PHYSICAL REVIEW B 88, 184303 (2013)

Irreversible thermodynamics of creep in crystalline solids

Y. Mishin,1,* J. A. Warren,2 R. F. Sekerka,3 and W. J. Boettinger2

1School of Physics, Astronomy and Computational Sciences, MSN 3F3, George Mason University, Fairfax, Virginia 22030, USA2Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

3Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA(Received 30 August 2013; revised manuscript received 19 October 2013; published 18 November 2013)

We develop an irreversible thermodynamics framework for the description of creep deformation in crystallinesolids by mechanisms that involve vacancy diffusion and lattice site generation and annihilation. The materialundergoing the creep deformation is treated as a nonhydrostatically stressed multicomponent solid medium withnonconserved lattice sites and inhomogeneities handled by employing gradient thermodynamics. Phase fieldsdescribe microstructure evolution, which gives rise to redistribution of vacancy sinks and sources in the materialduring the creep process. We derive a general expression for the entropy production rate and use it to identifyof the relevant fluxes and driving forces and to formulate phenomenological relations among them taking intoaccount symmetry properties of the material. As a simple application, we analyze a one-dimensional model of abicrystal in which the grain boundary acts as a sink and source of vacancies. The kinetic equations of the modeldescribe a creep deformation process accompanied by grain boundary migration and relative rigid translationsof the grains. They also demonstrate the effect of grain boundary migration induced by a vacancy concentrationgradient across the boundary.

DOI: 10.1103/PhysRevB.88.184303 PACS number(s): 66.30.!h, 61.72.!y, 62.20.Hg, 65.40.!b

I. INTRODUCTION

When subject to a high homologous temperature and asustained mechanical load below the yield strength, manymaterials exhibit a slow time-dependent plastic deformationcalled creep. This mode of deformation has been observedin different classes of materials ranging from metals andalloys to ceramics, polymers, and ice. While several creepdeformation mechanisms have been proposed over the years,we will focus in this work on mechanisms that require creationand annihilation of lattice sites.1 Such mechanisms include so-called diffusional creep in which the deformation is mediatedby vacancy diffusion through the lattice (Nabarro-Herringcreep)2,3 or along grain boundaries (GBs) (Coble creep),4,5

as well as creep by dislocation climb. A number of othermechanisms that do not necessarily involve site creation andannihilation, such as the thermally activated dislocation glide,will not be considered here.

Most of the models of creep developed so far havean ad hoc character: they are obtained by postulating aparticular mechanism and assuming a constitutive relationbetween the creep deformation rate and a chosen drivingforce. The development of a general and rigorous theoryof creep deformation requires at least the following threecomponents: (i) a thermodynamic model of a mechanicallystressed crystalline solid with nonconserved lattice sites, (ii) amodel of microstructure evolution that includes redistributionof vacancy sinks and sources and the motion of interfacesseparating different phases and/or grains, and (iii) a set ofkinetic equations derived from the entropy production rate6

and identification of the appropriate set of fluxes (includingthe creep deformation rate) and the conjugate driving forces.To our knowledge, a theory comprising all three componentshas not been developed to date.

Several theories involving one or two of the above compo-nents can be found in the literature. Svoboda et al.7,8 proposed

a creep model for multicomponent alloys with a continuousdistribution of vacancy sinks and sources. By contrast toprevious work, their kinetic equations have not been simplypostulated but rather derived from the maximum dissipationprinciple. The authors identified and clearly separated twocomponents of the creep deformation tensor, the volumedilation/contraction and the shear, and correctly establishedtheir decoupled character. However, their thermodynamictreatment of solid solutions is based on certain assumptionsand approximations that are not always justified. For example,they use the Gibbs-Duhem equation, which is valid onlyfor hydrostatically stressed systems and introduce so-called“generalized” chemical potentials, which include only thehydrostatic part of the stress tensor ! [see, e.g., Sekerka andCahn9 for criticism of using only the hydrostatic part of !(“solid pressure”) in solid-state thermodynamics]. In view ofthe nonuniqueness of chemical potentials of substitutionalcomponents in nonhydrostatic solids9–15 and the fact thatthe Gibbs free energy is no longer a useful thermodynamicpotential, development of thermodynamic models of stressedsolids should start from the first and second laws in theenergy-entropy representation16 and proceed with extremecare.

As such, a very general and rigorous thermodynamictreatment of multicomponent solids was developed by Larcheand Cahn10–12 as an extension of Gibbs’ thermodynamics16

to nonhydrostatic solid systems. Although their analysis isvalid for stressed solids with any number of substitutional andinterstitial components, it relies on the assumption that thelattice sites are conserved. The lattice conservation imposes theso-called “network constraint,” which penetrates through allthermodynamic equations. It is assumed that lattice sites can becreated or destroyed only at defects such as surfaces, interfaces,and dislocation cores. Such defective regions are excludedfrom the direct thermodynamic treatment and only enter thetheory through boundary conditions. Thus the question of

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

how the vacancy sinks and sources operate is essentiallyleft beyond the theory. Mullins and Sekerka13 proposed asimilar theory for multicomponent crystalline solids witha more general treatment of point defects based on theconcepts of extended variable sets. Their theory assumesthe conservation of lattice unit cells, which is similar tothe “network constraint.” Both Larche and Cahn10–12 andMullins and Sekerka13 analyzed equilibrium states of thesolid and did not study the irreversible thermodynamicsof creep deformation. (In Sec. 8.5, Larche and Cahn12 dodiscuss some creep problems, but they treat creep throughboundary conditions with perfect site conservation inside thelattice.)

Furthermore, these thermodynamic theories of solids10–13

are purely “classical” in which all thermodynamic propertiesdepend only on local thermodynamic densities16 but not theirgradients. Accordingly, transition regions between differentphases are treated as geometric surfaces of discontinuity16

endowed with certain postulated properties, such as theability (or inability) to support shear stresses or the capacity(or lack thereof) to generate or absorb vacancies. Existingcreep models7,8 are also classical and thus incapable ofdescribing the microstructure evolution as part of the creepprocess.

On the other hand, there are nonclassical (by nonclas-sical, we do not mean to imply that quantum mechanicsis used in the present paper) models of multicomponentfluid systems in which interfaces between phases are treatedvia the gradient thermodynamics approach17–20 also calledthe phase-field method (see, e.g., Ref. 21 and referencestherein). The fluid theories include rigorous derivations ofthe entropy production rate for the simultaneous processesof phase-field evolution, heat conduction, diffusion, and con-vective flows accompanied by viscous dissipation. However,extensions of such theories to solid materials are presentlylacking. The existing phase-field models of creep in solids22

describe creep deformation through a set of phase fieldsrelated to dislocations in specific slip systems. Such theoriesreproduce creep-controlled structural evolution in multiphasematerials without explicitly treating vacancies or the lattice.We should also mention the recently developed phase-fieldformulations involving large deformations,23–28 which donot treat creep deformation per se but provide a usefulformalism, some of which could be incorporated in creeptheories.

The goal of this paper is to develop a general irreversiblethermodynamics framework for the description of creepdeformation in solid materials by mechanisms involvingsite generation and annihilation and vacancy diffusion. Theproposed theory includes all three components (i)–(iii) men-tioned above. It can be viewed as a generalization of thenonclassical fluid theories21 to solid materials. Alternatively,it can be considered as a generalization of classical solid-statethermodynamics10–13 to nonclassical, nonequilibrium solidsystems with a nonconserved lattice.

In Secs. II and III, we introduce the kinematic descriptionof creep deformation and the balance relations that will beused in the rest of the paper. Section IV presents a ther-modynamic treatment of a nonclassical, nonhydrostaticallystressed multicomponent solid phase. We derive gradient

and time-dependent forms of the first and second laws forreversible thermodynamic processes in such a solid, alongwith a generalized form of the Gibbs-Duhem equation. Beforeproceeding to irreversible thermodynamics, we derive theconditions of full and constrained thermodynamic equilibriain the solid. These conditions constitute a generalization ofLarche and Cahn10–12 to nonclassical solids with continuouslydistributed nonconserved sites. The entropy production ratederived in Sec. VI serves as the starting point for the identi-fication of the relevant fluxes and forces and formulation ofphenomenological relations between them. We emphasize theimportance of symmetry properties of the material, formulatea set of phenomenological relations for isotropic materials,and outline possible extensions to lower-symmetry systemsby analyzing the tensor character of the fluxes and forces.The volume and shear components of the creep deformationrate7,8 emerge naturally from this analysis and are shown tobe coupled to different driving forces. To provide a simpleillustration of how the theory can be applied, we present aone-dimensional model of a bicrystal with a grain boundary(GB) acting as a sink and source of vacancies (Sec. VII).In the presence of vacancy oversaturation or under an appliedtensile stress, the kinetic equations describe creep deformationof the sample accompanied by GB migration and relative rigidtranslations of the grains. In Sec. VIII, we summarize the workand draw conclusions.

II. MASS AND SITE CONSERVATION LAWS ANDKINEMATICS OF DEFORMATION

We consider a crystalline solid composed of n substitutionalchemical species labeled i. The solid contains vacancies butnot interstitials, although this theory can be generalized toincorporate interstitials. We assume that there are no chemicalreactions among the species i. The crystalline structure isassumed to have a Bravais lattice, i.e., primitive lattice witha single basis site. The solid is subject to external potentialforces such as gravitational or electric (when the particles areelectrically charged as in ionic solids).

We start by formulating mass and particle conservationconditions satisfied by our system. Some of them are specificto a solid solution while others are equally valid for liquidsor gases. The substitutional lattice sites, referred to belowas simply sites, are generally not conserved. It is assumed,however, that we can still define a lattice velocity fieldvL(x,t). To this end, we assume that the solid contains animaginary network of sites which, on the time scale of ourobservations, are not destroyed by the creep process. Theseindestructible lattice sites will be called “markers.” (The term“marker” may sound somewhat confusing because of theassociation with the Kirkendall experiment29 in which themarkers were inert foreign objects intentionally embeddedin the lattice. In our case, the imaginary marker sites arephysically identical to other sites except for our knowledgethat they will “survive” the lattice site creation and annihilationduring the creep deformation process on a chosen time scale.)The lattice velocity vL(x,t), also referred to as the total latticevelocity, is defined as the velocity of a marker occupying thelocation x (relative to a fixed laboratory coordinate system)at a time t . We assume that the network of markers is

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dense enough to treat vL(x,t) as a continuous function ofcoordinates.

The number density ns(x,t) of the lattice sites per unitvolume satisfies the balance equation30

!ns

!t+ " · (nsvL) = rs, (1)

where rs(x,t) is the site generation rate (number of sites perunit volume per unit time). The sign of rs is positive for sitegeneration and negative for annihilation.1 This equation canbe rewritten as

dLns

dt+ ns" · vL = rs, (2)

where the lattice material time derivative dL/dt is defined by

dL

dt# !

!t+ vL·". (3)

The number density ni of each material species i obeys theparticle conservation law

!ni

!t+ " ·

!nivL + JL

i

"= 0, (4)

or

dLni

dt+ ni" · vL + " · JL

i = 0, (5)

where JLi # ni(vi ! vL) is the diffusion flux of species i

relative to the lattice and vi is its observed velocity relativeto the laboratory.

Since the markers are conserved during the deformationprocess, they can be used to define a deformation mappingx = x(x$,t) with respect to a chosen reference state, x$, of thematerial (see Fig. 1). This mapping defines the shape (or total)

x

y

x’

y’

F

F~

Reference state Deformed state

FIG. 1. (Color online) Dual-scale deformation of a solid materialwith site creation and annihilation. The marker sites (filled circles)and regular lattice sites (open circles) are connected by dashedlines to facilitate their tracking during the deformation. The shapedeformation gradient F is defined by the motion of the markers,whereas the local lattice deformation gradient F is defined by mappingof lattice sites in the vicinity of markers. Note that the deformationof the network of markers is different from the local latticedistortion.

deformation gradient31

F ##

!x!x$

$

t

(6)

and is related to the total lattice velocity (i.e., the velocity ofthe marker network) by

vL =#

!x!t

$

x$. (7)

If the material is crystalline, then besides F we can alsodefine another lattice deformation gradient F.32 To do so, weassume that for any lattice site we can identify instantaneousdirections of the crystallographic axes in its vicinity. Thisallows us to establish a local mapping between lattice vectors,y and y$, in the current and reference states, respectively (seeFig. 1). (The lattice vector mapping can break down in coreregions of lattice defects. It is assumed that such regionscomprise a negligibly small fraction of the material and do notoccur in the neighborhood of the markers.) The deformationgradient defined by

F ##

!y!y$

$

x$,t

(8)

represents local lattice distortions in the vicinity of a markersite x$. It should be emphasized that this definition of F doesnot imply conservation of sites in the vicinity of the marker.With time, some of the sites may disappear, but their locationsare then filled by other sites resulting in a self-reproducedlocal crystalline structure. This structure can be identified atany instant by examining the current atomic positions aroundthe marker and establishing their mapping on the referencecrystal structure. Since F is defined in a small vicinity ofevery marker site x$, we assume that it is independent ofy$ and is a continuous function of x$, i.e., F = F(x$,t). (Theability to describe lattice deformations by a single deformationgradient F relies on the assumption of a Bravais lattice ofthe crystal structure. Non-Bravais structures would requireadditional variables describing internal strains of the unitcell.)

Generally, F and F are two different tensors. In particular,the derivative

vL ##

!y!t

$

x$,y$(9)

defines the local lattice velocity vL due to elastic deformation,thermal expansion, and compositional strains. This velocityis generally different from the marker network velocity vL.The latter incorporates the same deformation effects as vL butadditionally includes the permanent deformation due to sitegeneration and annihilation.

Thus we introduce two different deformation gradientsbetween the same pair of deformed and reference states ofthe material: the shape deformation gradient F defined bythe marker-to-marker mapping, and the lattice deformationgradient F defined by local lattice mapping in the vicinityof every marker. The lattice site generation and annihilationduring the creep process produces permanent deformation

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leading to deviations of F from F. Experimentally, informationabout F could be obtained by x-ray diffraction measurementswhereas F could be simultaneously measured by dilatome-try. This type of measurement was used by Simmons andBalluffi33,34 to determine the equilibrium vacancy concen-tration in metals. This approach continues to be a powerfulexperimental tool for studying point-defect properties.35 Thisdual description of deformation is central to our theory and willbe employed for the calculations of the entropy production ratein the material and other kinetic characteristics of diffusionalcreep.

There is an important kinematic relation between thetwo velocities vL and vL, on one hand, and the latticesite production rate rs on the other. To derive it, return tothe site balance (2). This equation can be rewritten in theform

dLns

dt#

#!ns

!t

$

x$= rs ! ns"x · vL. (10)

On the other hand, the site density can be expressed as

ns = n$s

G, (11)

where G # det F and n$s is the lattice site density in the

reference state, assumed to be a constant. Using the Jacobiidentity31 it can be shown that

dLG

dt#

#!G

!t

$

x$= G"y · vL, (12)

where we used the local lattice mapping y = y(y$,x$,t) con-sidering the marker position x$ as a parameter. Applying thisrelation to Eq. (11), we have

dLns

dt#

#!ns

!t

$

x$= n$

s

%!(1/G)

!t

&

x$= !ns"y · vL. (13)

There is a subtle difference between Eqs. (10) and (13). InEq. (10), ns is the coarse-grained site density averaged over avolume containing a group of neighboring markers, whereasin Eq. (13), ns is a more detailed function of coordinatesnear a particular marker x$. Assuming that ns is a slowlyvarying function of coordinates on the scale of intermarkerdistances, we treat both densities as equal and their timederivatives in Eqs. (10) and (13) as identical. This immediatelygives

" · vL ! " · vL = rs

ns

. (14)

We dropped the subscripts of the divergence symbols, but itshould be remembered that the divergence of vL is taken locallywhereas the divergence of vL is coarse-grained over a volumecontaining multiple markers.

Equation (14) reflects the fact that the site generationcauses deviations of the total velocity divergence " · vL

from the local velocity divergence " · vL arising solelyfrom lattice distortions. In the absence of site generation,the two velocity fields are identical and Eq. (14) correctly

predicts rs = 0. We will show later that rs is the trace ofa more general tensor representing a more complete viewof the permanent deformation caused by site generation andannihilation.

III. BALANCE EQUATIONS

In this section, we summarize the momentum, energy andentropy balance relations that will be used in this work anddiscuss the assumptions and approximations underlying theserelations.

A. Momentum balance

For our multicomponent system, it is necessary to derivea consistent momentum balance equation. The standardmomentum equation for a single component solid, such astreated by Malvern,31 is no longer applicable because of themomentum carried by multicomponent diffusion. As shown inAppendix A, the correct momentum equation is

!

!t("v) + " ·

'n(

i=1

minivivi

)

= b + " · ! , (15)

where v is the barycentric velocity, " is the material density(mass per unit volume), b =

*ni=1 nibi is the external force

per unit volume, and " · ! is the force exerted by the stress !per unit volume of the material. We assume that the externalfields are conservative, so that the force per particle is

bi = !"#i , (16)

where #i are species-specific potential functions.Equation (15) can be rewritten with respect to the lattice

(see Appendix A):

"dLvL

dt= b + " · (! ! M) ! dLq

dt! q" · vL ! q · "vL,

(17)

where tensor M is given by

M #n(

i=1

mi

ni

JLi JL

i (18)

and vector

q #n(

i=1

miJLi = " (v ! vL) (19)

is the momentum density carried by the local center of massrelative to the lattice. Here, mi is the mass of particles ofspecies i. The derivative dLq/dt is the inertia force, whicharises due to the fact that the lattice and barycentric referencesare both noninertial.

B. Energy balance

The total energy e of the material per unit volume can beexpressed as

e = K + # + u, (20)

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where

K = 12

n(

i=1

mini |vi |2 (21)

is the macroscopic kinetic energy of the particles per unitdeformed volume,

# #n(

i=1

ni#i (22)

is potential energy in external fields per unit volume, andthe remaining term u is identified with internal energy perunit deformed volume. The latter includes the energy ofinteractions between the particles and the kinetic energy oftheir microscopic motion (e.g., lattice vibrations, molecularrotations, etc.), but excludes the macroscopic kinetic energydue to diffusion. The internal energy can be shown to satisfythe following balance equation with respect to the lattice (seeAppendix A):

dLu

dt+ u" · vL

= !" · JLu +

n(

i=1

bi · JLi + (! ! M) : "vL

!n(

i=1

+"

%mi

2n2i

!JL

i · JLi

"&+ mi

dLvi

dt

,· JL

i , (23)

where JLu is the internal energy flux relative to the lattice.

Equation (23) is valid for all, not necessarily reversible,process and expresses the first law of thermodynamics statingthat the change in internal energy equals the work done onthe system less the energy dissipated through its boundaries.As with the momentum balance relation (17), Eq. (23) isexact: it represents the internal energy balance without anyapproximations or assumptions other than the conservation ofenergy and the total energy ansatz (20).

We will also need the potential energy balance relation,

dL#

dt+ #" · vL = !

n(

i=1

bi · JLi !b · vL ! " ·

'n(

i=1

#iJLi

)

,

(24)

where the last term represents the divergence of the diffusiveflux of potential energy. This relation is also exact.

C. Entropy balance

The entropy balance is postulated in the form

dLs

dt+ s" · vL + " · JL

s = s, (25)

where s is entropy per unit deformed volume, JLs is the entropy

flux carried by the conduction of heat relative to the lattice, ands is the entropy production rate due to irreversible processes.

The goal of the subsequent development will be to computes. The common approach6 to achieving this goal is to calculatethe entropy rate (dLs/dt + s" · vL) and then rearrange theterms to form the divergence of fluxes that can be identifiedwith !" · JL

s . The remaining terms are then identified with

s. We will follow this route to derive s for a solid materialcontaining nonconserved lattice sites.

IV. LOCAL REVERSIBLE THERMODYNAMICS

A. The local equilibrium postulate

It is assumed that, although the entire solid material canbe away from equilibrium, its local internal energy, entropyand other thermodynamic variables are related to each othervia a fundamental equation of state describing reversibleprocesses. “Local” means here that this equation is followedonly by subsystems of the entire system that are smallenough to reach thermodynamic equilibrium before the entiresystem does, yet large enough to apply the full formalism ofthermodynamics. The locally equilibrium subsystems need notbe uniform and can be treated using the formalism of gradientthermodynamics.17–20

Relative to the moving lattice, the fundamental equation ispostulated in the functional form:

u = u(s,{ni},{$k},{"ni},{"$k},F). (26)

Here, $k (k = 1, . . . ,K) are nonconserved phase fields, "ni

and "$k are respective gradients, and F is the lattice deforma-tion gradient relative to a chosen reference state (see Sec. II).The phase fields $k can represent different thermodynamicphases of the material or be associated with different latticeorientations (grains) in a single-phase polycrystalline material.The gradients "$k and "ni are usually negligibly small insidethe bulk phases or grains but are important in the descriptionof interphase interfaces and GBs. The material regions whosethermodynamic description requires the gradients17–20 arereferred to as “nonclassical” as opposed to “classical” regions,which can be treated within the standard thermodynamics16 ofhomogeneous phases. Since u is a scalar while the gradients arevectors and F is a tensor, it is assumed that Eq. (26) satisfies therequired invariance under rotations of the coordinate system.It is also assumed to satisfy the frame-invariance (objectivity)requirement.31

When Eq. (26) is applied to different locations in the solid,it is assumed that the reference state used to describe the latticedeformation is the same for every location and is fixed onceand for all. For example, for a cubic crystal, the reference statecan be a perfectly cubic unit cell with a given (e.g., stress-free)lattice constant. This explains why properties of the referencestate, such as the reference volume per site, are not listedamong the variables of u.

B. The first and second laws of thermodynamics for localreversible processes

To derive the differential form of Eq. (26), let us firstconsider a uniform region containing a fixed number of latticesites. Suppose for the moment that the phase fields $k are notincluded. The standard differential form of the fundamentalequation for such a region is

dU = T dS +n(

i=1

MidNi + V (F!1 · ! ) · ·dF. (27)

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Here, U = uV , S = sV , and Ni = niV are the total internalenergy, entropy, and numbers of particles of the chemicalcomponents inside the region, respectively, V is its volume,T # !U/!S is temperature, and F!1 is the inverse of F. Thetensor ! is formally defined through the derivative !U/!F,

! # 1V

F · !U

!F, (28)

and has the meaning of the equilibrium Cauchy stress tensorin a uniform lattice. The Cauchy stress tensor is symmetricalas dictated by the angular-momentum balance in the absenceof body couples.31 As will be discussed later, ! is generallydifferent from the actual stress tensor ! in a nonuniform and/ornonequilibrium material. The obvious motivation behind thedefinition (28) is the standard form V $P · ·dF of the mechanicalwork term in continuum mechanics,31 V $ being the referencevolume of the region and P = J F!1 · ! the first Piola-Kirchhoff stress tensor. Finally, the derivative Mi # !U/!Ni

has the meaning of the diffusion potential of species i relativeto vacancies if the latter are treated as massless species. Ifonly the material particles are treated as species, Mi can beconsidered as simply the chemical potential of species i. Asdiscussed in the literature,12,13 both interpretations of Mi areequally legitimate and give the same results for all physicallyobservable quantities.

Equation (27) can be rewritten in terms of the volumedensities u, s, and ni :

du = T ds +n(

i=1

Midni + (F!1 · ! )

· · dF !'

u ! T s !n(

i=1

Mini

)dV

V. (29)

Using the identity31

dV

V= F!1 · ·dF, (30)

we obtain

du = T ds +n(

i=1

Midni + [F!1 · (! ! %I)] · ·dF, (31)

where I is the second rank unit tensor and

% # u ! T s !n(

i=1

Mini (32)

is the grand-canonical potential per unit volume.Equation (31) is the differential form of Eq. (26) for the

particular case of a uniform material without phase fields. Inthe presence of phase fields and the gradients "$k and "ni ,this equation becomes

du = T ds +n(

i=1

Midni +K(

k=1

!u

!$k

d$k +n(

i=1

!u

!"ni

· d"ni

+K(

k=1

!u

!"$k

· d"$k + [F!1 · (! ! %I)] · ·dF. (33)

Note that % appearing in the last term is now a nonclassicalquantity as it depends on the gradients through u [cf. Eq. (26)].

The gradient terms in Eq. (33) can be rearranged using theidentities

n(

i=1

!u

!"ni

· d"ni

=n(

i=1

" ·#

!u

!"ni

dni

$!

n(

i=1

#" · !u

!"ni

$dni

+n(

i=1

!u

!"ni

· (d"ni ! "dni) , (34)

K(

k=1

!u

!"$k

· d"$k

=K(

k=1

" ·#

!u

!"$k

d$k

$!

K(

k=1

#" · !u

!"$k

$d$k

+K(

k=1

!u

!"$k

· (d"$k ! "d$k) , (35)

where we recognize that the operations d and " may notcommute. Equation (33) finally becomes

du = T ds +n(

i=1

M%i dni +

K(

k=1

&%kd$k

+" ·'

n(

i=1

!u

!"ni

dni +K(

k=1

!u

!"$k

d$k

)

+n(

i=1

!u

!"ni

· (d"ni ! "dni)

+K(

k=1

!u

!"$k

· (d"$k ! "d$k) + [F!1 · (! ! %I)] ·· dF,

(36)

where36

M%i # !u

!ni

! " · !u

!"ni

= Mi ! " · !u

!"ni

(37)

is the nonclassical diffusion potential and

&%k # !u

!$k

! " · !u

!"$k

. (38)

Note that M%i and &%

k are variational derivatives37 of theinternal energy with respect to the concentrations ni and phasefields $k , respectively.

The obtained Eq. (36) is a formulation of the first andsecond laws of thermodynamics for local reversible processesin a lattice obeying the fundamental equation (26). It will serveas the starting point for several derivations performed below.

C. Generalized Gibbs-Duhem equation

By applying a partial Legendre transformation37 withrespect to s and ni , Eq. (36) can be transformed

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to

sdT +n(

i=1

nidM%i + d% !

K(

k=1

&%kd$k +

n(

i=1

d

#ni" · !u

!"ni

$! " ·

'n(

i=1

!u

!"ni

dni +K(

k=1

!u

!"$k

d$k

)

!n(

i=1

!u

!"ni

· (d"ni ! "dni) !K(

k=1

!u

!"$k

· (d"$k ! "d$k) ! (F!1 · (! ! %I)) · ·dF = 0. (39)

This equation can be viewed as a generalization of theGibbs-Duhem equation16 to a nonclassical solid subject tononhydrostatic mechanical stresses. In the particular case ofa hydrostatically stressed classical (no gradients) solid wehave ! = !pI, where p = !% is the equilibrium hydrostaticpressure, and Eq. (39) reduces to the standard Gibbs-Duhemequation16

sdT +n(

i=1

nidMi ! dp = 0. (40)

As an application of Eq. (39), suppose the differentialsrepresent infinitesimal differences between the values ofproperties at two nearby points x and x + dx at a fixed momentof time. Then dT = "T · dx, dM%

i = "M%i · dx, and similarly

for all other terms. In this particular case, the operators dand " commute, "d = d" = dx · "", and both sums in thethird line of Eq. (39) vanish. The remaining terms contain thecommon factor dx which cancels, giving

s"T +n(

i=1

ni"M%i !

K(

k=1

&%k"$k + " · (A% + %I)

! [F!1 · (! ! %I]) · ·(F&!" ) = 0, (41)

where

A% #'

n(

i=1

ni" · !u

!"ni

)

I !n(

i=1

!u

!"ni

"ni !K(

k=1

!u

!"$k

"$k

(42)

is a purely nonclassical second-rank tensor.Equation (41) is a gradient form of the generalized Gibbs-

Duhem equation (39). As will be shown later (see Sec. V),when the material reaches full thermodynamic equilibrium(including equilibrium with respect to site generation andannihilation), the first, third, and last terms in Eq. (41) vanishwhile the nonclassical chemical potentials satisfy the condition"M%

i ! bi = 0. It follows that under the full equilibriumconditions,

" · (A% + %I) + b = 0. (43)

Thus, in the absence of external fields, tensor (A% + %I) isdivergence free. In a one-dimensional system, this meansconservation of the quantity (A%

11 + %).The divergence-free character of (A% + %I) in the absence

of external fields originates from the property of the funda-mental equation (26) that the internal energy does not dependexplicitly on the position vector x. If it did, an additional term!u/!x would appear in Eq. (33) and eventually propagateto Eq. (43), so that the divergence of (A% + %I) would no

longer be zero. The mathematical procedure that producedthe divergence term in Eqs. (41) and (43) is essentiallyequivalent to a derivation of Noether’s theorem37 for a systemwith continuous translational symmetry. Applied fields bi

obviously destroy this symmetry and lead to a nonzerodivergence of (A% + %I) as indicated in Eq. (43).

D. Time-dependent form of the first and second laws

Returning to the general Eq. (36), we now consider the casewhere the differentials represent changes in time. Because theinternal energy u has been defined relative to the stationarylattice, its time evolution must be described by the latticematerial derivative dLu/dt defined by Eq. (5). We willtherefore interpret all differentials d in Eq. (36) as dL/dt .

The operators dL/dt and " do not commute, but it can beshown that38

dL

dt" ! " dL

dt= !"vL · ". (44)

As a result, the second line in Eq. (36) becomes

!n(

i=1

!u

!"ni

· "vL · "ni !K(

k=1

!u

!"$k

· "vL · "$k

and can be simplified to

!'

n(

i=1

!u

!"ni

"ni +K(

k=1

!u

!"$k

"$k

)

: "vL.

The last term in Eq. (36) can be transformed to

[F!1 · (! ! %I)] · ·dLFdt

= (! ! %I) : "vL,

where we used the identity39

dLFdt

= (vL&!" ) · F. (45)

Equation (36) becomes

dLu

dt= T

dLs

dt+

n(

i=1

M%i

dLni

dt+

K(

k=1

&%k

dL$k

dt

+" ·'

n(

i=1

!u

!"ni

dLni

dt+

K(

k=1

!u

!"$k

dL$k

dt

)

!'

n(

i=1

!u

!"ni

"ni +K(

k=1

!u

!"$k

"$k

)

: "vL

+ (! ! %I) : "vL. (46)

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

Note that this equation contains both the total (marker network)velocity and the local velocity of the lattice, the former comingfrom the material time derivatives and the latter from the locallattice deformation gradient.

The term with the chemical potentials can be furtherrearranged using the species conservation law, Eq. (5):

n(

i=1

M%i

dni

dt= !

n(

i=1

M%i " · JL

i !'

n(

i=1

M%i ni

)

" · vL

= !" ·'

n(

i=1

M%i JL

i

)

+n(

i=1

JLi · "M%

i

!'

n(

i=1

M%i ni

)

" · vL. (47)

For further calculations we need the energy and entropyrates to appear in the combinations (dLu/dt + u" · vL) and(dLs/dt + s" · vL), respectively. This is readily achieved byadding and subtracting u" · vL and s" · vL in Eq. (46). Aftersimple rearrangements, we arrive at the equation

dLu

dt+ u" · vL

= T

#dLs

dt+ s" · vL

$+

n(

i=1

JLi · "M%

i +K(

k=1

&%k

dL$k

dt

+" ·'

n(

i=1

!u

!"ni

dLni

dt+

K(

k=1

!u

!"$k

dL$k

dt!

n(

i=1

M%i JL

i

)

+(! + A%) : "vL ! (! ! %I) : R, (48)

where

R # ("vL ! "vL) . (49)

The tensor R represents the permanent part of the totaldeformation rate coming from the site generation and annihi-lation. According to Eq. (14) its trace,

Tr(R) = (" · vL ! " · vL) = rs

ns

, (50)

gives the site generation rate rs . However, the tensor Rcarries more information than rs as it reflects the possibleanisotropy in the generation of lattice sites. It differentiates,for example, between insertion of new lattice planes normalto a certain direction and creation of the same number of sitesby uniform “swelling” of the material. In fact, R captureseven a pure shear deformation rate in which new lattice planesare inserted parallel to one crystallographic orientation andsimultaneously removed parallel to another crystallographicorientation perpendicular to the first, so that the total numberof sites remains constant. One possible mechanism of thisprocess would be a concurrent climb of two perpendicularsets of edge dislocations, one inserting lattice planes andthe other eliminating perpendicular lattice planes. This couldbe accomplished by vacancy diffusion between the cores ofthe two dislocation sets without changing the net amount ofvacancies in the region.

Tensor R is related to the generalized creep strain-ratetensor "gc introduced by Svoboda et al.7,8 although the latter,by contrast to R, comprises both permanent and elastic parts

of the deformation. Similar to R, the tensor "gc includes boththe volume creep deformation by “swelling” or contractionand shear deformation arising due to orientational anisotropyof the microstructure or from nonhydrostatic components ofthe stress tensor.

It should be emphasized that Eq. (48) has been derivedfrom the fundamental Eq. (26) by a chain of mathematicaltransformations without any additional physical assumptionsor approximations other than the conservation and balanceequations of Secs. II and III. Equation (48) represents a time-dependent form of the first and second laws of thermodynamicsfor reversible processes in a continuous medium with thepostulated equation of state (26).

V. THE STATE OF EQUILIBRIUM

A. Derivation of equilibrium conditions

Before analyzing irreversible processes, we will derive theconditions of thermodynamic equilibrium of a multicompo-nent solid capable of site generation. This could be doneby requiring that the first-order variation of the total energyof a given material region enclosed in a rigid envelope bezero under the constraints of fixed entropy and fixed totalnumber of particles of every species. Instead of consideringinfinitesimal variations of the relevant parameters, we willreuse Eq. (48) by treating the rates of the reversible changes ofthose parameters as their variations. For example, the variation'$k can be formally considered to occur per unit time and berepresented by the material derivative dL$k/dt . Likewise, thevirtual lattice displacement 'xL can be thought of as occurringper unit time and be replaced by the lattice velocity vL.The macroscopic kinetic energy is a second-order variationand is excluded. This treatment is completely equivalent tothe virtual displacement method usually applied for findingthermodynamic equilibrium of continuous media.10,14,31

The equilibrium condition is- #

dLu

dt+ u" · vL

$dV +

- #dL#

dt+ #" · vL

$dV

!(

- #dLs

dt+ s" · vL

$dV = 0. (51)

The first integral is equivalent to- #

!u$

!t

$

x$dV $, (52)

u$ being internal energy per unit reference volume,40 andrepresents the rate of internal energy change of a given materialregion defined by lattice markers. Likewise, the second andthird integrals represent the rates of potential energy andentropy changes of the same material region. The entropyintegral has been added to impose the entropy constraintusing the Lagrange multiplier (. The required conservationof the total amount of each species will be enforced by zeronormal components of the diffusion fluxes at the boundary ofthe region and need not be imposed via additional Lagrangemultipliers.

Inserting the first integrand from Eq. (48), the divergenceterm becomes the surface integral over the boundary of the

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

region,

-n·

'n(

i=1

!u

!"ni

dLni

dt+

K(

k=1

!u

!"$k

dL$k

dt!

n(

i=1

M%i JL

i

)

dA,

(53)

n being a unit normal vector pointing outside the region anddA an increment of area. To ensure that the region is closed,the normal components of the diffusion fluxes will be takento be zero, n · JL

i = 0. Imposing also fixed boundary valuesof ni and $k , this surface integral vanishes. Furthermore, thevolume integral

-(! + A%) : "vLdV (54)

can be rewritten using the divergence theorem as-

n · (! + A%) · vLdA !-

" · (! + A%) · vLdV, (55)

where the surface integral is zero due to the boundary conditionvL = 0 (rigid boundary). Similarly, per Eq. (24), the potentialenergy integral contains the potential energy flux whichvanishes on the boundary, leaving

- #dL#

dt+ #" · vL

$dV = !

- 'n(

i=1

bi · JLi + b · vL

)

dV .

(56)

Combining the above equations, Eq. (51) becomes- #

dLs

dt+ s" · vL

$(T ! ()dV

+- n(

i=1

JLi · "(M%

i + #i)dV +- K(

k=1

&%k

dL$k

dtdV

!-

[" · (! + A%) + b] · vLdV !-

(! ! %I) : RdV = 0.

(57)

In the state of equilibrium, this relation must hold forany arbitrarily chosen region of the material. The integrandsare proportional to the entropy rate (dLs/dt + s" · vL), thephase-field rates dL$k/dt , the diffusion fluxes JL

i , the latticevelocity vL, and the creep deformation rate R, respectively.All these rates represent independent variations away from theequilibrium state. Assuming that they can take any arbitrarypositive or negative values, the coefficients multiplying theserates must be zero. We thus arrive at the following set ofnecessary conditions of equilibrium:

T = ( = const: thermal equilibrium, (58)

M%i + #i = const: chemical equilibrium, (59)

&%k = 0: phase-field equilibrium, (60)

" · (! + A%) + b = 0: mechanical equilibrium, (61)

! = %I: site generation equilibrium. (62)

B. Discussion of the equilibrium conditions

Equations (58)–(60) reproduce the well-known conditionsof thermal, chemical, and phase-field equilibrium: the unifor-mity of the temperature field, the constancy of the nonclassicalchemical potential M%

i plus the external potential #i for everyspecies, and vanishing variational derivative &%

k for everyphase field. The mechanical equilibrium condition could havebeen obtained from zero accelerations and zero diffusion fluxesin the momentum balance equation (17), giving " · ! + b = 0.Equation (61) shows that tensor (! + A%) plays the role of thenonclassical stress tensor. The latter has long been knownin fluid systems as the capillary tensor or Korteweg stress.41

In classical regions where the gradients of the chemicalcomposition and phase fields can be neglected and thusA% = 0, the mechanical equilibrium condition reduces to" · ! + b = 0, confirming that the tensor ! defined earlier byEq. (28) is indeed the equilibrium stress tensor in a classicalsolid.

Equation (62) is the condition of equilibrium with respectto site generation and annihilation, stating that tensor !must be diagonal: ! # !pI. This condition must be fulfilledeverywhere in the equilibrium material, including nonclassicalregions with significant gradients of ni and/or $k , suchas interface regions. However, the actual stress tensor insuch regions, (!pI + A%), remains nonhydrostatic due to thenonclassical contribution A%.

If Eq. (62) is satisfied, the mechanical equilibrium conditionbecomes

!"p + " · A% + b = 0 (63)

and in classical regions reduces to the standard hydrostaticequilibrium condition "p = b.31 Thus, in the presence ofefficient sinks and sources of vacancies capable of maintainingsite equilibrium, the solid behaves rheologically like a fluid.

Note that by inserting the obtained equilibrium conditions(58)–(60) and (62) in the generalized Gibbs-Duhem equation(41), we immediately recover Eq. (43) or its equivalent form(63). In other words, if all other equilibrium conditions aresatisfied, the mechanical equilibrium condition follows fromthe generalized Gibbs-Duhem equation (41). The reverse is nottrue: the mechanical equilibrium condition (61) can be satisfiedeven if the material has not reached complete equilibrium, inwhich case Eq. (43) is invalid.

According to Eq. (62), in equilibrium p = !%, i.e.,

u ! T s + p !n(

i=1

Mini = 0, (64)

in both classical and nonclassical regions. In classical regions,this relation has a clear thermodynamic meaning. In suchregions, the actual state of stress of the material is hydrostaticwith the pressure p. Under such conditions, one can uniquelydefine the chemical potentials µi of all chemical species aswell as the chemical potential µv of vacancies treated as ficti-tious massless species. (In nonhydrostatically stressed solids,chemical potentials of material species and vacancies cannotbe defined simultaneously due to the network constraint.10–12)The diffusion potentials Mi are then Mi = µi ! µv and the

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

left-hand side of Eq. (64) becomes

0 ='

u ! T s + p !n(

i=1

µini ! µvnv

)

+ µvns = µvns,

(65)

where nv # ns !*

i ni is the number density of vacanciesper unit volume and we used the Gibbs relation for hydrostaticsystems,16

u ! T s + p =n(

i=1

µini + µvnv. (66)

Thus Eq. (64) predicts that the equilibrium chemical potentialof vacancies in classical regions is zero:

µv = 0. (67)

This relation cannot be extended to nonclassical regions, e.g.,interfaces, where µv remains undefined.

It is important to recognize that the equilibrium condition(62) has been derived by considering independent variations ofall components of the creep deformation rate tensor R. Underreal conditions, the material’s microstructure can impose re-strictions on some of such variations. For example, the materialcan be only capable of site generation/annihilation by insertionor removal of lattice planes normal to a particular direction,

e.g., by growth or shrinkage of extrinsic stacking faults inthose planes. Alternatively, the site generation/annihilationcan occur exclusively by growth or dissolution of nanoporespermitting only isotropic “swelling” or contraction of thematerial. In all such cases, the material can reach a constrainedthermodynamic equilibrium with only some of the componentsof (! ! %I), or their linear combinations, being zero. In suchcases, the equilibrium stress tensor ! need not be hydrostatic.Under such constrained equilibrium conditions, Eq. (67) is nolonger valid, and furthermore, µv itself is undefined.

In the limiting case when the material does not containany sinks or sources of vacancies, R is identically zero andthe material can be equilibrated in any nonhydrostatic state ofstress. Equation (62) should be then removed from the list ofequilibrium conditions.

VI. IRREVERSIBLE THERMODYNAMICS

A. The entropy production rate

As indicated in Sec. III C, a route to the entropy productionis to (i) insert in Eq. (48) the internal energy rate (dLu/dt +u" · vL) from the energy balance equation (23), and (ii) splitthe obtained total entropy rate (dLs/dt + s" · vL) into theentropy flux !" · JL

s and entropy production rate s.Step (i) gives

T

#dLs

dt+ s" · vL

$+

n(

i=1

JLi ·

+"

%M%

i + mi

2n2i

(Ji · Ji)&

! bi + mi

dLvi

dt

,

+" ·'

JLu +

n(

i=1

!u

!"ni

dLni

dt+

K(

k=1

!u

!"$k

dL$k

dt!

n(

i=1

M%i JL

i

)

+K(

k=1

&%k

dL$k

dt! (! ! M ! ! ! A%) : "vL ! (! ! %I) : R = 0. (68)

Solving this equation for the total entropy rate,

dLs

dt+ s" · vL = !" ·

'JL

q

T

)

! 1T

n(

i=1

JLi ·

+"

%M%

i + mi

2n2i

(Ji · Ji)&

! bi + mi

dLvi

dt

,! 1

T

K(

k=1

&%k

dL$k

dt+ JL

q · " 1T

+ 1T

(! ! M ! ! ! A%) : "vL + 1T

(! ! %I) : R, (69)

where

JLq # JL

u +n(

i=1

!u

!"ni

dLni

dt+

K(

k=1

!u

!"$k

dL$k

dt!

n(

i=1

M%i JL

i (70)

is a heat flux relative to the lattice. The latter equals the total internal energy flux JLu less the internal energy transferred by

diffusion and by the motion of phase transformation fronts or GBs.Identifying the entropy flux

JLs #

JLq

T(71)

we finally obtain the entropy production rate:

s = ! 1T 2

"T · JLq ! 1

T

n(

i=1

+"

%M%

i + mi

2n2i

(Ji · Ji) + #i

&+ mi

dLvi

dt

,· JL

i

! 1T

K(

k=1

&%k

dL$k

dt+ 1

T(! ! M ! ! ! A%) : "vL + 1

T(! ! %I) : R. (72)

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

The individual terms of Eq. (72) describe the entropyproduction due to (i) heat conduction, (ii) diffusion driven bygradients of the nonclassical diffusion potentials M%

i , kineticenergy of diffusion mi(Ji · Ji)/2n2

i and external potentials#i , and by inertia forces mid

Lvi/dt , (iii) evolution ofthe phase fields, (iv) viscous dissipation by conversion ofthe strain rate to heat (e.g., generation of phonons), and(v) generation/annihilation of lattice sites. Each term can beinterpreted as the product of a driving force and a conjugategeneralized “flux,” the “fluxes” being JL

q (heat), JLi (diffusion),

dL$k/dt (phase-field evolution rate), "vL (the symmetricalpart of which is the deformation rate), and R (site generationrate).

Equation (72) represents the exact entropy production.For applications to slow processes such as creep, it can besimplified by neglecting the terms quadratic in diffusion fluxesand the inertia terms (see Appendix B). The approximate formof the entropy production rate, which will be used in the restof the paper, becomes

s = ! 1T 2

"T · JLq ! 1

T

n(

i=1

"(M%i + #i) · JL

i

! 1T

K(

k=1

&%k

dL$k

dt+ 1

T(! ! ! ! A%) : "vL

+ 1T

(! ! %I) : R. (73)

It is instructive to apply Eq. (73) to the state of ther-modynamic equilibrium, in which all driving forces mustvanish. Equating the driving forces to zero recovers thepreviously found conditions of thermal equilibrium (58),chemical equilibrium (59), phase-field equilibrium (60), andthe site generation equilibrium (62) (see Sec. V A). Thusthe fully equilibrated material is correctly predicted to behydrostatic. According to Eq. (73), in the absence of viscousdissipation, the dynamic stress tensor ! reduces to its staticvalue (! + A%) (see Sec. V B):

! = ! + A% (no viscous dissipation). (74)

We do not recover the mechanical equilibrium condition (61).However, the latter follows at once from the generalized Gibbs-Duhem equation (41) (see Sec. IV C) provided that all otherequilibrium conditions are satisfied.

It is interesting to note that if any of the components of(! ! %I) in the equilibrium state are nonzero due to restrictionson site generation, the condition of zero entropy productiondoes not recover the mechanical equilibrium condition. Thisshould not be surprising since s = 0 is only a necessarybut not sufficient condition of thermodynamic equilibrium.The absence of entropy production can be satisfied not onlyin the equilibrium state but also during (nearly) reversiblemechanical processes, such as propagation of elastic waveswith negligible dissipation.

In many situations, some of the driving forces appearing inEq. (73) can be negligibly small and the process can be drivenby the remaining forces. For example, on sufficiently shorttime scales, the site generation and diffusion processes can beneglected (R = 0, JL

i = 0) and the material can undergo fast(e.g., shock) deformation accompanied by viscous dissipation,

conduction of heat and possibly diffusionless phase trans-formations. As another example, for slow enough processes,one can neglect the viscous dissipation and assume thermaland mechanical equilibrium, leaving only diffusion, phasetransformations or GB motion, and site generation as thedominant processes. It is this latter regime that appears tobe most relevant to diffusional creep and will be discussed inmore detail later in Sec. VII.

B. Phenomenological relations

1. Material symmetry considerations

We will next postulate linear phenomenological relationsbetween the fluxes and forces appearing in the entropyproduction rate, Eq. (73). Generally, each flux can be linearlyrelated (coupled) to all forces entering this expression, andthe matrix of the linear coefficients must be symmetric by theOnsager reciprocal relations.42,43 It is known, however, thatsymmetry properties of the material44 can prevent couplingbetween certain fluxes and forces (Curie symmetry principle).In particular, if all properties of the material are isotropic,a flux can be caused only by forces having the sametensorial character. Quantities with four distinct types oftensorial character usually occur in expressions for the entropyproduction: scalars, polar vectors, axial vectors, and symmetrictraceless tensors of rank two. We will start by rearranging theterms in (73) according to their tensorial character. This steprequires only mathematical rearrangements in Eq. (73) anddoes not involve any assumptions regarding the symmetry ofthe material.

The phase-field rates dL$k/dt are scalars and the fluxes ofthe chemical components and heat are polar vectors. The forcesconjugate to these fluxes have the same tensorial character asthe fluxes. Thus we need not do anything about these terms.The remaining terms are double contractions of second ranktensors, which will be partitioned as follows.6 Each tensor Ais split in three parts:

A = 13 Tr(A)I + A(s) + A(a), (75)

where

A(s) = 12 (A + AT ) ! 1

3 Tr(A)I (76)

is the traceless symmetric part of A and

A(a) = 12 (A ! AT ) (77)

is the antisymmetric part of A. Applying this decompositionto two second-rank tensors A and B, it can be shown that

A : B = 13 Tr(A)Tr(B) + A(s) : B(s) + A(a) : B(a). (78)

The last term is equivalent to a dot product of two axialvectors.6 Thus the operation “:” only couples parts of thetensors that have the same tensorial character.

Applying this tensor decomposition and grouping to-gether the terms with the same tensor character, the entropy

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

production rate becomes

s = ! 1T

K(

k=1

&%k

dL$k

dt+ 1

T)" · vL ! 1

T(% ! *h) r

! 1T 2

"T · JLq ! 1

T

n(

i=1

"(M%i + #i) · JL

i

+ 1T

# : ("vL)(s) + 1T

(! ! *hI) : R(s)

! 1T

A(a)% : W, (79)

where r # rs/ns is the number of new sites generated per unittime per existing site and *h = (1/3)Tr(! ) is the “hydrostaticpart” of ! . In the above equation,

("vL)(s) = D ! 13 (" · vL) I (80)

is the total shear strain rate and

D = 12 ["vL + ("vL)T ] (81)

is the total deformation rate tensor.31 Tensor

W # ("vL)(a) = 12 ["vL ! ("vL)T ] (82)

is sometimes called the vorticity tensor and characterizes therate of lattice rotation.31 The symmetric part of the creepdeformation rate R is

R(s) = D ! D ! r

3I, (83)

where

D = 12 ["vL + ("vL)T ], (84)

and describes the rate of pure elastic deformation caused bythe creep process.

The scalar forces appearing in Eq. (79) include thenonclassical bulk viscosity stress

) = *h ! *h ! 13 Tr(A%) (85)

and the volume driving force for the site generation, (% ! *h).The tensor forces include the nonclassical viscous shear stress

# = ! ! ! ! A(s)% ! (*h ! *h) I (86)

and the driving force for the shear creep, (! ! *hI). Theindividual components of tensor A% are

A(s)% = !n(

i=1

#!u

!"ni

"ni

$(s)

!K(

k=1

#!u

!"$k

"$k

$(s)

, (87)

A(a)% = !n(

i=1

#!u

!"ni

"ni

$(a)

!K(

k=1

#!u

!"$k

"$k

$(a)

, (88)

Tr(A%) # 3

'n(

i=1

ni" · !u

!"ni

)

!n(

i=1

!u

!"ni

· "ni

!K(

k=1

!u

!"$k

· "$k. (89)

Note that the entropy production due to viscous dissipationis now split in three parts: the bulk viscosity )" · vL,the viscous shear # : ("vL)(s), and the rotational viscosityA(a)% : W. A similar splitting is used for fluid systems.6

The site generation is split in two parts: the volume part(% ! *h)r describing isotropic “swelling” or shrinkage ofthe material, and the shear part (! ! *hI) : R(s) describingshape deformation without changing the total number of sites.The latter process was discussed in the end of Sec. IV D. Notethat the antisymmetric part of tensor R does not appear in theentropy production due to the symmetry of the equilibriumCauchy stress ! .

2. Phenomenological relations for creep in isotropic materials

Each of the four lines in Eq. (79) contains terms withcontraction of tensors of the same tensor character. If thematerial is isotropic, only terms appearing in the same linecan couple with each other but not with terms in other lines.6

Furthermore, the phenomenological coefficients have to bescalars regardless of the tensor character of the fluxes andforces. [It is worth noting that this is true only when thesymmetric tensors are traceless. Symmetric tensors with atrace, such as the stress and small-strain tensors in elasticallyisotropic (e.g, cubic) materials, are linearly related withtwo phenomenological coefficients, e.g., the shear and bulkmoduli.45] This leads to the following phenomenologicalequations.

The scalar quantities appearing in the first line of Eq. (79)are coupled by the equations

dL$k

dt= ! 1

T

K(

m=1

Bkm&%m + 1

TBkv) ! 1

TBkr (% ! *h) ,

k = 1, . . . ,K,(90)

" · vL = 1T

K(

m=1

Bvm&%m + 1

TBvv) ! 1

TBvr (% ! *h) ,

r = ! 1T

K(

m=1

Brm&%m + 1

TBrv) ! 1

TBrr (% ! *h) .

By the Onsager relations,42,43 the (K + 2) ' (K + 2) matrixB is symmetric and must be positive definite. In particular,the diagonal coefficients Bkk , Bvv , and Brr must be positive.Generally, the site generation can be influenced by viscous dis-sipation, phase transformations and GB motion. Conversely,the phase-field evolution and viscosity can be influenced bysite generation.

The second line of Eq. (79) describes diffusion of thechemical species and heat. For simplicity, let us neglect thethermodiffusion cross-effects and decouple heat conductionfrom diffusion,

JLq = !Lqq

1T 2

"T , (91)

where Lqq > 0 is related to the heat conductivity coefficient +by + = Lqq/T 2. Then the diffusion equations form a separatesystem,

JLi = ! 1

T

n(

j=1

Lij"(M%j + #j ), i = 1, . . . ,n. (92)

The n ' n matrix L is symmetric and positive definite.In fluid systems in mechanical equilibrium, the chemical

potential gradients are linearly related to each other by the

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

Gibbs-Duhem equation.6,29 As a result, one of the gradients canbe eliminated. The complex solid systems considered here fol-low the generalized Gibbs-Duhem equation given by Eq. (41).Even in the absence of external fields (#j = 0), "M%

i arelinearly related only if the material is in thermal, phase-field,mechanical, and site-generation equilibrium (and thus in thehydrostatic state of stress). To keep the treatment general,we will treat the diffusion potential gradients as independentforces and the system of equations (92) as n ' n.

From the third line of Eq. (79), the shear viscosity rate andthe shear creep deformation rate are coupled by the equations

R(s) = 1TSrr (! ! *hI) + 1

TSrv# ,

(93)("vL)(s) = 1

TSvr (! ! *hI) + 1

TSvv# ,

where the matrix of coefficients is symmetric and positivedefinite. The diagonal coefficients Srr > 0 and Svv > 0 char-acterize the kinetics of shear creep deformation and shearviscosity, respectively, the latter being related to the viscositycoefficient , by Svv = 2T ,. Finally, it has been shown25 thatin isotropic materials A(a)% = 0. For such materials, the lastline of Eq. (79) involving the vorticity tensor W disappears.

As already mentioned, for slow processes such as creep itis reasonable to neglect the viscous dissipation and assume auniform temperature field and mechanical equilibrium. Theremaining phenomenological equations describe diffusion,phase-field evolution, site generation and creep deformation.Assuming for simplicity that the material is not subject toexternal fields, the obtained system of equations is

JLi = ! 1

T

n(

j=1

Lij"M%j , i = 1, . . . ,n, (94)

dL$k

dt= ! 1

T

K(

m=1

Bkm&%m ! 1

TBkr (% ! *h) , k = 1, . . . ,K,

(95)

r = ! 1T

K(

m=1

Brm&%m ! 1

TBrr (% ! *h) , (96)

R(s) = 1TSrr (! ! *hI) . (97)

Equations (96) and (97) clearly display the fundamentaldifference between the volume and shear components ofthe creep deformation. To simplify the discussion, supposethe material is at phase-field equilibrium, &%

m = 0. Then, byEq. (96) the site generation (and thus volume creep) ceaseswhen the driving force (% ! *h) turns to zero. The materialreaches equilibrium with respect to the net production andannihilation of sites. By contrast, Eq. (97) shows that the shearcreep never stops as long as a shear stress exists in the material.If Srr (= 0, the material continues to shear until it reaches ahydrostatic state of stress (if this is permitted by the boundaryconditions). As indicated earlier, this type of shear flow couldoccur, e.g., by the growth and dissolution of crystal planeswith different crystallographic orientations while preservingthe net number of sites. If this mechanism cannot operate,we have Srr = 0 and the material is only capable of isotropicsite generation causing volume expansion or contraction. As

already indicated, the tensor character of the creep deformationrate and its splitting into the volume and shear components wasidentified by Svoboda et al.7,8

Equations (94)–(97) also demonstrate that for an isotropicmaterial, diffusion is decoupled from creep deformation in thesense of irreversible thermodynamics. Diffusion can offer amechanism of creep (hence the term “diffusional” creep) andmay (or may not) kinetically control the total deformation rate.However, diffusion fluxes alone cannot cause creep deforma-tion and creep deformation cannot cause diffusion fluxes.

3. Example of phenomenological relations foranisotropic materials

The above equations rely on the assumption that the mate-rial is isotropic. While this assumption is adequate for fluids,polycrystalline materials can possess a lower symmetry due tothe crystallinity of the grains, orientational texture or certainfeatures of the microstructure. In such cases, the form of thephenomenological equations is established by (i) imposing therequirement of frame-invariance (objectivity) of the entropyproduction31 and (ii) analyzing the effects of the symmetryoperations available in the particular material on the individualterms in the entropy production. Symmetry operations actdifferently on fluxes and forces of different tensor character.Thus the tensor-split form of the entropy production given byEq. (79) can be taken as the starting point for this analysis.It should also be noted that the material deformations arepresented in Eq. (79) in the current (deformed) configuration.Large deformations can modify the material symmetry incomparison with the undeformed state. Thus the symmetryanalysis and derivation of phenomenological relations mayrequire reformulation of the entropy production with respectto the undeformed state. A detailed analysis of anisotropicmaterials lies beyond the scope of this paper and we willrestrict the discussion to one example.

In simple cases, the symmetry restrictions can be under-stood without resorting to rigorous analysis. For example,suppose the only mechanism of site generation and annihi-lation is the growth or dissolution of crystal planes normal toa certain crystallographic direction defined by a unit normaln. All other properties of the material related to diffusion andphase fields are assumed to remain fully isotropic. Retracingthe derivation of the entropy production for this particularcase, the site generation term becomes (*n ! %)Rn, where*n = n · ! · n is the normal stress on the growing or dissolvingcrystal planes and Rn = n · R · n is the normal creep rate (rateof permanent tension-compression parallel to n).

In this case, the site generation is represented by only theproduct of the scalar “flux” Rn and the scalar force (% ! *n).As such, this term belongs to the first line in Eq. (79) and cancouple to the scalar equations for the phase-field evolution. Thediffusion equations (94) remain unchanged but Eqs. (95)–(97)are replaced by

dL$k

dt= ! 1

T

K(

m=1

Bkm&%m ! 1

TBkr (% ! *n) , k = 1, . . . ,K,

(98)

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

Rn = ! 1T

K(

m=1

Brm&%m ! 1

TBrr (% ! *n) . (99)

Note that the separate shear creep equation (97) is nowredundant while the volume creep driven by (% ! *h) hasbeen replaced by uniaxial tension-compression creep drivenby (% ! *n).

VII. EXAMPLES OF APPLICATION

A. Model formulation

To illustrate the theory, we will apply it to a simpleone-dimensional system. Namely, we consider an elementalbicrystal with a symmetrical (e.g., [001] twist) GB. The grainsare treated as isotropic media and the entire bicrystal isassumed to possess the axial symmetry ()/m) around thecoordinate axis x1 normal to the GB plane. The system ischaracterized by a single phase field $ with the far-field values$ = 0 in one grain and $ = 1 in the other. This field can beinterpreted, e.g., as the angle of lattice rotation around thex1 axis normalized by the total lattice misorientation anglebetween the grains.

The lattice supports vacancies but not interstitials. Vacan-cies can be generated only within the GB region and only bythe growth or dissolution of lattice planes parallel to the GB.We neglect thermal expansion and the effect of vacancies onthe lattice parameter. Thus the latter can only be altered byelastic strains.

Elastic deformation of the lattice is described by theisotropic linear elasticity theory with Hooke’s law

-ij = 1 + .

E*ij ! 3

.

E'ij*h, (100)

where -ij is the small-strain tensor, E is the Young modulus,and . is Poisson’s ratio. Both E and . are considered constant.Although we use the small-strain approximation for elasticity,the total deformation of the material is allowed to be finite dueto the creep process. Since the deformations are assumed tobe slow and the viscous energy dissipation is neglected, thedynamic stress is identical to the static. The classical part ofthe stress will be denoted *ij without the tilde.

Due to the axial symmetry of the problem, the stresscomponents *11 (normal stress) and *22 = *33 (lateral stresses)depend only on the distance x along the x1 axis, the shear com-ponents being zero. Likewise, the normal strain component -11is a function of x, the lateral strains -22 = -33 are assumed tobe fixed, and the shear strains are zero. Under these conditions,knowing only the function *11(x) and using Hooke’s law onecan recover

*22(x) = *33(x) = .

1 ! .*11(x) + E

1 ! .-22 (101)

and

-11(x) = (1 + .)(1 ! 2.)E(1 ! .)

*11(x) ! 2.

(1 ! .)-22. (102)

The volume per site is

/ = /0 (1 + KT *h) , (103)

where /0 is the stress-free value of /, KT = 3(1 ! 2.)/E isthe isothermal compressibility and

*h(x) = (1 + .)3(1 ! .)

*11(x) + 2E

3(1 ! .)-22 (104)

is the hydrostatic part of the stress tensor.To describe thermodynamics of the solid, two adjustments

will be made with respect to the previous discussion. First,for practical convenience, all thermodynamic properties willbe described in terms of the Helmholtz free energy instead ofthe internal energy. All previous expressions for the entropyproduction remain unchanged, except that the derivatives ofthe internal energy density (e.g., !u/"$) taken previouslyat a fixed entropy are replaced by derivatives of the freeenergy density f (e.g., !f/"$) taken at a fixed temperature.46

Secondly, the fundamental equation for a specific materialusually comes from statistical-mechanical models and isformulated in terms of the site fractions of the componentsand thermodynamic properties (e.g., free energy) per site. Wewill therefore use the site fractions of atoms c and vacanciescv , keeping in mind that only one of them can be used asan independent variable (c + cv = 1). It is implicit in thistreatment that the GB structure is composed of sites and canbe obtained by an appropriate distortion of the lattice.

We postulate the fundamental equation of the solid in theform

f (T ,cv,") = 1/

[gc + gvcv + kT (c ln c + cv ln cv)] + w ($)

+ 120 ("$)2 + e (") . (105)

Here, g and gv are parameters of the ideal solution model foratoms and vacancies, k is Boltzmann’s factor,

w ($) = W$2 (1 ! $)2 (106)

is a double-well function with an amplitude W creating afree-energy barrier between the two lattice orientations, 0 is thegradient energy coefficient20 considered constant, and finallye (") is the elastic strain energy density of the lattice. Thelatter is quadratic in strains (and thus stresses) and will not bedetailed here since this term will be neglected. The expressionin the square brackets is the free energy of a uniform idealsolution per site. Note that this solution is treated classically,i.e., without a gradient term in c. By the symmetry of theproblem, the gradient "$ has only one nonzero component"x$.

This model is different from previous nonclassical interfacemodels with elasticity. Rottman47 proposed a Landau theoryof coherent phase boundaries and computed the interfacestress and other excess properties by including a gradientterm in strains. Johnson48 modeled a phase boundary betweentwo binary substitutional solutions using a gradient term incomposition. His model includes a compositional strain and,by contrast to Rottman’s work,47 treats the elastic strainenergy purely classically. Johnson carefully derives integralexpressions for the interface free energy, interface stress andinterface strain. While these workers were focused on theequilibrium state of the interface, Levitas49 recently proposeda time-dependent model with a single nonclassical orderparameter $ and elastic strain energy. Assuming mechanical

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equilibrium, he solved the phase-field evolution equation of theform !$/!t = !L&% and studied in detail the dynamics of theinterface stress at the nonequilibrium interface. His model doesnot include diffusion or site generation. Levitas proposed a fewother phase-field treatments of the interface stress.23–28 Phase-field models of coherent interfaces have also been studied inthe context of multivariant phase transformations.50,51

The subsequent calculations will be limited to first order instresses and strains. Thus the elastic strain energy term appear-ing in Eq. (105) and propagating to all other equations will beneglected. This approximation is sufficient for demonstratingsome simple results of the model.

From Eq. (105), we obtain the diffusion potential M ofatoms relative vacancies,

M = g ! gv + kT lnc

cv

, (107)

and thus the grand potential density

% = f ! Mc

/= 1

/(gv + kT ln cv) + w ($) + 1

20 ("x$)2 .

(108)

The variational derivative of f with respect to the phase fieldis given by the usual expression

&% = !f

!$! "x · !f

!"x$= w$ ($) ! 0"2

x$. (109)

Finally, the nonclassical tensor A% defined by Eq. (42) is A% =!0"$"$ and has only one nonzero component

A%11 = !0 ("x$)2 . (110)

B. The state of equilibrium

Before discussing the dynamics of creep deformation,we will find the state of thermodynamic equilibrium of thesystem. We assume that the system is already in thermalequilibrium and thus the temperature is uniform. The phase-field equilibrium condition &% = 0 reduces to the standardequation20

w ($) = 120("x$)2 (111)

predicting the phase-field profile

$(x) = 12 ! 1

2 tanhx

2*

0/2W. (112)

Using Eq. (111), the grand potential density (108) becomes

% = 1/

(gv + kT ln cv) + 0 ("x$)2 . (113)

The mechanical equilibrium condition (61) reduces to *11 +A%

11 = const, giving

*11(x) = *)11 + 0 ("x$)2 , (114)

where *)11 is the coordinate-independent normal stress inside

the grains. The site-generation equilibrium condition is % !*11 = 0 (see Sec. VI B3).

Using the above equations, we have

gv + kT ln cv = *)11 /, (115)

which can be rewritten as

kT lncv

c0v

= *)11 /, (116)

where c0v is the equilibrium vacancy concentration in the

absence of normal stress. The obtained Eq. (116) reproducesHerring’s relation for the effect of stresses on the vacancyconcentration in solids.3,52

Using Eq. (114), the equilibrium grand-potential densityacross the GB becomes

%(x) = *)11 + 0 ("x$)2 (117)

with %) = *)11 inside the grains.

The GB free energy 1 is computed as the excess of % overthe homogeneous grains:

1 =- )

!)[%(x) ! %)]dx = 0

- )

!)("x$)2 dx =

.0W

18.

(118)

The interface stress of the GB is isotropic, 222 = 233 # 2 , andis computed as the excess of *22. Using Eqs. (101) and (114),

*22(x) = .

1 ! .*)

11 + E

1 ! .-22 + 0

.

1 ! .("x$)2 , (119)

where only the last term contributes to the excess. Thus

2 =- )

!)

/*22(x) ! *)

22

0dx = 0

.

1 ! .

- )

!)("x$)2 dx

= .

1 ! .

.0W

18. (120)

We see that in this particular model 1 and 2 are proportionalto each other and independent of the stressed state of thegrains. They are generally different unless the materials isincompressible (. = 1/2).

We can also compute the GB excess volume E11 per unitarea as the excess of the strain component -11. Using Eq. (102),

-11(x) = (1 + .)(1 ! 2.)E(1 ! .)

*)11 ! 2.

(1 ! .)-22

+ 0(1 + .)(1 ! 2.)

E(1 ! .)("x$)2 , (121)

where only the last term contributes to the excess. Thus

E11 = 0(1 + .)(1 ! 2.)

E(1 ! .)

- )

!)("x$)2 dx = (1 + .)(1 ! 2.)

E(1 ! .)1 ,

(122)

where we used Eq. (118). In this model, the GB excess volumeis proportional to the GB free energy. For an incompressiblematerial (. = 1/2) we correctly obtain E11 = 0.

C. Dynamics of creep

1. Dynamic equations

We now consider irreversible processes involving vacancydiffusion, site generation, and GB motion. Due to the simpli-fied geometry of this example, we will obviously not be able to

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

model a real three-dimensional creep process taking place inpolycrystalline materials. However, several elementary stepsof this process can be reproduced and studied.

The dynamic equations of the system are based onEqs. (94), (98), and (99) adapted to this model. Neglectingall cross-effects, we have

JLx = !L

T"xM, (123)

!$

!t+ vL"x$ = !B

T

/w$ ($) ! 0"2

x$0, (124)

Rn = !Brr

T(% ! *11) (125)

with three kinetic coefficients L, B, and Brr . Here vL is thelattice velocity and Rn is the creep deformation rate (i.e., rateof the sample elongation or compression) in the x-direction.In keeping with the first-order approximation in stress adoptedhere, we replace the elastically deformed site volume / by itsstress-free value /0. In addition, Rn = "xvL ! -11 (-11 beingthe elastic tensile strain rate) will be approximated by simply"xvL. This approximation is applicable to steady-state creepunder a sustained load when the elastic deformation does notpractically change with time while the permanent deformationdue to creep increases and may reach tens of per cent. In thisregime, this approximation should work. Finally, we assumethat the system maintains mechanical equilibrium at all timesand thus Eq. (114) remains satisfied.

The diffusion equation (123) can be conveniently reformu-lated in terms of the vacancy flux JL

xv = !JLx and the vacancy

site fraction cv . Taking into account that cv + 1, we have

JLxv = !Dv"x

cv

/0, (126)

where Dv = k/0L/cv is the vacancy diffusion coefficientassumed to be constant. Rewriting also the continuity equation(4) in terms of cv , we finally obtain the vacancy diffusionequation

!cv

!t+ vL"xcv ! Dv"2

x cv = "xvL. (127)

The driving force for site generation is (% ! *11), which byEqs. (108) and (114) equals

kT

/0ln

cv

c0v

! *)11 + w ($) ! 1

20 ("x$)2 . (128)

The kinetic coefficient controlling the site generation ispostulated in the form

Brr

T= Brw ($) , (129)

where Br is a constant. This form ensures that site generationoccurs only within the GB region and not inside the grainswhere w ($) + 1. Thus the site generation equation (125)becomes

"xvL = !Brw ($)%kT

/0ln

cv

c0v

! *)11 + w ($) ! 1

20 ("x$)2

&.

(130)

The three equations (124), (127), and (130) with appropriateinitial and boundary conditions describe the entire dynamicsof our system in the deformed configuration.

2. Numerical examples

For numerical calculations it is convenient to nondi-mensionalize the above equations. We introduce the di-mensionless time 2 = 2WDvt/0, dimensionless coordinate3 = x

*2W/0, dimensionless lattice velocity , = !3L/!2 =

(*

0/2W/Dv)vL, and normalized vacancy concentration 4 =cv/c

0v . In terms of these variables, the equilibrium interface

thickness is approximately 53 , 1 and the diffusion timeacross the interface is approximately 52 , 1. The dynamicequations to be solved take the form

!$

!2= !,

!$

!3! 6$

%$

!2$2 ! 3$ + 1

"! !2$

!3 2

&, (131)

!4

!2= !,

!4

!3+ !24

!3 2+ 1

c0v

!,

!3, (132)

!,

!3= !6s$

2 (1 ! $)2+

ln 4 ! a* + aw

%$2 (1 ! $)2

!#

!$

!3

$2 &,. (133)

Here,

6$ = B0

DvT(134)

and

6s = Br0kT

2/0Dv

(135)

are dimensionless kinetic coefficients characterizing the ratesof the phase-field evolution and site generation, respectively,relative to diffusion. The two other dimensionless parame-ters, a* = *)

11 /0/kT and aw = W/0/kT , characterize thestrength of the applied stress and the phase-field barrier,respectively, relative to the thermal energy kT .

The system of equations (131)–(133) was solved numeri-cally on an interval 0 ! 3 ! L. The GB was initially placedat 3 = L/2 by solving Eq. (131) with , # 0 and the boundaryconditions

$(0,2 ) = 0, $(L,2 ) = 1. (136)

The obtained phase-field profile was very close to the infinite-system solution (112). The boundary conditions (136) weremaintained throughout the subsequent calculations. The equi-librium vacancy concentration was chosen to be c0

v = 10!4.This is an order of magnitude larger than typical experimentalvalues at the melting point of metals. However, the choice wasdictated by computational efficiency and was deemed to besufficient for qualitative demonstration of the effects.

For the velocity field ,(3,2 ), we used the initial condition,(3,0) = 0 and the boundary condition ,(0,2 ) = 0, whichfixes the position of the left end of the left grain. Forthe vacancy concentration field 4 (3,2 ), we used differentinitial conditions as specified below. Under these boundaryconditions, the system is open at its right end (3 = L) wherethe atoms as well as crystal planes are allowed to enter or leavethe system.

Example 1. We first consider a stress-free (a* = 0) bicrys-tal of length L = 800. The initial state is a uniform vacancyoversaturation with concentration 4 = 100. We impose azero-flux condition !4/!3 = 0 at the left end (3 = 0) and

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0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800

Vac

ancy

con

cent

ratio

n!

Distance "

10100

100010000

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

390 395 400 405 410

Vel

ocity

#.10

3

Distance "

10100

100010000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

390 395 400 405 410

Pha

se fi

eld

f

Distance "

1100

100010000

(a)

(b)

(c)

FIG. 2. (Color online) Dimensionless profiles of (a) vacancyconcentration 4 (initial value 4 = 100), (b) lattice velocity , and(c) phase field $. The time 2 is indicated in the legends. The modelparameters are 6$ = 1, 6s = 0.8, a* = 0 (no stress), and aw = 0.5.Note that the phase-field profile is drifting to the left (indicated bythe arrow) reflecting GB migration.

a fixed-concentration condition 4 (L,2 ) = 4 (L,0) at the rightend. In the absence of the GB or when the latter is unableto generate/eliminate sites (6s = 0), this initially uniformconcentration profile will not change with time. When 6s > 0,the GB starts to eliminate excess vacancies, creating a localconcentration minimum (see Fig. 2). With time, this minimumdeepens and widens as the vacancy concentration in theGB reaches its equilibrium value 4 = 1. This process isaccompanied by elimination of crystal planes in the GB region

resulting in shortening of both grains and thus a flow of theright grain to the left. This explains the uniform negativevelocity field on the right of the GB. The GB itself also movesto the left, although slower than the right grain. Since thevacancy concentration is small, vacancies from vast latticevolumes must be absorbed to eliminate even a single latticeplane. It is not surprising, therefore, that the GB displacementis much smaller than the width of the vacancy diffusion zone

(a)

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800V

acan

cy c

once

ntra

tion

Distance

10100

100010000

(b)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

390 395 400 405 410

Vel

ocity

#.10

3

Distance "

10100

100010000

(c)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

390 395 400 405 410

Pha

se fi

eld

f

Distance "

1100

100010000

FIG. 3. (Color online) Dimensionless profiles of (a) vacancyconcentration 4 (initial value 4 = 1), (b) lattice velocity ,, and(c) phase field $. The time 2 is indicated in the legends. Themodel parameters are 6$ = 1, 6s = 0.8, a* = 4.6 (tensile stress),and aw = 0.5. Note that the phase-field profile is drifting to the right(indicated by the arrow) reflecting GB migration.

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

around the boundary, which eventually reaches the size of thesample.

Example 2. Next we consider the same bicrystal (L = 800)subject to the same boundary conditions. Suppose it has beenequilibrated at zero value of the tensile stress. At a moment 2 =0, the stress is suddenly raised to a value a* = 4.6 (tension)corresponding to the new equilibrium vacancy concentration4 - 99.5. To reach it, the GB generates vacancies producinga concentration maximum that grows and widens with time(see Fig. 3). The vacancy generation occurs by embeddingextra crystal planes on either side of the GB, which results inthe motion of the right grain as well as the GB to the right.In this example, the application of the tensile stress causesthe growth of both grains by accretion of material in the GBregion, resulting in creep deformation of the sample. As in theprevious case, the GB displacement is small in comparisonwith the width of the diffusion zone due to the small vacancyconcentration.

Example 3. Suppose the bicrystal is stress free and avacancy concentration gradient has been created around theinitial GB position. Computationally, this has been achieved bycreating a linear vacancy concentration profile increasing from4 = 0 at 3 = 0 to 4 = 2 at 3 = L and keeping these boundaryvalues fixed (see Fig. 4). To amplify the concentration gradient,

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20 25 30 35 40

Vac

ancy

con

cent

ratio

n!

Distance "

01

1000100000

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

16 18 20 22 24

Pha

se fi

eld

f

Distance "

100002000070000

100000

FIG. 4. (Color online) Dimensionless profiles of (a) vacancyconcentration 4 and (b) phase field $. The time 2 is indicated inthe legends. The model parameters are 6$ = 1, 6s = 0.8, a* = 0,and aw = 0.5. The arrow indicates the trans-gradient induced GBmigration to the right.

(a)

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25 30 35 40

Vel

ocity

#.10

6

Distance "

1001000

10000100000

(b)

-1

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Vel

ocity

#.10

6

Time $.10%4

Grain boundary

Right grain

FIG. 5. (Color online) (a) Dimensionless profiles of the latticevelocity , and (b) the GB velocity ,GB and the velocity of the rightgrain as functions of time 2 . In (a), the time 2 is indicated in thelegends. The model parameters are 6$ = 1, 6s = 0.8, a* = 0, andaw = 0.5.

this calculation was performed in a smaller system withL = 40. Note that in its initial position at 3 = L/2, the GB seesthe equilibrium concentration 4 = 1. Thus this calculation isa test of the GB response to a vacancy concentration gradientaround the equilibrium value.

Due to the concentration gradient, the vacancies are initiallyoversaturated on the right of the GB and undersaturated onthe left. To approach equilibrium, excess vacancies must beeliminated by the GB on its right and generated on its left.This process is accompanied by elimination of crystal planeson the right and creation of new crystal planes on the left. As aresult, locally within the GB region, the left grain grows whilethe right grain shrinks, causing GB migration to the right. Thissite generation/annihilation process results in the positive peakof the lattice velocity in the GB region (see Fig. 5). The smallbump near the center of the peak is a nonclassical effect whichoriginates from the deviation of the system from phase-fieldequilibrium [the term multiplying aw in Eq. (133)]. The factthat the right grain has a negative velocity indicates that thenet vacancy balance is slightly shifted towards annihilation. Itis also observed that the height of the velocity peak decreaseswith time and drifts to the right together with the GB.

This example demonstrates an interesting effect in which aGB can be moved by a trans-gradient of vacancy concentration,a phenomenon which could be observable experimentally. To

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

provide an additional proof of this effect, the calculation wasrepeated with the opposite sign of the vacancy concentrationgradient but the same boundary values of the phase field. Asexpected, the gradient caused the GB to migrate to the left witha nearly identical magnitude of the velocity.

VIII. SUMMARY AND CONCLUSIONS

The proposed theory of creep takes classical solid-statethermodynamics10–13 as the starting point and generalizes it inat least two ways. First, we have lifted the “network constraint”and allowed lattice sites to be created or destroyed with arate that can be a continuous function of coordinates andin addition can depend on crystallographic direction. Thishas been achieved by introducing two different deformationgradients co-existing in the same material, one describinglocal lattice distortions due to elastic strains, compositionalstrain and thermal expansion, and the other describing the totaldeformation including the permanent distortion produced bythe creation and annihilation of lattice sites. The differencebetween the two represents the amount of creep deformation.Accordingly, its time derivative R defined by Eq. (49) isidentified with the creep deformation rate. Similar to recentwork7,8 and by contrast to other creep theories, the creepdeformation rate is a tensor that encapsulates both thevolume tension and compression due to the net productionor elimination of vacancies, and pure shear deformation byconcurrent site generation and annihilation without alteringthe total number of sites.

The particular formulation of the theory presented in thispaper relies on the assumption of a substitutional solid solutionwith a Bravais lattice. Accordingly, for a heterogeneousmaterial, its phases are assumed to be “coherent” with eachother, i.e., derivable from the same reference structure by affinedistortions. Furthermore, our treatment of the deformationgradient F as a continuous function of coordinates impliesthat interfaces between the phases are coherent. In the future,this version of the theory can be generalized to solids withinterstitials and nonBravais lattices, permitting a more generaltreatment of the structures of the phases and interphaseinterfaces. In the equilibrium limit, such a theory shouldreproduce the recently developed thermodynamic treatmentof coherent solid-solid interfaces.53,54

The tensor R reflects the symmetry of the material’smicrostructure and the operation of particular site generationmechanisms. We gave a few examples in which some of thecomponents of R are identically zero due to the absence ofcertain site generation mechanisms or presence of geometricrestrictions. In such cases, the material can be capable ofsupporting static shear stresses and can reach a (constrained)thermodynamic equilibrium in a nonhydrostatic state of stress.When R = 0, the theory reduces to the formulation inwhich the solid is subject to the “network constraint.” If allcomponents of R are nonzero, the equilibrium state has to behydrostatic. The ultimate equilibrium state of the material isuniform isotropic tension or compression.

The second generalization is the addition of phase fieldsand their gradients, along with gradients of concentrations ofthe chemical species. Owing to this nonclassical character,

the kinetic equations of the theory can automatically describethe evolution of microstructure as part of the creep deforma-tion process, eliminating the need to prescribe a particulardistribution of vacancy sinks and sources. For example, thesite creation and annihilation can be localized in GBs byappropriate choice of the phase-field dependence of the kineticcoefficient controlling the site generation rate. If the GBmoves, the vacancy sinks and sources will move togetherwith it.

The entropy production rate derived herein identifies severaldissipation mechanisms in the material: conduction of heat,diffusion of chemical species, evolution of the phase fields,viscous dissipation (e.g., by phonons), and finally site creationand annihilation. It also identifies the generalized forcesand generalized fluxes corresponding to different dissipationmechanisms. It particular, the creep deformation rate R isidentified as one such flux and the thermodynamic forcedriving the creep deformation is found to be (! ! %I) /T ,where % is the nonclassical grand potential density and !is the classical recoverable stress tensor. Diffusion is drivenby gradients of the nonclassical diffusion potentials M%

i andviscous dissipation by the deviation of the dynamic stresstensor ! from the nonclassical (Korteweg) stress (! + A%).The latter gives rise to interface stresses, which are thusautomatically included in this theory.

In formulating phenomenological relations between thefluxes and forces, we take into account the symmetry propertiesof the material.6 The symmetry analysis is prepared bypartitioning the entropy production into groups of termswith the same tensor character. The existence or absence ofcoupling between different groups is established by analyzingthe effect of the symmetry operations on the terms witha particular tensor character. The case of a fully isotropicmaterial is analyzed in greatest detail. The splitting of the creepdeformation rate R into the volume and shear componentsemerges as a result of this coupling analysis, with eachcomponent driven by a different thermodynamic force. Thecase of axial symmetry is also discussed as an example ofless symmetric materials. In this case, the volume and shearcomponents of R are inseparable and merge into a single tensiledeformation rate Rn = n · R · n, where n is the unit vectorparallel to the axis of symmetry. Rigorous analysis of othersymmetries relevant to particular classes of materials wouldbe an interesting direction for future work.

The obtained phenomenological equations can be used forformulating a set of kinetic equations describing the evolutionof the material during creep deformation. This requires inputin the form of a thermodynamic equation of state, coordinateand time dependencies of the kinetic coefficients and otherspecific properties of the material. While this theory awaitsapplications to real materials, it is illustrated in this paperby a simple one-dimensional example of a bicrystal with aGB acting as a sink and source of vacancies. The kineticequations have been formulated and solved numerically forthree different cases. The calculations demonstrate how thevacancy generation or absorption due to deviations fromvacancy equilibrium or caused by applied stresses can inducenot only creep deformation of the sample but also GBmigration (moving vacancy sink/source). The calculationsalso reveal an interesting effect of GB motion induced by

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

a vacancy concentration gradient across the boundary. Thistransgradient induced GB migration might occur in processessuch as radiation creep and deserves further study in thefuture.

ACKNOWLEDGMENTS

We are grateful to G. B. McFadden, J. E. Guyer, andJ. Odqvist for helpful discussions in the course of thisresearch. This work was supported by the National Institute ofStandards and Technology, Material Measurement Laboratory,the Materials Science and Engineering Division.

APPENDIX A: EXACT MOMENTUM AND ENERGYBALANCE RELATIONS

In this appendix, we derive the exact momentum and energybalance relations with respect to the lattice.

1. Momentum balance

Consider a fixed control region of a volume V . The rate ofthe total linear momentum P of the region is

dPdt

=-

V

!

!t("v) dV +

-

!V

n(

i=1

minivi (vi · n) dA, (A1)

where the second integral represents the momentum dissipa-tion through the boundaries. Here, n is a unit normal pointingoutside the region and dA is an increment of area of theboundary. All other notations have been defined in the maintext. Applying the divergence theorem,

dPdt

=-

V

%!

!t("v) + " · Z

&dV, (A2)

where

Z #n(

i=1

minivivi = "vLvL + vLq + qvL + M. (A3)

On the other hand, the total force acting on the region is

F =-

V

bdV +-

!V

n · !dA =-

V

(b + " · ! ) dV . (A4)

Writing down the Newton law dP/dt = F, we obtain themomentum balance equation in the barycentric formulation,

!

!t("v) + " · Z = b + " · ! . (A5)

Using the mass conservation relation

!"

!t= !" · ("v) , (A6)

it can be shown that

!

!t("v) = "

dLvL

dt+ dLq

dt! " · ("vLvL) ! vL" · q

! vL · "q. (A7)

Combining the above equations, we obtain the momentumbalance equation in the lattice representation:

"dLvL

dt= b + " · (! ! M) ! dLq

dt! q" · vL ! q · "vL.

(A8)

2. Energy balance

The internal energy density u was defined in the main textthrough the total energy ansatz (20). This internal energyappears in De Groot and Mazur,6 where it is denoted u%,but most of their discussion is focused on an approximateinternal energy defined by replacing K by only the barycentrickinetic energy "|v|2/2. Although this approximation greatlysimplifies all equations, our calculations will be based on thefull kinetic energy K .

The energy balance is governed by the energy conservationlaw,

!e$

!t+ " ·

!e$vL + JL

u + JLK

"= w, (A9)

where w is the rate of mechanical work per unit volume, e$ #e ! # is the energy without the potential energy since the latteris already included in w, JL

u is the internal energy flux and JLK

is the kinetic energy flux relative to the lattice. The internalenergy rate

dLu

dt+ u" · vL

= !" ·!JL

u + JLK

"+ w !

#dLK

dt+ K" · vL

$(A10)

can be derived by computing the terms appearing in the right-hand side.

To compute w, we consider a material region bounded bya set of moving markers. The total work rate on this regionincludes the work of volume forces and the work of stressacting on its boundary,

W =-

V

n(

i=1

nibi · vidV +-

!V

n · ! · vLdA

=-

V

1n(

i=1

nibi · vi + " · (! · vL)

2

dV, (A11)

from which

w =n(

i=1

bi · JLi + vL · (b + " · ! ) + ! : "vL· (A12)

The kinetic energy (21) can be split into three terms,

K = KL + Kd + q · vL, (A13)

where

KL = 12"|vL|2 (A14)

is the macroscopic kinetic energy of the lattice motion and

Kd =n(

i=1

mi

2ni

JLi · JL

i = 12

Tr(M) (A15)

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

is the kinetic energy of diffusion. Calculations show that

dLKL

dt+ KL" · vL = "

dLvL

dt· vL ! 1

2|vL|2" · q, (A16)

dLKd

dt+ Kd" · vL

= !n(

i=1

mi

2n2i

!JL

i · JLi

"" · JL

i +n(

i=1

miJLi · dLwi

dt, (A17)

dL

dt(q · vL) + (q · vL) (" · vL)

= q · dLvL

dt+ vL · dLq

dt+ (q · vL) (" · vL) , (A18)

where we denoted wi # vi ! vL. Summing up Eqs. (A16),(A17), and (A18), we obtain kinetic energy rate

dLK

dt+ K" · vL

= ("vL + q) · dLvL

dt+ vL · dLq

dt+ (q · vL) (" · vL)

!12|vL|2" · q !

n(

i=1

mi

2n2i

!JL

i · JLi

"" · JL

i

+n(

i=1

mi

ni

JLi · dLwi

dt. (A19)

For the lattice flux of kinetic energy, we have

JLK =

n(

i=1

12mini (vi · vi) (vi ! vL)

=n(

i=1

mi

2n2i

!JL

i · JLi

"JL

i + 12

q|vL|2 + M · vL. (A20)

Inserting Eqs. (A12), (A19), and (A20) in the right-handside of Eq. (A10), after lengthy calculations, we finally obtain

dLu

dt+ u" · vL = !" · JL

u +n(

i=1

bi · JLi + (! ! M) : "vL

!n(

i=1

+"

%mi

2n2i

!JL

i · JLi

"&+mi

dLvi

dt

,· JL

i .

(A21)

APPENDIX B: JUSTIFICATION OF THE APPROXIMATEFORM OF THE ENTROPY PRODUCTION

In the main text, we derived the exact expression for theentropy production rate (72). Deriving the linear constitutiverelations we assumed that the fluxes and forces were bothsmall. Under this linear approximation, all terms quadraticin fluxes and/or forces must be neglected. Tensor M definedby Eq. (18) is quadratic in the diffusion fluxes JL

i and canbe neglected. Further, the term mi(Ji · Ji)/2n2

i in the drivingforce of diffusion is also quadratic in diffusion fluxes and canalso be neglected.

Furthermore, for slow processes such as creep, the inertiaterms mid

Lvi/dt can be neglected after a short transient. Todemonstrate this, consider an isotropic material not subject

to external fields. Assuming a uniform temperature field,diffusion is decoupled from all other processes and is describedby the equations

JLi = ! 1

T

n(

j=1

Lij

#"M%

j + mj

dLvj

dt

$, i = 1, . . . ,n,

(B1)

where the n ' n matrix of kinetic coefficients L is symmetricand positive definite.

Rewrite (B1) as

JLi = ! 1

T

n(

j=1

Lij

#"M%

j + mj

dLwj

dt+ mj

dLvL

dt

$,

i = 1, . . . ,n (B2)

and consider the effect of each inertia term separately. Tounderstand the role of the first inertia term, suppose allother driving forces are zero. Neglecting also the cross-effectsamong the diffusion fluxes, the diffusion equations reduce to

wi = !miLii

T ni

dLwi

dt# !2i

dLwi

dt, i = 1, . . . ,n. (B3)

Assuming that

2i = miLii

T ni

(B4)

is a slow-varying function of time, Eq. (B3) have approxi-mately exponential solutions wi . exp(!t/2i) showing thatany initial acceleration of the particles relative to the latticedamps after a characteristic time 2i . Thus, for processesoccurring on time scales much longer than 2i , the inertia termsmjd

Lwj /dt can be neglected.To evaluate typical values of 2 , take one of the species,

say 1, and express the kinetic coefficient L11 through thediffusion coefficient D1 via L11 = n1D1/kB , which gives 2 =m1D1/(kBT ). Taking the molecular weight of 100 a.m.u., theupper bound of the diffusion coefficients in solids D1 = 10!9

m2/s and the temperature of 1000 K, we obtain 2 - 10!14 s. Atlower temperatures, 2 is even smaller. Thus the time scale ofdamping of the inertia terms is much smaller than the typicaltime scale of creep tests (many hours).

The inertia terms midLvL/dt originate from the accelerated

lattice motion due to applied mechanical stress as well as thesite generation and other relatively slow processes. Before thematerial reaches mechanical equilibrium, the lattice velocitiescan be very high, possibly comparable with the speed ofsound, and the inertia force mid

LvL/dt can be significant. Butthe subsequent creep deformation is a slow process in whichthe material maintains mechanical equilibrium and dLvL/dtreflects only the slow changes in the creep deformation rate.As a crude estimate, the magnitude of the lattice accelerationis related to variations in the creep deformation 0 by

dL ln |vL|dt

- d ln 0

dt. (B5)

During the steady-state creep, 0 remains nearly constantand depends only on the applied stress and temperature, sothat the inertia terms mid

LvL/dt can be neglected. During theprimary and tertiary stages, the right-hand side of Eq. (B5) stillremains small. For example, typical steady-state creep rates in

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MISHIN, WARREN, SEKERKA, AND BOETTINGER PHYSICAL REVIEW B 88, 184303 (2013)

metallic alloys are 0 - 10!6 to 10!3 s!1. During the primaryand tertiary stages, the rate changes by at most an order ofmagnitude over hundreds of hours. Thus, as an upper bound0 - 10!7 s!2, and thus dL ln |vL|/dt - 0.1 s!1.

To show that the inertia effects are negligible, we combinethe particle conservation law with the diffusion equation forspecies i,

JLi = ! 1

TLii

#"M%

j + mi

dLvL

dt

$, (B6)

in which we again neglected the cross effects among thefluxes. Treating the kinetic coefficient as a constant,

!ni

!t= ! 1

TLii"2M%

j ! " ·#

nivL ! 1T

Liimi

dLvL

dt

$.

(B7)

The ratio of the second term to the first inside the divergenceis on the order of

Liimi

T ni

dL ln |vL|dt

= 2i

dL ln |vL|dt

, (B8)

where 2i is the characteristic time (B4). The latter wasestimated to be ,10!14 s. Thus the inertia term in Eq. (B7)is more than ten orders of magnitude smaller than the normalterm nivL, reducing the diffusion equation to the usualform

!ni

!t= ! 1

TLii"2M%

j ! " · (nivL) . (B9)

These estimates justify the approximate form (73) of theentropy production for creep applications.

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30We follow the convention31 that the dot between vectors ortensors (e.g., a · b) denotes their inner product (contraction), whilejuxtaposition (e.g., ab) denotes their outer (dyadic) product. Twodots denote the double contractions a · ·b = Tr(a · b) and a : b =a · ·bT, where a and b are second-rank tensors and superscript T

denotes transposition. The differentiation operator " is treated asa vector. We use the notation a

&!" to show that " operates on thevector or tensor a appearing on its left.

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Eq. (3), the commutativity of !/!t and " and the vector identity"(vL · ") = vL · "" + "vL · ".

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dt= ( !F

!t)x$,y$ = !

!t[( !y

!y$ )x$,t ]x$,y$ = !!y$ [(

!y!t

)x$,y$ ]x$,t =( ! vL

!y$ )x$,t = ( ! vL

!y )x$,t (!y!y$ )x$,t = (vL

&!" y) · F. Recall our convention todrop the subscript y in the tensor "vL.

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IRREVERSIBLE THERMODYNAMICS OF CREEP IN . . . PHYSICAL REVIEW B 88, 184303 (2013)

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