Kinetic Equations for Particle Clusters Differing in Shape and the H-Theorem

Article Kinetic Equations for Particle Clusters Diﬀering in Shape and the H-theorem

Sergey Adzhiev 1,*, Janina Batishcheva 2, Igor Melikhov 1 and Victor Vedenyapin 2,3

1 Faculty of Chemistry, Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia 2 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4, Miusskaya sq., Moscow 125047, Russia 3 Department of Mathematics, Peoples’ Friendship University of Russia (RUDN-University), 6, Miklukho-Maklaya str., Moscow 117198, Russia * Correspondence: [email protected]; Tel.: +7-916-996-9017

Received: 21 May 2019; Accepted: 5 July 2019; Published: 22 July 2019

Abstract: The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.

Keywords: coagulation–fragmentation equations; Becker–Döring equations; H-theorem; entropy; conservation laws; Lyapunov functionals

1. Introduction The H-theorem was a subject of study of Ludwig Boltzmann [1]. He connected it with the entropy increase. Boltzmann devotes the first chapter to an equation that we now term as the spatially homogeneous Boltzmann equation with the dependence of the distribution function only on the magnitude of the velocity (on the square of the velocity or energy, which he calls “living force”). It is for this equation that Boltzmann proves the H-theorem. The second chapter of this work is remarkable for its simplicity and is called a “replacement of integrals by sums”—the simplest discrete models of the Boltzmann equation appear there. One is similar to the three velocities model, which we now term as the Godunov–Sultangazin model [2]. In the same chapter, the principle of maximum entropy appears, but only as a hint, as an example for one model. Thus, Boltzmann defines the simplest discrete model as what we now term as the Boltzmann extremals [3] (see also [4,5]). In [3–10], it was shown that the stationary solution can be obtained without solving the equation in different cases such as for discrete Boltzmann equations, for general chemical kinetics and for Liouville equations. It is also interesting to note that Boltzmann generalized his H-theorem for chemical kinetics in his work, but modern generalization of chemical

Physics 2019, 1, 229–252; doi:10.3390/physics1020019 www.mdpi.com/journal/physics Physics 2019, 1 230 kinetic equations was found 100 years later in [11–14]. The theory was also generalized to the quantum case; see [5,15,16] for review. Therefore, the classical H-theorem not only substantiates the second law of thermodynamics for the described systems but also provides information about the behavior of solutions. The proof of the H-theorem makes the behavior of solutions of equations clear, since it allows us to find out where they converge with time tending to infinity. This can be done without solving the equations and finding the Boltzmann extremal—the argument of the minimum of the H-function (the functional decreasing along the solutions), provided that the values of the linear conservation laws are fixed. The H-theorem ensures the stability of the obtained stationary solutions (Boltzmann extremals). We continue the line of Boltzmann’s work, trying to expand the class of equations for which the law of entropy increase can hold, and to investigate the conditions under which the H-theorem is valid. The H-function for the systems considered by Boltzmann is “minus” entropy. The interpretation of entropy as a thermodynamic potential, which increases for an isolated system, suggests a course of action in other situations: considering generalizations of the H-theorem for other models, the physical meaning of the H-function should be specified in each case. Regardless of the physical meaning, H-functions in mathematics are called Lyapunov functionals. In this paper, we consider a mathematical model designed to describe a supersaturated vapor or solution in which clusters of matter originate and grow. The task is to find out how a solid substance is formed from a multitude of molecules. What we call original particles are what the clusters consist of (these are, for example, molecules and identical nanoparticles), and we consider the problem of aggregating the original particles. The term “solid particle” as a synonym for the terms “cluster” and “aggregate of original particles” is not suitable, since the solid phase does not form immediately: aggregates from a small number of initial particles are not a solid phase. In this case, the equations should generalize the coagulation–fragmentation equations taking into account the shape, being a special case of the general equations of physicochemical kinetics (see Section2). Immediately, we note that in the considered models of formation of a solid substance the bonds between the original particles may not be chemical, but all the equations considered in the present work have the form of equations of physicochemical kinetics. Discrete coagulation–fragmentation equations describe the kinetics of cluster growth in which particles can coalesce by pairwise interaction, forming clusters with a larger number of molecules, and can fragment particles with a smaller number of molecules [6,17–19]:

m 1 dN(m,t) P = 1 − [C(l, m l)N(l, t)N(m l, t) F(l, m l)N(m, t)] dt 2 − − − − l=1 (1) +P ∞[C(l, m)N(l, t)N(m, t) F(l, m)N(m + l, t)], m 2. −l=1 − ≥

Here, N(m, t) is a numerical density (concentration) of particles consisting of m molecules at time moment t, C(l, m) is a kernel (a constant) of coagulation, and F(l, m) is a kernel (a constant) of fragmentation. Equation (1) was ﬁrst derived by Marian Smoluchowski [17] for F(l, m) 0 for all l, m (the ≡ Smoluchowski case). If only one original particle can attach to or separate from another particle, i.e., C(l, m) = F(l, m) = 0 when min l, m > 1, then Equation (1) represents the Becker–Döring equations: { } + dN(1,t) P = ∞[C(l, 1)N(l, t)N(1, t) F(l, 1)N(l + 1, t)] , dt − − h l=1 i dN(2,t) 1 ( ) 2( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( )] dt = 2 C 1, 1 N 1, t F 1, 1 N 2, t C 1, 2 N 1, t N 2, t F 1, 2 N 3, t , (2) dN(m t) − − − , = [C(1, m 1)N(1, t)N(m 1, t) F(1, m 1)N(m, t)] dt − − − − [C(1, m)N(1, t)N(m, t) F(1, m)N(m + 1, t)], m > 2. − − Physics 2019, 1 231

The conditions of the H-theorem for Equation (1) have been investigated in [18,19], and those for the Becker–Döring equations, Equation (2), in [20,21]. It is interesting to rewrite the results of [18–21] in terms of the Boltzmann extremals [1,4,5], which is the argument of the conditional minimum the H-function, provided that the constants of linear conservation laws are fixed. Namely, according to the H-theorem solutions of the equations converge to the Boltzmann extremals as soon as time tends to infinity. The description of the nucleation and growth of clusters using Equations (1) and (2) is greatly simplified. Usually, the coefficients in the equations of (1) and (2) do not take into account even the shape of the aggregates and the relief of their surfaces, and therefore such models do not provide the so-called magic numbers of atoms. In the Becker–Döring case, taking into account the shape of clusters should give the magical numbers of atoms [22], and, if properly generalized, equations in (1) describe the coalescence (and fragmentation) of not only the clusters, formed according to the Becker–Döring condition, but also the aggregates formed by aggregates [22,23]. Therefore, we propose, in addition to the number of original particles constituting a cluster, to take the shape as the other parameter. The objective of this paper is to consider evolutionary models that take into account the shapes of the clusters and investigate the H-theorem for them. This is done in order to create a theory of the emergence of such aggregates from the original particles. The results of this paper demonstrate a way to solve the issue of building the correct (physically justified) models of evolution of clusters having different shapes. This result is not affected by the fact that all the models considered in this paper are spatially homogeneous just as it was in the case of the Boltzmann equation and its discrete models [1,6]. Moreover, although we mean nanoparticles or molecules as the original particles, the results are suitable for any similar structures having a shape with the same interaction as described in the present work. The latter is due to the fact that we never distinguish between the sizes of the original particles (nano- or macro-). The layout of the paper is as follows. In the second section, we describe the general physicochemical kinetics system of equations and the conditions on the sections are determined when the H-theorem holds [6,7,11], with the results following. In Section 3.1, we derive the simplest models of the evolution of aggregates consisting of the original particles, which differ in shape. Here simplifying the model we reduce the number of parameters in description. In Section 3.2, we investigate the conditions on the coefficients of the equations for validity of the H-theorem. In Section 3.3 it is deduced, that for special but very important cases, the H-theorem is still satisfied: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are essential here. In this case, we obtain a simple equation for the limit stationary solution. It is the result of Sections 3.4 and 3.5. In addition, in Section 3.3 we connect the Boltzmann equation for the probability of state with the principle of detailed balance and obtain as for the coefficients of the rates of the processes. In the simplest models, we present the original particles in the shape of cubes. Of course, the molecules do not have a shape, and the modeling of the original particles in the shape of cubes means that for simplicity we restrict ourselves to only a simple Bravais cubic crystal lattice. This is done for ease of consideration and presentation when considering the simplest models and for illustrations. The main result of the present work (Sections 3.3–3.5), generally speaking, does not depend on the shape of the original particles: all that is required is that the original particles be the same size and shape. The rigorous formulation of our assumptions is provided in Section 3.3.

2. The Method and Deﬁnitions Here, we describe the general physicochemical kinetics and how the H-theorem holds [6,7,11]. Let there be a mixture of chemically or physically interacting substances or states with spatially homogeneous concentrations. Let us denote the concentration of the i-th substance or state, i = 1, 2, ... , n, at the time t as Ni(t). The equations for complex physicochemical processes in the general form are written as [6,7,11,12,24,25]: Physics 2019, 1 232

dN 1 X β i = (β α ) KαNα K Nβ , i = 1, 2, ... , n. (3) dt 2 i − i β − α (α,β) ∈= Here, Nα denotes the product Nα = Nα1 Nα2 ... Nαn , the summation is carried out over some 1 2 · · n ﬁnite set of multi-indices (α, β), and α = (α , α , ... , αn) and β = (β , β , ... , βn) are vectors with = 1 2 1 2 non-negative integer components. (α, β) corresponds to the elementary reaction or physical process:

α Kβ α S + α S + ... + αnSn β S + β S + ... + βnSn, (α, β) , (4) 1 1 2 2 → 1 1 2 2 ∈ = where in chemistry, Si is the chemical symbol of the i-th reacting substance, in physics, Si is i-th state, and Kα 0 are the coefficients of reaction rates (reaction constants) in chemistry or process velocity β ≥ in physics. The coefficients αi, βi are called stoichiometric coefficients in chemistry or number of states in physics. Without loss of generality, we can assume that the set is symmetric with respect = to permutations α and β. Meanwhile, some couples (α, β) may correspond to the zero reaction α β (processes) rate coefficients: Kβ = 0, despite that Kα > 0 (the irreversibility of reactions or processes is allowed). For clarity, let us consider an example of system of equations of physicochemical kinetics that describes one reversible reaction A + BC AB + C, where A, B, and C are symbols denoting some → atoms or complexes of atoms that are different between themselves, which do not change themselves during the reaction. We have that the set consists of two pairs: (α, β) and (β, α), where α = (1, 1, 0, 0) = and β = (0, 0, 1, 1). Their components have striking physical and chemical meaning. The first indexes (α1 = 1 and β1 = 0) correspond to the number of molecules of the substance A participating in the reaction, the second ones (α2 = 1 and β2 = 0) are the numbers of molecules BC, and the third and fourth ones (α3 = α4 = 0 and β3 = β4 = 1) are the numbers of molecules AB and C, respectively. We write the system of equations according to the general Equation (3): dN1 α β = (0 1) K N1N2 KαN3N4 , dt − β − dN2 α β = (0 1) K N1N2 KαN3N4 , dt − β − dN3 ( ) α β dt = 1 0 KβN1N2 KαN3N4 , − − β dN4 = (1 0) KαN N K N N , dt − β 1 2 − α 3 4 where N1(t), N2(t), N3(t), and N4(t) are the concentrations of molecules of substances A, BC, AB, and C, respectively, at a moment of time t, and the values of vectors α and β are written out above. There are three linear conservation laws: N1 + N3 = const, N2 + N3 = const, and N2 + N4 = const. The first of them formulates the conservation law of the number of atoms (or complexes of atoms) of the form A: N1+N3, the second one is the number of atoms of the form B, the second one is the number of atoms of the form B: N2 + N3, and the third law is the number of atoms of the form C: N2 + N4. It is easy to verify Pn that a linear operator: Iµ(t) = µiNi(t) = (µ, N) is conserved along the solutions of the system under i=1 consideration then and only then when the vector µ is orthogonal to all Boltzmann–Orlov–Moser–Bruno vectors [12,24–27] α β. In this example, the Boltzmann–Orlov–Moser–Bruno vector is unique up to a − factor: α β = (1, 1, 1, 1). − − − Thus, the stoichiometric coefficients are the numbers of molecules participating in the reaction (4). Si, i = 1, 2, ... , n, are the chemical symbols of the reacting molecules. Linear invariants of a system of physicochemical kinetics equations are the conservation laws of the number of atoms of each type entering into the composition of at least one of the substances Si participating in the reaction, or a linear combination of these linear conservation laws. More precisely, instead of atoms of each type, different complexes of atoms should be considered, which themselves do not change during the reaction. Note that if the number of molecules Si, i = 1, 2, ... , n, in an elementary reaction of the Equation (4) is preserved, then α = β , where α α + α + ... + αn. | | | | | | ≡ 1 2 Physics 2019, 1 233

Consider another example, when not only the composition and mass of clusters, but also shape of clusters is taken into account. We conﬁne ourselves to the case of aggregates consisting of no more than three original particles (molecules) and the shapes shown in Figure1. The original particles are cubes here. We introduce the notation for the cluster concentrations, as shown in Figure1: N1 is a concentration of individual molecules, N2 the aggregates of two molecules, and N31 and N32 are clusters of two shapes of the three molecules, and arrows indicate three possible reversible reactions. Then, the evolution equations for these quantities read as:

( ( ))( ( )) ( ( )) dN1 = 2 K 1, 1,1,1 1, 1,1,1 N 2 K 2, 1,1,2 N dt − (2,(1,1,2)) 1 − (1,(1,1,1))(1,(1,1,1)) 2 ( ( ))( ( )) ( ( )) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,1,3 N − (3,(1,1,3)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 31 ( ( ))( ( )) ( ( )) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,2,2 N , − (3,(1,2,2)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 32 ( ( ))( ( )) ( ( )) dN2 = K 1, 1,1,1 1, 1,1,1 N 2 K 2, 1,1,2 N dt (2,(1,1,2)) 1 − (1,(1,1,1))(1,(1,1,1)) 2 ( ( ))( ( )) ( ( )) (5) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,1,3 N − (3,(1,1,3)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 31 ( ( ))( ( )) ( ( )) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,2,2 N , − (3,(1,2,2)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 32 ( ( ))( ( )) ( ( )) dN31 = K 1, 1,1,1 2, 1,1,2 N N K 3, 1,1,3 N , dt (3,(1,1,3)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 31 ( ( ))( ( )) ( ( )) dN32 = K 1, 1,1,1 2, 1,1,2 N N K 3, 1,2,2 N . dt (3,(1,2,2)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 32 Instead of the first equation of (5), we can take the conservation law of the number of all molecules in the system: N + 2N + 3N + 3N N = const. 1 2 31 32 ≡ all ( ( ))( ( )) ( ( ))( ( )) ( ( ))( ( )) In (5), K 1, 1,1,1 1, 1,1,1 , K 1, 1,1,1 2, 1,1,2 , and K 1, 1,1,1 2, 1,1,2 are the constants (cross-sections, (2,(1,1,2)) (3,(1,1,3)) (3,(1,2,2)) ( ( )) ( ( )) ( ( )) frequency functions) of the coalescence, and K 2, 1,1,2 , K 3, 1,1,3 , and K 3, 1,2,2 (1,(1,1,1))(1,(1,1,1)) (1,(1,1,1))(2,(1,1,2)) (1,(1,1,1))(2,(1,1,2)) are frequencies of fragmentation. Hereinafter, we write the rate constants of the processes as coefficients in equations as in Equation (3). The meaning of the indices in the coefficients in Equation (5) are clarified in the next section, where we define the notion of a shape consisting from cubes and write out a more general model via Equations (8)–(10). Note that the systems in Equations (1) and (2) represent the physicochemical kinetics equations with an infinite number of reactions. The constants C(l, m) and F(l, m) in (1) and (2) are the rate coefficients of the processes if l , m, and, if l = m, then each of them gives a doubled reaction constant (in terms of Equation (3)). It is these constants (and not the coefficients of rates of the processes) that were introduced for the convenience of the writing of Equation (1). In [6,7,11], a classification of the equations of physicochemical kinetics (3) is given according to the entropy principle: S D E C. Here C denotes the class of all systems of Equation (3) with a finite ⊂ ⊂ ⊂ number of reactions. E is the class of systems for which the condition of semidetailed balance, or complex balance condition, or dynamical equilibrium is satisfied. This, as shown in [6,7,11,28], guarantees an increase of entropy. Boltzmann introduced the semidetailed balance condition for collisions in 1887 [28] and proved that it guarantees the positivity of the entropy production. The microscopic background for the semidetailed balance was found by Ernst Stueckelberg [29] in the Markov microkinetics of the intermediate compounds present in small amounts and whose concentrations are in quasiequilibrium with the main components. Under these microscopic assumptions, the semidetailed balance condition is just the balance equation for the Markov microkinetics according to the Michaelis–Menten–Stueckelberg theorem formulated by Gorban [30]. For chemical kinetics, this condition (known as the complex balance condition) was introduced by Horn and Jackson in 1972 [11]. D is the class of systems of Equation (3) with detailed balance, and S is the class of systems with symmetric reaction constants (i.e., α β systems for which Kβ = Kα). The concepts of detailed and semidetailed balance are explained below. Physics 2019, 1 FOR PEER REVIEW 5

under consideration then and only then when the vector μ is orthogonal to all Boltzmann–Orlov– Moser–Bruno vectors [12,24–27] α  β . In this example, the Boltzmann–Orlov–Moser–Bruno vector is unique up to a factor: α  β  1,1,1,1 . Thus, the stoichiometric coefficients are the numbers of molecules participating in the reaction

(4). Si , i  1,2,, n , are the chemical symbols of the reacting molecules. Linear invariants of a system of physicochemical kinetics equations are the conservation laws of the number of atoms of

each type entering into the composition of at least one of the substances Si participating in the reaction, or a linear combination of these linear conservation laws. More precisely, instead of atoms of each type, different complexes of atoms should be considered, which themselves do not change

during the reaction. Note that if the number of molecules Si , i  1,2,, n , in an elementary

reaction of the Equation (4) is preserved, then α  β , where α  1  2  n . Consider another example, when not only the composition and mass of clusters, but also shape of clusters is taken into account. We confine ourselves to the case of aggregates consisting of no more than three original particles (molecules) and the shapes shown in Figure 1. The original particles are

cubes here. We introduce the notation for the cluster concentrations, as shown in Figure 1: N1 is a Physics 2019, 1 234 concentration of individual molecules, N2 the aggregates of two molecules, and N31and N32 are clusters of two shapes of the three molecules, and arrows indicate three possible reversible reactions.

Figure 1. The. aggregates, consisting of no more than three original particles (cubes), and their concentrations. N is a concentration of individual molecules, N are aggregates of two molecules, Physics 2019, 1 FOR PEER REVIEW 1 2 7 Figureand N31 1.and TheN 32aggregates,are clusters consisting of two shapes of no of more the three than molecules. three original Thearrows particles indicate (cubes), three and possible their reversible reactions. concentrations. N1 is a concentration of individual molecules, N2 are aggregates of two 3. Results 3. Results molecules, and N31 and N32 are clusters of two shapes of the three molecules. The arrows indicate 3.1. The Simplest Models3.1. for Thethreethe Evolution Simplest possibleModels reversibleof Clusters for reactions. the Differing Evolution in Shape of Clusters Diﬀering in Shape

We call the original particlesThen,We call the theconstituting evolution original equations particles the cluster constituting for as these adjacent quantities the if cluster they read have as as: adjacent a common if they face. have a common face. We We assume that a cubeassume attaches that to a an cube aggregate attaches or to to an another aggregate original or to anotherparticle originalonly in such particle a w onlyay in such a way that it that it becomes adjacentbecomes at least adjacent to one other at least original to one particle. other original particle. In the first simplest modelIn the we first restrict simplest ourselves model weto the restrict investigation ourselves of to modeling the investigation the evolution of modeling the evolution of of clusters, differing inclusters, shape, di forffering the in Becker shape,–Döring for the case, Becker–Döring i.e., when case, only i.e.,thewhen original only particles the original particles coalesce coalesce with and fragmentizewith and from fragmentize the cluster from. the cluster. Each aggregate is characterizedEach aggregate by the is characterizednumber n of constituting by the number originaln of constituting particles. Moreover, original particles. Moreover, for for each cluster, we determineeach cluster, the we minimum determine rectangular the minimum parallelepiped rectangular parallelepiped in which this in cluster which is this cluster is placed so placed so that the facesthat of thethe facescubes of, its the components, cubes, its components, are parallel are to parallel the corresponding to the corresponding faces of facesthis of this parallelepiped. parallelepiped. Such recSuchtangular rectangular parallelepipeds parallelepipeds are characterized are characterized by length, by length, width, width, and height, and height, i.e., i.e., the numbers a1, 3 the numbers a , a , aa, a, defining, defining the the vector vector aaN3,, andand wewe considerconsider permutationspermutations ofof tripletstriplets ofof numbers (a , a , a ) as 1 2 2 3 3 ∈ 1 2 3 indistinguishable vectors. It is obvious that: numbers a1,a2,a3  as indistinguishable vectors. It is obvious that:

n a1 a2 a3. (6) n  a1  a2  a3 . ≤ · · (6)

It makes sense to considerIt makes not senseonly the to consider 3-dimensional not only aggregates the 3-dimensional of the original aggregates particles, of the but original particles, but also also 2-dimensional ones,2-dimensional since the growth ones, sinceof clusters the growth on the of surface clusters and on thethe surfacegrowth and of films the growth are of of ﬁlms are of interest. interest. Then, insteadThen, of cubes instead, we have of cubes, squares we havethat are squares connected that are to connectedeach other toso eachthat othertheir edges so that their edges coincide, coincide, and the analogueand the of condition analogue of(6) condition has the form: (6) has the form:

n  a  a . 1 2 n a1 a2.(7) (7)

≤ ·

In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of In the 1-dimensional case, however, we obtain that n = a1, which returns us to the system of d Equations (6) and (7). EquationsTherefore, (6)we andconsider (7). Therefore, aN , wewhere consider d  2a,3. Figured, where 2 showsd = 2, an 3. example Figure2 shows an example of ∈ of a vector a . Note a that vector in thea. Note 1-dimensional that in the case 1-dimensional (d = 1), we case get (dthe= 1),chains we get consisting the chains of consistingthe of the original original particles, andparticles, the breaking and theof one breaking bond of between one bond two between neighboring two neighboring original particles original particlesin the in the chain leads to chain leads to its breakingits breaking.. This means This meansthe processes the processes of agglomeration of agglomeration and fragmentation and fragmentation of chains of chains become essential become essential alongalong with with the theprocesses processes of coalesce of coalescencence and and fragmentation fragmentation of ofindividual individual original original particles to the chain. particles to the chain. TherefTherefore,ore, the 1 1-dimensional-dimensional case case of of the the evolution evolution of of clusters clusters of of different diﬀerent mass mass and shape under the and shape under the BeckerBecker–Döring–Döring constraint make makess sense to to consider only only when when the the growth growth of of chains occurs on the chains occurs on the surfaces, and and then then the the system system of of equations equation describings describing the the evolution evolution has has the the form of (2). form of (2).

Figure 2. The example of the vector , characterizing the shape of the particle consisting of n original particles. Here, a  1,2,3, n  5 .

d For each nN , we define a finite set Jn as the set of all vectors aN satisfying the condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a, where d , a Jn N . The set of all such pairs we denote by G .

Physics 2019, 1 FOR PEER REVIEW 7

Physics 2019, 1 FOR PEER REVIEW 7 3. Results

3. Results 3.1. The Simplest Models for the Evolution of Clusters Differing in Shape We call the original particles constituting the cluster as adjacent if they have a common face. 3.1. The Simplest Models for the Evolution of Clusters Differing in Shape Physics 2019, 1 FOR PEER REVIEWWe assume that a cube attaches to an aggregate or to another original particle7 only in such a way We call the original particles constituting the clusterthat itas becomes adjacent adjacent if they haveat least a commonto one other face. original particle. 3. Results We assume that a cube attaches to an aggregate or to anotherIn the original first simplest particle model only inwe such restrict a w ourselvesay to the investigation of modeling the evolution that it becomes adjacent at least to one other original particle. 3.1. The Simplest Models for ofthe clusters,Evolution differing of Clusters in Differing shape, forin Shape the Becker–Döring case, i.e., when only the original particles In the first simplest model we restrict ourselves tocoalesce the investigation with and fragmentize of modeling from the theevolution cluster . of clusters, differing in shape, We for call the the Becker original–Döring particles case,Each constituting i.e. aggregate, when onlythe is clcharacterized theuster original as adjacent particlesby the if th numberey have n a ofcommon constituting face. We original particles. Moreover, coalesce with and fragmentizeassume from that the clustera cube .attaches for to eachan aggregate cluster, weor to determine another original the minimum particle rectangularonly in such parallelepipeda way that it in which this cluster is Each aggregate is characterizedbecomes adjacent by the number at least tonplaced ofone constituting other so that original the original faces particle. of particles. the cubes Moreover,, its components, are parallel to the corresponding faces of this for each cluster, we determine In the the minimum first simplest rectangular modelparallelepiped. we parallelepiped restrict Such ourselves rec in tangularwhich to the this parallelepipedsinvestigation cluster is of are modeling characterized the evolution by length, width, and height, i.e., placed so that the faces of theof clusters,cubes, its differing components, in shape, are parallel for the to Becker–Döring the corresponding case, faces i.e., ofwhen this only the3 original particles the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of parallelepiped. Such rectangularcoalesce parallelepipeds with and fragmentize are characterized from the by cluster. length, width, and height, i.e., Each aggregate is characterized3numbers bya1 the,a2 ,numbera3  as indistinguishablen of constituting original vectors. particles. It is obvious Moreover, that: for the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of each cluster, we determine the minimum rectangular parallelepiped in which this cluster is placed so n  a  a  a . (6) numbers a1,a2,a3  as indistinguishablethat the faces vectors.of the Itcubes, is obvious its components, that: are parallel to the corresponding1 2 3 faces of this

parallelepiped. Such rectangular parallelepipeds are characterized by length, width, and height, i.e., n  a  a  a . It makes sense to consider not only(6 )the 3-dimensional aggregates of the original particles, but a a 1 a 2 3 a 3 the numbers 1 , 2 , 3 , also defining 2-dimensional the vector ones, Nsince, and the we growth consider of clusters permutations on the ofsurface triplets and of the growth of films are of It makes sense to consider not only the 3-dimensionalinterest aggregates. Then, instead of the of original cubes, we particles, have squares but that are connected to each other so that their edges numbers a1,a2 ,a3 as indistinguishable vectors. It is obvious that: also 2-dimensional ones, since the growth of clusters coincide,on the surface and the and analogue the growth of condition of films (are6) has of the form: interest. Then, instead of cubes, we have squares that are connected to each other so that their edges n  a1  a2  a3 . (6) n  a1  a2 . (7) coincide, and the analogue of condition (6) has the form: It makes sense to consider not only the 3-dimensional aggregates of the original particles, but In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of n  a1  a2 . (7) also 2-dimensional ones, since the growth of clusters on the surface and the growthd of films are of E quations (6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example In the 1-dimensional caseinterest., however, Then, weinstead obtain of thatcubes, n we a have, which squares returns that usare to connected the system to ofeach other so that their edges of a vector1 a . Note that in the 1-dimensional case (d = 1), we get the chains consisting of the coincide, and the analogued of condition (6) has the form: Equations (6) and (7). Therefore, we consider aN original, where particld  2es,,3. andFigure the 2 breaking shows an of example one bond between two neighboring original particles in the of a vector a . Note that in the 1-dimensional case chain(d = 1leads), we to get its thebreakingn chainsa1  .a This2 consisting. means the of theprocesses of agglomeration(7) and fragmentation of chains original particles, and the breaking of one bond betweenbecome two essential neighboring along original with the particles processes in theof coalescence and fragmentation of individual original In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of chain leads to its breaking. This means the processes ofpart agglomerationicles to the chain. and Thereffragmentationd ore, the 1of-dimensional chains case of the evolution of clusters of different mass become essential along withEquations the processes (6) and of (7).coalesce Therefore,andnce shapeand we fragmentation consider under the a Becker ofN individual, –whereDöring d constraintoriginal 2,3. Figure make 2s shows sense anto considerexample only when the growth of Physics 2019, 1 FOR PEER REVIEW 7 particles to the chain. Therefofore, a vector the 1- dimensionala . Note that case inchains the of 1-dimensionalthe occurs evolution on theof case clusters surfaces, (d = 1 of), we anddifferent get then the mass thechains system cons ofisting equation of thes original describing the evolution has the and shape under the Beckerparticles,–Döring and constraint the breaking make formsof sense one of bond (to2). consider between only two when neighboring the growth original of particles in the chain leads to its breaking. This means the processes of agglomeration and fragmentation of chains become 3. Resultschains occurs on the surfaces, and then the system of equations describing the evolution has the form of (2). essential along with the processes of coalescence and fragmentation of individual original particles to the chain. Therefore, the 1-dimensional case of the evolution of clusters of different mass and shape 3.1. The Simplest Models for the Evolution of Clusters Differing in Shape under the Becker–Döring constraint makes sense to consider only when the growth of chains occurs We call the original particles Physicsconstitutingon the2019 surfaces,, 1 the andcluster then as the adjacent system if of they equation have sa describing common face.the evolution has the form of (2). 235 We assume that a cube attachesPhysics to 2019 an , aggregate1 FOR PEER orREVIEW to another original particle only in such a way 8 that it becomes adjacent at least to one other original particle. In the first simplest model weC onsiderrestrict ourselves the evolution to the of investigation the concentrations of modeling (or numbers) the evolution of the clusters having the shape of clusters, differing in shapen, ,a for: theN n Becker,a,t ,– Döringt  0, case,. Without i.e., when loss only of generality, the original we particlesassume here that d  3 .       Figure 2. The example of the vector , characterizing the shape of the particle consisting of n coalesce with and fragmentize from the cluster. We need to write a system of equationsoriginal particles. of the typeHere, ofa the 1 ,physico2,3, nchemical 5 . kinetics with all Each aggregateFigure 2. isThe characterized exampleinteractions of theby vector theof the number form:, characterizing n of constituting the shape original of the particles. particle consisting Moreover, of n for each cluster, we determine the minimum rectangular parallelepiped in which this cluster is d original particles. Here, a  Figure1,2,3 2., Then  example5 . of the vectorFor eacha, characterizing n , we the define shape a of finite the particle set consistingn as the of setn original of all vectors a satisfying the placed so that the faces of the cubes, its components, are paralleln,a to the1, corresponding1,1,1Nn 1 faces,a  eofi  ,this J  (8) N particles. Here, a = (1, 2, 3), n = 5. d  3 d  2 parallelepiped. Such rectangular parallelepipedsFigure 2. Theare characterizedexample conditionof the by vector length, (6 ) awhen ,width characterizing , and height,((7 dthe) when shape i.e., of the ).particle Thus , consistinga shape isof described n by a pair n,a, where where n , i  0,1,2,3, e  0,0,0 is the zero vector, and e ( j 1,2,3) is a vector with For each nN , we defineN a finite set 3Jn 0as the set of all vectors daN satisfyingj thed the numbers a1 , a2 , a3 , defining theFor vectororiginal each particles.anN, we, andHere, define we a consider a finite1,,2 ,a3 set permutations, Jn(n)5as. theN set. Theof of triplets allset vectorsof all of sucha pairssatisfying we denote the by condition G . (6) condition (6) when d  3 ((7) when d ∈ 2 ). Thus, a shape is described by a pair n,3a , where∈ one inwhen j -thd place= 3 ((7) and when zerosd =in 2the). Thus, others. a shapeHere the is described vectors a by aN pair are(n ,sucha), where that nn,a,aG and(n) d. numbers a1,a2,a3  as indistinguishabled vectors. It is obvious that: ∈ ∈ ⊂ , a n . TheThe set set of of all all such such pairs pairs we we denote denote by by .. d J Nn 1,a  eFori  eachG , and n  inN any, we other define cases a finite,G the coefficientsset Jn as ofthe the set direct of all and vectors reverse a  reactionsN satisfying are the Considern  a the a evolution a of the concentrations (or numbers) of the clusters having the shape (n, a): assumedcondition to be zero. (6) 1when If 2i  d30. , then3 ((7) a whendoes notd  vary2 ). , Thus,but na shapeis( 6increased) is described by one by in aall pair four  ncases:,a , where N(n, a, t), t [0, + ). Without loss of generality, we assume here that d = 3. i  0,1,2,3. d It makes sense to consider not nonly the, a 3∈-dimensionaln ∞ . aggregatesThe set of all of suchthe original pairs we particles, denote by but . WeN needJ to writeN a system of equations of the type of theG physicochemical kinetics with all For brevity, we introduce the notation for the number of individual original particles: also 2-dimensional ones, since theinteractions growthConsider of clusters of thethe form: evolutionon the surface of the and concentrations the growth of (o filmsr numbers) are of of the clusters having the shape interest. Then, instead of cubesN,1 wet haveN1 squares,1,1,1,t that . are connected to each other so that their edges n,a : N n,a,t , t  0,(.n Without, a) + (1, loss(1, 1, of 1 ))generality,(n + 1, wea + assumeei), here that d  3 . (8) coincide, and the analogue of conditionThe system (6) has of the equations form: is: ↔ where n , i = 0, 1, 2, 3, e0 = (0, 0, 0) is the zero vector, and ej (j = 1, 2, 3) is a vector with one in j-th ∈ n  a  a 3. 3 (7) place anddN1 zeros t 1 in the2 others.1, 1,1,1 Here 1, 1,1,1 the vectors2 a  2, 1,1,1areei such that (n, a) and (n + 1, a + e ) , i  2 K N1  t  K∈ N 2 , 1,1,1∈ ei ,t  ∈ and in any other cases, the2, coe 1,1,1ffiecientsi  of the direct 1, 1,1,1and 1,reverse 1,1,1 reactions are assumed to be zero. If In the 1-dimensional case, however, wedt obtain thati1 n  a , which returns us to the system of i = 0, then a does not vary, but1 n is increased by one in all four cases: i = 0, 1, 2, 3. (9)  3d 1, 1,1,1 n ,a n,1 ae Equations (6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an examplei  ( ) For brevity, we introduceK the NtNntK notation1    for,a , the  number of individual Nn,  1 a original  ei ,t particles: , N1 t    n,1 aei  1, 1,1,1n ,a  ≡ of a vector a . Note that in the 1N-(dimensional1, (1, 1,n 1)20,at ) J.case n i  (d = 1), we get the chains consisting of the original particles, and the breaking of oneThe systembond between of equations two neighboring is: original particles in the chain leads to its breaking. This means the processes of agglomerationdN and n,,a fragmentation t of chains 3 F n,,a t become essential along with the processes of dNcoalesce(t) nceP and( 1,fragmentation(1,1,1))(1,(1,1,1)) of individual(2,(1,1,1 original)+e ) 1 = 2 K dt N 2(t) K i N(2, (1, 1, 1) + e , t) dt (2,(1,1,1)+e ) 1 (1,(1,1,1))(1,(1,1,1)) i particles to the chain. Therefore, the 1-dimensional3 case− ofi= the1 evolutioni of clusters of− different mass  + 1, 1,1,1nn3 1,a  eii  1, 1,1,1  1, a  e  (9) KP P P (1, N(1,1,1 t))( Nn,a) n 1,a  e , t  K(n+1,a+e ) N n, a ,t  and shape under the Becker–Döring constraint  make∞nn,,aa s senseK to consider1    onlyN (t) whenN(n, ia ,thet) growthK   ofi N(n + 1, a +e , t) , i0  (n+1,a+e ) 1 (1,(1,1,1))(n,a) i (10) chains occurs on the surfaces, and then the system−n=2 a of(n ) equationi=0 s describingi the evolution− has the ∈ 1, 1,1,1n ,a form of (2).     n,1 aei   Kn,1 ae NtNntK1    ,a , 1, 1,1,1n ,a Nn,  1 a  ei ,t ,  i dN(n,a,t)   (  )  dt = F n, a, t 3 ≡ P (1,(1,1,1))(n 1,a eni) 2 , 3 , ,a n . (1,(1,1,1))(n 1,a ei) K − − N (t)N(n 1, aJ e, t) K − − N(n, a, t) (n,a) 1 i (n,a) ≡ i=0 − − − (10) 1,1,1,1n,a Thus, here, K is( 1,the(1,1,1 cross))(n,a-)section (frequency (function,n+1,a+ei) reaction rate coefficient) of such n1,aei  K N1(t)N(n, a, t) K N(n + 1, a + ei, t) , − (n+1,a+ei) − (1,(1,1,1))(n,a) coalescence of the original particle 1,1,1,1n with= 2, 3,the... shape, a (n). , at which increases by the ∈   1,1,1,1n1,aei  value e , so the(1, ( shape1,1,1))(n ,a) n 1,a  e  is obtained; K is the frequency of Thus,i here, K  is the cross-sectioni  (frequencyn function,,a reaction rate coefficient) of such (n+1,a+ei) fragmentationcoalescence of the of the shape original particle into the(1, (original1, 1, 1)) withparticle the and shape the( nshape, a), at  whichn 1,aa increasesei  . by the value Figure 2. The example of the vector , characterizing the shape of the particle(1, ( consisting1,1,1))(n 1, aof ei)n Ife i, for so the each shape n(Nn + and1, a +everyei) is obtained;a,b JnK , we do − not− distinguishis the frequency the shape of fragmentations and of the original particles. Here, a  1,2,3, n  5 . (n,a) n,bshape, then (Enquations, a) into the (9) originaland (10) particle become and the theBecker shape–Döring(n 1, case,a e Equationi). (2). However, our task − − is to distinguishIf for each the n aggregatesand every consistinga, b of( n th),e we original dod not particles distinguish not the only shapes in mass,(n, a ) butand also(n, b in), then For each nN , we define a finite set Jn ∈as the set of all vectors∈ aN satisfying the shapeEquations. (9) and (10) become the Becker–Döring case, Equation (2). However, our task is to distinguish condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a , where Inthe the aggregates second consisting simplest of mo thedel original, we particles assume notthat, only on in mass, the contrary, but also in only shape. rectangular d , a Jn N . Theparallelepipeds set of all suchIn the pairs second can we coalesce denote simplest by with model, G .thei wer assumeidentical that, faces on, theso contrary, that the only result rectangular is a rectangular parallelepipeds parallelepipedcan coalesce again with. Thus their, clusters identical without faces, so voids that theare resultobtained, is a rectangulari.e., Equations parallelepiped (1) and (2) become again. Thus, equalities.clusters In the without 3-dimensional voids are obtained,case, we have: i.e., Equations (1) and (2) become equalities. In the 3-dimensional case, we have: n  a  a  a . 1n =2 a1 3a2 a3.(11) (11)

· · For eachFor each n , we, we define deﬁne a afinite ﬁnite set set I(n) asas the the set set of of all alltriples triples of natural of natural numbers numbers(a , a , a ) ∈ 1 2 3 satisfying the condition (11). Thus, due to the condition (11) in this model, a shape is described by a a1,a2,a3  satisfying the condition (11). Thus, due to the condition (11) in this model, a shape is

Physics 2019, 1 FOR PEER REVIEW 8

Consider the evolution of the concentrations (or numbers) of the clusters having the shape n,a: Nn,a,t , t 0,. Without loss of generality, we assume here that d  3 . We need to write a system of equations of the type of the physicochemical kinetics with all interactions of the form:

n,a 1,1,1,1 n 1,a  ei , (8)

where nN , i  0,1,2,3, e0  0,0,0 is the zero vector, and e j ( j 1,2,3) is a vector with 3 one in j -th place and zeros in the others. Here the vectors aN are such that n,aG and

n 1,a  ei G , and in any other cases, the coefficients of the direct and reverse reactions are assumed to be zero. If i  0 , then a does not vary, but n is increased by one in all four cases: i  0,1,2,3. For brevity, we introduce the notation for the number of individual original particles:

N1t  N1,1,1,1,t . The system of equations is:

3 dN1  t 1, 1,1,1 1, 1,1,1 2  2, 1,1,1ei   2K N1  t  K N 2 , 1,1,1 ei ,t   2, 1,1,1ei   1, 1,1,1 1, 1,1,1   dt i1

 3 (9) 1, 1,1,1n ,a n,1 aei  K NtNntK1    ,a ,  Nn,  1 a  ei ,t ,    n,1 aei  1, 1,1,1n ,a  n20a J n i  dN n,,a t F n,,a t dt 3  K1, 1,1,1nn 1,a  eii N t N n 1,a  e , t  K 1, 1,1,1  1, a  e  N n, a ,t    nn,,aa 1    i      i0  (10)

1, 1,1,1n ,a     n,1 aei   K NtNntK1    ,a ,  Nn,  1 a  ei ,t ,  n,1 aei  1, 1,1,1n ,a  n2 , 3 , ,a J n .

1,1,1,1n,a Thus,Physics here 2019, K, 1 FOR PEER REVIEW is the cross-section (frequency function, reaction rate coefficient) of such 7 n1,aei  coalescence of the original particle 1,1,1,1 with the shape , at which increases by the 1,1,1,1n1,ae  3. Results   i value ei , so the shape  n 1,a  ei  is obtained; Kn,a is the frequency of fragmentation3.1. The Simplest of the Models shape for the Evolution into the oforiginal Clusters particle Differing and in the Shape shape n 1,a  ei  . If for each nN and every a,b Jn , we do not distinguish the shapes and We call the original particles constituting the cluster as adjacent if they have a common face. n,bWe , thenassume Equations that a cube (9) and attaches (10) become to an aggregate the Becker or–Döring to another case, original Equation particle (2). However, only in oursuch task a w ay is tothat distinguish it becomes the adjacent aggregates at least consisting to one other of thoriginale original particle. particles not only in mass, but also in shape. In the first simplest model we restrict ourselves to the investigation of modeling the evolution ofIn clusters, the second differing simplestPhysics in shape 2019 mo,, 1 fordelFOR , the PEERwe Becker REVIEW assume–Döring that, case, on i.e. the, when contrary, only the only original rectangul particlesar 7 parallelepipedscoalesce with canand fragmentize coalesce with from thei ther cluster identical. faces, so that the result is a rectangular parallelepiped again. Thus, clusters without voids are obtained, i.e., Equations (1) and (2) become Each aggregate 3.is Resultscharacterized by the number n of constituting original particles. Moreover, equalities.for each In clusterthe 3-dimensional, we determine case , the we minimumhave: rectangular parallelepiped in which this cluster is placed so that thePhysics faces2019 of, 1 the cubes, its components, are parallel to the corresponding faces of this 236 3.1. The Simplest Modelsn  a for athe Evolutiona . of Clusters Differing in Shape (11) parallelepiped. Such rectangular parallelepipeds1 2are 3characterized by length, width, and height, i.e.,

We call the original particles3 constituting the cluster as adjacent if they have a common face. theFor numbers each atriple, a, we of, a numbers define, defining a( afinite ,thea , avector set) I( na) asN the3., Theand set concentrationwe of consider all triples permutations (or of the natural number) numbersof triplets of aggregates of having the 1 2We 3assume that1 2a cube3 ∈ attaches⊂ to an aggregate or to another original particle only in such a way shape (a1, a2, a3) in time moment t is denoted by N(a1, a2, a3, t). a1,numbersa2,a3  satisfying a1,a2,a 3thethat as condition indistinguishable it becomes (11 adjacent). Thus vectors., dueat least to It thetois obviousone condition other that original (11): in particle.this model , a shape is Deﬁne the function: ( In the first simplest model we restrict ourselves2, x = y ,to the investigation of modeling the evolution n  a1  a2 θˆa(3x., y) = (6) of clusters, differing in shape, for the Becker1,–xDöringy. case, i.e., when only the original particles , It makes sense tocoalesce consider with not and only fragmentize the 3-dimensional from the aggregates cluster. of the original particles, but We obtain the following evolution equation for the concentration of the original particles or shape also 2-dimensional ones, sinceEach theaggregate growth is of characterized clusters on the by surfacethe number and then of growth constituting of films original are of particles. Moreover, (1, 1, 1): interest. Then, insteadfor of eachcubes cluster, we have, we squares determine that theare connected minimum torectangular each other parallelepipedso that their edges in which this cluster is placed so that+ the faces of the cubes, its components, are parallel to the corresponding faces of this coincide, and the analoguedN(1, 1, of 1, tcondition) X∞ (6) has the form: parallelepiped.= Suchˆ( rec) tangular(b+1,1,1) parallelepipeds( + )are characterized(b,1,1)(1,1,1) ( by length,) ( width,) and height, i.e., θ l, 1 K( )( )N b 1, 1, 1, t K( + ) N b, 1, 1, t N 1, 1, 1, t . (12) dt b,1,1 1,1,1 − 3b 1,1,1 b=1 n  a1  a2 . (7) the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of

In the 1-dimensionalFornumbers the case evolution , however,a1,a2 of,a3 the we as concentration obtainindistinguishable that ofn  thea 1vectors. shape, which( 2,It returns 1,is 1obvious), we us have: to that the: system of d Equations (6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example ( ) ( )( ) ( ) dN 2,1,1,t = 1,1,1 1,1,1 2( n  a)1  a2 2,1,1a3 . ( ) (6) of a vector a . Note that in the 1-dimensionaldt K( case) (d =N 1),1, we 1, 1,gett theK( chains)( consisting)N 2, 1, 1, t of the 2,1,1 − 1,1,1 1,1,1 + original particles, and theIt breaking makeP s sense of one to(b +bondconsider2,1,1) between not only two the neighboring 3-dimensional(b,1,1)( 2,1,1original) aggregates particles of in the the original particles, but + ∞ θˆ(b, 2) K N(b + 2, 1, 1, t) K N(b, 1, 1, t)N(2, 1, 1, t) (b,1,1)(2,1,1) (b+2,1,1) (13) chain leads to its breakingalso 2.- dimensionalThisb=1 means the ones, processes since the of growthagglomeration of −clusters and on fragmentation the surface and of chains the growth of films are of + become essential along with theP processes(2, ofb+ 1,1coalesce) nce and fragmentation(2,b,1)(2,1,1 of) individual original interest+. Then∞ θˆ(, binstead, 1) K of cubesN, we(2, bhave+ 1, squares 1, t) K that are connectedN(2, b, 1, tto)N each(2, 1, o 1,thert) . so that their edges (2,b,1)(2,1,1) (2,b+1,1) particles to the chain.coincide, Therefore,b =and1 the the 1- dimensionalanalogue of conditioncase of the (6 evolution) has− the form: of clusters of different mass and shape under the Becker–Döring constraint makes sense to consider only when the growth of Writing down a similar equation for the evolutionn  a of a the concentration of an arbitrary shape chains occurs on the surfaces, and then the system of equations describing1 2 . the evolution has the (7) (a , a , a ), we obtain: form of (2). 1 2 3 In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of a1 1 d EquationsdN ((6a1), aand2,a3,t )(7). ThereforeP− , we(b ,considera2,a3)(a1 b ,aa2,a3) N( , where) d( 2,3. Figure) 2 shows an example dt = K − N b, a2, a3, t N a1 b, a2, a3, t (a1,a2,a3) − of a vector a . Note thatb = in1, the 1-dimensional case (d = 1), we get the chains consisting of the original particles, and theb breakinga1/2 of one bond between two neighboring original particles in the ≤ chain leads to its breaking. This(a 1means,a2,a3) the processes of agglomeration and fragmentation of chains K N(a1, a2, a3, t) become essential along with− the(b ,aprocesses2,a3)(a1 b,a2 ,ofa3) coalescence and fragmentation of individual(14) original + − P∞ (a1+b,a2,a3) particles to the chain. +Therefθ(ore,b, a1 the) K 1-dimensional N case(a1 + ofb ,thea2, aevolution3, t) of clusters of different mass (b,a2,a3)(a1,a2,a3) and shape under the Beckerb=1 –Döring constraint makes sense to consider only when the growth of (b,a2,a3)(a1,a2,a3) chains occurs on the surfaces,K and thenN ( theb, a 2 system, a3, t)N ( ofa1 , equationa2, a3, t) s describing the evolution has the Figure 2. The example of the vector , characterizing− (a1+b,a2,a3) the shape of the particle consisting of n form of (2). + ... , (a , a , a ) (n), n = 1, 2, ... original particles. Here, a  1,2,3, n  5 . 1 2 3 ∈

The last term here denotes the similar terms for a2 and a3, whichd is written for a1. To avoid a For each nN , we define a finite set Jn as the set of all vectors aN satisfying the repetition, such terms should be written out only for different numbers from a triplet (a1, a2, a3). In condition (6) whenEquations d  3 (12)–(14), ((7) when the coed ffi2cients). Thus represent, a shape the is reaction described (process) by a pair constants. n,a, where d , a JnTheN . second The set simplest of all such model pairs (14)we denote is a vectorial by G . version of the classical discrete Smoluchowski coagulation–fragmentation equations [17], where some compatibility condition of the vectors is demanded. The compatibility is that the only vectors that can merge are those which differ in one component.

If thereFigure is an 2.upper The examplelimit for ofn: then vectorM (M =,2, characterizing 3, ...), then we the have shape the of ﬁnite–dimensional the particle consisting system of n ≤ consisting fromoriginalM particles.equations, Here which, a  we1, call2,3 the, nM-th5 . approximation of Equations (9) and (10):

3 dN (t) P (1,(1,1,1))(1,(1,1,1)) (2,(1,1,1)+e ) 1 = 2 K N 2(t) K i N(2, (1, 1, 1) + e , t) d Fordt each nN (, 2,we(1,1,1 define)+e ) a finite1 set J(1,n(1,1,1 as))( the1,(1,1,1 set)) of all vectors aiN satisfying the − i=0 i − M 1 3 conditionP (6) PwhenP d (1,(31,1,1 ((7))()n ,awhen) d  2 ). Thus(n, +a1, shapea+e ) is described by a pair n,a, where − K N (t)N(n, a, t) K i N(n + 1, a + e , t) , (dn+1,a+e ) 1 (1,(1,1,1))(n,a) i −,n =a2a (n) i=0 . The seti of all such pairs− we denote by . ∈ J  N G dN(n,a,t) ( ) ( ) (15) dt = F n, a, t , n = 2, 3, ... , M 1, a n , 3 − ∈ dN(M,a,t) P (1,(1,1,1))(M 1,a e ) = K − − i N (t)N(M 1, a e , t) dt (M,a) 1 − − i i=0 (1,(1,1,1))(M 1,a e ) K − − i N(M, a, t) , a (M). − (M,a) ∈ Physics 2019, 1 FOR PEER REVIEW 7 Physics 2019, 1 FOR PEER REVIEW 7

3. Results 3. Results 3.1. The Simplest Models for the Evolution of Clusters Differing in Shape 3.1. The Simplest Models for the Evolution of Clusters Differing in Shape We call the original particles constituting the cluster as adjacent if they have a common face. We call the original particles constituting the cluster as adjacent if they have a common face. We assume that a cube attaches to an aggregate or to another original particle only in such a way We assume that a cube attaches to an aggregate or to another original particle only in such a way that it becomes adjacent at least to one other original particle. that it becomes adjacent at least to one other original particle. In the first simplest model we restrict ourselves to the investigation of modeling the evolution In the first simplest model we restrict ourselves to the investigation of modeling the evolution of clusters, differing in shape, for the Becker–Döring case, i.e., when only the original particles of clusters, differing in shape, for the Becker–Döring case, i.e., when only the original particles coalesce with and fragmentize from the cluster. coalesce with and fragmentize from the cluster. Each aggregate is characterized by the number n of constituting original particles. Moreover, Each aggregate is characterized by the number n of constituting original particles. Moreover, for each cluster, we determine the minimum rectangular parallelepiped in which this cluster is for each cluster, we determine the minimum rectangular parallelepiped in which this cluster is placed so that the faces of the cubes, its components, are parallel to the corresponding faces of this placed so that the faces of the cubes, its components, are parallel to the corresponding faces of this parallelepiped. Such rectangular parallelepipeds are characterized by length, width, and height, i.e., parallelepiped. Such rectangular parallelepipeds are characterized by length, width, and height, i.e., 3 the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of 3 the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of numbers a1,a2,a3  as indistinguishable vectors. It is obvious that: numbers a1,a2,a3  as indistinguishable vectors. It is obvious that: n  a1  a2  a3 . (6)

n  a  a  a Physics1 20192 , 13 FOR. PEER REVIEW (6) 8

It makes sense to consider not only the 3-dimensional aggregates of the original particles, but It makes sense to consider not only the 3-dimensional aggregates of the original particles, but also 2-dimensional ones, since the growth of clusters on the surface and the growth of films are of Consider the evolution of the concentrations (or numbers) of the clusters having the shape also 2-dimensional ones, since the growth of clusters on the surface and the growth of films are of interest. Then, instead of cubes, we have squares that are connected to each other so that their edges n,a: Nn,a,t , t 0,. Without loss of generality, we assume here that d  3 . interest. Then, instead of cubes, we have squares that are connected to each other so that their edges coincide, and the analogue of condition (6) has the form: coincide, and the analogue of condition (6) has the Weform: need to write a system of equations of the type of the physicochemical kinetics with all interactions of the form: n  a1  a2 . (7) n  a1  a2 . (7)

n,a 1,1,1,1 n 1,a Ine thei , 1-dimensional case, however,(8) we obtain that n  a1 , which returns us to the system of

In the 1-dimensional case, however, we obtain that n  a , which returns us to the system of d 1 Equations (6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example where dnN , i  0,1,2,3, e  0,0,0 is the zero vector, and e ( j 1,2,3) is a vector with Equations (6) and (7). Therefore, we consider aN , where d  2,3. Figure0 2 shows an example of a vector j a . Note that in the 1-dimensional case (d = 1), we get the chains consisting of the 3 of a vector a . Note that in the 1-dimensionalone casein j(-dth = place 1), we and get zeros the inchains the others. consisting Here of the the vectors original a Nparticl arees, such and that the breakingn,aG of andone bond between two neighboring original particles in the original particles, and the breaking of one bond between two neighboring original particles in the chain leads to its breaking. This means the processes of agglomeration and fragmentation of chains n 1,a  ei G , and in any other cases, the coefficients of the direct and reverse reactions are chain leads to its breaking. This means the processes of agglomeration and fragmentation of chains become essential along with the processes of coalescence and fragmentation of individual original assumed to be zero. If i  0 , then a does not vary, but n is increased by one in all four cases: become essential along with the processes of coalescence and fragmentation of individual original particles to the chain. Therefore, the 1-dimensional case of the evolution of clusters of different mass i  0,1,2,3. particles to the chain. Therefore, the 1-dimensional case of the evolution of clusters of different mass and shape under the Becker–Döring constraint makes sense to consider only when the growth of Physics 2019, 1 FOR PEER REVIEW 7 and shape under the Becker–Döring constraint makeFor s brevity,sense to weconsider introduce only whenthe notation the growth for of the numberchains occurs of individual on the surfaces, original and particles: then the system of equations describing the evolution has the chains occurs on the surfaces, and then the systemN1t  ofN equation1,1,1,1,st  describing. the evolution has the form of (2). form of (2). The system of equations3. Results is: 3 dN1  t 1, 1,1,1 1, 1,1,1 2  2, 1,1,1ei   23.1.K The Simplest Models N1  t for  the K Evolution of NClusters 2 , 1,1,1 Differing ei ,t in Shape   2, 1,1,1ei   1, 1,1,1 1, 1,1,1   dt i1

 3 We call the original particles constituting the cluster as adjacent if(9) they have a common face. 1, 1,1,1 n ,a n,1 ae We assume that a cube attaches to an aggregatei  or to another original particle only in such a way K NtNntK1    ,a ,  Nn,  1 a  ei ,t ,    n,1 aei  1, 1,1,1n ,a  n20a J n i  that it becomes adjacent at least to one other original particle. In the first simplest model we restrict ourselves to the investigation of modeling the evolution of clusters, dN differing n,,a tin shape, for the Becker–Döring case, i.e., when only the original particles F n,,a t Figure 2. The example of the vector , characterizing the shape of the particle consisting of n coalesce with anddt fragmentize from the cluster. Figure 2. The example of the vector , characterizing the shape of the particle consisting of n original particles. Here, a  1,2,3, n  5 . 3 Each aggregate is characterized by the number n of constituting original particles. Moreover, original particles. Here, a  1,2,3, n  5 .  1, 1,1,1nn 1,a  eii  1, 1,1,1  1, a  e  Knn,,aafor each N cluster1  t N, we n  determine1,a  ei , t  the  K minimum rectangular N n, a ,t  parallelepiped  in which this cluster is        d Physics 2019, 1 i0 For each nN , we define a finite(23710 ) set Jn as the set of all vectors aN satisfying the placed so thatd the faces of the cubes, its components, are parallel to the corresponding faces of this For each nN , we define a finite set Jn as the set of all  vectors1, 1,1,1n ,aaN satisfying the n,1 ae Kparallelepiped. NtNntK  Such,a , rec tangularcondition parallelepipedsi Nn, (6 ) 1 when a  are e ,t dcharacterized3 , ((7) when by length,d  2 ).width Thus, ,and a shape height, is i.e.described, by a pair n,a, where n,1 aei  1 1, 1,1,1n ,a i condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a, where d3d Then, the shape is describedthe by numbers a pair (n , aa)1,, where a2 , an3=, defining1, 2, 3, ... the, M, ,vectoraa J( na)NN ., . Weand The denote weset considerof theall such set permutations pairs we denote of triplets by G .of d ∈ ⊂ , a Jn N . The set of all such pairsof all we such denote pairs by by GM. . n2 , 3 , ,a J n . numbers a1,a2,a3  as indistinguishable vectors. It is obvious that: Instead of the ﬁrst equation of (15), we can write the conservation law of all original particles in 1,1,1,1n,a Thus, here, Kn1,ae  is the cross-section (frequency function, reaction rate coefficient) of such the system: i n  a1  a2  a3 . (6) M coalescence of the original particleX 1,X1,1,1 with the shape , at which increases by the (n N(n, a, t)) = const. It makes sense to consider not only1,1,1,1 then1 ,3a-dimensionale  aggregates of the original particles, but   · i value ei , so the shape  n 1n=,a1 a e(in) is obtained; Kn,a is the frequency of also 2-dimensional∈ ones, since the growth of clusters on the surface and the growth of films are of fragmentationFor the second of the model, shapeinterest the M-th. Theninto approximation the, instead original of ofcubesparticle (14), reads:we and have the squares shape thatn  1are,a connectedei  . to each other so that their edges coincide, and the analogue of condition (6) has the form: If for each nN and every a,b Jn , we do not distinguish the shapes and a1 1 dN(a1,a2,a3,t) P− (b,a2,a3)(a1 b,a2,a3) ( ) ( n  a  a . ) n,b , then Equationsdt (9)= and (10) becomeK the Becker− –DöringN b, a2 case,, a3, t EquationN a1 b,1 a(2).2, a2 3However,, t our task (7) (a1,a2,a3) − is to distinguish the aggregatesb = consisting1, of the original particles not only in mass, but also in In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of shape. b a1/2 ≤ d Equations ((a61,)a 2and,a3) (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example In the second simplest moKdel, we assumeN( a1that,, a2, a3 on, t) the contrary, only rectangular − (b,a2,a3)(a1 b,a2,a3) parallelepipeds can coalesceof a vector with+ theia . r N identicalote− that in faces the, 1so-dimensional that the result case (dis = a 1 ),rectangular we get the chains consisting of the P∞ (a1+b,a2,a3) (16) + original particles, Kand the breakingN(a of1 + oneb, a 2bond, a3, t )between two neighboring original particles in the parallelepiped again. Thus, clusters without voids(b,a2,a3 )(area1,a 2obtained,,a3) i.e., Equations (1) and (2) become equalities. In the 3-dimensionalchainb caseleads= 1,, we to have:its breaking . This means the processes of agglomeration and fragmentation of chains (becomea1 + b)a 2essentiala3 M along with the processes of coalescence and fragmentation of individual original ≤ particles(b,a2 ,ato3)( thea1,a 2chain.,a3n)  a Theref1  a2 ore,a3 . the 1-dimensional case of the evolution(11) of clusters of different mass K N(b, a2, a3, t)N(a1, a2, a3, t)

( + ) and− shapea1 b,a 2under,a3 the Becker–Döring constraint makes sense to consider only when the growth of For each , we define+ ... , a( afinite, a , a set) I(n), nas= the1, 2, set... , ofM . all triples of natural numbers chains occurs1 on2 3 the∈ surfaces, and then the system of equations describing the evolution has the a ,a ,a  satisfying the condition (11). Thus, due to the condition (11) in this model, a shape is 1 Then,2 3 instead of the equationform of for(2). the evolution of N(1,1,1,t) in (16), we can write the conservation law of all the original particles in the system:

a1aX2a3 M ≤ a1a2a3N(a1, a2, a3, t) = const. a1,a2,a3=1

Thus, we obtain the systems of Equations (9), (10) and (14), as well as their M-th approximations

(15) and (16). Now let us address the H-theorem point. We consider the H-theorem for a finite system of equations. This faces the same difficulties with Figure 2. The example of the vector , characterizing the shape of the particle consisting of n convergence of series for the infinite systems as discussed in [18–21]. It is important to point out original particles. Here, , n  5 . that the detailed study of the H-theorem in this case isa a problem1,2,3 for a separate paper. Nevertheless, our results involve the detailed balance and so are valid regardless of the finiteness of the systems d in question. For each nN , we define a finite set Jn as the set of all vectors aN satisfying the condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a, where 3.2. The Conditions of Detailed and Semidetailed Balanced , a Jn N . The set of all such pairs we denote by G . The condition of the detailed balance for the general system of equations of physicochemical kinetics (3) is actually the condition for the coefficients: let there exist at least one positive solution of the following system of equations:

β Kαξα = K ξβ, (α, β) . β α ∈ = This assumes that the rate of the direct reaction is equal to the rate of the reverse reaction for all the reactions. The number of equations in the detailed balance condition is equal to the number of reactions which is half of the number of pairs (α, β) in set , while the number of unknowns ξ equals = i n. The condition of detailed balance in the case of Equations (9) and (10) takes the following form: let there be at least one solution (ξ(n, a) > 0) of the system:

(1,(1,1,1))(n,a) (n+1,a+ei) K ξ1ξ(n, a) = K ξ(n + 1, a + ei), (17) (n+1,a+ei) (1,(1,1,1))(n,a) Physics 2019, 1 FOR PEER REVIEW 7

3. Results

3.1. The Simplest Models for the Evolution of Clusters Differing in Shape We call the original particles constituting the cluster as adjacent if they have a common face. We assume that a cube attaches to an aggregate or to another original particle only in such a way that it becomes adjacent at least to one other original particle. In the first simplest model we restrict ourselves to the investigation of modeling the evolution of clusters, differing in shape, for the Becker–Döring case, i.e., when only the original particles coalesce with and fragmentize from the cluster. Each aggregate is characterized by the number n of constituting original particles. Moreover, for each cluster, we determine the minimum rectangular parallelepiped in which this cluster is placed so that the faces of the cubes, its components, are parallel to the corresponding faces of this parallelepiped. Such rectangular parallelepipeds are characterized by length, width, and height, i.e., 3 the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of numbers a1,a2,a3  as indistinguishable vectors. It is obvious that:

n  a1  a2  a3 . (6)

It makes sense to consider not only the 3-dimensional aggregates of the original particles, but also 2-dimensional ones, since the growth of clusters on the surface and the growth of films are of interest. Then, instead of cubes, we have squares that are connected to each other so that their edges coincide, and the analogue of condition (6) has the form:

n  a1  a2 . (7)

In the 1-dimensional case, however, we obtain that n  a1 , which returns us to the system of d Equations (6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example of a vector a . Note that in the 1-dimensional case (d = 1), we get the chains consisting of the original particles, and the breaking of one bond between two neighboring original particles in the chain leads to its breaking. This means the processes of agglomeration and fragmentation of chains become essential along with the processes of coalescence and fragmentation of individual original particles to the chain. Therefore, the 1-dimensional case of the evolution of clusters of different mass and shape under the Becker–Döring constraint makes sense to consider only when the growth of chains occurs on the surfaces, and then the system of equations describing the evolution has the form of (2).

Figure 2. The example of the vector , characterizing the shape of the particle consisting of n original particles. Here, a  1,2,3, n  5 .

d For each nNPhysics, we define2019, 1 a finite set Jn as the set of all vectors aN satisfying the 238 condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a, where d , a Jn Nwhere. Theξ set> of0and all suchξ(n, apairs) > 0 we for denote all (n, aby) G.. 1 ∈ The condition of semidetailed balance [6,7,11,24,28,29] is a condition when there is at least one positive solution ξ of the following system of equations:

X α α X β β Kβξ = Kαξ . β β

Here, α is defined so that (α, β) with some β. The sum runs only over those multi-indices β α β ∈ = for which either Kβ , 0 or Kα , 0. The sum of the rates of all reactions with initial state α is equal to the sum of all rates of reactions with final state α. The number of equations in the condition of semidetailed balance is half of the number of different vectors α in pairs (α, β) . ∈ = The condition of semidetailed balance in the case of Equations (9) and (10) leads to the two types of relations:

X3 X3 (n,a) (1,(1,1,1))(n 1,a ei) ξ(n, a) K = K − − ξ1ξ(n 1, a ei) (18) (1,(1,1,1))(n 1,a ei) (n,a) − − i=0 − − i=0

and: 3 3 X (1,(1,1,1))(n 1,b) X (n,b+ej) ξ(n 1, b)ξ1 K − = K ξ n, b + ej . (19) − (n,b+ej) (1,(1,1,1))(n 1,b) j=0 j=0 −

Note that the relations of the form of (18) determine the recursion: ξ(n, a) expresses through ξ1 3 P (n,a) and ξ(n 1, ...) if K 0. (1,(1,1,1))(n 1,a e ) , − i=0 − − i The condition of detailed balance as well as the condition of semidetailed balance imposes restrictions on the cross-sections. If we consider these conditions for the M-th approximation of the Equations (9) and (10), i.e., for (15), then, this case contains some restrictions of the original case remaining unchanged. Therefore, it is advisable to consider examples of systems of the form (15). Consider an example when M = 4: each of shapes with n = 1 and n = 2 is unique, there are two shapes with n = 3, and there are three shapes with n = 4 in the 2-dimensional case and four shapes in the 3-dimensional case. Natural shapes from cubes with n 4 and reactions are shown in Figure3, and a similar scheme ≤ (graph) for our model for the 3- and 2-dimensional cases is shown in Figure4a,b, respectively. The naturalness is deﬁned as follows: the clusters have the same shape if they are transferred into each other by transformation of motion that does not change the orientation in space. The fact that the original particles have a reaction with a speciﬁc shape is not depicted here. Due to the condition of detailed balance in this example (M = 4), we have the following restriction on cross-sections for d = 3 (Figure4a):

( ( ))( ( )) ( ( ))( ( )) ( ( ))( ( )) ( ( ))( ( )) K 1, 1,1,1 3, 1,1,3 K 1, 1,1,1 2, 1,1,2 K 1, 1,1,1 3, 1,2,2 K 1, 1,1,1 2, 1,1,2 (4,(1,2,3)) (3,(1,1,3)) (4,(1,2,3)) (3,(1,2,2)) = , (20) ( ( )) ( ( )) ( ( )) ( ( )) K 4, 1,2,3 K 3, 1,1,3 K 4, 1,2,3 K 3, 1,2,2 (1,(1,1,1))(3,(1,1,3)) (1,(1,1,1))(2,(1,1,2)) (1,(1,1,1))(3,(1,2,2)) (1,(1,1,1))(2,(1,1,2))

which comes from the fact that starting from the node (2, (1, 1, 2)) one gets the node (4, (1, 2, 3)) in two diﬀerent ways. Restrictions on the cross-sections come in play as soon as the resulting graph is not a tree, i.e., contains various reaction paths. In the case of shapes, the graph in Figure3 is di ﬀerent, and the condition of detailed balance leads to the two relations (similar to (20)). The case of a stationary solution, when there are neither detailed nor semidetailed balances, is shown in Figure5, which corresponds not only to Figure3 but also to Figure4. Physics 2019, 1 FOR PEER REVIEW 12

Physics 2019, 1 239 Physics 2019, 1 FOR PEER REVIEW 12

Figure 3. The aggregates, consisting of no more than four original particles. . Figure 3. The aggregates, consisting of no more than four original particles.

Figure 3. The aggregates, consisting of no more than four original particles.

(a) (b)

FigureFigure 4. 4. TheThe shapesshapes ofof aggregates,aggregates, consistingconsisting ofof nono moremore thanthan fourfour originaloriginal particles,particles, forfor thethe ﬁrstfirst Physics 2019, 1 FOR PEER REVIEW 13 simplestsimplestd d-dimensional-dimensional model: model: ( a(a))d =d 3, (3 b), d(b=) 2.d  2 . (a) (b) Natural shapes from cubes with n  4 and reactions are shown in Figure 3, and a similar Figure 4. The shapes of aggregates, consisting of no more than four original particles, for the first scheme (graph) for our model for the 3- and 2-dimensional cases is shown in Figure 4a,b, respectively. simplest d-dimensional model: (a) d  3 , (b) d  2 . The naturalness is defined as follows: the clusters have the same shape if they are transferred into each other by transformation of motion that does not change the orientation in space. The fact that Natural shapes from cubes with n  4 and reactions are shown in Figure 3, and a similar the original particles have a reaction with a specific shape is not depicted here. scheme (graph) for our model for the 3- and 2-dimensional cases is shown in Figure 4a,b, respectively. Due to the condition of detailed balance in this example ( M  4 ), we have the following The naturalness is defined as follows: the clusters have the same shape if they are transferred into restriction on cross-sections for d  3 (Figure 4a): each other by transformation of motion that does not change the orientation in space. The fact that the original particles 1have,1,1,1 a3 ,reaction1,1,3 1with,1,1,1 a2 specific, 1,1,2 shape1,1, 1is,1 not3, 1, 2depicted,2 1,1 ,here.1,1 2, 1,1,2 K4,1,2,3 K3,1,1,3 K4,1,2,3 K3,1,2,2  M 4 , (20) Due to the conditionK 4,1,2,3 of detailedK 3,1 ,balance1,3 in thisK  4example,1,2,3 ( K3,1,),2, 2we have the following restriction on cross-sections1,1,1,1 3 ,for1,1, 3 d 13,1 ,1(Figure,1 2, 1,1,2 4a): 1,1,1,1 3, 1,2,2 1,1,1,1 2, 1,1,2 . which comes fromK the1,1, 1fact,1 3 ,that1,1,3 startingK 1,1,1,1 from2, 1,1 ,the2 nodeK 1 ,1,12,1,13,11,,2,2 K one1, 1gets,1,1  2the, 1,1 ,node2 4,1,2,3 in Figure 5. The minimum4,1,2,3 set of aggregates,3,1,1,3 for which the condition4,1,2,3 of detailed3,1, and2,2 semidetailed balance two different ways. 4Restrictions,1,2,3 on 3the,1,1 ,3cross-sections 4come,1,2,3 in play as 3soon,1,2,2 as the resulting, (20)graph is can be not valid,K and their concentrations.K DirectionsK of the arrows indicateK which of the direction of not a Figuretree, i.e., 5. containsThe minimum1,1,1 ,various1 3, 1 ,1set,3 reactionof aggregates,1,1,1,1 paths.2, 1,1, 2for which1, 1,the1,1 condition3, 1,2,2 of1, 1detailed,1,1 2, 1,1 ,2and semidetailed

the process prevails: coalescence with the original particle or fragmentation. Inbalance the case can ofbe shapes,not valid, the and graph their inconcentrations. Figure 3 is different, Directions and of the the arrows condition indicate of detailedwhich of balancethe whichdirection comes from of the the process fact thatprevails: starting coalescence from the with node the original2,1,1 ,particle2 one or gets fragmentation. the node 4,1,2,3 in leads to the two relations (similar to (20)). two differentLet us consider ways. Restrictions an example on for the this cross-sections situation, i.e., come a system in play of equations as soon as where the resulting only aggregates graph is shownnot a tree,The in Figure i.e.,case contains of5 area stationary considered. various solution, reaction Simply whenpaths. let us there record ar thee neither equations detailed by entering nor semidetailed notations forbalances, cluster is concentrations,shownIn the in Figurecase as of shown5, shapes, which in correspondsthe Figure graph5 (and in not asFigure it on wasly 3 to givenis Figuredifferent, in the 3 but exampleand also the to incondition Figure the introduction): 4. of detailedN balance2 is the numberleads toLet the of us aggregates two consider relations an of twoexample(similar original tofor (20)). this particles, situation,N31 andi.e., aN system32 are the of equations concentrations where of only aggregates aggregates of twoshown types in of Figure three 5 original are considered. particles, Simply and N4 letof us four. reco Then,rd the we equations obtain a by counterpart entering notations of (15): for cluster

concentrations, as shown in Figure 5 (and as it was given in the example in the introduction): N2 is

the number of aggregates of two original particles, N31 and N32 are the concentrations of

aggregates of two types of three original particles, and N4 of four. Then, we obtain a counterpart of (15):

dN 1, 1,1,1 1, 1,1,1 2, 1,1,2 2 KNKN 2  dt 2, 1,1,212 1, 1,1,1 1, 1,1,1 1, 1,1,1 2, 1,1,2 3, 1,1,3 KNNKN  3, 1,1,312 1, 1,1,1 2, 1,1,2 31 1, 1,1,1 2, 1,1,2 3, 1,2,2 KNNKN  , 3, 1,2,212 1, 1,1,1 2, 1,1,2 32

dN 1, 1,1,1 2, 1,1,2 3, 1,1,3 31 KNNK  N  dt 3, 1,1,3 12 1, 1,1,1 2, 1,1,2 31 1, 1,1,1 3, 1,1,3 4, 1,2,3 KNNKN, (21) 4, 1,2,3131 1, 1,1,1 3, 1,1,3 4

dN 1, 1,1,1 2, 1,1,2 3, 1,2,2 32 KNNKN  dt 3, 1,2,212 1, 1,1,1 2, 1,1,2 32 1, 1,1,1 3, 1,2,2 4, 1,2,3 KNNKN  , 4, 1,2,3132 1, 1,1,1 3, 1,2,2 4

dN 1, 1,1,1 3, 1,1,3 4, 1,2,3 4  K NN K N dt 4, 1,2,3131 1, 1,1,1 3, 1,1,3 4 1, 1,1,1 3, 1,2,2 4, 1,2,3 KNNKN  , 4, 1,2,3132 1, 1,1,1 3, 1,2,2 4

replacing the evolution equation of concentration N1 with the conservation law of the number of all original particles in the system:

N1  2N2  3N31  3N32  4N4  Nall  const . (22)

Physics 2019, 1 240

( ( ))( ( )) ( ( )) dN2 = K 1, 1,1,1 1, 1,1,1 N 2 K 2, 1,1,2 N dt (2,(1,1,2)) 1 − (1,(1,1,1))(1,(1,1,1)) 2 ( ( ))( ( )) ( ( )) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,1,3 N − (3,(1,1,3)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 31 ( ( ))( ( )) ( ( )) K 1, 1,1,1 2, 1,1,2 N N K 3, 1,2,2 N , − (3,(1,2,2)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 32 ( ( ))( ( )) ( ( )) dN31 = K 1, 1,1,1 2, 1,1,2 N N K 3, 1,1,3 N dt (3,(1,1,3)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 31 ( ( ))( ( )) ( ( )) K 1, 1,1,1 3, 1,1,3 N N K 4, 1,2,3 N , (21) − (4,(1,2,3)) 1 31 − (1,(1,1,1))(3,(1,1,3)) 4 ( ( ))( ( )) ( ( )) dN32 = K 1, 1,1,1 2, 1,1,2 N N K 3, 1,2,2 N dt (3,(1,2,2)) 1 2 − (1,(1,1,1))(2,(1,1,2)) 32 ( ( ))( ( )) ( ( )) K 1, 1,1,1 3, 1,2,2 N N K 4, 1,2,3 N , − (4,(1,2,3)) 1 32 − (1,(1,1,1))(3,(1,2,2)) 4 ( ( ))( ( )) ( ( )) dN4 = K 1, 1,1,1 3, 1,1,3 N N K 4, 1,2,3 N dt (4,(1,2,3)) 1 31 − (1,(1,1,1))(3,(1,1,3)) 4 ( ( ))( ( )) ( ( )) + K 1, 1,1,1 3, 1,2,2 N N K 4, 1,2,3 N , (4,(1,2,3)) 1 32 − (1,(1,1,1))(3,(1,2,2)) 4

replacing the evolution equation of concentration N1 with the conservation law of the number of all original particles in the system:

Physics 2019, 1 FOR PEER REVIEW 14 N + 2N + 3N + 3N + 4N N = const. (22) 1 2 31 32 4 ≡ all Obviously, the condition that the vector ξ   , , , ,  is a stationary solution of (21) Obviously, the condition that the vector ξ = 1(ξ12, ξ231, ξ3132, ξ324, ξ4) is a stationary solution of (21) andand (22 (22)) is isequivalent equivalent toto the the fact fact that that,, for for some some number number kkR,, thethe vectorvector ξξsatisﬁes satisfies the the system system ∈ (1,(11,1,1,1,1,))(13,3,(11,1,3,1,3)) (44,,1(,1,2,32,3)) KK4,1,2,3 ξ1ξ31  KK1,1,1,13,1,1,34 ξ =k, k, (4,(1,2,3)) 1 31 − (1,(1,1,1))(3,(1,1,3)) 4 − (1,(1,1,1))(3,(1,2,2)) (4,(1,2,3)) K 1,1,1,13,1,2,2 ξ ξ K K4,1,2,3  ξk,= k, (4,4,(11,2,3,2,3)) 1 321 32 −1,(11,,1(,11,1,13,))(1,23,,2(1,2,2 4 )) 4 (1,(1,1,1))(2,(1,1,2)) (3,(1,1,3)) (23) (23) K 1,1,1,12,1,1,2 ξ1ξ2 K3,1,1,3 ξ31 = k, K(3,3(1,1,3,1,1,3)) 12 −K1(,1,1(,11,1,1,12))(,12,,1,(21,1,231))  k,− (1,(1,1,1))(2,(1,1,2)) (3,(1,2,2)) K ξ ξ K ξ = k, (3,1,(1,2,21,1,1))2,1,1,2 1 2 3(,1,1,(21,1,1,2 ))(2,(1,1,2)) 32 K3,1,2,2 12  −K1,1,1,12,1,1,232  k,

andand:: (1,(1,1,1))(1,(1,1,1)) 2 (2,(1,1,2)) = K( ( )) ξ1 K( ( ))( ( ))ξ2 0. (24) 2,11,1,2,1,1,11,1,1,1 2 − 21,,1,1,1,11,2 1, 1,1,1 K   K   0 . (24) In Figure5, the directions of2,1 the,1,2 arrows indicate1 1, the1,1,1 direction1,1,1,1 2 of the process: coalescence prevails

withIn the Figure original 5, the particle directions or fragmentation. of the arrows If and indicate only ifthe all fourdirection reactions of the in Figure process:5 have coalescence the same prevailsmagnitude with ofthe “dominance”, original particle then or one fragmentation. obtains the stationaryIf and only solution if all four that reactions satisfies in (23). Figure Indeed, 5 have the thenumber same ofmagnitude arrows “up” of “dominance”, is equal to the numberthen one of obtain arrowss “down”the stationary and, therefore, solution the that number satisfies of original (23). Indeed,particles the is number preserved. of arrows For the aggregates“up” is equal from to the the original number particles, of arrows this “down” is also immediately and, therefore apparent, the numberfrom Figure of original5, where particles exactly is one preserved arrow comes. For the and ag goesgregates to each from vertex the original of the graph. particles The, directionsthis is also of immediatelythe arrows inapparent Figure5 from correspond Figure 5, to wherek > 0 inexactly (23). Theone detailedarrow comes balance and (the goes semidetailed to each vertex balance) of the is graph.the case The when directions the value of the of thearrows “predominance” in Figure 5 correspond of all reactions to k is zero:0 ink (23= 0.). The detailed balance (the semiIf thedetailed of value balanceξ1 is fixed,) is the then case (23) when becomes the a value linear of system. the “predominance The condition” of of detailed all reactions balance is of zero:Equation k  0 (20). is the condition that the determinant of this system is zero. In its turn, if the determinant of this system is zero, one obtains positive stationary solutions that If the of value 1 is fixed, then (23) becomes a linear system. The condition of detailed balance satisfy the detailed balance condition, which is the solution of Equations (23) and (24) with k = 0. By of Equation (20) is the condition that the determinant of this system is zero. virtue of the H-theorem, we obtain that such a stationary solution is unique and stable, if the value of In its turn, if the determinant of this system is zero, one obtains positive stationary solutions the constant N in (22) is fixed according to the initial data. that satisfy the detailedall balance condition, which is the solution of Equations (23) and (24) with If the determinant is not zero, then there are no stationary solutions satisfying the condition of the . By virtue of the H-theorem, we obtain that such a stationary solution is unique and stable, if detailed balance (except a trivial one when the vector ξ is zero). However, in this case, for each positive the value of the constant N in (22) is fixed according to the initial data. value of Nall in (22), we canall find the stationary solution (21). As soon as the system of (23) is solved, as a linearIf the system determinant for fixed is notξ1, thezero, value then of there the constant are no stationaryk is uniquely solutions determined satisfying by the the relationship condition of (24) thethrough detailed the balance value ξ(except1, and soa trivial the value one ξwhen1 is calculated the vector according is zero to). (22).However, in this case, for each positive value of in (22), we can find the stationary solution (21). As soon as the system of (23) is solved, as a linear system for fixed , the value of the constant k is uniquely determined by the relationship (24) through the value , and so the value is calculated according to (22). For Equation (14), the condition of semidetailed balance coincides with the condition of the detailed balance and has the form:

a b,a ,a  b,a ,a a ,a ,a  K 1 2 3 a  b,a ,a   K 2 3 1 2 3 b,a ,a a ,a ,a , b,a2 ,a3 a1 ,a2 ,a3  1 2 3 a1 b,a2 ,a3  2 3 1 2 3

a1 ,a2 b,a3  a1 ,b,a3 a1 ,a2 ,a3  Ka ,b,a a ,a ,a a1,a2  b,a3   Ka ,a b,a  a1,b,a3 a1,a2 ,a3 , 1 3 1 2 3 1 2 3 (25) a ,a ,a b a ,a ,ba ,a ,a  K 1 2 3 a ,a ,a  b  K 1 2 1 2 3 a ,a ,ba ,a ,a , a1 ,a2 ,ba1 ,a2 ,a3  1 2 3 a1 ,a2 ,a3 b 1 2 1 2 3

a1,a2 ,a3 In,n 1,2,,b 1,2,

In general, for equations of the type of coagulation–fragmentation equations, it is typical that the condition of semidetailed balance coincides with the condition of detailed balance. In the simplest models in Section 3.1, it is not yet clear what to take as a section, and it can be considered that the conditions of detailed or semidetailed balances are valid. Thus, we first need to consider the natural shapes of the original particles such as in Figure 3 and sections for them resulting from the physical models. Although we do not know how many shapes and how many ways there are of obtaining each of them from others, it is possible to prove that the condition of detailed balance for a specific physical model of the sections is fulfilled. First, we consider an anisotropic original particle; for these, the validity of condition of detailed balance is proven. Then,

Physics 2019, 1 FOR PEER REVIEW 8

n,a 1,1,1,1 n 1,a  ei , (8)

N1t  N1,1,1,1,t . The system of equations is:

1,1,1,1n,a Thus, here, K is the cross-section (frequency function, reaction rate coefficient) of such n1,aei  coalescence of the original particle 1,1,1,1 with the shape , at which increases by the 1,1,1,1n1,ae    i value ei , so the shape  n 1,a  ei  is obtained; Kn,a is the frequency of fragmentation of the shape into the original particle and the shape n 1,a  ei  . If for each nN and every a,b Jn , we do not distinguish the shapes and

n,b Physics, then 2019Equations, 1 (9) and (10) become the Becker–Döring case, Equation (2). However, our task 241 is to distinguish the aggregates consisting of the original particles not only in mass, but also in shape. In theFor second Equation simplest (14), the mo conditiondel, we of assume semidetailed that, balance on the coincides contrary, with only the rectangul conditionar of the parallelepipedsdetailed balance can coalesce and has the with form: their identical faces, so that the result is a rectangular parallelepiped again. Thus(a1+b, ,aclusters2,a3) without voids are obtained,(b,a2,a3)(a1,a 2i.e.,a3), Equations (1) and (2) become K ξ(a1 + b, a2, a3) = K ξ(b, a2, a3)ξ(a1, a2, a3), equalities. In the 3-dimensional(b,a2,a3)(a1 ,casea2,a3), we have: (a1+b,a2,a3) (a1,a2+b,a3) (a1,b,a3)(a1,a2,a3) K ξ(a1, a2 + b, a3) = K ξ(a1, b, a3)ξ(a1, a2, a3), (a1,b,a3)(a1,a2,a3) (a1,a2+b,a3) (25) (a1,a2,a3+b) n  a1  a2  a(3a1. ,a2,b)(a1,a2,a3) (11) K ξ(a1, a2, a3 + b) = K ξ(a1, a2, b)ξ(a1, a2, a3), (a1,a2,b)(a1,a2,a3) (a1,a2,a3+b) For each , we define a( afinite, a , a set) I(n), nas= the1, 2, set... , ofb = all1, 2, triples... of natural numbers 1 2 3 ∈ a ,a ,a  satisfying the condition (11). Thus, due to the condition (11) in this model, a shape is 1 2 3 In general, for equations of the type of coagulation–fragmentation equations, it is typical that the condition of semidetailed balance coincides with the condition of detailed balance. In the simplest models in Section 3.1, it is not yet clear what to take as a section, and it can be considered that the conditions of detailed or semidetailed balances are valid. Thus, we first need to consider the natural shapes of the original particles such as in Figure3 and sections for them resulting from the physical models. Although we do not know how many shapes and how many ways there are of obtaining each of them from others, it is possible to prove that the condition of detailed balance for a specific physical model of the sections is fulfilled. First, we consider an anisotropic original particle; for these, the validity of condition of detailed balance is proven. Then, we do a transition to the isotropic case. We prove that the condition of detailed balance is maintained. This gives a method of obtaining kinetic equations describing the evolution of clusters of different masses and shapes. After that, we prove the H-theorem under the condition of semidetailed balance (18) and (19). Then, the H-theorem is obtained under the condition of detailed balance, since Equations (18) and (19) are a consequence of the condition of Equation (17). For the second model, we consider the H-theorem under the condition of detailed balance (25), which coincides with the condition of semidetailed balance.

3.3. The Condition of Detailed Balance, the Arrhenius Law and the Model of the Evolution of Clusters Diﬀering in Shape The condition of detailed balance for the general system of physicochemical kinetics Equation (3) is: β Kαξα = K ξβ, (α, β) . (26) β α ∈ = If we (formally) put: 1 Ei ξi = exp , (27) Z −kBT Equation (26) can be rewritten in the following form: ! α β α β (α β, E) Kβ = KαZ| |−| | exp − , (α, β) , (28) kBT ∈ =

where E = (E1, E2, ... , En), Ei is the energy of the state Si participating in reactions of the form (4), kB is n P Ei the Boltzmann constant, T is the temperature of the medium, and Z = exp k T is the canonical i=1 − B partition function. Equation (27) is the Boltzmann equation for the probability of a given state. We connect the Boltzmann equation with the principle of the detailed balance and obtain that as soon as the principle of detailed equilibrium is satisﬁed, then the reaction constants can be written in the form: ! α β (α β, E) α | |−|2 | Kβ = AαβZ exp − , (α, β) , (29) 2kBT ∈ =

where Aαβ = Aβα are symmetric coefficients. In 1889, in “On the reaction rate of the inversion of cane sugar under the influence of acids”, Svante Arrhenius posited the relationship of the dependence of the rate constant of the chemical reaction on temperature [31], which today is called the Arrhenius equation or Arrhenius law (actually Physics 2019, 1 242 originally proposed by Vant Goff)[32,33]. The Arrhenius law for the system of equations of chemical kinetics (3) has the form:  Eα  α α  β  Kβ = Aβ exp , (α, β) . (30) −kBT  ∈ = α Here, Eβ is the activation energy of the reaction in Equation (4). The activation energy is the energy of molecules in a collision, the level of which is sufficient to carry out the chemical reaction of Equation α α β (4). Here, as is shown below, the coefficients Aβ are asymmetric: Aβ , Aα, unlike Aαβ = Aβα in Equation (29). From (30), we obtain that, for the equilibrium, the following relations for the reaction rate constants hold true: α α  α β  Kβ Aβ  Eβ Eα  = exp − , (α, β) . (31) β β  k T  Kα Aα − B  ∈ = Note that (31) is a more general condition than (30) in cases, where the rate constants of all reactions are given. Let for all (α, β) : ∈ = β Eα E = (β α, E), (32) β − α − i.e., the activation energy is an additive quantity. Then, relationship (31) takes the form: α α ! Kβ Aβ (α β, E) = exp − , (α, β) . (33) β β k T Kα Aα B ∈ =

α β α β Comparing Equations (28) and. (33), we obtain that Aβ/Aα = Z| |−| |. Generally, one can explain the relationship (32) and follow more closely the concept of the activation energy as soon as the concept of the so-called energy diagrams of reactions is used. α If the activation energy of reaction in Equation (4) Eβ EA (see Figure6), then the activation β ≡ energy of the inverse reaction is E = Eα Eβ + E , where Eα = (α,E) is the energy of the initial α − A complex α and Eβ = (β, E) is the energy of the ﬁnal product β. Therefore, relationship (32) is satisﬁed: α β E E = Eβ Eα = (β α, E) ∆H. This is the variation in enthalpy as a result of the reaction (see β − α − − ≡ Figure6).

Theorem 1. Let relationship (33) hold for all (α, β) . Let, then, that for the system: ∈ =

dN 1 X β i = (β α ) AαNα A Nβ , i = 1, 2, ... , n, (34) dt 2 i − i β − α (α,β) ∈=

α α β β there is a positive stationary solution η that satisﬁes the condition of detailed balance: Aβη = Aαη , (α, β) . ∈ = Then: (a) For Equation (3),

dN 1 X β i = (β α ) KαNα K Nβ , i = 1, 2, ... , n, dt 2 i − i β − α (α,β) ∈= Ei there is a stationary solution ξ: ξi = ηi exp , satisfying the detailed balance condition (26); − kBT (b) The H-function for Equation (3) diﬀers from the H-function for Equation (34):

n ! X Ni H(N) = Ni ln 1 , ηi − i=1 Physics 2019, 1 243

n P Ei by the amount k T Ni, i.e., has the form: i=1 B

n ! n ! n X Ni X Ni X Ei H(N) = Ni ln 1 = Ni ln 1 + Ni. (35) ξi − ηi − kBT i=1 i=1 i=1

The H-function (35) does not increase by the solution of Equation (3): dH 0. All stationary solutions of dt ≤ Equation (3) satisfy equalities of Equation (26); Physics 2019, 1 FOR PEER REVIEW P 16 (c) Equation (3) has n r conservation laws of the form µkN (t) = Ak = const (k = 1, ... , n r), where r − i i − is theFrom dimension (30), ofwe the obtain linear envelope that, for of thethe vectors equilibrium,α β (Boltzmann–Orlov–Moser–Bruno the following relations for the vectors reaction [16,26 ,rate27]), k P k− constantsand the vectors hold true:µ are orthogonal to all α β: µ (αi βi) = 0. The stationary solution of Equation (3) is − i − unique, all the constants Ak are ﬁxed, and it is given by: K α Aα  Eα  Eβ  β β exp  β α   β  Xβ  , α,β X  . (31)  k k  Ei k k N =Kξα expAα µ λ  kηBTexp + µ λ , (36) 0i i  i  ≡ i −k T i  B k k Note that (31) is a more general condition than (30) in cases where the rate constants of all reactionswhere λk are determinedgiven. by the Ak values; Let(d) Suchfor all a stationaryα,β   solution: exists if Ak is determined by the initial condition—a vector with non-negative P components N(0): Ak = µkN (0). A solution N(t) with this initial condition exists for all t > 0, is unique, i i α β and tends to the stationary solution (36). Eβ  Eα  β  α,E , (32)

i.e.,Proof. the TheactivationH-theorem energy for is the an chemicaladditive qu kineticsantity. equations Then, relationship with a finite (31) number takes the of reactionsform: under the condition of detailed balance wasα consideredα in [6,7]. By virtue of it, the Theorem 1 is obvious. Kβ Aβ  α  β,E   exp  , α,β   . (33) β β   β Kα Aα kBT α α β From (31) and the detailed balance conditions, Aβη = Aαη , we obtain the basic equation of thermodynamics: α β α  β Comparing Equations (28) and. (33), we obtain that Aβ Aα  Z . α Kβ Generally, one can explain= theα relationshipβ (β (32)α and) = (followβ α more) closely(β α the) concept of the kBT ln β Eβ Eα T , s , E T , s , activation energy as− soon asK αthe concept− of the− so-called− energy− diagrams− of reactions− is used. α If the activation energy of reaction in Equation (4) Eβ  EA (see Figure 6), then the activation where s = kB(ln η1, ln η2, ... , ln ηn). (β α, s) is entropy change as a result of the reaction. β − energyNow of the we inverse consider reaction the frequency is Eα  functionsEα  Eβ for E ourA , where case, analyze Eα  themα,E in is terms the energy of relationships of the initial (30) and (32), and check the fulfillment of the condition of detailed balance for them. complex α and Eβ  β,E is the energy of the final product β . Therefore, relationship (32) is Frequency functions in general form are unknown, but according to numerous observations, in α β satisfied:the case ofE gasesβ  E (vapors)α  Eβ  theyEα  lookβ  likeα,E (40) andН (41). This below, is the see variation [23]. In in this enthalpy section, as we a showresult that of the the reactioncondition (see of Figure detailed 6). balance is satisfied for such sections.

Figure 6. The energy. diagram of the reaction: EA is the activation energy of reaction, Eα = (α,E) is the energy of the initial complex α, Eβ = (β, E) is the energy of the ﬁnal product β, and ∆H Eβ Eα = ≡ − (β α, E) is the variation in enthalpy as a result of the reaction. Figure− 6. The energy diagram of the reaction: EA is the activation energy of reaction, Eα  α,E

is the energy of the initial complex α , Eβ  β,E is the energy of the final product β ,

Н EEβα  βαE  , is the variation in enthalpy as a result of the reaction.

Theorem 1. Let relationship (33) hold for all α,β   . Let, then, that for the system:

dNi 1 α α β β  i i Aβ N  Aα N , i 1,2,, n , (34) dt 2 α,β 

α α β β there is a positive stationary solution η that satisfies the condition of detailed balance: Aβ η  Aα η , α,β   .

Physics 2019, 1 244

In addition, we explain how the frequency functions are obtained from theoretical considerations. There are two main theories of molecular kinetics, namely, the theory of active collisions and the theory of an activated complex [32–37]. According to the first of these, so-called active collisions lead to chemical reactions—collisions of those particles for which the value of the energy of relative motion along the line of the centers of the colliding particles exceeds a certain value, the activation energy. The basic equations of the theory of active collisions were obtained by Max Trautz in 1916 [34] and William Lewis in 1918 [35]. A comparison of the total number of collisions and the number of active collisions shows that a large number of inactive collisions occur between the acts of reactions that require activation energy. Therefore, it is assumed that the particle velocity distribution functions are equilibrium or Maxwellian, and chemical transformations do not violate this statistical equilibrium. Then, within the framework of the theory of active collisions, the equations of chemical kinetics (for the case of pairing interactions) can be obtained from the Boltzmann equation [10] as in the transition to hydrodynamics in the case of a zero order approximation in the Chapman–Enskog method (transition to Euler equations) [6]. Indeed, we present the collision integral as a sum of all elastic (inactive) collisions and the integral of inelastic (active) collisions. Then, substituting the Maxwell distribution into such equations (written for each i-th type of particles), integrals of the first type vanish, and the second (after integrating both parts of the equation over the space of velocities) gives the right-hand side of the i-th equation of the system of (3). As a result, in spatially homogeneous cases, one obtains equations of (3). For simplicity, we restrict ourselves to the study of modeling the evolution of clusters, which differ in shape, for the Becker–Döring case, i.e., when only the original particles coalesce with and fragmentize from the aggregate. Thus, in our problem, the cross-section for coalescence is a shape with the original particle that can be modeled using the theory of active collisions (see below). When considering the fragmentation, one has to apply the theory of an activated complex: the original particles fragmentize from the aggregates not because they are knocked out by environmental particles by direct collision, but because the entire aggregate is in a disturbed state due to collisions with particles of the medium and emits those initial particles off their surface, the energy of which is greater than the activation energy of the process, the binding energy with the rest of the aggregate particles. The surface of an aggregate consisting of n original particles defines a set of “seats” for coalescence with a single original particle, which results in the aggregate consisting of the (n + 1) original particles. Let us introduce the notation for the combinatorial numbers, which arise in the consideration of shapes and transitions between them as a result of coalescence with or departure of a single original (1)X particle. Let a shape Y be obtained from a shape X by the addition of one original particle. Let CY denote the number of ways to obtain shape Y from shape X (the number of “seats” of shape X, which after coalescence with a single original particle gives shape Y). Here, the superscript “(1)” denotes the original particle. We denote a number of ways of obtaining X from Y in the result of the fragmentation of a single original particle from Y through CY . (1)X ( )( ) ( ) ( )( ) ( ) For example, from Figure1, we can see that C 1 1 = C 2 = 6 for d = 3 and C 1 1 = C 2 = 4 (2) (1)(1) (2) (1)(1) for d = 2, where“(2)” denotes dimmer, because a cube has six faces, i.e., six “seats”, and a square has four sides, i.e., four “seats” in the 2-dimensional case. ( )X Generally speaking, C 1 CY . Indeed, as can be seen from Figure1, for the reaction of Y , (1)X formation of one (left) of the shapes of aggregates, consisting of three original particles, as a result of coalescence an original particle with an aggregate of the two original particles, both these numbers are ( )X equal to two, and for the reaction of formation of the other (right) shape CY = 2, and C 1 = 8 for (1)X Y = (1)X = = d 3 (CY 4 for d 2). Let us consider the case of anisotropic particles. First, let us define what the anisotropy means using an example where the original particles are cubes, and then get a common definition of the Physics 2019, 1 245 anisotropic particles. Anisotropic shapes can be represented, for example, by means of the following mental model. We can represent anisotropic shapes as particles consisting not of cubes but of rectangular parallelepipeds, whose length, width, and height are different and the opposite faces are numbered “1” and “2”. The point is that all faces of original particles are different. These particles coalesce with a single original particle only so that faces of integrating original particles are the same by size and have different numbers: a face numbered “1” can merge with a face with the same size numbered “2” (or “2” ( )X vs. “1”). In this case, one obtains: C 1 = CY = 1. Y (1)X ( )X For shapes, consisting of such original particles, C 1 and CY are equal to 1. Here, we notice a Y (1)X ( )X feature where, for straight chains of molecules, they are equal to 2, while the relationship C 1 = CY Y (1)X is always satisfied. This gives a generalization of the definition of anisotropy from the case of cubes to ( )X the arbitrary shapes of the original particles. The definition is simple: C 1 = CY = 1. Y (1)X For the coalescence of an aggregate, consisting of n original particles, with a single original particle that results into the aggregate consisting of the (n + 1) original particles, each “seat” corresponds to an effective section of interaction. As a result, an original particle occupies the “seat” if its energy is sufficient for such a reaction. We use the assumption that all such sections can be considered identical, i.e., independent of the “seat”. We denote them through σ. Let us calculate the number of collisions of the “seat” with the flow of original particles falling onto the aggregates at a relative speed and consider the velocity distributions to be Maxwell distribution. Then, the corresponding average number of collisions of the original particles of mass m with aggregate 1/2 of n original particles is equal to σN u , where u 8kBT is the average of the absolute value of 1h| |i h| |i ≡ πµ the relative velocity of original particles and aggregates, µ = µ(n) m n is their reduced mass, and ≡ n+1 here N1 is the concentration, but not the number of original particles. Therefore, if all the collisions result in the attachment of the original particle to the aggregate, then the rate constant for the reaction of coalescence of a shape X with an original particle, which results in a shape Y, is equal to (1)X C σ u , (37) Y h| |i since the number of “seats” of shape X, resulting in shape Y, as soon as the original particle is attached, (1)X is equal to CY . According to the theory of active collisions, it is believed that only those original particles for which the kinetic energy of relative motion along the line of centers of the colliding particles is greater than (1)X the activation energy E1 enter the reaction. Therefore, to calculate the rate constant KY coalescence of shape X with the original particle, we obtain the value (37) multiplied by some function of E1/(kBT). This function has the form: g(E /(kBT)) exp( E /(kBT)), (38) 1 − 1 where g is a certain function, g(0) = 1, and significant dependence upon E1/(kBT) is included in the second factor. Therefore, by virtue of Equations (37) and (38), we have:

(1)X (1)X K = C A(n) exp( E (n)/(kBT)). (39) Y Y − 1

The activation energy E1 may depend on the mass of an aggregate, E1 = E1(n), but we suppose that there is no dependence on the shape. Note that A(n) σ u g(E /(kBT)) depends on n even if ≡ h| |i 1 E1 = const, since u depends on the µ = µ(n). h| |i (1)X Therefore, calculating the coalescence cross-sections KY , we restrict ourselves to the case of gases and the Maxwell–Smoluchowski model, since there are still no well-founded models for liquids. According to Maxwell, the collision cross-sections (for gases) are proportional to the surface area, and (1)X therefore a multiplicative coeﬃcient CY appears in Equations (37) and (39). Smoluchowski accepted that if only the fraction of molecules falling on a particle leads to their coalescence with the particle, Physics 2019, 1 246 then the rate of coalescence must also change the factor (38) [17]. From the point of view of the theory of active collisions, this factor returns us to the Arrhenius law in the relationship (39). If we neglect the diﬀerence of the function g from one (due to the fact that g(0) = 1), then only exp( E /(kBT)) in Equation (39) determines the fraction of active collisions. We assume that only the − 1 exponential factors in (30) determine the portion of active collisions. Then, Equation (34) describes the case with each collision leading to a reaction. The frequency of fragmentation of shape Y to the original particle and shape X is equal to [23]:

CY X(1)X Y kBT h i kBT Y K = χ exp E /kBT = χ C exp[ E /kBT], (40) (1)X h − j,XY h (1)X − XY j=1 where h is the Planck’s constant, χ is the correction factor, indices j = 1, 2, ... , CY number original (1)X particles, constituting shape Y, the fragmentation of each of which gives shape X, and Ej,XY is the activation energy of separation of the j-th original particles of such type. In the second equality in (40), we have taken into account that Ej,XY does not depend on j: Ej,XY = EXY is the activation energy for the fragmentation of the original particle from shape Y, a result of which is shape X. The theoretical substantiation of the expression of (40), as already noted, is given by the theory of the activated complex [32,33,36,37]. At the same time, the activation energy of fragmentation EXY is advisable to search in the form of the sum of activation energies required to break the connection with each of the KXY nearest neighbors (close-range interaction): EXY,k, k = 1, 2, ... , KXY, and overcome the attraction to all the original particles constituting the cluster (long-range interaction), the energy of which is E(n):

KXXY EXY = EXY,k + E(n). (41) k=1

E(n) is sought in the form: n 1/3 o E(n) = E 1 α / n + α0 , ∞ − ∞ which allows an extrapolation to the Gibbs–Thomson equation [23]. This equation forms the basis of the classical theory of nucleation. Here, E is the activation energy of overcoming the attraction to an ∞ infinitely large particle, and α and α0 are the coefficients describing the decrease in attraction due to ∞ the decrease of n. As is shown below, the form of the long-range interaction is completely irrelevant. So far, we considered a pair of arbitrary shapes, one obtained from another by coalescence or fragmentation of one original particle, and termed them X and Y. Now, consider a single arbitrary shape, which, in particular, may coincide with X or Y, and term it Z. Consider in a shape Z for each original particle, its component, the sum of the activation energies necessary for breaking the connection with each of the nearest neighbors. Let us sum them over all the original particles of shape Z and divide the obtained value by two, so that there are no duplicates in the summation of energies. Then, the energy calculated in this way can be called the energy of the close-range interaction of shape Z, EZ. Thus, for every shape Z, the energy of close-range interaction EZ is determined. Then, the first term in (41) is the difference between the energies of the close-range interaction of shapes Y and X. The second term, E(n), can be interpreted as the increment of energy of the long-range interaction in the transition from the shape, consisting of n original particles, to the shape consisting of (n + 1) original particles. One may not separate the energies of the close-range and long-range interactions. This corresponds to the fact that in Equation (41), the neighbors closest to the considered original particle of the cluster, consisting of n original particles, are all n 1 of its other original particles, i.e., K = n 1, and − XY − E(n) 0. ≡ Physics 2019, 1 247

Thus, generally speaking, for each shape Z we attribute the energy of the close-range interaction EZ, for which the following condition of the additivity is fulﬁlled:

E EX = E . (42) Y − XY Then, by virtue of Equations (40) and (42), we obtain that:

E E Y = Y Y X K(1)X C(1)XB exp − , (43) − kBT

kBT where B χ h = const. ≡ ( )X In virtue of Equations (39) and (43) for equilibrium constants K 1 /KY , the following form of Y (1)X relations (33) is valid: (1)X (1)X K C A(n) E E + E Y = Y exp 1 X Y . Y Y − − K C B kBT (1)X (1)X

If E1 = E1(n), then it is impossible to talk about the fulfillment of additivity Equation (32), and Theorem 1 cannot be directly applied. However, dependence E1(n) can be made in the pre-exponential factor A(n). As such, Theorem 1 is applicable, and the only thing that remains to be checked is the ( )X C 1 A(n) implementation of the detailed balance for the equilibrium constants Y , corresponding to the CY B (1)X ( )X equilibrium constants K 1 /KY . Y (1)X The condition of detailed balance for some model of shapes is satisfied if all possible relations are similar to Equation (20) for this model. We must prove that these relations are valid for any two (different) paths along the edges of the graph from the shapes (as in Figure3) such that their beginnings 2 3 r 1 Physics 2019, 1 FORand PEER their REVIEW ends coincide: X1 X2 X3 ... Xr 1 Xr and X1 X X ...22 X − Xr . → → → → − → → → → → → The number of fractions on the right and left sides of such relations is the same and equal to the 1X Y i.e., r 1. di Forfference anisotropic in the number original of particles original particles ( CY  inCX1rXand) inX the1, i.e., right(r and1). For left anisotropic parts of such original particles ( )X − (C 1 = CY ) in the right and left parts of such relations, there are identical expressions, and therefore relations, thereY are identical(1)X expressions, and therefore the condition of detailed balance is satisfied, even if B inthe (43 condition) depends of on detailed n . For balance isotropic is satisfied, particles, even it is if Btoin show (43) dependsthat the following on n. For isotropicrelations particles, it is are valid: to show that the following relations are valid:

1 X 1X 1 X 1 X 1 X 2 1 X 2r1 r 1   1 2 (1)X1  (1)rX12  (1)1Xr 1   (1)X1 (1)X (1)X − CX CX C CCX CC2 −C 3 C CX C 2 3 X2 Xr 3 X Xr X X2 X3r Xr    2  3  . X 2 X 3 X r ... X =X X r ... .(44) (44) X2 X3 Xr X2 X3 Xr C1X C1X C· 1X · C· C 2 ·C r1· · 1 2 C C r1 C1X1 1X C C 1X 2 C r 1 (1)X1 (1)X2 (1)Xr 1 (1)X1 (1)X (1)X − −

For each isotropicFor each shape isotropic X , shapewe considerX, we consider the set of the all set different of all diﬀ anisotropicerent anisotropic forms forms that thatgive give shape X. shape . WeWe denote denote the the cardinality cardinality of of this this set set as as M(X) .. Consider twoConsider isotropic two isotropicshapes: shapes: andX andY , Y where, where Y is is obtained obtained from fromX by the attachment by the of original (1)X particles (with C ways, and1X X is obtained from Y by fragmentation of an original particle by attachment of original particlesY (with C ways, and is obtained from by fragmentation CY Y (X) (Y) ( ) ways). ConsiderY a graph consisting of a set of vertices and all edges (reactions) of an original1 X particle by C ways). Consider a graph consisting of ∪ a set of vertices connecting vertices from1X a set (X) with vertices from (Y). The number of such edges, emanating ( )X XY from and the verticesall edges belonging (reactions) to connecting(X), is vertices(X) C 1from, and a set this numberX  with is vertices(Y) C Yfromfor the edges MM    Y M (1)X MY  . Thewhich number emanate of such from edges the vertices, emanating belonging from to the(Y ). vertices Therefore: belonging to , is 1X Y X C Y C ( ) M  Y , and this number is M  1X for( the) edges1 X = which( ) Yemanate from the vertices X CY Y C(1)X. (45) belonging to . Therefore: Note that it is namely due to the fact of the validity of the relations of the form of (45) that 1X Y the system of kinetic equationsMX C forY isotropic MY  shapesC1X . can be derived from the evolution(45) equations for

anisotropic shapes. For this, for each isotropic shape X all the anisotropic shapes from the set (X) Note that it is namely due to the fact of the validity of the relations of the form of (45) that the are identiﬁed, and the sum of all their concentrations is the concentration of shape X. Then, by virtue system of kinetic equations for isotropic shapes can be derived from the evolution equations for anisotropic shapes. For this, for each isotropic shape all the anisotropic shapes from the set are identified, and the sum of all their concentrations is the concentration of shape . Then, by virtue of (45) equations for isotropic forms with correct cross-sections (containing the correct values of the coefficients , ) are obtained from the evolution equations for anisotropic forms. In virtue of the relationship (45), Equation (44) is equivalent to:

2 r 1 MX  MX  MX  MX  MX  MX  1  2  r 1  1   2 3 , MX 2  MX 3  MX r  MX  MX  MX r 

which actually seems to be obvious. This means that the condition of detailed balance is valid for the kinetic equations of evolution of both anisotropic and isotropic shapes from the original particles. Therefore, the following is proven. Theorem 2. Let the Becker–Döring case be considered, and the frequency functions are defined by Equations (39) and (43), being customary for gases (i.e., calculated according to the theory of active collisions and the theory of an activated complex), namely:

1X 1X KY  CY Anexp E1n kBT,

Y Y  EY  EX  (46) K1X  C1X Bnexp   .  kBT 

Let the assumption of independence of values of the cross-sections from the “seats” formulated above be valid. Then the conditions of detailed balance and the H-theorem are fulfilled.

3.4. The Linear Conservation Laws and the Convergence to the Equilibrium

Physics 2019, 1 FOR PEER REVIEW 7

3. Results

3.1. The Simplest Models for the Evolution of Clusters Differing in Shape Physics 2019, 1 FOR PEER REVIEW We call the original particles constituting the cluster as adjacent8 if they have a common face. We assume that a cube attaches to an aggregate or to another original particle only in such a way Consider the evolution of the concentrations (or numbers) of the clusters having the shape that it becomes adjacent at least to one other original particle. n,a: Nn,a,t , t 0,. WithoutIn lossthe firstof generality, simplest modelwe assume we restrict here that ourselves d  3 .to the investigation of modeling the evolution We need to write a system ofof equations clusters, differing of the type in shape of the, physico for the chemical Becker–Döring kinetics case, with i.e. all, when only the original particles interactions of the form: coalesce with and fragmentize from the cluster. Each aggregate is characterized by the number n of constituting original particles. Moreover, n,a 1,1,1,1 n 1,a  ei , (8) Physics 2019, 1 FOR PEER REVIEW for each cluster, we determine the 22 minimum rectangular parallelepiped in which this cluster is

placed so that the faces of the cubes, its components, are parallel to the corresponding faces of this where nN , i  0,1,21,X3, e0 Y 0,0,0 is the zero vector, and e j ( j 1,2,3) is a vector with i.e., r 1 . For anisotropic original particles ( C  C ) in the right and left parts of such   Y 1Xparallelepiped. Such rectangular parallelepipeds3 are characterized by length, width, and height, i.e., one in j -th place and zeros in the others. Here the vectors aN are such that n,3aG and relations, there are identical expressions, and therefore the conditionthe numbers of detai a1led, a balance2 , a3 , definingis satisfied, the vector aN , and we consider permutations of triplets of even if B in (43) depends onn  1n,a. For e iisotropicG , and particles, in any otherit is to cases show, thethat coefficients the following of therelations direct and reverse reactions are Physics 2019, 1 numbers a1,a2,a3  as indistinguishable vectors. It is obvious that: 248 are valid: assumed to be zero. If i  0 , then a does not vary, but n is increased by one in all four cases:

i  0,1,2,3. 2 r1 n  a  a  a . (6) 1X1 1X 2 1X r1 1X1 1X 1X 1 2 3 C C C C 2 C 3 C X 2 X 3Forof (45) brevity, equationsX r we introduce forX isotropic Xthe forms notation withX r for correct the cross-sections number of individual (containing original the correct particles: values of the   ( ) 2  3  . (44) X 2 X 3 X r 1 X XY X It makeXsr sense to consider not only the 3-dimensional aggregates of the original particles, but C C coeﬃcientsC C , C ) are obtainedC fromr the1 evolution equations for anisotropic forms. 1X1 N1 1tX2 N1,1,11,1Xr,1tY . C1(1X)X C 2 1X 1 1alsoX 2-dimensional ones, since the growth of clusters on the surface and the growth of films are of

In virtue of the relationship (45), Equation (44) is equivalent to: The system of equations is: interest. Then, instead of cubes, we have squares that are connected to each other so that their edges For each isotropic shape X , we consider the set of all different anisotropic forms that give 3 coincide, and the analogue of condition ( 6) has the rform: dN t 1, 1,1,1 1, 1,1,1 2, 1,1,1 e 2 1 shape . We denote the cardinality of this1 set as M(X1) .  (X2)  2 (Xr 1)   (Xi 1 ) X X −  2K N1  t−  K N 2 , 1,1,1 ei ,t   2,  1,1,1ei  ... = 1, 1,1,1  1, 1,1,1  ...  , dt 2 3 n  a  a . (7) Consider two isotropic shapes: and Y i,( 1X where2) · (X3) is· obtained· (Xr) from (X )by · the(X ) · · (X1 r) 2

(9) 1X 3 1, 1,1,1n ,a n,1 ae attachment of original particles (with CY ways, and is obtainedIn the 1from-dimensional by fragmentat case, however,i ion we obtain that n  a1 , which returns us to the system of which actually seemsK ton, be1 ae obvious. NtNntK1    ,a ,  Nn,  1 a  ei ,t , Y     i  1, 1,1,1n ,a  n20a J n i  d of an original particle by C1X ways).This means Consider that the a condition graphEquations consisting of detailed (6) and of balance( 7). a Therefore set is of valid vertices, we for consider the kinetic a equationsN , where of evolution d  2,3 of. Figure 2 shows an example a MM XY   and all edges (reactions)both anisotropic connecting and vertices isotropicof from ashapesdN vector a set n,, froma M t. theX Note originalwith that vertices in particles. the from 1-dimensional case (d = 1), we get the chains consisting of the Therefore, the followingoriginal is proven. particles, Fand n the,,a tbreaking of one bond between two neighboring original particles in the MY  . The number of such edges, emanating from thechain vertices leadsdt to belonging its breaking to . This means, is the processes of agglomeration and fragmentation of chains 1X Theorem3 2.Y Let the Becker–Döring case be considered, and the frequency functions are defined by Equations (39) X C , and this number is Y C for1, 1,1,1 thenn  1,edgesa  eiibecome which essential emanat alonge from with the the  1, vertices 1,1,1 processes  1, a  e  of coalescence and fragmentation of individual original M Y M1XK N t N n 1,a  e , t  K N n, a ,t  and (43), being customarynn,,aa forpart gases1 icles (i.e., to calculated the chain. according iTheref ore, to the the theory 1-dimensional of active collisions case of and the theevolution theory of of clusters of different mass belonging to . Therefore: i0  (10) an activated complex), namely:and shape under the Becker–Döring constraint makes sense to consider only when the growth of 1, 1,1,1n ,a     n,1 aei   1XKY NtNntKchains  occurs ,a ,  on  the surfaces, Nn, and  then1 a  the e ,t system , of equations describing the evolution has the X C n,1 aeYi C 1 . 1, 1,1,1 n ,a (45) i M Y  ( )X M ( )X1X    E E 1 = 1 ( ) ( ( ) ( )) Y = Y ( ) Y X K C Aformn exp of (2).E n / kBT , K C B n exp − . (46) Y Y − 1 (1)X (1)X − k T Note that it is namely due to the fact of the validity of the relationsn2 ,of 3 ,the , aformJ of n ( .45) that the B system of kinetic equations for isotropic shapes can be derived from the evolution equations for Let the assumption1,1,1,1n,a of independence of values of the cross-sections from the “seats” formulated above be valid. Thus, here, K is the cross-section (frequency function, reaction rate coefficient) of such anisotropic shapes. For this, for eachThen isotropicthe conditionsn1,ae i shape of detailed all balance the anisotropic and the H-theorem shapes are from fulfilled. the set are identified, and coalescence the sum of of all the their original concentrati particleons  1 is, 1 the,1,1  concentration with the shape of shape , at. which increases by the 3.4. The Linear Conservation Laws and the Convergence to the1, Equilibrium1,1,1n1,ae  Then, by virtue of (45) equations for isotropic forms with correct  cross-sections (containing the i value ei , so the shape  n 1,a  ei  is obtained; Kn,a is the frequency of correct values of the coefficients The, consequence) are obtained of the Hfrom-theorem the evol forution the equations equations is for the convergence of solutions to the fragmentation of the shape into the original particle and the shape n 1,a  e . Boltzmann extremal [1,3], i.e., to the argument of the conditional minimum of the Hi -function, provided anisotropic forms. If that for the each constants nN ofand linear every conservation a,b Jn laws, we are do fixed. not distinguishThe role of linear the shape lawss remains and somewhat In virtue of the relationship (45), Equation (44) is equivalent to:Figure 2. The example of the vector , characterizing the shape of the particle consisting of n n,b ,mysterious, then Equations but they(9) and turn (1 out0) become to be regulators the Becker in–Döring various case, situations Equation [1,3 ,(2).6]. However, our task original2 particles.r Here1 , a  1,2,3, n  5 . MX1  isM  toX 2 distinguish MX ther 1  aggregatesMX1  consistingMX  of thMe  originalX  particles not only in mass, but also in  Lemma 1. The number (concentration)  of all original,particles is the only linear conservation law for the shape. 2 3 MX 2  MX 3 system inM EquationX r  (15):MX  MX  MX r  d In the second simplest modelFor, we each assume nN that,, we define on the a finite contrary, set J onlyn as rectangul the set ofar all vectors aN satisfying the XM X d  3 d  2 which actually seems toparallelepipeds be obvious. can coalesce withcondition their identical(6) when(n N faces(n, a,, t ))so((7 =) that constwhen the, result). Thusis a, arectangular shape is described by a pair n,a, where · d This means that the conditionparallelepiped of detailed again balance. Thus ,is clusters valid for without the, n =kinetica1 avoids(n )equations are obtained,. The of evolution set i.e. of, allEqua suchtions pairs (1) we and denote (2) become by . ∈ J  N G of both anisotropic and isotropicequalities. shapes In fromthe 3 -thedimensional original particles. case, we have: and for Equation (16): Therefore, the following is proven. M Xn  aX1  a2  a3 . (11) Theorem 2. Let the Becker–Döring case be considered, and the frequency functions are defined by (n N(a, t)) = const, · Equations (39) and (43), being customaryFor each for gases (i.e., ,we calculated define according a finite n to= set1 thea I (theoryn) as of the active set collisions of all triples of natural numbers ∈ and the theory of an activated complex), namely: a1,a2,ifa the3  conditionsatisfying of the detailed condition or semidetailed (11). Thus balance, due is to valid. the condition (11) in this model, a shape is 1X 1X KY  CY Anexp E1n kBT, Proof. We first consider the first of our models—the system in Equation (15). Reactions (8) can be Y Y  E  E  (46) Kwritten C in theBn formexp of (18):Y X . 1X 1X    kBT 

XM X XM X Let the assumption of independence of values of the cross-sections from the “seats”α formulated(n, a) above be βvalid.( n, a). (47) (n,a) ↔ (n,a) Then the conditions of detailed balance and the H-theorem are fulfilled.n= 1 a (n) n=1 a (n) ∈ ∈ 3.4. The Linear Conservation Laws and theThe Convergence linear functional to the Equilibrium for Equation (15) has the form:

XM X

IM = µ(n,a)N(n, a, t). (48) n=1 a (n) ∈ Physics 2019, 1 FOR PEER REVIEW 7

3. Results

Physics 2019, 1 FOR PEER REVIEW 3.1. The Simplest Models for the Evolution of Clusters Differing 7in Shape We call the original particles constituting the cluster as adjacent if they have a common face. 3. Results We assume that a cube attaches to an aggregate or to another original particle only in such a way that it becomes adjacent at least to one other original particle. 3.1. The Simplest Models for the Evolution of ClustersIn the Differing first simplest in Shape model we restrict ourselves to the investigation of modeling the evolution Physics 2019, 1 FOR PEER REVIEW 8 We call the original particles constitutingof clusters, the cluster differing as adjacentin shape ,if forthey the have Becker a common–Döring face. case, i.e., when only the original particles coalesce with and fragmentize from the cluster. Consider the evolution of the concentrationsWe assume (orthat numbers) a cube attaches of the clustersto an aggregate having torhe to shape another original particle only in such a way that it becomes adjacent at least to one other originalEach aggregate particle. is characterized by the number n of constituting original particles. Moreover, n,a: Nn,a,t , t 0,. Without loss of generality, we assume here that d  3 . In the first simplest model we restrictfor ourselves each cluster to ,the we investigation determine theof modeling minimum the rectangular evolution parallelepiped in which this cluster is We need to write a system of equations of the type of the physicochemical kinetics with all of clusters, differing in shape, for the placed Becker –soDöring that the case, faces i.e. of, whenthe cubes only, itsthe components, original particles are parallel to the corresponding faces of this interactions of the form: coalesce with and fragmentize from the clusterparallelepiped.. Such rectangular parallelepipeds are characterized by length, width, and height, i.e., 3 the numbers a , a , a3 , defining the vector aN , and we consider permutations of triplets of n,a 1,1,Each1,1 aggregaten 1,a is characterizedei , by the number( 8n) 1of constituting2 original particles. Moreover,

for each cluster , we determine the minimum rectangular parallelepiped in which this cluster is numbers a1,a2,a3  as indistinguishable vectors. It is obvious that: where nN , i  0,1,2,3, e0  0,0,0placed is the so zero that vector,the faces and of thee j cubes( j 1, ,its2,3 components,) is a vector are with parallel to the corresponding faces of this parallelepiped. Such rectangular3 parallelepipeds are characterized by length, widthn, and a height, a  a i.e.. , one in j -th place and zeros in the others. Here the vectors a are such that n,a  and 1 2 3 (6) N   G3

the numbers a1 , a2 , a3 , defining the vector aN , and we consider permutations of triplets of n 1,a  ei G , and in any other cases, the coefficients of the direct and reverseIt reactionsmakes sense are to consider not only the 3-dimensional aggregates of the original particles, but numbers a ,a ,a  as indistinguishable vectors. It is obvious that: assumed to be zero. If i  0 , then a does not vary1, but2 3n is increased by onealso in 2all-dimensional four cases: ones, since the growth of clusters on the surface and the growth of films are of interest. Then, instead of cubes, we have squares that are connected to each other so that their edges i  0,1,2,3. n  a  a  a coincide,1 and2 the3 .analogue of condition (6) has the form:(6)

For brevity, we introduce the notation for the number of individual original particles: It makes sense to consider not only the 3-dimensional aggregates of the original particles, but N1t  N1,1,1,1,t . n  a1  a2 . (7) The system of equations is: also 2-dimensional ones, since the growth of clusters on the surface and the growth of films are of interest. Then, instead of cubes, we have squaresIn the that 1- dimensionalare connected case to ,each however, other sowe that obtain their that edges n  a1 , which returns us to the system of dN t 3 d 1   1, 1,1,1 1, 1,1,1coincide, 2 and the 2, 1,1,1analogueei  of condition (E6quations) has the (form:6) and (7). Therefore, we consider aN , where d  2,3. Figure 2 shows an example  2K N1  t  K N 2 , 1,1,1 ei ,t   2, 1,1,1ei   1, 1,1,1 1, 1,1,1   dt i1 of a n vector a  a . . Note that in the 1-dimensional case( 7()d = 1), we get the chains consisting of the 1 2 (9)  3 original particl es, and the breaking of one bond between two neighboring original particles in the 1, 1,1,1n ,a n,1 aei  K NtNntK1   In the,a 1 ,-dimensional  case Nn,, however,  1 a chain ewei ,t  obtainleads , tothat its breakingn  a1 , which. This meansreturns the us processesto the system of agglomeration of and fragmentation of chains    n,1 aei  1, 1,1,1n ,a  n20a J n i  d Equations (6) and (7). Therefore, we considerbecome a essentialN , where along d with 2 ,the3. Figure processes 2 shows of coalesce an examplence and fragmentation of individual original dNof n a,, a vector t a . Note that in the 1-dimensionalparticles to case the ( chain.d = 1) ,Theref we getore, the the chains 1-dimensional consisting case of of the the evolution of clusters of different mass F n,,a t and shape under the Becker–Döring constraint makes sense to consider only when the growth of originaldt particles, and the breaking of one bond between two neighboring original particles in the chain leads to its breaking. This means thechains processes occurs of on agglomeration the surfaces, and fragmentation then the system of ofchains equation s describing the evolution has the 3 form of (2).  K1, 1,1,1nn 1,a  eii N tbecome N n  1,essentiala  e , t along  K 1,with 1,1,1 the  1, aprocesses  e  N n, a of ,t coalesce  nce and fragmentation of individual original   nn,,aa 1    i      i0  particles to the chain. Therefore, the 1-dimensional case (of10 the) evolution of clusters of different mass

1, 1,1,1n ,a and shape under the Becker–Döring constraint makes sense to consider only when the growth of     n,1 aei   K NtNntK1    ,a ,  Nn,  1 a  ei ,t ,  n,1 aei  chains occurs1, on 1,1,1  then ,a  surfaces, and then  the system of equations describing the evolution has the form of (2). n2 , 3 , ,a J n .

1,1,1,1n,a Thus, here, K is the cross-section (frequency function, reaction rate coefficient) of such n1,aei  coalescence of the original particle 1,1,1,1 with the shape , at which increases by the 1,1,1,1n1,ae  Figure 2. The example of the vector , characterizing the shape of the particle consisting of n   i value ei , so the shape  n 1,a  ei  is obtained; Kn,a is the frequency of Physics 2019, 1 original particles. Here, a  1,2,3, n  5 . 249 fragmentation of the shape into the original particle and the shape n 1,a  ei  . d If for each nN and every a,b Jn , we do not distinguish the shapeFors each nandN , we define a finite set Jn as the set of all vectors aN satisfying the It is preserved along the solutions of (15) if vector µ is orthogonal to all the n,b , then Equations (9) and (10) become theFigure Becker 2.– TheDöring example case, of Equation the vector (2). condition However,, characterizing (6 our) when task the shaped  3 of ((7 the) particlewhen d consisting 2 ). Thus of ,n a shape is described by a pair n,a , where Boltzmann–Orlov–Moser–Bruno vectors α β (see Section2 and Theorem 1c) [ 6,7,16,26,27]. Here, the   original particles. Here, a  1,2,3, n  5 . − d is to distinguish the aggregates consisting of the original particles not only in mass, buta alson in + = ( ) condition of orthogonality gives the relations, Jµ (n,a) Nµ(.1, The(1,1,1 set)) ofµ (alln+ 1,sucha+ei) pairs0, we where denoten, a by GM. shape. − ∈ and (n + 1, a + e ) M. All such relations lead to an arithmetic progression on n, and for any i ∈ d In the second simplest model, we For assumeshape each onethat,n gets:N on, we µ the define = contrary,n µ a finite only set, i.e., J rectangul then conservationas thear set of alllaw vectors of the numberaN of satisfying all original the particles, (n,a) · (1,(1,1,1)) parallelepipeds can coalesce with theiconditionr identicalM (6 ) faceswhen, sod  that3 ((7 the) when result d is 2a). rectangularThus, a shape is described by a pair n,a, where P P ( ( )) parallelepiped again. Thus, clusters without voids are obtained,n N nd, ai.e., t , ,E isqua thetions unique (1) and linear (2) conservation become law (up to a multiplicative constant). , n=a1a (n)· . The set of all such pairs we denote by . equalities. In the 3-dimensional case, we have: ∈ J N G Similarly, for Equation (16), for the conditions of orthogonality of vector µ to the n  a1  a2  a3 . (11) ( ) Boltzmann–Orlov–Moser–Bruno vectors, one obtains the following relations for any shape a , a , a 1 2 3 ∈ For each , we define a finite set I(n), nas= the1, 2, set... , ofM : all triples of natural numbers µ(a ,a ,a ) = µ(b,a ,a ) + µ(a b,a ,a ), a1,a2,a3  satisfying the condition (11). Thus, due to the condition (11) in this model1 2 ,3 a shape 2is3 1− 2 3 µ(a ,a ,a ) = µ(a ,b,a ) + µ(a ,a b,a ), 1 2 3 1 3 1 2− 3 µ(a ,a ,a ) = µ(a ,a ,b) + µ(a ,a ,a b), 1 2 3 1 2 1 2 3− where 1 b a1/2. From these by induction we get that µ(a ,a ,a ) = a1a2a3, i.e., the only linear ≤ ≤ 1 2 3 conservation law is the conservation law for the number of all original particles. It is clear that for more complex models than those represented by Equations (15) and (16), the uniqueness of the conservation law of the number of all original particles can be proven the same way.

3.5. The Boltzmann Principle for the Simplest Model of the Evolution of Clusters of Diﬀerent Masses and Shapes Consider the system of equations in (15). Let us show, for example, that the H-function:

M ! X X N(n, a, t) H = N(n, a, t) ln 1 , (49) ξ(n, a) − n=1 a (n) ∈ decreases by the nonstationary solutions of Equation (15) under the condition of detailed balance (3):

M dH = P P N(n,a,t) dN(n,a,t) dt ln ξ(n,a) dt n=1 a (n) ∈ 3 2 ! P N(2,(1,1,1)+e ,t) N (t) (1,(1,1,1))(1,(1,1,1)) N (t) N(2,(1,1,1)+e ,t) = ln i 2 ln 1 K ξ 2 1 i ξ(2,(1,1,1)+e ,t) ξ1 (2,(1,1,1)+e ) 1 ξ1 ξ(2,(1,1,1)+e ,t) i=0 i − i − i M 1 3 P P P N(n+1,a+e ,t) N (t) N(n,a,t) + − ln i ln 1 ln ξ(n+1,a+e ) ξ1 ξ(n,a) n=2 a (n) i=0 i − − ∈ (1,(1,1,1))(n,a) N1(t) N(n,a,t) N(n+1,a+ei,t) K ξ1ξ(n, a) ( ) ( + + ) 0. × (n+1,a+ei) ξ1 ξ n,a − ξ n 1,a ei ≤ The H-theorem for physicochemical kinetics equations with a ﬁnite number of reactions, under the condition of semidetailed balance was considered in [6,7,11]. Taking into account Lemma 1, we obtain the following formulation for the ﬁrst of our simplest models.

Theorem 3. Let the coeﬃcients for Equation (15) be such that there exists at least one positive solution ξ(n, a) of the system of Equations (18) and (19). Then: (a) H-function (49) decreases by the solutions of (15): dH 0. All stationary solutions of Equation (15) dt ≤ satisfy the equalities of Equations (18) and (19); PM P (b) Equation (15) has a unique linear integral (an invariant of the form IM = µ(n,a)N(n, a, t)), n=1 a (n) ∈ PM P and it is the conservation law of the number of all original particles: (n N(n, a, t)) Nall = const. n=1 a (n) · ≡ ∈ Physics 2019, 1 FOR PEER REVIEW 7

3. Results

numbers a1,a2,a3  as indistinguishable vectors. It is obvious that:

n  a1  a2  a3 . (6)

n  a1  a2 . (7)

Physics 2019, 1 250 Figure 2. The example of the vector , characterizing the shape of the particle consisting of n original particles. Here, a  1,2,3, n  5 . The stationary solution of the system of Equation (15) is unique if we ﬁx a constant Nall and is given by the d equationFor for eachthe Boltzmann nN , extremalswe define (the a argumentfinite set ofJ then minimum as the set of H-function, of all vectors provided aN that thesatisfying constants the of linear conservation laws are ﬁxed): condition (6) when d  3 ((7) when d  2 ). Thus, a shape is described by a pair n,a, where d , a Jn N . TheN Bset(n ,ofa) all= suchξ(n, a pairs) exp (weλn )denote, (n, aby) GM. , (50) ∀ ∈ where λ is uniquely determined by the value of Nall; (c) Such a stationary solution exists if the constant of the conservation law of the number of all initial particles is determined by the initial condition N(n, a, 0) 0 : (n, a) M . A solution N(n, a, t) 0 : (n, a) M ≥ ∈ ≥ ∈ with this initial condition exists for all t > 0, is unique, and tends to a stationary solution (50).

For the second of our simplest models, i.e,. for Equation (16), by Lemma 1, Theorem 3 is valid without changing, provided that there exists at least one positive solution of the system of Equation (25). Thus, we obtain what the arbitrary solution of Equations (15) and (16) tends toward as soon as time tends to inﬁnity. It is important to stress that this is deduced directly from the Boltzmann variational principle without solving the systems.

4. Conclusions The main results are as follows: (1) we proposed a new model (or even a group of models) describing the shape of crystals; (2) we clarified the conditions for the applicability of the H-theorem to this group of models; (3) it was found that the Arrhenius condition implies a detailed balance for the new proposed simple models and for the general case of models. Thus, we considered the issue of building correct (physically grounded) models of the evolution of cluster distribution functions for shapes based on the H-theorem (Boltzmann), which is being actively studied and is currently used in many mathematical problems of the natural sciences. We show that when building the models, it is necessary to monitor the fulfillment of the condition of detailed balance. For simplicity, in Section3, our study is limited to modeling the evolution of clusters that di ffer in shape for the Becker–Döring case, that is, when only the original particles coalesce with and are fragmented from the aggregate. It is interesting to generalize the result obtained (Theorem 2) to the case of the formation (and fragmentation) of aggregates consisting of agglomerates. The simplest models of evolution of clusters, differing by shapes, are proposed in Section 3.1. The significance of these models is that the form of equations is the same as in the case of more detailed models. It is of interest to generalize the results obtained to nonlinear systems with discrete time, in particular, even to construct discrete models of the Boltzmann equation with discrete time and to transfer the condition of semidetailed balance to discrete time. Consideration of the H-theorem for nonlinear systems with discrete time, in particular, even for the system of Becker–Döring equations, becomes an extremely important and urgent task, since computer modeling plays a key role in solving the fundamental problem of creating new materials [22,23]. In the linear case, when passing from continuous to discrete time, we have a transition from a Markov process to a Markov chain, and the H-theorem in this case is valid, as investigated earlier (see [8] and references therein). In the nonlinear case, it is valid in rare cases for explicit time discretization [8], and for the implicit case as investigated in [9]. These questions, as well as the results of the current work, clarify the point of constructing correct (physically grounded) kinetic models for the evolution of clusters differing in mass and other parameters. At the same time, both for experimenters and calculators who deal with problems of modeling the evolution of such structures (for example, the problem of their optimal synthesis [22,23,38]), is extremely important that the number of parameters (numbers or functions) remain minimal (we have A, B, and activation energy in (46)), since the addition of at least one parameter or correction factor significantly increases the computational complexity of the problem. Physics 2019, 1 251

Author Contributions: I.M. gave the general statement of the problem. All sections except Section 3.3 were written collectively during the authors’ discussions. V.V. linked the Boltzmann equation with the principle of detailed balance in Section 3.3. The remaining results in Section 3.3 were obtained by S.A. and J.B. Funding: This paper was supported by the Ministry of Education and Science of the Russian Federation on the program to improve the competitiveness of Peoples’ Friendship University of Russia (RUDN-University) “5–100” among the world’s leading research and educational centers in 2016–2020. Conﬂicts of Interest: The authors declare no conﬂicts of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

References

1. Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. In Kinetische Theorie II; Vieweg Verlag: Wiesbaden, Germany, 1970; pp. 115–125. [CrossRef] 2. Godunov, S.K.; Sultangazin, U.M. On discrete models of the kinetic Boltzmann equation. Russ. Math. Surv. 1971, 26, 1–56. [CrossRef] 3. Vedenyapin, V.V. Time averages and Boltzmann extremals. Dokl. Math. 2008, 78, 686–688. [CrossRef] 4. Adzhiev, S.Z.; Vedenyapin, V.V. Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model. Comput. Math. Math. Phys. 2011, 51, 1942–1952. [CrossRef] 5. Vedenyapin, V.V.; Adzhiev, S.Z. Entropy in the sense of Boltzmann and Poincaré. Russ. Math. Surv. 2014, 69, 995–1029. [CrossRef] 6. Sinitsyn, A.; Dulov, E.; Vedenyapin, V. Kinetic Boltzmann, Vlasov and Related Equations; Elsevier Science: Waltham, MA, USA, 2011. [CrossRef] 7. Batishcheva, Y.G.; Vedenyapin, V.V. The second law of thermodynamics for chemical kinetics. Mat. Model. 2005, 17, 106–110. 8. Adzhiev, S.Z.; Melikhov, I.V.; Vedenyapin, V.V. The H-theorem for the physico-chemical kinetic equations with explicit time discretization. Phys. A Stat. Mech. Appl. 2017, 481, 60–69. [CrossRef] 9. Adzhiev, S.Z.; Melikhov, I.V.; Vedenyapin, V.V. The H-theorem for the physico-chemical kinetic equations with discrete time and for their generalizations. Phys. A Stat. Mech. Appl. 2017, 480, 39–50. [CrossRef] 10. Adzhiev, S.Z.; Vedenyapin, V.V.; Volkov, Y.A.; Melikhov, I.V. Generalized Boltzmann-Type Equations for Aggregation in Gases. Comput. Math. Math. Phys. 2017, 57, 2017–2029. [CrossRef] 11. Horn, F.; Jackson, R. General mass action kinetics. Arch. Ration. Mech. Anal. 1972, 47, 87–116. [CrossRef] 12. Vol’pert, A.I.; Hudjaev, S.I. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics; Springer Netherlands: Dordrecht, The Netherlands, 1985. 13. Gorban, A.N.; Kaganovich, B.M.; Filippov, S.P.; Keiko, A.V.; Shamansky, V.A.; Shirkalin, I.A. Thermodynamic Equilibria and Extrema Analysis of Attainability Regions and Partial Equilibrium; Springer-Verlag: New York, NY, USA, 2006. 14. Physical and Chemical Processes in Gas Dynamics: Physical and Chemical Kinetics and Thermodynamics of Gases and Plasmas, Volume 2; Losev, S.A., Potapkin, B.V., Macheret, S.O., Chernyi, G.G., Eds.; American Institute of Aeronautics and Astronautics: Washington, DC, USA, 2004. 15. Vedenyapin, V.V.; Mingalev, I.V.; Mingalev, O.V. On discrete models of the quantum Boltzmann equation. Russ. Acad. Sci. Sb. Math. 1995, 80, 271–285. [CrossRef] 16. Vedenyapin, V.V.; Orlov, Y.N. Conservation laws for polynomial hamiltonians and for discrete models of the Boltzmann equation. Theor. Math. Phys. 1999, 121, 1516–1523. [CrossRef] 17. Smoluchowski, M. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Losungen. Z. Phys. Chem. 1917, 92, 129–168. [CrossRef] 18. Carr, J. Asymptotic behavior of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case. Proc. R. Soc. Edinb. Sect. A Math. 1992, 121, 231–244. [CrossRef] 19. Carr, J.; da Costa, F.P. Asymptotic behavior of solutions to the coagulation-fragmentation equations. I. Weak fragmentation. J. Stat. Phys. 1994, 77, 89–123. [CrossRef] 20. Ball, J.M.; Carr, J.; Penrose, O. The Becker-Döring cluster equations: Basic properties and asymptotic behavior of solutions. Commun. Math. Phys. 1986, 104, 657–692. [CrossRef] 21. Ball, J.M.; Carr, J. Asymptotic behavior of solutions to the Becker-Döring equations for arbitrary initial data. Proc. R. Soc. Edinb. Sect. A Math. 1988, 108, 109–116. [CrossRef] Physics 2019, 1 252

22. Melikhov, I.V. Physico-Chemical Evolution of Solids; BINOM: Moscow, Russia, 2006. 23. Melikhov, I.V. The evolutionary approach to creating nanostructures. Nanosyst. Phys. Chem. Math. 2010, 1, 148–155. 24. Malyshev, V.A.; Pirogov, S.A. Reversibility and irreversibility in stochastic chemical kinetics. Russ. Math. Surv. 2008, 63, 1–34. [CrossRef] 25. Gasnikov, A.V.; Gasnikova, E.V. On entropy-type functionals arising in stochastic chemical kinetics related to the concentration of the invariant measure and playing the role of Lyapunov functions in the dynamics of quasiaverages. Math. Notes 2013, 94, 854–861. [CrossRef] 26. Moser, J.K. Lectures on Hamiltonian Systems; American Mathematical Society; New York University: New York, NY, USA, 1968. 27. Bruno, A.D. The Restricted 3-Body Problem: Plane Periodic Orbits; Walter de Gruyter: Berlin, Germany, 1994. 28. Boltzmann, L. Neuer Beweis Zweier Sätze Über Das Wärmegleichgewicht Unter Mehratomigen Gasmolekülen; K.K. Hof-und Staatsdruckerei: Vienna, Austria, 1887. [CrossRef] 29. Stueckelberg, E.C.G. Theoreme H et unitarite de S. Helv. Phys. Acta 1952, 25, 577–580. [CrossRef] 30. Melikhov, I.V.; Rudin, V.N.; Kozlovskaya, E.D.; Adzhiev, S.Z.; Alekseeva, O.V. Morphological memory of polymers and their use in developing new materials technology. Theor. Found. Chem. Eng. 2016, 50, 260–269. [CrossRef] 31. Arrhenius, S. Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Z. Phys. Chem. 1889, 4U, 226–248. [CrossRef] 32. Stiller, W. Arrhenius Equation and Non-Equilibrium Kinetics: 100 Years Arrhenius Equation; B.G. Teubner Verlagsgesellschaft mbH: Stuttgart, Germany, 1989. 33. Laidler, K.J.; Klng, M.C. The Development of Transition-State Theory. J. Phys. Chem. 1983, 87, 2657–2664. [CrossRef] 34. Trautz, M. Das Gesetz der Reaktionsgeschwindigkeit und der Gleichgewichte in Gasen. Bestätigung der Additivität von Cv-3/2R. Neue Bestimmung der Integrationskonstanten und der Moleküldurchmesser. Z. Anorgan. Allgem. Chem. 1916, 96, 1–28. [CrossRef] 35. Lewis, W.C.McC. LXX.—Studies in catalysis. Part V. Quantitative expressions for the velocity, temperature-coeﬃcient, and eﬀect of the catalyst from the point of view of the radiation hypothesis. J. Chem. Soc. Trans. 1916, 109, 796–815. [CrossRef] 36. Kubasov, A.A. Chemical Kinetics and Catalysis, Part 2: Theoretical Foundations of Chemical Kinetics; Lomonosov Moscow State University: Moscow, Russia, 2005. 37. Belousov, N.V. Chemical Kinetics: The Electronic Course for Universities; IPK SFU: Krasnoyarsk, Russia, 2009. 38. Gorban, A.N.; Shahzad, M. The Michaelis–Menten– Stueckelberg Theorem. Entropy 2011, 13, 966–1019. [CrossRef]

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