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Abstract

For the characterization of porous structures, besides the specific surface area, the pore size distribution is of special interest. In this paper, the pore size distribution of mesoporous structures is calculated from the adsorption branch of the hysteresis loop, whereby only loops of type H1 are considered. The modeling is done only for cylindrical capillaries and the calculation of the pore size distribution is done by a combination of the percolation theory and the theory of capillary condensation. By applying percolation theory to the adsorption process, the pressure dependence of pore filling during capillary condensation is described. Because of a lack of strictly theoretically based equations for the adsorption process by capillary condensation, a mathematical adaptation to measured data is done by the Bernoulli equation. This is proposed by the authors for the first time for a description of adsorption–desorption processes. Thus a new, strictly thermodynamic method to calculate the mesopore size distribution is gained. As an example, the pore size distribution is calculated. The results coincide well with other evaluation methods.

Keywords

Mesopores capillary condensation percolation pore size distribution modeling

Introduction

Consider a thermodynamic system, a porous body, that is in contact with a gas (vapor) at pressure p and temperature T. The characteristic potential of the thermodynamic system is the free energy F. We consider the function F in the multidimensional thermodynamic space (multiformity) F (x, y, z, p, T, V), where V is the amount of condensate in the pore and x, y, z are the cartesian coordinates of an arbitrarily chosen pore. We will take a porous medium spatially statistically homogeneous in the sense that the probability of detecting a pore at any place with given characteristics does not depend on the coordinates (x, y, z).

We further will describe the thermodynamic system of a porous solid–gas (vapor) system in the framework of the metric geometry of equilibrium thermodynamics. In the first approximation, the geometry of a porous medium was described by the Bernoulli equation.

The governing parameter of the processes of condensation (adsorption) and evaporation (desorption) is the Laplace pressure. At a critical Laplace pressure characteristic of a pore of a given size, the core of the pore is in equilibrium with the bulk phase. For simplicity, we treat cylindrical pores only. If, for a pore with a certain radius value, the Laplace pressure changes from subcritical to supercritical, the pore is filled in a fluctuating way by capillary condensate, the “birth” of the filled pore and the “death” of the empty pore (adsorption) occur. When Laplace pressure decreases from supercritical to subcritical condensate evaporates from the core (desorption). The “birth” of the empty core and the “death” of the filled core occur. In thermodynamics, the processes of “birth”and “death” are described by the Bernoulli equation. In this case, the Bernoulli equation should be considered as the equation of state for the condensed phase in the core. This equation does not contain any assumptions or approximations.

One of the most important characteristics of porous media is the pore size distribution (PSD). In case of mesopores, the methods for obtaining PSD are based on experimental data of ad/desorption processes illustrated by the hysteresis loop. Here the PSD is calculated from the adsorption (desorption) branches of the hysteresis loop. The calculation of the PSD for mesoporous bodies is based on the capillary condensation theory and its determining equation, the Kelvin equation (Gregg and Sing, 1982). In this paper only the hysteresis loop H1 is considered. The H1 hysteresis loop is characteristic of adsorbents with a narrow range of uniform mesopores (Thommes et al., 2015). When analyzing the data of the adsorption experiment, the model of effective spherical or cylindrical pores is adopted as a first approximation (Adamson and Gast, 1997; Gregg and Sing, 1982).

In 1945 Wheeler proposed a method for calculating the PSD taking the model of an open cylindrical capillary as a basis (Wheeler, 1955). Subsequently, Wheeler’s approach was developed in the works of Barrett–Joyner–Halenda, Cranston–Inkley, and Dollimore–Heal. An analysis of these methods for calculating the PSD was given in Shull et al. (1948) and Barrett et al. (1951). Based on Wheeler’s approach, a generalized equation was proposed that makes it possible to obtain a PSD for pores of spherical, cylindrical, and slit forms (Shkolnikov and Sidorova, 2007).

At present, the formalism of percolation theory is successfully used to investigate adsorption hysteresis processes (Isichenko, 1992). In Isichenko’s (1992) work, processes of filling and emptying of nonwetting liquid from a nanoporous body for a spherical pore model are considered. However, due to the specific features of the thermodynamic system analyzed, the analysis of the hysteresis did not take into account the influence of a stable condensate film on the pore walls that occurs during the filling and emptying of the pores. This however is the determining factor in the Wheeler’s method.

The purpose of this paper is the theoretical justification of the method for calculating the PSD, which will be based on the formalism of percolation theory developed in Isichenko (1992) and the theory of capillary condensation on which the Wheeler’s method is based.

Theory

The most accepted theory of filling (emptying) pores with capillary condensate in mesopores is the percolation theory. In the framework of the theory of percolation, the filling of pores with a capillary condensate in a nanoporous body can be represented as a process of formation and growth of an infinite cluster of pores that are filled at a Laplace pressure. In this case, the surface tension forces are compensated by external pressure.

First, we define an open cylindrical capillary as the model pore. In addition, we assume that the length of the cylindrical pores l(R) is directly proportional to the radius R (proportionality factor n), and the active centers are distributed over the surface of the cylinder with a uniform, continuous density.

We assume that the PSD f(R) is spatially homogeneous. In the first approximation, we will not take into account the percolation effects associated with the scatter of the pore radius.

The process of capillary condensation is of fluctuation nature. In the first approximation, we assume that there is no interaction between different fluctuations. The independence of the processes happening in a certain pore from those in the adjacent pores can only be considered for the adsorption branch of the hysteresis loop (Gregg and Sing, 1982).

In percolation theory, when considering the process of interaction of gases (vapors) with a nanoporous adsorbent the determining factor is the susceptibility of the thermodynamic system calculated for the adsorption or desorption branches of the hysteresis loop (Isichenko, 1992).

We consider the process of filling mesopores by using the formalism of percolation theory (Isichenko, 1992). The value of the relative fraction of pores $θ (p)$ that are filled with a capillary condensate at a pressure $p$ is

θ (p) = \frac{1}{N_{0}} \int_{R (p)}^{\infty} f (R) d R

(1)

The value $θ (p)$ is analogous to the fraction of lattice sites introduced in the theory of percolation. This is the probability of finding a pore in the nanoporous body that is available for filling with a capillary condensate.

In equation (1) N₀ is the number of pores per unit volume; the lower limit of integration R(p) should be determined by setting the pressure in the bulk phase equal to the Laplace pressure determined for the given cluster.

When the pressure in the bulk phase is increased, pluralities of infinite clusters of filled pores are formed in the nanoporous body, each of which corresponds to a certain value of the Laplace pressure. The transformation of clusters of available pores into clusters of filled pores can be considered as their interaction. By analogy with the theory of percolation for the adsorption process, one can introduce a critical fraction of the accessible pores θ_c at which an infinite cluster of filled pores appears.

The possible value of the number of pores N in clusters at a given pressure in the bulk phase p can be conveniently described by the cluster distribution function in terms of the number of pores (Shkolnikov and Sidorova, 2007)

F (N) = C N^{- τ}

(2)

For 3D systems $τ =$ 2.2; C is the normalization coefficient. The number of pores in clusters (2) is (Isichenko, 1992)

N (ξ) \approx {| θ - θ_{c} |}^{- \frac{1}{τ}}

(3)

If the distribution function F(N) is known, the relative volume of all filled pores V(p) per unit volume of a porous body in a geometric cluster size ξ and pressure p is defined as follows

V (p) = \int_{1}^{N (ξ)} N F (N) ν (R) d R

(4)

The upper limit of integration is determined in percolation theory by formula (3), where ξ is the geometric cluster size (Isichenko, 1992). The quantity ν(R) is the pore volume in the cluster at a pressure p along the adsorption branch of the hysteresis loop.

Differentiating the integrals (1) and (4) along the limits of integration, taking into account the dependences (2) and (3), we obtain the following formula for the susceptibility of the adsorption system

ε = \frac{d}{dp} V_{cond} (p) \approx n {(θ - θ_{c})}^{β - 1} f (R) \frac{dR}{dp}

(5)

with V_cond(R): volume of capillary condensate; p: equilibrium pressure of the adsorbate in the bulk phase on the adsorption branch of the hysteresis loop;n: proportionality factor; θ and θ_c: proportion of available pores and the critical fraction of available pores; β: critical exponent of an infinite cluster of filled pores; for the 3D system β = 0.42 (Isichenko, 1992).

Then from equation (5) we obtain the following equation for calculating the PSD

f (R) \approx \frac{{(θ - θ_{c})}^{1 - β}}{n} \frac{d}{dp} V_{cond} (p) {(\frac{dR}{dp})}^{- 1}

(6)

Allowing for the comparison of quantitative and qualitative analysis of the differential functions f(R) for various adsorption systems, it is necessary to renormalize the function f(R) in the following form

φ (R) = \frac{f (R)}{\int_{R} f (R) d R}

(7)

So the differential function f(R) is normalized to the conditions

\int_{R} f (R) d R = V_{cond}

(8)

The use of the function $φ$ (R) as a result makes it possible to avoid the proportionality factor in equation (6). Thus, the calculation of the function $φ$ (R) simplifies both the calculation of the susceptibility of the adsorption system $ε$ and the derivative (dR/dp) from the experimental adsorption isotherm.

The susceptibility $ε$ can be calculated either analytically, if the equation of the adsorption branch of the hysteresis loop is known, or numerically from the experimental data describing the adsorption branch.

To calculate the pressure derivative of the function R(p), we use the equation for determining the radius of an open cylinder with capillary condensation (Brockhoff and Linsen, 1970)

R = t_{c} + \frac{γ V_{mol}}{RT} \frac{1}{[- \ln x_{a} - F (t)]} for t > t_{c}

(9)

with:

t_{c}

: critical thickness of the adsorbate film; t: thickness of the adsorbate film corresponding to the relative pressure x_а at t > t_c; γ: surface tension; V_mol: molar volume of liquid adsorbate at temperature T; x_а: relative pressure of adsorbate in the bulk phase at temperature T; F(t): function that determines the potential field of the adsorbent.

In the theory of polymolecular adsorption of Frenkel–Halsey–Hill (FHH), it is assumed in the first approximation that the adsorbate films and bulk liquid are identical in structure. For the function F(t) (Gregg and Sing, 1982) follows

F (t) = b t^{- μ}; μ = \frac{1}{α}

(10)

In equation (10), b and $α$ are the parameters of the equation of the adsorption isotherm of FHH.

Differentiating equation (9) with respect to $x_{а}$ , we obtain the following equation for the derivative of the function R(x_a)

\frac{dR}{d x_{a}} = \frac{γ V_{mol}}{RT} \frac{1}{x_{a} {(- \ln x_{a} - b t^{- μ})}^{2}} for t > t_{c}

(11)

Multiplying the left and right sides of equation (6) by the saturation pressure in the bulk phase p_s at temperature T, equation (6) should be rewritten as

f (R) \sim \frac{{(θ - θ_{c})}^{1 - β}}{n} \frac{d}{d x_{a}} V_{cond} (x_{a}) {(\frac{dR}{d x_{a}})}^{- 1}

(12)

To perform calculations using equation (12), it is necessary to know the functional dependence V_cond(x_a), that is the equation of the segment of the adsorption isotherm in the region of capillary condensation. At present, there is no strictly theoretically justified equation of adsorption in the region of capillary condensation.

However, it is obvious that for adsorption isotherms of type II (according to the BDDT classification) the section of isotherm in the region of capillary condensation has the form of a logistic function. Logistic functions are well known for a wide class of thermodynamic systems. In the general case, logistic processes are described by the Bernoulli equation (Korn and Korn, 1961; Prigogine, 1980).

For the adsorption isotherm, the Bernoulli equation has the form

\frac{dV (x_{a})}{d x_{a}} = g (x_{a}) V (x_{a}) - q (x_{a}) V {(x_{a})}^{2}

(13)

where g(x_a) and q(x_a) are the given real functions that are continuous on the segment

x_{0} \leq x \leq 1

, x₀ is the value of x for the extreme left point of the hysteresis loop. The initial conditions for integration of equation (13) are defined as the coordinates of the left starting point of the hysteresis loop.

The trivial integration of equation (13) determines the adsorption equation in the region of capillary condensation

V_{cond} (x_{a}) = \frac{g q^{- 1}}{1 + q^{- 1} \exp [- g m (x_{a} - x_{0})]}

(14)

In equation (14), the parameters m and x₀ are determined from the initial conditions of equation (13).

Differentiating equation (14) with respect to the variable x_a, we obtain

\frac{d V_{cond} (x_{a})}{d x_{a}} = \frac{{(\frac{g}{q})}^{2} \exp [- gm (x_{a} - x_{0})]}{{1 + q^{- 1} \exp [- gm (x_{a} - x_{0})]}^{2}}

(15)

Equations (7), (11), (12), and (15) will be taken as a basis for the calculation of the PSD from the experimental adsorption isotherm.

Results and discussion

As an example of calculating the PSD from equations (7), (11), (12), and (15) let us consider low-temperature adsorption of nitrogen on a sample of Davisil silica (Kuchma et al., 2006).

The isotherm is shown in Figure 1. This isotherm of adsorption will be described by the equation of FНН (Gregg and Sing, 1982)

t = \frac{V}{V_{m}} = {(\frac{b}{- \ln x_{a}})}^{α}

(16)

Figure 1.

Isotherm of adsorption (□)–desorption ( $•$ ) of nitrogen (77 K) on Davisil silica sample (Kuchma et al., 2006) plotted as adsorbed volume V versus relative pressure p/p_s.

In equation (16) V is the value of adsorption at the relative pressure of adsorbate in the bulk phase x_a and V_m the analogue value for monolayer filling. To determine the parameters b and α from the experimental data, the adsorption isotherm equation is rewritten in the following form (Gregg and Sing, 1982)

\ln V = α \ln b - α ln (- \ln x_{a})

(17)

Figure 2 shows the analyzed isotherm in coordinates of equation (17). Analyzing Figure 2, we can draw the following conclusions.

Figure 2.

Nitrogen adsorption isotherm in the coordinates of equation (17): Y = ln(−lnx_a) versus lnV, whereby x_a corresponds to the relative pressure of the adsorbate and V to the adsorbed volume.

The isotherm in the coordinates of equation (17) is linearized in the form of four segments:

In the region 0 < x ≤ x_m; 0 < a ≤ a_m monolayer formation takes place. For the examined sample, the right-hand point of the first segment determines the state of the saturated monolayer (x_m = 0.106; V_m = 73).

In the interval from V_m until 2.5 V_m pore volume filling takes place in cylindrical pores. This is the area of micropores. This results in x = 0.75; V = 182 for the silica sample.

To the right of point V = 182; x = 0.75 a stable condensate film is formed. The equation of state for a multilayer condensate film is given in equation (16) with parameters α = 0.48; b = 1.92. The extreme right-hand point with coordinates x_p = 0.79; V_p = 200 determines the state of the saturated multilayer. It is obvious in Figure 2 that the capillary condensation has two regions.

The saturated multilayer is the seed for a phase transition, the spontaneous capillary condensation. For x_p ≤ x < 1 and V_p ≤ V < V_max spontaneous capillary condensation is found. For the silica this means that to the right of the point with coordinates x > 0.8; V > 200 the adsorbate film loses its stability and the cores spontaneously fill with capillary condensate.

In order to carry out calculations using equations (14) and (15), it is necessary to determine (1) the validity of equation (13) for the isotherm and (2) the values of the parameters g and q, m, and x₀.

For the size of the volume of the capillary condensate in the pore, the distribution function $\frac{d V_{cond}}{d x_{a}} = f (V_{cond})$ was constructed.

Numerical differentiation was carried out by the Lagrange interpolation formula (Korn and Korn, 1961).

The volume of capillary condensate in the pore was calculated as follows: $V_{cond} = V - V_{p},$ with V_p the maximum volume of a stable condensate film. The differential dependence $\frac{d V_{cond}}{d x_{a}} = f (V_{cond)}$ thus obtained is used to analyze equation (13). Equation (13) is transformed to the form

\frac{1}{V_{cond}} \frac{d V_{cond}}{d x_{a}} = g - q V_{cond}

(18)

Figure 3 gives a graphical representation of equation (18) in coordinates [ $V_{cond}; Y = \frac{1}{V_{cond}} \frac{d V_{cond}}{d x_{a}}$ ].

Figure 3.

Dependence of the quantity Y with $Y = \frac{1}{V_{cond}} \frac{d V_{cond}}{dx}$ on the volume of the capillary condensate V_cond (equation (18)).

The analysis in Figure 3 shows that equation (18) is represented graphically by two straight-line segments. The intersection point has the coordinates V_cond = 400; Y = 1.25.

To determine the numerical values of the parameters m and x₀ in equation (14), the initial conditions for integrating equation (13) have to be defined. Theoretically, the initial condition for the integration of equation (13) is the point of loss of stability of the condensate film in the pore, which has the coordinates V_p = 200; x_р = 0.79. From the value V_p = 200, we calculate the volume of the capillary condensate. However, formally, the point characterizing the initial conditions of equation (13) should be shifted slightly to the left. In this case, the logical nature of the isotherm section will be more pronounced, which significantly improves the accuracy of the description of the isotherm section on which capillary condensation is observed. From these considerations the point V = 155; x = 0.72 was chosen as the initial conditions for integration of equation (13). Analyzing Figure 3 we obtain the numerical values of the parameters g and q of equation (18) g₁=3.9; q₁ = 7.12 × 10⁻³; $x_{01}$ = 0,72; $m_{1}$ = 11.65; g₂=4.1; q₂ = 0.008; $x_{02}$ = 0.72; $m_{2}$ = 6.4.

Now, we can proceed directly to the calculation of the PSD. The required thermophysical characteristics of the adsorbate are γ = 8.85 × 10⁻³N/m; V_mol = 34.6 × 10⁻⁶m³/mol; σ = 3.0 × 10⁻¹⁰m (Horvath, 1998; L’Air Liquide, 2002). Figure 4 shows the PSD function calculated with the equations above.

Figure 4.

Renormalized function of PSD φ(R) as a function of pore radius R.

Let us make a number of remarks on the character of the PSD presented in Figure 4. First, it should be noted that the calculation of the PSD, carried out analytically in this paper, determined the mesopore distribution in the interval $70 \leq R \times 10^{10} \leq 200$ , which, in general, coincides with the results presented by Kuchma et al. (2006). At the same time, the nature of this PSD fully corresponds to the thermodynamic features of the process of capillary condensation. The graph shown corresponds to a normal distribution with a practically zero eccentricity and a slight positive asymmetry. The similarity of the PSD for the adsorption system to a normal distribution can be considered as a hint that the adsorption process for an individual element of the porous medium is independent of those taking place in other pores. This corresponds to the concept of no pore blocking effect during adsorption (Gregg and Sing, 1982; Isichenko, 1992). However, the deviation of the PSD obtained for $R > 150 \times 10^{- 10} m$ from the normal distribution indicates that in this region pore blocking is also possible during adsorption.

Conclusion

It is shown that by combining the percolation theory and the theory of capillary condensation and taking into account the influence of the condensate on the pore walls, the PSD of a porous structure can be determined. For a description of adsorption–desorption processes, the usage of the Bernoulli equation was proposed by the authors for the first time. Based on the solution of the Bernoulli equation, a feature of the set of mesopores was established. The whole set of mesopores consists of two subsets of different dimensions. Consequently, the set of mesopores is a multifractal set. This is one of the main results of the article.

The second important result of this research is the thermodynamic analysis of the distribution function obtained. The PSD function is a spectral characteristic. Our article shows that the spectrum in the region of capillary condensation is a semifinite spectrum limited to the left. The article also shows that the asymmetry and eccentricity of the spectrum are associated with the pore blocking effect. Therefore, consideration of the adsorption process in the field of mesopores as a logistic process allowed us to obtain thermodynamically strictly (without any assumptions and approximations) the equation of state of capillary condensate in the core of the pores. This is a very important result to describe and explain the irreversible thermodynamics of adsorption (hysteresis loop). Writing the Bernoulli equation for the adsorption and desorption branches of the hysteresis loop, one can obtain an analytical expression for the nonholonomic parameter (degree of irreversibility) of the adsorption process. In this case, the nonholonomic parameter will be considered in the formalism of the metric geometry.

Using the nitrogen adsorption isotherm of a Davisil silica sample its PSD was calculated. A nearly normal distributed PSD is obtained. These results coincide with previous results and suggest that no pore blocking effects occur during the adsorption process in this porous structure.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

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