Network structures of polyhedral oligomeric silsesquioxane based nanocomposites: AMonte Carlo studyYu-Jane Sheng, Wei-Jung Lin, and Wen-Chang Chen Citation: The Journal of Chemical Physics 121, 9693 (2004); doi: 10.1063/1.1808124 View online: http://dx.doi.org/10.1063/1.1808124 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/121/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of the substituents on the thermal stability of hepta cyclopentyl, phenyl substitued - Polyhedral oligomericsilsesquioxane (hcp-POSS)/polystyrene (PS) nanocomposites AIP Conf. Proc. 1459, 247 (2012); 10.1063/1.4738458 Rheological investigation of interactions between sorbitol and polyhedral oligomeric silsesquioxane indevelopment of nanocomposites of isotactic polypropylene J. Rheol. 54, 761 (2010); 10.1122/1.3439695 The mechanical properties of crystalline cyclopentyl polyhedral oligomeric silsesquioxane J. Chem. Phys. 124, 214709 (2006); 10.1063/1.2208355 Monte Carlo simulation of structure and nanoscale interactions in polymer nanocomposites J. Chem. Phys. 121, 10814 (2004); 10.1063/1.1812752 A Monte Carlo study of metastable structures of the cyanoadamantane crystal J. Chem. Phys. 109, 6753 (1998); 10.1063/1.477321

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Network structures of polyhedral oligomeric silsesquioxane basednanocomposites: A Monte Carlo study

Yu-Jane ShengDepartment of Chemical Engineering, National Taiwan University, Taipei 106, Taiwanand Institute of Polymer Science and Engineering, National Taiwan University, Taipei 106, Taiwan

Wei-Jung LinDepartment of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan

Wen-Chang Chena)

Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwanand Institute of Polymer Science and Engineering, National Taiwan University, Taipei 106, Taiwan

~Received 3 June 2004; accepted 27 August 2004!

The network structures of polyhedral oligomeric silsesquioxane based nanocomposites are studiedby continuous-space Monte Carlo simulations. The nanoporous network contains intercubic poresand mesopores which can be clearly identified in this work. In terms of degree of cross linking andpore size distribution~PSD!, effects of linker length, tether rigidity, and number of reactive tethersare examined. It is found that the extent of cross linking as well as the intercubic pore size of thenetwork increases as linker length increases which are consistent with experimental findings.However, the mesopores appear to shift to a smaller radii regime for networks with longer linkers.Networks with rigid tethers contain lots of free linkers, thus, low cross linking density and narrowPSD are observed. On the other hand, reduction of the reactive tethers shows an insignificant effecton the degree of cross linking of the system. The fact that the intercubic pore size increases as thenumber of reactive tethers decreases causes the nanobuilding blocks to possess larger free volumesand distribute themselves more evenly throughout the system. As a result, it reduces the possibilityof forming large mesopores. ©2004 American Institute of Physics.@DOI: 10.1063/1.1808124#

I. INTRODUCTION

Well-defined nanobuilding blocks~NBB! have becomeone of the key components in nanomaterial science becausethey provide precise control of the architecture and function-ality. Various NBBs have been reported in the literature, in-cluding nanocubes, nanorods, nanotubes, nanospheres, nano-plates, and nanoprisms.1–6 Control of the macroscopicarchitecture, morphology, and properties could be achievedthrough the incorporation of the NBB.

Polyhedral oligomeric silsesquioxane~POSS! repre-sented one of the widely studied NBB for preparing ad-vanced nanocomposites recently.7–15 The general formula ofPOSS is@RSiO3/2#n consisted of a central core around 0.53nm with organic groupsR on the corner, in which octahedral(n58) is the mostly studied member of this family. Judi-cious organic-inorganic hybrid materials could be preparedby reacting the organic functional groupR on the corner withdesired monomer. The POSS functionality employed for pre-paring the hybrid materials include acrylic,16–18 epoxy,19–27

amine,28–30 vinyl,31 isocyanate,32 halide,33–36 andnorbonyl.37,38 The feasibility of controlling the arm number,arm length, and arm functionality makes POSS topologicallyideal for the preparation of nanocomposite materials. For ex-ample, monofunctionalized POSS was copolymerized withpolymers to result in organic-inorganic hybrid materials, in

which the POSS was endcapped39 or pendent.40,41 However,multifunctionalized POSS was copolymerized to formnetworks22,24,28,29,31or starlike structures.35,36The hybrid ma-terials using POSS as the nanocross linker achieved highcross linking density, high porosity, excellent thermal, andmechanical properties.

Although the POSS derived nanocomposites did showimpressive physical properties, the details of the microscopicstructure has not been fully explored yet. Computer simula-tions can provide a fundamental insight into the mechanism,the microstructure, and resulted properties of advanced nano-composites. Recently, Lammet al. performed Monte Carlosimulations on lattice to probe the microstructure and prop-erties of the cross linked POSS networks.42 Effects of linkerlength on the network properties, such as porosity, spatialdistribution of NBBs, and extent of cross linking were stud-ied. The work of Lammet al. obtained the cross linkingdensity fairly close to that reported by Laine andco-workers.31 However, the trend of decreasing cross linkingdensity with increasing linker length is in contrary to thatreported by Laine and co-workers.31 A possible explanationis proposed that the six-atom tether is the optimum length forbalancing the competition between steric hindrance andtether rigidity. Nevertheless, the fact that the functionalizedNBBs can only reside on discrete lattice sites also raisesquestions. Note that, normally, Monte Carlo simulations per-formed on lattice or in continuous space should providequalitatively similar results. However, it is also known that

a!Author to whom correspondence should be addressed. Fax: 886-2-23623040. Electronic mail: chenwc@ntu.edu.tw

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 19 15 NOVEMBER 2004

96930021-9606/2004/121(19)/9693/9/$22.00 © 2004 American Institute of Physics

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the limitations of the lattice model sometimes create arti-facts. Therefore, it would be justified to propose a modelcomparatively more realistic than the previous study to in-vestigate the correctness of the inconsistent trend. Further-more, the effect of tether rigidity and the number reactivelinkers on the size distribution of intercubic pores and meso-pores as well as the cross linking density of the cross linkedPOSS network has not been explored yet. Consequently, acontinuous space simulation for investigating the microstruc-ture of POSS based nanocomposites is developed. The simu-lation results will be compared with experimental works ofLaine and co-workers31 and the simulation study of Lamm,Chen, and Glotzer.42

A. Theoretical models and simulation

In the present study, the system consists of two kinds offundamental building blocks, i.e., NBB1 and NBB2. EachNBB is composed of a spherical cage and eight linker chains,as shown in Figs. 1 and 2. Each linker chain is of equallength L. The interactions between bonded beads are repre-sented by the infinite deep square-well potentials,43–45

Ui ,i 115H ` r ,s

0 s<r ,js

` js<r

, ~1!

wherer is the distance between two bonded linker beads ands is the diameter of a linker bead. Here, we choose that

j51.2 which mean that the bond length between bondedlinker beads stays freely betweens and 1.2s. Most of thenonbonded linker beads as well as cage-cage, cage-linkerbeads interact through purely volume exclusion potential.The only attractive interactions take place between thenearest-neighbor linker end beads of NBB1 and NBB2. Thepotential energy is the square-well potential, as shown in Eq.~2!,

Ua,b5H ` r ,s

2e s<r ,ls

0 ls<r

, ~2!

wherel is arbitrarily chosen to be 1.2. In this studye/kT510 wheree is the cross linked binding energy,k is theBoltzmann constant, andT is the system temperature. Thevalue of e/kT510 was set to assure cross link breakagerarely happens due to thermal motions. Herea andb denotethe end beads of the reactive tethers of NBB1 and NBB2,respectively.

To study the effect of linker rigidity on the microstruc-tures of the networks, bond angle constraint is added to thelinkers. The bond angle potential is set as follows,

Uu5ku~12cosu!2, ~3!

where ku is the bond angle strength andu is bond angle

FIG. 1. The scheme of hydrosilylation reaction ofvinyl- and hydrido-functionalized POSS and the result-ing porous structure.

9694 J. Chem. Phys., Vol. 121, No. 19, 15 November 2004 Sheng, Lin, and Chen

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between bond vectors ofrW i ,i 21 and rW i 11,i . Note thatku50denotes a fully flexible linker. In this work, cases forku50and 30 are studied.

The networks are prepared as follows. First, we place atotal of N5100 NBBs ~50 NBB1 and 50 NBB2! into thesimulation box in a random manner. The linkers are grownfrom the eight corners of the inner cube of the spherical cage,as shown in Fig. 2. The linker lengthL ranges from 1–3 inthis work. Therefore, the total number of linker beads andcage spheres sums up to about 900–2500 beads for the caseof L51 andL53, respectively. The volume fraction of thesystem is set to be 0.05. This volume fraction corresponds toabout 0.15 g/ml NBB concentration. Each system is pre-equilibrated for about (2 – 9)3108 Monte Carlo steps.

Three different attempted movements have been per-formed in the simulation process:~1! the bead displacement,~2! the translational displacement, and~3! the rotationalmovement of the NBBs. The bead displacement movement isperformed by randomly picking a linker bead and displacingit to a new position in the vicinity of the old position. Thetranslational displacement of the NBBs involves relocationsof the spherical cage and the attaching linkers as a whole tonew positions while keeping the internal structure of theNBB fixed. In the rotational moves, the internal structure ofthe chosen NBB is also unchanged. The new configurationsresulting from above-mentioned movements are accepted ac-cording to the standard Metropolis acceptance criterion.46 Aswe know the attempted process of movements 2 and 3 take alot more simulation time than that of movement 1, thereforethe frequency of performing these three steps is as8L:0.8:0.2. ForL53 system, on average, it requires about109 Monte Carlo steps for the production process or 250CPU hours on a Pentium III processor running at 1.7 GHz.

To measure the spatial arrangement of NBBs in the sys-tem, the pair correlation function,g(r ) of the spherical cage

FIG. 2. Schematic representation of the used molecularmodel. Two different sizes of linker beads each withthree tether lengths were demonstrated with the corre-sponding chemical structure.

FIG. 3. The snapshots of the network structures for the cases of~a! A3, ~b!B1 using continuous-space Monte Carlo simulation after reaction reachesequilibrium. The edge-length of each box is 150 Å.

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is calculated within the simulation. Our cage spheres all in-teract with purely volume exclusion potential and thusg de-pends only on the distance between the two particles,g isdefined as

g~r !5r~2!

r2, ~4!

wherer is the number density of the system andr~2! is thepair density function. Physically,g corresponds to the prob-ability ~relative to the bulk number density! of finding a sec-ond cage sphere a distancer from the center of a cagesphere. If NBBs are uncorrelated,g51. If g.1 then thepresence of the first NBB enhances the probability of findingthe second NBB in that region. The pair correlation functioncan be easily incorporated into simulation program.46

The geometric pore size distribution is calculated basedon the method developed by Gubbins and co-workers.47–49

Let Vpore(D) be the volume of the void space ‘‘coverable’’by spheres of radiusD or smaller. TheVpore(D) function canbe calculated by a Monte Carlo volume integration.46 Pointsare randomly chosen and by determining the largest coveringcircle for every point in the void volume,Vpore(D) can beobtained. It is a monotonically decreasing function ofr andis easily compared with the ‘‘cumulative pore volume’’curves often calculated in isothermal-based pore-size distri-bution ~PSD! methods.50 The derivative2dVpore(D)/dD isthe fraction of volume coverable by spheres of radiusD butnot D1dD and is a direct definition of pore size distribution.

In this study, our model system is intended to investigatethe experimental system,31 as shown in Fig. 1. The octavinyl-functionalized POSS, including vinyl (T8

vinyl) or vinyldimeth-ylsiloxy (Q8

vinyl), and octavinyl-functionalized POSS, includ-ing hydrido (T8

H) or hydridodimethylsiloxyl (Q8H), are cross

linked via hydrosilylation reaction catalyzed by platinumcomplex. Figure 2 shows our model of the NBB and linkers.The cage dimension is 5.4 Å along the body diagonal be-tween Si atoms.31 The bond length of tether is estimated tobe 1.6 Å and thus the diameter of cage sphere is 7.0 Å. Note

FIG. 4. Pair correlation functiong(r ) as a function ofr for A1–A3 andB1–B3.

TABLE I. The simulated microstructures ofA1–A3 andB1–B3 obtained the continuous-space model and theircomparison with the literature~Refs. 31 and 42!.

A1 A2 A3 B1 B2 B3

Number of atoms on tether backbone 2 4 6 6 12 18

Simulation parametersThe diametric ratio of cage to linker 4.5 4.5 4.5 1.5 1.5 1.5Number of linker beadsL 1 2 3 1 2 3

Mesopore size distribution~Å!From MC volume 15–50 15–40 10–40 10–65 10–60 10–55Experimental resulta 10–200 ¯ 10–40 10–40 ¯ ¯

Intercubic pore size~Å!Calculated fromg(r ) 8.5 9.4 10.6 12.4 17.7 22.9Experimental resultb 8 ¯ 10–12 10–12 ¯ ¯

Lattice model42¯ ¯ ¯ 11 ¯ ¯

Degree of cross linking~%!Simulated by continuous-space model 59% 76% 82% 91% 93% 95%Experimental resultc 44% 69% 81% 81% ¯ ¯

Lattice modeld ¯ ¯ ¯ 90% 86% 81%

aBy N2 sorption analysis~Ref. 31!.bObtained by positron annihilation lifetime spectroscopy, small-angle x-ray scattering, and wide-angle x-rayscattering analysis~Ref. 31!.

cObtained by13C and29Si NMR results~Ref. 31!.dReference 42.

9696 J. Chem. Phys., Vol. 121, No. 19, 15 November 2004 Sheng, Lin, and Chen

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that the linking bond length of Si–O, Si–C, and C–C are1.64, 1.88, and 1.55 Å, respectively. There are two cases ofdiametric ratios of cage to linker considered in the simula-tion. For the case of systemA, one linker bead (L51) rep-resentsonebackbone atom. The ratio of cage to linker diam-eter is estimated to be 4.5~7.0 Å/1.6 Å!. Network structure isformed by linkages the tethers with 2, 4, and 6 backboneatoms for the case ofA1 (L51), A2 (L52), and A3 (L53), respectively. For systems with even longer tetherlength, we have employed caseB of which one linker beadrepresentsthreebackbone atoms. CaseB is defined in a simi-lar way to that of Lamm, Chen, and Glotzer.42 Thus, the ratioof cage to linker diameter is 1.5 for caseB. Network struc-ture was formed by linkages at the corners by linkers with 3,6, and 9 atoms along backbones for the case ofB1 (L51),B2 (L52), andB3 (L53), respectively. By doing so, simu-lation time needed for systems with long linkers can begreatly reduced. Thus the structures of networks form bylinkages of tethers with 2~A1!, 4~A2!, 6~A3,B1!, 12~B2! and18~B3! backbone atoms are studied. Note that casesA3 andB1 represent the same system with tethers consisting of sixbackbone atoms and thus the simulation results can be com-pared.

II. RESULTS AND DISCUSSION

Continuous-space Monte Carlo simulations are per-formed to study the network structures of POSS based nano-composites. Effects of linker length, tether rigidity, and num-ber of reactive tethers on the pore size distribution anddegree of cross linking of the resulting networks are studied.

A. Effect of tether length on pore size distributionand degree of cross linking

Figures 3~a! and 3~b! show the snapshots of the networkstructures for the cases ofA3 andB1, respectively. The in-tercubic pores between NBBs and mesopores between NBBaggregates~as defined in Fig. 1! can be clearly observed inFig. 3. The intercubic pore between NBBs is estimated quan-titatively by the pair-correlation functiong(r ), which is thespatial distribution of NBBs as a function ofrg(r ) ofA1–A3 and B1–B3 are shown in Fig. 4. Physically,g(r )corresponds to the probability of finding a second NBB adistancer from the center of a NBB. Therefore, the first peakof g(r ) in Fig. 4 ~defined asr FP) represents the most prob-able distance between the central NBB and its nearest neigh-bors, which can be interpreted as the characteristic size of theintercubic pores. From Fig. 4, as expected, allg(r ) equalzero whenr is smaller than 7.0 Å, since the size of the NBBcage sphere is 7 Å. Ther FP of casesA1, A2, A3, B1, B2, andB3 are 8.5, 9.4, 10.6, 12.4, 17.7, and 22.9 Å, respectively, asalso listed in Table I. The results indicate that NBBs withlonger tethers possess a larger intercubic pore. The intercubicpore sizes ofA3 or B1 of our work agree well with thoseobtained from the experiment result~10–12 Å! ~Ref. 31! andthe lattice model~11 Å!.42 It is found that the peak ofg(r )broadens as tether length increases as can be seen in Fig. 4.This finding implies that tethers with longer length would notonly enlarge the distance between NBBs but also result in a

broader intercubic pore size distribution. Note that the in-tracubic pore in the present study is the size of the NBB cagesphere.

The understanding of mesopore size distribution is veryimportant for the applications of electronic applications, suchas low dielectric constant materials.10–15,30,31 To obtain aquantitative PSD, the cumulative pore volume,Vpore(D) iscalculated by using Monte Carlo volume integrationscheme.47–49 Figure 5~a! shows the variations ofVpore(D)/Vtotal ~defined as the volume fraction in pores ofdiameter larger thanD! versusD for casesA1, A2, andA3.The results are consistent with experimental observations

FIG. 5. ~a! Variation of P as a function of pore size using Monte Carlovolume integration.~b! The pore size distribution ofA1–A3.

9697J. Chem. Phys., Vol. 121, No. 19, 15 November 2004 Silsesquioxone based nanocomposites

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thatVpore(D) decreases monotonically with increasingD. Bytaking the derivative2dVpore(D)/dD, pore size distributionas a function of pore diameterD can be obtained as shown inFig. 5~b!. The most probable size of the mesopore reduces astether length increases. This is because as tether length in-creases the size of the intercubic pore increases leaving lessspace for the mesopore to develop. Hoebbelet al. alsoshowed that the length of the bridges between the cubes is acritical factor in determining porosity.51 The mesopores dis-tribute around 10–50 Å forA1; 10–40 Å for A2 and then10–40 Å for A3 in our work. Note that the mesopore sizedistribution of A1 and A3 reported in the literature31 are10–200 and 10–40 Å, respectively. The reported mesoporedistribution ofA1 up to 200 Å is beyond the present simu-lation scale of;100 Å. However, the trend of increasing thetether length resulting in smaller mesopore size found in ourwork is consistent with experimental results.

The degree of cross linking, or the extent of reaction, canbe obtained by calculating the ratio of cross linked linkerchain-ends to total reactive linker chain-ends and is listed inTable I. The degrees of cross linking density increase from

59% to 82% for the case ofA1–A3, and 91% to 95% forB1–B3, which are in fair agreement with the experimentalresults.31 A longer flexible linker has more degrees of free-dom to move around than a shorter one does. Thus, the re-active beads at chain-ends of the linkers can sweep out alarger volume for interacting with those of other linkers.Therefore, the degree of cross linking density increases withlonger linker length. However, a recent study which is basedon lattice model exhibits an opposite trend as the cross link-ing density of B1–B3 was found to be 90%, 86%, and81%.42 It was explained that six-atom tether may be the op-timum length for balancing the competition between sterichindrance and tether flexibility. The inconsistency betweentwo works might be attributed to the essential difference be-tween lattice and continuous-space model. Molecules in thelattice model cannot rotate and should reside on discrete lat-tice sites. The rotation motion of NBBs plays an importantrole in the arrangement of network structures. NBBs stackmore orderly without rotation, especially for system withshort tethers. Besides rotation, the usage of only one kind ofNBB in the previous paper could also affect the degree ofcross linking due to the self-cross linking between linkers ofthe same NBB. The degree of crosslinking ofA1 ~59%!found in this work is higher compared to that of the experi-mental result~44%!. It might be the consequence of neglect-ing the role of catalyst in the simulation. Note that as the

FIG. 6. Pair correlation functiong(r ) as a function ofr for A3 with flexibleand rigid tethers, respectively.

FIG. 7. The pore size distribution ofA3 with flexible and rigid tethers,respectively.

TABLE II. Simulated result of different rigidity and number of reactive tethers forA3.

n58, flexible

Rigidity Number of reactive tethers

n58, rigid n54 n52

Rigidity factor ku 0 3 0 0Mesopore size distribution~Å! 10–40 15–30 10–30 10–25Intercubic pore size~Å! calculated fromg(r ) 10.6 10.9 11.1 11.7Degree of crosslinking~%! 82 57 80.3 80.0

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network formation proceeds, it is becoming more difficult forthe free linker ends to adsorb onto the large catalytic Ptcomplex,31 especially for short linkers such asA1. Further-more, the solvent effect and linker compatibility, may alsoplay a key feature of nanoscale assembly. Those factors maybe of interest for future work.

As mentioned in the preceding section, in caseA, eachlinker bead represents 1 backbone atom while in caseB, eachlinker bead stands for three backbone atoms. Thus the struc-tures of networks form by linkages of tethers with 2~A1!,4~A2!, 6~A3,B1!, 12~B2!, and 18~B3! backbone atoms arestudied. It is found that systemsA andB demonstrate a con-sistent trend on showing the effect of the tether length on themesopore size, the intercubic pore size, and the degree ofcross linking. That is, asL increases, the mesopore size de-creases but the intercubic pore size and the degree of crosslinking increase. Therefore, systemsA andB are feasible forpredicting the assembly of NBBs. Note thatA3 andB1 cor-respond to the same system with six-atom tethers. However,there are differences in the predicted pore size distributionsand cross linking density forA3 andB1 cases. The degreesof cross linking are 82 and 91% forA3 andB1, respectively,while 82% is obtained from experimental report.31 The inter-cubic pore sizes are 10.6 and 12.6 Å forA3 andB1, respec-tively, while the range of 10–12 Å is reported in theliterature.31 These results indicate that systemA may be thebetter model for studying the POSS network. By careful in-spection of the snapshots ofA3 @Fig. 3~a!# and B1 @Fig.3~b!#, one could find thatB1 is composed of mostly linearstructure but only network structure is observed inA3. InB1, NBB1 is inclined to connect face to face with NBB2,linked by four linker chains on each NBB. Such linear struc-ture is probably due to using a larger linker sphere which isless realistic and thus large reactive volume is obtained. Theunexpected linear structure ofB1 was also probed by theg(r ). As shown in theB1 of Fig. 4, except forr FP512.4, anobvious peak appeared at 24 Å, which is twice the positionof first peak. However, it does not exist in the case ofA3.Therefore, although by using one linker bead to representthree backbone atoms can save large amount of computationtime, such simplified system may not be an appropriatechoice in predicting the system of short tethers.

B. Tether rigidity

In the above simulation, the tether is assumed to be com-pletely flexible, i.e., without any bond angle or torsionalangle constraints. However, the rigidity factor might be im-portant for the microstructure of nanocomposites. For ex-

ample, a recent report by Laine’s group30 showed a signifi-cant difference on the microstructure if two silsesquioxanecores with different rigidity were used for preparing nano-composites, octaglycidyldimethylsiloxyoctasilsesquioxane,and octa~ethylcyclohexylepoxide!dimethylsiloxy silsesqui-oxane. In this study, rigidity is introduced by imposing bondangle constraint upon linkers. As shown in Table II, the de-gree of crosslinking forA3 decreases from 82% to 57% asthe rigidity is added to the linkers. When linkers becomemore rigid, the collisions of reactive linker chain-ends hap-pen less frequently due to the limited orientation of linkers.Thus, degree of cross linking reduces accordingly. Linkerrigidity also enlarges the intercubic pore size from 10.6 to10.9 Å since now the linker tends to stretch outward, asshown in Fig. 6. Figure 7 shows the mesopore PSD ofA3 fordifferent rigidity. It can be seen clearly that networks withflexible linkers have a much broader pore size distribution. Itis because flexible linkers are free to extend or contract,therefore systems can form both large and small pores withgreater possibility. The results of low cross linking densityand narrow PSD of rigidA3 suggest that many free linkersexist and NBBs distribute more uniformly in space than flex-ible A3 does. With the incorporation of rigidity, this modelcan be further applied to POSS functionalized with liquidcrystailline tethers.52–56 The transition between nematic,smectic, and isotropic phases can be identified by the mo-lecular structures, which is dominated by the rigidity of theliquid-crystalline mesogen, composed of a soft spacer and ahighly rigid core.

FIG. 8. Schematic representation of POSS model with~a!2 and~b!4 reactivetethers, respectively.

FIG. 9. Pair correlation functiong(r ) as a function ofr for A3 with thereactive tether numbern52, 4, and 8, respectively.

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C. Number of reactive tethers

The number of reactive tethern of silsesquioxane core isnot necessary to be 8. It could be varied to 2~Ref. 57! or 4.58

In this work, we also intend to study the effect ofn on thenetwork structures. We choose the systemA3 with n52 and4 @as shown in Figs. 8~a! and 8~b!# and compare the resultswith that of n58. The pair-correlation functions ofA3 withdifferent number of reactive linkers,n52, 4, and 8 areshown in Fig. 9. The intercubic pore size increases as thenumber of reactive linkers decreases. Moreover, the intensityof the peak decreases asn decreases. The results suggest thata less ordered structure is formed for the cases of lessern,which is in agreement with our observations of the final net-work structures. Figure 10 shows the mesopore PSD ofA3with different reactive numbers,n52, 4, and 8. It is foundthat pores with size larger than 30 Å do not exist withinnetworks formed by NBBs with only 2 or 4 reactive tethers.The results imply that for the case ofn58, the POSSs aretightly cross linked to one another by at least six tethers~degree of cross linking3n), leaving larger voids. On theother hand, for the cases ofn52 and 4, on average, thenumbers of linked tethers between POSSs are 1.6 and 3.2. Itis expected that in this case NBBs possess larger free vol-umes and distribute themselves more evenly through out thesystem. Thus, it is less possible for large mesopores to form.As a result, the number of reactive linkers of the NBBs in-trinsically determined the final structure. Note that it is quiteinteresting to find that the degrees of cross linking of the case

of n52 and 4 are around 80%, which is almost the same asthat of n58. It seems that the ability for the reactive tethersto bind is not affected by the presence of the unreactivelinkers. Based on the results ofg(r ) and mesopore PSD, thenetwork structures for cases ofn52 and 4 seem to be rathersimilar.

III. CONCLUSIONS

In this work, we present a continuous-space Monte Carlosimulation to study the porous structure of POSS networks.The intercubic pore size is determined by the calculation ofthe pair correlation functiong(r ). Our results indicate thatNBB with a longer tether possesses a larger intercubic poreand a broader intercubic pore size distribution which are con-sistent with previous works. The mespore size, calculated bythe Monte Carlo volume integration method, decreases astether length increase. It is found that NBB with longer flex-ible linkers has more degrees of freedom to move aroundthan a shorter one does. Thus, the degrees of cross linkingdensity increase asL increases.

We have also introduced tether rigidity in this work byimposing bond angle constraint upon linkers. Rigidity causesNBBs to be less cross linked and enlarges the intercubic poresize. Also, networks with rigid linkers have much narrowerpore size distributions. In this work, the number of reactivetethers on a NBB has been varied to study the effect ofn onthe structures of the NBB networks. The intercubic pore sizeincreases as the number of reactive tethers decreases. Theresults suggest that a less ordered structure is formed for thecase of lessn. The NBBs are more tightly cross linked forn58, leaving much more large voids. On the other hand, fornetworks with less reactive linkers, NBBs possess larger freevolumes and disperse themselves more equally through outthe system. Therefore, the existence of large mesopores turnsout to be fairly unlikely.

ACKNOWLEDGMENTS

The authors thank the National Science Council and theDepartment of Economics Affairs of Taiwan for the financialsupports of this work.

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