Kutarov V. Vα., Dragan G. Sσ., & Schieferstein E.ρ
We examine the adsorption hysteresis for the multilayer adsorption with a hysteresis loop of the type H1 according to the IUPAC classification. To describe the adsorption branch of the hysteresis loop, the Frenkel - Halsey - Hill (FHH) equation has been used. The equation of the desorption branch of the hysteresis loop was obtained on the basis of joint solutions of the equation for the change in the Gibbs free energy function (taking into account the connectivity of pores) and the Laplace equation, written for the conditions of infinitely extended meniscus. The resulting equation was tested with the example of the nitrogen adsorption isotherm on the silica Davisil.
Keywords: isotherm, adsorption, desorption, hysteresis, connectivity.
Author α σ: Odesa I. I. Mechnikov National: University, Physics Research Institute, Pastera 27, Odesa, 65082, Ukraine,
ρ: UMSICHT, 3 Osterfelder Str., D46047 Oberhausen, Germany
An important characteristic of the dynamic equilibrium at the gas – solid interface is the adsorption isotherm. Its study can provide valuable information about the phase equilibrium at the interface in the nanoscale, and, most interestingly, about the structure and properties of the porous solid body. Specific properties of the isotherms can be employed as the capillary hysteresis characteristics which are used to retrieve the stochastic geometry of adsorbents and, in particular, the function of pores’ size distribution.
The most common method for calculating the distribution function involves a technique in which the model of a porous body is represented in the form of an open cylindrical capillary [1, 2]. At the same time, the adsorption - desorption processes in an open cylinder are studied insufficiently. For the condensation and evaporation processes in an open cylinder, there is no criterion of thermodynamic irreversibility, which would allow to predict the desorption branch of the hysteresis loop.
The aim of this work is to develop a theoretically rigorous and scientifically sound method of calculating the desorption conditions for a model of condensation and evaporation in open cylindrical capillaries.
We consider a capillary model in the form of open cylinder with the constant radius r, which exists in contact with the bulk phase (gas) at the temperature T and the chemical potential of the adsorbed phase μa. The variation of the Gibbs free energy upon transfer of dN moles of adsorbed substance to the bulk phase which is maintained at the same temperature and pressure is determined by the equation [3, 4]:
In the second term of the right-hand side of Eq. (1), ηa,d is the pore connectivity relative to the adsorption (a) or desorption (d) process , γ is the surface tension at the interface between the adsorbed film and bulk phase, dA is the variation of the interface area, μg is the chemical potential of the bulk (gaseous) phase.
The chemical potential of the adsorbed phase is determined from the Frenkel-Halsey-Hill (FHH) thermodynamic theory of the multilayer adsorption :
where μl is the chemical potential of the bulk liquid at the same temperature, is the potential that characterizes the interaction of the solid surface with molecules in the bulk phase, t = a/am is the relative thickness of the stable condensate film; a is the current amount of the adsorbed substance and am is the adsorption amount in a monolayer. The value of t is determined by the FHH  as:
where b and α are the FHH parameters, p and p0 are the equilibrium pressure and the saturation pressure of the adsorbate at temperature T, respectively. At the thermodynamic equilibrium :
where R is the universal gas constant.
For practical calculations, the pore connectivity can be taken in the form 
Here f is the porosity of a nanoporous body which is equal to the ratio of the pore volume to the total volume of the nanoporous body; rmin is the minimum pore radius for the range of the pore radius in which the capillary condensation and evaporation occur, χ is scaling parameter.
Generally, for any interface the following condition  holds:
where: r1 and r2 are the principal radii of curvature, dV is the change of the system volume and dA is the change of the system surface area.
For the interface between the adsorbed film and the bulk phase in the open cylinder equation (6) gives
where V1 is the molar volume of the liquid phase and r is the cylinder radius.
In what follows, the critical thickness of the film adsorbed at the internal surface of the cylindrical pore is denoted by tc. At t < tc, the FHH regime occurs, for which the film is stable; at higher t > tc the film becomes unstable and the core becomes filled due to the capillary condensation.
Having combined equations (1), (2), (4) and (7) and assuming the thermodynamic equilibrium condition dG = 0, one obtains the equation to describe the capillary condensation in the open cylinder:
Note that, assuming that the hysteresis takes place in the system considered, one should distinguish between the equilibrium pressure of the adsorbate on the adsorption branch of the hysteresis loop pa, which enters Eq. (8), and the equilibrium pressure of the adsorbate on the desorption branch of the hysteresis loop pd. At a pressure greater than the pressure in the bulk phase corresponding to the critical thickness of the adsorbed film, the film loses its stability, and the pore is spontaneously filled by capillary condensation mechanism.
Consider now the process of evaporation from the open cylindrical capillary. At the arbitrary point P of the surface of the meniscus (see Fig. 1), the two curvature radii of the meniscus are unequal and are determined according to the Laplace equation as 
where ω is the surface slope at the point P with respect to the positive direction of the X axis, . For the equilibrium meniscus on the desorption branch of the hysteresis loop, integrating the equations (1), (4), (6), (9), (10), (assuming dG = 0) one obtains the following equation:
where pd is the pressure in the bulk phase on the desorption branch of the hysteresis loop.
Now we suppose, as a first approximation, that the connectivity factor is constant. Then the integration of Eq. (11) on the desorption branch of the hysteresis loop yields:
The point which corresponds to the capillary evaporation on the desorption branch of the hysteresis loop is characterized by the parameters pd and tc. These values could be determined keeping in mind that the breakdown of the porous medium corresponds to the state when the meniscus becomes infinitely elongated, i.e. . In this breakdown point the equation (12) acquires the form
where the obvious substitution x = t − r is performed, see Fig. 1.
For further analysis, expression (13) should be converted to the form convenient for practical calculations. To this end, in the second term in the right hand side of Eq. (13), the function F(t) is expressed through parameters that define the adsorption branch of the hysteresis loop. This is conveniently done in the framework of FHH theory :
Then the integration of Eq. (13) with account for Eq. (14) yields:
where the parameters D, B and M were introduced as
, with t > tc (16)
(σ being the Wan der Waals diameter of the adsorbate molecule),
Equations (15) - (19) enable one to calculate the desorption branch of the hysteresis loop using the values obtained from the analysis of the adsorption branch.
For example, let us consider the low-temperature (T = 78 K) nitrogen adsorption on the silica Davisil F (see Fig. 2 ). The analyzed branch of the adsorption isotherms (Fig. 3) is represented in coordinates defined by the FHH equation (3) .
In the entire range of the relative pressure of the adsorbate in the bulk phase, the adsorption branch of the isotherm in coordinates ln (-lnx)-f(lnV) represents a broken straight line consisting of four segments. Coordinates of the right point of the first right segment (monolayer formation) are . The stable condensate film exists within the range With the parameters , equation (3) describes the adsorption branch of the hysteresis loop with a maximum relative deviation .
To the right from the point with abscissa = 0.81, the condensate film loses its stability, and the pores appear spontaneously filled by the capillary condensation. Thus, it is necessary to take into account the fact that the point quantifies the qualitative change in process of capillary condensation. In particular, it can be assumed that at the process of capillary condensation is retarded, compared with the process taking place at a relative pressure of the adsorbate in the bulk phase at .
Now we can proceed directly to the calculation of the desorption branch of the hysteresis loop via the formulas (15) – (19). When calculating the equations (15) – (19), we took the following values of the parameters of the adsorbed substance :
However, a priori, there are no reliable data on the changes of the connection coefficients and . The formula (5) proposed for calculation of the connection for the experimental interpretation of the pores as open cylindrical capillaries was not exposed Therefore, we start the calculation based on the experimental desorption branch of the hysteresis loop involving the values of the coefficient for the desorption process. To do this, we transform the formula (15) to the following form:
In this case, we assume that = 1. This assumption is based on the results obtained earlier . Now, by means of (20), we calculate the coefficient in the desorption process. It is necessary to take into account, that the character of the capillary condensation changes significantly near . Thus, the calculation according to formula (20) is only valid within the interval . In order to use the formula (5) one must specify the minimum pore radius in the capillary for condensation and evaporation. The relative pressure in the bulk phase corresponds to the pore radius , calculated by Eq. (8), which can be transformed to the form
Based on the results obtained earlier , it can be assumed that to the right of the point on the adsorption branch of the hysteresis loop, for pores with the core radius , the pore blocking is also possible. In order to determine the coefficient of pore connectivity characterizing the pore blocking on the adsorption branch of the hysteresis loop, we proceed as follows. The connectivity coefficient for the desorption branch of the hysteresis loop, calculated by formula (20), will be described by Eq. (5).
As a minimum we will take the pore radius corresponding to the value хр in the right neighborhood of the left point of the hysteresis loop. According to (21), the pore radius is equal to the minimum radius . Fig. 4 shows the dependence on the pore radius for . In such conditions, equation (5) can be written in the form
Further, we suppose that the behavior of the connectivity on the desorption branch of the hysteresis loop, , can be described by Eq. (5).
Fig. 4 describes the behavior of the coefficient within the range of the relative pressure of the adsorbent in the bulk phase corresponding to . At the same time, we calculate the coefficient by the formula (20) to the right of хс for the values of pore radius , provided that the connectivity value for the adsorption branch of the hysteresis loop is = 1. Thus, the values of calculated by Eq. (20) for and = 1 are = 0.99, and at , = 1,23. The obtained values of the coefficient are either significantly higher than the values of calculated via Eq. (22), or are physically meaningless. Therefore, we can conclude that at the region to the right of the point хс, the pore blocking occurs on the adsorption branch of the hysteresis loop and < 1.
Now we rewrite Eq. (23) in such a manner that the left-hand side value be determined by the formula for the radius of the pores :
The parameter M is determined depending on the ratio (19), which also depends on the coefficient in the adsorption process.
The right-hand side of (23) implicitly contains an unknown variable factor . As a result, Eq. (23) represents the transcendental equation for . The solution of this equation was performed numerically.
Fig. 4 shows the dependence of the coefficient on the pore radius for . This dependence is well described by the formula (5), which in this case looks as follows:
, rc < r < rmax . (24)
Equation (24) gives the possibility to calculate the coefficient for r within the interval . At the same time, a change in the curve behavior in Fig. 3 can be ascribed to the dependence on the magnitude of r which can be represented as a step function according to which and for , is determined by Eq. (24). However, there is no contradiction. From formula (24), it is easy to see that when , 0.95. During the calculation of the connectivity coefficient of the porous medium in the desorption process , the corresponding connectivity coefficient for the adsorption process, , is taken into account only via the parameter M by Eq. (18). For the values of pore radius , the value of M, with a slight change in the ratio , varies very insignificantly. It should be noted that the filling of pores in the adsorption step is a feature that is clearly visible in Fig. 2. And, therefore, the accurate calculation of the connectivity coefficient of the pore space justifies the process of the pore filling in the adsorption branch.
We now give some explanations relating the proportionality coefficients in formulas (22) and (24). These factors, according to formula (3), are determined by the porosity value f. Obviously, for the correct calculation, the local values of porosity should be taken associated to the leftmost interval of the pore radii corresponding to the processes of capillary condensation and evaporation. According to Eq. (5), values of the porosity determine the coefficients of proportionality of the formulas (22) and (24), and equal to 0.95 and 0.88 for the adsorption and desorption processes, respectively. Such a difference in the values of the local porosity determines special features of different branches of the hysteresis loop.
The information obtained on the behavior of the pore space connectivity in the adsorption and desorption processes allow one, using the formulas (15) – (19), to restore the desorption branch of the hysteresis loop on the basis of information on the adsorption branch. Calculations have shown that the system of equations (15) – (19) gives the results for the relative adsorbate pressure in the bulk phase for the desorption branch of the hysteresis loop without exceeding the maximum relative deviation.
As a result of this research, it has been shown that the obtained analytical expression satisfactorily describes the process of adsorption hysteresis for multilayer adsorption with a hysteresis loop of the type H1 (according to the IUPAC classification) for the nitrogen adsorption on Davisil. It is revealed that the adsorption and desorption branches of the hysteresis loop, which are determined from the joint solution of the Kelvin and Laplace equations, obey the control parameters in the form of the interface curvature in the direct and inverse processes and the magnitude of the potential field of the capillaries. Note that the average radii of curvature of the interface for the adsorption and desorption branches of the hysteresis loop are different. This difference is the determining quantity for the irreversibility of the adsorption process. It was found that the change in the connectivity of the pore space depends on the pore radius, and with increase of the pore radius, the connectivity of the pore space decreases. The connectivity of the pore space for the desorption branch of the hysteresis loop is less than for the adsorption branch of the hysteresis due to the effect of blocking the pores during desorption.
Fig. 1: Axial section of the meniscus in the open cylinder.
Fig.2: Adsorption-desorption isotherms of nitrogen on silica Davisil F
Fig. 3: The isotherm of the nitrogen adsorption by amorphous silica Davisil F in the coordinates of equation (3)
Fig. 4: Dependence of pore connectivity factor during adsorption (○) and desorption (Δ) on the radius of the pores.
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