Self-assembly of colloidal molecules due to self-generated flow
Ran Niu Thomas Speck Thomas Palberg





Institut für Physik, Johannes Gutenberg-Universität Mainz,
  Staudingerweg 7-9, 55128 Mainz, Germany


  The emergence of structure through aggregation is a fascinating topic and of both fundamental and practical interest. Here we demonstrate that self-generated solvent flow can be used to generate long-range attractions on the colloidal scale, with sub-pico Newton forces extending into the millimeter-range. We observe a rich dynamic behavior with the formation and fusion of small clusters resembling molecules, the dynamics of which is governed by an effective conservative energy that decays as 1/r. Breaking the flow symmetry, these clusters can be made active.


Self-assembly of colloidal molecules due to self-generated flow
    Ran Niu, Thomas Speck, Thomas Palberg
    
===============================================================





Colloidal particles acting as “big” artificial atoms have been instrumental in studying microscopic processes in condensed matter, from the kinetics of crystallization <cit.> to the vapor-liquid interface <cit.>. Due to their size, colloidal particles are observable directly in real space. Moreover, interactions are widely tunable, ranging from hard spheres to long-range repulsive, short-range attractive, and dipolar <cit.>. Consequently, colloidal particles can be assembled into a multitude of different structures: from clusters <cit.> and stable molecules <cit.> composed of a few particles to extended bulk structures like ionic binary crystals <cit.>. In addition, self-assembly into useful superstructures can be controlled by factors such as confinement <cit.> and particle shape <cit.>, which make colloids a versatile and fascinating form of matter <cit.>.

What is still missing are truly long-range attractions of like-charged (or uncharged) identical colloidal particles. There is much interest in the basic statistical physics of systems with such interactions, which play a role in gravitational collapse, two-dimensional elasticity, chemotactic collapse, quantum fluids, and atomic clusters <cit.>. One proposed realization are colloidal particles trapped at an interface <cit.> that experience screened, long-range attractions due to capillary fluctuations of the interface <cit.>. The attractive interactions then correspond to Newtonian gravity in two dimensions. Complex patterns are also known to arise for bacteria due to long-range chemotactic interactions <cit.>. Critical long-range Casimir forces have been reported for colloidal particles in a binary solvent <cit.>, which are tunable by temperature and surface chemistry. Finally, a recent theoretical proposal are catalytically active colloidal particles that interact through producing or consuming chemicals <cit.>. For simple diffusion the concentration profile of a chemical decays as inverse distance, implying long-range interactions that can be tuned through activity (how chemicals are produced or consumed) and mobility (how particles react to gradients).



Here, we implement long-range attractions through hydrodynamic flows coupling suspended particles <cit.>. We report on experiments using spherical ion exchange resin particles sedimented to the negatively charged substrate. The particles have diameters of 15μ m, for which Brownian diffusion is practically negligible on the experimental time scale. They interact due to self-generated local flow, resulting in an effective long-range 1/r attraction as expected for three-dimensional unscreened gravity. Our present understanding of the mechanism responsible for their aggregation can be summarized as follows <cit.>: By exchanging residual cationic impurities for stored hydrogen ions (Fig. <ref>a), the particles generate a concentration profile c that decays away from the particles. Different diffusion coefficients of the exchanged cations and the released ions locally generate diffusio-electric fields which retain overall electro-neutrality by slowing the outward drift of hydrogen ions and accelerating the inward drift of impurities.








The total system composed of solvent, ions, and colloidal particles is clearly out of thermal equilibrium through the free energy released by solvating the hydrogen ions, which drives the flow. However, appealing to a scale separation between particles and solvent, our crucial assumption will be that the motion of the particles themselves can be described by 

    _i = -∇_i U + _i,

where U=∑_i<ju(_i-_j) is a conservative potential, u(r) is the pair potential Eq. (<ref>), and _i models the noise with zero mean and correlations _i(t)_j^T(t')=21δ_ijδ(t-t').


The relative velocity of two particles with separation =_1-_2 is v(r)=(/r)·(_1-_2)δ(||-r)=2u'(r) after inserting Eq. (<ref>). Hence, we have direct access to the interactions u(r) through measuring a dynamic quantity, the approach velocity v(r) as shown in Fig. <ref>a. Approaching each other, the speed increases as expected but reaches a maximum at about r≃60μ m before it drops rapidly. 

This behavior can be rationalized by considering the generated flows in more detail. In the double layer of the substrate, the local fields generate electro-osmotic flow of the solvent since the negative substrate is screened by an excess of positive charges (Fig. <ref>b). The motion of the particles is determined by the slip velocity =-μ_p∇ c with ion concentration field c and phoretic mobility μ_p <cit.>, whereby our measured particle velocities are consistent with a constant mobility. For an isolated particle, the solvent flow is approaching symmetrically and no net motion results (the particle can be regarded as a sink in two dimensions; of course, the solvent is incompressible with backflow out of the plane of the substrate). For two particles, the hydrogen ion concentration is increased in the space between the particles, which reduces the gradient (with respect to the particles' surfaces) and thus the flow velocity. Hence, the flow on each particle now becomes asymmetric, resulting in an apparent attraction (Fig. <ref>c). Basically the same mechanism acts in the double layer of the particles but now leads to electro-phoretic motion. The particles themselves are (slightly) negatively charged due to the release of cations. The field generated by the concentration gradient of the other particle again generates a solvent flow, however, now particle and solvent taken together are force-free, which leads to a particle speed in the opposite direction of the field (Fig. <ref>d).

For purely diffusive ion motion one would expect the concentration to decay as c(r)∼1/r, but measurements of the pH reveal a more complicated behavior (these measurements had to be performed for a larger particle, see Supplemental Material <cit.>, but we expect the qualitative features to be the same for the smaller particles used here). While there is indeed a 1/r decay regime, closer to the ion exchange particle it changes to a slower decay that is well described as exponential. We speculate that this accumulation is caused by the local fields slowing down outward moving ions. We model this effect through a term resembling screening although we stress that it originates from the flow and not electrostatics. Combining both flows, the functional form of the potential reads

    u(r) = -/r + /re^-r/ξ

with three free parameters: the prefactors  and , and the screening length ξ. As shown in Fig. <ref>a, this function describes the experimental data very well. From the fit we obtain ≃6120μ m^3/s, ≃8805μ m^3/s, and ξ≃31.4μ m. The pair potential u(r) plotted in Fig. <ref>b has a minimum at r_0≃35μ m. As shown in the inset of Fig. <ref>b, r_0 agrees well with the maximum of the distributions of bond length r for dimers and trimers.


Usually, overdamped motion is described by a product of particle mobility and the gradient of the potential energy. Since we do not have access to these terms separately, we treat u(r) as an effective “energy” absorbing the phoretic mobility μ, with u(r) thus having units of a diffusion coefficient. Nevertheless, we can estimate physical energies employing the bare particle mobility μ_0, which quantifies the forces needed to move a single particle through the solvent with desired speed. For our particles in water its value is μ_0≃7.8μ m/(s· pN), yielding forces between particles of order 0.1pN. With u(r_0)≃-92.34μ m^2/s, the corresponding bond dissociation energy would thus be E_b=|u(r_0)|/μ_0≃7000kJ/mol.



The final ingredient for our theoretical model is an effective temperature, which we extract from the measured bond fluctuations. First, from the distribution of the bond length r for dimers we determine its variance Var(r)≃3.57μ m^2. Assuming that these vibrations are effectively equilibrated allows us to determine the analog of a temperature, ≈Var(r)u”(r_0)≃0.3μ m^2/s. We test this assumption for three particles, for which a quick calculation predicts that the harmonic bond fluctuations are 5/3 times larger <cit.>. The predicted value 5.95μ m^2 is only slightly smaller than the measured variance 6.25μ m^2, the difference of which is due to anharmonic higher-order vibrations.


Our suspension of ion exchange particles is not stationary but slowly collapses to a close-packed state due to the long-range interactions. Starting from a homogeneous density profile, during this process we observe the formation of colloidal clusters (“molecules”) with n particles. This assembly happens autonomously in contrast to prefabricated colloidal molecules <cit.>. While metastable, single clusters can be observed up to minutes, which allows in principle to study in detail different isomers and the “reactions” by which larger clusters form (more details are given as Supplemental Material <cit.>). In Fig. <ref>a we show the first few minutes of this process for a dilute suspension. The experiments were carried out with 60-90 particles within a field of view corresponding to an area fraction of approx. 0.25%. Without adjustable parameters, the observed dynamics are reproduced through the model described by Eq. (<ref>). In Fig. <ref>b we show a sequence of experimental snapshots for seven particles. We then perform simulations of the model using the extracted particle positions from the first experimental frame as initial positions. As shown in Fig. <ref>c, the simulations agree with the experiments on the same time scale. While for repeated simulation runs the positions differ due to the noise, the average behavior is consistent.



This fit-free quantitative agreement is corroborated by the time evolution of the fraction of clusters with weight n. From the analysis of the experiments, we extract the fraction N_n(t)/N of particles residing in clusters with n particles. The range of N is between 60 and 90, and to improve statistics we average over all experiments. We repeat the same analysis with the theoretical model for N=80, the comparison of which is shown in Figure <ref>. The qualitative behavior is that of irreversible aggregation as described through the Smoluchowski coagulation equation <cit.>, with a steady decrease of the monomer concentration and peaks for the n-mers that become flatter and shifted to later times for increasing weight n.


As demonstrated, for a one-component suspension of ion exchange particles in moderate flow the dynamics of the particles alone is effectively described through a conservative potential. However, for larger aggregates also the flow increases, leading to deviations from the predicted behavior (e.g. particles are lifted and pushed into the second layer). With even larger flows, particles are observed to spontaneously break the flow symmetry and become self-propelled. Another strategy is to explicitly break symmetry through mixtures of particles with different sizes or shapes, or mixtures of activated and passive particles <cit.>. Here we explore the consequences of adding anionic ion exchange particles of similar size but releasing OH^- ions, for which we again observe the assembly of low-weight clusters. This is shown exemplary in Fig. <ref> for two cationic particles and one anionic particle, which assemble into a trimer. As predicted in Ref. , the geometry together with the different mobilities/activities leads to a self-propelled complex. This can be seen through determining the center-of-mass speed of the three particles, which clearly shows a transition to a constant speed once the trimer has assembled.





To conclude, we have shown that ion exchange particles close to a charged substrate generate flows that lead to effectively conservative, long-range interactions. Autonomous flow-driven assembly through long-range attractions might enable novel non-equilibrium materials and strategies in self-assembly, in particular on intermediate scales for which thermal motion has become negligible but which cannot be manipulated directly. Moreover, through changing composition one may generate asymmetric flow. The resulting directed motion of colloidal particles in combination with volume exclusion leads to fascinating dynamic behavior ranging from clustering <cit.> and the formation of “living crystals” <cit.> to schooling and swarming <cit.>. Our results demonstrate how one can implement strategies to control and engineer interactions and directed motion on the same footing <cit.>, which is a step towards designing active particles that can perform dynamical tasks such as transport of cargo <cit.>. Concerning the size of the particles used here, we note that there is no conceptual barrier to using smaller particles, the main issue being the ion exchange rate and ion capacity of singe particles that determines the flow strength and the time over which the flow is being generated.





We acknowledge the DFG for funding within the priority program SPP 1726 (grant numbers PA 459/18-1 and SP 1382/3-1).











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@bib@innerbibempty

[Palberg(2014)]palb14
  author author T. Palberg, title title Crystallization
  kinetics of colloidal model suspensions: recent achievements and new
  perspectives, 10.1088/0953-8984/26/33/333101 journal journal J. Phys.: Condens. Matter volume 26, pages 333101 (year
  2014)NoStop
[Aarts et al.(2004)Aarts,
  Schmidt, and Lekkerkerker]aart04
  author author D. G. A. L.  Aarts, author M. Schmidt,  and author H. N. W. Lekkerkerker, title title Direct
  visual observation of thermal capillary waves, 10.1126/science.1097116 journal journal
  Science volume 304, pages 847–850
  (year 2004)NoStop
[Royall et al.(2003)Royall,
  Leunissen, and van Blaaderen]roya03
  author author C. P. Royall, author M. E. Leunissen,  and author A. van
  Blaaderen, title title A new
  colloidal model system to study long-range interactions quantitatively in
  real space, @noop  journal journal J.
  Phys. Condens. Matter volume 15, pages
  S3581 (year 2003)NoStop
[Yethiraj(2007)]yeth07
  author author A. Yethiraj, title title Tunable
  colloids: control of colloidal phase transitions with tunable
  interactions, 10.1039/B704251P journal
  journal Soft Matter volume 3,  pages 1099–1115 (year 2007)NoStop
[Meng et al.(2010)Meng,
  Arkus, Brenner, and Manoharan]meng10
  author author G. Meng, author N. Arkus,
  author M. P. Brenner,  and  author V. N. Manoharan,  title title The free-energy landscape of
  clusters of attractive hard spheres, 10.1126/science.1181263 journal journal
  Science volume 327, pages 560–563
  (year 2010)NoStop
[Perry et al.(2015)Perry,
  Holmes-Cerfon, Brenner, and Manoharan]perr15
  author author R. W. Perry, author M. C. Holmes-Cerfon, author M. P. Brenner,  and author V. N. Manoharan, title title
  Two-dimensional clusters of colloidal spheres: Ground states, excited
  states, and structural rearrangements, 10.1103/PhysRevLett.114.228301 journal journal
  Phys. Rev. Lett. volume 114, pages
  228301 (year 2015)NoStop
[Demirörs et al.(2015)Demirörs, Stiefelhagen, Vissers,
  Smallenburg, Dijkstra, Imhof, and van Blaaderen]demi15
  author author A. F. Demirörs, author J. C. P. Stiefelhagen, author T. Vissers, author F. Smallenburg, author M. Dijkstra, author A. Imhof,   and author A. van Blaaderen,  title title Long-ranged oppositely
  charged interactions for designing new types of colloidal clusters, 10.1103/PhysRevX.5.021012 journal journal Phys. Rev. X volume 5, pages
  021012 (year 2015)NoStop
[Manoharan et al.(2003)Manoharan, Elsesser, and Pine]mano03
  author author V. N. Manoharan, author M. T. Elsesser,  and author D. J. Pine, title title Dense packing and
  symmetry in small clusters of microspheres, 10.1126/science.1086189 journal journal
  Science volume 301, pages 483–487
  (year 2003)NoStop
[Sacanna et al.(2010)Sacanna, Irvine, Chaikin, and  Pine]saca10
  author author S. Sacanna, author W. T. M. Irvine, author P. M. Chaikin,
   and author D. J. Pine,  title title Lock and key colloids,  10.1038/nature08906 journal journal Nature volume 464, pages
  575–578 (year 2010)NoStop
[Leunissen et al.(2005)Leunissen, Christova, Hynninen,
  Royall, Campbell, Imhof,
  Dijkstra, van Roij, and van
  Blaaderen]leun05
  author author M. E. Leunissen, author C. G. Christova, author A.-P. Hynninen, author C. P. Royall, author A. I. Campbell, author A. Imhof,
  author M. Dijkstra, author R. van Roij,  and author A. van Blaaderen, title title Ionic colloidal crystals of oppositely
  charged particles, 10.1038/nature03946 journal journal Nature volume 437,  pages 235–240 (year 2005)NoStop
[de Nijs et al.(2015)de Nijs, Dussi, Smallenburg, Meeldijk, Groenendijk, Filion,
  Imhof, van Blaaderen, and Dijkstra]nijs15
  author author B. de Nijs, author S. Dussi,
  author F. Smallenburg, author J. D. Meeldijk, author
  D. J. Groenendijk, author
  L. Filion, author A. Imhof, author A. van Blaaderen,  and author M. Dijkstra, title title Entropy-driven formation of large icosahedral colloidal clusters by
  spherical confinement, 10.1038/nmat4072 journal journal Nature Mater. volume
  14, pages 56–60 (year 2015)NoStop
[Glotzer and Solomon(2007)]glot07
  author author S. C. Glotzer and author M. J. Solomon, title title Anisotropy of
  building blocks and their assembly into complex structures, 10.1038/nmat1949 journal journal
  Nature Mater. volume 6, pages
  557–562 (year 2007)NoStop
[Manoharan(2015)]mano15
  author author V. N. Manoharan, title title Colloidal
  matter: Packing, geometry, and entropy, 10.1126/science.1253751 journal journal
  Science volume 349 (year 2015),  10.1126/science.1253751NoStop
[Campa et al.(2009)Campa,
  Dauxois, and Ruffo]camp09
  author author A. Campa, author T. Dauxois,   and author S. Ruffo, title title Statistical mechanics and dynamics of
  solvable models with long-range interactions, 10.1016/j.physrep.2009.07.001 journal journal
  Phys. Rep. volume 480, pages 57–159
  (year 2009)NoStop
[Ghezzi and Earnshaw(1997)]ghez97
  author author F. Ghezzi and author J. C. Earnshaw, title title Formation of
  meso-structures in colloidal monolayers, @noop  journal journal J. Phys. Condens. Matter volume 9, pages L517 (year 1997)NoStop
[Bleibel et al.(2011)Bleibel, Dietrich, Domínguez, and  Oettel]blei11
  author author J. Bleibel, author S. Dietrich,
  author A. Domínguez,  and  author M. Oettel, title title Shock waves in capillary collapse of
  colloids: A model system for two-dimensional screened newtonian gravity,  10.1103/PhysRevLett.107.128302 journal
  journal Phys. Rev. Lett. volume 107,  pages 128302 (year 2011)NoStop
[Budrene and Berg(1991)]budr91
  author author E. O. Budrene and author H. C. Berg, title title Complex patterns
  formed by motile cells of escherichia coli, 10.1038/349630a0 journal journal Nature  volume 349, pages 630–633 (year 1991)NoStop
[Hertlein et al.(2008)Hertlein, Helden, Gambassi, Dietrich, and Bechinger]hert08
  author author C. Hertlein, author L. Helden,
  author A. Gambassi, author S. Dietrich,  and author C. Bechinger, title
  title Direct measurement of critical casimir
  forces, 10.1038/nature06443 journal
  journal Nature volume 451, pages 172–175 (year 2008)NoStop
[Soto and Golestanian(2014)]soto14
  author author R. Soto and author R. Golestanian, title title
  Self-assembly of catalytically active colloidal molecules: Tailoring
  activity through surface chemistry, 10.1103/PhysRevLett.112.068301 journal journal
  Phys. Rev. Lett. volume 112, pages
  068301 (year 2014)NoStop
[Soto and Golestanian(2015)]soto15
  author author R. Soto and author R. Golestanian, title title
  Self-assembly of active colloidal molecules with dynamic function, 10.1103/PhysRevE.91.052304 journal journal Phys. Rev. E volume 91, pages 052304 (year 2015)NoStop
[Banerjee et al.(2016)Banerjee, Williams, Azevedo, Helgeson, and Squires]bane16
  author author A. Banerjee, author I. Williams,
  author R. N. Azevedo, author M. E. Helgeson,  and author T. M. Squires, title title Soluto-inertial phenomena: Designing
  long-range, long-lasting, surface-specific interactions in suspensions,  10.1073/pnas.1604743113 journal journal Proc. Natl. Acad. Sci. U.S.A. volume
  113, pages 8612–8617 (year 2016)NoStop
[Feldmann et al.(2016)Feldmann, Maduar, Santer, Lomadze, Vinogradova, and Santer]feld16
  author author D. Feldmann, author S. R. Maduar, author M. Santer,
  author N. Lomadze, author O. I. Vinogradova,  and author S. Santer, title
  title Manipulation of small particles at solid liquid
  interface: light driven diffusioosmosis, 10.1038/srep36443 journal journal Sci. Rep.  volume 6, pages 36443 (year
  2016)NoStop
[Anderson(1989)]ande89
  author author J. L. Anderson, title title Colloid
  transport by interfacial forces, 10.1146/annurev.fl.21.010189.000425 journal journal Ann. Rev. Fluid Mech. volume 21,  pages 61–99 (year 1989)NoStop
[Niu et al.(2016)Niu,
  Kreissl, Brown, Rempfer,
  Botin, Holm, Palberg, and  de Graaf]niu16
  author author R. Niu, author P. Kreissl,
  author A. T. Brown, author G. Rempfer, author
  D. Botin, author C. Holm, author T. Palberg,  and author J. de Graaf, @noop  title Microfluidic
  pumping by micromolar salt concentrations,  (year 2016),  note arXiv:1610.00337NoStop
[sm()]sm
  @noop  note See Supplemental Material at xxx for a
  detailed description of the experiments and pH measurements, a calculation of
  the bond vibrations, and more details on the observed structures (citing
  references ) as well as supporting
  movies.Stop
[Ebbens et al.(2010)Ebbens,
  Jones, Ryan, Golestanian,  and Howse]ebbe10
  author author S. Ebbens, author R. A. L. Jones, author A. J. Ryan,
  author R. Golestanian,  and  author J. R. Howse, title title Self-assembled autonomous runners and
  tumblers, 10.1103/PhysRevE.82.015304 journal journal Phys. Rev. E volume
  82, pages 015304 (year 2010)NoStop
[Ni et al.(2017)Ni,
  Marini, Buttinoni, Wolf,  and Isa]ni17
  author author S. Ni, author E. Marini, author I. Buttinoni, author
  H. Wolf,  and author
  L. Isa, title title Rational design and fabrication of versatile active
  colloidal molecules, @noop  journal journal arXiv:1701.08061  (year 2017)NoStop
[Aldous(1999)]aldo99
  author author David Aldous, title title Deterministic
  and stochastic models for coalescence (aggregation and coagulation): A review
  of the mean-field theory for probabilists, 10.2307/3318611 journal journal Bernoulli  volume 5, pages 3–48 (year
  1999)NoStop
[Reinmüller et al.(2013)Reinmüller, Schöpe, and Palberg]rein13
  author author A. Reinmüller, author H. J. Schöpe,  and author T. Palberg, title title Self-organized
  cooperative swimming at low reynolds numbers, 10.1021/la3046466 journal journal Langmuir  volume 29, pages 1738–1742 (year 2013)NoStop
[Buttinoni et al.(2013)Buttinoni, Bialké, Kümmel,
  Löwen, Bechinger, and Speck]butt13
  author author I. Buttinoni, author J. Bialké,
  author F. Kümmel, author H. Löwen, author
  C. Bechinger,  and author
  T. Speck, title title Dynamical clustering and phase separation in suspensions
  of self-propelled colloidal particles, 10.1103/PhysRevLett.110.238301 journal journal
  Phys. Rev. Lett. volume 110, pages
  238301 (year 2013)NoStop
[Palacci et al.(2013)Palacci, Sacanna, Steinberg, Pine, and Chaikin]pala13
  author author J. Palacci, author S. Sacanna,
  author A. Preska Steinberg,
  author D. J. Pine,  and  author P. M. Chaikin,  title title Living crystals of
  light-activated colloidal surfers, 10.1126/science.1230020 journal journal
  Science volume 339, pages 936–940
  (year 2013)NoStop
[Yan et al.(2016)Yan,
  Han, Zhang, Xu, Luijten, and Granick]yan16
  author author J. Yan, author M. Han, author J. Zhang, author
  C. Xu, author E. Luijten,  and author S. Granick, title title Reconfiguring active particles by electrostatic imbalance, 10.1038/nmat4696 journal journal Nat.
  Mater. volume advance online publication (year 2016), 10.1038/nmat4696NoStop
[Zhang et al.(2016)Zhang,
  Yan, and Granick]zhan16
  author author J. Zhang, author J. Yan,  and  author S. Granick, title title Directed self-assembly pathways of
  active colloidal clusters, 10.1002/anie.201509978
  journal journal Angew. Chem. Int. Ed. volume 55, pages 5166–5169 (year
  2016)NoStop
[Sundararajan et al.(2008)Sundararajan, Lammert, Zudans,
  Crespi, and Sen]sund08
  author author S. Sundararajan, author P. E. Lammert, author A. W. Zudans,
  author V. H. Crespi,  and  author A. Sen, title title Catalytic motors for transport of
  colloidal cargo, 10.1021/nl072275j journal
  journal Nano Letters volume 8,  pages 1271–1276 (year 2008)NoStop
[Baraban et al.(2012)Baraban, Tasinkevych, Popescu,
  Sanchez, Dietrich, and Schmidt]bara12
  author author L. Baraban, author M. Tasinkevych, author M. N. Popescu, author S. Sanchez,
  author S. Dietrich,  and  author O. G. Schmidt,  title title Transport of cargo by
  catalytic janus micro-motors, 10.1039/C1SM06512B
  journal journal Soft Matter volume 8, pages 48–52 (year
  2012)NoStop
[Malins et al.(2013)Malins,
  Williams, Eggers, and Royall]mali13
  author author A. Malins, author S. R. Williams, author J. Eggers,   and author C. P. Royall,  title title Identification of structure
  in condensed matter with the topological cluster classification, http://dx.doi.org/10.1063/1.4832897 journal
  journal J. Chem. Phys. volume 139,  pages 234506 (year 2013)NoStop









  

    Supplementary Information
  







§ EXPERIMENTS


The particles are cationic ion exchange resin particles (CK10S, Mitsubishi Chemical Corporation, Japan) with diameter 2a=15.3±3.0μ m and counter ions Na^+. Prior to experiments, the particles were washed with 20 wt % hydrochloride acid solution to exchange the counter ions into H^+. Then we rinsed with doubly deionized water several times until solution pH of ∼ 7 and dried at 80^∘C for 2 h. The sample cell was built from a circular Perspex ring (inner diameter of 20mm, height of 1mm) fixed to microscopy slides by hydrolytically inert epoxy glue and dried for 24 h before use. The glass slides were washed with alkaline solution (Hellmanex®III, Hellma Analytics) for 30 min under sonication, and subsequently rinsed with tap water and deionized water for several times.

In a typical experiment, a weighted amount of particles was dispersed into doubly deionized water. Subsequently, 400 μ-liters of particle suspension was added into the sample cell. The cell was covered immediately to avoid contamination. Particles settled to the bottom of the cell within minutes. Movies were then taken at a frame rate of 0.5 Hz using an inverted scientific microscope (DMIRBE by Leica, Germany). Particles were tracked through extracting the perimeter using a home-written Python script. The velocity v(r) of two approaching particles was averaged over 160 pairs with any other particle at least 10 times their diameter away.

In a second set of experiments we have added anionic ion exchange resin particles (CA08S, Mitsubishi Chemical Corporation, Japan) with diameter 2a=15.1±0.3μ m and counter ions Cl^-. Before use, particles were washed with concentrated sodium hydroxide (NaOH) to exchange the counter ions into OH^- and subsequently washed with deionized water until the pH of the solution reached about 7. To distinguish cationic ion exchange resin and anionic ion exchange resin, tiny amount of pH indicator solution (pH 4-10, Sigma-Aldrich) was added to mark them redish and bluish, respectively.





§ HYDROGEN ION CONCENTRATION


We determine the pH profile around an ion exchange particle with diameter 45μ m using a mixture of Universal indicator solutions (1:3 volume ratio of pH 0-5 and pH 4-10, Sigma-Aldrich, Inc). As the concentration of proton decreases, the color ratio of blue-to-red decreases monotonically. Thus, measuring the blue-to-red color ratio at fixed pH, we get a calibration curve. Applying this calibration curve to every pixel around ion exchange particle, we determine the proton concentration. The color images were recorded using a consumer DSLR (D700, Nikon, Japen) mounted on an inverted scientific microscope (DMIRBE, Leica, Germany). Image recording starts ∼2s after the ion exchange particle gets into contact with the indicator solution. The measured concentration c(r,t) as a function of distance r from the particle center and time t after preparation is shown in Figure <ref>.



A prediction for the concentration profile is given by the solution of the diffusion equation

    ∂_tc(,t) = D_+∇^2c(,t) + δ()

with ion diffusion coefficient D_+ and assuming a constant rate  with which ions are released. The solution is

    c(r,t) = /4π D_+r(r/2√(D_+t))
      + c_∞

with background concentration c_∞. From the fit of this expression to the decay of the concentration profile we obtain a background pH≃5.45 and a diffusion coefficient D_+≃1300μ m^2/s, which is reasonable for hydrogen ions given the fact that they are hydrated. Strikingly, the qualitative behavior changes when approaching the particle and the concentration profile decays much slower than what is expected from the simple diffusion picture. In fact, it can be fitted well by an exponential decay. This implies that outward moving ions are slowed, presumably through local electric fields generated in conjunction with other ions. The resulting solvent flow is thus more complex than ∼∇ c∼1/r^2, which is captured by the form of our effective pair potential, see the discussion in the main text.





§ BOND VIBRATIONS IN THE TRIMER


While the suspension itself is non-stationary, for the bond vibrations we assume that an effective equilibrium has been reached described by the Boltzmann factor. We expand the energy

    U ≈ U_0 + 1/2∑_ij^2nH_ijδ x_iδ x_j = U_0 + 1/2∑_=1^2n_ q_^2,

where H⃗ is the 2n×2n Hessian matrix of second derivatives of the potential energy evaluated at the minimum energy configuration of the cluster with n particles. It has 2n eigenvalues _, of which three are zero corresponding to translation and rotation in two dimensions. The other eigenvalues are the spring constants of the vibrational modes.


For n=3, the minimum energy configuration is an equilateral triangle and the Hessian becomes H⃗=u”(r_0)A⃗ with constant matrix A⃗ (with non-zero eigenvalues 3/2,3/2,3). The harmonic fluctuations of any edge can easily be calculated as

    |_i-_j-r_0_ij|^2 = ∑_q_^2 =
    ∑_/_ = 5/3/u”(r_0)

as used in the main text.





§ METASTABLE MOLECULES


Figure <ref> shows snapshots of structural arrangements sorted by their molecular weight from n=3 to n=6. We only show structures that persisted for at least 100s. We distinguish isomers assigning a structural fingerprint counting the number n_b of particles with b direct bonds. For every particle configuration (experiments and simulations) we first determine clusters of mutually bonded particles, where a particle pair (i,j) forms a bond if the separation |_ij| is smaller than the cutoff of 40μ m. We then refine the bond network and only retain bonds between direct neighbors. A particle j is a direct neighbor of i if for all particles k with distance |_ik|<|_ij| the condition _ik·_jk>0 holds (the enclosed angle is less than π/2 radians) <cit.>. For each cluster we count the number of particles n_b with b bonds, the vector of which forms a “fingerprint” based on which we identify the isomeric structure of the molecule.

The structures to the left in Figure <ref> show the ground states minimizing the effective energy. From the eigenspectrum of vibrations we delineate (meta-)stable molecules (blue frames) from unstable molecules (red frames). For n=3 and n=4 the latter nevertheless occur with a non-vanishing probability since they are populated by reactions adding another monomer. We can identify these transition states easily by their “dangling bonds” with n_1>0. Transition states quickly rearrange into stable isomers for the same n. These are relatively long-lived – typically until another reaction occurs increasing the molecular weight – but we also observe spontaneous transitions between isomers (see Supplementary Movie 2 and Fig. <ref>). For n=6 we have four stable isomers. For short-range attractions counting only direct bonds, the structures II, III, and IV form the degenerate ground state manifold <cit.>. Due to the long-range attractions in our case, the degeneracy is lifted with the regular pentagon (I) representing the minimal energy configuration. While most transitions proceed through a slight shift of bonds, note that the transition IV→III involves breaking a bond with excited intermediate III^∗. Consequently, for the parallelogram (IV) we observe a higher population both in experiment and theory than one would expect from its energy alone.





