J. Phys. Chem. 1995, 99, 2408-241 1
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Density Profiles of Chemically Reacting Simple Fluids near Impenetrable Surfaces Orest Pizio+ Instituto de Quimica, UNAM Coyoachn, DF 04510 Mkxico Douglas Henderson* and Stefan Sokdowski’ Departamento de Fisica, Universidad Autbnoma Metropolitandktapalapa, Apartado Postal 55-534, 09340 DF Mkxico Received: October 4, 1994@
The density profiles of the particles of a chemically reacting hard sphere fluid near a hard wall are studied. The theory pioneered by Cummings and Stell is utilized to provide the description of a bulk fluid. Different models of the particle-wall interactions are assumed. It is shown that association in the bulk fluid produces different effects on the density profiles at low and high bulk fluid packing fractions. The particle-wall and particle-particle associating interactions are the competitive factors which determine the character of density profiles.
1. Introduction At present, the investigation of association phenomena and chemical reactions in fluids is one of the challenging areas of the liquid state theory. Different approaches have been developed to deal with the problem. One popular method is that of Wertheim,’-4 which has the fugacity expansion and multidensity formalism as key ingredients. Another method, closer to the standard liquid state theory, using density expansions, was constructed by Cummings and Stell.5-s One of the issues which has not yet been studied is the following. Most of the previous studies were focused on the bulk fluids, while the investigation of inhomogeneous chemically reacting, or “associating”, fluids is at its initial state. A lack of symmetry, compared with bulk fluids, and the need to incorporate strong finite-range interactions, which leads to the formation of bonds, make the problem complicated. It is evident that the presence of associative interactions in the bulk fluid will also influence its surface properties. Moreover, the surface properties (e.g., density profiles) will influence the association rate in the bulk. Thus, the problem, if stated rigorously, has to provide the mutually consistent influence of the surface properties and chemical association effects. There have been very few attempts to study inhomogeneous associative fluids. To solve the problem, different simplifications have been used. In order to introduce a certain degree of decoupling between the “inhomogeneity” and “reactivity”, the system of tangent hard spheres in a slitlike pore was s t ~ d i e d . ~ The simplifying factor in this case is the location of the bonding potential on the surface of hard spheres. The inhomogeneous fluid was considered at the singlet in order to eliminate the inhomogeneous pair correlation function. In this paper we shall consider a hard sphere chemically reacting fluid near an impenetrable surface. The treatment of the bulk fluid is given within the theory developed by Cummings and Ste11.5.6 The profiles will be studied within these approximations, which use the bulk direct correlation function
‘Permanent address: Institute for Physics of Condensed Matter, National Academy of Sciences of the Ukraine, 29001 1 Lviv. Ukraine. Permanent address: Computer Laboratory. Faculty of Chemistry, MCS University, 2003 1 Lublin. Poland. @Abstractpublished in Advance ACS Abstracts, January 15. 1995. 0022-365419512099-2408$09.00/0
(dcf). Before proceeding, let us discuss a few important issues, which do not depend on the type of theory which is applied. The model of inhomogeneous associative fluids is determined by the potential of the extemal field (particle-wall interactions) and the interparticle pair interactions. The form of the associative interactions determines what types of multimers are formed, i.e. the mechanism of steric saturation in the bulk fluid. While considering the reactive systems in pores, or in the presence of walls, it becomes evident that the geometrical restrictions prohibit some configurations of the particles; that is, the oneparticle potential participates in the mechanism of steric saturation. It does not influence, however, which types of multimers are formed. Another important comment is as follows. Stell and Z ~ O U ’ ~ - ’ ~ and Rasaiah and ZhuI6 developed approximations for the association parameter 1,which satisfy the law of mass action. It becomes possible, then, to determine the rate of chemical reaction K without solving the model with associative interactions. The only requirement is that the reference fluid, without associative interactions, must be described adequately. However, if one wants to investigate other properties besides K , it is necessary to possess the solution, either analytical or numerical, of the model. In the following section we present the procedure applied to investigate the profiles and discuss them in detail. Then the results are given and examined.
2. Theory Consider the model of chemically reacting hard spheres already ~ t u d i e d .We ~ would like to avoid unnecessary repetition and present only important details. The hard spheres of species A and B react A B +=AB with the formation of dimers AB. We denote the species A and B by the subscripts “1” and “2”. Each species has the same diameter. However, their interactions with the wall may be different. The bonding distance L is chosen to be 0.5, to provide dimerization only and simplify the calculations. Other values of L, for L < 0.5, can be studied in a similar manner. Because we have restricted our attention to dimers only, the case L > 0.5 is excluded. Without loss of generality, we use 0 (the hard sphere diameter) equal to unity. To determine the profiles, we need the direct correlation functions (dcf‘s) cY(r),which are given as
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J. Phys. Chem., Vol. 99, No. 8, 1995 2409
Density Profiles of Chemically Reacting Fluids 1
a 0.8
where qij(r) are Baxter factor functions which are known5 for the model under study. The factor functions qu(r) satisfy a differential-difference equation which contains a 6 function, reflecting the intramolecular coupling. The analytical solution5 utilizes the PercusYevick (PY) closure for dcf‘s. It follows then from eq 1 that the dcf‘s are
0.6 0.4
0.2 0
0
0.1
0.2
0.3
0.4
0.5 70
Figure 1. Dependence of the degree of association a on the packing fraction for different “isotherm” of t,given in the figure.
particle tends to zero while its size tends to infinity. The density profiles are then given by where TOis the packing fraction of the bulk system, TO= n(@lo ~20)/6,em is the bulk density of the component k, v = r$,L2/ 2 , v = d 2 L , a is the degree of association, and q’uJr) is the continuous part of the derivative of the factor function, which does not contain the 6 function and is known.5 The accuracy of the PY approximation for this model has been d i s c ~ s s e d . ~Of particular importance is the limit of complete association, i.e. the diatomic limit in the case L < 1/2. It is known that the dcf s for diatomic fluid have an exact long-range asymptotic form, which is not included in the PY approximation. For the states of incomplete association, this can be neg1e~ted.I~ The dcf‘s were calculated numerically, using qu(r) with an additional subroutine to yield the coefficients of qu(r). The solution of the model within PY approximation5 provides an adequate description of chemically reactive hard spheres in the bulk case. One crucial issue remains to be resolved, because the dcf‘s depend on the degree of association a. Stell and ZhouI3 proposed the zeroth-order approximation for determination of the association parameter 1,
+
where z is the sticky analogue of the temperature5 and is the cavity function of unassociated hard spheres. For L = 1 this approximation is very accurate, while for L < 1 it overestimates the degree of a~sociation.’~The parameter a, which follows from eq 3, has the form
where
and the definition of the profiles is standard, Qjk) = [hj(z) + ‘lei0 = gi(Z)ejo
Here c&) are the dcf‘s of the bulk fluid. The functions C&t) have the following form:
where the first term is the continuous part of C&t) and the second arises because of the intramolecular correlations of the bulk fluid. The function 0 is the step function. The HNCl and PY 1 approximations needed here are obtained by using the HNC or PY closures to eq 5. If the particle-wall correlations are described within the HNCl approximation, then the profiles are given by Qi(ZYQj0= exp[-Pujw(z>
2n
+
C
@M,
k
Figure 1 shows the dependence of the degree of association on the packing fraction, evaluated according to eq 4. It is evident that, in order to make quantitative conclusions about the validity of this approximation, one must possess the simulation results for the model under consideration. Recently, a simulational study of the chemically reacting fluids of this A simulation for class was presented for bulk inhomogeneous chemically reacting fluids would help to clarify this problem in future work. Our following treatment is based on 1,which is obtained from eq 4. In order to determine the cavity function of hard spheres, we applied the parametrized expression of Grundke and Henderson,21*22 similarly to Zhou and Stell.I4 The consideration of the inhomogeneity will be simplified and given within the singlet The planar, impenetrable wall is formed as a result of a limiting procedure in which the concentration of the wall
(7)
J-zdt
Ckj(z,t)kk(t) - l11 (9)
where uiw(z) is the potential of particle-wall interactions. Linearization of the exponent in eq 9 leads to the PY1 approximation for the profiles. We report results for both the HNCl and PY1 approximations. In both cases, the bulk functions Ckj(Z,t) are calculated in the PY approximation.
3. Results and Discussion Let us consider first the case of a hard wall.
We have then gl(z) = e&). The profiles presented in Figure 2 are calculated within the PY1 approximation. At lower densities of the bulk fluid, e.g. for ~0 = 0.1, the degree of association has a small influence on the contact values of density profiles. The difference between the profiles is manifested in the cusp of ei(z) at z = 0.5 for the highly dimerized fluid at t = 0.01 compared with the weakly dimerized fluid at t = 1. We can state that at low densities the diatomics have “enough
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1
1
i
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/ I
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0.5 0
0.5
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2
0
2.5
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Figure 2. Density profiles of particles near a hard wall calculated in the PY 1 approximation. The solid lines and dashed lines were evaluated for r = 0.01 and r = 1, respectively. The labels a and b correspond to 70 = 0.1 and 70 = 0.35, respectively. 5 ,
1
I
I 1
4 .
15 3
0
2
Figure 4. Adsorption isotherms for the particles in contact with a hard wall. The solid and dashed lines correspond to HNCl and PYl approximations, respectively. The label a denotes r = 0.01, and the label b denotes t = 1.
3
h
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Figure 3. Comparison of PY 1 and HNCl density profiles for the case of particles in contact with a hard wall. All the curves were evaluated for 7 0 = 0.35. The solid and dashed lines correspond to the PY 1 and HNCl approximations, respectively. For each set of solid and dashed lines, the line with higher contact value was obtained for t = 1, whereas the line with lower contact value was obtained for r = 0.01.
space” to occupy the surface layer in both cases. At higher densities (TO = 0.35), the contact value of the profile decreases with increasing degree of association. In addition, the second maximum of the profile shifts to larger distances from the wall, with increasing degree of association. The effect of the intramolecular structure of the highly dimerized fluid is more pronounced in the profiles, compared with the weakly dimerized case. In order to study the effect of the closure, we compare the HNCl data with the corresponding PY1 results in Figure 3. Both sets of the data differ in the vicinity of the wall, while at larger distances the profiles become similar. The difference between the HNCl and PY1 results increases at higher packing fractions. The effect of intramolecular coupling on the profiles is reflected in a similar way at lower densities, while at high packing fractions the HNCl profile has a more pronounced cusp compared with the PY 1 profile. It is known that the integrated characteristics, such as the adsorption isotherm,
are less influenced by the inaccuracies of the approximations than are the density profiles. Both approximations (HNC1 and PY1) lead to qualitatively similar behavior of the adsorption isotherms (Figure 4). The adsorption at the wall of the highly dimerized fluid increases more slowly as a function of bulk density, compared to the weakly dimerized fluid. At low densities, r, is larger for the highly dimerized fluid, while at higher densities, the effect is opposite. The density at which both curves cross, ecross, is characteristic for the model under study ( L = 0.5); for other bonding distances
- 0
0
0.5
1
1.5
Figure 5. Density profiles @ l ( z )(part a) and @ 2 ( z ) (part b) near a surface with attraction for particles “1”. The solid and dashed lines were evaluated for t = 0.01 and r = 1, respectively. The labels a and b correspond to ~0 = 0.1 and 170 = 0.35, respectively. All the curves were calculated using the HNC 1 approximation.
we would expect other vales of ecrosr. Both approximations (HNC1 and PY1) lead to similar values of ecross. It is worth noting that Holovko et a1.I0 came to similar conclusions for the contact value of the profile for the Wertheim model of dimerized hard spheres. However, their adsorption isotherms did not cross. This is probably due to the difference between their model and the one under study. Modification of the particle-wall interaction leads to the interesting changes of the density profiles. We shall assume that there is an attractive interaction between the wall and the particles “1” in the form
whereas the potential for the particles “2” is still given by eq 10. All our calculations have been carried out assuming that zo = (J = 1 and that PG = 3, i.e. for a rather weakly adsorbing wall. In this way we model “competition” between the particle-wall interactions and “reactivity”. The profiles are calculated within the HNC 1 approximation.
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0.4 .
a
‘
-0.2 0
b
I 0.2
0.4
0.6
PO
0.8
Figure 6. Adsorption isotherms rl (solid lines) and r2 (dashed lines) for a partially attractive adsorbing surface. The labels a and b denote z = 0.01 and t = 1, respectively. All the curves were calculated using the HNCl approximation.
The presence of attraction in eq 12 makes the contact values of el(z) larger; see Figure 5. As was seen previously in Figures 2 and 3, an increase in the degree of association decreases the contact values el(0). In this scale the effects of intramolecular coupling are not well-pronounced; the packing effects and ulw(z) determine the behavior of the profile. However, due to the effects of dimerization, the profiles e&) are strongly effected by the strength of intramolecular coupling. For a weakly dimerized fluid, the number of particles “2” in the surface layer is very small; see the curve evaluated for t = 1 and 70 = 0.1. For the fluid at a higher level of dimerization, z = 0.01 and 70 = 0.1, this number is mainly determined by the chemical reaction effect. At high densities, the packing effect becomes more important, but the influence of the chemical reaction is clearly seen. The latter effect influences filling of the surface layer, as well as the position of the second maximum of Q ~ ( z ) . Adsorption isotherms for this model are shown in Figure 6. Due to the chemical reaction effect, the values of r2 for a highly dimerized fluid are positive; for a weakly dimerized fluid we observe negative adsorption. As was seen previously (Le. in the absence of attractions), the values of rl become smaller with an increasing degree of association. The total effect r = rl r2 depends on the strength of attraction ( E ) and its range, as well as on the rate of chemical reaction (or on the parameter
+
t).
4. Conclusions The density profiles and adsorption isotherms exhibit some interesting effects. In the absence of an attractive external field,
the density profiles at contact and the resultant adsorption isotherms, are decreased for a highly associated fluid at high densities. We presume that this is due to the reduction in the number of particles and the lower “effective” density. This lowering of the “effective” density seems not to be so important at low densities and the resultant adsorption isotherm is increased for a highly associated fluid, resulting from the larger 0.5 (which is observed at high values of the profile for z densities also). As a result, there is a crossover in the values of the adsorption isotherm as a function of density. When one of the species is attracted to the wall, the adsorption of this species is increased, as one would expect. Again, the density profile of this species at contact is lowered for a highly associated fluid. The adsorption of the second species (with no direct attraction to the wall) is also strongly affected. For weakly associated fluids, the adsorption isotherm is negative, but becomes positive for a strongly associated fluid even without any direct attraction.
Acknowledgment. The authors wish to thank the CONACYT of MCxico (Grant No. 4186-E9405 and No. 3494-t and el Fondo para CBtedras Patrimoniales de Excelencia) for financial support. References and Notes (1) Wertheim, M. S. J . Stat. Phys. 1984, 35, 19. (2) Wertheim, M. S. J. Stat. Phys. 1984, 35, 35. (3) Wertheim, M. S. J . Stat. Phys. 1986, 42, 459. (4) Wertheim, M. S. J . Stat. Phys. 1986, 42, 417. ( 5 ) Cummings, P. T.; Stell, G. Mol. Phys. 1984, 51, 253. (6) Cummings, P. T.; Stell, G. Mol. Phys. 1985, 55, 33. (7) Cummings, P. T.; Stell, G. Mol. Phys. 1987, 60, 1315. (8) Lee, S. H.; Cummings, P. T.; Stell, G. Mol. Phys. 1987, 62, 65. (9) Kierlik, E.; Rosinberg, M. L. J. Chem. Phys. 1994, 100, 1716. (10) Holovko, M. F.; Vakarin, E. V. Preprint Inst. Cond. Matt. Phys.; Lviv. Ukraine. 1993. No. 93-15Y. Henderson, b. In Fundamentals of Inhomogeneous Fluids; HenD., Ed.; Marcel Dekker: New York, 1992; Chapter 4. Henderson, D.; Abraham, F. F.; Barker, J. A. Mol. Phys. 1976,31, Stell, G.; Zhou, Y. J. Chem. Phys. 1989, 91, 3618. Zhou, Y.; Stell, G. J. Chem. Phys. 1992, 96, 1504. Zhou, Y.; Stell, G. J. Chem. Phys. 1992, 96, 1507. Rasaiah, J. C.; Zhu, J. J. Chem. Phys. 1989, 92, 7554. Zhou, Y.; Stell, G. J. Chem. Phys. 1993, 98, 5777. Groot, R. D. J. Chem. Phys. 1992, 97, 3537. Smith, W. R.; Triska, B. J . Chem. Phys. 1994, 100, 3019. Busch. N. A,: Wertheim. M. S.: Chiew. Y. C.: Yarmush. M. L. J. Chem. Phys. 1994, 101, 3147. (21) Grundke, E. W., Henderson, D. Mol. Phys. 1972, 24, 269. (22) Henderson, D.; Grundke, E. W. J . Chem. Phys. 1975, 63, 601. JP942686L
