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Adsorption and Condensation in Pores Chaim Aharoni Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, Israel Received February 3, 1997. In Final Form: March 11, 1998 In sorption by porous materials, the condensation pressure is determined by the energetic properties of the pore walls surface as well as by the pore width. Equations analogous to the Kelvin equation are derived, in which the contact angle is replaced by a parameter related to the adsorption process that precedes condensation.
Introduction Sorption in a mesopore comprises two processes: “adsorption” at a peripheral region close to the pore walls in which the adsorption field induced by the solid surface is significant and “condensation” at the core of the pore, a region sufficiently distant from the pore walls in which the adsorption field is negligible. An important property of the condensation process is the fact that it takes place at a characteristic condensation pressure Pc, whereas adsorption takes place at any pressure greater than zero and increases continuously with the pressure. The resulting isotherm is characterized by a gradual increase of the amount sorbed as a function of the pressure until Pc is reached, and a sharp increase at Pc. Condensation is generally treated as a process independent of adsorption, and related to properties of the liquid-vapor interface, whereas adsorption is related to properties of the interface between the pore walls and the sorbate. The treatment of condensation is based on the fact that the liquid-vapor interface within the pore is concave, causing condensation at a pressure Pc smaller than the saturation pressure Ps. The ratio Pc/Ps is related by the Kelvin equation, to the surface energy at this interface, γlv, and to the radius of the core of the pore, rk
RTln
()
2νLγlvcosθ Pc )s rk P
(1)
where vL is the molar volume of the condensate, θ is the contact angle, and the subscripts l and v in γlv refer to liquid and vapor respectively, assuming that the condensate has liquidlike properties. However, the validity of the Kelvin equation does not contradict the fact that condensation in pores is also a sorption process that depends on the energetic properties of the solid-sorbate interface at the pore walls. The contact angle θ is determined by these properties and the presence of the parameter “cos θ ” in the Kelvin equation takes care of their effect on the condensation pressure. The contact angle can be determined experimentally with reasonable accuracy in systems involving nonporous surfaces, however the value of the contact angle relevant to a condensate occluded in a mesopore is difficult to determine, and the parameter “cos θ ” is often assumed to be equal to 1 in practical applications of the Kelvin equation.1 In the present paper condensation is treated as a sorption process determined by the energetic proper(1) Gregg S. J.; Sing K. S. W.,Adsorption Surface Area and Porosity, 2nd ed.; Academic: New York, 1982; pp 123-125.
ties of the solid-sorbate interface at the pore walls The parameter “cos θ ” of the Kelvin equation is replaced by a function of the surface coverage at the pore walls that can be estimated from experimental data, and the combined effect of adsorption coverage and pore width on condensation is assessed. Effect of the Energetic Properties of the Solid-Sorbate Interface on Adsorption and Condensation The driving force for the process of adsorption is the fact that it produces a decrease of the free energy at the solid surface. It is noted, however, that adsorption is associated with the formation of an adsorbate-vapor interface, and the existence of this interface contributes to the overall energy of the system and it attenuates the effect of adsorption on the decrease of the free energy of the surface. When condensation takes place, the adsorbate enters into contact with a liquid phase, the adsorbatevapor interface disappears, and an additional reduction of the overall surface energy occurs. The decrease of the free energy associated with the disappearance of the interface adsorbate-vapor can be considered as the driving force for condensation. A relation between the condensation pressure and the energetic properties of the pore walls surface, can be derived by applying a modified form of the Gibbs adsorption isotherm (see previous work2). We consider the change in free energy that takes place when nc moles of liquid condensate are occluded in a core space of radius rk and enter into contact with a pore wall surface of area A, covered with na moles of adsorbate. This energy change is given by
dG ) γ dA + µc dnc
(2)
where γ is the excess surface energy per unit area and µc is the excess free energy per mole of condensate. Integrating eq 2, holding the intensive terms γ and µc constant, applying total differentiation to the resulting expression, and subtracting eq 2 gives an equation analogous to the Gibbs adsorption isotherm
-dγ )
nc c dµ A
(3)
Equation 3 can be integrated using the limits of γ and µc corresponding to the initial and final states: γ changes (2) Aharoni, C. Langmuir 1997, 13, 1270.
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from γsav (surface covered by adsorbate in contact with a vapor phase) to γsal (surface covered by adsorbate in contact with a liquid phase). The corresponding values of µc are zero and µc . Noting that the quotient nc/A is constant in the case of condensation in a pore of constant radius, the equation obtained is
nc c µ A
γsav - γsal )
(4)
It is also noted that the excess free energy due to the presence of condensate, corresponds to the additional free energy required by the occluded condensate to leave the pore and become a free liquid. The change from the occluded liquid state to the free liquid state is associated with a change in the vapor pressure from Pc to Ps, and therefore µc is given by
µc ) RTln
Ps Pc
(5)
Combining eq 5 with eq 4, and noting that nc /A is related to rk, the hydraulic radius of the core of the pore, by
rk nc ) A 2νL
(6)
one obtains
RTln
()
2νL(γsav - γsal) Pc )s rk P
(7)
Relation between the Surface Energies at the Solid-Liquid and at the Liquid-Vapor Interfaces Equation 7 gives a relation between the condensation pressure and the radius of the core of the pore, it contains the parameter (γsav - γsal ) characteristic of the solidliquid interface. The Kelvin equation (eq 1) can be derived from this equation, noting that the ratio F between the parameter characteristic of the solid-liquid interface (γsav - γsal) and the parameter characteristic of the liquidvapor interface γlv, is equal to the cosine of the contact angle, cos θ, according to the Young-Dupre equation
γsav - γsal ) cos θ γlv
(8)
It can be shown that the ratio F can also be related to measurable properties of the solid -fluid system. Referring to the above-mentioned postulate, that the difference γsav - γsal results from the fact that adsorption is associated with the formation of an adsorbate - vapor interface, that disappears when the adsorbate is in contact with a condensate, and defining the excess surface energy per unit area due to the presence of this interface by γav, one obtains
γsav - γsal ) γav
(9)
for the case in which condensation takes place when the adsorbate coverage of the surface is complete. In the general case, condensation takes place before adsorption coverage is complete. At the condensation pressure Pc, the fraction σc of the pore walls surface is uncovered by adsorbate and the fraction (1 - σc) is covered (Figure 1). At the fraction of the surface covered by
Figure 1. Solid-sorbate interfaces in a pore. (a) Pore at Pc before condensation. (b) Pore at Pc after condensation. Key: (1) adsorbate; (2) vapor; (3) condensate; (4) interface: solidadsorbate-vapor (surface energy, γsav); (5) interface solidadsorbate-condensate (surface energy, γsal); (6) part of the surface covered by adsorbate (γsav - γsal ) γav); (7) part of the surface uncovered by adsorbate (γsav - γsal ) -γlv).
adsorbate, eq 9 is valid. At the fraction of the surface not covered by adsorbate, the condensate is not bonded to the solid surface, and its surface energy corresponds to that of a liquid-vapor interface and we have
γsav - γsal ) -γlv
(10)
For the overall surface, γsav - γsal is given by
γsav - γsal ) γav(1 - σc) - γlvσc
(11)
If we introduce the simplifying assumption that γav is approximately equal to γlv, eq 11 becomes
γsav - γsal ) 1 - 2σc γlv
(12)
Comparing with eq 8, one obtains
cos θ ) 1 - 2σc
(13)
Any of the parameters, cos θ or 1 - 2σc accounts for the ratio F. The parameter cos θ refers to a macroscopic state, caused by the surface forces, whereas 1 - 2σc refers to a molecular state that affects the surface forces.
Adsorption and Condensation in Pores
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Table 1. Relation between the Contact Angle and the Fraction of the Surface Uncovered by Adsorbate uncovered fraction σc
contact angle θ (degrees)
parameter F 1 - 2σc or cos θ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 36 53 66 78 90 102 114 127 143 180
1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0
A condensation equation based on the parameter σc is obtained by combining eq 12 with eq 7. This equation differs from the Kelvin equation in that the ratio F is expressed by a function of the adsorbate coverage instead of a function of the contact angle c
ln
2νLγlv(1 - 2σ ) Pc )s rkRT P
(14)
Table 1 gives values of θ corresponding to values of σc according to eq 13. At the limits in which the values of the parameters are predictable, the expected values are obtained. The state of maximum bonding between the condensate and the solid surface is given by σc ) 0 (maximum density of the adsorbed layer) and θ ) 0 (maximum contact). The state of no bonding is given by σc ) 1 (no adsorbate layer) and θ ) 180° (no contact). For the intermediate situation σc ) 0.5, the contact angle is 90°. Condensation Equations Containing Explicit Reference to Adsorption Parameters Equation 14 can also be written in terms of the parameters of the adsorption equation, that determines the coverage. An equation for monolayer adsorption, the Langmuir equation and an equation for multilayer adsorption, the BET equation, are taken as examples. The Langmuir equation can be written as
1 Θ P ) s s 1 - Θ P aP
[
]
(15)
where Θ is the fraction of the surface covered by adsorbate at the pressure P and a is a constant. Solving for Pc and Θc, the condensation pressure and the coverage at the condensation pressure, respectively, and combining with eq 14 noting that σc correspond to 1 - Θc, one obtains
2νLγlv ‚ ln Pr ) rkRT
1 aPs 1 Pr + aPs Pr -
(16)
where Pr ) Pc/Ps. The BET equation can be written as
nc cx ) nm (1 - x)(1 - x + cx)
(17)
where x is the ratio P/Ps, nm is the amount of adsorbate in a monolayer, and c is a constant. The derivation of the BET equation is based on the assumption that the part
Table 2. Numerical Values of the Parameter G Used in Equation 16 for Various Values of 1/aPs and Pr G ) (Pr - 1/aPs)/(Pr + 1/aPs) 1/aPs Pr
0.001
0.01
0.1
0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.980 0.990 0.993 0.995 0.996 0.997 0.997 0.998 0.998 0.998
0.818 0.905 0.935 0.951 0.961 0.967 0.972 0.975 0.978 0.980
0.000 0.333 0.500 0.600 0.667 0.714 0.750 0.778 0.800 0.818
0.000 0.091 0.167 0.231 0.286 0.333
Table 3. Numerical Values of the Parameter G Used in Equation 21 for Various Values of c and Pr G ) [Pr(c + 1) - 1]/[Pr(c - 1) + 1] c Pr
100
10
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.835 0.923 0.954 0.970 0.980 0.987 0.991 0.995 0.998 1.000
0.053 0.429 0.622 0.739 0.818 0.875 0.918 0.951 0.978 1.000
-0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000
of the surface si covered by i layers of adsorbate is related to the uncovered part of the surface s0 by
si ) cxis0
(18)
Equation 19 can be derived from eq 18
[
∞
∑0 si ) s0
]
1 - x + cx 1-x
(19)
It is noted that σc corresponds to s0/∑si at the condensation pressure Pc . Writing eq 19 for the conditions at the condensation pressure by replacing x by Pr and rearranging one obtains
σc )
1 - Pr 1 - Pr + cPr
(20)
and the condensation equation is given by
ln Pr ) -
2νLγlv Pr(c + 1) - 1 ‚ rkRT Pr(c - 1) + 1
(21)
Equations 16 and 21 are analogous to the Kelvin equation, the expression for F contains parameters of the adsorption isotherm and the condensation pressure. Numerical values of F used in eq 16 are calculated for various values of 1/Ps and Pr and given in Table 2. Numerical values of F used in eq 21 are given for various values of c and Pr in Table 3. Equations 16 and 21 are applicable to experimental data Studies aiming to assess the effects of adsorption on condensation by determining its effect on the contact angle on the basis of eq 1 are mentioned in the literature.1,3-5 (3) Derjagin, B. V., In Proceedings of the Second International Congress on Surface Activity; Butterworths: London, 1957; p 154.
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The radius of curvature of the liquid-vapor interface is shown to increase at the proximity of the pore walls in these studies, and the possibilities of defining the contact angle that should be introduced in eq 1 are discussed. The results do not lead to equations that can be used for the evaluation of the parameter cos θ from experimental data. Dependence of the Condensation Pressure on the Pore Size and on the Adsorption Coverage The combined effect of adsorbate coverage and pore size on the condensation pressure can be assessed by treating adsorption and condensation as competing processes. We consider a pore containing adsorbate in equilibrium with a vapor phase at a pressure P. In this case, a fraction of the pore walls surface, 1 - σ, is covered by one or more layers of adsorbate, and a fraction σ is uncovered. The system is subsequently submitted to a small increase of the pressure, which causes more sorption either by adsorption or condensation, and we postulate that the process selected is the one that produces a sorbate of lower fugacity. By plotting isothermal data as pressure against amount adsorbed, one obtains a representation of the fugacity of the adsorbate against the amount adsorbed. Such plots are always ascending. The curves A, in Figure 2 are examples of such plots. They were calculated for simplicity according to the Langmuir equation, eq 15, for various values of the parameter 1/aPs. The fugacity of a condensate in contact with a surface partially covered by adsorbate also depends on the fraction of the surface that is covered by adsorbate, and it decreases if this fraction increases (see eq 14). The dependence of the vapor pressure of the condensate on the adsorbate coverage was calculated according to eq 14, replacing the parameters Pc and σc by the variables P and σ, and noting that σ corresponds to 1 - Θ. The plots (curves C) are drawn for various values of the parameter 2vLγlv/rkRT. Thus, when the pressure is increased, the sorption process favored at low pressure and low adsorption coverage is further adsorption, and there is a value of the adsorption coverage corresponding to a pressure PC at which condensation is favored. A given sorption process is determined by a pair of curves A and C, and the point of intersection determines the condensation pressure and the corresponding adsorption coverage. We consider as an example the system characterized by the pair A1 and C2. Before the intersection point, the vapor pressure of the adsorbate is smaller than that of the condensate and at any pressure in that range sorption takes place as adsorption and not as condensation. At the intersection point (Θ ) Θc ) 0.975 and P/Ps ) Pc/Ps ) 0.39), the vapor pressure of the condensate equals that of the adsorbate. A small increase of the pressure at that point produces a small increase of adsorption at the proximity of the pore walls and complete filling of the core of the pore by condensation. The overall plot of the amount (4) Everett, D. H.; Haynes, J. M. Colloid Science; Chemical Society: London, 1973; Vol. 1, p 123. (5) White, L. R. J. Chem. Soc., Faraday Trans. 1 1977, 73, 390.
Figure 2. Plots of vapor pressures of adsorbate and condensate against adsorbate coverage. Plots of adsorbate: A1 (1/aPs ) 0.01), A2 (1/aPs ) 0.1), and A3 (1/aPs ) 0.2). Plots of condensate: C1 (2vLγlv/rkRT ) 0.05), C2 (2vLγlv/rkRT ) 1), and C3 (2vLγlv/rkRT ) 10).
sorbed versus pressure is an isotherm of type 4. It comprises a convex part corresponding to adsorption from 0 to Pc, and a sharp rise corresponding to condensation at Pc. If the C plot refers to a narrow pore (large value of the parameter 2vLγlv/rkRT), the parts of the isotherm corresponding to adsorption and to condensation may be indistinguishable and the type 1 shape is obtained. We consider the plot of the amount sorbed against the pressure for the pair C3 and A1. The segment representing adsorption at coverage smaller than Θc is almost vertical and would not be distinguishable from the segment representing condensation. Figure 2 also gives the possibility to assess the error obtained by assuming cos θ ) 1 in the Kelvin equation. Using this approximation is equivalent to estimating Pc/ Ps according to the point of intersection of the relevant curve c with the line Θ ) 1 instead of the appropriate curve A. The curve C2 cuts the line Θ ) 1 at Pc/Ps) 0.37, but it cuts the curves A1, A2, and A3 at the Pc/Ps values of 0.39, 0.51, and 0.60, respectively. The error is negligible in the case of strong adsorption and wide pore, and important in the case of weak adsorption and narrow pore. LA9701079
