Thermodynamically consistent incorporation of the Schneider
rate equations into two-phase models
Citation for published version (APA):
Hütter, M. (2001). Thermodynamically consistent incorporation of the Schneider rate equations into two-phase models. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 64(1), 011209/1-11.
https://doi.org/10.1103/PhysRevE.64.011209
DOI:
10.1103/PhysRevE.64.011209
Document status and date: Published: 01/01/2001 Document Version:
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Thermodynamically consistent incorporation of the Schneider rate equations
into two-phase models
Markus Hu¨tter*
ETH-Zu¨rich, Department of Materials, Institute of Polymers, Zu¨rich, Switzerland
共Received 24 February 2001; published 28 June 2001兲
We formulate a solid-liquid two-phase model including viscous stresses, heat conduction in the two phases, as well as heat exchange through the interface, and a phase change in the structure of nonequilibrium thermo-dynamics described by a general equation for the nonequilibrium reversible-irreversible coupling共GENERIC兲. The evolution of the microstructure is studied in terms of the Schneider rate equations introducing the nucle-ation rate and the radial growth rate of the solid phase. The applicnucle-ation of the GENERIC structure shows that this radial growth factor is not an additional, independent material function but is to be expressed in terms of the difference in the chemical potentials, in the temperatures, and in the pressures between the two phases. The contribution due to the pressure difference appears in conjunction with the surface tension in such a way, that a driving force results only if deviations from a generalized version of the Laplace equation occur. Further-more, it is found that for conditions under which the radial growth rate is zero, the nucleation rate must vanish. DOI: 10.1103/PhysRevE.64.011209 PACS number共s兲: 64.70.Dv, 44.35.⫹c, 64.60.⫺i, 81.30.Fb
I. INTRODUCTION
It is well accepted that the phase diagram of a material can be determined 共in principle兲 from a given thermody-namic potential. The use of equilibrium thermodythermody-namics, in this context in particular the extremum criteria 共and the cri-teria and constructions derived thereof兲, has shown to be of invaluable help. However, only little is known about the
dy-namics of the phase change and about its governing
equa-tions and criteria. This paper attempts to show how nonequi-librium formalisms can be used as a guideline to close this considerable gap. For this purpose, we unite a continuum two-phase model 关1兴 with the Schneider rate equations 关2兴, the latter giving a coarse-grained picture of the structural changes during phase transformation, in the framework of general equation for the nonequilibrium reversible-irreversible coupling 共GENERIC兲 关3,4兴. This paper is orga-nized as follows. First, a brief overview over the two-phase model and over the Schneider rate equations is given, before the essentials of the GENERIC formalism are outlined. Then, a model that unifies two-phase flow and the Schneider rate equations is incrementally developed using GENERIC as a guideline.
A. Two-phase flow
The description of two-phase flow adopted here consists of the hydrodynamic variables for both of the two phases, and in addition, of the volume fraction of one phase and of the amount of interface per unit volume. Hence, the micro-structure is characterized on a rather coarse level, having the
advantage that such a model is suitable for finite element simulations. A comprehensive introduction to this type of two-phase flow model can be found in the books of Ishii关1兴 and Drew and Passman关5兴. In the following, we consider the case of equal velocities of the two phases, thereby reducing the set of variables and of equations. This approximation is justified either by neglecting external forces and relative dif-fusion of the two phases or by assuming infinitely high in-terfacial friction between the two phases 关1,6兴. We may hence choose the following fields as independent variables for the solid(s)-liquid(l) two-phase system: the apparent mass densities␣(r) (␣⫽s,l), and the apparent internal en-ergy densities ⑀␣(r) (␣⫽s,l) where all apparent densities are the intrinsic densities per unit volume of the constituent times the volume fraction of the respective constituent, the total momentum density u(r) of the two phases, as well as the volume fraction of solid (r) and the amount of inter-face per unit volume (r). The velocity field v(r) is given by u(r)⫽关s(r)⫹l(r)兴v(r). Since we intend to also
in-clude the Schneider rate equations into the model and hence can follow the time evolution of the volume fraction, the mass transfer during phase change can be related to the change in volume fraction共in contrast to the linear constitu-tive equation on p.172 in Ref. 关1兴兲. If the rate of change of the volume fraction due to phase change is denoted by
˙兩pc, the rate of mass transfer is proportional to the phase
change rate,ˆ˙兩pc. The governing equations then read when
neglecting turbulent contributions 关1兴
ts⫹“•共vs兲⫽⫹ˆ˙兩pc, 共1兲
tl⫹“•共vl兲⫽⫺ˆ˙兩pc, 共2兲
tu⫹“•共vu兲⫽⫺“共ps⫹pl兲⫹“•共s⫹l兲⫹u, 共3兲
*Full address: ETH-Zu¨rich, Department of Materials, Institute of Polymers; ETH-Zentrum, Sonneggstr. 3, ML H 18, CH-8092 Zu¨r-ich, Switzerland. FAX: ⫹41 1 632 10 76. Email address: mhuetter@ifp.mat.ethz.ch
t⑀s⫹“•共v⑀s兲⫽⫺ps共“•v兲⫹s:共“v兲⫺“•qs⫹⑀s q ⫹⑀s⌫˙兩pc, 共4兲 t⑀l⫹“•共v⑀l兲⫽⫺pl共“•v兲⫹l:共“v兲⫺“•ql⫹⑀l q⫺⑀l⌫˙兩 pc. 共5兲
Let us briefly comment on the various contributions 共for more details the reader is referred to Refs.关1,6兴兲.
Using s⬅ andl⬅1⫺, the pressures p␣ are given by p␣⫽␣˜p␣, p˜␣ denoting the pressure in phase␣, and the viscous stresses and heat fluxes are expressed as
␣⫽␣␣关共“v兲⫹共“v兲T兴⫹2␣ˆ␣共“"v兲1, 共6兲 q␣⫽⫺␣␣•共“T␣兲, 共7兲
with the effective viscosities␣, the effective dilational vis-cosities ␣, ˆ␣⫽␣/2⫺␣/3, and the 共generally aniso-tropic兲 effective heat conduction tensors ␣ 共for details, see Ref.关6兴兲. The last term in Eq. 共6兲 is absent in Refs. 关1,6兴 but in analogy to classical, one-phase hydrodynamics one might wish to include it. The reader should notice that in the con-text of the two-phase models discussed here, all these phe-nomenological coefficients also depend on the microstruc-ture, i.e., on and 关1,6兴. A detailed discussion of the different phenomena included in the effective viscosity␣ is given, e.g., in Ref.关7兴. In particular, the constitutive assump-tion 共6兲 also holds for the rigid solid phase as long as the crystallites do not merge to form a single big crystal 关6,7兴. This peculiarity originates from the fact that the spatial reso-lution of the model presented here is larger than the size of the individual crystallites.
In the momentum balance Eq.共3兲, the term udenotes the effect of the surface tension given by 共see, e.g., Refs.
关1,6兴兲
u⫽“共H兲, 共8兲
where the coarse-grained mean curvature H depends on the microstructure, H(,). It is noteworthy that the interfacial momentum source共8兲 is such that it modifies the bulk pres-sure contribution in Eq.共3兲. Another effect of the interface is the possibility to exchange heat through the contact interface, hence there is a heat-flow proportional to the interfacial area. If ␣(0) denotes the microscopic heat conductivity, l␣(0) is a characteristic microscopic heat conduction length, and Ti
stands for the temperature of the interface, the interfacial energy change in phase ␣ due to interfacial heat transfer is given by Refs. 关1,6兴
⑀␣q⫽␣
(0)
l␣(0)共Ti⫺T␣兲. 共9兲
The rate of phase change affects not only the mass bal-ances共1兲,共2兲 as discussed above, but also the internal energy balances 共4兲,共5兲. The corresponding contributions are deter-mined by 关1,6兴
⑀␣⌫⫽ˆ h¯␣i⫺p˜␣i, 共10兲
where h¯␣idenotes the enthalpy per unit mass of phase␣, and
p
˜␣i is the pressure of phase ␣ at the interface. Since the
volume element, which defines the resolution of our model, is larger than the microstructures, it is necessary to introduce two different pressures, namely the bulk pressure p˜␣and the interface pressure p˜␣iin order not to lose too much detail. In general, the interfacial pressures are a subtle issue and have to be discussed for the specific problem at hand. For ex-ample, in liquid-solid two-phase systems one might assume
p
˜li⫽p˜l due to relatively fast pressure equilibration in the
liquid phase. As far as the solid phase is concerned, it is claimed in some references 关6–8兴 that, as long as the solid ‘‘crystals’’ are completely surrounded by liquid and there are no contacts between crystals, one has p˜si⫽p˜s. However, if
there is significant contact between the crystals or if the solid forms a continuous structure, additional pressure contribu-tions occur in the solid 关6–8兴. It is commonly assumed that the difference of the interfacial pressures relates to the mean curvature H of the interface and to the surface tension through p˜si⫺p˜li⫽H on which共8兲 is based. In the
follow-ing, fast pressure equilibration in the liquid phase and no contact between solid particles is assumed, i.e., p˜␣i⫽p˜␣. Similar to the pressures, one also needs to distinguish be-tween the bulk enthalpies h¯␣ and the interfacial enthalpies
h
¯␣i. It is well established that the difference between the
interfacial enthalpies 共and not the difference between their bulk counterparts兲 equals the latent heat L⫽h¯li⫺h¯si. A dis-cussion of possible closures for the interfacial enthalpies h¯␣i can be found in Refs.关1,6兴.
It is evident that the above set of Eqs. 共1兲–共5兲 is not closed for two reasons. First, in the above equations, the temperature of the interface enters, both directly trough ⑀␣q given in Eq.共9兲, as well as implicitly through(Ti). Since it
is desirable to have the interfacial temperature as a dynami-cal variable rather than as a constant, a corresponding equa-tion is required. In Ref.关1兴, such an equation is presented if some approximations can be made. It will be shown in Sec. II that the equation describing the dynamics of Ti 共or of an
equivalent variable兲 naturally arises when using the GE-NERIC formalism. Second, the above set of Eqs.共1兲–共5兲 is not closed because there are no evolution equations for the microstructural variables and. In order to close the sys-tem of equations, we here consider the Schneider rate equa-tions.
B. Schneider rate equations
In 1939–1941, Avrami studied the kinetics of phase change and developed the well-known Avrami equation 关9兴
to follow the time evolution of the volume fraction of the solid phase. Since this equation is an integral equation de-pending on the whole history of the process and is hence complicated to solve, Schneider, Ko¨ppel, and Berger 关2兴 transformed the integral equation into a set of four coupled differential equations for the variables兵⌿0,⌿1,⌿2,⌿3其
un-der quiescent conditions, where the physical volume fraction
and the amount of interface per unit volume can be expressed in terms of 兵⌿0,⌿1,⌿2,⌿3其. When
accommo-dating the original Schneider equations to flow conditions,
the convection mechanism for the variables
兵⌿0,⌿1,⌿2,⌿3其 needs special care. The explicit form of
the convection shall be addressed when formulating the model in the GENERIC framework. In the following, the dynamics of the structural variables is split into a convective and a phase change contribution, which are hereafter denoted by the subscripts ‘‘conv’’ and ‘‘pc,’’ respectively,
t⌿0⫽⌿˙0兩conv⫹⌿˙0兩pc, ⌿˙0兩pcªG共T兲⌿1, 共11兲
t⌿1⫽⌿˙1兩conv⫹⌿˙1兩pc, ⌿˙1兩pcªG共T兲⌿2, 共12兲
t⌿2⫽⌿˙2兩conv⫹⌿˙2兩pc, ⌿˙2兩pcªG共T兲⌿3, 共13兲
t⌿3⫽⌿˙3兩conv⫹⌿˙3兩pc, ⌿˙3兩pcª8␣共T兲, 共14兲
describing the growth of spherulitic structures, where G(T)
共in units of m s⫺1) is a radial growth rate, and ␣(T)关m⫺3 s⫺1兴 denotes the nucleation rate. The variables
兵⌿0,⌿1,⌿2,⌿3其 disregard the fact that, first, nuclei can be
swallowed by other growing crystallites, and, second, that different crystallites may impinge as crystallization proceeds. Hence, the above quantities are called ‘‘unrestricted.’’ ⌿0
denotes the unrestricted volume fraction, ⌿1 is the
unre-stricted surface per unit volume,⌿2is the total length of the
crystallites per unit volume, and⌿3represents the number of nuclei per unit volume. As in the two-phase model, the re-stricted 共physical兲 volume fraction and interfacial area in-stead of their unrestricted counterparts occur, it is essential to relate the two descriptions. There are two common relations between the real共i.e., restricted兲 and the unrestricted volume fraction of the form ⫽(⌿0) 关with d/d⌿0⬎0, (⌿0
⫽0)⫽0, and lim⌿ 0→⬁ (⌿0)⫽1], namely, ª1⫺e⫺⌿0, i.e., ⌿ 0⫽⫺ln共1⫺兲 Avrami 关9兴, 共15兲 ª ⌿0 1⫹⌿0 , i.e., ⌿0⫽ 1⫺ Tobin关10兴. 共16兲
The coupled Schneider rate equations can be mapped from the set兵⌿0,⌿1,⌿2,⌿3其 onto兵,,⌿2,⌿3其:
t⫽˙兩conv⫹˙兩pc, ˙兩pcªG共T兲, 共17兲 t⫽˙兩conv⫹˙兩pc, ˙兩pcªG共T兲L共,,⌿2兲, 共18兲 t⌿2⫽⌿˙2兩conv⫹⌿˙2兩pc, ⌿˙2兩pc⫽G共T兲⌿3, 共19兲 t⌿3⫽⌿˙3兩conv⫹⌿˙3兩pc, ⌿˙3兩pc⫽8␣共T兲, 共20兲 where ª
冉
d⌿0 d冊
⫺1 ⌿1, 共21兲 L共,,⌿2兲⫽冉
d⌿0 d冊
⫺1冉
⌿2⫺ d2⌿0 d2 2冊
. 共22兲The Schneider rate equations 共11兲–共14兲 as given above have been used by Eder and Janeschitz-Kriegl to study the crystallization of quiescent polymer melts 关11–15兴. Further-more, since the Schneider equations allow us to modify the phase change kinetics at different levels, it has even been possible to capture a number of essential phenomena occur-ring in the crystallization of sheared polymer melts关14,15兴. In their work, the 共modified兲 Schneider rate equations are solved simultaneously with a temperature equation including latent heat effects. In the procedure presented here this tem-perature equation is replaced by the internal energy balances and hence arises as a dependent equation. The above-mentioned applications of the Schneider rate equations clearly demonstrate their practical use and the strong need to relate them to a nonequilibrium context.
C. GENERIC structure
Recently, a general equation for the nonequilibrium reversible-irreversible coupling共GENERIC兲 has been devel-oped for describing nonequilibrium systems关3,4兴. When try-ing to formulate a model in the GENERIC framework, the first step is to choose the variables that describe the system. Similar to the procedure in equilibrium thermodynamics, the choice of variables must be such that they are independent and sufficient to capture the essential physics. Such a set of variables shall here be denoted by x. Note that x may have both discrete indices as well as continuous indices 共for field variables兲. According to GENERIC, the time evolution of the variables x can be written in the form
dx dt⫽L共x兲 ␦E ␦x⫹M共x兲 ␦S ␦x, 共23兲
where the two generators E and S are the total energy and entropy functionals in terms of the state variables x and L and M are certain matrices共operators兲. The matrix multipli-cations imply not only summations over discrete indices but may also include integration over continuous variables, and
␦/␦x typically implies functional rather than partial
also imposes certain conditions on the building blocks in Eq.
共23兲. First, Eq. 共23兲 is supplemented by the degeneracy
re-quirements
L共x兲␦S
␦x⫽0, 共24兲
M共x兲␦E
␦x⫽0. 共25兲
The requirement that the 共functional兲 derivative of the en-tropy lies in the null space of L represents the reversible nature of L. Hence, the functional form of S and L are con-strained such that the entropy is not affected by the operator generating the reversible dynamics, i.e., by L. On the other hand, the requirement that the functional derivative lies in the null space of M manifests that the total energy is not altered by the M contribution to the dynamics. In addition to these degeneracy requirements, L must be antisymmetric and fulfill the Jacobi identity, whereas M needs to be positive semidefinite and Onsager-Casimir symmetric. As a result of all these conditions one may easily show that the GENERIC Eq.共23兲 implies both the conservation of total energy as well as a nonnegative entropy production. The two contributions to the time evolution of x generated by the total energy E and the entropy S in Eq. 共23兲 are called the reversible and irre-versible contributions, respectively.
Both the complementary degeneracy requirements and the symmetry properties are essential for formulating proper L and M matrices when modeling concrete nonequilibrium problems. The list of systems, which have already been ex-pressed in the GENERIC form includes classical hydrody-namics, polymer kinetic theory共including hydrodynamic in-teraction, rigid constraints, reptation models, and polymer heat conductivity兲, chemical reactions, Boltzmann’s kinetic equation, and the Doi-Ohta model. These various applica-tions have shown that the two-generator idea and the degen-eracy requirements have strong implications. In order to jus-tify this approach, these elements of GENERIC, originally discovered by empirical observations, have been derived by projection operator formalisms 关16,17兴, which strongly en-courages us to use the GENERIC formalism.
II. GENERIC FORMULATION OF A TWO-PHASE MODEL
We here unify the two-phase model given by Eqs.共1兲–共5兲 with the Schneider rate Eqs. 共17兲–共20兲 in the GENERIC framework, thereby also constructing the appropriate equa-tions for the thermodynamic variables of the interface.
A. Set of variables and generating functionals The solid(s)-liquid(l) two-phase system is described in terms of the following variables. First, the two phases are characterized by the apparent mass densities␣(r) (␣⫽s,l), and the apparent internal energy densities ⑀␣(r) (␣⫽s,l) where all four apparent densities denote the intrinsic densi-ties per unit volume of the constituent times the volume frac-tion of the respective constituent. Second, the corresponding
quantities for the interface are the apparent interfacial mass density, i(r), and the apparent internal energy density, ⑀i(r). As previously discussed, it is sufficient under the equal velocity assumption to describe the flow by only one mo-mentum density, where in the following the total momo-mentum density u(r) accounts for both the momentum of the two phases as well as for the momentum of the interface, in con-trast to Sec. I A. And finally, according to the reformulated Schneider rate Eqs. 共17兲–共20兲, the microstructure shall be described by the volume fraction, the interfacial area per unit volume, by the unrestricted length⌿2 per unit volume,
and by the unrestricted number of crystallites ⌿3 per unit
volume. The set of variables x to describe the system is therefore
x⫽兵s,l,i,u,⑀s,⑀l,⑀i,,,⌿2,⌿3其. 共26兲
Natural expressions for the energy functional E and for the entropy functional S are obtained by the local equilibrium assumption. If the thermodynamics of the solid and of the liquid phase is characterized by the two branches ssand slof the entropy density and the thermodynamics of the interface is given by the entropy density si, the generating functionals E and S read E关x兴⫽
冕
冉
1 2 u2 s⫹l⫹i ⫹⑀s⫹⑀l⫹⑀i冊
d3r, 共27兲 S关x兴⫽冕
关ss共˜s,˜⑀s兲⫹共1⫺兲sl共˜l,˜⑀l兲⫹si共˜i,˜⑀i兲兴d3r 共28兲where the intrinsic quantities denoted by ‘‘⬃’’ for the indi-ces (s,l) are densities per volume of the respective phase
共and not with respect to the total volume of the volume
ele-ment兲 and for the index i are densities per amount of inter-face per unit volume. They can be expressed in terms of the corresponding apparent variables contained in x as
˜ s⫽˜s共s,兲⫽ s , 共29兲 ˜ l⫽˜l共l,兲⫽ l 1⫺, 共30兲 ˜ i⫽˜i共i,兲⫽ i , 共31兲 ⑀ ˜ s⫽˜⑀s共⑀s,兲⫽ ⑀s , 共32兲 ⑀ ˜ l⫽˜⑀l共⑀l,兲⫽ ⑀l 1⫺, 共33兲 ⑀ ˜ i⫽˜⑀i共⑀i,兲⫽ ⑀i . 共34兲
The following assumptions are implied in the above ex-pressions for the energy and entropy functionals. The current model includes neither the configurational entropy of the in-terface nor accounts for the relative arrangement of the crys-tallites. Disregarding the latter is in agreement with neglect-ing the relative diffusion of the two phases as discussed in Sec. I A. Furthermore, the pressure due to collisions between crystallites is not captured in the model presented here. In accordance with the comments on the interfacial pressures in Sec. I A, the current GENERIC model hence represents the situation p˜si⫽p˜s 共and p˜li⫽p˜l). Using the following
defini-tions for the temperatures, the chemical potentials per unit mass, and the pressures of the two phases and of the interface
共i.e., for ␣⫽s,l,i),
T␣共r兲⫽
冉
s␣共 ˜ ␣,˜⑀␣兲 ⑀˜ ␣冊
⫺1 , 共35兲 ␣共r兲 T␣共r兲⫽⫺ s␣共˜␣,˜⑀␣兲 ˜ ␣ , 共36兲 p ˜ ␣共r兲 T␣共r兲⫽s␣共˜␣,˜⑀␣兲⫺˜⑀␣ s␣共˜␣,˜⑀␣兲 ⑀˜␣ ⫺˜␣ s␣共˜␣,˜⑀␣兲 ˜␣ , 共37兲one finds for the functional derivatives of E and S
共38兲
where the velocity field v(r) is given by u(r)⫽关s(r)
⫹l(r)⫹i(r)兴v(r).
B. Reversible dynamics
In order to determine the reversible dynamics, we now construct the operator L. It has been discussed and illustrated in Ref.关4兴 that this operator is closely related to the genera-tor of space transformations on the field variables 共scalars, scalar densities, and vector densities兲. As an extension of classical one-phasic hydrodynamics关4兴, we propose the fol-lowing form for the operator L
with L su⫽⫺“s Lus⫽⫺s“, 共40a兲 L lu⫽⫺“l Lul⫽⫺l“, 共40b兲 L iu⫽⫺“i Lui⫽⫺i“, 共40c兲 Luu⫽⫺关“u⫹u“兴T, 共40d兲 L⑀ su⫽⫺“⑀s⫺ps“ Lu⑀s⫽⫺⑀s“⫺“ps, 共40e兲 L⑀ lu⫽⫺“⑀l⫺pl“ Lu⑀l⫽⫺⑀l“⫺“pl, 共40f兲 L⑀ iu⫽⫺“⑀i⫺ 2 3pi“ Lu⑀i⫽⫺⑀i“⫺ 2 3“pi, 共40g兲 Lu⫽⫺“⫹“ Lu⫽⫺“⫹“, 共40h兲 Lu⫽⫺“⫹ 23“ Lu⫽⫺“⫹ 2 3“, 共40i兲 L⌿ 2u⫽⫺“⌿2⫹ 1 3⌿2“ Lu⌿2⫽⫺⌿2“⫹ 1 3“⌿2, 共40j兲 L⌿ 3u⫽⫺“⌿3 Lu⌿3⫽⫺⌿3“, 共40k兲 and with the apparent pressures ps⫽˜ps, pl⫽(1⫺) p˜l,
and pi⫽˜pi. The derivatives act on all terms to the right of
them. The matrix L given by Eqs.共39兲, 共40兲 is antisymmetric and satisfies the Jacobi identity. Furthermore, the degeneracy
共24兲 is fulfilled due to the expressions for the pressures 共37兲.
The elements L
i,u and L⑀i,u, giving rise to the reversible
dynamics of the mass density 共in accordance with Refs.
关18,19兴兲 and the internal energy density of the interface 关which are absent in Eqs. 共1兲–共5兲兴 are justified on one hand
by analogy with the variables s andl and ⑀s and⑀l,
re-spectively, and on the other hand by the compatibility with the GENERIC antisymmetry and degeneracy requirements on L. It can be shown easily that L(r)•␦E/␦x(r) gives rise
to all reversible contributions in Eqs.共1兲–共5兲 and 共17兲–共20兲, i.e., to the convection mechanisms and the pressure contri-butions including a momentum source due to the pressure of the interface. If the surface tension is defined in terms of the pressure of the interface p˜i by⫽⫺p˜i, one obtains for
the momentum source term u⫽⫺23“pi⫽“(
2
3) rather than u⫽“(H) as given by Eq. 共8兲 where the curvature
H is involved. This discrepancy originates from the fact that
in the generating functionals E and S only the area of the interface enters, thereby not accounting for curvature. How-ever, for the specific case of spherical particles, the GE-NERIC expression for u coincides with Eq. 共8兲 since
H(,)⫽2
3/ 共see, e.g., p.178 in Ref. 关1兴兲. A further dif-ference between Eqs.共1兲–共5兲 and the GENERIC formulation presented here is that the latter accounts for a mass density of the interface that is absent in Eqs. 共1兲–共5兲, explaining why the momentum density as used in Eq.共27兲 includes also an interface contribution in contrast to the momentum density used in Eq. 共3兲.
Let us finally comment on the convection of the structural variables, i.e., on the form of the elements Lu, Lu, L⌿
2u, and L⌿
3u. The fact that all four structural variables have a different convection mechanism is motivated by the follow-ing argument. Considerfollow-ing the change in the structural vari-ables when simply blowing up the volume element isotropi-cally, it is easily seen that the volume fraction does not change, while the other quantities change since the surface area, the length, and the number scale differently than the volume element under blowing up. This analysis explains why the volume fraction is convected as a scalar in accor-dance with literature 共see, e.g., Refs. 关18–21兴兲, whereas the number density⌿3is naturally convected as a scalar density.
The ‘‘intermediate’’ variables and ⌿2 accordingly are
found to transform as indicated in the corresponding expres-sions in Eq. 共40兲. It should be mentioned that the amount of interface per unit volume is sometimes proposed to be convected as a scalar density共see, e.g., 关1,8,22兴兲 whereas the convection mechanism proposed above is in agreement with Ref. 关23兴. Furthermore, as the Doi-Ohta model for multi-phase flow shows, the shape of the interface enters the con-vection of 关23,24兴. However, since such detail is not in-cluded in the current description, the element Lugives a fair description of the convection of .
C. Irreversible dynamics
In this section, we construct the matrix M representing the irreversible effects in Eqs. 共1兲–共5兲 and 共17兲–共20兲, i.e., vis-cous stresses, heat conduction in the bulk of the two phases, and through the interface, and phase change. Due to the dif-ferent origin of the effects, M will be a sum of difdif-ferent contributions, each representing one specific phenomenon, i.e.,
M共r兲⫽M共r兲⫹M共r兲⫹Mq共r兲⫹Mpc共r兲, 共41兲
corresponding to viscous stresses, heat flow in the two phases, heat transport across the interface, and phase change, respectively. The criteria of GENERIC imposed on M are such that they can be verified for each contribution individu-ally in order to guarantee the compatibility with GENERIC of the total dissipative dynamics. Hence, in the following, we will show that each of the contributions is Onsager-Casimir symmetric, is positive-semidefinite and fulfills the degen-eracy requirement共25兲.
Before we proceed to the detailed discussion of the irre-versible contributions, we here briefly mention the general guideline along which the corresponding matrices M are constructed. First, one lists the variables that are involved. It may either be that the effect under consideration makes a contribution in the balance equation of these variables, or that the functional derivatives of the entropy with respect to these variables may help to construct meaningful driving forces for the effect under consideration. Then, all other el-ements in M that are not related to any of these variables should be set to zero. In order to simplify the notation, we only give the nonzero elements for each of the M contribu-tions in the following. For that purpose a subscript is
at-tached to the corresponding fគM contribution denoting in which rows共and columns兲 the nonzero elements are located.
1. Viscous stress
The matrix M that reproduces the viscous stress contri-butions in Eqs. 共3兲, 共4兲, 共5兲 with the closure 共6兲 can be de-termined by comparison with the classical one-phase hydro-dynamic case described in detail in Ref. 关4兴. Realizing that the contributions in the two-phase model due to viscous stresses are the sum of the single phase contributions, one finds with the aid of Ref.关4兴
M共r兲兵u,⑀s,⑀l其⫽
冉
Muu Mu⑀ s M u⑀l M⑀ su M ⑀s⑀s 0 M⑀ lu 0 M ⑀l⑀l 冊
, 共42兲 with Muu ⫽⫺关“共˘s⫹˘l兲“⫹1“•共˘s⫹˘l兲“兴T⫺2“共˘s⫹˘l兲“, 共43兲 Mu⑀s⫽“•˘s␥˙⫹“˘str␥˙ , 共44兲 M⑀ su ⫽⫺˘s␥˙•“⫺˘ str␥˙“, 共45兲 M⑀ s⑀s ⫽1 2˘s␥˙ :␥˙⫹ 1 2˘s共tr␥˙兲 2, 共46兲 Mu⑀l⫽“•˘l␥˙⫹“˘ltr␥˙ , 共47兲 M⑀ lu ⫽⫺˘l␥˙•“⫺˘ ltr␥˙“, 共48兲 M⑀ l⑀l ⫽1 2˘l␥˙ :␥˙⫹ 1 2˘l共tr␥˙兲 2, 共49兲 and ␥˙⫽“v⫹关“v兴T, 共50兲 ˘s⫽sTs, 共51兲 ˘ l⫽l共1⫺兲Tl, 共52兲 ˘s⫽ˆsTs, 共53兲 ˘ l⫽ˆl共1⫺兲Tl. 共54兲By calculating M(r)•␦S/␦x(r) one indeed recovers the
de-sired terms in Eqs.共3兲, 共4兲, 共5兲. Furthermore, one can show that the matrix M given by Eq. 共42兲 is symmetric, positive semidefinite and fulfills the degeneracy requirement共25兲.
2. Heat conduction: Bulk contribution
Analogously to the discussion for the viscous stress con-tribution, one can determine the irreversible contributions due to the bulk heat flows in Eqs.共4兲, 共5兲 using the closure
共7兲. The resulting matrix M共r兲兵⑀ s,⑀l其⫽⫺
冉
“•sTs2•“ 0 0 “•l共1⫺兲Tl 2•“冊
共55兲is symmetric, positive semidefinite, respects the degeneracy
共25兲 and by M(r)•␦S/␦x(r) leads to the desired heat flux
contributions.
3. Heat conduction: Interface contribution
Since the heat flux through the interface is specific to two-phase systems, we cannot here resort to one-phase hy-drodynamics to find the proper contributions⑀␣q given by Eq.
共9兲 to the Eqs. 共4兲, 共5兲. Since the expressions ⑀␣q contain the
temperatures Ts, Tl, and Ti, inspection of␦S/␦x suggests
that only the elements in the rows and columns⑀s, ⑀l, and
⑀i are nonzero. Indeed, the matrix Mq defined by
Mq共r兲兵⑀ s,⑀l,⑀i其 ⫽Ti
冉
s (0) ls (0)Ts 0 ⫺ s (0) ls (0)Ts 0 l (0) ll(0)Tl ⫺ l (0) ll(0)Tl ⫺s (0) ls(0)Ts ⫺ l (0) ll(0)Tl冉
s (0) ls(0)Ts⫹ l (0) ll(0)Tl冊
冊
共56兲is symmetric, positive semidefinite, respects the degeneracy requirement, and results by Mq(r)•␦S/␦x(r) in the desired
heat transfer contributions. In addition, the degeneracy re-quirement produced a corresponding contribution in the equation for the internal energy of the interface⑀i, such that the total energy is conserved by the interfacial heat transfer.
4. Phase change contributions
Here, we try to model the contributions to the evolution Eqs. 共1兲–共5兲, 共17兲–共20兲 arising due to phase change. It has been observed in many applications of the GENERIC frame-work共and is a fundamental essence thereof兲 that the appear-ances of one and the same phenomena in the different evo-lution equations are interwoven. In order to address this question for the phase change contribution when unifying the two-phase model with the Schneider rate equations, we first consider a wider class of closures for the phase change con-tributions than presented in Sec. I A in a twofold sense. First, we allow arbitrary sources for the phase transformation rates in the structural variables ⌽ of the form
generalizing the modified Schneider rate Eqs. 共17兲–共20兲, where we have used 兵⌽1,⌽2,⌽3,⌽4其 ⬅ 兵,,⌿2,⌿3其 to
simplify the notation. Second, the rate of phase change of any of the structural variables ⌽ may enter in any other balance equation in the sense
x
t⫽•••⫹⫽1
兺
4
a⌼共x兲. 共58兲
It is essential for the further procedure to notice that by virtue of Eqs.共57兲 and 共58兲, the set of vectors兵a1,a2,a3,a4其,
representing the effect of phase change in terms of
兵⌽1,⌽2,⌽3,⌽4其 on all variables x, is linearly independent
as inspection of their four last components shows
a1⫽
冉
⯗ 1 0 0 0冊
a2⫽冉
⯗ 0 1 0 0冊
a3⫽冉
⯗ 0 0 1 0冊
a4⫽冉
⯗ 0 0 0 1冊
共59兲representing the four Eqs. 共57兲. If we assume that the phase change terms can be formulated in GENERIC, one may write
⌽˙兩pc⫽⌼⫽b•
␦S
␦x 共⫽1,2,3,4兲. 共60兲
Then the matrix Mpc describing the phase change contribu-tions becomes due to Eq.共58兲
Mpc⫽
兺
⫽14
ab. 共61兲
In the following, we make the basic assumption that
Mpc•␦S/␦x is an ordinary matrix multiplication, i.e., that no
part of the operator acts as a derivative or as an integration. Thus, the vectors aand b are ‘‘normal’’ vectors. The task to formulate the phase change terms in GENERIC then be-comes a matter of linear algebra. Due to the fact that the vectors 兵a1,a2,a3,a4其 are linearly independent, one can
show in a mathematically rigorous manner that the matrix
Mpcis symmetric and positive semidefinite if and only if
Mpc⫽
兺
,⫽1 4 Aaa, 共62兲 with A⫽关A兴⭓0, 共63兲AT⫽A, i.e., A⫽A. 共64兲 Equation 共62兲 emphasizes that the driving forces for the phase change as introduced in Eq. 共60兲 are intimately coupled to the appearance of the phase change contributions in the evolution equations, namely to the vectors a, by
b⫽兺Aa. Furthermore, if we define
⌳(E)⬅a•␦E
␦x 共⫽1,2,3,4兲, 共65兲 ⌳(S)⬅a•␦S
␦x 共⫽1,2,3,4兲, 共66兲
the degeneracy requirement共25兲 and the equations of motion for the structural variables read
0⫽
兺
⫽1 4 A共x兲⌳(E)共x兲 共⫽1,2,3,4兲, 共67兲 ⌽˙兩pc⫽兺
⫽1 4 A共x兲⌳(S)共x兲 共⫽1,2,3,4兲. 共68兲The set of Eqs.共62兲–共68兲 contains the necessary and suf-ficient conditions for the phase change contributions in Eq.
共58兲 under the restrictions discussed after Eq. 共61兲. In the
following, we attempt to specify the matrix 关A兴 for the equations of motions 共1兲–共5兲. First, we identify the vectors
awith the corresponding contributions in Eqs.共1兲–共5兲. The fact that the evolution equations for the mass density of the interface and for the internal energy density of the interface, and hence, also their phase change contributions, are un-specified, leaves the corresponding components of the four vectors aundetermined, apart from the degeneracy require-ment 共67兲. However, as shown in the Appendix, the only physically meaningful choice of the corresponding compo-nents of ais such that⌳(E)⫽0. If one assumes that ⌿2and
⌿3 have no influence on the balance equations of iand⑀i 关i.e., if (a3)i⫽(a4)i⫽(a3)⑀i⫽(a4)⑀i⫽0兴, and using total
mass conservation关which by virtue of Eqs. 共1兲, 共2兲 results in (a1)
i⫽(a2)i⫽0], one concludes from ⌳
(E)⫽0 and Eqs.
共4兲, 共5兲 that (a1)⑀i⫽⫺(⑀s⌫⫺⑀l⌫) and (a2)⑀i⫽0. Since, by
do-ing so, all components of the four vectors aare determined, one finds
共69兲
⌳(S)⫽
冉
⑀s⌫冉
1 Ts⫺ 1 Ti冊
⫹⑀l ⌫冉
1 Ti⫺ 1 Tl冊
⫺ ˆ冉
s Ts⫺ l Tl冊
⫹冉
p ˜s Ts⫺ p ˜l Tl冊
⫺ Ti 0 0冊
. 共70兲By combining the evolution equation of the volume fraction
共17兲 with eq. 共68兲, ˙兩pc⫽G⫽共A11⌳ 1 (S)⫹A 12⌳2 (S)兲⬅共A˜ 11⌳1 (S)⫹A˜ 12⌳2 (S)兲, 共71兲
and by assuming the same thermodynamic driving forces for
˙兩pc(⫽⌽˙1兩pc), ˙兩pc(⫽⌽˙2兩pc), and for ⌿˙2兩pc(⫽⌽˙3兩pc) as
proposed by the Schneider Eqs. 共17兲–共19兲, the matrix A is found to be of the form 共the star symbols denote elements that are not yet determined兲
A⫽
冉
A˜11 A˜12 쐓 쐓 LA˜11 LA˜12 쐓 쐓 ⌿3A˜11 ⌿3A˜12 쐓 쐓 쐓 쐓 쐓 쐓冊
, 共72兲which, due to symmetry, becomes 共setting R⬅A˜11)
A⫽R共x兲
冉
2 L ⌿ 3 쐓 L L2 ⌿ 3L 쐓 ⌿3 ⌿3L 쐓 쐓 쐓 쐓 쐓 쐓冊
, 共73兲with the rate ‘‘constant’’ R(x)关K m4 J⫺1 s⫺1兴. The posi-tive semidefiniteness of A implies R(x)⭓0᭙ x. Through the explicit construction of the Choleski decomposition for the matrix A with the matrix elements already specified in Eq. 共73兲, it can be shown rigorously that the relation LA14
⫽A24must hold. The other four elements in the lower-right
corner, which do not contribute to the dynamics, remain un-determined except that A must be symmetric and positive semidefinite. The matrix A then takes the form
A⫽R共x兲
冉
2 L ⌿ 3 Q共 fគx兲 L L2 L⌿ 3 LQ共x兲 ⌿3 L⌿3 쐓 쐓 Q共x兲 LQ共x兲 쐓 쐓冊
, 共74兲whereQ(x) 关m⫺4兴 is an unspecified function. According to the phase change expressions for the structural variables given in the modified Schneider rate Eqs.共17兲–共20兲 and with Eq. 共68兲 one finds for the radial growth rate G and for the nucleation rate␣ G⫽R
冉
⌳1(S)⫹L ⌳2 (S)冊
, 共75兲 8␣⫽RQ冉
⌳1(S)⫹L ⌳2 (S)冊
. 共76兲Let us first comment on the expression 共75兲 for radial growth rate G. Using the functional derivatives of the en-tropy given in Eq.共38兲 and with the definition of the vector
⌳(S) 共66兲 one finds G R ⫽⌳1 (S)⫹L ⌳2 (S)⫽ˆ h¯ si
冉
1 Ts ⫺ 1 Ti冊
⫹ˆ h¯li冉
1 Ti ⫺ 1 Tl冊
⫺ˆ冉
s Ts⫺ l Tl冊
⫹ 1 1 Ti共关p˜s⫺p˜l兴⫺L兲, 共77兲where we have set the interfacial pressures equal to the bulk pressures, p˜␣i⫽p˜␣, in agreement with the discussion after Eq.共10兲 and the comments after Eq. 共32兲, and the last term in parentheses is the ‘‘Laplace’’ contribution. It is striking to see that the phase change is equally driven by the phase differences of all three intensive variables, namely tempera-ture, chemical potential, and pressure. In particular, the ‘‘Laplace’’ contribution in Eq.共77兲 is worth a comment. In a microscopic description of the interface, the Laplace equa-tion relates the pressure difference to the surface tension by equating the change in volume times the pressure difference to the change in surface times the surface tension. Since, according to the Schneider Eqs.共17兲, 共18兲, andL are the relevant quantities for change in volume fraction and in sur-face per volume, respectively, the last contribution in Eq.
共77兲 is the most natural and appropriate formulation of the
Laplace equation for the coarse grained description adopted in this paper. Furthermore, Eq. 共77兲 shows that the beyond-equilibrium situation in terms of a deviation from the recast Laplace equation is a driving force for phase change. The expression 共77兲 for the radial growth rate may be used to study the influence of the microstructure on the melting tem-perature T쐓. If the latter is defined by G⫽0 and Ts⫽Tl ⫽Ti⬅T쐓, the corresponding criterion reads
⫺ˆ共s⫺l兲⫹
冉
关p˜s⫺p˜l兴⫺L
冊
⫽0. 共78兲We notice that the microstructure enters through the prefac-torL/ 关m⫺1兴 of the surface tension, which becomes
tant for very small spherulites. If the independent variables in Eq.共78兲 are the melting temperature T쐓, the pressures p˜s
and p˜l, and the inverse length L/, Eq. 共78兲 predicts the change in the melting temperature when changing the spherulite size, i.e., the melting temperature depression共see also, e.g., Ref.关25兴兲. Finally, one should notice that as far as temperature and pressure is concerned, the corresponding quantities of the interface Ti and enter in Eq.共77兲, which
is in clear contrast to the chemical potential, where the influ-ence of the interfacial propertyiis absent. This asymmetry
originates from the fact that in the current model, the inter-facial mass densityiis not influenced by the phase change
of any of the structural variables, i.e., from (a)
i⫽0 ᭙.
However, in view of the procedure presented above, the ef-fect of phase change contributions in the balance equation for i on the radial growth rate 共77兲 may be elaborated straightforwardly.
A comparison of the expressions for the radial growth rate
G共75兲 and for the nucleation rate␣ 共76兲 results in
8␣⫽QG. 共79兲
It is of fundamental importance to notice that this does not necessarily mean that the nucleation rate␣ is essentially the same driving force as the radial growth rate G, since the function Q⫽Q(x) introduced in Eq. 共74兲 is an arbitrary function. Thus, the GENERIC formalism does not impose any constraint on the form of the nucleation rate except that the roots of the function G must also be roots of the function
␣ in order to have a nondiverging Q, and correspondingly nondiverging elements in the matrices A共74兲 and finally Mpc
共62兲. Physically speaking, this implies that for conditions
under which the radial growth rate is zero the nucleation rate must vanish. Hence, the distinction between nucleation and growth becomes obliterate under such ‘‘steady-state’’ condi-tions. It is worthwhile to notice that this constraint from the GENERIC formalism on␣ is much weaker than the restric-tions imposed on the form of G as given in Eqs.共75兲, 共77兲 since in the latter case the functionR must obey R⭓0. This is insofar a severe restriction, as in particular,R cannot ac-count for the different signs of the radial growth rate G in melting and crystallization conditions, respectively.
D. Final set of equations
We collect here the previously discussed building blocks of the GENERIC formalism in order to write down the final set of equations describing the two-phase flow, including the interface, in conjunction with the Schneider rate equations. At this point, the reader should notice the implications of the GENERIC structure on the equations for the mass density of the interface and for the internal energy density of the inter-face. First, the reversible contributions given by the elements
L
iuand L⑀iudefined in Eq. 共40兲 emerged not only due to
analogy with兵s,l其 and兵⑀s,⑀l其, respectively, but also due
to the GENERIC conditions imposed on L. Second, the irre-versible contributions due to heat transfer across the interface and due to phase change naturally共although not completely rigorously in the latter case兲 arose from the conditions
im-posed on M. Using the expressions for the functional deriva-tives of E and of S given in Eq.共38兲, and for the operators L
共39兲 and M 共41兲, the final and complete set of equations
describing the two-phase flow, including the dynamics of the interface and the Schneider rate equations, reads
ts⫹“•共vs兲⫽⫹ˆ G, 共80兲 tl⫹“•共vl兲⫽⫺ˆ G, 共81兲 ti⫹“•共vi兲⫽0, 共82兲 tu⫹“•共vu兲⫽⫺“共ps⫹pl兲⫹“•共s⫹l兲⫹“共 2 3兲, 共83兲 t⑀s⫹“•共v⑀s兲⫽⫺ps共“•v兲⫹s:共“v兲⫺“•qs⫹⑀s q⫹⑀s⌫ G, 共84兲 t⑀l⫹“•共v⑀l兲⫽⫺pl共“•v兲⫹l:共“v兲⫺“•ql⫹⑀lq⫺⑀l⌫G, 共85兲 t⑀i⫹“•共v⑀i兲⫽2 3共“•v兲⫺共⑀s q⫹⑀lq兲⫺共⑀s⌫⫺⑀l⌫兲G , 共86兲 t⫹v•“⫽G, 共87兲 t⫹v•“⫹1 3共“•v兲⫽GL共,,⌿2兲, 共88兲 t⌿2⫹“•共v⌿2兲⫺ 1 3⌿2共“•v兲⫽G⌿3, 共89兲 t⌿3⫹“•共v⌿3兲⫽8␣, 共90兲
where the expressions forL and G are given in Eqs. 共22兲 and
共77兲, respectively, and where the roots of the function G(x)
needs to be roots of ␣(x).
III. SUMMARY
The solid-liquid two-phase model given by the Eqs. 共1兲–
共5兲 has been united with the Schneider rate Eqs. 共17兲–共20兲
within the GENERIC framework of beyond-equilibrium thermodynamics in order to describe the dynamics of solidi-fication, resulting in Eqs. 共80兲–共90兲. The dynamic equations for the thermodynamic variables of the interface have been constructed using GENERIC as a guideline. It has been shown that dissipative contributions due to the viscous stresses and to the heat fluxes in both phases, as well as the heat exchange across the interface could be incorporated. Furthermore, according to the GENERIC formalism the ra-dial growth rate in the Schneider rate equations is not an additional material function but is expressed in terms of the phase differences in temperature, chemical potential, and pressure as shown in Eq.共77兲. It has been discussed that this expression for the radial growth factor naturally incorporates the dependence on the crystallite size due to the surface ten-sion in terms of a reformulated Laplace equation. Thus, it may be used to examine the dependence of the melting tem-perature on the microstructure. Finally, it is found that for conditions under which the radial growth rate is zero the nucleation rate must vanish.
ACKNOWLEDGMENT
The author thanks Hans Christian O¨ ttinger for helpful comments and discussions.
APPENDIX
In this appendix, it is shown why the phase change con-tributions in the equations for the mass densityiand for the internal energy density⑀i of the interface are most naturally chosen such that ⌳(E)⬅ fគ0. If we assume that there is no contribution in the equation for i and for ⑀i due to the
change in the structural variables⌿2and⌿3, it follows from the form of the functional derivative ␦E/␦x 共38兲, from the
two-phase flow Eqs.共1兲–共5兲 and from the definition of ⌳(E)
共65兲 that ⌳3
(E)⫽0 and ⌳ 4
(E)⫽0. In order to fulfill the
degen-eracy condition, four cases then need to be considered: Case 1: 兵⌳1(E)⫽0, ⌳2(E)⫽0其,
Case 2: 兵⌳1
(E)⫽0, ⌳ 2 (E)⫽0其
, Case 3: 兵⌳1(E)⫽0, ⌳2(E)⫽0其, Case 4: 兵⌳1(E)⫽0, ⌳2(E)⫽0其.
In the first case, one finds that the first column and row in the symmetric matrix A must be zero, resulting in Dt⌽1⫽Dt
⫽0. Accordingly, the second case leads to Dt⌽2⫽Dt⫽0.
Hence, the first two cases can be discarded. In the third case, the symmetric matrix A must be of the form
A⫽
冉
⌳2 (E) ⫺⌳ 1 (E) 쐓 쐓 ⫺⌳1 (E) 쐓 쐓 쐓 쐓 쐓 쐓 쐓 쐓 쐓 쐓 쐓冊
. 共A1兲However, according to the Schneider Eqs. 共17兲, 共18兲, the ratio Dt/Dt determined via Eq.共68兲 should only depend on the microstructure. It is very possible that this can be achieved but only with a very peculiar choice for the matrix element A22. The fourth case does not put any constraints on
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