the study of the rheological 
 properties of polyelectrolytes , which 
 in nature include nucleic acids , some proteins , and carbohydrates , 
 has important consequences with respect to both an understanding of 
 fundamental biological fluid dynamics and also emerging biotechnological 
 applications . recent industrial examples 
 of the use of either anionic or cationic synthetic polyelectrolytes ( carrying either negative or positive charges 
 along their polymer chains ) include applications in biomedical systems , drug delivery , the 
 food industry , water purification , nanotechnology , and surfactant 
 solutions . in this context , polyacrylamide ( pam ) 
 is a commonly used polyelectrolyte , 
 with applications that range from biology to industries 
 such as oil processing , purification 
 of natural water and wastewater , and 
 paper making . several literature studies 
 concerning different methods for the preparation of polyacrylamide 
 solutions and the influence on their rheological properties 
 of different parameters such as polymer concentration , solvent composition , 
 temperature , and aging of the sample have been reported . in these cases , the rheological properties of 
 pam , in common with other water - based solutions of polyelectrolytes , 
 are usually studied with conventional bulk rheometers and can be described by means of frequency - dependent complex 
 shear modulus g*( ) = g( ) 
 + ig( ) , where the real part ( i.e. , 
 the elastic modulus g( ) ) provides 
 a measure of the energy stored by the solution and the imaginary part 
 ( i.e. , the viscous modulus g( ) ) provides 
 a measure of the energy dissipated by the solution in response to 
 the work exerted by an external force . however , the measurement 
 of these quantities becomes complicated 
 at very low polyelectrolyte concentrations ( i.e. , a few parts per 
 million ) . under these circumstances , the rheological behavior of these 
 solutions that have very weak viscoelasticity is not fully understood , 
 in part because of the fact that the measurement of these systems 
 often falls below the range of sensitivity of classical bulk rheometers . for example , in multiphase systems 
 these small interaction forces can strongly influence the emulsion 
 morphology through the imposed flow as a result of the deformation , break up , and coalescence of the inclusions and through the formation of aggregates . as a consequence , there is importance in understanding 
 better these small interactions that occur in these low - concentration 
 polyelectrolyte solutions by using more sensitive microrheological 
 techniques . in the past decade , many such microrheology techniques have been developed and used to determine the rheological properties 
 of viscoelastic fluids , including those involving the use of video 
 particle tracking microrheology , magnetic 
 tweezers , and dynamic light scattering . among these methods , the technique of optical 
 tweezers , where a highly focused laser beam traps , in three dimensions , 
 micrometer - sized dielectric particles suspended in a fluid , has been extensively developed . most recently , a novel and simple analytical method for the analysis 
 of optical tweezing data for microrheology has been proposed by tassieri 
 et al . , greatly simplifying the application 
 of the technique . in this article , we adopt this new method to show the rheological characterization of 
 very low concentration 
 polyacrylamide solutions by means of optical tweezers measurements . we compare these results to conventional rheology 
 measurements , demonstrating the effectiveness of the technique in 
 exploring the very weak interactions between these polyelectrolyte 
 chains at low concentrations . the sensitivity 
 of the technique enables us not only to understand better the influence 
 of polymer chain interactions but also to demonstrate the roles of 
 salt concentration and aging in the rheological properties of these 
 solutions . in optical tweezing , we consider 
 the case of a micrometer - sized 
 spherical particle suspended in a fluid at thermal equilibrium where 
 it experiences random forces leading to brownian motion , driven by 
 the thermal fluctuations of the fluid s molecules . an analysis 
 of the particle s trajectory reveals information on the viscoelastic 
 properties of the suspending fluid , as demonstrated in the pioneering 
 work of mason and weitz that established 
 the field of microrheology . in the case when the probe s 
 fluctuations are constrained 
 by a stationary harmonic potential generated by optical tweezers , 
 one could write a generalized langevin equation1where m is the mass of the 
 particle , a(t ) is its acceleration , v(t ) is its velocity , and fr(t ) is the usual gaussian 
 white noise term , modeling stochastic thermal forces acting on the 
 particle . the integral term , which incorporates a generalized time - dependent 
 memory function , (t ) , represents viscous damping 
 by the fluid . here ,  is the optical trap stiffness that can 
 be easily determined by appealing to the principle of equipartition 
 of energy2where rj is the 
 time - independent 
 variance of the cartesian component ( j = x , y , z ) of the d - dimensional vector describing the particle s displacement 
 from the trap center , the origin of r. it has been shown that eq 1 can be solved 
 in terms of either the normalized mean - square displacement ( nmsd ) 
 ( ) = r()/2r or the normalized 
 position autocorrelation function ( npaf ) a( ) 
 = ( r(t ) r(t + )/r )  3where ( ) and ( ) 
 are the fourier transforms of ( ) 
 and a( ) , respectively . the inertia term ( m ) reported in the original 
 publications can be neglected here because 
 for micrometer - sized particles it becomes significant only above the 
 megahertz frequency range . equation 3 relates 
 the material s 
 complex shear modulus g*( ) to the observed 
 time - dependent bead trajectory r(t ) via the fourier transform of one of the related time - averaged 
 quantities ( i.e. , ( ) or a( ) ) . the evaluation of these fourier transforms , given only a finite set 
 of data points over a finite time window , is performed by means of 
 the analytical method recently proposed by tassieri et al . a 0.5 wt % stock polyacrylamide ( pam , mw 7000 kda , dryfloc by snf italia ) solution 
 was made in distilled water ( as the supporting solvent ) . solutions 
 were mixed for 48 h using magnetic stirrers at low speeds in order 
 to prevent mechanical degradation of the polymer chains . the stock 
 solution was then diluted by additions of distilled water in order 
 to obtain final concentrations of 25 , 50 , 125 , and 250 ppm ( w / w ) . 
 in this concentration range , the fluids show weak elasticity and a 
 viscosity slightly higher than those of the solvent . sodium chloride ( 1 , 2.5 , and 5 wt % ) was added to the final 
 solutions to investigate the effect of salt on fluids rheology . to 
 prevent degradation , the fluids were kept at 4 c until the measurement 
 was made . the rheological measurements 
 on the sample were carried out at 20 c by a stress - controlled 
 rotational mcr 301 rheometer ( anton paar instruments ) using double - gap 
 geometry . a preshear of 100 s is applied for 60 s to cancel possible loading effects . the oscillatory 
 tests are run in the range of linear viscoelasticity , as verified 
 by preliminary amplitude sweep tests . optical trapping was 
 achieved by means of a titanium  sapphire laser with a 5 w pump 
 ( verdi v5 laser ; coherent inc . ) , which provides up to 1 w at 830 nm . the tweezers use an inverted microscope , where the same objective 
 lens ( 100 , 1.3 numerical aperture , zeiss , plan - neofluar ) both 
 focuses the trapping beam and images the thermal fluctuations of a 
 single 5-m - diameter silica bead . samples are mounted on a motorized 
 microscope stage ( prior pro - scan ii ) . a cmos camera ( dalsa genie hm640 
 gige ) was used to collect high - speed images of a reduced field of 
 view . these images were processed in real time at 1 khz using 
 homemade labview ( national instruments ) single - particle - tracking software running on a personal computer . figure 1 shows the normalized position autocorrelation 
 function ( npaf ) and the normalized mean - square displacement ( nmsd ) 
 as a function of the lag time  for a single optically trapped 
 5-m - diameter silica bead suspended in two pam solutions at 
 concentrations of ( a ) 50 and ( b ) 250 ppm , respectively , with both 
 containing 1 wt % nacl . the inset of figure 1a shows a 10 s part of the entire displacement of the bead along 
 the x direction of the sample containing 50 ppm pam . it is interesting that neither npaf curve for these samples decays 
 as a single exponential , as would be expected for a newtonian fluid , but instead shows an intermediate shoulder ( or 
 a plateau for the nmsd curves ) before tending to zero ( or to one in 
 the case of the nmsds ) . such a shoulder ( or plateau ) is an interesting 
 feature not previously observed for optical tweezers , although interestingly 
 this has been predicted by a theoretical study of the dynamics of 
 an optically trapped particle suspended in a maxwell fluid . this feature is related to the material s 
 elastic plateau modulus , which constrains the bead fluctuations only 
 temporarily . as described above , the fluids linear viscoelastic 
 properties ( lve ) can be obtained from the analysis of the data shown 
 in figure 1 via eq 3 . npaf and 
 nmsd vs delay time  of a single optically trapped 
 5-m - diameter silica bead suspended in two water - based solutions 
 at pam concentrations of ( a ) 50 and ( b ) 250 ppm with 1 wt % nacl in 
 both solutions . the inset of panel a shows a 10 s part of the entire 
 displacement of the bead along the x direction . in figures 2 and 3 , we show a comparison between microrheology , 
 as measured by optical 
 tweezers , and bulk rheology , measured using a rotational rheometer , 
 for 50 and 250 ppm pam with 1 wt % nacl , the same fluids reported 
 in figure 1 . in figure 2 , the elastic ( g , circles ) and viscous ( g , triangles ) moduli are plotted as a function of 
 the oscillation frequency for the two different techniques . the two 
 methodologies show good agreement demonstrating that optical tweezing 
 provides a precise measurement of the material lve properties over 
 a much wider range of frequencies , giving more information on the 
 material characteristic relaxation times . it is interesting that both 
 measurements provide for the two fluids the same estimation of the 
 moduli slope and crossover frequency , with the latter being slightly 
 higher for the 250 ppm solution ( i.e. , the crossover frequency is 
 2.8 rad / s for the 50 ppm sample and 11 rad / s for the 250 ppm sample ) . elastic 
 ( g ) and viscous ( g ) 
 moduli as a function of the oscillation frequency measured 
 with optical tweezers ( ot ) and bulk rheometry ( br ) are compared for 
 ( a ) 50 and ( b ) 250 ppm pam solutions with 1 wt % nacl . in figure 3 , the same measurements 
 are shown 
 but instead the complex viscosity  * is plotted 
 against the frequency for both techniques ( optical tweezing , circles ; 
 bulk rheology , triangles ) ;  * shows shear thinning 
 trend down to a pseudo plateau value at frequencies higher than 10 
 rad / s , which is very close to the viscosity of water . at high frequencies , the solute molecules have no time to interact among each other , given 
 the low concentration , and the fluid viscosity almost matches that 
 of the solvent . in contrast , at low frequencies the viscosity increases , 
 showing non - newtonian behavior . in figure 3 , the steady - shear viscosity 
 (  ) , as measured using a bulk rotational rheometer , is also 
 reported as a function of the shear rate and is compared to the two 
 complex viscosity measurements ; the excellent superimposition among 
 the data indicates the applicability of the cox the data are also in good agreement 
 with previous measurements on similar 
 systems , reported only in the high - shear - rate range . frequency - dependent viscosity 
 measured with optical tweezers and 
 bulk rheometry , compared for ( a ) 50 and ( b ) 250 ppm pam solutions 
 with 1 wt % nacl . complex viscosities (  * ) , 
 plotted as a function of the frequency , are compared to shear viscosity 
 (  ) , plotted as a function of the imposed shear rate (  ) , 
 according to the cox figures 2 and 3 provide 
 a quantitative validation of the microrheology measurements performed 
 with optical tweezers in terms of both moduli and complex viscosity . 
 in the analysis reported in this article , we use optical tweezers 
 measurements to investigate the concentration functionality of the 
 lve properties of pam solutions , the effects of salt ( nacl ) addition 
 and fluid aging . figure 4 shows the concentration functionality 
 of lve properties 
 for four different fluids with pam concentration ranging from 25 to 
 250 ppm ; measurements of the complex viscosity (  * ) and loss modulus ( g ) for water are also 
 reported for comparison . in figure 4a , we report 
 the elastic modulus g. it can be seen that 
 in the intermediate frequency range ( ca . 1010 
 rad / s ) g shows a horizontal plateau , with 
 the modulus being a growing function of the polymer concentration . in contrast , the viscous modulus ( g ) , 
 reported in figure 4b , does not change substantially 
 with the pam concentration at high frequency . in the intermediate 
 frequency range , , galso shows a horizontal plateau , hence the horizontal 
 trend in both moduli suggests a gel - like rheological behavior of the 
 fluids in that frequency range . finally , the complex viscosity 
 (  * ) , calculated 
 from the same measurement data , is reported in figure 4c . in the low - frequency range , a significant increment is 
 observed as a function of the pam concentration , whereas at high frequencies 
 all of the curves collapse to the same plateau , corresponding to the 
 viscosity of water . in the thinning range of the complex viscosity 
 plot , it was possible to estimate the power law index , the measurements lead to the same value n  
 0.05 for all of the pam concentrations examined . a similar nondependency 
 of n from the pam concentration was previously observed 
 in a different concentration range , even if a higher value of n = 0.5 was reported . the discrepancy 
 can be likely attributed to different properties of the polyacrylamide 
 samples used ( i.e. , molecular weight , degree of hydrolysis , and salt 
 concentration ) . it is of note that the frequency where the fluids 
 collapse to newtonian behavior ( i.e. , the flow characteristic time 
 when the pam molecule interactions become negligible ) occurs at higher 
 frequencies as the pam concentration increases , as expected . overall 
 we can state that the concentration functionality of the lve properties 
 of pam solutions ( figure 4 ) is due to the increase 
 in the number of entanglements between the pam chains as shown in 
 the literature . concentration functionality of the linear 
 viscoelastic properties 
 ( ( a ) g , ( b ) g , and 
 ( c ) the complex viscosity ) of pam solutions at concentrations of 25 , 
 50 , 125 , and 250 ppm , respectively . further insight into the dependence of pam concentration 
 on fluid 
 viscosity can be obtained by comparing viscosity values at a fixed 
 frequency . we chose to compare data at  = 0.15 rad / s ( i.e. , 
 in the shear - thinning range , as indicated by the vertical line in 
 figure 4c ) . these viscosity values are plotted 
 as a function of pam concentration in figure 5 where a nonlinear dependence (   c ) is shown . this dependence is typical of polyelectrolyte 
 solutions in the semidilute entangled regime in a low - salt - concentration 
 limit . complex viscosity evaluated at 0.15 rad / s 
 vs pam concentration . 
 the continuous line is a guide for the gradient showing the prediction 
 for the semidilute entangled regime of polyelectrolytes (   c ) . , we report the 
 effect of salt ( nacl ) concentration on the lve properties of a pam 
 solution at a concentration of 125 ppm . it can be seen that the elastic 
 modulus of the pam solution shows nonmonotonic behavior ( figure 6a ) as a function of salt concentration . in fact , 
 by adding 1 wt % sodium chloride to the solution an initial increment 
 in the values of g is observed , with the 
 elastic moduli showing a constant plateau for frequencies lower than 
 10 rad / s , whereas the further addition of salt causes a decrease in 
 the g value in the same range of frequencies . the viscous modulus ( figure 6b ) shows a monotonic 
 decrease in the low - frequency range , whereas at high frequencies all 
 of the curves collapse together ( with the exception for the no - salt 
 fluid ( 0% ) that presents higher values and a lower slope ) . the influence 
 of the salt on the rheology is also reported in figure 6c in terms of the complex viscosity , combining the data of 
 the two moduli . this nonmonotonic effect of salt concentration 
 on g is possibly due to salt effects on the 
 inter- and intramolecular charges . when a polyelectrolyte is dissolved 
 in water in the absence of salt , the chain swells because of the mutual 
 repulsion of internal charges ( negative in our polyacrylamide samples ) 
 in order to minimize the electric intramolecular interactions . this 
 same mechanism hinders intermolecular interactions between chains 
 at these very low values of pam concentration ; this interpretation 
 is consistent with the fact that the sample without salt does not 
 show any gel - like behavior over the entire frequency range investigated 
 ( figure 6 ) . we observe that by adding 
 a small amount of nacl ( 1 wt % ) gel - like 
 behavior is exhibited ( figure 6 ) . this is due 
 to the fact that salt ions shield all intra- and intermolecular electric 
 interactions between and within the chains . such shielding of charges 
 favors the interaction between different chains , whereas it causes 
 chain shrinkage on the intramolecular level . thus , we conclude that 
 at low salt concentration the chain shrinkage is not sufficient to 
 prevent intermolecular interactions , thus allowing the gel - like behavior 
 that is found in the middle frequency range investigated ( figure 6 ) . in this scenario , further addition of salt generates 
 a stronger shrinkage of the pam chains with a consequent decrease 
 in the value of g resulting from the decrease 
 in the number of entanglements among the chains . similar weakening 
 in g due to the addition of salt has already 
 been observed in pam gels . viscoelastic moduli ( a ) g and ( b ) g and ( c ) complex 
 viscosity  * vs frequency of a pam solution 
 at a concentration of 125 ppm for 
 four different concentrations of salt ( nacl ) . in figure 7 , we report the changes , measured 
 over 5 weeks , of the viscoelastic properties of a pam solution at 
 a concentration of 125 ppm with 1 wt % nacl . it is clear that there 
 is a considerable drop in the viscoelastic properties within the first 
 3 weeks of sample preparation . once again , the main effect is measured 
 for the elastic modulus ( g ) , as reported 
 in figure 7a , and is also reflected in terms 
 of the complex viscosity (  * ) , as reported 
 in figure 7c , whereas the viscous modulus ( g ) shows only limited changes , mainly at low frequencies 
 ( figure 7b ) . after a period of about 
 3 weeks , the aging appears not to affect additional lve properties , 
 and the measurements were consistently reproducible . the effect of 
 aging on pam solutions was previously attributed to the weakness of 
 the entanglements between the pam chains . these interactions , unlike the stronger covalent bonds , could change 
 with time and modify the internal structure of the samples until some 
 equilibrium is eventually reached . aging of the viscoelastic properties of 
 a pam solution at a concentration 
 of 125 ppm and with 1 wt % nacl . a rheological characterization of polyacrylamide 
 solutions with 
 very low viscoelasticity was carried out both by optical tweezers 
 and bulk rheology . hence , we demonstrate the validity of the optical tweezers 
 as a tool for the rheological characterization of weakly viscoelastic 
 polyelectrolyte solutions . this technique allows the investigation 
 of much lower values of the elastic modulus ( below 10 pa ) over a wider range of frequency when compared to classical bulk 
 rheological measurements . furthermore , the concentration dependency 
 of the linear viscoelastic 
 properties of pam solutions was studied , and good agreement with literature 
 data was found . finally , we have investigated the influence of the 
 addition of salt and aging on the linear viscoelastic properties of 
 a pam solution . a nonmonotonic salt influence on very low concentration 
 pam solutions was found , depending on the salt concentration ; indeed , 
 the addition of salt can generate weak gel - like behavior at first 
 whereas further salt addition appears to weaken any pam network . we 
 also showed that pam samples change with time after preparation and 
 need at least 3 weeks to reach a stationary morphology . these results 
 are relevant to the characterization of polyelectrolyte solutions 
 with very low polymer concentrations , with applications in the food , 
 detergent , and pharmaceutical industries , where polyelectrolytes are 
 used to improve product stability and rheology .
recently , optical tweezing has been 
 used to provide a method for 
 microrheology addressed to measure the rheological properties of small 
 volumes of samples . in this work , 
 we corroborate this emerging field 
 of microrheology by using these optical methods for the characterization 
 of polyelectrolyte solutions with very low viscoelasticity . the influence 
 of polyelectrolyte ( i.e. , polyacrylamide , pam ) concentration , specifically 
 its aging , of the salt concentration is shown . 
 the close agreement 
 of the technique with classical bulk rheological measurements is demonstrated , 
 illustrating the advantages of the technique .