Predicting Adsorption on Bare and Modified Silica Surfaces

Article pubs.acs.org/JPCC

Predicting Adsorption on Bare and Modified Silica Surfaces Matthieu Lépinay,†,‡,∥ Lucile Broussous,† Christophe Licitra,‡ François Bertin,‡ Vincent Rouessac,∥ André Ayral,∥ and Benoit Coasne*,⊥ †

STMicroelectronics, 850, rue Jean Monnet, 38926 Crolles cedex, France Université Grenoble Alpes, F-38000 Grenoble, France, and CEA, LETI, MINATEC Campus, F-38054 Grenoble, France ∥ Institut Européen des Membranes, CNRS/ENSCM/Université de Montpellier, CC047, 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France ⊥ Multiscale Materials Science for Energy and Environment, UMI 3466 CNRS−MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States ‡

S Supporting Information *

ABSTRACT: We show that Derjaguin’s theory of adsorption can be used to predict adsorption on bare and modified surfaces using parameters available to simple experiments. Using experiment and molecular simulation of adsorption of various gases on hydroxylated, methylated, and trifluoromethylated silica, this simple parametrization of Derjaguin’s model allows predicting adsorption on any functionalized surface using a minimum set of parameters such as the heat of vaporization of the adsorbate and the Henry constant of the adsorption isotherm. This general yet simple scheme constitutes a powerful tool as it avoids having to carry out tedious and complex adsorption measurements.

1. INTRODUCTION Gas adsorption on bare and modified surfaces such as silicabased substrates is relevant to nanofluidics,1 nanolubrication,2 wetting,3 sensing,4 membrane separation,5 etc. In these applications, the process or device performance is governed by the interfacial properties of the film adsorbed at the surface. Adsorption of a given gas on bare and modified surfaces is also important in catalysis since the presence of a film at the surface of the catalyst hinders or enhances the yield and kinetics of the reaction. The thickness t(P) of the adsorbed film as a function of gas pressure P is also a key quantity for the characterization of surfaces and porous solids. For instance, this function is routinely used to determine specific surface area, microporosity, and mesoporosity in porous materials through techniques such as the Brunauer−Emmett−Teller (BET) or t-plot methods.6,7 The applications and phenomena above, which involve different adsorbate and surface types, show that a description of the thickness of the adsorbed film is needed to design efficient processes. In particular, there is strong motivation to develop a general approach to predict adsorption of different gases or liquids on a given substrate. While many approaches have succeeded in describing the thickness of the nitrogen film adsorbed at 77 K on silica surfaces (Halsey t-plot, Harkins and Jura t-plot, Frenkel−Halsey−Hill theory, etc.),6 a general model of the adsorption of any gas on different surfaces is still missing. In addition to parameters such as the temperature and the density of the adsorbed phase (assumed to be close to the liquid density of the adsorbate at the same temperature), two © XXXX American Chemical Society

other parameter types are needed to describe the change in the adsorbed amount with pressure: (1) an intrinsic property of the fluid describing its interfacial behavior and (2) a constant that is characteristic of the strength of the surface/adsorbate interactions. This is explicit in adsorption models such as the BET theory in which one needs to estimate the energies of the adsorbate molecule in the liquid phase and adsorbed at the surface. In this paper, we use the general framework of Derjaguin’s theory8,9 of adsorption and capillary condensation to describe gas adsorption on bare and modified surfaces. Using both experiments (ellipsometric porosimetry) and molecular simulations (atomistic Grand Canonical Monte Carlo) of the adsorption of adsorbates with various polarities (water, methanol, and toluene) on hydroxylated, methylated, and trifluoromethylated silicas, we show that Derjaguin’s thermodynamic model requires a very small set of parameters such as the heat of vaporization of the adsorbate and the Henry constant of the adsorbate/surface adsorption isotherm (i.e., the slope of the adsorption isotherm at very low pressures). Given the simplicity of this general method, it can be extended to other surface and adsorbate types. In particular, this model allows predicting pure component adsorption isotherms which can be used to determine coadsorption selectivities using theories such Received: November 24, 2014 Revised: February 18, 2015

A

DOI: 10.1021/jp511726a J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C as the ideal adsorbed solution theory (IAST).10 This theory, which requires pure component adsorption data such as those provided by the simple thermodynamic model reported in the present work, is known to accurately predict coadsorption isotherms. The remainder of this paper is organized as follows. Section 2 provides details about the experimental and computational details. Section 3 first reports the molecular simulation and experimental data for water, methanol, and toluene adsorption on hydroxylated, methylated, and trifluoromethylated silica surfaces. Section 3 then presents the simple model of fluid adsorption on bare and modified silica surfaces. This model and its parametrization are validated by comparing its predictions against experimental data for porous silicas. Section 4 reports concluding remarks.

desorption isotherms. For each gas partial pressure P/P0 (P0 is the bulk saturating vapor pressure of the adsorbate), the gas penetrates the material and equilibrium between the adsorbed phase and the gas phase is quickly reached. For these standard adsorption isotherms, after each gas pressure increase, a stabilization delay of 4 min was applied prior to any collection of ellipsometric and pressure data. To complete the adsorption isotherms with the desorption branch, a second stage consisted of a progressive decrease of the relative pressure of the gas with pumping out the vapor. Three adsorbates (water, methanol, and toluene) were used for both experiments and molecular simulations. These adsorbates were selected as they possess different polarities, sizes, and surface tensions (Table 1). Table 1. Properties of the Solvents Considered In This Work

2. EXPERIMENTAL AND COMPUTATIONAL METHODS The different substrates and adsorbates considered in this work are shown in Figure 1. In what follows, we first describe the

dipole moment (D) molar mass (g/mol) equivalent diameter13 (Å) enthalpy of vaporization14 (kJ/mol)

toluene

methanol

water

0.375 92.14 5.68 33.18

1.70 32.04 4.08 35.21

1.85 18.02 3.43 40.65

2.2. Molecular Simulation. 2.2.1. Grand Canonical Monte Carlo. Grand Canonical Monte Carlo (GCMC) simulations were used to simulate adsorption of water, methanol, and toluene on the different silica surfaces prepared in this work. The GCMC technique is a stochastic method that simulates a system having a constant volume V in equilibrium with an infinite reservoir of adsorbate molecules imposing its chemical potential μ and temperature T. The pressure of the reservoir is determined from its chemical potential μ and temperature T in order to obtain absolute adsorption isotherms as a function of pressure. For each substrate and adsorbate, the molecular simulations were performed at room temperature T = 300 K. For each pressure, the adsorbed amount is estimated from the average number of adsorbed molecules over at least 1000 GCMC configurations (taken after a first equilibration run during which the number of molecules and energy evolve to reach a plateau). For all gases and substrates, we have considered pressures ranging from 10−3P0 to 0.9P0, where P0 is the bulk saturating vapor pressure of the adsorbate. We did not attempt to obtain data at larger pressures as it would induce very large adsorbate uptakes at pressures close to the bulk saturating vapor pressure. Moreover, including larger pressures in our analysis would not bring insights as most of the adsorption phenomenon at such large pressures is driven by the gas/liquid properties of the adsorbate only. As for the low pressure range, all our molecular simulations were performed for pressures as low as 10−3P0. Considering the low bulk saturating vapor pressure of the different gases considered in this work (P0 ∼ 4.4 kPa for water, 18.7 kPa for methanol, and 4.2 kPa for toluene), the lowest pressures are already low enough to provide physical insights into the low adsorbed amount regime where the surface/adsorbate interactions play a crucial role. 2.2.2. Surface Models. Three realistic models of bare and modified silica were prepared: a hydroxylated surface, a methylated surface, and a trifluoromethylated surface. These materials were prepared from a block of bulk silica, which was obtained from crystalline silica using a “cook and quench” procedure, i.e., classical molecular dynamics at high temperature. The crystalline silica (cristobalite) cell was melted at 2000 K by means of molecular dynamics simulation, using the

Figure 1. (left) Atomistic models of the hydroxylated, methylated, and trifluoromethylated surfaces. A block of amorphous silica is cut and set in contact with reservoirs along the z direction to create two free interfaces. Red and orange spheres are the O and Si atoms. Blue sticks are Si dangling bonds at the surfaces which are then saturated with OH, CH3, or CF3 groups (center). (right) The three adsorbates considered are water, methanol, and toluene. Red, orange, gray, yellow, and white spheres are the O, Si, C, F, and H atoms. Pink spheres are the CH3 groups in methanol and toluene which are treated as a united atom.

experimental methods as well as the samples considered. We then present the molecular simulation techniques and interaction models used to describe water, methanol, and toluene adsorption on hydroxylated, methylated, and trifluoromethylated silica surfaces. 2.1. Experimental Methods. The optical ellipsometric porosimetry (EP) measurements were done on silica materials to determine adsorption of water, methanol, and toluene. Measurements were performed on spin-on deposited porous silicas which have been obtained from sol−gel chemistry. These samples have been described elsewhere.11,12 The silica precursor for both films was tetraethoxysilane (TEOS). The thin layers were deposited by spin-coating on silicon wafers. The thicknesses of the silica layers, which were measured by spectroscopic ellipsometry, are 512 and 724 nm, respectively. In situ ellipsometric measurements of adsorption are performed using a spectroscopic ellipsometer and a pumping system. The vacuum chamber is outgassed before the introduction of the adsorbate in the chamber. The EP apparatuses consisted of two multiwavelength Sopra EP12 and Semilab GES5E ellipsometers. The first apparatus has an automated vacuum system, while the second apparatus is connected to a manually activated pumping system (Drytel 1025, Alcatel). Prior to the measurements, materials were outgassed up to 200 °C for 5 min. The closed cell containing the sample was then pumped down to vacuum (8 × 10−2 mbar) to empty the porosity before measuring any gas adsorption− B


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The Journal of Physical Chemistry C DL_POLY Classics package15 and the CHIK potential.16 The silica was then quenched to obtain a silica amorphous structure: 300 ps molecular dynamics runs allowed the system to relax, with the temperature being lowered by 10% for each run, until room temperature (300 K) was reached.17,18 A silica portion (approximately 3 × 3 × 3 nm3) was then cut to fit in a 3 × 3 × 10 nm3 simulation box with periodic boundary conditions. The oxygen dangling bonds at the upper and lower surfaces along the z surface were then passivated with H atoms to obtain hydroxylated surfaces (resulting in an OH surface density of 6.8/nm2). In order to obtain methylated and trifluoromethylated surfaces, the Si−OH groups were replaced with SiCH3 and SiCF3 groups. For each type of modified surface chemistry, we considered two substitution ratios: η = 0.7 and 1. Finally, the surfaces were allowed to relax by means of molecular dynamics simulation prior to simulating adsorption of the three chosen adsorbates.19 2.2.3. Solvent Models and Molecular Interactions. Water was modeled as a rigid molecule using the simple point-charge model.20,21 The HOH angle is 109.47°, and the rigid OH bond length is 1.0 Å. The hydrogen and oxygen atoms carry partial charges of +0.41e and −0.82e, respectively. The oxygen atom is the center of a Lennard-Jones (LJ) interaction potential. The methanol molecule was described with the coarse-grained model developed by Schnabel,22 where the hydrogen atoms of the methyl group are omitted so that CH3 is treated as a single site in order to decrease the computational time without compromising the physical description of the molecule. The distance between the CH3 group and the oxygen atom of the OH group and between the hydrogen and oxygen atoms of the OH group are 1.4246 and 0.94510 Å, respectively. The CH3OH angle is equal to 108.53°. Each of the three sites interacts through the Coulombic interaction, while the repulsion/dispersion interaction is described using a LennardJones (LJ) potential. The partial charges for CH3, O, and H are 0.24746e, −0.67874e, and 0.43128e, respectively. Toluene was described using the OPLS model by Jorgensen et al.23 with the geometry described by Snurr et al.24 This model is coarsegrained with one site for the CH3 group while all other atoms are explicitly described. Bond lengths are C−C = 1.40 Å, C−H = 1.08 Å, and C−CH3 = 1.51 Å, and all angles are equal to 120°. The partial charges for hydrogen and carbon atoms are 0.115e and −0.115e, respectively (the partial charge for the CH3 group is also +0.115e to ensure electroneutrality of the molecule). The interactions between each site of the solvent molecules and the silicon and oxygen of silica, as well as with the atoms of the CH3 and CF3 groups, are the sum of the Coulombic and dispersion interactions with a short-range repulsive contribution. The geometry of these molecules, as well as the interaction potential parameters obtained from the literature, are given in Tables S1 and S2 of the Supporting Information. We emphasize that the adsorption isotherms obtained from Grand Canonical Monte Carlo simulations were not fitted against the experimental results. In particular, the interaction potentials which were used to model adsorption of different gases on bare and modified silicas were derived from simple parameters available in the literature. Parameters such as the partial charges carried by the atoms of the adsorbate molecules and the silica surfaces are taken from previous works in which they were estimated to reproduce simple properties such as the liquid density, equation of state, etc. Other parameters are the cross-atom interaction parameters such as the dispersion and

repulsive interaction parameters which were obtained from physical combining rules (Lorentz−Berthelot combining rules). Finally, while using another set of force fields would necessarily lead to different numbers, we believe that the results reported in the present work would remain qualitatively valid as the force fields used in the first place are known to provide a reasonable picture of silica surfaces and the fluid properties of the different adsorbates considered.

3. RESULTS AND DISCUSSION Ellipsometric porosimetry (EP) measurements were performed to probe water, methanol, and toluene adsorption on two silica surfaces which have been described elsewhere.11,12 In our atomistic simulations of water, methanol, and toluene adsorption, realistic models of silica surfaces were prepared: (1) hydroxylated, (2) methylated, and (3) trifluoromethylated. As described in section 2, the two last surfaces were obtained from the hydroxylated surface by replacing the OH groups by CH3 and CF3 groups. All these materials were prepared from a block of bulk amorphous silica.25 For each type of modified surface chemistry, we considered two substitution ratios: η = 0.7 and η = 1. Molecular simulations for planar substrates cannot capture the curvature effects inherent to the porosity of the experimental samples (pore filling). However, by restricting ourselves to the low pressure range, the comparison between the experimental and simulation data is fully relevant since such curvature effects are negligible at low pressure. Such an approximation is at the basis of the t-plot method6,26 in which one determines pore surface roughness and microporosity by comparing adsorption data with reference data obtained for a planar, nonporous reference substrate (see ref 27 for a recent discussion of curvature effects on the validity of the t-plot method). The simulated adsorption data obtained in the present paper for planar surfaces and the thermodynamic model described below are particularly useful as they allow determining t-curves (reference data used in the t-plot model) for any adsorbate/adsorbent system. Moreover, this model will be validated at the end of this section by comparing its predictions against experimental adsorption data for porous silicas. Figure 2 shows the adsorption isotherms for water, methanol, and toluene at room temperature on the bare and modified silica surfaces. The results for both substitution ratios η = 0.7 and 1 are reported for the methylated and trifluoromethylated surfaces. Figure 2 also reports the experimental EP measurements on porous silicas. We also show data for mesoporous MCM-41 and SBA-15 whose surface is made up of hydroxylated silica.28−31 The experimental data have been converted in units of micromoles per square meter (μmol/m2) by normalization to their specific surface area. For all samples, the latter was obtained from the N2 adsorption isotherm at 77 K using the BET model. The uncertainty over the specific surface area determined from the BET method makes the quantitative comparison between the experimental and simulated data difficult. Such a comparison is in itself uncertain as the experimental samples exhibit some mesoporosity (in contrast to the planar substrates considered in the molecular simulations). As a result, the experimental data cannot be used to quantitatively validate our molecular simulations because pore filling at low pressure makes the experimental adsorbed amounts larger than what would be observed for planar (or macroporous) substrates. On the other hand, as discussed below, the trends observed in such experimental data show that C


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and SBA-15 samples. The data for methanol on hydroxylated surfaces are also in agreement with the experiments performed by Mikhail et al. for different oxide surfaces.32 The simulation data also agree with our EP measurements for hydroxylated samples although the simulated adsorbed amounts for toluene slightly overestimate the experimental data. Considering the large size of this molecule (∼0.8 nm), such a discrepancy is thought to be due to limited pore access in the experimental sample. As expected, the experimental data for the methylated sample, whose surface is hydrophobic, are close to the simulated data for the methylated and trifluoromethylated surfaces. For all adsorbates, adsorption overall increases upon increasing the polarity of the surface: trifluoromethylated, methylated, hydroxylated. In particular, almost no adsorption occurs on the trifluoromethylated surfaces due to the very hydrophobic nature of this substrate. While the adsorbed amount for a given adsorbate increases upon increasing the surface polarity, water adsorbed on the methylated surface is an exception. At low pressures, 90° corresponds to nonwetting. The spreading coefficient S = γsv − (γsl + γlv) allows describing the different situations; S > 0 corresponds to complete wetting while S < 0 corresponds to partial wetting. While this theory is very powerful, it is often not convenient from a practical point of view as estimating S is a difficult task. This comes from the fact that there is no simple and direct experiment to estimate γsv and γsl independently. The situation is even more complex as gas adsorption occurs (to minimize the free energy of the solid surface) so that γsv is given by Gibbs’ adsorption isotherm:

Figure 2. Water (top), methanol (middle), and toluene (bottom) adsorption isotherms on bare and modified silica. The simulated adsorption isotherms are (black symbols) hydroxylated, (red symbols) methylated, and (blue symbols) trifluoromethylated. For the methylated and trifluoromethylated surfaces, the filled and empty symbols are for η = 0.7 and 1, respectively. + and × are EP experimental data for the two silica samples. Purple × signs are for hydroxylated MCM-41/SBA-15,28−31 while purple + signs are for oxide surfaces.26

our molecular simulations capture the effect of surface grafting on the adsorption of different polar gases. For the sake of comparing the experimental and simulation data, the experimental data in Figure 2 have been plotted in the low pressure range only so that the data are representative of the adsorption regime only. For all gases, the simulated data for hydroxylated silica are consistent with the experimental adsorbed amounts, which show the ability of our model to describe, at least qualitatively, the physics at play. In particular, the simulated adsorbed amounts for water, toluene, and methanol on hydroxylated silica are in reasonable agreement with the data for MCM-41 D


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Table 2. Parameters Extracted from the Fit of the Adsorption Isotherms against the Derjaguin and FHH Models, and Also the Normalized Henry Constants KH′ = KHP0 hydroxylated

methylated (η = 0.7)

methylated (η = 1)

trifluoromethylated (η = 0.7)

trifluoromethylated (η = 1)

γsv = γ0 −

∫0

adsorbate

S (mJ/m2)

ξ (nm)

n

K (nm)

KHP0

water methanol toluene water methanol toluene water methanol toluene water methanol toluene water methanol toluene

68.3 94.9 45.6 96.3 91 20.2 106 56.8 9.7 36.6 16 2 44 33.2 0.2

0.0739 0.1934 0.2281 0.0759 0.1575 0.1768 0.0433 0.1207 0.077 0.0225 0.0233 0.0179 0.0249 0.024 0.0021

0.924 0.876 0.904 1.21 2.08 0.456 1.58 1.22 0.609 1.29 2.58 0.486 1.15 1.67 0.5139

0.155 0.560 0.749 0.183 0.321 0.455 0.145 0.235 0.167 0.0670 0.0506 0.0411 0.0892 0.0860 0.0037

0.5543 0.98299 0.21046 1.60932 1.34146 0.03779 2.05626 1.12064 0.04977 0.5257 0.48046 4.50 × 10−4 0.77363 0.49256 6.00 × 10−5

P

Γ(P)kBT d(ln P)

W (t ) = −

(1)

12πt

2

π (t ) = −

with

A d W (t ) = − SLV3 dt 6πt (4)

where γ0 is the surface tension of the solid surface (in vacuum), kB is Boltzmann’s constant, T is the temperature, P is the gas pressure, and Γ(P) is the adsorbed amount per unit of surface area. In order to derive a simple model of gas adsorption on bare and modified silica surfaces, we restrict ourselves to wetting situations (S > 0). Let t be the thickness of the liquid film (L) adsorbed on the solid surface (S). This film is in contact with a gas reservoir which imposes its temperature T and chemical potential μ. Assuming the gas behaves as an ideal gas, the pressure P is such that μ − μ0 = kBT ln(P/P0), where μ0 is the chemical potential corresponding to the bulk saturating vapor pressure P0. In the framework of Gibbs’ dividing surface formalism, gas adsorption on a wetting surface can be described using Derjaguin’s theory of adsorption and capillary condensation,8 which is derived from the grand free energy Ω of the system described above: Ω(t ) = −PVG − PLVL + γSLA + γLGA + AW (t )

ASLV

where ASLV is the Hamaker constant. In this case, eq 4 leads to the celebrated Frenkel−Halsey−Hill (FHH) equation:34 ⎡ ⎢ ASLV t=⎢ ⎢ 6π (ρL − ρG )kBT ln ⎣

() P P0

⎤1/3 ⎛ P ⎞−1/3 ⎥ ⎥ ∼ ln⎜ ⎟ ⎝ P0 ⎠ ⎥ ⎦

(5)

In practice, because molecular interactions are often the sum of different contributions (dispersion, multipolar interactions, short-range repulsion), experimental data do not follow the 1/3 power law in eq 5, and the following empirical equation is preferred: t = K ln(P/P0)−1/n, where K and n are parameters fitted against the experimental data. n is usually between 2 and 3, while K depends on the solid/adsorbate interaction strength. The most famous version of eq 5 is the Halsey t-plot35 in which n = 3 and K = 0.354 × 5/2.303 = 0.769 nm (see ref 36 for a discussion on the use of the FHH equation to describe gas adsorption on silica). When different force types are involved (not only nonpermanent dipole−dipole dispersion interactions), it is sometimes useful to consider that W(t) is an exponentially decaying function:9,37 W(t) = S exp(−t/ξ), where S is the spreading coefficient introduced above and ξ is a correlation length related to the range of the physical interactions responsible for adsorption. When the latter form is used, the film thickness t becomes (see derivation in the Supporting Information)

(2)

where A is the solid surface, PL is the pressure within the adsorbed phase, and VL = At and VG = V − VL are the volumes of the adsorbed and gas phases, respectively. γLG and γSL are the gas−adsorbate and solid−adsorbate surface tensions. W(t) in eq 2 is the interface potential which describes adsorption at the surface as it accounts for the interaction between the adsorbate molecule and solid surface. W(t) is related to the disjoining or solvation pressure π(t) which is often invoked to describe adsorption phenomena and deformation of porous materials:

⎛ξ ⎛ P ⎞⎞ t = −ξ ln⎜⎜ (ρG − ρL )kBT ln⎜ ⎟⎟⎟ ⎝ P0 ⎠⎠ ⎝S

⎛P⎞ d W (t ) π (t ) = − = PG − PL = (ρG − ρL )kBT ln⎜ ⎟ dt ⎝ P0 ⎠

(6)

We note that this model, which is based on Derjaguin’s theory of adsorption, exhibits some similarity with the polarization model of De Boer and Zwikker.38,39 In this model, the authors derived an equation somewhat similar to eq 6 by writing that the chemical potential of the adsorbed film having a thickness t is equal to the chemical potential of the liquid phase corrected for the surface interaction which is assumed to scale as ∼exp(−t). The use of such an exponentially decaying function to describe the effective adsorbate/surface

(3)

where ρG and ρL are the gas and liquid number densities of the adsorbate. The film thickness t at a given pressure can be obtained by minimizing Ω with respect to t: dΩ/dt = 0. The right-hand side of eq 3 is obtained by integrating the Gibbs− Duhem relationship dP = ρ dμ between P(μ) and P0(μ0). For dispersion forces, the wall-fluid interatomic potential scales as 1/r6. When integrated over a surface, one gets E


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The Journal of Physical Chemistry C interaction is consistent with the fact that thin films can be stabilized by short-ranged polar surface forces.3,40 Even in the case of an apolar adsorbate (argon), it was found that an exponentially decaying interaction better describes adsorption than the usual 1/t2 decay corresponding to van der Waals interactions.41 This exponential form for the adsorbate/surface interaction can be justified as follows. First, the adsorbate/ surface interaction necessarily deviates from the purely dispersive form since it corresponds to an effective interaction that encompasses different terms: dispersion interactions including higher order terms in the multipolar expansion of the electrostatic interaction and polarizability interactions arising from the interaction between the adsorbate and the electrostatic field created by the partial charges of the substrate atoms. Moreover, restricting surface interactions to two-body interactions such as the dispersion interactions is known to misrepresent such an effective interaction due to missing manybody interactions.42 Finally, the effective adsorbate/surface interaction must also include a short-range repulsive interaction when surface and adsorbate atoms become close enough to experience strong electron cloud repulsion.43 Owing to these different terms, the effective surface interaction between apolar or polar gases and surfaces is often described using a simple exponentially decaying function.37 In this function, ξ represents the typical length over which this effective interaction decays. In fact, the exponentially decaying function (with the spreading parameter S as prefactor) can be shown to verify the conditions that the surface interaction goes to zero when the film becomes very thick and converges toward S when the adsorbed film disappears (“drying”, i.e., t → 0).37,41 The simulated adsorption isotherms for water, methanol, and toluene on bare and modified silicas were fitted with eq 6. As shown in Figure 3, in all cases, eq 6 describes accurately the simulated adsorption data to pressures at least equal to 0.4P0. This result is confirmed by the insets in Figure 3 in which the same fits are shown when the data are plotted as a function of ln(−ln P/P0) instead of P/P0; as expected from Derjaguin’s model described in eq 6, the film thickness varies linearly with ln(−ln P/P0). These results confirm the ability of Derjaguin’s theory to describe, at least in the low pressure range ( 0.95, are given in Table 2. The Henry constants KH′ normalized to the saturation pressure P0 are also given in Table 2. In order to find simple expressions for the parameters S and ξ, we plotted these two quantities as a function of quantities that are characteristic of the adsorbate interfacial properties and of the adsorbate/adsorbent interaction strength. Considering that S describes the spreading of the adsorbate on the solid surface, it has to be related to the affinity of the substrate for the adsorbate. Figure 4 shows S as a function of the Henry constant KH for the different surfaces and adsorbates. KH, which corresponds to the slope of the adsorption isotherm in the low pressure range, is characteristic of the adsorbate/surface interaction strength. Instead of KH, we use KH′ = KHP0, which corresponds to the slope of the adsorption isotherm when plotted as a function of the relative vapor pressure P/P0. While KH is the Henry constant for the adsorption isotherm when the adsorbed amounts are plotted as a function of the absolute

Figure 3. Film thickness t as a function of relative pressure P/P0 for water (top), methanol (middle), and toluene (bottom) adsorbed on bare and modified silica. The symbols are the simulated data obtained from GCMC simulations, while the lines are fits against the thermodynamic model corresponding to eq 6. The black symbols are for the hydroxylated surface, the red symbols are for the methylated surface, and the blue symbols are for the trifluoromethylated surface. For the methylated and trifluoromethylated surfaces, the filled and empty symbols are for η = 0.7 and 1, respectively. Similarly, the solid and dashed lines for these two surfaces are for η = 0.7 and 1, respectively. For each adsorbate, the inset shows the same fits when the data are plotted as a function of ln(−ln P/P0) instead of P/P0. As expected from eq 6, the film thicknesses follow a linear relationship when plotted as a function of ln(−ln P/P0).

pressure (in pascal for instance), KH′ = KHP0 is the Henry constant for the adsorption isotherm when the adsorbed amounts are plotted as a function of the relative pressure P/P0. Both constants represent the affinity of the adsorbate for the surface. However, by using KH′ instead of KH, one corrects for the effect of the saturating vapor pressure. In other words, when comparing different adsorbates, KH′ allows correcting for the F


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Figure 4. Spreading coefficient S as a function of Henry constant KH′ = KHP0 for water (circles), methanol (squares), and toluene (triangles) adsorbed on bare and modified silica. The black, red, and blue symbols are for hydroxylated, methylated, and trifluoromethylated surfaces. For the methylated and trifluoromethylated surfaces, the open and closed symbols are for η = 0.7 and 1.

Figure 5. Interaction length ξ as a function of ΔHvap for water (circles), methanol (squares), and toluene (triangles) adsorbed at 300 K on bare and modified silica. The black, red, and blue symbols are for hydroxylated, methylated, and trifluoromethylated surfaces. For the methylated and trifluoromethylated surfaces, the open and closed symbols are for η = 0.7 and 1. The inset shows the parameter K from the FHH model as a function of the interaction length ξ in Derjaguin’s model.

intrinsic saturating vapor pressure of each adsorbate. In fact, using P/P0 and KH′ instead of P and KH allows correcting for the bulk reference state of the adsorbate at the temperature of the adsorption isotherm. Figure 4 shows that S varies linearly with KH′ with a slope which is only weakly dependent on the surface type. In contrast, the intercept y0 of these linear plots seems to be related to the solid surface energy; y0 increases from trifluoromethylated, to methylated, to hydroxylated silica. These results show that S can be estimated from the slope of the adsorption isotherm in the low pressure range. This result is consistent with the work by Lecloux and Pirard,44 who showed that the standard adsorption isotherm for t-plot measurements has to be chosen according to the strength of the adsorbent− adsorbate interactions through the CBET constant. The parameter ξ, which describes the range of the interaction responsible for adsorption, was plotted as a function of several quantities characteristic of the adsorbate interfacial properties. Figure 5 shows ξ as a function of the heat of vaporization ΔHvap of the adsorbate for the different surfaces and gases. Depending on the hydrophobicity/hydrophilicity of the surface, ξ exhibits different behaviors upon varying ΔHvap.45 For the two hydrophilic surfaces (hydroxylated surface and methylated surface with η = 0.7), ξ decreases linearly with increasing the adsorbate polarity. Considering that ΔHvap increases with increasing the liquid polarity, this is consistent with the idea that adsorption of polar adsorbates is mainly ruled by electrostatic interactions between the adsorbate molecule and the surface. Low-energy surfaces interact with molecules through an interaction that is short-range (small ξ) compared to that for high-energy surfaces (large ξ). For the two trifluoromethylated surfaces, ξ is only weakly sensitive to the adsorbate polarity. This is due to the fact that adsorption of polar molecules on such hydrophobic surfaces is not favorable. Finally, due to the complexity of adsorption on the fully methylated surface (where water molecules can get adsorbed in between the modified groups at the surface), ξ depends in a nontrivial way on ΔHvap. We note that the analysis above should be restricted to wetting situations; indeed, Derjaguin’s theory which is used to derive the simple model above is only relevant when a homogeneous film wets the surface. The insert

in Figure 5 shows the consistency between the FHH model and Derjaguin’s theory; there is a direct correlation between the constant K in the FHH equation and the interaction length ξ in Derjaguin’s theory. Considering that ξ describes the range of the intermolecular interaction responsible for adsorption, it is reasonable to assume that, for a given surface, it scales with an intrinsic property of the fluid (in contrast, for a given fluid, ξ must scale with the surface energy of the host substrate). This idea is supported by our data which show that, for a given surface, ξ is related to the heat of vaporization of the adsorbate. In order to validate our model, we have compared its predictions with experimental measurements for water, methanol, and toluene on porous silicas. The parameters S and ξ derived using the strategy above were used to predict from Derjaguin’s model (eq 2) adsorption in a cylindrical pore7,41 of a given pore radius R. With this geometry, VG = πL(R − t)2, VL = πL(R2 − (R − t)2), and VS are the pressure and volume of the gas, adsorbed, and solid phases, respectively. ALG = 2πL(R − e) and ASL = 2πLR are the gas-adsorbed phase and solid-adsorbed phase surface areas, respectively. We solved eq 2 for the cylindrical geometry with the parameters obtained from molecular simulations using the strategy above. Figure 6 compares for each gas (water, toluene, methanol) the predictions of our model with experimental measurements on hydroxylated silicas. We also compare our model with experimental data for mesoporous MCM-41.28−30 For the sake of comparison, the theoretical and experimental data have been normalized to the maximum adsorbed amount. For all gases, the simulated data are in good agreement with the experimental adsorbed amounts, which shows the ability of our model to describe the physics at play. In particular, the simulated adsorbed amounts prior to capillary condensation are in good agreement with the experimental data, which shows that our derivation of the parameters S and ξ provide a realistic description of the adsorbate/surface interactions. Moreover, for all gases, the theoretical capillary condensation/evaporation pressures are in good agreement with the experimental data. The model also reproduces quantitatively the relative height of G


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modified silica surfaces. The simple model reported in this paper allows predicting adsorption on surfaces with different affinities for the adsorbate using a few parameters available in the literature. For gas molecule and surface chemistry leading to complete wetting, adsorbed film thicknesses can be accurately predicted from a parameter characteristic of the interfacial behavior of the gas and a parameter describing the strength of the adsorbent/adsorbate interactions. Moreover, once the parameters for a given adsorbate/adsorbent system are known, adsorption and condensation in porous materials having the same surface chemistry can be accurately described using Derjaguin’s theory. We note that this model can be straightforwardly extended to liquid adsorption on surfaces. The possibility to describe adsorption on different surfaces provides a simple tool to predict and design optimal materials and processes by playing with variables such as surface chemistry, pore size, heat of adsorption, adsorbate capacity, or specific surface.

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ASSOCIATED CONTENT

* Supporting Information S

Molecular simulation parameters and additional data. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1-617-324-4357.

Figure 6. Theoretical (lines) and experimental (symbols) adsorption isotherms at room temperature: (top) water in a porous silica layer12 (○) and in MCM-4131 with D = 2.5 nm (●); (bottom) methanol in MCM-4129 with D = 2.22 nm (□) and toluene in MCM-4130 with D = 4.5 nm (△).

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been carried out within the framework of the ICoME2 LabEx (ANR-11-LABX-0053) and the A*MIDEX projects (ANR-11-IDEX-0001-02) cofunded by the French program “Investissements d’Avenir” which is managed by the ANR, the French National Research Agency.

the hysteresis defined as the ratio of the adsorbed amount prior to capillary condensation (or evaporation) to the adsorbed amount when the pore is filled. The fact that the model always predicts a capillary hysteresis loop can be explained as follows. Condensation is a metastable transition which occurs as fluctuations become large enough, causing the cylindrical film adsorbed at the pore surface to collapse. While condensation is assumed to occur at the end of the metastability region (spinodal curve corresponding to the limit of film stability) in the theoretical approach, condensation in the experiments can occur prior to the end of this region due to larger fluctuations. As a result, when capillary condensation is reversible for the experimental samples, comparison with the model should be made for the desorption branch only as it occurs at equilibrium (pore emptying through the pore opening toward the external environment).

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REFERENCES

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4. CONCLUSION Experiments and molecular simulations were used to investigate the adsorption of different adsorbates (water, methanol, and toluene) on hydroxylated, methylated, and trifluoromethylated silicas. We show that Derjarguin’s theory of adsorption and capillary condensation can be used to describe the adsorption of various gases on bare and modified surfaces using a minimum set of parameters such as the heat of vaporization of the adsorbate and the Henry constant of the adsorbate/surface adsorption isotherm. This model can be easily extended to other surface and adsorbate types. The results reported in this paper shed light on the role of surface energy and adsorbate polarity in adsorption on bare and H


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Predicting Adsorption on Bare and Modified Silica Surfaces

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