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Morphological Transitions during the Formation of Templated Mesoporous Materials: Theoretical Modeling N. Gov,*,† Itamar Borukhov,‡,§ and D. Goldfarb† Department of Chemical Physics and Department of Materials and Interfaces, The Weizmann Institute of Science, P.O. Box 26, RehoVot, Israel 76100 ReceiVed August 21, 2005. In Final Form: NoVember 10, 2005 We put forward a theoretical model for the morphological transitions of templated mesoporous materials. These materials consist of a mixture of surfactant molecules and inorganic compounds which evolve dynamically upon mixing to form different morphologies depending on the composition and conditions at which mixing occurs. Our theoretical analysis is based on the assumption that adsorption of the inorganic compounds onto mesoscopic assemblies of surfactant molecules changes the effective interactions between the surfactant molecules, consequently lowering the spontaneous curvature of the surfactant layer and inducing morphological changes in the system. On the basis of a mean field phase diagram, we are able to follow the trajectories of the system starting with different initial conditions, and predict the final morphology of the product. In a typical scenario, the reduction in the spontaneous curvature leads first to a smooth transition from compact spherical micelles to elongated wormlike micelles. In the second stage, the layer of inorganic material coating the micelles gives rise to attractive inter-micellar interactions that eventually induce a collapse of the system into a closely packed hexagonal array of coated cylinders. Other pathways may lead to different structures including disordered bicontinuous and ordered cubic phases. The model is in good qualitative agreement with experimental observations.
I. Introduction Ordered mesoporous materials that are characterized by high surface area and a narrow pore size distribution can be prepared on the basis of organic surfactant molecules that self-assemble in solution into micellar aggregates.1-8 These aggregates serve as templates for the polymerization of inorganic oxide precursors which in turn modify the morphology of the template structures. In the final product the surfactant molecules are encapsulated within a newly formed inorganic structure. When removed, the surfactants leave voids whose size, arrangement in space, and symmetry are reminiscent of the template structure. This procedure turned out to be extremely productive since different combinations of inorganic precursors and surfactant mesostructures yield numerous mesoporous materials with a wide range of physical and chemical properties. Since the first reports of their synthesis in 1992,1 many different ordered mesoporous materials have been synthesized with a variety of compositions and morphologies using different types of templates ranging from charged alkylammonium salts to neutral block copolymers, as summarized in a few recent reviews.2-8 * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Chemical Physics. ‡ Department of Materials and Interfaces. § Current Address: Compugen, Ltd., 72 Pinchas Rosen St., Tel Aviv 69512, Israel. (1) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Lenowicz, M. E.; Kresge, C. T.; Schmidt, 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, 10834-10843. Kresge, C. T.; Lenowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Science 1992, 359, 710-712. (2) Schu¨th, F. Chem. Mater. 2001, 13, 3184-3195. (3) Sayari, A.; Hamoudi, S. Chem. Mater. 2001, 13, 3151-3168. (4) Patarin, J; Lebeau, B; Zana, R. Curr. Opin. Colloid Interface Sci. 2002, 7, 107-115. (5) Soler-Illia, G. J. D.; Sanchez, C.; Lebeau, B.; Patarin J. Chem. ReV. 2002, 102, 4093-4138. (6) Stein, A. AdV. Mater. 2003, 15, 763-775. (7) Linssen, T.; Cassiers, K.; Cool, P. AdV. Colloid Interface Sci. 2003, 103, 121-147. (8) Palmqvist, A. E. C. Curr. Opin. Colloid Interface Sci. 2003, 8, 145-155.
In general, two types of reactions can be distinguished. (i) In the first type, the concentration of the organic template molecules is high, the template is pre-assembled as a lyotropic liquid crystal, and the formation mechanism is referred to as the true liquid crystal (TLC) mechanism.9,10 (ii) In the second type, the concentration of the template molecules is relatively low, and the final structure results from a cooperative self-assembly process (CSA) involving both the inorganic precursors and the organic molecules.11 In fact, the CSA mechanism may play a role in the first family of syntheses as well. Although there is a preexisting liquid crystalline phase, it may be destroyed during the addition of the organosilane because of the alcohol produced by its hydrolysis. The evaporation of the alcohol then produces a liquid crystalline phase, with a structure dictated by the new composition. The second type of reactions is more commonly used because of the larger variability in the final product. Examples include MCM-411 and SBA-15,12 perhaps the most extensively studied mesoporous materials. Unraveling the detailed formation mechanism of the CSA route has therefore been the focus of a number of in situ experimental investigations,4 where different methods probe the system on different length scales. The molecular level can be studied using a variety of in situ spectroscopic methods such as electron paramagnetic resonance (EPR),13-19 nuclear (9) Attard, G. S.; Glyde, J. C.; Go¨ltner, C. G. Nature 1995, 378, 366. (10) Raimondi, H. E.; Seddon, J. M. Liq. Cryst. 1999, 26, 305-339. (11) Firouzi, A.; Kumar, D.; Bull, L. M.; Besier, T.; Sieger, P.; Huo, Q.; Walker, S. A.; Zasadzinski, J. A.; Glinka, C.; Nicol, J.; Margolese, D.; Stucky, G. D.; Chmelka, B. F. Science 1995, 267, 1138. (12) Zhao, D.; Huo, Q.; Feng, J.; Chmelka, B. F.; Stucky, G. D. J. Am. Chem. Soc. 1998, 120, 6024. (13) Zhang, J.; Luz, Z.; Zimmermann, H.; Goldfarb, D. J. Phys. Chem. B 2000, 104, 279. (14) Zhang, J.; Goldfarb, D. Mesoporous Mater. 2001, 48, 143-149. (15) Zhang, J.; Carl, P. J.; Zimmermann H.; Goldfarb, D. J. Phys. Chem. B 2002, 106, 5382. (16) Ruthstein, S.; Frydman, V.; Kababya, S.; Landau, M.; Goldfarb, D. J. Phys. Chem. B 2003, 107, 1739. (17) Ruthstein, S.; Frydman, V.; Goldfarb, D. J. Phys. Chem. B 2004, 108, 9016. (18) Galarneau, A.; Renzo, F. D.; Fajula, F.; Mollo, L.; Fubini, B.; Ottaviani, M. F. J. Colloid Interface Sci. 1998, 201, 105-107.
10.1021/la052272r CCC: $33.50 © 2006 American Chemical Society Published on Web 12/10/2005
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magnetic resonance (NMR),20-23 fluorescence probing techniques,24 and infrared (IR) spectroscopy.26,25 The mesoscopic scale, in which the structures evolve prior to the formation of long range order, is examined by cryogenic transmission electron microscopy (TEM)27 and small-angle X-ray scattering (SAXS).23,28 In particular, the appearance of long range order can be probed by in situ SAXS. The third scale is the macroscopic length scale, which characterizes the morphology of the system in its final form, and can be probed, e.g., by scanning electron microscopy (SEM).29 It is well established that the surfactant, the inorganic precursor, and the thermal processing determine the rate at which the mesoporous material is produced and its final structure.4,9,11,20,22,29 Since silica precursors are predominantly used in these systems, we will focus in the following on silica composites, but it should be emphasized that similar arguments apply also to other composites based on titanium (TiO2), zirconium (ZrO2), aluminum (Al2O3), etc. Initially the surfactant solution is in a dilute spherical or wormlike micellar phase to which silica precursors, such as sodium silicate or tetramethyloxysilane (TMOS), are introduced together with an acid or base that catalyzes the hydrolysis and polymerization of the silica. With the exception of a few studies,24 it is generally accepted that the initial stage involves adsorption of silicate ions at the micellar interface, driven either by charge matching30 or hydrogen bonding.12 The next step involves two possibilities: (i) The silicate adsorption leads to rearrangement of the original micellar morphology, mainly lengthening the micelles,27 followed by condensation of the silicate-covered micelles into ordered or disordered collapsed phases. (ii) Alternatively, silicate adsorption may not change the morphology of the micelles, but rather reduce the inter-micellar repulsion and cause aggregation into larger particles and precipitation of a disordered phase, which then may rearrange to form an ordered phase.22,23 While the number of experimental in situ studies of the formation of mesoporous materials has been growing, so far there has been no systematic theoretical account of the formation mechanism. Nevertheless, there have been several qualitative suggestions regarding the mechanism,20,31-33 implemented in various phenomenological and simulation calculations. For example, simple geometrical arguments were used to describe the effect of silicate ion binding on the micellar curvature and the emerging structure.20,31 Such arguments rely on the surfactant (19) Ottaviani, M. F.; Moscatelli, A.; Desplantier-Giscard, D.; Di Renzo, F.; Kooyman, P. J.; Alonso, B.; Galarneau, A. J. Phys. Chem. B 2004, 108, 12123. (20) Firouzi, A.; Atef, F.; Oertli, A. G.; Stucky, G. D.; Chmelka, B. F. J. Am. Chem. Soc. 1997, 119, 3596. (21) Melosh, N. A.; Lipic, P.; Bates, F. S.; Wudl, F.; Stucky, G. D.; Fredrickson, G. H.; Chmelka, B. F. Macromolecules 1999, 32, 4332-4342. (22) Flodstro¨m, K.; Wennerstro¨m, H.; Alfredsson, V. Langmuir 2004, 20, 680. (23) Flodstro¨m, K.; Teixeira, C. V.; Amenitsch, H.; Alfredsson, V.; Linde´n, M. Langmuir 2004, 20, 4885. (24) Frasch, J.; Lebeau, B.; Soluard, M.; Patarin, J.; Zana, R. Langmuir 2000, 16, 9049. (25) Holmes, S. M.; Zholobenko, V. L.; Thursfield, A.; Plaisted, R. J.; Cundy, C. S.; Dwyer, J. J. Chem. Soc.: Faraday Trans. 1998, 94, 2025. (26) Calabro, D. C.; Valyocsik, E. W.; Ryan, F. X. Microporous Mater. 1996, 7, 243. (27) Regev, O. Langmuir 1996, 12, 4940. (28) Linde´n, M.; Schunk, S. A.; Schu¨th, F. Angew. Chem., Int. Ed. Engl. 1998, 37, 6, 821. (29) Yu, C.; Fan, J.; Tian, B.; Zhao, D. Chem. Mater. 2004, 16, 889. (30) Monnier, A.; Schu¨th, F.; Huo, Q.; Kumar, D.; Margolese, D.; Maxwell, R. S.; Stucky, G. D.; Krishnamurty, M.; Petroff, P.; Firouzi, A.; Janicke, M.; Chmelka, B. F. Science 1993, 261, 1299-1303. (31) Huo, Q.; Margolese, D. I.; Stucky, G. D. Chem. Mater. 1996, 8, 11471160. (32) Siperstein, F. R.; Gubbins, K. E. Mol. Simulat. 2001, 27, 339-352. (33) Firouzi, A.; Stucky, G. D.; Chmelka, B. F. Synthesis of Porous Materials; Occelli, M. L., Kessler, H., Eds.; Marcel Dekker: New York, 1996; pp 379-389.
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Figure 1. Schematic view of the main mechanisms that lead to morphological changes following the addition of silicate ions to a micellar surfactant solution. (a) Adsorption of silicate anions in any oligomeric state (filled circles) onto the surfactant headgroups (empty circles) reduces the spontaneous curvature of the surfactant layer. As a consequence, spherical micelles elongate and become cylindrical (b). (c) The adsorbed silicate also induces attractive interactions between micelles by direct polymerization between the silicate on neighboring micelles (black circles). (d) Adsorption of silicate ions onto the hydrophilic chains of block-copolymers can also reduce the spontaneous curvature.
packing parameter g ) V/alc, where V is the effective volume of the hydrophobic chain, a is the mean aggregate surface area per hydrophilic headgroup, and lc is the effective hydrophobic chain length.34,35 Coarse-grained Monte Carlo simulations36 of the process of silicate buildup give more microscopic information, although limited to two-dimensions. Our approach is more quantitative and general, so that we can describe a wider range of phenomena. In this work, we describe theoretically two effects of silicate adsorption and present a quantitative model, based on a specific mechanism, that allows us to predict the morphology of these materials. First, we relate the changes in the morphology of the surfactant micelles to a reduction in the spontaneous curvature of the surfactant layer due to adsorbed silicate. The reduction in the spontaneous curvature of the surfactant layer can be attributed to two effects (Figure 1a,b):11 (i) silicate neutralizes the electrostatic repulsion among surfactant headgroups and (ii) silicate polymerizes to form silicate oligomers which pull together the headgroups. The reduction in the effective area per headgroup, (34) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. (35) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic: London, 1990. (36) Bhattacharya, A.; Mahanti, S. D. J. Phys.: Cond. Mater. 2001, 13, 14131428.
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Figure 2. Geometrical estimation of the spontaneous curvature, cj, of a surfactant layer. The headgroup radius is a0 and the hydrophobic tail is of length d and radius r0.
a, as compared to the unchanged hydrophobic chain length, results in a decrease in the spontaneous curvature of the surfactant. The geometry of a schematic surfactant (Figure 2) shows that increasing the headgroup area increases the curvature and viceversa. We assume that the bare surfactant has a positive spontaneous curvature, such that it forms isolated spherical and wormlike micelles. Reduction in the spontaneous curvature then leads to increase in the length of the wormlike micelles and further morphological changes.37 In the case of block copolymers, the headgroup part of the surfactant is replaced by a hydrophilic chain (Figure 1d), whose radius of gyration in the water determines the effective headgroup area. When silicate ions adsorb to the hydrophilic chain by hydrogen bonding, the hydration level of the chain is effectively reduced,17 and the chain collapses to a smaller radius.38 Furthermore, the silicate ions can cross-link the chains of neighboring block-copolymers (Figure 1d). The result of these two processes is again a reduction in the effective headgroup size and in the spontaneous curvature. The second effect we explore is the enhancement of intermicellar attractive interactions due to adsorbed silicate. This occurs both indirectly by screening the repulsive surfactantsurfactant electrostatic interactions and directly by contributing to attractive interactions of the van der Waals type and as a result of silicate polymerization (Figure 1c). The increased attraction eventually leads to the collapse of the micelles into a condensed phase, as shown in Monte Carlo simulations,36 of model twodimensional systems. Our treatment relies on a mean-field model for the thermodynamic equilibrium phase diagram of the surfactant aggregates, assuming that the amount of adsorbed silicate determines the spontaneous curvature and the effective surfactant-surfactant interaction. We present a phase diagram that depends on the surfactant concentration and the fraction of surfactants that are covered by silicates. This phase diagram was calculated under conditions of thermal equilibrium, while we deal with an inherently out-of-equilibrium system: the amount of adsorbed and polymerized silicate changes irreversibly as a function of time. As a result, the physical system moves along a specific trajectory in the phase diagram, depending on its initial conditions. The dynamic aspects of the irreversible processes of silicate adsorption and polymerization are not treated directly within this model, and the rate of motion along the trajectory is not determined. Nevertheless, the surfactant molecules at each point along the trajectory are arranged at the morphology calculated for that (37) Zilman, A.; Safran, S. A.; Sottmann, T.; Strey, R. Langmuir 2004, 20, 2199. (38) Ruthstein, S.; Schmidt, Y.; Talmon, Y.; Goldfarb, D. Manuscript in preparation, 2005.
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point under the assumption of thermodynamic equilibrium, if the following conditions apply. (i) The surfactant-silicate assemblies remain fluid in the sense that molecules are free to rearrange within individual structures, and whole aggregates can fuse and separate from each other. It has been shown that prior to filtration and drying the silica network formed at the micellar interface is liquidlike and can therefore change its morphology rather easily. This changes after the hydrothermal stage, where the silicate polymerization is enhanced forming a more stable silica wall.39 (ii) The rate of the irreversible processes is slow compared to the rate at which the surfactant molecules are able to adapt and change the morphology of their aggregates. Indeed, processes that involve preparation of mesoporous materials using very fast reaction rates may be outside the scope of this model. Concerning the kinetics of the silicate adsorption, there have been a number of studies which followed in situ the kinetics of the formation of mesoporous materials. These showed that the structure is formed within a few to tens of minutes, depending on the specific material formed and the reaction conditions. For example in situ XRD studies of the formation of MCM-41 with tetraethoxysiliane as a silica source showed that the hexagonal order is formed within 3 min.40 In situ EPR measurements on MCM-41 revealed two stages in the formation of MCM-41, the first lasting about 12 min.41 In these preparations, the silica source used was TEOS, if sodium silicate is used as a source the reaction is significantly slower and can take a few hours.18 The formation of SBA-15, prepared in acidic solution with a block copolymer as a template is significantly slower and ranges between 40 and 120 min depending on the reaction conditions.16,23 By comparison, the rate of micellar and surfactant redistribution and morphology changes at a rate that inversely depends on the overall density.42 Still, the typical lifetime of a micelle is in the millisecond range,42 so we can regard the process of silicate adsorption as slow in comparison.
II. Thermodynamic Model We begin by analyzing the morphological changes that result from the reduction in the spontaneous curvature of the micelles. The spontaneous curvature represents the tendency of the surfactant layer to curve (see mathematical formulation below) and therefore plays a central role in determining the favorable shape of surfactant aggregates. At low surfactant concentrations, the majority of micelles are spherical in shape.43 However, as the spontaneous curvature is reduced below a critical value, the system undergoes a continuous transition from spherical to cylindrical micelles (see, e.g., Figure 3a and Appendix A). Another transition occurs at even lower values of the spontaneous curvatures from isolated cylindrical to inter-connected branched networks. These transitions were studied quantitatively in the situation where the spontaneous curvature decreases as the temperature increases37 (see also eq 15 below). In the present study, we follow a similar scenario except that here the spontaneous curvature reduces as a result of silicatete adsorption rather than by raising the temperature.37 The process of silicate polymerization is a central one for the formation of the mesoporous materials. An example of the (39) Baute, D.; Frydman, V.; Zimmermann, H.; Kababya, S.; Goldfarb, D. J. Phys. Chem. B 2005, 109, 7807-7816. (40) Linde´n, M.; Schunk, S. A.; Schu¨th, F. Angew. Chem., Int. Ed. Engl. 1998, 37, 821-823. (41) Zhang, J.; Luz, Z.; Goldfarb, D. J. Phys. Chem. B 1997, 101, 7087. (42) Ilgenfritz, G.; Schneider R.; Grell, E.; Lewitzki, E.; Ruf, H. Langmuir 2004, 20, 1620-1630. (43) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces and Membranes. Frontiers in Physics; Addison-Wesley Publishing Company: Reading, MA, 1994; Vol. 90.
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where κ and κj are the bending rigidity and splay modulus, respectively, and c1 and c2 are the principal curvatures. For a cylinder of radius R, the principal curvatures are c1 ) 1/R and c2 ) 0, whereas for a sphere, c1 ) c2 ) 1/R. The spontaneous curvature, cj, is a measure of the layer tendency to curve and in our approach is a function of the fraction of surfactant molecules that are covered by adsorbed silicate, the surface coverage φs. For simplicity, we assume that the dependence of the spontaneous curvature on the amount of adsorbed silicate is linear (see Appendix B).
cj ) c0(1 - φs)
(2)
where c0 is the bare spontaneous curvature of a surfactant layer. Typically, c0 = 1/d where d is of the order of the length of a surfactant chain. For simplicity, we have assumed that when the surfactant layer is fully covered it prefers to be flat, cj = 0. This is a natural choice since a fully polymerized silicate layer will favor a flat configuration. However, this is not necessarily always the case (see, e.g., Appendix B), and our approach can be easily generalized to cases where the spontaneous curvature at full coverage is nonzero. Furthermore, we neglect any dependence of κ and κj on φs. This assumption is valid for a low degree of polymerization, such that the surfactant-silicate molecules remain fluid and the bending modulus is still dominated by the response of the un-altered hydrophobic chain. Typical values for the bending rigidity κ are comparable or higher than the thermal energy kBT.43,37 The splay modulus κj is typically negative and of the order of the bending rigidity. The reduction in the spontaneous curvature increases the energy of the end-caps of cylindrical micelles37,44
κ e(φs) ) Ae - 2π (1 - φs) kBT Figure 3. (a) Typical phase diagram for the surfactant-silicate system, as functions of the silicate adsorption coverage, φs, and the surfactant volume fraction, φ (see text for details), using s ) -1.75. (b) An example of the condensation transition described by the spinodal of the free energy including the silicate-induced micellar attractions, using s ) -0.25.
(3)
and lowers the energy of 3-fold junctions, given by
κ j(φs) ) Aj + 4π (1 - φs) kBT
(4)
where e and j are expressed in units of kBT, Ae ) (3πκ + 2πκj)/kBT, and Aj ) -(2πκ + 2πκj)/kBT.37 The reason for these energy changes are straightforward: reducing the spontaneous curvature makes it less energetically favorable to form the highly curved end-caps, whereas it makes it more favorable to form the relatively flat 3-fold junctions.44 The free energy of the system can be written in terms of the overall surfactant volume fraction, φ, and the volume fractions of end-caps, φe, and junctions, φj37
importance of this polymerization process appears in ref 20, where the formation of oligomeric silicate anions is shown to be a prerequisite for the entire mesoporous phases to appear. Furthermore, it shows that the adsorbed silicate may be in the form of larger objects than single ions. In our model, this distinction does not play a role, as shown schematically in Figure 1. Note also that in our model we assume that the system is well above the CMC point, so that we can work with basic units composed of spherical or cylindrical micelles and not individual surfactant molecules. This is the standard scheme developed in ref 37. In the discussion below, we assume that the silicate oligomers adsorb uniformly over the surface of the micelles. However, as we discuss in Appendix A, the silicate molecules adsorb preferentially to low curvature regions, i.e., to the linear sections of the cylindrical micelles (Figure 1b). For clarity, and since this effect does not affect the qualitative behavior of the system in an essential manner, its discussion is left to Appendix A. The tendency of the surfactant layer to curve can be quantified by writing down the bending energy (per unit area)43
F(φ)/kBT ) (1 - φ) ln(1 - φ) + φe(ln φe - 1) + φj(ln φj 1 3 1) + φee + φjj - φe ln φ - φj ln φ (5) 2 2
κ ub ) (c1 + c2 - cj)2 + κjc1c2 2
(44) Tlusty, T.; Safran, S. A.; Strey, R. Phys. ReV. Lett. 2000, 84, 1244; J. Phys.: Condens. Matter 2000, 12, A253.
(1)
The first term represents the self-avoidance of the surfactant aggregates. The next two terms represent the entropies of the end-caps and junctions, respectively, while the following two terms are due to the excess energies associated with end-caps and junctions as given by eqs 3,4. Finally, the last two terms represent the loss in relative translational entropy of surfactant molecules that are are constrained by the end-caps and junctions formations.37 For clarity, we do not take into account at first
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inter-micellar interactions beyond excluded volume interactions. These will be discussed later on (see eqs 11 and 12 below). The above free energy can be minimized with respect to φe and φj yielding
φj ) φ3/2e-j φe ) φ1/2e-e
(6)
allowing us to re-express eq 5 as
F(φ)/kBT ) (1 - φ) ln(1 - φ) - φ1/2e-e - φ3/2e-j (7) Note that in this expression the free energy depends on the parameters κ, κj, and φs, through the cap and junction energies eqs 3 and 4. In all of the calculations given below, we used κ ) kBT and κj ) -0.46kBT. From the above equations, one observes that each end-cap or junction lowers the free energy of the system by kBT, in analogy to point defects in ordered systems.45 The negative sign of the last term in eq 7 may lead to an instability by which the system phase separates into a junction-rich and a junction-poor phase. We will see below that this is indeed the case and discuss its implication to the morphology of the emerging structures. It is clear from eq 3 that silicate adsorption increases the cap energy of the micelles. An increase in cap energy makes longer cylindrical micelles more favorable. The point beyond which the typical micellar length L is larger than the micellar radius d (see Figure 1),37 L/d ) 2φ/φe = 2φ1/2eg(φs), occurs at a critical surfactant volume fraction
1 φc1/2 ) e-e(φs) 2
(8)
or, equivalently, at a critical silicate coverage
φs,c ) 1 -
log(4φ) + 2Ae 4π(κ/kBT)
(9)
Expression 9 defines a geometric transition line, which is plotted as a dashed curve in Figure 3a. Note that at high surfactant volume fractions, φc J (1/4) exp(-2Ae + 4πκ/kBT) ∼ 20%, there are no spherical micelles, even in the absence of silicate ions. At higher silicate coverage, there is a thermodynamic transition from cylinders to a connected network.37 This transition is driven by the reduced junction energy, as the spontaneous curvature decreases. The spinodal line of this transition, the line of instability where the second derivative of the free-energy, eq 7, with respect to the surfactant concentration, φ, vanishes, ∂2F/∂f 2 ) 0, is plotted in Figure 3a (top solid curve). We calculate this transition line numerically. We assume here that silicate adsorption and polymerization are irreversible processes so that the amount of adsorbed silicate, φs, is not a variational quantity and is determined by the adsorption and polymerization rates. Another crossover line, the percolation transition, is also plotted in Figure 3a (dashed-dotted line). It is given by37
φ ) (1/3)ej-e
(10)
which coincides with the spinodal line in the limit of low surfactant concentrations (top left corner of Figure 3a). At high surfactant (45) Zilman, A. G.; Safran, S. A. Phys. ReV. E 2002, 66, 051107.
concentrations, it signals a geometric transition to a connected network and is not associated with a thermodynamic phase transition. Note that at high silicate coverage, φs f 1, where the spontaneous curvature vanishes (eq 2), a phase of lamellar vesicles is expected to appear, as is the case with surfactant-lipid mixtures.46 A rough expectation is that this phase appears at spontaneous curvatures cjd j 0.25,44 corresponding here to φs J 0.75 (Figure 3a). Additional phases that are expected to occur at very high surfactant volume fractions include ordered hexagonal and lamellar phases, even in the absence of adsorbed silicate.47 So far we have considered surfactant-silicate systems with no direct interactions, beyond excluded volume. As mentioned in the Introduction, the adsorbed silicate induces an effective short-range attraction between the surfactant molecules (Appendix B). The strength of this effective attraction depends on the charge of the silicate ions and the strength of their polymerization, both of which are dependent on the conditions of the solution (salt concentration, pH, and temperature). The simplest description of this effect is to introduce a mean-field attraction between the surfactant molecules due to the adsorbed silicate. This is described by a new two-body interaction term in the free energy (7)
∆F/kBT ) V0φ2φs2
(11)
where V0 < 0 describes the attraction between surfactant molecules covered by silicate, where φφs is the volume fraction of covered surfactant molecules. We assumed here that the attraction is only between surfactant molecules that are covered by adsorbed silicate (Figure 1c), which are then able to form a polymerized linkage between them. Alternatively, we can treat the junction formed by two cylindrical micelles due to the adsorbed silicate, as an effective “cross-linker”48 (Figure 1c). The extra terms in the free energy, eq 7, due to the presence of 4-fold junctions, of volume fraction φ4, is
∆F/kBT ) φ4(ln φ4 - 1) + φ4s - 2φ4 ln(φφs)
(12)
where the first term represents the entropy of such junctions, the second is the energy of creating a junction between micelles due to the silicate polymerization, s < 0, and the last term is the reduction in the entropy of the surfactants that are confined to 4-fold junctions. Minimizing the free energy with respect to φ4, we find φ4 ) (φφs)2e-s and ∆F/kBT ) - 2(φφs)2e-s. Hence, we recover eq 11, with V0 ) -2e-s. Let us stress here again that terms of the form φ2φs and φ 2 which describe interactions between a surfactant molecule that has no silicate adsorbed and another that is either adsorbed with silicate or not are not included. In these cases, there is no polymerization possible across the two micelles, and we neglect here any other sources of attractive interactions (such as electrostatic, etc.). We note in passing that depending on the exact conditions cross-links can lead to either network-like structures or to micellar bundles.49-51 In particular, two new morphologies that might be of particular interest are the two-dimensional raft-like aggregates (46) Kozlov, M. M.; Lichtenberg, D.; Andelman, D. J. Phys. Chem. B 1997, 101, 6600. (47) Blin, J. L.; Lesieur, P.; Ste´be´, M. J. Langmuir 2004, 20, 491. (48) Zilman, A. G.; Safran, S. A. Europhys. Lett. 2003, 63, 139. (49) Borukhov, I.; Bruinsma, R. F.; Gelbart, W. M.; Liu, A. J. Phys. ReV. Lett. 2001, 86, 2182-2185. Borukhov, I.; Lee, K.-C.; Bruinsma, R. F.; Gelbart, W. M.; Liu, A. J.; Stevens, M. J. J. Chem. Phys. 2002, 117, 462-480. (50) Borukhov, I.; Bruinsma R. F. Phys. ReV. Lett. 2001, 87, 158101. (51) Borukhov, I.; Bruinsma, R. F.; Gelbart, W. M.; Liu, A. J. Proc. Natl. Acad. Sci. 2005, 102, 3673-3678.
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that are formed by filamentous actin in the presence of multivalent counterions50,52 and the three-dimensional cubatic phase.51 The modified spinodal line calculated from the full free energy, F + ∆F (eqs 7 and 12), is presented in Figure 3a (bottom solid curve). The silicate-induced attraction shifts the spinodal (instability) line to lower silicate coverage as compared with the attraction-free spinodal line (top solid curve in Figure 3a). The reason being that the direct silicate-induced attraction (eq 12) is more efficient in inducing inter-micellar attractions than the entropy of the 3-fold junctions. This spinodal line describes now a coexistence region between a low-density phase of isolated micelles and a high-density phase of “bundles” of attached micelles.48 This was observed experimentally by cryo-TEM.38 The coexistence lines are marked as horizontal lines in Figure 3b since, under our assumptions, silicate coverage is not affected by the change in the morphology. When the morphology of the micelles in the dilute phase is that of isolated long cylinders, we expect the corresponding bundles (or precipitates) to have a close-packed, hexagonal symmetry. The uniformity of this hexagonal phase is determined by the length scale over which the micelles are oriented along the same direction. This length-scale is determined by the average length of the micelles at the point where the system first reaches the transition line. As long as branching in the dilute phase is rare, the length distribution is exponential and long cylinders prevail. When branching is dominant, the average length of the straight cylinders is decreased, and the hexagonal phase is of low quality. Finally, if the percolation of branching occurs before the spinodal is reached, then a bicontinuous network occurs, which has no hexagonal symmetry at all. Such bicontinuous structures can be amorphous or ordered in a cubic-type phase. Note that in the condensed phase the system is above the percolation line, so we could expect it to be always bicontinuous. This, however, is not the case since we reached this point in phase space by forming micellar bundles, which once formed are assumed irreversible and their individual morphology is fixed. If this is not true, and the formed silicate layer is fluid enough to allow major rearrangement of the surfactant, then more branching can appear over time, and the quality of the hexagonal phase will deteriorate. Finally, let us mention another transition that occurs in solutions of cylindrical micelles: the isotropic-nematic transition.51,53 This transition was found to occur for a solution of rodlike objects when their volume fraction increases above a critical value, given by the ratio of the rod diameter, 2d, to length, L: φN = R(2d/L), where R is a dimensionless coefficient of order unity. In the present system, the ability of wormlike micelles to branch is detrimental to the nematic transition, as it tends to make each micelle less rodlike and more spherical (tree-like). To allow for branching, the average length of a micelle
L h ) 2φ/φe
(13)
has to be replaced with the average length of a directed segment37
L hd )
φ φe/2 + φj/3
(14)
Using these lengths (eqs 13 and 14) in the criterion for the nematic transition, we calculate the two different Isotropic/Nematic (52) Wong, G. C. L.; Lin, A.; Tang, J. X.; Li, Y.; Janmey, P. A.; Safinya, C. R. Phys. ReV. Lett. 2003, 91, 018103. (53) Onsager, L. Ann. N. Y. Acad. Sci. 1949, 51, 627-659.
Figure 4. Trajectories (vertical solid arrows) of silicate adsorption process in the phase diagram for the surfactant-silicate system, as a function of the silicate adsorption fraction φs and the surfactant fraction φ. The transition lines, using s ) -0.8, are solid line, condensation spinodal; dash-dot line, percolation transition; dashed line, nematic transition; horizontal dotted line, lamellar transition. Also marked is the minimum silicate fraction that blocks further growth after condensation s,min (horizontal dash-dot line). The two vertical dashed lines mark the region where the hexagonal phase is best produced, due to condensation of cylindrical micelles.
transition curves, which are plotted in Figure 3a as dotted (eq 13, no branching) and dashed-dotted curves (eq 14, including branching). At low silicate coverage, the differences between the two curves are small but as silicate coverage increases, the number of 3-fold junctions grows, the effective segment length decreases, and the isotropic/nematic transition is pushed to higher surfactant concentrations. This, of course, will affect the morphology dramatically since in the nematic phase the cylindrical micelles are globally oriented and condensation is more likely to produce high quality hexagonal phases.
III. Discussion and Comparison to Experiments We now proceed to compare our model with the experimental data. We assume that once silicate adsorption starts the system develops continuously along vertical trajectories in the phase diagram (vertical arrows in Figure 4). The properties of such a trajectory depend on the initial conditions, the silicate adsorption rate, etc. and determine the final morphology. Evidence supporting this view of gradual morphological change comes from recent cryo-TEM pictures, in which the reaction mixture of SBA-15 was quenched by rapid freezing at different times following addition of silicate ions.38 The images demonstrate that as time progresses, the micelles tend to elongate and above a certain critical time they collapse to form packed condensed phases. Threadlike micelles were also observed by cryo-TEM approximately 3 min into the reaction of MCM-41.27 The main pathways the system may follow, and the resulting morphologies are described below. (i) Lamellar Phase. We start at the lowest surfactant concentrations (trajectory A in Figure 4). Here, the trajectory may take the system into the lamellar phase before condensation to the hexagonal phase occurs. This explains the observations in ref 11 that, for surfactant concentrations lower than some critical value, a lamellar phase was formed. The lower critical surfactant concentration for the appearance of the hexagonal phase (marked by the leftmost vertical dashed line in Figure 4)
Morphological Transitions in Mesoporous Materials
Figure 5. Schematic view of the cylindrical micelles in a hexagonal stack shown here for the case of block-copolymers. (a) At high surfactant concentrations, φs < φs,min (trajectory C in Figure 4), the adsorbed silicate (shaded region) does not saturate the hydrophilic corona at the time of bundle condensation. (b) At low surfactant concentrations, φs > φs,min (trajectory B in Figure 4), a saturated adsorbed silicate layer forms.
is determined by the point at which the horizontal line marking the onset of the lamellar phase crosses the spinodal line. (ii) Saturated Hexagonal Phase. At higher, but relatively low, surfactant concentrations, the trajectory transforms the system from spherical micelles to long cylindrical (wormlike) micelles followed by a transition into the condensed phase, which is most likely to have hexagonal symmetry (trajectory B in Figure 4). Once the hexagonal phase is formed, silicate adsorption may either continue by diffusion throughout the hexagonal stack (Figure 5a) or be blocked (Figure 5b), depending on the structure and density of the formed silicate layer. The latter depends on the amount of silicate that adsorbed into the micelles when the system reaches the condensation line, φs,c. Let us define φs,min as the critical value of adsorbed silicate, above which further adsorption is blocked in the bundled phase (Figure 5b). For φs,c J φs,min, the thickness of the final silicate structure is given by the silicate coverage at the time of condensation. From Figure 4 (trajectory B), we predict that there will be a region where the final silicate thickness decreases with increasing initial surfactant concentration. Indeed, the experimental data supports this prediction: as the surfactant fraction is raised from 5% to 20%, the measured thickness of the silicate layer decreases.47 (iii) Unsaturated Hexagonal Phase. At higher surfactant concentrations, the condensation transition occurs at lower values of the silicate coverage, φs,c < φs,min (trajectory C in Figure 4). In this case, silicate continues to diffuse into the condensed hexagonal stack (Figure 5a). However, once silicate coverage reaches the saturated value φs,min, diffusion is not expected to continue, and we conclude that at higher surfactant concentrations, the thickness of the final silicate layer reaches a constant value, φs,min.
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(iv) Disordered Bicontinuous or Ordered Cubic Phase. At even higher surfactant concentrations, the micelles are above the percolation line when condensation occurs (trajectory D in Figure 4). At the onset of condensation, the micelles form a percolated bicontinuous network phase with a high concentration of random (disordered) junctions along the cylinders. We therefore expect a disordered bicontinuous phase to replace the hexagonal phase at surfactant concentrations above the critical surfactant concentration where the percolation line crosses the spinodal line (rightmost vertical dashed line in Figure 4), as is indeed observed experimentally.47 A bicontinuous phase may also evolve into an ordered cubic geometry provided that the system is fluid enough to allow rearrangement of the branches during or after the condensation occurs. (v) Disordered Solid, Bicontinuous, or Cubic Phase. When the silicate-induced attraction is strong enough, the spinodal line of phase separation will occur at low silicate adsorption levels and will cross into the low concentration regime of spherical micelles. In this case, the spherical micelles will condense into close-packed aggregates at low levels of adsorbed silicate, φs,c < φs,min, similar to trajectory C in Figure 4. In these aggregates, the adsorbed silicate level is low and silicate adsorption continues within the aggregates. Concurrently, the micelles in the aggregate will elongate and align to form hexagonal bundles of cylinders, since the silicate remains highly fluid at these low adsorption levels. Only when the silicate adsorption level approaches the percolation line (Figure 3a) does the problem of micellar branching arise. The quality of the hexagonal phase will again depend on the fluidity of the structures, which determines whether branching and micellar rearrangement is allowed. Such a scenario is suggested by the experiments of Flo¨dstrom et al.22,23 Note that one of the most basic features of the hexagonal mesoporous materials is also naturally explained within our model; the ends of the cylinders that form the hexagonal bundles are open, that is free of any significant layer of silicate. This property is key to the applicability of these materials since it allows access to their inner spaces. This feature follows naturally from our treatment of Appendix A, where the asymmetric silicate adsorption on the cylindrical and end-cap surfaces is derived (Figure 6b). In the model we presented above, we included excluded volume repulsion and a contact-type attraction between silicate-covered surfactant molecules, that describes the polymerization of the silicon. We could add to our model electrostatic interactions between the micelles. If these are highly screened, then they will just act to modify the interactions considered so far but not change the overall behavior. If the electrostatic interactions are longrange and dominant over the polymerization energy, then the phase diagram may change. Since the silicate-covered micelles in our system are overall neutral, it is reasonable that the electrostatic interactions are not dominant. Finally, let us discuss briefly the effect of temperature on the resulting morphologies. To compare the temperature dependence of the morphology with experiments, we need to model the temperature dependence of the spontaneous curvature of the surfactant. Qualitatively, we expect that increased temperature will result in an overall lowering of the spontaneous curvature, similar to eq 237
cj ) c0(1 - φs)(1 - T/T*)
(15)
where T* is a surfactant-dependent parameter, which is likely to depend also on the level of silicate adsorption. This is equivalent to an effective enhancement of φs. We therefore expect that at higher temperatures bundle formation will occur at lower silicate
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in cryo-TEM experiments,38,54 one may be able to plot the dynamics along the predicted trajectories in the phase diagram (Figure 4) for a particular choice of reaction components. Acknowledgment. We thank S. Ruthstein for useful discussions and access to her unpublished data. We thank S. Safran and A. Zilman for fruitful discussions. N.G. is grateful to the EU SoftComp NoE grant and the Robert Rees Fund for Applied Research for their support. D.G. acknowledges the support of the center of excellence “Origin of ordering and functionality in mesostructures hybrid materials.” supported by the Israel Science Foundation (Grant No. 800301-1) I.B. acknowledges the support of the German Israel Foundation.
Appendix A: Nonuniform Adsorption onto Cylindrical Micelles
Figure 6. (a) Volume fraction of adsorbed silicate on the cylindrical part φ1 as a function of their chemical potential µs for χb ) 0.2 (solid line), χb ) 1 (dashed curve), and χb ) 10 (dash-dot). (b) Volume fraction of adsorbed inorganic molecules on the spherical part φ2 as a function of φ1 for the same values of χb. The dotted curve is φ2 ) φ1
adsorption levels, corresponding to thinner silicate shells and larger pore diameters. This is indeed the experimentally observed trend.47 For lack of quantitative parameters, we do not attempt a more exact comparison.
IV. Conclusions In this article, we have put forward a novel approach for predicting the morphologies of mesoporous materials that are formed when polymerizable silica precursors are added to a solution of surfactant molecules. Our approach is based on a mean-field calculation describing the phase diagram of surfactant aggregates in the presence of the adsorbed silica molecules. On the basis of this phase diagram, we were able to predict the final morphology in different scenarios, as a function of the surfactant properties (such as the bending moduli), its initial concentration, the properties of the inorganic molecules (such as their degree of polymerization), and the temperature. Our results are in good qualitative agreement with experimental findings. The analysis presented here is only preliminary and oversimplified and can be extended in several ways. First, to extract quantitative predictions from this model, further knowledge of the physical characteristics of the interactions of the adsorbed species and the surfactant molecules is required. In particular, the dependence of the spontaneous curvature on the adsorption of the inorganic molecules is a key parameter which determines the pathway that the system will follow and its final morphology. Second, our approach did not treat long-range electrostatic interactions explicitly, which may modify the phase diagram dramatically. It would be interesting to recalculate the phase diagram in the presence of electrostatic interactions and explore how they affect the self-assembly process. Third, in our approach, we did not treat explicitly the dynamics of the adsorption of the silicate and the rate and degree of oligomerization. We hope that our study will encourage both experimental and theoretical studies of these systems, such as more elaborate simulations36 for example. For example, by following the morphology observed
In this appendix, we study the partitioning of adsorbed precursors between the cylindrical body and the spherical end caps of the micellar aggregates. Our analysis is based on the observation that when specific precursors, such as silicate or aluminate, adsorb onto a surfactant aggregate they may lower the spontaneous curvature of the surfactant layer. The bending energy of the surfactant layer can then be reduced by inducing spherical micelles to elongate and transform into wormlike micelles. Furthermore, because of the lower curvature at the cylindrical parts, as compared to the spherical end caps, it is energetically more favorable for the precursors to adsorb onto the cylindrical parts rather than onto the end caps. Our aim in this appendix is to quantify the partitioning of the precursors within the micelles and characterize the degree of micelle elongation. This more elaborate analysis is consistent with the simpler model presented in the main body of this paper. For this purpose, we adopt a different formalism than in the main body of the article, which allows us to follow the changes in the length distribution of the micelles upon silicate adsorption. However, since this approach does not provide a convenient framework for describing micellar branches, we did not implement it in the main body of the article. We model the micelles as sphero-cylinders consisting of two spherical end-caps of radius d and a central cylindrical part of the same radius and length L (see Figure 1b). The end-caps contain n = 4πd2/πa02 surfactant molecules, a0 being the radius of a surfactant headgroup. Similarly, the cylindrical part contains l ) 2πdL/πa02 surfactant molecules. The free energy of a collection of micelles of varying lengths can be written as
F)
∑l Fl{log Fl - 1 + l[R(φ1) - sφ1 - µsφ1] + n[b(φ2) sφ2 - µsφ2]} + λ[∑(l + n)Fl - Fm] (16) l
where Fl is the concentration of micelles containing l + n surfactants (in units of 1/da02). The first term in the summation is the translational entropy of the micelles. The second term is the contribution from the cylindrical parts where φ1 is the volume fraction of inorganic molecules adsorbed onto the cylindrical part, s their binding energy, and µs their chemical potential. From eq 1 for the bending energy, it follows that
R(φ) )
χb 2 φ + φ log φ + (1 - φ) log(1 - φ) 2
(17)
where χb ) κπa02/d2 is a dimensionless parameter that character(54) Pevzner, S.; Regev, O.; Lind, A.; Linde´n, M. J. Am. Chem. Soc. 2003, 125, 652.
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izes the contribution of the bending energy per adsorption site. The next terms are the entropies of silicate-adsorbed molecules and empty sites, respectively. Similarly, the third term in eq 16 is the contribution from the end-caps where φ2 is now the volume fraction of silicate adsorbed onto the spherical caps and
b(φ) )
χb (1 + φ)2 + χG + φ log φ + (1 - φ) log (1 - φ) 2 (18)
where χG ) κjπa02/d2 is typically negative. Finally, the last term in eq 16 fixes the total amount of surfactant molecules in micelles, where λ is a Lagrange multiplier coupled to the constraint and Fm is the total concentration of surfactants in micelles. The free energy, eq 16, can be minimized with respect to the micelle size distribution, Fl, the adsorbed volume fractions φ1 and φ2 and the Lagrange multiplier, λ. The resulting equations describe the equilibrium state of the system. Although we are interested in the kinetics of the process, it is important to realize that the equilibrium behavior determines to what states the system is driven to. The equations for the adsorbed volume fractions are
µs + s ) χbφ1 + log[φ1/(1 - φ1)] ) χb(1 + φ2) + log[φ2/(1 - φ2)] (19) Note that the relations between µs, φ1, and φ2 depend only on the dimensionless parameter χb and are independent of the amount of surfactants in the system, the micellar size distribution, etc. Typical dependence of φ1 on µs and between φ2 and φ1 is shown in Figure 6 for different values of χb. The adsorption isotherms for φ1 and φ2 have the form of a Frumkin isotherm with repulsive interactions that are due to the bending energy, eq 1. For χb , 1, the adsorption isotherm reduces to an ideal Langmuir isotherm. The dimensionless parameter χb determines the asymmetry in adsorption onto the cylinders (φ1) and the less favorable spherical end caps (φ2). Adsorption does not depend on χG as long as the number of micelles does not change. The equations for Fl and λ become after a slight rearrangement
Fl ) e-(a+∆b)n-al Fm )
(
(20)
)
n 1 -(a+∆b)n + e a a2
(21)
where a ) λ - χbφ12 + log(1 - φ1) replaces λ as the Lagrange multiplier
∆b )
(
)
( )
χb 1 - φ1 + χG + χb(φ12 - φ22) - log 2 1 - φ2
(22)
is the effective end-cap energy, and φ1 and φ2 are related through eq 19. For the length distribution, eq 20, the average micelle length is 〈l〉 ) 1/a. Note that the dependence on χb, χG, and µs (or, equivalently φ1) enters into eqs 20 and 21 only through ∆b (eq 22), which represents basically the free energy difference of a surfactant molecule in the spherical caps with respect to the cylindrical part. Typical behavior is shown in Figure 7a for the same values of χb as in Figure 6 and for χG ) -χb/2. When ∆b increases, the surfactant molecules have a lower free energy in the cylindrical part. The micellar size distribution is then determined by the competition between the translational entropy of the micelles,
Figure 7. Effective end-cap energy ∆b (a) and average micelle length 〈l〉 (b) as functions of φ1.
Figure 8. Crossover point where 〈l〉 ) n as a function of the overall surfactant volume fraction Fm.
favoring a large number of small micelles, and the end-cap energy that drives the system toward a small number of long micelles. This competition leads to micelle elongation as shown in Figure 7b. The crossover from spherical to cylindrical micelles is continuous rather than a sharp (first order) phase transition. The crossover occurs when 〈l〉 = n and is plotted in Figure 8 as a function of the surfactant concentration Fm for the same values of χb and χG as before. As the surfactant concentration increases, the entropy of the micelles decreases and the tendency toward fewer longer micelles increases. This can be easily demonstrated by inserting the condition 〈l〉 = n in eq 21 which yields
1 ∆b* ) [log 2n2 - 1 - log Fm] n
(23)
The horizontal dotted line in Figure 7a is the value if ∆b* for Fm ) 10-4. It is now clear that as χb increases in magnitude the crossover should shift to lower values of φ1 since now ∆b grows faster. As can be seen in Figure 8 this is indeed the case.
Appendix B: The Effect of Adsorption on the Spontaneous Curvature Let us consider the equilibrium headgroup separation a0 and its dependence on the silicate coverage φs. A simple description of the effective surfactant-surfactant interactions is to include
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some attractive potential and a repulsive electrostatic potential
V=
q2 1 2 + kr r 2
(24)
where q is the effective headgroup charge, is the dielectric constant of the solution, and k is an elastic coefficient which characterizes the interactions between surfactant molecules. It includes contributions from van der Waals interactions and the hydrophobicity-induced confinement of the surfactant molecules. Minimizing eq 24, the equilibrium headgroup separation becomes a0 = (q2/k)1/3. If the adsorbed silicate reduces the effective positive charge per surfactant headgroup, due to its opposite (negative) charge, q f q(1 - φs), the result will be a reduction in the equilibrium separation, a0 f a0(1 - φs)2/3 = a0(1 - 2φs/3). Note that the headgroup hard-core, as well as the hard-core radius of the hydrophobic chain r0, impose a lower cutoff on a0, that
can be incorporated in eq 24. For amphiphilic polymers, the elastic constant k is a function of the radius of gyration, Rg, of the hydrophilic block, k = kBT/Rg2. In this case, the equilibrium headgroup separation becomes a0 = (Rg2q2/kBT)1/3. The latter can be expressed as a0 = (Rg2lbq2/e2)1/3, where e is the elementary charge and lb ) e2/kBT = 7 Å for an aqueous solution at room temperature. In the limit of low curvature, cj , 1/a0, one can extract from the geometrical description shown in Figure 2, an estimate for the spontaneous curvature, cj = (a0 - r0)/(r0d), where r0 is the radius of the hydrophobic chain and d its length. Combined with the above estimate for the equilibrium headgroup separation, a0, we obtain a linear dependence of the spontaneous curvature on the silicon coverage, φs, eq 2, as long as the linear dependence of a0 on φs holds. LA052272R
