Pore Structure Simulation of Gels with a Binary Monomer Size Distribution

Journal of Sol-Gel Science and Technology 34, 273–280, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Pore Structure Simulation of Gels with a Binary MonomerSize Distribution

J. PRIMERALaboratoire des verres, UMR CNRS 5587, Universite Montpellier II Place E. Bataillon,

34095 Montpellier cedex, France; Departamento de fisica, FEC, LUZ. Maracaibo, Venezuela

T. WOIGNIERLaboratoire des Colloıdes, Verres et Nanocomposites UMR CNRS 5587, Universite Montpellier II, Place E.

Bataillon, 34095 Montpellier cedex, France; IRD-UR Seqbio-Pole de Recherche Agronomique de la Martinique,97232 Le Lamentin, France

A. HASMY∗

Laboratorio de Fısica Estadıstica de Sistemas Desordenados, Centro de Fısica, IVIC, Apdo.21827,Caracas 1020A, Venezuela

[email protected]

Received June 1, 2004; Accepted December 22, 2004

Abstract. The pore size distribution in silica gels can be tailored by the addition of silica soot particles duringthe gel formation. We introduce a numerical model in order to simulate the structure of this “composite gel”. Thealgorithm is based on Diffusion-Limited Cluster-Cluster Aggregation model with an initial binary distribution ofmonomer sizes. The textural properties of the simulated gels are calculated using a simple triangulation method.Nitrogen adsorption-desorption experiments show that with the powder addition the mean pore size is shifted to-wards larger pore size and the specific surface area decreases. Numerical results of the mean pore size, specificsurface area, and particles are in good agreement with experimental data. Because of these textural propertiesthis new type of gels and aerogels has larger permeability and interesting properties as host matrix. The compos-ite gels and the numerical model could also be helpful to simulate the natural allophanic gel found in volcanicsoils.

Keywords: composite gels, porous medium, cluster-cluster aggregation, allophanic soil

1. Introduction

Silica aerogels are fascinating materials because oftheir peculiar physical properties such as a very lowsound velocity (<100 m/s) [1], a large specific sur-face area (>500 m2/ g) (2), a low thermal conductivity(20 mW · m−1 · K−1) [3], and a fractal structure [4].These peculiar features are essentially due to the very

∗To whom correspondence should be addressed.

large pore volume which can be easily tailored from 0%to 99.5%. Pore volume is controlled by the gel synthe-sis conditions [6] (i.e. the siloxane monomer content),or by an additional sintering step [7]. The potentialapplications of the aerogels are catalyst supports [8],insulaters [3], cosmic dust capters [9], and glass pre-cursors [7].

Another possible interest for these porous materialscould be as a host matrix. The large pore volume canbe used as a sponge to incorporate chemical species in

274 Primera, Woignier and Hasmy

such a way to get a two phases material. The chemicalspecies are initially processed in a liquid form but canbe dried after the pore filling. This idea has been alreadyproposed for example to prepare binary silicate glasses[10], confining host matrix for liquid crystals [11, 12],and durable glass-ceramics for the radioactive wastescontainment [13].

However, because of their poor mechanical proper-ties, the capillary stresses during the pore filling by aliquid induce cracks and can destroy the aerogel ma-trix [7]. One solution could be the improvement of theaerogel mechanical properties by the increase of thebulk density (decrease of the pore volume).

Usually the bulk density is increased by a highermonomer content and in that case the pore size distri-bution is shifted towards the small pore size [14, 15].Consequently, because the capillary stresses magni-tude depends on the inverse of the pore size (Laplace’slaw), the capillary stresses increase and the aerogelnetwork cracks although the improvement of the me-chanical features. Different ways has been proposed toincrease the mechanical properties of solid network:by aging [16, 17], addition of alkoxide solution af-ter gelation [18], dissolution and reprecipitation ofsilica driven by differences in solubility [19]. How-ever, it has been shown that it is possible to adjustthe aerogel bulk density by the addition of silica pow-der in the monomer solution, just before gelation. Theaerosil addition increases the bulk density and there-fore the mechanical properties, but also affects the ag-gregation process, the aerogel structure [20] and thepore size distribution [21]. The mean pore diameterincreases and this “composite aerogel” is able to re-sist the capillary stresses during the pore filling by aliquid.

To understand the effect of the powder additionon the aerogel structure and properties, in this pa-per we performed simulations of pure aerogel and”composite aerogel” networks by using the Diffusion-Limited Cluster-cluster Aggregation (DLCA) model[22, 23]. The pore size distribution of the resultingstructures are calculated by means of a triangula-tion method. We also calculate the specific surfacearea and the different numerical results will be dis-cussed and validated by comparison with experimentaldata.

These composite gels and the associated numericalmodel could be a new approach to simulate and under-stand the properties of natural gels of allophane issuedfrom the lixiviation of volcanic glasses [24].

2. Experimental Techniques

2.1. Gel Synthesis

Two sets of materials have been prepared: pure aerogel(Ax) and composite aerogel (C Ax), where x representsthe silica content in the parent solution expressed ing/cm3.

The pure aerogels synthesis is already describedin the literature [6, 7]. In summary, silica aerogel ismade from an alcogel issued from hydrolysis and poly-condensation reactions of organosilicate compounds(tetraethoxysilane TEOS, molar ratio H2O/TEOS = 4)dissolved in alcohol [6]; x , the silica content is ad-justed by the ratio monomer/alcohol in the solution.After gelation the solid network (gel) is full of alcoholand, according to a procedure previously reported [7],is transformed into aerogel by a ethanol supercriticaldrying process at 18 MPa and 310◦C.

To prepare the composite aerogels [20, 21], TEOSwas first mixed with HCI 0.01 N (molar ratioH2O/TEOS = 15) and the hydrolysed solution ofTEOS (13 wt.%) was mixed with fumed silica parti-cles (aerosil OX50) under stirring. For the compos-ite gels x (the total silica content) is adjusted by theaerosil/(aerosil + TEOS) weigth ratio which varies be-tween 0 and 70%. The goal of the silica soot additionis to control the mean pore size [20, 21]. The pH ofthe sol was then adjusted to 4.5 with NH4OH (0.1 N)which leads to gelation in a few minutes. The gelationis followed by the ethanol supercritical drying.

For these two sets of samples x , the silica weightcontent covers the range 0.1–0.4 g/cm3.

2.2. Textural Characterisation

From the weight and the geometrical dimensions, theaerogels bulk density ρb of the aerogels was deter-mined. The specific volume Vp of the gel is calculatedfrom ρb and ρs , the bulk and skeletal density of theporous body (ρs = 2.2 g/cm3)

Vp = 1/ρb − 1/ρs (1)

The specific surface area (S) was determinedvia N2 adsorption-desorption experiments (BETanalysis).

The hydraulic diameter Dh [25] which can be asso-ciated to the mean pore size is derived from the textural

Pore Structure Simulation of Gels with a Binary Monomer Size Distribution 275

features Vp and S:

Dh = 4Vp/S (2)

3. The Model

3.1. Aerogel Simulation

It has been shown that the Diffusion-Limited Cluster-cluster Aggregation (DLCA) model can describe quitewell the structure of porous systems obtained via sol-gel techniques [26]. In this work we considered the on-lattice version of the DLCA model. The model consistson a Monte Carlo algorithm which builds clusters ona lattice within a cubic box of edge length L . In ourcase we considered L = 35. Initially, the sites are ran-domly (but uniformly in space) on the lattice sites upto volume fraction or monomer concentration CM =N/L3. Periodic boundary conditions at the box sidesare assumed. All monomers are allowed to undergo aBrownian motion, i.e. they perform a translation mo-tion by one unit step in any of the 6 directions ±1, ±1,±1 chosen at random. The monomers (or aggregates)will irreversibly stick together when they come intocontact with another monomer (or aggregate). Aggre-gates of monomers are also able to perform a Brownianmotion and, if a collision occurs, will stick to the en-countered monomer or aggregate. In DLCA, all theclusters diffuse rigidly with a diffusivity constant in-versely proportional to their gyration radius. The pro-cedure is repeated until all monomers and clusters forma single connected network.

To simulate the composite aerogel structure, we con-sider two different sizes for the initial cubic monomers.The larger monomers represent the aerosil particleswith a size seven times larger than the one of the unitcell (small monomers). Such factor corresponds to ourrough estimation of the ratio between aerosil parti-cle size and monomer size of the sol-gel experiment.The aggregation process takes place as in the standardDLCA algorithm explained above. Figure 1 shows atwo-dimensional illustration of a pure and compositegels modeling at different stages of the aggregationprocess.

3.2. Pore Size Distribution

For the calculation of the pore size distribution we usea two-dimensional triangulation method in the pore

Figure 1. A two-dimensional illustration of a pure (top) and com-posite gels (bottom) modelisation at different stage of the aggregationprocess.

space. Details of the method are described elsewhere[27]. The main steps of the method are as follows:

(i) A large number of points are randomly put in thepore space and at each point the triangulation pro-cedure is applied.

(ii) From a given point (being the common vertex 1of the further triangles), the first triangle is builtby drawing a straight line in the x direction, from1 up to the intersection with the pore surface.

(iii) This line corresponds to the first side of this trian-gle, and the end line point defines its vertex 2.

(iv) The vertex 3 is obtained in a similar way, but nowthe straight line starts with an angle α respect tothe first side. The end point of this second line(second side of the triangle) corresponds to vertex3. In this study, we have chosen α = 1 radian.

(v) The third side closing the triangle is obtained bydefining a line from 2 to 3.

(vi) For a second triangle, the side obtained by the line1 to 3 of the first triangle will correspond to thefirst side of the new triangle.

(vii) The procedure is repeated 360 times around thepoint 1.

The summation of the triangle areas then constitutesa good estimate of the cross section of the pore. Theequivalent diameter is calculated supposing a circlewith an area equal to the sum over the triangle areas.The total pore space of the sample is measured by visit-ing all random points initially generated and placed inthe void space of the sample. Measuring the frequency


of the appearance of a particular pore diameter yieldsthe pore size distribution. The pore size distribution, isnormalised considering that the integral of the curve isequal to the sample porosity.

The “specific surface areas”∑

of the simulatedporous systems are calculated as follows:

∑=

∑Ni=1

∑6j=1 Si j

N(3)

where j denotes the nearest neighbours of an occupiedsite i (6 in a cubic lattice), Si j = 1 if site j is empty,and Si j = 0 if not.

Thus,∑

is the normalised number of free surfacesgenerated by the aggregation of the N cubic particles.∑

can vary between 6 (for an assembly of non con-nected particles) and 6N−1/3 (for a non porous cubicaggregate), reflecting this last value the external bound-ary of the cube under consideration, when periodicboundary conditions are not considered. All numericalresults reported in this work consist of a large averagenumber of samples, never smaller than 50 on the mostdemanding CPU time conditions.

4. Results and Discussion

In this section we will determine the textural properties(the mean pore size and the specific surface area) andcompare the numerical results to the experimental datafor the two different sets of simulated structures andaerogels.

4.1. Experimental Data

As explained in the introduction it is possible to de-crease the pore volume by the increase of the monomercontent in the solution.

Figure 2 shows the specific surface area and the asso-ciated Dh versus the silica content for the pure aerogel.The specific surface area decreases slowly from 400to 300 m2/g and the Dh decreases as the silica con-tent increases. It has been previously shown that, forthis kind of gel [5], whatever the monomer concen-tration, the aggregation process is the same and theclusters grow by addition of small particles. The par-ticle size is in the range of 1–2 nm. However, as thegel point is approached, the clusters stick together tobuild the gel network. Afterwards, the clusters cannotgrow further. The size of the clusters is obviously lim-ited by their number density. This kind of mechanism

Figure 2. Experimental specific surface area S(�) and hydraulicdiameter Dh (�) as a function of the silica weight content for the setof pure gels.

explained why the pore size distribution narrows andwhy the specific surface area does not change stronglywith the monomer concentration. The specific surfacearea (which depends on the particle size) is not affectedbecause the particle morphology is not changed. Onthe other hand, the pore volume which depend on thecluster size (the larger pores are the pores between theclusters), decrease with the monomer content and thepore size distribution is shifted towards lower pore sizes[15].

In the case of the composite gels, Fig. 3 puts in ev-idence the strong influence of the aerosil addition onthe specific surface area and the hydraulic diameter.USAXS (ultra small angle X-ray scattering) data showthat the cluster formation is affected by the particlescontent because the aggregation mechanism is changed[16]. Because of the strong reactivity of TEOS with theaerosil particles, for high soot content a large part ofthe TEOS is consumed and the polymeric clusters can-not grow freely. The structure of the composite gel isthat of two embedded networks: the polymeric (pure)gel and the particulate network. The polymeric struc-ture progressively disappears and the textural proper-ties of the composite material depends on the aerosilcontent. Because of the larger size of the aerosil par-ticle, the spaces between them tailor the meso-and-macroporosity.

4.2. Numerical Results

Figures 4 and 5 show the evolution of the pore sizedistributions versus the TEOS and aerosil content re-


Figure 3. Experimental specific surface area S (�) and the hy-draulic diameter Dh (�) as a function of the silica weight content forthe set of composite gels.

spectively calculated by the DLCA model for the twosets of gels: pure and composite gels. The simulationresults are in a qualitative agreement with the experi-mental data. The pore size distribution evolves in op-posite direction when the silica content is increased bythe monomer content or by the addition of large silicaparticles.

In such a way to quantitatively compare the simu-lation results to experimental data, the mean pore size

Figure 4. Pore sizes distribution for different silica weight content obtained from simulation of the pure gels.

of the simulated structure (Dmean) is extracted from thepore size distribution.

Dmean =∑

i

fi Di

/∑

i

fi (4)

Where fi is the frequency of appearence for the porediameter Di . The mean pore size is a dimensionlessnumber. The relation between the mean pore size andthe hydraulic diameter Dh is:

Dh = aDmean (5)

Where the subunit a is a length.From the model we are able to calculate

∑the “spe-

cific surface area” (see also Section 2) of the numeri-cal gel. This quantity is dimensionless, the relation be-tween

∑and S the experimental specific surface area

is:

∑= ρs a S (6)

Figures 6 and 7 show the mean pore size and the∑

evolution versus the silica content for the two sets ofsimulated porous network. Clearly the model is able toqualitatively account for the two different experimentalresults: decrease of the mean pore size with the TEOScontent (pure gel) and the increase of the mean pore sizewith increasing the aerosil content (composite gel).


Figure 5. Pore sizes distribution for different silica weight content obtained from simulation for composite gel samples.

Figure 6. Calculated specific surface area∑

( � ) and mean diam-eter Dmean(�) as a function of the silica weight content for the set ofpure gels.

These figures show also that, for the standard DLCA(pure aerogel),

∑depends only slightly on silica con-

tent (i.e.TEOS content). In fact, the variation is lowerthan 10%. On the other hand, for DLCA gels withmonomers of two sizes (composite aerogels),

∑is

notably affected by the addition of silica particles(i.e.aerosil), and decreases by a factor close to 2.5 overthe range of silica content investigated. The qualitativeagreement between the experimental data (Figs. 1 and2) and the numerical results (Figs. 6 and 7) gives agood confidence in the pertinence of the model and thenumerical method applied.

Figure 7. Calculated specific surface area∑

(� ) and the diameterDmean(�) as a function of the silica weight content for composite gelsamples.

On a quantitative point of view, from Dh (experi-ments) and Dmean (simulations) on one hand, and from∑

and S on the other hand, it is possible to indepen-dently evaluate a, the subunit size (see Eqs. (5) and(6)). In Figs. 8 and 9 the a evolution drawn from (5)and (6) for the two sets of samples (pure and compos-ite aerogels) are plotted. Note that for both cases theresulting value for a is quite similar.

We observe that for the pure gel, a is constant(2–4 nm) over the whole range of composition, andin a good agreement with the SANS results alreadypublished on the same kind of samples [4]. The subunit


Figure 8. Evolution of the subunit size a extracted from the ratioS/

∑(�) and from the ratio Dh /Dmean(�) as a function of the silica

weight content for pure gels.

Figure 9. Subunit size a extracted from the ratio S/∑

(�) and fromthe ratio Dh/Dmean(�) as a function of the silica weight content forthe set of composite gels.

size calculated from the data of the composite samplesis also in the range of a few nanometers (3–6 nm) (ex-cept for the high loading percent). These results arealso in a close agreement with the experimental SAXSdata [16].

This higher a value for the composite aerogels setcompared to the pure aerogels can be explained by the“Ostwald ripening effect”. According to ller [28] theprimary particles grow in size and decrease in numberas highly soluble small particles dissolve and reprecip-itate on larger. Because of the high water content inthe “composite” solution compared to the pure solu-tion, the silica solubility increases, the Ostwald ripen-ing is favoured and consequently the size of the subunitsincreases.

5. Conclusion

The numerical results and the experimental data showclearly that the addition of large particle in the gelling

solution allows to tailor the textural gel properties suchas the specific surface, the pore size distribution andconsequently the permeability of the porous samples.The increase of the pore size with higher aerogel den-sity and thus the permeability improvement [29] asso-ciated with a mechanical strengthening confirms thatthis kind of porous network can be used as a host matrixfor different chemical species (i.e. nuclear wastes [29]).The good agreement between simulations and experi-ments shows that composite aerogels can be simulatedby the DLCA algorithm with an initial binary distribu-tion of monomer sizes.

The triangulation method implemented in this workprovides a simple scheme to compute the pore sizedistribution of the aerogel networks. Such an algo-rithm and a DLCA modelisation could be usefulto predict textural properties of aerogels as a func-tion of synthesis parameters. This would complementthe measure of pore size distribution of real aero-gel samples, since standard experimental techniques(Hg porosimetry, N2, adsorption-desorption, thermo-porometry) sometimes fail because of the artefactscoming from the poor mechanical properties of the gels[30].

Finally, composite gels and the associated numer-ical simulation could be a new approach to simulateand calculate the physical properties of other com-posite material with a gel phase like for example, theallophanic soils. These soils are interesting in termsof mitigation of the greenhouse effect (C sequestra-tion) and the knowledge of the physical soils fea-tures are of importance. Allophane is a natural amor-phous silicate which exhibits physical properties veryclose to those of synthetic silica gel (high specificsurface area, large pore volume and water content,fractal structure of the aggregates) [31]. One descrip-tion of the allophanic soils [32] is that of a veryopen lattice types structure with coarser particles andthe gel-like allophane occupying the space betweenthem. So, the composite gels approach could be help-ful to simply simulate the complicated allophanic soilsstructure.

Acknowledgments

We would like to thank the PCP Venezuelan-Frenchexchange program and Fundaciencia-Mochima fortheir support, and Cecalcula (Venezuela) for computerfacilities.


References

1. M. Gronauer and J. Fricke, Acustica 59, 169 (1986).2. G.A. Nicolaon and S.J. Teichner, Bull. Soc. Chim. France 5,

1906 (1968); US Patent No. 3, 672, 833 1972.3. J. Fricke, J. Non-Cryst. Solids 147/148, 356 (1992).4. D.W. Schaefer and K.D. Keefer, Phys. Rev. Lett. 56, 2199

(1986).5. T. Woignier, J. Phalippou. J. Pelous, and E. Courtens, J. Non-

Cryst. Solids 121, 198 (1990).6. J.F. Brinker and G.W. Scherer, Sol-Gel Science (Academic Press,

1990).7. T. Woignier, J. Phalippou, and M. Prassas, J. Mater. Sci. 25, 3117

(1990).8. S.J. Teichner, G.A. Vicarini, and G.E.E. Gardes, Adv. Coll. In-

terface Sci. 5, 245 (1976).9. P. Tsou, J. Non-Cryst. Solids 186, 415 (1995).

10. J. Bouaziz, D. Bourret, and R Sempere, J. Non-Cryst. Solids 82,225 (1986).

11. T. Bellini, N. Clark, and D.W. Schaefer, Phys. Rev. Lett. 74,2740 (1995).

12. S. Kralj, G. Lahajnar, A. Zidansk, N. Vrbancic, M. Vilfan, R.Blink, and M. Kosek, Phys. Rev. E. 48, 340 (1993).

13. T. Woignier, J. Reynes, J. Phalippou, and J.L. Dussossoy, J. Non-Cryst. Solids 225, 353 (1998).

14. A.B. Jarzebski,J. Lorenc, Y. Aristov, and N. Lisistza, J. Non-Cryst. Solids 190, 198 (1995).

15. J. Phalippou, A. Ayral, T. Woignier, M. Pauthe, J.F. Quinson,and A. Lechatelut, Europhys. Lett. 14, 249 (1991).

16. S.A. Pardenek, J.W. Flemming, and L.C. Klein in UltrastructureProcessing of Advanced Ceramics, edited by J.D. Mackenzie andD.R. Ulrich (N.Y. Wiley, 1988) p. 379.

17. G.W. Scherer, S.A. Pardenek, and R.M. Swiatek, J. Non-Cryst.Solids 107(1), 14 (1988).

18. M.A. Einarsurd, J. Non-Cryst. Solids 225, 1 (1998).19. T. Mizuno, H. Nagata, and S.J. Manabe, J. Non-Cryst. Solids

100, 236 (1988).20. T. Woignier, C. Mariere, P. Dieudonne, J. Primera, M. Lamy,

and J. Phalippou, J. Non-Cryst. Solids 285, 175 (2001)21. M. Toki. S. Miyashita, T. Takeuchi, S. Kande, and A. Kochi.

Non-Cryst. Solids 100, 479 (1988).22. P. Meakin, Phys. Rev. Lett. 51, 1119 (1983).23. M. Kolb, R. Botet, and R. Jullien, Phys. Rev. Lett. 51, 1123

(1983).24. P. Quantin, J. Baiesdent, A. Bouleau, M. Delaune, and C .Feller,

Geoderma. 50, 125 (1991)25. G.W. Scherer, J. Non-Cryst. Solids 215, 155 (1997).26. A. Hasmy, E. Anglaret, M. Foret, J. Pelous, and R. Jullien, Phys.

Rev. B 50, 6006 (1994).27. J. Primera, A. Hasmy, and T. Woignier, J. Sol-Gel Sci. Techn 26,

671 (2003)28. R.K. Her, “The chemistry of silica”, Wiley N.Y. (1979)29. J. Reynes, T. Woignier, and J. Phalippou. J. Non-Cryst. Solids

285, 353 (2001).30. G.W. Scherer, D. Smith, and D. Stein, J. Non-Cryst. Solids 186,

309 (1995).31. T. Woignier, E Braudeau, J. Non-Cryst. Solids Submitted to.32. K.B. Wallace, Geotechnique 23(4), 203 (1973)

Pore Structure Simulation of Gels with a Binary Monomer Size Distribution

Documents