Ostwald ripening in porous materials

Quimica nova

Pascova, R.¹  Gutzow, Iwan¹  Slezov, Vitali V.²  Schmelzer, Jürn³  Möller, Jörg³ 

[1] Bulgarian Academy of Sciences, Bulgaria;Kharkov Institute of Physics and Technology, Kharkov, Ukraine;Universität Rostock, Rostock, Germany

关键词: phase transformations in solids;    elastic strains;    coarsening. ;     ;    1. INTRODUCTION The present work was originated by attempts toexplain theoretically results of Gutzow andPascova1 on coarsening of a segregating silver chloride phase in highly viscousglass-forming melts. In these experiments;    after an initial periodof coarsening of an ensemble of clusters;    wherecluster number;    N;    and average cluster size;    á;    Rñ;    remainedpractically constant for extended periods oftime;    t.In order to understand such kind of behavior;    the possibleeffect of elastic strains on cluster growth and coarsening wasconsidered. However in following such line of thinking;    difficultiesoccured. In one of the first investigations of Lifshitz and Slezov onthe effect of elastic strains on coarsening the conclusion wasdrawn that elastic stresses may lead only to quantitativemodi-fications but not to a qualitative change of the basic features of the process. Therefore;    the problem was to analyse under which conditions such conclusionis true and;    vice versa;    under which conditions may be not valid. For this purpose;    different models of evolution of elastic strains were analyzed as a first step. ;    2. MODELS OF ELASTIC STRAINS IN CLUSTER GROWTHThe most widely employed model in consideringelastic strains in phase transformations in solids is directed tothe analysis of elastic strains resulting from a misfitbetween ambient and newly evolving solidphases3;    4. The same kind of strains is also of considerable importance for theunderstanding of crystallization processes in glasses;    in particular;    the preferential surface crystallization in systems wherethe differences between volume per particle in the glass andthe crystalline phase are particularlyhigh5. Provided;    in the bulk of a solid;    a cluster of a new phase is formed;    the totalenergy of elastic deformations;    F(e);    E;    and Poisson's number;    g);    cclustertheir volume concentration;    Dµ;    is the differencein the chemical potencial per particle between ambient and newly evolving phases;    respectively;    s the specific interfacial energy andA the surface area of a cluster.An inspection of equation (4) shows that the effect of elastic strains on clustergrowth for this model of evolution of strains does not depend on the size ofthe cluster. Therefore;    strains of the type considered so far (and analyzedoriginally also by Lifshitz and Slezov) cannot affect the coarsening behaviorqualitatively.The situation may become;    however;    quite different if elastic strains in segregationprocesses in multicomponent solutions are considered. Suppose;    one of the componentsof a binary solution segregates and has a diffusion coefficient D considerablylarger as compared with the ambient phase particles. In this case;    elastic strainsin segregation evolve resembling the deformation behavior of an elastic spring.For elastically deformed springs;    the force is proportional to the elongation;    the energy is proportional to the elongation squared.If the initial volume of a cluster;    when such type ofstrains begin to act;    is denoted asVo;    the influence of elastic strains increases with increasingcluster size and may;    consequently;    also qualitatively change the coarseningbehavior. Therefore;    the problem arises to develop a theory of cluster growthand coarsening under the influence of such qualitatively different;    non-linearin the cluster volume;    types of elastic strains. ;    3. COARSENING AT NON-LINEAR INCREASE OF THE ENERGY OF ELASTICSTRAINS: A FIRST APPROACHThe first approach to the description of coarseningunder the influence of elastic strains was based on athermodynamic analysis of the process of first-order phasetransformations6. It results in the derivation of differential equations describingthe evolution of the average cluster size;    á;    Rñ;    and the number of clusters;    N;    in the system (see Schmelzer(1985)7;    Schmelzer;    Gutzow(1988)8;    Schmelzer;    GutzowPascova9;    10;    Gutzow;    D their diffusion coefficient;    kB Boltzmann's constant andT the absolute temperature;    á;    nñ;    the number of segregating particles in a cluster of sizeá;    Rñ;    . The quantity Zreflects specific properties of the system under consideration.Quite generally;    the relation Z <;    - 1 holds. The absolute value ofthis parameter increases rapidly in the course of the coarsening process. Above equations allow to describe the whole coarseningprocess including its initial stages. With respect to the influence ofelastic strains;    the following consequences can be drawn from these results: •;    •;    If F(e)= 0 (absence of elastic strains) or F(e)= eV;    elastic strains do not modify thecoarsening process qualitatively. The Lifshitz-Slezov results for á;    R(t)ñ;    and N(t) are obtained as special cases.•;    If elastic strains result in energies of elastic deformations growingmore rapidly than linear with the cluster volume;    a qualitative change ofthe coarsening behavior occurs.Note that the mechanism of inhibition of clustergrowth discussed here is due to cluster - matrix interactions. Lateran alternative mechanism of inhibition of coarsening wasdeveloped by Kawasaki and Enomoto12. This mechanism;    notconsidered here;    is due to elastic field interactions of different clusters.The theoretical results obtained allow to give asatisfactory explanation of the experiments of Pascova and Gutzowand similar dependencies obtained later. It has additionaladvantage connected with its simplicity and straightforward applicability. The theory is however limited in its scope;    it givesonly expressions for the average cluster size and the numberof clusters in the system. Thus;    a detailed study ofcoarsening processes (description of the evolution of the clustersize distribution function and related quantities);    whencluster-matrix interactions result in deformation energies growingmore rapidly than linear with the volume of a cluster;    is ofinterest. The results may be of importance for the understandingof coarsening in porous materials (vycor glasses;    zeolithes;    etc) or segregation in higly viscous melts and polymers. Different approaches in this respect are summarized in the subsequent sections(for details see also Schmelzer;    Mö;    ller (1992)13;    Slezov;    Schmelzer;    Mö;    ller (1993)14;    Schmelzer;    Mö;    ller;    Slezov (1995)15). ;    4. OSTWALD RIPENING IN A SYSTEM OF HARD PORES OF EQUAL SIZEROWe consider first the case that an ensemble ofclusters evolves in a system of pores of equal sizeRo. The matrix is absolutely rigid;    i.e.;    the growth of the clusters isterminated immediately;    the timescale is measured in appropriately chosen dimensionless units(cf.13). The evolution of the cluster size distribution;    f(R;    t);    assuming that initiallya Lifshitz-Slezov distribution;    PLS;    is established in the system;    R1(R;    Rñ;    = Ro. The method is equally well applicable for any arbitrary initial distribution.The final state is the same independent of the initial conditions. ;    5. OSTWALD RIPENING IN A SYSTEM OF WEAK PORESAs a next limiting case;    allowing an analytical treatment;    it is assumed that the inhibiting effect of elastic strainsincreases only slowly with an increasing size of the cluster. Forthis case;    the growth equation is writtenas13;    we obtain;    in a good approximation;    the parameters u1 and u2 areslowly varying functions of time. This way;    also the function P(u)varies slowly with time. The way of variation depends hereby on the type ofresponse of the matrix. ;    6. COARSENING IN A SYSTEM OF NON DEFORMABLE PORES WITH A GIVENPORE-SIZE DISTRIBUTION;   

DOI  :  10.1590/S0100-40421998000400030

学科分类：化学（综合）

来源: Sociedade Brasileira de Quimica

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