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Temperature effects during Ostwald ripening Giridhar Madras a) Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India Benjamin J. McCoy Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 Temperature influences Ostwald ripening through its effect on interfacial energy, growth rate coefficients, and equilibrium solubility. We have applied a distribution kinetics model to examine such temperature effects. The model accounts for the Gibbs–Thomson influence that favors growth of larger particles, and the dissolution of unstable particles smaller than critical nucleus size. Scaled equations for the particle size distribution and solution concentration as functions of time are solved numerically. Moments of the distribution show the temporal evolution of number and mass concentration, average particle size, and polydispersity index. Parametric and asymptotic trends are plotted and discussed in relation to reported observations. Temperature programming is proposed as a potential method to control the size distribution during the phase transition. We also explore how two crystal polymorphs can be separated by a temperature program based on different interfacial properties of the crystal forms. I. INTRODUCTION Particle growth during phase transitions in materials and pharmaceutical processing is influenced by kinetics and ther- modynamics through temperature effects. The effect of tem- perature on interfacial energy, diffusion and growth rate co- efficients, and equilibrium solubility at the microstructural level influences crystal or grain properties during the phase transition. Thus temperature is a potential control parameter that can be manipulated to optimize product properties and manufacturing methods. Ostwald ripening is the last stage of a condensation transition from gas to liquid or from liquid to solid. 1–3 During ripening of a distribution of particles, the Gibbs–Thomson effect determines that smaller particles are more soluble than larger particles. 4 Smaller particles can shrink to their critical nucleus size and rapidly vanish be- cause of the thermodynamic instability of subcritical clusters. 5 This denucleation process leads to a diminution in the number of particles, and a consequent asymptotic power- law evolution to a monodisperse distribution, ultimately con- sisting of a single large particle. 6 All the participating pro- cesses are affected by the temperature as the system proceeds toward its asymptotic behavior. Our aim is to explore the possibility that temperature programming can provide a way to tailor the particle distribution during ripening. Among the earliest models for the particle size distribu- tion were those of Lifschitz and Slyozov 7 ~LS! and Wagner 8 ~W!, whose approximations included assuming the monomer concentration is constant at its equilibrium value. Marqusee and Ross 9 expanded on the LSW model by showing it rep- resents the leading terms in a series for the long time solu- tion. Venzl 10 solved the governing first-order nonlinear dif- ferential equation numerically, assuming that clusters vanished at a rate varying exponentially or inversely with time. Bhakta and Ruckenstein 11 more recently based a sto- chastic theory of ripening on a discrete microscopic continu- ity equation that generalized the LSW differential equation with rate constants assumed independent of particle size. We have recently 4–6 formulated a new approach to Ostwald rip- ening ~or isothermal recrystallization! that accounts for the evolution of the particle size distribution expressed in terms of the particle mass, rather than its radius. The distribution- kinetics approach with single monomer addition and disso- ciation is reversible and is generally applicable to growth, dissolution, or ripening phenomena. Denucleation of un- stable clusters ensures that the cluster number decreases as required for a realistic model of ripening. We have shown 12 how the LSW model 7,8 and subsequent enhancements of the model by other investigators 9,10,13–17 correctly depict the time dependence of particle number concentration and aver- age particle size, but often approximate the higher moments of the particle size distribution. Particles with more than one crystal structure, 18 or poly- morphs, provide an example for investigation. Polymorphs with different shapes have different surface properties, which influences the growth rate of crystal faces and shape the crys- tal habit. 19 Two polymorphs affected differently by tempera- ture will respond differently to temperature varying with time. One polymorph may be more stable at a given tempera- ture than another, and thus the more stable form would grow faster whereas the less stable form would grow slower. Con- trolling the particle size distributions can potentially be op- timized by applying a temperature program. Our theory ignores temperature gradients within the melt-particle system, which should be valid when the Prandtl number is small ~Pr!1!. Heat conduction in the presence of an imposed linear temperature gradient has been considered a! Author to whom correspondence should be addressed. Tel.: 91-080-309-2321; Fax: 91-080-360-0683; Electronic mail: [email protected]
Transcript
Page 1: Temperature effects during Ostwald ripening - [email protected] - Indian

Temperature effects during Ostwald ripeningGiridhar Madrasa)

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

Benjamin J. McCoyDepartment of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803

Temperature influences Ostwald ripening through its effect on interfacial energy, growth ratecoefficients, and equilibrium solubility. We have applied a distribution kinetics model to examinesuch temperature effects. The model accounts for the Gibbs–Thomson influence that favors growthof larger particles, and the dissolution of unstable particles smaller than critical nucleus size. Scaledequations for the particle size distribution and solution concentration as functions of time are solvednumerically. Moments of the distribution show the temporal evolution of number and massconcentration, average particle size, and polydispersity index. Parametric and asymptotic trends areplotted and discussed in relation to reported observations. Temperature programming is proposed asa potential method to control the size distribution during the phase transition. We also explore howtwo crystal polymorphs can be separated by a temperature program based on different interfacialproperties of the crystal forms.

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I. INTRODUCTION

Particle growth during phase transitions in materials apharmaceutical processing is influenced by kinetics and tmodynamics through temperature effects. The effect of teperature on interfacial energy, diffusion and growth rateefficients, and equilibrium solubility at the microstructurlevel influences crystal or grain properties during the phtransition. Thus temperature is a potential control paramthat can be manipulated to optimize product propertiesmanufacturing methods. Ostwald ripening is the last staga condensation transition from gas to liquid or from liquidsolid.1–3 During ripening of a distribution of particles, thGibbs–Thomson effect determines that smaller particlesmore soluble than larger particles.4 Smaller particles canshrink to their critical nucleus size and rapidly vanish bcause of the thermodynamic instability of subcriticclusters.5 This denucleation process leads to a diminutionthe number of particles, and a consequent asymptotic polaw evolution to a monodisperse distribution, ultimately cosisting of a single large particle.6 All the participating pro-cesses are affected by the temperature as the system protoward its asymptotic behavior. Our aim is to explore tpossibility that temperature programming can provide a wto tailor the particle distribution during ripening.

Among the earliest models for the particle size distribtion were those of Lifschitz and Slyozov7 ~LS! and Wagner8

~W!, whose approximations included assuming the monoconcentration is constant at its equilibrium value. Marquand Ross9 expanded on the LSW model by showing it reresents the leading terms in a series for the long time stion. Venzl10 solved the governing first-order nonlinear d

a!Author to whom correspondence should be addressed.91-080-309-2321; Fax: 91-080-360-0683; Electronic [email protected]

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ferential equation numerically, assuming that clustvanished at a rate varying exponentially or inversely wtime. Bhakta and Ruckenstein11 more recently based a stochastic theory of ripening on a discrete microscopic contiity equation that generalized the LSW differential equatiwith rate constants assumed independent of particle sizehave recently4–6 formulated a new approach to Ostwald riening ~or isothermal recrystallization! that accounts for theevolution of the particle size distribution expressed in terof the particle mass, rather than its radius. The distributikinetics approach with single monomer addition and disciation is reversible and is generally applicable to growdissolution, or ripening phenomena. Denucleation of ustable clusters ensures that the cluster number decreasrequired for a realistic model of ripening. We have show12

how the LSW model7,8 and subsequent enhancements ofmodel by other investigators9,10,13–17 correctly depict thetime dependence of particle number concentration and aage particle size, but often approximate the higher momeof the particle size distribution.

Particles with more than one crystal structure,18 or poly-morphs, provide an example for investigation. Polymorpwith different shapes have different surface properties, whinfluences the growth rate of crystal faces and shape the ctal habit.19 Two polymorphs affected differently by temperature will respond differently to temperature varying witime. One polymorph may be more stable at a given tempture than another, and thus the more stable form would gfaster whereas the less stable form would grow slower. Ctrolling the particle size distributions can potentially be otimized by applying a temperature program.

Our theory ignores temperature gradients within tmelt-particle system, which should be valid when the Prannumber is small~Pr!1!. Heat conduction in the presencean imposed linear temperature gradient has been consid

l.:

Page 2: Temperature effects during Ostwald ripening - [email protected] - Indian

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by Snyderet al.20 in a numerical simulation of particle coarsening.

The temperature effects incorporated into the presmodel include the diffusion-influenced growth coefficienthe Gibbs–Thomson effect of particle curvature on equirium solubility, the phase-transition energy~heat of solidifi-cation or vaporization!, the critical nucleus size, and interfacial energy~surface tension!. The dissolution rate coefficienis related to the growth rate coefficient by microscopicversibility, thereby determining its temperature dependenThe absolute temperature is scaled by a reference tempture, which for gas–liquid systems is the critical temperatTc .

In an earlier paper21 we presented a crystal growttheory with temperature effects, whereas in the present wwe focus on Ostwald ripening. Ripening is caused byvarying curvature of different interfaces, and thus canimportant whenever a distribution of particle sizes exisThe consequent interfacial energy~or Gibbs–Thomson! ef-fect is sensitive to temperature, thus offering an opportunto control cluster or grain size by temperature programmiFor two polymorphs, any of the kinetic or thermodynamparameters in the model might have different values, buteffect of the interfacial energy coefficient is a propertyparticular interest. In what follows, we examine the tempeture dependence of the parameters that influence ripenWe begin by discussing the elements of distribution kinethrough population dynamics~Sec. II!, then propose anasymptotic solution to the dimensionless population dynaics equation~Sec. III!, present and discuss the results of tnumerical analysis of the population dynamics equat~Sec. IV!, and finally provide comparisons with experimenobservations along with conclusions~Sec. V!.

II. DISTRIBUTION KINETICS

The size distribution is defined byc(x,t)dx, represent-ing the concentration of clusters~crystals, droplets, particles!at time t in the differential mass range (x,x1dx). Momentsare defined as integrals over the mass,

c~n!~ t !5E0

`

c~x,t !xn dx. ~2.1!

The zeroth moment,c(0)(t), and the first moment,c(1)(t),are the time-dependent molar~or number! concentration ofclusters and the cluster mass concentration~mass/volume!,respectively. The ratio of the two is the average cluster mcavg5c(1)/c(0). The variance,cvar5c(2)/c(0)2@cavg#2, andthe polydispersity index,cpd5c(2)c(0)/c(1)2, are measures othe polydispersity. The molar concentration,m(0)(t), of sol-ute monomer of molecular weightxm is the zeroth momenof the monomer distribution,m(x,t)5m(0)(t)d(x2xm).

The deposition or condensation process by which momers of massx85xm are reversibly added to or dissociatefrom a cluster of massx can be written as the reactionlikprocess,22,23

C~x!1M ~x8! �kg~x!

kd~x!

C~x1x8!, ~2.2!

nt,-

-e.ra-e

rkee.

y.

ef-g.s

-

nl

s,

-

whereC(x) is the cluster of massx and M (x85xm) is themonomer. The mass balance equations governing the cludistribution,c(x,t), and the monomer distribution,m(x,t),are

]c~x,t !/]t52kg~x!c~x,t !E0

`

m~x8,t !dx8

1E0

x

kg~x2x8!c~x2x8,t !m~x8,t !dx8

2kd~x!c~x,t !1Ex

`

kd~x8!c~x8,t !

3d~x2~x82xm!!dx82Id~x2x* ! ~2.3!

and

]m~x,t !/]t52m~x,t !E0

`

kg~x8!c~x8,t !dx8

1Ex

`

kd~x8!c~x8,t !d~x2xm!dx8

1Id~x2x* !x* /xm . ~2.4!

Nucleation of clusters of massx* at rateI are source termsor, in case of ripening, sink terms for denucleation, whioccurs when clusters shrink to their critical size,x* , andthen spontaneously vanish. The difference between ordindissolution due to concentration driving forces and total dintegration due to thermodynamic instability is thus undscored. This is the key distinction between crystal growt21

alone and the present discussion of growth with ripening.ordinary particle growth or dissolution, we would setI 50.Initial conditions for Eqs.~2.3! and ~2.4! are c(x,t50)5c0(x) and m(x,t50)5m0

(0)d(x2xm). The mass balancefollows from Eqs.~2.3!–~2.4!, and can be expressed in termof mass concentrations,

xmm0~0!1c0

~1!5xmm~0!~ t !1c~1!~ t !. ~2.4a!

The size distribution changes according to Eq.~2.3!,which becomes, when the integrations over the Dirac disbutions are performed, the finite-difference differential eqution,

]c~x,t !/]t52kg~x!c~x,t !m~0!~ t !

1kg~x2xmc~x2xm ,t !m~0!~ t !

2kd~x!c~x,t !1kd~x1xm!c~x1xm ,t !

2Id~x2x* !. ~2.5!

Equation ~2.5! can be expanded forxm!x to convert thedifferences into differentials, yielding the customary~ap-proximate! continuity equation applied to particle growtand ripening.4,6,15

At equilibrium, ]c/]t50 andI 50, so that Eq.~2.5! im-plies

kd~x!5meq~0!kg~x!, ~2.6!

Page 3: Temperature effects during Ostwald ripening - [email protected] - Indian

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which is a statement of microscopic reversibility~detailedbalance!. With rate coefficients for a cluster of mass,x, givenan expression forkg(x), one can calculatekd(x).

A monomer that attaches to a cluster may diffuthrough the solution to react at the cluster surface. Sdiffusion-controlled reactions have a rate coefficierepresented24 by

kg54pDmr c , ~2.7!

where the cluster radius is related to its massx by r c

5(3x/4prc)1/3, in terms of the cluster mass densityrc ,

which we assume to be constant with temperature. As uin kinetics, the temperature dependence of the growth~addi-tion, aggregation! rate is quite weak relative to dissolutio~dissociation, scission! rate. For the large range of tempertures in some ceramic processing methods, however, dsion effects can be significant, so we assume an activaenergy for the growth coefficient to account for activatdiffusion,

Dm5D0 exp~2E/RT!;

thus

kg~x!5gxl exp~2E/RT!, ~2.8!

whereE is the activation energy,R is the gas constant, andl51/3, theng54pD0(3/4prc)

1/3. The 1/3 power onx thusrepresents diffusion-controlled ripening, the primary issueprevious work.13–15 When growth is limited by monomeattachment and dissociation at the cluster surface, thecoefficient is proportional to the cluster surface area,kg

}r c2, so that we can writekg proportional tox2/3; thus in Eq.

~2.8!, l52/3 for surface-controlled ripening.6 If the deposi-tion is independent of the surface area, thenkg varies asx0.Other expressions for the rate coefficients that are applicto cluster growth may be realistic for complex and combinrate processes.

The temperature dependence for growth and ripenininfluenced by the thermodynamic properties. The interfacurvature effect is prescribed by the Gibbs–Thomson eqtion expressed in terms ofm`

(0) , the equilibrium solubility~or vapor pressure! of a plane surface,

meq~0!5m`

~0! exp~V! ~2.9!

with

V52sxm /r crckBT, ~2.10!

wherexm /rc is monomer molar volume,s is the interfacialenergy,kB is Boltzmann’s constant, andT is temperature.Asymptotic models1,14 always linearize the Gibbs–Thomsoequation, an approximation not valid for small particles ding early stages of ripening or at low temperatures25 accord-ing to Eq.~2.10!.

The critical nucleus radius at a given solute concention m(0) is

r * 52sxm /@rckBT ln~m~0!/m`~0!!#. ~2.11!

Surface tension for the gas–liquid interface decreases nelinearly with temperature,26 thus we takes5s0(12T/Tc),whereTc is the critical~or reference! temperature, causings

ht

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te

led

isl

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to vanish at the critical point. For liquid–solid interfaces, ttemperature dependence ofs may be represented by a mocomplex function. The temperature dependence of the elibrium solubility is given by

m`~0!5m` exp~2DH/RT!, ~2.12!

whereDH is the molar energy of the phase transition, am` is the flat-surface equilibrium solubility at largeT.

We define scaled dimensionless quantities for the mand temperature relationships,

C5cxm /m` , C~n!5c~n!/m`xmn , j5x/xm ,

u5tgm`xml , S5m~0!/m` ,

Seq5S exp~h/Q2V!, V5w~Q2121!j1/3,

w5~3xm/4prc!21/32s0xm /rckBTc , ~2.13!

Q5T/Tc , J5I /~gm`2 xm

l !, h5DH/RTc ,

e5E/RTc .

Note thatj is the number of monomers in the cluster andQis the reduced temperature (0,Q,1). The ratioS is definedrelative to the high temperature solubilitym` , rather than tothe plane-surface solubilitym`

(0) as in our earlier isothermawork.4,6 The supersaturation ratio defined asSeq5m(0)/meq

(0)

evolves to unity at thermodynamic equilibrium. The scalnumber ~or moles! of particles,C(0)5c(0)/m` , is also inunits of the solubilitym` . The Gibbs–Thomson factorV,Eq. ~2.13!, is expressed in terms of a scaled interfacial eergy,w. Substituting these expressions in Eqs.~2.3! and~2.4!yields the dimensionless equations,

]C~j,u!/]u5S~u!exp~2e/Q!@2jlC~j,u!

1~j21!lC~j21,u!#2jl exp@2~h

1e!/Q#exp@w~Q2121!j21/3#C~j,u!

1~j11!l exp@2~h1e!/Q#exp@w~Q21

21!~j11!21/3#C~j11,u!2Jd~j2j* !

~2.14!

and

dS~u!/du5exp~2e/Q!@2S~u!1exp~2h/Q!

3exp@w~Q2121!~Cavg!21/3##C~l!1Jj* .

~2.15!

The initial conditions areS(u50)5S0 and C(j,u50)5C0(j). Because the rate coefficients are related by micscopic reversibility in Eq.~2.4!, Eq. ~2.15! provides the re-quired thermodynamic equilibrium,m(0)5meq

(0) , whendS/du50 andJ50. The number of monomers in the criticanucleus is

j* 5@w~Q2121!/~ ln S1h/Q!#3, ~2.16!

which varies with time because of the time dependenceQ~u! andS(u).

From Eq.~2.1! the scaled moments are

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1692 J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 G. Madras and B. J. McCoy

FIG. 8. Effect of exponential temperature-rate parametera on the time evolution of~a! S ~solid line! and supersaturation ratioSeq ~dotted line!, ~b! particlenumber concentrationC(0), ~c! particle average massCavg, ~d! polydispersityCpd. The parameters in the calculations areQ50.510.9(exp(2au)),C0

(0)51, C0avg5100,S055, w51, l50, e50.01,h51.

anopnla

ingtureing.

based on distribution kinetics, provides a way to understand compute how the particle-size distribution and its prerties develop in time. Such a theory is useful in understaing observed behavior during phase transitions and in p

Downloaded 09 Mar 2004 to 203.200.43.195. Redistribution subject to AI

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ning effective and efficient processes for manufacturmaterials. The current article has explored the temperaeffects on the last stage of condensation, Ostwald ripenThe proposed model allows key physical properties~interfa-

FIG. 9. Evolution of two polymorphs, A~solid line! and B ~dotted line!, as a function of the temperature-rate parametera for ~a! particle numberconcentrations,~b! particle mass concentrations, and~c! particle average masses. The parameters in the calculations areu50.0510.9(exp(2au)),CA0

(0)5CB0(0)51, CA0

avg5CB0avg5100,wA51, wB52, S055, l50, e50.01,h51.

P license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Page 11: Temperature effects during Ostwald ripening - [email protected] - Indian

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1693J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 Temperature effects during Ostwald ripening

cial energy, heat of condensation, activation energy, GibThomson curvature effect on growth rate and on the critnucleus size! to be incorporated into the quantitative evalation of an evolving size distribution. The distributionkinetics approach begins with the population dynamics eqtion that describes the dependence of the size-distribufunctions on time and particle mass. The equation incluparticle growth and dissolution kinetics, as well as thenucleation rate for particles that have shrunk to their critinucleus size. An accompanying equation for the solutconcentration affords a mass balance for particle massdissolved solute. The mass moments of the distributwhich is solved numerically in scaled~dimensionless! form,yield the particle number concentration~zeroth moment! andmass concentration~first moment!, and hence average paticle mass. As a measure of the distribution’s width, the podispersity index is based on the second mass moment. Aprevious work we find asymptotic power-law temporal bhavior for decreasing particle concentration and increasaverage particle size.

With temperature as an active parameter in the moone can determine not only the influence of different teperatures on ripening, but also the effect of temperaturegramming. Changing temperature with time can potentiacontrol particulate size distributions based on realistic meling of crystallization processes during cooling or heatiWe have presented results for linear and for exponential tperature processes, and have demonstrated enhanceding during cooling. Temperature programs further providmethod to distinguish and possibly separate polymorphs~dif-ferent structural forms with the same crystalline compotion!. If two polymorphs have different interfacial energiewhich could cause different shapes for the two forms, ththe computations suggest that ripening may manifest difent dynamics, and hence separation. Increasing the temture according to an exponential program demonstratesthe polymorph with larger interfacial energy will essentiadissolve away, while the polymorph with smaller energycreases in mass.

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1A. Baldan, J. Mater. Sci.37, 2171~2002!.2A. Baldan, J. Mater. Sci.37, 2379~2002!.3P. W. Voorhees, Annu. Rev. Mater. Sci.22, 197 ~1992!.4G. Madras and B. J. McCoy, J. Chem. Phys.115, 6699~2001!.5G. Madras and B. J. McCoy, J. Chem. Phys.117, 6607~2002!.6G. Madras and B. J. McCoy, J. Chem. Phys.117, 8042~2002!.7I. M. Lifshitz and J. Slyozhov, J. Phys. Chem. Solids19, 35 ~1961!.8C. Wagner, Z. Elektrochem.65, 243 ~1961!.9J. A. Marqusee and J. Ross, J. Chem. Phys.79, 373 ~1983!.

10G. Venzl, Ber. Bunsenges. Phys. Chem.87, 318 ~1983!.11A. Bhakta and E. Ruckenstein, J. Chem. Phys.103, 7120~1995!.12G. Madras and B. J. McCoy, Chem. Eng. Sci.~to be published!.13H. Gratz, J. Mater. Sci. Lett.18, 1637~1999!.14K. Binder, Phys. Rev. B15, 4425~1977!.15H. Xia and M. Zinke-Allmang, Physica A261, 176 ~1998!.16M. Zinke-Allmang, L. C. Feldman, and M. H. Grabow, Surf. Sci. Rep.16,

377 ~1992!.17G. R. Carlow and M. Zinke-Allmang, Surf. Sci.328, 311 ~1995!.18R. Mohan, K.-K. Koo, C. Strege, and A. S. Myerson, Ind. Eng. Che

Res.40, 6111~2001!.19D. B. Patience and J. B. Rawlings, AIChE J.47, 2125~2001!.20V. A. Snyder, N. Akaiwa, J. Alkemper, and P. W. Voorhees, Metall. Mat

Trans. A30A, 2341~1999!.21G. Madras and B. J. McCoy, Acta. Mater.51, 2031~2003!.22B. J. McCoy, J. Colloid Interface Sci.228, 64 ~2000!.23G. Madras and B. J. McCoy, J. Cryst. Growth243, 204 ~2002!.24D. F. Calef and J. M. Deutch, Annu. Rev. Phys. Chem.34, 493 ~1983!.25M. Strobel, K.-H. Heinig, and W. Moller, Phys. Rev. B64, 245422~2001!.26R. C. Reid, J. M. Prausnitz, and T. K. Sherwood,The Properties of Gases

and Liquids~McGraw-Hill, New York, 1977!, p. 612.27S. Olive, U. Grafe, and I. Steinbach, Comput. Mater. Sci.7, 94 ~1996!.28R. H. Perry and D. W. Green,Perry’s Chemical Engineers’ Handbook, 7th

ed. ~McGraw-Hill, New York, 1997!, Table 2-224.29J. M. Rousseaux, P. Weisbecker, H. Muhr, and E. Plasari, Ind. Eng. Ch

Res.41, 6059~2002!.30S. L. Girshick and C.-P. Chiu, J. Chem. Phys.93, 1273~1990!.31C. H. Yang and H. Liu, J. Chem. Phys.84, 416 ~1986!.32W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Nu-

merical Recipes in C: The Art of Scientific Computing~Cambridge Uni-versity Press, New York, 1993!.

33A. J. Ardell, Mater. Sci. Eng.A238, 108 ~1997!.34S. E. Offerman, N. H. van Dijk, J. Sietsma, S. Grigull, E. M. Lauridsen,

Margulies, H. F. Poulsen, M. Th. Rekveldt, and S. van der Zwaag, Scie298, 1003~2002!.

35M. Militzer, Science298, 975 ~2002!.36X. D. Zhang, P. Bonniwell, H. L. Fraser, W. A. Baeslack, D. J. Evans,

Ginter, T. Bayha, and B. Cornell, Mater. Sci. Eng., A343, 210 ~2003!.37D. N. Seidman, E. A. Marquis, and D. C. Dunand, Acta Mater.50, 4121

~2002!.

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