Surface Diffusion of Adsorbed Molecules in Porous Media: Monolayer

Ind. Eng. Chem. Res. 2001, 40, 4005-4031

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REVIEWS Surface Diffusion of Adsorbed Molecules in Porous Media: Monolayer, Multilayer, and Capillary Condensation Regimes Jeong-Gil Choi,† D. D. Do,*,‡ and H. D. Do‡ Department of Chemical Engineering, Division of Chemical and Polymer Engineering, Hannam University, Taejon 306-791, South Korea, and Department of Chemical Engineering, The University of Queensland, St Lucia QLD 4072, Australia

This review provides an overview of surface diffusion and capillary condensate flow in porous media. Emphasis has been placed on the distinction between purely surface diffusion, multilayer surface diffusion, and capillary condensate flow. 1. Introduction The surface diffusion of adsorbed molecules in porous media has long been considered one of the most interesting subjects associated with solid surfaces because surface diffusion contributes significantly to mass transport in porous media, especially in separation and many other industrial processes. For example, it was reported that, for microporous catalysts, surface diffusion can account for more than 50% of the total mass flow rate.1,2 For a better understanding of surface diffusion, it is necessary to distinguish surface flow from other modes of flow, including gas-phase flow and capillary condensate flow, which are the main types of mass transport in mesoporous or macroporous media. Because the diffusion of adsorbed molecules on solid surfaces is assumed to be an activated process involving molecules jumping from one site to neighboring sites, several important variables can influence the mass transfer rates through adsorptive porous materials, including surface concentration, operating temperature, gas species, and pore structure. Judging from the effects of these parameters on the flow rates, a number of theoretical models have been proposed in the literature. Carman and co-workers found that surface diffusivity increases rapidly as monolayer coverage is approached.3,4 Beyond monolayer coverage, the diffusivity showed a decline and then increased again abruptly in the capillary condensation region. Therefore, with respect to surface concentration, three different regimes can be identified: a monolayer region, a multilayer region, and a capillary condensation region. Multilayer adsorption occurs when the gas molecules adsorb in several layers on the surface. For further increases in gas relative pressure, capillary condensation takes place. However, for heterogeneous surfaces, capillary condensation starts * To whom the correspondence should be addressed: D. D. Do, Department of Chemical Engineering, University of Queensland, St. Lucia, Qld 4072, Australia. Phone: +61-7-3365-4154. Fax: +61-7-3365-2789. E-mail: duongd@ cheque.uq.edu.au. † Hannam University. ‡ The University of Queensland.

to occur on some parts of the surface while multilayering is still occurring on others, leading to the coexistence of the multilayer-adsorbed and capillary-condensed phases in the pores. Because these are very complicated phenomena of mass transport, very little has been understood concerning movements of multilayer-adsorbed molecules and capillary condensates. Furthermore, systematic comprehensive investigations for the above-mentioned three movements have not yet been reported, despite the availability of reports treating each subject separately. Starting with this background, this review is aimed at providing updated information on all aspects of these three transport phenomena by systematically treating movements of monolayer-adsorbed molecules, multilayer-adsorbed molecules, and capillary condensates. Finally, diffusion in microporous materials, i.e., zeolites, will also be briefly elucidated, including pure- and multicomponent diffusion processes. 2. Classification of Adsorption In general, the phenomenon of adsorption can be classified in different ways.5 It can be classified as physical or chemical adsorption on the basis of the magnitude of the heat of adsorption. Although such an approach is widely used because of its convenience, it is not very precise. Adsorption can also be classified as mobile or immobile adsorption from the molecular mobility point of view. Alternatively, it can be classified as monolayer or multilayer depending on the magnitude of the relative pressure. We now briefly discuss these three different classification schemes. 2.1. Physical Adsorption and Chemisorption. Because adsorption occurs through interactions between a solid and molecules in the fluid phase, it is important to distinguish between two different types of adsorption: chemisorption and physical adsorption (physisorption). For chemisorption, there is a direct chemical bond between the adsorbate and the surface, whereas for physisorption, no direct chemical bond is formed, but rather, the adsorbate is held by physical (i.e., van der Waals and electrostatic) forces. Typical chemisorption

10.1021/ie010195z CCC: $20.00 © 2001 American Chemical Society Published on Web 08/21/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

and physisorption energy ranges are 15-100 and 2-10 kcal/mol, respectively, for simple molecules. On a more fundamental level, the electrons of the chemisorbed molecules are shared with the surface. As a result, the electronic structure of the adsorbate is significantly perturbed, but the electronic structure of the surface is perturbed to a lesser extent. However, for physisorption, the surface does not share electrons with the adsorbate, so that the electronic structure of the adsorbate is perturbed to a much lesser extent. Although it is not possible to clearly discriminate between physisorption and chemisorption in every case, there are some important distinguishing features between them. For example, chemisorption is necessarily confined to a monolayer, whereas physisorption generally occurs for multilayers at relatively high pressures. A physisorbed molecule usually keeps its identity and, on desorption, returns to its original form. However, a chemisorbed molecule loses its identity on reaction or dissociation and cannot be recovered by desorption. The energy of chemisorption is of the same order of magnitude as the energy change in a comparable chemical reaction. Physisorption is always exothermic, and the energy involved is not much larger than the energy of condensation of the adsorbate. 2.2. Mobile and Immobile Adsorption. As mentioned earlier, from a mobility standpoint, adsorption types can also be classified as mobile or immobile. In the case of mobile adsorption, a molecule can move around while it is adsorbed on the surface, that is, this molecule remains in the adsorbed state all of the time. For immobile adsorption, a molecule does not leave its adsorbed location until it desorbs and returns to the fluid phase. It is important to note that localized adsorption behaves like immobile adsorption, whereas a nonlocalized molecule might be mobile. In general, adsorbed molecules take up fixed locations, but in some cases, they are free to move from position to position. 2.3. Monolayer and Multilayer Adsorption and Capillary Condensation. The amount of physically adsorbed gas molecules always decreases monotonically as temperature increases. The amount is usually associated with the relative pressure, P/P0, where P and P0 are the partial vapor pressure of a component in the system and the saturation vapor pressure at the same temperature, respectively. For example, when the relative pressure is about 0.01 or less, the amount of physical adsorption for nonporous materials can be considered negligible. However, this is not true for microporous solids such as active carbon. For P/P0 ≈ 0.1, the amount adsorbed corresponds to a monolayer. The monolayer capacity is usually defined as either the chemisorbed amount required to occupy all surface sorption sites or the physisorbed amount required to cover the surface. As the pressure increases progressively, multilayer adsorption occurs until a bulk liquid is reached at P/P0 ) 1.0. The approximate values of relative pressure for multilayer adsorption range from 0.1 to 0.3, beyond which pores are filled with a liquidlike phase. This phase transition is known as capillary condensation. It is noteworthy that these concepts should apply only for conditions below the critical point. The amount of gas adsorbed, q, by a given mass of solid, m, is dependent on the equilibrium pressure, P; the temperature, T; and the nature of the gas-solid system

q/m ) f(P,T,system)

(1)

Figure 1. Five types of adsorption isotherms described by Brunauer (ref 6).

For a constant temperature, eq 1 can be written as

q/m ) f(P)T

(2)

and if the temperature is below the critical temperature, it can be written as

q/m ) f(P/P0)T

(3)

Equations 2 and 3 represent the adsorption isotherm, which is the relationship between the amount adsorbed and the relative pressure at a given temperature. The IUPAC 1985 classification of physisorption isotherms is shown in Figure 1.6 Type I is often referred to as the Langmuir type because it corresponds to monolayer adsorption isotherms. Type II is the common S-shaped isotherm and the uptake at the top of the rise in the curve represents the completion of the monolayer and the beginning of the formation of the multilayer. Types IV and V are observed in multilayer adsorption on highly porous adsorbents, and the flattening of the isotherms at near-saturation pressure is ascribed to complete filling of all pores. 3. Transport of an Adsorbable Gas 3.1. Theoretical Aspects of Flow. Broadly speaking, there are two different types of mass transport: (1) mass transport in microporous media and (2) mass transport in mesoporous or macroporous media. According to the IUPAC classification scheme, pores are divided into three categories on the basis of size:

micropores

d < 2 nm

mesopores

2 nm < d < 50 nm

macropores 50 nm < d This division is based on the difference in the types of forces controlling adsorption behavior. In the micropore range, the overlapping surface forces contributed by

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4007 Table 1. Transport Regimes in Porous Media transport type

pore diameter (nm)

viscous flow molecular diffusion Knudsen diffusion micropore (configurational) diffusion

>20 >10 2-100 0; otherwise, H(x) ) 0 (28) and the parameter λ is the ratio of two rate constants

λ)

Figure 5. Plots of surface diffusivity versus concentration (ref 7).

of the second term in the denominator of eq 23. Figure 5 presents plots of the surface diffusivity as a function of surface concentration, showing a comparison between the theories of Higashi et al.40 and Yang et al.7 Whereas Higashi et al.’s theory gives an infinite value at θ ) 1, Yang et al.’s theory predicts a slower rise in diffusivity with respect to θ and gives a finite value at θ ) 1. Correlations using the models of Higashi et al. and Yang et al. were made for experimental data for propane, butane, and CF2Cl2 on different solids (silica glass, Spheron, and Carbolac matrix) at different temperatures, as shown in the literature. 4.2.3. Suzuki and Fujii’s Model. Suzuki and Fujii investigated the surface diffusion of propionic acid using activated carbon and suggested the following equation for the surface concentration dependence of the surface diffusivity43

Ds ) (aq)n Ds0

(24)

where a and n are constants such that the isosteric heat of adsorption Qs and the isotherm q(C) are described by

Qs ) -Q0 ln(aq)

(25)

and

q)

1 1 1 + KC kC1/n

(26)

where Q0 is constant. Equation 24 correlated well with the experimental data; however, the trends in the variation of surface diffusivity with respect to the amount adsorbed were different from the previous models of Higashi et al. and Yang et al. As monolayer coverage is approached, the models of Higashi et al. and Yang et al. show a rapid increase in diffusivity, whereas the model of Suzuki and Fujii exhibits a slow increase. 4.2.4. Chen and Yang’s Model. Recently, Chen and Yang, who used a transition state theory, proposed the

kb km

(29)

where kb is the rate constant for blockage and km is the rate constant for forward migration. The parameter λ indicates the degree of blockage by other molecules that have already occupied sites. Thus, for λ ) 0, eq 27 reduces to Higashi et al.’s model. However, depending on the nonzero values of λ, the surface diffusivity varies with the surface concentration. It increases with increasing surface concentration for small λ, whereas it decreases with increasing concentration for large λ. Chen and Yang reported that the values of the blockage parameter λ for a small molecule (ethane or propane) in zeolite A and a large molecule (triethylamine) in zeolite 13X were observed to be ∼0.2 and 10, respectively, implying that the surface diffusivity of triethylamine decreases with the concentration because of a large value of λ.44 4.2.5. Kapoor and Yang’s Model. For activated carbon systems, it is frequently observed that the surface diffusivity increases very rapidly with the surface concentration. These unusual features in activated carbons could not be explained using the abovementioned models of Higashi et al., Yang et al., and Chen and Yang. It was suggested in the literature that this rapid increase in surface diffusivity might be due to the heterogeneity of the systems, as first investigated by Siedel and Carl.47 They assumed that the surface diffusivity was independent of adsorbate concentration on a homogeneous surface and that the concentration dependence of surface diffusivity was due to surface heterogeneity. Their approach predicts zero surface diffusivity at zero concentration, which is not consistent with the experimental4,48,49 and theoretical23,40,42 results reported by other researchers. Later, Kapoor and Yang derived the following equations for surface diffusivity by assumpting that the surface was composed of a series of parallel paths with uniform but different energies and that surface transport proceeded in the direction of the parallel paths50

Ds e2sθ - 1 )1+ s for a1 ) 1/2 s(2θ-1) Ds0 e -e

(30)

and

(

Ds e2sθ - 1 2s )1+ s -s s Ds0 e - e e - es(2θ-1)

)

for a1 ) 1 (31)

Here, the parameters a1 and s represent the mean ratio

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4013

of the activation energy to the adsorption energy and the degree of surface heterogeneity in a given system, respectively. The fractional surface coverage θ is related to the pressure through the Unilan equation

(

1 + b1esP 1 ln θ) 2s 1 + b1e-sP

)

(32)

where b1 is the mean adsorption affinity constant with b1 ) b∞ exp(Em/RT), Em is the mean adsorption energy, and the heterogeneity parameter s is 1.732ζ/RT, with ζ being the square root of the variance of the energy distribution. Although the surface diffusivity increases rapidly with θ greater than 1/2, it is still comparable to that of HIO model for θ < 1/2. Nonetheless, all of the experimental data for hydrocarbons in activated carbon could not be explained using this parallel path model (PPM) because of the insufficient degree in the increase of surface diffusivity. Again, in 1990, Kapoor and Yang proposed another more complicated model based on the effective medium approximation (EMA) to obtain the overall surface diffusivity.51 There are two different EMA expressions according to the different dimension of diffusion: one-dimensional EMA (EMA-1D) and twodimensional EMA (EMA-2D). Whereas EMA-1D shows an even greater increase in the surface diffusivity with concentration, the behavior of EMA-1D is similar to that of PPM. 4.2.6. Statistical Mechanical Approach. Several researchers have carried out statistical mechanical treatments for the purpose of studying surface diffusion.52-54 Among these, Hwang and Kammermeyer derived an explicit expression for the dependence of surface diffusivity on temperature and molecular weight using partition functions53

Ds )

βTxT -E/RT e xM

(33)

Equation 33 can be applied to the condition of zero concentration at which the partition between two phases is linear. Hwang and Kammermeyer observed good agreement between theory and experimental data obtained using various gases including helium, hydrogen, oxygen, argon, and carbon dioxide. 4.2.7. Diffusion in the Knudsen Regime with a Potential Field. In gas diffusion in the Knudsen regime where the gas molecules collide with the walls, the collisions of gas molecules are influenced by the potential energy between the molecules and the surface of a pore. Therefore, the total permeability is written as55

B)

K1

[

]

1 + R1(e*/kT - 1) xMT 1 + β1*/kT

(34)

where K1 ) gλ1(8/πR)1/2; g and λ1 are the geometrical factor and mean free path of the molecules, respectively; β1 and R1 are constants; k is the Boltzmann constant; and * is the effective potential energy between a gas molecule and the solid surface. The first term on the right-hand side of eq 34 shows Knudsen flow (gas-phase flow), and the second term indicates surface flow. By setting * equal to zero in eq 34, the permeability for Knudsen flow can be obtained. Consequently, the difference between nonzero potential field and zero potential field is the permeability contributed by the surface

flow. This approach has also been investigated by Weaver and Metzner,56 Roybal and Sandler,57 and Nicholson and Petropoulos.58-62 4.2.8. Structural Approach. In 1996, Do presented a model for the surface diffusion of ethane and propane in heterogeneous activated carbon and addressed the effect of structure on the transport surface diffusivity. The structure of activated carbon is complex and is considered to be a combination of amorphous and graphitic regions. Because of the nonexistence of a definite surface available for diffusion, it might not be appropriate to use the concept of “surface diffusion” in the case of activated carbon. In this model, it is assumed that adsorbate molecules penetrate (adsorb) into a graphitic layer unit, diffuse through it, and evaporate (desorb) out of it.45 Do’s model gives rise to an apparent surface diffusivity that is an explicit function of temperature and concentration of the adsorbed species. For example, in the case of the Langmuir isotherm, Do proposed the following equation for the dependence of surface diffusivity on surface concentration

Ds )

Ds0 (1 - θ)2

(35)

Experimental data for the surface diffusion of ethane and propane on activated carbon were correlated using this model and found to exhibit a stronger dependence on the adsorbed concentration than that given by the Darken relation, Ds ) Ds0/(1 - θ). 4.3. Effect of Surface Concentration. As pointed out earlier, the surface diffusivity (Ds) is a strong function of the surface concentration and usually increases with increasing amount adsorbed. Figures 6 and 7 present typical experimental data for the equilibrium isotherms and the corresponding concentration dependence of surface diffusivity for various adsorbates on porous Vycor glass.25 A number of theoretical models describing the concentration dependence of the surface diffusivity are presented in the literature, and a brief introduction was already given in the previous section. Therefore, in this section, other factors that are also influenced by the variation of surface concentration are discussed in relation to the surface diffusivity. Ross and Good used the following equation to explain the surface diffusivity of n-butane on Spheron 6(2700)48

Ds ) Ds1 exp(∆S/R) exp(-E/RT)

(36)

where ∆S is the entropy of activation and E is the energy of activation. From this equation, it can be seen that the change in surface diffusivity Ds is associated with the variation in the entropy of surface molecules. As the surface concentration increases, the entropy of the surface species decreases. The relationship between surface coverage and entropy might explain the dependence of the surface diffusivity on the surface coverage. Carman and Raal reported that, for less than a monolayer coverage, the surface diffusivity increases rapidly with increasing surface concentration.4 They considered that this increase was due to the variation in the heat of adsorption and attributed the surface diffusion to the most loosely bound molecules, implying that, at low coverage, high-energy sites are first occupied by molecules that have a very low mobility. For further increases in the surface coverage, lower-energy sites are

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Figure 6. Equilibrium isotherms (top) and concentration dependence of Ds (bottom) for the diffusion of various gases on porous Vycor glass (ref 49).

progressively filled, and these loosely bound molecules diffuse at a faster rate. Gilliland et al. also observed that there was a strong relationship between surface diffusivity and surface concentration.21,49 They ascribed this behavior to a decrease in the heat of adsorption with increasing surface coverage. Later, this result was used to provide a general correlation of the diffusivity data for different systems with plots of log D versus (-∆H). Using the assumption that the driving force for transport is the gradient of the chemical potential,46,63 Aharoni derived expressions for surface flux in nonhomogeneous porous media where the energy of adsorption varies.64 Sun and Meunier proposed a model using the gradient of chemical potential as the driving force to describe pore and surface diffusivities in a microporous particle.65 4.4. Effect of Operating Temperature. As shown in Table 4, a several experimental reports in the literature show that the surface diffusivity of monolayeradsorbed molecules can be also expressed as a function of the operating temperature; see, for example, refs 10, 21, and 66. In fact, surface diffusion is an activated process in which the surface diffusivity varies with temperature according to an equation of the Arrhenius form that can be written as

Ds ) Ds0 exp(-E/RT)

(37)

where E is the activation energy of surface diffusion. Gilliland et al. suggested that the activation energy of surface diffusion, E, is associated with the energy of

Figure 7. Equilibrium isotherms (top) and concentration dependence of Ds for the diffusion of various gases on porous Vycor glass (ref 10).

adsorption, Q, through

E ) a2Q

(38)

where a2 is a constant that depends on the adsorbateadsorbent system.49 After correlating the diffusivities of adsorbed species in various systems, Sladek et al. found that a2 ) 1 or 1/2 for physically adsorbed molecules, depending on the type of bond between the adsorbate and adsorbent.21 According to eq 37, the surface diffusion coefficient increases exponentially with temperature. Taking the logarithm of eq 37, one obtains

ln Ds ) ln Ds0 - E/RT

(39)

which suggests that the plot of the logarithm of the diffusivity versus the reciprocal temperature should yield a straight line with a slope of -E/R and an intercept of log Ds. However, it is sometimes observed that the plot of eq 39 is curved slightly upward over the temperature range between 350 and 1000 K, as shown in Figure 8. This unusual behavior has been a controversial subject, giving rise to a number of hypotheses concerning the temperature dependence of gas diffusion. For example, several investigators found a similar result for the temperature effect using He gas in glass.67-73 Judging from most discussions for the deviation shown in Figure 8, it has been agreed that a temperature-dependent term should be included in the preexponential of the Arrhenius equation.67,69-73 This relationship is given by

Ds0 ) Ds0*Tn

(40)

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4015 Table 5. Effects of Gas Species on Surface Diffusivities of Monolayer-Adsorbed Molecules

Figure 8. Temperature dependence of helium diffusivity in vitreous silica (ref 67). Table 4. Effects of Operating Temperature on Surface Diffusivities of Monolayer-Adsorbed Molecules for Different Solid-Gas Systems solid

adsorbate

temp (K)

θ

Ds × 105 (cm2/s)

ref

silica alumina cracking cat. Vycor glass Vycor glass silica alumina cracking cat. Spheron 6 (2700°) carbon black Linde silica Linde silica Vycor glass Vycor glass Carbolac Carbolac Carbolac Carbolac Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass Graphon Graphon alumina alumina alumina alumina carbon carbon carbon

C2H6 C2H6 C 2H 6 C 2H 6 C 3H 8 C3H8 n-C4H10 n-C4H10 n-C4H10 n-C4H10 i-C4H10 i-C4H10 CO2 CO2 CO2 CO2 SO2 SO2 CO2 CO2 C3H6 C3H6 C3H6 C2H6 C2H6 C 2H 6 C 3H 6 C3H6 C3H6 C3H6 C3H6 C2H4 C2H4 C3H6 C3H6 C3H6 C3H6 C3H6

323 400 323 343 323 400 303 323 259 323 273 323 240 252 273 293 288 303 195 223 273 298 313 273 298 323 273 298 323 298 323 283 303 283 303 273 283 297

0.13-0.53 0.14-0.41 0.04-0.26 0.03-0.17 0.37-0.84 0.27-0.67 0.63-0.89 0.38-0.78 0.14-0.99 0.07-0.71 0.07-0.61 0.02-0.25 P2). As mentioned earlier, the capillary condensa-

+

]

RTFappx2

l(Pt - P2)

CRStτ p (l - z)l(P1 - P2) 2

(61)

where l is the length of the capillary. As P2 increases, the entire pore is slowly filled with capillary condensate. Thus, mode 3 is reached, and capillary condensation occurs downstream (P2 > Pt and t2 < r)

Bt3 )

[

]

(r - t1)2 P1 (r - t2)2 P2 KRTF ln ln P0 P0 Mµ(P1 - P2) r2 r2 (62)

When the upstream pressure increases further, the upstream end of the pore becomes filled with bulk condensate. At the same time, if the downstream pressure remains smaller than the capillary condensation pressure (P2 < Pt), mode 4 is observed with a meniscus located somewhere inside the capillary

l(Pt - P2) Bt4 ) (Bg1 + Bs) (l - z)(P1 - P2)

(63)

Mode 5 is the limiting case of mode 4, representing the state when the whole pore is filled with condensate (P2 > Pt and t2 < r)

Bt5 )

2 P2 - KRTF (r - t2) ln 2 P0 Mµ(P1 - P2) r

(64)

Finally, mode 6 represents the case in which the entire pore is filled with bulk condensate (t2 > r) and the condensate flow obeys the Hagen-Poiseuille equation

Bt6 )

FNπr4 8Mτµ

(65)

The above permeability equations for each mode were derived macroscopically on the basis of cylindrical capillary structures so that the above treatment is restricted to the idealized case of a single pore. Because

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4023 Table 10. Effect of Pore Structure on Permeability of Capillary Condensates solid

adsorbate

pore radius (Å)

temp (K)

permeabilitya (mol m-1 s-1 Pa-1)

relative pressure at Bmaxb

γ-alumina Vycor glass Vycor glass Vycor glass Linde silica Carbolac Linde silica Carbolac

propylene Freon-113 Freon-113 water vapor CF2Cl2 CF2Cl2 CF2Cl2 CF2Cl2

15 21 21 21 64 11 64 11

236 292 315 343 240 240 252 252

1.48 × 10-10 7.41 × 10-10 2.17 × 10-10 1.18 × 10-9 6.87 × 10-9 1.45 × 10-9 4.01 × 10-9 8.92 × 10-10

0.77 0.56 0.59 0.59

a

ref 120 113 113 113 4 4 4 4

Maximum permeability or permeability for all of the pores filled with capillary condensate. b Bmax ) maximum permeability.

actual porous materials have wide pore size distributions, a new model should address the influence of the pore structure on the determination of the permeability of condensable vapors through porous media. 6.3. Effect of Pore Structure. Uhlhorn et al. studied the capillary condensation of propylene on supported γ-alumina films at 263 K.120 On the basis of the upstream pressure between 265 and 359 kPa and the saturation vapor pressure of propylene (P0 ) 423 kPa), values of the relative pressure (Pr) were calculated to range from 0.63 to 0.85. According to the six different possible flow modes suggested by Rhim and Hwang who used Vycor glass as the porous medium, theoretical capillary condensation was expected to start at an upstream pressure of 284 kPa (Pr ) 0.67, mode 2).112 However, Uhlhorn et al. observed capillary condensation at the higher upstream pressure of 326 kPa (Pr ) 0.77, mode 5), at which the maximum permeability was observed.120 They ascribed this apparent discrepancy to the different pore structure, that is, the slit shape of the pores in the γ-alumina matrix shown in the literature.122,123 They explained that the slit-shaped pores of the γ-alumina are gradually filled with layers of adsorbate until bulk condensate forms at the upstream end of the pore (when the layer thickness has become larger than the pore half width). In their data, this point was obtained at Pr ) 0.77, corresponding to the maximum value in the permeability. Therefore, using the γ-alumina films, they concluded that mode 2 did not exist in slit-shaped pores in the adsorption mode. Uhlhorn et al. also investigated the effect of promoter on the capillary condensation of propylene on supported γ-alumina films.120 Using magnesia-modified γ-alumina thin films, a similar result was observed for the relative pressure where the capillary condensation started, but the main difference was the magnitude of the permeability. The permeability of propylene was decreased drastically by a factor of about 20. They suspected that this difference was attributed to the high loading of magnesia, which decreased the pore volume and possibly the pore size. From these results, it can be seen that the pore structure of the adsorbent is important in the movement of capillary condensates, causing variations in the permeability of adsorbates. Table 10 shows the effect of pore size on the capillary condensation of CF2Cl2 on two different porous media (Linde silica and Carbolac). At two operating temperatures (240 and 252 K), Linde silica had about 5 times higher permeabilities than Carbolac. Because the pore radius of Linde silica ()64 Å) is larger than that of Carbolac ()11 Å), these results indicated that a large pore size results in high permeability in capillary condensation of porous media. Tamon et al. studied the flow mechanism of adsorbates through porous media in the presence of capillary condensation and found that

Figure 15. Effect of mean pore radius on network permeability (ref 116).

the flow of capillary condensates was more resistant in finer pores.23 Moreover, it appears that the permeability of capillary condensable vapors might be more influenced by the pore size than by the operating temperature. Kainourgiakis et al. introduced a network approach to modeling of the transport of condensable vapors in mesoporous structures.116 They showed that such permeability is affected by material structural parameters, in particular, the average pore radius. Figure 15 shows the effect of the mean pore radius on network permeability. It can be seen that both the magnitude and the location of the peak as a function of the relative pressure were definitely influenced by the pore radius. Smaller pore radii were found to lead to reduced flow rates in the individual pores and to a permeability maximum, which was ascribed to the existence of greater resistance to the flow of the capillary condensate in the smaller pores. 6.4. Effect of Adsorbable Gas Species. Rhim and Hwang obtained the permeabilities of three different vapors (ethane, n-butane, and carbon dioxide) through Vycor porous glass at various pressures.112 A maximum permeability was observed as the relative mean pressure (Pm/P0) for all of the gas vapors increased, indicating the occurrence of capillary condensation. However, the relative mean pressure (Pm/P0) at the maximum permeability (Bmax) was observed to be dependent on the gas species (Table 11). Furthermore, the value of Pm/P0 at the maximum permeability was also influenced by the pressure difference between the two sides of the porous medium (∆P/P0). For instance, for the flow of ethane at 298 K, the value of Pm/P0 at Bmax increased with decreasing ∆P/P0 within the pressure difference range between 0.17 and 0.66. Similar results were also found for the permeability of carbon dioxide with different values of ∆P/P0 ranging from 0.054 to 0.211.

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Table 11. Effect of Adsorbable Gas Species on Permeability of Capillary Condensates

solid Vycor glass Vycor glass Vycor glass Vycor glass Vycor glass

adsorbate CO2 n-butane CO2 ethane ethane

pore relative radius permeabilitya pressure (Å) (mol m-1 s-1 Pa-1) at Bmaxb 273 273 298 298 298

0.88 × 1.88 × 10-10 0.43 × 10-12 0.84 × 10-12 1.09 × 10-12 10-12

0.80 0.51 0.79 0.65c 0.52d

ref 112 112 112 112 112

a Maximum permeability or permeability for all of the pores filled with capillary condensate. b Bmax ) maximum permeability. c ∆P/P ) 0.17. d ∆P/P ) 0.66. 0 0

Figure 18. Permeability of propylene and helium through a supported γ-alumina thin film at 263 K (ref 120).

Figure 16. Permeability of ethane at 298 K through porous Vycor glass (ref 112).

Figure 17. Permeability of carbon dioxide through Vycor glass (ref 112).

However, the dependence of the value of Pm/P0 on ∆P/ P0 at Bmax was much stronger for ethane than for carbon dioxide (Figures 16 and 17). The maximum permeabilities of ethane were higher than those of carbon dioxide. Although Rhim and Hwang did not explain the difference in permeability of the different gas species, their results showed a clear effect of gas species on permeability. Because all of the experimental conditions were the same for the permeability measurements, they considered that one of the reasons for the permeability difference might be differences in the physical properties of the gas species such as viscosity. However, at 273 K, the maximum permeability of n-butane was about three times higher than that of carbon dioxide. Because the values of viscosity for n-butane, carbon dioxide, and ethane are 1.5 × 10-5, 1.01 × 10-5, and 0.94 × 10-5 Pa

s, respectively, these results show the negative and positive dependence of viscosity on permeability. Debye and Cleland studied the dependence of viscosity on the flow rates of normal paraffins in porous Vycor glass.124 They obtained results for both positive and negative deviations from the viscosity dependence predicted by Poiseuille’s law for liquids flowing in pores and explained these observations by assuming a slipping adsorbed layer of some thickness, with the thickness giving rise to negative deviations and the slip to positive deviations. In the case of the results of Rhim and Hwang, in addition to the effect of viscosity, another possible reason for the different permeabilities of n-butane and carbon dioxide could be the use of different ∆P/P0 values despite the unavailability of ∆P/P0 in their publication. 6.5. Effect of Operating Temperature. Rhim and Hwang measured the permeability of carbon dioxide using Vycor glass at different temperatures.112 They observed that the maximum permeability occurs at Pm/ P0 ) ∼0.8 for four different values of ∆P/P0, implying that the location of the relative pressure observed at the maximum permeability might not be influenced by the operating temperature. However, the values of maximum permeability were different depending on the temperature. The permeability obtained at 273 K was almost twice as high as that obtained at 298 K. In particular, in the case of CO2 permeation using Vycor glass, the critical temperature (304 K) is very important. As shown in Figure 17, above the critical temperature, the permeability remained almost constant, but below the critical temperature, it exhibited a maximum with respect to pressure and increased with decreasing operating temperature before the saturation vapor pressure was reached. A similar temperature effect was also shown in the transport of propylene on a supported γ-alumina film.120 The permeability at 263 K increased monotonically with increasing relative pressure, so the highest permeability was measured to be 1.0 × 10-10 mol m-1 s-1 Pa-1 at saturation pressure (Pr ) 1.0) (Figure 18). However, at 236 K, the permeability plotted as a function of relative pressure showed a maximum peak at Pr ) 0.77 (Figure 19). This result indicates that capillary condensation took place at this lower temperature. Although the critical temperature of propylene is 364.85 K, the result observed at 263 K did not show a maximum in permeability. This implies that the permeability of propylene was more strongly influenced by the operating temper-


Figure 19. Permeability of a supported γ-alumina thin film for nitrogen and propylene at 236 K as a function of the relative pressure of propylene.

ature in this propylene-alumina system. The value of maximum permeability at 236 K was about 1.5 times higher than that of the highest permeability obtained at 263 K (1.48 × 10-10 mol m-1 s-1 Pa-1). The monotonic increase in the propylene permeability at 263 K was considered to be associated with multilayer diffusion. When most of the pores were filled with condensate, a maximum was reached. After this maximum, the permeability dropped because the permeability for the liquid flow is much lower than that for the vapor flow.10,11,20,23,113,114 Carman and Raal observed the permeabilities of different systems at different temperatures in which all of the pores were filled with capillary condensate.4 For CF2Cl2-Linde silica, the permeabilities obtained at 240 and 251.5 K were 6.87 × 10-9 and 4.01 × 10-9 mol m-1 s-1 Pa-1, respectively. Similarly, for SO2-Linde silica permeabilities of 2.16 × 10-8 and 1.42 × 10-8 mol m-1 s-1 Pa-1 were measured at temperatures of 263 and 273 K, respectively. Another effect of temperature on permeability was reported for the CF2Cl2-Carbolac system, which had permeabilities of 1.45 × 10-9 and 8.92 × 10-10 mol m-1 s-1 Pa-1 at 240 and 251.5 K, respectively. All of these results indicate that lower temperatures gave rise to the higher permeabilities, again exhibiting a clear effect of temperature on permeability. Using a network model for the permeability of condensable vapors through mesoporous media, Kainourgiakis et al. observed the effect of temperature on network permeability.116 As shown in Figure 20, the permeability decreased along the entire relative pressure range with increasing temperature, which was in agreement with previous results.125 These results show that the occurrence of capillary condensation might have been influenced by the operating temperature. As the temperature varies, the maximum peak permeability can change, whereas the location of the maximum peak (relative pressure) remains constant. Nonetheless, as shown in Table 12, the relative pressure at maximum permeability varies for different solid-gas systems. Generally speaking, the flow rates in all possible flow regimes (Knudsen, surface, capillary condensation, and Poiseuille) depend on the temperature either in a straightforward manner or through the combined variation in the physical properties of the vapor-solid system (mainly adsorbed amount, saturation pressure, and viscosity). For example, al-

Figure 20. Effect of temperature on network permeability (ref 116). Table 12. Effect of Operating Temperature on Permeability of Capillary Condensates for Different Solid-Gas Systems

solid

adsorbate

γ-alumina γ-alumina Vycor glass Vycor glass Vycor glass Vycor glass Linde silica Linde silica Linde silica Linde silica Carbolac Carbolac

propylene propylene CO2 CO2 Freon-113 Freon-113 CF2Cl2 CF2Cl2 SO2 SO2 CF2Cl2 CF2Cl2

relative temp permeabilitya pressure -1 -1 -1 (K) (mol m s Pa ) at Bmaxb 236 263 273 298 292 315 240 252 263 273 240 252

1.48 × 10-10 1.00 × 10-10 0.88 × 10-12 0.43 × 10-12 7.41 × 10-10 2.17 × 10-10 6.87 × 10-9 4.01 × 10-9 2.16 × 10-8 1.42 × 10-8 1.45 × 10-9 8.92 × 10-10

0.77 1.00 0.80 0.79 0.56 0.59

ref 120 120 112 112 113 113 4 4 4 4 4 4

a Maximum permeability or permeability for all of the pores filled with capillary condensate. b Bmax ) maximum permeability.

though the viscosity of a liquid decreases with increasing temperature, the degree of reduction in permeability is more pronounced in the capillary condensation regime. The variation in permeability with temperature was considered to be associated with a change in the characteristic diffusion length and maybe also with a variation of the layer thickness.124 Therefore, it is important to decide what temperature should be used in the transport of a condensable gas. 7. Diffusion in Zeolites 7.1.Overview. Because of their importance in the areas of adsorption and catalysis, zeolites have been attractive to many researchers, and the diffusion of molecules in zeolites has been extensively investigated in recent decades.126,127 Diffusion in micropores such as zeolites or microporous carbons is generally known to be dominated by the interactions between the diffusing molecule and the pore walls.25 Thus, in small pores, steric and other effects associated with the proximity of the pore walls become important, and diffusion is an activated process. Because the diffusing molecules are within the force field of the pore walls, it is reasonable to consider the fluid within the pore as a single adsorbed phase. Consequently, diffusion in small-pore zeolites has many features similar to surface diffusion, and the same diffusion mechanism can be also used for modeling zeolitic diffusion. Diffusion within this regime is sometimes called configurational diffusion, intracrystalline diffusion, or simply micropore diffusion.

4026


Several distinct features are often experimentally observed for zeolitic diffusion: (1) very small diffusion coefficients,128 (2) high activation energies,127 (3) a strong concentration dependences,129-131 and (4) much slower flow of counterdiffusion compared to that of single-component diffusion.132,133 Very brief descriptions of these features are given in this segment. Because of the small pore size, the rate of diffusion in zeolites becomes relatively slow, and diffusion in zeolites is more complex than Knudsen diffusion or bulk diffusion. Representative diffusion coefficients in zeolites reported in the literature range from 10-14 to 10-5 cm2/s.128 For comparison, typical values of diffusion coefficients for bulk and Knudsen diffusions of gases are known to be in the ranges of 10-1 and 10-3 cm2/s, respectively. In general, the apparent activation energy of zeolitic diffusion increases with increasing size of the diffusing molecules. Also the smaller the diffusion coefficient, the higher the activation energy. Barrer and Brook reported large variations in diffusivity in zeolites (increases or decreases by as much as 2 orders of magnitude), indicating that astrong dependence of diffusivity on concentration was present.129 Although little work has been done on counterdiffusion or codiffusion in zeolites, an important consistent result obtained from previous studies is that counterdiffusion in binary systems is much slower than that in single-component systems.132,134-136 7.2. Theory. The flux J (mol m-2 s-1) through microporous materials such as zeolites, glasses, and microporous carbons has been phenomenologically observed to increase with increasing temperature according to an Arrhenius type of relation

J ∝ J0e-Ez/RT

(66)

where Ez (kJ/mol) is the activation energy, which ranges from 2 to 40 kJ/mol depending on pore size and gas molecule size.25,137-141 Here, the activation energy is an apparent value because it consists of two contributions: the activation energy for micropore diffusion (E1) and the isosteric heat of adsorption (qs) of the gas molecules on the solid surface. Barrer proposed the following equation for intracrystalline flux through a zeolite142

J)

s1k3 -E1/RT (θ1 - θn) e n-1

(67)

where s1 and k3 are the number of pores per unit crosssectional area and the preexponential constant for the jumping rate, respectively; n is the diffusion step number in the pore; θ1 and θn are the occupation degrees at the pore entrance and the pore end, respectively; and θ ) n1/n∞, where n1 is the amount adsorbed at a certain pressure and n∞ is the amount adsorbed at saturation. For Henry’s law, the occupation degree and the Henry’s constant can be written as

θ)

1 KP n∞ s

K ) K0eq /RT

(68) (69)

respectively, where K0 is a preexponential constant. From the above equations (eqs 67-69), it can be seen that, for Henry sorption, the flux depends linearly on

the pressure difference and the apparent activation energy, Ez, is a combination of E1 and -qs. More details and other descriptions are presented in the literature.75,143,144 7.3. Pure- and Multicomponent Diffusion. Since Barrer and co-workers pioneering research in zeolitic diffusion, many other workers have also been investigating diffusional behaviors in natural and synthetic zeolites using different methods.36,127,132,145-149 However, most of these studies have been exclusively focused on pure-component diffusion, notwithstanding the fact that multicomponent diffusion is known to be much more important for practical purposes. Thus, very little has been made in developing a fundamental understanding of multicomponent diffusion in zeolites.133,134,150,151 Furthermore, some contradictory experimental results concerning the effects of system properties on the diffusional behavior have been found. Therefore, more reliable experimental data on the zeolitic diffusion and its concentration and temperature dependences are essential for the development of existing diffusion theory for zeolites. In 1941, Barrer first attempted to model diffusion in zeolites using a kinetic theory145 in which the diffusion rate of a molecule depends on the probability of its occupying an vacant site. Later, in 1949, Barrer and Jost147 proposed a model using the concepts of irreversible thermodynamics, which was extended to twocomponent systems by Habgood.135 Riekert136 and Palekar and Rajadhyakhsha152-154 studied one- and twocomponent diffusion using a lattice model. Theodorou and Wei155 proposed a model using a random walk on a square two-dimensional grid, which was developed by Tsikoyiannis156 to estimate the diffusional fluxes using first-order correlation functions. Qureshi and Wei studied one- and two-component diffusion in zeolite ZSM-5 both theoretically and experimentally.133,134 Wei and coworkers used a stochastic (microscopic) and computational approach to formulate single- and multicomponent theories.133,157 Yang and co-workers also suggested a method for predicting multicomponent diffusivities from single-component diffusivities using a kinetic theory.151,158,159 In this review, one example of the estimation of multicomponent diffusivities proposed by Yang and co-workers is introduced for zeolitic diffusion n

[1 Dii ) Di0

(1 - ωij)θj]j*i ∑ j)1 n

1-

Dij ) Di0

(1 - ωij)θj ∑ j)1

[(1 - ωij)θi]j*i

1-

(70)

n

(71)

(1 - ωij)θj ∑ j)1

where ωij is the ratio of the sticking probability of molecule i on adsorbed molecule j to the sticking probability of molecule i on a vacant site and n is the number of components. Although these multicomponent diffusivities were derived using the principles of irreversible thermodynamics, which required pure-component diffusivities and mixed-gas isotherms,151,158 eqs 70 and 71 are, for binary diffusion (n ) 2), the same as


those obtained using a random-walk kinetic theory and are the simpler model to use.160 8. Concluding Remarks Because of the nonexistence of well-organized studies on the diffusion of adsorbed molecules in porous media, the main objective of this review was to systematically discuss the motions of monolayer-adsorbed molecules, multilayer-adsorbed molecules, and capillary condensates, which are often observed in mass transport in porous media. In fact, even though some information on each subject is available separately, no comprehensive review had been published to date. Therefore, this review was aimed at covering these three different movements with the probable effects of system variables on the flow rates. Moreover, for each type of mass transport, the theoretical considerations proposed in the literature have also been reviewed. In particular, very complicated phenomena of mass transport in the regimes of multilayer adsorption and capillary condensation resulted in a lack of sufficient information available so that this review covered the limited data in these two transport systems. Finally, the diffusion process in zeolites, one of the most important microporous materials, was briefly described as well. All in all, despite long investigations over more than 50 years, more theoretical and experimental observations are still needed for a better understanding of the diffusion phenomena of adsorbed molecules in porous media. Accordingly, based on this review, the following recommendations are proposed for future research: 1. The diffusion process of adsorbed multilayers should be investigated more thoroughly. 2. More information on the transport of capillary condensates should be obtained both experimentally and theoretically. 3. Fully satisfactory flow mechanisms in adsorbed multilayers and capillary condensates should be unambiguously elucidated. 4. The methods for predicting multicomponent diffusivities in zeolites should be studied in greater depth and clarified. Acknowledgment J.-G.C. appreciates the Korea Research Foundation for financial support under the project name 2000 Year Hak-Jin University Professor Overseas Study”. Support from the Australian Research Council is also gratefully acknowledged. Nomenclature a, a0 ) constants defined in eqs 25 and 52, respectively a1 ) mean ratio of activation energy to the adsorption energy a2 ) constant defined in eq 38 b ) constant defined in eq 10 b1 ) mean adsorption affinity constant B ) permeability, mol m-1 s-1 Pa-1 Bv, Bk ) permeabilities defined in eqs 6 and 8, mol m-1 s-1 Pa-1 Bt ) apparent permeability, mol m-1 s-1 Pa-1 c0 ) concentration, mol/m3 C ) concentration, mol/m3 C1 ) constant characteristic of the porous media Ca ) constant defined in eq 50 CB ) BET constant

d ) pore size, nm Ds ) surface diffusion coefficient, m2/s De ) effective diffusion coefficient, m2/s Ds0, Ds1 ) parameter defined in eqs 22 and 36, respectively, m2/s Ds0* ) constant defined in eq 40 Ds0# ) surface diffusivity at zero surface coverage, m2/s Dt, Dp ) overall effective pore diffusivity and combined pore diffusivity, respectively, m2/s Dii, Dij ) multicomponent surface diffusion coefficient, m2/s ∆E ) effective energy, J/mol ∆E1, ∆E2 ) activation energies for surface diffusion on the first and second layers, respectively, J/mol E ) activation energy, J/mol Em ) mean adsorption energy, J/mol Ea0, Ea1, Es1 ) differential heat of adsorption, heat of vaporization, and activation energy for migrating all layers above the first layer, respectively, J/mol Ez ) activation energy defined in eq 66, J/mol g ) geometrical factor h ) properties of porous media defined in eq 10 H(x) ) Heaviside function, 1 for x > 0 and 0 for x e 0 J ) molar flux, mol m-2 s-1 J0 ) constant defined in eq 66, mol m-2 s-1 Jv, Jk, Jt, Js, Jc ) molar fluxes defined in eqs 5, 7, 9, 11, and 14, respectively k ) Boltzmann constant, J/K k1, k2 ) parameters defined in eqs 47 and 53, respectively k3 ) preexponential constant for the jumping rate kb, km, ka, kd ) rate constants K ) constant defined in eq 26 K1 ) constant defined in eq 34 K(c) ) slope of the adsorption isotherm Kn ) Knudsen number KC ) Kozeny constant K0 ) preexponential constant defined in eq 69 Ks ) dimensionless adsorption equilibrium constant l ) length of capillary, m L ) thickness, m MW, M ) molecular weights, kg n ) diffusion step number in pore n1, n∞ ) amount adsorbed at a certain pressure and amount adsorbed at saturation, respectively, mol/kg P, P0 ) partial vapor pressure and saturation vapor pressure, respectively, Pa Pd ) dimensionless pressure, P/Pref Pm ) mean pressure, Pa Pref ) reference pressure, Pa Pr ) relative pressure Pt, Pc,eff ) capillary condensation pressure and effective capillary pressure, respectively, Pa P1, P2 ) upstream and downstream pressures, respectively, Pa q ) amount of gas adsorbed, mol/kg qm ) amount of monolayer-adsorbed gas, mol/kg Q ) energy of adsorption, J/mol Qs ) isosteric heat of adsorption, J/mol Q0 ) constant defined in eq 25 Qt ) total quantity of the diffusing species, mol/m2 r ) radius of cylindrical capillary, m R ) gas constant, J mol-1K-1 s ) degree of the surface heterogeneity s1 ) pore number per unit cross-sectional area ∆S ) entropy of activation, J mol-1K-1 St ) specific surface area, m2/g SC ) specific surface area of capillary-condensed phase, m2/g t0, t1, t2, t3 ) film thicknesses, m t ) time, s T ) temperature, K u* ) average velocity of capillary condensate, m/s

4028


Vm ) molar volume, m3/mol VC, VT ) volume of capillary-condensed phase and total pore volume, respectively, m3/kg Vt ) specific volume of adsorbed layer, m3/kg x ) relative pressure (P/P0) y ) nonintegral parameter defined in eq 54 z ) distance coordinate, m Greek Letters R ) ratio of rate constants defined in eq 56 R1 ) constant defined in eq 34 Rd ) constant β, β1, β2 ) constants defined in eq 33, 34, and 57, respectively βd ) constant λ1 ) mean free path, nm λ ) blockage parameter defined in eq 29 , T ) porosities * ) effective potential energy between a gas molecule and the solid surface, J/mol τ ) tortuosity factor τt ) time of oscillation of the molecules in the adsorbed state, s τ0, τ1 ) mean holding times for migration in the first layer and in all layers above the first layer, respectively, s µ ) gas viscosity, Pa s µL ) viscosity in liquid state, Pa s F ) density, kg/m3 Fapp ) apparent density, kg/m3 σ ) interfacial tension, J/m2 φ ) contact angle γ ) geometric constant of pore structure v ) vibration frequency v1, v2 ) vibration frequencies of vacant site and of occupied site, respectively δ ) average distance, m θ, θt ) surface coverages θe ) effective surface coverage θ1, θn ) occupation degrees at the pore entrance and the pore end, respectively ζ ) square root of variance for energy distribution ωij ) ratio of sticking probability of molecule i on adsorbed molecule j to sticking probability of molecule i on a vacant site

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Received for review February 28, 2001 Revised manuscript received June 19, 2001 Accepted June 22, 2001 IE010195Z

Surface Diffusion of Adsorbed Molecules in Porous Media: Monolayer

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