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Reversed Ostwald Ripening Vol.46, No.4, 1983 J. Soc. Photogr. Sci. Technol. Japan 研 究 Reversed Ostwald Ripening Tadao SUGIMOTO* Research Laboratories, Ashigara, Fuji Photo Film Co., Ltd. Minamiashigara, Kanagawa 250-01 (Received March 10; Accept for Publication June 20, 1983) Abstract It was theoretically deduced that smaller particles might grow at the expense of larger particles during aging in a closed dispersion under some specific conditions. This theoretical prediction was verified with some mixed emulsions composed of two kinds of monodisperse AgBr particles; e.g., the growth of small cubic AgBr particles at the expense of larger octahedral AgBr particles during aging at a low pBr, or the growth of small octahedral particles at the expense of larger cubic particles during aging at a high pBr. As a result, the size distribution became narrower. This phenomenon was named " Reversed Ostwald Ripening." The reversed Ostwald ripening was a relatively rapid process observed at the early stage of aging until the small particles reached the equilibrium form. Then, it switched to the normal Ostwald ripen- inc. durina which the large particles grew by the dissolution of the small particles. 1. Introduction 2. Theory In the previous papers (1-4), the surface As has been derived in Refs. 1 and 2, the chemical potentials of {100} and {111} faces surface chemical potentials of a {100} face of a tetradecahedral microcrstal were intro- and a {111} face of a tetradecahedral parti- duced to describe the concepts of the equi- cle are expressed as librium form and the steady form and the relationships between the two forms. The (1) theories were experimentally verified with where G is the surface free energy of the monodisperse AgBr particles. particle, n the mole number of the constit- In the present article, it is to be shown uent molecules of the particle and rioo and that an apparently anomalous phenomenon ri i i the central distances (distances from the named " Reversed Ostwald Ripening " can be predicted from the aforementioned the- particle center) to the {100} face and the {111} face of the particle respectively (1, 2). ories. This is the reversed process of the There are two kinds of tetradecahedra bound Ostwald ripening; i.e., the growth of small by {100} and {111} faces: type A (1/√3≦ particles by dissolution of larger particle in γ111/γ100≦2/√3) and type B (2/√3≦ γ111/γ100 a closed system. This appears to be char- ≦ √3). as illustrated in Fig.1. The sur- acteristic of polyhedral particles and it will face chemical potentials of these tetrade- be actually demonstrated with some mixed cahedra are given in Table I (1, 2). The dispersions composed of two types of mono- chemical potentials are related with the solu- disperse AgBr particles. bility products of the respective faces as (2) *Research Laboratories, Ashigara, Fuji Photo Film Co., Ltd. Minamiashigara, Kanagawa 250-01 where Ksp is the solubility product of an ―306― Vol.46, No.4, 1983 Reversed Ostwald Ripening ―307― Fig. 1 An octahedron, tetradecahedra A, M, and B, and a cube. Table I Surface chemical potentials of {100} and {111} faces for tetradecahedra. infinite-sized particle. The solubility pro- expense of the coigns of both Lub and IIoct, ducts can be replaced by the respective whereas the {111} faces of "Oct may remain solubilities. virtually ungrown. In this case, the volume Now let us consider a dispersion of mono- of each particle of Icub may increase owing disperse AgBr particles of a cubic or an to the dissolution of the {100} faces of IIoct, octahedral shape. Since the specific surface though the self-recrystallization which con- energy ratio of a {111} face to a {100} face currently occurs within Icub will not contri- of AgBr, ƒÐ111/ƒÐ100, is mostly located between bute to the volume increase of Lib. Con- 1/•ã3 and •ã3 , the surface chemical poten- sequently, it will be possible for the smaller tials or the solubility products of coigns of particles Icub to grow by the dissolution of the cubes and the octahedra are of infinite the larger particles IIoct This is the revers- values. In addition, if the growth rate con- ed Ostwald ripening. stant of {100} faces is much larger than that Eventually, as a result of the self-recrys- of {111} faces, monodisperse cubic particles tallization and the voluminal growth, parti- in a closed system may attain the equilib- cles Icub will reach the equilibrium form, rium form much earlier than monodisperse where the solubilities of the {100} and {111} octahedral particles in a similar closed sys- faces of Icub are equal. If the solubility levels tem, since the rate-determining step of the of the coigns of Icub and IIoct, i.e., the {111} self-recrystallization is expected to be mostly faces of Icub and the {100} faces of IIoct, are the growth process of {100} faces of the assumed to be kept almost equal during the cubes or of {111} faces of the octahedra by dissolution for simplicity reasons, the solu- the very rapid dissolution of coigns of the bility of {100} faces of IIoct will agree with cubes or the octahedra (1, 2). the equal solubilities of {100} and {111} faces Therefore, if we consider a mixed disper- of Icub at the equilibrium points of Lib. Since sion of ideally monodisperse smaller cubic the driving force of particle growth of Icub particles (Lib) and monodisperse larger octa- will disappear at the this moment, only that hedral particles (IIoct) in such a condition, the of the slow reaction-controlled growth of {100} faces of Icub may rapidly grow at the the {111} faces of Iloct will be retained. This ―308― Tadao SUGIMOTO J. Soc.Photogr. Sci. Technol. Japan is the end of the reversed Ostwald ripening (5) and the beginning of the normal Ostwald ripening as well. and Now, the {100} and {111} faces of Icub will turn to be dissolved due to the growth of (6) the {111} faces of IIoct. At the same time, the {100} faces of IIoct will turn to grow be- Here, ƒÐ100 and ƒÐ111 are the specific surface cause of the rising solubility level of the energies of the {100} and {111} faces, re- {100} and {111} faces of Icub due to the sim- spectively, and Vm is the molar volume of ultaneous dissolution of both kinds of faces. the solid. Thus, it follows that In other words, particles IIoct will return to an octahedral shape in this stage. This re- (7) latively slow process is the normal Ostwald On the other hand, it holds from the mass ripening. In this stage, the solubilities of balance that the {100} and {111} faces of Icub and that of the {100} faces of IIoct will be all kept equal, (8) since they will be balanced with bulk con- where N and N' are the particle numbers centration of solute owing to the very high of Icub and IIoct respectively, and v and v' dissolution rate constants of these faces. are the mean particle volumes of Icub and Thus, the equilibrium shape of the dissolv- IIoct respectively. Here, both N and N' re- ing particles Lub will be retained during the main constant for the whole processes. normal Ostwald ripening. Since Icub and IIoct are of type B and type A In the light of this survey, the theory of respectively, v and v' are given by the reversed Ostwald ripening is to be de- (9) scribed for a mixed dispersion of smaller and particles Icub and larger particles IIoct, where the particles are bound by {100} (10) and/or {111} faces, 6111/610o is between (see Ref. 2). Thus, substituting Eqs. (9) and 1/•ã3 and 2/•ã3 , the growth rate constant (10) into Eq. (8), one obtains of the {100} faces is much greater than that of the {111} faces, and the dissolution rates of both kinds of faces are diffusion-cont- (11) rolled. Also, any symbol indicating IIoct will be denoted by a prime as ƒÁ'100,ƒÁ'111, ƒÊ'100,ƒÊ'111, where the following approximation was used: etc., in order to distinct from the symbols (12) for Icub which are free of primes. Here, ƒÁ'111/ since ƒÁ100•âƒÁ111 as we have assumed in the of Icub and ƒÁ'111/ƒÁ'100 of IIoct are designatedƒÁ'100 beginning. by p and p', respectively: Substituting Eq. (7) into Eq. (11) by using (3) Eq. (12), one obtains I. The Reversed Ostwald Ripening Stage (13) (i)2/√3≦p≦ √3: where In this p range, particles Icub remain in the range of type B, whereas particles IIoct (14) are of type A. The surface chemical poten- tial of the {111} faces of Icub,ƒÊ111, and that and of the {100} faces of IIoct, ƒÊ'100, are assumed (15) to be kept equal: By using the boundary condition of ƒÁ100=a100. (4) at p=•ã3, one obtains the solution of the where differential equation (13) as Vol.46, No.4, 1983 Reversed Ostwald Ripening ―309― a) (16) where a100 is the ƒÁ100 of the initial cubes of of Icub. Thus, the surface chemical potentials of the {100} and {111} faces of Icub are obtain as (17) and b) (18) Also, the mean particle volume of Icub is given by (19) from Eqs. (9) and (16). Meanwhile, the surface chemical potentials of the {100} and {111} faces of IIoct and the mean particle volume of IIoct are expressed as (20) Fig. 2 Diagrams of f values (a) and particle vol- umes(b) as functions of p for particles I and II in a mixed dispersion.
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