Langmuir 1987,3, 369-372
369
The Role of Adsorbent Heterogeneity in Adsorption from Binary Liquid Mixtures? Shivaji Sircar Air Products and Chemicals, Inc., Allentown, Pennsylvania 18105 Received October 24, 1986 A parametric study of the role of adsorbent heterogeneity in adsorption from ideal binary liquid mixtures of equal adsorbate sizes is carried out, using simple models for the local homogeneous adsorption isotherm and a site selectivity distribution function to describe heterogeneity. It shows that the detailed structure of the selectivity distribution function is not critical in describing overall isotherm shapes on the heterogeneous adsorbent. Adsorbent heterogeneity has a much more pronounced effect on the shape of the surface excess isotherm than on the heat of immersion as a function of liquid composition.
Introduction Most adsorbents of practical interest are energetically heterogeneous. They contain a distribution of adsorption sites of varying adsorbateadsorbent interaction energies which may be different for different adsorbates. Consequently, the selectivity of adsorption of the components of a liquid mixture can vary from site to site. This factor must be taken into account in describing adsorption from a liquid mixture. Unfortunately, the site selectivity distribution in an adsorbent cannot be experimentally measured. The experimentally measured equilibrium adsorption property for a liquid mixture is the overall surface excess isotherm which is an integrated effect of the equilibrium adsorption characteristic of the individual sites. It is, however, possible t~ develop simple mathematical models of adsorbent surface heterogeneity and derive analytical expressions for the overall surface excess isotherm which can be used for data interpretation. This paper reviews some of the recent developments in this area for adsorption of an ideal binary liquid mixture of equal adsorbate sizes and discusses some of the key results of practical interest. Mathematical Model One simple model for adsorption on a heterogeneous adsorbent assumes that the adsorbent surface is composed of a patchwise distribution of homogeneous sites, each site being characterized by a selectivity of adsorption for component 1 of the binary liquid mixture. The overall surface excess isotherm for that component is then obtained by the integral of the surface excess contributions of each site:’ N,e(X,) = L y n I e ( S , X l ) h ( Sd)S
constant T (1)
selectivity between S and S + dS for component 1. The limits of the integrals in eq 1 and 2 are between the lowest (S,) and the highest (S,) values of site selectivities of component 1 on the heterogeneous surface. The cumulative site selectivity distribution is given by
where f ( S ) is the fraction of adsorption sites having a selectivity between S, and S for component 1. An analytic expression for N l e ( X l )can be obtained by evaluating the integral of eq 1 by using analytical functions for n(S)
dS = 1
X(S) d S represents the fraction of adsorption sites having ‘Presented at the ”Kiselev Memorial Symposium”, 60th Colloid
and Surface Science Symposium, Atlanta, GA, June 15-18,1986; K. S. W.Sing and R. A. Pierotti, Chairmen.
X(S) = -SneWaS n! 0 < S < a;n = integer; CY
>0
Uniform Distribution X(S) = l/(S, - S,)
S,
< S < SH
(4b)
These two distribution functions have strikingly different shapes. The r distribution has a skewed-Gaussian-like shape and the uniform distribution has a symmetric-rectangular shape. S varies between zero (S, = 0) and infiiity (1) b s , S.;Olivier, J. P. On Physical Adsorption; Wiley: New York, 1964; Chapter 4. (2) Sircar, S.;Myers, A. L. J . Phys. Chem. 1970, 70, 2828. (3) Larionov, 0.G.; Myers, A. L. Chem. Eng. Sci. 1971, 26, 1025. (4) Sircar, S.J . Chem. SOC.,Faraday Trans. 1 1983, 79. 2085. (6) Sircar, S.Surf. Sci. 1984, 148,478.
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370 Langmuir, Vol. 3, No. 3, 1987
Sircar
Table I. Properties of the Selectivity Distribution Functions mean (d (n + U/a ( S H + sL)/2
r uniform
dispersion ( u )
*
(=a/r) l / ( n 1)ll2 ( S H - SL)/3l/'(sH
skewness (sK) 2/(n
+
(n + l)l/Z/ff ( S H - SL)/2 x 31/2
+
0
+ SL)
Table 11. Characteristics of the Surface Excess Isotherm and the Heat of Immersion Function
'[(l
(SM= m) for the I'distribution. These limits are the two extremes of physically possible adsorption behavior on the surface representing sites having infinite selectivities for components 2 and 1, respectively. S varies between SL (=S,) and S H (=SM) for the uniform distribution. The mean ( p ) , the dispersion (a), and the skewness (sK) of the two distributions are given in Table I. A parameter (9 = a l p ) can be defined to describe the degree of heterogeneity of the adsorbent. 9 = 0 (a = 0) represents a homogeneous adsorbent. Both distributions themtake the form of a Dirac 6 function and the parameters of X(S) approach the limits of n, a m; p = n/a for the r and SL= SH= p for the uniform distribution. Adsorbent heterogeneity increases as 9 increases. The maximum heterogeneity by these models correspond to n = 0 (9= 1)for the and SL= 0 (9= 0.577) for the uniform distribution.
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Surface Excess Isotherms Equations 3 and 4a or 4b can be substituted into eq 1 and integrated to ~ b t a i n ~ ? ~ Distribution
Nle = m[X, - 6eeE,+,(6)]
(54
Uniform Distribution x 2
Xl(SH - S L )
In
(
") ]
SHX1+ SLX, + xz
(5b)
where 0 = a(Xz/Xl)and E,+1(6) is the exponential integral of order ( n + 1). Equations 5a and 5b represent the overall surface excess isotherms for heterogeneous adsorbents by the I' and uniform site selectivity distribution models, respectively. The parameters of these equations are m, n, and CY for the I' distribution and m,SH,and SLfor the uniform distribution. They can also be described in terms of m, p, and 9 by using the relationships given in Table I. They reduce to eq 3 at the limit of 9 0. The slopes of these isotherms at the two boundaries ( X , 0, X , 1)are given in Table 11. It may be seen that the isotherm slopes at X1 0 for both models depend only on the mean selectivity of the distribution and they are the same as the isotherm slope of the homogeneous adsorbent. The slopes of these isotherms at x1 1,on the other hand, depend both on p and the degree of adsorbent heterogeneity (*I. Thus, the adsorbent heterogeneity has a more pronounced effect on the isotherm shape at higher concentrations of X,.
-- -
-
-
Heat of Immersion The other experimentally measured thermodynamic property for adsorption from liquids is the heat of immersion of the adsorbent into the liquid. The dimen-
+ 31/2'J')/(l
- 31/z'P)]
sionless heat of immersion for a homogeneous adsorbent corresponding to the surface excess isotherm of eq 3 is given by6
Q and Qi are the heat of immersions of the adsorbent into a liquid mixture of composition Xi and the pure liquid i, respectively. For a heterogeneous adsorbent, eq 6 and 4a or 4b can be substituted into an integral of the form of eq 1 to obtain I' Distribution F* = 1 - 6e'E,+l(8)
(74
Uniform Distribution
where F* is the dimensionless heat of immersion of the heterogeneous adsorbent defined in the same way as F in eq 6. The slopes of F*(X,) at the limits of X , 0 and X1 1 for eq 7a and 7b are given in Table 11. Again, it may be seen that the slopes at X, 0 depend only on the mean selectivity and they are the same as that for the homogeneous adsorbent. The slope of F*(Xl) at X1 1,on the other hand, depends both on p and 9. Thus, the effect of adsorbent heterogeneity on the heat of immersion of the adsorbent into a liquid mixture is manifested only at high X1.
-
-
-
-
Model Isotherms Figure 1shows several surface excess isotherms plotted as Nle/m vs. X1 generated by using eq 3, 5a, and 5b. A mean selectivity of p = 1.5 was used for these curves. Curve a represents the isotherms for the homogeneous adsorbent (9= 0). The solid lines of curves b and c represent the isotherms for the r distribution using 9 = 0.500 and 0.707, respectively. It may be seen that the shape of the surface excess isotherm changes drastically as the adsorbent heterogeneity increases for a given 1.1. The value of Nledecreases for a given X1 and the maximum of the isotherm occurs at a lower X1for a heterogeneous adsorbent. For a sufficiently high value of 9,the isotherm becomes "S" shaped. The initial portion of the isotherm (X, 0) is, however, not affected by 9 because of the reason given above. This analysis demonstrates the interesting point that an S-shaped isotherm for an ideal liquid mixture of equal
-
(6) Sircar, S.; Novosad, J.; Myers, A. L. Ind. Eng. Chem. Fundam. 1972, 11, 249.
Langmuir, Vol. 3, No. 3, 1987 371
Adsorption from Binary Liquid Mixtures
0.10
t E
> .z
.L‘;
0.05
0.2
z
0.4
0.6
0.8
1.0
x 1-
Figure 3. Model surface excess isotherms for p = 10.0: (a) homogeneous; (b) \k = 0.707 (I’), \k = 0.553 (uniform).
0
0.2
0.4
0.6
Figure 1. Model surface excess isotherms for p = 1.5: (a) homogeneous; (b) \k = 0.500 (I’), B = 0.450(uniform);(c) B = 0.707 (I’), B = 0.545 (uniform).
X1-r
Figure 4. Model heat of immersion as a function of liquid composition: (a) homogeneous for p = 10.0; (b) corresponding to curve b of Figure 3.
-
X1-
Figure 2. Model heat of immersion as a function of liquid composition: (a) homogeneous for p = 1.5; (c) corresponding to curve c of Figure 1.
adsorbate sizes can be caused due to adsorbent heterogeneity. This has been experimentally observed.’ The homogeneous model cannot explain such S-shaped behavior. The dashed lines of curves b and c represent the best fit of the solid lines by the uniform distribution model using p = 1.5, \k = 0.45 for curve b and p = 1.45, \k = 0.545 for curve c. This shows that despite the drastic difference in the detailed shape of the two selectivity distribution functions, they can be used to describe surface excess isotherms (both U and S shapes) equally well using approximately the same values of p. In other words, the detailed shape of the site selectivity distribution function is not critical in determining the overall surface excess isotherms for a given local homogeneous isotherm. Figure 2 shows plots of F” vs. X1Corresponding to curves a and c of Figure 1using either eq 7a or eq 7b. It may be seen that adsorbent heterogeneity has very little effect on the variation of the heat of immersion of the adsorbent as a function of X1. The curve for a homogeneous adsorbent (curve a) and that for a very heterogeneous adsorbent (curve c) are not much different even though the corresponding surface excess isotherms were drastically different. F* for a heterogeneous adsorbent is slightly (7) Kipling, J . J.; Tester, D. A. J. Chem. SOC.1962, 4123.
lower than F for a homogeneous adsorbent for a given X,. The initial portion (X, 0) for the F” vs. X1plot is independent of adsorbent heterogeneity because of the reason given earlier. Figure 3 shows the surface excess isotherms generated from eq 3,5a, and 5b for a large mean selectivity ( p = 10) of adsorption for component 1. Curve a is for a homogeneous adsorbent. Curve b is generated by using \k = 0.707 and 0.553 for I’ and uniform distributions, respectively. This figure demonstrates that although Nleis reduced for any given X1, the effect of adsorbent heterogeneity on the surface excess isotherm is much less pronounced when p is large. In this case, it may not be possible to change a homogeneous U-shaped isotherm to an S-shaped isotherm by adsorbent heterogeneity alone. Figure 4 shows the F” vs. X1plots corresponding to isotherm curves a and b of Figure 3. Again, the effect of adsorbent heterogeneity on the variation of heat of immersion as a function of XI is weak. The initial slope of the F*(Xl) plot is large because of the large value of p. Figure 5 shows the I’ and uniform site selectivity distribution functions, A(S), corresponding to the surface excess isotherms b and c of Figure 1. The figure plots the ratio of A(S)to its maximum value (A), against the ratio S / p . The dashed lines correspond to the isotherm curve b and the solid lines correspond to the isotherm curve c. The detailed structures of these distribution functions are quite different. The uniform distributions have symmetric rectangular shapes while the r distributions are very much skewed toward the high selectivity ends. The plots show that a considerable fraction of sites have selectivity less than unity ( S / p < 0.667) indicating that these sites preferentially adsorb component 2 of the mixture. This is interesting because the overall isotherms of curve b is U shaped. Thus, the sites selective toward component 1
372
Langmuir 1987, 3, 372-377
of the mixture dominate the overall isotherm shape in this case. The fraction of site with S < 1 is more for the isotherm of curve c which results in an S-shaped isotherm at large XI. The cumulative fraction of sites having S less than unity [ f ( l )can ] be obtained by using eq 2a, 4a, and 4b as r Distribution r ( n + 1, a ) f(1) = 1 (84 n! Uniform Distribution
where r(n + 1, a)is an incomplete F function.* Equations 8a and 8b were used to calculate f ( 1 ) for curves b and c of Figure 5. They were 0.278 and 0.286 for curve b and 0.385 and 0.336 for curve c, using the r and the uniform distributions, respectively. Thus, both distributions provided very similar f ( 1 ) values for isotherms b and c of Figure 1 despite their different shapes. In summary, this study shows the following: (a) Adsorbent heterogeneity plays a significant role in determining the shape of the surface excess isotherm for adsorption of binary liquid mixtures. (b) Simple mathematical models based on the concept of a site selectivity distribution on a heterogeneous surface can be adequately used to account for adsorbent heterogeneity. The detailed structure of the distribution function is not critical. The choice of the model, therefore, (8) “Handbook of Mathematical Functions”; Abramowitz, M., Stegun,
A., Eds. Applied Mathematics Ser. 55; National Bureau of Standards, U.S. Government Printing Office: Washington, DC, 1972.
1
3
2
4
5
SOlP
Figure 5. r and uniform site selectivity distribution functions: (solid line) correspondingto isotherms c of Figure 1; (dashed lines) corresponding to isotherms b of Figure 1. depends on its mathematical simplicity. (c) The effect of adsorbent heterogeneity is more pronounced when the mean of the site selectivity distribution is low. An S-shaped isotherm for adsorption of an ideal liquid mixture of equal adsorbate sizes can be caused by adsorbent heterogeneity. (d) The effect of adsorbent heterogeneity on the heat of immersion of a liquid mixture is much less pronounced than that on the surface excess isotherm. Consequently, the heat of immersion cannot be used to evaluate the degree of adsorbent heterogeneity.
Counterion Binding on Mixed Micelles: Effect of Surfactant Structure James F. Rathman and John F. Scamehorn* School of Chemical Engineering and Materials Science, Institute for Applied Surfactant Research, University of Oklahoma, Norman, Oklahoma 73019 Received April 11, 1986. I n Final Form: October 28, 1986 Previous work has shown that an electrostatic model can be used to accurately predict the fractional counterion binding on mixed ionic/nonionic micelles. New results show that this model can be successfully applied to a wide variety of surfactant mixtures. In particular, it is shown that mixed micelles containing nonethoxylated nonionic surfactants such as phosphine oxides, amine oxides, and sulfoxides at appropriate pH conditions exhibit binding behavior similar to that of mixtures containing polyethoxylates. The following ionic surfactants were studied in mixtures with a polydisperse nonylphenol polyethoxylate as the nonionic surfactant: dodecylpyridinium chloride, hexadecylpyridinium chloride, sodium dodecyl sulfate, and sodium octylbenzenesulfonate. The electrostatic model describes the binding for these systems very well. The following nonionic Surfactants were studied in mixtures with the cationic surfactant hexadecylpyridinium chloride: a monodisperse dodecyl polyethoxylate alcohol, two polydisperse nonylphenol polyethoxylates, dodecyldimethylphosphine oxide, dodecyldimethylamine oxide, and decyl methyl sulfoxide. The electrostatic model accurately describes the binding on all systems, although the model is slightly less accurate for the sulfoxide mixtures. These results are consistent with the concept that ionic/nonionic surfactant interactions and nonidealities of mixing in micelles are primarily of electrostatic origin, with specific chemical interactions, if present, being of secondary importance. Introduction The association of counterions with mixed micelles composed of ionic and nonionic surfactants is of great
* To whom correspondence should be addressed. 0743-7463/87/2403-0372$01.50/0
interest due to the development of applications which take advantage of the synergistic behavior of such systems. Despite this, there are still very few reported data in this area. It has been shown that the fractional counterion binding on binary mixed ionic/nonionic surfactant micelles can be 0 1987 American Chemical Society
