American Mineralogist, Volume 59, pages 1286-1298, 1974
EquilibriumAcrossa GoherentInterface
Thermodynamic
in a StressedCrystal
Prennr-YvesF. RouNl
Departmentol Earth and PlanetarySciences,
M ossachusetts I nstitu:teol T echnology,
Cambridge,Massrchusetts02I 39
Abstract
Migrating twin boundaries, boundaries in crystals undergoing displacive phase transformations, coherent precipitates, and coherent exsolution lamellae-all provide examples of coherent interfaces propagating in a stressed crystalline framework. The motion of such
boundaries is often reversible or characterized by a sufficiently small hysteresis, so that they
can be considered at equilibrium within the stressedsolid; the presenl paper is concerned with
the conditions of such equilibrium. It is emphasized, following I. W. Gibbs, that a solid
under stress is in equilibrium only if the motion of at least one chemical component (solid
component) of the solid through the framework is considered an impossible process; other
chemical components able to migrate are fluid components. After the additional 'constraint of
coherency' for the migration of the interface is examined and expressed mathematically, the
condition of equilibrium can be established, showing the exact dependence of equilibrium on
normal and shear stress along the interface, temperature, and chemical potentials of the
fluid components. The general condition is then applied to the following special cases: (1)
coherent transformations in which the 'stress'free transformation strain' is much larger than
any elastic strain, (2) twinning, and (3) in-finitesimal transformation strain. The migration
of some low-angle tilt or kink boundaries can be analogous to that of coherent boundaries,
and their equilibrium, if reached, is ruled by the same equations. However, growth across
most high-angle grain boundaries of a polycrystalline aggregate is not constrained in the same
manner; as a consequenceit is argued thbt in presence of a shear stress such boundary cannot
be in reversible equilibrium with respect both to growth and to shear.
fnfuoduction
The general problem of thermodynamicequilibrium of stressedsolidshas concernedmany authors
becauseof its possible applicationsto mineralogy,
petrology,geophysics,and materialssciences.The
topic has recently been reviewed by Paterson
(1973). Gibbs (1906) originally derived the condition of external equilibrium of a stressedsolid in
contact with a fluid, and introducedthe distinction
betweensolid and fluid componentsof a solid. These
conceptsare essentialto any study of the thermodynamicequilibriumof a stressedsolid and, in particular,to thepresentwork; becausetheir significance
is often not fully appreciated,Gibbs' results will be
summarized, and the constraint of solid behauior
defined.
'Present address: Department of
Geology and Erindale
College, University of Toronto, Mississauga, Ontario, Canada.L5L 1C6.
At equilibriuma solid-fluidboundarydoesnot and
cannot support any shear stress;'therehave therefore been attempts to generalizeGibbs' results to
boundarieswhich supportshearstresses(seePaterson, 1973). Coherent interfacesare examplesof
such boundaries;an understandingof their equilibrium has important implications for such problems
as twin-boundary migration, precipitation hardening, solid exsolution, and other phase transformations within crystals (or glassesat temperatures
sufficientlybelow their glasstransition).
The equilibriumconditionsfor coherentinterfaces
derived in this paper are quite general; they do
not assumethat the transformation strain or the
elasticstrain are infinitesimal,nor do they assume
However,the
that the transformationis isochemical.
thicknessof the interfacial region is neglectedand
the surfaceenergyis not considered;the applicability
of the resultsdependson the validity of theselatter
assumptions.
1286
THERMODYNAMIC
EQUILIBRIUM
ACROSS A COHERENT INTERFACE
Constraintof Solid Behavior
Gibbs (1906, pp. 184-21.8)considereda stressed
elasticsolid able to dissolveinto or crystallizefrom
a fluid. Let the fluid in contactwith the solid at one
point of its boundary be at a pressureP. External
equilibriumis obtainedwhen the solid neither dissolvesnor grows spontaneouslyat its boundary with
the fluid. If the solid is of uniform composition and
if molar quantitiesare used,the chemicalpotential
p of its component in the fluid at pressure P is
givenby (Gibbs,Eq. 385)
p:E_rS+PV
(1)
where E, ,S and Z are respectivelymolar energy,
entropy, and volume of the solid in the equilibrium
state considered, and 7 is the temperature. The
significanceof this result is better understood when
considering,as doesGibbs, the following specialcase.
Threefluid pressuresP', P", and Pa are eachapplied
on two oppositefacesof a rectangularparallelepiped
of the solid, assumedhomogeneous.
The solid is then
under homogeneous
stress,of principal components
-P', -P' and -P'. The condition of equilibrium
with respectto growth or solution becomesfor each
pair of faces(Gibbs' Eqs. 393-395)
E-rSJ-P'7:r'
E - rS * P'V : r'
E-rS!psV:p,3
where ;r1, p.2,ps are the values of the chemical potential of the componentof the solid in each fluid.
If migration of the component of the solid were
possible, either through the solid itself or through
some external channel,this componentwould indeedmigratefrom the fluids at the highestpressures
to the fluid at the lowest pressure.Consequently,
equilibrium of a non-hydrostaticallystressedsolid
must alwaysbe understoodto be limited by the constraint that such migration is not a possiblemechanism. This constraintis called here the constraint of
solid behauior.We note (Kamb, 1961) that a chemical potential for the componentof the solid is not
definedwithin the solid itself.
Gibbswent on to examine(p. 215) the caseof a
solid which also containsfluid componentsa'8, etc.
Recentwriters,e.g. L| Oriani, and Darken (1966)
or Paterson(1973), refer to theseas mobile components. In Gibbs' words, fluid componentsare
such "that there are no passiveresistancesto the
motion of the fluid componentsexceptsuchasvanish
1287
with the velocity of the motion." For these components Gibbs shows that a chemical potential can
be defined and is constant throughout the solid and
the fluids when the system is in equilibrium. As
Gibbs' definition makes clear, a chemical component
o is fluid (and therefore a chemical potential pd
must exist and have a constant value throughout the
system at equilibrium) if a can move independently,
through the solid, regardless of whether or not the
corresponding concentration or concentration variation of ,a in the solid can be detected.
When a solid contains fluid components, the constraint of solid behavior is that at least one chemical
component of the solid is not free to moDe through
the solid. Gibbs called such a component a solid
component.z The very definitions of fluid and solid
components imply that there exists a three-dimensional framework which is conserved and with respect to which motion of chemical species may or
may not occur, independently of possibletranslations
or deformations of the framework itself.
Recognition of the existenceof a solid component
and its identification are essentialto any consideration of equilibrium of a non-hydrostatically stressed
solid. In this work solid behavior is always assumed.
In practice, for crystals, this means that mechanisms
such as unlimited motion of dislocations or NabarroHerring diffusion creep do not occur, or are so slow,
compared to boundary migration or diffusion of fluid
components, that they can be neglected. We shall
also suppose the fluidity of the fluid components to
be perfect, "leaving it to be determined by experiment how far and in what cases these suppositions
are realized" (Gibbs, 1906, p. 215) .
Coherent Transformations and Coherent Interfaces
A coherent transformation in a solid is a phase
transformation in which the three-dimensional
framework is preserved without rupture. This definition does not exclude the possibility that some
'Not all atomic,ionic, or molecularspecieswhich can
move may be able to do so independently;there may be
relations of dependencebetweenthem, such as site constraintsor constraintsof electroneutrality(Thompson,1969).
alA set of independentfluid componentscan nevertheless
ways be defined.Conversely,a solid componentdoes not
to any one atomic,ionic, or molecnecessarily
correspond
ular speciesunable to diffuse through the lattice of the
solid. For example, two substitutionalspecies,a and p,
may both be able to diffusethrough the lattice but be tied
by a siteconstraint.
1288
PIERRE-YVES F. ROBIN
atomic rearrangementmay occur, allowing some
chemicalspeciesto changeposition or migrate. A
coherenttransformationis thereforenot necessarily
displacive (Buerger, 1948) although displacive
transformationsare coherent.In the presentpaper,
attention is concentrated on crystalline solids.
Generalizationto amorphouschain polymersis possible and may have applicationsto transition and
at low temperatures.
in glasses
exsolutionphenomena
Let us call the two possiblestatesof a crystal
'phase' a and'phase' b.3 The transformationof a
crystal from phasea to phase b can be visualized
as the sweepingthroughthe crystal of one or more
boundariesseparatingdomainsof b from domains
that are still a. Such a boundary is a coherent
boundary, or coherent interface. In the presenceof
in a and b may arisefrom
suchan interface,stresses
the very co-existenceof the two pltasesand from
the constraintthat the two latticesbe matchedacross
the interface;stressesmay also be imposed at the
externalboundariesof the crystal.
There are cases where, instead of sweeping
through the whole crystal, a coherent interface
reachesan equilibrium position. Let us look at a
few examples.
MechanicalTwinning
It is only when their stateis affectedby nonscalar
parameterssuch as electricfield, magneticfield, or
stressthat twins must be consideredto be distinct
phases,as their molar energiesmay then differ.
twinMineralogistsare familiar with stress-induced
ning in calcite. Mechanical twinning is found in
many other minerals (DeVore, 1970). The migration of Dauphin6 twin boundariesin quartz under
stressis particularly well documentedby Frondel
(1945), Thomasand Wooster(1951), and Tullis
and Tullis (1972). When non-uniform stressesare
applied to the external boundariesof a crystal in
which mechanical twinning is possible, the two
s The term phase is often taken to designate only a state
of a substancein which all thermodynamic parameters are
uniform throughout. Becausestresses,in particular, are not
assumed uniform here, the term phase simply refers to a
portion of the crystal characterizedby a unique equation
of state and by continuous thermodynamic properties.
This broader definition is consistent with usage in the
metallurgical or geological (e.9., Wang, 1968) literature.
By contrast, 'crystal' designatesthe framework which is
conserved,regardlessof which state it is in; such usage is
implicit in many studies of displacive or order-disorder
transitions.
twins may coexist within the crystal, separatedby
coherenttwin boundaries.As with otherphasetransformations,the extent to which equilibrium is actually reachedby the migratingtwin boundarymust
be determinedin each case and will depend on
factorssuch as time, temperature,and what level of
disequilibriumis consideredsignificant.For Dauappearsto vanish
phin6twinningin quartz,hysteresis
at temperaturesnearingthe a-B transition (Young'
reversiblemigration of twin
1962). Stress-induced
boundariesis also observedin metals.Basinskyand
Christian (I954a) show how such observedmigration can explain the rubberlike elasticityof lowtetragonaltwinnedphases
temperatureface-centered
in indium-thalliumand gold-cadmiumalloys.
I sochemicaI Coherent Transitions
Displacive, semi-reconstructive,and order-disorder transformations(Buerger, 1948) can all be
classifiedas isochemicalcoherenttransformations.
A few examplesare given by Coe (1970). Well
known are the highJow transitions in crystalline
phasesof silica and in the feldspars,and the transition of calciteI to calciteII at high pressure(Bridgman, 1939). Which,and to what extent,martensitic
reactionsare coherenttransitionsis a subjectof debate in the metallurgicalliterature'The W-L-R and
B-M crystallographictheoriesof martensitictransformations(Weschler,Lieberman,and Read,1953;
Bowlesand Mackenzie,L954) in effectassumecoherency and have successfullyexplained many
Yet electronmicrosfeaturesof thesetransformations.
copy of Fe-Ni-Calloysrevealsthat growthof martensite is accompaniedby the formation of large numbers of dislocations,and the boundarieshave been
describedas semi-coherent(e.g.,seeOwen,Schoen,
and Srinivasan,1970). ln some systems,however,
the reactionshowsremarkablylittle hysteresis:20oC
or lessin Au-Cd (Changand Read,1951) ; between
2"C and 4'C in In-Th (Burkart and Read, 1953;
Basinskyand Christian, I954b); it seemsaccepted
that these 'thermoelasticmartensites'(Owen e/ al,
1970) form by truly coherent reactions.The inon the temperatureat which some
fluenceof stresses
of thesereactionsoccur has been exploredexperitransitionin quartz (Coe and
mentally: for the ,a-.,8
Paterson,1'969);and for the martensiticreactionin
In-Tl (Burkart and Read, 1953) and Fe-Ni alloys
(Pateland Cohen,1953). Wang (1968) has also
qualitativelyexplainedthe observedsmearingout
of the calcite I-calcite II transition in limestones
THERMODYNAMIC
EQUILIBRIUM
over a wide range of confining pressureby the inhomogeneous
Stresses
arisingin such aggregate.
According to Wang and Meltzer (1973), the high
absorptionof acousticwavesin theselimestonesover
the same range of pressurealso indicates an extremely rapid migration of .the phase boundariesin
responseto the alternatingstress;their interpretation suggeststhat calcite I-II boundarieswould be
very closeto their equilibriumpositionsunder static
conditions.
ACROSS A COHERENT
INTERFACE
1289
In summary,known examplesof coherentinterfacesare many, and more are likely to be found with
the increasinguse of transmissionelectron microscopes.It must be emphasizedagain,however,that
whetheror not a coherentinterfaceis ableto migrate
to an equilibriumposition must be justifiedin each
caseor be explicitlyacknowledged
as an assumption.
Method
Equilibrium Criterion
CoherentPrecipitates
A coherentprecipitateis a domainwithin a crystal
which differs from its host by its composition and,
in general,by its stress-freelattice dimensions,but
yet has retainedthe continuityof its latticewith the
latticeof the host.A non-hydrostaticand often nonUniform stress-fieldis generally present. Torthe extent that thesestresses
are maintainedwithout creep
and without the correspondingdestructionof the
lattice framework, the equilibrium reached implies
the existenceof a solid component,commonto both
phases,and of at leastone fluid component.
A fully coherent precipitate is one which has
maintainedcoherencyon all its boundarieswith the
host. Examplesof fully coherentprecipitateshave
been describedmostly for metals. The G.P. zones
(Guinier-Preston) in Al-Ag, Al-Zn, and Al-Cu
alloys, cobalt particles in Cu-Co alloys and y, in
nickel-basealloys are cited by Kelly and Nicholson
(1963). A partially coherentprecipitateis one for
which coherencyis maintainedonly along a specific
crystallographicorientation. In metals, the Widmanstattenstructuresof aluminum alloys (Kelly
and Nicholson,1963) are well-knownexamples.
There is no general review of the nature of the
boundariesof exsolution lamellae in oxides and
silicates.Fully coherent submicroscopicexsolution
lamellaeof augitein pigeonitehave been observed
by transmissionelectron microscopyin lunar and
terrestrial pyroxenes (Christie et al, I97l). The
angular relationships between clinopyroxene host
and exsolutionlamellaereportedby Robinsoner al
(1971) are also often consistentwith the assumption of coherencyof the boundaries(Robin, 1974b).
Champnessand Lorimer (1973) observecalciumrich submicroscopic
coherentprecipitatesin an orthopyroxenefrom the Stillwatercomplex,for which
they proposethe name of G.P. zones,by analogy
with metals. Cryptoperthite lamellae in alkali feldspar are also generallycoherent(Robin, 1974a).
Gibbs demonstrated
Eq. 1 by using the following
minimum energycriterion:
"For the equilibriumof any isolatedsystemit
is necessaryand sufficientthat, in all possible
variationsin the state of the systemwhich do
not alter its entropy,the variationof its energy
shalleithervanishor be positive."
As there have been argumentsin the literature
regardingthe definitionand use of energyfunctions
other than internal energy for solids under stress
(seePaterson,1973), we will use the samecriterion
as Gibbs. As a result. it will be found that one of
the conditionsof equilibriumis that the temperature
be uniform throughoutthe system.This is, of course
a result which could be acceptedand bypassedby
using the Helmholtz energy.The advantagewould
be minor, however, and Gibbs' treatment will be
followedinstead.
Let us then considera portion of crystal which
containsa coherentinterfaceat equilibriumseparating phasesa and b. Isolate this volume by a "fixed
envelopewhich is impermeableto matter and to
heat, and to which the solid is firmly attached
whereverthey meet" (Gibbs, 1906, p. 187). We
assumethat stresses,compositions,energyand entropy densities,etc., althoughnot necessarilyuniform, vary continuouslywithin each phase.
Statesand CoordinateSystems
During a migration of the interface a portion of
the lattice changesphase, and the corresponding
strainis not necessarily
infinitesimal.In order to describesuchfinite strainone mustcarefullydistinguish
betweenthe identification of lattice points and their
location.Identificationof points is accomplished
by
the use of a referencestate and a referencecoordinate system,'location of thesepoints in the actual
coherentequilibrium state or in variational departures from the equilibrium state is done by their
coordinatesin the equilibrium coordinate system.
PIERRE-YVES F, ROBIN
I29O
Both coordinate systems are assumed rectangular in the equilibriumstateof a small volume of phase
which is equal to unity in the referencestate. All
cartesianand of samehandedness.
units of mass,volumes,and areasare chosensmall
point
in
the
or
of
a
vector
a
of
The coordinates
prime)
regarded
enough so that propertiescan be considereduni(noted
can
be
a
by
reference state
form within them.
reference
Therefore,
the
mainly as a'tagging'system.
fn an actual state,the position of a point of the
physical
reality
and
have
any
state does not have to
may differ from its equilibriumposition
particular
framework
any
for
can be chosen as is convenient
amount Exa,and, consequently,so
by
a
variational
with
preserve
continuity
problem; it must, however,
strain,
Era,7(see the virtual deformthe
local
to
convenient
may
the actual state. It is sometimes
1951,p. 44-50). The entropy
Murnaghan,
ations
of
system
choose the reference state and its coordinate
vary
by Es', and therefore,from
may
also
density
to coincide with the equilibrium state and coordinate
variation is given by
internal
energy
3,
the
Eq.
and
system; the distinction between identification
position of points must be maintained, however.
Physical quantities like energy, entropy, chemical
concentrations exist only in an actual state; their
(Einstein'ssummationconventionon repeatedsubdensities, however, can be defined with respect to
volumes of the lattice equal to unity in the reference scriptsis usedin this paper) which we rewrite as
state, in which case these densitieswill also be noted
(4)
6et : o;i Dx;,i * T ds,
by a primed lower-caseletter. Force vectors also only
exist in an actual state, and their components are As Gibbs (1906, p. 186) pointedout, o;7is the ith
defined in the equilibrium coordinates; in defining component,in the equilibrium coordinates,of the
stressesor the stress tensor. however. the unit areas force acting (in the equilibriumstate) on a surface
elementwhich would be equalto a unit area in the
used and their orientations are the ones of the lattice
referencestate and would be perpendicularto the I
in the referencestate.
A point in the solid, identified by its reference axis of the referencecoordinate system. (If the
referencestate coincideswith the equilibriumstate,
state coordinates, xi', has in the equilibrium state the
coordinates xi, (i : l, 2, 3). We can define the or only differs from the equilibrium state by an
infinitesimalstrain,o;yis the stressas definedin inquantities
,"': (u*),",,* (#)"'
0xo
,., : -od
Q)
The set of xi,i is a description of the local deformation or strain between the reference and the equilibrium states.If the reference state and coordinates are
chosen so as to coincide with the equilibrium state
and coordinates, then at equilibriurrt, xr,j = 'Eri(Kronecker's8).
Fundamental Equation ol State
Let us first consider a crystal which does not contain any fluid component. Under conditions where
we can neglect body forces, the energy density of
the crystal is only a function of the state of strain
and of the entropy content. We thus write the fundamental equation of state of each phase as
e'" :
e ' " ( x r . n "s, ' " )
e,b:
e,b(xo,ru
s r, u )
(3)
where e'and s'are respectivelyenergy and entropy
densities, or, more precisely, energy and entropy
finitesimalstrainelasticity),
Besides displacements and entropy density
changesthroughout each phase, there is another
possible variation of the system from its equilibrium state: the interfacemay migrate through the
solid, its migration correspondingto a changein
phase at the interface and preservingcoherency.
We need to examinethis last variation carefully.
Geometryof a Migrating CoherentInterface
Figure la representsportions of phasesa and b
and a portion of interfacein the equilibriumstate.
Theseportions are chosensmall enoughfor stresses
and entropy to be and to remain uniform within
eachphaseand for the variationalmigration of the
interfaceto be uniform. Figure 1c is the configuration after such a migration.As Figure 1 shows,this
migration can be decomposedinto the following
sequence:
( 1) An imaginarycut is made alongthe interface.
(2) Points in a are moved by a distanceEr4o,
points in b are moved by a distanceEr;b
THERMODYNAMIC
EQUILIBRIUM
ACROSS A COHERENT INTERFACE
1291
(Fig. 1b). The relative displacementof
b with respectto a is
6xru-6xr':Q/P'
(3) A volume of one phase is transformed
into a volume of the other phasehaving
the same mass, thereby reestablishing
(o)
(b )
(c)
coherency.Points which are not affected
by this phasechangeare not moved (Fig.
Flo. l. Migration of a coherent interface in the actual
1c).
state. Lines have been drawn in a and b to help visualization; they can be interpreted as the traces of two sets of
We canmakethe followingremarks:
(i) The direction of Q'P' is imposedby the lattice planes preservedin the transformation.
statesof strain of the lattice in a and b
and is thereforenot an independentpaventionon vr and 8g is suchthat 8l'is positivewhen
rametersubjectto variations.
b growsat the expenseof a.
(ii) The length of Q?, is proportionalto the
Equations 5 and 6 expresscompletely the conmass transferred from a to b. Let O be
straint of coherency.They show how the growth of
the intersection of the direction of Q'P'
each phase and the relative displacementsof a
with the new position of the interface
and b at a particular point of the interfaceare func(Fig. 1c). We must have IOP'l/lOQ'l =
tions of only one independentparameter,8g.
p"/pb,whetepo and pDare molar densities
In the rest of the paper vr is called the charof a and b, respectively,in the equilibacteristic Dector4of the coherent boundary; in genrium state.
eral, if the interface is curved and elasticstrains are
We want to normalize variations lo a unit uarianot uniform within each phase,vr is a function of
tion. Figure L representsconfigurationsof the actual
positionon thd interface.
state.Let us choosean elementof area of interface
in the actualstatewhich has an areaof unity in the
Conditions of Equilibrium
referencestate. Let us then consider a particular
The total changein energyof the systemfor an
migration in which one mole is transformed from
variation from the equilibrium state is now
arbitrary
phasea to phaseb over this chosenarea elernent;
zeroor positive.Using Eq. 4, this statestated
to
be
we call v4 the correspondingparticular vector Q'P'.
ment
is
written
For an arbitrary migration we have, therefore,
Dxno-
6x;o :
$g vn
(5)
DE :
I
(o;i" lxi.i I
T 6s")dV'
JV'd
where 8g is the arbitrary parameter, positive or negative; Eg is in fact the number of moles transformed
from a to b across one unit area of interface in the
referencestate.
The position at equilibrium of the interface within
the lattice can be 'mapped' in the reference state.
The variational growth of b at the expense of a
(or vice-versa) is mapped in the reference state as
a displacement of the interface by a distance El,
perpendicular to it. From the definition of va, Dl',
for the same arbitrary migration as in Eq. 5, is given
by
61, :
6g/ p,
(6)
where p' is the molar density of the region swept by
the interface in the reference state. The sisn con-
o
+ T6s'u)dv'
+ I (on,u6x,,0,
Jv'o
+ I
(e'u- e") Dl/dA' > o
(7)
J A'
where V' and A' ate respectively volume and area
in the referencestate.
aThe transformation of the region swept by the interface is an inoariant plane strain (see Wayman, 1964). ln
the transformation the length and orientation of any vector
parallel to the interface are not affected,and OP'is transformed into OQ'(Figure 1). OP'and OQ'are colinearand
their orientation is therefore that of the only nondegenerate
characteristic Dector (or eigenoector) of the transformation;
the corresponding eigenvalueis p"/ po.
r292
PIERRE-YVES F, ROBIN
howNot all variationsin Eq. 7 are independent,
ever:
( 1) The total entropyis constrainedto remain
constant:
ur:-1""a ; ; , i 6 x ;d" V ' -
ff
+
F
+
- s/")6l/ dA' : o.
/ , G"
(8)
Eqs. 7 and 8 can be satisfiedfor any arbitrary
transfer of entropy only if (Gibbs, 1906, p. 186,
Eq.364)
(9)
Z : constantthroughout the system
(2) Yariational displacements,,8x4,are continuous throughoutthe volumesof a and
b and are zero wheneverthe crystal meets
the fixed envelope. Green's divergence
theoremgivestherefore
e,' ; ls,b - 7s,b --J,, \---;.6sdA') 0
t
Jv
'"
(2) oei" a' jo *
t
ff
D x ; ,d] V ' I
"n?
-
:
.[^
*
-
[^
I
o , , , , bD x ,duv ' .
Jv'"
-onio ''ru.
a n 1i ' i " :
_
-eru ' TSru
pp
(14)
er" _
Tsr" - T ; v ' : g
(15)
on the interface.
If molar energies and entropies are used, as p'
is a molar density, Eq. 15 becomes
(,Eu-
oour|,xn,',dV'
TSo) -
(E -
Fu-
6gdA'
(10)
o]nti\
a x , bd A '
tl
I
inter-
rS") -
Tavn :
g
(16a)
or
o ' o ia ' i o v i
(oolr't" +
the
Ta is the lth component in the equilibrium
coordinates of the force on o across an
element of area of interface equal to unity
in the referencestate.
(3) The last condition of equilibrium, and
the most important result of tbis analysis,
is
rv,o
I
0 on
( 13 )
Tt :
a i i . i 6 x td
" V' ,
where n';o and n'iD are unit vectors normal
to the interface in the reference state,
drawn outward from a and b; clearly t' f
=-n'jb. By using Eq. 5 the sum of the
two above integrals can be rewritten as
oujb a' jb =
Eq. 13 is the condition of mechanical
equilibrium on the interface. All componentsof the stresstensor need not be
equal. It is convenient to define To, the
stresson the interface.
n ' , u6 x , u d A '
[^,o,1
-
(11)
face.
and similarly
o x , ,dl v ' :
In,,o,',
Ts,"6t; r.'; vr)\
The variationalterms in each integral of Eq. 11
are now independent. The equilibrium conditions
are thereforeobtainedby equatingthe Eulerian of
eachintegralto zero:
(12)
(I) oni,i: 0, throughouta andb.
This is the usual equationof mechanical
equilibrium in the absenceof body forces.
6x;,id V' :
oo7n'i"6xn"
dA'
Iu,"on,r
I^,
-
o , , , i6 x nduv '
(o,",o'," + oJ n'ib) 6xnb
dA'
+ /
dv'
/ ar'"dv' + J|v , o 6s'o
Jv'a
1",,
oai,i6x;"dV'-
I
or,,o,6xiodV'.
Jv'b
(3) From Eq. 6, El' is not independentbut a
functionof 69.
Therefore,by insertingEqs. 6, 8, 9, and 10 into
Eq. 7, the total variation of energyof the systemis
F" -
T'vn:
g
(16b)
where F", Fu aremolar Helmholtz energies.Tn and vo
are vectorsin the equilibrium coordinates,and can be
resolved into componentsnormal and tangential to
the interfaceat that point:
Vr =
€IIif
T;:
-Pn;*
"Yt'
rt,
wheren, is a unit vector normal to the interfaceoutward from a in the equilibrium state.Then
THERMODYNAMIC
T;v; :
EQUILIBRIUM
ACROSS A COHERENT INTERFACE
-Pe l'Yir;'
If the referencestate is chosento coincide with the
actual coherentequilibrium state,the definition of vi
makes
,:
Vu- V"
where V", Vu are molar volumes in the equilibrium
state.Then the equilibrium condition becomes
(Eu-TSu*Pvb1
-(E
- ?'S"+ PV")-r;rr:
o
(t6c)
where, it is recalled, -P and ri are respectivelythe
normal stressand the componentsof the tangential
stresson a acrossthe interface.
In spite of their familiar look, the expressionsin
parentheses
shouldnot be confusedwith the expression
of G, or pr, defined for phases under hydrostatic
pressure:(i) P is only one componentof a nonhydrostatic stresstensor;(ii) equality of the two parentheses
doesnot ensureequilibrium, unlessTnro: 0; (iii) in
the most generalcaseof equilibrium, when the interface is curved and stresseswithin eachphaseare not
uniform (e.g.,coherentboundarieswithin grains of a
stressed polycrystalline aggregate), the terms in
parentheses
do not maintain a constantvaluealongthe
interface.
Existence of Fluid Components
In developingEq. 16 it was assumedthat there
wasno fluid componentin the solid.As notedearlier,
Gibbs ( 1906,p. 2I5-2I8) pointsout that a chemical
potential can be defined in the solid for each fluid
componentby (Gibbs'Eq. 467)
de' : o.iidxo,i* Z ds,*
E p" ap," (17)
and demonstrates
that pr,"is constantthroughoutthe
systemat equilibrium; p'o is the molar density of ,a,
that is, the number of moles of component,a in
the equilibriumstate in a volume of solid equal to
unity in the referencestate. The demonstrationis
in fact formally identicalto the demonstrationthat
the temperatureis uniform. It is sufficientto treat
p'" Tikes', and p" like ? in Eqs. 4, 7, 8, and 9. The
final equilibrium equation therefore becomes
(Eb-
TSb-
E p" xu'") -
( E - r S "-
1r"*"'")
- T,uo:s
(tsa)
wbere X"'", Xb," ate ratios of the molar density of
1293
fluid componerrt,a to the molar densityof the solid
component in c and b respectively. Only independent fluid componentsare included in the summation.
If the referencestate is chosen to coincide with
the equilibrium state (or if it only differs from the
equilibriumstateby an infinitesimalstrain), and Ti
and v.; are decomposedinto normal and tangential
components,
Eq. 18abecomes
( E u-
T S b+ P V b -
1
r"*',") - (E -
TS"
+ PV" -
- ' y n r o: 0
(18b)
1r*"'1
In the most generalcase, only temperature,T,
and chemicalpotentialsof the fluid components,/ia,
have a constant value throushout a volume at
equilibrium.
Case of Finite Transformation Strain
Let us considerhere a situation in which the stressfree transformation strain is finite and much larger
than any elasticstrain ofeither phase.In practicesuch
a transitionis not coherentunlessa compositionplane
existsover which the two phasescan be matchedwith
stresses
which do not exceedthe strengthof themineral.
Let us therefore assumethat we are dealing with a
coherentinterfacealong such a plane, at equilibrium
at a temperatureT. The characteristicvector of the
transformationacrossthis plane is chiefly determined
by the stress-freetransformationstrain.
We first restrict our discussionto a solid with no
fluid component.At the temperattre T, a and b would
alsobe in equilibrium if both wereunder a hydrostatic
pressureP". If F"'u, Fu'', /"'', /b'e are the molar
Helmholtz energiesand molar volumesat P' and T,
this equilibrium is describedby the equationof hydrostatic equilibrium
(Fo'"- F"'")+ P'(70''- V"'"):o
(19)
When a and b are at equilibrium acrossthe coherent
interface under stress, the equilibrium condition
(Eq. 16c)is
( F u ' "- F " ' ) + ( a F u - a F ' ) + p ( V u ' "- V " ' ' )
(20)
+ P(A Vb- AV') - 7r : 0
whereAF", aF', aV" andaVbaremolarstrainenergies
and volume changeswhen going from hydrostatic
pressureP" to the coherentsituation;7 is the absolute
valueof the tangentialcomponent,7', of the characteristic vector, and r is the resolvedshear stressin the
PIERRE-YVES F. ROBIN
1294
directionof to (hence"yir; : 7r). Becausethe transformation strain is much larger than the elastic
strains,as soon as r is of the sameorder of magnitude
as the stressesrequired to maintain coherencythe
term yr is much larger than the elasticstrain energy
termsAF', lF', and"alsotian Pd,I", Pa4u;atJbrtiii,
(AFb - AF") and P(aVb - aV"\ are
the differences
negligiblecomparedto ?2. Consequently,subtracting
Eq. 19 from Eq. 20,the equilibrium condition reduces
to
fect shear-induced pigeonite lamellae (Robin,
1,97k').
An analogousargument leads to an equilibrium
condition similar to Eq. 20) :
(Fu'"+ Pfu'") - (F'" t PV'") - tr
: I u"(xu." - f'")
d
Cose of Finite Simple Shear and No Fluid
Component
(21)
In this caseVo : Vo : V, and(seeFig. 2)
(P - P') AV - It : O,
where AZ is now the molar volume difference
A7:
fb'"-
V'"-V'-
tr.
Eq. l6b becomes
F u- F -
Vrtant:
V tan,!
/e r\
Eq. 2l can be differentiatedwith respectto T and
gives
o
(27)
Eq.24 becomes
-P" A7'
(**-fr^v-tfrr:o
(26)
Vtant!.
t:
By combiningEq. 19 with Eq. 16b insteadof Eq.
16c,Eq. 2l canalsobe written
T ; u ;:
(25)
(;/.,
(28)
€qui,.
Eqs 27 and 28 are identical to the results of Coe
(1970),exceptfor a differentsignconventionon A^9.
(22)
Finite StrainTwinning
(using the Clausius-Clapeyronequation for dP"fdT,
and neglecting dy/dT). Therefore the following
relations must be satisfied for the maintenanceof
equilibrium:
at constant 7:
(9., equi,.
#:
P: (#)"
at constant
*,,,.
: -f;'
H,
Twins are at equilibrium with each other under
any hydrostatic pressure, and in particular when
the hydrostaticpressureis equal to the normal component of stressP. If the stress-freetransformation
strain (a finite simple shear) satisfiesthe assumpe3) tions of this section,the equilibriumcondition (Eq.
21) becomes
(24)
7r=0
or
Patel and Cohen (1953) combinedEqs. 23 and
r-20
Q9)
24 to explainthe effectsof hydrostaticcompression,
In its equilibrium position, such a twin boundary
uniaxial compression,and uniaxial tension on the
not support any resolved shear stress.This
in
iron-nickel
iron-nickeldoes
and
martensiticreaction
mean that other shear stresscomponents
not
does
carbonalloys.
The finite resolved shear stresswhich is
zero.
are
Finite Strain with Fluid Components
to move a twin boundary is thus a
required
often
(7r is a molar energy)
disequilibrium
by
large
of
the
measure
Coherenttransformationscharacterized
before migrating.
can
support
A
boundary
that the
finite strains may also involve fluid components.
(Pbca)
to
possiblecase,for example,is the ortho
Infinitesimal Transformation Strain
clino (P21/c) transition in the pyroxene quadriIf the transformationstrain is very small, the
lateral; formation of shear-inducedlamellae of
clinopyroxene in orthopyroxene is reported by referencestatecan be chosensuchthat the strain of
severalauthors (e.g., Coe and Miiller, 1973) and, the lattice with respectto the referencestate is inat temperatureswhere cation diffusion is sufficiently finitesimalin any actual state of the system.The
rapid, such clinopyroxene lamellae could become characteristicvector of the transformationis (see
richer than the host in Fe and Ca, becomingin ef- Appendix)
THERMODYNAMIC EQUILIBRIUM ACROSSA COHERENT INTERFACE
vr :
7f2(e;n' -
- (erro ero")noo
f(2 Ae;;n;o -
vl :
1295
er*")nr"l
(30)
Aelpn;")
whete ea1a,
ealbate the symmetricstrain tensorswith
respectto the referencestate, and Aeor.is therefore
the total transformationstrain acrossthe interface.
The scalar product Taq may be expressedas
TiYi :
V(2r, Aerin;" f
(31a)
P Aepp)
If the coordinatesystemis suchthat its l-axis is perpendicular to the interface at the point under consideration,
Aezz :
Dt :
Tr:
A€sa :
A€ze :
V AE'-, D2 :
ab
6n" :
0, Aelol
2V Ae12,us
Cn" :
6rr, T2:
: A€rr,
: 2V Ae,s
Cn
Ts:
db
:
ctz
, d :
6n
:
612,
6
cB :
ote
and therefore
T;u; :
V(o, Ae1 I
2o' Aer, *
2oB A1s),
(31b)
Frc. 2. Characteristic vector, for finite simple shear, The
vector vr has only a tangential component, of modulus 7.
The transformation is more easily characterized by its
shear angle ry'. When a molar volume 7 changes phase
across a unit area of interface, lQ'P'l = 1 = V tan {';
henceEq.26.
or, in matrix notation,
T;Dt:
V(a, Ae, * ouAesf o6Ae6) (3lc)
We recognizethe decomposition-Pe *ys4 obtained
previously. The equilibrium condition follows immediatelyby insertingEq. 31 into Eqs. 16 or 18.
Previous Results and Discussion
In spite of their many possibleapplications,only
a few attempts to obtain results analogousto the
ones presentedhere can be found in the literature.
The works of Patel and Cohen (1953) and of Coe
(1970), for finite transformationstrain and no fluid
component,have already been mentioned.Oriani
(1966) consideredthe equilibrium of a coherent
spherical precipitate of an isotropic phase B in an
isotropic matrix ,a, the stress-freetransformation
strain being itself infinitesimaland isotropic. Because
of the symmetry of the situation, pressureis hydrostatic and uniform in B, and the characteristicvector
of the transformation is everywherenormal to the
interface.Oriani's problem is thus not difierent from
that of finding the equilibrium betweena solid matrix
a and a spherical inclusion of fluid B which can
dissolvethe solid. Mclellan (1970) and Paterson
(1973) give the condition of equilibrium of a coherentinterface,when the strain is infinitesimaland
there are no fluid components,as
F'-- F -
V o nA
r e;r:9.
(32)
Although F;q. 32 appears to ignore the fact that
the stressesare in general not equal on both sides
of the interface,the last term does reduce to Eq.
31 when the proper coherencyconditionsfor ,Aeai
are taken into account. Fletcher (I973) also
reached the proper equilibrium condition for infinitesimal strain.
Kamb (1959) and Paterson(1973) give the fol'grain boundary'
lowing equilibrium equation for a
supportinga shearstress:
1Eu-rSu+pvu)-(E- rs'+ Pv"):0
(33)
where P is, as before, the normal component of
stresson the interface.
Although Kamb (1959) doesnot detail his derivation, the only stated assumptionsare that "there
the shear
is no slippage at the interface and
interface
.
.
. ," conthe
across
stressis continuous
interface'
a
coherent
by
fulfilled
ditions which are
There is also no assumptionwhich would exclude
a coherentboundary in Paterson'sderivation' The
discrepancybetweenEqs. 33 and 16 thereforerequiresexamination.
The very notion of. equilibrium with respect to
a possible grain growtft requires that such grain
growth be reversible.That is, if the state of the
r296
PIERRE.YYES F, ROBIN
systemis alteredby a small variationalgrowth of,
say, phaseb, at the expenseof a, the reversevariation must bring the systemto its original state. In
particular,in presenceof a shear stress,points of
one phasemust have recoveredtheir original position with respectto points of the other phase. If
this requirementwere not met, the sum of the two
variationswould amount to a slip; slip then being
a possiblemechanism,equilibriumwould requireno
shear componentof stressalong the interface.On
the contrary, when the above requirementof reversibilityis met, it can be verifiedthat any variational migration of the boundary must occur in a
way similar to that describedin Figure 1, and can
thereforebe describedby a characteristicvector v4.
At this point it is important to recognize that
the migrationof an interface,as in Figures 1 or 2,
is indeed not restrictedto coherentinterfaces.We
may, for example,considera row of parallel edgedislocationsoriented as in Figure 3. When conditions are such that thesedislocationscannot climb
and do not move independentlyfrom one another,
this boundarymigrateslike the coherentboundaries
of FiguresI or 2. Growth of kink bandsconstitutes
anotherpossibleexample.
Therefore, in order to obtain Eq. 33, Kamb
(1959) and Paterson(1973) had to implicitly assume that for a grain boundarysupportinga shear
stressthe characteristicvector va w?Sperpendicular
to the interface(and thereforayt:0
in Eq. 16).
This is particularly clear in Paterson'sderivation,
in which an imaginary,inert, and permeablematrix
t
I
I
I
r-- -
\t
\
I
I
I
i
l--
I
I
applies the stress to the solid. To calculate the
changein potential energyof the matrix for a variational growth of the crystal,Patersonassumesthat
points of the matrix are moved along a direction
perpendicularto the interfaceto meet the new surface of the crystal. If contact between the solid
and the matrix is assumedto be reestablished
by
moving points of the matrix along some different
direction, a different value for the change in potential energyresults;yet, there is no a p'riori reason
to choosebetweenthesetwo assumptions.
Kamb (1959) and Paterson(1973) soughtto
apply Eq. 33 to the grain boundariesin polycrystalline aggregates.
However, contrary to the subgrain
or the kink boundariesenvisagedabove, random
high-anglegrainboundariesgenerallyhaveno mechanism which could constraintheir migration in presence of a shearstressto be reversiblein the same
manner.Indeed,becauseof the generallycomplete
disruption of the lattices along the boundary, the
position of the a lattice has no way of influencing
(or of being influencedby) the position of the b
lattice. It is possibleto conceiveequilibrium with
respectto growth while at the sametime the boundary is sliding irreversibly.In reality, however,sliding
along the boundary is likely to occur much faster
than any phasegrowth acrossit; growth, or equilibrium with respectto it, would thereforeobtain when
the grain boundary no longer supportsany shear
stress,in which case Gibbs' equilibrium condition
applies.
Except for theories of preferred orientations,
Gibbs equilibrium condition has so far not been
used in mineralogicalproblems.By contrast,equilibrium conditions for coherent interfaces have
found applicationsin various problems of phase
changesin the solid state (Patel and Cohen, 1953;
Coe, 1970; Robin, I974a,c) and are likely to find
more asthe microscopicstudyof interfacesdevelops.
Appendix
The infinitesimal strain with respect to the reference state
can be characterizedby the symmetric strain tensors
t(*,,1 * xi,u)
e n i u:
6n,i
(A1)
eoi":i@n,I +xi,I)-6oi
FIc. 3. Wall of gliding edge dislocations.The full lines
are traces of glide planes. Simultaneous gliding of the
dislocations results in a shear deformation of the framework, essentially similar to the interface migrations presentedin Figures I and 2.
vectoris
The characteristic
ai:
v(xi,bi-xt.x)ni"
or, using Eq. A1,
uo :
Vlz(e.,,u -
"n,")n,"
(xi,oo-
xt,i)nfl
(A2)
THERMODYNAMIC
EQUILIBRIUM
The fact that the transformation strain leaves the plane of
the interface invariant is expressedby
@ , , 1- x , , I ) t n : o
(A3)
where tr is any vector tangent to the interface at the point
under consideration. Also. the molar volume chanse is
v;n;o:
(A4)
V(en,u-enn")
The term (xt.tu - xt,t") ntn - kr of Eq. A2 can be
evaluated.From Eq. 43, k+tr - 0. Also
- (x;.1- xi,i)n1"n;":
kntr;"
#o,n:,
eii' -
:
aiio, from Eq. ,A4.
Therefore &r is normal to the interface and equal to
l<,:(ee1,b-erlo)n,"
Inserting into Eq. A2, we obtain finally
vn :
V121eoro-
-
(eo*u -
"nr")o,"
er,.o)no"]
(A5) : (30)
Acknowledgments
This work was carried out at the MassachusettsInstitute
of Technology with the support of grants from the National
Science Foundation (GA-31900) and from the U.S. Geological Survey (14-08-0001-13229); it is a part of the
author's doctoral work done under the supervision of
W. F. Brace. At various stages of its preparation the
manuscript greatly benefitted from the comments and
criticisms of W. F. Brace, R. S. Coe, W. B. Kamb, and
D. L. Kohlstedt; discussionswith R. M Stesky were also
quite useful.
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