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Spinodal Decomposition in Binary Gases

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Spinodal Decomposition in Binary Gases VOLUME 78, NUMBER 18 PHYSICAL REVIEW LETTERS 5MAY 1997 Spinodal Decomposition in Binary Gases S. Bastea and J. L. Lebowitz Department of Physics and Mathematics, Rutgers University, Piscataway, New Jersey 08855-0849 (Received 10 December 1996) We carried out three-dimensional simulations, with about 1.4 3 106 particles, of phase segregation in a low density binary fluid mixture, described mesoscopically by energy and momentum conserving Boltzmann-Vlasov equations. Using a combination of direct simulation Monte Carlo for the short range collisions and a version of particle-in-cell evolution for the smooth long range interaction, we found dynamical scaling after the ratio of the interface thickness to the domain size is less than ,0.1. The a 1 scaling length Rstd grows at late times like t , with a ­ 1 for critical quenches and a ­ 3 for off- critical ones. We also measured the variation of temperature, total particle density, and hydrodynamic velocity during the segregation process. [S0031-9007(97)03106-2] PACS numbers: 64.75.+g, 47.70.Nd, 68.10.–m The process of phase segregation through which a sys- coexistence region they will describe gas-gas segregation tem evolves towards equilibrium following a temperature [14] into two phases, one rich in particles of type 1 and quench from a homogeneous phase into a two phase re- the other rich in particles of type 2. (Examples of gas gion of its phase diagram has been of continuing interest mixtures that have a miscibility gap are helium-hydrogen, during the last decades [1], but many problems still re- helium-nitrogen, neon-xenon, etc. [14].) We believe main to be solved. This is particularly so for fluids,when that the model contains the essential features of phase particle, momentum, and energy densities are conserved separation in general binary fluid mixtures. locally; these are currently the focus of both numerical To simulate our system we modeled the Boltzmann studies [2–7] and microgravity experiments [8,9]. collisional part using a stochastic algorithm due to Bird In this Letter we present computer simulations of [15], known as direct simulation Monte Carlo (DSMC), spinodal decomposition in a three-dimensional mixture while for the Vlasov part we used the particle-to-grid- of two kinds of particles that we label 1 and 2 using a weighting method, well known in plasma physics [16]. novel microscopic dynamics and computational scheme. In the DSMC method the physical space is divided into The particles interact with each other through short range cells containing typically tens of particles. The main interactions modeled here by hard spheres having the ingredients of this procedure are the alternation of free same mass m and diameter d. Particles of different kinds flow over a time interval Dt and representative collisions interact also through a long range repulsive Kac potential, 3 among pairs of particles sharing the same cell. In the Vsrd ­ g Usgrd. The equilibrium properties of such particle-to-grid-weighting algorithm the particle densities a system are well understood; there is even a rigorous are computed on a spatial grid through some weighting proof of a phase transition at low temperatures to an depending on the particle position; then the Vlasov forces immiscible state [10], which in the limit g ! 0 [11] is are calculated on the same grid. Finally, the forces at the described by the mean-field theory. When the density n position of each particle are interpolated from the forces is low enough, nd3 ø 1, and the potential sufficiently on the grid. The coupling of these methods, which have long ranged, ng23 ¿ 1, the free energy of the system been extensively used individually, made possible our is well approximated by F k T n r$ ln n r$ 1 ­ B f 1s d 1s d simulations of phase segregation with 1.4 3 106 particles, n r$ ln n r$ dr 1 V jr$ 2 r$ j n r$ n r$ dr$ dr$ and 2s d 2s dg s 1 2 d 1s 1Rd 2s 2d 1 2 with only modest computational resources: a typical run the g ! 0 critical temperature, which should be an upper g R 0 1 took about 32 CPU hours on a 233 MHz Alpha Station. bound for Tc at g.0, is given by kBTc ­ 2 n Usrd dr$. It appears that this method can be extended to the study In this regime the dynamical evolution of the system R of the effects of phase segregation on inhomogeneous should be well described by two coupled Boltzmann- hydrodynamical flows of practical importance [17]. Vlasov equations: Since one of our main interests was the late time ≠fi ≠fi F$i ≠fi 1 y?$ 1 ? ­Jffi,f1 1f2g, (1) hydrodynamical regime, a delicate balance had to be struck ≠t ≠r$ m ≠y$ between the size of the system, the range of the potential, i ­ 1, 2, where fisr$, y$ , td are the one-particle distribu- the temperature, and the particle density, making sure that 0 0 0 tion functions, F$ isr$, td ­ 2= Vsjr$ 2 r$ jdnjsr$ d dr$ , each of the methods is used within its range of validity and n r$0 f r$0, y, t dy with i, j 1, 2, i fi j, and that their combination remains computer manageable. On js d ­ js $ d $ R ­ J f, g is the Boltzmann collision operator for hard core the one hand the potential must be reasonably long ranged f g R interactions [12]. Kinetic equations of this type have been so that the Vlasov description is physically appropriate proposed in [13], and if the system is quenched inside the and numerically sound, and on the other hand it must 0031-9007y97y78(18)y3499(4)$10.00 © 1997 The American Physical Society 3499 VOLUME 78, NUMBER 18 PHYSICAL REVIEW LETTERS 5MAY 1997 have a range much smaller than the size of the system. We first present results for critical quenches (equal This restriction made necessary the use of two spatial volume fractions of the two phases). Three different po- grids: a somewhat coarse one for the collisions and a finer tential ranges were used, and for each one of them 10–12 grid for the long range potential. It also imposed the quenches were performed. The domain size was probed use of quadratic spline interpolation for the calculation of using the pair correlation function, Csr$, td ­ V 21 3 grid quantities and a ten-point difference scheme for the k x$ fsx$, tdfsx$ 1 r$, tdl, where f ­ sn1 2 n2dysn1 1 calculation of the forces [16]. n d is the local order parameter, n and n are the local P2 1 2 Our results were obtained using a system with 1 382 400 particle densities, V is the volume, and the average particles in a cube with periodic boundary conditions. We is over the different runs. We determined Csr$, td by also studied smaller systems to identify unavoidable finite- an inverse Fourier transform of the structure function size effects. The interaction potential used was Gaussian, Ssk$, td ­ jfÄ sk$, tdj2; the order parameter fsr$, td and its 3 2 2x2 Ä $ Usxd ­ ap 2 e , a.0, but there is no reason to be- Fourier transform fsk, td were computed on a 64 3 lieve that different repulsive potentials would qualitatively 64 3 64 cubic grid. The first zero of the spherically change the results. All quenches were performed at a to- averaged correlation function, Csr, td, was used as a tal particle density nd3 . 0.01 and an initial temperature measure of the typical domain size, Rstd. The data are 0 averages of the independent runs. T0, T0yTc ­ 0.5. The initial conditions for each run were random positions for all particles and velocities distributed Assuming the existence of a single characteristic length according to a Maxwellian of constant temperature. (In the scale, the dynamical scaling prediction [1] for the late DSMC evolution, as in the Boltzmann-Vlasov equations, time spherically averaged correlation function is Csr, td. the hard cores only enter in determining the collision cross CfryRstdg. In Fig. 1 the correlation function for potential 21 sections.) The total energy of the system was very well range g ­ 0.4 is plotted starting at t ­ 160, showing conserved by the dynamics. This meant that the kinetic that the system is well within the scaling regime. energy and hence the temperature increased as the system Simple dimensional analysis of the hydrodynamical segregated, but at late times it changed very slowly on the evolution equations in the limit of large domain sizes, time scale of our simulations. We indicate the final tem- appropriate for late times, yields a linear growth law for the domain sizes [19], Rstd ~ ssyhdt, when R is below perature T in Figs. 1–4. The effective number of particles 2 1 2 2 in the range of the potential was about 100–500. Rh ­ h yrs and a t 3 law, Rstd ~ ssyrd 3 t 3 , for R above In the following we compare results of our simula- Rh [20,21]; s is the surface tension coefficient, h is the tions with available theoretical and experimental work and shear viscosity, and r is the density. We measured s check various assumptions made in the former, e.g., the ne- using Laplace’s law and h by studying the decay of a glect of density and temperature variations. We are also sinusoidal velocity profile. The time evolution of Rstd in currently investigating both formal and rigorous Chapman- our simulations is at late times (see Fig.
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