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 potential 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 volumes(b) as functions of p for particles I

and II in a mixed dispersion. For the (21) upper diagram (a), directions of changes and in f's are shown with arrows (•©fast deposition, •© slow deposition, •© dis-

(22) solution), and the solid lines (-) and the broken lines (---) indicate f values where ƒÁ'111=a'111=const. from Eq. (12). of {111} and {100} faces, respectively. In the range of 2/•ã3•…p•…•ã3, ƒÊ111 and I(100) means the {100} faces of Icub , and

are both kept equal and going down ƒÊ'100 at I(111), II (100), II (111) are identified like- the same time. ƒÊ100 also slowly drops where- wise. For the lower diagram (b), [M] de- as ui11 remains almost unchanged. The par- signates the intermediate form between ticle volume v is increased, whereas v' is types A and B of teradecahedra (p = diminished (see Fig. 2). 2/•ã3) as shown in Fig. 1.

(ii)α ≦p≦2/√3: Since particles Icub are now in the range of type A, ƒÊ111 is given by

(23)

Since particles IIoct remain in the range of (24) type A, ƒÊ'100 is given by ―310― Tadao SUGIMOTO J. Soc. Photogr. Sci. Technol. Japan

It follows from μ111=μ'100 that tionship:

(25)

where (27) (26) where the relation of Eq. (12) is used again. On the other hand, applying the relation Combination of Eqs. (12), (25), (26) and (27) of Eq. (8) to Icub and IIoct both of which are yields of type A, one obtains the following rela-

(28)

The integrated form of Eq. (28) is

(29)

where (36)

(30) and

(31) (37) The solution of Eq. (29) is numerically obtained. From the mass balance of Eq. (8) it follows Thus, ƒÊ100, ƒÊ111 and v of Icub are given by that

(32)

(33) (38) Combining Eqs. (36), (37) and (38), one ob- (34) tains

while ƒÊ'100, ƒÊ'111 and v' for IIoct are of the same forms as Eqs. (20), (21) and (22), respectively. In this stage of the reversed Ostwald ripen- (39) ing, ƒÊ111 and ƒÊ'100 are are still kept lowered, while where turns to be elevated and finally becomesƒÊ100 equal to ƒÊ111 and ƒÊ'100. ƒÊ'111 remains almost un- (40) changed. v is still kept increased, whereas v' is decreased (see Fig. 2). The solution of Eq. (39) for ƒÁ'100 is

II. The Normal Ostwald Ripening Stage (41)

(p=ƒ¿) where ƒÁ' is the ƒÁ'100 at p'e. Here p'e is the p' In this stage, the following relation holds: value at the end of the reversed Ostwald (35) ripening. Thus, it holds that On the other hand, since the total volume Vol.46, No.4, 1983 Reversed Ostwald Ripening ―311―

haviors of ƒÊ100, ƒÊ100, v, ƒÊ'100, ƒÊ'111 and v' are shown of Icub and IIoct is constant, the following in Fig. 2. Here, we set ƒ¿=0.78 and N= N', relation holds (see Eq. (10)): and f is the dimensionless form of it whose unit is 2ƒÐ100Vm/a. In Fig. 2, the small particles Icub rapidly grow by the dissolution of (42) the {100} faces of the large particles IIoct

Combination of Eqs. (36), (37), (42) and the until Icub and IIoct reach the points 2 and 2', relation of ƒÁ'111= a'111 at p'e give respectively (the reversed Ostwald ripening). In the course of the reversed Ostwald ripen- (43) ing, ƒÊ'100 is assumed to remain equal to ƒÊ111 as represented by the points 1' and 1. The Also, p'e is given by deposition of solute dissolved from {111} (44) faces of Icub to {100} faces of Icub does not result in the volume change of each parti- It follows from Eqs. (41), (43) and (44) that cle of Icub (self-recrystallization). Then they turn to the slow process of the normal Ost- (45) wald ripening during which the large particles IIoct grow at the expense of the small Therefore, one obtains the following rela- particles Icub. This process lasts until par- tions for IIoct: ticles Icub are completely dissolved while

particles IIoct are left as sharp-edged oct- (46) ahedra. Then, the octahedral particles will undergo the self-recrystallization to approach again the equilibrium of p' =ƒ¿.

(47) 3. Examples (48) A mixed system composed of smaller cubic particles (ƒÁ100=0.247ƒÊm) of AgBr and equiv- where ƒÁ'100 is given by Eq. (45). alent moles of larger octahedral particles The corresponding quantities for Icub are (ƒÁ111= 0.327ƒÊm) was employed as an example given from Eqs. (35), (42) and (46) by of Fig. 2. The dispersion was aged for 60 min with agitation under the condition of

[AgBr] = 0.1 M, [NH3] =0.5 M, [NH4NO3] = 0.1 (49) M, pH = 8.48, [gelatin] =1 wt % , pBr =1.40 and temp =50•Ž. In this case, a is equal to 0.78, and the growth of the {100} faces is diffusion-controlled while that of the {111} faces (50) is reaction-controlled, and the dissolutions where ƒÁ'100 and v' are given by Eqs. (45) and of both faces are diffusion-controlled (1-4). (48), respectively. The electron micrographs of the original In the stage of normal Ostwald ripening, and the aged mixtures are shown in the 100, ƒÊ111 and ƒÊ'100 are all equal and ƒÊ are kept Fig. 3. The corresponding histograms of going up at the same time. p value of Icub the size-distributions obtained with a Coulter remains constant at ƒ¿. ƒÊ111 is still almost Counter are exhibited in Fig. 4. These fig- unchanged. v turns to decrease while v' ures verify the reversed Ostwald ripening: turns to increase (see Fig. 2). i.e., the larger octahedral particles have If cubes of a100 = a and a111=•ã 3 a and oct- shrunk while the smaller cubes have grown, ahedra of a'100 = 3a and a'111=•ã3 a are used resulting in the narrowing of the size dis- as starting particles of Icub and IIoct respec- tribution. tively, then the particle volume ratio of the When a mixture of larger cubic AgBr par- starting particles are v'0/v0=4.5, and the be- ticles and smaller octahedral particles was ―312― Tadao SUGIMOTO J. Soc. Photogr. Sci. Technol. Japan

aged at a high pBr such as pBr 2.5 under a similar condition, the growth of the smaller octahedra by the dissolution of the larger cubes was observed, as had been readily anticipated.

Fig. 4 Histograms of size distribution of the particles shown in Fig. 3. Original distribution (- - -) ; distribution after 60-min aging

(-). Equivalent sphere = sphere with the equivalent particle volume.

K/Y=solubility product of a {111} face. K4=solubility product of an infinite-sized

particle. N=particle number of particles L. in a

system.

n=mole number of constituent molecules of

a particle.

p=Y111/r100

q=Z/r111=•ã 3-1/p r100=central distance to {100} faces (distance

from the particle center to its own

{100} faces). r111 =central distance to {111} faces.

v=particles volume

v0=initial particle volume

Vm=molar volume of solid

z=•ã3 r111r100

IX= 6iii/Gioo

Fig. 3 Electron micrographs of a mixture of large p100=surface chemical potential of a {100} octahedral particles and small cubic ones face. before and after 60-min aging. (s111=surface chemical potential of a {111} (a) original mixture; (b) after 60-min face. aging(temp=50•Ž, [NH3]=0.5 M, [NH4NO3] o100=specific surface energy of {100} faces. =1.0 M, [gelatinl=1 wt %). a111=specific surface energy of {111} faces.

4. Notations References

1) Sugimoto, T., The Fall Meeting of the Society

of Photographic Science and Technology of a100=initial value of rioo. Japan, November, 1981. a111=initial value of Tull. 2) Sugimoto, T., J. Colloid Interface Sci. 91, 51 (1983). 3) Sugimoto, T., The Annual Meeting of the So- ciety of Photographic Science and Technology G=surface free energy of a particle. of Japan, May, 1981. g=a100/r1000 4) Sugimoto, T., J. Colloid Interface Sci., 93, 461 K/P°=solubility product of a {100} face. (1983).