arXiv:cond-mat/0603391v3 [cond-mat.soft] 8 Sep 2006 irsoi ovn atce.Freape yia col- 1 typical diameter a of example, loid For . solvent microscopic length-scale aldsletdnmc ncmue iuain be- simulations computer enormous the considering in when apparent dynamics comes solvent tailed forces. inter-colloid of de- sum be pairwise usually cannot a and into character, interactions composed the many-body hydrodynamic a of the of Moreover, behavior are mo- hydrodynamic 2]. 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The Enschede, AE, 7500 217, Box PO 3 rpriso colloidal of properties opttoa ipyis nvriyo Twente, of University Biophysics, Computational µ nwtrdiffuses water in m .T Padding T. J. coarse-graining Dtd uut2,2018) 21, August (Dated: time 10 and wa- ,2 3 2, 1, n .A Louis A. A. and c 5,wihcntk noacuthge re terms order higher account Dynam- into take Stokesian The can is which form. methods [5], analytical these ics approximate of an successful by many most HI in- BD, that the of approaches clude implementation computational the applied their of have to representation authors [1] simple a tensor added Oseen who [4], Cammon 2]. [1, suspension the of upne atce.TeeH a alo ssol as slowly as off fall may HI 1 These particles. the suspended between to (HI) leads interactions as which hydrodynamic – – equations long-ranged Stokes solvent Navier the the by through described completely transport it momentum popular, the Brownian neglects very to simplicity, understandably its proportional is to friction dynamics due particle local Although, a colloidal velocity. as their the well of as displacement positions, molecules random solvent the a with induce collisions that [3], (BD) assumes dynamics which Brownian through account into taken different suspensions. first of colloidal we describe variety of to so, wide dynamics devised the do the been have we of that techniques before subset simulation But a discuss scheme. briefly such one detail /r einn ihtepoern oko ra n Mc- and Ermak of work pioneering the with Beginning ttesmls ee,teeet ftesletcnbe can solvent the of effects the level, simplest the At n a ulttvl ffc h yaia behavior dynamical the affect qualitatively can and , ucin n eae efdffso and diffusion self related and functions n ipiiy h ai agvnequation Langevin basic the simplicity, e a acltos ycnrs,teecal- these contrast, By calculations. cal hsclyrlvn hsclprocesses. physical relevant physically e ie n togyoeetmtn tat it overestimating strongly and times edmninesnmesi h correct the in numbers dimensionless se -clso hspolmi emphasized. is problem this of h-scales osdrtosaepoel ae into taken properly are considerations c fcrflytnn h simulation the tuning carefully of nce gshmssc sLtieBoltzmann Lattice as such schemes ng ecmie ffcso rwinand Brownian of effects combined he nmn htteei osc thing such no is there that mind in g ubrP,wihdsrbsterel- the describes which Pe, number t sle iesols hydrodynamic dimensionless esolves e ubrK,wihdsrbsthe describes which Kn, number sen r slqi-ieo a-ie h Mach the gas-like, or liquid-like is ort dMlclrDnmc n Stochas- and Dynamics Molecular id c sfiiesz ffcsadsolvent and effects size finite as uch lct uocreainfnto in function auto-correlation elocity tr formto nophysical onto method our of eters eecpdtgte omaximize to together telescoped e a,CmrdeC21W UK 1EW, CB2 Cambridge oad, prac fieta ffcs the effects, inertial of mportance d abig B E,UK 0EL, CB3 Cambridge ad, 1 suspensions: l 2 in a multipole expansion of the HI interactions. Al- though some more sophisticated recent implementations of Stokesian Dynamics have achieved an (N log N) scal- ing with the number of colloids [6], thisO method is still relatively slow, and becomes difficult to implement in complex boundary conditions. Another way to solve for the HI is by direct numer- ical simulation (DNS) methods, where the solid parti- cles are described by explicit boundary conditions for the Navier-Stokes equations [7]. These methods are better adapted to non-Brownian particles, but can still be ap- plied to understand the effects of HI on colloidal dynam- ics. The fluid particle dynamics (FPD) method of Tanaka and Araki [8], and a related method by Yamamoto et al. [9], are two important recent approaches to solving the Navier Stokes equations that go a step beyond stan- dard DNS, and simplify the problem of boundary con- ditions by using a smoothed step function at the colloid surfaces. In principle can be added to these methods [10]. The difficulty of including complex boundary condi- tions in DNS approaches has stimulated the use of lattice- gas based techniques to resolve the Navier Stokes equa- tions. These methods exploit the fact that only a few con- ditions, such as (local) energy and momentum conserva- tion, need to be satisfied to allow the correct (thermo) hy- drodynamics to emerge in the continuum limit. Greatly simplified particle collision rules, easily amenable to ef- FIG. 1: Schematic picture depicting how a colloidal ficient computer simulation, are therefore possible, and dispersion can be coarse-grained by replacing the solvent complex boundary conditions, such as those that char- with a particle-based hydrodynamics solver such as Lattice acterize moving colloids, are easier to treat than in DNS Boltzmann (LB), Dissipative Particle Dynamics (DPD), the methods. The most popular of these techniques is Lat- Lowe-Anderson thermostat, or Stochastic Rotation Dynamics tice Boltzmann (LB) where a linearized and pre-averaged (SRD). Each method introduces an effective coarse-graining Boltzmann equation is discretized and solved on a lat- length-scale that is chosen to be smaller than those of the tice [11]. Ladd has pioneered the application of this mesoscopic colloids but much larger than the natural length- method to solid-fluid suspensions [12, 13]. This is il- scales of a microscopic solvent. By obeying local momentum lustrated schematically in Fig. 1. The solid particles conservation, all four methods reproduce Navier-Stokes hy- drodynamics on larger length scales. For LB thermal fluctua- are modeled as hollow spheres, propagated with New- tions must be added in separately, but these emerge naturally tonian dynamics, and coupled to the LB solvent through for the other three methods. In this paper we focus on SRD, a bounce-back collision rule on the surface. The fluctu- but many of the lessons learned should also apply to other ations that lead to Brownian motion were modeled by particle based coarse-graining methods. adding a stochastic term to the stress tensor. Recently this has been criticized and an improved method to in- clude Brownian noise that does not suffer from some lat- momentum and therefore generates the correct Navier tice induced artifacts was suggested [14]. A number of Stokes hydrodynamics in the continuum limit. other groups have recently derived alternative ways of In fact, these two methodological advances can be sep- coupling a colloidal particle to a LB fluid [15, 16, 17], in arated. Soft potentials such as those postulated for in- part to simulate the dynamics of charged systems. novation 1) may indeed arise from careful equilibrium The discrete nature of lattice based methods can also coarse-graining of complex fluids [20, 21, 22, 23]. How- bring disadvantages, particularly when treating fluids ever, their proper statistical mechanical interpretation, with more than one length-scale. Dissipative Particle Dy- even for equilibrium properties, is quite subtle. For ex- namics (DPD) [18, 19] is a popular off-lattice alternative ample, a potential that generates the correct pair struc- that includes hydrodynamics and Brownian fluctuations. ture will not normally reproduce the correct virial pres- It can be viewed as an extension of standard Newtonian sure due to so called representability problems [24]. When MD techniques, but with two important innovations: used in dynamical simulations, the correct application of 1) soft potentials that allow large time-steps and rapid effective potentials is even more difficult to properly de- equilibration; rive. For polymer dynamics, for instance, uncrossabil- 2) a Galilean invariant thermostat that locally conserves ity constraints must be re-introduced to prevent coarse- 3 grained polymers from passing through one another [25]. with a certain probability, exchange its relative velocity Depletion interactions in a multi component solution also with another that lies within a certain radius. One can depend on the relative rates of depletant diffusion coef- imagine other collision rules as well. As long as they lo- ficients and particle flow velocities [26, 27]. At present, cally conserve momentum and energy, just as the lattice therefore, the statistical mechanical origins of innovation gas methods do, they generate the correct Navier Stokes 1) of DPD, the use of soft potentials out of equilibrium, hydrodynamics, although for SRD it is necessary to in- are at best obscure. We are convinced that a correct clude a grid-shift procedure to enforce Galilean invari- microscopic derivation of the coarse-grained DPD repre- ance, something first pointed out by Ihle and Kroll [38]. sentation of the dynamics, if this can indeed be done, An important advantage of SRD is that its simplified will show that the interpretation of such soft potentials dynamics have allowed the analytic calculation of several depends on dynamic as well as static (phase-space) av- transport coefficients [39, 40, 41], greatly facilitating its erages. Viewing the DPD particles as static “clumps” use. In the rest of this paper we will concentrate on the of underlying fluid is almost certainly incorrect. It may, SRD method, although many of our general conclusions in fact, be more fruitful to abandon simple analogies to and derivations should be easily extendible to the Lowe- the potential energy of a Hamiltonian system, and in- Anderson thermostat and related methods summarized stead view the interactions as a kind of coarse-grained in Fig. 1. self-energy [28]. SRD can be applied to directly simulate flow, as done Innovation 2), on the other hand, can be put on firmer for example in refs [42, 43], but its stochastic nature statistical mechanical footing, see e.g. [29], and can be means that a noise average must be performed to cal- usefully employed to study the dynamics of complex sys- culate flow lines, and this may make it less efficient tems with other types of inter-particle interactions [30]. than pre-averaged methods like LB. Where SRD becomes The main advantage of the DPD thermostat is that, by more attractive is for the simulation of complex particles preserving momentum conservation, the hydrodynamic embedded in a solvent. This is because in addition to interactions that intrinsically arise from microcanonical long-ranged HI, it also naturally contains Brownian fluc- MD are preserved for calculations in the canonical en- tuations, and both typically need to be resolved for a semble. Other thermostats typically screen the hydro- proper statistical mechanical treatment of the dynamics dynamic interactions beyond a certain length-scale [31]. of mesoscopic suspended particles. For example, SRD For weak damping this may not be a problem, but for has been used to study polymers under flow [44, 45, 46], strong damping it could be. the effect of hydrodynamics on protein folding and poly- Simulating only the colloids with DPD ignores the mer collapse [47], and the conformations of vesicles under dominant solvent hydrodynamics. While the solvent flow [48]. In each of the previously mentioned examples, could be treated directly by DPD, as suggested in Fig. 1 the suspended particles are coupled to the solvent by par- (see also [32]), this method is quite computationally ex- ticipating in the collision step. pensive because the “solvent” particles interact through A less coarse-grained coupling can be achieved by al- pairwise potentials. lowing direct collisions, obeying Newton’s laws of mo- In an important paper [33], Lowe took these ideas fur- tion, between the SRD solvent and the suspended parti- ther and combined the local momentum conservation of cles. Such an approach is important for systems, such as the DPD thermostat with the stochastic nature of the colloidal suspensions, where the solvent and colloid time Anderson thermostat to derive a coarse-grained scheme, and length-scales need to be clearly separated. Male- now called the Lowe-Anderson thermostat, which is much vanets and Kapral [49] derived such a hybrid algorithm more efficient than treating the solvent with full DPD. It that combined a full MD scheme of the solute-solute and has recently been applied, for example, to polymer dy- solute-solvent interactions, while treating the solvent- namics [34]. solvent interactions via SRD. Early applications were to Independently, Malevanets and Kapral [35] derived a a two-dimensional many-particle system [50], and to the method now called stochastic rotation dynamics (SRD) aggregation of colloidal particles [51]. or also multiple particle collision dynamics (we choose We have recently extended this approach, and applied the former nomenclature here). In many ways it resem- it to the sedimentation of up to 800 hard sphere (HS) like bles the Lowe-Anderson thermostat [33], or the much colloids as a function of volume fraction, for a number of older direct simulation Monte Carlo (DSMC) method of different values of the Peclet number [52]. To achieve Bird [36, 37]. For all three of these methods, the parti- these simulations we adapted the method of Malevanets cles are ideal, and move in a continuous space, subject and Kapral [49] in a number of ways that were only briefly to Newton’s laws of motion. At discrete time-steps, a described in ref. [52]. In the current paper we provide coarse-grained collision step allows particles to exchange more in depth analysis of the simulation method we used, momentum. In SRD, space is partitioned into a rectan- and describe some potential pitfalls. In particular we fo- gular grid, and at discrete time-steps the particles inside cus on the different time and length-scales that arise from each cell exchange momentum by rotating their velocity our coarse-graining approach, as well as the role of dimen- vectors relative to the center of mass velocity of the cell sionless hydrodynamic numbers that express the relative (see Fig. 1). Similarly, in Lowe’s method, a particle can, importance of competing physical phenomena. Very re- 4 cently, another similar study of colloidal sedimentation by accurately integrating Newton’s equations of motion, and aggregation has been carried out [53]. Some of our dvi results and analysis are similar, but some are different, mf = fi, (1) and we will mention these where appropriate. dt dri After the introductory overview above, we begin in sec- = v . (2) dt i tion II by carefully describing the properties of a pure SRD fluid, focusing on simple derivations that highlight ri and vi are the position and velocity of fluid particle i, the dominant involved for each transport coeffi- respectively while fi is the total (external) force on par- cient. Section III explains how to implement the coupling ticle i, which may come from an external field such as of a colloidal particle to an SRD solvent bath, and shows gravity, or fixed boundary conditions such as hard walls, how to avoid spurious depletion interactions and how to or moving boundary conditions such as suspended col- understand lubrication forces. In Section IV we analyze loids. The direct forces between pairs of fluid particles how optimizing the efficiency of particle based coarse- are, however, neglected in the propagation step. Herein graining schemes affects different dimensionless hydrody- lies the main advantage – the origin of the efficiency – namic numbers, such as the Schmidt number, the Mach of SRD. Instead of directly treating the interactions be- number, the Reynolds number, the Knudsen number, and tween the fluid particles, a coarse-grained collision step the Peclet number. Section VI describes the hierarchy is performed at each time-step ∆tc: First, space is par- 3 of time-scales that determine the physics of a colloidal titioned into cubic cells of volume a0. Next, for each suspension, and compares these to the compressed hier- cell, the particle velocities relative to the center of mass archy in our coarse-grained method. In section VII we velocity vcm of the cell are rotated: tackle the question of how to map from a coarse-grained v vcm + R (v vcm) . (3) simulation, optimized for computational efficiency, to a i 7→ i − real colloidal system. Finally after a Conclusions section, R is a rotation matrix which rotates velocities by a fixed we include a few appendices that discuss amongst other angle α around a randomly oriented axis. The aim of things how to thermostat the SRD system (Appendix I); the collision step is to transfer momentum between the why the popular Langevin equation without memory ef- fluid particles while conserving the total momentum and fects does a remarkably poor job of capturing both the energy of each cell. Both the collision and the streaming long and the short time dynamics of the colloidal veloc- step conserve phase-space volume, and it has been shown ity auto-correlation function (Appendix II); and, finally, that the single particle velocity distribution evolves to a how various physical properties and dimensionless num- Maxwell-Boltzmann distribution [35]. bers scale for an SRD simulation, and how these compare The rotation procedure can thus be viewed as a coarse- to a colloid of radius 10 nm or 1 µm in H20 (Appendix graining of particle collisions over space and time. Be- III). cause mass, momentum, and energy are conserved lo- cally, the correct hydrodynamic (Navier Stokes) equa- tions are captured in the continuum limit, including the effect of thermal noise [35]. II. PROPERTIES OF A PURE SRD SOLVENT At low temperatures or small collision times ∆tc, the transport coefficients of SRD, as originally formu- lated [35], show anomalies caused by the fact that fluid In SRD, the solvent is represented by a large number particles in a given cell can remain in that cell and partic- Nf of particles of mass mf . Here and in the following, ipate in several collision steps [38]. Under these circum- we will call these “fluid” particles, with the caveat that, stances the assumption of molecular chaos and Galilean however tempting, they should not be viewed as some invariance are incorrect. However, this anomaly can be kind of composite particles or clusters made up of the cured by applying a random shift of the cell coordinates underlying (molecular) fluid. Instead, SRD should be in- before the collision step [38, 39]. terpreted as a Navier Stokes solver that includes thermal It should also be noted that the collision step in SRD noise. The particles are merely a convenient computa- does not locally conserve angular momentum. As a con- tional device to facilitate the coarse-graining of the fluid sequence, the stress tensor σ is not, in general, a symmet- properties. ric function of the derivatives of the flow field (although it is still rotationally symmetric) [41]. The asymmetric part can be interpreted as a viscous stress associated with the vorticity v of the velocity field v(r). The stress ∇× A. Simulation method for a pure solvent tensor will therefore depend on the amount of vorticity in the flow field. However, the total force on a fluid el- ement is determined not by the stress tensor itself, but In the first (propagation) step of the algorithm, the by its divergence σ, which is what enters the Navier positions and velocities of the fluid particles are propa- Stokes equations.∇ Taking · the divergence causes the ex- gated for a time ∆tc (the time between collision steps) plicit vorticity dependence to drop out (the gradient of a 5 curl is zero). In principle, the stress tensor could be made symmetric by applying random rotations as well as ran- TABLE I: Units and simulation parameters for an SRD fluid. The parameters listed in the table all need to be fixed inde- dom translations to the cell. Or, alternatively, angular pendently to determine a simulation. momentum can be explicitly conserved by dynamically adapting the collision rule, as done by Ryder et al [54], Basic Units Derived Units who found no significant differences in fluid properties mf (although there is, of course, less flexibility in choosing a0 = length t0 = a0 = time kB T simulation parameters). This result has been confirmed r kB T = energy 2 by some recent theoretical calculations [55], that demon- a0 kB T strate that the asymmetry of the stress tensor has only a m = mass D0 = = a0 = diffusion constant f t0 s mf few consequences, such as a correction to the sound wave 2 associated with viscous dissipation of longi- a0 kB T ν0 = = a0 = kinematic tudinal waves. These are not important for the t0 s mf fluid properties we are trying to model, and so, on bal- ance, we chose not to implement possible fixes to improve γmf mf kBT η0 = = 2 = viscosity on angular momentum conservation. t0a0 p a0

Independent fluid simulation parameters

B. Transport coefficients and the dimensionless γ = average number of particles per cell mean-free path ∆tc = SRD collision time step

α = SRD rotation angle The simplicity of SRD collisions has facilitated the an- alytical calculation of many transport coefficients [39, L = box length 40, 41, 55]. These analytical expressions are particu- larly useful because they enable us to efficiently tune the viscosity and other properties of the fluid, without the Independent colloid simulation parameters need for trial and error simulations. In this section we ∆tMD = MD time step will summarize a number of these transport coefficients, where possible giving a simple derivation of the dominant σcc = colloid-colloid collision diameter Eq. (16) physics. ǫcc = colloid-colloid energy scale Eq. (16)

σcf = colloid-fluid collision diameter Eq. (17)

1. Units and the dimensionless mean-free path ǫcf = colloid-fluid energy scale Eq. (17)

Nc = number of colloids In this paper we will use the following units: lengths Mc = colloid mass will be in units of cell-size a0, energies in units of kBT and masses in units of mf (This corresponds to setting a0 = 1, kBT = 1 and mf = 1). Time, for example, is expressed in units of t0 = a0 mf /kBT , the number density nf = 3 γ/a0 and other derived units can be found in table I. We find it instructivep to express the transport coefficients and other parameters of the SRD fluid in terms of the 2. Fluid self-diffusion constant dimensionless mean-free path

∆t k T ∆t A simple back of the envelope estimate of the self- λ = c B = c , (4) a0 s mf t0 diffusion constant Df of a fluid particle can be obtained from a random-walk picture. In a unit of time t0, a particle will experience 1/λ collisions, in between which which provides a measure of the average fraction of a cell it moves an average distance λa0. Similarly, a heav- size that a fluid particle travels between collisions. ier (tagged) particle of mass Mt, which exchanges mo- This particular choice of units helps highlight the basic mentum with the fluid by participating in the coarse- physics of the coarse-graining method. The (nontrivial) grained collision step, will move an average distance question of how to map them on to the units of real lM = ∆tc kBT/Mt = λa0/√Mt between collisions. By physical system will be discussed in section VII. viewing this motion as a random walk of step-size l , p M 6 the diffusion coefficient follows: apart. All the above suggests that some care must be taken when interpreting the dynamics of heavier parti- D m f λ, (5) cles that couple through the coarse-grained collision step, D0 ∼ Mt especially when two or more heavy particles are in close 2 proximity. expressed in units of D0 = a0/t0 = a0 kB T/mf . The diffusion coefficient Df for a pure fluid particle of mass p mf is therefore given by Df /D0 λ. 3. Kinematic viscosity A more systematic derivation≈ of the diffusion coeffi- cient of a fluid particle, but still within a random col- lision approximation, results in the following expres- The spread of a velocity fluctuation δv(r) in a fluid sion [39, 56]: can be described by a diffusion equation [57]:

∂δv(r) 2 Df 3 γ 1 = ν δv(r) (8) = λ . (6) ∂t ∇ D0 2(1 cos(α)) γ 1 − 2  −  −   where ν is the kinematic viscosity, which determines the The dependence on γ is weak. If, for example, we rate at which momentum or vorticity “diffuses away”. take α = π/2, the value used in this paper, then 2 The units of kinematic viscosity are ν0 = a0/t0 = limγ→∞ Df /D0 = λ, the same as Eq.(5). a k T/m which are the same as those for particle A similar expression can be derived for the self- 0 B f self diffusion i.e. D0 = ν0. diffusion coefficient of a heavier tagged particle of mass pMomentum is transported through two mechanisms: M [39, 56]: t 1) By particles streaming between collision steps, lead- ing to a “kinetic” contribution to the kinematic viscos- γ + Mt Dt λmf 3 mf 1 = . (7) ity νkin. Since for this gas-like contribution the momen- D0 Mt 2(1 cos(α)) γ − 2 tum is transported by particle motion, we expect ν " − ! # kin to scale like the particle self-diffusion coefficient Df , i.e. Note that this equation does not quite reduce to Eq. (6) ν /ν0 λ. kin ∼ for Mt/mf = 1 (because of slightly different approxi- 2) By momentum being re-distributed among the parti- mations), but the relative discrepancy decreases with in- cles of each cell during the collision step, resulting in a creasing γ. “collisional” contribution to the kinematic viscosity νcol. While Eq. (6) is accurate for larger mean-free paths, This mimics the way momentum is transferred due to where the random collision approximation is expected to inter-particle collisions, and would be the dominant con- be valid, it begins to show deviations from simulations for tribution in a dense fluid such as water at standard tem- λ < 0.6 [56], when longer-time kinetic correlations begin perature and . Again a simple random-walk ar- to∼ develop. Ripoll et al [56] argue that these correlations gument explains the expected scaling in SRD: Each col- induce interactions of a hydrodynamic nature that en- lision step distributes momentum among particles in a hance the diffusion coefficient for a fluid particle. For cell, making a step-size that scales like a0. Since there are 3 example, for λ =0.1 and α = 4 π, they measured a fluid 1/λ collision steps per unit time t0, this suggests that the self-diffusion constant Df that is about 25% larger than collisional contribution to the kinematic viscosity should the value found from Eq. (6). The enhancement is even scale as νcol/ν0 1/λ. more pronounced for heavier particles: for the same sim- Accurate analytical∼ expressions for the kinematic vis- ulation parameters they found that Dt was enhanced by cosity ν = νkin + νcol of SRD have been derived [40, 55], about 75% over the prediction of Eq. (7) when Mt > 10. and these can be rewritten in the following dimensionless We note that coupling a large particle to the solvent∼ form: through participation in the coarse-grained collision step νkin λ ν leads to a diffusion coefficient which scales with mass as = fkin(γ, α) (9) D /D0 1/M , whereas if one couples a colloid of ra- ν0 3 t ∼ t dius Rc to the solvent through direct MD collisions, one νcol 1 ν 1 = fcol(γ, α). (10) expects Dt/D0 a0/Rc (mf /Mt) 3 . Moreover, it has ν0 18λ been shown [56]∼ that the∼ effective hydrodynamic particle where the dependence on the collisional angle α and fluid radius is approximately given by a 2a0/(πγ). For any reasonable average number of fluid particles≈ per cell, the number density γ is subsumed in the following two fac- effective hydrodynamic radius a of the heavier particle is tors: therefore much less than the collision cell size a0. On the 15γ 3 f ν (γ, α)= , other hand, the hydrodynamic field is accurately resolved kin (γ 1+e−γ )(4 2 cos(α) 2 cos(2α)) − 2 down to a scale comparable to a0 (vide infra). What this − − − (11) implies is that this coupling method will yield correct hydrodynamic interactions only at large distances, when the colloids are more than several hydrodynamic radii f ν (γ, α)=(1 cos(α))(1 1/γ +e−γ/γ). (12) col − − 7

These factors only depend weakly on γ for the typical and the density γ can be varied independently. More- parameters used in simulations. For example, at α = over, the collisional contribution to the viscosity adds a π/2 and γ = 5, the angle and number density we use in new dimension, allowing a much wider range of physical ν this paper, they take the values fkin(5,π/2)=1.620 and fluids to be modeled than just simple gases. ν fcol(5,π/2)=0.801. For this choice of collision angle α, they monotonically converge to fkin = fcol = 1 in the limit of large γ. III. COLLOID SIMULATION METHOD

4. Shear viscosity Malevanets and Kapral [49] first showed how to imple- ment a hybrid MD scheme that couples a set of colloids Suppose the fluid is sheared in the x direction, with the to a bath of SRD particles. In this section, we expand on gradient of the average flow field in the y direction. Two their method, describing in detail the implementation we neighboring fluid elements with different y-coordinates used in ref. [52]. We restrict ourselves to HS like colloids will then experience a friction force in the x-direction, as with steep interparticle repulsions, although attractions expressed by the xy component of the stress tensor σ, between colloids can easily be added on. The colloid- which for a simple liquid is linearly proportional to the colloid and colloid-fluid interactions, ϕcc(r) and ϕcf (r) instantaneous flow field gradient, respectively, are integrated via a normal MD procedure, while the fluid-fluid interactions are coarse-grained with ∂vx(y) σxy = η . (13) SRD. Because the number of fluid particles vastly out- ∂y numbers the number of HS colloids, treating their in- teractions approximately via SRD greatly speeds up the The coefficient of proportionality η is called the shear simulation. viscosity, and is it related to the kinematic viscosity by 3 η = ρf ν, where ρf = mf γ/a0 is the fluid mass density. From Eqs. 9–12 it follows that the two contributions to the shear viscosity can be written in dimensionless form A. Colloid-colloid and colloid-solvent interactions as:

ηkin λγ ν Although it is possible to implement an event-driven = fkin(γ, α) (14) η0 3 dynamics of HS colloids in an SRD solvent [60], here we ηcol γ ν approximate pure HS colloids by steep repulsive interac- = fcol(γ, α). (15) η0 18λ tions of the WCA form [61]:

2 where η0 = m/a0t0 = mf kB T/a is the unit of shear 48 24 0 σcc σcc 1 1/24 viscosity. 4ǫcc + (r 2 σcc) ϕcc(r)= r − r 4 ≤ In contrast to the expressionsp for the diffusion of a    0    (r > 21/24σ ). fluid particle or a tagged particle, Eqs. (9) - (15) com-  cc (16) pare quantitatively to simulations over a wide range of Similarly, the colloid-fluid interaction takes the WCA parameters [39, 40, 56]. For the parameters we used in form: our simulations in [52, 58], i.e. λ =0.1, α = π/2,γ = 5, the collisional contribution to the viscosity dominates: 12 6 σcf σcf 1 1/6 νkin = 0.054ν0 and νcol = 0.45ν0. This is typical for 4ǫ + (r 2 σ ) ϕ (r)= cf r − r 4 ≤ cf λ 1, where ν and η can be taken to a good first ap- cf   ≪      1/6 proximation by the collisional contribution only. In fact  0 (r> 2 σcf ). throughout this paper we will mainly focus on this small (17)  λ limit. The mass mf of a fluid particle is typically much smaller It is also instructive to compare the expressions de- than the mass Mc of a colloid, so that the average ther- rived in this section to what one would expect for simple mal velocity of the fluid particles is larger than that gases, where, as famously first derived and demonstrated of the colloid particles by a factor Mc/mf . For this experimentally by Maxwell in the 1860’s [59], the shear reason the time-step ∆tMD is usually restricted by the p viscosity is independent of density. This result differs fluid-colloid interaction (17), allowing fairly large ex- from the kinetic (gas-like) contribution to the viscosity ponents n for the colloid-colloid interaction ϕcc(r) = 2n n in Eq. (14), because in a real dilute gas the mean-free 4ǫcc (σcc/r) (σcc/r) +1/4 . We choose n = 24, − path scales as λ 1/γ, canceling the dominant density which makes the colloid-colloid potential steep, more like ∝   dependence in ηkin γλ. The same argument explains hard-spheres, while still soft enough to allow the time- why the self-diffusion∝ and kinematic viscosity of the SRD step to be set by the colloid-solvent interaction. fluid are, to first order, independent of γ, while in a gas The positions and velocities of the colloidal spheres are they would scale as 1/γ. In SRD, the mean-free path λ propagated through the Velocity Verlet algorithm [62] 8   with a time step ∆tMD:    Fi (t) 2  Ri (t + ∆tMD) = Ri (t)+ Vi (t) ∆tMD + ∆tMD(18),  2Mc   Fi (t)+ Fi (t + ∆tMD)  Vi (t + ∆tMD) = Vi (t)+ ∆tMD.  2Mc  (19)     Ri and Vi are the position and velocity of colloid i, re-  spectively. Fi is the total force on that colloid, exerted by   the fluid particles, an external field, such as gravity, ex-   ternal potentials such as repulsive walls, as well as other  colloids within the range of the interaction potential (16). The positions r and velocities v of SRD particles are FIG. 2: Schematic picture depicting how a fluid in- i i teracts with a colloid, imparting both linear and angular mo- updated by similarly solving Newton’s Eqns. (1,2) every mentum. Near the colloidal surface, here represented by the time-step ∆tMD, and with the SRD procedure of Eq. (3) shaded region, there may be a steric stabilization layer, or a every time-step ∆tc. double-layer made up of co- and counter-ions. In SRD, the Choosing both ∆tMD and ∆tc as large as possible en- detailed manner in which a fluid particle interacts with this hances the efficiency of a simulation. To first order, each boundary layer is represented by a coarse-grained stick or slip timestep is determined by different physics ∆tMD by the boundary condition. steepness of the potentials, and ∆tc by the desired fluid properties, and so there is some freedom in choosing their relative values. We used ∆tc/∆tMD = 4 for our simula- distributions: tions of sedimentation [52], but other authors have used P (v ) v exp βv2 (20) ratios of 50 [49] or even an order of magnitude larger n ∝ n − n 2 than that [53]. Later in the paper we will revisit this P (vt) exp βvt ,  (21) question, linking the time-steps to various dimensionless ∝ − hydrodynamic numbers and Brownian time-scales. so that the colloid acts as an additional  thermostat [66]. Such stochastic boundary conditions have been used for colloidal particles by Inoue et al. [50] and Hecht et al. [53]. We have systematically studied several implementations B. Stick and slip boundary conditions of stick boundary conditions for spherical colloids [58], and derived a version of the stochastic boundary condi- Because the surface of a colloid is never perfectly tions which reproduces linear and angular velocity corre- smooth, collisions with fluid particles transfer angular lation functions that agree with Enskog theory for short as well as linear momentum. As demonstrated in Fig. 2, times, and hydrodynamic mode-coupling theory for long the exact molecular details of the colloid-fluid interac- times. We argue that the stochastic rule of Eq. (20) is tions may be very complex, and mediated via co- and more like a real physical colloid – where fluid-surface counter-ions, grafted polymer brushes etc.... However, interactions are mediated by steric stabilizing layers or on the time and length-scales over which our hybrid MD- local co- and counter-ion concentrations – than bounce- SRD method coarse-grains the solvent, these interactions back rules are [58]. can be approximated by stick boundary conditions: the Nevertheless, in this paper, many examples will be for tangential velocity of the fluid, relative to the surface of radial interactions such as those described in Eq. (17). the colloid, is zero at the surface of the colloid [63, 64]. These do not transfer angular momentum to a spherical For most situations, this boundary condition should be colloid, and so induce effective slip boundary conditions. sufficient, although in some cases, such as a non-wetting For many of the hydrodynamic effects we will discuss here surface, large slip-lengths may occur [65]. the difference with stick boundary conditions is quanti- In computer simulations, stick boundary conditions tative, not qualitative, and also well understood. may be implemented by bounce-back rules, where both parallel and perpendicular components of the relative ve- locity are reversed upon a collision with a surface. These C. Depletion and lubrication forces have been applied by Lamura et al [42], who needed to modify the bounce-back rules slightly to properly repro- 1. Spurious depletion forces induced by the fluid duce stick boundaries for a Poiseuille flow geometry. Stick boundaries can also be modeled by a stochastic We would like to issue a warning that the additional rule. After a collision, the relative tangential velocity fluid degrees of freedom may inadvertently introduce de- vt and relative normal velocity vn are taken from the pletion forces between the colloids. Because the number 9 density of SRD particles is much higher than that of the 1 colloids, even a small overlap between two colloids can lead to enormous attractions. 0 For low colloid the equilibrium depletion in-

teraction between any two colloids caused by the presence ) 3 of the ideal fluid particles is given by [67, 68]: cf -1 σ T B

Φdepl(d)= nf kB T [Vexcl(d) Vexcl( )] , (22) k − ∞ f -2 3 where nf = γ/a0 is the number density of fluid particles / (n and Vexcl(d) is the (free) volume excluded to the fluid by depl -3

the presence of two colloids separated by a distance d. Φ The latter is given by hard sphere interaction -4 WCA interaction, ε = 2.5 k T 3 B Vexcl(d)= d r 1 exp [ βϕ (r r1) βϕ (r r2)] , { − − cf − − cf − } Z (23) -5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 where r1 r2 = d. An example is given in Fig. 3 where | − | σ we have plotted the resulting depletion potential for the d / cf colloid-solvent interaction (17), with ǫcc = ǫcf =2.5kBT as routinely used in our simulations, as well as the deple- FIG. 3: (Color online) Effective depletion potentials induced tion interaction resulting from a truly HS colloid-solvent between two colloids by the SRD fluid particles, for HS fluid- interaction. The latter can easily be calculated analyti- colloid (solid line) and WCA fluid-colloid (dashed line) in- cally, with the result teractions taken from Eq. (17). The interparticle distance is measured in units of σcf . Whether these attractive poten- 3 tials have important effects or not depends on the choice of HS 4 3 3d d diameter σcc in the bare colloid-colloid potential (16). Φdepl(d) = nf kB T πσcf 1 + 3 − 3 − 4σcf 16σcf !

for d< 2σcf . (24) is not a problem for equilibrium properties, it will in-

For the pure HS interactions, one could take σcc 2σcf troduce errors for non-equilibrium properties. Finally, and the depletion forces would have no effect. But≥ for when external fields drive the colloid, small but persis- interactions such as those used in Eqs. (16) and (17), tent anisotropies in the solvent density around a colloid the softer repulsions mean that inter-colloid distances may occur [27]. Although these density variations are less than σcc are regularly sampled. A more stringent (and should be) small, the resulting variations in deple- criterion of σcc must therefore be used to avoid spuri- tion interactions can be large. ous depletion effects. As an example, we re-analyze the To avoid these problems, we routinely choose the simulations of ref. [51], where a WCA form with n = 6 colloid-fluid interaction range σcf slightly below half the was used for the fluid-colloid interactions, and a normal colloid diameter σcc/2. More precisely, we ensure that 1 the colloid-colloid interaction equals 2.5kBT at a distance Lennard Jones n = 6 potential with σcf 2 σcc was used for the colloid-colloid interactions. The≥ authors found d where the depletion interactions have become zero, i.e., differences between “” calculations without SRD at a distance of twice the colloid-solvent interaction cut- particles, and a hybrid scheme coupling the colloids to an off radius. Smaller distances will consequently be rare, SRD solvent. These were correctly attributed to “solvent and adequately dealt with by the compensation poten- induced pressure”, which we quantify here as depletion tial. This solution may be a more realistic representa- 1 tion anyhow, since in practice for charge and even for interactions. For one set of their parameters, σcf = 2 σcc, the effect is mainly to soften the repulsion, but for their sterically stabilized colloids, the effective colloid-colloid other parameter set: σcf =0.65σcc, depletion attractions diameter σcc is expected to be larger than twice the ef- induce an effective attractive well-depth of over 40kBT ! fective colloid-fluid diameter σcf . This is particularly so Since the depletion potentials can be calculated an- for charged colloids at large Debye screening lengths. alytically, one might try counteracting them by intro- ducing a compensating repulsive potential of the form: 2. Lubrication forces depend on surface details Φcomp = Φdepl between the colloids. However, there are three problems− with this approach: Firstly, at higher colloid packing fractions, three and higher order inter- When two surfaces approach one another, they must actions may also need to be added, and these are very displace the fluid between them, while if they move apart, difficult to calculate. Secondly, the depletion interactions fluid must flow into the space freed up between the sur- are not instantaneous, and in fact only converge to their faces. At very short inter-surface distances, this results equilibrium average algebraically in time [26]. While this in so-called lubrication forces, which are repulsive for col- 10 loids approaching each other, and attractive for colloids 2.0 moving apart [1, 2, 63]. These forces are expected to be φ particularly important for driven dense colloidal suspen- c = 0.228 sions; see e.g. [69] for a recent review. φ = 0.108 1.5 c An additional advantage of our choice of diameters σci φ = 0.054 above is that more fluid particles will fit in the space c φ between two colloids, and consequently the lubrication c = 0.027 forces will be more accurately represented. It should be 1.0

kept in mind that for colloids, the exact nature of the g(r) short-range lubrication forces will depend on physical de- tails of the surface, such as its roughness, or presence of a grafted polymeric stabilizing layer [70, 71, 72]. For perfectly smooth colloids, analytic limiting expressions 0.5 for the lubrication forces can be derived [63], showing a divergence at short distances. We have confirmed that SRD resolves these lubrication forces down to surpris- 0.0 ingly low interparticle distances. But, at some point, this 0 1 2 3 will break down, depending of course on the choice of sim- r/σ ulation parameters (such as σcf /a0, λ, and γ), as well as cc the details of the particular type of colloidal particles that one wishes to model [70, 71, 72]. An explicit analytic cor- FIG. 4: (Color online) Colloid radial distribution functions rection could be applied to properly resolve these forces g(r) for various colloid volume fractions φc. The simulations for very small distances, as was recently implemented for were performed with BD (black lines) and SRD (red lines) Lattice Boltzmann [73]. However, in this paper, we will simulations, and are virtually indistinguishable within statis- tical errors. assume that our choice of σcf is small enough for SRD to sufficiently resolve lubrication forces. The lack of a com- plete divergence at very short distances may be a bet- tions ter model of what happens for real colloids anyway. For dense suspensions under strong shear, explicit lubrication 1 φ = πn σ3 , (25) force corrections as well as other short-ranged colloid- c 6 c cc colloid interactions arising from surface details such as polymer coats will almost certainly need to be put in by where the subscript in φc refers to volume fraction based hand, see e.g. ref. [72] for further discussion of this subtle on the colloid-colloid interaction, to distinguished it from problem. the volume fraction φ based on the colloid hydrodynamic radius [74]. From Fig. 4 is clear that the two simula- tions are indistinguishable within statistical errors, as 1 D. Test of static properties required. Even though σcf < 2 σcc, we still found it necessary to include an explicit compensating potential without which the g(r) generated by SRD is increased The equilibrium properties of a statistical mechanical noticeably at contact when compared to BD simulations. system should be independent of the detailed dynamics Finally, we note that any simulation will show finite by which it explores phase space. Therefore one requisite size effects. These may be thermodynamic as well as condition imposed on our coarse-grained dynamics is that dynamic. If there are thermodynamic finite-size effects it reproduces the correct static properties of an ensemble (for example due to a long correlation length caused by of colloids. proximity to a critical point), then a correct simulation An obvious property to measure is the radial distribu- technique will show the same behaviour regardless of the tion function g(r) [61]. In Fig. 4 we depict some g(r)s underlying dynamics. obtained from SRD simulations at a number of differ- ent densities, and compare these to similar simulations with standard Brownian dynamics. The colloids interact IV. DIMENSIONLESS NUMBERS via potentials of the form (16) – (17) with σcf = 2a0, σcc = 4.3a0, and ǫcc = ǫcf = 2.5kBT . For the SRD we 1 used γ = 5, α = 2 π, ∆tMD = 0.025t0 and ∆tc = 0.1t0, Different regimes of hydrodynamic behavior can be implying λ = 0.1. For the BD we used a background characterized by a series of dimensionless numbers that friction calculated from the effective hydrodynamic ra- indicate the relative strengths of competing physical pro- dius (vide infra) and a time-step equal to the ∆tMD used cesses [57]. If two distinct physical systems can be de- above. The colloids were placed in a box of dimensions scribed by the same set of hydrodynamic numbers, then 3 32 32 96a0, and the number of particles was varied their flow behavior will be governed by the same physics, from× 64× to 540 in order to achieve different packing frac- even if their time and length scales differ by orders of 11

than the fluid diffusion coefficient, i.e. Dcol Df to TABLE II: Dimensionless hydrodynamic numbers for col- achieve the correct time-scale separation, as we≪ will see loidal suspensions. in Section VI. collisional momentum transport ν Schmidt Sc = Eq. (26) kinetic momentum transport Df 1. Dependence of Schmidt number on simulation

flow velocity vs parameters Mach Ma = Eq. (28) sound velocity cs From Eqs. (6) – (12) it follows that the Schmidt num- inertial forces vsa Reynolds Re = Eq. (29) ber can be rewritten as viscous forces ν 1 1 mean free path λfree Sc + , (27) Knudsen Kn = Eq. (34) ≈ 3 18λ2 particle size a where we have ignored the dependence on γ and α since convective transport vsa Peclet Pe = Eq. (35) the dominant scaling is with the dimensionless mean-free diffusive transport Dcol path λ. For larger λ, where the kinetic contributions to the viscosity dominate, the Sc number is small, and the dynamics is gas-like. Because both the fluid diffu- sion coefficient (6) and the kinetic contribution to the magnitude. In other words, a macroscopic boulder sedi- viscosity (9) scale linearly with λ, the only way to ob- tain a large Sc number is in the limit λ 1 where the menting in viscous molten magma and a mesoscopic col- ≪ loid sedimenting in water may show similar behavior if collisional contribution to ν dominates. they share key hydrodynamic dimensionless numbers. It Low Sc numbers may be a more general characteris- is therefore very instructive to analyze where a system de- tic of particle-based coarse-graining methods. For com- scribed by SRD sits in “hydrodynamic parameter space”. putational reasons, the number of particles is normally In this section we review this parameter space, one “di- greatly reduced compared to the solvent one is modeling. mension” at a time, by exploring the hydrodynamic num- Thus the average inter-particle separation and mean-free bers listed in Table II. path are substantially increased, typically leading to a smaller kinematic viscosity and a larger fluid diffusion coefficient Df . With DPD, for example, one typically A. Schmidt number finds Sc 1 [29, 33]. It is therefore difficult to achieve large Sc≈ numbers, as in a real liquid, without sacrific- ing computational efficiency. But this may not always The Schmidt number be necessary. As long as momentum transport is clearly ν Sc = (26) faster than mass transport, the solvent should behave in Df a liquid-like fashion. With this argument in mind, and also because the Sc number does not directly enter the is important for characterizing a pure fluid. It expresses Navier Stokes equations we want to solve, we use λ =0.1 the rate of diffusive momentum transfer, measured by in this paper and in ref. [52], which leads to a Sc 5 the kinematic viscosity ν, relative to the rate of diffusive for γ = 5 and α = π/2. This should be sufficient≈ for mass transfer, measured by the fluid particle self-diffusion the problems we study. For other systems, such as those coefficient Df . For a gas, momentum transport is domi- where the suspended particles are coupled to the solvent nated by mass diffusion, so that Sc 1, whereas for a liq- ≈ through the collision step, more care may be needed to uid, momentum transport is dominated by inter-particle ensure that the Sc number is indeed large enough [56]. collisions, and Sc 1. For low Schmidt numbers, the dynamics of SRD≫ is indeed “gas-like”, while for larger Sc, the dynamics shows collective behavior reminiscent of B. Mach number the hydrodynamics of a liquid [56]. This distinction will be particularly important for simulating the dynamics of embedded particles that couple to the solvent through The Mach number measures the ratio participation in the collision step. For particles that vs Ma = , (28) couple directly, either through a potential, or through cf bounce-back or stochastic boundary conditions, this dis- tinction is less important because the fluid-particle dif- between vs, the speed of solvent or colloid flow, and fusion coefficient Df does not directly enter the Navier cf = (5/3)(kBT/mf ), the . In contrast Stokes equations. Of course other physical properties can to the Schmidt number, which is an intrinsic property of be affected. For example, we expect that the self diffu- the solvent,p it depends directly on flow velocity. The sion coefficient of a colloid Dcol should be much smaller Ma number measures compressibility effects [57] since 12 sound speed is related to the compressibility of a liq- the same as the kinematic time 3 uid. Because cf in many liquids is of order 10 m/s, the Ma numbers for physical colloidal systems are extremely 2 σcf small under normally achievable flow conditions. Just τν = = Re tS (31) as for the Sc number, however, particle based coarse- ν graining schemes drastically lower the Ma number. The particle mass mf is typically much greater than the mass it takes momentum to diffuse over that distance, i.e. of a molecule of the underlying fluid, resulting in lower Re=τν /tS 1, then the particle will feel vorticity ef- ≈ velocities, and moreover, due to the lower density, colli- fects from its own motion a distance σcf away, leading to sions also occur less frequently. These effects mean that non-linear inertial contributions to its motion. Since hy- the speed of sound is much lower in a coarse-grained sys- drodynamic interactions can decay as slowly as 1/r, their tem than it is in the underlying physical fluid. Or, in influence can be non-negligible. If, on the other hand, Re other words, particle based coarse-graining systems are 1, then vorticity will have diffused away and the par- ≪ typically much more compressible than the solvents they ticle will only feel very weak hydrodynamic effects from model. its own motion. 2 Ma number effects typically scale with Ma [57], and so Exactly when inertial finite Re effects become signif- the Ma number does not need to be nearly as small as for icant depends on the physical system under investiga- a realistic fluid to still be in the correct regime of hydro- tion. For example, in a pipe, where the length-scale a in dynamic parameter space. This is convenient, because to Eq. (29) is its diameter, the transition from simpler lami- lower the Ma number, one would need to integrate over nar to more complex turbulent flow is at a pipe Reynolds longer fluid particle trajectories to allow, for example, number of Re 2000 [57, 75]. On the other hand, for a a colloidal particle to flow over a given distance, making single spherical≈ particle, a non-linear dependence of the the simulation computationally more expensive. So there friction ξ on the velocity vs, induced by inertial effects, is a compromise between small Ma numbers and compu- starts to become noticeable for a particle Reynolds num- tational efficiency. We limit our Ma numbers to values ber of Re 1 [57], while deviations in the symmetry such that Ma 0.1, but it might be possible, in some of the streamlines≈ around a rotating sphere have been ≤ situations, to double or triple that limit without causing observed in calculations for Re 0.1 [76]. undue error. For example, incompressible hydrodynam- ≈ For typical colloidal suspensions, where the particle ics is used for aerodynamic flows up to such Ma numbers diameter is on the order of a few µm down to a few since the errors are expected to scale as 1/(1 Ma2) [57]. nm, the particle Reynolds number is rarely more than When working in units of m and k T , the− only way to f B 10−3. In this so-called Stokes regime viscous forces dom- keep the Ma number below our upper limit is to restrict inate and inertial effects can be completely ignored. The the maximum flow velocity to v < 0.1c . The flow veloc- s f Navier-Stokes equations can be replaced by the linear ity itself is, of course, determined∼ by the external fields, Stokes equations [57] so that analytic solutions are easier but also by other parameters of the system. to obtain [63]. However, some of the resulting behavior is non-intuitive for those used to hydrodynamic effects on a macroscopic scale. For example, as famously explained C. Reynolds number by Purcell in a talk entitled “Life at low Reynolds num- bers” [77], many simple processes in biology occur on The Reynolds number is one of the most important di- small length scales, well into the Stokes regime. The mensionless numbers characterizing hydrodynamic flows. conditions that bacteria, typically a few µm long, expe- Mathematically, it measures the relative importance of rience in water are more akin to those humans would ex- the non-linear terms in the Navier-Stokes equations [57]. perience in extremely thick molasses. Similarly, colloids, Physically, it determines the relative importance of iner- polymers, vesicles and other small suspended objects are tial over viscous forces, and can be expressed as: all subject to the same physics, their motion dominated by viscous forces. For example, if for a colloid sediment- v a Re = s (29) ing at 1 µm/s, gravity were instantaneously turned off, ν then the Stokes equations suggest that it would come to a complete halt in a distance significantly less than one A,˚ where a is a length-scale, in our case, the hydrodynamic reflecting the irrelevance of inertial forces in this low Re radius of a colloid, i.e. a σ , and v is a flow velocity. ≈ cf s number regime. It should be kept in mind that when the For a spherical particle in a flow, the following heuristic Stokes regime is reached because of small length scales (as argument helps clarify the physics behind the Reynolds opposed to very large such as those found for Stokes time number: If the volcanic lava flows), then thermal fluctuations are also σcf important. These will drive diffusive behavior [78]. In tS = , (30) vs many ways SRD is ideally suited for this regime of small particles because the thermal fluctuations are naturally it takes a particle to advect over its own radius is about included [49]. 13

1. Dependence of the Reynolds number on simulation For large Knudsen numbers Kn > 10 continuum Navier- parameters Stokes equations completely break∼ down, but even for much smaller Knudsen numbers, significant rarefaction From Eqs. (28) and (29), it follows that the Reynolds effects are seen. For example, for flow in a pipe, where a number for a colloid of hydrodynamic radius a σcf can in Eq. (34) is taken to be the pipe radius, important non- be written as: ≈ continuum effects are seen when Kn > 0.1. In fact it is exactly for these conditions of modest∼ Kn numbers, when 5 σ ν0 corrections to the Navier Stokes become noticeable, that Re = Ma cf . (32) 3 a0 ν DSMC approaches are often used [36, 37]. SRD, which r     is related to DSMC, may also be expected to work well Equating the hydrodynamic radius a to σcf from the for such flows. It could find important applications in fluid-colloid WCA interaction of Eq. (17) is not quite microfluidics and other micromechanical devices where correct, as we will see in section V, but is a good enough Kn effects are expected to play a role [80, 81]. approximation in light of the fact that these hydrody- Colloidal dispersions normally have very small Kn namic numbers are qualitative rather than quantitative numbers because the mean-free path of most liquid sol- indicators. vents is very small. For water at standard temperature In order to keep the Reynolds number low, one must and pressure λfree 3 A.˚ Just as found for the other di- ≈ either use small particles, or very viscous fluids, or low mensionless numbers, coarse-graining typically leads to velocities vs. The latter condition is commensurate with larger Kn numbers because of the increase of the mean- a low Ma number, which is also desirable. For small free path. Making the Kn number smaller also typically ν enough λ (and ignoring for simplicty the factor fcol(γ, α) increases the computational cost because a larger number in Eq. (10)) the Re number then scales as of collisions need to be calculated. In our simulations, we keep Kn 0.05 for spheres. This rough criterion is based σ ≤ Re 23 Ma cf λ. (33) on the observation that for small Kn numbers the fric- ≈ a0 tion coefficient on a sphere is expected to be decreased by a factor 1 α Kn, where α is a material dependent Again, we see that a smaller mean-free path λ, which constant of order− 1 [63], so that we expect Kn number enhances the collisional viscosity, also helps bring the Re effects to be of the same order as other coarse-graining number down. This parameter choice is also consistent errors. There are two ways to achieve small Kn numbers: with a larger Sc number. Thus larger Sc and smaller Ma one is by increasing σcf /a0, the other is by decreasing λ. numbers both help keep the Re number low for SRD. The second condition is commensurate with a large Sc In principle a large viscosity can also be obtained for number or a small Re number. large λ, which enhances the kinetic viscosity, but this In ref. [53], the Kn numbers for colloids were Kn 0.5 choice also lowers the Sc number and raises the Knudsen and Kn 0.8, depending on their colloid-solvent≈ cou- number, which, as we will see in the next sub-section, is pling. Such≈ large Kn numbers should have a significant not desirable. effect on particle friction coefficients, and this may ex- Just as was found for the Ma number, it is relatively plain why these authors find that volume fraction φ has speaking more expensive computationally to simulate for less of an effect on the sedimentation velocity than we low Re numbers because the flow velocity must be kept do [52]. low, which means longer simulation times are necessary to reach time-scales where the suspended particles or fluid flows have moved a significant distance. We there- E. Peclet number fore compromise and normally keep Re < 0.2, which is similar to the choice made for LB simulations∼ [14, 79]. The Peclet number Pe measures the relative strength of For many situations related to the flow of colloids, this convective transport to diffusive transport. For example, should be sufficiently stringent. for a colloid of radius a, travelling at an average velocity vs, the Pe number is defined as: D. Knudsen number v a Pe = s (35) Dcol The Knudsen number Kn measures rarefaction effects, where D is the colloid diffusion coefficient. Just as for and can be written as col the Re number, the Pe number can be interpreted as a λ ratio of a diffusive to a convective time-scale, but now the Kn = free (34) a former time-scale is not for the diffusion of momentum but rather it is given by the colloid diffusion time where a is a characteristic length-scale of the fluid flow, 2 and λfree is the mean-free path of the fluid or gas. For a σcf τD = (36) colloid in SRD one could take a σ and λfree λa0. ≈ cf ≈ Dcol 14 which measures how long it takes for a colloid to diffuse over a distance σcf . We again take a σcf so that, using 1.0 Eq. (30), the Pe number can be written≈ as:

τD ) 0.8 Pe = . (37) E tS ξ

If Pe 1 then the colloid moves convectively over a dis- - 1/ ≫ ξ 0.6 tance much larger than its radius σcf in the time τD that it diffuses over that same distance. Brownian fluctuations (1/ cf are expected to be less important in this regime. For Pe 0.4

1, on the other hand, the opposite is the case, and πησ ≪the main transport mechanism is diffusive (note that on 4 2 long enough time-scales (t > τD/Pe ) convection will al- 0.2 ways eventually “outrun” diffusion [78]). It is sometimes thought that for low Pe numbers hydrodynamic effects can be safely ignored, but this is not always true. For ex- 0.0 ample, we found that the reduction of average sedimenta- 0.00 0.05 0.10 0.15 σ tion velocity with particle volume fraction, famously first cf / L explained by Batchelor [82], is independent of Pe number down to Pe = 0.1 at least [52]. FIG. 5: Finite system size scaling of the Stokes friction ξS on a sphere of size σcf = 2 in a flow of velocity vs = 0.01 a0/t0. The correction factor f defined in Eq. (42) can be extracted 1. Dependence of Peclet number on simulation parameters from measured friction ξ (drag force/velocity) and estimated Enskog friction ξE defined in Eq. (40). It compares well to The highest Pe number achievable in simulation is lim- theory, Eq. (43). ited by the constraints on the Ma and Re numbers. For example the Ma number sets an upper limit on the maxi- solved due to Kn number and discretization effects. De- mum Pe number by limiting v . From Eqs. (29) and (35), s creasing the box-size L falls foul of the long-ranged na- it follows that the Peclet number can be re-written in ture of the HI, and therefore, just as for Coulomb inter- terms of the Reynolds number as actions [62], finite size effects such as those induced by 2 periodic images must be treated with special care. ν ν σ Pe = Re 6πγ cf Re (38) Increasing the flow velocity may also be desirable since Dcol ≈ ν0 a0     more Stokes times τS = σcf /vs can be achieved for the same number of SRD collision steps, thus increasing com- where we have approximated D k T/(6πησ ). col B cf putational efficiency. The Ma number gives one con- This shows that for a given constraint≈ on the Re num- straint on v , but usually the more stringent constraint ber, increasing γ or σ increases the range of accessi- s cf comes from keeping the Re number low to prevent un- ble Pe numbers. Similarly, when the kinematic viscosity wanted inertial effects. is dominated by the collisional contribution, decreasing the dimensionless mean-free path λ will also increase the max −2 maximum Pe number allowed since Pe λ Re A. Finite-size effects λ−1Ma. However, these changes increase the∼ computa-∼ tional cost of the simulation. 1. Finite-size correction to the friction

V. FINITE-SIZE, DISCRETIZATION, AND The friction coefficient ξ can be extracted from the INERTIAL EFFECTS Stokes drag Fd on a fixed colloid in fluid flow:

Fd = ξv∞ 4πηav∞ (39) The cost of an SRD simulation scales almost linearly − ≡− with the number of fluid particles Nf in the system, and where v∞ is the flow field at large distances. The pre- this contribution is usually much larger than the cost of factor 4 comes from using slip boundary conditions; it including the colloidal degrees of freedom. To optimize would be 6 for stick-boundary conditions, as verified the efficiency of a simulation, one would therefore like in [58]. In principle, since η is accurately known from to keep Nf as small as possible. This objective can be theory for SRD, this expression can be used to extract achieved by keeping σcf /a0 and the box-size L small. Un- the hydrodynamic radius a, which is not necessarily the fortunately both these choices are constrained by the er- same as σcf , from a simulation, as we did in [52]. rors they introduce. Reducing σcf /a0 means that short- The hydrodynamic radius a can also be directly cal- ranged hydrodynamic fields become less accurately re- culated from theory. To derive this, it is important to 15 recognize that there are two sources of friction [49]. The first comes from the local Brownian collisions with the 1 10 small particles, and can be calculated by a simplified En- σ = 6 skog/Boltzmann type kinetic theory [83]: cf σ cf = 4 1/2 8 2πkBTMcmf 2 ξE = nf σcf , (40) v) 3 M + mf σ = 2   πη cf which is here adapted for slip boundary conditions. A re- 0 / (4

d 10 lated expression for stick boundary conditions, including

F σ = 1 that for rotational frictions, is described in [58], where it cf was shown that the short-time exponential decays of the linear and angular velocity auto-correlation functions are σ = 0.5 quantitatively described by Enskog theory. cf L/σ = 16 The second contribution to the friction, ξS , comes from cf integrating the Stokes solution to the hydrodynamic field -1 10 -3 -2 -1 0 1 over the surface of the particle, defined here as r = σcf . 10 10 10 10 10 These two contributions to the friction should be added in parallel to obtain the total friction [83, 84] (see also Re Appendix B): FIG. 6: Drag force Fd divided by 4πηv for various colloid sizes 1 1 1 and Reynolds numbers (in all cases the box size L = 16σcf ). = + . (41) For small Reynolds numbers this ratio converges to the effec- ξ ξS ξE tive hydrodynamic radius a. Dashed lines are theoretical pre- dictions from combining Stokes and Enskog frictions in (41). In contrast to the Enskog friction, which is local, we expect substantial box-size effects on the Stokes friction ξS since it depends on long-ranged hydrodynamic effects. These can be expressed in terms of a correction factor the theory. These are most likely due to Kn number and f(σcf /L) that should go to 1 for very large systems : discretization effects, to be discussed in the next sub- section. −1 ξS =4πησcf f (σcf /L). (42) For the smallest spheres the mass ratio Mc/mf may also change the measured friction if instead of fixing the To measure this correction factor we plot, in Fig.5, the sphere we were to let it move freely. For σcf = a0, form 4πησ (1/ξ 1/ξ ) for various system box sizes for cf − E Mc/mf 21, which is small enough to have a significant which we have measured ξ from the simulation, and es- effect [87].≈ For larger spheres, this is not expected to be timated ξ from Eq. (40). As expected, the correction E a problem. For example Mc/mf 168 for σcf =2a0 and factor tends to 1 for smaller σ /L. More detailed calcu- ≈ cf Mc/mf 1340 for σcf =4a0. lations [85, 86], taking into account the effect of periodic ≈ boundaries, suggests that to lowest order in σcf /L the correction factor should scale as 2. Effective hydrodynamic radius σ f(σ /L) 1+2.837 cf . (43) cf ≈ L From the calculated or measured friction we can also obtain the effective hydrodynamic radius aeff , which is Indeed, a least squares fit of the data to this form gives continuum concept, from a comparison of the microscopic a slope of c 2.9, close to the theoretical value and in frictions from Eq. (41) with Eq. (39): agreement with≈ similar Lattice Boltzmann simulations of a single colloidal sphere [15]. ξSξE With these ingredients in hand, we can calculate the aeff = (44) 4πη (ξS + ξE) theoretical expected friction from Eqns. (40) - (43). We know from previous work that the Enskog contribution at (for stick-boundaries the 4π should be replaced by 6π). short times and the hydrodynamic contribution at long We find aeff (σcf =6)=6.11a0, aeff (σcf =4)=3.78a0, times quantitatively reproduce the rotational and trans- and aeff (σcf =2)=1.55a0. The effective hydrodynamic lational velocity auto-correlation functions (VACF) for radius is increased by the finite-size effects, but lowered σcf /a0 = 2 5 [58], and so we expect that the friction by the Enskog contribution which is added in parallel. coefficients should− also be accurately described by these Because this latter contribution is relatively more impor- theories. We test this further in Fig. 6 for a number of tant for small colloids, aeff < σcf for smaller σcf . For different values of σcf /a0, and find excellent agreement σcf = 6a0 the Enskog contribution is smaller than the with theory for Re 1 and σ /a0 2. For σ /a0 =1 finite-size effects, and so the effective hydrodynamic ra- ≤ cf ≥ cf and below, on the other hand, we find deviations from dius is larger than σcf . Obviously this latter effect de- 16

a) σ = 0.5 b) σ = 1 pends on the box-size. In an infinite box aeff < σcf for all cf cf values of σcf , due to the Enskog contribution. Note that it is the effective hydrodynamic radius aeff which sets the 6 long-ranged hydrodynamic fields, as we will see below. 4

3. Turning off long-ranged hydrodynamics 2

Hydrodynamic forces can be turned off in SRD by reg- 0 ularly randomizing the absolute fluid particle velocities c) σ = 2 d) σ = 4 (with for example a naive Langevin thermostat). For cf cf particles embedded in an SRD solvent through partic- 6 ipation in the collision step, this trick can be used to compare the effects of hydrodynamics to a purely Brow- 4 nian simulation[40]. The Yeomans group has successfully applied this idea in a study of polymer collapse and pro- 2 tein folding [47]. However for the case of the colloids embedded through 0 direct solvent collisions, turning off the hydrodynamic forces by randomizing the velocities greatly enhances the -4 -2 0 2 4 -4 -2 0 2 4 friction because, as is clear from Eqns. (40) - (42), the two contributions add in parallel. Without long-ranged FIG. 7: Magnified difference fields 10[v(r) − vSt(r)]/v∞ for hydrodynamics, the friction would be entirely dominated different colloidal sizes (the axes are scaled by σcf ). In all 2 cases Re ≈ 0.1, and box size L = 16σcf . by the Enskog contribution (40) which scales with σcf , and can be much larger than the hydrodynamic contri- bution which scales as σcf . Another way of stating this would be: By locally conserving momentum, SRD al- lows the development of long-ranged hydrodynamic fluid velocity correlations that greatly reduce the friction felt with theory for these smallest sphere sizes. We also note by a larger colloidal particle compared to the friction it that using a = σ instead of the more accurate value would feel from a purely random Brownian heat-bath at cf of the effective hydrodynamic radius a calculated from the same temperature and number density n = γ/a3. f 0 theory results in significantly larger deviations between measured and theoretical flow-fields. This independently B. Discretization effects on flow-field and friction confirms the values of the effective hydrodynamic radius.

For pure Stokes flow (Re = 0), the velocity around a The increased accuracy from using larger σcf /a0 comes fixed slip-boundary sphere of (effective) hydrodynamic at a sharp increase in computational cost. To make sure radius a can be exactly calculated: that the finite size effects in each simulation of Fig. 7 are approximately the same, the box-size was scaled as a a L/σ . This means that doubling the colloid size leads vSt(r)= v∞ 1 v∞ ˆrˆr (45) cf − 2r − · 2r to an eightfold increase in the number of fluid particles.   Moreover, the maximum velocity of the fluid must go where v∞ is the velocity field far away from the sphere, down linearly in colloid size to keep the Re number, de- and r the vector pointing from the center of the sphere to fined in Eq. (29), constant. If in addition we keep the a position inside the fluid, with corresponding unit vector number of Stokes times fixed, meaning that the fluid flows ˆr = r/r. a certain multiple of the colloid radius or the box size, In Fig. 7 we plot the difference between the measured then a larger particle also means the fluid needs to flow field and the theoretical expected field (45) for four dif- over a proportionally longer total distance. The over- ferent colloid radius to cell size ratios σ /a , using the cf 0 all computational costs for this calculation then scales at values of the effective hydrodynamic radius a calculated least as (σ /a )5, which is quite steep. For that reason, from combining Eq. (41) and Eq. 39. As expected, the cf 0 we advocate using smaller colloids wherever possible. field is more accurately reproduced as the colloid radius becomes larger with respect to the cell size a0. This improvement arises because SRD discretization and hy- In most of our simulations we choose σcf = 2, which drodynamic Knudsen number effects become smaller. In leads to a small relative error in the full velocity field and Fig. 7 we observe quite large deviations in the hydrody- for which we can fully explain the observed friction (as namic field for σcf 1. These discretization effects may seen in the previous subsection). This size is similar to explain why the measured≤ frictions in Fig. 6 do not agree what is commonly used in LB [14, 79]. 17

a) Re = 0.08 b) Re = 0.08 15 VI. THE HIERARCHY OF TIME SCALES

A. Colloidal time-scales 10 Many different time-scales govern the physics of a col- 5 loid of mass M embedded in a solvent [1]. The most im- portant ones are summarized in table III, and discussed 0 in more detail below.

c) Re = 0.4 d) Re = 2 15 1. Fluid time-scales

10 The shortest of these is the solvent collision time τcol over which fluid molecules interact with each other. For 5 a typical molecular fluid, τcol is on the order of a few tens of fs, and any MD scheme for a molecular fluid must use a discretization time-step tMD τcol to properly integrate 0 the equations of motion. ≪ -5 0 5 -5 0 5 The next time-scale up is the solvent relaxation time τf , which measures how fast the solvent VACF decays. −14 −13 FIG. 8: (a) Fluid flow-field around a fixed colloid with σ = 2 For a typical molecular liquid, τf 10 10 s. cf ≈ − and L = 64a0 for Re = 0.08. (b)-(d) Difference field 10[v(r)− (For water, at room temperature, for example, it is on vSt(r)]/v∞ for Re = 0.08, 0.4, and 2, respectively. the order of 50 femtoseconds)

2. Hydrodynamic time-scales for colloids

Hydrodynamic interactions propagate by momentum sonic time C. Inertial effects on flow-field and friction (vorticity) diffusion, and also by sound. The a tcs = . (46) cs

One of the challenges in SRD is to keep the Re number it takes a sound wave to travel the radius of a colloid is down. It is virtually impossible to reach the extremely typically very small, on the order of 1 ns for a colloid of low Re number (Stokes) regime of realistic colloids, but radius a = 1µm. For that reason, sound effects are of- that is not necessary either. In Fig. 6 we see that the fric- ten ignored for colloidal suspensions under the assump- tion only begins to noticeably vary from the Stokes limit tion that they will have dissipated so quickly that they at Re 1 (at least on a logarithmic scale). We expect have no noticeably influence on the dynamics. However, the flow-field≈ itself to be more sensitive to finite Re num- some experiments [88] and theory [89] do find effects from ber effects. To study this directly, we examine, in Fig. 8, sound waves on colloidal hydrodynamics, so that this is- the flow-field around a fixed colloid of size σcf = 2a0 in sue is not completely settled yet. a box of L = 32a0 for for different values of Re. The dif- The kinematic time, τν , defined in Eq. (31) (and Ta- ferences with the Stokes flow field of Eq. (45) are shown ble III) as the time for momentum (vorticity) to diffuse in Fig. 8(b)-(d) for Re = 0.08, 0.4, and 2 respectively. In over the radius of a colloid, is particularly important for all cases we used a hydrodynamic radius of a =1.55 for hydrodynamics. It sets the time-scale over which hy- the theoretical comparison, as explained in the previous drodynamic interactions develop. For a colloid of radius sub-section. The lengths of the vectors in Figs. (b)-(d) 1µm, τ 10−6 s, which is much faster the colloidal ν ≈ are multiplied by 10 for clarity. We observe that the rel- diffusion time τD. ative errors increase with Re. They are on the order of When studying problems with a finite flow velocity, 5% for Re=0.08, and increase to something on the order another hydrodynamic time-scale emerges. The Stokes of 40% for Re=2. This is exactly what is expected, of time tS, defined in Eq. (30) (and Table III) as the time for course, as we are moving away from the Stokes regime a colloid to advect over its own radius, can be related to with which these flow lines are being compared. Since the kinematic time by the relation τν = Re tS. Because we also expect effects on the order of a few % from Kn colloidal particles are in the Stokes regime where Re 1, ≪ number, Ma number and various finite size effects, we we find τν tS. argue that keeping Re 0.2 should be good enough for Simulation≪ and analytical methods based on the Os- most of the applications≤ we have in mind. een tensor and its generalizations, e.g. [4, 5], implicitly 18

TABLE III: Time-scales relevant for colloidal suspensions

Solvent time-scales

−15 Solvent collision time over which solvent molecules interact τcol ≈ 10 s

−14 −13 Solvent relaxation time over which solvent velocity correlations decay τf ≈ 10 − 10 s

Hydrodynamic time-scales a Sonic time over which sound propagates one colloidal radius tcs = Eq. (46) cs a2 Kinematic time over which momentum (vorticity) diffuses one colloidal radius τν = Eq. (31) ν a τD Stokes time over which a colloid convects over its own radius tS = = Eq. (30) vs Pe Brownian time-scales

Fokker-Planck time over which force-force correlations decay τFP

Mc Enskog relaxation time over which short-time colloid velocity correlations decay τE = Eq. (53) ξE

Mc Brownian relaxation time over which colloid velocity correlations decay in Langevin Eq. τB = Eq. (47) ξS a2 Colloid diffusion time over which a colloid diffuses over its radius τD = Eq. (36) Dcol Ordering of time-scales for colloidal particles

τcol < τf ,τFP < τE, tcs < τB < τν < τD, tS

assume that the hydrodynamic interactions develop in- time for a colloid to lose memory of its velocity. How- stantaneously, i.e. that τν 0. For the low Re number ever, as discussed in Appendix B, this picture, based on regime of colloidal dispersions≈ this is indeed a good ap- the Langevin equation, is in fact incorrect for colloids. proximation. Of course it must be kept in mind that τν Nevertheless, τB has the advantage that it can easily be is a diffusive time-scale, so that the distance over which calculated and so we will use it as a crude upper bound it propagates grows as √t. Thus the approximation of on the colloid velocity de-correlation time. instantaneous hydrodynamics must be interpreted with Perhaps the most important time-scale relevant for some care for effects on larger length-scales. Brownian motion is the diffusion time τD, described in Eq. (36) as the time for a colloidal particle to diffuse over its radius. For a colloid of radius 1µm, τD 5 s, 3 ≈ 3. Brownian time-scales for colloids and even though τD a , it remains much larger than most microscopic time-scales∼ in the mesoscopic colloidal The fastest time-scale relevant to the colloidal Brown- regime. ian motion is the Fokker-Planck time τF P defined in [1] as the time over which the force-force correlation function decays. It is related to τf , the time over which the fluid B. Time-scales for coarse-grained simulation looses memory of its velocity because the forces on the colloid are caused by collisions with the fluid particles, In the previous subsection we saw that the relevant velocity differences de-correlate on a time scale of order time-scales for a single colloid in a solvent can span as τf . many as 15 orders of magnitude. Clearly it would be The next time-scale in this series is the Brownian time: impossible to bridge all the time-scales of a physical col- loidal system – from the molecular τcol to the mesoscopic

Mc τD – in a single simulation. Thankfully, it is not neces- τB = (47) sary to exactly reproduce each of the different time-scales ξS in order to achieve a correct coarse-graining of colloidal where ξS = 6πηa is the Stokes friction for stick bound- dynamics. As long as they are clearly separated, the cor- ary conditions. It is often claimed that this measures the rect physics should still emerge. 19

the efficiency of a simulation while still retaining key physical features. −15 τ 10 s col − fluid collision −14 τ 10 s f − fluid de−correlation −13 τ 10 s FP − Fokker−Planck 1. SRD fluid time-scales −12 10 s −11 10 s τ −3 τ −10 cs −sonic 10 s FP − Fokker−Planck In SRD the physical time-scale τcol is coarse-grained 10 s −2 τ −9 10 s −sonic out, and the effect of the collisions calculated in an aver- 10 s cs Telescoping down −1 τ age way every time-step ∆t . The time-scale τ on which −8 10 s ν −kinematic c f 10 s −0 −7 10 s τ −colloid the velocity correlations decay can be quite easily calcu- 10 s τ D B −Browinian (Langevin) lated from a random-collision approximation. Following −6 τ 10 s ν −kinematic 2 [56]: τf λt0/ ln[1 3 (1 cos α)(1 1/γ)]. For our pa- ≈− 1 − − − rameters, α = π,γ = 5, we find τ 0.76∆t =0.076t0. 2 f ≈ c −1 However, it should be kept in mind that for small λ the 10 s −0 exponential decay with t/τ turns over to a slower alge- 10 s τ −colloid diffusion f D braic decay at larger t [56]. The amplitude of this “tail” is, however, quite small.

FIG. 9: (Color online) Schematic depiction of our strategy 2. Hydrodynamic time-scales for simulation for coarse-graining across the hierarchy of time-scales for a colloid (here the example taken is for a colloid of radius 1 µm in H20). We first identify the relevant time-scales, and then For our choice of units cs = 5/3 a0/t0, so that the sonic time of Eq. (46) reduces to telescope them down to a hierarchy which is compacted to p maximise simulation efficiency, but sufficiently separated to σcf correctly resolve the underlying physical behaviour. tcs 0.775 , (48) ≈ a0 which is independent of λ or γ. In the limit of small λ, the ratio of the kinematic time Since we take the view that the “solvent” particles are really a Navier Stokes solver with noise, what is needed τν to ∆tc can be simplified to the following form: 2 to reproduce Brownian behavior is first of all that the col- τν σcf loid experiences random kicks from the solvent on a short 18 2 (49) ∆tc ≈ a enough time-scale. By dramatically reducing the num- 0 so that the condition τ , τ τ is very easy to fulfil. ber of “molecules” in the solvent, the number of kicks f F P ≪ ν per unit of time is similarly reduced. Here we identify Furthermore, under the same approximations, the ratio τ /t 23σ λ so that for λ too small the kinematic the Fokker-Planck time τF P as the time-scale on which ν cs ≈ cf the colloid experiences random Brownian motion. For time becomes faster than the sonic time. For the simu- Brownian motion it doesn’t really matter how the kicks lation parameters used in [52] (and in Fig. 10), we find τ /τ 5. are produced, they could be completely uncorrelated, but ν cs ≈ since we also require that the solvent transports momen- tum, solvent particles must have some kind of correla- 3. Brownian time-scales for simulation tion, which in our case is represented by the time τf . Other colloid relaxation times should be much longer than this. We therefore require first of all that τ τ We expect the Fokker Planck time τF P to scale as ∆tc, F P ≪ B and τ τ . since this is roughly equivalent to the time τf over which f ≪ B Secondly, the colloid’s VACF should decay to zero well the fluid velocities will have randomized. In Fig. 10 we before it has diffused or convected over its own radius. plot the force-force correlation function A proper separation of time-scales in a coarse-graining F (t)F (0) C (t)= h i. (50) scheme then requires that τB τD as well as τB tS F F F 2 for systems where convection is≪ important. ≪ h i Finally, for the correct hydrodynamics to emerge we Its short time behavior is dominated by the random forces, and the initial decay time gives a good estimate of require that τF P , τf τν τD. But this separation no ≪ ≪ τ . As can be seen in the inset of Fig. 10, τ 0.09t0, longer needs to be over many orders of magnitude. As ar- F P F P ≈ gued in [52], one order of magnitude separation between which is indeed on the order of ∆tc or τf . If we make the reasonable approximation that σ time-scales should be sufficient for most applications. We cc ≈ illustrate this approach in Fig. 9. 2σcf , then the ratio of the Brownian time τB to the kine- In the next few subsections we discuss in more quanti- matic time τν simplifies to: tative detail how our coarse-graining strategy telescopes τB 2ρc (51) down the hierarchy of time-scales in order to maximize τν ≈ 9ρf 20

1.0 φ = 0.02 0.02 1.0 0.8 0.8 φ = 0.01 0.6 φ = 0.02 0.6 0.4 0.02 φ = 0.04 (t) 0.0 0.1 0.2 0.3 C (t) VV VV ) 0.4 D(t) 0.01 φ C CFF(t) D( exp(-t/τ ) 0.2 B exp(-ξ t/M) E 0.01 0.00 0.02 0.04 0.06 0.0 φ

-0.2 0.00 0 1 2 3 4 5 6 0 200 400 600 800 1000 τ τ t / B t/ B

FIG. 10: (Color online) Normalized VACF for a single col- FIG. 11: Time dependent self-diffusion coefficient D(t) for loid with σcf = 2a0 and L = 32a0: measured (solid line), different volume fractions φ, achieved by varying the number Enskog short time prediction (dot-dashed line), and Brown- of colloids in a box of fixed volume. The inset shows the ian approximation (dashed line). The normalized force-force self-diffusion coefficient (D = lim t →∞D(t) as a function of auto-correlation CFF (t) (dotted line) decays on a much faster volume fraction, together with an analytical prediction D = time scale. The inset shows a magnification of the short-time D0(1 − 1.1795φ) [90] valid in the low-density limit. regime. Note that the colloid exhibits ballistic motion on a time-scale ∼ τf . The time-scales τFP ≈ 0.09, τB ≈ 2.5 and τD ≈ 200 are clearly separated by at least an order of magni- tude, as required. If we assume that λ 1 and, for simplicity, that ξ ξS, i.e. we ignore the Enskog≪ contribution, then the diffusion≈ time scales as: so that the ratio of τB to τν is the same as for a real 2 3 3 physical system. Since buoyancy requirements mean that σcf 6πησcf γ σcf γ σcf τD = t0 τB , normally ρc ρf , the time-scales are ordered as τB < τν . D ≈ k T ≈ λ a ≈ 4λ2 a ≈ col B  0   0  Moreover, since the ratio τν /∆tc 1, independently of (55) γ or even λ (as long as λ 1), the≫ condition that τ F P so that τB τD is not hard to fulfil. τ is also not hard to fulfil≪ in an SRD simulation. ≪ ≪ B It is also instructive to examine the ratio of the diffu- These time-scales are illustrated in Fig. 10 where we sion time τ to the kinematic time τ : plot the normalized time correlation function: D ν

V (t)V (0) τD ν γ σcf = 0.06 . (56) CV V (t)= h 2 i, (52) 2 V τν Dcol ≈ λ a0 h i   with V the Cartesian velocity coordinate of a single col- In general we advocate keeping λ small to increase the Sc loid. After the initial non-Markovian quadratic decay, number Sc = ν/Df , and since another obvious constraint the subsequent short-time exponential decay is not given is D D , there is not too much difficulty achieving Enskog time col f by τB but instead by the the desired≪ separation of time-scales τ τ . ν ≪ D Mc As a concrete example of how these time-scales are τE = (53) ξE separated in a simulation, consider the parameters used in Fig. 10 for which we find τf = 0.076t0, τF P = 0.09t0 where ξE is the Enskog friction which can be calculated τB =2.5t0, τν =8t0 and τD = 200t0. But more generally, from kinetic theory [58, 83], see Eq. (40), and generally what the analysis of this section shows is that obtaining τE < τB. The physical origins of this behavior are de- the correct hierarchy of time-scales: scribed in more detail in Appendix B. The colloid diffusion coefficient is directly related to τ , τ τ , τ < τ τ ,t (57) the friction by the Stokes-Einstein relation: f F P ≪ E B ν ≪ D S

kBT is virtually guaranteed once the conditions on dimension- Dcol = . (54) ξ less numbers, detailed in Section IV, are fulfilled. 21

12 VII. MAPPING BETWEEN A PHYSICAL AND 0.10 φ = 0.02 COARSE-GRAINED SYSTEMS: NO FREE 10 LUNCH? 0.05 2 ~t σ 2 The ultimate goal of any coarse-graining procedure is 8 2 cf > τ to correctly describe physical phenomena in the natural 2 0.00 B world. As we have argued in this paper, it is impossible 6 to bridge, in a single simulation, all the time and length- 0 1 2 scales relevant to a colloid suspended in a solvent. Com- promises must be made. Nevertheless, a fruitful mapping 4

<(x(t)-x(0)) from a coarse-grained simulation to a physical system is possible, and greatly facilitated by expressing the phys- 2 ical properties of interest in dimensionless terms. The best way to illustrate this is with some examples. τ 0 D

0 20 40 60 80 100 A. Example 1: Mapping to diffusive and Stokes τ time-scales t / B For many dynamic phenomena in colloidal dispersions, FIG. 12: Mean-square-displacement of a σcf = 2a0 colloid the most important time-scale is the diffusion time τ . (solid line). The dotted line is a fit to the long time limit. The D inset shows a magnification of the short-time regime, high- To map a coarse-grained simulation onto a real physi- lighting the initial ballistic regime (dashed line is hV (0)i2t2), cal system one could therefore equate the diffusion time which rapidly turns over to the diffusive regime. and the colloid radius a of the simulation to that of the real physical system. For example, take the simulation parameters from table VI in Appendix C for a = 2a0, C. Measurements of diffusion and compare these to the properties of colloids of radius a = 1µm and a = 10 nm from table V. For the larger colloids, τD = 5s, and equating this to the simulation τD To further test the hybrid MD-SRD coarse-graining means that a0 =0.5µm and t0 =0.02 s. Performing the −6 scheme, we plot in Fig. 11 the time-dependent diffusion same exercise for the smaller colloid, where τD =5 10 −8 × coefficient D(t), defined as: s, results in a0 = 5 nm and t0 =2 10 s. The first thing that becomes obvious from this× exercise is that “physi- t cal” time is set here not by the coarse-grained simulation, D(t)= dt′ V (t′)V (0) (58) 0 h i but by the particular system that one is comparing to. Z In other words, many different physical systems could be for a number of different volume fractions φ = mapped to the same simulation. 4 3 If the system is also subjected to flow, then a second 3 N/V πσcf . Note that the convergence slows with in- creasing volume fraction φ. The infinite time integral time-scale emerges, the Stokes time tS = Pe τD. As long gives the diffusion coefficient, i.e. Dcol = limt→∞ D(t). as the physical Pe number is achievable without compro- In the limit of low densities the diffusion coefficient has mising the simulation quality, then this time-scale can si- been predicted to take the form D(φ)= D0(1 cφ) with multaneously be set equal to that of the physical system. 1.1795 for slip boundary spheres [90], and this− provides Behavior that depends primarily on these two time-scales a good fit at the lower volume fractions. should then be correctly rendered by the SRD hybrid MD In Figure 12 we plot the mean-square displacement of scheme described in this paper. a Cartesian component of the colloidal particle position. As highlighted in the inset, at short times there is an initial ballistic regime due to motion at an average mean B. Example 2: Mapping to kinematic time-scales 2 square velocity V (0) = kBT/M, resulting in a mean- h i 2 2 square displacement (X(t) X(0)) = (kB T/M)t . At By fixing τD or tS, as in example 1, other time-scales times t τ , the motionh is clearly− diffusive,i as expected, are not correctly reproduced. For the large and small col- ≫ ν with a linear dependence on t and a slope of 2Dcol, where loid systems described above, we would find τν =0.17 s −7 Dcol is given by the infinite time limit of Eq. (58). On and 1.7 10 s respectively, which contrasts with the × −6 −10 the scale of the plot the asymptotic regime is reached well physical values of τν = 10 s and 10 s shown in Ta- before the time τD over which the particle has diffused, ble V. In other words, fixing τD results in values of τν on average, over its radius. In Appendix B the behavior that are too large by orders of magnitude. But of course of D(t) and the related mean-square displacement are these much larger values of τν are by design, because discussed in more detail for time-scales t τ . the SRD simulation is optimized by compacting the very ≪ D 22 large physical hierarchy of time-scales. If we are inter- value. For either choice of time-scales, other properties, ested in processes dominated by τD, having different τν such as the fluid self-diffusion constant or the number doesn’t matter, so long as τν τD. density, will have different values from the underlying On the other hand, if we want≪ to describe processes physical system. that occur on much smaller time-scales, for example long- Setting the mass in this way means that the unit of 2 −2 time tails in colloidal VACFs, then simulation times τν thermal energy kBT0 = mf a0t0 would also be very could be mapped onto the physical τν instead. For the large, leading to the appearance of very low tempera- same a = 2a0 simulation parameters used above, the tures. However, because temperature only determines a = 1µm system now maps onto a0 = 0.5µm, and behavior through the way it scales potentials, it is quite −7 t0 = 1.3 10 s with of course a strongly underesti- easy to simply re-normalize the effective energy scales × −5 mated diffusion time: τD = 3 10 s instead of 5 s. and obtain the same behavior as for the physical sys- Similarly for the a = 10 nm system,× the mapping is to tem. Alternatively one could set the temperature equal −11 −9 a0 = 5 nm, t0 =1.3 10 s and τD =3 10 s instead to that of a physical system, but that would mean that of 5 10−6 s. For many× processes on time-scales× t < τ , the masses would again differ significantly from the real × D this underestimate of τD doesn’t really matter. It is only physical system. At any rate, for the same simulation when time-scales are mixed that extra care must be em- (say of sedimentation at a given value of Pe) the values ployed in interpreting the simulation results. And, if the of mass or temperature depend on the physical system processes can be described in terms of different dimen- one compares to. This apparent ambiguity (or flexibil- sionless times, then it should normally still be possible ity) simply reflects the fact that the dominant effects of to disentangle the simulation results and correctly map temperature and mass come into play as relative ratios onto a real physical system. and not as absolute values. Another lesson from these examples of mapping to dif- ferent time-scales comes from comparing the kinematic viscosity of a physical system, which say takes the value D. Example 4: Mapping to attractive potentials: ν = 106µm2/s for water, to its value in SRD simula- problems with length-scales? tions. If for the a = 2a0 SRD system described above we map time onto τD, as in Example 1, then for the 2 So far we have only treated one length-scale in the a = 1µm system ν = 6µm /s, while for the a = 10 nm problem, the radius a. Implicitly we have therefore as- system ν =6 10−3µm2/s. As expected, both are much × sumed that the colloid-colloid interaction does not have smaller than the true value in water. If we instead map another intrinsic length-scale of its own. For hard-sphere the times to τν , then of course the kinematic viscosity colloids this is strictly true, and for steep-repulsions such takes the same value as for the physical system (since as the WCA form of Eq. (16) it is also a fairly good we also set the length-scales equal to the physical length- assumption that the exact choice of the exponent n in scales). This apparent ambiguity in defining a property this equation does not significantly affect the dynamical like the kinematic viscosity is inherent in many coarse- properties [61]. Similarly, there is some freedom to set graining schemes. We take the view that the precise value the repulsive fluid-colloid potential of Eq. (17) in such a of ν is not important, the only thing that matters is that way as to optimize the simulation parameters, in par- it is large enough to ensure a proper separation of times- ticular ∆tMD. The physical liquid-colloid interaction scales and the correct regime of hydrodynamic numbers. would obviously have a much more complex form, but SRD coarse-grains out such details. A similar argument can be made for colloid and fluid interactions with hard C. Example 3: Mapping mass and temperature walls, needed, for example, to study problems with con- finement, for which, inter alia, SRD is particularly well In the examples above, we have only discussed setting suited. lengths and times. This leaves either the mass mf or tem- Things become more complicated for colloids with an perature (thermal energy) kBT still to be set. Because for explicit attractive interaction, such as DLVO or a de- many dynamic properties the absolute mass is effectively pletion potential [2]. These potentials introduce a new an irrelevant variable (although relative masses may not length-scale, the range of the attraction. The ratio of be) there is some freedom to choose. One possibility this length to the hard-core diameter helps determine would be to set Mc, the mass of the colloid, equal to the equilibrium properties of the fluid [21, 22]. In a the physical mass. For an a = 1µm neutrally buoyant physical system this ratio may be quite small. Keep- −14 colloid this sets mf = 2.5 10 g, (equal to about ing the ratio the same in the simulations then leads to 8 × 8 10 water molecules) while for an a = 10 nm colloid potentials that are very steep on the scale of σcc. The × −20 we find mf = 2.5 10 g (equal to about 800 water MD time-step ∆tMD may need to be very small in or- molecules). By construction× this procedure produces the der to properly integrate the MD equations of motion. correct physical ρf . If we fix time with τν , this will there- Such a small ∆tMD can make a simulation very ineffi- fore also generate the correct shear viscosity η. If instead cient, as was found in [53] who followed this strategy we fix τ , then η will be much smaller than the physical and were forced to use ∆t /∆t 455. However, in D MD c ≈ 23 that case the DLVO potential dominates the behaviour, VIII. CONCLUSION so that SRD was only included to roughly resolve the long range hydrodynamics[91]. In general, we advocate adapt- Correctly rendering the principal Brownian and hy- ing the potential to maximize simulation efficiency, while drodynamic properties of colloids in solution is a chal- preserving key physical properties such as the topology lenging task for computer simulation. In this article we of the phase diagram. For example, the attractive en- have explored how treating the solvent with SRD, which ergy scale can be set by the dimensionless reduced second coarse-grains the collisions between solvent particles over virial coefficient, which gives an excellent approximation both time and space, leads to an efficient solution of the for how far one is from the liquid-liquid critical point [92]. thermo-hydrodynamic equations of the solvent, in the When the attractive interaction range is less than about external field provided by the colloids. Although it is 30% of the colloid hard-core diameter, the colloidal “liq- impossible to simulate the entire range of time-scales en- uid” phase becomes metastable with respect to the fluid- countered in real colloidal suspensions, which can span solid phase-transition. At low colloid packing fraction φc as much as 15 orders of magnitude, we argue that this this leads to so-called “energetic fluid” behavior [21]. To is also not necessary. Instead, by recognizing firstly that simulate a colloidal suspension in this “energetic fluid” hydrodynamic effects are governed by a set of dimen- regime, having a range of less than 30% of a should suf- sionless numbers, and secondly that the full hierarchy fice. In this way adding attractions does not have to make of time-scales can be telescoped down to a much more the simulation much less efficient. Moreover the effects manageable range while still being properly physically of these constraints, placed by the colloid-colloid interac- separated, we demonstrate that the key Brownian and tion on ∆tMD, are to some degree mitigated by the fact hydrodynamic properties of a colloidal suspension can that it is usually the colloid-fluid interaction which sets indeed be captured by our SRD based coarse-graining this time-scale. scheme. In particular, to simulate in the colloidal regime, the There are still a few final subtleties to mention. First modeler should ensure that the dimensionless hydrody- of all, using small σ /a in the simulations may greatly cf 0 namic numbers such as the Mach, Reynolds and Knud- increase efficiency, but it also leads to a relatively larger sen numbers are small enough ( 1), and the Schmidt contribution of the Enskog friction to the total friction number is large enough ( 1). While≪ these numbers are in Eq. (41). For physical colloids in a molecular solvent constrained by approximate≫ limits, the Peclet number, the exact magnitude of the Enskog friction is still an which measures the relative strength of the convective open question, but undoubtedly for larger colloids it is transport to the diffusive transport of the colloids, can almost negligible. Some of the Enskog effects can simply be either larger or smaller than 1, depending on physi- be absorbed into an effective hydrodynamic radius a, but cal properties such as the colloidal size and the strength others must be interpreted with some care. of the external field that one wants to study. It turns Another regime where we might expect to see devia- out that if the hydrodynamic numbers above are chosen tions beyond a simple re-scaling of time-scales would be correctly, it is usually not so difficult to also satisfy the at very short times. For example, when colloids form a proper separation of the time-scales. crystal, their oscillations can be de-composed into lattice We explored how to reach the desired regime of hydro- phonons. All but the longest wave-length longitudinal dynamic numbers and time-scales by tuning the simula- oscillations are overdamped due to the viscous drag and tion parameters. The two most important parameters are backflow effects of the solvent [93]. In SRD, however, the dimensionless mean-free path λ which measures what some of the very short wavelength oscillations might be fraction of a cell size a0 an SRD particle travels between preserved due to the fact that the motion on those time- collisions performed at every time-step ∆tc, and the ratio scales is still ballistic. These may have an impact on the of the colloid radius σcf to the SRD cell size a0. The gen- interpretation of SRD simulations of microrheology. eral picture that emerges is that the collision time ∆tc must be chosen so that λ is small since this helps keep In summary, these examples demonstrate that as long the Schmidt number high, and both the Knudsen and as the coarse-grained simulation produces the correct hy- Reynolds numbers low. Of course for computational effi- drodynamic and Brownian behavior then there is consid- ciency, the collision time should not be too small either. erable freedom in assigning real physical values to the pa- Similarly although a larger colloid radius means that the rameters in the simulation. Attractive interactions may hydrodynamic fields are more accurately rendered, this introduce a new length-scale, but a judicious choice of a should be tempered by the fact that the simulation ef- model potential should still reproduce the correct phys- ficiency drops off rapidly with increasing colloid radius. ical behavior. The same simulation may therefore be We find that choosing σcf /a0 2, which may seem small, ≈ mapped onto multiple different physical systems. Con- already leads to accurate hydrodynamic properties. versely, for the same physical system, different choices A number of subtleties occur at the fluid-colloid inter- can be made for the time-scales that are mapped to phys- face, which may induce spurious depletion interactions ical values, but by design, not all time-scales can be si- between the colloids, but these can be avoided by care- multaneously resolved. ful choice of potential parameters for slip boundary con- 24 ditions. Stick boundary conditions can also be imple- either stick or slip conditions. This method is therefore mented, and we advocate using stochastic bounce-back particularly promising for simulations in the fields of bio- rules to achieve this coarse-graining over the fluid-colloid and nanofluidics, where small meso particles flow in con- interactions. strained geometries, possibly confined by soft fluctuating The simplicity of the SRD solvent facilitates the calcu- walls. We are currently exploring these possibilities. lation of the the short-time behavior of the VACF from kinetic theory. We argue that for times shorter than the sonic time tcs = a/cs, where hydrodynamic collective modes will not yet have fully developed, the decay of the Acknowledgments VACF is typically much more rapid than the prediction of the simple Langevin equation. The ensuing Enskog JTP thanks the EPSRC and IMPACT Faraday, and contribution to the total friction should be added in par- AAL thanks the Royal Society (London) for financial allel to the microscopic hydrodynamic friction, and from support. We thank W. Briels, H. L¨owen, J. Lopez Lopez, the sum an analytic expression for the effective hydrody- J. San´e, A. Mayhew Seers, I. Pagonabarraga, A. Wilber, namic radius aeff can be derived which is consistent with and A. Wysocki for very helpful conversations, and W. independent measurements of the long-ranged hydrody- den Otter for a careful reading of the manuscript. namic fields. Although we have shown how to correctly render the principal Brownian and hydrodynamic behavior of col- loids in solution there is, as always, no such thing as a APPENDIX A: THERMOSTATING OF SRD free lunch. The price to be paid is an inevitable con- UNDER FLOW sequence of compressing together the hierarchy of time and length-scales: not all the simulation parameters can When an external field is applied to the fluid (or to ob- be simultaneously mapped onto a physical system of in- jects embedded in the fluid), energy is pumped into the terest. However the cost of the “lunch” can be haggled system and, as a consequence, the average temperature down by recognizing that only a few physical parameters will rise. To prevent this from happening, the system are usually important. For example, one could map the must be coupled to a thermostat. In order not to influ- simulation onto real physical time, say through τD and ence the average flow, thermostating requires a local and tS, or alternatively through τν ; there is some freedom Galilean invariant definition of temperature. (or ambiguity) in the choice of which time scale to use. The correspondence with physical reality becomes more We achieve this by relating the instantaneous local complex when different physical time-scales mix, but in temperature in a cell to the mean square deviation of the practice they can often be disentangled when both the fluid particle velocities from the center of mass velocity coarse-grained simulation and the physical system are ex- of that cell. To minimize interference of the thermostat pressed in terms of dimensionless numbers and ratios. In with the dynamics, we choose to measure the overall tem- other words, there is no substitute for careful physical perature as: insight which is, as always, priceless. mf 2 A number of lessons can be drawn from this exercise kB Tmeas = (vi vcm,c) (A1) Nfree − that may be relevant for other coarse-graining schemes. cell c i∈c X X SRD is closely related to Lattice-Boltzmann and to the 3(Ncell c 1) (Ncell c > 1) Lowe-Anderson thermostat, and many of the conclusions Nfree = − (A2) 0 (N 1) above should transfer in an obvious way. For dissipative cell c ( cell c X ≤ particle dynamics, we argue that the physical interpreta- tion is facilitated by recognizing that, just as for SRD, the In Eq. (A2), Ncell c is the instantaneous number of fluid DPD particles should not be viewed as “clumps of fluid” particles within cell c. Three degrees of freedom must be but rather as a convenient computational tool to solve the subtracted for fixing the center of mass velocity of each underlying thermo-hydrodynamic equations. All these cell (note that this implies that the local temperature is methods must correctly resolve the time-scale hierarchy, not defined in cells containing 0 or 1 particle). During and satisfy the relevant hydrodynamic numbers, in order the simulations, the temperature Tmeas is measured every to reproduce the right underlying physics. ∆tc (this can be done very efficiently because the relative Our measurements of the VACF and the discussion velocities are readily available in the collision step rou- in Appendix B show explicitly that Brownian Dynam- tine). The thermostat then acts by rescaling all relative ics simulations, which are based on the simple Langevin velocities vi vcm,c by a factor T/Tmeas. This strict equation, do not correctly capture either the long or the enforcement− of the overall temperature may be relaxed short-time decay of colloidal VACFs. by allowing appropriate fluctuations,p but in view of the A major advantage of SRD is the relative ease with very large number of fluid particles (typically 106) this which solutes and external boundary conditions (walls) will not have a measurable effect on the dynamics of the are introduced. Boundaries may be hard or soft, with fluid particles. 25

APPENDIX B: THE LANGEVIN EQUATION performed by Hauge [95] and Hinch [96] (and others ap- AND MEMORY EFFECTS parently much earlier [97]). For ρc = ρf , i.e. a neutrally buoyant particle, these expressions can be written as Within the Brownian approximation each Cartesian component of the colloid velocity V is described by a ∞ 1 2 simple Langevin equation of the form: H 2 kBT 1 x exp [ x(t/τν )] V (t)V (0) = dx x− x2 h i 3 Mc 3π 0 1+ 3 + 9 ! dV R Z Mc = ξV + F (t) (B1) (B5) dt − where we have chosen to use an integral form as in [98]. F R(t)F R(t′) = 2k Tξδ(t t′) (B2) B − Under the assumption that the sound effects have dis- sipated away, the only hydrodynamic time-scale is the without any memory effect in the friction coefficient ξ, kinematic time τν defined in Eq. (31). The VACF must or in the random force F R(t). This Langevin equation therefore scale with t/τν , independently of colloid size. fundamentally arises from the assumption that the force- Eq. (B5) is written in a way to emphasize this scaling. kicks are Markovian: each step is independent of any pre- vious behavior. Of course on a very short time-scale this By expanding the denominator it is not hard to see is not true, but since Brownian motion is self-similar, the that at longer times the well-known algebraic tail results: Markovian assumptions underlying Eq. (B1) might be ex- pected to be accurate on longer time-scales. Inspection of Fig. 10 shows that the force-force correlation func- k T 1 τ 3/2 k T tion (50) does indeed decay on a time-scale τ which lim V (t)V (0) H B ν = B F P →∞ 3/2 t h i ≈ Mc 9√π t 12mγ (πνt) is much faster than other Brownian time-scales in the   system. (B6) From the Langevin Eq. (B1) it follows that the velocity with the same pre-factor that was found from mode- auto-correlation function (VACF) takes the form coupling theory [99]. The integral of Eq. (B5) is L kB T V (t)V (0) = exp( t/τB) (B3) h i Mc − ∞ kB T where τ = M /ξ. The VACF is directly related to the dt V (t)V (0) H = , (B7) B c h i 6πηa diffusion constant through the Green-Kubo relation: Z0

kB T Dcol = lim dt V (t)V (0) = (B4) which means that all the hydrodynamic contributions to t→∞ h i ξ the friction ξ come from V (t)V (0) H . The integral con- Z 1 2 h i and Eq. (B3) has the attractive property that its integral verges slowly, as (t/τν ) , because of the long-time tail. The VACF can easily be related to the mean-square dis- gives the correct Stokes-Einstein form for Dcol, and that its t = 0 limit gives the expected value from equipar- placement, and deviations from purely diffusive motion are still observable for t τ , and have been measured tition. It is therefore a popular pedagogical device for ≫ ν introducing Brownian motion. Notwithstanding its se- in experiments [98, 100, 101]. ductive simplicity, we will argue that this Langevin ap- Although the last equality in Eq. (B6) shows that the proach strongly underestimates the initial decay rate of VACF tail amplitude is independent of colloid size, it the VACF, and badly overestimates its long-time decay should be kept in mind that the algebraic decay does not 2 rate. set in until t τν = a /ν. Thus it is the relative (not absolute) effect≈ of the tail that is the same for all colloids. For example, at t = τν , the VACF will have decayed to 1. VACF at long times 4% of its initial value of kBT/Mc, independently of col- loid mass Mc. Even though this tail amplitude may seem At longer times, as first shown in MD simulations by small, about half the weight of the integral (B7) that de- Alder and Wainwright [94], the VACF shows an algebraic termines the friction is from t > τν . The dominance of decay that is much slower than the exponential decay of the non-Markovian hydrodynamic∼ behavior in determin- Eq. (B3). The difference is due to the breakdown of the ing the friction for colloids does not appear to hold for Markovian assumption for times t τF P . Memory ef- microscopic fluids, where, for example for LJ liquids near fects arise because the momentum that≫ a colloidal parti- the triple point, it is thought that the long-time tail only cle transfers to the fluid is conserved, with dynamics de- adds a few % to the friction coefficient [102]. However, scribed by the Navier Stokes equations, resulting in long- this is not due to the simple Langevin equation (B1) be- ranged flow patterns that affect a colloid on much longer ing more accurate, but rather to the modified short-time time-scales than Eq.(B2) suggests. Calculations taking behavior of the VACF, caused by fluid correlations (a into account the full time-dependent hydrodynamics were different non-Markovian effect). 26

1.00 that suggested by the simple Langevin equation. And even after the sound wave has decayed, V (t)V (0) H de- cays much more rapidly (faster than simpleh exponential)i than Eq. (B3) for times t < τB . SRD simulations con- firm this more rapid short∼ time decay. For example, in 0.10 Fig 13 we show a direct comparison of the measured VACF to the continuum approach, where V (t)V (0) = (t) H cs h i

VV V (t)V (0) + V (t)V (0) . For short times the decay

C hpredicted fromi soundh agreesi reasonable well, and for long CVV(t) 0.01 times the full hydrodynamic theory quantitatively fits the Enskog translation simulation data. -3/2 mode-coupling t A closer look at the short-time behavior in our SRD Full hydrodynamic VACF simulations, both for slip boundaries as in Fig. 10 and Full hydrodynamic + sound for stick-boundaries as in Fig 13 or in [58], shows that 0.00 a better fit to the short-time exponential decay of the 0.00 0.01 0.10 1.00 10.00 VACF is given by t/tν E kB T V (t)V (0) = exp( t/τE) , (B11) h i Mc − FIG. 13: (Color online) Normalized VACF for a = 2a0 and stick boundary conditions (from [58]), compared to the where the Enskog time (53) scales as: short-time Enskog result (B11), and the full hydrodynamic VACF (B5) by itself, and with the VACF (B9) from sound Mc 0.5 a added. We also show the asymptotic long-time hydrodynamic τE = 0.645tcs (B12) tail from mode-coupling theory ξE ≈ kBT/m ≈ for heavy (stick boundary)p colloids in an SRD fluid. Note 2. VACF at short times that the Enskog decay time is very close to the decay 2 time 3 tcs of the sound wave (B9). That the SRD Enskog time should scale as t is perhaps not too surprising, At short times the hydrodynamic contribution to the cs because that is the natural time-scale of the SRD fluid. VACF (B5) reduces to However, the fact that the pre-factors in the exponen- tial of Eqs (B9) and (B11) are virtually the same may H 2 kB T lim V (t)V (0) = (B8) be accidental. Although in a real liquid the increased t→0h i 3 Mc compressibility factor means τE and τcs are lower than 1 their values for an ideal gas, the two times change at because, within a continuum description, 3 of the energy is dissipated as a sound wave at velocity cs, slightly different rates so that we don’t expect the same equivalence in physical systems. cs 1 kBT In contrast to the sound wave, the integral over the V (t)V (0) = exp [ (3/2)(t/tcs)] h i 3 Mc − · Enskog VACF leads to a finite value: √3 t √3 t ∞ k T cos √3 sin dt V (t)V (0) E = B (B13) " 2 tcs ! − 2 tcs !# h i ξ Z0 E (B9) which quantitatively explains the friction we measured in our simulations, as discussed in section V. In con- where the sonic time tcs = a/cs is defined in Table III. The sound wave doesn’t contribute to the friction or dif- trast to the VACF from sound, the Enskog result cannot fusion since simply be added to the VACF from Eq. (B5) because ∞ this would violate the t = 0 limit of the VACF, set by cs equipartition. However, because the Enskog contribution dt V (t)V (0) =0. (B10) < 0 h i occurs on times t tcs, where sound and other collective Z modes of (compressible)∼ hydrodynamics are not yet fully For a more extended discussion of the role of sound on developed, it may be that Eqs. (B5) and (B9), derived hydrodynamics see e.g. ref. [89]. from continuum theory, should be modified on these short Because t τ , the sound-wave contribution to the cs ≪ ν time-scales. Since the dominant contribution to the in- VACF decays much faster than the hydrodynamic contri- tegral in Eq. (B7) is for longer times, the exact form of H bution V (t)V (0) or even than the Langevin approxi- the hydrodynamic contribution (B5) on these short times h i L 9 < mation V (t)V (0) (recall that τν = 2 τB for neutrally (t tcs) does not have much influence on the hydrody- buoyanth colloids.)i Therefore, in the continuum picture, namic∼ part of the friction. Thus the approximation of the short-time decay of the VACF is much faster than the short-time VACF by the Enskog result of Eq. (B11), 27 and the longer-time (t τE , but still t < τν )) VACF by Eq. (B5) may be a fairly≫ reasonable empirical approach. This approximation, together with the Green-Kubo rela- 0.25 tion (B4), then justifies, a-posteori, the parallel addition of frictions in Eq. (41). 0.20 Just as was found for the long-time behavior, the short- time behavior of the VACF displays important devia- 0.15

tions from the simple Langevin picture. We find that the D(t) VACF decays much faster, on the order of the sonic time

τcs rather than the Brownian time τB. From the tables 0.10 Measured D(t) V and VI one can see how large these differences can be. D(t) from full hydrodynamics For example, for colloids with a = 1µm, the short-time H −9 0.05 D (t) including sound Enskog or sonic decay times are of order 10 s, while τB is of order 10−7 s, so that the decay of the VACF will be D(t) from Langevin dominated by < V (t)V (0) >H , with its non-exponential 0.00 behavior, over almost the entire time regime. 2.00 4.00 6.00 An example of the effect of the Enskog contribution on t/tν the friction or the mean-square displacement is given in Fig. 14. The integral of the VACF, D(t) from Eq. (58), FIG. 14: (Color online) Integral D(t) of the normalized VACF which is directly related to the mean-square displace- from Fig. 13 (solid line), compared to the integral of Eq. (B5) ment [61], is extracted from the simulations described (dashed line), and including the contribution from sound as in in Fig 13, and compared to Eq. (B14) (dotted line). The simple Langevin form from the integral of Eq. (B3) (dash-dotted line), is calculated with ξ = H t 6πησcf in order to have the same asymptotic value as D (t). DH (t)= dt′ V (t′)V (0) H + V (t′)V (0) cs (B14) Because in the plot we integrate the normalized VACF, the 2 0 h i h i Z t →∞ limit of D(t) for the two theoretical approaches is 9 t0   in our units, independent of colloid size. The measured D(t) calculated using Eqs. (B5) and (B9). Both show ex- has a larger asymptotic value due to the initial effect of the 1 actly the same (t/τν ) 2 scaling for t > τν , but the simu- Enskog friction. lated D(t) is larger because the initial∼ Enskog contribu- tion means that D(t) DH (t)+ D , for t τ , with ≈ E ≫ E DE = kBT/ξE defined in Eq. (B13). Table IV compares the properties of a pure SRD fluid, The slow algebraic convergence to the final diffusion in the limit of small mean-free path λ, to the properties coefficient, or related friction, also helps explain the scal- of water. ing of the finite size effects of Eq. (43). A finite box of Table V summarizes the main scaling and values for a length L means that the measured D(t) is cut off roughly number of different properties of two different size neu- 1 trally buoyant colloids in water. To approximate the En- at t = (L2/a2)τ , and because dtD(t) (τ /t) 2 for L ν ∼ ν skog friction we used a simplified form valid for stick t > τν the effect on the diffusion constant can be writ- R boundary conditions in an ideal fluid, as in [58], which ten∼ as D D0(1 c(L/a)) which explains the scaling form of Eq.≈ (43).− Fig. 14 also demonstrates how much ignores the compressibility and other expected complex- more rapidly the D(t) from the Langevin Eq. (B3) con- ities of real water. These predictions should therefore be taken with a grain of salt. For simplicity, we have ignored verges to its final result D = kBT/ξ when compared to the simulations and hydrodynamic theories. any potential Enskog effects on the diffusion constant or time-scales. In summary, even though the simple Langevin equa- Table VI shows a very similar comparison for two dif- tion (B1), without memory, may have pedagogical merit, ferent size colloids in an SRD fluid. For simplicity we as- it does a poor job of describing the VACF for colloids in sume that there are no finite-size effects, which explains solution. small differences when compared to similar parameters used in the text of the main article. However, in contrast to Table V, we do include the Enskog contribution when APPENDIX C: COMPARISON OF SRD TO calculating the total friction and related properties. PHYSICAL PARAMETERS

At the end of the day one would like to compare a coarse-grained simulation to a real physical colloidal dis- persion. To help facilitate such comparisons, we have summarized a number of physical parameters and dimen- sionless numbers for colloids in water, and also colloids in an SRD fluid. 28

TABLE IV: Summary of the physical properties of a 3D SRD fluid in the regime of small reduced mean-free path λ. In the first column we show how these parameters scale for SRD. To highlight the scaling with the reduced mean-free pathλ and the number of particles per cell γ, we only show the collisional contribution to the viscosity in Eqs. (9) - (15), which dominates for ν small λ, and also ignore the factor fcol(γ, α). In the second column we list the parameter values for our choice of simulation parameters in this paper and in refs [52, 58], and in the last column we compare to the parameter values for H20 at standard temperature and pressure. For the SRD the units are in terms of mf , a0 and kB T , so that the time unit is t0 = a0 mf /kB T . 1 p scaling in SRD value for λ = 0.1, γ = 5, α = 2 π value for H20

mf mf −12 g ρf mass density γ 3 5 3 1 × 10 3 a0 a0 µm −3 −3 10 −3 nf number density γ a0 5 a0 3.35 × 10 µm

5kB T a0 9 µm cs speed of sound 1.29 1.48 × 10 s 3mf t0 s

λfree mean free path λfree = λa0 0.1a0 ≈ 3 A˚

γ mf mf −6 g ηf shear viscosity ≈ ≈ 2.50 1 × 10 18λ a0t0 a0t0 µm s 2 2 2 1 a0 a0 6 µm ν kinematic viscosity ≈ ≈ 0.50 1 × 10 18λ t0 t0 s 2 2 2 a0 a0 3 µm Df self diffusion constant ≈ λ ≈ 0.1 2.35 × 10 t0 t0 s ν 1 ν ν Sc Schmidt number ≈ 2 ≈ 5 ≈ 425 Df 18λ Df Df 29

TABLE V: Physical parameters, dimensionless hydrodynamic TABLE VI: Physical parameters, dimensionless hydrody- numbers and time-scales for spherical colloids of radius a = namic numbers and time-scales for a stick boundary colloid 0.01µm (10nm) and a = 1µminH20 at standard temperature in an SRD fluid. In the first column, we take the same small and pressure, moving at a velocity vS = 10µm/s. The mass λ limit as in Table IV to highlight the dominant scaling. In density ρc of the colloid is taken to be the same as that of wa- the other columns more accurate values for ν/ν0 are used. ter (the colloid is neutrally buoyant), and the hydrodynamic The hydrodynamic numbers and times were estimated with a radius a is taken to be the same as the physical radius. Where, length-scale a ≈ σcf ≈ σcc/2. The colloids are neutrally buoy- 3 for notational clarity, the units are not explicitly shown, then ant, ρc = ρf = γmf /a0, and their velocity is vS = 0.05a0/t0, length is measured in µm, time in s, and mass in g. i.e. Ma=0.039. Units as in Table IV.

Physical parameters a = 10nm a = 1µm Physical parameters σcf = 2 a0 σcf = 10 a0

4 3 m 4 4 3 −12 g −18 −12 M ≈ πγσ f ≈ 168m 2.09 × 10 m M ≈ πa × 10 3 4.19 × 10 g 4.19 × 10 g cf 3 f f 3 µm 3 a0

ξE 2 2 ξE ≈ 1.6 × 10 a ≈ 1.6 ≈ 1.6 × 10 ≈ 8.6 σcf λ ≈ 1.8 ≈ 9.0 ξS ξS −5 −7 g −5 g γσcf mf mf ξS = 6πηa ≈ 1.9 × 10 a ≈ 1.9 × 10 ≈ 1.9 × 10 ξS = 6πηa ≈ ≈ 96 ≈ 478 s s λ t0 t0 2 2 2 2 kB T 0.2 µm µm kBT λ a0 a0 Dcol ≈ ≈ ≈ 20 ≈ 0.2 Dcol ≈ ≈ ≈ 0.0165 ≈ 0.00236 6πηa a s s 6πηa γσcf t0 t0

Hydrodynamic numbers a = 10nm a = 1µm Hydrodynamic numbers σcf = 2 a0 σcf = 10 a0

vSa 2 2 Pe = ≈ 5vS a ≈ 0.005 ≈ 50 vS a γσcf Dcol Pe = ≈ 1.29Ma ≈ 6.2 ≈ 212 Dcol λ vS a −6 −7 −5 Re = ≈ 10 vS a ≈ 10 ≈ 10 vSa ν Re = ≈ 23 Ma λσcf ≈ 0.2 ≈ 1 ν λfree 0.3 −3 Kn = ≈ × 10 ≈ 0.03 ≈ 0.0003 λfree λ a a Kn = ≈ ≈ 0.05 ≈ 0.01 a σcf vS −10 −10 −10 Ma = ≈ 6.76 × 10 vS ≈ 6.8 × 10 ≈ 6.8 × 10 vS cs Ma = ≈ 0.039 ≈ 0.039 cs

Time-scales a = 10nm a = 1µm Time-scales σcf = 2 a0 σcf = 10 a0

2 2 3 a 3 −6 a γσcf 4 τD = ≈ 5a ≈ 5 × 10 s ≈ 5 s τD = ≈ ≈ 242t0 ≈ 4.2 × 10 t0 Dcol Dcol λ a τν τD a τν τD σcf tS = = = = 0.001 s = 0.1 s tS = = = ≈ ≈ 31t0 ≈ 155t0 vS Re Pe vs Re Pe 1.29Ma 2 2 a −6 2 −10 −6 a 2 τν = ≈ 10 a ≈ 10 s ≈ 10 s τν = ≈ 18 λσ ≈ 8.0t0 ≈ 200t0 ν ν cf M 2 −11 −7 M 2 τB = = τν ≈ 2.2 × 10 s ≈ 2.2 × 10 s τB = ≈ τv ≈ 1.75t0 ≈ 44t0 ξS 9 ξS 9 a −10 −12 −10 M tcs = ≈ 6.7 × 10 a ≈ 6.7 × 10 s ≈ 6.7 × 10 s 0 0 cs τE = ≈ 0.5σcf ≈ 0.98t ≈ 4.9t ξE σcf tcs = ≈ 0.78σcf ≈ 1.55t0 ≈ 7.8t0 cs 30

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