Percolation models of adsorption–desorption equilibria and kinetics for systems with hysteresis

Colloids and Surfaces A: Physicochem. Eng. Aspects 300 (2007) 191–203

Percolation models of adsorption–desorption equilibria and kinetics for systems with hysteresis ˇ osˇ a,∗ , Pavol Rajniak a , Frantiˇsek Stˇ ˇ ep´anek b Miroslav So´ a

Department of Chemical and Biochemical Engineering, Slovak Technical University, Radlinsk´eho 9, 812 37 Bratislava, Slovakia b Department of Chemical Engineering, Imperial College, Prince Consort Road, London SW7 2AZ, United Kingdom Received 20 June 2006; received in revised form 19 October 2006; accepted 23 October 2006 Available online 29 October 2006

Abstract Characterisation of internal structure of porous adsorbents was made using simplified 2D and 3D lattices of spherical pore sites and cylindrical pore connections based on primary adsorption and primary desorption equilibrium data. Two basic models of adsorption–desorption hysteresis in porous sorbents, i.e. the independent pore model and the pore blocking model, were developed and tested. Very good agreement with experimental equilibrium data for the systems Vycor glass–nitrogen was obtained by simultaneous optimisation of the lattice size and the lattice connectivity and by incorporating the pore-blocking assumption into the 3D percolation model. The pore blocking model predicts correctly also the equilibrium scanning curves and cycles inside the hysteresis loop. Evaluated connectivity and size distributions for both, the pore sites and the pore connections, were subsequently used also for reconstruction of the lattices and for modelling of adsorption–desorption kinetics of the same experimental system. Theoretical models predict correctly main qualitative features observed experimentally, i.e. maxima and minima of the mass transfer coefficient. © 2006 Elsevier B.V. All rights reserved. Keywords: Characterisation; Percolation; Modelling; Adsorption; Desorption; Hysteresis; Equilibrium; Kinetics

1. Introduction The problem of predicting adsorption–desorption equilibria and mass transfer rates of adsorbable vapours in porous media in the range of pressure where capillary condensation occurs is a significant one for the design and operation of adsorbers, dryers, catalytic reactors, and so on. The effective understanding and design of porous adsorbents or catalyst supports depends upon the ability to measure and evaluate experimentally the pore size distribution and the topology of the pore network. Recently, new methods based on the use of percolation theory to analyse adsorption–desorption hysteresis were proposed for determination of the pore size distribution and connectivity of porous solids [1–6]. For adsorption and desorption of condensable vapours, the vapour in the adsorbent exists as adsorbed molecules on the solid surface, condensate in fine pores and as

∗ Corresponding author. Current address: Institute for Chemical and Bioengineering, Department of Chemistry and Applied Bioscience, ETH Zurich, 8093 Zurich, Switzerland. Tel.: +41 44 633 4659; fax: +41 44 632 1082. ˇ osˇ). E-mail address: [email protected] (M. So´

0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.10.056

vapour in the voids. During desorption, as the relative pressure is reduced, systems in which capillary condensation occurs, generally show hysteresis, that is, in a particular pressure range more vapour remains adsorbed than during the initial adsorption process. Classical explanations of hysteresis based on single pores [7,8] cannot satisfactory explain some important experimental observations, such as the higher-order adsorption–desorption scanning curves. Models that treat the pore system as an interconnected network have been developed more recently. These models attribute hysteresis to pore blocking where the emptying of a large pore filled with capillary liquid has to be preceded by the emptying of its smaller neighbours [1,2,4,5,9–11]. Hence, the primary desorption is a connectivity-related phenomenon. Such phenomena can be described by percolation theory [12]. Like in adsorption–desorption equilibria, hysteresis also exists for the kinetics in the capillary condensation region. Hysteresis in both equilibria and kinetics are illustrated in Fig. 1a and b. The concentration dependence of diffusivity exhibits a maximum during adsorption and a minimum during desorption. Various experimental methods used to determine the mass transport rates in the capillary condensation regime [13–16] and theoretical approaches explaining the experimental results

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Nomenclature a ci (τ) C dji Lji N NI NS p P P0 q r R Ri Rji t T V VL x x

amount adsorbed (cm3 g−1 ) adsorbate concentration connectivity relative diffusivity for the connection between pores i and j length of the cylindrical pore connection (m) number of pore sites in each row or column of the N × N lattice number of internal pore sites of the lattice NI = (N − 2) × (N − 2) number of surface pore sites of the lattice NS = 4 × (N − 1) probability that the bond is filled by capillary condensate vapour pressure (Pa) saturation vapour pressure (Pa) probability that the site is filled by capillary condensate bond or site radius of curvature (nm) gas constant (J mol−1 K−1 ) radius of the spherical pore site (m) radius of the cylindrical pore connection (m) adsorbed film thickness (nm) temperature (K) part of pore volume filled or emptied during a sorption step (m3 ) molar volume of the liquid adsorbate (m3 mol−1 ) relative pressure increment of relative pressure for a sorption step

Greek symbols γ surface tension (N m−1 ) ϕ basic mass transport coefficient (m3 s−1 ) τ time (s) Subscripts A adsorption D desorption i, j indexes for pore connections and pore sites s surface Superscripts L lower limiting point T percolation threshold U upper limiting point

[17–20] have appeared in the literature. Literature reports on hysteresis in the kinetics of adsorption–desorption include the study of capillary condensation flow of toluene in Vycor glass [13], gravimetric measurements of the kinetics of isothermal adsorption and desorption of isopropanol in Vycor glass [21], multilayer diffusion and capillary condensation of propylene in supported alumina films [22], permeabilities in Vycor glass [23], and isothermal transport of liquids in partially saturated

Fig. 1. Experimental hysteresis-dependent adsorption–desorption equilibria and kinetics. (a) Equilibria: L, lower limiting point; U, upper limiting point of the hysteresis loop and (b) kinetics; (—) adsorption; (– – –) desorption.

packed beds of glass spheres [24]. Theoretical models used in these studies represent the classic continuum approach [25] and the complexities caused by the network effects were not considered. In our recent works [6,26,27] network models were formulated for predicting effective Fickian diffusivities of condensable vapours in porous media where capillary condensation and adsorption–desorption hysteresis occur. The models combine the equilibrium theory based on the pore-blocking interpretation of hysteresis in the interconnected network of pores [1,26–29] and the percolation model of mass transport in the network with randomly interspersed regions for capillary condensation and surface flow. A new predictive model based on properties of Bethe lattices was proposed [26] to account for the existence of liquid-filled “blind” pores that results in a maximum and subsequent decrease in the total diffusion rate. For desorption, a new “shell and core” representation of the network model was proposed. Information from adsorption–desorption equilibria was needed to compute the thickness of the shell in which desorption/evaporation occurs for concentrations higher than the percolation threshold. However, there are several questionable assumptions in the analytical models presented in our previous contribution [26], the distinct boundary between the shell and the core, the use of effective medium approximation (EMA) and the use of Bethe tree for mass transfer during desorption. This contribution presents basic properties of a percolation model based on square and cubic lattice structures. First, connectivity and distributions of spherical pore sites and cylindrical pore connections are evaluated by comparing numerical solutions of the model to experimental equilibrium data for adsorption–desorption equilibria of nitrogen on Vycor glass.

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Evaluated connectivity, size distributions and lattices are subsequently used also for modelling of mass transport during adsorption and desorption and comparison with kinetic experiments for the same experimental system. In addition, the final pore size distribution (PSD) obtained using above mentioned simplified model of the porous structure was compared with PSD obtained using reconstructed porous media technique [30]. 2. Experimental 2.1. Materials Vapour of liquid nitrogen at its boiling temperature of 77 K was used as adsorptive in sorption experiments. Vycor glass, a porous glass, which has been widely used as a model material in studies of properties of fluids and molecules in highly confined geometries [13,23,31], was used as adsorbent for experimental study with nitrogen vapour. The large internal surface area of Vycor effectively adsorbs molecules at low ambient vapour pressures, while the large pore volume effectively absorbs bulk fluids by capillary condensation at higher ambient vapour pressures [24]. 2.2. Apparatus and procedure Equilibrium and kinetic adsorption–desorption experimental data were obtained using a Carlo Erba Sorptomatic 1900 apparatus. A complete adsorption–desorption isotherm, which usually takes hours of operation, is obtained automatically. In all measurements, the sample initially at equilibrium was subject to a sudden small change in partial pressure of adsorptive and the changes of amount adsorbed during adsorption or desorption were continually recorded. Heat effects during the sorption measurements were minimized by allowing only small step changes in relative pressure during each measurement. Because the sorption rates were independent of the pellet size, we can assume that the total sorption rate was controlled by the mass transport processes within micro-particles. Successive adsorption and desorption were conducted by changing the composition of the adsorptive and allowing adequate time to establish equilibrium [26]. There was no influence by external diffusion, because pure nitrogen was used in the experiments. 3. Theoretical considerations 3.1. Basic models of adsorption-desorption hysteresis A basic equation for any model of adsorption–desorption hysteresis is the macroscopic Kelvin equation for capillary condensation: ln

P −2γVL 1 = P0 RT r

(1)

which relates the vapour (condensation) pressure P to the characteristic dimension of the pore connection or pore site, r. The hysteresis of capillary condensable vapour has several possible origins. A classical hysteresis theory, the ‘indepen-

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dent pore model’ [7], regards the difference between the relative vapour pressures at which ink-bottle pores fill and empty as the prime factor causing hysteresis. There is a single relative vapour pressure corresponding to the filling of the pore site with radius, Rads . On the other hand, there is a different single relative vapour pressure corresponding to the emptying of the pore connection with radius Rdes < Rads . More elusive is the ‘pore-blocking model’ [1,4,5,32]. A pore filled by capillary condensate cannot empty until at least one of its neighbours has emptied as a consequence of smaller dimension of the neighbours or corresponding connections. The incorporation of the pore-blocking effect extends the independent pore theory. There is still a single pressure for filling but several for emptying, one for each connection to the neighbouring pores. The pore-blocking effect depends crucially on the interconnections and the interconnectedness of the pore space of the porous material. Both basic models of hysteresis were tested using the 2D and 3D lattices and compared with experimental data in this contribution. 3.2. Theory of adsorption–desorption equilibria in systems with hysteresis Percolation theory is the theory of connected structures formed by random links on a lattice or tree. The classical percolation theory focuses on two main problems, the bond percolation problem, and the site percolation problem. The primary adsorption process in the capillary condensation region can be treated as a classic site percolation problem. On the other hand, the desorption process for systems with hysteresis is the bond percolation problem. During adsorption, the probability that a pore site fills at some value of x is q, and is unaffected by interconnectivity effects. Pores with different dimensions are randomly distributed in the pore space. All pores are equally accessible and connectivity of the pores plays no role. Because, for geometric and thermodynamic reasons, all of the windows to a given pore site may fill with condensate before the site fills. However, even if the pore site becomes isolated from the bulk vapour, it can still fill by condensing vapour from adjacent pores that can then refill from the bulk vapour. All that matters is the effective size of the pore sites. However, percolation properties are of particular interest for study of the primary adsorption kinetics [26,27]. During desorption, at some value of the relative vapour pressure, x, there is an associated probability, p, that condensate can evaporate from the bond (connection) and so empty the site (cavity) of the pore. The condition for such bond actually emptying must also include the probability that it is connected via conductive bonds to the edge of the network. During desorption from ideal (infinite) system starting with a filled system, p is progressively reduced until at some critical (threshold) probability, pT , the system starts to empty. Thereafter, pores continue to empty as they become connected and more and more windows are conductive. The position of the percolation threshold is affected by the connectivity C, which is defined as the average number of windows per pore site. The explanation of the sharp knee on the desorption isotherm where desorption commences, shown

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Fig. 2. Theoretical hysteresis-dependent adsorption–desorption equilibria for infinite systems.

in Fig. 2, is that at this point the connection-making process proceeds in a cascade-like manner into the bulk of the material and empty pores are no longer restricted to the surface of the adsorbent. Disagreement between theory and experiment exists for the case of real primary desorption, Fig. 1a or 3, since the real desorption isotherm is not horizontal and desorption does occur. Most of the theoretical and experimental studies indicate that the finite size of the micro-particles of the porous material and corresponding desorption from the surface of the micro-particles is the main reason of the disagreement. Alternative theoretical explanations (decompression or nucleation) of this experimental fact were discussed in works [1,5]. Empirical correction functions were proposed for computation of the real amount adsorbed during primary desorption, as well as during higherorder desorption processes. More sophisticated model to explain and unify equilibrium and kinetics of vaporisation from surface of micro-particles based on ‘shell and core’ approach was developed in [26]. 3.3. Theory of adsorption–desorption kinetics in systems with hysteresis We consider the adsorption–desorption process of a condensable vapour in a porous adsorbent. In Fig. 1a the adsorption–desorption equilibria for the process are shown. At relative pressure below point L, the lower closure point of the hysteresis loop, only surface adsorption occurs. Mass transport

Fig. 3. Experimental adsorption–desorption equilibrium data for the system Vycor glass–nitrogen at 77 K: (䊉) experimental adsorption, () experimental desorption.

rates in this region are the same for both adsorption and desorption. Above point U, the upper closure point of the hysteresis loop, all pores become filled with capillary condensate. Transport of liquid condensate in the region above point U occurs by hydraulic pressure. The study of the kinetics in this region is out of scope of this work. In this contribution we are mainly interested in the mass transport rate in the hysteresis region of simultaneous surface diffusion and capillary condensation, that is, in the region between points L and U. Mass transport in the capillary condensation region is a complex phenomenon. The conditions under which capillary condensation occurs are also those under which significant surface diffusion is expected. Vapour phase transport is in this case usually orders of magnitude smaller than the surface flow because of the relatively small amount of molecules in the vapour phase compared to that in the adsorbed phase [13,21]. It is now generally recognized that the capillary suction accompanying condensation is a significant accelerator of the mass transfer [13,23,26,27,33]. Any capillary condensate volume elements create a short-circuit effect leading to a reduction in the length of the diffusion path and a corresponding increase in the effective diffusivity. So, the first effect of the capillary condensation is to increase the mass transport rate when some pores are filled with condensate. However, in real porous adsorbents, there exist blind (dead-end) pores. These blind pores are accessible for filling by surface adsorption and capillary condensation; but after filling with capillary condensate, they do not increase the total mass transport rate, but on the contrary, decrease it, because they do not conduct the flow. The fraction of pores that belongs to blind clusters will increase with increasing pressure. Therefore, in real porous adsorbents with the presence of the blind pores, all three main mechanisms (surface diffusion, capillary condensation, and liquid flow of capillary condensate) may already be operative simultaneously below point U. Subsequently a maximum in the total mass transfer rate exists during adsorption at a relative pressure x < xU , as shown in Fig. 1b. Physical situation is different for the case of desorption and evaporation of the condensed adsorbate. While the condensation generally increases the mass transport rate, evaporation is a slower process and one can expect a decrease in the total mass transport rate with an increasing role of evaporation from pores. At high relative pressure x < xU desorption occurs only from widest pores at the external surface of the finite micro-particles of adsorbent. Large pores in the core of the micro-particle cannot empty in this step, because are blocked by smaller neighbouring pores and narrow pore connections. Apparent mass transport and equilibration is relatively fast, because the length of mass transport path is short. The boundary of the phase separation penetrates along the widest pores and connections into the micro-particle volume. In this step the mass transport path increases and desorption rate decreases with decreasing pressure. At the relative pressure xT corresponding to percolation threshold for desorption, there is a sufficient number of pore connections in which adsorbate is below its condensation pressure (and thus is present as either metastable liquid or vapour) and desorption from the bulk of adsorbent starts. Mass trans-

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port path is complicated and long and desorption rate is very slow. Next decrease of pressure causes that higher and higher number of vapour-filled pore connections exists and apparent mass transport rate again increases. Subsequently a minimum in the total mass transfer rate exists during desorption at a relative pressure xL < x < xU , as shown in Fig. 1b. 3.4. Models of porous structure For the modelling of processes inside the porous structure different types of lattice models can be used. In previous works of our group [4,5,26,27,34] the Bethe tree structure was used for the interconnectedness of pore sites. The main attraction of Bethe tree structure is that the description of its behaviour when it is randomly filled or emptied can be carried out analytically. Bethe tree can give simplified expressions for several properties of the porous medium, while retaining most features of percolation theory [1,28,32,35]. However, the Bethe tree models are advantageous only for the simplest cases, e.g. for one component isothermal system, otherwise the analytical solutions (if obtainable) would be too awkward. Main theoretical drawback of the Bethe lattice is that it does not allow for reconnections among the pores. A more realistic network is obtained if the pores are distributed on a lattice with reconnections, such as square (2D) or cubic (3D) lattices. Theoretical part of this work was carried out by the method of numerical simulations of adsorption and desorption on the lattice models of different dimensions [36]. We used the alternative in which spherical pore sites are connected to each other by cylindrical connections. The pore sites are placed at the nodes and the connections along the bonds of the lattice. Small version (8 × 8) of the lattice is shown in Fig. 4.

The spherical pore sites are connected to each other by cylindrical connections. There are also “blind” pores in the lattice, which are accessible for filling by surface adsorption and capillary condensation, but after filling with capillary condensation, they do not conduct the flow [26]. As was mentioned above the PSD obtained from just described method was compared with PSD extracted from the simulation of the capillary condensation in 3D digitally represented porous media. Since this method is described in details in our parallel work [30], we omit its description and only results will be discussed. 3.5. Simulation algorithm for generation of porous structure Various explicit functions were tested for size distributions of pore sites and pore connections and generation of porous structure models [36]. The generation of the pore sites for any experimental adsorption–desorption systems is simpler and obviously is the same for both, the independent pore model or the pore blocking model. In this work the size distribution function of spherical pore cavities is evaluated from the primary adsorption isotherm using the method of Broekhof and de Boer [37]. The pore sites volume is calculated following its characteristic radius and it is assumed that the pore volume resides entirely in the pore sites (volume of pore connections is assumed negligible). The amount adsorbed is increasing with increasing of relative pressure. Below the capillary condensation the amount adsorbed is related to the thickness of adsorbed film on the pore surface. The film thickness was determined by the value of the relative pressure using empirical relation [8, p. 135]: 

5 t = 0.31377 − ln(x)

Fig. 4. Square (2D) lattice of spherical pore sites and cylindrical pore connections as a model of porous adsorbent; s: surface pores; 26, 29 = blind pores.

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1/3 (2)

At the lower limiting point–starting point for hysteresis loop, L (see Fig. 1 or 3) the capillary condensation commences in finest pores. For these pores the sum of thickness of adsorbed film and inner core must be lower than the radius calculated from Eq. (1). The amount adsorbed in such pores is computed from volume of entire pore site assuming that the site is completely filled with capillary condensate. As the pressure is progressively increased, wider and wider pores are filled until the relative pressure xU corresponding to the upper limiting point U is reached. It is presumed that at U all pores are already filled with capillary condensate and any adsorption beyond this pressure is associated only with the change of menisci freely accessible to the vapour. On the other hand, the generation of the pore connections is more complicated and is different for both basic models of hysteresis. For the independent pore model, we use basically the same algorithm as for primary adsorption. There is a single relative pressure (adsorption radius) computed from Eq. (1) for filling and a single relative pressure (desorption radius) for emptying. The situation for adsorbed film on the surface of the pore is the same for desorption as for adsorption. After the evaporation of the liquid from the pore, there still remains an adsorbed film that is equivalent to the value of relative pressure [7]. Size distri-

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bution of pore connections is irrelevant and cannot be evaluated for the independent pore model. The size distribution function of cylindrical pore connections was evaluated iteratively from the primary desorption isotherm by employing ideas of the pore blocking mechanism of hysteresis. An arbitrary sharp distribution function for connections, which overlaps the cavity size distribution function, was used as initial approximation. This initial function was then iteratively improved by fitting to desorption experimental data by employing percolation theory for primary desorption [1,32]. The procedure was based on the assumption, that radii of all connections of a cavity are smaller than the cavity radius. The pore blocking effect depends on both, radius of a pore and connection of the pore with the vapour phase (existence of connection between pore and surface). Whether a pore remains full or empty depends upon whether at least one of its windows is both connected to the vapour, and can also allow a capillary meniscus, computed from Kelvin equation, to pass. If the filled network of pores is emptied by desorption, after the evaporation of the liquid from the pore site, there still remains an adsorbed film in both, the pore site and in pore connections, corresponding to the value of relative pressure [8]. First, the ideal lattices (without defects, every internal pore having four/six connections depend on whether 2D or 3D lattice is used) were treated. The radius of each pore site was generated first, using subroutine RNWIB from IMSL library, which generates pseudo-random numbers from a Weibull distribution. Once the pore sites were generated, four/six pore connections for each pore site were generated using subroutine RNSTA, which generates pseudo-random numbers from a stable distribution. A lattice with a connectivity of three (in the case of 2D lattice) could be obtained by removing at random 25% of the bonds. The important constraint for the generation of the pore connections was that the radius associated with the pore site must be greater than the radius of any of its connections. Negligible volume of each connection was supposed. More details about the fitting procedure and about generation of different lattices contain our recent work [6,36]. 3.6. Simulation algorithm for modelling of mass transport The lattices obtained by fitting to equilibrium data were regenerated and used also for modelling of mass transfer during adsorption and desorption. Additional assumptions for the mass transport modelling in the lattices are: 1. Parameter ϕ = (diffusivity × cross-section/length) is a basic mass transfer coefficient in each connection for lattice with connections of constant lengths and constant cross-sections. Relative value of conductance dji of a connection ji between sites i and j is given only by the transport mechanism (surface diffusion, capillary condensation) in the connection and is independent on the connection radius Rji and length Lji . 2. Vi is volume of pore site to be filled or to be emptied during each adsorption or desorption step, and is calculated for the difference x between starting and final relative pressure of the step.

3. Adsorbate concentration ci (τ) in the ith pore site (as well as concentrations cj (τ) in the neighbouring sites j) is defined as fraction of Vi , which is filled at time τ. 4. The thickness of adsorption film, t, in the pores and connections for pressures lower than critical pressure from Kelvin equation is computed from empirical relation (2). 5. Condensation in the cylindrical pore connection with radius Rji starts if Rji − t <

4.7 ln(1/x)

(3)

6. Condensation in the spherical pore site with radius Ri starts if 9.4 (4) Ri − t < ln(1/x) Eqs. (3) and (4) are alternative forms of Kelvin equation valid for condensation of nitrogen at 77 K [8]. 7. Adsorbate concentration in the pore sites at the edges of the lattice (surface sites) is always in equilibrium with the relative pressure at the lattice surface. 8. Total amount adsorbed in the lattice can be computed by summation of all pore volumes filled by surface adsorption or capillary condensation. In other words, the densities of adsorption film and condensate are equal. 9. An “effective Fickian diffusivity” in the lattice can be evaluated by fitting the computed uptake curve to the solution of the diffusion equation (Fick’s law). Then the mathematical model of mass transport in 2D square lattice presented in Fig. 4 contains: • transient mass balances for internal pore sites: dci (τ)  Vi = ϕdji [cj (τ) − ci (τ)], i = 1, NI dτ

(5)

j

• boundary conditions (adsorbate concentrations in surface pores) for adsorption: cs = 1,

s = 1, NS

(6a)

• boundary conditions for desorption: cs = 0,

s = 1, NS

(6b)

Thickness of adsorbed phase t was used as dependent variable in balance Eq. (5) after simple transformation and the system of differential equations was integrated numerically. Simulations for different adsorption steps were performed similarly as real experiments, i.e. for an increment x of relative pressure around the lattice surface the mass balances were integrated until reaching equilibrium in all sites. During the integration the Kelvin equation for connections Eq. (3) and for pore sites Eq. (4) was permanently tested. After reaching the condensation pressure in any connection ji, the relative diffusivity dji was increased to arrange acceleration of the mass transport process via suction. Subsequently, after reaching the Kelvin pressure in the ith site, the diffusivities dji of all connections of the ith

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site were again increased to further amplify the mass transfer process. The adsorption algorithm for lattices with blind pores (for C < 4) was more complicated, because the existence and growth of blind clusters were tested during simulations. Diffusivities dji = 0 for connections between blind pores were employed and the mass transport process was stopped in the pores. Desorption algorithm was further complicated by the pore blocking in the network. Whether a pore is emptying during the desorption step depends upon whether at least one of its connections is both connected to the vapour, and can also allow a capillary meniscus to pass, Eq. (2). The same relative diffusivities of pore connections were employed for both, condensation and evaporation. Then the only factor reducing the total desorption rate was prolongation of the mass transport path by pore blocking. After evaporation of the liquid from the pore site, there still remains an adsorbed film in both, the pore site and in pore connections, corresponding to the value of relative pressure and next emptying occurs via surface diffusion. 4. Results and discussion 4.1. Adsorption–desorption equilibria and porous structure Basic qualitative properties of 2D and 3D lattice percolation models were studied theoretically prior to application to experimental equilibrium data.

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First, the model capability to predict hysteresis qualitatively was tested. In Fig. 5 the visualisation of both, the primary adsorption and the primary desorption process is presented. Distributions of pores of the 20 × 20 × 20 lattice filled with capillary condensate are compared for relative pressure inside the main hysteresis below (Fig. 5a and b) and above (Fig. 5c and d) the percolation threshold. Analogous results for 2D lattices were presented in [6]. As manifestation of the hysteresis effect, the number of filled pore sites is higher for desorption. Other interesting features and differences can be also emphasized. First, unlike the random distribution of filled pore sites for adsorption, as shown in Fig. 5a and c, the distribution for desorption is significantly more “organised”. As it is shown in Fig. 5d, for relative pressure above the percolation threshold, x = 0.62, there are small clusters of empty sites at the edges of the lattice and a compact core of the filled sites. For relative pressure below the threshold, x = 0.5, one can still observe small clusters of filled pores in the network, however, there are several chains of empty (connecting) pores in both directions, Fig. 5b. Fractions of filled sites for desorption, significantly higher than for adsorption, exhibit the adsorption–desorption hysteresis. Next, as the second test of the model, the impact of lattice size on the shape of equilibrium isotherms was investigated. The square lattices with different lattice sizes (10 × 10, 30 × 30, 50 × 50, 100 × 100) were tested. Fig. 6 shows that results for the 2D lattice 100 × 100 pores are still different than those for the lattice 50 × 50. As expected and reported before [3,6], increasing

Fig. 5. Distribution of pores filled with capillary condensation for the same relative pressure inside the main hysteresis loop. 3D lattice with 20 × 20 × 20 pore sites and connectivity C = 5.8. (a) Primary adsorption, x = 0.5; (b) primary desorption, x = 0.5; (c) primary adsorption, x = 0.62; (d) primary desorption, x = 0.62.

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Fig. 6. Impact of lattice size on the shape of equilibrium isotherms. Lattice dimensions: 10 × 10 (—), 30 × 30 (– – –), 50 × 50 (· · ·), 100 × 100 (– · –).

size of lattice extends the horizontal part of theoretical primary desorption branch of hysteresis loop between upper limiting point and percolation threshold. Subsequently, the position of percolation threshold (the rounded knee) of primary desorption is moving to lower relative pressures. Moreover, the rounded knee is sharper for bigger lattices. Unlike the infinite Bethe tree structures, the finite size of the square lattice better reflects the finite size of micro-particles in real porous materials. Similar effect of the lattice size was observed also for 3D cubic geometry. The third test of the lattice model is its capability to predict experimentally observed secondary processes. For equilibrium cycles operating inside the main hysteresis loop within closure points L and U, the resulting equilibrium path will depend on the “history” of the process, i.e. on the position of starting point. The detailed description of such scanning curves is in works [5,8,9]. An infinite number of different equilibrium paths can be obtained depending on the position of starting point. The cycles of these kinds were found also experimentally for system silica gel–water vapour [5]. Other works treat the equilibrium scanning curves inside reconstructed pore structure [38], or during the pressure swing adsorption process [34]. In Fig. 7 comparison of predicted scanning curves with pore blocking model and independent pore model for 2D lattice geometry is shown. The solution with independent pore model, Fig. 7a is qualitatively different from experimental results published elsewhere [1,5,34]. On the other hand, pore-blocking model, Fig. 7b, successfully predicts the shape and position of different scanning curves. All the tests prove that the simple network models can be used for computing of theoretical adsorption–desorption equilibria for systems with hysteresis. The experimental system Vycor glass–nitrogen was chosen for comparison with theoretical predictions of the lattice models. Experimental data presented in Fig. 3 show significant hysteresis loop of Type-H2 (IUPAC classification). The data are compared with theoretical predictions of the square and cubic lattice models in Fig. 8. Very good prediction of the primary adsorption isotherm was obtained using both, 2D and 3D lattices. For the primary desorption isotherm, the independent pore model gives wrong shape of hysteresis loop, as already shown in Fig. 7a. Better agreement with experimental data was obtained by pore blocking model and 2D lattices, which fairly predicts also the primary desorption isotherm. The model exhibits slow decrease of adsorbed amount between the upper limiting point and perco-

Fig. 7. Scanning curves for system Vycor glass–nitrogen: (a) independent pore model and (b) pore blocking model.

lation threshold and rapid decrease of adsorbed amount below the percolation threshold. However, the position of theoretical percolation threshold is at lower relative pressure than the experimentally observed. It indicates, that the connectivity of the real network can be higher than four, the maximum value for selected 2D network. Best agreement between experimental and computed desorption equilibrium curve was obtained by employing 3D lattice model 20 × 20 × 20 with average connectivity C = 5.8. Evaluated size distribution functions for pore sites and pore connections are presented in Fig. 9 for both, 2D and 3D lattices. Note, that the size distributions for the pore sites are almost identical for both, square and cubic lattices, because the

Fig. 8. Experimental () and computed hysteresis dependent adsorption– desorption equilibria for systems Vycor glass–nitrogen: (—) primary adsorption and desorption for pore blocking model with 20 × 20 × 20 cubic lattice; (– – –) primary adsorption and desorption for pore blocking model with 23 × 23 square lattice.

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Fig. 9. Histograms of optimized distribution functions fitted to the experimental data: (a) pore sites for 23 × 23 square lattice, (b) pore connections for 23 × 23 square lattice, (c) pore sites for 20 × 20 × 20 cubic lattice, and (d) pore connections for 20 × 20 × 20 cubic lattice.

primary adsorption isotherm is not impacted by the distribution of pore connections with negligible volume. Since this description of the structure of the porous media is still simple in the following we calculate the capillary condensation in the digitally reconstructed porous media [30,38]. A direct comparison between experimentally measured PSD for porous Vycor glass, and PSD evaluated using the virtual capillary condensation method [30], is shown in Fig. 10. The computer simulation results have been obtained for a Gaussian correlated porous medium with a porosity of 0.35 and correlation length of 2.0 nm. A two-dimensional cross-section of the Gaussian porous medium used for the simulations is shown in Fig. 11. Gaussian correlated porous media are relatively simple to generate, yet they are considered a good approximation of the real structure of porous Vycor glass [39,40]. The method for obtaining PSD from reconstructed porous media by the virtual capillary condensation method is fully described in our related publication [30]. Let us now discuss the results presented in Fig. 10. As can be seen from figure, the maxima of the PSD measured experimentally (same as in Fig. 9c) and that obtained

from reconstructed porous media algorithm agree quantitatively and there is also a qualitative agreement between the measured and the simulated PSD. The two main differences are that: (i) the experimentally measured PSD does not contain any pores below 1.75 nm, while the simulated PSD contains pores down to 1.0 nm; (ii) the proportion of pores larger than approximately 2.75 nm is somewhat larger in the experimental case. The former difference can be explained by the fact that during the evaluation of PSD from the experimental data, no capillary condensation was assumed to occur below the lower closure point of the hysteresis loop. The amount adsorbed below the lower closure point was attributed solely to surface adsorption and therefore not contributing to PSD. On the other hand – as is also discussed in [30] – there can be regions on the capillary condensation curve that are reversible (i.e. they do not contribute to hysteresis). These regions, essentially dead-ends of round pores, are responsible for the non-zero pore volume on

Fig. 10. Pore size distribution of Vycor glass evaluated from experiments (nitrogen sorption at 77 K) and calculated by the virtual capillary condensation method using Gaussian correlated porous media as a model Vycor glass.

Fig. 11. Cross-section of a Gaussian correlated porous medium with porosity 0.35 and pore-space correlation length of 2.0 nm, which has been used for the calculation of PSD plotted in Fig. 10.

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the simulation PSD for pores below 1.75 nm. The discrepancy in the range of larger pores is most likely due to the fact that the Gaussian porous media do not tend to have as broad a distribution of pore sizes as may occur in real Vycor glass. This difference could in principle be rectified by generating ‘twotier’ Gaussian porous media that would contain a combination of pores with different correlation lengths. That way a wider range of pore sizes would be generated. Overall, however, one can conclude that the agreement between the experimental and computer simulation PSD is good. 4.2. Adsorption–desorption kinetics Kinetic properties of system Vycor glass–nitrogen were studied only using the square lattices. In all figures the experimental and theoretical diffusivities are normalised. The data are rationed against the diffusivity at the lower closure point D0 . First, we tested basic qualitative properties of the kinetic models. The 2D lattices obtained by fitting to equilibrium data were regenerated and used for modelling of mass transfer during adsorption and desorption. Impact of the lattice size on evaluated Fickian diffusivities during adsorption is presented in Fig. 12. Relative diffusivities employed in adsorption simulations were dji = 1 for surface diffusion, dji = 2 for condensation in the connection ji, and dji = 5 for condensation in the site i. As observed experimentally [26], the

Fig. 12. Impact of lattice size on the computed effective diffusivity for adsorption. (a) Lattice dimensions () 50 × 50, (䊉) 30 × 30; (b) lattice dimension 20 × 20, average values and error bars evaluated from 10 simulations.

Fig. 13. Impact of lattice size on the computed effective diffusivity for desorption. (a) Lattice dimensions () 30 × 30, (䊉) 20 × 20; (b) lattice dimension 20 × 20, average values and error bars evaluated from 10 simulations.

concentration dependence of the normalized diffusivity exhibits a maximum for adsorption. The maximum value increases with the lattice size as manifestation of the increasing mass transport path for larger lattices. Standard deviations 4–12% were evaluated from 10 simulations with randomly generated 20 × 20 lattices. Impact of the lattice size on desorption simulations is presented in Fig. 13. The concentration dependence of the normalized diffusivity exhibits a minimum. Again, the difference between the minimum and maximum diffusivity increases with the lattice size. Standard deviations 6–22% were evaluated from 10 simulations with randomly generated 20 × 20 lattices. Relative diffusivities employed in desorption simulations were dji = 1 for surface diffusion and dji = 5 for evaporation from any adjoining site i or j. This is a significant difference comparing to our previous work [26] in which we assumed lower pore diffusivities for evaporation than for condensation or surface diffusion. The process of evaporation from an individual pore site, which is typically in metastable state, can be very quick. Simulations confirm that the reason of the slow desorption close to percolation threshold is long and complicated mass transfer path in the lattice with many blocked connections and pores. Next, some interesting features of the mass transport during adsorption and desorption in the lattice from Fig. 4 are discussed. Time dependence of the adsorbed film thickness, t, during adsorption inside the pores is shown in Fig. 14a and b. As one can see in Fig. 14a much slower filling of pores by the surface diffusion comparing to rapid and robust capillary condensation

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Fig. 14. Time dependence of adsorbed film thickness during adsorption inside the pores from Fig. 4. (a) Complete transients for central pore 16 and its four neighbours; (b) detailed inset for filling of the pores at the onset of capillary condensation.

illustrated by practically vertical lines is observed. Because the capillary condensation commences almost simultaneously in all four neighbours (10, 15, 17, 22) of the pore 16, there exists a local minimum of the film thickness in the pore 16 as manifestation of the adsorbate suction by its neighbours, as illustrated in Fig. 14b. Interesting behaviour is theoretically predicted also during desorption in the same subsystems of pores, as presented in Fig. 15. Comparing Figs. 14a and 15a we can see that capillary condensation starts in the pore 10 and then subsequently in pores 17, 15, 22 and 16. On the other hand, the evaporation from the same pores occurs in the order: 10, 16, 15, 22 and 17 as consequence of pore blocking in the network. Fig. 15b shows existence of maxima on the time dependencies of the film thickness for pores 10 and 4 when large amount of adsorptive evaporating from pore 16 passes through the pores to the lattice surface. Experimental data of the normalised Fickian diffusivity for desorption for system Vycor glass–nitrogen are shown in Fig. 16. Comparisons to the theoretical predictions by desorption model of this work, as well as by “shell and core” model [26], are presented in the figure. Both theoretical models are able to predict well, qualitatively, the experimental concentration dependence of the diffusivity. Comparing to our previous work [26] the quantitative agreement was significantly improved, too. Less satisfactory prediction of the minimum position by the 2D lattice model is consequence of its unsatisfactory prediction of the primary desorption curve, as already discussed and presented in

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Fig. 15. Time dependence of adsorbed film thickness during desorption inside the pores from Fig. 4. (a) Complete transients for central pore 16 and its four neighbours; (b) detailed inset for emptying of the pores.

Fig. 16. Experimental diffusivity data (䊉) vs. theoretical predictions by using lattice model (—) and “shell and core” model (– – –).

Fig. 8. We expect that the position will be correct after employing the optimised cubic lattice (Fig. 9c and d). 5. Conclusions Percolation lattice models were developed for modelling of adsorption–desorption equilibria and kinetics for systems with hysteresis. The main results are as follows: (1) Satisfactory agreement with experimental data for system Vycor glass–nitrogen was obtained by incorporating the pore blocking assumption into the percolation model.

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(2) Parametric study of the lattice size was performed showing that the 2D lattice with size 30 × 30 or 3D lattice with 20 × 20 × 20 are sufficient to get satisfactorily smooth curves of primary adsorption and desorption equilibrium curves. (3) To get better fit of the model to the experimental data, one needs simultaneous optimisation of the lattice size and lattice connectivity as well as testing of different distribution functions for pore sites and pore connections. Higher flexibility and reality of the model was achieved by employing cubic (3D) lattices. (4) The percolation lattice models containing the pore blocking assumption were also successful for modelling of equilibrium scanning curves inside the hysteresis loop. (5) Evaluated connectivity and size distributions for both, the pore sites and the pore connections can be subsequently used also for modelling of adsorption–desorption kinetics of the same experimental system. (6) Kinetic models satisfactorily predict maximum of the Fickian diffusivity for adsorption as well as minimum for desorption. (7) Significant improvement of the quantitative agreement with experimental data for desorption was achieved assuming higher pore diffusivities for evaporation from pores. Simulations indicate that the tortuous path for the emptying of the core of the sorbent micro-particles is the main reason of the slow desorption close to the percolation threshold. (8) Unsatisfactory prediction of the primary desorption equilibrium by the simple 2D model causes also incorrect position of predicted minimum on the concentration dependence of normalised Fickian diffusivity. Both problems can be solved by employing cubic lattices. (9) Adsorption model predicts interesting theoretical transient behaviour in the selected pores with possible local minima of the adsorbed film thickness as a consequence of suction of adsorbate by neighbouring pores in which capillary condensation occurs. (10) Desorption model predicts local maxima of the film thickness as consequence of evaporation of adsorbate from neighbouring pore. (11) Filling or emptying of the same pores can start in very different order, because of pore blocking during desorption. (12) Models proposed in this work represent next step in unification of theories for equilibrium and kinetics for systems with hysteresis. Information on the adsorption–desorption equilibria are used for prediction of kinetic behaviour. References [1] G. Mason, Determination of the pore-size distributions and pore-space interconnectivity of Vycor porous-glass from adsorption desorption hysteresis capillary condensation isotherms, Proc. R. Soc. Lond. 41 (1988) 453–486. [2] N.A. Seaton, Determination of the connectivity of porous solids from nitrogen sorption measurements, Chem. Eng. Sci. 46 (1991) 1895–1909. [3] H. Liu, L. Zhang, N.A. Seaton, Determination of the connectivity of porous solids from nitrogen sorption measurements. 2. Generalization, Chem. Eng. Sci. 47 (1992) 4393–4404.

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