J . Phys. Chem. 1989, 93, 4132-4138
4132
citation of the catalyst at low temperature and pressure to inhibit the total oxidation reaction. It remains to be seen whether or not a more selective and active catalyst can be found in this direction. Our work has provided information on the nature of the participation of the catalyst in the oxidation process. This study has shown that the valence state of the transition-metal surface ions and their reducing properties are responsible for electron transfer from the active catalyst adsorption center to the adsorbed mol-
ecule(s). 0-and O2lattice species and surface anion radicals seem to be involved in the oxidative dehydrogenation of hydrocarbons on titanium polytungstate due to their reducing properties. These observations are in agreement with thermally activated processes reported for CH4 oxidation.'S6-* Registry No. TiOz, 7440-32-6; H4(SiWI2Oa), 12027-38-2; CH4, 7482-8; CO, 630-08-0; C 0 2 , 124-38-9; HzO,7732-18-5.
Surface Tension of Simple Mixtures: Comparison between Theory and Experiment B. S. Almeida* Departamento de Engenharia Qurmica do IST e Centro de Qdmica Estrutural, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
and M. M. Telo da Gama Departamento de F k c a da FCUL e Centro de F h c a da MatPria Condensada, Au. ProJ Gama Pinto, 2, 1699 Lisboa Codex, Portugal (Received: February 22, 1988; In Final Form: August 22, 1988)
We report a study of the surface tension of both pure and mixed fluids of simple molecules using a microscopic mean-field theory (MFT). The pure components are modeled by Lennard-Jones potentials with two different sets of intermolecular parameters reported in the literature. One of these yields results that are in much better agreement with the experimental data. We have also studied the dependence of the interfacial properties of binary mixtures on the cross-interaction parameters and it was found that (for mixtures of krypton, ethene, and ethane) reasonable agreement with experiment can only be obtained when the Lorentz-Berthelot combining rule is relaxed.
1. Introduction
Although in the past few years considerable effort has been made by theoreticians to understand the surface behavior of pure fluids and mixtures, in most cases the calculations were carried out for model systems that bear little resemblance to real fluids. I n general's2 the choice of potential parameters was made for theoretical reasons, preventing a direct comparison with real mixtures. As a matter of fact, the number of experimental results for surface properties of relatively simple binary mixtures has not been a b ~ n d a n t ~and . ~ the first measurements for a series of mixtures have only appeared r e ~ e n t l y . ~Consequently .~ our study is the first systematic comparison between the results of a microscopic theory and experiment. In this paper we apply mean-field theory (MFT) to predict the surface tension of krypton, ethane, and ethene and their binary mixtures. The experimental results7~* for these systems cover the whole range of composition and a fairly wide temperature range ( 1 16-1 35 K) allowing comparisons with theoretical predictions over an unprecedented range of intermolecular and thermodynamic parameters. A previous comparison of the results of this theory (1) Telo da Gama, M . M.; Evans, R. Mol. Phys. 1983.48, 229. Telo da Gama, M. M.; Evans, R. Mol. Phys. 1983, 48, 251. (2) Lee, D.J.: Telo da Gama, M. M.; Gubbins, K. E. J . Phys. Chem. 1985, 89, 1514. ( 3 ) Sprow, F. B.; Prausnitz, J. M . Trans. Faraday SOC.1966, 62, 1097. (4) Fuks, S.; Bellemans, A. Physica 1966, 32, 594. ( 5 ) Almeida, B. S. Ph.D. Thesis, Technical University of Lisbon, 1986. (6) Nadler, K. C.; Zollweg, J. A.; Street, W. B.; McLure, I. A . J . Colloid Interface Sci., in press. (7) Soares, V . A. M.; Almeida, B.S.; McLure. I. A,; Higgins, R. A. Fluid Phase Equilib. 1986, 32, 9 . (8) Almeida, B. S.; Soares, V . A. M.; McLure, I. A,; Calado, J. C. G . J . Chem. Soc., Faraday Trans. I , submitted for publication.
0022-3654/89/2093-4132$01.50/0
and experiment was restricted to mixtures of Ar + Kr9 and Ar + CH4.9 (We will comment on the results of this study in section 5.) On the other hand Lee et a1.I0 carried out a detailed comparison between the results of M F T and molecular dynamics for model mixtures of argon and krypton. The agreement was fair in both cases. In a different study Lee et aL2 have also applied M F T to calculate the surface tension, adsorption, and density profiles of binary mixtures of Lennard-Jones fluids in which one of the components is a supercritical vapor. The agreement with the results of molecular dynamics for these systems was also found to be satisfactory. Earlier studies by Falls et al.," Carey et a1.,I2and Telo da Gama and EvansI3 used the square gradient approximation to calculate the interfacial properties of several binary mixtures but these results have been superseded by the results of MFT. The latter takes into account the long-range nature of the intermolecular interactions, which is important, for example, in determining the equilibrium thickness of wetting 1 a ~ e r s . l In ~ addition, M F T is easier to implement, which makes its practical application quite feasible. Other theories (the so-called classical thermodynamic theories) have been used in the study of interfaces, namely corresponding states, regular solution, and lattice theory. However, an accurate prediction of the'experimental results for the surface tension of (9) Thurtell, J. H.; Chapman, W. G.; Nadler, K. C., preprint. (10) Lee, D.J.; Telo da Gama, M . M.; Gubbins, K. E. Mol. Phys. 1984, 53, 1 1 13. ( 1 1) Falls, A. H.; Scriven, L. E.; David, H.
T.J . Chem. Phys.
1983, 78,
7300. (12) Carey, B. S.; Scriven, L. E.; David, H. T . AIChE J . 1980, 26, 705. (13) Telo da Gama, M. M.; Evans, R. Mol. Phys. 1980, 41, 1091. (14) Telo da Gama, M . M.; Sullivan, D.In Fluid Interfacial Phenomena; Croxton, C. A,, Ed.; Wiley: London 1986, Chapter 2, p 45.
0 1 9 8 9 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4133
Surface Tension of Simple Mixtures simple liquids and their mixtures is possible only by adjusting one or more empirical parameters of unknown physical significance.8 For this reason these theories are often difficult to justify theoretically. This paper is arranged as follows. In section 2 we give a short account of the theory. The comparison between the predicted values for the surface tension of pure fluids with experimental results is made in section 3. In section 4 we discuss the effects of deviations from the Lorentz-Berthelot combining rule on the surface tension of binary mixtures. These deviations are quantified by empirical parameters which can be fitted to thermodynamic properties of the bulk phase. We then compare the theoretical surface tensions obtained with these parameters to experimental data. Finally, in section 5 we make some concluding remarks. 2. Theory The mean-field grand potential functional Q for an inhomogeneous binary mixture in the absence of an external field can be written as follows:’5
TABLE I: L-J Parameters c and u for Krypton, Ethane, and Ethene Calculated from the Critical Constants and from the Second Virial Coefficients
Kr
critical const
c/k, K
virial coeff
elk, K
u, A u, A
166.3 3.632 164.5 3.60
C~HL 242.6 4.271 243.0 3.954
C& 224.3 4.090 199.2 4.523
properties are given by simple analytic expressions.’* The equilibrium densities of the coexisting liquid and vapor phases can be calculated at each temperature and composition by solving the simultaneous equations’ which express constancy of the pressure and chemical potentials in both phases. We obtain then the limiting values pi(liquid) and pi(vapor) which provide the boundary conditions to solve eq 2 for the density profiles pi(r) 2 pi(z) ( z being the normal to the planar interface). These equations can be solved numerically by an iterative process. The (k 1)th profile p;+’(z) is related to the kth profile by
+
where pi is the number density of species i, p i is the chemical potential of this species, u - is~ the attractive interaction, Vis the volume of the system, and) is the free energy density of a uniform mixture characterized by the repulsive interactions. The equilibrium densities p: are obtained through the minimization of the grand potential. Setting the derivative of Q with respect to pi equal to zero yields a set of coupled integral equations
The density profiles in each iteration can be obtained through the inversion of the hard-sphere chemical potential. (Hard-sphere thermodynamic calculations are done in the compressibility Percus-Yevick approximation.13J9) This iteration scheme is, in the majority of cases, rapidly convergent, yielding accurate values for the densities and the surface tensions. The Gibbs adsorption equation for the surface tension y of a binary mixture, at constant temperature, is
where p p is the chemical potential of species i in the repulsive reference fluid. Then, from the equilibrium grand potential Q [ { p p ) ] we can derive the surface tension y for a planar interface
ri,the adsorption of component i with respect to an arbitrary dividing surface, zG, is defined as
n+pv 7
’
7
(3)
where A is the area of the interface and p the equilibrium bulk pressure. Equation 1 for Q is written assuming that (i) the free energy density of the repulsive reference system can be treated locally, (ii) the attractive part of the potential is treated in a mean-field approximation. We use a Lennard-Jones potential
fh
(4) divided, according to Weeks, Chandler, and Andersen (WCA), into a reference potential ui:(r) and an attractive perturbation uijB(r)
=0
u.,a(r) IJ = -eij = uikJ(r)
otherwise for r 4 2’l6uij otherwise
where cij and uij are the usual Lennard-Jones parameters. The repulsive system is approximated by an equivalent hardsphere fluid with diameters chosen according to the BarkerHenderson pre~cription.’~.’’The attractive part of the potential is treated in a mean-field fashion and the bulk thermodynamic ( I 5) Telo da Gama, M. M.; Thurtell, J. H. J . Chem. SOC.,Faraday Tram.
172 1. (16) Verlet, L.; Weiss. J. Phys. Reu. A 1972, 5 , 939. (17) Lee, L. L.; Levesque, D. Mol. Phys. 1973, 26, 1351.
2 1986,82,
where ppfl is the bulk density of species i in each phase. Widom20 has shown that the surface tension given by mean-field theories, such as the one used in this work, satisfies the Gibbs adsorption equation (eq 6).
3. Results for the Pure Components The prediction of the thermodynamic properties of pure fluids depends strongly on the appropriate choice of parameters t and u. Two sets of values for e and u are presented in Table I for krypton, ethane, and ethene-one obtained from second virial coefficients data2I and the other based upon the critical constants reported in the compilation of Ambrose and Townsend.22 For the latter we take argon as the reference fluid with the well-established parameters e/k = 119.8 K and u = 3.405 A. In Figures 1 and 2 we compare the density p and the surface tension y obtained using both sets of potential parameters with the experimental results for krypton, ethane, and ethene as a function of temperature. The circles and stars are experimental (18) Tarazona, P.; Tela da Gama, M. M.; Evans, R. Mol. Phys. 1983,49, 283. (19) Tela da Gama, M . M.; Evans, R. Faraday Symp. Chem. SOC.1981, 16, 45. (20) Widom, B. Physicu A (Amsterdam) 1979, 95A, 1 . (21) Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: London 1967; p 1 1 10. (22) Ambrose, D.; Townsend, R. Vapour-Liquid Critical Properties; National Physics Laboratory: Teddington, U.K., 1978. (23) Albuquerque, G.M . N.; Calado, J. C. G.; Nunes da Ponte, M . L.; Staveley, L. A. K. Cryogenics 1980, 20, 416. (24) Haynes, W. H.; Hiza, M. J. J . Chem. Thermodyn. 1977, 9, 179. (25) Menes, F.; Dorfmuller, T.; Bigeleisen, T. J . Chem. Phys. 1970, 53, 2689.
4134
Almeida and Telo da Gama
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 2.5
1
I
I
I
I
TABLE 11: 5. Values Obtained through the Fitting of
I
mixture + C2H6 + C2H4 C2H4 + C2H6
0
P / g cm4
0
2.4
E 0.959
Kr Kr
0 0
0
0.939
0.984
0
2.3
2.2
2.1
------- - - - - -- - - - - - - - - - - - b
0.7
x
I
0.6
x
t
0.5
0.4 15 ” 0.3
1
u 0.5
0.0
T/K
1.0
XK,
Figure 1. Density versus temperature for krypton (a), ethane (b), and ethene (c). The solid lines correspond to L-J parameters obtained from critical constants and the dashed lines to the parameters derived from second virial coefficients. The symbols represent the experimental results: 0,krypton;23*, ethane;240 , ethene.25
29
F/mN w’
Figure 3. Surface tension versus liquid composition for krypton + ethane at T = 116 K. MFT results with ( = 0.959 (-) and with 5 = 1 (---), and experimental points (*).
the former set is remarkably better. The maximum deviation between the experimental and the theoretical surface tension occurs for krypton at the upper end of the temperature range and it is of the order of 4%.
4. Results for the Mixtures The prediction of excess properties of real mixtures depends not only on the values of the L-J parameters for each component but also, and quite substantially, on the unlike interaction parameters. For t,, and glI we chose the set of values derived from the critical constants. The parameters til and ulJare defined by the equations
27
25
23
61,
=
Il(cii
+ u,j)/2
(8)
Figure 2. Surface tension versus temperature for (b), and ethene (c). The solid lines correspond to L-J parameters obtained from critical constants and the dashed lines to the parameters derived from second virial coefficients. The symbols represent the experimental result^:^^' 0,krypton; *, ethane; e. ethene.
The Lorentz-Berthelot combining rule ( 4 = 1, 7 = 1) does not yield reasonable predictions for the excess properties of real mixtures. In general, with a proper choice of .$ (keeping 17 = 1 ) all excess properties may be predicted with good accuracy.26 We took for i the values obtained by fitting the excess Gibbs energy at the equimolar concentration Glj2E calculated by using the Frish-Longuet-Higgins-Widom equation of state2’ to the exThese values for the mixtures perimental results of Calado et krypton ethane, krypton ethene, and ethane ethene are used for the fits were shown in Table 11. The values of determined at 116 K for the mixtures with krypton and at 162 K for the hydrocarbon mixture. We assumed to be temperature independent. The surface tensions (of both pure fluids and mixtures) are therefore obtained without fitting for any interfacial property. Figures 3 and 4,5 and 6, and 7 and 8 show the surface tension versus composition at two temperatures for the mixtures krypton + ethane, krypton + ethene, and ethene + ethane, respectively.
values, the solid lines are calculated with L-J parameters from the critical constants, and the dashed lines with the corresponding parameters are obtained from the second virial coefficients. For kyrpton the latter set of t and CT values lead to a slightly better agreement; for ethane and ethene the agreement achieved with
( 2 6 ) Bohn, M.; Lago, S.; Fisher, J.; Kohler, F. Fluid Phase Equilib. 1985, 23, 1 3 7 . ( 2 7 ) Longuet-Higgins, H.; Widom. B. Mol. Phys. 1964, 8, 549. ( 2 8 ) Calado. J . C G.; Azevedo, E. J. S. G.; Soares, V . A . M . Chem. Eng. Conrniun. 1980, 5. 149.
19
+
115
I
I
I
I
I
119
123
127
131
135
139
T/ K krypton, (a), ethane
+
+
-
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4135
Surface Tension of Simple Mixtures
Y/mN m-l 25
.
23
0.0
1.0
0.5 XC2H4
Figure 7. Surface tension versus liquid composition for ethene + ethane at T = 117 K. MFT results with 5 = 0.984 (-) and with f = 1 (---), and experimental points (*).
21
19
17 23 15
..
c
I
I
0.0
I
I
I
I
I
I
I 1 .o
0.5
XK,
Figure 4. Surface tension versus liquid composition for krypton + ethane at T = 124 K. MFT results with f = 0.959 (-) and with f = 1 (---), and experimental points (*).
..-.
22
( 1 1 1 1 1 1 1 1 1 1 0.0
1.0
0.5 XC*H,
Figure 8. Surface tension versus liquid composition for ethene + ethane at T = 135 K. MFT results with f = 0.984 (-), with f = 1 (---), and experimental points (*).
V/m N
0
YE/"
m-l
-
0.4
- 0.8
- 1.2 16
c
'1
- 1.6
I l r l r l l l l I l
0.0
1 .o
0.5
XKr
- 2.0 +
Figure 5. Surface tension versus liquid composition for krypton ethene at T = I16 K. MFT results with f = 0.939 (-) and with = 1 (---),
and experimental points (*).
1 .o
0.5
XK,
Figure 9. Excess surface tension versus liquid composition for krypton + ethane at T = 116 K. MFT results with 5 = 0.959 (-) and experi-
mental points (*).
24
V/mN m22
20
18
16
0.0
0 .o
1.0
0.5
XKI
Figure 6. Surface tension versus liquid composition for krypton + ethene at T = 124 K. MFT results with f = 0.939 (-) and with f = I (---), and experimental points (*).
The stars are the experimental values* and the lines represent the results from M F T (dashed lines, using { = I ; solid lines, using
fitted 4's). The behavior of the hydrocarbon mixture differs from the two other mixtures which have krypton as one of the components. For the first system the calculated surface tensions are lower than the experimental values; the situation is reversed for the last two mixtures. In general, the predictions which use E = 1 lead to almost ideal mixing and for the Kr + C2H6the theoretical surface tensions are even higher than the ideal value defined as xlyl + x2y2. The negative deviation from ideality, which is the usual b e h a v i ~ r ~ *for ~ , ~mixtures -' of relatively simple molecules, is predicted only with fitted E's. In Figures 9, 10, and 1 1 the excess surface tension yE,defined as the difference between the actual y and the ideal value, for the mixtures Kr + C 2 H 6 , Kr + C2H4, and C2H4 + C 2 H 6 , respectively, is shown as a function of composition. The solid curves represent the predictions with fitted {'s and the error bars enclose the experimental results.* The agreement between theory and experiment for the hydrocarbon mixture is excellent. For the remaining systems the agreement is poorer, although the asymmetry of the yEcurve for Kr + C2H4 is well predicted by MFT. A small reduction in the values of E, however, will improve significantly the agreement between theory and experiment since the excess surface tension is extremely sensitive to this parameter.
4136 The Journal of Physical Chemistry, Vol. 93, No. 10, 1989
Almeida and Telo da Gama 0.512 I
0
I
YYmN m-l
-
p*
1.0
0.307
0.205
- 2.0
I
I
I
0.102
- 3.0
0.0
-12.5
-7.5
-2.5
2.5
7.5
12.1
r*=z/u
1 .o
0.5
0.0
XKI
Figure 10. Excess surface tension versus liquid composition for krypton + ethene at T = 116 K. MFT results with 5 = 0.939 (-) and experimental points (*). I
I
I
I
I
I
I
I
VE/mNm-’
0.377
I
0.075
0.0
- 0.4 I
I
I
l
1 0.5
I
I
1 .o xc2
H4
TABLE 111: Theoretical and Experimental Values for the Excess Surface Tension at the Equimolar Concentration and at Two Different Temperatures of the Mixtures: Krypton Ethane, Krypton + Ethene, and Ethene + Ethane (rEin m N m”)
+
+
ClH4
C,H4
+ C2H6
.
I -2.5
I
-12.5
-7.5
O
Figure 11. Excess surface tension versus liquid composition for ethene + ethane at T = 117 K. MFT results with 5 = 0.984 (-) and experimental points (*).
mixture Kr C2H6
1
1
2.5
7.5
12.5
2+=2/l7
0.0
f
I
T
0.151
Kr
I
T/K
YEhiFI
YEcxpt
116 124 116 124 1 I7 135
-0.94 -0.90 -1.77 -1.70 -0.39 -0.39
-1.94 -2.83 -2.4 -3.3 4.28 -0.31
As a matter of fact excellent agreement is found for the mixture Kr + C2H6 with E = 0.935 and for Kr + C,H4 with E = 0.915. These values of 5 yield significant enrichments of the surface layer with the component of lower surface tension. The influence of temperature on the excess surface tension is also studied. In Table I11 we compare the theoretical yE for the equimolar concentration with the experimental values determined at two different temperatures. M F T predicts a decrease (in absolute value) of the excess surface tension with increasing temperature for the systems under study with the exception of the hydrocarbon mixture where the excess tension remains constant in this small temperature range. The same dependence was obtained with the square gradient theory applied by Telo da Gama et a1.I3 to the mixture argon +
15)
Figure 12. Density profiles for the vapor-liquid interface of krypton ( I ) + ethane (2) mixture at T = 116 K: (a) xKr= 0.1; (b) xKr= 0.5
krypton. The experimental results, however, do not consistently yield yE’sthat decrease with increasing T. Only a few of these measurements of yE, however, are unambiguous as far as the temperature dependence of yE is concerned. Among those, the results of Nadler et aL6 for Ar Kr indicate an increase in (the absolute value of) yE as T increases while the measurements of H i g g i n ~for ~ ~methane perfluoromethane indicate a decrease in yE when the temperature increases. In fact, the excess tension for simple mixtures is only a few percent of the total surface tension and this is of the order of the experimental errors (0.1 mN m-l), preventing definite conclusions to be drawn in most cases. In Figures 12-14 we plot the results for the equilibrium density profiles of the mixtures Kr + C2H6, Kr + C2H4,and C2H4+ C2H6 a t two bulk liquid compositions, xI = 0.1 and x1 = 0.5, and a t the temperature T = 116 K for the systems with krypton and T = 117 K for the ethene ethane mixture. The krypton profiles are nonmonotonic for the two systems at both compositions. They exhibit a maximum at the interface, yielding a large positive adsorption of krypton (the component with the lower surface tension) at the surface. By contrast, for the mixture ethene ethane, the profiles of both components are essentially monotonic in agreement with what we would have anticipated on the basis of the similarity of their surface tensions. In fact for low compositions of ethene the C2H4 profile still has a small “bump” at the interface which disappears at larger compositions.
+
+
+
-+
(29) Higgins, R. A. MSc. Thesis, University of Sheffield, U.K., 1985.
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4137
Surface Tension of Simple Mixtures 0.584
0.714
0.451
0.571
\
P*
P* 0.429
0.338
0.288
0.225
I 0.143
0.113
1
0.0
,
I
I
-7.5
-2.5
I
I
2.5
7.5
I
I
I
1
z*=z/(T
, ., 0.390
0.413
0.312
0.330
P*
P"
0.234
0.248
0.157
0.185
0.078
0.082
0.0
.5
1
a*=zlo I
(b)
!1
I
Figure 13. Density profiles for the vapor-liquid interface of krypton ( I ) + ethene (2) mixture at T = 116 K: (a) xKl = 0.1; (b) xKr = 0.5.
Figure 14. Density profiles for the vapor-liquid interface of ethene (1) + ethane (2) mixture at T = 117 K: (a) X C = ~0.1;~(b)~xCZH,= 0.5.
The Gibbs adsorption equation (6) for a binary mixture may be written in the following form
experiment depends, crucially, on the choice of the molecular parameters. Other authors9 predicted good results for mixtures of Ar + Kr using a slightly different version of this theory and parameters errand uti which were fitted to the phase diagrams of the pure components a t each T. By contrast we used temperature-independent values for t and u (derived from the critical constants) which allowed the simultaneous prediction of the density and surface tension for the pure components. In addition, agreement between the predicted and experimental excess surface tensions for the mixtures was found to depend strongly on the cross-interaction parameter, ell. This effect is related to the enrichment of the surface layer in the component with the lower surface tension. The lower the value of til (Le., &),the larger the absolute value of the excess surface tension and the richer the surface layer in that component. We found that reasonable values of yEcould be predicted with values of $, fitted to bulk properties (at one temperature and composition). The dependence yE on the temperature, on the other hand, is not fully understood. The theory predicts that lyEldecreases as T increases for all the systems investigated. (Thurtell et al? found that lyEl increases as T increases for mixtures of Ar + Kr but the effect is due to the temperature dependence of their potential parameters.) Real mixtures, however, seem to exhibit both types of behavior-there are experimental reports of lyElincreasing with temperature (Kr + CzH6, Kr + C2H4, C2H4 + C2H6,Ar + Kr) while for other systems the opposite behavior is found (CH4 + CF4). In some cases yEis of the same order of magnitude as the experimental error, which makes the experimental situation anything but clear.
(s),
= -r2,1
(9)
where rZql is the adsorption of component 2 relative to component 1; Le., by choosing the dividing surface so that the adsorption of is related to rl and r2through30 component 1 is zero, r2,1
rz,l= r2- rl-P2" - Pzl PI"
- PI'
We have calculated the derivative of the tension y with respect to the chemical potential w2 from
(a),(z),/(a), =
(11)
and checked that the numerical results do indeed satisfy eq 9 provided that the density profiles are obtained with sufficient accuracy (1 part in IO6). 5. Conclusions The MFT investigated in this paper provides a useful method to predict surface tensions of pure fluids and mixtures with quasi-spherical molecules. The agreement between theory and (30) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: Bristol, U.K., 1966.
4138
J . Phys. Chem. 1989, 93, 4138-4142
Finally, let us make some remarks concerning the theory. First of all it is clear that the choice of interactions is very important. The Lennard-Jones potentials used in this work are only approximate models for interactions between spherical molecules. Even in this case the effects of three-body forces seem to be rather important in determining the interfacial proper tie^.^' Putting this aside, there are certainly other anisotropic pair interactions that could be important. For the systems considered in this paper, (31) Miyazaki, K . ; Baker, J. A,; Pound, J. M. J . Chem. Phys. 3364.
1976, 64,
quadrupole-quadrupole interactions could be rather important since both ethane and ethene have large quadrupole moments. We are now starting to investigate these effects. Last but not least, mean-field theory is based on several (more or less) drastic approximations such as the neglect of capillary waves. The effect of the latter on the surface tension is expected to be negligible but the effects of shorter ranged correlations on y could be significant. The inclusion of such correlations in the theory would however make it extremely complicated, and for practical purposes the simplest microscopic theory is probably the most useful.
Characterization of the Porous Structure of Agglomerated Microspheres by Spectroscopy
NMR
Wm. Curtis Conner,* Edward L. Weist, Department of Chemical Engineering, University of Massachusetts, Amherst. Massachusetts 01003
Taro Ito,' and Jacques Fraissard Laboratoire de Chimie des Surfaces, associP au CNRS. UA 70, UniversitZ Pierre et Marie Curie, 75252 Paris Cedex OS, France (Received: July 1 1 , 1988; In Final Form: December 12, 1988)
The pore structure created by the compression of a series of nonporous silica spheres was characterized by using xenon as a probe molecule, in the same manner that many workers use xenon-129 as probe for zeolite cages and channels. Although the materials used in this study had pores an order of magnitude larger than zeolites, the chemical shift measured by nuclear magnetic resonance spectroscopy of the adsorbed xenon atoms was as high as that found for zeolites. The results were analyzed by comparison with the void sizes as determined by mercury porosimetry. The chemical shift was interpreted as the result of a fast exchange between adsorbed xenon atoms and free xenon atoms in the pores of the adsorbate.
Introduction The use of xenon as a probe molecule for studying the pore structure of zeolites has become a very popular method since the introduction of the technique by Fraissard and co-workers.'q2 The chemical shift from the nuclear magnetic resonance spectra of xenon-129 adsorbed in zeolites is indicative of the size of the cages and channels within the zeolite. The observed chemical shift is dependent upon a number of factors: collisions of xenon atoms with each other or the walls, the presence of cations or metal atoms within the supercages, etc. These interactions of the xenon atom are transmitted directly to the electron environment around the xenon nucleus and thus affect the chemical shift as determined by NMR spectroscopy. Recently, xenon has become accepted as a general probe molecule for an extended series of microporous materials.2-8 Until now, xenon has been used to probe the structure of solids such as zeolites2-* and organic polymer^^^'^ exclusively. In this work, we studied the chemical shift of xenon adsorbed in nonzeolitic porous materials. For zeolites, the measured chemical shift of adsorbed xenon increases with the amount of xenon in the framework due to the increase in xenon-xenon interactions a t the higher pressures.'s2 For example, the chemical shift of xenon adsorbed on sodium-" (Si/AI = 1.35) zeolite increased from 70 to 105 ppm as the number of adsorbed xenon atoms increased from 10" to lo2' atoms per gram of zeolite.' The value of the chemical shift extrapolated to zero xenon concentration is characteristic of the size of the channels or cages in the zeolite. As the size of the voids decreases, the chemical shift becomes greater due to the increase of the interactions of the adsorbed xenon with the walls of the pores. Extrapolation to zero xenon concentration of the chemical shift Present address: Research Institute for Catalysis, Hokkaido University, Sapporo 060, Japan.
0022-3654/89/2093-4138$0 1.50/0
of xenon in faujasites, which have 1.3-nm-diameter cages, is around 60 ppm from that of xenon gas at zero pressure. Spectra of zeolites with varying pore dimensions have been studied, and in all cases larger pores resulted in a decrease in the chemical shifts. The pores of the materials used in this study were an order of magnitude greater in size than the largest studied in zeolites, Le., 15-30 nm in pore radius. If the chemical shift for Xe within these pore networks were based on an extrapolation to these dimensions of the results for zeolite studies, the chemical shifts were expected to be less than 10 ppm. This presumes that the chemical shifts depended on the same Xe-wall and Xe-Xe interactions found for zeolites.
Experimental Section The pore structure to be studied was created by compression up to 17.5 tons/im2 of Degussa Aerosils 200 and 380, fumed silica spheres of 12- and 7-nm diameters. Aerosils are in the form of nonporous spheres, the surface of which contains on average three silanol groups (Si-OH) per 10 nm2." The compression of the Ito, T.; Fraissard, J. J. Chem. Phys. 1982, 76, 5225. Fraissard, J . ; Ito, T. Zeolites, in press. Davis, M. E.; Saldarriaga, C.; Montes, C.; Hanson, B. J . Phys. Chem. 1988, 92, 2557. (4) Shoemaker, R.; Apple, T . J . Phys. Chem. 1987, 91, 4024. (5) Ryoo, R.; Liu, S.-B.;de Menorval, L. C.; Takegoshi, K.; Chmelka, B.; Trecoske, M.; Pines, A. J . Phys. Chem. 1987, 91, 6575. (6) Johnson, D. W.; Griffiths, L. Zeolites 1987, 7, 484. (7) Scharpf, E. W.; Crecely, R. W.; Gates, B. C.; Dybowski, C. J . Phys. Chem. 1986, 90, 9. (8) Ripmeester, J. J . Magn. Reson. 1984, 56, 247. (9) Stengle, T.R.; Williamson, K. L. Macromolecules 1987, 20, 1428. (10) Sefcik, M. D.; Scheafer, J.; Desa, J. A. E.; Yelon, W. B. Polym. Prepr. ( A m . Chem Soc.. Diu. Polym. Chem.) 1983, 24, 85.
1989 American Chemical Society
