Temperature Inhomogeneities Simulated with Multiparticle-Collision Dynamics Daniel Lüsebrink and Marisol Ripoll

Temperature inhomogeneities simulated with multiparticle-collision dynamics Daniel Lüsebrink and Marisol Ripoll

Citation: J. Chem. Phys. 136, 084106 (2012); doi: 10.1063/1.3687168 View online: http://dx.doi.org/10.1063/1.3687168 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i8 Published by the American Institute of Physics.

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Temperature inhomogeneities simulated with multiparticle-collision dynamics Daniel Lüsebrink and Marisol Ripolla) Theoretical Soft-Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Jülich, 52425 Jülich, Germany (Received 28 October 2011; accepted 2 February 2012; published online 27 February 2012)

The mesoscopic simulation technique known as multiparticle collision dynamics is presented as a very appropriate method to simulate complex systems in the presence of temperature inhomogeneities. Three different methods to impose the temperature gradient are compared and characterized in the parameter landscape. Two methods include the interaction of the system with confining walls. The third method considers open boundary conditions by imposing energy fluxes. The transport of energy characterizing the thermal diffusivity is also investigated. The dependence of this transport coefficient on the method parameters and the accuracy of existing analytical theories is discussed. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.3687168]

I. INTRODUCTION not the energy, what means that they are restricted to be used in isothermal conditions. In the LB method particle densities The influence of temperature inhomogeneities is of high move in the nodes of a lattice with discretized velocities, such relevance for many systems in nature and industry. Ex- that mass and momentum fulfil local conservation laws.21 In amples range from separation techniques,1, 2 microfluidic order to include the internal energy flux several routes have applications,3, 4 or even conditions that might have facilitated been explored25–28 without one that has clearly shown to be the origin of life.5, 6 The physical phenomena related with the optimal one. Examples are to consider higher complexity the presence of temperature gradients are thermophoresis or of the lattices to increase the symmetries, to add counter-terms transport of mass, and transport of energy. Apart from vari- to cancel spurious anisotropic operators, or to account for hy- ous experimental techniques,7 and analytical theories,8, 9 the brid descriptions. These approaches have recently been suc- development of computers and simulation techniques has al- cessfully employed to investigate systems such as phase sep- ready contributed to gain deeper understanding and to in- aration of binary fluids29 or Rayleigh Taylor instabilities.30 In crease the applicability of these phenomena.10–12 standard DPD, particles positions and velocities are updated A large part of the existing simulations that consider the according to Newton equations of motion by considering con- presence of temperature gradients are performed with molec- servative, dissipative, and stochastic interactions. Mass and ular dynamics (MD) with essentially an atomistic descrip- momentum are then conserved quantities and thermal fluctu- tion of the system,13–15 or alternatively with direct simula- ations are taken into account. An energy conserving extension tion Monte Carlo method (DSMC), which is a particle-based, of the method (DPD+e) has been proposed31, 32 by consider- numerical scheme for solving the nonlinear Boltzmann equa- ing an internal energy variable for each particle, where only tion for hard spheres.16–18 Nevertheless, reproducing the prop- the heat conduction contribution needs to be specified.33–35 erties of most soft matter systems such as colloidal disper- It has been shown that DPD+e can sustain transport of en- sions or polymer solutions is a task beyond the possibilities of ergy and temperature gradients,33, 36, 37 although precise con- simulations based in these approaches. The large separation servation requires integration algorithms even more involved of the relevant length and time scales of solute and solvent than in standard DPD.38–40 Recent applications of this model requires the formulation of effective coarse-grained descrip- have been found in convection,41 multicomponet systems,42 tions of the solvent and the solute. An important requirement and nanocomposities.43 of such descriptions is the inclusion of hydrodynamic inter- A more recent approach has been proposed by Male- actions which are mediated by the solvent and are non-local vanets and Kapral44, 45 with the method multiparticle collision both in time and space. In recent years, several particle-based dynamics (MPC), also known as stochastic rotation dynam- hydrodynamic mesoscopic techniques have been proposed.19 ics. This method conserves in its basic implementation mass, Mass and momentum local conservation are necessary re- momentum, and energy, including also thermal fluctuations. quirements for hydrodynamic interactions to be correctly This simplicity presents MPC as a promising tool for the in- modeled. Two of the most well-known and extended hydro- vestigation of the effect of temperature inhomogeneities in dynamic mesoscopic simulation techniques are lattice Boltz- complex systems. In this work, we include essentially for the mann (LB) (Refs. 20, 21) and dissipative dynamics dynamics first time temperature gradients with the MPC method, care- (DPD).22–24 In the most extended versions of both these meth- fully analyzing the implementation in confinement and with ods mass and momentum are locally conserved quantities but open boundary conditions.46 We also investigate the transport of heat characterizing the thermal diffusivity coefficient as a a)Electronic mail: [email protected]. function of the MPC parameters.

II. METHOD mer solutions,50, 54 or more recently to fulfil the fluctu- ation theorem.55 In isothermal conditions, MPC has al- A. Multiparticle collision dynamics ready been applied to a large number of systems like col- MPC is a particle-based method, in which the solvent is loidal solutions,56, 57 rod-like colloids,58, 59 polyelectrolyte represented by N point particles.47, 48 Each particle i = (1,..., solutions,60 or nanoswimmers.61–63 N) has a constant mass mi, a variable position ri, and a variable velocity vi. The mass of the particles is typically the same for B. Temperature profile establishment all of them m, while positions and velocities are continuously distributed in the phase space. The MPC dynamics takes place We simulate a three-dimensional system, with periodic in two alternating steps. In the streaming step all particles bal- boundary conditions in two spatial directions, and a temper- listically propagate ignoring their neighbours during a certain ature difference imposed at the boundaries of the third direc- collision time h, tion, z. The boundary temperatures are Tc at the cold bath, and Th at the hot bath (with Tc < Th). Various possible implemen- ri (t + h) = ri (t) + hvi (t). (1) tations of the thermal baths are discussed in Sec. III. For clar- ity, we explain before some general aspects of the temperature In the collision step the simulation box is divided in cubic profile establishment. Initially the system has a uniform tem- collision boxes of size a. The interaction is performed among perature, which is generally chosen to be T = (T + T )/2. particles in the same collision box through the center of mass 0 h c After a short equilibration time, the system in contact with the velocity v that considers all particles j which are located cm, i thermal baths displays a steady temperature distribution that in the same collision box as particle i at time t, for not too large temperature differences is linear, i,t mj vj (t) T − T = j = + h c vcm,i(t) i,t . (2) T (z) Tc z, (4) j mj Lz The actual collision is defined as a rotation by an angle α with Lz the distance between the two baths. An example of around a random direction of the relative velocity of the par- the simulated temperature profile can be seen in Fig. 1, with = = = ticle to the collision box center of mass. The particle velocity Lz Ly Lx 40. The temperature is computed as the after the collision is then, average kinetic energy per particle, with averages calculated within layers parallel to the walls and therefore perpendicu- + = + R − vi (t h) vcm,i(t) (α)[vi (t) vcm,i(t)], (3) lar to the temperature gradient direction. The actual average system temperature shows small deviations from T in the with R(α) the stochastic rotation matrix. This simple colli- 0 case that the total energy is not a fixed quantity, when it will sion rule implies that each particle changes the magnitude and be time dependent T(t). In these cases, all particle velocities the direction of its velocity, such that the total mass, momen- can globally be rescaled to enforce precisely this constraint, tum, and also kinetic energy are conserved in each collision with a frequency to be determined in each application. We box before and after the collision. This ensures the presence choose as the reference value T = 1, the average tempera- of hydrodynamic interactions, together with the sustainabil- ture in absence of fluctuations, namely, T . Standard values ity of temperature gradients, and thermal fluctuations. In or- 0 chosen for the boundary temperatures are T = 0.9 and T der to preserve Galilean invariance, and to enhance collisional c h = 1.1. These values result in a relative temperature differ- transport random shift needs to be additionally considered.49 ence (T − T )/T = 0.2. Considering as a reference value the This simply means that the superimposed regular lattice that h c room temperature T = 300 K, this relative difference would defines the collision boxes has an origin that shifts in each correspond to a temperature difference of 60 ◦C. collision between 0 and a in each direction. The reference units are usually chosen to be the particle mass m, the collision box size a, and the equilibrium temper- 1.1 ature T, that are typically set to one. This corresponds to mea- ρ ρ– 2 (z)/ sure length as xˆ = x/a and time as tˆ = t kB T/ma , with kB the Boltzmann constant. The parameters that will determine the properties of the MPC are then the collision time h,the rotation angle α, and the number particle density ρ = Na3/V, 1.0 with V the volume of the simulation box. The range of parameters with large values of α and small values of h has shown to display a liquid-like behavior characterized by large values of the Schmidt number Sc = ν/D with ν the kinematic viscosity – 50, 51 T(z)/T and D the diffusion coefficient. Local angular momentum 0.9 conservation is though violated in this standard formulation. An appropriate modification of the method has already been 0 10 20 30 40 proposed,52, 53 for cases in which this is of relevance. z The MPC method has already been extensively tested, FIG. 1. Example of a temperature profile (circles) and the corresponding showing, for example, to reproduce the Navier-Stokes particle number density profile (squares). The symbols report the simulated equation,44 to include hydrodynamic interactions in poly- values and the lines correspond respectively to Eqs. (4) and (5).

Together with the temperature proﬁle, the solvent den- walls, such that the effect of virtual particles is considered.64 sity also becomes position dependent as a result of a constant In cases where there are nw interacting particles in a collision pressure p throughout the system. The MPC ﬂuid is known to box intersecting with the wall with nw <ρ, ρ − nw virtual behave with the ideal gas equation of state, pV = NkBT.The particles will be considered. These particles are chosen with a position dependent particle number density ρ(z) = N(z)a3/V momenta drawn from a Maxwell-Boltzmann distribution with

is then, zero mean velocity and variance (ρ − nw) kB T/m, such that p the values for temperature and density match those values in ρ(z) = . (5) kB T (z) the bulk. The implementation of the temperature gradient is If the temperature profile in Eq. (4) is considered, the pressure achieved by considering the walls as thermal baths and impos- can be calculated taking into account that the total number ing temperatures T and T , respectively, in the cold and hot of particles in the simulation box is constant, i.e., d3rρ(z) c h V walls. The corresponding densities ρ and ρ are then given = N, what leads to c h by Eq. (5). Therefore, the interaction with the virtual parti- k (T − T ) p = ρ B h c .(6)cles in the hot wall provides energy to the system, while the ln (Th/Tc) interaction with the virtual particles in the cold wall absorbs Although Eq. (5) corresponds to a inversely linear depen- averagely an equivalent amount of energy. The temperature dence with the position, in the limit of small temperature dif- distribution of the solvent between the walls nicely reaches a ferences, the density profile can be approximated to a linear linear profile. The temperature of the fluid close to the walls is function, though not the same as the wall temperature as can be seen in Fig. 2(a). This temperature jump is an effect related although 1 T − T L ρ(z) = ρ 1 − h c z − z (7) decoupled to the slip velocity of particles close to a wall in 67, 68 T Lz 2 the presence of solvent flow. In order to characterize the as can be seen in Fig. 1. temperature jump, we define the dimensionless quantity TJ, T (0) − T T = c . (8) J − III. BOUNDARY CONDITIONS T Tc The solvent temperatures at the walls, T(0) and T(L ) can be A. Walls with virtual particles z characterized from a linear fit. In case of perfect match T(0) Systems confined between parallel walls at homogeneous = Tc, and TJ = 0 by construction. Meanwhile, in the limiting temperature conditions have been largely simulated with the case of maximum mismatch, there would be no temperature MPC solvent.64–66 In order to obtain stick boundary condi- variation upon contact with the heat bath, this is T (0) = T tions, i.e., walls and neighbouring particles with equal veloc- and TJ = 1. ities, two modifications have been included in the algorithm. Figure 2(b) shows TJ as a function of the solvent param- First in the streaming step bounce-back is considered. This eters, that increases for decreasing α and increasing h.The means that particles reaching one of the walls revert their di- dependence of TJ with the system size is further analyzed rection and velocity. In the collision step the consideration of for a parameter set with large temperature mismatch values. random shift translates in partially filled collision boxes at the Results are displayed in Fig. 2(c) where it can be seen that

0.1 h 1 1.1 – 1.0 T(z)/T 1.0 (a) (b) T 0.9 J 0 z 20 40 0.5 1.0 TJ 0.5 (c) 0.0 0.0 0.00 0.03 0.06 0 45 90 135 180 a/Lz α

FIG. 2. (a) Temperature proﬁle with h = 0.5 and walls with virtual particles. Open circles are simulation results, solid lines are the temperatures imposed at the walls, and dashed lines the actual boundary temperatures T(0) and T(Lz). Otherwise stated the employed parameters are h = 0.1, α = 120, ρ = 5, a cubic size L = 40, Tc = 0.9, and Th = 1.1. (b) Dimensionless temperature jump TJ in Eq. (8) as a function of α (circles, down-axis), and as a function of h (squares, up-axis). (c) Dependence of TJ with the inverse of the system length Lz in the temperature gradient direction, with α = 30 and Lx = 20 = Ly.

increasing Lz noticeable decreases TJ. Eventually, it can be h extrapolated that T → 0forL →∞. On the other hand, we 0.1 1 J z 0.5 have checked that TJ does not depend on the neutral directions L , L , and no significant differences are observed by choos- x y 0.95 ing the hot layer instead of the cold in the definition of T in J T – Eq. (8). Finally, trial simulations imposing a starting configu- J T(z)/T ration with TJ = 0, show the same final stationary state with = TJ 0. 0.90 Similar effects have already been observed in simulations performed with other methods as DSMC (Ref. 17) or LB,69, 70 0510z and are known in the context of rarefied gases.67, 68 The com- mon physical characteristic in these systems is that the average particle mean free path λ varies due to the presence of 0.0 0 45 90 135 180 the wall, such that it is effectively smaller in the neighbour- α hood of a wall than in the bulk region. Considering this variation, kinetic theory predicts the existence of a layer of width FIG. 3. TJ for walls with thermostats as a function of α,andh. Symbols λ with nonlinear temperature profile,67, 68 what defines the and parameters similar to Fig. 2. The inset shows the detail of a temperature profile close to the wall for with α = 30, h = 0.1, ρ = 5, and Lz = 40. kinetic boundary layer, also known as the Knudsen layer. The overall relevance of such layer effect is then related with the system size, via the Knudsen number which is defined as plement thermostats in the boundary slabs next to the walls. Kn = λ/L being L a characteristic system size, which in our The limiting temperatures Tc and Th are now enforced in the case is the distance between walls√ Lz. The mean free path in cells neighbouring the walls by rescaling the kinetic energies MPC is usually defined as λ = h kB T/m. This is though not of such solvent particles in each simulation step. The temper- an appropriate value to quantify Kn, since a more physical ature distribution is now linear sufficiently far away from the definition should regard the increase of λ with decreasing col- walls as can be seen in the inset of Fig. 3. A shoulder-like be- lision angle,71 and decreasing density, what would account for havior appears though close to the boundaries, what implies the divergence in the limit α → 0, or ρ → 0, where no colli- the presence of a temperature jump. To quantify this behavior, sions occur. All our results in Figs. 2(b) and 2(c) could then we employ the definition of TJ in Eq. (8) with T(0) calculated be explained as direct relation between of TJ and Kn similar from a linear fit to the temperature profile by excluding the to the behavior of rarefied gases.67, 68 A small difference could first slabs at the boundaries, as shown in the inset of Fig. 3. have been expected by choosing the hot layer in the definition Similarly to the previous case, TJ is dependent on the method of TJ in Eq. (8) since the width of the Knudsen layer depends parameters, as shown in Fig. 3. As it could be expected, the on the temperature. It seems though that the differentiation is data show a similar trend, and TJ becomes more significant smaller than the statistical precision of the presented simula- for large values of h or smaller α given a certain value of tion results. the system size. This is in agreement with the dependence of In the DPD-heat conduction model33, 34 arrested particles the Knudsen number. The deviation of the wall temperature is interchange internal energy, such that in spite of the absence though much less pronounced than for the virtual particles in of particle motion, local temperature variations are possible. Fig. 2(b). It can therefore be concluded that the dependence of Simulations with this model have shown to display a temper- the temperature jump does not only depend exclusively on the ature gradient without any jump in the proximity of the walls. Knudsen number but also on the specific boundary conditions This observation also supports the explanation of the MPC that provide different energy accommodation coefficients.68 temperature jump at the walls in terms of the variation of the Knudsen number. A different system where a surface tem- 1. Comparison of wall implementations perature drop has been experimentally observed is a strongly heated spherical nanoparticle in a thermalized fluid.72 In this We can conclude that both these methods produce sat- case, the fluid has radially varying temperature around the isfactory linear profiles away from the walls, apart from the particle which displays a jump relative to the particle tem- temperature jump at the boundaries. Therefore, they can be perature. This has been justified as a manifestation of the in- employed to study the effects of temperature gradients in the terfacial thermal resistance, caused by the mismatch of the presence of confining walls. Furthermore, the parameters for thermal properties between the solid and liquid, and by the which the temperature jump are smallest, namely, small colli- strength of the interfacial bonding, although the dependence sion times h, and large α values, are the parameters for which of the Knudsen number has not been yet analyzed. the MPC algorithm has been mostly employed, since they are also the values for which the Schmidt number is larger and where MPC behaves as a liquid-like solvent.50, 51, 54 In order to compare the two methods for implement- B. Walls with thermostats ing walls at different temperatures, several issues have to be To investigate other routes that more accurately accom- considered. First, we have characterized that the temperature modate the temperature of the solvent particles close to the jump is considerably more pronounced for the walls with vir- walls, we disregard the virtual particles at the walls, and im- tual particles than for the thermostat ones. In practice, the

effective temperature gradient will be calculated with the ac- (a) 0.6 (b) cold tual boundary temperatures, and given that in both cases the 1.1 hot – profiles are nicely linear, no further consequences are ex- T(z)/T 0.4 pected unless the precise boundary behavior is of interest. 1.0 f(v) From the computational point of view, it should be noted that 0.2 virtual particles at the walls constitute an additional, but not 0.9 0 really significant effort since just three stochastic velocities 0 10 20 30 40 0 1 2 3 4 per collision box are required. This computational cost is ap- z v proximately the same as the thermalization of the layers in the FIG. 5. (a) Temperature profile obtained from the velocity exchange algo- proximity of the wall. With both techniques the boundary lay- rithm with h = 0.1, α = 120, and ρ = 10. Symbols correspond to the mea- ers have to be disregarded. In the case of the virtual particles sured temperatures, dashed-line is the estimated temperature profile from one layer will be sufficient, while for the thermostated walls it Eq. (10). (b) Velocity squared distribution in Eq. (9) for the temperatures = = will be necessary to not account for at least two layers. Con- Tc 0.9 and Th 1.1, typically used for the cold and hot baths. sequently, the walls with virtual particles will be in general better suited to study temperature gradients in the presence of hot layer. We employ the modification proposed by Müller- confining walls, although a more precise consideration should Plathe11 and since then mostly employed with MD. The idea be performed in each particular application of the methods. consists in determining the hottest particles in the cold slab and the coldest particles in the hot slab, and then exchange C. Periodic boundary conditions: Velocity their velocities, as a type of a Maxwell’s demon. This method exchange algorithm has the advantage that the total energy E and momentum P of the system are exactly conserved. The temperature profile ob- In order to study problems without confinement, the im- tained with the velocity exchange algorithm in Fig. 5(a) shows plementation of periodic boundary conditions is a convenient a nice linear profile between the cold and hot slabs. This in- route. In the presence of a temperature gradient the periodic- dicates that the velocity exchange algorithm provides a useful ity is obtained by defining a cold layer at one extreme of the tool to study the effect of temperature gradients with open simulation box and a hot layer in the center. In this way the boundary conditions in combination with the MPC solvent. simulation box is divided in two halves with increasing tem- On the other hand, a temperature jump between the boundary perature towards the center, as has been sketched in Fig. 4. layer and the bulk has been observed for some solvent pa- Similar than in the presence of hard walls, there are dif- rameters. Although now there are no walls implemented, in ferent ways to enforce the cold and hot layers to have differ- the boundary layers the average particle collision frequency ent temperatures than the bulk. One could consider the pres- is varied, and consequently the mean free path, what defines a ence of virtual particles at fixed temperatures, or thermalize corresponding Knudsen layer. For this case, we have not pre- the particles in such layers. In fact, previous simulations with cisely quantified the effect but the observed trend is similar MPC and a temperature gradient have been very briefly re- to before, namely, the temperature jump increases for small 71 ported by Pooley and Yeomans. They employ a simulation values of α and large values of h. Therefore, and also con- = box of size Lz 100, and thermostat as the cold bath the layer sidering the artificial exchange of energy between the hot = = between z 0 and z 20, and as the hot bath the layer be- and cold slabs, these are usually disregarded from the anal- = = tween z 50 and z 70. They do not report any temperature ysis, similarly as it is done with the two previous boundary jump although with the employed parameters we would ex- conditions. pect them to be present. In order to discuss how strong is the impact of such un- Here we follow a different route in which the bath tem- physical transformation, we characterize the distribution of peratures at the boundaries are not directly imposed, but a particle velocity squared f(v). In the presence of a temper- consequence of an energy flux. The original method was in- ature gradient the velocity distribution P(v) can be approx- 10 troduced by Hafskjold et al. and employed in the first sim- imated with Maxwell-Boltzmann, although strictly speaking 73–75 ulations with MD and temperature gradients. They pro- it will deviate from it.76 f(v) can then be calculated as f(v)dv posed to transfer fixed amounts of energy from the cold to the 2 = v P(v)dv, where denotes the integration over the po- lar and azimuthal angles,

t 3/2

m mv

o 2

= −

f (v) 4π v exp . (9)

H o t C C o l d l o C 2πkB T 2kB T This is the so-called Maxwell’s speed function. In Fig. 5(b), f(v) is displayed for the two temperatures typically used as cold and hot baths. It can be observed that the difference between the two distributions is not so large, such that the exchange of velocity is easily absorbed by the new distribution. Additionally, the number of particles in each layer is large FIG. 4. Illustration of the periodic simulation box in the presence of a tem- enough, such that the velocity distribution is not strongly per- perature gradient. turbed. Although the energy ﬂux is locally applied in each

exchange, the energy is distributed on average over the whole (small collision times h, and large rotation angle α).50, 51 This slab. As already discussed, for some parameters this veloc- was attributed to the breakdown of the validity of the molec- ity exchange is enough to produce a boundary temperature ular chaos approximation employed in the theory. Molecular jump, and it will also then perturb the propagation of hydro- chaos assumes the absence of particle correlations, which are dynamic interactions. The disruption is though not expected the origin of the building up of the hydrodynamic interactions. to be dramatic in the standard parameter regime since in each Similar to other transport properties the analytical expres- velocity exchange only a very small percentage of particles sion of the thermal diffusivity kT can be expressed as the sum = kin + col are affected. This artificial effect in the propagation of hydro- of a kinetic and a collision contribution kT kT kT that dynamic interactions at the exchange layers has to be consid- read ered together with other three known coexisting effects. First are the standard finite size effects, this is the effect of the peri- kB Th d kkin = − 1 odic images and more important the truncation of the hydro- T 2m 1 − cos(α) dynamic propagation spectra.35 The third effect is the particular symmetry of these non-equilibrium simulations with the 2d 7 − d 1 + − csc2(α/2) , two subsequent temperature gradients in opposite directions. ρ 5 4 Consequently, it will be of particular importance to discuss a2 1 1 the effect of the boundary layers in each application of the kcol = 1 − [1 − cos(α)]. (11) method. T h 3(d + 2)ρ ρ The velocity exchange algorithm with both MD and = = MPC, controls the magnitude of the temperature gradient by These expressions are valid for d 2ord 3 dimensions. In previous works, k has been measured in equilibrium sim- tuning the time between velocity exchanges τ ex and the num- T ulations by two types of measurements. One is performed by ber of particles nex in each exchange. To our knowledge, the determination of the temperature gradient as a function of the characterizing the dynamic structure factor, where informa- simulation parameters has only been quantified by perform- tion about thermal diffusivity and the sound attenuation is 79–82 ing the corresponding simulations. In the Appendix, we pro- obtained. Independent measurements of kT are obtained pose an argument by which the temperature gradient can be by the quantification of the correlation of the entropy den- approximated by sity. These values are in good agreement with the theoretical predictions, for the employed values of h. Alternatively, sim- 1 nex ln(cρaA) ulations in the presence of a temperature gradient have been |∇T |= kB T , (10) 2 τexκT A reported by Pooley and Yeomans71 in which measurements of

with κT the thermal conductivity, A = LxLy the cold/hot layer kT agree well with the analytical expressions for not too small 3/2 area, and the constant c = 4π(m/2πkB T ) .InFig.5(a) both values of the rotation angle α and the employed parameters. simulated and estimated temperature profiles are displayed, In this work we obtain simulation measurements of the showing that Eq. (10) gives a reasonable estimation of the thermal diffusivity in a very broad range of parameters. These temperature gradient. The deviation can be attributed to the simulations are performed outside equilibrium, in the pres- number of approximations performed in calculating Eq. (10). ence of steady temperature gradients induced with the three In practice, the values of the temperature gradient, mean tem- types of boundary conditions discussed in Sec. III. These re- perature, system size, and solvent parameters will be de- sults allow us to survey the accuracy of the analytical expres- termined by the simulation requirements, and the exchange sions where both, collisional and kinetic contributions, are value τ ex/nex will be estimated. The temperature gradient re- simultaneously taken into account, and compare the perfor- laxes in between the velocity exchanges and since a stationary mance of the different boundary conditions employed in this temperature profile is required, τ ex is chosen as small as pos- work. In the simulations, the energy exchange E between sible. It would, for example, not be reasonable to perform 100 the cold and the hot baths can be exactly quantified. E is exchanges every 100 steps instead of one exchange in every given by the difference of kinetic energies of the solvent be- step. fore and after the contact with each heat bath. The energy exchange in the hot and the cold bath are exactly the same by construction in the case of the velocity exchange periodic IV. THERMAL DIFFUSIVITY boundary conditions. When the temperature gradient is sim- In a series of papers Ihle, Tüzel, and Kroll have stud- ulated in combination with the walls, both quantities are not ied in great detail the transport properties of the MPC exactly the same, but only on average. The flux of energy jq solvent.49, 77–82 By using a discrete-time projection operator in the system is then obtained as technique, they calculate Green-Kubo relations to character- E ize the MPC transport coefficients. The shear viscosity has j = , (12) q A t been measured by them and others50, 79, 80, 83, 84 showing in all cases very good agreement between analytical theory and with A the area, and t is the time interval considered to quan- simulation results for the whole range of MPC parameters. tify E.TheFourier law states that the heat flux is propor- On the other hand, the simulation measurements of the self- tional to the negative temperature gradient, diffusion coefficient show a noticeable discrepancy for the MPC parameter region where the Schmidt number is larger jq =−κT ∇T, (13)

4 0.2 (a) (b) k k T T 0.2 Δ k 0.1 ρ = 5 T 0.3 ρ = 5 0.3 –– 3 kT 0.0 k 0.2 0.2 0.1 -0.1 T 0.1 0.2 :h ρ = 5 Δ k ρ 2 ρ = 20 0.1 T = 5 0.1 –– ρ = 20 kT 45 90α 135 180 h 0.1 0.2 0.0 1 ρ ρ = 20 = 20 (a) (b) 0 -0.1 0 45 90 135 0.0 0.5 1.0 1.5 2.0 0 45 90 135 0.0 0.5 1.0 1.5 2.0 α h α h

FIG. 7. Relative deviation of the simulated thermal diffusivity k with FIG. 6. Thermal diffusivity for two values of ρ.(a)k as a function of α T, sim T respect to the analytical approach k , k /k ≡ (k − k )/k a) for h = 0.1. (b) k as a function of h for α = 120. The insets are a zoom- T, an T T T, sim T, an T, an T as a function of α (b) as a function of h. Simulation values are those presented in for large values of α and small h, respectively. Lines correspond to the in Fig. 6 for walls and thermostats. Open symbols and dashed lines employ analytical approach in Eq. (11) and symbols to simulation results. Continuous both analytical contributions in Eq. (11), while solid symbols and solid lines lines correspond to ρ = 5 and dashes lines to ρ = 20. Triangles refer to take into account the collisional contribution in Eq. (15). simulations with walls and thermostats, circles to walls with virtual particles, and squares to the velocity exchange algorithm. calculated,

with κT the thermal conductivity, which is simply related to 2 col a 1 −ρ the thermal diffusivity kT by k = 1 + e (ln ρ − 1) T h 3(d + 2)ρ

κT = ρcpkT , (14) 1 1 1 − − + O [1 − cos(α)]. (15) 2 3 where cp is the specific heat per particle at constant pressure, ρ ρ ρ which is cp = (d + 2)kB/2 since MPC describes an ideal fluid. The simulations have been performed with 105 time steps for Figure 7 quantifies as well the relative deviation with respect equilibration and 105 time steps to obtain a good averaged to this expression. Equation (15) shows to worsen the pre- value of E. The simulation box is cubic with size L = 40 diction of Eq. (11) for ρ = 5 and not to change it appre- when using walls and doubled in the temperature gradient di- ciably for ρ = 20. The worsening could be explained since rection when employing the velocity exchange algorithm. Eq. (15) is reported in Ref. 81 to be a better approach for 85 The thermal diffusivity kT simulation results for two val- large values of ρ. Interestingly, the deviation shows to de- ues of the number density ρ are displayed in Fig. 6 together crease significantly with increasing density, and for ρ = 20 with the corresponding analytical predictions in Eq. (11),as it monotonously decreases with decreasing h. This could in- function of the rotation angle α and the collision time h.Com- dicate that the origin of the deviations is a combination of parison among the three temperature gradient implementa- two effects. One is the breakdown of the molecular chaos ap- tions is performed for the smaller number density. The agree- proximation, similar to what is already quantified for the self- ment is in general very good. Some deviations among the diffusion coefficient.51, 82 And the other effect would be the three sets of data can only be observed in the case of very large influence of the density fluctuations at smaller densities. In a collision times, where the results obtained with the velocity recent work,86 Ihle derives the transport equations for MPC exchange algorithm differ from the results obtained in the directly from the Liouville equation by means of a Chapman- presence of the walls and the analytical results. From this we Enskog expansion, where particle number fluctuations are au- could conclude that the velocity exchange algorithm with pe- tomatically part of the derivation. This is in contrast with pre- riodic boundary conditions may display some artifacts when vious approximations where the particle number fluctuations large collision times are employed. This could be related to were included by averaging over an assumed Poisson distribu- the presence of periodic boundary conditions or to the veloc- tion of the particle positions. Unfortunately, quantitative com- ity exchange algorithm. Nonetheless, in most applications the parison is not possible since only expressions in two dimen- ◦ employed collision times are much smaller, such that this will sions and with a fixed collision angle α = 90 are provided in in general not be a problem. that work. In spite of the deviations, the analytical values are The accuracy of the analytical prediction in Eq. (11) is reasonable, and the energy transport can be easily character- quantified more precisely in Fig. 7 by displaying the relative ized within the MPC solvent. deviation of the simulated thermal diffusivity kT, sim with respect to the analytical approach k . Although the trend is T, an V. DISCUSSION AND CONCLUSIONS not monotonous, it can be said that in general the prediction is better for systems with large ρ, large α, and large h.For The implementation of temperature gradients with the small values of α and h the deviations decrease, probably due MPC solvent in the presence of walls at fixed different tem- to a cancellation of errors. The analytical approach in Eq. (11) peratures results in the desired linear profiles. The tempera- shows to underestimate the simulated values up to 8% for ρ tures at the walls show a temperature jump that depend on the = 20, but up to 20% for ρ = 5. Ihle et al. already reported in Knudsen number and the boundary conditions, such that can Ref. 81 that the expressions for the heat diffusivity in Eq. (11) be minimized by the use of adequate parameters. In case of were of O(1/ρ). In that work, higher order terms were open boundary conditions, the velocity exchange algorithm is

implemented. An argument to estimate the temperature gradi- vmin corresponds to the coldest particle in the hot slab. In gen- ent in terms of the exchange and simulation parameters valid eral, we can approximate vmin 0, as can be seen in Fig. 5(b). for various simulation methods is presented and probed for vmax corresponds to the hottest particle in the cold slab. In MPC. The energy transport can be easily characterized within order to get a rough estimation of such velocity in terms of the MPC solvent, and shows to agree reasonably well with ex- the input parameters, we first approximate the temperature isting analytical values obtained by means of kinetic theory. Tc T in Eq. (9). Second, since we are in the tail of the dis- 2 Another relevant aspect in the applicability of the method tribution, we neglect the correction given by the factor vmax is the separation of the solvent characteristic times of energy in Eq. (9), such that and mass propagation. This is important when the system 2 mvmax does not only contain linearly varying temperatures, but also f (vmax ) c exp − . (A3) more general temperature inhomogeneities. This is the case of 2kB T a hot Brownian particle that carries with it a radially symmet- And third we assume that in such a cold layer there are ρaA ric temperature distribution,72, 87, 88 where a temperature drop particles, or similarly that the density is the same as the aver- at the particle surface has also been reported. For a spherical age density ρc ρ, and that there will always be at least one particle of radius σ , the characteristic thermal energy propa- particle with the temperature at the tail of the distribution, 2 gation time can be expressed as τ k ∼ σ /kT. The particle mass 2 1 diffusion time is then τ D ∼ σ /Ds, where the self-diffusion co- f (v ) . (A4) max ρaA efficient is Ds kB T/6πησ assuming the Stokes equation, stick boundary conditions, and being η the shear viscosity. In From Eqs. (A3) and (A4), we obtain our rough estimation of order to obtain an estimation of these characteristic times the vmax, transport coefficients can be safely approximated by the exist- 2k T 82 2 B ing kinetic theory studies. Parameters standardly employed vmax ln(cρaA). (A5) −2 m in applications allow us to calculate τ k/τ D ∼ 10 . This indi- cates that the important time scale separation can be properly Therefore, comparing the heat flux jq in Eqs. (12) and (13), taken into account by MPC, as already shown in the simula- and substituting Eq. (A5) in Eq. (A2) and then in Eq. (A1),we tion of a thermophoretic nanoswimmer.89 All these results put get that Eq. (10) provides the estimated temperature gradient. the MPC method forward as a promising and attractive tool to The resulting expression in Eq. 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