Kinetics and Thermodynamics of the Phase Transformation in the Nanocrystalline Substance—Gas Phase System

Abstract

This study presents a model of the reaction of a nanocrystalline substance within the gas phase, where diffusion of gas reactants in the volume of the nanocrystallites is a rate-limiting step. According to the model calculations carried out, the rate of diffusion across the phase boundary located on the nanocrystallite surface limits the rate of the process. It was stated that in chemical processes with a phase transformation, the critical concentration of absorbate depends on two factors: the specific active surface area of the adsorbent and the difference in chemical potentials between the gas phase and the equilibrium potential at which the phase transformation occurs. When the actual adsorbate potential in the gas phase is much greater than the equilibrium potential of the nanocrystallite with the largest specific active surface, nanocrystallites undergo phase changes in the order according to their specific active surfaces from the largest to the smallest. In a process where the actual adsorbate potential is close to an equilibrium one, nanocrystallites undergo phase transformation in the order of their specific active surface from the smallest to the largest.

1. Introduction

There is an extensive body of literature on diffusion processes in various types of macroscopic, coarsely crystalline structures [1,2,3,4,5,6,7].

The interaction of ammonia, hydrogen, and nitrogen with an iron surface was studied in order to learn the elementary stages of the catalytic synthesis of ammonia [8,9,10,11,12,13,14,15,16,17,18]. The phase diagram for an iron-nitrogen system was first presented by Jack [19,20]. Lehrer conducted research on the thermodynamic equilibrium in a coarsely crystalline iron-ammonia-hydrogen system. Based on the results obtained, he determined the limits of the individual phases of the iron nitrides, depending on the temperature and composition of the gaseous ammonia-hydrogen mixtures—Lehrer diagram [21]. Thermodynamic studies on nitrogen and carbon in iron and steel were conducted, e.g., by Kunze, Mittemeijer [22,23] and others [24].

The kinetics of the nitriding process of the iron catalyst have been described in previous works [25,26,27,28]. It was found that the concentration of nitrogen in the volume of iron nanocrystallites at a given time in the reaction depends on the ratio of nanocrystallite active surface area S to volume V of nanocrystallite, i.e., on the specific active surface area, A = S/V. The critical concentration necessary for the phase transition of iron-iron nitride will be achieved faster in small nanocrystallites than in large ones. Thus, under kinetic conditions, where the rate of the process is limited by the rate of dissociative gas adsorption over the nanocrystallite surface, iron nanocrystallites undergo phase transformation into iron nitride Fe₄N in the order of their sizes from the smallest to largest, along with the progress of the nitriding process. The concentration gradient in the volume of nanocrystallites practically does not occur, and after exceeding the critical concentration, nanocrystallites undergo phase transformation in their entire volume [27].

A model of the reaction between the nanocrystalline solid phase and the gas phase has been proposed [27], in which the rate of the process is limited by the rate of dissociative gas adsorption over the nanocrystallite’s surface, and the diffusion is fast enough for the equilibrium between gas adsorbed on the surface and gas dissolved in the volume of nanocrystallites to be established—a reaction model in the adsorption area.

Methods for determining the mass and size distribution of nanocrystallites (grain mass distribution, GMD; grain size distribution, GSD) have been proposed [26,27,28]. Using the methods, the relative size distribution of iron nanocrystallites, by their specific active surface area, was determined in the industrial ammonia synthesis catalyst [26,28], with greater accuracy compared to previous, commonly used methods. It was observed that the tested catalyst was characterized by a bimodal size distribution of iron nanocrystallites.

In the process of the nitriding of nanocrystalline iron, stationary states were identified [29,30], in which the rate of catalytic decomposition of ammonia r_d = const, the rate of nitriding reaction r_n = 0, and the maximum conversion m_n of nanocrystalline material was a function of temperature and the nitriding potential P = p_NH3/p_H2^3/2 (p_NH3, p_H2—partial pressures of ammonia and hydrogen, respectively) in the gas phase. In steady states, there may be single-phase systems (one crystallographic phase—α-Fe(N) or γ’-Fe₄N) as well as two-phase systems (α-Fe(N) + γ’-Fe₄N). It has been observed that in stationary states established at increasingly higher nitriding potentials, along with increasing the conversion degree, nanocrystallites undergo a phase transformation in the reverse order compared to kinetic measurements, i.e., according to size from the largest to the smallest [29]. It was found that the sum of the chemical potential of nitrogen and the surface energy balances the deformation potential of the iron crystal sub-lattice at the phase transformation from α-Fe(N) to γ’-Fe₄N [29].

In steady-state conditions in a two-phase system, the occurrence of hysteresis was observed for the dependence of the degree of conversion on the nitriding potential for nanocrystalline iron nitriding and the reduction of formed nitrides [30]. Depending on the direction of the reaction (nitriding or reduction), nanocrystallites undergo a phase transition at different gas-phase chemical potentials. It was found [30] that the chemical potential of the gas phase was equal to the potential of nitrogen in the iron/iron nitride nanocrystallites. Critical nitriding potentials at which nanocrystals of various sizes undergo phase transformation in the processes of nitriding P₀ⁿ and reduction in the nitrides formed P₀^r were determined to be P₀ⁿ > P₀^r.

The mechanisms of the nanocrystalline iron nitriding and reduction processes for the obtained nitrides were explained together with the thermodynamic foundations of the hysteresis phenomenon [30]. It was found that the coexistence of two crystallographic phases side-by-side (α-Fe(N) + γ’-Fe₄N) is consistent with the Gibbs phase rule after taking into account the parameter relating to the size of the nanocrystallite as an additional degree of freedom. The critical nitriding potentials in the processes of nanocrystalline iron nitriding and nitride reduction are directly proportional to the specific active surface area of nanocrystallites.

The adsorption kinetic model describes a system well if the adsorption rate is much lower than the rate of the phenomena occurring in the volume of the nanocrystallite. This paper presents a new model of the reaction of a nanocrystalline substance within the gas phase in a case where the reaction rate-limiting step is the rate of diffusion in the solid phase.

2. Results and Discussion

The solid-phase diffusion process was described in detail in Crank’s extensive study [1]. Based on the relationships presented there, it is possible to describe the kinetics of processes between the gas and solid phases in relation to processes occurring in nanocrystalline materials. It was assumed that:

-

The diffusion coefficient D does not depend on the concentration of absorbate and its value can be expressed by the relationship:

D = D₀ exp(−E_a/RT)

(1)

where E_a—activation energy of the diffusion process, R—gas constant, T—temperature, and D₀—pre-exponential coefficient;

-

Between gas phase and a solid state there are established chemical equilibrium states, at which the chemical potentials of sorbate in the gas phase μ^g, adsorbate on the crystallite surface μ^s, and absorbate in the crystallite volume μ^b are equal:

μ^g = μ^s = μ^b

(2a)

wherein

μ^s = E_s⁻ − RT ln (K^2D ln P_B)

(2b)

and

μ^g = 2μ^b

(2c)

wherein μ does not depend on crystallite sizes.

In a system with an unlimited solubility of a sorbate, an equilibrium state is established between the surface of the nanocrystallite and the gas phase; therefore, the molar surface concentration a^s of the adsorbate and the molar bulk concentration a^b of the absorbate correspond to the chemical potential P of this component in the gas phase:

ln a^s = K_ad lnP

(3a)

ln a^b = K_ab lnP

(3b)

ln a^s/ln a^b = K_seg lnP

(3c)

where K_ads—equilibrium constant of the adsorption process, and K_abs—equilibrium constant of the absorption process.

The free enthalpy values in the state of chemical equilibrium are expressed by the equations:

G^g/a^g = G^s/a^s = G^b/a^b

(4a)

lnK^g/a^g = lnK^s/a^s = lnK^b/a^b

(4b)

where a—activity of a component, G—free enthalpy, R—gas constant, T—temperature, K—equilibrium constant; superscripts g, s, and b—correspond to gas, solid, and bulk, respectively.

-

The process occurs in the kinetic area and the rate-limiting step is the adsorbate diffusion rate from the surface to the volume of nanocrystalline absorbent;

-

At low values of the concentration of the absorbate solution in the absorbent, Equation (5) is satisfied:

x^b = n_ab^b/n^b = K_ab^* P

(5)

where n_ab^b—moles of absorbate in crystallite volume, and n^b—sum of the moles of absorbate and absorbent.

During the process of creating dispersed structures, the surface endoenergetic effect associated with the creation of a free α phase surface ΔE^s = E_s⁻ a_A^s,max/a_A^b is balanced by the sum of the exoenergetic effects ΔG^b + ΔG^2D (Figure 1). The creation of two-dimensional structures on the resulting surface, as an exothermic process, can be represented as the vector ΔG^2D. Thus, the sum of energies associated with surface transformations can be expressed as ΔG^s = ΔE^s − ΔG^2D. Both the free enthalpy of the α solution formation ΔG^b and the free enthalpy of creating 2D structures ΔG^2D are functions of the chemical potential of the gas component. As a result of establishing equilibrium in the dispersion process, smaller crystallites are formed with a specific active surface area under constant formation conditions of temperature and chemical potential of component B.

The relationship describing ΔG^2D for the adsorption process is as follows:

ΔG^2D,ads = −RTa_A^s,max ln (K^adsln P_B) + ΔE^s

(6a)

where R—gas constant, T—temperature, and K^ads—adsorption process equilibrium constant related to activities of a component.

Similarly, one can derive relationships describing ΔG^2D for other processes:

-

Segregation:

ΔG^2D,seg = −RTa_A^s,max ln (K^ads/K^abs ln P_B) + ΔE^s

(6b)

where K^abs—absorption process equilibrium constant related to activities of a component.

-

Wetting:

ΔG^2D,wet = −RTa_A^s,max ln (K^sublK^adsln P_B) + ΔE^s

(6c)

where K^subl—sublimation process equilibrium constant related to activities of a component.

In general, we obtain:

ΔG^2D = −RTa_A^s,max ln (K^2Dln P_B) + ΔE^s

(6d)

where K^2D—equilibrium constant of the process of forming 2D monolayer of component B on the surface of phase α-A(B).

The free enthalpy of solution formation ΔG_B^b is given by:

ΔG_B^b = −RTa_B^b lnP_B

(7)

After substituting Equations (6d) and (7):

ΔG_B^b + ΔG^2D = −RT(a_B^b lnP_B − a_A^s,max ln (K^2Dln P_B))

(8)

According to the energy balance, smaller crystallites (blue color in Figure 1) require more energy to maintain their nanostructure when compared to larger crystallites (red color in Figure 1). Due to the different values of active specific surface areas and different ratios of surface area to volume for different crystallites, the free enthalpy connected with the surface phenomena ΔG^2D and connected with phenomena in the bulk of crystallite ΔG_B^b for the total energy of the crystallite will differ and the impact on ΔG^2D will be greater for smaller crystallites.

Using a graph of energy changes for the process of dispersion of a coarsely crystalline substance and the formation of surface structures of the α-A(B) phase in thermodynamic equilibrium (Figure 1), the slope of the line representing the change in the energy of the system together with the change in the P_B potential is determined by the relationship:

$$\frac{{∆E}_A^s-{∆G}^{2D}}{{∆G}_B^b+{∆G}^{2D}}=\frac{2E_{s-}{lna}_A^{s,max}}{-RT{lna}_B^{b}ln{P}_B-RTln{a}_A^{s,max}ln{K}^{2D}}=\frac{2E_{s-}lnA}{{{\mu }_B^{b}lna}_B^b+{\mu }_B^{s}lnA}$$

(9)

2.1. Kinetics of the Reaction in Solid Solution

Based on the literature data [31], the actual diffusion coefficients for carbon, nitrogen, and oxygen in metals in the temperature range of 200–500 °C are within the range of 10⁻¹⁴–10⁻¹² m²/s. A constant value of the diffusion coefficient in the volume of the sorbate nanocrystallite D = 10⁻¹³ m²/s was assumed for model calculations.

In order to determine the effect of the diffusion rate in the volume of nanocrystallite on the process rate in a solution with unlimited solubility of absorbate, model calculations of concentrations in the volume of a spherical nanocrystallite (with radius r_max in spheres with different radii r ∈ (0, r_max>) at constant gas-phase potential and surface concentration of adsorbent were performed using the equation proposed by Crank [1] and here modified to the following form:

$$\frac{x^b\left(r\right)}{x^s}=\frac{r_{\max }}{r}{\displaystyle \sum _{n=0}^n\left\{\mathrm{erfc}\frac{\left(2n+1\right)r_{\max }-r}{2\sqrt{\mathrm{Dt}}}-\mathrm{erfc}\frac{\left(2n+1\right)r_{\max }+r}{2\sqrt{\mathrm{Dt}}}\right\}}$$

(10)

where t—time, x^b—molar concentration of absorbate, and n—number of spheres in nanocrystallite (assumed n = r_max/a), a—lattice constant.

Figure 2 shows the results for crystallites of various sizes at a constant molar concentration of adsorbate on the surface for a defined, constant maximum potential P_max, assuming that ΔG_seg = 0.

From Figure 2a it can be concluded that for nanocrystallites of r = 40 nm, concentrations within their whole volume equilibrate and reach maximum values in less than 1 s (red, dashed line in Figure 2a). If we more precisely consider the process of equilibration of those concentrations in nanocrystallites of r = 40 nm (Figure 2b), we may state (based on model calculation results) that the time needed to equilibrate the concentration in the volume of such nanocrystallite is 0.01 s (red, dashed line in Figure 2b).

That model value r = 40 nm is based on the results of our measurements using scanning electron microscopy. According to these results, the largest iron nanocrystallites in the tested samples had diameters of approximately 80 nm. Our intention is to determine the values of characteristic parameters primarily for the largest nanocrystallites as those representing the end of the chemical reaction—if these large nanocrystallites undergo a phase transformation over time, smaller ones will undergo this transformation faster.

Based on the assumptions made so far, the schematic dependence of the absorbate concentration in nanocrystallite volume x^b on the process time and nanocrystallite radius was presented at a constant surface concentration of adsorbate x^s and with a diffusion coefficient of D = 10⁻¹⁹ m²/s at the interface composed of single layer of atoms of solid phase (Figure 3).

The surface concentration will be greater than the equilibrium concentration determined by the equilibrium between the surface area and the volume of the nanocrystallite. The recrystallization of the substrate phase to the product phase will take place after reaching the average surface and volume concentration of sorbate in the nanocrystallite, corresponding to the critical concentration in its entire volume. The rate at which the nanocrystallite saturates depends on the difference in surface concentration and the concentration in volume of the nanocrystallite.

In the solid phase, a constant level of absorbate concentration is quickly established at a constant maximum during the experiment at the adsorbate potential in the gas phase P_max.

Extensive research on diffusion and a thorough arrangement of various equations and models describing this process can be found in the work [32]. This work extends the scope of application of the presented models to nonlinear multicomponent diffusion, taking into account the kinetics of the chemical reactions in various states of matter. In the current work, we focused on processes involving chemical reactions in the solid phase, which is an attempt to answer one of the questions posed in the conclusions by the authors of the work [32] on diffusion and chemical reactions in such systems. Additionally, we assumed that the rate-limiting step of the process is the passage of atoms adsorbed on the surface into the volume of iron nanocrystallites through a single surface layer of iron atoms (the diffusion of the dissolved component in the volume of nanocrystallites is much faster and a product layer is not formed on the disappearing core as in the shrinking core model) [33].

The average molar concentration of a substance dissolved in the entire volume of nanocrystallite $\overline{x^b}\left(t\right)$ as a function of time was calculated using Equation proposed by Crank [1]:

$$\frac{\overline{x^b}\left(t\right)}{x^s}=6{\left(\frac{\mathrm{Dt}}{r_{\max }^2}\right)}^{\frac{1}{2}}\left\{{\pi }^{-\frac{1}{2}}+2{\displaystyle \sum _{n=1}^n\mathrm{ierfc}\frac{{\mathrm{nr}}_{\max }}{\sqrt{\mathrm{Dt}}}}\right\}-3\frac{\mathrm{Dt}}{r_{\max }^2}$$

(11)

Using Equation (11), the dependence of the average molar concentration of absorbate in a nanocrystallite with a radius of 40 nm on the time with diffusion coefficients D = 10⁻²⁰–10⁻¹³ m²/s is presented in Figure 4a,b and for different temperatures in Figure 4c.

It was found that for a diffusion coefficient of 10⁻¹³ m²/s (typical for coarse-crystalline materials) in nanocrystalline materials, the concentration of absorbate and the concentration of adsorbate on the surface of the nanocrystallite were equalized in a very short period of time (Figure 4a). For a diffusion coefficient of 10⁻¹⁹ m²/s, the steady state absorbate concentration was reached over a period of about 2000 s (Figure 4b).

A significant difference in the values of diffusion coefficients leads to the conclusion that the stage limiting the rate of the process is the rate of diffusion of the adsorbate from the surface across the phase interface into the nanocrystallite volume. The adsorbate must overcome not only the energy barrier related to the vibrations of the lattice but also the barrier related to the dissolution of the absorbate in the crystallite volume. Equation (1) will change its form:

D = D₀ exp[−(E_a + ΔG_dis)/RT]

(12)

where E_a—activation energy of diffusion through the crystallite surface, and ΔG_dis = −ΔG_seg—enthalpy of dissolution of the adsorbate in the volume of nanocrystallite.

The effect of the change in E_a + ΔG_dis and the temperature on the change in concentration over time is shown in Figure 4b.

The rate of the chemical reaction between the gaseous substance and one nanocrystalline solid of α phase can be described taking into account the influence of three parameters on the course of the chemical reaction: temperature, chemical potential of the gas phase, and nanocrystallite size distribution:

$$\frac{d\overline{x^b}\left(t\right)}{dt}=\frac{\partial \overline{x^b}\left(t\right)}{\partial T}\frac{dT}{dt}+\frac{\partial \overline{x^b}\left(t\right)}{\partial P\left(t\right)}\frac{dP\left(t\right)}{dt}+\frac{\partial \overline{x^b}\left(t\right)}{\partial r}\frac{dr}{dt}$$

(13)

For a single nanocrystallite, the reaction rate can be described as:

$${\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)}_i=A_i\left[P\left(t\right)\right]k_0\exp \left(\frac{-E}{RT}\right)$$

(14)

From Equations (12) and (13) and taking into account Equation (14), we obtain:

$$\begin{array}{l}{\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)}_i=A_i{D}_0\exp \left(-\frac{E_a+\Delta G_{dis}}{RT}\right)\frac{E_a+\Delta G_{dis}}{R{T}^2}P\left(t\right)\frac{dT}{dt}+A_i\frac{d{P}_{\max }}{dt}+\\ +A_i{D}_0\exp \left(-\frac{E_a+\Delta G_{dis}}{RT}\right)P\left(t\right)\frac{dr}{dt}\end{array}$$

(15)

The measured reaction rate is the sum of the process rates on individual i-th nanocrystallites in the sample:

$$\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)={\displaystyle {\sum }_{i=\min }^{i=\max }{n}_i^b{\left[\frac{d\overline{x^b}\left(t\right)}{dt}\right]}_i}={\displaystyle {\sum }_{i=\min }^{i=\max }{n}_i^b{\left[\frac{d{n}^b\left(t\right)}{n_i^{b,\alpha }dt}\right]}_i}={\displaystyle {\sum }_{i=\min }^{i=\max }f\left(GMD\right){\left[\frac{d{n}^b\left(t\right)}{dt}\right]}_i}$$

(16)

The actual system consists of a set of crystallites described by the probability density distribution of their sizes characterized by the specific surface area of the crystallite A = S/V. It was assumed in this paper that this is a bimodal distribution consistent with experimental tests [25,26,28,29,30] and model calculations [29,30] for the nanocrystalline iron-ammonia-hydrogen system.

The dependence of the total concentration of the absorbate in the volume of sample of nanocrystallites x^b on temperature, chemical potential of the gas phase, and crystallite size can be described by continuous functions as follows:

$$x^b={\displaystyle \underset{T0}{\overset{T}{\int }}{\displaystyle \underset{P0}{\overset{P}{\int }}{\displaystyle \underset{r0}{\overset{r}{\int }}\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)dTdP\left(t\right)dr}}}$$

(17)

In the area of one α phase and at constant temperature as well as at constant potential of the gas phase, the averaged reaction rate is as follows:

$${\left(\frac{\Delta \overline{x^b}\left(t\right)}{\Delta t}\right)}_i=n_i^b\left(\frac{{\displaystyle \underset{x^b=0}{\overset{x^b}{\int }}{\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)}_i^{\alpha }}}{n_i^{b,\alpha }\Delta t_i}+C_1+C_2\right)$$

(18)

where C₁, and C₂—constants taking into account the influence of temperature and gas-phase potential on the reaction rate.

2.2. Kinetics and Thermodynamics of a Solution at Different Constant Potentials of the Gas Phase

Assuming ΔG_seg = 0, the chemical compositions of the surface and volume at equilibrium are equal to each other. The dependence of concentration in volume x^b on the time, based on Equation (11) for different constant potentials (0.25, 0.50, 0.75, and 1.00), is shown in Figure 5.

The gas-phase potential determines the surface concentration related to the specific surface area. In the kinetic region of the process, the concentrations in the crystallite volume are a function of the gas-phase potential and the crystallite radius. Crystallites reach a certain concentration in the order of their radii from the smallest to the largest. After an infinitely long time, the state of chemical equilibrium is established, where Equation (2) is satisfied for each crystallite size. The maximum concentration of the adsorbate in the surface layer x^s_max and the absorbate in the crystallite volume x^b_max are equal to each other and are a function of the gas-phase potential only.

2.3. Thermodynamics and Kinetics of the System with Phase Transformation (Two-Phase System)

With the chemical potential of the adsorbate in the gas phase P₀(A)_i and depending on the specific active surface area of the i-th nanocrystallite at the critical concentration, x^b,cri, the i-th nanocrystallite of the α phase undergoes a phase change α→β in its entire volume. At this potential, the phase β is formed with the concentration x_i^b,β,min and the maximum concentration of defects x_D^max (Figure 6). The defect concentration is equal to the difference between the maximum absorbate concentration in the β phase and the minimum absorbate concentration in this phase after the transition from the α phase:

x_D^β = x_i^β,cri − x_i^β,min = −x^β

(19)

At constant chemical potentials and after an infinitely long time, the state of chemical equilibrium is established, characterized by a specific equilibrium potential of the gas phase, activities in the solid phase on the surface and in the volume, process temperature, and specific active surface (Figure 6). The phase transformation of the α phase occurs when the critical values are taken into account in Equation (3):

ln a^s,cri = K_ad lnP₀

(20)

ln a^b,cri = K_ab lnP₀

(21)

where a^s,cri—critical activity at the surface, a^b,cri—critical activity in crystallite volume, and P₀—equilibrium potential of the gas phase.

Assuming that the reference conditions are the process conditions (constant temperature and zero chemical potential of the gas phase), the change in free enthalpies can be replaced by the values of free enthalpies of sorbate in the individual phases:

$$K_{ab}^{\alpha ,cri}=\frac{\ln a^{b,\alpha ,cri}}{\ln P_0\left(A_i\right)}=\exp {\left(-\frac{\Delta G^{\alpha ,cri}}{RT}\right)}_i$$

(22)

For small concentrations, a simplification can be used:

$$K_{ab}^{*\alpha ,cri}=\frac{x^{\alpha ,cri}}{P_0\left(A_i\right)}$$

(23)

The ratio of the equilibrium constants under phase transition conditions is as follows:

$$\begin{array}{l}\frac{K_{ab}^{\alpha ,cri}}{K_{ab}^{*\alpha ,cri}}=\frac{P_0\left(A_i\right)\ln a^{b,\alpha ,cri}}{x^{\alpha ,cri}\ln P_0\left(A_i\right)}=\frac{P_0\left(A_i\right)\ln \left(x^{b,\alpha ,cri}f\right)}{x^{\alpha ,cri}\ln P_0\left(A_i\right)}=\frac{P_0\left(A_i\right)}{\ln P_0\left(A_i\right)}\frac{\ln x^{b,\alpha ,cri}+\ln f}{x^{\alpha ,cri}}=\frac{P_0\left(A_i\right)}{\ln P_0\left(A_i\right)}\left(\frac{\ln x^{b,\alpha ,cri}}{x^{\alpha ,cri}}+\frac{\ln f}{x^{\alpha ,cri}}\right)\\ \\ K_{ab}^{*\alpha ,cri}=\frac{x^{\alpha ,cri}\ln P_0\left(A_i\right)}{P_0\left(A_i\right)\ln a^{b,\alpha ,cri}}\exp {\left(-\frac{\Delta G^{\alpha ,cri}}{RT}\right)}_i\end{array}$$

(24)

$$K^{\beta ,\min \prime }=\frac{\ln a^{\beta ,\min }}{\ln P_0\left(A_i\right)}=\exp {\left(-\frac{\Delta G^{\alpha \to \beta }-\Delta G^{\alpha ,cri}}{RT}\right)}_i$$

(25)

According to the balance (26), the phase transformation takes place at a constant energy of the system, i.e., the sum of the free enthalpies: of deformation of the crystal lattice in the α phase volume in the absorption process ΔG^α→β, of a component dissolved in the nanocrystallite volume ΔG^b, of a chemisorbed component on the nanocrystallite surface ΔG^s; and the change in surface energy of nanocrystallite ΔE is equal to zero and Equations (26)–(28) are also satisfied [29]:

−ΔG^α→β + ΔG^b + ΔG^s + ΔE = 0

(26)

For low gas-phase potentials:

$$\Delta G^b=x^{\alpha ,cri,b}\left(\Delta G_{seg}^{b,\beta }-\Delta G_{seg}^{b,\alpha }\right)$$

(27)

$$\Delta G^s=\Delta x^{a,s}\Delta G_{dis}^{\beta }=-\Delta x^{a,s}\Delta G_{seg}^{\beta }$$

(28)

Due to the interactions between the atoms adsorbed on the adsorbent surface, for high concentrations of the adsorbate, Equation (28) will take the form:

$$\Delta G^s=\Delta a^{a,s}\Delta G_{dis}^{\beta }=-\Delta a^{a,s}\Delta G_{seg}^{\beta }$$

(29)

The change in surface energy at high adsorbate concentrations is expressed as:

$$\Delta E=a^{\beta ,free}\Delta E^{\alpha \to \beta ,s}$$

(30)

wherein

$$\Delta E^{\alpha \to \beta ,s}=E^{free}\left(A-a^s\right)+E^{occup}{a}^s$$

(31)

Based on the Equation (8), the sum of ΔG^b + ΔG^2D was calculated for crystallites with specific active surfaces in the range of 0.075–0.300 nm⁻¹ (Figure 7). The graph also shows the values of ΔG^2D and ΔG^b for individual crystallites calculated on the basis of the Equations (4b) and (5), respectively, with a gas-phase potential of 0.002 and segregation free enthalpy ΔG_seg = −20 kJ.

The proportion of free enthalpy related to the crystallite surface of the total free enthalpy of the system is 97% for a crystallite with a radius of 20 nm. Thus, it can be assumed that smaller crystallites can be treated as nanocrystallites, and larger crystallites will have properties corresponding to coarse-crystalline substances. Based on Equation (6d), knowing the enthalpy balance for at least two crystallites and extrapolating the obtained values to the conditions of coarse-crystalline substances (green line in Figure 7), it is possible to determine the value of the enthalpy ΔE^s related to the process of coarse-crystalline substance dispersion.

The values of the expression ΔG^b + ΔG^2D were determined on the basis of Equation (8) for crystallites with specific active surfaces in the range of 0.075–0.300 nm⁻¹ (Figure 8) with a gas-phase potential in the range of 0.25–0.75 and segregation free enthalpy ΔG_seg = −20 kJ.

For crystallites with different A, the expression ΔG^2D/ΔG^b will have a different value: the smaller the value of A, the smaller the ratio of ΔG^2D/ΔG^b. As a result of establishing equilibrium, smaller crystallites are transformed under the higher chemical potential of the gas components rather than larger crystallites.

In terms of surface tension, usually the joules (J) per square meter of surface area (m²) unit is used [34,35]. According to the definitions adopted in this work, unit J/mol of surface [36] is used for the molar chemical potential of the surface where the surface area is expressed in terms of mole of surface atoms which corresponds to the usual unit for surface tension. The energy for creating a clean surface ΔE^s is then given by:

ΔE^s = E_s⁻ A = −A RTlnx^s

(32)

where E_s⁻—molar chemical potential of free surface, and f—activity coefficient.

The total surface energy ΣΔE_A^s, defined for a specific surface and volume of crystallite and taking into account the fraction of free and occupied crystallite surface, is a function of surface concentration adsorbate and is equal to [27]:

$$\Sigma \Delta E^s=a^{s,\max }\left[E_{s^{-}}\left(a^{s,\max }-a^s\right)+E_{s^{-}}^{ad}{a}^s\right]=A\left[E_{s^{-}}\left(a^{s,\max }-a^s\right)+E_{s^{-}}^{ad}{a}^s\right]$$

(33)

where Es−ad—molar chemical potential of occupied surface.

After substituting Equations (6d), (7) and (32):

$$-x^{b}RT\ln P-a^{s,\max }RT\ln K^{2D}+a^{s,\max }\left[E_{s^{-}}\left(a^{s,\max }-a^s\right)+E_{s^{-}}^{ad}{a}^s\right]=0$$

(34)

This balance applies to a crystallite and only to the extent of restructuring its surface. When a^s,max is changed, the active specific surface area changes despite the constant x^b. Equation (34) represents the minimum system energy at a given volume of crystallites and their specific surface.

The dependence of the energy change needed for the phase transformation for the S/V value of the i-th crystallite on the potential can be approximated by a linear function:

E(S/V) = aP + b

(35)

When ΔG_seg is less than or greater than zero, according to the McLean–Langmuir equation at equilibrium, x^s_max will be different from x^b_max, and moreover, these concentrations will be a function of the crystallite radius and the potential P₀^α,max(A_i):

$$\begin{array}{l}{\left[\frac{K^{\alpha }\ln P_0^{\alpha ,\max }\left(A_i\right)}{A_i-K^{\alpha }\ln P_0^{\alpha ,\max }\left(A_i\right)}\right]}_i={\left[\frac{\ln a_{\max }^{b,\alpha }}{1-\ln a_{\max }^{b,\alpha }}\exp \left(-\frac{\Delta G_{seg}^{\alpha }}{RT}\right)\right]}_i\\ {\left[\frac{K^{\alpha }\ln P_0^{\alpha ,\max }\left(A_i\right)}{A_i-K^{\alpha }\ln P_0^{\alpha ,\max }\left(A_i\right)}\right]}_i={\left[\ln x_{\max }^{b,\alpha }\exp \left(-\frac{\Delta G_{seg}^{\alpha }}{RT}\right)\right]}_i\end{array}$$

(36)

After calculating the logarithm of the maximal activities of the absorbate $\ln a_{\max }^b=\ln P_0^{\alpha ,\max }\left(A_i\right)K^{\alpha ,abs}$ and the adsorbate $\ln a_{\max }^s=\ln P_0^{\alpha ,\max }\left(A_i\right)K^{\alpha ,ads}$ in the α phase, the value of the free segregation enthalpy ΔG_seg^α can be determined for each crystallite in the system. In the coordinate system ln$a_{\max }^b$ = f(ln$a_{\max }^s$) = f(lnP₀^max), the relation $a_{\max }^b$/$a_{\max }^s$ is a line with slope K^α,ab.

If we consider a system containing nanocrystallites with specific active surfaces A_i ∈ <A_min–A_max>, nanocrystallites will undergo phase transformation in their entire volume at the potentials P₀(A)_i ∈ <P₀(A_min)–P₀(A_max)>. The value of ΔE is directly proportional to the specific active surface area, and therefore, the fractions of larger crystallites will undergo a phase change at lower P₀(A_i) potentials and smaller x^s and x^b concentrations than smaller crystallites (Figure 9)—in equilibrium states established with increasing gas-phase potential, crystallites will reach maximum concentrations and undergo phase changes in the order of their radius from the largest to the smallest, and the α and β phase crystallites will be present in the sample next to each other at a specific potential P₀(A_i) ≈ P₀(A_min).

With (P(t) − P₀(A_min) → 0, the rate of crystallite conversion with r_max = 10 nm is lower and occurs at higher gas-phase potentials than the crystallite conversion rate with r_max = 40 nm. The chemical potential at which phase transformation of i-th crystallite occurs can be expressed by the relationship:

lnP₀(A)_i = ln P₀(A_min) + A_ilnx^s_cri/lnx^b_cri = ln P₀(A_min) − A_iK^seg

(37)

Under the conditions where the potential P(t) >> P₀(A_min), all crystallites can undergo a phase change (Figure 10) and the rate of crystallite conversion with r_max = 10 nm is higher and occurs at lower gas-phase potentials than the crystallite conversion rate with r_max = 40 nm.

Taking into account the phase transformation and the formation of the second nanocrystalline phase β for a single nanocrystallite, the total average reaction rate can be expressed by the equation:

$${\left(\frac{\Delta \overline{x^b}\left(t\right)}{\Delta t}\right)}_i=n_i^b\left(\frac{{\displaystyle \underset{x^b=0}{\overset{x^{cri}}{\int }}{\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)}_i^{\alpha }}}{n_i^{b,\alpha }\Delta t_i}+\frac{{\left(x^{\beta ,\min }-x^{\alpha ,cri}\right)}_i}{n_i^{b,\alpha }\Delta t_i}+\frac{{\displaystyle \underset{x^{\beta ,\min }}{\overset{x^{\beta ,\max }}{\int }}{\left(\frac{d\overline{x^b}\left(t\right)}{dt}\right)}_i^{\beta }}}{n_i^{b,\alpha }\left[\Delta t_i+t\left(x^{\beta }\right)\right]}+C_1+C_2\right)$$

(38)

After the phase transition, the concentration difference is expressed only by the second component of the right side of the Equation (38).

2.4. Process in the Isothermal System and the Linear Relationship P = f (t)

Equation (13) takes into account the influence of three parameters on the reaction rate. Assuming that the system is isothermal, Equation (13) boils down to the form:

$$\frac{d\overline{x^b}\left(t\right)}{dt}=C_1+\frac{\partial \overline{x^b}\left(t\right)}{\partial P\left(t\right)}\frac{dP\left(t\right)}{dt}+\frac{\partial \overline{x^b}\left(t\right)}{\partial r}\frac{dr}{dt}$$

(39)

Assuming that the adsorbate concentration in the gas phase x^g(t) varies according to the linear model:

$$P\left(t\right)=x^g\left(t\right)=kt$$

(40)

where k—proportional coefficient, t—time, and adsorbate surface concentration changes over time according to Equation (3a).

Any changes in absorbate concentration x^b as a function of radius, when the concentration changes in adsorbate are described by Equations (3a) and (40), can be calculated using Equation (41) (for the initial conditions, the concentration in the nanocrystallite volume is zero and the concentration on its surface is a function of time) [1]:

$$\frac{x^b\left(r\right)}{A}=k\left(t-\frac{r_{\max }^2-r^2}{6D}\right)-\frac{2k{r}_{\max }^3}{D{\pi }^{3}r}{\displaystyle \sum _{n=1}^n\frac{{\left(-1\right)}^n}{n^3}\exp \left(-D{n}^2{\pi }^{2}t/r_{\max }^2\right)\sin \frac{n\pi r}{r_{\max }}}$$

(41)

The average concentration of the solute in the entire volume of nanocrystallite during the reaction with the gas phase can be determined using Equation (42) [1]:

$$\frac{\overline{x^b}\left(t\right)}{A}=\left(t-\frac{r_{\max }^2}{15D}\right)+\frac{6r_{\max }^2}{{\pi }^{4}D}{\displaystyle \sum _{n=1}^n\frac{1}{n^4}\exp \left(-D{n}^2{\pi }^{2}t/r_{\max }^2\right)}$$

(42)

Calculation results for nanocrystallites with radii of 10 nm (A = 0.3000 nm⁻¹) and 40 nm (A = 0.0750 nm⁻¹) and for a diffusion coefficient of 10⁻¹⁹ m²/s are presented in Figure 11.

Assuming that P(t) >> P₀(A), the smaller crystallite (r_max = 10 nm) reaches the critical concentration in a shorter time and undergoes a phase change compared to the larger crystallite (r_max = 40 nm).

2.5. Process in the Isothermal System and the Exponential Relationship P = f (t)

Assuming that the adsorbate concentration in the gas phase x^g(t) varies according to the leaching model with perfect mixing:

$$P\left(t\right)=x^g\left(t\right)=x_0^g\left[1-\exp \left(-\frac{v}{V_r}t\right)\right]$$

(43)

where v—gas flow rate, V_r—reactor volume, x₀^g—maximum concentration in the gas phase, and the adsorbate surface concentration changes over time according to Equation (3a).

Any changes in absorbate concentration x^b as a function of radius, when the concentration changes in adsorbate are described by Equations (3a) and (43), can be calculated using Equation (44) (for the initial conditions, the concentration in the nanocrystallite volume is zero and the concentration on its surface is a function of time) [1]:

$$\frac{x^b\left(r\right)}{A}=1-\frac{r_{\max }}{r}\exp \left(-\frac{v}{V_r}t\right)\frac{\sin \left[{\left(\frac{v}{V_r}{r}_{\max }^2/D\right)}^{\frac{1}{2}}r/r_{\max }\right]}{\sin {\left(\frac{v}{V_r}{r}_{\max }^2/D\right)}^{\frac{1}{2}}}-\frac{2\frac{v}{V_r}{r}_{\max }^3}{\pi \mathrm{Dr}}{\displaystyle \sum _{n=1}^n{\left(-1\right)}^n\frac{\exp \left(-{\mathrm{Dn}}^2{\pi }^{2}t/r_{\max }^2\right)}{n\left(n^2{\pi }^2-\frac{v}{V_r}{r}_{\max }^2/D\right)}\sin \frac{rn\pi }{r_{\max }}}$$

(44)

The average concentration of the solute in the entire volume of nanocrystallite during the reaction in the gas phase can be determined using Equation (45) [1]:

$$\frac{\overline{x^b}\left(t\right)}{A}=1-\frac{3D}{\frac{v}{V_r}{r}_{\max }^2}\exp \left(-\frac{v}{V_r}t\right)\left[1-{\left(\frac{\frac{v}{V_r}{r}_{\max }^2}{D}\right)}^{\frac{1}{2}}\cot {\left(\frac{\frac{v}{V_r}{r}_{\max }^2}{D}\right)}^{\frac{1}{2}}\right]+\frac{6\frac{v}{V_r}{r}_{\max }^2}{{\pi }^{2}D}{\displaystyle \sum _{n=1}^n\frac{\exp \left(-{\mathrm{Dn}}^2{\pi }^{2}t/r_{\max }^2\right)}{n^2\left(n^2{\pi }^2-\frac{v}{V_r}{r}_{\max }^2/D\right)}}$$

(45)

Calculation results derived from Equation (44) for 10 nm (A = 0.3000 nm⁻¹) and 40 nm nanocrystallites (A = 0.0750 nm⁻¹), with a diffusion coefficient of 10⁻¹⁹ m²/s and various values of the parameter v/V_r = <0.0003–0.0300>, are shown in Figure 12.

The results presented in Figure 12 are presented as the change in ln $\overline{x^b}\left(t\right)/A$ depending on ln P(t) = ln x^g (Figure 13).

After differentiating over time, the relationships shown in Figure 12a for v/V_r = 0.0030 and two crystallites; the average rate of the absorption process was determined according to Equation (19) and presented in Figure 14.

The rate of the absorption process at the time of the phase change for crystallites of 10 and 40 nm is, respectively, 0.0021 and 0.0009 t⁻¹.

After summing the results presented in Figure 12 for individual crystallites in the range of 10–40 nm (every 1 nm) and according to the crystallite shares given by the bimodal mass distribution, a total and integral diagram of concentration changes in the entire crystallite system is obtained for the kinetic (Figure 15a) and equilibrium regions (Figure 15b).

To sum up, in the current work, we deal with a chemical reaction in a solid phase—nitriding reaction of nanocrystallite iron, for which we have assumed a probable mechanism of reaction with the diffusion resistance of an extremely thin surface layer (literally surface layer—we consider the change in the thermodynamic states of nitrogen after passing through a single layer of iron atoms on the nanocrystallite surface)—model No. 1. Since this is the rate-limiting step of the entire process, the equalization of concentrations (i.e., eliminating concentration gradients in the volume of iron nanocrystallites) should occur faster, and we will not observe a growing layer of the product but rather an equal concentration in the volume of nanocrystallites. Then, after exceeding the critical concentration, the phase transformation of the substrate into the product will occur immediately in the entire volume of the nanocrystallite. If the product layer does not grow in the proposed model, the associated diffusion resistance does not change (increase) as the reaction progresses, as occurs in the shrinking core model (model No. 2) [33]. The adsorption model of the reaction is also known (model No. 3) [27], in which the rate of the process is limited by the rate of adsorption and dissociation of the gaseous substrate on the nanocrystallite surface. It was also assumed that there is no concentration gradient in the volume of nanocrystallites (an equilibrium is established between the adsorbate on the surface and the substrate dissolved in the volume of nanocrystallites), and the substrate–product phase transformation will occur in the entire volume of nanocrystallites. So, to sum up, we may compare those models and conclude that there are some similarities (in model No. 1 and in model No. 3 there is no concentration gradient in the volume of nanocrystallites; in model No. 1 and in model No. 2 the process rate is limited by the diffusion rate) but also differences (in model No. 1, the process rate is limited by the diffusion rate and in model No. 3 by the rate of gas adsorption on the surface of the nanocrystallites; in model No. 1, diffusion only concerns the passage of adsorbate atoms from the surface through an unchanging interface into the nanocrystallite volume, where the concentrations are equalized, and in model No. 2, the diffusion rate changes due to the surface layer of the product growing as the reaction progresses, limiting the possibility of contacting the gaseous substrates and those remaining in the solid phase).

The proposed model can be used to describe any reaction system in which a phase transformation may occur as a result of exceeding a certain critical concentration in the volume of nanoparticles. Examples include nitriding, carburizing, and oxidation reactions of transition metal nanoctystalites and reduction in the resulting products, e.g., with hydrogen. Experimental results for the reactions we obtained earlier inspired us to develop the current reaction model. Another example may be the growth of a solid clathrate hydrate phase upon exposure of ice to hydrate-forming gases, such as methane or carbon dioxide described in a scientific paper by Englezos et al. [37].

3. Conclusions

According to the model calculations carried out, nanocrystallite volume concentrations level out very quickly. That justifies the assumption for further calculations that at the time of transformation the average concentration in the nanocrystallite volume is equal to the concentration of the adsorbate in the absorbent.

If the adsorption process does not limit the reaction rate of the gaseous component with the nanocrystalline absorbent, the surface concentration of the adsorbate is directly proportional to the logarithm of the adsorbate potential in the gas phase.

The rate of diffusion across the phase boundary limits the rate of the process. With perfect mixing of the gas component in a reactor volume, the sorption process rate depends on the ratio of gas flow through the reactor volume, i.e., the ratio of the logarithm of the chemical potential of the gas adsorbate to the volume of nanocrystallite.

In processes in which phase transformation takes place, the critical concentration of absorbate depends on the specific active surface area of the adsorbent and the difference in chemical potentials of the gas-phase and equilibrium potentials at which the phase transformation occurs. When the adsorbate potential in the gas phase is greater than the equilibrium potential of the nanocrystallite with the largest specific active surface, nanocrystallites undergo phase changes in the order according to their specific active surfaces from the largest to the smallest. In a process in which the adsorbate potential P(t) changes according to the relationship P(t)_i = P₀(A)_i, nanocrystallites undergo phase transformation in order of their specific active surface from the smallest to the largest.