radial patterns in mesoporous silica

Communications

636 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1999 0935-9648/99/0806-0636 $ 17.50+.50/0 Adv. Mater. 1999, 11, No. 8

types of lasing guests, such as rare earth complexes or con-jugated polymers. The requirements concerning homoge-neity and material shape can be fulfilled. These other typesof light-emitting species are now being investigated for in-corporation into mesoporous hosts in order to construct la-ser materials. This new class of materials offers many possi-ble variations according to the selection of luminescentspecies or coadsorbates to extend the wavelength range ofemission and excitation. The self-assembled host, espe-cially, allows appreciable control of mesoscale dimensions,symmetry and orientational ordering of guest-host struc-tures. Energy transfer might be exploited for sensitizing adye or for exciting narrow-line emitters, such as rare-earthcomplexes. Saturable absorbers could also be incorporatedin the composite to make ultrafast lasers. These compositesprovide versatile opportunities for the construction of effi-cient and tunable new laser materials. Further options inusing mesoporous host materials for lasing guests are touse them as highly sensitive sensors[25] where the lasingwavelength is varied, for example, by gas adsorption.

Received: January 18, 1999

±[1] C. T. Kresge, M. E. Leonowicz, W. J. Roth, J. C. Vartuli, J. S. Beck,

Nature 1992, 359, 710.[2] See e.g.: D. Zhao et al., Science 1998, 279, 548.[3] Q. Huo, D. Zhao, J. Feng, K. Weston, S. K. Buratto, G. D. Stucky, S.

Schacht, F. Schüth, Adv. Mater. 1997, 9, 974.[4] C. Gojon, B. Dureault, N. Hovnanian, C. Guizard, Sens. Actuators B

1997, 38, 154.[5] R. Leon, D. Margolese, G. Stucky, P. M. Petroff, Phys. Rev. B 1995, 52,

R2285.[6] D. M. Antonelli, J. Y. Ying, Angew. Chem., Int. Ed. Engl. 1995, 34,

2015. P. V. Braun, P. Osenar, S. I. Stupp, Nature 1996, 380, 325. P. Yang,D. Zhao, D. Margolese, B. F. Chmelka, G. D. Stucky, Nature, in press.

[7] C. G. Wu, T. Bein, Science 1994, 266, 1013.[8] P. J. Bruinsma, A. Y. Kim, J. Liu, S. Baskaran, Chem. Mater. 1997, 9,

2507.[9] P. Yang, D. Zhao, B. F. Chmelka, G. D. Stucky, Chem. Mater. 1998, 10,

2033.[10] H. Tamai, M. Ikeuchi, S. Kojima, H. Yasuda, Adv. Mater. 1997, 9, 55.[11] H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, G. A. Ozin,

Nature 1996, 379, 703. H. Yang, N. Coombs, I. Sokolov, G. A. Ozin, Na-ture 1996, 381, 589. I. A. Aksay et al., Science 1996, 273, 892. M. Oga-wa, J. Am. Chem. Soc. 1994, 116, 7941. S. H. Tolbert, T. E. Schäffer, J.Feng, P. K. Hansma, G. D. Stucky, Chem. Mater. 1997, 9, 1962.

[12] M. Trau et al., Nature 1997, 390, 674. P. Yang et al., Science 1999, 282,2244.

[13] S. Mann, G. Ozin, Nature 1996, 382, 313.[14] V. G. Kozlov, V. BulovicÂ, P. E. Burrows, R. S. Forrest, Nature 1997,

389, 362.[15] M. Berggren, A. Dodabalapur, R. E. Slusher, Z. Bao, Nature 1997,

389, 466.[16] M. D. McGehee et al., Appl. Phys. Lett. 1998, 72, 1536. F. Hide et al.,

Science 1996, 273, 1833. N. Tessler, G. J. Denton, R. H. Friend, Nature1996, 382, 695.

[17] D. Shamrakov, P. Reisfeld, Chem. Phys. Lett. 1993, 213, 47.[18] D. Lo, J. E. Parris, J. L. Lawless, Appl. Phys. B 1992, 55, 365.[19] G. Ihlein, F. Schüth, O. Krauss, U. Vietze, F. Laeri, Adv. Mater. 1998,

10, 1117.[20] D. Zhao, P. Yang, N. Melosh, J. Feng, B. F. Chmelka, G. D. Stucky,

Adv. Mater. 1998, 10, 1380.[21] F. Marlow et al., Micropor. Mater. 1996, 6, 43.[22] This method has been used for microcrystals, for example: K. Hoff-

mann, F. Marlow, J. Caro, Adv. Mater. 1997, 9, 567.[23] Here, we use the mean intensity of the 10 ns pulses to describe the

pump radiation. So, 100 kW/cm2 corresponds to 1 mJ/(cm2×pulse).[24] See e.g.: M. D. McGehee et al., Phys. Rev. B 1998, 58, 7035.[25] T. A. Dickinson, J. White, J. S. Kauer, D. R. Walt, Nature 1996, 382, 697.

Radial Patterns in Mesoporous Silica**

By Igor Sokolov, Hong Yang, Geoffrey A. Ozin,* andCharles T. Kresge

It has been found that in a surfactant-templated quies-cent aqueous acidic synthesis of hexagonal mesoporous sil-ica, three basic shapes can be formed with increasing pH,respectively fibers, discoids and spheres.[1±6] Intriguing ra-dial patterns that have been observed on the surface ofsome of these shapes have defied explanation. It is consid-ered that morphogenesis of mesoporous silica shapes is ini-tiated by topological defects in a hexagonal silicate liquidcrystal seed.[1±6] For instance, under conditions that favor a2p-line disclination and a discoid shape for the nucleatingsilicate mesophase, the director field whirls concentricallyand coaxially around the defect line. The surface meso-structure and zeta potential of this embryonic silicate dis-coid serves to direct the silicification and growth of meso-porous silica to a discoid shape.

Herein we show that the classical theory of elasticity forliquid crystals[7] can be used to minimize the bulk and sur-face free energy of a hexagonal silicate mesophase contain-ing a 2p-line disclination defect. These calculations revealthe emergence of radial patterns that bear a striking resem-blance to those observed on the surface of mesoporous sil-ica discoids. It is noted that silicification and contraction ofthe silicate mesophase can impose radial and longitudinalcompressive stresses on the discoid. The compressive strainmay be dissipated by corrugation of the disclination mak-ing it more visible. Depending on the relative rates of poly-merization of the silicate core and corona, the strain mayalso be released by deforming the flat discoid to create pro-truding or sunken discoid shapes.

An insight into the mode of formation of hexagonalmesoporous silica fiber, discoid and sphere shapes has beenobtained from scanning and transmission electron (SEM,TEM), atomic force (AFM), and polarization optical mi-croscopy (POM). These techniques define the relation be-tween the pattern of channel director fields, optical bire-fringence and morphology, and hence provide clues aboutthe growth process of a particular form.[1±5]

Morphokinetic, dynamic light scattering, TEM and AFMstudies of the formation of different mesoporous silica

±

[*] Prof. G. A. Ozin, Dr. I. Sokolov, Dr. H. YangMaterials Chemistry Research GroupChemistry Department, University of Toronto80 St. George StreetToronto, Ontario, M5S 3H6 (Canada)

Dr. C. T. KresgeMobil Technology CompanyPaulsboro, New Jersey 08066-0480 (USA)

[**] GAO is indebted to the Canada Council for the award of an Issac Wal-ton Foundation Research Fellowship 1995±97 that was held during thisresearch. GAO is also deeply grateful to the Mobil Technology Com-pany for financial support of this research. Technical discussions withProf. Raymond Kapral proved to be most enlightening.

shapes have shown the process begins with the self-assem-bly of a ~50 nm diameter silicate liquid crystal seed. Thisgrows by accreting silicate micelles from solution, rigidifiesby polymerization of the silicate and arrives at a particularshape.[6] The pH of the growth medium determines the sur-face charge on the seed and silicifying mesophase, the an-choring of silicate micelles to the seed and growing meso-phase, and the rate of formation of a particularmesoporous silica shape.[6]

We analyzed the SEMs of about eighty discoids andfound that about 10 % display periodic radial patterns, butwith no particular maximum in the distribution of the num-ber of observed rays N, apart from N = 0. Qualitatively, asN increases the number of observed patterned discoids de-creases. The distribution has a cut-off of N ~ 60±70 rays,where three out of eighty discoids fall in this range, how-ever, one case of N ~ 140 rays was observed. SEM, TEM,and POM images that portray representative radial pat-terns on the surface of discoid shaped mesoporous silicaare shown in Figure 1.

The main objective of this paper is to provide a theoreti-cal model that can describe the origin of periodic radialpatterns on the surface of mesoporous silica discoids.

To theoretically describe the occurrence of the afore-mentioned surface patterning process in mesoporous silica,one can consider the free energy of different shapes. Forthe liquid crystal stage, the free energy density, with respectto a non-deformed nematic, is given by the theory of elasti-city (Eq. 1),[7] where n® is a unit vector of the director-fieldof the nematic, and K1,2,3 are the elastic modulae, that de-note splay, twist, and bend deformations, respectively.

(1)

When polymerization of the silicate causes the shape tobecome rigid, the free energy density will be described byfive different elastic modulae l1±5,[6] (Eq. 2) where uij is thedistortion tensor. In linear order Equation 3 applies, whereui is the displacement along the xi-axis.

(2)

(3)

During the liquid crystal stage, Equation 1 determines theenergetically most favorable shapes. After rigidification,Equation 2 describes the free energy density of the shapes.Between these two extremes, both Equations 1 and 2 are im-portant. In what follows, we shall rigorously study the ener-getics of the silicate liquid crystal stage while the spatio-tem-poral effects of silicification will be discussed qualitatively.

We shall consider the classical model of nematics withsome modification. The standard model[7] deals with a liq-uid crystal without a boundary. In the case of a hexagonalsilicate mesophase in solution the boundary exists, andmoreover, it is very important for the study of shape.Therefore, we have to consider the contribution of the sur-face tension to the overall shape. It means that the freeenergy Equation 2 should include the surface energyterm.[8] To obtain the free energy Fd, one needs to integratethe energy density Fd of Equation 1 over the volume of theliquid crystal, taking into account the surface tensionenergy (Eq 4), where s is the surface tension constant, andn® is the unit vector of the director-field.

(4)

Let us now simplify Equation 4 by reference to the ex-perimental data. First, we have rarely observed bending ortwisting of the mesoporous silica discoid shapes in the z-di-rection. K2 is the modulus responsible for such bending/twisting. Therefore, we shall set K2 = 0. Furthermore, fromTEM observations, one observes a concentric arrangementof channels around the main rotation axis of the discoids,that is, the director-field is in the plane of the discoid per-pendicular to the z-axis, Figure 2.

With this information, we can reduce the problem to 2Done. For such geometry of the director field, it is conveni-ent to use a polar coordinate system. We shall use the nota-tion shown in Figure 3.

Adv. Mater. 1999, 11, No. 8 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1999 0935-9648/99/0806-0637 $ 17.50+.50/0 637

Communications

Fig. 1. Representative SEM, TEM, and POMimages (left to right) that display radial pat-terns on the surface of a discoid shapedmesoporous silica.

CMYBC

MY

B

Communications


Using these notations to define the director field, onecan rewrite the free energy as shown in Equation 5, whereC is a constant, proportional to the z-dimension of the dis-coid, H and the sum of K1 and K3.

(5)

(6)

Equation 6 also holds, where R is the radius of the dis-coid and R0 is the smallest radius of the liquid crystal dis-coid which is still described by this theory. We set it to thediameter of one micellar rod namely ~5 nm. Typically theradius of the discoid is about 5 mm with a height H ~ 1 mm.Taking some representative values for s ~ 10±3 N/m andK ~ 10±11 N we arrive at a typical value of D ~ 100. Further-more, because of the process of rigidification one should

expect the elastic moduli K to increase relative to that ofthe silicate liquid crystal. Therefore it is more realistic toassume D < 100.

To find the energetically favorable configurations oneneeds to minimize the function in Equation 5 which leadsto Equation 7.

(7)

It should be noted that solutions of this equation withoutsurface tension, that is D = 0, are known to be cylindricallysymmetrical (C2½n±1½ symmetry group) around the z-axissubject to the restriction shown in Equation 8, where n iscalled the Frank index of disclination, n = 0, ± 1/2, ± 1,± 3/2... This index corresponds to 2½n ± 1½ number of raysin the configuration of the director field. Such rays origi-nate at the z-axis. Two examples with n = 1 and 5/2 areshown in Figure 4.

Fig. 4. Configurations of the director field for the Frank index n = 1(C = p/2) and 5/2, without the surface tension term.

(8)

It is easy to see that the configuration with n = 1 (C =p/2) is still the solution of Equation 7 with surface tension.To find the other solutions with n ¹ 1, we note that the inte-gral of motion of Equation 7 is given by Equation 9.

(9)

The solutions with n < 1 do not correspond to a roundshape and will not be considered in this paper. Following astandard technique, we can find the solution in parametricform (Eq. 10), where the angle C0 is the angle C at thepoint of the director-field near the ray, Figure 5.

(10)

The constant Q is determined from the following bound-ary condition by Equation 11. The angle C0 can range from

Fig. 2. Schematic depicting the channel architecture of a discoid shapedhexagonal silicate mesophase containing a 2p-line disclination.

Fig. 3. Polar coordinate notation for the liquid crystal, where n® is the vectorof the director field.

CMYB

CM

YB

p/2 to 0, where the latter is considered usually to be thecase for D = 0.

(11)

Fig. 5. Definition of C0 in Equation 10.

Here we say a few words about the choice of C0. First,the solution with C0 = 0 does not exist for non-zero D. Thisis easy to see from Equation 9. If in that equationC = C0 = 0, then C¢ must be infinite, which is not physicallymeaningful. Nevertheless, in the framework of the presentapproach we shall consider that C0 can be very small.

Figure 6 shows the dependence of the free energy of theliquid crystal with n = 5/2 (a = 0.5) as a function of C0 fordifferent D = 0, 10, 20, 30. One can see that the case ofsmall C0 is energetically unfavorable. However, if D = 0then the energy very weakly depends on the angle. How-ever, if the surface tension is not equal to zero the freeenergy increases exponentially as C0 approaches zero. Thismeans that the value C0 = 0 is energetically unfavorableprovided the surface tension exists, i.e., D ¹ 0. However,this conclusion is not so straightforward because the config-uration with C0 = 0 is essentially a break of the directorfield. This may mean that the classical theory of elasticityused here is not longer valid. Consequently, if a defect withC0 = 0 was created, by supplying a sufficient amount ofenergy, then such a defect could be stable. Hereafter, weshall not consider such possibility.

Fig. 6. Free energy Fd/C of the discoid shaped liquid crystal as a function ofC0 for different D = 0, 10, 20, 30.

In Figure 7 we present numerical solutions of Equa-tions 9 and 11. The structure shown in Figure 7a has the

same energy as the one shown in Figure 4. Inspection ofthese illustrations shows the effect of squeezing or round-ing the shape envelope. Figure 7b is presented to show theform of a potentially possible solution with very small C0.It is qualitatively the same as the one in Figure 7a. Theonly difference is that the rays are more pronounced. Fig-ure 7c shows a solution with 8 rays, n = 5. A comparison ofthese calculated radial patterns for the discoid mesophase,with radial patterns observed on the surface of mesoporoussilica discoid shapes, Figures 1 and 9, reveals that they arerather similar.

Fig. 7. Lines of the director field for discoid shaped silicate liquid crystalswith: a) n = 5/2, D = 10, C0 = p/3 (this corresponds to the same free energyas in Figure 4, the n = 5/2 case), b) n = 5/2, D = 10, C0 = p/100 000, c) n = 5,D = 10, C0 = p/4.

A graphical presentation of the discoid simulations in 3Dbears a close resemblance to those observed experimen-tally, Figure 8. Two aspects of the 3D simulation are note-worthy. First, despite the fact that the discoid is flat it givesthe impression of being corrugated and rounded betweenthe rays. This is a visual artifact. Secondly, experimentallythe discoids often have a sunken shape, which is absent inthe simulation. We discuss these additional effects in thetext that follows.

We have shown that radial features observed on the sur-face of mesoporous silica discoids can appear as patterns atthe silicate liquid crystal stage of formation of the shapes.Let us discuss the spatio-temporal effects of the next stage,that is silicification of the silicate mesophase. First, as wementioned above the surface tension constant D is propor-tional to the radius of the crystal. Therefore during thegrowth D increases. As one can see from Figure 6, to main-tain the same free energy, the crystal should decrease its C0

ultimately to give a smooth round shape, which corre-sponds to C0 = p/2. On the other hand polymerization ofthe silicate mesophase tends to quench this process throughrigidification. So in the final stage of growth one should ex-pect to see a frozen replica of the radial pattern in the sili-cate mesophase.


Communications

CMYBC

MY

B

Communications


The radial pattern of the discoid shaped silicate meso-phase may be perturbed by compression during silicifica-tion. This kind of compression has been observed experi-mentally and can be explained chemically in terms ofreplacing two silicon hydroxyl groups by a silicon±oxygen±silicon bridge bond that occurs during silicification. The di-ameter of silicate micellar rods has been observed to de-crease by up to 60±70 % of its original value before silicifi-cation. A similar polymerization induced contractionshould occur along the longitudinal direction of the micellerods. In general however, these two contractions can berather different because of an additional factor relating tothe electrical double layer between the silicate micellarrods, which will contribute to the distance between the sili-cified channels. Hence the thickness of the double layer isexpected to change during the silicification process therebyresulting in different radial and longitudinal contractions.Therefore we shall consider two different polymerization-induced contractions namely radial gR and longitudinal gL

defined as in Equation 12, where rafter,before (Lafter,before)are radii (lengths) of a silicate micelle rod after and beforethe rigidification, respectively, and Rbefore is its radial loca-tion in the discoid, see Figure 9.

rafter = gR(Rbefore) rbefore

Lafter = gL(Lbefore) Lbefore (12)

Consider the possible deformations of the silicate liquidcrystal discoid that can appear as a result of the contractionexpressed in Equation 12. We shall consider all deforma-tions of an originally flat-faced cylinder that consists of infi-nitely rigid and radially concentric silicate micellar rods.The assumption of infinite rigidity is justified by the factthat intra-rod binding involves a balance of electrostaticand hydrophobic interactions, whereas inter-rod interac-tions are van der Waals in nature and much weaker. There-fore, the silicate micellar rods can more readily shift rel-ative to one another rather than be deformed. As a resultinternal stresses will be relieved on deforming the discoidwhile maintaining the integrity of the rods. Thus internaldeformation of the rods may then be described only byEquation 12. Also for simplicity we shall consider deforma-tion of a discoid containing no rays.

The degree of contraction depends upon the relativerates of polymerization of the core and corona regions of

the silicate mesophase, which isexpected to depend on pH, timeand temperature. This effect cancause a change in the shape of thediscoid. Let us consider this ideain detail. Because of the longitu-dinal contraction the length ofevery silicate micellar rod de-creases according to Equation 12.Based on the aforementioned as-sumptions the length of each rod

can be equated to 2pR, where R is the radial location of therod in the discoid. Therefore we can rewrite the longitudi-nal part of Equation 12 as Equation 13.

Rafter = gL(Rbefore) Rbefore (13)

Because the rods are considered to be infinitely rigid thisrelation expresses the radial contraction of the location ofthe rod in the discoid.

If the distance between the radial position of any tworods, R1 and R2, increases then to keep the volume betweenthese radii unchanged [H(R2 ± R1)]before = [H(R2 ± R1)]after,and some of the rods from above and below will be drawninwards. If the opposite occurs the rods from above and be-low will be squeezed out. Consequently, the height of thediscoid in the region of the rods will decrease for the firstcase and increase for the second one. These definitions arepresented in Figure 9.

This model is modified by the radial part of Equation 12because the height H should change proportionally to theradial contraction factor, Hafter = gR(Rbefore) Hbefore. Alsothe radial location of the rod in the discoid tends to de-crease as shown in Equation 14, where Rafter is distinct toRafter in Equation 13.

(14)

Consequently, the volume between two channels,[H (R2 ± R1)]before that was unchanged has now becomegR #(Rbefore) Hbefore (R2after ± R1after). Therefore, the condi-tion of constant volume between radii R1 and R2 is ex-pressed as in Equation 15.

gR (Rbefore) Hbefore (R2 after ± R1 after) =(R2 after ± R1 after) Hafter (15)

To derive the formula for the height after contraction asa function of the height before contraction, let us writeEquation 15 in differential form. To do this we consider[(R2 ± R1)]before º dR. From Equations 13 and 14 one de-rives Equation 16, where g¢L(r) is the derivative with re-spect to r.

Fig. 8. Observed and calculated radially patterned discoids.

CMYB

CM

YB

dRafter = gR (Rbefore) dRdRafter = (gL(Rbefore) + g¢L (Rbefore) Rbefore) dR (16)

Putting Equation 16 into Equation 15 and taking into ac-count Equation 12 we arrive at the sought after solution(Eq. 17).

Hafter

g2R

Rbefore

gL Rbefore gL

Rbefore Rbefore

Hbefore (17)

Rafter = gL (Rbefore)Rbefore

To apply this formula to a concrete example we have toassume something about the dependency of gR(r) and gL(r)on r. Let us consider the simplest scenario of a linear con-traction of the rods in the growing discoid. Then the func-tions gR(r) and gL(r) can be defined by Equation 18, wherermax is the radius of the discoid and a, b are dimensionlessconstants.

gR r 1 a r rmaxgL r 1 b r rmax

or (18)

gR r 1 a r rmaxgL r 1 b r rmax

From chemical considerations one can conclude that0 < b < 0.4. Because of the contribution of the electricaldouble layer to the inter-rod contraction the restriction ona is much weaker, 0 < a < 1. Figure 10 shows 3D simula-tions based on the first set of Equations 18 with a = 0.4,b = 0.2, Figure 10a, and with a = 0.5, b = 0.4, Figure 10b.This respectively yields protruding and sunken mesoporoussilica discoid shapes. Simulations based on the second setof Equations 18 look qualitatively the same.

It is not obvious which case gives rise to protruding orsunken radially patterned discoid shapes. For simplicity letus define a sunken shape for Hafter(r = 0) < Hafter(r = rmax),otherwise it is protruding. Together with Equation 17 thisleads to restrictions on a and b. Figure 11 presents numeri-

cal solutions for the first set of Equations 18, Figure 11a,and for the second set, Figure 11b. From an experimentalpoint of view it is expected that the core will be more con-tracted than the corona region because the discoid growsand polymerizes from the center. This situation is describedby the second set of Equations 18, Figure 11b. One can seethat the protruding case is the most prevalent which is con-sistent with the experimental observations. Deformationsof this type, for radially patterned mesoporous silica dis-coids, have been observed experimentally, Figure 12.


Communications

Fig. 9. Illustration of the contraction of a discoid shaped silicate mesophase due to inhomogeneous polymerization.

Fig. 10. Simulation of the polymerization-induced transformation of an initi-ally planar and radially patterned discoid shaped silicate mesophase to yielda) protruding and b) sunken mesoporous silica discoids. The calculation isbased upon a) the first set of Equations 18 with a = 0.4, b = 0.2, b) the firstset of Equations 18 but with a = 0.5, b = 0.4.

a)

b)

CMYBC

MY

B

Communications


The microscopy images shown in Figure 12 provide com-pelling evidence for the generality of radial patterning inmesoporous silica discoids, gyroids and spirals. Experimen-tal and theoretical studies are planned to delve more deep-ly into the origin of these fascinating patterns in a range ofmesoporous inorganic fibers, films and shapes.

Received: September 21, 1998Final version: February 3, 1999

±[1] G. A. Ozin, H. Yang, N. Coombs, Nature 1997, 386, 692.[2] G. A. Ozin, N. Coombs, I. Sokolov, H. Yang, Adv. Mater. 1997, 9, 662.[3] G. A. Ozin, H. Yang, N. Coombs, I. Sokolov, J. Mater. Chem. 1998, 8,

743.[4] G. A. Ozin, C. T. Kresge, H. Yang, Adv. Mater. 1998, 10, 883.[5] G. A. Ozin, C. T. Kresge, H. Yang, Stud. Surf. Sci. Catal. 1998, 117,

119.[6] G. A. Ozin, C. T. Kresge, S. M. Yang, H. Yang, N. Coombs, I. Sokolov,

Adv. Mater. 1999, 11, 52.[7] L. D. Landau, E. M. Lifshitz, Theory of Elasticity, Pergamon, New

York 1972.[8] A. L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth, Wi-

ley, New York 1995.

Fig. 11. Area for the set of parameters a and b giving protruding and sunkenshapes: a) corresponds to the first set of Equations 18, and b) to the secondset.

Fig. 12. Representative SEM images of hex-agonal mesoporous silica discoid, gyroidand spiral shapes that show radial patternsand evidence of compressive distortions in-duced by silicate polymerization.

CMYB

CM

YB

radial patterns in mesoporous silica

Documents