# Molecular Modeling of Supercritical Processes and the Lattice—Gas Model

## Abstract

**:**

## 1. Introduction

_{2}—diethylene glycol monoethyl ether (ethylcarbitol)” [99], the phase equilibrium of the propylene glycol–propane/butane system and the solubility of propylene glycol in supercritical propane–butane mixture [100], the solubility of ammonium palmitate in supercritical carbon dioxide [101], the solubility of bio-diesel fuel components (methyl esters of stearic and palmitic acid) in supercritical carbon dioxide [102], and in many other systems.

## 2. Molecular Level

#### 2.1. Ideal Systems

_{i}is the concentration of molecules, measured as the number of i-type molecules in a unit volume; ${k}_{ij}{}^{0}$ is the rate constant pre-exponential factor; ${E}_{ij}$ is the reaction’s energy of activation between i and j reagents; $\beta =1/{k}_{B}T$, ${k}_{B}$ is the Boltzmann constant, and T is the temperature. In Expression (1), for a heterogeneous process, the area which does not change during the process is expressed in terms of the concentration of adsorbed particles θ

_{i}[35], which determines the fraction of the surface occupied by component i: ${U}_{ij}={k}_{ij}{\theta}_{i}{\theta}_{j}$ (the product n

_{i}n

_{j}is replaced by the product θ

_{i}θ

_{j}). To calculate the rate constants, the theory of absolute reaction rates is used [31]. This theory expresses the rate constants of the elementary steps with the partition functions of the reactants and the activated complex (AC) of the stage. Equation (1) assumes that the stage of chemical transformation is slow, the particles move completely independently, and an equilibrium distribution of components is realized in space. This means two important things: there is no intermolecular interaction in the system and there are no diffusion inhibitions at the micro- and macro-levels.

_{i}. However, the rate of the stage slows down with the increased pressure, even if the SCF component is inert. This pressure effect is due to the filling of the SCF of the volume of the system and a decrease in the probability of approaching the reagents, as well as the fact that, at high molecular densities, the formation of different associates around each reagent is inevitable.

#### 2.2. Non-Ideal Systems and the Lattice-Gas Model

_{i}(or its number density). Then, the normalization condition will be written as $\sum _{j=1}^{s}{\mathsf{\theta}}_{i}}=1$ and the value $\theta ={\displaystyle \sum _{j=1}^{s-1}{\mathsf{\theta}}_{i}}$ is the complete occupancy of a lattice system by all i components of the system, 1 ≤ i ≤ s − 1. The quantity is the fraction of free sites (recall that vacancies are not thermodynamic characteristics). The ratio ${\mathrm{x}}_{i}={\mathsf{\theta}}_{i}/\mathsf{\theta}$ is the mole fraction of component i.

_{ij}. The analogous parameter of any particle with a neighboring vacancy is equal to zero. The probability of finding a pair of particles ij at neighboring nodes is characterized by the value θ

_{ij}—this is the pair distribution function of particles. Such functions are needed to describe the probability that reactants i and j will meet in dense phases so that a chemical reaction can occur.

_{fg}

^{AB}is written in the theory of non-ideal reaction systems [66,106] as the following expression (subscripts f and g indicate the numbers of sites where reagents A and B are located):

_{ij}represents the interaction parameter of the AC reaction for an i-type particle with a neighboring particle of type j.

_{fh}

^{ij}= θ

_{fh}

^{ij}/θ

_{f}

^{i}is the conditional probability of finding particles j next to particles i. Here, the numbers of neighboring sites (subscripts) are introduced only to indicate differences in positions on the lattice of the reactants: θ

_{fh}

^{ij}= θ

_{ij}and t

_{fh}

^{ij}= t

_{ij}. In non-ideal systems, θ

_{ij}≠ θ

_{i}θ

_{j}, which corresponds to the correlated distribution of components in space. The case of equality θ

_{ij}= θ

_{i}θ

_{j}corresponds to the chaotic distribution of components, which is typical for ideal systems (see Formula (1)).

_{AB}instead of the product θ

_{A}θ

_{B}) and the heights of activation barriers through the functions Λ

_{fg}

^{AB}. Functions S

_{i}take into account the influence of neighbors on the magnitude of activation barriers through the difference in interaction parameters due to the influence of the neighboring particle j (via ${\mathsf{\delta}\mathsf{\epsilon}}_{ij}={\mathsf{\epsilon}}^{*}{}_{ij}-{\mathsf{\epsilon}}_{ij}$). The exponential factor with ${\mathsf{\beta}\mathsf{\epsilon}}_{fg}^{AB}$ in Formula (2) is necessary for the transition at low system densities from Formula (4) to Expression (1) for the law of mass action, as for an ideal system [66].

_{ij}that characterizes the probability that two particles i and j can be on neighboring sites. The calculation of pair functions θ

_{ij}in non-ideal reaction systems is always carried out in some particular approximation because the problem cannot be solved exactly [64,65,66]. In this case, the so-called quasi-chemical approximation (QCA) is used [23,64,65,66]. Historically, it was the first approximation in which the effects of direct correlations between interacting molecules were taken into account. There, each pair of neighboring molecules is considered independent of other molecules in the system. The function θ

_{ij}depends on the interaction energy of molecules and concentrations of components.

_{ij}> 0; analogously, i and j particles repulse one another when ε

_{ij}< 0.

_{i}represents the partial pressures in the gas phase (1 ≤ i ≤ s − 1); θ

_{i}is the degree of filling the surface with particles i; a

_{i}= a

_{i}

^{0}exp(βQ

_{i}), a

_{i}

^{0}is the pre-exponential of Henry’s constant; Q

_{i}is the binding energy of particle i with the surface; and ${x}_{ij}=\mathrm{exp}(-{\mathsf{\beta}\mathsf{\epsilon}}_{ij})-1$.

## 3. Thermodynamics and Kinetics

- The second law of thermodynamics and connection between models of equilibrium and kinetics;
- A self-consistence of equilibrium and kinetics in ideal systems;
- A self-consistence of equilibrium and kinetics in non-ideal systems;
- The equations of a state for non-ideal systems and their connection with kinetic models;
- Why it is impossible to use factors of activity for the AC in kinetic models;
- Thermodynamic parameters of the critical area and the requirement of technologies.

#### 3.1. The Second Law of Thermodynamics and Connection between Models of Equilibrium and Kinetics

#### 3.2. Self-Consistence of Equilibrium and Kinetics in Ideal Systems

_{i}and A

_{j}in brackets denote different reacting particles, and the values of ν

_{i}and ν

_{j}are equal to the negative and positive values of the stoichiometric coefficient (the sign of the coefficient is determined by their location: on the left or right side of the equation). The constants k

_{1}and k

_{2}are the reaction rate constants in the forward and backward directions. Numerically, they are equal to the reaction rate at single values of the concentration of each of their reagents in the forward direction.

_{1}/k

_{2}is the equilibrium constant of the stage.

_{A}= K

_{A}θ

_{A}, where K

_{A}is the desorption rate constant and θ

_{A}is the fraction of the occupied surface. The adsorption rate on free surface areas (V is the symbol of vacancies) will be written as U

_{V}= K

_{V}Pθ

_{V}, where K

_{V}is the adsorption rate constant, P is the pressure in the gas phase, and θ

_{V}is the fraction of the free surface (θ

_{V}= 1 − θ

_{A}).

_{A}= U

_{V}, it follows that K

_{A}θ

_{A}= K

_{V}Pθ

_{V}, or θ

_{A}= K

_{V}P(1 − θ

_{A})/KA or θ

_{A}= K

_{V}P/(K

_{A}+ K

_{V}P), and θ

_{A}= a

_{1}P/(1 + a

_{1}P), a

_{1}= K

_{V}/K

_{A}is the adsorption coefficient without dissociation. If the adsorption process proceeds with the dissociation of gas phase molecules, then the rates of desorption and adsorption will be rewritten as U

_{AA}= K

_{AA}θ

_{AA}= K

_{AA}(θ

_{A})

^{2}and U

_{VV}= K

_{VV}Pθ

_{VV}= K

_{VV}P(1 − θ

_{A})

^{2}. Hence, θ

_{A}= a

_{2}P

^{1/2}/(1 + a

_{2}P

^{1/2}) is the Langmuir isotherm of dissociating molecules and a

_{2}= (K

_{VV}/K

_{AA})

^{1/2}is the adsorption coefficient in the presence of dissociation.

_{A}= y/(1 + y), where y = y

_{1}= a

_{1}P for adsorption without dissociation and y = y

_{2}= a

_{2}P

^{1/2}in the presence of dissociation. It follows that in the coordinates θ

_{A}= θ

_{A}(y) both dependencies behave equivalently. The coincidence of both dependences θ

_{A}= θ

_{A}(y) means that the equilibrium adsorption of dissociating molecules does not depend on what occurs first: dissociation of molecules in the gas phase or their adsorption. Both of these equilibrium dependences are obtained from the condition that the second law of thermodynamics according to Clausius is satisfied—from equalizing the velocities of the oppositely directed velocities of the stages. As a result, the four different rates of elementary stages give one equilibrium concentration dependence.

_{i}are the chemical potential of the standard state and the molar volume concentration of component i. However, in kinetic models for non-ideal systems it is impossible to use activities or activity coefficients [115] as natural thermodynamic replacements of concentrations (see below).

#### 3.3. Self-Consistence of Equilibrium and Kinetics in Non-Ideal Systems

_{A}= θ

_{A}(y) in non-ideal reaction systems will be considered for adsorption and desorption processes with (m = 2) and without (m = 1) dissociation. Let us consider the equalities of the rates in both directions, U

_{A}= U

_{V}and U

_{AA}= U

_{VV}, expressed by Formulas (2) and (3) in the QCA.

_{2}molecules in the form of two routes: the dissociation of A

_{2}molecules into atoms A in the bulk phase followed by their adsorption or the adsorption of A

_{2}molecules followed by the dissociation process. Both routes are described by the rates of stages in the forward (adsorption) and reverse (desorption) directions. At equilibrium, adsorption isotherms for A

_{2}molecules and A atoms should be obtained. Moreover, the degree of surface filling θ

_{A}must be the same regardless of the route, i.e., the final equilibrium state does not depend on the way the equilibrium is reached. This fact reflects the concept of Clausius twice: the equilibrium state itself follows from the equality of the rates of the stages and for different routes, it receives a single mathematical dependence for the degree of surface filling on pressure.

_{A}corresponding to the equalities of the rates of the stages for chemisorption (ε

_{AA}< 0), without dissociation (Figure 1a) and in the presence of dissociation (Figure 1b). Here, V

_{i}= U

_{i}/K

_{i}, V

_{ii}= U

_{ii}/K

_{ii}. Each field consists of three curves for the logarithm of the concentration factors for rates of non-dissociative (Figure 1a) and dissociative desorption (Figure 1b), adsorption (curve 1), desorption (curve 2), and the logarithm of the isotherm of adsorption (curve 3, right y-axis). As in ideal systems (Section 3.2), both curves 3 are identical for non-ideal systems.

#### 3.4. The Equations of a State for Non-Ideal Systems and Their Connection with Kinetic Models

#### 3.5. Why It Is Impossible to Use Factors of Activity for AC in Kinetic Models

_{ij}. As a result of this averaging procedure, the nature of the effective interaction between particles changes: instead of repulsion, their effective attraction appears (which distorts the physical nature of the system). When the density of the system changes, the particles behave as in the case of a first-order phase transition with a nonmonotonic concentration dependence of the rate of the desorption stage.

#### 3.6. Thermodynamic Parameters of the Critical Area and the Requirement of Technologies

_{i}on concentration components i in the vicinity of the coordinates of the critical point (P

_{cr}, T

_{cr}): ${({\partial}^{}{\mu}_{i}/\partial {n}_{i}{}^{})}_{{P}_{cr,}{T}_{cr}}=0$ and ${({\partial}^{2}{\mu}_{i}/\partial {n}_{i}{}^{2})}_{{P}_{cr,}{T}_{cr}}=0$. This peculiarity to the same extent concerns both the average field and the QCA approach in the LGM. It must be valid in any calculation method.

_{2}. On the basis of the gas-dynamic model of the impulse jet expansion of a van der Waals gas, a strategy experiment on the determination of the parameters of the “coil–globule” transition of the polymer chain in SC carbon dioxide was developed. To use the condition of constant isochoric heat capacity outside the near-critical point in modeling, it is necessary to determine the structure of the near-critical region.

_{2}in the near-critical region, the conditions of expansion corresponding to the model was determined. An experimental design (geometry and dimensions of the basic elements of the installation, and the duration of the impulse valve) was developed. Possible variants of the experiment and its data processing were discussed.

_{cr}, T

_{cr}) and the presence of large fluctuations of parameters on the supercritical isotherm, to ensure the stability of obtaining the target product.

## 4. Model of the Effective Pair Potential

#### 4.1. Internal Motions of Particles

^{3}, where the value κ is the root-mean-square displacement of the molecule from the center; it can be estimated from the theory of harmonic motions in a solid for temperatures above the Debye temperature. That is, the value of κ is found from the parameters of the paired Lennard–Jones interaction potential [139]. In the general case, for any fluid densities θ, the available volume can be estimated as the ratio V(θ)/V(θ = 0) = L(θ)

^{3}, where L(θ) = t

_{AV}+ κ t

_{AA}, where the function t

_{AA}, characterizing the conditional probability of finding two neighboring molecules A, is defined in Section 2.2. More accurate estimates of the available volume V(θ)/V(θ = 0) are possible if a geometric model [139] is used that refines different positions of neighboring molecules [139].

_{0}(1 − uT), where the function u reflects the vibrational motion of molecules. Previously, this form of dependence of the interaction parameter was considered as a convenient fitting function [119,120,121,146,147,148]. This approach [133] makes it possible to express the function u in terms of potential functions without introducing additional parameters.

#### 4.2. Vapor–Liquid Systems

_{0}(isolated dimer potential well depth) [23]. The values of these parameters are actively used in calculations for any density using the Monte Carlo and molecular dynamics methods. They were determined from experimental data for low-density gases (described up to the second virial coefficient). With increasing fluid density, it is necessary to add triple interactions [23,137,149,150]. Therefore, to use the effective pair potential (ε

_{ef}) of the Lennard–Jones type in the LGM for any densities [151,152,153], the following function was used:

_{tr}) for nearest neighbors (Δ

_{1r}—Kronecker symbol) in the form d

_{tr}= 0.2(z − 1)ε

_{3}/ε

_{0}[66,154], where ε

_{3}is the triple interaction parameter. The t

_{AA}function reflects the presence of a third particle A nearby (it is defined in Section 2.2). For simplicity, it is assumed that the contributions from concentration and temperature are taken into account separately. For a quantitative description of experimental systems, it is possible to involve contributions from several coordination spheres. A similar structure to Equation (10) for effective pair potentials is also preserved for multicomponent mixtures [155].

^{3}is the reduced number density, for argon in the volume (σ = 0.34 nm and ε

_{0}= 119K [23]). The comparison was made for the virial expansion (curves 2–4) [156] and for the LGM [151] at T = 162 K [157]. Figure 4a shows that at least five terms of the virial expansion are required to agree with the experimental data and with the LGM.

_{c}, where ρ

_{c}is the density at the critical point. Throughout the region, there is agreement between the experimental data [158] and the calculation by the LGM, including the value of the critical parameter β equal to 0.37.

_{2}binary mixture [159]. Curves for different compositions of the mole fraction of nitrogen are shown in Figure 4b.

## 5. Influence of SCFs on Equilibrium and Kinetic Characteristics

_{2}molecules, low molecular weight alkanes, alcohols, freons, water, etc. Such solvents are multicomponent mixtures of low molecular weight substances. To model SC systems, it is necessary to be able to calculate the bulk properties of SCFs and their contacts with the surfaces of solids (non-porous and porous). For polymer matrices, the dissolution of SCF molecules through open surfaces is possible. All these properties of SCF systems can be taken into account within the framework of the unified LGM technique [66,106] discussed above.

#### 5.1. Effect of SCFs on the Characteristics of Adsorption Processes

_{2}) has been extensively studied [171,172]. CO

_{2}molecules are a solvent for some polymers, combining many important technological factors such as environmental friendliness, low cost, ease of removal from the polymer, incombustibility, etc.

_{A}= 0.05 (1), 0.5 (2), and 0.85 (3). A comparison of Figure 6a,b leads to the conclusion that, on the strongly binding surface, component A is displaced from adsorption sites at higher SCF pressures more slowly and over a broader range of pressures of component SCF than for the weakly binding surface. The same sort of curves will apply for chemisorption—values of partial pressure change (the range of pressure of the basic component A at chemisorption and physical adsorption differs by approximately 4–6 orders of magnitude) and the course of concentration dependences remains similar.

#### 5.2. Effect of an SCF on the Concentration Dependence of the Rate of a Reaction

_{AB}

^{ef}for a bimolecular reaction [170] as

_{12}= U

_{AB}/(k

_{AB}

^{0}θ

_{A}θ

_{B}) = exp(–βE

_{AB}

^{ef})

_{AB}

^{ef}= E

_{AB}+ε

_{AB}− β

^{−1}ln(θ

_{AB}/θ

_{A}θ

_{B}) − (z − 1)ln(S

_{A}S

_{B}).

_{AB}

^{ef}= E

_{AB}. At high fluid densities, when the proportion of reagents A and B is small, and the proportion of SCF is large (θ

_{C}>> θ

_{A}+ θ

_{B}), then

_{AB}

^{ef}≈ E

_{AB}− (z − 1)(δε

_{AC}+ δε

_{BC}) = E

_{AB}+ (z − 1)(1 − α)(ε

_{AC}+ ε

_{BC}).

_{ij}= α is used for both reagents.

_{AB}

^{ef}. If α < 1, then the presence of the SCF associates around the reagents increases E

_{AB}

^{ef}. If α > 1, then the SCF associates decrease E

_{AB}

^{ef}. The difference in values of ΔE

_{AB}(θ) = E

_{AB}

^{ef}− E

_{AB}indicates the effect of the SCF on the activation energy of the stage.

_{AB}

^{ef}with varying pressure and temperature. In the case of a large value of E

_{AB}

^{ef}, the influence of the SCF is small if the rate of the chemical reaction itself remains in the region of the kinetic regime. However, due to the addition of a large amount of inert SCF molecules, with an increase in the pressure in the system, the nature of the flow of the bimolecular reaction can change and move from the kinetic regime to the diffusion regime.

_{ij}

^{ef}for chemical reactions and transport stages was carried out in [170]. The contributions of the total density of the system and the role of intermolecular interactions at different temperatures of the SCF of the system were discussed. The theory of non-ideal reaction systems allows the SCF process to separate into failure contributions from pressure and temperature. In situations where the contribution of temperature predominates, the differences between the values of E

_{ij}

^{ef}and E

_{ij}are small, so Equation (1) can be used. If the contribution from pressure is predominant, then the differences between E

_{ij}

^{ef}and E

_{ij}limit the region of the SCF for concentrations where the mixture can be considered an ideal one.

_{cr}) [174].

_{A}= θ

_{B}) is considered, in which the reaction between molecules A and C is also possible (C is the SCF component). The proportion of component C varies and it is demonstrated how the effective activation energy E

_{AC}(ef) changes with respect to the energy E

_{AC}of the same reaction in an ideal system as a function of the total amount of substance θ = θ

_{A}+ θ

_{B}+ θ

_{C}.

_{AB}(ef) = E

_{AB}at any θ

_{C}.)

#### 5.3. Effect of an SCF on the Concentration Dependence of the Dissipative Coefficients

_{i}* = z

_{fg}* U

_{iV}/θ

_{f}

^{i},

_{fg}* is the number of possible hops to nearest-neighbor sites g for the fth cell along the direction in which the label moves. The expression for w

_{i}

_{V}is given by Formula (2). This refers to the bimolecular hop of molecule i, i + V → V + i, in which the first “reactant” is a moving type i molecule, and the second “reactant” is the vacancy into which the molecule i is transferred. The activation energy of this process is E

_{i}

_{V}= 0 for a bulk phase and E

_{i}

_{V}> 0 for the surface migration step. With the growth of the full density of a system, the fraction of free volume decreases and the factor of self-diffusion of any particle decreases.

_{iV}(ef) = E

_{iV}− β

^{−}

^{1}ln(θ

_{iV}/θ

_{i}θ

_{V}) − (z − 1)ln(S

_{i}S

_{V}),

_{iV}= E

_{iV}(ef) − E

_{iV}= (z − 1)(1 − 2α)ε

_{AC}. For α = 0.5, ΔE

_{i}

_{V}= 0 and the decrease in the self-diffusion coefficient is only due to the decrease in the free volume fraction.

_{A}) in the SCF bulk (E

_{i}

_{V}= 0). The effect of the density of the supercritical component was studied as a function of the density θ

_{C}varying between zero and 1 − θ

_{A}for two main component coverages, θ

_{A}= 0.01 and 0.1.

_{C}curves is similar to the behavior of the curves shown in Figure 9. The concentration dependence of the self-diffusion coefficients depends strongly on the nature of the supercritical component. The stronger the interaction between SCF molecules, the greater the extent to which diffusion slows down with an increasing SCF concentration. The self-diffusion coefficient decreases as the temperature is raised.

_{fg}for spherical molecules of comparable sizes is expressed as follows [25]:

_{i}is the mole fraction of component i and θ is the total coverage of the system.

^{1/2}and depends linearly on the density. For high densities, we have an exponential temperature dependence, as in Eyring’s conventional model [31]. Equation (14) allows viscosity to be calculated for any composition of a multicomponent mixture.

_{A}= 0.01 and 0.1 is smaller than in the case of the fixed amounts of component A. The concentration dependences of viscosity are normalized to the viscosity of component A in a rarefied gas. The calculations were carried out at a fixed ε

_{AA}value for component A and a decreasing ε

_{0}value or an increasing γ

_{A}= ε

_{AA}/ε

_{CC}ratio. An increase in γ

_{A}leads to an increase in the energy of the lateral interactions between the main component A and the SCF. As a result, the viscosity of the system decreases. Therefore, by changing the SCF, it is possible to vary the viscosity of the system in a fairly wide range. This range is temperature-dependent: it widens with increasing temperature.

## 6. LGM and Dissipative Coefficients

_{m}/η

_{m0}(where the denominator η

_{m0}is a normalization factor to the properties of a rarefied gas) on the reduced density ρ

_{rm}, obtained for the Ar − N

_{2}binary mixture for three values of τ according to this empirical formula and to the LGM (thick lines are the calculation of the LGM [160]). Here, the reduced temperature τ = T/T

_{cr}= 0.75, 1, and 3, where T

_{cr}is the critical temperature reflecting the change in the value as a function of the composition of the mixture.

## 7. Extension of the Models

#### 7.1. Nonspherical Potential Functions

_{2}molecule, has noticeable differences in shape from a spherical one. The ratio of its long and short axes is 1.38. As an example, Figure 15 shows the compressibility factor χ = PV/(PV)

_{s}of CO

_{2}molecules [183], where the product of pressure and specific volume (PV)

_{s}refers to normal conditions. The calculations were carried out for different potential models: curves 1–3 are given for a single-site model; curve 4 reflects the difference in the shape of the hard sphere of the molecule. (For comparison, curve 2 is given for the strong contribution of the triple interaction).

_{2}molecules (T

_{cr}= 304 K). Calculations of the separation curve of CO

_{2}molecules demonstrate a number of differences depending on the interaction potential; their consideration depends on the goal of modeling the processes with their participation.

#### 7.2. Water Molecules

_{2}O)

_{f}+ V

_{g}= V

_{f}+ (H

_{2}O)

_{g}, in which the numbers of the neighboring sites f and g are used as indices and the symbol Vg denotes a vacant site with the number g, i.e., the diffusion of water molecules is a special case of a bimolecular reaction occurring on the f and g sites.

#### 7.3. Kinetic Equations

_{2}and ozone on coke-like deposits on a surface of Pt–Re/γ-Al

_{2}O

_{3}bimetallic reforming catalysts [199].

## 8. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a_{i0} | pre-exponential factor of the Henry constant for molecules of type i |

d_{tr} | triple interaction parameter associated with its energy ε_{3} |

D_{i}* | partial self-diffusion coefficient for molecules of type i |

E_{ij} | reaction’s energy of activation between i and j reagents |

E_{A}(ef)
| effective activation energy of desorption |

k_{B} | Boltzmann constant |

k_{ij} | rate constant of elementary reaction i + j → products |

k_{ij}^{0} | rate constant pre-exponential factor for elementary reaction i + j → products |

k_{1} and k_{2} | reaction rate constants in the forward and backward directions |

K | equilibrium constant of the stage |

M | number of sites in the system |

n_{i} | concentration of i-type molecules |

Q_{i} | energy of i-particle bonding with the surface |

Q | amount of heat |

Q_{s} | statistical sum of the system |

P | pressure |

P_{i} | partial pressure of i-type molecules |

P({γ_{f}^{i}},τ)
| probability of finding the system at the time τ in a state {γ_{f}^{i}}. For the sake of brevity, this state is denoted as {I} ≡ {γ_{f}^{i}} |

s | number of occupation states of any cell or site |

S | entropy |

S_{m} | molecular property in flow |

T | temperature |

t_{fh}^{ij} = θ_{fh}^{ij}/θ_{f}^{i} | function of the conventional probability of j particles being near i particles (fh represents the numbers of sites containing these particles) |

U | internal energy |

U_{ij} | rate of an elementary stage of a bimolecular reaction i + j → products |

U_{f}^{i}(α)
| rate of the elementary single-site stage i ↔ b with number α in the site f |

U_{fg}^{ij}(α)
| rate of the elementary two- site stage i + j_{α} ↔ b + d_{α} with number α in two sites fg |

V | volume of the system |

u | contribution from the vibrational motion of molecules to energy parameters |

W_{α}({I} → {II})
| probability of the elementary process α which resulted at time τ in the transfer of the system from the initial state {I} to the final state {II} |

x_{i} = θ_{i}/θ | mole fraction of component i among all molecules of the mixture. |

z | nearest neighbors of any site or cell |

z_{fg}* | the number of possible hops to nearest-neighbor sites g for the fth cell along the direction in which the label moves |

Z | compressibility factor |

α | number of stages in the total process |

α_{i} | activity coefficient of i-type reagents |

α_{ij}* | denotes the activity coefficients of ACs |

α_{ij} = ε_{ij}*/ε_{ij}, for simplicity α_{ij} = α
| is used for both reagents |

γ_{f}^{i} | variable determined the occupation state of site with number f (1 ≤ f ≤ M) by particle of type i (1 ≤ i ≤ s) |

ε_{ij} | parameter of this interaction between ij pairs of neighboring particles |

ε_{ij}*
| interaction parameter for reaction AC using i-type particles and neighboring j-type particles |

η | shear viscosity coefficient |

κ | heat capacity coefficient |

κ_{D} | mean square displacement of particles in a solid in the harmonic approximation |

λ | average cell size |

${\mathsf{\Lambda}}_{fg}^{AB}$ | non-ideality function for the two-site stage |

${\mathsf{\Lambda}}_{f}^{i}$ | non-ideality function for a one-site stage |

μ_{i} | chemical potential of component i |

μ_{i}^{0} | chemical potential of the standard state for component i |

ρ | mean free path of a particle |

θ_{i} | concentration of particles type i in the (surface or bulk)) system |

θ = ∑_{I = 1}^{s−1}θ_{i} | complete occupancy of a lattice system by all i components of the system, 1 ≤ i ≤ s − 1 |

θ_{fh}^{ij} | probability of two particles i and j being on nearest neighboring sites f and h (for homogeneous system θ_{ij}, is the pair particle distribution function) |

ν_{i} | stoichiometric coefficient |

## Appendix A. Kinetics Equation in the LGM

_{f}

^{i}, where f is the cell number, 1 ≤ f ≤ M, subscript i denotes the state of occupation of the cell with number f, 1 ≤ f ≤ s, and s is the number of the states of cell occupation including a vacancy (M is the number of sites) [66]. For the two components of the lattice systems (s = 2) of any site of the lattice structure corresponding to the one-component system for which i = A or V (vacancies). If the site f contains an adsorbed particle A, then γ

_{f}

^{A}= 1 and γ

_{f}

^{V}= 0; if the cell is free, then there is a vacancy, so γ

_{f}

^{A}= 0 and γ

_{f}

^{V}= 1. The random variables γ

_{f}

^{i}are subject to the following relations:

_{f}

^{i}γ

_{f}

^{j}= Δ

_{ij}γ

_{f}

^{i}, where Δ

_{ij}is the Kronecker symbol, which means that any site is unavoidably occupied by one, but only one, particle.

_{f}

^{i}} = γ

_{1}

^{i}, γ

_{2}

^{j}, …, γ

_{M}

^{n}the complete set (or full list) of values γ

_{f}

^{i}of all lattice sites, which uniquely determine the complete configuration of the locations of the particles on the lattice at time τ, and, by P({γ

_{f}

^{i}},τ), the probability of finding the system at this time in a state {γ

_{f}

^{i}}. For the sake of brevity, this state is denoted as {I} ≡ {γ

_{f}

^{i}}. Let the common studied process consist of many stages and through α we denote the number of elementary stages in the process. The master equation for the evolution of the full distribution function of the system in a state {I}, due to the implementation of the elementary processes α in condensed phases, has the following form (the so-called Glauber-type equation) [66,163,205,206]:

_{α}are subject to the condition of detailed balance:

_{α}({I} → {II}) exp(–βH({I})) = W

_{α}({II} → {I}) exp(–βH({II})),

_{f}

^{i}}, τ → ∞) = exp(–βH({γ

_{f}

^{i}}))/Q

_{s}; here, Q

_{s}is the statistical sum of the system [66]. Expressions for W

_{α}({I} → {II}) are constructed with all the molecular features of the system taken into account [68].

_{f}

^{i}},τ), the evolution of the system is described using a shortened method of defining it by time distribution function (correlators) determined by

_{f}

^{i}= <γ

_{f}

^{i}>) and second (θ

_{fg}

^{ij}= <γ

_{f}

^{i}γ

_{g}

^{j}>) correlations in the general form can be written as

_{f}

^{i}(α) is the rate of the elementary single-site processes i ↔ b (here h ∈ z

_{f}), U

_{fg}

^{ij}(α), and α is the rate of the elementary two-site processes i + j

_{α}↔ b + d

_{α}(h ∈ z) on the nearest sites; the second term in P

_{fg}

^{ij}describes the stage i + m ↔ b + c on neighboring sites f and h (and the term of P

_{gf}

^{ji}describes the stages on sites g and h and similar stages on sites f and h) [66,106]. All the rates of the elementary stages U

_{f}

^{i}(α) and U

_{fg}

^{ij}(α) are calculated in the framework of the theory of absolute reaction rates for non-ideal reaction systems written in the quasi-chemical approximation of the interparticle interaction. The rates of two-site stages U

_{fg}

^{ij}(α) have the form of Equations (2) and (3), and the rates of single-site stages U

_{f}

^{i}(α) in the QCA are expressed as

_{m}in these planes can be written as S

_{m}(x = +ρ) = S

_{m}(x = 0) ± ρdS

_{m}/dx, where the symbol S

_{m}means the concentration, the momentum (in the direction y, for example) or the energy of the particles moving along the axis X. The flow of quantity S

_{m}through plane 0 consists of two oppositely directed movements of particles from the planes x = ±ρ.

_{m}through the selected plane is calculated, where S

_{m}is modified by the following variables: (1) the number of molecules—for calculation of self-diffusion coefficient D

_{i}* and mass transfer coefficients D

_{ij}, (2) the number of impulses—for the calculation of the shear η and bulk viscosity ξ, and (3) the amount of energy—for the calculation of the thermal conductivity κ. There are two channels of transfer of the property S

_{m}: η = η

_{1}+ η

_{2}, κ = κ

_{1}+ κ

_{2}: (1) the transfer of molecules via a separated plane—the calculation of the coefficients D

_{i}*, D

_{ij}, η

_{1}, and κ

_{1}; (2) transfer of the property (momentum and energy) through collisions—the calculation of the coefficients η

_{2}, ξ, and κ

_{2}[68]. The results of calculating the dissipative coefficients are also shown in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13.

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**Figure 1.**Concentration factors (V

_{i}and V

_{ii}) in rates of non-dissociative (

**a**) and dissociative (

**b**) adsorption (curves 1) and desorption (curves 2) for a non-ideal chemisorption (ε

_{AA}< 0) system (V

_{i}= U

_{i}/K

_{i}, V

_{ii}= U

_{ii}/K

_{ii}). Curves 3 (right y-axis) correspond to the equilibrium isotherm.

**Figure 2.**Concentration dependence of E

_{A}(ef) for monomolecular desorption at 300 K calculated in the case of (1) fast and (2) slow elementary stage [123].

**Figure 3.**A supercritical area on the phase diagram of the van-der-Waals substances (A-MSC-SC-B-A) for which thermodynamics parameters are not desirable for technological processes.

**Figure 4.**(

**a**) Concentration dependences of the argon compressibility factor in [151]. Experimental values at 162 K [157] (dots); LGM calculations (1); calculations using the virial equations in [156]; (2) with regard to the second virial coefficient; (3) with regard to the second and third virial coefficients; (4) with regard to the second–fourth virial coefficients. The inset shows the phase diagram of argon. Symbols represent experimental values from [158]; the solid line represents calculations in [155] for d

_{tr}= 0.15, u = 0.00075. (

**b**) Phase diagrams of Ar, N

_{2}, and an Ar–N

_{2}mixture at different compositions with nitrogen mole fractions x

_{N2}= (1) 0, (2) 0.2, (3) 0.4, (4) 0.6, (5) 0.8, (6) 1.0. Symbols represent experimental values from [159]; solid lines represent calculations in [160].

**Figure 5.**Generalized compressibility factors of dense gases that obey the law of corresponding states [161].

**Figure 6.**Dependences of a system’s pressure on the concentration of ammonia in a binary ammonia–nitrogen mixture at different temperatures. Experimental values from [168] are on the left; calculated values are on the right. Temperature values on the right lines from bottom to top: 90, 110, 120, 130, 140, 170, and two lines around 180, and 210°.

**Figure 7.**Dependence of adsorption of the basic component A in binary mixture of component A and component SCF for rising pressure of SCF for fixed concentration component A corresponding to θ

_{A}= 0.05 (1), 0.5 (2), and 0.85 (3) for weak (

**a**) and strong (

**b**) adsorption [173].

**Figure 8.**Dependences E

_{AC}on the degree of occupancy θ at ε

_{CC}= 0.38 (energy, kcal/mol), ε

_{AA}= 4ε

_{CC}, and ε

_{BB}= 2.4ε

_{CC}for T = 1.1T

_{c}; E

_{12}= (1) 3, (2) 10, (3) 20; and T = 2.5T

_{c}, E

_{12}= (4) 3, (5) 10, (6) 20. Curves correspond to reagent concentrations x

_{A}= x

_{B}= 0.01-mole fractions.

**Figure 9.**Self-diffusion coefficient of component A (D

_{A}) as a function of θ = (θ

_{A}+ θ

_{C}) at γ

_{A}= ε

_{AA}/ε

_{CC}= (1) 1, (2) 1.6, (3) 3.2, and (4) 6.4 for τ = 1.15. Solid lines: θ

_{A}= 0.01; dotted lines: θ

_{A}= 0.1.

**Figure 10.**Viscosity as a function of θ at γ

_{A}= (1) 1, (2) 1.6, (3) 3.2, and (4) 6.4 for τ =1.15. Solid lines: θ

_{A}= 0.01; dotted lines: θ

_{A}= 0.1.

**Figure 11.**(

**a**) Dependences of O

_{2}shear viscosity η on pressure P at different gas temperatures: (1) 289, (2) 328 K. Dots are experimental values from [177]. (

**b**) Analogous dependences for low densities. (

**c**) Generalized diagram of reduced shear viscosity η/η

_{0}, depending on reduced pressure P

_{r}at different reduced temperatures Tr. Dots are experimental values from [23,26].

**Figure 12.**(

**a**) Heat capacity coefficients of argon at (1) 273 and (2) 400 K. Dots represent experimental values [158,177]; lines represent calculations [152]. (

**b**) Experimental data on the generalized heat capacity coefficient in [26]. Dots represent experimental values; lines represent calculations.

**Figure 13.**(

**a**) Generalized diagrams of the self-diffusion coefficient for the p/p

_{C}< 4 range of pressures. Lines represent calculations in [160]; dots represent experimental values from [26] at τ = (1) 1, (2) 1.1, (3) 1.2, (4) 1.3, (5) 1.4, (6) 1.6, (7) 2, and (8) 3. (

**b**) Generalized diagrams of the mutual diffusion coefficients of an Ar–N

_{2}mixture with ${x}_{{N}_{2}}$ = 0, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0-mole fractions at τ = (1) 1, (2) 1.3, and (3) 3 in the p/pc < 4 range of pressures. Series of curves calculated for each τ value are displayed from the bottom up as the concentration of nitrogen grows [160]. Dots represent experimental data recalculated according to the rules in [26].

**Figure 14.**Logarithmic dependences ${\eta}_{m}^{}/{\eta}_{m}^{0}$ on reduced density ρ

_{rm}, obtained for a binary Ar–N

_{2}mixture at different reduced temperatures τ: (1) 0.75, (2) 1, and (3) 3; thick lines represent the calculated values from [160]; thin lines represent experimental values from [26].

**Figure 15.**Comparison of experimental [184] (dots) and calculated lines of compressibility factors for CO

_{2}molecules; calculations were made for (1, 2, 4) T = 273 and (3) 373 K.

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Tovbin, Y.K.
Molecular Modeling of Supercritical Processes and the Lattice—Gas Model. *Processes* **2023**, *11*, 2541.
https://doi.org/10.3390/pr11092541

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Tovbin YK.
Molecular Modeling of Supercritical Processes and the Lattice—Gas Model. *Processes*. 2023; 11(9):2541.
https://doi.org/10.3390/pr11092541

**Chicago/Turabian Style**

Tovbin, Yuri Konstantinovich.
2023. "Molecular Modeling of Supercritical Processes and the Lattice—Gas Model" *Processes* 11, no. 9: 2541.
https://doi.org/10.3390/pr11092541