Association of atoms : 1. Spontaneous 2. Successive grow (Markovs chains): A 1 – atom or molecule, A i – cluster of i-size. A 1 + A 1 = A 2 A 2 + A 1 = A 3 A 3 + A 1 = A A i-1 + A 1 = A i 3. Aggregation of i- and k-size clusters. A i-k + A k = A i These mechanisms of processes could be describe with the help of such reaction (reaction of i-order) i A 1 = A i Frenkel: for system of V volume: N = N 1 + 2N 2 + 3N iN i and N i / N 1 i = Q i /O 1 i N 1 and N i – number of atoms and I-size aggregates Q 1 and Q i – full statistic sums

Following of law of conservation of energy = + where - full energy, - potential, - kinetic energy of system. So Q = Q p. Q q Following of law of conservation of momentum Q p(i) = Q p(1) i and N i / N 1 i = O q(i) / O q(1) i Following of Gibbs canonical distribution Ψ 1 Q q(1) = V. exp( ) kT where v i – volume available for atom in cluster, Ψ 1 and Ψ j - potential energies of molecular interaction for atom and i-size cluster C= N/V and for crystals n 1 = N 1 /N o and n i = N i /N o i v i i-1 iψ 1 – ψ j C i = C 1 i exp( ) and i! kT where N o – number of internodes in crystal i 1 v i i-1 iψ 1 - ψ j N i = N 1 i exp( )where v i = v o. i i! V i-1 kTv o – volume occupied by atom in crystal lattice i i i-1 iψ 1 – ψ j n i = n 1 i exp( ) i! kT i v i i-1 ψ j Q q(i) = V exp( ) i! kT

Investigation of silver colloidal particles in gelatin and in AgBr crystals N m = 1 Electron microscopic photo of ultra fine section of ultra fine-grain emulsion: crystal AgBr (1), colloidal particle (2) and layer of gelatin-Br 2. (3).

m 1 v m m-1 m. ψ 1 - ψ j 1 = N m = N 1 m exp( ) m! V m-1 kT m m! V m-1 1/m ψ j – ψ 1 and N 1 = ( ) exp( ) v m m-1 kT As a result of permutation : m i (m!) i/m i i-1 /m ψ j – ψ j N 1 = v o i/m-1. v 1-i/m. exp( ) i! m i-i/m kT ψ - ψ i = 2 σv / r where σ – surface tension (free surface energy of Gibbs), r – radius of nuclei For spherical particles r = (3v o i/(4 )) 1/3 ψ i = 2/3 (4π) 1/3. σ. (3v o ) 2/3. i –1/3 = U. i –1/3 U = 2(4 ) 1/3 σ. (3v o ) 2/3 / 3 = const for = 400 mJ/m 2, v o = А 3, U = eV ψ 1 = U, а ψ = 0 I Let the dependence ψ j – j is continuous. j=1 In that case i i 3 ψ j = ψ j dj = --- U (i 2/3 -1) j=1 1 2 Dependence between ψ i (potential energy of clusters building) and i (size of cluster).

The rules of formation of photolytic arising new phase of silver in ultra-fine-grain Lippmann photo-emulsions are investigated, with the help of chemical thermodynamics the interrelation between the sizes of arising colloidal particles of silver, size of AgBr microcrystals, where they are formed, and size of formed oligoatomic silver clusters is formulated. V=const, m 3 >m 2 >m 1 m=const, V 3 >V 2 >V 1 (m!) i/m i i-1 v o i/m-1 U i m 1 i 1 N i = ( ) exp(---- [ ]) i! m i-1/m V kT m j=1 j 1/3 j=1 j 1/3

Surface tension of silver in vacuum Calculated the value of a surface tension of silver for small colloidal particles and clusters from 1 to 400 is determined as 860 mJ/m 2 According Stransky-Kaischev theory the potential energy of cluster formation j can be determine as i j = S, where S is a clusters surface. For spherical aggregate this equation can be transform in: j=1 S = U.. i 2/3, where U = (4 ) 1/3 (3v o ) 2/3 = 3, m 2, by v o = 17,06 A 3 – molar volume (volume occupied of one atom in crystal lattice). Energy of formation of i-size aggregates can be equal to value i i, calculated by method of Huckel or three-particle potential i i 1 - j = i i. j=1 After substitution = i i / U (i - i 2/3 ) we can calculate surface tension of silver as a function of aggregates size i :

Surface tension of silver in H 2 0 and Gelatine Determination of critical size r cr. by vacuum deposition and electron microscopic investigations of particles sizes distribution. Investigation of surface tension as a function of critical size 2.. v o where – surface tension, E = E red – E ox = v o – molar volume (17,06 A 3 ), F. r cr. F – Faraday constant 9, A* s* mol -1 1 – Ag (pure) in H 2 O = 720 mJ/m Ag in Gelatine = 670 mJ/m 2

Resume: In microcrystals of silver bromide in balance with colloidal particle there is a "dissolved" part of silver as oligoatomic (molecular) clusters. If to consider the polydispersiveness of microcrystals in a photolayer there should be two maxima in distribution of silver particles: colloidal particles of silver and clusters undercritical size (molecular aggregates). At reduction of the size of colloidal particles (Ag m ) the concentration of molecular clusters should be increased. Thus not only the concentration of atoms N 1 is increased, but also there is a relative integration of particles in molecular distribution. At growth of Ag m there is the return process accompanying with sharp reduction of molecular clusters' concentration (and their practically full disappearance in equilibrium with massive particles of silver Ag m ). Change of the sizes of silver bromide microcrystals also exerts influence on coloring centers distribution. At a constant exposition in the greater crystal (on the size) is formed larger silver colloidal particle due to absorption of the greater number of quantums. That results to decreasing of concentration of the primary (molecular) centers. At constant size Ag m the reduction of the sizes of silver bromide crystal conducts to reduction of number of the primary centers. However, as a whole for a photographic layer (at a constancy of surface concentration of silver salts) occurs the increase of number of AgBr crystals (and consequently, numbers of Ag m ), that results to increase of general number of primary centers in all layer. It is necessary to note, that change only one parameter V does not exert so strong influence on character of centers distribution, as size Ag m. Thermodynamic analysis of Ag-AgBr system has shown, that between all components of system, such as primary centers, secondary centers and a silver halide crystal there is an interaction, and change of a condition of one of its components results in change of others.