Langmuir 1999, 15, 3321-3333
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A Model for the Adsorption Equilibria of Solutes with Multiple Adsorption Orientations Xuezhi Jin,† Zidu Ma,‡ Julian Talbot,§ and Nien-Hwa Linda Wang*,| Innovative Design Engineering Associates, Inc., 1795 Boston Post Road, Guilford, Connecticut 06437, Bioanalytical Systems, Inc., 2701 Kent Avenue, West Lafayette, Indiana 47906, Department of Chemistry and Biochemistry, Duquesne University, Pittsburgh Pennsylvania 15282-1503, and School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283 Received March 27, 1998. In Final Form: February 5, 1999 Both direct and indirect experimental evidence indicates that nonspherical adsorbates adopt multiple out-of-plane orientations on solid substrates. Few existing models account explicity for this effect on the adsorption equilibrium and kinetics, particularly when steric blocking effects are significant. In this work, a quantitative model is developed to account explicitly for the effects of steric blocking, solute shape, affinities of the different adsorption orientations, and solute concentration on the intrinsic equilibrium behavior. The model accurately correlates experimental isotherms in both gas/solid and liquid/solid systems. The fitted adsorption capacity and aspect ratio of the solutes are consistent with the solute size, shape, and specific surface area of the adsorbent. The proposed model describes quantitatively the tendency at low surface coverage for a solute to adsorb in an orientation with the largest surface contact area (energetically favored), while at a high coverage, most adsorbed molecules are orientated with a smaller contact area (sterically favored). The transition from side-on to end-on average orientation occurs over a large range of coverage if the aspect ratio is small and the affinity of the side-on orientation is much higher than that of the end-on orientation. The model also predicts that the apparent linear isotherm parameter estimated experimentally can increase with decreasing surface coverage, resulting in a strongly nonlinear Scatchard plot. This model could be valuable both in fundamental mechanistic interpretation of adsorption processes and for accurate experimental data correlation.
1. Introduction Adsorption is a key step in chromatography, heterogeneous catalysis, crystallization, electrochemical reactions, biological processes, and colloidal phenomena in general.1-5 Adsorption-based separation technologies are now widely used as basic analytical tools and large-scale separation/purification methods.6-11 For these applications, understanding the fundamental mechanism and predicting the kinetic and isotherm behavior of the adsorption process are two key issues in model-based design, optimization, scale-up, and process control. Numerous adsorption models and equations have been * To whom correspondence should be addressed. † Innovative Design Engineering Associates, Inc. ‡ Bioanalytical Systems, Inc. § Duquesne University. | Purdue University. (1) Masel, R. I. Principles of Adsorption and Reaction on Solid Surfaces; Wiley: New York, 1996. (2) Oscik, J. Adsorption; PWN-Polish Scientific Publishers: Warsaw, Poland, 1982. (3) Gregg, S. J.; Sing., K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (4) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. (5) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, 1987. (6) Wankat, P. C. Rate-Controlled Separations; Elsevier Applied Science: New York, 1990. (7) Wankat, P. C. Large Scale Adsorption and Chromatography; CRC Press: Boca Raton, FL, 1986; Vols. I and II. (8) Guiochon, G.; Shirazi, S.G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, MA, 1994. (9) Dechow, F. J. Separation and Purification Techniques in Biotechnology; Noyes: Park Ridge, NJ, 1989. (10) Noll, K. E.; Gounaris, V.; Hou, W.-S. Adsorption Technology for Air and Water Pollution Control; Lewis: Chelsea, MI, 1992. (11) Scope, R. K. Protein Purification; Springer-Verlag: New York, 1982.
proposed12-20 and most of them either do not consider the molecular shape at all or are based on the assumption that solutes of the same kind cover the same amount of surface area. However, most chemical species are nonspherical and can adsorb on a surface with different orientations with respect to the surface normal and, hence, different contact areas. This, in turn, leads to differences in the adsorption and desorption rate constants, ka and kd (and hence the equilibrium constant, K), steric blocking effects, and the adsorption capacity for each adsorption orientation. Additionally, competition among the orientations changes with surface coverage and results in a change of average orientation with increasing solution-phase concentration. Because of these features, the overall adsorption kinetics and equilibrium can differ both quantitatively and qualitatively from systems with a unique adsorption orientation. Therefore, a satisfactory model should account for both the effects of multiple adsorption orientations and steric blocking. Evidence for multiple adsorption orientations comes from both direct and indirect experimental obsevations of many different systems (Table 1).21-31 In these systems, (12) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (13) Freundlich, H. Colloid and Capillary Chemistry; Mathuen: London, 1926; pp 110-134. (14) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (15) Fowler, R. H.; Guggenhein, E. A. Statistical Thermodynamics. Statistical Mechanics for Students of Physics and Chemistry; Macmillan: New York, 1939. (16) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (17) Kopaciewicz, W.; Rounds, M. A.; Fausnaugh, J.; Regnier, F. E. J. Chromatogr. 1983, 266, 3. (18) Jin, X.; Tarjus, G.; Talbot, J. J. Phys. A: Math. Gen. 1994, 27, 1195. (19) Jin, X.; Talbot, J.; Wang, N.-H. L. AIChE J. 1994, 40, 1685. (20) Talbot, J.; Jin, X.; Wang, N.-H. L. Langmuir 1994, 10, 1663.
10.1021/la980350n CCC: $18.00 © 1999 American Chemical Society Published on Web 04/09/1999
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Table 1. Key Literature References on Adsorbate Orientation (A) Small Solutes author
adsorbate
sorbent
technique
eqm
Tabony et al. (1980)21 Meehan et al. (1980)22 Boddenberg and Grosse (1986)23 Vernov and Steele (1991)24
benzene benzene benzene
graphite graphite graphite and boron nitride graphite
NMR neutron diffraction 2H NMR
X X X
side-on at low coverage; end-on at high coverage side-on at low coverage side-on at low coverage
computer simulations
X
energetic justification of side-on to end-on transition; 10% end-on
benzene
observations
(B) Large Solutes author
adsorbate
sorbent
Morrissey and Han (1978)25 γ-globulin
PS lattices
Norde and Lyklema (1978)26 HPA, RNase
PS lattices
Soderquist and Walton (1980)27 Fair and Jamieson (1980)28 (1983)29
Van Dulm et al. Mazsaroff et al. (1988)30 Waldmann-Meyer and Knippel (1992)31
HSA, γ-globulin, fibrinogen BSA, HSA, γ-globulin HPA IgG albumin, IgG
PP silicone PS lattices
technique
eqm kinetics
quasielastic light scattering batch eqm.
X
circular dichroism batch eqm. photon correlation
X
PS-charged radio label ion exchanger chromatography glass PS electrophoresis
synergistic or repulsive latteral interactions of adsorbed molecules in addition to the aforementioned energetic and steric effects can also play a role in the observed multiple orientations. For large solutes, conformational changes upon adsorption further complicates the process. These additional factors are not considered here. The goal of this study is to understand how the binding energies and steric blocking effects of multiple orientations affect adsorption equilibrium. To describe a system of adsorbed solutes with multiple orientations, we propose to treat the adsorbed solutes of each individual adsorption orientation as different species which possess different energetic and steric parameters. In this work, our focus is on the development of a quantitative model to describe the adsorption equilibria for systems with two adsorption orientations. The model accounts explicitly for the effects of (a) steric blocking, (b) solute shape, (c) affinity of each adsorption orientation, and (d) solute concentration in the solution phase. The model is tested by comparison to experimental isotherm data ranging from gas/solid systems to liquid/solid systems which are difficult to explain or fit with available adsorption models. In all the cases, the model accurately correlates the experimental data and the resulting parameters are physically consistent. Using this model, we are able to explain some experimental results that have not been well-understood previously such as, at a low surface coverage, a solute tends to adsorb in the orientation with the largest surface contact area (ener(21) Tabony, J.; White, J. W. Surf. Sci. 1980, 95, L282. (22) Meehan, P.; Rayment, T.; Thomas, R. K.; Bomchil, G.; White, J. W. J. Chem. Soc., Faraday Trans. 1 1980, 76, 2011. (23) Boddenberg, B.; Grosse, R. Z. Naturforsch. 1986, 41A, 1361. (24) Vernov, A.; Steele, W. A. Langmuir 1991, 7, 2817. (25) Morrissey, B. W.; Han, C. C. J. Collid Interface Sci. 1978, 65 (3), 423. (26) Norde, W.; Lyklema, J. J. Colloid Interface Sci. 1978, 66 (2), 257. (27) Soderquist, M. E.; Walton, A. G. J. Colloid Interface Sci. 1980, 75 (2), 386. (28) Fair, B. D.; Jamieson, A. M. J. Colloid Interface Sci. 1980, 77 (2), 525. (29) Van Dulm, P.; Norde, W. J. Colloid Interface Sci. 1983, 91 (1), 248. (30) Mazsaroff, I.; Cook, S.; Regnier, F. E. J. Chromatogr. 1988, 443, 119. (31) Waldmann-Meyer, H.; Knippel, E. J. Colloid Interface Sci. 1992, 148 (2), 508.
X X
X X X X
observations side-on to end-on from low to high surface coverage steps or kinks in q(c); conformational change kink in q(c); overshoot in q(t); side-on: HSA, fibrinogen; end-on: γ-globulin two steps in q(c) overshoot in q(t) side-on to end-on IgG: end-on; albumin: side-on in PS, end-on in glass; kink in q(c)
getically favored), while at a high coverage, most adsorbed molecules have an orientation which requires less surface area (sterically favored). 2. Theory Consider a system containing solutes with n possible adsorption orientations. If the adsorption and desorption rate constants corresponding to orientation i are k′a,i and kd,i, then the net adsorption rate associated with this orientation is
dFi ) k′a,icφi - kd,iFi dt
(1)
where Fi is the surface density of adsorbed solutes in the ith orientation and c is the concentration of solutes in the solution immediately above the adsorption surface. The function φi describes the steric blocking effect associated with the ith orientation. Equation 1 is similar to the m θi) term is Langmuir model,12 except that the (1 - ∑i-1 replaced by the function φi. Here, θi is the coverage of component i. If we assume that there are no aggregation or dissociation reactions among the adsorbed solutes, k′a,i and kd,i can be considered as constants (independent of F) at a fixed temperature and solvent strength. Therefore, the adsorption kinetics, as well as the equilibrium isotherm (which is a special case of the former at dFi/dt ) 0), can be described once the set of {φi} is known. The available surface function, φi, represents the average probability of successfully inserting a hard core molecule in an existing adsorption configuration.32 It is defined as the fraction of the surface on which placement of the centers of additional particles will not result in an overlap with existing solutes (e.g., the white space in Figure 1, parts b and 1 c.19,32,33 This concept has been extended to describe adsorption on a random site surface (RSS),34 adsorption processes with desorption,19 and a multicomponent system with different sizes of solutes.20 Besides the total surface density (or coverage θ), φi also depends on the distribution of solutes in the adsorption (32) Widom, B. J. Chem. Phys. 1996, 44, 3888. (33) Schaaf, P.; Talbot, J. J. Chem. Phys. 1989, 91, 4401. (34) Jin, X.; Wang, N.-H. L.; Tarjus, G.; Talbot, J. J. Phys. Chem. 1993, 97, 4256.
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configuration which is directly related to the manner in which the solutes are introduced onto the surface (irreversibly or reversibly), to the size ratios and shapes of different adsorption orientations (which are determined by the physical dimensions of the adsorbing species), and to the fractions of solutes of different orientations, which are affected both explicitly by the {k′a,i} and {kd,i} and implicitly by the {φi} itself. Exact descriptions of φ for adsorption kinectics and equilibrium states exist only for one-dimensional systems. However, an approximate available surface function at equilibrium, φi, for a multicomponent system in two dimensions, can be obtained from the scaled particle theory (SPT):20,35-37
∑FjAj) Si Ai∑Fj + ∑FjSj Ai(∑FjSj)2 2π 1 - ∑FjAj 4π(1 - ∑FjAi)2
The geometrical parameters for these three models are listed in Table 2. In deriving the φi for disk-ellipse model, an approximate equation, S ≈ π (a + b), has been used for the circumference of the ellipse, instead of using the exact equation of e ) (a - b)/(a + b), where a and b are the lengths of major and minor axes of the ellipsoid and e ) (a - b)/(a + b) S2 ) π(a + b)(1 + (e2/4) + (e4/64) + ...).
(1) Disk (1)-disk (2) projections: φ1 ) (1 - θ1 - θ2) ×
[
exp -
[
]]
(3)
[
]]
(4)
3θ1 + β-1(β-1 + 2)θ2 θ1 + β-1θ2 1 - θ 1 - θ2 1 - θ 1 - θ2
2
and
ln φeq i ) ln(1 -
φ2 ) (1 - θ1 - θ2) ×
[
(2)
where Si and Ai denote the perimeter and cross-sectional area of the ith component (the ith orientation in this application), respectively. Comparison with numerically exact computer simulations has shown that the {φeq i } for a two-component equilibrium mixture of hard disks obtained from SPT are accurate up to θ ) 0.5, and that for a large coverage range, the kinetics calculated by using {φeq i } for {φi} in eq 1 are very close to the results of a computer simulation of an adsorption-desorption process.20 Here, we assume that the same conclusions apply to nonspherical particles such as ellipses and rods. Therefore, given the shapes of the adsorbates, we can derive the {φeq i } from eq 2 and then, as a good approximation, replace {φi} in eq 1 to calculate the kinetics and equilibrium isotherm for an adsorbate with multiple orientations. We focus primarily on the equilibrium properties of the model in this paper. The experimental and simulation studies referred to previously indicate a continuous orientational distribution of the adsorbed molecules with respect to the surface normal. The application of SPT is greatly simplified, however, by considering just two representative orientations: vertical with a disklike projected contact area and horizontal with an elliptical (for an ellipsoid) or a rectangular (for a rod) contact area. We will refer to these as the “end-on” (first) and “side-on” (second) orientations, respectively. Although the SPT can, in principle, be extended to allow for a continuous out-of-plane orientational distribution, additional approximations would be required. Note that the SPT does not place any restriction on the in-plane orientation which is implicitly continuous in the theory. 2.1. Available Surface Functions for Three Model Systems. In this work, we consider three models for systems with two orientations: (1) the projections of both orientations are disks of different sizes; (2) the end-on is a disk while the side-on is an ellipse; (3) the end-on is a disk and the side-on is a rectangle. While the latter two correspond to ellipse or rodlike solutes, the first is an imaginary, although simple case. (35) Helfand, E.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1961, 34 (3), 1037. (36) Boublik, T. Mol. Phys. 1975, 29, 421. (37) Talbot, J. Ph.D. Thesis, Southampton University, U.K., 1985; J. Chem. Phys. 1997, 106, 4696.
exp -
3θ2 + β(β + 2)θ1 θ2 + βθ1 1 - θ 1 - θ2 1 - θ 1 - θ2
2
(2) Disk (1)-ellipse (2) projections: φ2 ) (1 - θ1 - θ2) ×
[
-1
exp -
3θ1 + (2β + 1)θ2 1 - θ 1 - θ2
[
(
)
1 + β-1 θ2 2 1 - θ1 - θ2
θ1 +
]] 2
(5)
and φ2 ) (1 - θ1 - θ2) ×
[
exp -
(
)
β β-1 + 2 + θ2 2 2 1 - θ1 - θ2
θ1(1 + 2β) +
[
θ1xβ + (xβ + 1/xβ)θ2 1 - θ1 - θ 2
]] 2
(6)
(3) Disk (1)-rectangle (2) projections: φ1 ) (1 - θ1 - θ2) ×
[
exp -
(
3θ1 + β-1 1 +
π θ 4 2
)
1 - θ1 - θ2
-
[
(
)
1 + β-1 θ2 2 1 - θ1 - θ2
θ1 +
]] 2
(7)
and φ2 ) (1 - θ1 - θ2) ×
[
4 2 (1 + 2β)θ1 + 1 + β + 2 + β-1 θ2 π π exp 1 - θ 1 - θ2
[
(
)
(
)
2 1 1 θ1xβ + xβ + θ2 2 xπ xβ 1 - θ 1 - θ2
]] 2
(8)
where β is the ratio of the longest axis of the body to the shortest (see Table 2). To illustrate the differences in φ for symmetric and asymmetric solutes, the equilibrium configuration for a disk-disk system (β ) 2) is shown in Figure 1a. The fraction of the surface that is available for the center of the small disk (φ1) is represented by the white area in Figure 1b and that for the large disk (φ2) is represented by the white area in Figure 1c. A comparison
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Table 2. Geometrical Properties and Parameters for the Three Modelsa
a Note: S is the circumrference; A is the cross-sectional area of orientation i. β is the aspect ratio. b The exact value is S ) π(a + b)(e2/4 i i 2 + e4/64 + ‚‚‚), where e ) (a - b)/(a + b).
2.2. Kinetic and Isotherm Expressions for the TwoOrientation Models. By substituting φi into eq 1, we have the following kinetic expressions:
dF dF1 dF2 ) + dt dt dt
(9)
dF1 ) k′a,1cφ1(θ1,θ2) - kd,1F1 dt
(10)
dF2 ) k′a,2cφ2(θ1,θ2) - kd,2F2 dt
(11)
where
and
Unfortunately, Fi cannot normally be obtained directly from experiments; usually, the measurable quantity is the amount adsorbed per unit mass or volume of the sorbent, q. Without a knowledge of the specific surface area of the sorbent and the molecular dimension of the adsorbate, conversion from q to F or from q to θ cannot be made. Therefore, further assumptions are needed in order to apply eqs 10 and 11. First, we reason that there must exist a constant conversion factor, A h , so that
h Fi qi ) A
(12)
h Fm,i, where qm,i is the maximum As a special case, qm,i ) A adsorption capacity of species i. The corresponding coverage, θi,∞ ) aiFm,i, is the maximum possible coverage for species i and is commonly called the jamming limit surface coverage. We next introduce the normalized surface coverage θ/i :
θ/i )
Figure 1. Illustration of the concept of available surface functions for a system with two adsorption orientations: (a) An equilibrium configuration with K1 ) 0.25, K2 ) 2, β ) 2.0, θ1 ) 0.0848, and θ2 ) 0.1913. (b) Same as (a), except that the white areas are available for the adsorption of the end-on (first) orientation. (c) In this configuration, the white areas represent the available surface areas for the side-on (second) adsorption orientation.
of Figure 1, parts b and c, clearly shows that φ2 is much smaller than φ1.
θi Fi qi ) ) θi,∞ θm,i qm,i
(13)
Clearly, eq 13 indicates that the adsorption fraction relative to the maximum is independent of A h . If θi,∞ is known, then θi can be calculated as θi ) (qi/qm,i)θi,∞. We note that the value of θi,∞ depends on factors such as the reversibility of the process, the adsorption site density, and the molecular shape.34 The values of θ∞ for irreversible adsorption on a continuous surface (CS)39 and on the random site surface (RSS)34 are known. In the former case, (38) Jin, X. Computer Simulation and Theoretical Analysis of Adsorption Kinetics and Equilibria with Applications. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1994. (39) Evans, J. W. Phys. Rev. Lett. 1989, 62, 2641.
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θ∞ ) 0.547 for spherical particles. For a system with desorption, computer simulation reveals the value of θ∞ for disks on the CS to be the maximum nonoverlap packing fraction (0.9069).38 For other shapes, no information is available. However, if we assume that θ∞ equals the maximum nonoverlapping packing fractions, then we have θ∞ ) 0.9069 for ellipses and θ∞ ) 1 for rectangles. At this point, we need to address the accuracy of the SPT equations. We know that at θ ) θ∞, φ should be zero. According to the SPT equation, however, only at ∑ni θi ) 1 is φi ) 0 which is physically impossible for disks or ellipses. Therefore, the SPT itself is an approximate theory which assumes θ∞ ) 1 for all shapes. For this reason and because the actual θ∞ (for reversible adsorption) may be in the range of 0.9069-1, we also simply take the value of θi,∞ for each orientation to be unity. In other words, since θ/i ) θi/(θi,∞ ) 1) ) θi, we may regard the θi in the expansion of φi as θ/i (in the remaining text, we use θi to represent θ/i ). In this way, we finally can express the isotherms as
qm,1K1cφ1(q1/qm,1,q2/qm,2) - q1 ) 0
(14)
qm,2K2cφ2(q1/qm,1,q2/qm,2) - q2 ) 0
(15)
and
where
K1 )
ka,1 kd,1
and
K2 )
ka,2 kd,2
(16)
are the equilibrium constants for the two orientations, respectively. In the above expressions, we do not consider β as an independent variable, since once qm,1 and qm,2 are given, β is fixed. For the disk-disk, disk-ellipse, and disk-rod models, the corresponding β are (qm,1/qm,2)1/2, (qm,1/qm,2), and (πqm,1/4qm,2), respectively. Thus, the kinetics of the two orientation models are characterized by six parameters (qm,1, qm,2, ka,1, ka,2, kd,1, and kd,2), while four are required for the isotherms (qm,1, qm,2, K1, K2). 3. Analysis of the Isotherm Behavior To simplify the analysis, we define the following parameters:
C* ) qm,1K1c
(17)
and
Kr )
K2 K1
(19)
and
qeq 2 -
qm,2 K C*φ2 ) 0 qm,1 r
Since there is no explicit expression for qeq(C*), a numerical iteration has to be used to calculate the qeq at a given C*. We used the IMSL subroutine, DNEQNF,40 to eq search for qeq 1 and q2 . We note that, for a spherical solute adsorbing on a surface, the equilibrium constants, Ki, for the orientations may be different but their φi and qm,i are the same. For such a system, the isotherm can be expressed as n
(18)
where C* may be regarded as a scaled concentration which reflects the combined effect of bulk-phase concentration, adsorption capacity, and affinity of the first orientation, and Kr is the affinity ratio of the side-on orientation to the end-on orientation. Then, the isotherm equations become
qeq 1 - C*φ1 ) 0
Figure 2. Isotherms calculated at qm,1 )1, Kr ) 10, and β ) 2 for the disk-disk (solid), disk-rod (diamond), and diskellipse (dashed) models. (a) The total equilibrium adsorption amount, qeq, is shown versus the normalized concentration C*; (b) equilibrium adsorption amount for the first orientation, qeq 1 , versus C*; (c) equilibrium adsorption amount for the second orientation, qeq 2 , versus C*.
(20)
qeq ) (
∑i Kr,i)φ1C* ) KφC*
(21)
where Kr,i is defined as Ki/K1. Equation 21 indicates that, for a spherical solute, the presence of heterogeneity does not change the form of its isotherm. 3.1. Comparison of the Three Models. As discussed before, the shape of the solute can guide us in the choice of φi for the system. However, it is helpful to know to what degree the three models are different. For this, we calculated the isotherms at qm,1 ) 1, β ) 2, and Kr ) 10. The results shown in Figure 2 indicate that with the same β the three models are quite different. At low C*, the order (40) IMSL, MATH/LIBRARYs FORTRAN Subroutines for Mathematical Applications, version 1.0; IMSL, Inc.: Houston, TX, 1987.
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Figure 3. Same as Figure 2 except that the parameters for the isotherms are qm,1 ) 1, Kr ) 10, and (qm,1/qm,2) ) 2.
of qeq is disk-ellipse > disk-rod > disk-disk, whereas the order reverses at high C* (Figure 2a). In almost the entire range of C*, the order of qeq 1 is disk-disk > disk-rod * > disk-ellipse while the opposite is true for qeq 2 versus C (Figure 2, parts b and c). In other words, at the same β, eq the disk-disk model has the highest ratio of qeq 1 /q2 , while the disk-ellipse has the lowest ratio. The area ratios (A2/ A1) ) (qm,1/qm,2) for the three models also have the same order; that is, disk-disk > disk-rod > disk-ellipse (see Table 2). We suspected that the differences among the three models might be reduced if the same value of qm,1/qm,2 were used. Therefore, we calculated the isotherms at the qm,1/qm,2 ) 2 and Kr ) 10. As shown in Figure 3 in this * case, the order for qeq versus C* and qeq 1 versus C is diskrod > disk-ellipse >disk-disk while the opposite is true * for qeq 2 versus C . However, as expected, the differences among the three isotherms, especially in the qeq versus C* plot, are much smaller than when β is fixed. To further verify this observation, we examined a number of isotherms with a wide range of parameters. In all the cases, the difference among the three models with the same qm,1/ qm,2 is smaller than for those with the same β. Differences between the isotherms remain small even with qm,1/qm,2 ) 4 and Kr ) 40. Therefore, qm,1/qm,2 is a better indicator than β in characterizing the relative steric blocking effect of the two adsorption orientations. For practical purposes, any one of the models could be used to correlate experimental data with a similar accuracy. The results presented below are from the disk-ellipse model unless noted otherwise.
Figure 4. Isotherms of the ellipsoid model for systems with qm,1 )1 and β ) 2. The values of Kr for the four isotherm lines are 1, 5, 20, and 100. (a) Total adsorbed amount, qeq, versus C*. The value of Kr increases from top to bottom at the high C*. (b) qeq 1 (the end-on orientation) versus C*. The value of Kr increases from top to bottom. (c) qeq 2 (the side-on orientation) versus C*. The Kr order is opposite that in (b). (d) The same data as that in (a) but in the form a of Scatchard plot. The value of Kr decreases for the lines from top to bottom (at low qeq).
3.2. Effect of Affinity Ratio Kr. To illustrate the effect of Kr, the isotherms for systems with qm,1 ) 1, β ) 2, and Kr ) 1, 5, 20, and 100 were evaluated. From Figure 4a-e we observe that qeq increases with increasing Kr at small C*, while for large C* the trend is reversed. qeq 1 , as well as the adsorbed mole fraction of the end-on orientation, increases monotonically with C* at any Kr, whereas qeq 2 has a local maximum. All of these properties have, as their root cause, a competition between energetic and steric effects. At low coverages, steric effects are relatively unimportant and the isotherm is controlled by the energetic factor, Kr. Thus,
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Langmuir, Vol. 15, No. 9, 1999 3327
multiple adsorption orientations, one may observe a strong concentration dependence of the apparent equilibrium constant K, that is, the lower the concentration, the higher the apparent K value. 3.3. Effect of Solute Aspect Ratio β. To demonstrate the effect of the solute aspect ratio on the isotherms, we calculated the isotherms while fixing qm,1 ) 1 and Kr ) 10 but at various values of β (1, 1.25, 2.0, 3.0, and 4.0). The reason we used Kr > 1 is that in most adsorption modes (such as hydrophobic and ionic interaction) the “side-on” orientation has a larger contact area and its equilibrium constant should be larger than that of a solute in an “end-on” orientation. The results are plotted in Figure 5a-e. Again, the effect of β is different at low and high surface coverages. As mentioned earlier, at a low coverage (or C) qeq is mainly determined by the value of Kr. For moderate values of Kr, eq both qeq 1 and q2 are significant. Therefore, we may define an apparent adsorption capacity as qap m ≈ 0.5(qm,1 + qm,2) ) 0.5qm,1(1 + 1/β). Obviously, qap m decreases with increasing β, as does qeq (see Figure 5a). In the high-coverage (or C*) range, the side-on orientation is sterically disfavored and an increase of β will fureq decreases, ther hinder its adsorption. Therefore, qeq 2 /q eq increases with an increase in β (see /q whereas qeq 1 Figure 5b,c. With a large β and moderate Kr, and at high eq can be virtually negligible and, as a coverage, qeq 2 /q eq result, q is mainly determined by qeq 1 . In other words, the system behaves as if it had a single (end-on) orientation. The effect of β on the deviation of the isotherm from the Langmuir model is reflected in the Scatchard plot by an increased curvature as β increases (see Figure 5d). 4. Experimental Isotherm Data Correlation and Discussion
Figure 5. Effect of β on the isotherm. qm,1 and Kr for the isotherms are fixed at 1 and 10. The corresponding values of β for the five isotherm lines are 1, 1.25, 2.0, 3.0, and 4.0. From top to bottom, the value of β decreases in (b) and increases in (a) (at low C*), (c), and (d).
the side-on orientation, which is energetically favored when Kr > 1, dominates at low coverages. At sufficiently high surface coverages, however, the side-on orientation is sterically disfavored and the molecules in the end-on orientation prevail. This competition leads directly to the maximum in qeq 2 . When K g 10, the maximum is broad and the transition from predominantly side-on orientation to predominantly end-on orientation occurs only when C* is very large (Figure 4b). To expose most clearly the differences between the SPT and the Langmuir isotherms, we show the two in the form of a Scatchard plot. In this representation, the Langmuir isotherm is linear whereas there is strong curvature in the SPT plots, which increases with increasing Kr. This result indicates that, in estimation of an adsorption equilibrium isotherm of an asymmetric solute with
To test the validity of the model, we have applied eqs 21 and 22 to correlate the isotherms of six different systems. Both the two-disks and rod forms of the twoorientation model are used in the correlation and indicated as T-DISKS and T-ROD under the column of models of Table 3. The isotherm data for the adsorption of acetone, propionitrile, p-cresol, and p-chlorophenol on an activated carbon, which has a specific surface area of 1000 m2/g, are taken from Radke and Prausnitz.41,42 The data of benzene (gas) adsorption on a silica gel and pentane (gas) adsorption on an activated carbon at T ) 273.1, 298.15, and 323.15 K, come from Myers and Sircar43 and Kuro-Oka et al.,44 respectively. For the purpose of comparison, the data were also fitted by three other well-known models: the single-component Langmuir equation (LANG),12 the double-component Langmuir equation (DLANG), and a single-orientation model incorporating the steric blocking effect (SPT-1).20 The isotherms of these models are as follows:
(a) LANG qeq )
qmKc 1 + Kc
(22)
(41) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (42) Radke, C. J.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1972, 11, 445. (43) Sircar, S.; Myers, A. L. AIChE J. 1973, 19, 159. (44) Kuro-Oka, M.; Suzuki, T.; Nitta, T.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 588.
3328 Langmuir, Vol. 15, No. 9, 1999
Jin et al.
Table 3. Molecular Information and Parameters Obtained from Isotherm Correlations (A) Liquid/Solid Systems formula MW dimension (Å × Å × Å)
adsorbate
fitting parameters T (K)
models
qm,1 (mmol g-1)
T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG.
4.070 4.260 1.581 4.573 1.729 6.882 7.252 2.727 7.763 2.704 7.322 7.641 2.194 5.064 2.506 7.019 9.198 2.097 5.272 2.844
acetone
CH3COCH3 64 3.71 × 3.11 × 2.09 βmol ≈ 1.78
298.15
propionitrile
CH3CH2CN 55 4.85 × 3.02 × 2.91 βmol ≈ 1.67
298.15
p-cresol
CH3C6H4OH 108 6.37 × 4.79 × 1.78 βmol ≈ 3.57
298.15
p-chlorophenol
ClC6H4OH 129 5.71 × 4.79 × 1.27 βmol ≈ 4.50
298.15
β
K1
3.460 4.876 0.152a
35.93 32.42 39.57 24.14 50.75 17.95 16.22 25.22 13.19 33.10 1681 1334 188.1 10871 11284 2899 2511 186 14430 10496
3.854 7.713 0.133a
1.560 1.765 1.434a
1.524 1.694 1.487a
K2 571.1 436.4 2386
428.1 327.4 2075
46152 43912 62047
91072 89471 106171
dev % 3.46 2.88 3.50 90.77 126.2 3.35 3.98 3.31 96.57 119.0 1.02 0.88 4.89 18.50 42.93 0.85 0.75 3.49 16.16 44.91
effective diameter 6.86 Å
5.28 Å
5.12 Å
5.23 Å
(B) Gas/Solid Systems formula MW dimension (Å × Å × Å)
adsorbate benzene
fitting parameters
C6H6 84 4.79 × 4.79 × 1.27 βmol ≈ 3.77
pentane
models
303.15
T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG. T-DISKS D-ROD DLANG. SPT-1 LANG.
9.027 9.010 3.497 9.178 4.057 7.567 7.510 2.339 7.500 4.224 7.553 7.570 2.961 7.455 4.142 7.268 7.312 3.090 7.353 3.876
273.15
pentane
CH3(CH2)6CH3 72 5.72 × 4.46 × 2.73 βmol ≈ 2.09
298.15
pentane
a
T (K)
qm,1 (mmol g-1)
323.15
β
K1
2.32 3.37 0.819a
1.289 1.309 0.678 1.052 1.417 635.9 726.9 15.20 503.7 262.9 79.08 78.03 9.533 73.48 40.00 20.36 19.71 4.864 15.56 11.62
1.582 1.923 2.30a
1.704 2.038 1.348a
1.975 2.641 1.154a
K2
dev %
18.47 20.36 33.17
2.58 2.69 1.98 10.48 20.59 0.85 0.88 3.87 5.73 27.51 0.73 0.59 4.66 20.25 95.24 2.33 1.90 6.52 60.05 183.68
6225 11302 2453
692.9 655.9 1222
410.0 375.4 811.6
This is the value of qm,2 for the double Langmuir model.
(b) DLANG qeq )
qm,2K2c qm,1K1c + 1 + K1c 1 + K2c
(23)
qeq ) qmKcφ
(24)
(c) SPT-1
where
(
φ(θ) ) (1 - θ) exp -
)
3θ - 2θ2 (1 - θ)2
(25)
Equation 25 may be recovered from eqs 3 or 4 by setting β ) 1. DLANG is based on the assumption that the sorbent surface has two types of adsorption sites and the separation
between them is at least one solute diameter. Therefore, a solute can access the two kinds of sites independently with capacities (qm,1 or qm,2) and equilibrium constants (K1 or K2). The adsorption behavior for each type of site can be described by the Langmuir equation. Many other adsorption models can fit the data on activated carbon better than LANG and DLANG. However, they are more complex and less widely used than LANG and DLANG. For this reason, the comparison here is focused on the two well-known models and the singleorientation model (SPT-1), which is a limiting case of the present model. In the fitting procedure all the isotherm parameters were allowed to vary freely until the best set of parameters satisfying the least-squares criterion was found. The results are summarized in Table 3a,b and plotted in Figures 6-10. For clarity, only fits of the disk-rod (solid
Adsorption Equilibria of Solutes
Figure 6. Fits of various models to experimental isotherm data (points) of acetone on activated carbon. The data are from Radke and Prausnitz. The solid lines represent the twoorientation model (rod form); dashed and broken lines show fits of SPT-1 and Langmuir, respectively. (a) is the normal isotherm plot (amount adsorbed, q, versus solute concentration, c or P, and (b) is the corresponding Scatchard plot (q/c versus q).
Langmuir, Vol. 15, No. 9, 1999 3329
Figure 8. Fits of vaious models to experimental isotherm data of p-chlorophenol on activated carbon. The same data source and convention as those in Figure 6.
Figure 9. Fits of various models to isotherm data of benzene (gas) adsorption on silica gel. The data from Sircar and Myers. The convention is the same as that in Figure 6. Figure 7. Fits of various models to isotherm data of p-cresol on activated carbon. The same data source and convention as those in Figure 6.
line), the SPT-1 (dashed line), and the LANG (broken line) are plotted in the figures. In the tables, the second column lists the molecular dimensions of the solutes which are estimates from the Insight II simulation package.45 The approximate aspect ratios, reported in the same column, are estimated by dividing the longest dimension by the shortest dimension of the molecules. The effective molecular diameter, σmol, is estimated as the cubic root of the products of the molecular dimensions. The last column in Table 3A lists the effective solute diameters, σeff, which are calculated from the specific surface area of the (45) Biosym, Insight IIsReference Guide, version 2.2.0; Biosym Technologies: San Diego, CA, 1993.
adsorbent (A h ), the adsorption capacity for the end-on orientation (qm,1), and the assumption of θ∞ ) 0.9069:
σeff )
x
4θ∞A h πqm,1NA
(26)
where NA is Avogadro’s number. From the tables and Figures 6-10, it is apparent that both SPT-1 and, particularly, the Langmuir equation correlate with the data very poorly. The two-orientation model, however, provides an excellent fit which is within the experimental error for most systems. The poor fit of the Langmuir equation is also apparent in the corresponding Scatchard plots: the Langmuir equation results in a straight line but the data are highly nonlinear in such a plot (except in Figure 10 where the q/P has been plotted on a logarithmic scale in order to clearly separate the three plots at different temperatures).
3330 Langmuir, Vol. 15, No. 9, 1999
Figure 10. Fits of various models to isotherm data of pentane (gas) adsorption on silica gel at T ) 273.1, 298.15, and 323.15 K. The data are from Kuro-Oka et al. The convention is the same as that in Figure 6.
Certainly, the two-orientation model is expected to fit better than the one-orientation model, since the former has more adjustable parameters. However, as reported in Table 3A,B, all the other seven sets of data, except for benzene, are correlated better or much better by the twoorientation model than by DLANG, which has the same number of adjustable parameters. As reported in Table 3A, the values of σeff for the systems of acetone, propionitrile, p-cresol, and p-chlorophenol appear to be close to, but slightly larger than, σmol. In other words, the correlated adsorption capacities are close to, but slightly smaller than, that predicted from the σmol and A h . This suggests that (1) not all the area of the sorbent is available for the adsorption of these solutes and (2) multilayer adsorption is unlikely in these systems. The specific area of activated carbon, which has a wide range of pore sizes, is estimated by the method of nitrogen adsorption.3,46,47 Some micropores, which are penetrable by N2, may not be accessible by the larger solutes considered here. Furthermore, hydration of the hydrophilic group (which all four solutes possess) increases the effective size, which further hinders entrance of these solutes into the small hydrophobic pores of the activated carbon. As a result, the actual area accessible by the solutes will be less than A h. The specific area is unknown for the other sorbents and no comparison between σeff and σmol can be made. However, the value of qm,1 for pentane at three different temperatures does support the physical consistency of the model. h and By definition, qm depends only on the accessible A σmol of the solute. We expect qm to be independent of (46) Barrett, E. P.; Joyner, L. G.: Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373. (47) Broekhoff, J. C. P. In Preparation of Catalysts, II; Delmon, B., Grange, P., Jacobs, P., Poncelot, G., Eds.; Elsevier: New York, 1979; p 663.
Jin et al.
temperature when σmol and the accessible A h are themselves temperature-independent. If the conformation of the molecular chain extends at high temperature, qm is expected to decrease with T. From Table 3b we see that indeed the qm,1 values at the three different temperatures are similar, and they decrease slightly as T increases. From Table 3 we see that although SPT-1 fits the data poorly, the resulting qm, in most cases, is very close to qm,1 from the two-orientation model. As discussed before, at a high surface coverage, if K2 is not much greater than K1, then qm is mainly due to the end-on orientation. Therefore, the qm from the SPT-1 is more or less representative of the end-on orientation. Indeed, besides the qm,1, the values of K from the SPT-1 are also very close to the values of K1 obtained with the two-orientation model, except for p-cresol and p-chlorophenol (in these two cases, the values of K2 are about 30 times those of K1). On the other hand, at low surface coverages, the difference between φ1 and φ2 is relatively small. In this case, when Kr is very large (such as for p-cresol and p-chlorophenol), q reflects mainly the adsorption behavior of the side-on orientation. This suggests we may be able to estimate the adsorption behavior of each orientation independently by using the SPT-1 equation to correlate separately the low- and high-coverage isotherm data. Comparing the values of βmol with the β from the correlation (with the rod model), we see that for the gassolid systems, the two are quite consistent. For example, the values of βmol and β for benzene are 3.77 and 3.37, respectively. The βmol for pentane is 2.09, while the correlated β at the three temperatures are 1.923, 2.038, and 2.641, respectively. The increase of β with temperature may be due to a conformational change; that is, the molecular chain becomes more extended at higher temperatures. For the liquid-solid systems, large discrepancies exist between the βmol and β. For example, β for acetone and propionitrile are much larger than βmol, whereas for p-cresol and p-chlorophenol the reverse is true. Many factors could be responsible for the discrepancy including (a) For a liquid-solid system, the solvent may also adsorb weakly on the solid surface, which is not accounted for in the model. (b) A solute may have more than two adsorption orientations. The projected area of each orientation depends on the degree of tilt. (c) As mentioned before, a certain fraction of the micropores of a porous adsorbent, such as activated carbon, may be accessible to the “sideon” but not to the “end-on” orientation. Furthermore, the K for the solutes adsorbing in those micropores could be much larger than the K for the “side-on” orientation due to wall effects. Obviously, isotherms cannot be precisely predicted by the model in the presence of one or more of these factors, although they may still be well-correlated by the model. In this case, however, the fitting parameters are empirical. Actually, most of the factors mentioned above can be included in our model. For example, to include competition of solvent molecules in low-affinity systems, we can take the solvent as an additional species and use the multicomponent adsorption form of this model. Similarly, to account for additional adsorption orientations, the general form of eq 2 can be used, while to consider factor (c), we can use the following equation:
qeq ) (qm,1K1φ1 + qm,2K2φ2 + qm,SKSφS)C
(27)
In this case, the total adsorbed amount is the sum of molecules adsorbing in large pores, qeq L ) qm,1K1φ1 + qm,2K2φ2, which can be modeled by the two-orientation
Adsorption Equilibria of Solutes
Langmuir, Vol. 15, No. 9, 1999 3331
from aggregation of the hydrocarbon chains of surfactants or formation of multiple layers on the surface. Full understanding of the interplay between steric blocking and aggregation effects on the adsorption behavior of surfactants would require more advanced theories than the model presented here.
Figure 11. The Van’t Hoff equation (30) applied to the pentane/ activated carbon system. The straight lines are least-squares fits to the correlated values of the side-on (K2) and end-on (K1) equilibrium constants.
model, and those in micropores, qeq S , which can be modeled by the SPT-1 equation with an independent capacity, qeq S , and equilibrium constants KS. Finally, we examine the correlated K1 and K2. From Table 3, we see that, for all the systems, as expected, K2 is greater than K1. We have also examined the temperature dependence of the equilibrium constants for the system pentane/activated carbon. The following integrated form of the Van’t Hoff equation should describe the data if the associated enthalpy of adsorption, ∆H, is constant over the given temperature range:
ln
K(T2) K(T1)
)-
(
)
1 ∆H 1 R T2 T1
(28)
Figure 11 shows a plot of ln(K1) and ln(K2) versus 1/T. For the end-on equilibrium constant, K1, the slope and intercept of a linear least-squares fit are 6357 and -17, respectively. For the side-on equilibrium constant, K2, the same parameters are 6076 and -13. These results are somewhat disappointing if we interpret the slope as -∆H/ R, since one might expect |∆H2|, > |∆H1|. This fit is a more stringent test of the model. However, one cannot have great confidence in the numerical values since only three temperatures are examined and there is considerable deviation from linearity in the case of K2. More extensive temperature-dependent data would permit a more thorough evaluation of the model. K1 and K2 of p-chlorophenol are about twice those of p-cresol, suggesting that Cl- interacts with the activated carbon more strongly than H3C-. This is consistent with the observation that toluene has a lower affinity than chlorobenzene.48 The two-orientation model elucidates also the sterric effects on the adsorption of large asymmetric molecules such as surfactants.49,50 The adsorption isotherms of many surfactants show two plateaus, instead of the single plateau of a Langmuir model or the two-orientation model (Figures 2a-5a). Experimental and theoretical studies in the literature indicate that the first plateau is due to monolayer adsorption and the second plateau is due to formation of bilayers on the adsorbent surface. Within the monolayer region, the two-orientation model accounts for the shift in the average orientation from “side-on” at low coverages to “end-on” at high coverages. However, this model does not take into account any contributions (48) Snoeyink, V. L. In Water Quality and Treatment, American Water Works Association, 4th ed.; McGraw-Hill: New York, 1990; Chapter 1. (49) Lajtar, L.; Narkiewicz-Michalek, J.; Rudzinski, W. Langmuir 1994, 10, 3754. (50) Somasunda ran, P.; Kunjappu, J. T.; Kumar, Ch. V.; Turro, N. J.; Barton, J. K. Langmuir 1989, 5, 215.
5. Summary and Conclusions An asymmetric solute on a surface can have a continuous orientation with respect to the surface normal. The apparent equilibrium isotherm and intrinsic adsorption kinetics will depend on a combination of the adsorption behavior of each individual orientation. In this work, a quantitative model has been developed to describe such a system. In the model, the adsorbed solutes with different orientations are treated as different species at the same bulk concentration. The steric blocking effect of each orientation is accounted for by its individual available surface function which is in turn derived from the scaled particle theory, in which monolayer adsorption of hard convex particles is assumed. A systematic analysis of the isotherm behavior for systems with two adsorption orientations is carried out. The key properties of the model are as follows: (a) Although the expressions of φi are different for solutes with disk-disk, disk-ellipse, and disk-rod shapes, the resulting isotherms are very similar when qm,1/qm,2 is used as a common indicator to characterize the relative steric blocking effect. The three expressions have almost the same correlation power. Therefore, for data correlation, anyone of them can be used. (b) The geometric parameters, qm,1 and β1 of the model, can be, in principle, calculated from a knowledge of the solute shape, dimension, and the specific adsorption surface area. These parameters may be regarded as nonadjustable. (c) At a low surface coverage, the ratio of the amount adsorbed from the two orientations is mainly determined by their relative affinity (energetic aspect). The side-on orientation is more favored in this case. On the other hand, because of the large steric disadvantage of the side-on orientation at a high coverage, in most cases, the end-on orientation becomes the dominant one at a high concentration. (d) As the concentration increases, the transition from side-on dominance to end-on dominance depends on the values of β and the affinity ratio, Kr. For a large β and small Kr, this transition may occur over a small range of coverage or solution concentration, and vice versa. (e) For an asymmetric solute with multiple adsorption orientations, the Scatchard plot is nonlinear and the nonlinearity increases with an increasing aspect ratio and affinity ratio. The model has been used to correlate experimental isotherms of six systems. The results are better than those obtained with some existing models with the same number of parameters, such as the double Langmuir equations. In most cases, moreover, the correlated parameters are consistent with the solute size, shape, and specific surface area of the sorbents. Acknowledgment. This work was supported by grants from NSF, GER-9024174, the Peterson Foundation, and the Showalter Foundation. Notation A h ) specific adsorption area of a sorbent Ai ) cross-sectional area of a solute with orientation i c ) solute concentration in solution phase C* ) defined as qm,1K1c K ) adsorption equilibrium constant
3332 Langmuir, Vol. 15, No. 9, 1999
Jin et al.
K* ) a constant defined as kac/kd K1 ) adsorption equilibrium constant of the end-on orientation K/1 ) defined as qm,1kac/kd,1 K2 ) adsorption equilibrium constant of the side-on orientation Kr ) defined as K2/K1 kr,i ) defined as Ki/K1 ka ) adsorption rate constant ka,i ) adsorption rate constant of the ith orientation ka,r ) ratio of adsorption rate constants between the endon and side-on kd ) desorption rate constant kd,i ) desorption rate constant of the ith orientation kd,r ) ratio of desorption rate constants between the endon and side-on q ) amount of solutes adsorbed in solid surface qap m ) apparent adsorption capacity qi ) amount solutes adsorbed in solid with orientation i qm ) maximum adsorption capacity qm,i ) maximum adsorption capacity with orientation i Si ) circumference of a solute with orientation i T ) temperature t ) real adsorption time t* ) defined as qm,1ka,1t Greek Letters βmol ) aspect ratio of a solute calculated from its molecular structure β ) aspect ratio of a solute with two orientations defined in Table 2 φ ) available surface function φeq ) available surface function at equilibrium φeq i ) available surface function at equilibrium for the ith orientation φi ) available surface function for the ith orientation F ) solute surface density Fi ) surface density of adsorbed solutes of the ith orientation σ ) solute diameter σeff ) effective solute diameter calculated from its correlated qm σmol ) solute diameter calculated directly from its molecular structure θ ) total surface coverage θ* ) normalized surface coverage with respect to θ∞ θeq ) surface coverage at equilibrium θeq i ) surface coverage at equilibrium with orientation i θi ) surface coverage associated to the ith orientation θi,∞ ) jamming limit surface coverage associated to the ith orientation θ∞ ) jamming limit surface coverage Subscripts 1 ) the first (or end-on) adsorption orientation 2 ) the second (or side-on) adsorption orientation a ) adsorption d ) desportion i ) the ith adsorption orientation m ) maximum
πσ22F2 4
∑FjSj ) 3θ
S2
∑Fj + A1(
2π
∑FjSj)2 ) θ 4π
A2(
2
(3) (4)
+ β(β + 2)θ1
(5)
+ β-1θ2
1
∑FjSj)2 ) βθ 4π
1
(6)
+ θ2
(7)
Substituting eqs 3, 4, and 6 into eq 2 in the text, we have
θ1 ) (1 - θ1 - θ2) ×
[
exp -
[
]]
3θ1β-1(β-1 + 2)θ2 θ1 + β-1θ2 1 - θ1 - θ2 1 - θ 1 - θ2
2
(8)
Similarly, substituting eqs 3, 5, and 7 into eq 2 results in
θ2 ) (1 - θ1 - θ2) ×
[
exp -
[
]]
θ2 + βθ1 3θ2β(β + 2)θ1 1 - θ1 - θ2 1 - θ 1 - θ2
2
(9)
For an ellipsoidal solute, the corresponding terms of eq 3 become
∑Fj +
A1
∑Fj +
A2
∑FjSj ) 3θ
S1
2π
+ (2β-1 + 1)θ2
∑FjSj ) (1 + 2β)θ 2π
∑FjSj)2 ) (θ
2
4π
A2(
1
S2
A1(
∑FjSj)2 ) 4π
1
+
( (
(
θ1xβ +
(10)
(
)
β β-1 + 2 + θ2 2 2 (11)
)) ))
1 + β-1 θ2 2
+
[
-1
exp -
(1)
or
θ2 ) A2F2 )
A2 -
φ1 ) (1 - θ1 - θ2) ×
2
πσ1 F1 4
(j ) 1, 2) ∑FjAj ) 1 - θ1 - θ2 S1∑FjSj ) 3θ1 + β-1(β-1 + 2)θ2 A1 - ∑Fj + 2π 1-
2
xβ + 1xβ θ2 2
(12)
2
(13)
The corresponding φ1 and φ2 are
Appendix: Derivation of Oi from SPT For the two-disk model, the relation between the surface coverage, φi, and particle density, Fi, for an individual orientation is
θ1 ) A1F1 )
Each term in eq 2 can be expressed in terms of surface coverages and the size ratio of the two adsorption orientations, β, that is,
3θ1 + (2β + 1)θ2 1 - θ 1 - θ2
[
)
]] 2
(14)
and
φ2 ) (1 - θ1 - θ2) ×
[
exp -
(
)
β β-1 + 2 + θ2 2 2 1 - θ 1 - θ2
θ1 + (1 + 2β) +
[
(
)
xβ + 1xβ θ2 2 1 - θ1 - θ2
θ2xβ +
(2)
(
1 + β-1 θ2 2 1 - θ1 - θ2
θ1 +
]] 2
(15)
Adsorption Equilibria of Solutes
Langmuir, Vol. 15, No. 9, 1999 3333
For a rodlike solute, the corresponding terms of eq 3 become
∑Fj +
A1
∑Fj +
A2
∑FjSj ) 3θ
S1
2π
1
(
+ β-1 1 +
π θ 4 2
)
∑FjSj )
2π 4 2 (1 + 2β)θ1 + 1 + (β + 2 + β-1) θ2 (17) π π A1(
∑FjSj)
2
4π
∑FjSj) 4π
2
)
[
exp -
[
(
)
1 + β1 π θ2 θ1 + θ2 4 2 1 - θ 1 - θ2 1 - θ1 - θ2
(
3θ1 + β1 1 +
)
]] 2
(20)
and
S2
((
A2(
(16)
φ1 ) (1 - θ1 - θ2) ×
( ((
) )
( (
)) ) ))
1 + β-1 θ2 ) θ1 + 2
)(
2
2 1 1 xβ + θ1xβ + θ2 2 xπ xβ
The corresponding φ1 and φ2 are
2
(18)
φ2 ) (1 - θ1 - θ2) ×
[
4 2 (1 + 2β)θ1 + 1 + (β + 2 + β-1)θ2 π π exp 1 - θ 1 - θ2
(19) LA980350N
(
[
(
)
(
))
1 1 2 θ1xβ + xβ + θ2 2 xπ xβ 1 - θ1 - θ2
]] 2
(21)
