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Journal of Thermodynamics
Volume 2011 (2011), Article ID 940385, 9 pages
Research Article

Equilibrium Conditions at a Solid-Solid Interface

Los Alamos National Laboratory, Theoretical Division, MS-B221, Los Alamos, NM 87545, USA

Received 22 July 2010; Revised 3 December 2010; Accepted 5 January 2011

Academic Editor: Mohammad Al-Nimr

Copyright © 2011 JeeYeon N. Plohr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We derive the thermodynamic conditions necessary for two elastoplastic solid phases to coexist in equilibrium. Beyond temperature, velocity, and traction continuity, these conditions require continuity of a generalization of the specific Gibbs free energy. We express this quantity in the Eulerian frame as well as the Lagrangian frame. We also show that two approaches in deriving the equilibrium conditions, one on the continuum level and the other on the atomistic scale, yield the same results. Finally, we discuss two possible interpretations for the Gibbs free energy, which lead to distinct generalizations, except in the case of inviscid fluids, where they coincide.

1. Introduction

Our goal is to derive the conditions that characterize the thermodynamic equilibrium of two coexisting solid phases for an elastoplastic material. These conditions answer the question: what are the proper boundary conditions at an interface between two solids in equilibrium? This question arises in many applications, such as those involving a transformation from one solid phase to another.

For inviscid fluids, the well-known conditions due to Gibbs provide a complete answer to the corresponding question. For solids, however, only approximate conditions are in widespread use. These approximate conditions are obtained by “extending” the conditions for fluids [1, 2]. Such extensions are not correct because the behavior of a solid is more complex than that of a fluid, being that the stress in a solid tensor, not simply a scalar.

In the present work, we follow a different approach. We examine the underlying principles that hold at a solid-solid interface, and we reformulate two different approaches to deriving the necessary conditions in the Lagrangian frame. The first approach [3] considers a solid to be a continuum; the second approach [4] views solids on atomistic scale. In reformulating each approach, we avoid assuming that either the transformation strain or the elastic strain is infinitesimal. However, the thickness of the interfacial region is neglected, and surface energy is not considered.

In addition to deriving them for the Lagrangian frame, we derive the equilibrium conditions for the Eulerian frame. In particular, we arrive at a transparent formula for the proper generalization of the Gibbs free energy. Deriving the equilibrium conditions on the continuum level in the Eulerian frame is a prerequisite for comparing the two Lagrangian approaches. We show that these two approaches produce identical thermodynamic equilibrium conditions.

One issue that we highlight is how best to define the Gibbs free energy for solids. We review how the Gibbs free energy for fluids is defined in order to guide the choice of definition for solids. Rather surprisingly, we find that two possible definitions of Gibbs free energy, one as a quantity that must be continuous across a material interface in equilibrium and the other as the chemical potential, yield different formulae for solids.

At the conclusion, we will have a set of necessary conditions for a solid-solid interface to be in equilibrium. In some applications, however, these conditions do not provide a set of boundary conditions that is complete, that is, fully determines the evolution of the interface. We assert that this situation is not a failure for the necessary conditions, but rather is inherent for solid-solid interfaces and that material-specific interface conditions (e.g., a constitutive relation) are needed as supplemental boundary conditions.

Remark 1. We employ notation that is standard in continuum mechanics instead of that used in [3, 4]. In rewriting the derivations from [4], we use specific quantities (i.e., specific volume, specific energy, etc.) instead of molar densities.

2. Solid Interface Conditions Derived from the Rankine-Hugoniot Jump Conditions [3]

The continuum-level thermodynamic equilibrium conditions that hold at an interface between two solids can be derived from the Rankine-Hugoniot jump conditions. These conditions constrain the two states on either side of a discontinuity (which can represent a shock wave or a material interface) in conformance with conservation of mass, momentum, and energy.

2.1. Inviscid Fluid Dynamics

As motivation, we first review the Rankine-Hugoniot jump conditions for inviscid fluid dynamics. Here, 𝜌 is the mass density, 𝑢𝑖 is the particle velocity, 𝜀 is the specific internal energy, 𝑠 is the specific entropy, 𝑝 is the pressure, 𝑣=𝜌1 is the specific volume, 𝑞𝑖 is the specific heat flux, 𝑇 is the temperature, 𝑠𝑛 is the interface propagation speed, 𝑚 is the mass flux through the interface, and 𝑛𝑖 is the unit vector normal to the interface, pointing from the side labeled − to the side labeled +. Also, for a state quantity 𝐴, define Δ𝐴=𝐴+𝐴 and 𝐴=1/2(𝐴+𝐴+). The well-known Rankine-Hugoniot jump conditions corresponding to mass, momentum, and energy conservation are [57]𝜌𝑛𝑖𝑢𝑖𝑠𝑛=𝜌+𝑛𝑖𝑢𝑖+𝑠𝑛𝑚𝑛=𝑚,(1)𝑚Δ𝑢+𝑛Δ𝑝=0,(2)Δ𝜀+𝑝Δ𝑣+Δ𝑖𝑞𝑖=0.(3) We also have the entropy inequality that𝑛𝑚Δ𝑠+Δ𝑖𝑞𝑖𝑇0.(4) Notice that the mass jump condition implies that𝑛𝑚Δ𝑣=Δ𝑖𝑢𝑖.(5)

Remark 2. We allow for heat flow across the interface so that the temperature can become continuous. When the temperature is uniform, the heat flux is zero.

A necessary condition for thermal equilibrium at the interface is that the temperature is continuous. Subtracting the entropy jump condition, multiplied by the common value of temperature, from the energy jump condition gives𝑚Δ(𝜀𝑇𝑠)+𝑝Δ𝑣0.(6) Since𝑛𝑝Δ𝑣=Δ(𝑝𝑣)𝑣Δ𝑝=Δ(𝑝𝑣)+𝑚𝑣Δ𝑖𝑢𝑖=Δ(𝑝𝑣)+𝑚21𝑣Δ𝑣=Δ(𝑝𝑣)+2𝑚2Δ𝑣2,(7) this inequality is equivalent to1𝑚Δ𝜀𝑇𝑠+𝑝𝑣+2𝑚2𝑣20.(8) In thermodynamic equilibrium, where 𝑚0 and no entropy is generated at the interface (i.e., the inequality becomes equality),Δ[]𝜀𝑇𝑠+𝑝𝑣=0.(9) We recognize the quantity in brackets as the specific Gibbs free energy 𝑔=𝜀𝑇𝑠+𝑝𝑣. The thermodynamic equilibrium conditions are thereforeΔ𝑛Δ𝑇=0,(10)𝑖𝑢𝑖=0,(11)Δ𝑝=0,(12)Δ𝑔=0,(13) where (11) and (12) derive from (5) and (2), respectively, in the limit as 𝑚0.

2.2. Lagrangian Elasticity

We turn now to elasticity. In the notation, we employ for Lagrangian continuum mechanics, 𝐹𝑖𝛼 is the deformation gradient, 𝜌0 is the reference mass density (𝜌0=𝜌𝐽, where 𝐽=det𝐹), 𝑉𝑖 is the particle velocity, 𝑃𝑖𝛽 is the first Piola-Kirchhoff stress tensor, 𝑄𝛾 is the specific heat flux, 𝑆𝑁 is the interface propagation speed, and 𝑁𝛼 is the vector normal to the interface. The Rankine-Hugoniot jump conditions corresponding to continuity, momentum and energy conservation, and entropy balance are [8]𝑆𝑁Δ𝐹𝑖𝛼=Δ𝑉𝑖𝑁𝛼,(14)𝜌0𝑆𝑁Δ𝑉𝑖𝑃=Δ𝑖𝛽𝑁𝛽,(15)𝜌0𝑆𝑁Δ𝜀=Δ𝑉𝑖𝑃𝑖𝛽𝑁𝛽𝑁Δ𝛾𝑄𝛾,(16)𝜌0𝑆𝑁𝑁Δ𝑆+Δ𝛾𝑄𝛾𝑇0.(17)

Remark 3. Because 𝑁𝛼 and 𝑛𝑖 are related through 𝑛𝑗=𝐽𝑁𝛽(𝐹1)𝛽𝑗, our assumed normalization 𝑛𝑖𝑛𝑖=1 precludes setting 𝑁𝛼𝑁𝛼=1.

The energy jump condition, combined with the mass jump condition, is𝜌0𝑆𝑁𝑣Δ𝜀0𝑁𝛾𝑁𝛾Δ𝐹𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽𝑁+Δ𝛾𝑄𝛾=0,(18) where 𝑣0=𝜌01. Subtracting the entropy jump condition, multiplied by the common value of temperature, we obtain𝜌0𝑆𝑁𝑣Δ(𝜀𝑇𝑠)0𝑁𝛾𝑁𝛾Δ𝐹𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽0.(19) AsΔ𝐹𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽𝐹=Δ𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽𝐹𝑖𝛼𝑁𝛼𝑃Δ𝑖𝛽𝑁𝛽𝐹=Δ𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽+𝜌0𝑆𝑁𝐹𝑖𝛼𝑁𝛼Δ𝑉𝑖𝐹=Δ𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽𝜌0𝑆2𝑁𝑁𝛾𝑁𝛾𝐹𝑖𝛼𝑁𝛼𝐹Δ𝑖𝛽𝑁𝛽𝐹=Δ𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽12𝜌0𝑆2𝑁𝑁𝛾𝑁𝛾Δ𝑁𝛼𝐶𝛼𝛽𝑁𝛽,(20) where 𝐶𝛼𝛽=𝐹𝑖𝛼𝐹𝑖𝛽 is the Cauchy-Green tensor, we find that𝜌0𝑆𝑁Δ1𝜀𝑇𝑠𝑁𝛾𝑁𝛾×𝑣0𝐹𝑖𝛼𝑁𝛼𝑃𝑖𝛽𝑁𝛽12𝑆𝑁2𝑁𝛾𝑁𝛾𝑁𝛼𝐶𝛼𝛽𝑁𝛽0.(21) In thermodynamic equilibrium, where 𝑆𝑁0 and no entropy is generated at the interface,Δ𝜀𝑇𝑠𝑣0𝐹𝑖𝛼𝑁𝛼𝑁𝛾𝑁𝛾𝑃𝑖𝛽𝑁𝛽=0.(22) As we shall soon justify, we identify the quantity in brackets,̃𝑔=𝜀𝑇𝑠𝑣0𝐹𝑖𝛼𝑁𝛼𝑁𝛾𝑁𝛾𝑃𝑖𝛽𝑁𝛽,(23) as the generalized specific Gibbs free energy. The thermodynamic equilibrium conditions are thereforeΔ𝑇=0,(24)Δ𝑉𝑖Δ𝑃=0,(25)𝑖𝛽𝑁𝛽=0,(26)Δ̃𝑔=0,(27) where (25) and (26) derive from (14) and (15), respectively, in the 𝑆𝑁0 limit.

Remark 4. Notice that in contrast to the case of inviscid fluid dynamics, the tangential velocity jump is constrained to be zero; that is, there is no slip at the interface. If the continuity equation is not enforced at the interface, the two materials can slip relative to each other, in a manner controlled, for example, by a friction law.

Define the Eshelby energy-momentum tensor𝑀𝛼𝛽=𝜌0(𝜀𝑇𝑠)𝛿𝛼𝛽𝐹𝑖𝛼𝑃𝑖𝛽.(28) Then, the generalized specific Gibbs free energy is related to the normal-normal component of 𝑀𝛼𝛽̃𝑔=𝑣0𝑁𝛼𝑀𝛼𝛽𝑁𝛽𝑁𝛾𝑁𝛾.(29) We may also express the Eshelby energy-momentum tensor in terms of the symmetric Piola-Kirchhoff tensor 𝑆𝛼𝛾, which is defined by the relationship 𝑃𝑖𝛽=𝐹𝑖𝛼𝑆𝛼𝛽𝑀𝛼𝛽=𝜌0(𝜀𝑇𝑠)𝛿𝛼𝛽𝐶𝛼𝛾𝑆𝛾𝛽.(30) When 𝑆𝛼𝛽=𝐽𝑝(𝐶1)𝛼𝛽, as it is for an inviscid fluid, the tensor 𝑣0𝑀𝛼𝛽 reduces to 𝑔𝛿𝛼𝛽, so that ̃𝑔 reduces to 𝑔. This result motivates calling ̃𝑔 the generalized specific Gibbs free energy.

2.3. Eulerian Elasticity

Next, we derive the equivalent form of the thermodynamic equilibrium conditions in the Eulerian frame. Recall that the Cauchy stress is𝜎𝑖𝑗=𝐽1𝐹𝑖𝛼𝑆𝛼𝛽𝐹𝑗𝛽=𝐽1𝑃𝑖𝛽𝐹𝑗𝛽,(31) and that the spatial normal 𝑛𝑖 corresponding to 𝑁𝛾 is𝑛𝑗=𝐽𝑁𝛽𝐹1𝛽𝑗.(32) Therefore,𝑃𝑖𝛽𝑁𝛽=𝜎𝑖𝑗𝑛𝑗.(33) Moreover, 𝐹𝑖𝛼𝑁𝛼=𝐽1𝑏𝑖𝑗𝑛𝑗 (where 𝑏𝑖𝑗=𝐹𝑖𝛼𝐹𝑗𝛼 is the inverse Finger tensor), and 𝑁𝛾𝑁𝛾=𝐽2𝑛𝑝𝑏𝑝𝑞𝑛𝑞, so that𝑣0𝐹𝑖𝛼𝑁𝛼𝑁𝛾𝑁𝛾𝑛=𝑣𝑘𝑏𝑘𝑖𝑛𝑝𝑏𝑝𝑞𝑛𝑞.(34) Thus,𝑛̃𝑔=𝜀𝑇𝑠𝑣𝑘𝑏𝑘𝑖𝑛𝑝𝑏𝑝𝑞𝑛𝑞𝜎𝑖𝑗𝑛𝑗,(35) and the thermodynamic equilibrium conditions areΔ𝑇=0,(36)Δ𝑢𝑖Δ𝜎=0,(37)𝑖𝑗𝑛𝑗=0,(38)Δ̃𝑔=0.(39)

Let us introduce the notation𝐽𝑖𝑛=𝐽𝑘𝑏𝑘𝑖𝑛𝑝𝑏𝑝𝑞𝑛𝑞=𝐹𝑖𝛼𝑁𝛼𝑁𝛾𝑁𝛾.(40) This notation is natural because the normal component of 𝐽𝑖 is 𝐽𝑛𝑖𝐽𝑖=𝐽. In terms of this notation, the generalized specific Gibbs free energy simplifies tõ𝑔=𝜀𝑇𝑠𝑣0𝐽𝑖𝜎𝑖𝑗𝑛𝑗,(41) and the thermodynamic equilibrium condition isΔ𝜀𝑇𝑠𝑣0𝐽𝑖𝜎𝑖𝑗𝑛𝑗=0.(42)

When the Cauchy stress is decomposed as𝜎𝑖𝑗=𝑝𝛿𝑖𝑗+𝑠𝑖𝑗,(43) where 𝑝=(1/3)𝜎𝑘𝑘 is the mean pressure and 𝑠𝑖𝑗 is the deviatoric stress, then the generalized specific Gibbs free energy takes the form̃𝑔=𝜀𝑇𝑠+𝑝𝑣𝑣0𝐽𝑖𝑠𝑖𝑗𝑛𝑗.(44) Thus, the thermodynamic equilibrium condition isΔ𝜀𝑇𝑠+𝑝𝑣𝑣0𝐽𝑖𝑠𝑖𝑗𝑛𝑗=0.(45) If the Cauchy stress is an isotropic pressure, that is, 𝑠𝑖𝑗=0, then ̃𝑔 reduces to 𝜀𝑇𝑠+𝑝𝑣=𝑔.

More generally, we may derive the approximate form of the generalized specific Gibbs free energy in the small-strain limit. If 𝜖𝑖𝑗 denotes the infinitesimal strain, then 𝑏𝑖𝑗 is related to 𝜖𝑖𝑗 through𝑏𝑖𝑗𝛿𝑖𝑗+2𝜖𝑖𝑗(46) to within second order in the displacement gradient. As a result,𝑣0𝐽𝑖𝑠𝑖𝑗𝑛𝑗𝑣(12𝜖𝑛𝑛)𝑠𝑛𝑛+2𝑛𝑖𝑠𝑖𝑗𝜖𝑗𝑘𝑛𝑘,(47) where 𝑠𝑛𝑛=𝑛𝑖𝑠𝑖𝑗𝑛𝑗 and 𝜖𝑛𝑛=𝑛𝑖𝜖𝑖𝑗𝑛𝑗. Thus, with 𝜎𝑛𝑛=𝑛𝑖𝜎𝑖𝑗𝑛𝑗=𝑝+𝑠𝑛𝑛,̃𝑔𝜀𝑇𝑠𝜎𝑛𝑛𝑛𝑣2𝑣𝑖𝑠𝑖𝑗𝜖𝑗𝑘𝑛𝑘𝑠𝑛𝑛𝜖𝑛𝑛.(48) Notice that the quantity in brackets vanishes if 𝑠𝑖𝑗 is diagonal.

Remark 5. The generalized Gibbs free energy for solids of Sections 2.2 and 2.3 are based on the conception that the Gibbs free energy is a quantity that is continuous at the phase boundary at the equilibrium. Another interpretation of the Gibbs free energy is that it is the proportionality coefficient between the increase in the total internal energy and the extra mass added to the system, that is, chemical potential. We examine this issue in the discussion.

2.4. Plasticity

Deriving the proper equilibrium conditions from the Rankine-Hugoniot jump conditions that include the effects of plasticity requires the governing equations for the plasticity to be in conservative form. Unlike elasticity, the plastic response of a material depends on the type of a material, the deformation rate, and so forth; a universal form of the plastic flow equation is not available. In this paper, we consider one important class of plasticity, namely, viscoplasticity. In the following, we present the flow rule and hardening law for viscoplasticity in conservative form. We refer the readers to [8] for more details.

First of all, we decompose the deformation gradient as a product𝐹𝑖𝛼=𝐹𝑒𝑖𝑎𝐹𝑝𝑎𝛼,(49) where (𝐹𝑝)𝑎𝛼, called the plastic part of the deformation gradient, is a matrix of the same type as the gradient of a map from the undeformed (Lagrangian) frame to an intermediate reference frame in which the material is in a stress-free state, and (𝐹𝑒)𝑖𝑎, the elastic part of the deformation gradient, is a matrix of the same type as the gradient of a map from the intermediate reference frame to the deformed (Eulerian) frame. In addition, we define the plastic strain 𝐸𝑝𝛼𝛽 to be𝐸𝑝𝛼𝛽1=2𝐹𝑝𝑎𝛼𝐹𝑝𝑎𝛽𝛿𝛼𝛽.(50)

In general, the flow rule can be written aṡ𝐸𝑝𝛼𝛽=Λ𝛼𝛽𝐸𝛾𝛿,𝐸𝑝𝛾𝛿,,𝐾,𝑆(51) where 𝐾 stands for extra internal variables such as a hardening parameter. The hardening law takes the forṁ𝐸𝐾=𝐻𝛾𝛿,𝐸𝑝𝛾𝛿.,𝐾,𝑆(52) Both of these equations are in conservative form. The corresponding jump conditions are𝑆𝑁Δ𝐸𝑝𝛼𝛽=0,𝑆𝑁Δ𝐾=0.(53) When the interface is moving at the nonzero speed, we conclude that the jump in 𝐸𝑝𝛼𝛽 and 𝐾 vanish. Taking the limit 𝑆𝑁0, we conclude thatΔ𝐸𝑝𝛼𝛽=0,Δ𝐾=0,(54) at the interface. Together with (24)–(27), and (54) must be satisfied at the interface between viscoplastic materials in equilibrium.

3. Thermodynamic Equilibrium across a Coherent Interface [4]

In this section, we derive the thermodynamic equilibrium conditions across a coherent interface. A coherent transformation in a solid is a phase transformation in which the three-dimensional lattice is preserved without rupture, and a coherent interface is an interface between the different phases of a coherent phase transformation. The treatment here is microscopic; that is, the phase transformation is viewed at the atomic level. In this view, lattice vectors parallel to the interface remain unchanged by the phase transformation.

3.1. Fundamental Equation of State

Let us say that phase 𝑎 is transformed into phase 𝑏. Let 𝑥𝑖 (𝑖=1,2,3) denote the spatial coordinates and 𝑋𝛼 (𝛼=1,2,3) the reference coordinates. We also use the superscripts 𝑎 and 𝑏 to distinguish between the phases 𝑎 and 𝑏.

In terms of the deformation gradients 𝐹𝑖𝛼𝑎 and 𝐹𝑖𝛼𝑏 of phases 𝑎 and 𝑏, the fundamental equations of state of the two phases are 𝜀𝑎=𝜀𝑎𝐹𝑖𝛼𝑎,𝑠𝑎,𝜀(55)𝑏=𝜀𝑏𝐹𝑖𝛼𝑏,𝑠𝑏.(56) Infinitesimal variations in the specific internal energies are given by the equations 𝛿𝜀𝑎=𝜕𝜀𝑎𝜕𝐹𝑖𝛼𝑎𝛿𝐹𝑖𝛼𝑎+𝜕𝜀𝑎𝜕𝑠𝑎𝛿𝑠𝑎,𝛿𝜀𝑏=𝜕𝜀𝑏𝜕𝐹𝑖𝛼𝑏𝛿𝐹𝑖𝛼𝑏+𝜕𝜀𝑏𝜕𝑠𝑏𝛿𝑠𝑏,(57) which can be rewritten as𝛿𝜀𝑎=𝑣0𝑃𝑖𝛼𝑎𝛿𝐹𝑎𝑖𝛼+𝑇𝛿𝑠𝑎,𝛿𝜀𝑏=𝑣0𝑃𝑖𝛼𝑏𝛿𝐹𝑏𝑖𝛼+𝑇𝛿𝑠𝑏.(58)

3.2. Geometry of a Migrating Coherent Interface

The phase transformation of solid can be thought of as the migration of the interface between the two phases. The migration of the interface can be decomposed into the following steps. (1)An imaginary cut is made along the interface. (2)Points in 𝑎 are moved by a distance 𝛿𝑥𝑖𝑎, and points in 𝑏 are moved by a distance 𝛿𝑥𝑖𝑏. (3)A volume of phase 𝑎 is transformed into phase 𝑏, reestablishing the contact of the two phases.

These steps, which are illustrated in Figure 1, help in constructing the mathematical formalism, but they are not necessarily physical. Note that only the difference of the displacements 𝛿𝑥𝑖𝑏𝛿𝑥𝑖𝑎 matters and that this vector is determined by the difference between the lattice vectors of the two phases. If phase 𝑏 has the larger specific volume, the two phases move apart in step (2) so as to make room for phase 𝑎 to turn into phase 𝑏 in step (3); otherwise, if phase 𝑎 has the larger specific volume, parts of the two phases overlap after step (2) to accommodate the shrinkage in step (3).

Figure 1: Schematic migration of the phase boundary during a phase transformation.

We can write the relative displacement of 𝑏 with respect to 𝑎 as𝛿𝑥𝑖𝑏𝛿𝑥𝑖𝑎=𝛿𝑚𝑤𝑖,(59) where 𝛿𝑚 is the mass transformed from phase 𝑎 to phase 𝑏 across a unit area of the interface in the reference frame. Thus, 𝛿𝑚=𝑛𝑚𝐼, where 𝑛 is the number of atomic layers that are transformed and 𝑚𝐼 is the mass of a single atomic layer at the interface with unit reference are (which is the same for the two phases). We can infer that𝑤𝑖=𝑎𝑖𝑏𝑎𝑖𝑎𝑚𝐼,(60) where 𝑎𝑖 is a lattice vector in thermodynamic equilibrium that changes during the phase transformation and, therefore, is not parallel to the interface. If the two phases have rectangular crystal structures, 𝑎𝑖 is the lattice vector that is orthogonal to the interface in the reference state (assuming the reference frame is chosen to be an orthogonal one).

From these definitions, we find that the interface moves by a distance 𝛿 in the reference frame such that𝛿=𝛿𝑚𝜌0=𝑣0𝛿𝑚,(61) where 𝜌0 is the reference mass density and 𝑣0 is the reference specific volume (𝑣0=𝜌01).

As an illustration, we apply the result to a simple case where the lattice structures of two phases are orthogonal, and the lattice constant of phase 𝑏 is twice longer than that of phase 𝑎. (Refer to Figure 2.) Let us choose the reference frame that has the same structure as the phase 𝑎 homogeneously. Suppose that one layer of phase 𝑎 is transformed into phase 𝑏. Then, we have the relative displacement as 𝛿𝑥𝑏𝛿𝑥𝑎=𝑎𝑎. On the other hand, 𝛿𝑚=𝑚𝐼 and 𝑤=𝑎𝑎/𝑚𝐼 so that we have the relation (59). (We have dropped the superscript 𝑖 labeling a component of a vector, for only the normal component matters in this case.) As only one layer transforms, the distance in the reference frame that the interface moved is 𝑎𝑎. It is also confirmed from the right-hand side of (61) that 𝛿𝑚/𝜌0=𝑚𝐼/(𝑚𝐼/𝑎𝑎)=𝑎𝑎.

Figure 2: Simple example of a phase transformation.
3.3. Conditions of Thermodynamic Equilibrium

Assume that the system comprising phases 𝑎 and 𝑏 is in thermodynamic equilibrium. In particular, take the system to be stationary and assume that the temperature is constant throughout the system, since otherwise heat would flow. A fundamental property of a thermodynamic system in thermodynamic equilibrium is that, for an infinitesimal variation away from thermodynamic equilibrium state, with the entropy held fixed, the variation of the energy of the system vanishes𝛿𝐸=0withentropyxed,(62) where𝛿𝐸=𝑉0𝑎𝛿𝜀𝑎𝜌0𝑑𝑉+𝑉0𝑏𝛿𝜀𝑏𝜌0𝑑𝑉+𝑉𝜀𝑏𝜀𝑎𝜌0𝑑𝑉.(63) Here, 𝑉0𝑎 and 𝑉0𝑏 are the total volumes of phase 𝑎 and 𝑏 in the reference frame, and 𝑉 is the volume of the region swept by the phase transformation. We proceed to derive conditions for thermodynamic equilibrium from this principle.

Using (58) for the internal energy variation, the total energy variation with the phase transformation becomes𝛿𝐸=𝑉0𝑎𝑣0𝑃𝑎𝑖𝛼𝛿𝐹𝑎𝑖𝛼+𝑇𝛿𝑠𝑎𝜌0+𝑑𝑉𝑉0𝑏𝑣0𝑃𝑏𝑖𝛼𝛿𝐹𝑏𝑖𝛼+𝑇𝛿𝑠𝑏𝜌0+𝑑𝑉𝑉𝜀𝑏𝜀𝑎𝜌0𝑑𝑉.(64) In the region swept by the phase transformation, the volume element 𝑑𝑉 can be replaced by 𝛿𝑑𝐴 or 𝑣0𝛿𝑚𝑑𝐴, where 𝛿 and 𝛿𝑚 are defined in the previous section. With this replacement, (64) is rewritten as𝛿𝐸=𝑉0𝑎𝑣0𝑃𝑎𝑖𝛼𝛿𝐹𝑎𝑖𝛼+𝑇𝛿𝑠𝑎𝜌0+𝑑𝑉𝑉0𝑏𝑣0𝑃𝑏𝑖𝛼𝛿𝐹𝑏𝑖𝛼+𝑇𝛿𝑠𝑏𝜌0+𝑑𝑉𝐴𝜀𝑏𝜀𝑎𝛿𝑚𝑑𝐴,(65) where 𝐴 is the area at the interface between two phases.

On the right-hand side of (65), the variations in the displacement and the entropy are not independent. As (62) is valid only when the entropy is held fixed, we impose that𝑉0𝑎𝛿𝑠𝑎𝜌0𝑑𝑉+𝑉0𝑏𝛿𝑠𝑏𝜌0𝑑𝑉+𝐴𝑠𝑏𝑠𝑎𝛿𝑚𝑑𝐴=0.(66)

We also rewrite the energy variation associated with the displacements by using the following identities:𝑉0𝑎𝑃𝑎𝑖𝛼𝛿𝐹𝑎𝑖𝛼=𝑑𝑉𝐴𝑃𝑎𝑖𝛼𝑁𝛼𝑎𝛿𝑥𝑖𝑎𝑑𝐴𝑉0𝑎𝑃𝑎𝑖𝛼,𝛼𝛿𝑥𝑖𝑎𝑑𝑉,(67) and similarly𝑉0𝑏𝑃𝑏𝑖𝛼𝛿𝐹𝑏𝑖𝛼=𝑑𝑉𝐴𝑃𝑏𝑖𝛼𝑁𝛼𝑏𝛿𝑥𝑖𝑏𝑑𝐴𝑉0𝑏𝑃𝑏𝑖𝛼,𝛼𝛿𝑥𝑖𝑏𝑑𝑉,(68) where 𝑁𝛼𝑎 and 𝑁𝛼𝑏 are unit vectors normal to the interface in the reference state, drawn outward from 𝑎 and 𝑏, respectively, so that 𝑁𝛼𝑎=𝑁𝛼𝑏. Strictly speaking, the surface integrals in the first terms of the right hand side in (67) and (68) should be calculated over the whole surface boundaries of phase 𝑎 and 𝑏. By assuming that the displacement is negligible on the surface away from the phase boundary (e.g., surrounded by a rigid wall), we reduce the integration area to the interface between two phases, 𝐴.

Therefore, with the aid of (59), we write the sum of the foregoing two integrals as𝑉0𝑎𝑃𝑎𝑖𝛼𝛿𝐹𝑎𝑖𝛼𝑑𝑉+𝑉0𝑏𝑃𝑏𝑖𝛼𝛿𝐹𝑏𝑖𝛼𝑑𝑉=𝐴𝑃𝑎𝑖𝛼𝑁𝛼𝑎𝑤𝑖+𝛿𝑚𝑑𝐴𝐴𝑃𝑎𝑖𝛼𝑁𝛼𝑎+𝑃𝑏𝑖𝛼𝑁𝛼𝑏𝛿𝑥𝑖𝑏𝑑𝐴𝑉0𝑎𝑃𝑎𝑖𝛼,𝛼𝛿𝑥𝑖𝑎𝑑𝑉𝑉0𝑏𝑃𝑏𝑖𝛼,𝛼𝛿𝑥𝑖𝑏𝑑𝑉.(69) Using (66) and (69), the total energy variation (65) is𝛿𝐸=𝑉0𝑎𝑃𝑎𝑖𝛼,𝛼𝛿𝑥𝑖𝑎𝑑𝑉𝑉0𝑏𝑃𝑏𝑖𝛼,𝛼𝛿𝑥𝑖𝑏+𝑑𝑉𝐴𝑃𝑎𝑖𝛼𝑁𝛼𝑎+𝑃𝑎𝑖𝛼𝑁𝛼𝑏𝛿𝑥𝑖𝑏+𝑑𝐴𝐴𝜀𝑏𝑇𝑠𝑏(𝜀𝑎𝑇𝑠𝑎)𝑤𝑖𝑃𝑎𝑖𝛼𝑁𝛼𝑎𝛿𝑚𝑑𝐴.(70)

The variational factors in each integral of (70) are independent. The thermodynamic equilibrium condition (62) therefore implies that each integrand is zero.(1)The volume integrals yield 𝑃𝑖𝛼,𝛼=0throughoutthephasesaandb,(71) which is the usual equation of mechanical equilibrium in the absence of body forces. Equivalently, 𝜎𝑖𝑗,𝑗=0throughoutthephasesaandb.(72) In deriving (72), we have invoked (31) and the Piola identity, [𝐽(𝐹1)𝛼𝑖],𝛼=0. (2)The first area integral yields 𝑃𝑎𝑖𝛼𝑁𝛼𝑎+𝑃𝑏𝑖𝛼𝑁𝛼𝑏=0(73) on the interface. Equivalently, by (33), 𝜎𝑖𝑗𝑎𝑛𝑗𝑎+𝜎𝑖𝑗𝑏𝑛𝑗𝑏=0.(74) These equations mean that the traction (i.e., the normal stress) 𝑡𝑖=𝜎𝑖𝑗𝑛𝑗,(75) applied to the interface is continuous, that is, 𝑡𝑖𝑎=𝑡𝑖𝑏. (3)Perhaps most importantly, the second area integral yields 𝜀𝑏𝑇𝑠𝑏(𝜀𝑎𝑇𝑠𝑎)𝑤𝑖𝑃𝑖𝛼𝑁𝛼𝑎=0,(76) at the interface. Equivalently, 𝜀𝑏𝑇𝑠𝑏(𝜀𝑎𝑇𝑠𝑎)𝑤𝑖𝜎𝑖𝑗𝑛𝑗=0.(77) To understand this condition better, we write 𝜎𝑖𝑗=𝑝𝛿𝑖𝑗+𝑠𝑖𝑗 and find that 𝜀𝑏𝑇𝑠𝑏(𝜀𝑎𝑇𝑠𝑎)+𝑝𝑤𝑖𝑛𝑖𝑤𝑖𝑠𝑖𝑗𝑛𝑗=0.(78) From the definition of 𝑤𝑖, we derive that 𝑛𝑖𝑤𝑖=𝑛𝑖𝑎𝑖𝑏𝑎𝑖𝑎𝑚𝐼=𝑣𝑏𝑣𝑎,(79) where 𝑣𝑎 and 𝑣𝑏 are the specific volumes of phases 𝑎 and 𝑏, respectively. Therefore, 𝜀𝑏𝑇𝑠𝑏+𝑝𝑣𝑏(𝜀𝑎𝑇𝑠𝑎+𝑝𝑣𝑎)𝑤𝑖𝑠𝑖𝑗𝑛𝑗=0,(80) or 𝑔𝑏𝑔𝑎𝑤𝑖𝑠𝑖𝑗𝑛𝑗=0,(81) where 𝑔 is the usual specific Gibbs free energy. The extra term 𝑤𝑖𝑠𝑖𝑗𝑛𝑗 accounts for the contribution from the tangential components of the displacement and stress.

4. Equivalence of the Derived Thermodynamic Equilibrium Conditions

In this section, we show the equivalence of the two sets of thermodynamic equilibrium conditions that have been derived in Sections 2 and 3.(a)The temperature continuity condition, (24) or (36), of Section 2 corresponds to the somewhat stronger condition of temperature constancy assumed in Section 3. (b)The velocity continuity condition, (25) or (37), of Section 2 corresponds to the somewhat stronger condition of stationarity in Section 3. (c)The traction continuity conditions (26) and (73) involving the Piola-Kirchhoff stress are identical, as are the corresponding conditions (38) and (74) involving the Cauchy stress. (d)We compare (42) and (77) under the assumption that the phase transformation is coherent. Evidently, these equations are equivalent if we show that 𝑤𝑖=𝑣0Δ𝐽𝑖.(82) By the coherence of the phase transformation, there exists a lattice vector 𝑎𝛼0 in the reference frame that maps, under 𝐹𝑖𝛼𝑎 and 𝐹𝑖𝛼𝑏, to the spatial lattice vectors 𝑎𝑖𝑎 and 𝑎𝑖𝑏, respectively. By definition of 𝑤𝑖, 𝑤𝑖=𝑎𝑖𝑏𝑎𝑖𝑎𝑚𝐼=Δ𝐹𝑖𝛼𝑎𝛼0𝑚𝐼.(83) Recall that the deformation gradient 𝐹𝑖𝛼=𝜕𝑥𝑖/𝜕𝑋𝛼 satisfies 𝜖𝛼𝛽𝛾𝐹𝑖𝛽,𝛾=0,(84) when it is smooth and satisfies the analogous jump conditions 𝜖𝛼𝛽𝛾Δ𝐹𝑖𝛽𝑁𝛾=0,(85) at discontinuities. In other words, Δ𝐹𝑖𝛼𝑁𝛼. The definition of 𝐽𝑖=𝐹𝑖𝛼𝑁𝛼/(𝑁𝛾𝑁𝛾) determines the proportionality factor Δ𝐹𝑖𝛼=Δ𝐽𝑖𝑁𝛼.(86) As a result, 𝑤𝑖=Δ𝐽𝑖𝑁𝛼𝑎𝛼0𝑚𝐼=𝑣0Δ𝐽𝑖.(87)

5. Summary and Discussion

5.1. Equilibrium Conditions

In the subject of solid-solid phase transformations, the question—what are the correct thermodynamic equilibrium conditions between solid phases?—has not been answered satisfactorily. For inviscid fluids, the answer is continuity of (a) temperature, (b) velocity, (c) pressure, and (d) specific Gibbs free energy. The situation is more complicated for solids. The first issue concerns properly generalizing the Gibbs free energy when the stress tensor is not simply a pressure. Certainly, the term replacing 𝑝𝑣 in the usual Gibbs free energy should reduce to 𝑝𝑣 when the stress tensor is 𝜎𝑖𝑗=𝑝𝛿𝑖𝑗. Yet, there are many combinations of stress, strain, and specific volume that satisfy this criterion. The second issue is whether there are enough conditions to determine the states of the two phases.

We have derived the following thermodynamic equilibrium conditions at the interface between two solid phases: (1)continuity of the temperature,(2)continuity of the velocity,(3)continuity of the traction, (4)continuity of the generalized specific Gibbs free energy.

We assert that these results address the two issues mentioned above.

In constructing the thermodynamic equilibrium conditions for two solid phases, we have paid close attention to the origin of these conditions for the case of fluids. The fluid conditions are so physically intuitive that we often take them for granted. However, we recognize that the underlying physical principles are the conservation laws and the entropy inequality. These principles generate a complete set of conditions connecting the two phases. For fluids, these principles yield the commonly accepted thermodynamic equilibrium conditions.

For solids, we have derived the corresponding conditions in Section 2. The derivation yields a particular generalization of the specific Gibbs free energy𝑔=𝜀𝑇𝑠+𝑝𝑣(88) is generalized tõ𝑔=𝜀𝑇𝑠+𝑝𝑣𝑣0𝐽𝑖𝑠𝑖𝑗𝑛𝑗.(89)

Perhaps surprisingly, the generalized specific Gibbs free energy is independent of the tangential-tangential components of stress. Because no thermodynamic equilibrium conditions involve stress components other than the tractions (𝑡𝑖=𝜎𝑖𝑗𝑛𝑗), it might be thought that our derivation is incomplete. Nonetheless, the conditions that we derived are necessary and sufficient to conserve mass, momentum, and energy and to satisfy the entropy inequality. If these are all of the physical principles we require to be satisfied between two phases, these conditions are all we get. We conclude that any further conditions, if needed, must be materials-specific constitutive relations.

For example, these further conditions might specify the interfacial interaction between the phases. Do they slip freely past each other, or are they fixed together? Such a question, the answer to which reflects a material property, lies outside the scope of general thermodynamic arguments.

Remark 6. A similar issue arises in any physical problem that treats an interface. In interfacial instability problems, like Rayleigh-Taylor or Richtmyer-Meshkov flows for solids, modeling requires assumptions concerning the properties of the material interface [9, 10].

We emphasize that the derivation in Section 3 makes stronger assumptions (global thermodynamic equilibrium) and applies only to a very special type of phase transformation, a coherent transformation. In general, a phase transformation can link very different crystal structures, which might not be coherent. Even leaving aside melting and gasification, many types of phase transformation between solid phases are not coherent. Thus, the derivation in Section 2 is more general. Nonetheless, the detailed picture in Section 3 helps clarify the physics behind the thermodynamic equilibrium condition.

5.2. Gibbs Free Energy

Before closing the discussion, we want to bring your attention to the interpretation of the Gibbs free energy. The generalized Gibbs free energy defined in (89) is interpreted as a quantity that should be continuous at the phase interface at the equilibrium (chemical potential). On the other hand, the Gibbs free energy can also be viewed as an increase in total internal energy with the addition of a unit mass to the thermodynamic system. In this section, we derive the Gibbs free energy based on the latter notion to find the result different from (89). For concreteness of the argument, we assume the thermoelasticity.

The equation of state for a thermoelastic material is𝑢=𝑢𝑠,𝐸𝛼𝛽,(90) where 𝑢 is specific internal energy, 𝑠 is specific entropy and 𝐸𝛼𝛽 is Cauchy-Green Lagrangian strain tensor. Then, the fundamental thermodynamic identity for thermoelastic solids is𝜌𝑑𝑢=𝜌𝑇𝑑𝑠+𝐽1𝑆𝛼𝛽𝑑𝐸𝛽𝛼,(91) where 𝑆𝛼𝛽=𝜌0𝜕𝑢/𝜕𝐸𝛼𝛽 is the symmetric Piola-Kirchhoff stress tensor and 𝐽=(det𝐶)1/2 is the Jacobian of 𝐹, which is such that 𝜌=𝐽1𝜌0. From (31), 𝜎𝑖𝑗=𝐽1𝐹𝑖𝛼𝑆𝛼𝛽𝐹𝛽𝑗.

To better understand the work term 𝐽1𝑆𝛼𝛽𝑑𝐸𝛽𝛼 in (91), we decompose the strain tensor 𝐸 into the volumetric part and shear part. Let𝐶𝛼𝛽=𝐽2/3𝐶𝛼𝛽,𝐸𝛼𝛽1=2𝐶𝛼𝛽𝛿𝛼𝛽.(92) Note that det𝐶=1, that is, 𝐶 is a shear strain.

Using these definitions, the differential of 𝐸𝛼𝛽 is𝑑𝐸𝛼𝛽=13𝐽1𝐶𝛼𝛽𝑑𝐽+𝐽2/3𝑑𝐸𝛼𝛽,(93) and therefore,𝐽1𝑆𝛼𝛽𝑑𝐸𝛽𝛼=𝐽1𝑝𝑑𝐽+𝐽1/3𝑆𝛼𝛽𝑑𝐸𝛽𝛼,(94) where 𝑝=1/3tr𝜎 is the mean pressure.

In terms of the Cauchy stress, this equation becomes𝐽1𝑆𝛼𝛽𝑑𝐸𝛽𝛼=𝐽1𝑝𝑑𝐽+𝐽2/3𝐹1𝛼𝑖(dev𝜎)𝑖𝑗𝐹𝑇𝛽𝑗𝑑𝐸𝛽𝛼,(95) by observing that (𝐶1)𝛼𝛽𝑑𝐸𝛽𝛼=0 so that 𝑆𝛼𝛽 on the right-hand side in (94) can be replaced by 𝑆𝛼𝛽1/3𝑆𝛾𝛿𝐶𝛿𝛾(𝐶1)𝛼𝛽, which equals 𝐽(𝐹1)𝛼𝑖(dev𝜎)𝑖𝑗(𝐹𝑇)𝑗𝛽.

Hence, we get the fundamental thermodynamic identity in Eulerian frame𝜌𝑑𝑢=𝜌𝑇𝑑𝑠𝜌𝑝𝑑𝑣+𝐽2/3(dev𝜎)𝑖𝑗𝐹𝑇𝑗𝛼𝑑𝐸𝛼𝛽𝐹1𝛽𝑖.(96)

Now, note that an external quantity 𝑄 of thermodynamic system satisfies 𝑄=𝑚𝑞, where 𝑚 is total mass of the system and 𝑞 is corresponding specific quantity of 𝑄. Hence, we write the internal energy, entropy, and volume as 𝑆𝑈(totalinternalenergy)=𝑚𝑢,(totalentropy)=𝑚𝑠,𝑉(totalvolume)=𝑚𝑣.(97) Then, the fundamental thermodynamic identity (96), multiplied by 𝑉 is𝑑𝑈𝑢𝑑𝑚=𝑇(𝑑𝑆𝑠𝑑𝑚)𝑝(𝑑𝑉𝑣𝑑𝑚)+𝑉𝐽2/3(𝜎)𝑖𝑗𝐹𝑇𝑗𝛼𝑑𝐸𝛼𝛽𝐹1𝛽𝑖,(98) so that𝑑𝑈=𝑇𝑑𝑆𝑝𝑑𝑉+𝑉𝐽2/3(𝜎)𝑖𝑗𝐹𝑇𝑗𝛼𝑑𝐸𝛼𝛽𝐹1𝛽𝑖+𝑢𝑇𝑠+𝑝𝑣𝑑𝑚.(99) Thus, we are led to identify the coefficient of 𝑑𝑚 as the specific Gibb free energy𝑔=𝑇𝜕𝑈|||𝜕𝑚𝑆,𝑉,𝐸=𝑢𝑇𝑠+𝑝𝑣.(100) Note that there is no tensorial dependency and instead only pressure and volume enter the formula in contrast to (89). For fluids, the two interpretations yield the same definition.


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