics of the wetting film drainage was obtained to determine the disjoining pressure by
monitoring the changes in the film thickness using the interference methods. Parkinson and
Ralston [91] followed a method similar to Nakamura and Uchida by releasing a tiny air bubble
towards a titania surface in an aqueous solution. The kinetics of the film drainage was captured
from the interference fringes by means of a high-speed camera at 1000 hz. The film drainage
kinetics were described by the Taylor equation [103],
dh Fh
b (1.8)
dt 6R2
where F is a sum of forces acting on the air bubble. It contains both the hydrodynamic and
b
surface forces. The surface force between an air bubble and a solid surface across a thin liquid
film can be acquired when dh/dt is obtained.
12 |
Virginia Tech | A multi-wavelength interferometry technique offers an alternative solution for the thickness
measurement of the thin liquid film between an air bubble and a solid surface. Pushkarova and
Horn [104, 105] modified the surface force apparatus (SFA) to measure the interfacial profiles of
the thin liquid films between an air bubble and a silver-coated mica surface by monitoring the
reflected FECO (Fringes of Equal Chromatic Order). The air bubble was manually fixed on a
customized Teflon block with a pin hole on the surface. The use of the multi-wavelength
interferometry technique for the thickness measurement of the thin liquid film exhibits a high
resolution of the film thickness below 30 nm.
With the invention of the atomic force microscopy (AFM), some investigators applied the
AFM technique to measure the interaction directly between an air bubble and a solid particle in a
liquid. Many attempts have been taken for a force measurement between an air bubble and a
particle in water [46, 48, 106-108]. Ducker et al. [46] might be the first to measure the force
between an air bubble and a spherical particle using AFM. The AFM force measurement was
conducted by attaching a particle on the tip of a cantilever and fixing an air bubble on a
hydrophobic surface. The interaction force was recorded by monitoring the deflection of the
cantilever when the air bubble approached towards the particle. A follow-up experiment was
conducted by Preuss and Butt [48], showing a repulsion due to the electrostatic double-layer
force between the air bubble and the particle. When the bubble was driven against the
hydrophobic sphere, the force jumped to an adhesion at a large separation distance.
However, the direct measurement of the interaction force between an air bubble and a solid
particle was often accompanied with deformation of the air bubble under both the hydrodynamic
and surface forces. Therefore, the force obtained directly using the AFM technique is composed
of both the hydrodynamic and surface forces due to intermolecular interaction. A solution was
proposed by Chan et al. [109-112], who derived a mathematical model to predict the dynamic
interaction between an air bubble and a rigid solid surface in a liquid.
Manor et al. [113, 114] measured the interaction force between an air bubble and a solid
surface directly using the AFM technique by attaching an air bubble on the surface of the
cantilever. The disjoining pressure in a thin liquid (or the surface force between the bubble and
the solid surface) was extracted by fitting the experimentally measured force with the theoretical
prediction using Chan's model on the basis of the lubrication theory [109]. The lubrication model
13 |
Virginia Tech | quantitatively predicted the film profiles of the thin liquid film. However, the film profiles varied
significantly when the interfacial boundary condition was slightly changed. Krasowska et al.
[115] also applied the AFM technique to study the interaction force between an air bubble and a
titania particle in a KCl solution. The particle was glued on the AFM tip, while the air bubble
was fixed on the hydrophobic surface and positioned at the bottom of the fluid cell. Using Chan's
model on the dynamic interaction between an air bubble and a particle, the raw AFM data for the
bubble-particle interaction was interpreted to extract the disjoining pressure in the thin liquid
film and the projected spatiotemporal profiles of the thin liquid films.
When comparing the dynamic methods with the static methods in determining the disjoining
pressure in a wetting film, the dynamic methods exhibit the applicability in determining the
disjoining pressures in wetting films with a variety of configurations. However, the dynamic
methods are often influenced by the fluid stresses during the course of the interaction. Therefore,
the determination of the disjoining pressure was not possible without an accurate fluid model for
the dynamic methods. Additionally, the dynamic methods showed an ability of determining a
negative disjoining pressure in an unstable wetting film, which played a significant role in
understanding the fundamentals of the coagulation processes.
1.3 Dissertation Outline
The objective of the present study is to determine the interaction forces between air bubbles
and solid surfaces in water. This work is aimed to measure the negative disjoining pressures in
the wetting films of water formed on the hydrophobic solid surfaces and discuss the origin of the
attractive pressure that cannot be predicted by the DLVO theory.
The present work is focused on studying both the surface and hydrodynamic forces operating
between air bubbles and solid surfaces during the course of the bubble-particle interaction. In
Chapter 2, a modified thin film pressure balance (TFPB) technique has been used to monitor the
changes in the thickness profiles of the thin liquid film between an air bubble and a solid surface.
A mathematical model based on the lubrication theory has been derived to determine the
disjoining pressures (or surface forces) in wetting films.
In Chapters 3 and 4, the role of collectors and the effect of bubble size in the stability of
wetting films have been studied by measuring the drainage rates of the wetting films formed on
14 |
Virginia Tech | the surface hydrophobic gold surfaces. The surface hydrophobicity was controlled by varying the
collector concentration and the contact time for hydrophobization. The results will be useful for
better understanding the factors affecting flotation efficiency.
In Chapter 5, the asymmetric hydrophobic interactions between bubble and gold are predicted
from the symmetric hydrophobic interactions between two air bubbles and between two gold
surfaces. It has been found that the asymmetric interactions can be predicted using the geometric
mean combining rule, which is used for molecular interactions such as van der Waals interaction.
In Chapter 6, the Frumkin-Derjaguin isotherm is used to predict the stability of wetting films
due to electrical double-layer interactions. The results show that wetting films becomes stable
when the double-layer forces are repulsive and unstable when the double-layer forces are
attractive.
In Chapter 7, a new scientific instrument that can be used to directly measure for the first time
the interaction forces between a deformable object such as air bubble and a solid surface. The
instrument, which is referred to as force apparatus for deformable surfaces (FADS), consists of
two optical systems: one for determining temporal and spatial film profiles, and the other for
monitoring the deflection of a cantilever for direct surface force measurements.
In Chapter 8, the experimental results obtained using the FADS developed in Chapter 7 have
been analyzed by the numerical methods. A model has been derived on the basis of the
lubrication theory and the DLVO theory. The results presented in this chapter are for stable
wetting films due to positive disjoining pressures.
Chapter 9 is the same as Chapter 8 except that the theoretical analysis has been made for the
unstable wetting films of water formed on hydrophilic surfaces.
In Chapter 10, the FADS has been used to study the interaction between an air bubble and
gold surfaces hydrophobized by potassium ethyl xanthate (KEX). The results show that the
disjoining pressure is negative due to the presence of the hydrophobic force not considered in the
DLVO theory.
In Chapter 11, the results obtained in the present work are summarized and conclusions are
drawn, which suggestions for future research presented at the end.
15 |
Virginia Tech | Chapter 2. Effect of Hydrophobicity on the Stability of
the Wetting Films of Water Formed on Gold Surfaces
ABSTRACT
We have developed a methodology that can be used to determine disjoining pressures (Π) in
both stable and unstable wetting films from the spatial and temporal profiles of dynamic wetting
films. The results show that wetting films drain initially by the capillary pressure created by the
changes in curvature at the air/water interface and subsequently by the disjoining pressure
created by surface forces. The drainage rate of the film formed on a gold surface with a receding
contact angle (θ) of 17o decreases with film thickness due to a corresponding increase in positive
r
Π, resulting in the formation of a stable film. The wetting film formed on a hydrophobic gold
with θ = 81o drains much faster due to the presence of negative Π in the film, resulting in film
r
rupture. Analysis of the experimental data using the Frumkin-Derjaguin isotherm suggests that
short-range hydrophobic forces are responsible for film rupture and long-range hydrophobic
forces accelerate film thinning.
25 |
Virginia Tech | 2.1 Introduction
Wetting films form on surfaces by vapor deposition or when particles and bubbles suspended
in liquids encounter each other under dynamic conditions. Thus, wetting films are ubiquitous and
play important roles in daily lives and many industrial processes, such as flotation, foam and
emulsion destabilization, food processing, self-cleaning materials design, heat transfer, corrosion
protection, etc. The liquid in a wetting film drains by gravity and/or capillary pressure initially.
When the film thickness is sufficiently reduced, the drainage rate and the stability of the thin
liquid film (TLF) are also controlled by the pressure created due to the surface forces acting
between the walls confining the liquid. The pressure in a TLF is referred to as disjoining (or
wedging apart) pressure (Π), which is defined as the change in excess Gibbs free energy per unit
area of a flat film (G) with film thickness (h) [1, 2],
(h)(G/h) (2.1)
p,T,
S
at constant pressure (p), temperature (T), and chemical potential of solutes (μ) [1]. The TLF is
s
stable when ∂Π/∂h < 0. The concept of disjoining pressure, introduced by Derjaguin [3] in 1935,
served as a basis for modeling the stability of not only the wetting films but also colloidal
suspensions.
As is well known, TLFs are stable on hydrophilic surfaces if the liquid is water. On the
surface of a well-cleaned hydrophilic quartz, thick wetting films known as β films are formed [1,
4]. As the mineral substrate becomes less hydrophilic by heating or methylation, the film
becomes thinner and ruptures spontaneously, forming lenses of water on the surface [5]. At
equilibrium, the lenses form contact angles (θ ) on a partially hydrophobic surface, with the
o
surface covered by a thinner film known as α film [6]. Thus, the rupture of wetting films
represents a β to α phase transition.
That β films are substantially thicker than expected from the DLVO theory lead to a
suggestion that the “structural forces” not considered in the theory may be present in the wetting
films formed on hydrophilic surfaces. The term was first introduced by Derjaguin and Kussakov
[4], who considered that the boundary layers of water have structures different from that of the
bulk water. As two surfaces approach each other, the boundary layers overlap and give rise to
short-range repulsive forces. The hydration forces measured between silica and glass surfaces in
26 |
Virginia Tech | water are regarded as structural forces [7-9], while others view that they are caused by the silica
hairs formed on the surface [10].
Thermodynamically, a wetting film can rupture when the θ of water on a substrate is greater
o
than zero. In view of the Frumkin-Derjaguin isotherm [11, 12], Laskowski and Kitchener [13]
wrote: “The criterion for a contact angle to develop is that films in a certain range of thickness
would be subject to a negative disjoining pressure,” hinting the presence of a long-range
attractive force in unstable wetting films. Derjaguin and Churaev [14] showed later that the
Frumkin-Derjaguin isotherm can explain the increase in θ of quartz up to 15-16o with the
o
decrease in the electrostatic component of disjoining pressure that can be controlled by pH and
electrolyte (KCl) concentrations. The authors noted, however, that larger increases in θ cannot
o
be explained without recognizing the actions of hydrophobic attraction forces [14-16].
Blake and Kitchener [17] found that the wetting films of water formed on methylated silica
were meta-stable due to the large repulsive double-layer forces present in the films. In the
presence of 8.6 x 10-3 M KCl, however, the film ruptured at 64 nm. This observation was
attributed to a double-layer compression, which in turn brought the film to the domain of
“hydrophobic force”. In this regard, Blake and Kitchener considered the hydrophobic force a
shorter-range force than the double-layer force.
Israelachvili and Pashley [18] were the first to directly measure the short-range hydrophobic
force between two curved mica surfaces using the surface force apparatus (SFA). Since then,
many other investigators reported the measurement of both short- and long-range hydrophobic
forces using SFA and atomic force microscope (AFM). The measured forces are often
represented in the following form [19],
F/R C exp(h/D )C exp(h/D ) (2.2)
1 1 2 2
where F/R is the force normalized by the radius of curvature of the macroscopic surface(s) used
for the measurement, h the closest separation distance between the macroscopic surfaces, and C ,
1
C , D and D are fitting parameters. The first and second exponential terms of Eq. (2.2)
2 1 2
represent the short- and long-range hydrophobic forces, respectively, with D being in the ranges
1
of 1-2.8 nm at h < 10 nm and D in the range of 4.5-50 nm at longer separations [19-21]. Some of
2
the experimental data show two distinct regions where short- and long-range forces are
27 |
Virginia Tech | predominant, while other data can be fitted to a single-exponential function. Still other
investigators used a power law to represent hydrophobic forces [22, 23],
F/R K /6h2 (2.3)
Eq. (2.3) is of the same form as the van der Waals-dispersion force, with K being the only
adjustable parameter.
Eriksson et al. [24] derived a theoretical model for hydrophobic force based on the idea that
the intervening water between two surfaces is progressively more ordered with decreasing h,
which is consistent with the concept of the structural force. The model suggests an increase in
order parameter with decreasing h and corresponding decreases in enthalpy and free energy per
unit area. The authors related the free energy changes to surface forces using the Derjaguin
approximation [25] and modeled hydrophobic surface forces in the middle and long ranges at h >
2 nm. Other investigators attributed the long-range attractive forces to the electrostatic attractions
between charged [26-28] and uncharged [29] domains, pre-existing nano-bubbles [30, 31],
cavitation [32], and others. It appeared for a while that the long-range attractions are indeed due
to the nano-bubbles and/or cavitation. Typical evidences for this possibility included the
discontinuities (or ‘steps’) in force vs. distance curves, disappearance of attractive forces in
degassed water, and actual observation of nano-bubbles on hydrophobic surfaces. More recent
studies showed, however, that long-range attractions can still be observed in degassed solutions
[33, 34], and that the steps can be avoided by modifying the conditions and procedures involved
in force measurement [35].
Another difficulty for accepting the hydrophobic force as a structural force is that molecular
dynamic (MD) simulations invariably showed that structuring of water is limited to only a few
layers of solvent molecules, while the long-range attractions are observed in experiment at h ≈ 80
nm. It should be noted, however, that the excess free energies per molecule related to the long-
range attractions are in the range of 10-5 to 10-3 kT, while the energy of a hydrogen bond is about
7 kT [24]. Therefore, it would be a challenge to detect the structural changes responsible for the
long-range attractions using the MD simulations, which would have difficulties in detecting
changes in energy below approximately 0.1 kT per molecule due to limitations in accuracy.
28 |
Virginia Tech | The most recent view on the origin of hydrophobic forces has been presented by Hammer et
al. [36]. According to these authors, the attractive forces measured in the range of 1.5-15 nm are
‘pure’ but still not well-understood long-range forces possibly due to the enhanced Hamaker
constant, while the attractions below 1.5 nm may be due to ‘pure’ short-range hydrophobic
forces related to water structuring effect. On the other hand, the long-range forces observed at h >
15-20 nm may be due to the electrostatic attractions between charged patches and/or bridging
vapor cavities.
Studies of wetting films provided useful information for better understanding the surface
forces present in TLFs. It was found that the repulsive structural forces measured in the wetting
films of water on silica and glass surfaces are similar to the hydration forces measured in direct
surface force measurements [7, 8]. Also, Churaev [16, 37] suggested that the attractive structural
forces found in the wetting films on partially hydrophobic silica surface are short-range
hydrophobic forces. It should be pointed out, however, that this conclusion was drawn by
extrapolating the positive disjoining pressures measured on metastable films rather than from the
negative disjoining pressures measured directly in experiment. The bubble-against-plate
technique developed originally by Derjaguin and Kussakov can be used to measure Π > 0 [38].
However, it has been a challenge to directly measure Π < 0 due to several reasons, including fast
kinetics of film thinning, deformation of air/water interface, and complex interactions between
hydrodynamic and surface forces. In 1969, Laskowski and Kitchener [13] wrote: “There is no
theory leading to even approximate calculation of negative disjoining pressures on hydrophobic
surfaces.” More recently, many investigators reported the measurement of the surface forces
involved in bubble-particle interactions using SFA and AFM, with inconsistent results [39-42].
Horn and his co-workers [43, 44] studied the interactions between mica surface and mercury
drop in a 0.1 mM KCl solution by recording the spatial and temporal profiles of the intervening
films using a modified SFA and a video camera. The authors compared the results with the
profiles obtained from the theoretical predictions made using the Reynolds lubrication theory,
with excellent agreements. The predictions involved the use of the disjoining pressures
calculated using the DLVO theory from the known potentials at the mercury/solution and
mica/solution interfaces. In the mica-mercury system studied, the attractive van der Waals force
was minimal as compared to the double layer forces.
29 |
Virginia Tech | In the present work, we used the bubble-against-plate technique to directly measure both the
positive and negative disjoining pressures in the wetting films of water formed on gold surfaces.
Using a high-speed video camera, we recorded the Newton rings as a function of time to obtain
spatial and temporal film profiles of the wetting films and extracted appropriate hydrodynamic
information that can be used to obtain disjoining pressure isotherms, Π(h), using an analytical
expression derived in the present work. The expression was derived from the Reynolds
lubrication theory and has a provision to study the effect of slip, which has been a topic of
interest and debate in recent years. From the experimentally obtained Π(h) isotherms, it was
possible to obtain the information on hydrophobic disjoining pressure (Π ), which plays a
h
dominant role at contact angles above approximately 40o [16]. The methodology developed here
is a step forward from an earlier approach, in which the Reynolds lubrication approximation [45]
was used to determine the hydrophobic disjoining pressures at different contact angles [46].
2.2 Model Derivation for Disjoining Pressure
When an air bubble is pressed against a plate lying horizontally in water, a thin liquid film is
formed in between. The pressure of the liquid in the film becomes higher than that of the bulk
liquid due to the curvature difference, causing the film to thin. As the film thinning continues,
disjoining pressure (Π) in the film also begins to play a role. If it is positive (or repulsive), the
film thinning is retarded. In the present system, the repulsive disjoining pressure arises from
double-layer interaction and increases exponentially with decreasing h. Thus, there can be a
point where the excess pressure due to curvature change (p )becomes equal to Π, when the film
cur
thinning stops at an equilibrium thickness (h ). If a water film is formed on a substantially
e
hydrophobic surface, the disjoining pressure can be negative (or attractive) and can facilitate the
film thinning process. In this case, the film ruptures catastrophically at a critical thickness (h ).
c
Thermodynamically, the rupture occurs when contact angle is greater than zero. Thus, the
process of film thinning is controlled initially by p (or capillary pressure), while Π plays a
cur
significant role during later stages.
The film thinning process can be described by considering the force balance in the fluid using
the Navier-Stokes equation, which is highly nonlinear and difficult to solve analytically. For
TLFs and at low fluid velocities, it can be reduced to the linear Stokes' equation,
30 |
Virginia Tech | p 2u (2.4)
where p is the excess pressure in the film relative to the pressure in the far field, u the fluid
velocity, and µ the fluid viscosity.
By combining Eq. (2.4) with the continuity equation for film thinning, one obtains the
Reynolds lubrication equation [47],
1 h h 1 p
U rh3 (2.5)
2 u r t 12r r r
where h the film thickness, r the radial position, and U is the slip velocity at the interface. Eq.
u
(2.5) can be reduced to an approximate form as follows [45],
dh 2h3p
dt 3R2
(2.6)
under conditions of flat films and no slip at phase boundaries, i.e., U = 0. Eq. (2.6) has been used
u
widely to explain the kinetics of film thinning [45, 48].
Eq. [5] can be integrated twice to obtain
r 1 r h p
p 12r dr dr p(r ) (2.7)
rrh3
r0 t r
r0
By integrating Eq. (2.7) again under the boundary conditions that p(r ) 0 and
p/r 0, one obtains
r0
r 1 r h
p12 r dr dr (2.8)
rrh3 r0 t
A normal stress balance across the interface gives a relation between the excess pressure,
disjoining pressure, and curvature pressures as
p p (2.9)
cur
For a weakly-changing fluid height, i.e., h/r << 1, the curvature pressure becomes,
2 h
p r (2.10)
cur R r r r
31 |
Virginia Tech | where R is the radius of the bubble in the far field, and γ is the air/water interfacial tension.
Substituting Eqs. (2.8) and (2.10) into Eq. (2.9), one obtains an expression for the disjoining
pressure
2 h r 1 r h
r 12 r dr dr (2.11)
R r r r rrh3 r0 t
which is a function of h(r,t) only. Thus, a disjoining pressure isotherm, Π(h), can be numerically
determined if the spatial and temporal profiles of a wetting film can be obtained in experiment.
2.3 Experiment
2.3.1 Materials
Gold-coated glass plates (CA134, EMF) were used as substrates for wetting films. They were
cleaned in a boiling piranha solution (7:3 by volume of H SO :H O ) for 1 hour, rinsed with
2 4 2 2
Millipore water (>18.2 MΩ/cm), and then dried in a nitrogen gas stream [35]. The freshly-
cleaned gold substrate exhibited r.m.s. roughness of 1-2 nm as determined from AFM images.
The gold plate obtained in this manner exhibited an equilibrium contact angle (θ ) of 42° with
o
receding (θ) and advancing (θ ) angles of 17o and 60°, respectively, as measured by using the
r a
dynamic sessile drop technique.
A gold plate obtained in the manner described above was rendered hydrophobic by immersing
it in a 10-5 M potassium amyl xanthate (KAX) solution for 10 min, followed by rinsing with
Millipore water and drying in a nitrogen gas stream. The xanthate-treated gold exhibited contact
angles of θ = 81o, θ = 91°, and θ = 99°. The hydrophobizing agent had been purified twice just
r o a
before use by dissolving a technical grade KAX (>90%, TCI America) in acetone (HPLC grade,
Fisher Sci.) and re-crystallizing in diethyl ether (99.999%, Sigma-Aldrich).
2.3.2 Procedure
Both the xanthate-treated and untreated gold plates were used as substrates for wetting films
of water. The kinetics of film thinning was monitored using the thin film pressure balance (TFPB)
technique described previously [46]. In each experiment, a gold plate was placed on a film
32 |
Virginia Tech | Figure 2.1 Schematics of the wetting film holder placed in the thin film pressure balance
(TFPB) apparatus. The method of studying wetting films used in the present
work is similar to the bubble-against-plate technique of Derjaguin and
Kussakov [4] except that the spatial and temporal profiles of film is obtained
by recording the Newton rings with a high-speed video camera to obtain
information on both stable and unstable wetting films. The syringe pump is
used to control the radius of the thin liquid film between the air bubble and
plate.
holder with a 2-mm inner diameter as shown in Figure 2.1. The film holder was then placed in a
vapor-saturated chamber (to avoid evaporation of the film), which in turn was placed on an
inverted microscope (Olympus IX51) equipped with a tilting stage (M-044.00, Polytec PI). A
monochromatic light source with a center wavelength of 526 nm was directed to the wetting film
so that the reflected light can produce optical interference patterns (Newton rings) as curvature of
the air/water interface changes with time.
Initially, the excess water in the film holder was removed by means of a piston pump until the
first Newton ring appeared on the camera screen. The film was then allowed to thin
spontaneously, while recording the interference patterns as a function of time using a CCD
camera (Fastcam 512PCI, Photron) at 60 frames per second. The data recorded in this manner
were used to obtain the temporal and spatial profiles of wetting films, which were then used to
obtain the values of radial flow velocities and the values of h/r and h/t that are necessary
to determine Π using Eq. (2.11).
33 |
Virginia Tech | Film thinning experiments were conducted with wetting films smaller than 20 μm in radius,
so that dimple formation could be avoided. The film radii were controlled by means of the piston
pump, while the tilting stage was used to ensure that the films were disposed horizontally.
Dimples are formed when the liquid near the edge of a film drains substantially faster than at the
center, which is the case with large films [49]. Eq. (2.11) can be used for the data obtained from
both dimpled and flat films. With the former, only the data collected in the barrier rims can be
used to calculate Π. With the latter, the data obtained in the whole area of a film can be used. It
may be noteworthy that the images of the wetting films obtained in the present work were
smooth and showed no irregularities.
2.4 Result
Figure 2.2 compares the temporal profiles of the wetting films formed on the gold surfaces
with and without hydrophobization by KAX. The untreated substrate exhibited θ = 17o, while
r
the treated one exhibited θ = 81o. Other investigators also reported non-zero contact angles for
r
gold surfaces cleaned using various methods [50]. This may be attributed to the abstraction of
impurities from the air owing to the large Hamaker constant of gold. Note that the wetting film
formed on the hydrophobized gold surface thinned much faster than that formed on the untreated
surface. With the latter, an equilibrium film thickness (h ) of 105 nm was reached in 11 s, while
e
with the former the film thinned to a thickness of 29 nm in 0.8 s and ruptured. The values of h
e
and the critical rupture thickness (h ) obtained in the present work are within the range of values
c
reported in the literature [51, 52].
34 |
Virginia Tech | Figure 2.3 shows the temporal changes in the average radial velocity, u(r,t), of the liquid in the
wetting films. The values were obtained using the continuity equation,
h 1 (rhu)
(2.12)
t r r
under nonslip boundary conditions on both interfaces. Eq. (2.12) was used to determine u(r,t)
from the velocities of film thinning (∂h/∂t) obtained from the temporal profiles (h(r,t)) of the
films presented in Figure 2.2. Since the profiles are axial-symmetric, the radial velocity, u(0,t), at
the center of the film is zero.
During film thinning, u increased with r, reached a maximum, and then decreased as r
increased further. In the film formed on the untreated gold surface, u decreased steadily as the
film became thinner, which can be attributed to the increase in positive (or repulsive) disjoining
pressure and, hence, a decrease in p with decreasing film thickness. At t = 11 s, u was almost
zero, indicating that the film reached an equilibrium thickness, at which disjoining pressure was
equal to the capillary pressure and p ≈ 0.
In the wetting film formed on the KAX-treated hydrophobic gold, u increased with time at r <
0.02-0.03 mm. The acceleration of the film thinning kinetics observed in this region may be due
to the presence of a negative (attractive) disjoining pressure in the film. Note here that the
acceleration increased with decreasing film thickness particularly at h below approximately 100
nm. At the radial distances above approximately 0.03 mm, the film behaved similarly as that
formed on the untreated gold surface. In this region, the films were too thick for the negative
disjoining pressure to play a role in film thinning, particularly during the initial stages of the film
thinning process.
From the radial velocity gradients in the vertical (z) direction, the shear rates, ∂u(z)/∂z, at the
solid/liquid interface were obtained, with the results plotted in Figure 2.4. The calculations were
made under the nonslip boundary condition at both the air/water and solid/water interfaces. On
the untreated gold surface (Fig. 2.4a), the shear rate decreased steadily with decreasing film
thickness and reached zero at the equilibrium film thickness.
The shear rates calculated for the film formed on the hydrophobic gold (Fig. 2.4b) were much
higher. However, the shear rates obtained during the initial stages of the film thinning process
were about the same as those calculated for the film formed on the hydrophilic gold (Fig. 2.4a).
38 |
Virginia Tech | Figure 2.5 Comparison of the excess pressures (p) in the thin films of water formed on
the gold surfaces with a) θ = 17o and b) θ = 81o. The large values of p in b)
r r
are due to the negative disjoining pressure in the film and are responsible for
the high drainage rates.
For example, the shear rate was 300 s-1 at t = 0 s, which was the same as obtained for the film
formed on the untreated gold. This finding suggests that the initial film thinning process is
controlled by the excess pressure created due to curvature change (p ). As the film thinning
cur
continued, the shear rate increased with time and reached a maximum of 5500 s-1 at t = 0.8 s. The
increase in shear rate may be attributed to the increase in the negative disjoining pressure with
decreasing film thickness.
Eqs. (2.5) and (2.6) show that under the nonslip condition, i.e.,U = 0, the kinetics of film
u
thinning varies with the excess pressure (p) in a wetting film. Figure 2.5 shows the values of p
plotted vs. the radial distance (r) as obtained using Eq. (2.8). At t = 0 s, the p in the wetting film
formed on the surface of untreated gold was 50 N/m2 at the center, which is close to the value of
39 |
Virginia Tech | dramatically to 2000 N/m2 at t = 0.7 s, when the local film thickness at the center reached 44 nm.
It can be stated, therefore, that the large value of p and the high pressure gradient were
responsible for the fast film thinning kinetics observed in Figure 2.2.
The driving force for film thinning is the excess hydrodynamic pressure (p) in the film, which
in turn varies with the pressure change due to curvature (p ) and the disjoining pressure in the
cur
film (Π), as shown in Eq. (2.9). Figure 6 shows the changes in p, p and Π with film thickness
cur
(h), as obtained using Eqs. (2.8), (2.10) and (2.11), respectively. In these calculations, the values
of γ = 72 mN/m and R = 2 mm were used. In the wetting films formed on both the untreated and
treated gold surfaces, p increased with decreasing film thickness and reached a plateau value of
cur
72 N/m2. In the film formed on the untreated surface with θ = 17o, the p at the center of the film
r
increased slightly as the film thinned from approximately 300 to 200 nm, where p was dominated
by p . As the film drained further, p decreased sharply due to the corresponding increase in the
cur
positive disjoining pressure. In the wetting film formed on the KAX-treated gold surface with θ
r
= 81o, however, p increased sharply with decreasing film thickness due to the increase in the
negative disjoining pressure. Obviously, the sharp increase in p was responsible for the fast
kinetics of film thinning shown in Fig. 2.2b. Note here that at h < 100 nm, p is negligibly small
cur
in magnitude as compared to Π. Thus, film thinning is controlled initially by the curvature
pressure and subsequently by the negative disjoining pressure.
2.5 Discussion
The results presented in the foregoing section show that the driving force for the drainage of
wetting films is the excess pressure (p) in the films. Initially, p is dominated by the pressure
associated with the changes in curvature (p ) of the air/water interface and then by the
cur
disjoining pressure (Π) created by the surface forces in the film.
Figure 2.7 shows the changes in these parameters with the thickness (h) of the wetting films
formed on the untreated gold surfaces with θ = 17o and the KAX-treated gold with θ = 81o. In
r r
the former, Π may consist of two components in accordance to the DLVO theory
(2.13)
d e
i.e., the dispersion force component,
41 |
Virginia Tech | A
132
d 6h3
(2.14)
and the electrostatic force component due to double-layer interaction,
2
0 2 2 cosech(h)2 coth(h)
e 2sinh(h) 1 2 1 2
(2.15)
In Eq. [14], A is the Hamaker constant for the wetting film of water formed on gold. Eq. [15] is
132
the Hogg–Healey–Fuerstenau (HHF) approximation [53], in which ε is the permittivity in
0
vacuum, dielectric constant of water, ψ and ψ the double-layer potentials at the solid/water
1 2
and air/water interfaces, respectively, and κ is the reciprocal Debye length. In this
communication, the subscripts 1, 2, and 3 represent solid, gas, and water phases, respectively.
Other investigators also used the HHF approximation to study bubble-particle interactions [42,
54, 55].
In using Eq. [14], the value of A (= -14.8x10-20 J) was used as reported by Tabor et al. [56].
132
These investigators used an AFM to directly measure the repulsive van der Waals force between
air bubble and bare gold surface in water. The A value was close to those that can be
132
calculated using the geometric mean combining rule with the Hamaker constants for gold (10-
40x10-20 J) from the full Lifshitz equation and for air bubbles (3.7x10-20 J) in water [57].
In using Eq. [15], ψ was taken to be the same as the -potential of the micro-spheres of gold
1
in distilled water, which was reported to be -40 mV [46]. On the other hand, the values of ψ = -
2
29 mV and -1 = 42 nm were obtained by fitting the Π values (circles), obtained from the
temporal film profiles shown in Figure 2a using Eq. (2.11), to Eqs. (2.13)-(2.15). As shown in
Fig. 2.7a, the fit is reasonable. The value of -29 mV obtained for the air/water interface from the
fitting procedure is close to the -potentials of argon bubbles in distilled water as reported by
Usui et al. [58]. These authors calculated the -potentials from the Dorn potentials measured
using bubble columns. They found that unlike the case with glass beads, the -potentials of
bubbles become less negative with increasing size, which was attributed to the likelihood that the
surface charges are displaced backward resulting in a weaker electric dipole and hence a reduced
Dorn potential [59]. Likewise, the charges at the air/water interface of a wetting film may
traverse toward the edge of the film and give rise to the same effect. Note here that the Debye
42 |
Virginia Tech | Figure 2.7 Disjoining pressure isotherms (Π(h)) for due to different surface forces in
the wetting films formed on gold surfaces with a) θ = 17o and b) θ = 81o.
r r
The circles in a) shows the values of Π obtained using Eq. (2.11) from the
film profiles in Figure 2.2a. The electrostatic disjoining pressure (Π ) was
e
obtained by subtracting the van der Waals disjoining pressure (Π ) from
d
Π. The equilibrium film thickness (h ) occurs when Π is equal to the
e
curvature pressure (p ). The triangles in b) represent the values of Π
cur
obtained using Eq. (2.11) from the film profiles in Figure 2.2b. The
dashed line in b) shows Π(h) represented by Eq. (2.17). The hydrophobic
disjoining pressure (Π ) was obtained by subtracting Π and Π from Π(h).
h e d
length obtained from the curve fitting exercise was relatively small for the interactions in
Millipore water. In a previous work, one of us fitted the force curves obtained between glass
43 |
Virginia Tech | sphere and silica plate in Nanopure water using AFM with -1 = 42 nm [22].
Note in Figure 2.7a that both the electrostatic and van der Waals force components of the
disjoining pressure were repulsive, with the former being much larger. Also shown in the figure
are the changes in curvature pressure (p ) with h. At the crossover point between the p vs. h
cur cur
and the Π vs. h curves, Π is equal to p . At this point, the excess pressure in the film (p)
cur
becomes zero according to Eq. (2.6), when the film thinning process stops, resulting in the
formation of a stable film at an equilibrium thickness (h ). As shown, the value of h = 105 nm
e e
obtained in experiment agrees well with the predictions from the DLVO theory.
Figure 2.7b shows the results obtained with the KAX-treated gold with θ = 81o. The triangles
r
represent the Π values obtained from the temporal and spatial film profiles obtained in
experiment using Eq. (2.11). As shown, Π is negative and becomes increasingly so with
decreasing film thickness. Also shown are the isotherms of Π and Π drawn by assuming that
e d
both ψ and A do not change due to xanthate adsorption. Several investigators showed indeed
1 131
that the -potentials of sulfide minerals do not change significantly upon xanthate adsorption
[60]. That Π < 0 while both Π and Π are positive show that there should be an attractive force,
e d
i.e., hydrophobic force, in the wetting film under consideration. Thus, one may write the
extended [61] (or modified [62]) DLVO theory
(2.16)
d e h
in which Π represents the contribution from the hydrophobic force. Since the values of Π, Π
h d
and Π are known, one can back-calculate the values of Π using Eq. (2.16). In the present work,
e h
we represent the hydrophobic disjoining pressure as follows,
C h
exp (2.17)
h 2D D
where C and D (decay length) are constants for the surface forces measured between two
cylinders or between sphere and flat surfaces. We adjusted these two parameters to obtain the
values of Π that can best fit the Π values obtained from the film profiles. It was found that C = -
h
1.55 mN/m and D = 36 nm can best fit the experimental data. The Π and Π isotherms obtained
h
in this manner are given as dotted and solid lines, respectively, in Figure 2.7b. Owing to the large
negative Π, p increases beyond p according to Eq. (2.9), and hence the film thinning process
cur
44 |
Virginia Tech | accelerates until the film ruptures. The large decay length obtained in the present work might be
partly due to the fact that we did not consider the capillary wave mechanism [63, 64]. Manica et
al. [43] showed, however that it is not necessary to invoke capillary waves to predict the time at
which the aqueous film between mica and mercury collapses. The C and D values obtained from
the data presented in Figure 2.7b are comparable to those reported in the literature for the
hydrophobic interactions between two macroscopic surfaces [35, 65, 66].
Thermodynamically, wetting films rupture when contact angle (θ) is greater than zero, or the
wetting tension (γ - γ ) is less than the surface tension of water (γ ). For the process of bubble-
12 13 23
particle attachment, receding angle (θ) would be more relevant than the equilibrium (θ ) or
r o
advancing (θ ) angles. In the present work, two gold surfaces with θ = 42o and 91o have been
a o
studied. Only with the latter, we observed film rupture, which can be explained in view of the
Frumkin and Derjaguin isotherm [11, 12],
h
0
cos 1(1/ )(h)dh (2.18)
o 23
where h is the thickness of the film in equilibrium with the meniscus of water. From Eq. (2.2),
0
we can obtain an expression for the hydrophobic disjoining pressure as follows,
1 C h C h
h
2 D1 1exp
D
1 D2 2exp
D
2
(2.19)
where the parameters with subscripts 1 and 2 represent those for the short- and long-range
hydrophobic forces, respectively. One can substitute Eq (2.19) into Eq. (2.16) along with Eqs.
(2.14) and (2.15) to obtain Π(h), which can then be substituted into Eq. (2.18) for integration. By
integrating it from infinity to h , one obtains the following
0
cos 1G/
o 23
1 A 2 exp(h )2 2 C h C h
1 132 1 2 0 1 2 1 exp( 0 ) 2 exp( 0 )
12h2 o exp(2h )1 2 D 2 D
23 0 0 1 2
(2.20)
From Eq. (2.1),
45 |
Virginia Tech | h
o
G (h)dh
(2.21)
(cos 1)
23 0
where ΔG is the Gibbs free energy change per unit area associated with thinning of a wetting
film from infinity to a distance h .
0
For the KAX-treated gold surface, A is negative; therefore, the first term in the bracket
132
representing the free energy change per unit area due to van der Waals interaction (ΔG ) is
d
positive and hence is not conducive to film thinning and rupture. The second term in the bracket
represents the free energy change due to electrostatic interaction (ΔG ). It can be negative, and
e
its magnitude can be larger than that of ΔG if and have opposite signs or have the same
d 1 2
sign but of large difference in magnitudes. The fourth term represents the free energy change due
to the long-range hydrophobic force (ΔG ), which is negative as C < 0. However, its magnitude
h 2
is smaller than that of ΔG >0; therefore, the film remains stable. Only the short-range
d
hydrophobic pressure can become large enough at smaller film thickness to overcome the
repulsive van der Waals pressure and cause the film to rupture. At h < h , the magnitude of ΔG
c h
becomes larger than that of ΔG .
d
In the present work, we used Eq. (2.20) to determine the C and D parameters for the short-
1 1
range hydrophobic force. We know the values of C and D from curve fitting the data presented
2 2
in Figure 2.7b to Eq. (2.17). (The C and D parameters of this equation can be considered equal to
C and D , respectively, as the experimental data show only the long-range part of Π .) We also
2 2 h
know the values of A , , , θ , and γ . Although we do not have the value of h , we know
132 1, 2 0 23 0
that it is the film thickness at Π = 0. Thus, we have two equations and three unknowns. In the
present work, we estimated the values of C and D to be -0.642 N/m and 2.3 nm, respectively,
1 1
under the assumption that h < 1 nm. The value of h is found to be 0.56 nm, which is equivalent
0 0
to approximately two layers of water molecules adsorbed on the hydrophobic surface of θ = 91o
o
at equilibrium.
In the present work, it was possible to measure only the long-range part of the hydrophobic
disjoining pressure as shown in Figure 2.7b. The reason is that the wetting film has already
ruptured catastrophically by the time the attractive short-range hydrophobic pressure becomes
strong enough to overcome the repulsive van der Waals pressure. It would, therefore, be
46 |
Virginia Tech | impossible to measure the short range hydrophobic force in the wetting film formed on strongly
hydrophobic surface. On the other hand, the theoretical analysis carried out in view of the
Frumkin-Derjaguin theory of wetting suggests that short-range hydrophobic force should exist in
wetting films if the film is to rupture.
As shown in Figure 2.7b, Π extends to h ≈ 150 nm, which is beyond the range of
h
hydrophobic forces observed between hydrophobic surfaces of solids. On the other hand, the
studies conducted on foam films (or between two bubbles) showed that hydrophobic force
affects drainage rates at film thickness as large as 250 nm [67]. This observation was given an
explanation that the air-side of the air/water interface is more hydrophobic than the best-known
hydrophobic substances known to date in view of the interfacial tensions involved [68]. Note in
Figure 2.6b that the excess pressure (p) in the wetting film on the hydrophobized gold surface
begins to increase at h ≈ 150 nm where Π begins to appear. Recognizing that it is the excess
h
pressure p that is the driving force for film drainage, it would be reasonable to suggest that the
long-range hydrophobic force is responsible for the accelerated film drainage observed in the
present work, while the short-range hydrophobic force is responsible for the film rupture at h =
c
29 nm.
Above discussions suggest that hydrophobic force can affect both film drainage and rupture.
However, the accelerated drainage rate observed with the KAX-treated hydrophobic gold could
also be explained by the slip at the solid/liquid interface [69, 70]. It has already been shown that
the no-slip boundary condition can be used for the air/water interface of the wetting film formed
in pure water [46, 49, 71, 72]. In this regard, we may need to consider the slip only at the
solid/water interface. In general, slip velocity varies with shear rate, i.e., (cid:1847) = (cid:1854)(cid:2011)̇, where b is the
(cid:3048)
slip length and (cid:2011)̇ is the shear rate. Assuming that the horizontal liquid flow velocity, u(z), is the
sum of pressure-driven flow, u (z), and slip-driven flow, u (z), one can readily derive the
p u
following relation
1 p hz
u(z)u (z)u (z) (z2 zh)U (2.22)
p u 2r u h
where U is the slip velocity at the solid/water interface. It can be readily shown that
u
47 |
Virginia Tech | b p h2
U (2.23)
u 2r bh
Eqs. (2.22) and (2.23) were used to calculate the ratios between u and u at t = 0.7 s and h = 44
p u
nm, which are plotted in Figure 2.8. The calculations were made at three arbitrarily chosen slip
lengths at the solid/liquid interface: b = 10, 20 and 40 nm. The ratio becomes smaller at the
center than near the edge, indicating that slip becomes more important as film becomes thinner.
At b = 10 nm, the ratio is approximately 2 at the center, indicating that the pressure-driven flow
(u ) is more important than the slip-driven flow (u ) in this region. The slip plays as important a
p u
role as the excess pressure (p) at b = 40 nm, in which case the ratio becomes 1.
Assuming that the slip at the solid/water interface affects the flow velocities in thin films, Eq.
Figure 2.8 The ratios between the pressure-driven velocities (u ) and the slip-driven
p
velocities (u ) along the radial direction at t = 0.7 s and h = 44 nm as measured at
u
the center of the film (r = 0). The number on each curve represents the arbitrary
slip lengths in nm used in the calculation.
48 |
Virginia Tech | (2.11) can be modified using the continuity equation to obtain,
2 h r 12(bh) r h
r r dr dr (2.24)
R r r r rrh3(4bh) r0 t
which can be used to predict Π from slip length and the vertical thinning velocity (∂h/∂t). With
Eq. (2.24), we calculated Π vs. h curves at b = 0, 20, 40, and 1000 nm, which were then used to
obtain the Π vs. h curves shown in Figure 9 in the same manner as employed for obtaining the
h
Π vs. h curves given in Figure 7b. Under the no-slip boundary condition, the decay length (D) of
h
Π was 36 nm. As b increased to 20 nm, D decreased to 35 nm. Doubling b to 40 nm caused a
h
decrease in D to 34 nm. Even at b = 1000 nm, D decreased only to 28 nm. These findings
suggest that it is difficult to explain the accelerated film drainage with slip alone. It is necessary
to recognize the presence of the long-range hydrophobic force in wetting films.
Even if slip can explain increased drainage rate to some extent, it would be difficult to explain
the rupture of the wetting films formed on hydrophobic surfaces. As has already been noted in a
foregoing paragraph, wetting films can rupture when there is a short-range hydrophobic force
that can overcome the repulsive van der Waals force. Knowing that the rupture can occur when
θ > 0, it may be reasonable to expect to see a short-range hydrophobic force as long as contact
angle is greater than zero. At high contact angles, a long-range hydrophobic force may also be
present and help expedite the kinetics of film thinning. The solid/water interface may slip at high
contact angle, but the data obtained in the present work show that its effect is small as compared
with the effect of the long-range hydrophobic force. As shown in Figure 2.6b, the negative
disjoining pressure causes the excess pressure (p) to increase. If the disjoining pressure is
positive, on the other hand, the p decreases with decreasing film thickness. It is possible,
nevertheless, that slip can be an important factor at contact angles much higher than studied in
the present work and when a surface has a much higher degree of roughness that would help trap
vapor phase. Further investigation is needed to study the effects of contact angles in greater
detail.
In the present work, we assumed that there is no slip at the air/water interfaces of the wetting
films studied in the present work. This assumption was based on the observation that the thinning
kinetics of the wetting films formed in the absence of surfactant can be modeled using the
Reynolds lubrication theory under the no-slip boundary condition [46, 49, 71, 72]. It is well
49 |
Virginia Tech | Figure 2.9 Hydrophobic disjoining pressure isotherms (Π (h)) obtained at different slip
h
lengths (b). The number on each curve represents the slip length. The decay
lengths (D) of the isotherms decrease with increasing b; however, the negative
hydrophobic pressure remains long-range even at b = 1000 nm.
known, on the other hand, that the surface of an air bubble rising in surfactant-free water is
mobile [73]. Apparently, the air/water interface in the confined space of a wetting film behaves
differently from the same in free space. Parkinson and Ralston [71] reported an interesting
observation that a bubble rising toward a particle goes through a transition from the full-slip to
no-slip boundary conditions.
In the present work, we used two gold surfaces, one with θ = 17o and the other with θ = 81o.
r r
A question to be raised here may be why the former behaved as if it was a hydrophilic surface.
An easy answer may be that the substrate was not hydrophobic enough to create a significant
hydrophobic force. According to Churaev [16], the DLVO theory is applicable for colloids with
contact angles in a narrow range of contact angles, i.e., between 20o to 40o. One should include
50 |
Virginia Tech | the contributions from the structural forces, i.e., hydration force at θ < 20o and the hydrophobic
force at θ > 40o. Further, Derjaguin and Churaev [14] noted that “The formation of large contact
angles cannot be explained without including the structural forces in the calculation of wetting.”
The methodology developed in the present work allows direct measurement of disjoining
pressures on flat surfaces without a need to use curved surfaces. For many biological and mineral
materials, it is difficult to prepare curved surfaces. The methodology should also be useful for
studying the surface chemistry of deformable materials such as oil, bitumen, foam films, etc.
Furthermore, it can measure the disjoining pressures of unstable films, which has been a
challenge for a long time. It is the information on the negative disjoining pressure that is useful
for designing new materials with super-hydrophobic, self-cleaning, corrosion resistant, and high-
efficiency heat transfer surfaces. In addition, one can readily convert measured disjoining
pressures to surface forces using the Derjaguin approximation [25].
2.6 Summary and Conclusion
The stability of wetting films of water formed on gold surfaces was studied by monitoring the
spatial and temporal profiles (h(r,t)) of the film. The film radii were kept small (<20 µm) to
avoid dimple formation, so that the data obtained across the film can be used to determine the
disjoining pressure isotherms (Π(h)) in the thin liquid film (TLF) between a flat substrate and an
air bubble. The hydrodynamic information derived from the profiles and arbitrary slip lengths (b)
were then used as input to an analytical expression for disjoining pressure (Π) derived from the
Reynolds lubrication theory. The disjoining pressures in the water film formed on a gold surface
with a receding contact angle (θ) of 17o showed repulsive disjoining pressures, while those in
r
the film formed on a gold with θ = 81o showed negative disjoining pressures. The results were
r
analyzed to determine the electrostatic (Π ) and hydrophobic (Π ) components of Π. The
e h
hydrophobic disjoining pressure is represented as a double-exponential function, with decay
lengths (D) of 2.3 and 36 nm under the no-slip boundary condition. The experimental data
obtained in the present work suggest that the short-range hydrophobic force is responsible for the
rupture of wetting films formed on the hydrophobic surface, while the long-range hydrophobic
force is responsible for the accelerated film thinning. It was found that the slip length has
minimal impact on the decay length.
51 |
Virginia Tech | Chapter 3. A Fundamental Study on the Role of Collector
in the Kinetics of Bubble-Particle Interaction
ABSTRACT
As an air bubble approaches a gold surface, a thin liquid film (TLF) is formed in between.
The excess pressure (p) of the liquid in the film determines the rate of film thinning, which is of
critical importance in flotation. In the present work, we have measured the kinetics of film
thinning using a modified thin film pressure balance (TFPB) technique by monitoring the
interference patterns using a high speed camera. The results were analyzed using the Reynolds
lubrication theory. It has been found that initially a TLF thins by the curvature pressure (p )
cur
created due to the bubble deformation and subsequently by the disjoining pressure (Π) created by
the surface forces in a wetting film. The results show that Π > 0 on a hydrophilic surface and Π <
0 on a surface hydrophobized by xanthate. Thus, the role of xanthate in flotation is to create the
hydrophobic force that can overcome the repulsive force present in wetting films. It has been
found also that the use of small bubbles is more effective for increasing the kinetics of film
thinning and hence flotation rate.
58 |
Virginia Tech | 3.1 Introduction
In flotation, bubbles and particles collide with each other before hydrophobic particles are
selectively collected on the surface of air bubbles. During the initial stages of collision, a bubble
deforms and produces a thin liquid film (TLF) between the bubble and particle, which is referred
to as wetting film. The curvature change associated with the deformation creates a capillary
pressure and hence causes the film to drain. As the film thins due to drainage to a thickness
below ~200 nm, surface forces begin to control the drainage process. On a hydrophobic surface,
the film becomes unstable and ruptures, allowing bubble-particle adhesion to occur. On a
hydrophilic surface, the film remains stable, denying bubble-particle adhesion.
Many investigators studied the mechanisms by which wetting films become unstable when
mineral particles are hydrophobized by collector coating. It has been suggested that the films are
destabilized by the negative disjoining pressure () [1] caused by the hydrophobic force present
in the film [2]. In 1982, Israelachvili and Pashley [3] reported the first direct measurement of the
hydrophobic force, albeit in the TLFs between two hydrophobic surfaces rather than in wetting
films, using the surface force apparatus (SFA). It was shown later that hydrophobic forces can
also be measured using the atomic force microscope (AFM) [4]. More recently, we have
measured the negative disjoining pressures in wetting films using a modified thin film pressure
balance (TFPB) technique, in which the spatial and temporal profiles of an air bubble
approaching a flat hydrophobic surface was monitored by means of a high-speed camera [5].
Disjoining pressure is defined as the change in excess Gibbs free energy per unit area of a flat
TLF (G) with thickness (h) [6, 7],
(h)(G/h)
p,T, S (3.1)
at constant pressure (p), temperature (T), and chemical potential of solutes (μ) . Thus, a TLF
s
should be stable thermodynamically when ∂Π/∂h < 0. The concept of disjoining pressure is
useful for analyzing the stability of colloids, foams and wetting films. According to the DLVO
theory,
(3.2)
e d
59 |
Virginia Tech | where is the disjoining pressure due to electrostatic interaction between overlapping electrical
e
double layers and is the disjoining pressure due to the van der Waals dispersion force present
d
in a TLF. Eq. (3.2) is applicable when there are no other forces in the film, which may be the
case when the contact angles () of interacting surfaces are in the range of 20o to 40o [8].
At higher contact angles, Eq. (3.2) may be extended to include the contribution from the
hydrophobic force ( ) as follow,
h
(3.3)
e d h
In flotation, both and are usually positive (or repulsive); therefore, the use of Eq. (3.2)
e d
would not give a negative disjoining pressure that is required for bubble-particle adhesion. Eq.
(3.3), on the other hand, allows to become negative when , which is negative, can override
h
the effects of and . For bubble-particle interactions, is always positive and can vary
e d d e
with pH and the type and the amount of a collector used for flotation. In most cases, the surface
charges of the bubbles and particles are of the same sign (usually negative at alkaline pH);
therefore, both and are repulsive. Under these conditions, it is necessary that < 0 for
e d h
bubble-particle adhesion to occur.
The Derjaguin approximation [9] relates to the hydrophobic force (F ) as follows,
h h
F
h 2 dh (3.4)
R h h
where R is the radius of curvature of the hydrophobic surfaces interacting with each other. The
hydrophobic forces measured in experiment are commonly represented using the following
relation,
F h
h Cexp( ) (3.5)
R D
where C and D (decay length) are fitting parameters. Form Eqs. (4) and (5), one can obtain the
following relation,
C h
exp (3.6)
h 2D D
60 |
Virginia Tech | While the electrical double-layer and van der Waals forces are well understood, there is no
consensus on the origin of the hydrophobic force. Thus, one can determine and using well-
e d
defined theoretical expressions, while Eq. (3.6) is empirical. According to Rabinovich and
Derjaguin [10], hydrophobic force originates from the structural changes in the overlapping
boundary layers of water as two hydrophobic surfaces approach each other. Eriksson et al. [11]
derived an exponential force law based on the basis of the same concept. However, other
investigators showed evidences that the ‘hydrophobic force’ is an artifact due to the bubbles or
cavities present in TFLs. A recent view on the subject is that while the short-range attractions
observed at separations below ~20 nm are ‘true’ hydrophobic force, longer-range attractions may
not be related to surface hydrophobicity [12]. It has been shown, on the other hand, that the range
of hydrophobic force increases with increasing chain length of n-alkane homologues [13] and
water contact angle [14, 15].
As is well known, hydrocarbon chains of collector molecules associate with each other to
form self-assembled monolayers or hemi-micelles [16, 17]. The driving force for the molecular-
scale hydrophobic interaction is due to the configurational rearrangement of water molecules as
two hydrophobic species come into contact, which entails entropy increase. It has been shown,
however, that the hydrophobic interaction at macroscopic scale involves a decrease in both
entropy and enthalpy, with the enthalpy change (H) being slightly larger in magnitude than the
change in the entropy term (TS) [18]. These findings suggest that some type of structure may
be forming in the vicinity of hydrophobic surfaces in support of the structural theory of the
hydrophobic force as originally proposed by Derjaguin and his colleagues [19, 20].
In the present work, we have measured the negative disjoining pressures ( ) present in the
h
wetting films formed on hydrophobic gold surfaces using the modified TFPB technique
described above. Potassium amyl xanthate (KAX) was used for the hydrophobization of gold.
The results will be used to discuss the basic role of collectors in flotation.
61 |
Virginia Tech | 3.2 Experiment
3.2.1 Materials
Gold-coated glass plates (CA134, EMF) were used as substrates for wetting films. They were
cleaned in a piranha solution (7:3 by volume of H SO :H O ) for 10 minutes at 120 ºC, rinsed
2 4 2 2
with Millipore water (>18.2 MΩ/cm), and then dried in a nitrogen gas stream. The freshly-
cleaned gold substrate exhibited rms roughness of 1-2 nm as determined from AFM images. The
gold plate obtained in this manner exhibited an equilibrium contact angle (θ ) of 42° with
o
receding (θ) and advancing (θ ) angles of 17o and 60°, respectively, as measured by using the
r a
dynamic sessile drop technique.
A gold plate obtained in the manner described above was rendered hydrophobic by immersing
it in a 10-5 M potassium amyl xanthate (KAX) solution, followed by rinsing with Millipore water
and drying in a nitrogen gas stream. The hydrophobizing agent had been purified twice just
before use by dissolving a technical grade KAX (>90%, TCI America) in acetone (HPLC grade,
Fisher Sci.) and re-crystallizing in diethyl ether (99.999%, Sigma-Aldrich). KAX solution is
prepared freshly just before use.
3.2.2 Thin Film Pressure Balance
Both the xanthate-treated and untreated gold plates were used as substrates for wetting films
of water. The kinetics of film thinning was monitored using the modified thin film pressure
balance (TFPB) technique [21]. The interference patterns (or Newton rings) are recorded using a
high-speed CCD camera (Fastcam 512PCI, Photron) at 120 frames per second. A monochromic
light with a center wavelength of 546 nm was obtained by passing the light source from a
mercury short-arc lamp through a basspass interference filter (10 nm Bandwidth, Edmund
Optics).
Initially, the excess water in the film holder was removed by means of a piston pump until the
Newton ring was appeared on the camera screen. The film was then allowed to thin
spontaneously, while recording the interference patterns as a function of time. The interference
fringes behaved perfectly in axial-symmetric manner; therefore, we analyzed the data in
62 |
Virginia Tech | were used to obtain the information on the rate of film drainage and determine the disjoining
pressure () in the thin film.
Film thinning experiments were conducted using both small and large films of radii (r) in the
f
range of 10-20 μm and 60-70 μm, respectively. In general, larger films thin with dimples and
smaller films do so without dimples. The film radii were controlled by means of a piston pump.
3.3 Result and Discussion
Figure 3.1 compares the temporal profiles of the wetting film formed on (a) a freshly cleaned
gold surface with a receding contact angle (θ) of 17o and (b) a gold surface hydrophobized with
r
potassium amyl xanthate (KAX) with θ = 81o. The hydrophobic gold surface was prepared by
r
immersing it in a 10-5 M KAX solution for 10 min. The xanthate-treated gold surface exhibited
equilibrium, advancing and receding angles of 91o, 99o and 81o, respectively. As shown, the film
Figure 3.2 Minimum film thickness (h ) vs. time (t) plots for the non-dimpled wetting films
min
formed on the gold surfaces with and without KAX treatment. The number on
each curve represents the receding contact angle (). Untreated gold showed =
r r
17o.
64 |
Virginia Tech | of water formed on the xanthate-treated hydrophobic gold surface was much faster than that on
the bare gold surface. On the untreated bare gold surface, the film thinned gradually, and reached
an equilibrium thickness (h ) of 103 nm in 12 s. The film formed on the xanthate-treated gold
e
surface thinned much faster and ruptured within milliseconds at the critical rupture thickness (h )
c
of 40 nm.
Figure 3.2 shows a set of film thickness vs. time plots made for the wetting films formed on
gold-coated plates immersed in a 10-5 M KAX solution for different contact times. The film
thicknesses used in these plots were the minimum thicknesses (h ) of the films without dimples.
min
The receding contact angles (θ) increased from 73o to 81o as the contact time was increased from
r
2 to 10 minutes. After the 60 minutes of contact time, θ decreased to 78o most probably due to
r
the adsorption of xanthate in multi-layers. As shown, the film thinning kinetics increased with
increasing θ.
r
The liquid in a wetting film thins when the pressure of the liquid in the film is higher than that
in the far field. A normal stress balance across a horizontal wetting film gives the following
relation,
p p
(3.7)
cur
where p is the excess pressure in the film, p the pressure due to the changes in curvature, and Π
cur
is the disjoining pressure due to the surface forces acting between the air/liquid and solid/liquid
interfaces of a wetting film.
The excess hydrodynamic pressure could be obtained from the linear Stoke’s equation with
continuity equation assuming no slip boundary conditions at both the air/water and solid/water
interfaces. It has already been shown that the no-slip boundary condition can be used for the
air/water interface in the wetting film of pure water [23-25]. When the water wets a hydrophilic
solid surface, there is no question that the no-slip condition can be used [26]. However, slip may
occur when the water flows over a hydrophobic solid surface [27]. Also, the slip velocity (or slip
length) correlates well with surface roughness and shear rate [28, 29]. The shear rates measured
in the present work are much smaller than those employed for the measurement of slip lengths on
hydrophobic surfaces using AFM or SFA. Therefore, we assume that the no slip boundary
condition holds regardless of contact angle.
65 |
Virginia Tech | 2 h
p r (3.9)
cur R r r r
where R is the radius of the bubble in the far field and γ is the air/water interfacial tension.
Substituting Eqs. (3.8) and (3.9) into Eq. (3.7), one obtains an expression for Π
2 h r 1 r h
r 12 r dr dr (3.10)
R r r r rrh3 r0 t
which is a function of h(r,t) only. Eq. (3.10) can be used to determine a disjoining pressure
isotherm, Π(h), numerically if the spatial and temporal profiles of a wetting film can be
determined experimentally using the method used in the present work.
Figure 3.4 The excess pressure (p) in the thin wetting films of water formed on the gold
surfaces with (a) θ = 17o and (b) θ = 81o. The dramatic increase of p with
r r
increasing receding contact angle is responsible for the increased drainage rate
observed at the higher contact angle as shown in Figures 3.1 and 3.2.
67 |
Virginia Tech | Figure 3.5 Comparison of the disjoining pressures (Π) in the thin wetting films of water
formed on the gold surfaces with (a) θ = 17o and (b) θ = 81o. The negative
r r
disjoining pressure observed at the higher contact angle is responsible for the high
drainage rate observed at the higher contact angle. The negative disjoining pressure
increases with time due to decreasing film thickness.
Figure 3.3 compares the time evolution of the curvature pressure (p ) vs. radial distance (r)
cur
plots in the thin wetting films of water formed on the gold surfaces with and without
hydrophobization with KAX. As shown, p increases from the far field to the center. As the
cur
film thinning continues, the p at the center becomes larger, causing the film to become flatter
cur
with time. Note here that the values of p in the thin films formed on the xanthate-treated and -
cur
untreated gold surfaces are about the same. The slight differences observed are due to the
difference in the size of the bubbles used in the measurements,
68 |
Virginia Tech | to t = 0.2 s and then decreased to zero at t = 15 s. On the xanthate-treated gold surface with θ =
r
81o, p continually increased with time. At t = 0.583 s, p reached a value of 630 N/m2 at the film
center. At such a high pressure, the film ruptures catastrophically. The sharp increase in p is due
to the increase in the negative disjoining pressure (or attractive hydrophobic force), as will be
shown below.
Figure 3.5 compares the disjoining pressure (Π) developed in the thin wetting film formed on
the gold surfaces with and without hydrophobization with KAX. It shows that Π > 0 in the thin
liquid film formed on the bare gold surface, while Π < 0 in the film formed on the hydrophobic
surface. According to Eq. (3.7), p becomes more positive and hence the liquid film thins faster
when Π < 0. It is believed that, the negative disjoining pressure is due to the hydrophobic force.
In wetting films, the disjoining pressure due to the van der Waals force is always positive. In the
wetting films studied in the present work, both the air/water and solid/water interfaces are
negatively charged [21]; therefore, disjoining pressure due to double-layer interaction is also
repulsive. Therefore, the negative disjoining pressure can arise, i.e., Π becomes less than zero,
only in the presence of hydrophobic force in the wetting film. It is the role of collector (xanthate
in the present system) to create the hydrophobic force.
As shown above, film drainage is controlled initially by p at h > 200 nm and then by the
cur
negative disjoining pressure at smaller film thicknesses. When Π > 0, the thin film of water
formed between bubble and particle is stable and reaches an equilibrium film thickness (h ).
eq
Under this condition, no flotation is possible.
Above we have discussed the results of our drainage experiments conducted using wetting
films of smaller radii. These films do not form dimples and remain approximately spherical even
at the center of the film. Large diameter films (r > 50μm), which can be formed by withdrawing
the liquid quickly using the syringe pump, bounce up at the center and form convex-shaped
dimples as shown in Figure 3.6a. The torus-shaped water film surrounding a dimple is referred
to as barrier rim. In this case, h occurs at the base of a barrier rim.
min
Figure 3.6 compares the temporal and spatial profiles of the dimpled wetting films formed on
the gold surfaces with and without hydrophobization with KAX. On the hydrophilic surface, a
dimple was formed initially at h = 360 nm on the barrier rim located at r = 70 μm. As the film
min
thinning continued, the dimple disappeared gradually, forming a flat equilibrium film at h = 103
e
70 |
Virginia Tech | Figure 3.7 Comparison of the thinning rates obtained for dimpled and non-dimpled wetting
films formed on gold surfaces. When the gold substrate was treated with 10-5 M
KAX, the receding contact angle () increased from 17o to 81o. h represents the
r min
minimum film thickness in a wetting film.
nm. On the gold surface with θ = 81o, however, the film was flat initially and then formed a
r
dimple as the rate of film thinning accelerated due to the appearance of the negative disjoining
pressure (or attractive hydrophobic force) with decreasing film thickness. A dimple began to
form at 0.2 s, although the film remained close-to-flat. The dimple became more convex with
time, forming a well-developed barrier rim. At t = 0.916 s, the film ruptured catastrophically at
h (or h ) = 120 nm.
min c
Figure 3.7 compares the h vs. time plots for the dimpled and non-dimpled wetting films
min
formed on the gold surfaces with and without hydrophobizaton. As was increased from 17o to
r
81odue to xanthate coating, the wetting films thinned faster for both dimpled and non-dimpled
cases. Note also that non-dimpled films thinned faster than dimpled films on both the hydrophilic
and hydrophobic surfaces. In general, smaller bubbles are less likely to form dimples due to the
higher Laplace pressures, which vary with the square of bubble radius. The results presented in
Fig. 3.7 suggest, therefore, that smaller air bubbles should more readily attach themselves onto
71 |
Virginia Tech | hydrophobic surfaces and give rise to faster flotation kinetics. Of course, smaller bubbles also
give higher surface area flux (S ), which is another benefit of using smaller bubbles for flotation
b
[30, 31] .
3.4 Conclusion
We have studied the kinetics of film thinning by monitoring the temporal and spatial profiles
of the wetting films formed on the gold surfaces treated with potassium amyl xanthate (KAX).
The results have been analyzed using the Reynolds lubrication theory to determine the excess
pressures (p) in the films. It has been found that the excess pressure is substantially higher in the
wetting films formed on hydrophobic surfaces, which explains the fast kinetics of film thinning
observed. The high excess pressure is largely due to the negative hydrophobic disjoining
pressure ( ) present in the film. On the hydrophilic surface, the excess pressure is low due to
h
the presence of the positive double-layer and van der Waals disjoining pressures. Thus, the role
of collector is to improve the kinetics of film thinning and to destabilize wetting films, so that
they can rupture expeditiously and form bubble particle aggregated. The results obtained in the
present work suggest also that smaller bubbles can attach on hydrophobic surfaces more readily
than larger bubbles.
3.5 References
[1] Laskowski, J., Kitchener, J.A.,"The hydrophilic--hydrophobic transition on silica" J.
Colloid Interface Sci. 29 (1969) 670-679.
[2] Blake, T.D., Kitchener, J.A.,"Stability of aqueous films on hydrophobic methylated
silica" J. Chem. Soc., Faraday Trans. 1 68 (1972) 1435-1442.
[3] Israelachvili, J., Pashley, R.,"The hydrophobic interaction is long range, decaying
exponentially with distance" Nature 300 (1982) 341-342.
[4] Rabinovich, Y.I., Yoon, R.H.,"Use of Atomic-Force Microscope for the Measurements of
Hydrophobic Forces between Silanated Silica Plate and Glass Sphere" Langmuir 10
(1994) 1903-1909.
[5] Pan, L., Jung, S., Yoon, R.H.,"Effect of hydrophobicity on the stability of the wetting
films of water formed on gold surfaces" J. Colloid Interface Sci. 361 (2011) 321-330.
72 |
Virginia Tech | Chapter 4. Effect of Bubble Size on the Rate of Wetting
Film Drainage
ABSTRACT
The effect of bubble size on the drainage rate of wetting films has been studied by
monitoring the spatiotemporal profiles of the fast-evolving wetting films. The study was
conducted by recording the interference fringes of the wetting films by means of a high-speed
camera, calculating film thicknesses from the fringes offline, and reconstructing the film profiles
with a nano-scale resolution. The profiles were then analyzed on the basis of the Reynolds
lubrication theory to determine the contributions from the hydrodynamic and surface forces to
the film thinning process.
It was found that the rate of film drainage is controlled initially by the curvature pressure
and subsequently by the disjoining pressure when the film thickness reaches approximately 200
nm. As a result, the overall lifetime of the wetting film formed by a smaller bubble is shorter
than that for a large bubble, which provides an explanation for the facts that smaller bubbles
have shorter induction times for bubble-particle interaction and are more efficient in flotation.
75 |
Virginia Tech | 4.1 Introduction
In froth flotation, solid-solid separation is achieved by selectively attaching hydrophobic
particles on the surface of the air bubbles and carrying them to the top of a flotation cell, leaving
the hydrophilic particles behind. During the initial stages of the bubble-particle attachment,
particles undergo close encounters with air bubbles by colliding and sliding on the surface. As a
particle approaches sufficiently close to the surface of an air bubble, a thin liquid film (TLF) is
formed, which is referred to wetting film. The water in a TLF is confined between two surfaces,
i.e., solid and air (or vapor). Unlike the thin water film confined between two solid surfaces, one
of the two surfaces, i.e., air bubble, deforms, and hence create a surface tension (or Laplace)
pressure in the film. The excess pressure due to the Laplace pressure causes the film to thin. As
the TLF becomes thinner, the two surfaces interact with each other via surface forces, creating a
disjoining pressure (). The disjoining pressure due to surface forces plays a dominant role at
separations below 300 nm.
Of all the physical and chemical parameters that control the flotation rate and the mineral
recovery, the collectors (or hydrophobizing agents) are considered to be one of the most
important factors that determine the flotation behaviors. Various reagents were used to render
the mineral hydrophobic, varying with the types of the minerals. Typically, short-chain alkyl
xanthates and thionocarbamates are used to hydrophobize sulfide minerals and precious metals.
For the flotation of silica and other oxide minerals, long-chain amines are commonly used.
Regardless of the collector type, the role of collectors is to render a desired mineral hydrophobic
and destabilize the TLF so that it can rupture for bubble-particle attachment.
Laskowski and Kitchener [1] suggested that the disjoining pressure in the film should be
negative for the wetting film to rupture. The authors recognized, however, that according to the
DLVO theory the disjoining pressure should be positive, which led to the suggestion that the
negative disjoining pressure may arise from the long-range hydrophobic effect. Blake and
Kitchener [2] conducted the measurement of thickness of wetting films on methylated silica and
found that the TLF ruptures at thickness much larger than predicted by the DLVO theory, which
led to a suggestion that ‘hydrophobic force’ may be present in the wetting film formed on the
methylated surface. The Blake and Kitchener is credited for the first recognition of hydrophobic
76 |
Virginia Tech | force. However, it was not until 1982, when Israelachvili and Pashley [3] reported the first direct
measurement of long-range hydrophobic force with a decay length of 1.1 nm between mica
surfaces in CTAB solutions using a surface force apparatus (SFA). Many follow-up experiments
[4-13] showed longer-range hydrophobic forces.
It has been shown in Chapter 2 and 3, a modified thin film pressure balance (TFPB) technique
was used to show that long-range attractions exist in the wetting films formed on the xanthate-
treated gold surfaces. In this technique, the spatiotemporal thickness profiles of the wetting films
were monitored by means of a high-speed camera. Based on these results, it was suggested that
the role of collector, such as xanthate, is to create a negative hydrophobic force in a wetting film,
which facilitated the film drainage and destabilized the wetting film.
However, the disjoining pressure only act when the thickness of the TLFs were reduced to
~300 nm. At film thickness above 300 nm, surface tension pressure (or Laplace pressure) was
the dominant factor in the film drainage. Earlier work showed that flotation rate and recovery
increased when smaller air bubbles were used [14, 15]. Anfruns and Kitchener [16] studied the
capture rate of a single air bubble in a particle suspension of aqueous solution. The result showed
that the use of smaller bubbles improved the collection efficiency of fine particles. Ahmed and
Jameson [17] examined the role of the bubble size on the flotation rate of fine particles in
turbulent condition. They showed that the flotation rate of fine particles was substantially
improved when the smaller bubbles were used.
The benefit of using smaller bubbles for flotation has been attributed to the increase in the
probability of the collision with decreasing bubble size. Until 1980s-1990s, many investigators
relied on measuring induction times to determine the probability of bubble-particle attachment.
They found that the bubble size influenced the attachment efficiency substantially [14, 18]. Two
empirical models for predicting the attachment efficiency were developed later by Dobby and
Finch Dobby and Finch [19]and Yoon and Luttrell Yoon and Luttrell [14]. Both models used the
induction time to predict the attachment efficiency. Note also that neither of these models
considers the fundamentals of the thin film drainage and the deformation of air bubbles when
colliding with the particles. As suggested by Humeres et al. [20], theoretical approaches for
estimating the probability of attachment (P ) have not been successful. It is, therefore, important
a
77 |
Virginia Tech | to study the fundamentals of the wetting film drainage during the process of bubble-particle
attachment.
It is well known that wetting films drain initially by the Laplace pressure and subsequently by
the disjoining pressure when the film thickness is below 300 nm. In this chapter, the kinetics of
wetting film drainage has been measured using different sizes of air bubbles. The experiments
were conducted on both the hydrophilic and hydrophobic gold surfaces. The bubble sizes were
controlled by changing the radius (R ) of the capillary tube. The kinetics studies were conducted
o
by measuring the temporal and spatial profiles of the wetting films using the modified thin film
pressure balance (TFPB) technique. The film profiles were analyzed on the basis of the Reynolds
lubrication theory to calculate the curvature pressure (or local Laplace pressure) and the
disjoining pressure in TLFs. The results are particularly useful for predicting the wetting film
drainage, flotation rate and consequently the attachment efficiency.
4.2 Experimental
4.2.1 Materials
The gold substrates were prepared by depositing a thin layer of gold on flat silicon surfaces in
a vacuum chamber. The deposition was operated using the E-beam physical vapor deposition
technique (PVD-250, Kurt J. Lesker) in a class-100 cleanroom. Single-side polished silicon
wafers (orientation: <100>, University wafer, Inc.) were used as the base substrates. They were
cleaned in a boiling piranha solution (70:30 by volume H SO :H O ) for 5 minutes, rinsed
2 4 2 2
thoroughly with water, and deoxidized using buffered oxide etch (BOE) solution for seconds.
The removal of the silicon oxide layer enhances the bonding between the deposition metal layers
and the silicon surface. The deposition process was operated in a 10−6 torr vacuum chamber at
room temperature. The deposition process follows two steps: a 50 Å thick titanium adhesion
layer, and followed by a 500 Å thick gold layer.
The gold coated wafers were diced into the square pieces with a dimension of 0.5” × 0.5”.
The dicing process was carried out using a micro-auto dicing saw. Shortly after the dicing
process, the square substrates were cleaned in a water-based ultrasonic bath to remove the
residual silicon particles. It was followed by immersing the gold substrates in a mild piranha
78 |
Virginia Tech | solution for 2 minutes, rinsed thoroughly with water and dried in a nitrogen gas stream. The
gold substrates behaved a perfect mirror, and exhibited a root mean square roughness of below
0.5 nm over an area of 1× 1 µm2, as shown by the AFM images. The dynamic contact angles are
measured using the dynamic sessile drop technique. The gold plate obtained in this manner
exhibited the equilibrium (θ ), advancing (θ ) and receding (θ) water contact angle of 42o, 60o
o a r
and 17o, respectively. All the experiments were conducted using the ultrapure water with the
resistance of above 18.2 MΩ•cm produced by Direct-Q water purification system (Millipore).
All the glassware were soaked in the base bath (a saturated potassium hydroxide in isopropanol)
overnight, and rinsed thoroughly with pure water.
The hydrophobic gold surfaces were prepared by immersing the substrates in a 10−5 M
potassium amyl xanthate (KAX) solution for 10 minutes. After the desired immersion time, they
were temporally stored in pure water. The hydrophobic gold surfaces exhibited the contact
angles of θ = 91 o, θ = 99 o and θ = 81o. KAX was purified by dissolving a technical grade
o a r
KAX (>90%, TCI America) in acetone (HPLC grade, Fisher Sci.), and recrystallized in diethyl
ether (99.999%, Sigma-Aldrich). The KAX aqueous solution was freshly prepared before use to
minimize the oxidation effect.
4.2.2 Thin Film Pressure Balance
The kinetics of film thinning was studied using the modified thin film pressure balance (TFPB)
technique [21]. In the modified TFPB technique, the liquid between vapor phase and solid
surface was sucked out by a screw-type pump. Once the water film was drained to a thickness
below 1 µm thickness, it began to deform and drained spontaneously by the surface tension
pressure (or Laplace pressure). The dynamic film drainage was monitored by recording the
interference fringes by means of a high speed camera (HiSpec 4, Fastec Imaging). The obtained
fringes showed the temporal and spatial profiles of the TLFs, which can be used to obtain the
hydrodynamic pressure and the disjoining pressure in the wetting film. The screw-type pump
was custom made with the Teflon head and the brass nuts. The screw connection was sealed with
the vacuum grease, and the Teflon head was wrapped with the FEP film to ensure the good
impermeability.
79 |
Virginia Tech | the capillaries determined the radius of the air bubbles. In the present work, we used three
different sizes of air bubbles with radii (R ) of 1.5, 2.0 and 3.9 mm, respectively.
o
4.3 Results
4.3.1 Hydrophilic Surfaces
Figure 4.1 compares the spatial and temporal profiles of the wetting films formed on the
hydrophilic gold surfaces using three different sizes of air bubbles. The results were compared
using air bubbles with R = (a) 1.5 mm, (b) 2.0 mm and (c) 3.9 mm, respectively. In the present
o
work, the film size, i.e., the radius of the flat film, was controlled in the range of 10-30 µm. In
general, a dimple was developed when the film size was large. With a small film, the dimple
formation was prevented. At R = 1.5 mm, film thinned from the minimum thickness (h ) of
o min
400 nm to 300 nm in 0.155 s. The film continued thinning, and reached the equilibrium thickness
(h ) of 95 nm. At R = 2.0 mm, the film thinned slower than at R = 1.5 mm. It took 0.284 s for
e o o
the film to reach 300 nm. At t = 6.016 s, the film was in equilibrium with h = 110 nm. Figure
e
4.1c) shows the spatiotemporal thickness profiles of the wetting films at R = 3.9 mm. The result
o
showed that the film reached equilibrium in 8.975 s at h = 147 nm. In an equilibrated wetting
e
film, the opposing disjoining pressure was balanced by the surface tension pressure created by
the curvature changes. Note that the wetting film using the bigger size of the air bubbles
exhibited a thicker equilibrium film thickness than the film using the smaller sized bubble. It is
suggested that disjoining pressure in a wetting film formed on a hydrophilic gold surface
increased repulsively as decreasing the equilibrium film thickness.
Figure 4.2 compares the drainage rate of the wetting films formed on the hydrophilic gold
surfaces using the different sizes of air bubbles. The plot shows a set of the minimum film
thickness (h ) vs. time (t). As shown, film thinned faster at R = 1.5 mm than at R = 2.0 mm.
min o o
As the bubble size increased, the film thinning was retarded. The faster thinning kinetics
obtained using the smaller bubbles were due to the larger curvature pressure.
The drainage of the TLFs between two macroscopic surfaces was commonly described by the
Reynolds lubrication theory. For a thin film between two flat surfaces, the Reynolds
approximation is commonly used to describe the rate of film thinning (dh/dt),
81 |
Virginia Tech | Figure 4.2 Minimum film thickness (h ) vs. time (t) for the wetting films formed on the
min
bare gold surfaces using air bubbles with radius (R ) of 1.5, 2.0 and 3.9 mm,
o
respectively. The smaller bubble thins faster.
dh 2h3P
(4.1)
dt 3R2
f
where h is the film thickness, µ the liquid viscosity, R the radius of the flat film, and ∆P is the
f
pressure difference between the flat film and the outer film. ∆P is given as follows,
2
P (4.2)
R
o
where R is the radius of the bubble, γ the surface tension of the liquid, and Π is the disjoining
o
pressure. 2γ/R represents the pressure contributed from the Laplace pressure. By combining eq.
o
(4.1) with eq. (4.2), one could easily conclude that dh/dt depends on ∆P, which inversely on R .
o
The limitation of using eq. (4.1) to describe the wetting film drainage was that the film was
assumed to be flat. In order for a qualitative study of both the curvature pressure and the
disjoining pressure at a curved interface, Reynolds lubrication equation was used to describe the
thin film drainage,
82 |
Virginia Tech | dh 1 p
rh3 (4.3)
dt 12r r r
where r is the radial position of the film. Eq. (4.3) was derived from the linear Stokes equabtion
with the continuity equation. Eq. (4.3) assumed that no slip boundary conditions held at both the
air/water and the solid/water interfaces. One of these two assumptions has been experimentally
confirmed that no-slip boundary condition holds for the air/water interface in wetting films of
pure water [22, 23]. When the water flows over the hydrophobic surface, the slippage might
occur. However, it has been shown that the slip velocity (or slip length) correlated with the
surface roughness and the shear rate. Since the shear rate of the liquid in the wetting film (a max
of 5000 s−1) is much smaller than the value obtained for the measurement of the slip lengths
using the AFM and SFA, one might assume that no slip boundary condition holds regardless of
contact angle.
Thus, one could obtain the expressions for p by integrating the eq. (4.3) twice with eq. (4.4)
[24],
r 1 r h
p12 r dr dr (4.4)
rrh3 r0 t
The boundary condition for solving the Reynolds lubrication equation is a normal stress balance
at an air/water interface,
(4.5)
p p
cur
where p is the curvature pressure created by the curvature changes, and Π is the disjoining
cur
pressure due to the surface forces between the air/liquid and solid/liquid interfaces confining a
thin film.
The curvature pressure (p ) can be obtained from the following relation,
cur
2 h
p r (4.6)
cur R r r r
0
83 |
Virginia Tech | where R is the radius of an air bubble in the far field. Eq. (4.6) is an expression for the curvature
o
pressure due to the changes in the curvature of the air/water interface.
Substituting eqs. (4.4) and (4.6) into (4.5), one can obtain an expression for Π,
2 h r 1 r h
r 12 r dr dr (4.7)
R r r r rrh3 r0 t
0
Using Eq. (4.7), the disjoining pressure can be numerically determined from the spatial and
temporal profiles of the wetting films.
Figure 4.3 shows a comparison of the curvature pressure (p ) vs. radial distance (r) in the
cur
wetting films obtained at a) R = 1.5 mm and b) R = 3.9 mm. It was shown that p increased
o o cur
from the far field to the center of the film. As shown in Eq. (4.4), the gradient of the excess
pressure (∂p/∂r) controlled the film drainage. Once the curvature pressure gradient was
Figure 4.3 Changes in curvature pressure (p ) with radial distance (r) in wetting films
cur
formed on bare gold surfaces with R = 1.5 mm and 3.9 mm. p increases with
o cur
time as the film becomes flat. The smaller bubble has a larger curvature pressure
than the larger bubbles. The larger curvature pressure using the smaller sizes of
the air bubbles is responsible for faster thinning kinetics.
84 |
Virginia Tech | developed along the radial direction, the film was thinning spontaneously. p was small at t = 0,
cur
and it became larger as film thinned. It was clearly shown that the curvature pressure was much
larger at R = 1.5 mm than those obtained at R = 3.9 mm. At equilibrium, p reached a
o o cur
maximum value of 92 N/m2 at R = 1.5 mm, while the maximum of p was equal to 25 N/m2 at
o cur
R = 3.9 mm. According to eq. (4.4), the driving pressure p was mainly contributed from p ,
o cur
when Π ≈ 0. Therefore, the use of the smaller bubbles creates a higher curvature pressure, which
is favorable for the wetting film drainage.
Figure 4.4 shows the disjoining pressure isotherm, Π(h) vs. h, in the wetting film formed on a
hydrophilic gold surface. The disjoining pressure data were obtained at the flat film using eq.
(5.7). It was found that a positive disjoining pressure increased with decreasing the film
Figure 4.4 Disjoining pressure isotherm in wetting films formed on the bare gold surfaces
using the air bubbles in radius of 1.5, 2.0 and 3.9 mm, respectively. Π and Π
d e
represent the disjoining pressure due to the van der Waals dispersion force and
electrical double layer force. Π is the total disjoining pressure. The smaller
t
bubble has a larger Laplace pressure, which gives a thinner equilibrium film
thickness.
85 |
Virginia Tech | thickness, and reached the maximum in equilibrium. According to the DLVO theory, the film
reached the equilibrium, when the opposing pressure (in this case, the repulsive disjoining
pressure) was equivalent to the curvature pressure. Since the smaller bubble had the larger
curvature pressure, one could obtain a smaller h for a smaller bubble. At R = 3.9 mm, the film
e o
reached equilibrium at h = 147 nm where Π = 25 N/m2. At R = 1.5 mm, h = 90 nm where Π =
e o e
92 N/m2.
The obtained disjoining pressure could be expressed in accordance to the classical DLVO
theory,
(4.8)
t d e
where Π is the disjoining pressure contributed from the van der Waals dispersion force, and Π
d e
is the disjoining pressure due to the electric double-layer force. The disjoining pressure due to
the van der Waals force, Π , is given by,
d
A
132 (4.9)
d 6h3
and the electric double-layer disjoining pressure, Π , could be represented as,
d
2
0 2 2 cosech(h)2 coth(h) (4.10)
e 2sinh(h) 1 2 1 2
in which A is Hamaker constant for the wetting film of water formed on the gold surface. In
132
using eq. (4.9), A = −14.8 × 10−20 J was used as reported by Tabor et al from the direct force
132
measurement between an air bubble and a gold surface [25]. In Eq. (4.10), ε is the permittivity
o
in vacuum, ε is dielectric constant of water, ψ and ψ are the double-layer potentials at the
1 2
gold/water and air/water interfaces, respectively, and κ is the reciprocal Debye length. In this
communication, the subscript 1, 2, and 3 represent solid, vapor and liquid phase, respectively.
86 |
Virginia Tech | ting film was calculated using the Hogg–Healey–Fuerstenau (HHF) approximation [26]. In using
eq. (4.10), ψ was taken to be the same as the ζ -potential of the gold spheres in pure water,
1
which was reported to be -40 mV [24]. By fitting the obtained disjoining pressure with the
theoretical DLVO curve (Π), one could obtain the best-fit values of ψ = -29 mV and κ−1 = 42
t 2
nm. As shown in Figure 4.4, the dash line represents Π, having a close fit with the measured
t
disjoining pressure obtained from the use of the different sizes of the air bubbles. The value of ψ
2
(= -29 mV) is close to the ζ -potentials of the argon bubbles in distilled water as reported by Usui
et al [27]. Note that the Debye length obtained from the curve fitting was relatively small,
compared to the values in pure water reported in the colloidal textbooks. However, many
investigators obtained the similar results in pure water as what we found in the present work,
showing that κ−1 ≈ 42 nm.
The obtained Π using the different sizes of air bubbles were scattered along the theoretical Π
t
Figure 4.6 Minimum film thickness (h ) vs. time (t) of the wetting films formed on the
min
xanthate-treated gold surfaces using different sizes of air bubbles with radius of 1.5,
2.0 and 3.9 mm, respectively.
88 |
Virginia Tech | predicted by the DLVO theory. It was found that the disjoining pressure was independent on the
bubble size, while on the film thickness only. As shown, the disjoining pressure in the wetting
film formed on the hydrophilic gold surface increased with decreasing the film thickness.
4.3.2 Hydrophobic Surfaces
Above we have shown the results of the wetting film drainage on the hydrophilic gold surface.
It has been found that the film was drained by the curvature pressure initially, and subsequently
by the disjoining pressure. The use of the small bubble was favorable for the film thinning. We
have also shown that on the hydrophilic gold surface, the disjoining pressure became more
repulsive as decreasing the film thickness. As the disjoining pressure became equivalent to the
Figure 4.7 Changes in disjoining pressure (Π) with radial distance (r) in the wetting films
formed on the xanthate-treated gold surfaces at R = 1.5 mm and 3.9 mm for
o
various times after film formation. Π < 0 was observed when θ = 91o, and the
o
negative Π increases with time as the film thins. The varying profiles of Π
obtained using different sizes of the air bubbles was corresponding to the spatial
thickness profiles only.
89 |
Virginia Tech | Figure 4.8 A comparison of curvature pressure (p ) vs. radial distance (r) in the wetting films
cur
formed on the xanthate-treated gold surfaces using the air bubbles with R = 1.5
o
mm and 3.9 mm.
curvature pressure, the film reached the equilibrium.
Figure 4.5 compares the spatiotemporal profiles of the wetting films formed on the
hydrophobic gold surfaces using three different sizes of air bubbles. The gold surfaces were
rendered hydrophobic by potassium amyl xanthate with θ = 91◦. At R = 1.5 mm, the film
o o
thinned to the minimum thickness of 300, 200, and 100 nm in 0.095, 0.205 and 0.420 s,
respectively. As the bubble size increased to 2.0 mm, the rate of the film thinning was retarded.
The time for film reaching the minimum thickness of 300, 200, and 100 nm increased to 0.11,
0.25 and 0.55 s. At R = 3.9 mm, it took 0.71 s for h = 100 nm. Comparing the film profiles
o min
obtained on the hydrophobic gold surfaces with those obtained on the hydrophilic gold surfaces,
the film thinned much faster when the gold surfaces were treated hydrophobic. It was believed
that xanthate adsorption on gold surfaces created a hydrophobic force, facilitating the thinning
kinetics and destabilizing the wetting film [28].
90 |
Virginia Tech | Figure 4.6 compares the rate of the wetting film drainage on the xanthate-treated gold surfaces
at R = 1.5, 2.0 and 3.9 mm, respectively. As shown, the film thinning was initially drained by
o
the curvature pressure only. As the minimum film thickness was below 100 nm, rate of film
thinning no longer decayed exponentially with time. Instead, the film thinning was accelerating
due to the attractive force. As h < 100 nm, the film became unstable and ruptured at a critical
rupture thickness (h ). It was found that the use of the small bubbles thinned fast above 100 nm.
c
As h < 100 nm, the slopes for the film thinning curves using the different sizes of bubbles were
similar.
We compared both p and Π in the wetting film formed on a hydrophobic gold surface.
cur
Figure 4.7 shows the evolution of the spatial profiles of p in the wetting film formed on the
cur
hydrophobic gold surface at R = a) 1.5 mm and b) 3.9 mm. As shown, the value of p was
o cur
much bigger using a small air bubble (Ro = 1.5 mm) than a larger bubble (R = 3.9 mm). The
o
larger p drove the film thinning faster, and ruptured the film in a shorter time. At R = 1.5 mm,
cur o
p increased with time, and reached a plateau value of 92 N/m2 at the center of the film. In the
cur
same manner, p increased with time at R = 3.9 mm, while the maximum value of p at the
cur o cur
center was only 25 N/m2. The higher p obtained using the smaller bubble was responsible for
cur
the faster thinning kinetics, as shown in Figure 4.5.
Figure 4.8 compares the disjoining pressure (Π) developed in the wetting film formed on the
xanthate-treated gold surfaces. It showed that Π <0 in the wetting film formed on the gold
surface treated by xanthate. Π became more negative as the film thinned, and reached a value of
-700 N/m2 at the center at t = 0.42 s for R = 1.5 mm. The Π vs. r obtained at R = 3.9 mm
o o
showed the similar results with that obtained at R = 1.5 mm. Note that, Π became much larger
o
than the p at t > 0.205 s for R = 1.5 mm, or at t > 0.4 s for R = 3.9 mm, indicating that the
cur o o
film thinning was initially controlled by p , and subsequently by Π.
cur
Figure 4.9 shows the disjoining pressure isotherm in the wetting film formed on a
hydrophobic gold surface treated by xanthate. The plot shows Π obtained from the
spatiotemporal thickness profiles of the wetting films at R = 1.5, 2.0 and 3.9 mm, respectively.
o
As shown, the disjoining pressure is negative and become more negative as the film thickness
decreased. In a wetting film, the disjoining pressure due to the van der Waals dispersion force
was always positive. The zeta potential of gold sphere was not changed after the xanthate
91 |
Virginia Tech | treatment; therefore, the electric double-layer force was repulsive. Only the hydrophobic force
can destabilize the wetting film. Thus, one might consider an inclusion of the hydrophobic force
in the extended DLVO theory [29, 30],
(4.11)
t d e h
in which Π represents the hydrophobic disjoining pressure. Since the value of Π, Π and Π are
h t d e
known, one might back-calculate the values of Π using Eq. (4.11). The hydrophobic force was
h
represented as a power law [31, 32],
Figure 4.9 Disjoining pressure isotherm in the wetting film formed on xanthate-treated gold
surfaces using different sizes of the air bubbles with radius of 1.5, 2.0 and 3.9
mm, respectively. Π < 0 was found on the hydrophobic gold surface, regardless
of the size of the air bubbles. The hydrophobic disjoining pressure was
represented as a power law. The best fit was obtained with an inclusion of the
hydrophobic force with force constant K = 1.1 × 10−17 J.
132
92 |
Virginia Tech | K
132 (4.12)
h 6h3
Eq. (4.12) is of the same form as the van der Waals-dispersion force, and K is the hydrophobic
132
force constant. K = 1.1 × 10−17 J was obtained from a best fit of the theoretical disjoining
132
pressure curve with the experimental data. We have shown that the hydrophobic force dominated
over the electrical double-layer force and the van der Waals force in wetting films. The net
attraction brought the film to be sequenced, and ruptured afterwards. The results showed that the
possibility of bubble-particle attachment was enhanced using the smaller sizes of the air bubbles
and the collectors.
4.4 Conclusions
The effect of bubble size on the wetting film drainage was studied using the modified thin
film pressure balance (TFPB) technique. Effect of bubble size was studied using different sizes
of capillary tubes. The spatiotemporal film profiles were extracted from the interference fringes
and used to determine both the hydrodynamic pressure and the disjoining pressure in wetting
films.
It has been found that the film thinning was initially controlled by the curvature pressure (p )
cur
due to the deformation of the air bubble, and subsequently by the disjoining pressure (Π) created
by surface force in wetting films. The results showed that the use of small bubbles increased the
thinning kinetics of the wetting films formed on both the hydrophilic and the xanthate-treated
hydrophobic gold surfaces. On a hydrophilic surface, p was balanced by Π arising from the
cur
van der Waals dispersion force and the electrostatic double-layer force. On a xanthate-treated
gold surface, the film thinned expeditiously, leading to a rupture where the film became unstable.
The short lifetime of the wetting films using the smaller air bubbles were attributed to a higher
curvature pressure. The present results suggested that the use of the small bubble was favorable
for the wetting film drainage, resulting in an increase of the attachment efficiency. It was also
found that a negative disjoining pressure was present in wetting films due to the presence of the
hydrophobic force.
93 |
Virginia Tech | Chapter 5. Predicting the Asymmetric Hydrophobic
Interactions in Wetting Films from the Symmetric
Hydrophobic Interactions in Colloid and Foam films
ABSTRACT
A modified thin film pressure balance (TFPB) technique has been used to determine the
disjoining pressures () in the wetting films of water formed on the surfaces of different
hydrophobicities. After subtracting the contributions from the double-layer and van der Waals
forces, the hydrophobic disjoining pressures were determined and converted to asymmetric
hydrophobic interaction forces using the Derjaguin approximation. It has been found that the
asymmetric force constants (K ) can be predicted from the symmetric hydrophobic force
132
constants (K ) between hydrophobic surfaces of identical hydrophobicity and the symmetric
131
hydrophobic force constant (K ) between two air bubbles (K = 5.3 × 10-17 J) in pure water
232 232
using the geometric mean combining rule. This finding suggests that hydrophobic forces may be
of molecular origin, which is in agreement with the results of the recent thermodynamic studies
that hydrophobic forces are the consequence of the water molecules forming H-bonded structures
in the thin liquid films confined between hydrophobic surfaces (Wang et al., J. of Colloid and
Interface Sci. 379 (2012) 114-120).
97 |
Virginia Tech | 5.1 Introduction
The DLVO theory is named after Derjaguin and Laudau, Verwey and Overbeek, and has been
used as the guiding principle in colloid chemistry for more than 70 years [1, 2]. The early
application of the theory was found for wetting films by Derjaguin and his colleagues [3, 4], who
showed that a wetting film, i.e., a thin liquid film between a solid surface and an air bubble, was
distinct from the bulk liquid. In a wetting film, both the van der Waals-dispersion force and the
electrical double-layer force are repulsive for many systems. Therefore, both of the DLVO forces
are stabilizing wetting films when the surface is hydrophilic [5, 6]. When the surface becomes
hydrophobic, however, the film becomes unstable and ruptures to form a contact angle.
Laskowski and Kitchener [7] suggested that for a contact angle to form on a solid surface, the
disjoining pressure () of the wetting film must be negative. It is now known that the negative
disjoining pressure arises from the hydrophobic force, which has been a topic of contentious
discussion for a long time.
Rabinovich and Derjaguin [8] suggested that the hydrophobic force is a structural force
created due to overlapping boundary layers of two approaching surfaces. The concept of
structural force was first conceived to explain in the experimental data obtained from wetting
film studies conducted on hydrophilic surfaces. It was found that the equilibrium film
thicknesses formed on hydrophilic surfaces were much thicker than predicted by the DLVO
theory [9, 10]. As the solid surfaces became more hydrophobic, however, wetting films became
less stable and ruptured. Israelachvili and Pashley [11] were the first to measure an attractive
hydrophobic force with a decay length of 1.1 nm between two hydrophobized mica surfaces with
θ = 64o. Many follow-up experiments were conducted, showing that both short- and long-range
hydrophobic forces were present in the TLFs formed between two hydrophobic surfaces [12-15].
The origin of the hydrophobic force, however, has been controversial after more than two
decades of contentious debates. Some investigators suggested that the hydrophobic force
originates from the changes in water structure [16], while others showed that it is caused by the
correlation of charged patches [17, 18]. Some researchers believe that the long-range
hydrophobic force was an artifact caused by the pre-existing nanobubbles on hydrophobic
surfaces [19, 20]. Wang and Yoon [21] showed that hydrophobic interaction entails a enthalpy
98 |
Virginia Tech | decrease, which has been attributed to the formation the low density liquids (LDLs) species in
the TLFs confined between hydrophobic surfaces.
Most discussions regarding the hydrophobic interaction were based on the force data obtained
between two symmetric surfaces. The surface force data obtained with asymmetric hydrophobic
surfaces exhibited unexpected results. Claesson et al. [22] conducted surface force measurement
between a bare mica surface and a hydrophobic mica surface. The latter was prepared by
depositing the dimethyldictylammonium (DDOA+) ions by means of the Langmuir-Blodgett
technique. It was found that a stronger attraction was present in TLFs between a bare mica
surface and a hydrophobic mica surface than those conducted between two hydrophobic mica
surfaces. They suggested that the attraction was an electrostatic mechanism, in which the
overlaps of the oppositely charged double layers created a net attractive force. Tsao et al. [23]
reported results between a bare mica and a dieicosyldimethylammonium treated hydrophobic
mica surface across a thin liquid of water. These authors suggested, however, that the long-range
attraction is due to the local “amphiphilic structure” rather than the double-layer interaction.
Meyer et al. [17] obtained results that were similar to those reported by Tsao et al. with mica
surfaces coated with a long-chain ammonium salt, and attributed the results to changes in
amphiphilic structure. Rabinovich and Yoon [24] conducted asymmetric force measurements
between a hydrophilic glass sphere on one side and a robust hydrophobic glass surface on the
other. The silica surface and glass sphere were treated with octadecyltrichlorosilane and
trimethylchlorosilane. The results showed that the forces measured at h < 5 nm deviated from the
DLVO theory, which was contributed to a short-range hydrophobic force. In a follow-up paper
by Yoon et al. [25], force measurement was conducted between a hydrophobic glass sphere with
θ = 109◦ and a silica plate with the varying hydrophobicities. It was found that the hydrophobic
force became stronger as the solid hydrophobicity increased.
We recently measured the disjoining pressure in the TLFs of water between an air bubble and
a hydrophobic gold surface [26]. The gold surface was hydrophobized using potassium amyl
xanthate (KAX), which is commonly used as collector for the flotation of sulfide minerals and
precious metals. It was found that the disjoining pressure in a wetting film was repulsive on a
bare gold surface. It became attractive when the gold surfaces were rendered hydrophobic by
KAX. It was found that the hydrophobic forces calculated from the negative disjoining pressures
were somewhat stronger than the asymmetric hydrophobic forces measured between two gold
99 |
Virginia Tech | surfaces with identical hydrophobicity. This finding qualitatively explained by the fact that air
bubble is more hydrophobic than hydrophobic gold in view of the interfacial tensions involved.
In the present work, a series of AFM surface force measurements using symmetric
hydrophobic surfaces, and a series of disjoining pressure measurements in wetting films using
the modified TFPB technique were conducted. The disjoining pressure were converted to the
corresponding hydrophobic forces using the Derjaguin approximation [27]. A set of symmetric
and asymmetric hydrophobic forces are then compared to see if the geometric mean combining
rule is applicable. As is well known, this rule is applicable for predicting molecular forces such
as van der Waals force. If it is found that the geometric combining rule is applicable for
hydrophobic interactions, the hydrophobic interaction may be considered a molecular force. It
has been already been shown that the combining rule can be applicable for predicting the
asymmetric hydrophobic interactions between two solid surfaces from the symmetric
hydrophobic interactions between two like surfaces [25].
5.2 Materials and Methods
5.2.1 Materials
Potassium ethyl xanthate (KEX, > 90%, TCI America) was used to hydrophobize the gold
surfaces. KEX was purified by dissolving in acetone, and recrystallized in diethyl ether twice
before use. All experiments were conducted using the ultrapure water obtained from a Direct-Q3
water purification system (EMD Millipore). The ultrapure water has a resistivity of 18.2 MΩ•cm
and < 10 ppb of total organic carbon.
5.2.2 Substrate preparation
The gold substrates were prepared by depositing a thin layer of gold on a flat silicon surface
using the physical vapor deposition technique. The deposition was performed using the PVD-250
system (Kurt J. Lesker) in a class-100 cleanroom. Single-side polished silicon wafers (orientation:
< 100 >, University wafer, Inc.) were used as the base substrates. They were cleaned in a boiling
piranha solution (70:30 by volume H SO :H O ) for 10 minutes, rinsed thoroughly with water,
2 4 2 2
and deoxidized using buffered oxide etch (BOE) solution for 10 seconds. The deposition process
100 |
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