Langmuir 1995,11, 2121-2124
2121
Computer Simulation of Two-Phase Equilibrium of Methane in Narrow-Slit Wetted Pores E. M. Piotrovskaya" Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, Pennsylvania 16802
E . N . Brodskaya Department of Chemistry, St. Petersburg State University, Petrodvoretz, St. Petersburg, 198904, Russia Received November 8, 1994. In Final Form: February 27, 1995@ Grand canonicalMonte Carlo simulationswere carried out for the estimation of the two-phaseequilibrium
of methane in narrow-slit pores with wetted walls. The pore width varied from 4 up to 12 molecular diameters, and the systems were investigated at 111.1 K. It was shown that for the graphite-like pores the vapor-liquid coexistence at the concave meniscus existed for pore widths not less than 10 molecular diameters. For the weaker adsorbate-adsorbent potential, vapor-liquid coexistence was found for the smaller values of pore widths. The dependence of the surface tension of the concave meniscus on its
curvature was estimated.
The investigations of strongly nonuniform systems including small adsorption systems are of great scientific interest. At present these investigations are being approached both theoretically1 and by computer simulation methods.2-6 Direct experimental measurements in small systems are extremely difficult because these systems as a rule are nonuniform everywhere. One of the serious problems arising during such investigations is connected with the changes of the two-phase equilibrium coexistence with changes in system sizes. We have discussed earlier7 the dependence of the conditions of liquid-vapor coexistence of a Lennard- Jones fluid in thin pores of different sizes and shapes with absolutely hard walls. The investigations were done by the grand canonical Monte Carlo (MC)method. I t allowed us to calculate Laplace pressure in such pores and to estimate the dependence ofthe surface tension of a convex meniscus, y , on its curvature, R,. However, from the practical point of view, the adsorbing systems, that is, fluid systems in thin pores with wetted walls, are of particular interest. The concave meniscus is being formed in these pores at the liquid-vapor equilibrium, and the dependence of the surface tension on the meniscus curvature has to differ in character from the same y(R,) dependence for the convex meniscus. It is possible to say that if the convex meniscus is analogous to a drop, then the concave meniscus is analogous to a bubble. Among the numerous works on computer simulation of the adsorption systems we shall mention only t h o ~ e in ~ which - ~ the capillary condensation of LennardJones fluids in pores was studied by the grand canonical MC method. A principal goal of the present investigation is to find out the dependence of the conditions of the two-phase
* Permanent address: Department of Chemistry, St. Petersburg State University, Petrodvoretz, St. Petersburg, 198904,Russia. Abstract published in Advance ACS Abstracts, May 15,1995. (1)Ball, P. C.; Evans, R. Mol. Phys. 1988,63,195. (2) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987,62,215. (3)Heffelfinger, G. S.;van Swol, F.;Gubbins, K. E. Mol. Phys. 1987, @
61,1381. (4)Jiang, S.;Rhykerd, S. L.; Gubbins,K. E. Mol. Phys. 1993,79,373. (5) Sokolovski, S.;Fisher, J. Phys. Reu. A 1990,41,6866. (6)Sokolovski, S. Phys. Rev. A 1991,44,3732. (7)Brodskaya, E.N.;Piotrovskaya, E. M. Langmuir 1994,10,1837.
liquid-vapor equilibrium in the slit-adsorbing pores on the pore width. The adsorbate-adsorbent interaction parameters correspond approximately to the methanegraphite system, as well a s to the model system for which methane was also taken as an adsorbate, while the adsorbate-adsorbent interaction were twice as weak. The systems of this type are considered to be the most simple for the description of adsorbate-adsorbate and adsorbateadsorbent interactions. At the same time the methanegraphite system appears to be very important from the practical point of view, because it can be used to model, in the first approximation, the behavior of methane in coals.
Simulations The method used for these investigations is the grand canonical MC method, which allows one to investigate open systems at the given volume (V), temperature (T), and chemical potential of the component (p) (here this component is methane). It means that a t a given value of the chemical potential it is possible to obtain only one phase in the system. In our case it is either liquid or vapor. To estimate the conditions oftwo-phase equilibria, that is, to obtain the value of the chemical potential for we have used the liquid-vapor coexistence point (peoex), the method proposed in ref 2 and used by us earlier.7 It is connected with the separate calculations of the grand thermodynamic potential for liquid and gas phases over a wide range of chemical potentials. The dependence of on the chemical the grand thermodynamic potential (9) potential p and temperature T may be defined a s follows
where N is the number of particles, U is the total energy of the system, and A is the surface area. Using these equations, 9 can be derived by integrating eq 1 along the chemical potential between two arbitrary points of the adsorption isotherm and along the temperature (according to eq 2) between two adsorption isotherms at a given chemical potential. At a very low pressure, where one
0743-746319512411-2121$09.00/0 0 1995 American Chemical Society
Piotrovskaya and Brodskaya
2122 Langmuir, Vol. 11, No. 6, 1995 Table 1. Parameters of the Potentials CHd-CHd
system I1
CHd-graphite
ECH4/kIIE, K
~ C H nm ~ ,
148.1
0.373
EcH4-dk,
K
1158.0
U C C H ~ - Cnm ,
mi-&,K
0.224
578.0
UCHr-S,
nm
0.224
can use the approximation of an ideal gas, we have /Aid@)
= -KTWd(,)
(3)
To be able to deal with eq 3 it is necessary to calculate the adsorption isotherm a t a supercritical temperature, before calculating the adsorption isotherms a t the temperature of the investigation. The dependence of S2@) for the liquid was estimated using eqs 1 and 2, and the assumption of the gas ideality was used to obtain the dependence of S2@) for the gas. The intersection point of the gas and liquid branches of Q@) gives the value of the chemical potential for the coexistence ofliquid-vaporpc,, in the pore a t the given temperature. As it was already pointed out above, the present work deals with the system of methane in thin slit pores of graphite, as well as with the system where the methanewall interaction is one-half that of the methane-graphite interaction. Let us call this pore system 11. The methanemethane interactions are described by a truncated Lennard-Jones potential [@(r)](the cutoff distance is rc,@(r) = 0 a t r > r,)
@(r)= 4€[(u/r)12-
(4)
where E and u are the parameters of the potential, and r is the intermolecular distance. In our calculations we consider the adsorbent to be a continuum with a constant density; thus the adsorbateadsorbent interactions are described by a (9-3) potential @AS(rAS)
= (3fi/2)EAS[(aAS/rAS)9 - (uAS/rAS)31 ( 5 )
where EAS and UAS are the parameters of the adsorbateadsorbent potential. The values of the potential parameters (eqs 4 and 5) for both systems are given in Table 1. All calculations in this work were held in the reduced units, so that the parameters E C H , and UCH, of the potential given by eq 4 were taken equal to unity. It means that the results can be used for any Lennard-Jones fluid adsorbed in a pore for which the depth of the adsorption mimimum in @AS is approximately either 8 or 4 times deeper than the depth of the potential well for m u . The slit pores we deal with in this work are formed by two planar semiinfinite solid surfaces parallel to the xy plane. The sizes of the basic MC cells in the x and y directions were equal to I, = ly = 4.4,6.6,8.8,13.2, or 17.6 (in units of UCHJ. Periodic boundary conditions (PBC) were used for the systems in these directions. The slit spacings (pore widths) changed from H = 4.0 up to H = 12.0. The calculations were carried out a t the temperature T = 0.75 (for methane it is 111.1 K), the supercritical temperature was chosen as T = 1.2 (for methane it is 177.7 K). The length of the MC chains was about 6-106 configurations, and averaging was performed over the last 3.10-6 configurations. Two-Phase Equilibria The adsorption isothermsp(2\r)ofmethane in a graphitelike pore a t the temperatures T = 0.75 and 1.2 are given in Figure 1for the pore width H = 10.0 (I, = Zy = 6.6). The adsorption isotherms for system I1 are given in Figure 2 a t H = 10.0 (I, = ly = 6.6) for the same temperatures. Similar isotherms have been obtained for both systems
0 -17
_----/ -18
-16
-14
-13
.I2
-11
.lo
, -0
-8
P Figure 1. Adsorption isotherms of methane in a slit graphite pore of the width H = 10.0 at supercritical temperature, T = 1.2 (I), and the temperature of investigation, T = 0.75 (11).
I
roo4
60
1
0 -16
1
-16
-14
.13
-12
-11
-10
-0
-8
P
Figure 2. Adsorption isotherms of the system I1 for the pore of widthH = 10.0 at supercritical temperature, T = 1.2 (I),and the temperature of investigation, T = 0.75 (11).
for the pore widths H = 4.0, 6.0, 8.0, and 12.0. The supercritical isotherms a t T = 1.2 lie below the subcritical ones in the range of high densities. The similar behavior of these dependences was shown also in ref 2, where capillary condensation of a Lennard-Jones fluid in a cylindrical capillary with adsorbing walls has been investigated. A different situation was observed in the calculations of adsorption isotherms in slit and cylindrical capillaries with absolutely hard walls.' In this case, the supercritical isotherms lie above the isotherms for the lower temperature. As seen from Figures 1 and 2, the lower temperature isotherm has two branches, but unlike the situation for capillaries with absolutely hard walls, we almost do not observe the hysteresis loop for the system of methane in graphite pores a t T = 0.75. Similar results were obtained in ref 4, where it was shown that for the same system a wide hysteresis loop was observed a t a lower temperature of T = 0.4, but with a temperature increase, the loop became more narrow and practically disappeared a t T = 0.7. According to the method described above (eqs 1-3) for both systems under investigation, the chemical potentials for vapor-liquid coexistence (pcoe,)were calculated for different pore widths. The results are given in Table 2. As seen from the table, with the decrease in the graphite decreases pore width from H = 12.0to 6.0, the value of,ucoex frompcoe,= -10.32 (at H = 12.0) to -10.90 (at H = 6.0). The only deviation is seen in the results for H = 4.0; in this case we obtained ,uco,,= - 10.49, and this result will be discussed later. According to the results for system 11, we found that the values ofpUcoex change very slightly with
Two-Phase Equilibrium of Methane
Langmuir, Vol. 11, No. 6, 1995 2123
Table 2. The Results of Calculationsa H
Pcoex
12.0 10.0 8.0 6.0 4.0
-10.32 -10.46 -10.65 -10.90 -10.49
12.0 10.0 8.0 6.0 4.0 12.0 6.0
R. Y CH1-graphite 2.5 f 0.5 1.15 f 0.20 0.86 f 0.20 1.5 f 0.5
lPl 0.459 0.576
system I1 12.5 f 1.5 10.0 f 1.0 10.0 f 1.0
4t
-9.80 1.5 f 0.15 0.120 0.150 -9.88 -9.93 -9.95 -9.55 Lennard-Jones Fluid in Slit Unwetted Pores -9.40 5.0 f 0.5 1.47 f 0.15 0.30 0.86 f 0.15 0.43 -9.26 2.0 f 0.5
In order to obtain the data in real units it is necessary to multiply H a n d R,by 0.373nm, pcWxby 2.044.10-21J , y by 14.69 mN/m, and 1A.P by 39.38~10~ Pa. Then, for example, the first line of Table 2 is the following: H, 4.48 nm; ,y,, 2 1 . 1 ~ 1 0J-; ~Ra, ~ 0.93 f 0.19 nm; y 16.9 f 2.9 mN/m; lhpl 18.08 Pa.
the decrease in pore width, from -9.80 (at H = 12.0) to -9.95 (at H = 6.0). It is possible to say that for the pores with H = 10.0, 8.0, and 6.0,the value of pcoexis almost constant in the range -9.90 f0.05. Nevertheless for the pore H = 4.0 the value of p,,,, jumps to a value of -9.55. In order to check our results and to be sure that they do not depend on the sizes of the basic MC cell, extra calculations were carried out for the pores of widths H = 10.0, 6.0, and 4.0 with the basic MC cell size enlarged in thex and y directions. For H = 10.0 the calculations were held for I, = ly = 4.4,6.6,and 8.8; for H = 6.0, I, = ly = 6.6and 8.8; and for H = 4.0, I, = ly = 8.8, 13.2, and 17.6. It is shown that there is no dependence ofthe value ofpcoex on the size of the basic MC cell. In the case of adsorption systems, especially for strong adsorbate-adsorbent interactions characteristic of the systems under investigation, the structure of the adsorbed layers near solid walls has to play an important role for the capillary condensation phenomena. It was shown earliel.5,6that for strong adsorbate-adsorbent interactions there exist several (at least three) fixed layers near the solid walls, and the density of these layers is substantially higher than the density of a liquid state. In order to check these previous estimations, we made the calculations of the local density profiles for the systems under investigation. The most typical examples of such density profiles are given for the pore with H = 10.0 (1, = ly = 6.6)for the system of methane in a graphite pore (Figure 3) and system I1 (Figure 3). As seen from Figure 3 for the system of methane in the graphite pore, we obtain three well-defined adsorbed layers of high density near each graphite wall, and only in the middle of the pore do we get a density close to that of liquid methane at T = 0.75. Due to the fact that only liquid-like layers in the pore can form the concave meniscus, it is possible to say that the phenomenon of capillary condensation occurs in no other pores but those with H = 12.0 and 10.0, and for these pores concave menisci are formed. In the rest of the pores from H = 4.0 up t o 8.0 there exist only adsorbed layers. Besides, the overlapping of the adsorbed layers in the pore with H = 4.0 can explain the situation why the value ofpcoex for this pore does not follow the general dependence ofp,,, on H for other pores. For system I1we get a different situation. As it is seen from Figure 3b, there are only two adsorbed layers near the walls. Although the first layer still has a rather high density, the density of the second one is just a little higher than the density of the bulk liquid methane. Thus, one can suppose that for system
0 '
0
J
, ~ \'
I
I
4
5
.._4'
1
I
2
3
z Figure 3. Local density profile of methane in a slit graphite pore (I) and of system I1 (11)at the temperature T = 0.75.The pore width for both systems is H = 10.0. I1 the phenomenon of capillary condensation with the formation of the concave meniscus is possible for the pores of H = 12.0, 10.0, and 8.0.
Laplace Pressure and Surface Tension It is possible to estimate the Laplace pressure, that is, the pressure difference in the phases separated by the curved meniscus, for the pores in which menisci are formed. For the concave meniscus this value is negative. In the case of slit adsorption pores, one has to suppose the formation of a concave meniscus of a cylindrical form. It is possible to obtain the vapor pressure Ps from the calculated values of pcoexby assuming the ideality of the vapor phase (Table 2). The pressure Pa is the pressure of a hypothetical liquid phase a t the same temperature and chemical potential as the fluid in the pore. The calculation of Pa is done as in ref 7. We just integrated the equation of state of a uniform liquid dP = eadp, where ea is the liquid density. Then, assuming the liquid is incompressible, one obtains
where PI is the pressure of the uniform liquid phase a t T = 0.75 and the chemical potential p1. The calculations of P1 were done in ref 7 for a Lennard-Jones liquid according to the vinal equation of state. For the chosen chemical potential p1= -9.50, we obtained PI = 0.228 f 0.032 and ea= 0.838 f 0.004.The calculations o f P and PB allow us to estimate the Laplace pressure AP = Pa Ps. The values of IAPI for the systems under investigation are given in Table 2. Now it is possible to estimate the surface tension of the meniscus ( y ) for the systems of methane in graphite pores using the Laplace equation
AP=
yc,
(7)
where C, is the average curvature of the meniscus. For the concave cylindrical meniscus, C , is connected with its radius of curvature as follows: C , = -llR,. So,for the calculations of the surface tension y , we need to know the value of the radius of curvature radius R,. As mentioned above (concerning Figure 3), in the adsorption systems a thin layer near a wall, consisting of several monolayers, has a pronounced layer structure. Thus, for the system
Piotrovskaya and Brodskaya
2124 Langmuir, Vol. 11, No. 6, 1995
methane-graphite there exist a t least three monolayers near each wall (Figure 3), and a meniscus is formed in the rest of the volume on these layers for the case of total wetting. It is evident that for H = 10.0 the radius R, of this meniscus is not larger than 2.0 (Figure 3). We assume that the uncertainty in the estimations of R, is approximately 0.5, and then for H = 10.0 we get R, = 1.5 f 0.5 and for H = 12.0,R, = 2.5 f 0.5. These estimations ofR, are in good agreement with the results ofref 3, where the values of R, for a cylindrical pore radius R = 5.0 changed from 2.66 to 3.36 in the temperature range from 0.7 to 0.85. It is necessary to stress that the adsorption field in ref 3 was weaker than the one we used for the methane-graphite system. We have calculated the surface tension y s for the values of R, above according to eq 7 and they are given in Table 2. In the same table the results for the surface tension of the convex meniscus are also given. It is obvious that the surface tension decreases with the decrease in the radius of curvature of the concave meniscus from 2.5 to 1.5, but it is a little higher than the surface tension of the convex meniscus in the same range ofR, values. We can state that the values of y, practically coincidefor a concave meniscus withR, = 1.5 and a convex meniscus with R, = 2.0. For system 11,in which adsorbate-adsorbent interactions are half that of methane-graphite system, we assume that due to the decrease of the adsorption potential the system will be characterized by partial wetting and the meniscus which forms will be flatter. The form of the density profile in such systems (Figure 3b) also favors such a n assumption. In this case it is very difficult to use any approximations concerning the meniscus radius of curvature R,. That is why for system I1 we consider the reverse problem. We can assume on the basis of all the preceding calculations of the surface tension that we can estimate the surface tension of a flat surface as y = 1.5 f0.15. We can take this value of y as the highest possible limit for the surface tension of the meniscus. Then, according to eq 7, it is not difficult to estimate the radius of curvature R, for the pores of different widths (Table 2). The radii of curvature for the menisci in the case of partial wetting appeared to be much larger than in the case of almost total wetting (methane in graphite pores). So, according to the given data it is possible to conclude that for the convexmeniscus the dependence of the surface tension on the curvature becomes important when the
radii of curvature are less than five molecular diameters, while for the concave meniscus this dependence appears for the smaller radii of curvature. This correlates well with the general thermodynamical approach to the dependence of ys(Rs).8First of all, the Tolman asymptotic equation is to be valid for large values of R,
where 6 is the so-called Tolman’s length, which is equal to the distance between a n equimolecular surface and the surface of tension. The second result from the general thermodynamics is that y, 0 as R, 0.8 Taking into account different signs of C , for concave and convex menisci, these arguments bring one to the conclusion that the function y,(R,) for the concave meniscus has to lie higher than the similar function for the convex meniscus. And that isjust the result we obtained in our calculations. It is necessary to point out that the adsorption field has a strong influence on the position of the phase equilibrium coexistence point, and as expected, there is also a strong dependence of the radius of curvature of the meniscus on the adsorption field. Thus, decreasing the adsorption field by a factor of 2 brings about an approximate 5-fold increase in the radius of curvature. Certainly all our considerations about the values of the radius of curvature ofthe meniscus R, are based only on several assumptions. To calculate more precise R, values and to assess the approximations used by us to obtain the values of the radius of curvature of the meniscus, it is necessary to do special computer simulations of closed systems in a canonical ensemble. It will allow us to calculate the value of R, rather precisely in a direct computer experiment. These calculations a t present have been done only for the system in the pore with absolutely hard w a l k g Certainly, we are performing the calculations for systems with adsorbing walls with different values of adsorbate-adsorbent interactions.
-
-
Acknowledgment. This work supported by grant R22000 of the International Science Foundation. LA940885Q (8) Rusanov, A. I. Phaselgeichgewichte und Grenzfliichenerscheinungen; Akademie-Verlag: Berlin, 1978; p 4659. (9) Brodskaya, E. N.; Piotrovskaya, E. M. Russ. J. Phys. Chem. Submitted.
