Wall Thickness Prediction in Precipitated Precursors of Mesoporous

ARTICLE pubs.acs.org/JPCC

Wall Thickness Prediction in Precipitated Precursors of Mesoporous Materials Agnes Grandjean,* Guillaume Toquer, and Thomas Zemb Institut de Chimie Separative de Marcoule, UMR5257, CEA-CNRS-UM2-ENSCM, BP17171, 30207 Bagnols sur Ceze, France

bS Supporting Information ABSTRACT: We describe a parameter-free analytical model based on molecular force balance to quantitatively explain the wall thickness of silica-based mesoporous materials obtained by a sol
gel route. Simple synthesis routes were proposed 20 years ago that led to a welldeveloped class of porous materials with mesoscale pores (i.e., between 2 and 50 nm). The general route is a micelle-templated precipitation of silicate-based polymers. To optimize thermal stability and efficient resistance to leaching, the wall thickness must be as large as possible, whereas the microporosity has to be as low as possible. Experimental attempts to control wall thickness have appeared in more than 100 publications, but a clear general predictive model is not yet available. Here, we propose a rational evaluation of wall thickness based on molecular force balance that minimizes the free energy between adjacent micelles. The force balance takes into account the following three main uncoupled driving forces: repulsive electrostatic, repulsive hydration, and attractive van der Waals. We argue that our predictive model is an efficient guide for mesoporous material formulations.

’ INTRODUCTION The first step in the preparation of mesoporous materials1 involves the formation of a lyotropic surfactant-based liquid crystal that is used as a template to form apolar, self-assembled semirigid cylinders formed by elongated surfactant micelles as shown in Figure 1. A sol
gel process occurs in the aqueous phase between the templating cylinders. Tuning the distance between the cylinders during the inorganic polymerization is then crucial for controlling the final wall thickness. The diameter of the templating cylinders is linked to the final pore size, whereas the wall thickness is more difficult to control a priori. Still unexplained are observations that the wall thickness systematically decreases as the surfactant chain length increases,2 whereas the final wall thickness does not depend significantly on the nature and concentration of added salts.3
5 In addition, final wall thicknesses are larger for nonionic surfactants that provide systematically thicker and denser walls. Mesostructures used as templates exist in a wet dispersion of the precursor material, which is subsequently aged, washed, and calcined to remove the surfactant-based template. Many experimental conditions, including temperature, surfactant concentration, pH, and the nature and volume of the headgroups of surfactants that self-organize into giant cylindrical micelles, have been tested in the synthesis of mesoporous materials. The similarity between the observed mesophase of inorganic
surfactant composites and those observed in aqueous surfactant solutions (containing inactive analogues of the reactants) suggests that the final structures in both cases result from the same driving forces.6
8 Although inorganic polymerization is a nonequilibrium process, gelation is assumed to occur in equilibrium systems. At the r 2011 American Chemical Society

Figure 1. Hexagonal arrays of cylindrical pores. Geometrical parameters used in the force balance model: apolar radius, rap; polar head size, rh; center-to-center distance, a; inter-reticular distance, d10; wall thickness, t; pore radius, r = rap þ rh.

time scale of colloidal force balance systems, a biphasic system is at equilibrium with the excess solvent when attractive and repulsive mechanisms are balanced. In the first example of small-angle X-ray scattering (SAXS) experiments following the growth of precipitates,9 the occurrence of a first-order transition of self-organized cylindrical templates demonstrated that, between cylinders, the material is a gel rather than a solid. To predict the wall thickness of a mesoporous final structure, the surface force balance is used by considering interactions Received: January 16, 2011 Revised: May 4, 2011 Published: May 23, 2011 11525

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between adjacent aggregates. This approach was pioneered by Le Neveu et al.10,11 for bilayers and has been used for the maximum swelling of spontaneous vesicles made from ionic surfactants12 and for microporous zeolites.13 This general colloidal force balance approach was also the first to predict lipid liposome swelling obtained for solvent exchange.14 This approach is now the basis for all equations of state determined for colloids of biological interest, such as lipids or DNA complexes.15 It was extended to colloidal crystals by Versmold et al.,16 as well as to polymer capsules17 and swollen liquid-crystalline lamellar phases.18 To the best of our knowledge, an extension toward a predictive model for a micellar solution containing silicates has not yet been proposed because of the difficulty in considering hydrophobic edge
edge interactions. Therefore, for the case of cylindrical, gel-separated, elongated micelles containing a significant ion concentration, we propose herein a new approach that separates the molecular interactions into three additive pressure terms. The pressures are the derivatives of the free energy with respect to distance and consist of two repulsive terms (i.e., hydration and electrostatic) and one attractive term (i.e., van der Waals19).

’ THEORETICAL BASIS The local microstructure considered here is a hexagonal array of cylindrical pores for both experimental data and calculations. The following main independent geometrical values to be determined are shown in Figure 1: the apolar radius rap, the polar head size rh of the surfactant, and the center-to-center distance between adjacent cylinders a. A direct and precise determination of a is provided by small-angle X-ray scattering (SAXS) measurements. The inter-reticular distance of a twodimensional hexagonal phase is defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE1Þ dhk ¼ h2 þ k2 þ hk and is determined from the first-order Bragg peak position (q10) because d10 ¼

2π q10

ðE2Þ

Therefore, the distance a between the axes of neighboring cylinders (i.e., the hexagonal lattice parameter) is given by 2 a ¼ pffiffiffi 3d10

ðE3Þ

In addition, gas adsorption/desorption experiments play a significant role in the characterization of mesoporous materials. These experiments probe the surface properties and determine the specific surface area (defined as the total available open area for nitrogen adsorption), pore volume, and pore size distribution. Until recently, the most accurate method suitable for pore size calculation through adsorption experiments was based on geometrical considerations of an infinite hexagonal array of cylindrical pores.20
22 Using this method, the pore diameter, 2r, can be obtained from the mesopore volume Vp (determined from adsorption isotherm experiments) and the lattice parameter d10 (given by SAXS data) as follows !1=2 FVp 2r ¼ cd10 ðE4Þ 1 þ FVp

where sffiffiffiffiffiffiffiffiffiffi 8 c ¼ pffiffiffi 3π

ðE5Þ

and F is the density of the pore walls. Combining the SAXS and adsorption methods allows one to determine the thickness t of the wall simply from t = a
2r. Another method for pore size determination is based on the use of the invariant, that is, the second moment of the scattering intensity integrated over the largest possible angular range.6,9 This method is useful when both the thickness and the roughness of the transition region between pore and walls have to be determined. In this method, the adsorption isotherm is not required. Only the absolute scattering intensity, in addition to the knowledge of the electron density contrast between the wall and template, is required. The availability of a swivelling monocrystal becomes essential because form factors without radial averaging need to considered. To the best of our knowledge, this method was reported only once, and the wall/pore interface was shown to be extremely flat in that case.6 Considering the long-range interactions between macroscopic particles and surfaces in liquids, there are three main important forces:23 the attractive van der Waals or dispersion interaction;19 the repulsive, electrostatic term;24 and the hydration force linked to a nonelectrostatic binding enthalpy of solvent and solute to the interface. For an analytical description of the repulsive forces, the hydration interactions are added to the electrostatics of the molecular interactions [i.e., standard DLVO (Deryagin
Landau
Verwey
Overbeek) approach]. Hydration forces have thus been considered in comparison with electrostatic and dispersion forces. In the vast majority of cases, they dominate over DLVO in the last 0.5 nm.23 Hydration forces have been cited in a force balance over 5000 times in the past 30 years. No general predictive theory exists; however, there are two basic points that are common to all theories and experiments. First, the exponential decay reported in all examples is between 0.18 and 0.22 nm. Second, the contact pressure is directly related to the enthalpy of water adsorption to the surface of the colloid (the cylindrical micellar surface in our case). The attractive term is treated as a generalized van der Waals interaction that is valid for solids25 and for liquids.26 This molecular force balance approach sums supramolecular interactions, assuming that they are uncoupled, and searches for a minimum in the potential. This approach has been used previously to understand phase behavior and the stability of molecular systems organized along a local hexagonal order.27,28 In the case of two adjacent cylinders, the distance between them, and thus the wall thickness t, is given by the minimum value of the potential energy sum per unit length L, assuming that the three dominant interactions (electrostatic, van der Waals, and hydration) can be added EðtÞ ¼ EES ðtÞ þ EvdW ðtÞ þ Eh ðtÞ

ðE6Þ

An example of a force balance calculation is shown in Figure 2. Colloidal force balance assumes additive colloidal interactions and neglects fluctuation-enhanced electrostatic and related interactions. Here, we give the simplest generic expression of the interaction potentials. Differentiation with respect to distance with constant topology expresses the colloidal force balance in 11526

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Then, according to this description, the analytical expression for the electrostatic interaction as a function of the wall thickness t is given by EES ðtÞ 1 ¼ jðtÞ L LB ν ¼

Figure 2. Typical example of a force balance calculation with the dominant interaction potentials for the hydration, van der Waals, and electrostatic forces. The equilibrium interaxial distance is given by the minimum value of the total potential energy (i.e., the sum of the three interactions).

units of force. Zero net force defines the equilibrium distance between adjacent colloids during precipitation and growth. In terms of free energy, equilibrium means the existence of a common tangent. We first consider the electrostatic interaction between two highly charged cylinders. The lengthwise electrostatic energy of cylinders as a function of the wall thickness t can be expressed by the linearized Poisson
Boltzmann equation EES ðtÞ Zeff ¼ jðtÞ L L

ðE7Þ

where Zeff is the effective charge of the cylinders, a is the interaxial distance between the two cylinders, and j(t) is the electrostatic potential. In the case of highly charged systems, the effective charge introduced by Manning is the dominant interaction29 ξeff ¼

Zeff 1 LB ¼ z L

ðE8Þ

where LB is the Bjerrum length, which is equal to 7.2 Å at room temperature in water;24 L is the cylinder length; and z is the charge of the counterions. When divalent or monovalent ions are dominant, the reduced effective charge, ξeff, is about 10 times lower than the structural charge. Most ions are confined around the cylinders, leaving the interaction corresponding to the effective charge.30 Then, the reduced electrostatic potential as a function of wall thickness t is given by31 jðtÞ ¼

2ξeff K0 fk½t þ 2ðrap þ rh Þg f½kðrap þ rh ÞK1 ½kðrap þ rh Þg2

ðE9Þ

where K0 and K1 are the zeroth- and first-order Bessel functions, respectively; rap and rh have previously been defined; and k is the inverse of the Debye length. This latter value (k) is related to the ionic strength I and, thus, to the concentrations Ci and the charges zi of the ions in solution pffiffi 1 k ¼ 0:33 I ¼ 0:33 2

∑i Ci zi 2

ðE10Þ

K0 fk½t þ 2ðrap þ rh Þg 1 2 LB z z fkðrap þ rh ÞK1 ½kðrap þ rh Þg2

ðE11Þ

The experimental parameters are the geometrical parameters rap and rh and also the inverse Debye length k (linked to the ionic strength) and the charge of ions in solution z. The repulsive hydration force has been evidenced by Parsegian et al. using the so-called osmotic stress technique.32 In the case of parallel cylinders, the corresponding energy per unit length L as a function of the wall thickness t is expressed as Q 2   λ Eh ðtÞ 1 t 0 ðE12Þ ¼ exp
L λ 4π kT where the two experimental parameters are the decay length λ and the osmotic compressibility Π0 (i.e., the pressure extrapolated to contact). These two repulsive mechanisms are counterbalanced by the van der Waals attractive potential between apolar cylinders (assimilated to oil) interacting through an aqueous medium. The free energy per unit length L corresponding to the van der Waals attraction as a function of wall thickness t is given by33 rffiffiffiffiffi rap EvdW ðtÞ 2 ¼
A pffiffiffi ðE13Þ L 12 2ðt þ 2rh Þ3=2 where the experimental parameter is the Hamaker constant of oil interacting through water (A) and rap and rh are the geometric parameters previously defined. The experimental parameters in the expression for colloidal interactions can be deduced from the experimental determination of phase diagrams and osmotic equilibria or from other experiments related to derivatives of the free energy expression. Consequently, experimental determination of forces or potentials between colloids, as well as determination of distances during equilibrium by scattering and/or gas adsorption, provides a satisfactory estimate of structural quantities. The numerical values required to evaluate a given wall thickness include the apolar radius, the polar head size, the ionic strength, the Hamaker constant, the decay length, and the osmotic pressure compressibility. The apolar radius (rap) is simply estimated from the hydrocarbon chain length by using the empirical Tanford formula34 rap ¼ 0:154 þ 0:1265n

ðE14Þ

where n is the number of carbon atoms in the hydrocarbon chain. The polar head size (rh) is estimated from partial molar volumes derived from density measurements and is combined with SAXS
SANS (small-angle neutron scattering) absolutescaled scattering cross-section determinations.35 The volumes of the different headgroups are obtained by SAXS or SANS experiments.36 For ionic surfactants, the mean radius is calculated by assuming a spherical headgroup. For nonionic polyethoxy surfactants, the volume36 and the surface area37 of the (CH2
CH2
O)j headgroup can be estimated from the SAXS 11527

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The Journal of Physical Chemistry C measurements as a function of j (the number of CH2
CH2
O units). In the first approximation, the mean head size can then be estimated by dividing the known volume of hydrated heads by the area of the chain/headgroup interface. The head size has been estimated to be about equal to j. The ionic strength is estimated only for ionic surfactants where the electrostatic energy has to be taken into account. Under alkaline conditions, it has been previously demonstrated that the anionic silicate species I
and cationic surfactant Sþ (here, ammonium Nþ) cooperatively organize to form hexagonal, lamellar, or cubic structures. Some groups have shown that, depending on the nature and the concentration of the surfactant and silica sources, negatively charged silica species are concentrated around the surfactant of micelles.38 At high pH and high silica/surfactant ratios, the charged silica species are preferentially adsorbed at the micellar interface and have much stronger interactions with the charged interfaces than monovalent counterions (Br
or Cl
).7 In our calculations for the case of trimethylammonium surfactant, the concentration used for ionic strength was assumed to be equal to the silica concentration. This approximation is based on experimental observations such as dynamic light scattering and rheology measurements.39 Further evidence for this assumption is provided by multinuclear magnetic resonance spectroscopy, SAXS, and polarized optical microscopy,40 as well as a phase diagram analysis of the cetyltrimethylammonium bromide (CTAB)/tetramethylammonium silicate (TMAS)/water41 system. It was shown that negatively charged silica species promote a sphere-to-rod transition of cetyltrimethylammonium chloride (CTAC) micelles. Additional evidence for silicate
surfactant cooperativity is the formation of a microporous zeolite structure with a short hydrocarbon chain (C6 or C8), although these short-chain surfactants do not generally form mesophases in binary surfactants.42 Recently, molecular simulation studies have shown the existence of strong interactions between cationic surfactant and anionic silica species adsorbed on the micelle surface.43 Values of the Hamaker constant can be found in reference books.19,33 For oil cylinders of hydrocarbon chains across water, the Hamaker constant has been assumed to be equal to 0.6kT.44 For fluorocarbon oil cylinders across the water, a higher value is obviously expected.45 The least characterized values taken from the literature are the contact hydration pressure and the exact decay length of the hydration force. They can be determined by direct extrapolation of the hydration force to zero water, as pioneered by Lyle and Tiddy,46 or by integration of SFA (surface force apparatus) or AFM (atomic force microscopy) measurements, as proposed by Parsegian.19 It is difficult to define the reference plane taken as the origin of distances for the contact hydration pressure. The contact must be defined consistently for the different mechanisms involved. In early work on lyotropic liquid crystals, the socalled neutral plane (i.e. the plane where neither lateral extension nor compression occurs during dilution) was often chosen for convenience. In the case of cylindrical aggregates of surfactants, the natural choice is the plane between the apolar core of the micelles and the surfactant headgroup. The decay length and the osmotic compressibility depend on the polar head size. In the case of dimethylammonium ions, Π0 and λ have been measured over a wide temperature range.47 Other experimental data have been measured by Simon et al.48 in the case of surfactants with trimethylammonium phosphate headgroups. Here, the osmotic pressure at contact, Π0, and the

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Table 1. Experimental Parameters Used for the Model headgroup size (Å)

(CH3)3Nþ (C2H5)3Nþ NH3 2.9 3.4 1

(CH2
CH2
O)j ∼j

nature

ionic

ionic

nonionic nonionic

Π0 (Pa)

4  108

4  108

2  109

2  109

λ (Å) Hamaker

1.9 0.6

1.9 0.6

2.0 0.6

2.0 0.6

constant (kBT)

delay length scale were determined over different temperature ranges. An extrapolation of these different experimental data (i.e., with different temperature ranges but similar headgroups) indicates agreement for the osmotic pressure at 25 °C. Nevertheless, the decay length data given by Simon et al. are underestimated because of variations in the area per headgroup during osmotic compression, which should have been taken into account to determine real decay distances instead of apparent decay distances. Charged amphiphiles bring surfaces together, which corresponds to dehydration (i.e., counterion condensation) and, therefore, to a decrease in headgroup area. As a result, the distance between interfaces varies more than the value derived from interaxial distances alone.47 Based on the available experimental data, Π0 and λ for trimethylammonium polar heads at 25 °C were determined to be 4  108 Pa and 1.9 Å, respectively. In the case of nonionic polar headgroups, experimental data on Π0 and λ are scarce. The parameter values are higher and reported as 2  109 Pa and 2.0 Å.49 All of these numerical inputs taken from separate experiments used for the calculations are summarized in Table 1.

’ RESULTS AND DISCUSSION A comparison of the model with experimental data is given here, along with several examples, to better explain the variation in the wall thickness with the experimental process. Effect of Surfactant Chain Length. An unexpected result recognized early by Beck et al. in one of the seminal papers in this field2 is the decrease in the wall thickness when the templating surfactant chain length increases (all other conditions being equal). This is a priori surprising because the hexagonal-phase lattice parameter of a homologous series of surfactant chain lengths should increase linearly with the chain length when the water/surfactant mole ratio remains constant. The radius of the hydrophobic core is known from SAXS data to increase by 2
2.5 Å for each pair of CH2 groups added. Thus, the interaxial distance and wall thickness might be expected to increase. Beck et al. obtained a surprising decrease in wall thickness when varying the hydrophobic chain length from 8 to 16 methylene groups. To study the effect of surfactant chain length on the structure and pore dimensions of MCM41, Beck et al. used quaternary ammonium chloride surfactants with different chain lengths, that is, CnH2nþ1(CH3)3Nþ, where n varied from 8 to 16. It should be noted that the silica concentration in the surfactant
water solution exhibits small variations according to the surfactant used; consequently, we considered these effects in our calculation as explained before. Another example given by Jarionec et al.20 is based on alkyltrimethylammonium bromide with a chain length of 8
16 carbon atoms. Here, the silica concentration and surfactant concentration did not vary with the surfactant chain length. All 11528

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Figure 3. Effect of the templating surfactant chain length n on the comparison between experimental and expected values: (b) experimental data from ref 20 and (—) calculations from this model, (O) experimental data from ref 2 and (


) calculations from this model, and (2) experimental data from ref 50 and (
3
) calculations from this model.

parameters were maintained constant except for the chain length. The ionic strength was therefore different for these two experiments from that in previous studies.2,20 By identifying the minimum intercylinder potential from these parameters, an expected wall thickness can be calculated. The last example was also given by Jarionec et al.50 and used a mixture of triethylammonium and trimethylammonium surfactants with chain lengths from 12 to 20. The effect of this mixture on the headgroup size was considered in our calculations. Values predicted by the molecular force balance are compared to the experimental values in Figure 3. First, the parameter-free model proposed here properly predicts the decrease of wall thickness with the surfactant chain length except for the point at C16 from Beck et al.,2 which seems to be an artifact when compared with the other experimental values. Moreover, we argue that the difference in the measured wall thickness between the two comparable experimental systems from the previous studies2,20 arises from the distinct silica-to-surfactant concentration ratios in the media. In the case of a higher silica-to-surfactant ratio, the wall thickness is smaller. We explain qualitatively the origin of the surprising wallthinning effect. Increasing the radius of the template does not influence the hydration or electrostatic effects because the van der Waals attraction increases. More precisely, the decrease in wall thickness with chain length is mainly attributed to the variation in van der Waals potential as shown in Figure 4. Indeed, the van der Waals attractive potential between apolar cylinders interacting through an aqueous medium increases with the apolar surfactant size (rap) and leads to a decrease of the wall thickness (see eq E14). Increasing this apolar surfactant size has no effect on the hydration repulsive potential (see eq E13) and only a weak effect on the electrostatic repulsive potential (see eq E12). Effect of Osmotic Pressure. In the case of a nonionic surfactant, the osmotic pressure of the hydration energy is about 1 order of magnitude higher than in the case of an ionic surfactant. Moreover, the electrostatic potential term is obviously equal to zero. Increasing the osmotic pressure at contact, Π0, leads to an increase in the expected wall thickness. Consequently, the use of a nonionic surfactant will lead to a qualitative increase in the wall thickness. The molecular force balance approach explains the thicker wall obtained with nonionic surfactants

Figure 4. Individual contributions of each energy term to two calculations: (a) effect of chain length CnH2nþ1 for n = 8 (solid lines) and n = 20 (dotted lines) and (b) effect of ionic strength for I = 0.2 (solid lines) and I = 3 (dotted lines).

as experimentally observed in the case of polymers (or copolymers). Moreover, as given by the van der Waals energy equation, the polar headgroup also affects the position of the minimum intercylinder potential. A larger headgroup size leads to an increase in the expected equilibrium wall thickness as observed experimentally.51 Qualitatively, the molecular force balance obtained through the model predicts the increase of the wall thickness from using a nonionic surfactant or a larger polar head size as in the case of copolymers. Some authors have used the same alkyl chains as those described previously in the case of alkyltrimethylammonium but with different headgroups. Pinnavaia et al.52 used dodecylamine (C12 amine) and Stebe et al. used dodecylethoxy [C12
(CH2
CH2
O)8].53 It is then possible to compare these experimental data in terms of wall thickness for ionic surfactants with the same chain length. These comparisons are shown in Figure 5a, where the experimental wall thickness is plotted as a function of the value expected from the parameter-free approach proposed here. First, close agreement is found in all cases between the experimental and calculated data. It is important to note that, as expected, the use of a nonionic surfactant leads to a higher wall thickness. However, this result is offset by the effect of the headgroup size. In the case of an amine polar head, the small size implies a wall thickness of the same order of magnitude as in the ionic quaternary ammonium polar head case. Here, the higher osmotic pressure cancels the effect of the small size of this headgroup. However, when an ethoxy complex is used as the polar head, both the higher osmotic pressure and the greater 11529

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Figure 5. Expected wall thickness for identical chain lengths: (a) Effect of the nature of the surfactant polar headgroup on wall thickness. Data from previous studies for ammonium2,20 (þ), (C2H4
O)853 (b), and amine52 (O). (b) Effect of headgroup size for a nonionic surfactant. Data from ref 51 (b) and calculations (—) from this model.

Figure 6. Effect of ionic strength on wall thickness: (a) ionic surfactant [(b) experimental data from ref 3 and (—) calculations from this model] and (b) nonionic surfactant [experimental data from (b,O) ref 51 and (2) ref 53 and calculations from this model (—,


)].

polar head size involve an increase of the wall thickness. In these cases, the molecular force balance is still effective in predicting the wall thickness variations with the size and nature of the headgroup. In the case of nonionic templating structures such as poly(ethylene oxide), (EO)j, it is relatively easy to change the polar head size without changing the apolar radius of the chain. According to the calculation of the force charge balance, increasing the polar head size leads to an increase in the wall thickness. This observation could also qualitatively explain the larger wall thickness in the case where copolymers are the template. Experimentally, Prouzet et al.51 synthesized mesoporous silica using nonionic poly(ethylene oxide)- (PEO-) based surfactants, varying the EO chains. The surfactants used were Tergical 15-S-N with j = 9, 12, and 15 [CH3
(CH2)15
(CH2CH2O)j
OH]. A small increase was observed in the wall thickness with increasing PEO chain length as predicted by the molecular force balance (Figure 5b). Effect of Salt. The first result concerning the length-scale effect seems to show a non-negligible effect of the salt concentration on the wall thickness value for ionic surfactants. Attempts to formulate the continuous phase, in which precipitation occurs, have been made using more or less surface-active salts interacting with the headgroups.3
5 This effect on the wall thickness was evaluated in the colloidal force balance approach by varying the surface charge and ionic strength. Only a small variation in the wall thickness was observed at high alkalinity, which is consistent with the model prediction. Use of different counterions such as Br
or Cl
in the starting surfactant has very little effect on the

wall thickness.4 Hofmeister anions can affect the formation of mesoporous silica, but the current understanding of this effect is limited because of the chemical complexity of the systems.54 Shi et al.3 synthesized mesoporous silica with NaCl concentrations ranging from 0.2 to 2.0 mol/L while maintaining all other parameters (e.g., silica/surfactant concentrations, nature of surfactant) constant. Here, the ionic surfactant used was cetyltrimethylammonium bromide (C16). Using the same parameters as in the last case (except for the ionic strength, which is linked to the ionic concentration), we identify the minimum intercylinder potential and thus predict a subsequent variation of the wall thickness. The values of these results are compared to the experimental values in Figure 6a. As expected, the addition of salt to the system leads to a significant drop in the electrostatic energy and can be neglected for highly concentrated solutions. This leads naturally to a decrease in the wall thickness. At high salt concentrations, the effect is no longer noticeable because the electrostatic potential becomes negligible. This prediction agrees with experimental data as shown in Figure 6a. In Figure 4b, we report the contributions of each energy term to the two calculations with two different ionic strengths. In the case of an ionic surfactant, this effect is well predicted by the molecular force balance through the electrostatic term. In the case of a nonionic surfactant S0, a neutral silicate species I0 is suggested to interact with the micellar aggregate through hydrogen bonding between the hydroxyl groups of the hydrolyzed silicate species and the polar surfactant headgroup. This is a neutral route, and an electrostatic term is needed in the molecular 11530

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thickness for a given formulation but also predicts quantitatively the variation of interaxial distances due to the replacement of the templating surfactant forming the hexagonal phase in the oil/ surfactant/water ternary phase diagram. To summarize, Figure 7 plots the experimental thickness versus the expected value for all data reported in this article.

Figure 7. Wall thickness measured versus wall thickness expected from molecular force balance using experimental data collected from different experiments used in precipitating precursors of mesoporous materials.

force balance. The presence of salts and an increase of the I0/S0 ratio seem to lead to a small decrease in the wall thickness.51,53 As shown in Figure 6b, this effect is small and could also be attributed to experimental error. The simple molecular force balance described here does not predict this behavior, which is one of the restrictions of this model. Nevertheless, this effect can be explained by the presence of salt, which strongly modifies the surface and bulk properties of surfactant solutions.54,55 Effect of the Hamaker Constant. Until recently, all of the examples given here have concerned hydrogenated saturated chains. It is known from surface force experiments that the Hamaker constants are around 1.75kT instead of 0.6kT for apolar domains interacting through water.45 A series of experiments was performed to synthesize mesoporous materials with a fluorinated surfactant53 or a more complex sulfofluorinated surfactant.56 To compare the effect of fluorocarbon chains instead of hydrocarbon chains, data from Stebe et al.53 are interesting because all parameters are held constant except for the nature of the apolar core and the headgroup size. Stebe et al. used RF8(EO)9 as the hydrocarbon chain and RH12(EO)8 as the fluorocarbon chain (RF8 = C8F17C2H4 and RH12 = C12H25 have the same volume). The final wall thicknesses were measured in both cases, and a decrease in wall thickness was observed when using fluorocarbon chains in the templating cylindrical micelles, even when the headgroup was comparatively larger. Experimental data from this reference are too poor to precisely determine the wall thicknesses of these two samples. Nevertheless, they can be estimated at about 22 Å for a hydrocarbon chain and 13 Å for a fluorocarbon chain. This observation is qualitatively explained from the molecular force balance model because of the higher values of the Hamaker constant for fluorocarbon chains. Prediction from the colloidal force balance without any adjustable parameters leads to a wall thickness of 18 Å for hydrocarbon chains and 16 Å for fluorocarbon chains. The absolute difference between the experimental and calculated data arises both from the difficulty in properly measuring the wall thickness values and from the indirect determination of the Hamaker constant (derived mainly from AFM measurements in the case of the fluorocarbon chains). Nevertheless, an increase of the Hamaker constant leads qualitatively to a decrease in wall thickness in agreement with the experimental data. Finally, considering all of the preceding examples, the use of the colloidal force balance not only rationalizes the obtained wall

’ CONCLUSIONS The simple analytical model based on molecular force balance proposed here explains qualitatively, and in some simple cases quantitatively, the wall thickness of silica-based mesoporous materials obtained by a sol
gel route. In this analytical model with no free parameters, three main uncoupled driving forces are considered: repulsive electrostatic, repulsive hydration, and attractive van der Waals. Beyond the case of precursors of mesoporous materials demonstrated in this work, our simple model is a real and efficient guide for the formulation of material precursors with targeted distances required for an optimized lifespan. ’ ASSOCIATED CONTENT

bS

Supporting Information. Evaluation of the variation of the expected wall thickness with changes in the headgroup size, osmotic compressibility, decay length, and Hamaker constant. All parameters except one were fixed, and the wall thickness was calculated as described in this article. This information is available free of charge via the Internet at http://pubs.acs.org

’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ33 4 66 79 66 22. Fax: þ33 4 66 79 76 11. E-mail: agnes. [email protected].

’ ACKNOWLEDGMENT We thank Luc Belloni for helpful discussions and Frederic Ne for access to unpublished data. We also thank the Commissariat a l’Energie Atomique (CEA) and the Ecole Nationale Superieure de Chimie Montpellier (ENSCM) for their financial support. Part of this work was performed within the framework of COST network D43 “Nanoscience for nanotechnology”. ’ REFERENCES (1) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (2) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T. W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (3) Yu, J.; Shi, J. L.; Chen, H. R.; Yan, J. N.; Yan, D. S. Microporous Mesoporous Mater. 2001, 46, 153. (4) Lin, H. P.; Cheng, S. F.; Mou, C. Y. Microporous Mater. 1997, 10, 111. (5) Coustel, N.; Direnzo, F.; Fajula, F. J. Chem. Soc., Chem. Commun. 1994, 967. (6) Ne, F.; Zemb, T. J. Appl. Crystallogr. 2003, 36, 1013. (7) Linden, M.; Schacht, S.; Schuth, F.; Steel, A.; Unger, K. K. J. Porous Mater. 1998, 5, 177. (8) Renzo, F. D.; Galarneau, A.; Desplantier-Giscard, D.; Mastrantuono, L. Chim. Ind. 1999, 81, 587. (9) Ne, F.; Testard, F.; Zemb, T.; Grillo, I. Langmuir 2003, 19, 8503. 11531

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