5712 | Soft Matter, 2015, 11, 5712--5718 This journal is©The Royal Society of Chemistry 2015

Cite this: SoftMatter, 2015,

11, 5712

Effects of hydrodynamic interactions on thecrystallization of passive and activecolloidal systems

Shuxian Li, Huijun Jiang and Zhonghuai Hou*

Effects of hydrodynamic interactions (HI) on the crystallization of a two-dimensional suspension of colloidal

particles have been investigated, by applying a multiscale simulation method combining multiparticle

collision dynamics for solvent particles with standard molecular dynamics for the colloids. For a passive

system, we find that HI slightly shifts the freezing point to a smaller density, while the equilibrium structure

remains nearly unchanged for a given global order parameter. For an active system, however, HI can

significantly shift the freezing density to a higher value and the freezing transition becomes more

continuous compared to its passive counterpart. This HI-induced shift becomes more remarkable with

increasing propelling force. In addition, HI may also enhance the structural heterogeneities in an active

system. For both passive and active systems, it is shown that HI can accelerate the relaxation process to

their final steady state.

I. Introduction

Colloidal suspensions consist of typically micrometre-sizedspheres, and are playing an increasingly important role as modelsystems to study a variety of phenomena in condensed matterphysics, such as crystallization, melting, defect dynamics, phaseseparation and glass transition.1–6 It is crucial to understand indetail the static and dynamic behavior of colloidal suspensionsso that one can improve artificial material properties or developnew and reliable assembly processes. Very recently, with a hugedevelopment of chemical and optical techniques, active colloidalsystems have received considerable attention due to theirinteresting dynamics and possible applications, includingnon-equilibrium transport phenomena, pattern formation,microswimmers, nanomachines, and drug delivery. In contrastto conventional passive colloidal particles which are in thermalequilibrium with the surrounding solvent and undergo Brownianmotion, active colloidal particles consume energy at the indivi-dual scale in order to generate self-propelled motion, which canbe achieved by, for example, chemical reaction catalyzed on itsown surface,7–9 or self-thermophoresis due to the photothermaleffect.10,11 Actually, this has opened up new strategies to explorethe structural and phase behavior of self-propelled colloidalparticles in a non-equilibrium system. For instance, it has beenshown in a number of experimental and theoretical work that

particle activity affects significantly the phase separation anddynamic clustering in active colloidal suspensions.12–16 Com-pared to the passive counterpart, active colloidal particles haveto crystallize at higher density accompanied by pronouncedstructural heterogeneities.17 In addition, particle activity canspeed up the relaxation dynamics of a dense suspension of self-propelled hard spheres by orders of magnitude, such that theglass transition shifts to higher packing fractions toward therandom-close-packing limit,18 to list just a few.

On the other hand, one notes that typical experiments ofcolloidal systems take place in hydrodynamic environments,wherein long-range hydrodynamic interactions (HI) betweenthe particles cannot be ignored. It has been shown that HIcould play rather important or even subtle roles in phasetransition or collective behavior of both hard and soft colloidalsuspensions. For example, Furukawa and Tanaka showed thatHI can significantly promote gelation or lower the colloidalvolume fraction for percolation as compared to their absence.19

Radu and Schilling studied the crystal nucleation process ofsuspensions of hard spheres, finding that HI can speed up thenucleation process by increasing the kinetic pre-factor withoutaffecting the size, shape and structure of nuclei.20 Interestingly,Roehm et al. showed that HI might also lead to a significantreduction of the crystal growth velocity in a suspension ofcolloids with Yukawa interactions, in contrast to the case ofhard spheres.21 More importantly, in contrast to equilibriumstates which depend only on the underlying free energy land-scape, the steady state properties in non-equilibrium systemsdepend strongly on dynamic factors especially long-range solvent

Department of Chemical Physics, Hefei National Laboratory for Physical Sciences at

Microscales, iChEM, University of Science and Technology of China, Hefei,

Anhui 230026, China. E-mail: hzhlj@ustc.edu.cn

Received 1st April 2015,Accepted 8th June 2015

DOI: 10.1039/c5sm00768b

www.rsc.org/softmatter

Soft Matter

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hydrodynamics. It was shown that HI strongly suppresses motilityinduced phase separation,22 and determines collective motion andphase behavior of active colloids in quasi-two-dimensional con-finement.23 Goto and Tanaka revealed that HI can lead to very richphase behaviors of rotating hard disks at zero temperature, includ-ing a fluid state, a hexatic state, a glass state, etc.24 These studiesstrongly suggest that the interplay between HI and activity maysignificantly influence the collective dynamics of active matter andsurely deserves more extensive work.

In the present paper, we try to address such an issue byinvestigating the impacts of HI on the crystallization process ofa two dimensional suspension of colloids with an adjustabledriven force. We note that colloidal particles in our modelsystem are different from the ‘‘force-free’’ agents described bySaintillan and Shelley,25 where the paradigmatic active particlesare mainly about biological microswimmers. These ‘‘force free’’particles suspended in a viscous fluid, such as swimmingbacteria and microscopic algae, are commonly modeled aspusher or pullers. The net force on a particle vanishes in theStokes limit with a low Reynolds number, when the particle istypically neutrally buoyant, and an extensile (contractile) dipoleflow appears around this pusher (puller) particle. Moreovermost bacteria move in a run-and-tumble way over large timescales. The present work, however, is relevant for ‘‘synthetic’’colloidal particles, such as latex colloids, silica colloids and thelike. Self-propelled (active) colloids can be obtained in this caseby chemical reactions at the surface (Janus particles),7–9,26

asymmetric magnetic response, or self-thermophoresis due tothe photothermal effect.10,11,27 These colloids, whose energysupply (e.g. chemical reaction, electric or magnetic field) isunceasing, are clearly ‘‘forced particles’’, where the forcesoriginate in a solvent environment including both thermalfluctuation characterized by Brownian bath and energy supplydominated by their directed motion at short times.8 One canadjust the driven force by changing the external electric ormagnetic fields or by introducing alternative chemical reactions.In the present work, we assume that the active particles aredriven by forces with constant magnitudes along stochasticrotational directions. This kind of scheme has been widely usedto study the Brownian motion of self-propelled particles.

The non-equilibrium, nonlinear, nonlocal, non-instantaneousnature of HI makes analytical approaches to this problem verydifficult and thus numerical simulations are expected to play acrucial role. Here we employ a hybrid mesoscopic approach to thisproblem. To account for HI, we use the multi-particle collisiondynamics (MPCD) approach introduced by Malevanets andKapral,28 which has been widely used to study the effects of HIin many systems (for a recent review see ref. 29 and 30). Moleculardynamics (MD) simulation is used to address the interactionbetween the colloid particles. With this hybrid MD-MPCD method,we can make an easy assessment of the effects of HI. As alreadyshown in ref. 17, the critical density for crystallization of a suspen-sion of active colloids is shifted to a higher value compared to thatof passive particles. In this paper, we find that HI can further shiftthe critical density to a much larger value for active particles, whilein contrast, it slightly shifts the freezing density to a smaller value

for a passive system. In addition, the influence of HI becomesstronger with the increment of the driving force. Furthermore, HIseems not to influence the equilibrium structure of a passivesystem, while it may lead to more pronounced structural hetero-geneity of an active system. Our results demonstrate that HI andactivity cooperatively determines the crystallization process ofactive colloidal systems.

II. Models and methods

To account for the long-range hydrodynamic effects, here weadopt the mesoscopic MPCD approach to simulate thedynamics of the surrounding small solvent particles. In MPCD,Ns point solvent particles with mass m, representing coarsegrained real molecules, free stream and undergo effectivecollisions at discrete time intervals DtMPC, accounting for theeffects of many real collisions during this time interval. Duringthe streaming step, the coordinates xi(t) of the solvent particlesare updated according to

xi(t + DtMPC) = xi(t) + vi(t)DtMPC, (1)

where vi(t) is the velocity of solvent particle i. In the collisionstep, all the solvent particles are sorted into cubic cells with sizea0 and then the velocities of solvent particles in a certain cellare updated according to a stochastic rotation

vi(t + DtMPC) = vCM(t) + P(vi(t) � vCM(t)), (2)

where vCM is the center-of-mass velocity of particles in thecurrent cell and P is a rotation matrix which rotates velocitiesby an angle a around an axis generated randomly for each cellat each time step. In the present two dimensional calculationswe keep a = p/2, if not otherwise specified. The scheme conservesmass, momentum and energy, and thus the long-range HI canbe properly taken into account. Note that a random shift of theentire computational grid before every collision step should beperformed to restore the Galilean invariance.31 Throughout thepresent paper we employ the cell size a0, the mass of solventparticle m and kBT to be the unit of length, mass and energy,respectively, where T is the temperature of the solvent bath. Withthese units, the time unit reads t ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimsa02=kBT

p.

We consider a two-dimensional suspension of N activecolloidal particles, with position Ri(t) and velocity Vi(t) forparticle i at time t, in hydrodynamic environments. The inter-action between the particles are modeled by the widely usedYukawa potential

U Rij

� �¼ G

exp �kRij

� �Rij

; (3)

where Rij = |Ri � Rj| represents the distance between particle iand j, and k denotes the inverse screening length. G = U0a0/kBTis a dimensionless coupling strength, where U0 is the barepotential strength.17 Note that the potential is soft, such thatthe particle size is not well-defined. But with increasing G, therepulsive interaction among adjacent particles increases corres-ponding to a larger effective particle size and a larger volume

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fraction for fixed number density. The dynamics of the colloidalparticles are described by the following equations,

dRiðtÞdt

¼ ViðtÞ;

MdViðtÞdt

¼ �Xj

riU Rij

� �þ f ni; (4)

where M is the particle mass and ri refers to the gradient withrespect to Ri. f denotes a constant propelling force which is exertedon particle i in the direction ni(t) = (cosfi(t), sinfi(t)). Note that theparticles are passive for f = 0 and active for f 4 0. The direction ofthis external force changes randomly in a free diffusion manner,

_fi(t) = 2DrZi(t), (5)

where Dr is the rotational diffusion coefficient and Zi(t) is theGaussian white noise with zero mean hZi(t)i = 0 and unitvariance hZi(t)Zj (t0)i = 2Drdijd(t � t0).

In the MD-MPCD hybrid scheme, the dynamics of thecolloidal particles are simulated by using eqn (3) to (5) withtime step DtMD. The solvent dynamics are simulated by usingeqn (1) and (2) with time step DtMPC. The coupling between thecolloidal particles and the solvent particles is realized by the waythat the colloid takes part in the collision step eqn (2) as a pointparticle within its cell with the center-of-mass velocity to be

VCM ¼mPNsol

i¼1vi þM

PNcol

i¼1Vi

NsolmþNcolM; (6)

where Nsol and Ncol are the numbers of solvent and colloidalparticles in the current collision cell, respectively.

We note that the approach described here has already beenused in many studies as an efficient and reliable mesoscopicsimulation method.20,32–36 Particularly, it is convenient to turn offHI, which facilitates us to address its specific role. Since the keyfeature of HI is the local conservation of momentum and energy,the basic idea of switching-off HI is to randomly interchange thevelocities of all particles inside a cell after each collision step.37,38

Consequently, the velocity correlations which result in HI dis-appear while the equilibrium Maxwell–Boltzmann distributionand local friction are maintained. Running simulations under thesame initial conditions and parameter values but with HI presentor absent, greatly facilitate pinpointing the effect of HI.

To keep the system temperature T, we use the thermostat byrescaling the solvent velocities at the cell level as proposed inref. 39. MD equations for the colloidal particles are integratedusing the time-reversible velocity Verlet algorithm with a time stepDtMD = 0.002t. The time interval between succeeding collisionsteps is DtMPC = 0.1t, and the rotational diffusion coefficient isfixed at Dr = 3.5t�1. We set the mass of a colloidal particle M to beequal to the average mass of solvent particles per cell, i.e., M = mrs,with rs = 10 being the average number of solvent particles percollision cell. The inverse screening length is k = 3.5a0

�1, which ismainly influenced by the ion concentration in the solvent, and wecut off the potential at r = 2a0 in this study. The effective couplingstrength G and the external force f are free variable parameters.

III. Results and discussion

Our simulations are carried out for N = 900 particles in a twodimensional box under periodic boundary conditions. The

scale of the simulation box is set to Lx

�Ly ¼

ffiffiffi3p

=2 with Ly =30a0 so that the suspension can crystallize into the hexagonalcrystal without any defects. We start from a random initialconfiguration where both particle positions and self-propelledforce orientations are uniformly distributed. The initial velo-cities of all particles were Gaussian distributed with zero meanand variance kBT/m (or kBT/M) for solvent (or colloidal) parti-cles. For the data collection, the system evolves for at least 105tto make sure that each particle has moved on an average adistance several times larger than its own diameter such that anew steady state can be established. To characterize the structure,we use a global bond-orientational order parameter40 which isgiven by

c6 �1

N

XNj¼1

1

N0

XN0

k¼1exp i6yjk� ������

�����2* +; (7)

where j runs over all particles of the system and k runs over theN0 = 6 nearest neighbors of the jth particle. yij is the anglebetween the bond vector connecting particle j to k and apredefined axis (e.g., x). The angle bracket h�i denotes theensemble average. Note that c6 = 0 in a disordered phasewhereas in a perfect crystal lattice c6 = 1.

A. Passive colloidal system

For comparison, we first investigate a system of passive parti-cles, i.e., f = 0. In Fig. 1, the variation of c6 as a function of thecoupling strength G is shown, with and without HI considered.As expected, there is a clear transition between disorderedliquid and ordered crystal phases with increasing G. Fourtypical configuration snapshots of the system without HI arealso shown in the insets of Fig. 1 for disordered liquid states(c6 = 0, 0.04, (a), (b)), a transition state (c6 = 0.37, (c)) and an

Fig. 1 Dependence of the global order parameter c6 on the couplingstrength G for a passive system with HI (J, red dash line) and without HI(&, black solid line). The insets show four typical snapshots for (a) c6 = 0,(b) c6 = 0.04, (c) c6 = 0.37 and (d) c6 = 0.72, respectively. The color paletteis drawn according to the local bond order parameter %q6 around eachcolloidal particle.

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ordered crystalline state (c6 = 0.72 (d)), respectively. The colorswithin these snapshots are drawn according to the local orderparameter %q6(i) for particle i defined as following

�q6ðiÞ ¼ Re1

6

X6j¼1

q6ðiÞq6�ð jÞ (8)

where q6ðiÞ ¼P6j¼1

exp i6yij� �

. In the transition state with c6 = 0.37,

large crystalline clusters (dark regime) with relatively large %q6 canbe apparently observed.

We now turn to the role of HI in the passive system. Asshown in Fig. 1, HI slightly shifts the transition point, G*, to asmaller value of G. This seems to indicate that HI can helpstabilize the crystalline cluster for G close to the transitionpoint. To further demonstrate this point, we have drawn thedependence of c6 on time t for G = 170 in Fig. 2, wherein HI isinitially on but it is turned off after the system has reached thestable solid state. Clearly, after HI is off, the solid state losesstability with c6 decreasing rather quickly and the systemchanges to the liquid state. In addition, if we turned on HIagain, the system will relax back to the solid state again asdemonstrated in this figure. Typical snapshots of the systembefore and after the switch-off of HI are depicted for illustration.

Although HI can slightly shift the transition point, we findthat the equilibrium structure of the system for given c6 showslittle difference with HI being present or not. This is demon-strated in Fig. 3(a), where the distributions P(%q6) of the localorder parameter %q6 for four different values of c6 are depicted.As illustrated in ref. 17, such a distribution characterizes thestructural fluctuation of the system. It is obvious that P(%q6) isalmost the same with or without HI, if the value of the globalorder parameter c6 is the same.

To get more insight, we have also investigated the dynamicalbehavior, especially the relaxation process of the system to equili-brium. For this purpose, we have calculated the self-intermediatescattering function

Fs qm; tð Þ ¼ 1

N

XNj¼1

exp iqm � rjðtÞ � rjð0Þ� �

; (9)

where qm is the wave vector corresponding to the first peak in thestructure factor and qm = |qm|. Typical Fs(q

m,t) for c6 = 0.04,c6 = 0.3, c6 = 0.47, and c6 = 0.72 are shown in Fig. 3(b).Interestingly, one finds that HI can remarkably accelerate therelaxation process for c6 = 0.04 and c6 = 0.3, where the systemlies at the liquid side, although the final equilibrium structures arethe same as already demonstrated in Fig. 3(a). For large c6,however, the system is more solid-like such that the particles aredynamically arrested, the scattering function Fs(q

m,t) show compli-cated features with plateaus and long tails. In this latter case,the role of HI seems to be more subtle and one can hardly say itaccelerates the relaxation or not.

B. Active colloidal system

After we have investigated the passive system, in this section,we study the effects of HI on the freezing transition of activecolloidal systems, where the force f 4 0. In Fig. 4(a), we haveplotted the order parameter c6 as functions of G for differentvalues of f, with or without HI. First of all, the transition pointG* shifts to larger values with increasing particle activity fwithout HI, which is consistent with the results reported inref. 17. Interestingly, we find that such a shift is furtherremarkably enhanced when HI is present. This is drasticallyin contrast to the passive system as shown in Fig. 1, where HIseems to slightly reduce the value of G*. In addition, thepresence of HI makes the transition less sharp for activesystems. To show this, we plot the relative width of the transition

Fig. 2 Typical time series of c6 in the passive system for G = 170. HI isswitched off at t = 2 � 104 and on at t = 4 � 104. Three typical snapshotsare shown in the insets.

Fig. 3 (a) Distribution function P( %q6) and (b) self-intermediate scatteringfunctions Fs(q

m,t) in a passive system for c6 = 0.04 (black), 0.3 (red), 0.47(green), and 0.72 (blue). Dashed and solid lines indicate ones with andwithout HI, respectively.

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region DG = G1 � G0, where the order parameter c6 = 0.5 at G1

and c6 = 0.04 at G0, as functions of the external force f inFig. 4(b). Clearly, if HI is not taken into account, DG remainsnearly constant and small with the variation of f, implying thatthe transition is almost discontinuous. When HI is present,however, the transition becomes more continuous and the widthDG increases gradually with increasing external force f. Thislatter fact makes it ambiguous to define the transition point, anddifferent criteria used may lead to different G*.17 Since we onlywant to address the role of HI here, we have applied a simplerule to locate G*, where the fluctuation of c6, s2(G) = hc6

2i �hc6i2, reaches the maximum. In Fig. 4(c), the dependence of thefreezing point G* on the external force f is shown. Clearly, G*increases much faster with f when HI is considered, and HIinduced shift of the transition point DG* also increases remarkablywith f as shown in the inset.

In Fig. 5, the change in c6 with time t upon the switch on oroff of HI for f = 10 and G = 240 is depicted, together with typical

snapshots. For these parameters, finally the system is in anordered crystal state without HI, but in an unordered liquidstate if HI is present. Clearly, the crystal state loses stabilityshortly after the switch-on of HI and turn to the liquid state, thelatter can also change back to the crystal state quickly if we furtherturn off the HI. Thus HI seems to destroy the ordered state in anactive system, quite different to its role in a passive system.

Similar to the case of the passive system, we have alsoinvestigated the distribution function P(%q6) for typical fixedvalues of c6 as shown in Fig. 6(a) for f = 10. Quite different fromthe passive system where f = 0, we find that the distribution

Fig. 4 (a) Freezing curves of active colloidal systems with force f = 5(black), 8 (red), 10 (green) and f = 15 (blue). The intersections of the twohorizontal lines with the freezing curves define the value of G1 wherec6 = 0.5 and G0 where c6 = 0.04. (b) Relative width of the transition regionDG = G1 � G0 as a function of the self-propelled force f. (c) Dependence ofthe transition point G* on the force f. The inset shows the HI-induced shiftof G* as a function of f & (solid line) and J (dashed line), the ones withand without HI, respectively.

Fig. 5 Typical time series of c6 in an active system with f = 10 for G = 240.HI is switched off at t = 2 � 104 and on at t = 4 � 104. Three typicalsnapshots are shown in the insets.

Fig. 6 (a) Distribution function P( %q6) and (b) self-intermediate scatteringfunctions Fs(q

m,t) in an active colloidal system with f = 10 for c6 = 0.06(black), 0.2 (red), 0.5 (green), and 0.72 (blue). Dashed and solid linesindicate the ones with and without HI, respectively.

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shifts considerably when HI is present, particularly for rela-tively large c6 = 0.5 and 0.72. This indicates that the activeparticles are locally more ordered when HI is present, if theglobal order parameter c6 is the same. Since c6 characterizes aglobal coherent order, while %q6 is defined locally, the peak-shiftto a higher value of %q6 in the distribution P(%q6) thus indicates amore heterogeneous structure with small bubbles of liquidregions remaining.17 The corresponding relaxation dynamicsis shown in Fig. 6(b), where Fs(q

m,t) is presented as functions oftime. Similar to the case of the passive system, one can see thatHI again accelerates the relaxation dynamics in the liquidregion. Interesting features are observed for c6 = 0.2, whereFs(q

m,t) decays very slowly without HI, while it decays muchfaster when HI is present. The physical mechanism behind thisenhanced relaxation dynamics is not clear at the present stageand may deserve more extensive studies.

IV. Discussion and conclusion

Here, we would like to point out that our model system actuallyconsists of disks moving in a 2-dimensional (2D) fluid. Inprinciple, our predictions may be tested by experiments oncolloidal particles on a (quasi) two dimensional substrate.26,27

Since HI scales differently in 2D and 3D, one should be carefulin applying the results directly to more realistic 3D cases and itwould be interesting to simulate a corresponding 3D model.Nevertheless, we think the physical picture that HI can induce adrastic shift of the transition point for an active system while ithardly influence the transition in a passive one, may be alsoapplicable in 3D. We also note that the present results may benot applicable to ‘‘force-free’’ biological microswimmers asdiscussed above in the Introduction section.

In conclusion, we have studied crystallization of a two-dimensional suspension of colloidal particles interacting viasoft Yukawa potential by using a hybrid MD-MPCD simulationmethod which facilitates us to investigate the specific effects oflong range HI. We have compared the results for passive systemsand active systems, where in the latter case the colloidal particlesare driven by an external force that changes direction randomlyin a tumbling manner. Interestingly, we found that the role of HIis quite different in an active system from that in a passivesystem. Firstly, HI can significantly shift the freezing density ofan active system to much higher values, while in contrast, itslightly shifts the freezing density to a smaller value for a passivesystem. In addition, HI does not change much the equilibriumstructure of a passive system, while it may lead to more pro-nounced structural heterogeneities in an active system. Further-more, the effects of HI become more remarkable with increasingdriving force, and freezing transition becomes less sharp whenHI takes effect. We also found that the relaxation process may beaccelerated by HI for both active and passive systems. It wouldbe interesting to figure out why HI plays such subtle roles andwhy the role is so different for passive and active systems, whichunfortunately is not available at the current stage and shoulddefinitely deserve more future work. We hope our work can open

more perspectives on the study of collective behavior of activesystems, in particular, the important roles of HI.

Acknowledgements

This work is supported by National Basic Research Program ofChina (2013CB834606), by National Science Foundation ofChina (21125313, 21473165, 21403204), and by the FundamentalResearch Funds for the Central Universities (WK2060030018,2340000034).

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