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Langmuir 1997, 13, 6726-6730
Some Aspects of Fluid Behavior in Unwetted Micropores from Computer Simulations Elena N. Brodskaya and Elena M. Piotrovskaya* Department of Chemistry, St. Petersburg State University, Universitetsky pr. 2, St. Petersburg 198904, Russia Received April 28, 1997. In Final Form: June 30, 1997X The behavior of a Stockmayer fluid in unwetted slit and cylindrical capillaries is discussed and compared to that of a Lennard-Jones fluid. For a Lennard-Jones fluid in a slit capillary, a convex meniscus is observed in the direct computer simulations. The influence of the isothermal compressibility on the density in micropores is studied.
1. Introduction In a previous paper,1 the dependence of the Laplace pressure of a Lennard-Jones fluid in several narrow unwetted pores upon the size and the form of such pores was discussed. It was shown that the surface tension of the convex meniscii formed in these pores depends upon the average curvature of the dividing surface but not on its form. In the present work, we consider several questions that appeared in the computer simulations of fluids in unwetted pores. First, it is interesting to determine whether this conclusion is correct for fluids other than the Lennard-Jones ones. For this purpose the behavior of a Stockmayer fluid in cylindrical and slitlike capillaries has been investigated. Secondly, in ref 1, the meniscus itself was not simulated and its radius was estimated on the basis of some approximations. To check the correctness of these approximations, it is of special interest to simulate a two-phase system in a pore with a meniscus dividing the phases in order to estimate its radius of curvature directly. Thirdly, the adsorption isotherms in unwetted pores showed an unusual behavior of the density with changing temperature in which an increase of temperature higher than the critical temperature caused an increase of density. Some explanation of such a dependence of density on temperature is presented in this work. Investigations of the two-phase equilibria of a polar fluid in several pores are discussed in section 2. The results of simulations of the meniscus of a Lennard-Jones fluid in a slit pore are given in section 3, and the behavior of the pore-fluid density upon changes of temperature is considered in section 4. In all cases, these simulations were performed using grand canonical and canonical Monte Carlo (MC) methods. 2. Phase Equilibria of a Stockmayer Fluid in Pores There are several papers on computer simulations of the two-phase liquid-vapor equilibria of a Stockmayer fluid.2-6 The values of the chemical potential of a X
Abstract published in Advance ACS Abstracts, October 15, 1997.
(1) Brodskaya, E. N.; Piotrovskaya, E. M. Langmuir 1994, 10 (6), 635. (2) Van Leeuwen, M. E.; Smit, B.; Hendriks, E. M. Mol. Phys. 1993, 78, 271. (3) Eggebrecht, J.; Gubbins, K. E.; Thompson, S. M. J. Chem. Phys. 1987, 86, 2299. (4) Shreve, A. P.; Walton, J. P. R. B.; Gubbins, K. E. J. Chem. Phys. 1986, 85, 2178. (5) Han, K.-K.; Cushman, J. H.; Diestler, D. J. J. Chem. Phys. 1993, 96, 7867. (6) Han, K.-K.; Cushman, J. H.; Diestler, D. J. Mol. Phys. 1993, 79, 537.
S0743-7463(97)00429-0 CCC: $14.00
Stockmayer fluid were calculated in ref 5 by Monte Carlo (MC) methods. The conditions for the liquid-vapor coexistence of a Stockmayer fluid at the flat surface were investigated in refs 2 and 3 by molecular dynamics (MD) at different values of the molecular dipole moment and temperature. Small droplets of a Stockmayer fluid were investigated in ref 4, where it was shown that the surface tension decreases with a decrease of the radius of a droplet. The changes in the structure of the adsorbed Stockmayer fluid in slitlike pores were considered in ref 6 by grand canonical MC methods for several pore widths at various values of the chemical potential. The width of the pore changed from 2.5 up to 4 molecular diameters. In the present work the vapor-liquid coexistence for a Stockmayer fluid was considered in narrow unwetted cylindrical and slitlike capillaries. The intermolecular potential for a Stockmayer fluid consists of a LennardJones interaction of two spherical atoms ΦLJ(r) plus the interaction of two point dipoles d1 and d2 located in the centers of atoms. It can be written as follows:
Φ(rij) ) ΦLJ(rij) + di‚dj/rij3 - 3(di‚rij)(dj‚rij)/rij5 (1) where ΦLJ is the Lennard-Jones potential
ΦLJ(r) ) 4[(σ/r)12 - (σ/r)6]
(2)
di is the dipole moment vector of molecule i, and rij is the vector distance between molecules i and j. In this study, the value of the dipole moment was set equal to 2.0(σ3)1/2. The Stockmayer potential is long range, as are all potentials used for the description of polar molecules. But as shown in computer simulations of bulk polar liquids (methanol, water),7,8 taking account of long range interactions, for example, with the help of the Ewald summation method, does not significantly give better results for the thermodynamic properties of the systems but does slow the running of the computer program. That is why we used in this work an ordinary spherical cutoff procedure for the potential but enlarged the cutoff radius by a factor of two in comparison with that for simple bulk liquids. The calculations are carried out by grand canonical MC methods for given values of (µVT), where µ is the chemical potential, V is the volume and T is the temperature. The length of the MC chains for averaging was not less than 3 × 106 configurations. Here narrow cylindrical pores of radius R ) 3.0σ and 6.0σ as well as slitlike pores of width H ) 6.0σ with absolutely hard unwetted walls are (7) Mezei, M. Mol. Simul. 1992, 9, 257. (8) Gotlib, I. Y.; Piotrovskaya, E. M. Russ. J. Phys. Chem. 1996, 70 (1), 142.
© 1997 American Chemical Society
Fluid Behavior in Unwetted Micropores
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Figure 1. Isotherms of a Stockmayer fluid in an unwetted pore with H ) 6.0σ at T2 ) 2.5/k and T1 ) 1.5/k. Figure 3. Dependence of the grand thermodynamic potential Ω on the chemical potential µ for a Stockmayer fluid in unwetted slit pores of width H ) 6.0σ. Table 1. Basic Results for a Stockmayer Fluid
Figure 2. Isotherms of a Lennard-Jones fluid in an unwetted pore with H ) 6.0σ at T2 ) 1.2/k and T1 ) 0.75/k.
considered. The axis of the cylinder coincided with the x axis, and periodic boundary conditions were used in this direction. The size of the basic MC cell in the x direction was l ) 8.8σ, and the cutoff radius for the potentials was rc ) 4.4σ. For slit pores, periodic boundary conditions were used along both the x and y axes. One of the goals of this work is the investigation of the liquid-vapor coexistence of a Stockmayer fluid in narrow unwetted pores. A procedure for doing this was proposed in ref 99 and used in our earlier work1 on the computer simulations of vapor-liquid coexistence in pores. The main idea of this method is connected with the determination of the dependence of the grand thermodynamic potential Ω on the chemical potential separately for the isotherms of gas and of liquid. The intersection of two branches of the isotherm for Ω corresponds to the point of phase equilibrium and gives the value of µcoex. In order to find Ω(µ), it is necessary to consider systems at two temperatures, one of which should be supercritical. For the Stockmayer systems in micropores a supercritical temperature T2 was chosen equal to 2.5/k, and the temperature of investigation T1 was equal to 1.5/k, as in refs 3 and 4. The isotherms of a Stockmayer fluid in an unwetted slit pore with H ) 6.0σ are shown in Figure 1. The supercritical isotherm at T ) 2.5/k lies above the isotherm at T ) 1.5/k, similar to the behavior of the LennardJones liquid in unwetted pores (Figure 2).1 For adsorption (9) Peterson, D. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215.
R (σ)
µ(1)coex ()
Fβ (σ-3)
Pβ (σ-3)
PR (σ-3)
Rs (σ)
∞5 6.0 3.0
-6.60 -6.24 -5.74
0.013 0.016 0.022
0.020 0.023 0.033
0.020 0.301 0.695
∞ 4.9 2.1
systems,9,10 the supercritical isotherms usually lie lower than the subcritical ones. The isotherms for the unwetted slit pore with H ) 6.0σ coincide with those for the cylindrical capillary with R ) 6.0σ, and they are very similar for both the Lennard-Jones and the Stockmayer fluids. The existence of a hysteresis loop in the adsorption isotherm when adsorption and desorption branches do not coincide is often discussed in the literature.11-15 In ref 15, for example, it is supposed that the appearance of hysteresis in computer simulations of homogeneous adsorption pores is the result of the insufficient length of calculations. The results of the present work confirm that for unwetted pores metastability of the desorption branch practically does not exist. However, to get reliable results for the chemical potential of the two-phase coexistence in pores it is better to use the procedure described above than to calculate directly the desorption branch of the isotherm. The results of the calculation of Ω are shown in Figure 3 for the slit pore with H ) 6.0σ. The values of the configurational part of the chemical potential at the coexistence point µ(1)coex for all the systems investigated at T ) 1.5/k are given in Table 1. The values of the standard state chemical potential µ0 are equal to -24.864 at T1 ) 1.5/k, and -45.266 at T2 ) 2.5/k. The value of µ(1)coex for the bulk phases was estimated from the vapor pressure assuming vapor ideality3 and is also cited in Table 1. The chemical potentials in pores are higher for the narrower pores, which is the evident effect of the increase of the vapor pressure with increasing curvature of the convex meniscus. It should be noted that the results for (10) Brodskaya, E. N.; Piotrovskaya, E. M. Langmuir 1995, 11 (6), 642. (11) Heffelfinger, G. S.; van Swol, F. B.; Gubbins, K. E. J. Chem. Phys. 1988, 89, 5202. (12) Papadopoulou, A.; van Swol, F.; Marconi, U. M. B. J. Chem. Phys. 1992, 97, 6942. (13) Evans, R.; Marconi, U. M. B.; Tarasona, P. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1763. (14) Walton, J. P. R. B.; Quirke, N. Mol. Simul. 1984, 2, 361; Chem. Phys. Lett. 1986, 129, 382. (15) Shoen, M.; Rhykerd, C. L., Jr.; Cushman, J. H.; Diestler, D. G. Mol. Phys. 1989, 66, 1171.
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Brodskaya and Piotrovskaya
the cases of R ) 6.0σ and H ) 6.0σ coincide perfectly. On the basis of µ(1)coex, the vapor density Fβ and pressure Pβ above the convex meniscus were estimated. These values are also given in Table 1. It is necessary to point out that the changes in the chemical potentials with changes in the size of the pore are very similar for Lennard-Jones and Stockmayer fluids. For example, the difference in the chemical potentials for the capillaries with R ) 6.0σ and 3.0σ is 0.5 for a Stockmayer fluid and 0.56 for a Lennard-Jones fluid.1 In a similar experiment for a Lennard-Jones fluid1 the surface tension γ was evaluated on the basis of the Laplace equation with PR- Pβ ) ∆P:
∆P ) 2γ/Rs
(3)
where PR and Pβ are the pressures of the bulk liquid and vapor phases separated by a curved meniscus. (These phases can be viewed as two bulk phases in two different containers connected by the capillary under investigation.) In eq 3, Rs is the radius of the surface of tension of the meniscus. Here, we shall use eq 3 to estimate the radius Rs of the dividing surface using the value of the surface tension of the flat surface γ ) 0.686/σ2 (see ref 3). In the first approximation, we neglect the dependence of the surface tension on the curvature of the dividing surface. As mentioned above, the pressures of the ideal gas above the meniscus Pβ are given in Table 1. The pressure in the liquid phase PR can be obtained from
PR ) P1 + FR(µR - µ1)
(4)
where µR ) µcoex and P1 and µ1 are the pressure and chemical potential in some reference state such as a twophase equilibrium at the flat surface which was investigated in ref 3. PR is also given in Table 1, and it should be noted that FR is equal to 0.77/σ3 (see ref 3). According to eq 3, the radius of the surface of tension Rs of the meniscus in the capillary with the radius 3.0σ is equal to 2.1σ, whereas in the capillary with the radius 6.0σ it is equal to 4.9σ (Table 1). It should be pointed out that these results almost coincide with those for the Lennard-Jones fluid in similar capillaries.1 The dependence of the surface tension on the curvature of the meniscus produce smaller values of Rs than estimated above. So, it is possible to consider these values as the upper boundaries of the radii of the surface of tension for a Stockmayer fluid in the given capillaries. The closeness of the values of these radii of the meniscus both for Lennard-Jones and Stockmayer fluids confirms the similarity of the surface structure of these liquids. Besides, on the grounds of the coincidence of the conditions of two-phase coexistence for the slit pore with H ) 6.0σ and the cylindrical capillary with R ) 6.0σ, it is possible to conclude that the average curvature of the meniscus is the same for these two cases and that the surface tension of the polar fluid depends only on the curvature but not the form of the meniscus. The same conclusion was obtained for a Lennard-Jones fluid in ref 1. 3. Meniscus for the Lennard-Jones Fluid in an Unwetted Slitlike Capillary It is evident that direct computer experiments that determine the form and curvature radius of the meniscus are needed to check the approximate estimations of these quantities. The form of the concave meniscus for wetted pores has been simulated for slit16,1816-18 and cylindrical17 capillaries. In ref 16, the unwetted pore of width 21.6σ (16) Saville, G. J. Chem. Soc., Faraday Trans. 2 1977, 73, 1122.
Figure 4. Form of the meniscus of a Lennard-Jones fluid in an unwetted slit capillary of width H ) 6.0σ at T ) 0.75/k.
was considered, but the main attention was paid to the determination of the wetting angle. The present work is aimed at simulations of the two-phase vapor-liquid system in an unwetted slit pore in order to find the form of the convex meniscus. As in all previous simulations of meniscii we consider a Lennard-Jones fluid with the potential of eq 2. A number of conditions are to be met to model such a system. First, it is obligatory to model a closed system, e.g. to perform the MC simulations in a canonical (NVT) ensemble. Second, the number of molecules in the system under investigation has to be large enough to obtain a stable two-phase liquid-vapor system in the capillary. In the present work, the temperature was set equal to 0.75 /k and the number of particles in the basic MC cell was N ) 576. Periodic boundary conditions were used in the x and y directions, and in the z direction, the system was confined between two hard walls. So, the sizes of the basic MC cell in three directions were lx ) 35.2σ, ly ) 4.4σ, and lz ) H ) 6.0σ. All the molecules in the initial configuration were situated in the central part of the cell, forming a slab with a density of about 0.8σ-3 and a width of 24.0σ in the x direction and 6.0σ in the z direction. During the averaging of the system, a liquid-vapor interface was formed in the x direction and a liquid-solid interface was formed in the z direction. In these directions the local density changes monotonously. Third, the length of the MC chains must be very large to obtain reliable local densities, especially in the vicinity of the meniscus. The length of the MC chain was 50 × 106 configurations. In order to obtain the local density F(x,z) the capillary was divided into thin slabs of the following dimensions: ly along the y direction and 0.1σ along the x and z directions. As a result of this procedure we estimated the local density of the fluid both at the liquid-vapor and liquid-solid interfaces and inside the dense fluid. The behavior of the local density in the vicinity of the meniscus is of special interest. To characterize the form of the meniscus, we used the following condition:
F(x,y) ) 0.5(FR + Fβ)
(5)
According to this equation we obtained the meniscus shown in Figure 4. It should be noticed that the density fluctuations in the layers near the walls are rather large, but in the central part of the meniscus they are reasonable, and here the meniscus is defined quite well. Clearly, the (17) Nijmeijer, M. J. P.; Bruin, C.; Bakker, A. F.; van Leeuwen, J. M. J. Physica A. 1989, 160, 166. (18) Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 61, 1381.
Fluid Behavior in Unwetted Micropores
Langmuir, Vol. 13, No. 25, 1997 6729
form of the meniscus was not ideally cylindrical, but in its central part it is very close to a cylinder with the radius R ) 2.0σ. The largest deviations of the form of the meniscus from cylindrical are found in the layers near the walls of width 1.0σ. Thus, the direct calculations of the radius of the meniscus in the computer simulations almost coincide with our approximate estimations of this value from ref 1, which confirms the correctness of the latter.
Table 2. Parameters of the Potentials for Adsorption Systems a/k (K) /k (K)
σ (nm)
I
II
σa (nm)
148.1
0.373
1158.0
579.0
0.224
Table 3. Average Density G of a Lennard-Jones Fluid at µ ) -9.0E, T1 ) 0.75E/k, and T2 ) 1.2E/k and a Stockmayer Fluid at µ ) -26.0E, T1 ) 1.5E/k, and T2 ) 2.5E/k Lennard-Jones fluid
4. Dependence of Density in Capillaries on Temperature The unusual behavior of the average density with an increase of temperature was found during the investigations of phase coexistence in unwetted pores. The density of both Lennard-Jones and Stockmayer fluids in such pores increases with the heating of fluids to supercritical temperatures (Figures 1 and 2). The dependence of the density of a bulk system on temperature at constant pressure is directly connected with the thermal coefficient R according to the definition
R ) (1/V)(∂V/∂T)p ) -(1/F)(∂F/∂T)p
bulk 0.8725 0.8503
0.8024 0.8920
0.9072 0.9008
0.8715 0.8728
unwetted slit H ) 6.0σ
unwetted cylinder R ) 6.0σ
1.0300 1.0093
1.0643 1.1561
(6)
In the case of a fluid in pores the average density in the pore is to be used in eq 6, and the derivatives are calculated at the constant value of the pore size, that is the width of the slit pore or the radius of the cylinder. The fact that the average density in unwetted pores increases with an increase of temperature seems to be in contradiction with the ordinary behavior of simple fluids. It is true that the thermal coefficient of bulk liquids at constant pressure is positive (it is (∂F/∂T)p < 0) except for rare exceptions. However, it is necessary to mention that in the present simulations the density changes were calculated at constant chemical potential, not at constant pressure. It is easy to obtain the following equation from thermodynamics:
(∂F/∂T)µ ) (∂F/∂T)p + sFκ
T1 T2
unwetted slit H ) 6.0σ
Stockmayer fluid
adsorption slit H ) 10.0σ I II
(7)
where s is the entropy per unit volume and κ ) F-1(∂F/∂p)T is the coefficient of the isothermal compressibility. As mentioned above, all the derivatives for the average density in eq 7 are to be taken at constant pore size. Obviously, the second term in the right part of eq 7 is always positive. As the compressibility of the liquid away from the critical region is small, then the contribution of this term is also small, and the sign of the derivative (∂F/ ∂T)µ has to coincide with the sign of (∂F/∂T)p. However at supercritical temperatures the compressibility is increasing and as a result the contribution of the second term to eq 7 may be of importance for the determination of the sign of (∂F/∂T)µ. To reveal the role of the compressibility in the changes of density with temperature, the calculations of the local density in pores and for the bulk system were carried out at the temperatures T1 and T2 at given values of the chemical potential. The MC simulations of a Lennard-Jones fluid with the intermolecular potential given by eq 2 were carried out earlier for the unwetted1 and wetted10 pores. In the latter case the Lennard-Jones fluid corresponded to methane in a slitlike pore with smooth walls. The interactions between these walls and the adsorbate were described by the potential Φa(z), which gives the dependence of the potential energy of the adsorbate molecules on the distance
Figure 5. Density profiles of a Lennard-Jones fluid in wetted pores of type I of width H ) 10.0σ at T1 ) 0.75/k and T2 ) 1.2/k and chemical potential µ ) -9.0.
from the flat wall of the adsorbent:
Φa(z) ) (3 x3/2)a[(σa/z)9 - (σa/z)3]
(8)
where a and σa have the same meaning as and σ for the potential (eq 2). The parameters of intermolecular and adsorption potentials are presented in Table 2. The values of and σ are given for methane, with the larger value of a ) 1158.0k corresponding approximately to the interaction of methane with the graphite surface (let us name such pores as pores of type I and the pores with the weaker adsorption field as pores of type II). Slitlike pores of width from H ) 4.0σ to H ) 12.0σ were investigated. The “dead” volume near the adsorption walls was not taken into consideration in the calculations of the average densities in this case. The calculations were carried out at values of the chemical potential equal to -9.0 for a Lennard-Jones fluid and -26.0 for a Stockmayer fluid. These values correspond to the liquid branch of the isotherms (Figures 1 and 2). The values of the average density for some systems are given in Table 3. Some data from refs 1 and 10 are also included in this table. The density of the bulk fluid decreases with the increase of the temperature as is seen from this table. This change is relatively small for the given temperature interval (for methane it is δT ) T2 - T1 ) 66.6). This indicates the considerable contribution of the second term in eq 7, which partially compensates the negative value of the derivative (∂F/∂T)p. In the wetted pores (of graphite type) the role of this term increases, but still the sign of eq 7 is defined by the first term. In the pores with the weaker adsorption field (pore II) the density hardly changes with temperature. In the unwetted pores, the increase of fluid density
6730 Langmuir, Vol. 13, No. 25, 1997
Figure 6. Density profiles of a Lennard-Jones fluid in wetted pores of type II of width H ) 10.0σ at T1 ) 0.75/k and T2 ) 1.2 /k and chemical potential µ ) -9.0.
Figure 7. Density profiles of a Lennard-Jones fluid in unwetted pores of width H ) 6.0σ at T1 ) 0.75/k and T2 ) 1.2/k and chemical potential µ ) -9.0.
with increasing temperature is already observed. Therefore, the compressibility factor for these systems begins to play a decisive role. It is evidently connected with the strong inhomogeneity of the systems in pores characterized by a density lower than that of the corresponding bulk liquid at subcritical temperatures. The changes of the local densities in pores with temperature are shown in Figures 5-9. It is seen that, in pore I (Figure 5), the height of the peaks near the walls decreases substantially with the increase of the temperature, and this produces a decrease of the average density in this pore. In pore II with a weaker adsorption field (Figure 6), the peak of the first adsorbed layer with an increase of the temperature is moved a bit closer to the wall, whereas the rest of the peaks become a bit lower and wider, and the average density almost does not change. In the unwetted pores (Figures 7-9) the behavior of the local density in the middle part of each pore is almost constant with the increase of the temperature, while the local density near the walls increases substantially. This
Brodskaya and Piotrovskaya
Figure 8. Density profiles of a Stockmayer fluid in unwetted pores of width H ) 6.0σ at T1 ) 1.5/k and T2 ) 2.5/k and chemical potential µ ) -30.0.
Figure 9. Density profiles of a Stockmayer fluid in unwetted pores of width H ) 6.0σ at T1 ) 1.5/k and chemical potentials µ ) -30.0 (1) and µ ) -26.0 (2).
explains the increase of the density in the pore as a whole. These changes are similar to the ones that take place in pores with an increase of pressure, as is seen from Figure 9, where the density profiles of a Stockmayer fluid in unwetted pores are shown for two values of the chemical potential. The increase of the chemical potential corresponding to an increase of the pressure in the liquid produces an increase of the local density near the walls of the pore. It is possible to state on the basis of these results that, even for simple fluids (such as Lennard-Jones and Stockmayer fluids) in narrow pores in the supercritical region, a change of the sign of the thermal coefficient is being observed which is caused by the increase of the contribution of the fluid compressibility. Acknowledgment. The present work was supported by the International Science Foundation (grant R22000). LA970429C
