2137
Langmuir 1996,11, 2137-2142
Critical Depletion of a Pure Fluid in Controlled-Pore Glass. Experimental Results and Grand Canonical Ensemble Monte Carlo Simulation Matthias Thommes and Gerhard H. Findenegg" I. -N.-Stranski-Institutfur Physikalische und Theoretische Chemie, Technische Universitat Berlin, Strasse des 17.Juni 112, 0-10623 Berlin, Germany
Martin Schoen Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstrasse 36, 0-10623 Berlin, Germany Received November 14, 1994. I n Final Form: March 9, 1995@ The effect of confined geometry on the critical adsorption of a fluid has been studied by measuring the physisorption of sulfur hexafluoride (SF6) in a mesoporous controlled-pore glass (CPG, mean pore width 31 nm) along an isochoric path at the critical density e, in the one-phaseregion above the critical temperature of the fluid (15 K 2 T - T, 2 0.3 K). Whereas the surface excess amount of a fluid against a single planar adsorbing surface is expected to increase in a monotonic way along e = Be for T T,, it is found for the present system that the excess amount of fluid in the pore reaches a maximum at T - T, % 1.5 K and then decreases sharply on further approaching Tc along the critical isochore. Close to the critical point (T T, < 0.3 K) the mean density of the pore fluid even tends to values lower than the density of the bulk fluid. A grand canonical ensemble (GCE) Monte Carlo simulation of a fluid between two parallel walls (wallto-wall distance 20 molecular diameters) in equilibrium with a bulk reservoir at the critical density suggests that this effect can be attributed to a depletion in the core region of the pore as T approaches T,.
-
1. Introduction The criticality of fluids in pores of mesoscopic size has attracted much attention in recent y e a r ~ , l - mainly ~ because of the underlying interplay of two relevant scales, viz., the pore size D and the correlation length E of density fluctuations in the fluid. Away from the critical point, E is of the order of a few molecular diameters, but as the critical point is approached E increases without limit in the bulk fluid. Accordingly, there exists a near-critical region in the temperature-density (T-e) diagram of the fluid in which E > D, while 5 < D outside this region (see Figure 1). In the inner region the criticality of the fluid will be affected by the confinement of the pore space. Pore condensation represents a first-order phase transition from agaslike to a liquidlike state of the pore fluid a t equi1ibr;um with a reservoir of bulk gas.l The locus of states T p ( e of ) the bulk gas a t which condensation occurs in pores of given size and shape is called the pore condensation line. On the basis of the Kelvin equation and other classical theories of pore condensation6the pore condensation line would extend up to the bulk critical temperature T, and terminate in the bulk critical point C(T,,ec). However, finite-size scaling arguments lead to the conjecture2t h a t a fluid confined between two parallel semi-infinite walls may reach criticality a t a temperature Tcplower than T,. In such a n ideal slitpore the correlation length can grow without limit in the directions parallel to the walls but not perpendicular to the wall, where E is Abstract published in Advance A C S Abstracts, May 1, 1995. (1) Evans, R. In Liquides auz Interfaces; Les Houches 1988 Session XLVIII; Course 1; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; North Holland: Amsterdam, 1990. Evans, R. J.Phys. Condens. Matter 1990,2, 8989. (2)Fisher, M. E.; Nakanishi, H. J. Chem. Phys. 1981, 75, 5857. Nakanishi, H.; Fisher, M. E. J. Chem. Phys. 1983, 78, 3279. (3) Binder, K.; Landau, D. P. J. Chem. Phys. 1992,96, 1444. (4) Tovbin, Y. K.; Votyakov, E. V. Langmuir 1993, 9, 2652. (5) Ball, P. C.; Evans, R. Langmuir 1989, 5, 714. (6) Gregg, S. J.;Sing, K. S. W. Adsorption, SurfaceArea andPorosity; Academic Press: London, 1982. @
0743-746319512411-2137$09.00/0
Pk
Qc
Q-
Figure 1. Temperature-density (T-e) diagram of a fluid showing the coexistence curve To@)with the critical point C (0). Away from the critical point the correlation length 5 is smaller than the pore size D of mesopores but 5 =- D in a nearcritical region. The pore condensation line Tp(e)terminates at a pore critical point CP (A). The dashed vertical lines in the one-phase region of the bulk fluid represent (i) an isochore intersecting the pore condensation line, (ii) an isochore above the density @k of the point CP, and (iii)the critical isochore at e = ec,which is studied in the present work.
limited by the wall-to-wall distance D. As a consequence, pore criticality is shifted by a temperature increment AT, = Tc- Tcp=- 0 such t h a t &ATc)= cD, where c is a constant for a given pore geometry and strength of fluid-wall interaction.2 Accordingly, the pore condensation line will terminate in a pore critical point CP(T,,,ek) a t which this condition is met. (In the schematic T-e diagram of Figure 1 we have arbitrarily assumed c = 1 for simplicity.) In a preceding paper' Thommes and Findenegg (TF) have mapped the pore condensation line of sulfur hexafluoride (SF6) in two mesoporous glass materials (CPG) of well-defined narrow pore-size distribution. In order to test the prediction of a critical-point shift for these systems, TF measured the temperature dependence of the sorbed amount offluid nualong several isochoric paths (i.e., paths (7) Thommes, M.; Findenegg, G. H. Langmuir 1994, 10, 4270.
0 1995 American Chemical Society
2138 Langmuir, Vol. 11, No. 6, 1995 of constant bulk density e) a t reduced densities gig, between 0.5 and 0.8. The experimental isochores a t bulk densities up to a value @k were found to exhibit pore condensation a t some temperature Tp(q), but the isochores a t g > @k showed a continuous increase of na as the temperature was lowered down to the respective gasAiquid coexistence temperature To(@) (cf. Figure 1). From these results the pore critical temperature Tcpwas derived. TF found that the pore critical temperature of SF6 in the two CPG materials with D = 31 and 24 nm mean pore size is shifted by increments AT, of 0.5 and 0.9 K, respectively. Furthermore, the entire coexistence curve of gaslike and liquidlike states of the pore fluid is shifted to higher mean densities relative to the coexistence curve of the bulk fluid. The study of Thommes and Findenegg has shown that the critical state of the pore fluid coexists with a bulk fluid of a density @k much lower than the critical density g c (gklg, x 0.75 for SFdCPG-350). In the present paper we investigate the inverse situation, i.e., the state of the pore fluid when the bulk fluid is in the inner region (6 > D)of Figure 1, where fluctuations of sizes greater than D are suppressed in the pore. To our knowledge no experimental study of this problem has been published up to now. The present work emerged from an investigation of the critical adsorption of SF6 on a finely divided (colloidal) graphite substrate (Vulcan-3G graphitized carbon black).8 According to the Fisher-de Gennes (FdG) theory of critical a d s ~ r p t i o nthe , ~ density profile @)(z) of a near-critical fluid against a flat semi-infinite wall is a slowly decaying algebraic function of the scaled distance x = 215, where z is the distance from the surface and 5 is again the correlation length of critical fluctuations in the bulk fluid. Asymptotically close to T, the correlation length increases as a power law 5 = tat-"in t = IT - T,I/Tcfor t 0, where is a critical amplitude and Y a critical exponent (Y = 0.63 for 3D Ising-like systems). Experimentally, one measures the overall surface excess amount
-
r
-
where A, is the surface area of the adsorbing surface and e = e(% c-1 is the bulk density of the fluid. The scaling form of the FdG density profile and the power laws for the correlation length and the order parameter of the fluid imply a n asymptotic power law for the surface excess along the critical isochore, namely, r = rot-("-@). Here, j3 is the critical exponent of the order parameter (/3 = 0.31 for 3D Ising-like systems) and rais the critical amplitude of r. However, such a behavior was not found by Thommes et aL8 for the SFdVulcan-3G system: Well above T, the adsorption increases weakly (asto be expected on the basis of the FdG power law), but close to T, it falls off sharply in contradiction with the expected behavior. This result was confirmed by repeating the measurements under microgravity conditions on the free-flying platform EURECA. A preliminary analysis of these microgravity results is presented elsewhere.1° It was argued that the unexpected adsorption behavior of the SFdVulcan-3G system near T , may be caused by confinement effects of the fluid in the colloidal absorbent, in which the wallto-wall distance between neighboring carbon particles may play a role similar to that of the pore size D in a porous material. In order to test this conjecture we have now (8) Thommes, M.; Findenegg,G. H.;Lewandowski, H.Ber.Bunsenges. Phys. Chem. 1994,98,477. (9) Fisher, M. E.; de Gennes, P. G. C. R.Acad. Sci. B 1978,287,207. (10)Thommes, M.;Findenegg, G. H. Adu. Space Res. 1996,16, 83.
Thommes et al. measured the critical isochore of SF6 in a porous glass, which, unlike the colloidal graphite, comprises a rigid interconnected mesopore system. Results of this experimental study are presented in section 3. In section 4 we report results obtained by grand canonical ensemble Monte Carlo (GCEMC) simulations of a fluid between two parallel walls. These computer simulations offer a qualitative microscopic rationale of the experimental findings, a s will be explained in the Discussion. 2. Experimental Technique 2.1. Materials. A controlled-poreglass (CPG-10)manufac-
tured by CPG Inc. (New Jersey) and received from Fluka (NeuUlm, Germany) was used as the adsorbent. CPG-10 materials are silica glasses comprising an interconnected network of mesopores of approximately cylindrical geometry and a narrow pore-size distribution. (It is quoted that over 80% of the pores have diameters within 5% of the mean value.) The material used in this study (designated as CPG-350)has a nominal pore diameter of 31.3nm and a specific surface area of 67 m2/g(values given by the manufacturer); its specific pore volume is 1.10cm3/g (derived from pore condensation studies with isopentane near room temperature). Sulfur hexafluoride (SF6)with a specified purity of 99.995% was used as the adsorptive. The critical temperature of our sample was determined by turbidity measurements and was 318.703 & 0.010 K. The critical density and critical pressure of SFe were taken from the work of Watanabe et a1.l1 ( g , = 0.740 kg dm-3, P, = 3.759 MPa). 2.2. Procedure. The sorptionmeasurements were performed by a volumetric method described in the preceding paper.7 The two-cell apparatus consists of a reference cell of fured volume containing the fluid at the density of the experimental isochore and a sorption cell of variable volume containing the fluid in contact with ca. 3.8 g of the CPG material. The experimental setup is housed in a three-stage high-precisionthermostat (HPT), which yields a temperature control within 1 mK over a period of days. An experimental run is started at a temperature TIca. 15 K above the critical temperature T, by adjusting the fluid density in the two cells. As temperature is now lowered to a value 2'2 < TI, a pressure difference (Ap)between the two cells builds up due to the sorption of the fluid in the substrate. This pressure difference is monitored by a sensitive differential pressure transducer. In order to restore Ap = 0 (thus restoring the original density e of the fluid reservoir), the volume of the sorption cell is changed from the original value VI to a new value VZby means of a piston-cylinder device. Accordingly, changes of the sorption exeess amount of the fluid for a given temperature increment, Anu, are measured in terms of the corresponding volume changes
A complete sorption scan is obtained by loweringthe temperature stepwise from T I down to the critical temperature and recording
the volume increments for the individual temperature increments. Scans without substrate in the sorption cell (blank measurements) were made to account for the inherent asymmetry of the two-cell system and for S P U ~ ~ O Utemperature S gradients in the experimental setup. These effectswere taken into account by a baseline volume correction AVb(T) = V(T) - V(T1), where V(T)represents the volume of the sorption cell in the blank measurement. Figure 2 shows raw data of sorption measurements for SFs in CPG-350 at two near-critical densities of the fluid (@/ec= 0.995 and 0.999) and two blank measurements at nearly the same density (e/@,= 0.998). The baseline correction Avb(T) exhibits a linear temperature dependence down to T - T, 1K (Figure 2a). The derivative d(AVb)/dT of this linear region can be accounted for by the combined effect of the thermal expansivity and compressibility of the hydraulic oil in the piston-cylinder device.' Noticeable deviations from linearity of the baseline are seen at temperatures T - T, < 1 K (see Figure 2b). These deviations are attributed to uncompensated spurious temper-
-
(11) Watanabe, K.; Watanabe, H . ; Oguchi, K. Proc. Symp. Thermophys. Properties 1977, 7 , 489.
Critical Depletion of a Pure Fluid 0.61
Langmuir, Vol. 11, No. 6, 1995 2139
.
I N I
E d
0
E
=t
3c
U
-01
I+ .
0
.
2
,
,
I
4
, G
oi
, , , 0 1 0 1 2 1 4
, , , ,
0
I
I
-0.61 0
1;
118
0
I
1
I
I
I
I
I
2
L
6
8
10
12
14
16
T-Tc IK Figure 3. Critical adsorption of SF6 in controlled-pore glass (CPG-350) of mean pore width 31 nm (v,e/@,= 0.995; B, @/ec = 0,999)and analogous resultsfor SF6onVdcan 3-Ggraphitized carbon black (0,e/@,= 1.010).
(b) 1
2
3
T-Tc I K Figure 2. AV(T) curves of the sorption isochores of SFdCPG350 (v,m) and blank measurements without adsorbent (A, x ) at near-critical densities (0.99 < e/@,< 1.00): (a)Temperature range 14.5 K L T - T, L 0.45 K the dashed line shows the extrapolation of the linear region of the two blank measurements. (b) Enlarged representation of the near-critical region. ature gradients on the inner stage of the HPT, amplified by the high compressibility of the fluid in its near-critical region. The error bars exhibited in Figure 2b represent maximum conceivable errors as may result from improper settings of HPT. In reality, the agreement between independent blank measurements was much better than this maximum error if the settings of HPT were chosen properly. The same arguments apply for the error of the sorption measurements. The sorption measurements with CPG-350yield volume changes AV,(T) significantly greater than the blank measurements, with a pronounced maximum near T - T, = 1.5 K and a sharp decrease of AV, on further approaching T,(see Figure 2). The sorption excess amount of the fluid n'(T) at a temperature T is then derived from AV,(T? and the corresponding baseline correction Avb(T) by eq 3,
n'(T1), the value of nu at the initial temperature T I = 60 "C at the experimental density e , was taken from the 60 "C surface excess isotherm of the SFdCPG-350 system, which had been measured by a high-pressure microbalance technique. In preceding studies of pore condensation of SF6 in CPG materials7 and of the critical adsorption of SF6 on graphitized carbon black,8 AVs had generally been much greater than the corresponding baseline correction A&. In the present work, these two quantities are more similar in magnitude and thus the bracket in eq 3 is more affected by errors in the baseline correction than in the former studies. However,Figure 2b showsthat the steep decrease of the AV,(T) curve of the sorption isochores starts already at temperatures at which the baseline Avb(T) is still almost linear; and even at T - T, < 1 K the temperature dependence of Avb is much smaller (1 order of magnitude or more) than that of the sorption term AV,. At temperatures T - T, < 0.3 K the results become affected significantly by a stratification of the fluid due to the gravitational field. But above this near-critical regime the findings concerning the temperature dependence of n'(T) are not affected seriously by experimental artifacts.
3. Results Figure 3 shows the results for the critical isochore of the system SFdCPG-350 as derived from the raw data in
5
I-
- - -.- - - - -
04
06
08
1.0
12
p I kg d n i 3
Figure 4. Temperature-density diagram for SFdCPG-350,
showing the mean densities of the pore fluid along the isochores (@/ec= 0.54,0.64,0.78, and 1.00). The pore coexistence curve (full line) and the coexistence curve of the bulk fluid (dashed curve) are also shown. The results for the critical isochore ( 0 ) were obtained in the present work, those for subcritical densities (0) are taken from ref 7 .
Figure 2. The sorption is expressed as surface excess amount per unit surface area, nalA,. Away from the critical temperature the results for the two near-critical densities agree within experimental accuracy. In both data sets naIA,exhibits a maximum near T - T , x 1.5K and a very pronounced decrease as T approaches T, further. In the near-critical region the data points on the el@,= 0.995 isochore fall more and more below those on the @/ec= 0.999 isochore; yet this difference is not significantly greater than the estimated error limits of this region. On both isochores nalA,becomes even negative close to T, (not shown in Figure 3). Results for the critical adsorption of SFe on Vulcan-3G graphitized carbon black obtained by Thommes et a1.8 are also shown in Figure 3 for comparison. The values of n'/A, in CPG are somewhat lower than those on graphitized carbon, as to be expected ffom the fact that the attractive fluid-wall interaction for SFs with silica glass is weaker than with graphite.' From the measured excess amount nuthe mean density epof the pore fluid was derived by the relation (4)
where npis the amount of fluid in the pore volume up and e is again the density of the bulk fluid reservoir. Figure 4 shows a plot of the mean density of the pore fluid along the critical isochore (e/@, = 1)and along three subcritical isochores of the bulk fluid, using results from ref 7. Along
Thommes et al.
2140 Langmuir, Vol. 11, No. 6, 1995 the isochores at @/ec = 0.54 and 0.64 pore filling proceeds via multilayer adsorption a t the pore wall followed by pore condensation in the core. (In Figure 4 the pore coexistence curve represents the locus of the mean densities of the coexistent gas-like and liquid-like states of the pore fluid.) The bulk isochore at el@, = 0.78 corresponds to a weakly supercritical isochore of the pore fluid, where pore filling proceeds in a continuous way. Along the critical isochore of the bulk fluid, e/@,= 1,the density ofthe pore fluid increases weakly with decreasing temperature down to T - T , x 1.5 K, but closer to T, it falls off sharply, indicating a depletion of the pores in the critical region of the bulk fluid.
4. Monte Carlo Simulations By Monte Carlo (MC) simulations we attempt to elucidate microscopic details of the experimentally observed decrease of eP along the critical isochore for T T, discussed in section 3. In the experiments, the fluid in the mesoporous substrate is in thermodynamic equilibrium with a bulk fluid reservoir. This suggests temperature T , chemical potential p , and volume V as appropriate state variables to define the thermodynamic states of the fluid in the pore and the reservoir. Accordingly, the grand canonical ensemble (GCE) is most apt for computational studies of this phenomenon.12 Since the depletion phenomenon was found not only with the porous glass but also with a colloidal graphite substrate (cf. Figure 3),it was conjectured that it should not depend on details of the pore geometry. Therefore, we have chosen a rather simple model system, namely, a slitlike pore with a wallto-wall distance corresponding to 20 molecular diameters, instead of cylindrical pores with mean pore width of 30 nm, as in the CPG material. Nevertheless, the model is believed to catch the main aspect of the experimental situation, uiz., a fluid in a pore of mesoscopic size coexisting with a bulk fluid near bulk criticality. 4.1. The Model. We employ a model in which a fluid of spherical molecules is confined between two plane parallel solid surfaces (walls) separated by a distance sz along the z-axis of the (Cartesian)coordinate system. The walls are smooth in the lateral (xy)directions, lacking any structure. The configurational energy of the model system is taken as a sum of pairwise fluid-fluid (Mand fluid-wall (fw) interactions
-
u = u, + Ufw(l’+
(5)
where the superscripts denote interactions between fluid molecules with the lower (1)and the upper (2) wall. We take the Lennard-Jones (U) 12-6 potential for the fluidfluid interactions
where cffisthe well depth, u the molecular diameter, and rij = Iri - rjl denotes the distance between molecules i and j . We take N
(7) i=l
We model fluid-wall interactions by assuming each wall to consist of a n infinite number of planes separated by a spacing A; each layer is composed of N , atoms smeared (12) Schoen, M. Computer Simulation of Condensed Phases in Complex Geometries; Springer: Heidelberg, 1993.
out over the area s2 of the plane. If the wall atoms are of the same size u as the fluid molecules and interact with a fluid molecule via a U 12-6 potential one has13
+ 0.61 i ) (T
;(ziik’
with d , = N$s2. The first two terms on the right-hand side of eq 8 result from eq 6 by averaging u(rJ over xi and yi while holding ri (the position of a fluid molecule in the slit) fixed. The third term on the right-hand side of eq 8 arises from a summation of the attractive interaction over a quasi-infinite number of crystallographic planes (assuming A = u for simplicity). The above potential was introduced originally for the interaction of molecules with a graphite surface13and may not be entirely appropriate for silica glass. However, as we shall see, the details of the long-range attractive interactions do not affect the results of the present study in a qualitative way.14 To enable a quantitative discussion it is useful to introduce the parameter
as a measure of the relative strength of fluid-wall and fluid-fluid interactions. For the results presented in this paper we choose d , = 0 . 7 8 2 7 ~ - ~In. this case a fluid-wall interaction with e f =~csis strong enough to induce a layer structure of the fluid a t the fluidlwall interface. Details of the GCEMC simulation will be presented e1~ewhere.l~ 4.2. Monte Carlo Results. Since we are treating the fluid in the pore as a thermodynamically open system in virtual contact with a bulk reservoir, we need to determine the relevant chemical potentials corresponding to the experimental condition Q eca t various temperatures T 2 T,, where ec and T, are now the critical density and critical temperature of the bulk U fluid. The values of p(T T J are obtained in a trial-and-error fashion from a series of GCEMC runs in which p is varied at fixed T until the bulk density e = (N)Nagrees with e,*=(N)dN = 0.36 to within 52.5%. This value for &and the value of T,*=kgTIcR = 1.36 are adopted from the work of Hansen and Verlet.15 Once the values forp are known for a set of temperatures in the bulk, we use these values in GCEMC simulations of the pore fluid and determine its local density via
-
where (N(z))is the average number of fluid molecules located in a thin layer of thickness, Az* = Az0-l = 0.02 about z. The GCEMC simulations are carried out for a mesoscopic pore of a width s,* = .sZu-l= 20. Figure 5 shows a plot of q(l)*(z)as a function of position between the lower and upper wall for f = 0.8607 ( e f ~ / = c ~0.7) for two temperatures, F = 3 (far above the critical temperature) and F = 1.36 x Tc*. In view of the symmetry of the density profile with respect to the plane a t z = 0, symmetrically equivalent points in the regions z > 0 and (13)Steele, W. A. Surf. Sci. 1973,36,317;The Interaction of Gases with Solid Surfaces, Pergamon Press: Oxford, 1974; Chapter 2. (14) Schoen, M.;Thommes, M.; Findenegg, G. H. Submitted to Phys. Reu. E. (15)Hansen, J. P.; Verlet, L. Phys. Rev. 1969, 184, 151.
Critical Depletion of a Pure Fluid
Langmuir, Vol. 11, No. 6, 1995 2141 ”’-
I
I
0.4
0 -05 05
-0.4
-03
-ai
-02
0
1.2 T,*
16
20
24
T*
28
32
Figure 6. Overall average density ep* as a function of temperature T* for the fluidwall potential of eq 8, with f = 0.7377 (A)andf= 0.8607 (0).Also shown are results obtained for the potential of eq 8 without the 21-3term, withf= 0.9836 (0). Note that the results for this truncated potential fall between the two versions of the full potential. At r* = 3.00, the three results coincide.
03
5. Discussion 0.2
I -030
, -0,2
-0.1
0
ZIS, Figure 5. Local density e(’)*(z)of the pore fluid as a function of position z between lower and upper walls for sT = 20 and f = 0.8607: q = 1.36 (0);T* = 3.00 (0).The horizontal solid line represents the fluid density e: = 0.365 of the bulk reservoir. (a)Entire regionz < 0 (wall region and core region). The vertical line indicates the range of direct fluid-wall interactions defined as the distance zJk)at which u(zik))has decayed to 10.01of its value at the potential minimum (see eq 8). (b) Enlarged representation of the core region. z < 0 have been averaged and only the lower half of the pore space (-0.5 Izls, I0.0) is shown. Close to the wall, e(l)*(z) is a n oscillatory function of z , reflecting the
formation of individual layers of adsorbed molecules.12 Note that at P = 1.36 x T,*three individual layers are clearly displayed. At !P = 3 only two peaks in @l)*(z)are discernible, and they are less pronounced than those a t the lower temperature, as to be expected on account of the higher kinetic energy of fluid molecules as T increases. These features are taken as evidence that the fluid-wall interaction is sufficiently strong to cause complete wetting of the wall by the fluid. Well above the critical temperature Tc*,the fluid in the core region between the walls appears to be homogeneous, i.e., @)*(z)is nearly constant and equal to the bulk density @:in the entire core region (see Figure 5b). On the other hand, a t the near-critical temperature (P= 1.36) the local density e(l)*(z)decreases with increasing z in the core region and reaches a value distinctly lower than e,* near the center of the pore (zls, = 0). At this temperature the local density in the core region of the pore is depleted with respect to the density of the bulk reservoir. With regard to the range of the direct fluid-wall interactions (see Figure 5), one may distinguish between a wall region where layering of fluid molecules is reflected by the oscillatory behavior of e(%) and acore region where the pore fluid assumes the bulk critical density a t sufficiently high temperatures. For sz* = 20 the core region dominates. It is therefore instructive to investigate the effect of density depletion on the overall average density ep*of the pore fluid which is analogous to the experimentally accessible mean density defined by eq 4. In Figure 6 we show results for ep*obtained by integration of the e(l)*(z)profiles for the fluid-wall potential eq 8 with two different values off, corresponding to ~fw/Eff=0.6 and c d ~ f=f 0.7 (see Figure 5).
Qualitatively, the mean pore density ep*shown in Figure 6 exhibits a strikingly similar temperature dependence as the experimental results in Figures 3 and 4,although the temperature range over which density depletion is observed is much wider than in the experiment. Also shown in Figure 6 are results obtained for the potential of eq 8 without the z ~ term, - ~ with f = 0.9836 ( E % I E ~=~ 0.8). This 10-4fluid-wall interaction potential is often used in adsorption studies by Monte Carlo methods.16 Note that the results for this truncated potential are in qualitative agreement with the results obtained for the full potential of eq 8; this indicates that the results are not affected significantly by details of the long-range attractive fluidwall interaction. From the density profiles of Figure 5 it is evident that the decay of the overall average density ep*at temperatures near T,is caused by a n increasing tendency to remove molecules from the core region of the pore. At the same time the density in the vicinity of the walls increases a s T decreases. Therefore this phenomenon is intrinsically different from “ordinary” drying, where $l)*(z) would decrease mainly near the wall, not in the core regi0n.l’ Incidentally, the (weakly)negative value of ep*a t the highest temperature exhibited in Figure 6 is to be attributed to the wall region where the excludedvolume effect causes a negative deviation from the bulk density and thus a negative surface excess. Such a negative adsorption has indeed been found experimentally in high-pressurehigh-temperature adsorption studies of gases.ls In conclusion, the GCEMC simulations presented in section 4 yield results in qualitative agreement with the observed decrease of nu and epalong the critical isochore of the SFdCPG-350 system near T,. However, two important questions remain to be answered: (i) Do the results of the simulation reflect the same physical phenomenon as the experiments or do they have different causes? (ii) What is the physical explanation of the observed depletion? As mentioned above, the experimentally observed depletion takes place rather abruptly in the true nearcritical region of the bulk fluid a t (T- TJT, < 5 x (16) Bottani, E. J.; Bakaev, V. A.Langmuir 1994,10, 1550. (17)Density profiles exhibiting a decrease of the local density near a wall were reported by Marconi, who analyzed the adsorption of fluids confined in pores in the proximity of the bulk critical point by means of Landau phenomenological models and nonlocal mean-free-energy functionals. Marini Bettolo Marconi, U. Phys. Reu. A 1988,38,6267. See also: Jamnik, A.; Bratko, D. Chem. Phys. Lett. 1993,203,465. (18)Malbrunst, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Langmuir 1992,8, 577.
Thommes et al.
2142 Langmuir, Vol. 11, No. 6, 1995 whereas in the simulations it extends over a much wider temperature range. To understand the origin of this difference we have analyzed the Monte Carlo results for the uniform Lennard-Jones fluid which is in virtual contact with the pore fluid a t given values of temperature and chemical potential. According to the Ornstein-Zernike theory, the total correlation function h(r)of (homogeneous) fluids becomes much longer ranged than the direct correlation function c(r)near the critical point, and the correlation length E mentioned in section 1is a measure ofthe range of h(r).19Accordingly, an analysis of the range of h(r) as it results from the MC simulations for the homogeneous fluid can tell over what temperature range one may expect to see the influence of critical behavior in our simulations. Results of this analysis will be presented in a detailed account of the MC simulations in ref 14. These results show that h(r) as obtained by MC indeed becomes long ranged well above T,: At P = 1.60, where the depletion phenomenon begins to show up in the MC simulation (see Figure 61, h(r)already exhibits a noticeable flat tail extending t o rla > 5. At T* = 1.40, corresponding to (P - T,*)/T,* x 3 x this long-range tail is very pronounced. Thus, in our simulations the temperature range affected by the growing size of fluctuations is much wider than in the experimental system. The fact t h a t in the simulations depletion commences a t much larger T T, than in the real system is consistent with these findings. Therefore, we conclude t h a t the simulations are indeed reflecting the same physical phenomenon as the experiments. What is the origin of the observed depletion effect? Intuitively, it may be attributed to the peculiar situation of an equilibrium of a bulk fluid near its critical point with a pore fluid which is well away from its critical point. Well above T , the mean density Qp of the pore fluid is distinctly greater than the bulk density e,, and a decrease (19) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982; Chapter 9.2.
in temperature causes a further weak increase of Qp due to adsorption near the walls. However, as T,is approached the compressibility of the bulk fluid reservoir increases greatly. In order to maintain mechanical equilibrium between the pore fluid and the bulk reservoir, the pore fluid expands and thus its mean density decreases. Accordingly, critical depletion is driven by the criticality of the bulk fluid, while the thermodynamic properties of the pore fluid remain smooth nonsingular functions of temperature in this region. This intuitive picture suggests t h a t this effect should be independent of pore geometry and may, in principle, occur with all kinds of porous adsorbents including strongly agglomerated colloidal powders like graphitized carbon black. In the forthcoming paper14it will be discussed in more detail that this depletion phenomenon does not depend on the precise form of the long-range fluid-wall interaction potential and is indeed a consequence of restricted density fluctuations due to confinement of the fluid by the walls. To amplify this aspect we shall present an analysis based on a van der Waals model which indicates that the wellknown phenomena of restricted fluctuations and critical point shifts in confined geometries, and the novel depletion effect are interrelated. It will also be shown that the density depletion depends on the net strength ofthe fluidwall interaction as well as on pore size. For example, if sL and f (eq 9) are chosen improperly, density depletion may not occur at all.
Acknowledgment. This work was supported by the Deutsche Agentur fiir Raumfahrtangelegenheiten (DARA) under Grant 50 WM 9115. We are also grateful to the Scientific Council of Hochstleistungsrechenzentrum (HLRZ) a t Forschungszentrum Julich for the allowance of computer time on the CRAY Y-MP1832.M.S. thanks the Deutsche Forschungsgemeinschafl (DFG) for a Heisenberg fellowship (Scho 525/5-1). LA940925Q
