Ind. Eng. Chem. Res. 1998, 37, 3221-3229
3221
Adsorption and Heats of Immersion of n-Alkanes on Model Silica Gel Peter A. Gordon and Eduardo D. Glandt* Department of Chemical Engineering, University of Pennsylvania, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393
Grand canonical ensemble Monte Carlo simulations are performed with configurational bias to investigate the adsorption behavior of n-alkanes on a model microporous silica gel. The qualitative shape of the adsorption isotherm is seen to change with increasing alkane chain length. A transition from continuous pore filling to layering of molecules at the surface is observed between n-octane and n-dodecane. In addition, heats of immersion are calculated for the model adsorbents in alkanes via computer simulation. This quantity, which to our knowledge has not been calculated in this manner previously, provides a convenient point of contact between simulation and experiment. Agreement between experimental and calculated heats is quite good, provided careful accounting is made of the accessible pore volume and surface area. Introduction The interfacial thermodynamic properties of alkanes, such as the wetting, adhesion, and adsorption behavior, have been long-standing subjects of research. Results from modeling studies of alkanes in inhomogeneous environments, however, have lagged experimental observation. This is partly because theoretical approaches are complicated by the molecular architecture of chain fluids. Approaches have included self-consistent field theories,1 scaling theories,2 density functional theories,3 and computer simulation. Early efforts have focused on lattice systems, but in many cases have been extended to the continuum. Recently, computer simulation methods have been applied increasingly to the study the behavior of chain molecules with solid microporous adsorbents.4-9 The canonical ensemble has been used in the examination of the conformational and energetic properties of the adsorbed phase. To generate adsorption isotherms in the canonical ensemble, however, the adsorbed-phase chemical potential must be determined at each loading and referenced to the corresponding bulk fluid at equilibrium with the confined phase. This has been done using the chain increment method in conjunction with canonical simulations of chain molecules,7,10 but is a somewhat indirect and cumbersome technique to measure isotherms. More recently, configuration bias sampling has been extended to the grand canonical ensemble, which allows the direct measurement of adsorption isotherms of chain molecules via molecular simulation.9 Even with the ability to probe adsorption phenomena directly with computer simulation, we note that it is often difficult to find experimental data with which to compare theoretical predictions in a rigorous manner. For instance, computer simulation investigations on the energetics and locations of adsorption sites available to n-butane through decane have been conducted on a variety of all-silica zeolites.5,6 These studies reported * To whom correspondence should be addressed. Phone: (215) 898-6928. Fax: (215) 573-2093. E-mail: eglandt@ seas.upenn.edu.
observations concerning the conformational behavior of molecules adsorbed in various-sized channels and saw interesting variations in the isosteric heats of adsorption as a function of adsorbent pore size, but could not be compared to the appropriate experimental data. One reason for the lack of adsorption data in the literature for n-alkanes on industrially important microporous adsorbents is found in the difficulties associated with achieving the low pressures where adsorbent pore spaces first fill. From an experimental standpoint, it is much easier to work with these systems in the liquid phase (i.e., at pressures above saturation). Computer simulation techniques involving adsorption, on the other hand, are much easier to implement and yield more information when working with low-pressure vapors. Clearly, it would be advantageous to examine properties accessible via both experimental and simulation techniques. The heat of immersion is such a property. The heat of immersion is readily measured experimentally11 and can yield insight into the nature of the adsorbate-surface interaction. To our knowledge, this quantity has not been calculated using computer simulation, but holds promise in making contact between simulation and experiment in systems where other experimental data are not available. In this paper, we investigate adsorption of four n-alkanes (n-butane through dodecane) on model silica gel to examine the basic features of adsorption in these systems. We examine adsorption isotherms, isosteric heats of adsorption, and adsorbate structure in the pores of the silica gel. We also calculate the heats of immersion and compare the results to available experimental observations. Background The equilibrium adsorption behavior of chain molecules in inhomogeneous environments, both on surfaces and in confined geometries, is known to depend on the complex relationship between enthalpic and entropic driving forces. Attractive adsorbate-solid interactions favor adsorption of chains near a surface, but an entropic penalty associated with chains ordering in a restricted geometry opposes this arrangement. Which
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3222 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
effect is more important, however, may depend on a number of factors. For instance, in ellipsometry studies12 of n-octane on surface-oxidized silicon wafers, linear isotherms were observed up to a reduced vapor pressure of approximately P/P0 ) 0.7. The relatively low rate of uptake of the surface suggests weak lateral interactions and parallel orientation of molecules to the surface. On similar surfaces, Schlangen and co-workers13 have observed invariance in the uptake per unit area of n-heptane, octane, and nonane with respect to relative pressure. In addition, multilayer formation (and ultimately surface wetting) was seen only at relative pressures above 0.8, at loadings that correspond closely to a monolayer of chains oriented perpendicular to the surface. It was thus postulated that as loading increases during submonolayer coverage, adsorbed chains reorient themselves to configurations perpendicular to the surface. These experimental observations were supported by self-consistent field (SCF) theory predictions of lattice chain octamers adsorbing on an attractive surface.14 It is worth noting that homogeneous adsorbate models (where all chain segments were identical) were not able to reproduce the experimental isotherms even qualitatively. Heterogeneous models, where end segments possess a stronger surface adsorption strength than those of the interior, were necessary to reproduce the observed results. Chain molecules confined between slit pores with purely repulsive15,16 and with repulsive and attractive7,8 interactions have been considered by a number of workers. These systems exhibit a range of behaviors, depending on the conditions. When the wall-adsorbate interaction is purely repulsive, the main effect of confinement is to decrease the conformational entropy of the chain molecules. At low loadings, this causes a depletion of chains near the walls. At higher loadings, packing effects cause enhancement of the density near the walls. The magnitude of this depletion/enhancement effect is decreased as confinement increases. Vega et al. found that, in the limit of strongly attractive pore walls, freely jointed Lennard-Jones chains in narrow pores, and particularly their end segments, show a strong preference for the surface.7 Alkane molecules have a restricted set of favorable internal conformations, manifested as trans- and gaucheorientations of carbon bonds. The above observations suggest that, compared to freely jointed chains, alkanes are expected to experience a lower entropic penalty upon adsorption, but that their greater degree of rigidity also will make them pack less efficiently near a surface. The methyl end-groups in alkanes have a larger surface interaction energy than the interior segments and should exhibit some preference for adsorption near the surface. The degree of confinement imposed by a microporous adsorbent is expected to alter the adsorption behavior substantially, when compared to that observed at free surfaces. Heats of Immersion The heat of immersion of an adsorbent in a liquid can be defined as the net heat evolved when the outgassed solid is immersed in the liquid. By convention, the heat of immersion is reported as a positive quantity, although immersion is an exothermic process.
The heat of immersion is sometimes used in the estimation of the surface area of adsorbents. Since the amount of heat released by a given adsorbent is proportional to its surface area, determination of the heat of immersion of a sample of known surface area in a particular liquid permits the estimation of the surface area of other samples. The underlying assumption is that the chemical nature of the surface is unchanged from one sample to the other. The proportionality between surface area and heat evolved has been shown to hold even in the presence of extensive microporosity. Although the heightened adsorption potential in micropores will increase the heat of immersion,17 this increase is actually a reflection of the increased surface area available to the admolecules. In the case of micropores, this method for the determination of surface area is free from any models, unlike the traditional nitrogen adsorption Brunauer-Emmett-Teller (BET) technique. Using the fact that the enthalpy of immersion is a state property, it is possible to set up a heat balance following two separate paths describing the immersion of the solid in the liquid.18 In the first path, n moles of vapor are condensed (step a1) and the solid is then immersed in the liquid (step a2). The enthalpy change in this sequence is
∆H1 ) ∆HR1 + ∆HR2 ) -n∆Hvap - ∆Himm
(1)
In an alternative path, n moles of vapor are exposed to the solid until equilibrium is reached, where n1 moles are adsorbed (step b1), the remaining vapor is condensed (step b2), and the solid is finally immersed in the remaining liquid (step b3). In this case, the enthalpy change can be expressed as
∆H2 ) ∆Hb1 + ∆Hb2 + ∆Hb3 ) -∆Hsv - (n - n1)∆Hvap - q (2) where ∆Hsv is the enthalpy change associated with the evacuated solid coming to equilibrium with the saturated vapor. Equating and rearranging eqs 1 and 2 for ∆Himm. yields
∆Himm. ) (∆Hsv - n1∆Hvap) + q
(3)
The group in parentheses in eq 3 is termed the net integral heat of adsorption. In practice, the quantity of heat released in step b3, where the solid covered by an adsorbed film is exposed to the remaining fluid, is small and can be safely neglected. Thus, the heat of immersion is approximated by the net integral heat of adsorption. The net integral heat of adsorption can be related to the differential heat of adsorption via
∆Hsv )
∫0n qd dnx 1
(4)
where we note that this process is carried out at constant bulk volume, as opposed to bulk pressure (necessitating the use of the differential, not the isosteric heat of adsorption). The differential heat of adsorption
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3223 Table 1. Parameters Used in n-Alkane Potential σ(CH3) ) σ(CH2) ) 3.93 Å; (CH3)/k ) 114 K; (CH2)/k ) 47K kθ ) 62 500 K/rad2; θeq ) 114° a1/k ) 355.03 K; a2/k ) -68.19 K; a3/k ) 791.32 K
Lennard-Jones bond bending torsion
is related to the isosteric heat, which we measure in computer simulations, by
qd ) qst +
P Fbulk
(5)
Therefore, if we measure the isosteric heat of adsorption over the isotherm from zero to full coverage, we have
∆Himm )
[
∫01 qst(T,θ) - F
]
P dθ - ∆Hvap. (6) bulk(T,θ)
where Θ is the fraction of the pore filled with adsorbate. Equation 6 gives us a working expression that can be used to calculate the heat of immersion. Models and Methods A number of models have been proposed to describe n-alkanes.19-21 Many models adopt a “bead segment” description of the chain, in which each methyl group is represented as a single Lennard-Jones site. Alternatively, anisotropic models that incorporate the influence of methyl hydrogens have been shown to be an improvement when considering diffusional processes.22 In addition, intramolecular bond bending and torsional potentials are often employed to describe the conformational behavior of the chain. Even for similar potential descriptions, model parameters reported in the literature can vary greatly.23 The adsorbate models employed in this work are given by Siepmann and co-workers, who fit their model parameters to describe experimentally observed vaporliquid coexistence curves.24 The n-alkane chain consists of n segments separated by a fixed bond distance d ) 1.54 Å. The intermolecular potential energy between two alkane chains is given by a Lennard-Jones 12-6 interaction, which includes interactions between all sites on adjacent chains, and intrachain interactions between sites separated by four or more bonds. The potential is truncated at 13.8 Å and no tail corrections are employed. The intramolecular potential functional form is given by 3 1 Uintra(θi,φi) ) kθ(θi - θeq)2 + aj(1 + cos(jφi)) 2 j)1
∑
(7)
where θi and φi are the angle between segments (i - 1, i, i + 1) and the torsional angle between segments ri,i+1 and ri+2,i+3, respectively. The parameters employed in the alkane potential are listed in Table 1. We note here that we employ the Lennard-Jones CH3 parameters in all results presented in nondimensionalized form. Second Virial Coefficients As a test of the models employed, we examine the second virial coefficients of this alkane model over a broad temperature range. Calculation of the second virial coefficient involves integrating the Mayer f-
Figure 1. Comparison of calculated and experimental virial coefficients for n-alkanes. The points correspond to Monte Carlo integration and lines are experimental data from the literature.28 (0) n-Butane simulation; (______) n-butane experiment from Huff et al.; (O) n-hexane simulation; (- - -) n-hexane experiment from Al-Bizreh et al., (4) n-octane simulation; (-‚-‚) n-octane experiment from Al-Bizreh et al.
function over all possible relative positions and orientations of two molecules:
B2(T) )
∫(exp(-βU(r1,r2,ω1,ω2{φ1,θ1},{φ2,θ2}))-1) ×
dr1 dr2 dω1 dω2 d{φ1,θ1} d{φ2,θ2} (8)
For molecules with intramolecular degrees of freedom, the integration proceeds over all possible relative internal conformations of the molecules as well. Several methods have been proposed for numerically integrating this function.25,26 We perform a Monte Carlo integration of eq 8, using configurational biasing to generate internal chain conformations. The integral is broken into sections according to the separation between centers of mass of the two molecules, and each section is numerically evaluated until convergence (judged through some reasonable criterion) is obtained. The organization of the numerical calculation method is similar in spirit to that employed by Calles and Myers in the evaluation of B2 for simple molecules with orientational degrees of freedom.27 The results are shown in Figure 1 and compared with experimental data from several sources.28 At higher temperatures, agreement between the experimental and calculated coefficients are excellent. At lower temperatures, the discrepancy between the two increases, particularly as the alkane chain length increases. However, the agreement is sufficient to demonstrate the suitability of the potential models employed in our work. The adsorbent description is the same as that used in our previous simulations on silica gel,29,30 namely, the composite sphere approximation. The model is based on simulation studies originally performed by MacElroy and Raghavan31 and later Kaminsky and Monson.32 It consists of a collection of spherical adsorbent particles in an equilibrium hard sphere configuration. Each microsphere is modeled as a continuum of Lennard-Jones 12-6 interaction sites. The interac-
3224 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
tion of a Lennard-Jones adsorbate particle with one of these composite spheres has been shown to be given by32
UCS(d) )
((
16 π F R3 3 sf s
d6 +
)
21 4 2 R6 12 d R + 3d2R4 + σ 5 3 sf (d2 - R2)9 σsf6
)
(d2 - R2)3
(9)
In this expression, d is the distance between the adsorbate and the center of the composite sphere, σsf and sf are the solid-fluid collision diameter and well depth, respectively, and Fs is the number density of Lennard-Jones sites comprising the adsorbent. The interaction energy diverges as d approaches R, the effective hard core radius of the composite sphere. The physical parameters used here are the same as those determined by Kaminsky and Monson, namely, R ) 1.347 nm and Fs ) 44/nm3. The hard sphere packing fraction is η ) 0.386. Using Delaunay tetrahedra to form a tessellation of the adsorbent volume,30 we find that a pore can be characterized by a spherical cavity with an average diameter of approximately 20 Å. In simulations of alkanes, we ignore interactions between the adsorbate and polar end groups bound to the surface of the microspheres. The gas-solid Lennard-Jones interaction parameters are handled through Lorentz-Berthelot mixing rules. Grand canonical Monte Carlo (GCMC) simulations are performed with configurational biasing. The general theory behind this method is discussed in more detail in several works.20,33-34 Here, we provide only a description of a biased move involving chain molecule insertions and refer the reader to other sources for particular details on the implementation of the method.9,35 In an insertion attempt, segments of a chain molecule are “grown” sequentially within the system volume. The location of the first segment is chosen randomly, as in the normal GCMC insertion scheme. For the next segment, a set of k trial positions are generated, each the bond length distance d from the center of the first chain segment. A particular trial position is generated with a probability proportional to the intramolecular energy associated with that trial segment position. Among the k trial positions, a candidate at position ri is selected according to the probability:
pi )
exp(-βUi,ext(ri))
(10)
k
exp(-βUi,ext(rj)) ∑ j)1 where Ui,ext(rj) is the total intermolecular energy of the ith trial segment at position rj (including Lennard-Jones and external gas-solid interactions). The chain growth process is then repeated, growing from the center of the previously chosen segment position, until growth of the chain is completed. The trial configuration is accepted according to
(
pins ) min 1,
zV 〈Wigext〉(N
)
Wnew
+ 1)
(11)
Figure 2. Ideal gas Rosenbluth factors for n-alkanes as a function of reduced temperature. At a given temperature, the quantity 〈WIGext〉 is a required input for GCMC simulations employing the configurational bias sampling algorithm and is calculated as the ensemble-averaged Rosenbluth factor for a single n-alkane chain in the bulk. (0) n-octane; (4) n-hexane; (O) n-decane; (3) ndodecane. Lines are provided as a guide to the eye.
where z is the activity of the adsorbate and Wnew is the Rosenbluth factor for the inserted chain, Nseg
k
∑exp(-βUi,ext(rj))) ∏ i)2 j)1 (
Wnew ) exp(-βU1,ext)
kNseg
(12)
Moves for deletions and regrowth of polymer chains follow analogous procedures. Finally, it is important to recognize that the bulk reservoir in equilibrium with the simulation box is not the same as the conventional GCMC reservoir. In the unbiased GCMC scheme, the reservoir is a system of gas molecules at the same temperature and chemical potential as the simulation box. At low enough densities, the gas may be treated as ideal. In this case, the reservoir is approximated as a system of ideal chains, which possess only internal (bond-bending and torsional) interactions, as opposed to ideal gas chains (which possess additional intrachain nonbonded Lennard-Jones interactions). This implies that the reservoir chemical potential is shifted with respect to the chemical potential of the real fluid. The magnitude of this shift can be calculated as the average Rosenbluth factor (eq 12) for a single ideal gas chain in the bulk at a given temperature and is denoted as 〈Wext ig 〉 in eq 11. This quantity is only a function of temperature and is plotted for several alkanes in Figure 2. In a given simulation, the number of trial orientations generated per segment k was chosen as the same as the number of segments in the alkane chain. Depending upon the density, between 1 and 5 × 106 moves were attempted in generating a Markov chain. A minimum of the first 25% of the configurations were used for equilibration and not included in calculating averages. Uncertainties were estimated using subblock averages.
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3225
Figure 3. Adsorption isotherms of various n-alkanes on model silica gel at T ) 311 K. (O) n-butane; (0) n-hexane; (4) n-octane; (]) n-dodecane.
Figure 4. Adsorption isotherms of (O) n-butane and (0) n-hexane on Davison silica gel at 303 K.36 The gel sample used in the study had a BET surface area of 670 m2/g, a pore diameter of 0.43 cm3/ g, and an average pore diameter of 22 Å.
Results and Discussion Isotherms. Isotherms for n-butane, hexane, octane, and dodecane were computed at 311 K and are plotted in Figure 3. Note that the horizontal axis is logarithmic. In a linear plot, these isotherms would have a concavedown initial curvature. All isotherms exhibit type I loading behavior except n-dodecane. For the latter case, the inset of the graph clearly indicates a plateau region at approximately 0.35 mmol/g of adsorbent. A plateau region in the isotherm has also been found in similar simulations of n-C14H30. This “knee” is characteristic of type II isotherm behavior and is usually indicative of mesoporous or nonporous solids. The plateau of such isotherms corresponds to the completion of a monolayer of coverage on the surface. The change in shape of the isotherm from type I to II as the alkane chain length increases shows how adsorbate molecules placed in the same adsorbent will sense very different environments. As the alkane chain length increases, we see an increased layering tendency of molecules on the surface, which must be induced by the increased entropic resistance toward adsorption in the small pores of the silica gel. Limited adsorption isotherm data in the literature are available on these systems, but we note that Al-Sahhaf et al. measured adsorption isotherms of butane, pentane, and hexane between 283 and 303 K on a similar high surface area silica gel.36 The gel employed in this study had a BET surface area of 670 m2/g, a pore volume of 0.43 cm3/g, and an average pore diameter of 22 Å. Isotherms for butane and hexane at 303 K from this work are depicted in Figure 4 and exhibit loading behavior that is qualitatively similar to our calculated isotherms. Discrepancies between the experiment and our model may stem from the differences in the microstructural features of the adsorbent and the slightly different temperature of the isotherms. Although we are aware of no experimental data for adsorption isotherms on silica gel for alkanes longer than n-heptane,37 we can make comparisons to studies of similar systems. In particular, investigations of Lennard-Jones polymer chains adsorbed in graphitic slit
pores by Vega and co-workers7 offer an interesting comparison to our results. In their work, adsorption of 20-mer chains at a supercritical temperature of T* ) 5.0 is examined in a regime of strong adsorbate-wall interactions and a very high degree of confinement (the pore width is 5.5σff, the collision diameter of the polymer bead). The chains are modeled as Lennard-Jones segments bound together through a harmonic spring potential. They find that the density of the pore is significantly enhanced compared to that at the bulk phase at low loadings because of the strong gas-solid interactions. At higher loadings, however, the bulk and pore densities in equilibrium are comparable, reflecting the difficulty of forcing the chains into the narrow pore. We note here that the Lennard-Jones chains used in Vega’s work are significantly more confined than those in our study. At the conditions of their simulations, the ratio of the squared radius of gyration of the chains to the width of the pore is approximately unity. In our work, this ratio is approximately 0.2. This explains why no plateaus are observed in the isotherms of Vega et al.; their studies focus on pores too small to observe layering effects of polymers at the surface of the adsorbent. Studies of polymers in confined environments commonly employ scaling concepts as characterization tools. They have been used to examine the concentration profiles of polymer chains in both good and bad solvents adsorbed at a surface and confined between plates.2,38 To explore scaling behaviors, however, it must be possible to examine properties of the polymer with respect to nondimensional parameters that characterize the system. For instance, for polymer chains with only repulsive interactions, scaling arguments predict that the bulk radius of gyration of a single chain with n segments confined between two walls separated by a distance d can be written as
〈Rgyr(d)〉 〈Rgyr,bulk〉
)
(
d
〈Rgyr,bulk〉
)
-0.25
(13)
3226 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
where Rgyr and Rgyr,bulk are the radius of gyration of the confined polymer chain and the chain in the bulk, respectively. Van Giessen and Szleifer have confirmed eq 13 via simulation to good accuracy.8 For more complicated systems, scaling behaviors may be more difficult to detect through computer simulation. This was illustrated by the same workers, who investigated a model triblock copolymer confined between adsorbing walls. The polymer had the structure AnBmAn, where the wall-A interactions were attractive and the wall-B interactions were repulsive.8 In this case, the properties of the system are expected to be dependent on a number of parameters, including the strength of the wall-A segment interactions, strength of the A-A interactions, ratio of A to B segments in the polymer chain, total chain length, and degree of confinement between the plates. Due to the number of parameters, no scaling behaviors were detected through their simulations. Given the similar complexity of our system and the computational burden associated with generating our adsorption data, it appears impractical to gather enough adsorption data with respect to carbon chain length to investigate scaling laws here. Alkanes adsorbing on modified flat silica surfaces display a prewetting transition. The uptake of alkanes on such a surface decreases, relative to bare silica, due to screening of the interactions by the chemisorbed methyl groups.13 In modeling studies, Schlangen and Koopal14 observed the prewetting transition for octane on an alkylated surface. Their model had similar surface interactions for the interior and end segments of the chain. This is in contrast to our alkane model, where the energetic well depth of the interior and end chain segments are 47 and 114 K, respectively. In addition, the interfacial tension of octane on an alkylated surface is very close to the critical surface tension for wetting. Thus, small temperature changes or surface modifications were found to be capable of causing a transition from complete to partial wetting as well. We did not expect to see a prewetting transition in our system, because of the geometry, the stronger gas-solid interactions, and high degree of confinement of the fluid. Adsorbate Structure. For chain molecules, adsorbate structure is conveniently examined through pair correlation functions and orientational order parameters. In Figure 5 we present solid-fluid pair correlation functions for n-dodecane at several loadings. The correlations are computed between each chain segment of the adsorbate and the silica microspheres. At low loadings, the correlation functions exhibit a strong initial peak approximately 4 Å in width. Increased loading brings about the growth of the second peak, indicating the filling of the interior of the pore spaces. The orientational order parameter gives a measure of the alignment of admolecules with respect to the surface of the adsorbent. We denote this as s(θ), defined as
3 1 s(θ) ) 〈cos2 θ〉 2 2
(14)
where θ is the angle between the vector connecting two adjoining chain segments and the vector normal to the surface of the microsphere. This function ranges in values from -0.5 to 1.0, where the minimum value corresponds to chain segments lying parallel to the surface and the maximum implies orientation normal to the surface. The angle is averaged over all chain
Figure 5. Solid-fluid pair correlation functions for n-dodecane at various loadings at T ) 311 K. The correlation function is between the centers of the silica gel microspheres and each chain segment of the alkane. (______) 0.08 mmol/g adsorbent; (- - -) 0.36 mmol/g; (-‚-‚) 1.02 mmol/g.
Figure 6. Orientational order parameter for n-dodecane on model silica gel at 311 K. The loadings correspond to the pair correlation functions of Figure 5.
segment bonds as a function of distance from the surface. This function is shown in Figure 6 for adsorbed n-dodecane at several loadings. At distances corresponding to the first adsorbed layer, s(θ) is near its minimum, reflecting parallel orientation of chain segments to the surface. The chains in this layer adopt this orientation to maximize the number of contacts with the surface, which provides energetic stability. At larger distances from the surface, the orientation parameter rises steeply, but shows a more shallow minimum again at a distance corresponding to the second layer. Chain segments in the second layer clearly are less strongly affected by the surface, although there is still a slight preference toward packing parallel to the surface. The order parameter then rises to slightly positive values and slowly decays toward zero. The
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3227
Figure 7. Isosteric heats of adsorption vs adsorbate loading for n-alkanes on model silica gel at T ) 311 K. Points correspond to heats calculated by ensemble fluctuations, and lines correspond to heats calculated through numerical differentiation of eq 16. (O) n-butane; (0) n-hexane; (4) n-octane; (]) n-dodecane.
decay toward zero is expected, as the orientation between solid and adsorbate become uncorrelated. As loading increases, we see relatively little variation in these curves. These observations are in contrast to those of Schlangen et al., who saw reorientation of alkane chains perpendicular to the surface near the completion of a monolayer of coverage.13,14 We attribute these differences to the strong enthalpic driving force for adsorption in the small pores of the model silica gel, although entropic effects are bound to play a role as well. Finally, we note that similar orientational behavior has been observed for alkanes and model polymer chains confined in slitlike pores.22,39 Isosteric Heats of Adsorption. Figure 7 depicts the isosteric heats of adsorption as a function of adsorbate loading. The isosteric heat of adsorption can be defined as
qst ) Hbulk -
( ) ∂Uads ∂Nads
(15)
T,Vads
where Hbulk is the molar enthalpy of the bulk. The second term may be approximated by fitting the adsorbate energy to a polynomial function of adsorbate density, and numerically differentiating,40
( ) ∂Uads ∂Nads
( )
) Uads + Fads
T,Vads
∂Uads ∂Fads
(16)
T,Vads
We have also calculated the heat of adsorption using the ensemble fluctuation method, as41
qst ) Hbulk -
〈UadsNads〉 - 〈Uads〉〈Nads〉 〈Nads2〉 - 〈Nads〉2
(17)
where 〈X〉 is the equilibrium average of X. Both methods yield similar results, although the heats derived through numerical differentiation are much smoother functions of loading. This is especially evident
at higher loadings, where the lower insertion and deletion acceptance rates in the GCMC simulations create large uncertainties in the heats obtained through the fluctuation method. In general, at low loadings, the magnitude of the heats decrease with increasing loading, behavior typical of a heterogeneous adsorbent. In all cases, the heats of adsorption level off at a loading that corresponds to roughly a monolayer of coverage. The approximately constant values of the heat of adsorption at higher loadings appear to be at least 50% larger than the corresponding bulk heats of vaporization. This suggests that the effect of the confining environment, whose average pore diameter is approximately 20 Å, is significant. In comparison, the heats of adsorption measured through the differentiation of adsorption isotherms by Al-Sahhaf et al. predicts very similar heats for both hexane and butane. This observation runs counter to the expectation that the isosteric heat of adsorption will increase as the chain number increases. Heats of ImmersionsEstimation of Physical Parameters. To calculate the heat of immersion, estimates of the bulk density, specific surface area, and pore volume for the model adsorbent must be made. For the bulk density, we adopt the value of F ) 1.129 g/cm3 measured by Gangwal et al., which was used in the design of the model adsorbent. The latter two quantities are readily obtained experimentally, but values measured on the same sample can vary significantly, depending on the nature of the test probe, particularly when the adsorbent has developed microporosity. For instance, Gonzalez-Martin and co-workers42 found that surface area measurements of carbon blacks obtained from applying the BET equation to nitrogen and pnitrophenol adsorption, and using the Harkins/Jura equation (which estimates the statistical thickness of the adsorbed layer), can vary by as much as 50%. Similar variations are observed in the pore volume as well. It is therefore imperative to calculate the accessible surface area and pore volume to the adsorbate under consideration. For our model, the estimation of these quantities is complicated by the fact that there is a region of excluded volume around the composite microspheres that compose the model gel. Thus, a molecule adsorbed near the surface will reside at some equilibrium separation from the surface and perceive a different accessible surface area and pore volume in the system. We can estimate these parameters with the use of the random media model of Rikvold and Stell.43 In that work, expressions for the surface area and porosity are developed for collections of sphere assemblies as a function of packing fraction and degree of matrix particle interpenetrability. These equations are adaptable to probe molecules of nonzero radius. As an example, we can approximate a helium porosimetry experiment with a hard sphere probe (employing an equivalent hard sphere radius of the size of helium;44 r ) 1.09 Å) in our microporous silica gel (with packing fraction η ) 0.386 and Rc ) 13.47 Å). This yields a porosity of 0.513, which is very close to 0.49, the experimentally observed porosity of the xerogel upon which the Kaminsky and Monson’s model is based.31,32,45 Using this approach, we find that the n-alkane models perceive a pore volume of 0.47 cm3/g and surface area of 875 m2/g. We note that this specific surface area is approximately 25-30% higher than nitrogen adsorption BET values on xerogels, but this
3228 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 2. Parameters Used in Heats of Immersion Calculations for n-Alkanes at 311 K48,49 species
Fliq. (g/cm3)
∆Hvap. (kcal/mol)
W0 (mmol/g of solid)
n-butane n-hexane n-octane n-dodecane
0.57 0.64 0.68 0.74
4.7 7.4 9.8 13.7
4.62 3.50 2.80 2.05
Figure 8. Heats of immersion for various n-alkanes as a function of alkane chain length. Experimental data is also presented on two different samples of silica gel, both from Dubinin et al.46 (b) calculated heats of immersion; (0) 2-MSK silica gel; (4) KSK-2 silica gel. Physical characteristics of the samples are also shown in the figure.
method is known to underpredict surface area in the presence of micropores. We use these numbers, in conjunction with the values listed in Table 2, for the bulk liquid density, enthalpy of vaporization, and maximum loading of the model silica gel pore, to evaluate eq 6 for the heat of immersion. Figure 8 shows the calculated heats of immersion for four different alkanes as a function of carbon chain length. Immersion heats determined calorimetrically from Dubinin et al.46 of two different samples of silica gel in alkanes are also shown for comparison. The surface area and maximum pore diameter of each sample is also presented in the figure. Given the uncertainties in our assumptions, we find suprisingly good agreement between the numbers obtained by simulation and experiment. In both experiment and simulation, the heats of immersion are not strong functions of carbon chain length. We note that for hexane and heptane, the heats of immersion measured by Dubinin et al. do not change significantly between the two samples. Thus, the effect of decreasing the average pore size on the heat of immersion per unit area appears to be negligible. Heats of immersion of carbon blacks in alkanes were also found to have a similar insensitivity to alkane chain length.47 Conclusions The adsorption loading behavior of n-alkanes in a model microporous silica gel was examined using configurational bias grand canonical Monte Carlo simulations. This technique has been demonstrated to allow
direct and efficient computation of isotherms of chain molecules. The technique developed by Smit9 holds considerable promise for further investigations in similar systems, such as in examining the influence of chain branching, or in the case of alcohols, hydrophililc end groups on adsorption in microporous materials. Isotherms of n-alkane chains varying in length from n-butane to n-dodecane were found to undergo a change from continuous micropore filling to layering at the surface as the chain length increases. We attribute this behavior to the increasing entropic resistance toward adsorption with increasing chain length. The observed plateau in the isotherm should become more pronounced as the chain length increases. The isosteric heats of adsorption show similar behavior with increasing alkane chain length, with pronounced enhancement of the heats at low coverage. Calculations of the heats of immersion were performed via computer simulation, with good agreement between existing experimental data. However, it is extremely important to refer the immersion heat to the accessible surface area of the adsorbent. While it takes a complete isotherm to generate a single value of the heat of immersion, the method can be applied to any variety of adsorbents and fluids and can be easily extended to fluid mixtures. The adsorption of polar-nonpolar mixtures would be of interest, although no attempt to do so was made in this work. We note, however, that Bering and coworkers37 considered the system n-heptane-watersilica gel and found that adsorption of either component in the presence of the other did not have a significant effect on the adsorption behavior of the system. It was concluded that n-heptane and water formed a “microheterogeneous” system, with adsorption of each component occurring on different adsorption sites in the gel. Acknowledgment This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences. Literature Cited (1) Scheutjens, J. M. H. M.; Fleer, G. J. Statistical theory of the adsorption of interacting chain molecules. 1. Partition function, segment density distribution and adsorption isotherms. J. Phys. Chem. 1979, 83, 1619-1635. (2) de Gennes, P. G. Polymer solutions near an interface. 1. Adsorption and depletion layers. Macromolecules 1981, 14, 16371644. (3) Sen, S.; Cohen, J. M.; McCoy, J. D.; Curro, J. The structure of a rotational isomeric state alkane melt near a hard-wall. J. Chem. Phys. 1994, 101, 9010-9015. (4) Raghavan, K.; MacElroy, J. M. D. Molecular-dynamics simulations of adsorbed alkanes in silica micropores at low-tomoderate loadings. Mol. Simul. 1995, 15, 1-33. (5) Bates, S. P.; van Well, W. J. M.; van Santen, R. A.; Smit, B. Energetics of n-alkanes in zeolites: A configurational-bias Monte Carlo investigation into pore size dependence. J. Am. Chem. Soc. 1996, 118, 6753-6759. (6) Bates, S. P.; van Well, W. J. M.; van Santen, R. A.; Smit, B. Location and conformation of N-alkanes in zeolites: An analysis of configurational-bias Monte Carlo calculations. J. Phys. Chem. 1996, 100, 17573-17581. (7) Vega, L. F.; Panagiotopoulos, A. Z.; Gubbins, K. E. Chemicalpotentials and adsorption-isotherms of polymers confined between parallel plates. Chem. Eng. Sci. 1994, 49, 2921-2929. (8) van Giessen, A. E.; Szleifer, I. Monte Carlo simulations of chain molecules in confined environments. J. Chem. Phys. 1995, 102, 9069-9076.
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Received for review February 10, 1998 Revised manuscript received April 27, 1998 Accepted April 29, 1998 IE9800804
