PHYSICAL REVIEW E 83, 041504 (2011)
Phase-field model of solid-liquid phase transition with density difference and latent heat in velocity and elastic fields Kyohei Takae and Akira Onuki Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Received 3 December 2010; published 25 April 2011) We present a phase-field model of solid-liquid transitions with inhomogeneous temperature in one-component systems, including hydrodynamics and elasticity. Our model can describe plastic deformations at large elastic strains. We use it to investigate the melting of a solid domain, accounting for the latent heat effect, where there appears a velocity field in liquid and an elastic field in solid. We present simulation results in two dimensions for three cases of melting. First, a solid domain is placed on a heated wall, which melts mostly near the solid-liquid-wall contact region. Second, a solid domain is suspended in a warmer liquid under shear flow, which rotates as a whole because of elasticity and melts gradually. Cooling of the surrounding liquid is accelerated by convection. Third, a solid rod is under high compression in liquid, where slips appear from the solid-liquid interface, leading to a plastic deformation. Subsequently, melting starts in the plastically deformed areas, eventually resulting in the fracture of the rod into pieces. In these phase-transition processes, the interface temperature is kept nearly equal to the coexisting temperature Tcx (p) away from the heated wall, but this local equilibrium is not attained near the the contact region. We also examine a first-order liquid-liquid phase transition under heating from a boundary in one-component liquids. DOI: 10.1103/PhysRevE.83.041504
PACS number(s): 64.70.D−, 44.35.+c, 83.50.−v, 64.70.Ja
I. INTRODUCTION
In solid-liquid phase transitions, various nonequilibrium patterns have been observed, where the dynamics is governed by diffusion of heat and/or composition [1]. To reproduce such patterns numerically, phase-field models have been used extensively [2–6], where a space-time-dependent phase field φ(r,t) takes different values in solid and liquid, varying smoothly across diffuse interfaces. In original papers [2–4], φ represents a coarse-grained structural order and is a nonconserved variable. In a recent model, the order parameter is a density variable varying on atomic length scales [5,6], which can then reproduce polycrystal states with grain boundaries. In the phase-field approach, as a merit, any surface boundary conditions need not be imposed explicitly in simulations. While hydrodynamic flow in liquid has been neglected in most theories of crystal growth, there have been a number of observations of strong couplings between hydrodynamics and phase changes [7]. Though still insufficient, some attempts have been made to include the velocity field into theory [8–13]. For example, dendritic growth was examined in applied flow [8,11–13]. Beckermann et al. [9] presented a phase-field model for binary mixtures with hydrodynamics to examine the growth of solid domains in liquid. Anderson et al. [10] presented a general theory of solidification with hydrodynamics, where the solid was treated as an extremely viscous liquid. Regarding the origin of the hydrodynamic couplings, we note the following: (i) The density and concentration differences between the two phases spontaneously induce a velocity field at a moving solid-liquid interface. (ii) We may apply a forced flow from outside to enhance transport of mass and/or heat. (iii) The latent heat released or absorbed upon phase changes produces an inhomogeneous temperature field, leading to a complicated flow. (iv) We may heat or cool a system containing solid and liquid from a boundary 1539-3755/2011/83(4)/041504(14)
wall to induce melting or solidification in an inhomogeneous environment, also leading to a complicated flow. Thus, the hydrodynamic effects are crucial in a variety of situations during solidification or melting. Moreover, not enough attention has been paid to the elastic effects in the solid region at solid-liquid transitions. For example, we experience that ice cubes undergo solid body motions during melting on a heated wall or in stirred warm water. On the other hand, attempts to construct hydrodynamic equations for solids have been made for a long time [14–18]. Phase-field calculations including elasticity were performed on the surface instability in epitaxial film growth [19,20]. For solid-solid phase transitions, various phase-field models have been used to investigate strain-induced phase ordering [21]. In this paper, we present a phase-field model including both hydrodynamics and elasticity, for simplicity, in onecomponent systems. It should be based on well-defined thermodynamics to properly account for the latent heat, the density change, and the temperature and pressure deviations upon phase changes. In our model, we will also use a theory of nonlinear elasticity applicable for large elastic strains [22]. Our model can then describe plastic deformations and melting of highly strained solids. We also mention phase transitions in one-component liquids between two liquid phases with different microscopic structures [23–25]. In particular, in triphenyl phosphite, Tanaka’s group [25] studied the phase-ordering dynamics between two liquid phases with a large viscosity difference. To interpret their data, they introduced a nonconserved order parameter representing microscopic structural order, while the order parameter in usual liquid-liquid phase separation in binary mixtures is the composition. We should also develop a phase-field model of liquid-liquid phase transitions in onecomponent systems, where the structural order parameter is
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©2011 American Physical Society
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PHYSICAL REVIEW E 83, 041504 (2011)
coupled to the hydrodynamic variables. In this paper, we claim that our model of solid-liquid transitions should be applicable also to liquid-liquid phase transitions if the shear modulus in the model elastic energy is set equal to zero. Various mesoscopic (coarse-grained) simulation methods have also been used to investigate two-fluid hydrodynamics at gas-liquid or liquid-liquid transitions, where the interface has a finite thickness and the gradient stress tensor is a new entity in the generalized hydrodynamic equations. See a review in Ref. [26]. We note that the Ginzburg-Landau formalism is not appropriate when the temperature is inhomogeneous. In such nonequilibrium situations, we should start with an entropy functional determined by well-defined dynamic variables (the order parameter and the hydrodynamic variables). Recently, one of the present authors developed an entropy formalism for compressible fluids with inhomogeneous temperature, which is called the dynamic van der Waals model [27]. In its applications [28], it was used to investigate evaporation of a liquid droplet on a heated wall and spreading of a liquid film on a cooled or heated wall, where the latent heat effects are crucial. The organization of this paper is as follows. We will present a phase-field model for solid-liquid transitions including the hydrodynamics and the elasticity in Sec. II. The thermodynamics of solid-liquid transitions will be discussed in Sec. III. The background of our numerical analysis will be given in Sec. IV. As applications of our model, we will investigate melting of a solid domain on a heated substrate in Sec. V, melting of a solid domain in a warmer liquid under shear flow in Sec. VI, and plastic deformation and melting of a compressed solid rod in liquid in Sec. VII. Numerical results for the case of a liquid-liquid transition will also be given in Sec. V. II. PHASE-FIELD MODEL
In the entropy formalism, we discuss thermodynamics and dynamics for the phase field φ and the hydrodynamic variables in two dimensions. The bulk values of the phase field φ will be given by φ = 0 (liquid) = 1 (solid).
(2.1)
In the interface region, φ takes intermediate values between 0 and 1. In this diffuse interface description, we need not to impose the interface boundary conditions as a merit of simulation. To include the solid elasticity into our theory, we will introduce a shear modulus G(φ)(∝ φ 2 ) and strains. If we set G = 0, our model describes the dynamics at liquid-liquid phase transitions. A. Gradient entropy and elastic energy
We start with presenting the entropy formalism to treat phase transitions in inhomogeneous temperature. We first introduce an entropy density including a gradient contribution as [3,27] Sˆ = S(n,e,φ) − 12 C|∇φ|2 ,
(2.2)
where S(n,e,φ) is a function of the number density n, the internal energy density e, and φ. In this work C is a positive constant (which may depend on φ more generally). The temperature T and the chemical potential μ are defined by 1/T = ∂S/∂e and μ/T = −∂S/∂n as functions of n, e, and φ. The derivative with respect to φ is written as ∂S/∂φ = −/T . The differential form of S then reads dS = (de − μ dn − dφ)/T ,
(2.3)
which yields T , μ, and as functions of n, e, and φ. Neglecting the gradient energy density [27], we assume the total energy density in the form ρ (2.4) eT = e + |v|2 + G(φ,n)(e2 ,e3 ), 2 where ρ = mn is the mass density with m being the molecular mass, v is the velocity field, and the last term is the elastic energy density. The shear modulus G = G(φ,n) can depend on φ and n. It is zero for φ = 0 (in liquid) and is a positive constant G0 (n) for φ = 1 (in solid) for solid-liquid transitions. In our theory, e2 and e3 represent anisotropic elastic strains. For small elastic deformations in two-dimensional solids, we may introduce the displacement vector u = (ux ,uy ) and express e2 and e3 as e2 = ∇x ux − ∇y uy ,
e3 = ∇x uy + ∇y ux .
(2.5)
Hereafter ∇x = ∂/∂x and ∇y = ∂/∂y. In the isotropic linear elasticity [30], is bilinear as (2.6) = 12 e22 + e32 ). However, the linear elasticity does not hold for large strains where dislocations are formed. In our previous paper [22], we proposed to use a periodic form of with respect to e2 and e3 in the plastic regime. In our simulation, we will use the following nonlinear form, 1 [3 − cos(2π e2 ) − cos(2π e+ ) − cos(2π e− )], (2.7) 6π 2 √ where e± = ( 3e3 ± e2 )/2. This form has the hexagonal symmetry being invariant with respect to rotations of the reference frame by ±π/3. For small e2 and e3 , the linear elasticity form (2.5) follows from the Taylor expansion = e2 /2 − π 2 e4 /8 + · · ·, where e2 = e22 + e32 . In equilibrium at homogeneous T , we may introduce the Helmholtz free energy 1 (2.8) F = d r f + G + T C|∇φ|2 , 2 =
where f (n,T ,φ) = e − T S is the Helmholtz free-energy density. As a function of n, T , and φ, it satisfies df = −S dT + μ dn + dφ,
(2.9)
which follows from Eq. (2.3). Then F is a functional of n and φ at each given T . Here we introduce the generalized thermodynamic force ˆ = δF /δφ associated with φ. From Eqs. (2.8) and (2.9) we obtain ˆ = − T C ∇ 2 φ + G ,
(2.10)
where G = ∂G/∂φ is the derivative with respect to φ at fixed n. It follows the equilibrium condition ˆ = 0. However, it is
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nonvanishing out of equilibrium. We will use ˆ as defined above in the dynamic equation of φ [see Eq. (2.20) below]. In this paper, we neglect the gradient part of the energy density (∝|∇φ|2 ) and the entropic part of the elastic free energy to make the simplest theory. Their inclusion into our theoretical scheme is straightforward. It is worth noting that both the gradient entropy and the gradient energy were included in our previous paper [27]. B. Dynamic equations
In our problem the fundamental dynamic variables are φ, ρ = mn, ρv, eT , e2 , and e3 . We follow the principle of nonnegative entropy production to set up the dynamic equations for these variables. The conserved variables ρ, ρv, and eT obey ∂ρ + ∇ · (ρv) = 0, ∂t
(2.11)
↔ ∂ρv ↔ ↔ + ∇ · (ρvv+ − σ e ) = ∇· σ v , ∂t
(2.12)
↔ ∂eT ↔ ↔ + ∇ · [eT v + ( − σ e ) · v] = ∇ · [ σ v ·v + λ∇T ]. ∂t (2.13)
These equations are of the same forms as the hydrodynamic equations of one-component liquids [29]. In our case, however, the stress tensor contains additional contributions. From Eq. (2.12) the total stress tensor consists of four parts as ↔
↔
↔
↔
total = ρvv+ − σ e − σ v .
(2.14)
↔
The second part = { ij } is composed of the pressure and the gradient stress as
ij = pδij + T C(∇i φ∇j φ − 12 |∇φ|2 δij ).
(2.15)
In p we exclude the elastic contribution as p = nμ − e + ST = n
∂f − f. ∂n
(2.16)
↔
The third part σ e = {σeij } is the elastic stress tensor, ∂G δij + G ij , σeij = G − n ∂n
(2.17)
where the strain components ij are written as xx = − yy =
∂ ∂ , xy = yx = , ∂e2 ∂e3
(2.18)
↔
in two dimensions. The fourth part σ v = {σvij } is the viscous stress tensor expressed in terms of the shear viscosity η and the bulk viscosity ηB as σvij = η(∇i vj + ∇j vi ) + (ηB − η)δij ∇ · v,
(2.19)
in two dimensions. In the energy equation (2.13), λ is the thermal conductivity. In Eqs. (2.11)– (2.13), we are writing the dissipative terms on the right-hand sides. The dynamic equations of φ, e2 , and e3 are highly nontrivial. In terms of ˆ in Eq. (2.10), φ is assumed to obey ∂φ ˆ + v · ∇φ = −, ∂t
(2.20)
where is the kinetic coefficient. From Eq. (2.5) the strains e2 and e3 obey ∂e2 /∂t ∼ = ∇x vx − ∇y vy and ∂e3 /∂t ∼ = ∇x vy + ∇y vx for small deviations from an equilibrium solid state. Further adding the convection terms, but neglecting dissipative terms, we assume ∂e2 + v · ∇e2 − ∇x vx + ∇y vy = 0, (2.21) ∂t ∂e3 + v · ∇e3 − ∇x vy − ∇y vx = 0. (2.22) ∂t From Eqs. (2.3) and (2.16) the pressure p satisfies the thermodynamic differential relation, d(p/T ) = −ed(1/T ) + nd(μ/T ) − (/T )dφ.
(2.23) ↔
Including the gradient contributions (∝C), the stress tensor in Eq. (2.15) is made to satisfy ↔
1 μ = −e∇ + n∇ − ∇· T T T
− C∇ 2 φ ∇φ, T
(2.24)
where the elastic contributions (∝G) are not included. We have determined the gradient part of ij in Eq. (2.15) such that the above relation holds. ˆ Starting with Eqs. (2.2) and (2.3) we have T ∂ S/∂t = ∂e/∂t − μ ∂n/∂t − ∂φ/∂t − C∇φ · ∇(∂φ/∂t). Further using the dynamic equations so far and Eq. (2.24), we may derive the dynamic equation for Sˆ in the form [29] λ ∂ ˆ ˆ = ∇ · C∇φ ˆ S + ∇ · (Sv) + ∇T ∂t T +(˙ θ + ˙v + ˙φ )/T , (2.25) where ˙v , ˙θ , and ˙φ are the heat production rates, λ ˙θ = (∇T )2 , ˙v = σij ∇i vj , ˙φ = ˆ 2 . T ij
(2.26)
These arise from the heat conduction, the viscous dissipation, and the order parameter relaxation, respectively. In Eq. (2.25), the elastic contributions cancel to vanish except in ˙φ [see Eq. (2.10)], owing to the reversible form of Eqs. (2.21) and (2.22). Without heat input from outside, the above form ensures establishment of equilibrium where ∇i T = ∇i vj = ˆ = 0. The counterpart of Eq. (2.25) for the gas-liquid transition was derived in our previous work [27]. Finally, we examine the linear dynamics for small deviations around an equilibrium, unstrained solid (φ = 1). Here we may introduce the displacement vector u as in Eq. (2.5). In the linear elastic theory [30], the density deviation δn = n − ns = δρ/m is related to the dilation strain e1 as [31] −δn/ns ∼ = e1 ≡ ∇x ux + ∇y uy ,
(2.27)
where ns is the average solid density. If the convective terms on the left-hand sides are neglected, Eqs. (2.21) and (2.22) simply lead to the equality of the lattice velocity and the mass velocity, ∂u = v. (2.28) ∂t This is consistent with the linearized form of the continuity equation (2.11). Then Eq. (2.12) yields the equation of motion
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PHYSICAL REVIEW E 83, 041504 (2011) ↔
ρ ∂v/∂t = ∇ · ( σ L + σv ) for isotropic solids, where σv is ↔ the viscous stress (2.19) [30] and σ L = {σLij } is the linear elastic stress tensor. In terms of the shear modulus G and the isothermal compressibility Ks , it is written as σLij = G(∇i uj + ∇j ui ) + Ks−1 − G δij ∇ · u. (2.29) However, there can be a difference between the lattice velocity and mass velocity in the presence of vacancies [14–17] and/or dislocations [18].
We suppose a reference equilibrium state at T = T0 and p = p0 = pcx (T0 ), where liquid and solid coexist macroscopically and the chemical potential takes a common value μ = μ0 = μcx (T0 ). In this section, we assume no anisotropic strain (e2 = e3 = 0). The quantities in the reference liquid will be denoted with the subscript 0, while those in the reference solid with the subscript 0s. For example, the number and energy densities in the reference liquid (solid) are written as n0 and e0 (n0s and e0s ), respectively. The entropy densities in the reference two phases are written as (3.1)
where α = (φ = 0) or s (φ = 1). We treat liquid and solid states close to the reference liquid and solid. A. Model entropy
To perform simulation, we use an approximate form of the entropy density S = S(n,e,φ) valid for 0 φ 1. It needs to take a maximum as a function of n, e, and φ to ensure the existence of equilibrium. It also needs to smoothly connect the entropy S = S(n,e,0) for liquid states close to the reference liquid and the entropy Ss = S(n,e,1) for solid states close to the reference solid. In this paper, we use the following form: S=
τ2 ζ2 p0 + e − μ0 n − − C0 − W (φ), T0 2T0 K0 2
B0 e − e0 − (n − n0 ) − a1 θ (φ), T0 C0 T0 C0 ζ = n − n0 − a2 n0 θ (φ),
(3.3) (3.4)
where B0 , a1 , and a2 are constants [32]. Here W (φ) in S is a double-well function of φ with a minimum at 0 and 1, while θ (φ) in τ and ζ monotonically increases with increasing φ from 0 to 1. They satisfy W (0) = W (1) = 0, θ (0) = 0,
θ (1) = 1,
W (0) = W (1) = 0, θ (0) = θ (1) = 0.
(3.6)
θ (φ) = φ (3 − 2φ),
(3.7)
2
where A is a positive constant. Requiring τ = ζ = 0 also in the reference solid (φ = 1), we obtain a1 = [e0s − e0 − B0 (n0s − n0 )]/T0 C0 ,
(3.8)
a2 = (n0s − n0 )/n0 .
(3.9)
(3.5)
T = T0 /(1 − τ ), μ = [μ0 − B0 τ + ζ /K0 ]/(1 − τ ),
(3.10) (3.11)
= T [W (φ) − hθ (φ)],
(3.12)
in terms of τ , ζ , and φ. The quantity h in is a linear combination of τ and ζ defined as a2 n0 h = a1 C0 τ + ζ. (3.13) T 0 K0 Here = 0 for φ = 0 (liquid) and φ = 1 (solid) from Eq. (3.5). The pressure p in Eq. (2.16) becomes p τ n0 p0 + (e0 − B0 n0 ) + ζ = T T0 T0 T 0 K0 τ2 ζ2 + C0 − W + hθ. + (3.14) 2T0 K0 2 The expression for the Helmholtz free-energy density f (n,T ,φ) follows from f/T = e/T − S = nμ/T − p/T . Note that τ and ζ are field variables as well as T and μ, which exhibit no discontinuity between the coexisting two phases. This is possible owing to the last terms (∝θ ) in Eqs. (3.3) and (3.4). In liquid (solid) states close to the reference liquid (solid), the entropy density is given by S = S(n,e,0) [Ss = S(n,e,1)]. Setting φ = 0 or 1, we obtain T0 Sα = p0 + e − μ0 n − (n − n0α )2 /2K0
(3.2)
where K0 and C0 are positive constants. In the right-hand side, the first term is also rewritten as S0α + [e − e0α − μ0 (n − n0α )]/T0 from Eq. (3.1). The second and third terms are of second order for φ = 0 and 1. We introduce two variables τ and ζ by τ=
W (φ) = 12 Aφ 2 (1 − φ)2 ,
From Eq. (2.3) we express T , μ, and as
III. THERMODYNAMICS
S0α = (p0 + e0α − μ0 n0α )/T0 ,
Hereafter W (φ) = dW (φ)/dφ and θ (φ) = dθ (φ)/dφ. We adopt the following simple forms of W and θ ,
−[e − e0α − B0 (n − n0α )]2 /2T0 C0 ,
(3.15)
where α = or s. Thus S and Ss contain the terms up to second order in the deviations n − n0α and e − e0α [21,33]. Both for φ = 0 and 1, we find simple expressions, n = n0α + ζ, e = e0α + B0 ζ + T0 C0 τ,
(3.16)
where α = for φ = 0 or s for φ = 1. Also for φ = 0 and 1, we may calculate the constant-volume specific heat CV = (∂e/∂T )n (per unit volume) and the isothermal compressibility KT = (∂n/∂p)T /n. From Eqs. (3.3), (3.4), and (3.14) they are expressed in terms of the constants C0 and K0 as CV = C0 T02 /T 2 ,
KT = K0 T0 /n2 T .
(3.17)
Thus, CV is commonly equal to C0 , while KT is equal to K0 /n20α in the reference liquid (α = ) and solid (α = s). Also both in liquid and solid states, it holds the relation ∂e e + p T ∂p − B0 = = , (3.18) ∂n T n n ∂T n
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where use is made of the Maxwell relation n2 (∂σ/∂n)T = −(∂p/∂T )n , with σ = S/n being the entropy per particle. Since e − e0α − B0 (n − n0α ) ∼ = CV (T − T0 ), the variable τ represents the reduced temperature deviation [21,33].
C. Surface tension
B. Two-phase coexistence
We examine two-phase coexistence in equilibrium when the solid is unstrained (e2 = e3 = 0). For homogeneous T and μ or for homogeneous τ and ζ , the bulk equilibrium phase is obtained by minimization of f − μn = −p, where f = e − T S is the Helmholtz free-energy density. Due to the last term in the right-hand side of Eq. (3.14), the liquid phase φ = 0 is realized for h < 0 and the solid phase φ = 1 for h > 0. Thus, the parameter h represents the distance from the equilibrium coexistence curve in the τ -ζ plane. We examine the thermodynamics in equilibrium two-phase coexistence, where h = C0 a1 τ + (a2 n0 /T0 K0 )ζ = 0.
(3.19)
We consider a planar interface between an unstrained solid (e2 = e3 = 0) and a liquid varying along the z axis. From the homogeneity of τ and ζ in Eqs. (3.3) and (3.4) across the interface, the profiles of e(z) and n(z) along the interface normal are expressed as a2 ζ + (e0s − e0 )θ (z), e(z) = e0 + B0 − a1 K0 n(z) = n0 + ζ + (n0s − n0 )θ (z),
where θ (z) = θ [φ(z)]. From Eqs. (2.10) and (3.12) the interface equation for φ = φ(z) becomes T W − T Cd 2 φ/dz2 = 0, which is integrated to give
1 dφ 2 (3.29) W (φ) = C
. 2 dz Using Eq. (3.6) we find the interface profile, φ(z) = 1/(1 + ez/ξ ),
The coexistence line in the τ -ζ plane is expressed as (∂ζ /∂τ )cx = −T0 K0 C0 a1 /a2 n0 ,
(3.20)
where (∂ · · ·/∂ · · ·)cx represents the derivative in two-phase coexistence. In the pressure in Eq. (3.14), we eliminate ζ using Eq. (3.19) to obtain the coexistence pressure pcx (T ) in the form pcx − p0 = T0 (B1 τ + B2 τ 2 /2)/(1 − τ ) B2 (T − T0 )2 , = B1 (T − T0 ) + 2T where B1 and B2 are constants defined by
(3.21)
B1 = (e0 + p0 − B0 n0 )/T0 − C0 a1 /a2 , B2 = C0 + T0 K0 (C0 a1 /a2 n0 )2 .
(3.22) (3.23)
The derivative (∂p/∂T )cx along the coexistence curve is written as ∂p = B1 + B2 (τ − τ 2 /2). (3.24) ∂T cx Therefore B1 is equal to the value of (∂p/∂T )cx in the reference state. From Eq. (3.16) the differences of the density and the energy density between solid and liquid are independent of τ as n = ns − n = a2 n0 and e = es − e = B0 a2 n0 + T0 C0 a1 . We also examine the behavior of the particle volume v = 1/n and the entropy σ = S/n per particle in the two phases. The volume difference v ≡ v − vs = n/n ns per particle is written as v = a2 n0 /(n0 + ζ )(n0s + ζ ),
(3.25)
where ζ = (∂ζ /∂τ )cx τ . Some calculations yield the Clapeyron-Clausius relation between v and the entropy difference is σ = σ − σs = S /n − Ss /ns , ∂p σ = v. (3.26) ∂T cx The latent heat per particle is q = T σ . Its value in the reference state (τ = 0) is written as q0 = B1 T0 a2 /(1 + a2 ).
(3.27)
(3.28)
(3.30)
which tends to 1 (solid) as z → −∞ and to 0 (liquid) as z → ∞. The interface thickness ξ is of the form ξ = (C/A)1/2 .
(3.31)
The surface tension γ is given by γ = dz[p0 − p(z)]. From Eq. (3.14) we obtain 1 1 (3.32) γ =T dφ 2CW (φ) = T Aξ. 6 0 Thus the surface tension between unstrained solid and liquid is isotropic in this paper, while the anisotropy of the surface tension arising from the crystal structure is crucial in real crystal growth. IV. NUMERICAL METHOD A. Integrating the entropy equation
We performed two-dimensional simulations by integrating Eqs.(2.11), (2.12), (2.20), (2.21), (2.22), and (2.25) in Secs. V–VII. The system size is 800 × 400 in Secs. V and VI and 400 × 400 in Sec. VII. The simulation mesh length is x = ξ = (C/A)1/2 [see Eq. (3.31)]. In the horizontal x direction (0 < x < L), the periodic boundary condition is imposed. At the bottom y = 0 and the top y = H , the no-slip condition is imposed. In shear flow with rate γ˙ , vx is equal to the boundary velocities ±γ˙ H /2 at the bottom and top. In addition, we set ∂φ/∂z = 0 at y = 0 and H , neglecting φ-dependent surface entropy and energy. In our analysis, we integrated the entropy equation (2.25) for Sˆ in Eq. (2.2) not using the energy equation (2.13) as in our previous simulations [28]. In the right-hand side of Eq. (2.25), the heat production rates in Eq. (2.26) appear explicitly, so that the space integral of the time increment ˆ ˆ S(r,t + t) − S(r,t) is non-negative definite without applied heat flow. Notice that the total entropy is maximized at equilibrium with varying n, e, and φ in a closed system. Thus, with our entropy method, the gradients ∇T and ∇i vj and the thermodynamic force ˆ for φ tend to vanish at long
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times in the whole space (including the interface regions) without applied heat flow. This serves to give rise to smooth variations of the temperature and velocity near the interface even in nonequilibrium [28]. It is worth noting that many authors have encountered a parasitic flow around a curved interface in numerically solving the hydrodynamic equations in two-phase states with inhomogeneous temperature [34,35]. It is an artificial flow, since its magnitude depends on the discretization method. With our integration method, however, we can treat only very small systems, whose length remains on the order of a ˚ In the future, we should few hundred nanometers for ξ ∼ 3 A. use adaptive mesh techniques to treat larger samples, which have been used in some phase-field calculations [8,11,36].
where η and ηs are the liquid and solid viscosities. Let ν = η /mn0 be the liquid kinematic viscosity. We measure space and time in units of ξ and t0 = 0.4ξ 2 /ν .
In our simulation, the viscosity ratio Mv ≡ ηs /η is unity for solid-liquid transition, while it is 5 or 50 for liquid-liquid transition in Sec. V B. The simulation time mesh t is 0.01t0 for Mv = 1 and 5, while it is t = 0.0025t0 for Mv = 50. The thermal conductivity λ in Eq. (3.13) and the kinetic coefficient in Eq. (3.20) are given by λ = 1.79ν0 C0 (1 + 3φ 2 ), = 0.16/n0 kB T0 t0 ,
B. Parameter values
The parameters in the free energy are taken as follows. The coefficient A in Eq. (3.6) is given by A = 0.4n0 kB ,
(4.1)
so C = 0.4n0 kB ξ 2 from Eq. (3.31) and γ = kB T n0 ξ/15 from Eq. (3.32). The thermodynamic quantities in the reference liquid are given by
a1 = −0.27.
V. MELTING OF A SEMICIRCULAR DOMAIN ON A HEATED SUBSTRATE
(4.2)
In Eq. (3.17), C0 is the constant-volume specific heat CV and K0 is the isothermal compressibility KT multiplied by n20 in the reference liquid. Instead of specifying B0 in Eq. (3.18), we give αp0 in Eq. (4.2), which is the isobaric thermal expansion coefficient αp = −(∂n/∂T )p /n in the reference liquid. The isobaric specific heat Cp = T n(∂σ/∂T )p is equal to 9.17n0 kB = 1.1C0 from the thermodynamic identity T αp2 = (Cp − CV )KT [21] and the derivative (∂p/∂T )n = αp /KT in Eq. (3.18) is equal to 3.6n0 kB in the reference liquid. The reference entropy density, internal energy, and pressure are S0 = 7kB n0 , e0 = n0 kB T0 , and p0 = 0.01n0 kB T0 . We fix the coefficient a1 in Eq. (3.8) as in Eq. (4.2). On the other hand, the parameter a2 = n0s /n0 − 1 in Eq. (3.9) is set equal to 0.2 and −0.2. The solid is denser than the liquid for a2 > 0, while the liquid is denser than the solid for a2 < 0 as for water. Then B1 [= (∂p/∂T )cx in the reference state] in Eq. (3.22) is 14.9n0 for a2 = 0.2 and −7.7n0 for a2 = −0.2, The latent heat q0 in Eq. (3.27) is 2.5kB T0 for a2 = 0.2 and 1.9kB T0 for a2 = −0.2. For water, q0 /T0 ∼ = 2.65kB and a2 ∼ = −0.1. For solid-liquid transitions, the shear modulus is assumed to change as G = G0 n0 kB T0 φ 2 .
(4.3)
where G0 is a parameter representing the solid elasticity. We set G0 = 10 in Secs. VA and VI and G0 = 1 in Sec. VII, while we set G0 = 0 in Sec.V B for the liquid-liquid phase transition. The shear and bulk viscosities are given by η(φ) = ηB (φ) = η + (ηs − η )φ 2 ,
(4.6)
so the Prandtl number Pr (=ν Cp /λ) is 1/1.63 in liquid. The thermal conductivity of the solid is assumed to be four times larger than that of the liquid, which is the case for ice and water. Integrating the momentum equation (2.12), we further need to fix the molecular mass m, which appears in the mass density ρ = mn and the sound velocity c = [(∂p/∂ρ)σ ]1/2 . In this paper, we set mξ 2 /kB T0 t02 = 0.025. Then c = 4.15(kB T0 /m)1/2 = 26.2ξ/t0 in the reference liquid.
C0 = 8.34kB n0 , K0 = 0.064n0 /kB T0 , αp0 = 0.23/T0 ,
(4.5)
(4.4)
A. Solid-liquid phase transition
Initially, a solid semicircle with radius R = 200ξ was placed on the substrate y = 0 in liquid, where φ = 1 inside the domain, φ = 0 outside it, and τ = ζ = e2 = e3 = 0 in the whole cell. The initial density profile was n = n0 [1 + a2 θ (φ)]. The system then evolved obeying the dynamic equations, as described in Sec. IV, with the boundary temperatures at z = 0 and H being fixed at T0 . Subsequently, small relaxations followed near the interface in short times (∼t0 ). After an equilibration time of 100t0 , the bottom temperature was raised to 1.05T0 with the top temperature kept at T0 . We set t = 0 at this bottom heating. In Fig. 1, the domain shapes and the velocity field v are shown at four times for the two cases a2 = ±0.2. Hereafter the region with φ > 1/2 is treated as solid (and is shown in blue). The solid density ns is higher or lower than the liquid density n depending on the sign of a2 . As a marked feature, the melting takes place mostly near the heated wall and a velocity field is induced in the surrounding liquid (as in the case of drop evaporation on a heated substrate [28]). For a2 = 0.2 in the left-hand plates, the flow is from the solid to the liquid near the bottom. For a2 = −0.2 in the right-hand plates, on the contrary, it is from the liquid to the solid. At the last time t = 5000t0 in the two cases, the solid domain is detached from the wall and is very slowly moving as a whole, which is 0.0005ξ/t0 (upward) for a2 = 0.2 and −0.0012ξ/t0 (downward) for a2 = −0.2. Owing to the elasticity, the solid velocity is nearly uniform within the domain and is much smaller than the liquid velocity. Here, the strains e2 and e3 remain of order 10−3 and are well in the linear elasticity regime. The resultant shear stress Ge2 ∼ 10−2 n0 kB T0 is nevertheless sufficient to
041504-6
PHASE-FIELD MODEL OF SOLID-LIQUID PHASE . . .
B
PHYSICAL REVIEW E 83, 041504 (2011)
t = 500 t 0
t = 2000 t 0
t = 500 t 0
t = 2000 t 0
t = 3600 t 0
t = 5000 t 0
t = 3600 t 0
t = 5000 t 0
A
C
0.008 ξ / t 0
0.008 ξ / t 0
solid: a2 = 0.2
solid: a2 = - 0.2
FIG. 1. (Color online) Solid domain on a heated wall and velocity field (arrows) for a2 = 0.2 (left-hand four plates) and for a2 = −0.2 (right-hand four plates) in the region 1/2 < x/L < 7/8 and 0 < y/H < 3/4 at t/t0 = 500, 2000, 3600, and 5000. Arrows below the panels correspond to a velocity of 0.008ξ/t0 . Melting mostly takes place close to the heated wall, where the velocity is from solid to liquid for a2 = 0.2 (left-hand side) and from liquid to solid for a2 = −0.2 (right-hand side).
realize the solid body motion. At the interface, it is balanced with the viscous stress η D in the surrounding liquid, where D ∼ 10−3 t0−1 is the typical velocity gradient on the liquid side. Using the mass conservation at the interface, we may express the local melting flux J as J = ns (vs − vint ) = n (v − vint ),
(5.1)
where ns (∼ = n0s ) and n (∼ = n0 ) are the solid and liquid densities, vs and v are the solid and liquid velocities in the normal direction (from solid to liquid) near the interface, and vint is the interface velocity. In the present case, we may neglect vs compared to v . Then v and vint are related as v = (1 − ns /n )vint ∼ = −a2 vint ,
(5.2)
in terms of a2 in Eq. (3.9). Since vint < 0 during melting, Eq. (5.2) explains the direction of the liquid velocity near the bottom in Fig. 1. In Fig. 2, we plot v , vint , and J = n (v − vint ) for a2 = ±0.2, to confirm Eq. (5.2). It shows that melting occurs close to the bottom. We also notice that |vint | is considerably larger than |v |, which is obviously due to the small size of a2 . The particle number in the solid region is defined as Ns (t) = d r nφ, (5.3) which decreases in time during melting. In the left-hand panel of Fig. 3, we show the melted particle number Ns (0) − Ns (t) for a2 = ±0.2. We can see that the melting is considerably slowed down upon detachment of the solid domain from the wall.
The right-hand panel of Fig. 3 displays the average temperature T and the temperature T at three points A, B, and C for a2 = 0.2 and at point B for a2 = −0.2. As marked in the left-hand bottom panel of Fig. 1, (x/L,y/H ) = (0.5,0.2), (0.5,0.7), and (0.8,0.2), at points A, B, and C, respectively. Point A is at the middle of the solid domain, where the temperature increases up to a maximum about 1.025T0 for t 2.5 × 103 t0 but decreases afterward due to cooling by latent heat. The heat conduction into the solid is weakened as the constricted part of the solid is narrowed. At point B in the liquid above the solid domain, we show the temperature for
a2 = 0.2
0.04
a2 = -0.2
0.04
J 0.02
v
0
0
vint
-0.02 -0.04
J
0.02
v
0
-0.02
0.1
0.2
y /H
0.3
-0.04
vint 0
0.1
0.2
0.3
y /H
FIG. 2. (Color online) Liquid velocity v , interface velocity vint , and melting flux J = n (v − vint ) in the normal direction along the interface in units of ξ/t0 or n0 ξ/t0 at t = 2000t0 , where a2 = 0.2 (left-hand side) and a2 = −0.2 (right-hand side) for a solid-liquid transition. Melting occurs mostly in the constricted part close to the heated wall at y = 0.
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KYOHEI TAKAE AND AKIRA ONUKI
PHYSICAL REVIEW E 83, 041504 (2011)
solid: a2 = 0.2
1.03
10-3(N(0) - N(t)) 3
a2 = 0.2
T / T0
4 1.02
a2 = 0.2
T / T0
t = 2000 t 0
A C
2 1.01
a2 = -0.2
< T > / T0
1
B 1
0
0
1
2
3
10-3 t / t0
4
0
1
2
3
10-3 t / t
4
5
0.8
0.7
0.6
x /L
the two cases a2 = ±0.2. There, the deviation δT = T − T0 arises from the pressure change δp = p − p0 adiabatically (at constant entropy σ per particle) as (5.4)
where (∂T /∂p)σ ∼ 0.03/kB n0 . Thus δT at point B is positive for a2 = 0.2 and negative for a2 = −0.2 (see Fig. 5 below for the sign of δp). Note that the thermal diffusion length (DT t)1/2 ∼ (t/t0 )1/2 ξ is still shorter than the cell height H = 400ξ in our simulation time, where DT = λ/Cp (∼ = 6.5ν0 ) is the thermal diffusivity. For example, at t = 2000t0 , it is of order 70ξ . In the upper plates of Fig. 4, the temperature profiles (viewed from above) are displayed for a2 = 0.2 at t/t0 = 2 × 103 and 5 × 103 before and after the detachment of the domain from the heated wall at y = 0. As discussed below in Eq. (5.4), the temperature change is adiabatic and small in the liquid region above the solid domain (around point B in Fig. 1). At these two times, the temperature exhibits a steep gradient near the wall, while it is nearly flat along the interface far from the wall. After the detachment, the temperature gradient is mostly supported by a narrow liquid layer between the domain and the wall. Namely, the interface is divided into a constricted part close to the wall with a large temperature gradient and a body part far from the wall with a nearly homogeneous temperature. Away from the wall, the local equilibrium should hold and the interface temperature should be close to the melting temperature Tcx (p) at a nearly homogeneous pressure p = p(t), as in the simulation of an evaporating droplet on a heated wall [28]. Note also that the Laplace pressure difference γ /R outside and inside the domain is very small in our simulation, since Eq. (3.32) gives γ /Rn0 kB T0 = 0.07ξ/R,
1.04
1.02
1.02
1
1
0
0 0.4
(5.5)
which is of order 3 × 10−4 for R/ξ ∼ 200. Recall that we have introduced the parameter h in Eq. (3.13), which represents the distance from the coexistence
0.8
0.8
0
FIG. 3. (Color online) Left-hand side: Melted particle number Ns (0) − Ns (t) multiplied by 10−3 for a2 = ±0.2. Arrows indicate the time at which the solid domain is detached from the wall. Right-hand side: T (r,t)/T0 at three points A, B, and C for a2 = 0.2 and at point B for a2 = −0.2 (lowest curve), as marked in the left-hand bottom panel in Fig. 1. Shown also is the space average T (t) divided by T0 for a2 = 0.2.
δT ∼ = (∂T /∂p)σ δp,
1.04
0.4
B (a2 = - 0.2)
5
T / T0
t = 5000 t 0
2 10 h / n0 kB
0.8
0.5
0. 4
x /L
0. 6
3 10 h / n0 kB
t = 2000t 0
0. 8 0
0.6
0.5
x /L
4 0 -4 -8 0. 6 0. 4 0. 2
4 0 -4 -8 -12 0. 2
0.7
0. 2
y /H
0. 4
x /L
y /H
t = 5000t 0
0. 4 0. 6
0. 8 0
0. 2
0. 6
y /H
FIG. 4. (Color online) Top: Temperature in the region 0.5 < x/L < 0.8 and 0 < y/H < 1 around a solid domain (in blue) in liquid (in red), where a2 = 0.2. See the corresponding domain shapes in Fig. 1. At t = 2000t0 (left-hand side), heat from the wall is being used to melt the solid attached to the wall. At t = 5000t0 (right-hand side), the solid domain (in blue) is detached from the wall and its interface temperature becomes a constant. Bottom: Field variable h in Eq. (3.13) along the interface using the same data, which is −1 multiplied by 100n−1 0 at t = 2000t0 (left-hand side) and by 1000n0 at t = 5000t0 (right-hand side). While h < 0 in the constricted part (in red), h ∼ = 0 in the body part (in blue).
curve. In the lower plates of Fig. 4, we plot h along the interface viewing it from below. This quantity is continuous across the interface and is well defined on the interface. In the constricted part at t/t0 = 2000, h is slightly negative down to −0.1n0 kB as y → 0 and its gradient along the interface is ∼−10% of that of τ . In the body part at t/t0 = 2000 and 5000, h is nearly zero and is at most of order 10−3 n0 kB . We also examine the pressure. Right after the heating of the lower wall at t = 0, the liquid next to the wall expands to emit a high-pressure sound pulse [37,38]. It travels throughout the cell on the acoustic time scale H /c ∼ 15t0 to damp after several traversals due to the viscosities in Eq. (2.19), where c is the sound velocity. On longer time scales, the pressure should gradually increase for a2 > 0 and decrease for a2 < 0, as the melting proceeds at fixed cell volume. In Fig. 5, we illustrate these behaviors by plotting the average pressure on the top plate pH (t) defined by pH (t) =
L
dx p(x,H,t)/L.
(5.6)
0
The curves for a2 = ±0.2 begin to separate for t > ∼ 40t0 , while there is almost no difference in the very early stage t < 20t0 . ∼ For a2 = 0.2, pH (t) increases in time on the average on both short and long times. For a2 = −0.2, it is nearly constant on
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PHASE-FIELD MODEL OF SOLID-LIQUID PHASE . . .
PHYSICAL REVIEW E 83, 041504 (2011)
(pH- p0) / n0 kBT0 0.012
t = 2000 t 0
t = 5000 t 0
0.06
0.009
0.04
a2=0.2
a2=0.2
0.006
0.02
0.003
0
a2=-0.2 a2=-0.2
0
-0.02
-0.003 0
100
t / t0
200
0.02 ξ / t 0
-0.04 0
1000
fluid: Mv = 5
a2 = 0.2
2000
t / t0
t = 2000 t 0
t = 5000 t 0
FIG. 5. (Color online) Pressure change pH (t) − p0 on the upper plate y = H divided by n0 kB T0 . On a short time scale (0 < t/t0 < 200) (left-hand side) the pressure change occurs due to traversals of acoustic pulses, while on a long time scale (0 < t/t0 < 2000) it is caused by a decrease in the solid region (melting). The curves are very different for a2 = 0.2 and −0.2.
the average in the time interval 40t0 < ∼t < ∼180t0 and decreases afterwards. Thermoacoustic effects caused by heated boundaries have been studied mostly in one-phase states [37,38]. In two-phase situations, a volume expansion occurs also near the interface due to the first-order phase transition. In the present melting case, we estimate it per each particle from the solid to the liquid. Obviously, the liquid near the interface expands by −1 ∼ v = n−1 − ns = a2 /n0s due to the solid-liquid density difference, but it also shrinks by (∂v/∂σ )p σ since it is cooled by the latent heat absorption by the particle [see Eqs. (3.25) and (3.26)]. Using the Maxwell relation (∂v/∂σ )p = (∂T /∂p)σ and the Clapeyron-Clausius relation (3.26), the net volume change per particle is given by the difference ∂T (v)melt = v − σ ∂p σ (5.7) = v(1 − αL ), where αL = (∂T /∂p)σ (∂p/∂T )cx in the second line [39]. For the parameter values adopted, αL is equal to 0.37 for a2 = 0.2 and to −0.19 for a2 = −0.2. The curve of a2 = −0.2 in the time interval 40 < ∼t < ∼ 180 in Fig. 5 is produced by a balance between sound waves from the expansion in the liquid next to the heated wall and those from the shrinkage in the liquid next to the interface. B. Liquid-liquid phase transition in one-component liquids
We are interested in how the melting behavior is altered without elasticity but with latent heat [25]. Thus, by setting G = 0, we also performed simulation in the same geometry with a2 = 0.2. The other parameters were unchanged. Then the system phase separates into a viscous liquid phase I and a more viscous liquid phase II. The phase field φ is zero in phase I and is unity in phase II. The entropy per particle σ is higher and the density n is lower in phase I than in phase II. We shall see a significant velocity field around a droplet of phase II heated from below even for very large viscosity ratio Mv = ηs /η . In Eq. (2.19), η is the shear viscosity of phase I and ηs is that of phase II.
0.008 ξ / t 0 fluid: Mv = 50
a2 = 0.2
FIG. 6. (Color online) Droplet shapes and velocity field at liquidliquid transition with G = 0 on a heated wall at t/t0 = 2000 and 5000, where 1/2 < x/L < 7/8 and 0 < y/H < 3/4. The viscosity ratio Mv is 5 (top) and 50 (bottom). The projected part is moving upward. The typical velocity field is of order 0.02ξ/t0 for Mv = 5 and 0.008ξ/t0 for Mv = 50.
As in the previous section, we initially placed a semicircular droplet of phase II with radius R = 200ξ in a liquid of phase I on a substrate. Figure 6 displays the droplet profiles (where φ > 1/2) and the velocity field v at t/t0 = 2000 and 5000. The upper and lower plates correspond to Mv = 5 and 50, respectively. The droplet is considerably elongated vertically for Mv = 5 and at t/t0 = 5000 (top right-hand side), while the other droplet shapes are not much different from those in Fig. 1. The arrows below the left-hand-side plates indicate the typical velocity vc of order 0.02ξ/t0 for Mv = 5 and of order 0.008ξ/t0 for Mv = 50. A circulating velocity field is produced around the projected part of the droplet, which moves slowly as a whole even for Mv = 50. The velocity gradient is of order vc /R within the droplet, while it is much larger in the surrounding liquid. In this manner, the tangential component of the viscous stress tensor can be continuous across the interface even for Mv = 50. In Fig. 7, we show the velocities in the normal direction along the interface at t = 2000t0 as in Fig. 2. That is, vI and vII are those in phase I and II, vint is the interface velocity, and J is the flux from phase II to phase I. As in Eq. (5.1) the mass balance yields J = nII (vII − vint ) = nI (vI − vint ),
(5.8)
where nI and nII are the densities in the two phases. For Mv = 5, the three velocities vI , vII , and vint nearly coincide for y/H > 0.15 and J is significant only in the region ∼ y/H < ∼ 0.1. For Mv = 50, vI and vII are considerably smaller and J is even more localized near the bottom. Very close to the bottom (y/H < 0.03), vI and vII are appreciably different ∼ from each other and their magnitudes are much smaller than
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KYOHEI TAKAE AND AKIRA ONUKI
PHYSICAL REVIEW E 83, 041504 (2011)
0.06
fluid: Mv =5 0.04
J 0.02 0
v
v
vint
-0.02 -0.04
0
0.1
0.2
0.3
0.4
0.5
y /H 0.06
fluid: Mv =50 0.04
J
0.02
v 0
v vint
-0.02 -0.04
0
0.1
0.2
0.3
0.4
temperature profiles at the corresponding times, demonstrating a considerable cooling of the liquid particularly in the regions where the melted liquid has been convected. The temperature in the solid is almost homogeneous, while it exhibits a large gradient in the surrounding liquid. In fact, the temperature at the solid center is lower than the interface temperature (∼ =T0 here) by 0.026T0 , 0.005T0 , and 0.005T0 at γ˙ t = 0.2, 2, and 4, respectively. In Fig. 9, we plot the solid particle number Ns (t) defined in Eq. (5.3) in the left-hand side and the temperature at three points A, B, and C and its space average in the right-hand side, where (x/L,y/H ) = (0.25,0.8) at point A, (0.75,0.8) at point B, and (0.5,0.5) at point C. We can see that the temperature at the solid center C approaches the interface temperature (∼ =T0 ) on the time scale of the thermal diffusion time (∼R 2 /DT ∼ 2000t0 ) and that the temperature at point B is lower than that at point A due to the convective transport of the melted particles. In our situation, the Reynolds number Re = R 2 γ˙ /ν and the Peclet number Pe = R 2 γ˙ /Dth are ∼5 and 3, respectively, in the early stage, where R = 100ξ is the domain radius and Dth ∼ = 0.7ξ 2 /t0 is the thermal diffisivity. In experiments using large particles or domains, these numbers are mostly much larger than unity. For Pe 1, the thermal diffusion length remains of order R/Pe1/2 in the directions perpendicular to the flow (in the direction making an angle of 3π/4 with respect to the x axis here) [21,40].
0.5
VII. A SOLID ROD UNDER HIGH COMPRESSION IN LIQUID: PLASTIC DEFORMATION, MELTING, AND FRACTURE
y /H FIG. 7. (Color online) Velocities vI and vII in two phases I and II, interface velocity vint , and flux J = n (v − vint ) from phase II to phase I in the normal direction along the interface in units of ξ/t0 or n0 ξ/t0 , where t = 2000t0 and a2 = 0.2. Here Mv = 5 (top) and 50 (bottom).
|vint |. We confirm that Eq. (5.8) holds excellently for these two cases. VI. MELTING OF A SOLID DOMAIN IN A SHEARED WARMER LIQUID
As the second example, on a 800 × 400 lattice, we initially placed a 240×240 solid square in liquid with a2 = 0.2. The temperature was 0.95T0 in the solid and 1.1T0 in the liquid. For t > 0, the top and bottom boundaries were kept insulating; that is, we assumed ∂T /∂y = 0 at y = 0 and L. The periodic boundary condition was imposed along the x axis. The top and bottom boundaries were moved with velocities ±γ˙ H /2 with γ˙ = 2 × 10−4 t0−1 . For this shear rate, the viscous heating in the liquid is negligible during the simulation (t < 4/γ˙ ). Nevertheless, the applied shear flow could accelerate melting of the solid and cooling of the liquid. The upper plates of Fig. 8 display the solid shapes at γ˙ t = 0.2, 2, and 4 from the left-hand side in the middle region of the cell 1/4 < x/L < 3/4 and 0 < y/H < 1, where the solid is melting and rotating as a whole. The strains e2 and e3 in the solid are of order 10−4 at the solid center and of order 10−3 near the interface. The lower plates of Fig. 8 give the
We have so far treated weakly strained solids, where the linear elasticity relations (2.5) and (2.6) hold well [though we have used the nonlinear form (2.7) in our simulation]. However, dislocations and slips appear at high applied strain. As demonstrated in our previous papers [22], our nonlinear elasticity model in Eq. (2.7) can describe such defect proliferation. Hence, as the third example, we followed the time evolution of a uniaxially compressed solid rod in liquid with a2 = 0.2 on a 400 × 400 cubic lattice, so L = H = 400ξ here. The rod with a 100 × 400 rectangular shape was under compression, e2 = 0.2,
e3 = 0,
(7.1)
at t = 0. The shear modulus was taken to be G = n0 kB T0 φ 2 or G0 = 1 in Eq. (4.3). Then initial elastic energy density was ∼0.02n0 kB T0 in the solid. The initial temperature was T = T0 . The top and bottom boundaries were kept insulating or ∂T /∂z = 0. In the upper four panels in Fig. 10, we illustrate the inception of slip formation at t/t0 = 580, 620, 660, and 700 taking place in the lower right-hand part of the rod. Arrows indicate the displacement vector u(r,t) at point r and at time t defined as [41] t u(r,t) = dt v(r,t ), (7.2) 0
so this quantity can be calculated even in liquid. Here we neglect the difference between the lattice velocity and the mass
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PHASE-FIELD MODEL OF SOLID-LIQUID PHASE . . .
PHYSICAL REVIEW E 83, 041504 (2011)
t = 20000 t 0
t = 10000 t 0
t = 1000 t 0
A
B
C
t = 1000 t 0
t = 10000 t 0
t = 20000 t 0
T / T0
T / T0
T / T0
1.1
1.1 1.05 1 0.95
1.1
1.05
1.05
1
1
1
1
1
0.8 0.8
0.8
0.6 0.6
y /H
0.4 0.2 0
0.3
0.4
0.5
0.6
y /H
0.7
0.4 0.2 0.3
0
0.4
0.5
x /L
x /L
0.6
0.7
0.6
y /H
0.4 0.2 0
0.3
0.4
0.5
0.6
0.7
x /L
FIG. 8. (Color online) Top: A square-shaped solid domain in warmer liquid under shear flow with γ˙ = 2 × 10−4 t0−1 at γ˙ t = 0.2,2, and 4 from the left-hand side. It gradually melts and rotates as a solid body. Melted particles are convected to cool the surrounding liquid. Bottom: Corresponding temperature profiles, which is lowest in the directions making angles of π/4 and 5π/4 with respect to the x axis.
velocity, as discussed around Eqs. (2.28) and (2.29). We can see that an edge dislocation appears from the solid-liquid interface and glides into the solid in the direction of 3π/4 with respect to the x axis. The maximum displacement |u| on the slip is 1.0ξ , 1.4ξ , 1.6ξ , and 2.9ξ at these four times. In the lower three plates of Fig. 10, we show subsequent solid shapes at t/t0 = 800, 1200, and 4000. At t/t0 = 800, the plastic deformation has almost ended and the maximum slip displacement is ∼10ξ , but the solid and liquid are still moving with the maximum of the velocity being 0.2ξ/t0 at the slip lines. Remarkably, melting begins to take place around the slip areas (plastically deformed regions) at (b) t/t0 = 1200, eventually resulting in 1.10
6
10-4 Ns
A
1.08
< T > / T0
1.06
5
T / T0
1.04 4
eel = G(φ)(e2 ,e3 ),
B
(7.3)
1.02 1.00
3
C
0.98 0.96
2
fracture into five unstrained solid domains at (c) t/t0 = 4000. In Fig. 11, we enlarge the lower part of the rod in the snapshot (a) at t/t0 = 800 and display the velocity field v in the left-hand panel and the plastic displacement u in the right-hand panel by arrows. In Fig. 12, we give the solid particle number Ns (t) in Eq. (5.3) on the left-hand side and the pressure deviation p − p0 on the right-hand side. These curves indicate that the slip areas start to melt at t ∼ 1000t0 . After this time, the melting rate −dNs (t)/dt increases suddenly and the pressure deviation is roughly proportional to the melted particle number Ns (0) − Ns (t). The latter increases in time for a2 = 0.2 as in Fig. 5. In the upper plate of Fig. 13, we show the time evolution of the elastic energy density,
0
0.5
1
10
-4 t
1.5
/ t0
2
0.94
0
0.5
1
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2
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FIG. 9. (Color online) Left-hand side: Solid particle number Ns (t) vs time. Right-hand side: Temperature T divided by T0 at three points A, B, and C and space average T /T0 (see the top of the right-hand panel in Fig. 8 for their locations).
at two points A and B (see Fig. 10 for A and B) together with the space average eel . Upon a plastic deformation, eel decreases to zero at point A far from the slips, but it increases to high values with large fluctuations due to defects at point B in the slip area. In the inset, the average kinetic energy density eK = mnv 2 /2 increases during plastic deformation, while the average interface energy density eint = C|∇φ|2 /2 remains constant in time. Point A is in the bulk solid far from the slips, so the elastic energy density there decreases to zero abruptly for t > 800t0 . At point B, it increases after the plastic deformation ∼ but decreases on melting.
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5.0
- p0 n0 kBT0
-4
10 Ns 4.8 0.10 4.6 4.4
t=580t0
t=620t0
t=660t0
0.05
t=700t0 4.2 4.0
0.00 0
1
2
3
4
0
1
2
3
4
-3 10 t / t 0
-3
10 t / t 0
FIG. 12. (Color online) Time evolution of the melted particle number Ns (t) multiplied by 10−4 (left-hand side) and the average pressure deviation p − p0 divided by n0 kB T0 (right-hand side). Melting in the plastically deformed areas begins at approximately t = 1000t0 . The pressure increases with melting for a2 = 0.2. A
A
B
B
B
(a) t=800 t 0
(b) t=1200 t 0
(c) t=4000 t 0
FIG. 10. (Color online) Top: Inception of slip formation from a solid-liquid interface with initial strain e2 = 0.2 at t/t0 = 580, 620, 660, and 700 in the right-hand lower part of the rod 0.60 < x/L < 0.64 and 0.01 < y/L < 0.05, where a slip is growing. Arrows are displacements from t = 0 in Eq. (7.1). Bottom: Subsequent overall rod profiles in the region 0.25 < x/L < 0.75 and 0 < y/L < 1. Multiple slips are created at t/t0 = 800 in (a), while melting in plastically deformed regions begins at t/t0 = 1200 in (b), resulting in fracture at t/t0 = 4000 in (c). Arrows represent the velocity field v. Its maximum is 0.2ξ/t0 in (a), while the system is almost at rest in (b) and (c).
upon the plastic deformation but are cooled with melting due to latent heat. The temperature at point A slightly increases due to the pressure change adiabatically (as point B in Fig. 3). Figure 14 gives the profiles of the elastic energy density eel in Eq. (7.3) in the upper plates and the temperature profiles in the lower plates in the lower part of the rod at three times (a) t/t0 = 800, (b) 1200, and (c) 4000. In (a), eel increases in the slip regions and vanishes far from them, while T also increases in the slip areas but is nearly unchanged far from them. In (b) and (c), melting takes place in the slip regions, Energy densities /10-2 n0 kBT0
A
In the lower plate of Fig. 13, the time evolution of the temperature is plotted at points A and B together with the space average T . The slip regions including point B are heated t=800 t 0 Velocity
v
Displacement
eel
0.06
< ek >
4 0.03
3
B
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0 700
800
900
1000
t/t0
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< eel >
A
0 0
u
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(a) (b) 10
(b)
(a)
5
2 -3
3
4
t / t0
1.03
T / T0
1.02 1.01
A
1.00
< T > / T0
B
0.99 0.98 0.3
0.4
0.2 ξ / t 0
0.5
x/L
0.6
0.7
0.4
10 ξ
0.5
0
0.6
1
-3
2
3
4
10 t / t 0
x/L
FIG. 11. (Color online) Expanded illustrations of plastic deformation at t = 800t0 before melting in the lower part of the left-hand bottom panel (a) in Fig. 10. Left-hand side: Velocity field v in the region 0.25 < x/L < 0.75 and 0 < y/L < 0.5. Its maximum is 0.2ξ/t0 . Right-hand side: Displacement vector u in the region 0.35 < x/L < 0.65 and 0 < y/L < 0.3 (more expanded). Its maximum is 10ξ . The maximum of |e2 | is 8.0 and that of |e3 | is 3.0.
FIG. 13. (Color online) Top: Time evolution of elastic energy density eel = G at two points A and B and average eel . See the lower plates of Fig. 10 for A and B. The inset gives the average kinetic energy density eK and average interface energy density eint . These energy densities are measured in units of 10−2 n0 kB T0 . Bottom: Time evolution of T /T0 at two points A and B and average T /T0 .
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102 eel / n0 kBT0 (a)
(b) 0.3
4 3 2 1 0
0.2
(c)
4 3 2 1 0
0.3 0.2
0.1
0.2
y/H
0.1
0.4
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y/H
y/H 0.4
4 3 2 1 0
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x/L
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(b)
(c)
1.02 1.01 1.00 0.99
1.01 1.00 0.99 0.98
1.01 1.00 0.99 0.98
0.3
0.3 0. 2
0. 2 0.4
0.5
x/L
0.1
y/H
0.6
0.4
0.5
x/L
0.1 0.6
0.3 0. 2
y /H 0.4
0.5
x/L
0.1
y/H
0. 6
FIG. 14. (Color online) Time evolution of elastic energy density eel in units of 10−2 n0 kB T0 (top) and T /T0 (bottom), where (a) t/t0 = 800 before melting, (b) 1200 during melting, and (c) 4000 after fracture. Shown is the lower part of the rod in the region 0.3 < x/L < 0.7 and 0 < y/H < 0.3 (see the lower plates in Fig. 10). Here φ > 1/2 in the solid region (in blue) and φ < 1/2 in the liquid region (in red).
leading to an decrease of eel and a latent-heat cooling in the slip areas. VIII. SUMMARY AND REMARKS
We have presented a phase-field model of solid-liquid transitions in the entropy formalism including hydrodynamics and nonlinear elasticity. Integrating the dynamic equations in this model, we have presented three numerical examples of melting in two dimensions. They demonstrate utility of our model in describing melting phenomena in complicated situations, though we can treat only very small systems at present. We summarize our main results: (i) In Sec. II, the total entropy density Sˆ in Eq. (2.2) depends on the phase field φ, the number density n, and the internal energy density e, and contains a gradient part, while the total energy density consists of e, the kinetic energy eK , and the elastic energy eel = G(φ)(e2 ,e3 ). The elasticity is introduced using strain fields e2 and e3 representing anisotropic elastic deformations. Using the principle of non-negative entropy production, we have constructed dynamic equations for these variables. They describe phase-transition dynamics accounting for the hydrodynamic and elastic effects. If the shear modulus G vanishes, we obtain the dynamic equations applicable to liquid-liquid phase transitions in one-component liquids. (ii) In Sec. III, we have proposed a model entropy density S in Eq. (3.2), which contains first- and second-order deviations of n and e and assumes a maximum as a function of n, e, and φ in equilibrium. Here we treat thermodynamic states close to a reference two-phase state. (iii) After explaining our simulation method in Sec. IV, we have presented our simulation results in Secs. V–VII. In
Sec. V, melting of a solid domain on a heated wall has been examined in Figs. 1–5. The heat supplied from the wall is absorbed into the latent heat of a thin liquid layer growing between the solid domain and the wall. Also simulation in the same geometry has been performed at a liquid-liquid transition in one-component liquids in Figs. 6 and 7, where the velocity field is appreciable in a projected part of the domain even if the viscosity of the domain is much larger than that of the surrounding liquid. In these examples, the melting is highly localized on the interface very close to the heated wall. Away from the wall, the interface temperature is nearly equal to the coexisting temperature Tcx (p) as in Fig. 4. (iv) In Sec. VI, a solid domain melts to cool the surrounding liquid. This process has been accelerated by applied shear flow. Small elastic strains (∼10−3 here) of the domain enable its solid body rotation in shear flow. As in Figs. 8 and 9, the temperature is almost homogeneous within the domain, while it exhibits a large gradient in the surrounding liquid. (v) In Sec. VII, we have simulated a plastic deformation of a highly compressed rod and its subsequent time evolution. In such plastically deformed slip areas, defects are proliferated, leading to transient increases in the temperature and the elastic energy. Afterward, melting and cooling (by latent heat) take place in these areas. This defect-induced melting eventually results in fracture of the solid, releasing the elastic energy stored. We note that premelting around defects (including grain boundaries) has been studied using a more microscopic phase-field model [42]. Molecular-dynamics simulations of premelting around defects should be highly informative. Finally, we make some remarks. There can be a number of problems to be studied in our scheme such as dendrite formation in shear flow and/or heat flow, spinodal decomposition and nucleation, epitaxial growth, and recrystallization.
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In particular, it is of great interest to investigate kinetics with large strains or even with dislocations. The relevance of dislocations is well recognized in the growth of epitaxial films. In liquid-liquid phase transitions also, we should further investigate the effects of latent heat and shear flow in phase ordering.
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ACKNOWLEDGMENTS
This work was supported by KAKENHI (Grant-in-Aid for Scientific Research) on Priority Area Soft Matter Physics from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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