Non-Monotonic Mpemba Effect in Binary Molecular Suspensions

EPJ Web of Conferences 249, 09005 (2021) https://doi.org/10.1051/epjconf/202124909005 Powders and Grains 2021

Non-monotonic Mpemba effect in binary molecular suspensions

Rubén Gómez González1,∗ and Vicente Garzó2,∗∗ 1Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain 2Departamento de Física and Instituto de Computación Cientíﬁca Avanzada (ICCAEx), Universidad de Extremadura, E-06071 Badajoz, Spain

Abstract. The Mpemba effect is a phenomenon in which an initially hotter sample cools sooner. In this paper, we show the emergence of a non-monotonic Mpemba-like effect in a molecular binary mixture immersed in a viscous gas. Namely, a crossover in the temperature evolution when at least one of the samples presents non-monotonic relaxation. The influence of the bath on the dynamics of the particles is modeled via a viscous drag force plus a stochastic Langevin-like term. Each component of the mixture interchanges energy with the bath depending on the mechanical properties of its particles. This discrimination causes the coupling between the time evolution of temperature with that of the partial temperatures of each component. The non-monotonic Mpemba effect—and its inverse and mixed counterparts—stems from this coupling. In order to obtain analytical results, the velocity distribution functions of each component are approximated by considering multitempera- ture Maxwellian distributions. The theoretical results derived from the Enskog kinetic theory show an excellent agreement with direct simulation Monte Carlo (DMSC) data.

1 Introduction models employed to describe the emergence of Mpemba- like effect (and its inverse counterpart, namely when the Under certain conditions, a sample of water at a hot- two initial temperatures are below the asymptotic one) ter temperature freezes faster. This counterintuitive phe- in granular [8–11]—i.e. with inelastic collisions— and nomenon is known in literature as the Mpemba effect ac- molecular gases [12, 13]. In the former case, inelasticity cording to E. B. Mpemba [1], who was the first researcher of collisions couples the evolution of the (granular) tem- to make rigorous observations. Although several mecha- perature with that of non-hydrodynamic variables–such as nisms have been proposed to explain this effect in water the fourth cumulant or kurtosis and/or the rotational-to- [2], there are still concerns regarding the true nature of it translational temperature ratio. This coupling is the reason [3]. of the occurrence of the Mpemba effect in granular gases Leaving aside the origin of the Mpemba effect in wa- during the relaxation towards the steady state. On the other ter, it is clear that to observe this effect, the time evolution hand, no inelasticity couples the temperature with any ki- of temperature must be coupled to that of microscopic—or netic variable in the case of molecular gases, therefore kinetic—variables. For this reason, the Mpemba effect is other mechanisms must be considered. For example, in considered to be a memory effect [4]. This kind of out- a recent paper [12], the molecular gas is driven by means of-equilibrium processes have also been observed in vari- of a non-linear drag force plus a stochastic force. The non- ous systems such as clathrate hydrates [5], polymers [6], linearity in the velocity dependence of the drag force turns and colloids [7]. In these systems, a Mpemba-like effect out in the coupling between the temperature T—seen as a emerges when two samples at different initial temperatures measure of the amount of kinetic energy—with the fourth evolve in such a way that at a given time tc the relaxation cumulant a2. Nevertheless, in concordance with the granu- curves for the temperatures cross each other. Thus, freez- lar case [14], a2 is assumed to be small. As a result, initial ing is avoided and a straightforward explanation can be temperatures must be chosen to be close enough to each provided. other for the emergence of the Mpemba effect. However, there remain a lot of variables that have in- A stronger Mpemba-like effect has been recently re- fluence on the temperature evolution. As a consequence, ported in the case of a mixture of molecular gases [13]. In setting the initial conditions giving rise to a crossing of the this paper, the mixture is in contact with a thermal reser- cooling curves can result in an arduous task. For this rea- voir. As usual [15], in the context of the Enskog kinetic son, from an analytical point of view, simple models have theory, the influence of the surrounding fluid on the solid been usually considered to gain some insight into the prob- particles is modeled via an external force composed by two lem. This is particularly the case of kinetic-theory-based terms: (i) a viscous drag force proportional to the (instan- ∗e-mail: [email protected] taneous) velocity of the solid particles and (ii) a stochastic ∗∗e-mail: [email protected] Langevin-like term. While the first term tries to mimic A video is available at https://doi.org/10.48448/x39j-0504 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 249, 09005 (2021) https://doi.org/10.1051/epjconf/202124909005 Powders and Grains 2021

the loss of energy due to frictional forces through the drag all the relevant information of the system is enclosed in coefficients γi (i = 1, 2), the second term simulates the the one-particle velocity distribution functions fi(r, v, t) of energy transmission due to instantaneous and random col- species i. For moderate densities and homogeneous states, lisions with the bath. In this work, in accordance with the functions fi obey the set of coupled Enskog kinetic the results obtained in lattice-Boltzmann simulations [16], equations [19]: the drag coefficients γi are chosen for each species. This differentiation in the interaction with the background gas ∂ f X2 i + F f = J [r, v| f , f ], i = 1, 2 (1) causes the coupling between the evolution equations of ∂t i i i j i j j=1 the temperature T(t) and the partial temperatures Ti(t) of where Ji j[ fi, f j] is the Enskog collision operator and the each component. This coupling allows that, with an ap- operator Fi represents the influence of the bath on the dy- propriate choice of the initial values, the Mpemba effect namics of the particles. For low-Reynolds numbers, this (and its inverse equivalent) arises in the relaxation towards force term is usually represented by a drag force plus a equilibrium—defined in terms of a Maxwellian distribu- Fokker–Planck collision operator [20]. In this way, the tion at the background temperature Tex. Moreover, the use Enskog equation reads [21] of the partial temperatures as the control parameter per- mits more flexibility in the selection of the initial condi- ∂ f ∂ γ k T ∂2 f X2 i − γ · v f − i B ex i J f , f , tions. Namely, temperature curves can cross even when i i 2 = i j[ i j] (2) ∂t ∂v mi ∂v the initial differences are of the same order than the tem- j=1 perature themselves (large Mpemba effect). Furthermore, where γi are the drag coefficients and kB is the Boltz- a Maxwellian approximation was used to estimate the par- mann constant. Here, according to the simulation results tial production rates ξi—which measure rate of energy in- in gas-solid flows reported by Yin and Sundaresan [16], terchanged in collisions among particles of species i with the coefficients γi can be written as γi = γ Ri, where j. Thus, no cumulants are needed and the effect is easily P 0 γ = 36ηg/ ρ(σ + σ ) , ρ = mini is the total mass understood. 0 1 2 i density, ni is the number density of the component i, and The main goal of the present work is to extend the ηg is the viscosity of the solvent. Moreover, dimensionless results obtained in [13] to those situations where the functions Ri depend on the mole fraction xi = ni/(n + n ), temperature presents a non-monotonic relaxation through 1 2 mass m1/m2 and size σ1/σ2 ratios, and the partial vol- an appropiately choice of the initial conditions. The 3 ume fractions φi = (π/6)niσ , related with the total one by non-monotonic Mpemba-like effect appears when at least i φ = φ1 + φ2. More details of this kind of Langevin-like one of the two samples whose temperature curves cross models can be found in Refs. [21, 22]. presents non-monotonic relaxation. Namely, when the For homogeneous situations, all the relevant informa- | − | temperature difference T(t) Tex —of at least one of the tion of the binary mixture is provided by the partial tem- samples—increases at the early stages of the evolution peratures Ti(t) of the component i—or, equivalently, by the but, at a given time, T(t) reaches the equilibrium value temperature ratio θ(t) = T1(t)/T2(t) and the (total) temper- Tex. To fulfil this goal, we consider the Enskog kinetic ature T(t) = x1T1(t) + x2T2(t) of the mixture. The partial theory in combination with the Fokker–Planck suspen- temperatures are defined as sion model described above. Since the theoretical pre- Z dictions derived in this work are based on the use of the 1 2 Ti = dv miv fi(v). (3) Maxwellian approach, a comparison between theory and 3kBni simulations is convenient to assess the reliability of the theory. Here, we compare our theoretical results against The evolution equations for the temperature ratio θ ∗ the numerical solution of the kinetic equations by means and the (reduced) temperature T = T/Tex can be obtained of the DSMC method conveniently adapted to the suspen- by multiplying both sides of the Enskog equation (2) by 2 sion model [17, 18]. miv and integrating over velocity. They are given by [13]

2 Enskog kinetic theory Let us consider an ensemble of hard spheres of masses m1 and m2 and diameters σ1 and σ2 immersed in a viscous gas at temperature Tex. At a kinetic-theory level,

∗ ∗ ∂ x1γ θ + x2γ ∗ ∗ ∗ − ∗ 1 2 ≡ ∗ T = 2 x1γ1 + x2γ2 2T Φ, (4) ∂t 1 + x1(θ − 1) √ s ∂ 1 + x (θ − 1) 16 π T ∗ µ µ (µ + µ θ) ∗ − ∗ 1 − ∗ − ∗ 12 21 12 21 − ∗ θ = 2 γ1 γ2θ ∗ 2θ(γ1 γ2) + χ12 (1 θ)(x1θ + x2) , (5) ∂t T 3 2 1 + x1(θ − 1) √ ∗ 2 where we have introduced the scaled time t = 4(n1 + n2)/(σ1 + σ2) 4kBTex(m1 + m2)t. Here, µi j = mi/(mi + m j), χ12 is the pair correlation function, 8R 8k T γ∗ = i , and T ∗ = B ex . (6) i p ∗ 3 ex 2 2 2Tex(n1 + n2)(σ1 + σ2) (m1 + m2)(σ1 + σ2) γ0

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In order to write Eq 5, the partial production rates analysis of the latter situation. However, the determina- Z tion of some criterion on whether or not temperature evo- mi 2 ξi = − dv v Ji j[ fi, f j], i , j (7) lution behaves monotonically with the temperature ratio 3nikBTi falls outside the purposes of this work. have been calculated by approaching the distribution func- Hence, to test the emergence of the Mpemba effect we perform the derivative of Φ with respect to θ for fixed T ∗. tions fi to their Maxwellian forms: The result is !3/2 ! m m v2 ! ∗ ∗ i i ∂Φ x1 x2(γ − γ ) fi(v, t) → ni exp − . (8) = 2T ∗ 2 1 , (9) 2πkBTi(t) 2kBTi(t) 2 ∂θ T ∗ (x2 + x1θ) More details can be found in Refs. [13, 23]. ∗ ∗ which is always a positive (negative) function if γ2 > γ1 If we freely let the mixture evolve, it will asymptot- (γ∗ < γ∗). Therefore, the initial values must satisfy the ically achieve the equilibrium state where equipartition 2 1 eq eq following conditions holds, i.e. T1 = T2 = Teq = Tex. However, during the T ∗ − T ∗ time evolution towards this final state, partial temperatures 0,A 0,B ∗ ∗ > 0, γ1 > γ2, are not equal (T1(t) , T2(t)). This symmetry breaking and θ0,A − θ0,B the fact that γ1 , γ2 (because R1 , R2) causes the emer- T ∗ − T ∗ 0,A 0,B ∗ ∗ gence of the Mpemba effect in the mixture. Note that in < 0, γ1 < γ2. (10) θ0,A − θ0,B the case that γ1 = γ2, total temperature evolution decou- ples from that of the partial temperatures and the Mpemba Same conditions were obtained in the linear analysis de- effect dissapears. veloped in Ref. [13]. However, Eq. 10 does not constrain the region the initial conditions must belong to. In other words, differences between Φ(T ∗ , θ ) and Φ(T ∗ , θ ) 3 Results and discussion 0,A 0,A 0,B 0,B must be chosen to be large enough for the occurrence of In this section, we consider temperature crossings where a the Mpemba effect.

non-monotonic relaxation is present. For this reason, the non-linearity of the evolution equations 4-5 plays a crucial 1.2

(a) role. The linear counterpart has been previously analysed in Ref. [13] where a perfect agreement between exact the- ∗ 1.1 oretical expressions—for the (scaled) crossing time tc— and DSMC simulations has been found. Unfortunately, *

1.0

unlike the linear case, we are lack of simple analytical ex- T pressions in the far-from-equilibrium framework. Thus, more qualitative analysis is required to establish the necessary but no suﬃcient conditions for the ocurrence of the 0.9 Mpemba eﬀect. Let us consider two homogeneous states A and B far 0.8

away from equilibrium. The time evolution of these states (b) ∗ is fully determined by their initial temperatures T0,A and 1.04 ∗ T0,B and their temperature ratios θ0,A and θ0,B. Thus, we can establish a necessary condition—for a crossover 1.02 in the temperature relaxation—based on the selection of *

1.00

∗ ∗ T the initial slopes Φ(T0,A, θ0,A) and Φ(T0,B, θ0,B) of each state through their dependence on the initial values T ∗ and 0 0.98 θ0. Under this assumption, a necessary condition for the ∗ ∗ Mpemba eﬀect is Φ(T0,B, θ0,B) > Φ(T0,A, θ0,A), where we 0.96 have considered the state A to be initially hotter than B, ∗ ∗ namely T0,A > T0,B. 0.94 Next step is to ensure that the function Φ(T ∗, θ) exhibit 0.0 0.2 0.4 0.6 0.8 1.0 a monotonic dependence on θ, so that, with an appropriate t*

selection of the temperature ratios θ0,A and θ0,B, we can ∗ ensure the temperatures T ∗ and T ∗ will get closer or away Figure 1. Evolution of the (reduced) temperature T over the A B time t∗ for m /m = 10, σ /σ = 1, φ = 0.1, T ∗ = 1, and from each other at the early stages. Only the ﬁrst option is 1 2 1 2 ex x1 = 0.5. Solid lines represent theoretical values and symbols considered here as a simple way to attain the Mpemba ef- DSMC data. From top to bottom, panel (a) corresponds to the fect. This will be always the case for those systems that be- NMME (cooling) and to the NMIME (heating) and panel (b) to have monotonically and for those with non-monotonic re- the MME. laxation but in which temperature evolution presents only one relative extreme value. The aim of this paper is the

3 EPJ Web of Conferences 249, 09005 (2021) https://doi.org/10.1051/epjconf/202124909005 Powders and Grains 2021

Table 1. T ∗ Initial values of the (reduced) temperatures 0 and was supported by the predoctoral fellowship of the Spanish temperature ratios θ0 used to generate the relaxation curves government, Grant No. BES-2017-079725. shown in Fig. 1. References Panel (a) Panel (b) ∗ ∗ Color of lines and symbols T0 θ0 T0 θ0 [1] E.B. Mpemba, D.G. Osborne, Phys. Educ. 4, 172- NMME MME 175 (1969) Red 1.2 1.1 1.05 0.6 [2] M. Jeng, AM. J. Phys. 74, 514-522 (2006) Blue 1.1 2.1 [3] H.C. Burridge, O. Hallstadius, Proc. Royal. Soc. A NMIME — 476, 20190829 (2020) Red 0.8 0.9 [4] N.C. Keim, J.D. Paulsen, Z. Zeravcic, S. Sastry, S.R. Blue 0.9 0.4 0.95 1.8 Nagel, Rev. Mod. Phys. 91, 035002 (2019) [5] Y. Ahn, H. Kang, D. Koh, H. Lee, Korean J. Chem. To illustrate the occurrence of the non-monotonic Eng. 33, 1903-1907 (2016) Mpemba effect, we consider the following approach for [6] C. Hu, J. Li, S. Huang, H. Li, C. Luo, J. Chen, the pair correlation functions [24] S. Jiang, L. An, Cryst. Growth Des. 18, 5757-5762 (2018) 2 !2 1 3φ σiσ j M2 2φ σiσ j M2 χ , [7] A. Kumar, J. Bechhoefer, Nature 584, 64-68 (2020) i j = + 2 + 3 1 − φ (1 − φ) (σi + σ j)M3 (1 − φ) (σi + σ j)M3 [8] A. Lasanta, F.V. Reyes, A. Prados, A. Santos, Phys. (11) Rev. Lett. 119, 148001 (2017) where M = P x σ`. ` i i i [9] A. Torrente, M.A. López-Castaño, A. Lasanta, F.V. The non-monotonic Mpemba effect for a cooling, a Reyes, A. Prados, A. Santos, Phys. Rev. E 99, mixed heating and a transition are plotted in Fig. 1. More 060901(R) (2019) specifically, the mixture under consideration is composed [10] A. Biswas, V.V. Prasad, O. Raz, R. Rajesh, Phys. of equally distributed (x = 1 ) hard spheres of same di- 1 2 Rev. E 102, 012906 (2020) ameters (σ = σ ) and different masses (10m = m ). 1 2 1 2 [11] S. Takada, H. Hayakawa, A. Santos, arXiv: In addition, we consider a moderate dense system (φ = 2011.00812 (2020) 0.1). Lines are the theoretical results derived from the Enskog equation and symbols represent the DSMC data. [12] A. Santos, A. Prados, Phys. Fluids 32, 072010 (2020) ∗ ∗ [13] R. Gómez González, N. Khalil, V. Garzó, For this parameter space, γ1 = 0.445 and γ2 = 4.451 ∗ ∗ arXiv:2010.14215 (2020) and, consequently, the ratio T0,A − T0,B / θ0,A − θ0,B is chosen to be less than 0 in concordance with Eq. 10. [14] V. Garzó, Granular Gaseous Flows (Springer Nature For the Mpemba effect to occur, slopes Φ(T ∗, θ) are Switzerland, Basel, 2019) independently selected for the cooling and the heating [15] D.L. Koch, R.J. Hill, Annu. Rev. Fluid Mech. 33, cases (see table 1 for more details). Fig. 1(a) shows 619-647 (2001) the non-monotonic Mpemba effect (NMME)—and its in- [16] X. Yin, S. Sundaresan, AIChE 55, 1352-1368 (2009) verse counterpart (NMIME)—to emerge even when the [17] G.A. Bird, Molecular Gas Dynamics and the Direct initial temperature differences are around 10% (large non- Simulation of Gas Flows (Oxford University Press, monotonic Mpemba effect). On the other hand, in Fig. 1(b) Oxford, 1994) the system is initiated at two different temperatures: one [18] J.M. Montanero, V. Garzó, Granul. Matter 4, 17-24 cooler and the other hotter than Tex. Because of the (2002) non-linearity of the evolution equation of the tempera- [19] S. Chapman, T.G. Cowling, The Mathematical The- ture, a crossover is still possible. This curious effect is ory of Non-Uniform Gases (Cambridge University the so-called mixed Mpemba effect (MME) [11]. More- Press, Cambridge, 1970) over, Fig. 1 highlights an excellent agreement between the [20] T.P.C. van Noije, M.H. Ernst, Granul. Matter 1, 57- Enskog theory and the DSMC simulations. This excel- 64 (1998) lent agreement ensures the accuracy of the Maxwellian [21] R. Gómez González, N. Khalil, V. Garzó, Phys. Rev. approximation (Eq. 8) to model the velocity distribution E 101, 012904 (2020) function in these out-of-equilibrium systems. [22] H. Hayakawa, S. Takada, V. Garzó, Phys. Rev. E 96, The present work has been supported by the Spanish 069904 (2017) government through Grant No. FIS2016-76359-P and by the Junta de Extremadura (Spain) through Grant Nos. IB16013 and [23] E. Goldman, Phys. Fluids 10, 1928 (1967) GR18079, partially financed by “Fondo Europeo de Desarrollo [24] T. Boublík, J. Chem. Phys. 53, 471-472 (1970) Regional” funds. The research of Rubén Gómez González