Polymer Reviews

This article was downloaded by:[Mark, James] On: 26 October 2007 Access Details: [subscription number 783513023] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Polymer Reviews Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713597276

Some Simulations and Theoretical Studies on Poly(dimethylsiloxane)

Sizhu Z. Wu a; James E. Mark b a Key Laboratory of Beijing on "Preparation and Processing of Novel Polymer Materials", College of Materials Science & Engineering, Beijing University of Chemical Technology, Beijing, China b Department of Chemistry and the Polymer Research Center, The University of Cincinnati, Cincinnati, OH, USA Online Publication Date: 01 October 2007 To cite this Article: Wu, Sizhu Z. and Mark, James E. (2007) 'Some Simulations and Theoretical Studies on Poly(dimethylsiloxane)', Polymer Reviews, 47:4, 463 - 485 To link to this article: DOI: 10.1080/15583720701638179 URL: http://dx.doi.org/10.1080/15583720701638179

PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

Downloaded By: [Mark, James] At: 12:40 26 October 2007

Polymer Reviews, 47:463–485, 2007 Copyright # Taylor & Francis Group, LLC ISSN 1558-3724 print/1558-3716 online DOI: 10.1080/15583720701638179

Some Simulations and Theoretical Studies on Poly(dimethylsiloxane) SIZHU Z. WU1 AND JAMES E. MARK2 1

Key Laboratory of Beijing on “Preparation and Processing of Novel Polymer Materials”, College of Materials Science & Engineering, Beijing University of Chemical Technology, Beijing, China 2 Department of Chemistry and the Polymer Research Center, The University of Cincinnati, Cincinnati, OH, USA This review describes somewhat personalized choices of simulations and theoretical studies that have been carried out on poly(dimethylsiloxane) (PDMS) [-Si(CH3)2O-]. It starts from the simplest systems and proceeds to more complex ones. The specific order of topics is (i) single chains (structure, statistical properties, single-molecule deformations), (ii) uncross-linked bulk polymer (thermodynamics, structure, crystallization, permeability), (iii) unfilled elastomeric networks (cross linking, end linking, stress-strain isotherms, sol extractions, cyclic trapping), and (iv) filled elastomers (reinforcement from spherical filler particles, and ellipsoidal fillers, both oriented and unoriented). Keywords poly(dimethylsiloxane), polymer simulations, Monte Carlo methods, molecular dynamics, rotational isomeric state theory, mechanical properties

Single Chains Structure The focus of this review is really on the statistical properties of long chains of PDMS, but it may be useful to begin by including some basic structural information on siloxanes in general. Aspects of the basic structure, some as fundamental as the nature of the Si-O bond, can be found in a number of publications, particularly review articles.1 – 3 There have also been numerous studies of conformational preferences and arrangements of the chains in the crystalline state.4 – 9 One of the most intriguing aspects of the siloxane backbone is the ability of the Si-O-Si bond angle to invert through the linear (1808) state,10 – 13 analogous of the inversion of the ammonia “umbrella” through the planar state.14 Polysiloxanes can do this inversion at little cost in energy, and this

Received 8 June 2007; Accepted 3 August 2007. Address correspondence to James E. Mark, Department of Chemistry and Polymer Research Center, The University of Cincinnati, Cincinnati, OH 45221-0172, USA. E-mail: [email protected]

463


464

S. Z. Wu and J. E. Mark

“floppiness” indicates the chain is almost freely-jointed at these oxygen atoms along the backbone. With regard to chemical reactions, there have been extensive studies of both the polymerization process (by ring-opening of the cyclic trimer),15 and chain degradation (by thermal decomposition).16

Rotational Isomeric State Modeling In rotational isomeric state models, the continuum of rotations occurring about skeletal bonds is replaced by a small number (generally three) of rotational states that are judiciously chosen.4,8,17,18 Preferences among these states is then characterized by Boltzmann factors as statistical weights, with the required energies obtained by either potential energy calculations or by interpreting available conformation-dependent properties in terms of the models. Multiplication of matrices containing these statistical weights is then used to generate the partition function and related thermodynamic quantities, and multiplications of similar matrices containing structural information is then used to predict or interpret various properties of the chains.4,8,19 Examples of such properties are end-to-end distances, radii of gyration, dipole moments, optical anisotropies, etc., all as unperturbed by intramolecular excluded volume interactions between chain segments.20 An important extension of these ideas is the use of this model to generate distributions of end-to-end distances, instead of simply their averages.18,21 The same statistical weights were used in Monte Carlo simulations to generate representative chains, and their end-to-end distances r were calculated. The corresponding distribution function was obtained by accumulating large numbers of these Monte Carlo chains with end-to-end vectors within various range intervals, and dividing these numbers by the total number of the chains, N. The distances were then placed into a histogram to produce the desired end-to-end vector probability distribution function P(r) or P(r/nlo), where n is the number of skeletal bonds of length lo. The histogram generally consisted of 20 equally spaced intervals over the allowed range 0 , r/nlo , 1. The function P(r/nlo) was smoothed using the IMSL cubic spline subroutine CSINT.22 These distributions are very useful for chains that can not be described by the Gaussian limit, specifically chains that are too short, too stiff, or stretched too close to the limits of their extensibility. Some typical results are shown in Fig. 1.21 They document how bad the Gaussian distribution is for short chains of polyethylene (PE) and PDMS particularly in the region of high extension that is critical for an understanding of ultimate properties.

Stretching of Single Chains Molecular dynamics simulations have been carried out to characterize thermodynamic and conformational changes involved in stretching a PDMS chain in the melt,23 or polymer chains in general.24 – 26 Such studies are important not only for rubberlike elasticity,27,28 but also for interpreting single-molecule experiments such as pulling a reptating chain out of a network or gel,29 as described further below.


Simulations and Theoretical Studies on PDMS

465

Figure 1. Comparisons among the rotational isomeric state distribution functions for the end-toend distance r for polyethylene and poly(dimethylsiloxane) (PDMS) chains having n ¼ 20 skeletal bonds of length l, and the Gaussian approximation for the PDMS distribution.

Uncross-Linked Bulk Polymers Thermodynamics and Structure There have been extensive studies of PDMS melts, particularly with regard to intermolecular pair correlation functions,30 chain dimensions and intermolecular structures,31 and surface tensions.32

Crystallization There is now considerable interest in using simulations for characterizing crystallization in copolymeric materials. Of particular interest is a model capable of simulating chain ordering in copolymers composed of two comonomers, at least one of which is crystallizable.33 Typically, the chains are placed in parallel, two-dimensional arrangements. Neighboring chains are then searched for like-sequence matches in order to estimate extents of crystallinity. Chains stacked in arbitrary registrations are taken to model quenched samples. Annealed samples, on the other hand, are modeled by sliding the chains past one another longitudinally to search for the largest possible matching densities.34 The longitudinal movement of the chains relative to one another, out of register, approximately models the lateral searching of sequences in copolymeric chains during annealing.35,36 Two examples of elastomers studied by such simulations are the polysiloxane chemical copolymers poly(diphenylsiloxane-co-dimethylsiloxane),37 – 40 and the polysiloxane stereochemical copolymers poly[methyl(3,3,3-trifluoropropyl)siloxane].40,41 Some illustrative results for the stereochemical copolymer can be briefly described. The goal was to increase the isotacticity of the polymer to the extent that crystallization could be achieved, and this was done by increasing the amount of the stereoregular cis cylic trimer in the mixture of cyclic trimers being polymerized. The larger the fraction


466


of the Sˇ trimer, and the greater crystallinity Sˇ and this is paralleled by the increases in long sequences shown by the simulation results presented in Fig. 2. Permeability PDMS is an extraordinarily unusual polymer, as evidenced by the numerous unusual properties it exhibits.17,42 Its almost unprecedentedly high permeability, is the most intriguing and probably also the most important (due to its importance in gas-separation membranes, drug-delivery systems, extended applications of agricultural chemicals, etc.). The high permeability suggests large free volumes, but the molecular origin of such increased free volumes is unknown.43,44 There have been two conjectures over the years, but neither was apparently ever pursued. Flory suggested that “. . . the irregularity of the cross section of the chain obstructs efficient packing of the polymer chains in the bulk”.45 The irregularity would be the very bulky Si(CH3)2 skeletal groups alternating with the compact O atoms. One of the present authors provided the alternative suggestion that the packing problems could be due to the “. . . alternating large and more normal bond angles”.46 More specifically, the bond angle u is approximately 1438 at O atoms and is 1108 at the Si atoms, and this difference Du ¼ 338 winds the low-energy trans form of PDMS into closed polygons after around 11 repeat units (rather than into a planar zigzag form, or some other arrangement that propagates along a rectilinear axis). Molecular dynamics simulations have been carried out in an attempt to predict values of the permeability D of PDMS, particularly in comparison to that of amorphous PE as a bench-mark material.43,44 The goal was to estimate values of D for these two polymers and then for PE chains modified in steps to have some of the structural features of the PDMS chain, for example its alternating values of skeletal bond angles. Simulations on such “hybridized” PE chains at least qualitatively established which features of PDMS were most important in giving it its very high permeabilities. Making the skeletal bond angles in PE alternate in the way they do in PDMS gave a relatively large increase in D. This supports the idea that the closed polygons arising from such bond angle alternation can cause problems with efficient packing of chains in the bulk state.46 In contrast, making the two parts of the PE repeat unit different in size, as they are in PDMS, had very little effect. This may have been anticipated, in hindsight, since there

Figure 2. Results showing the increase in lengths of “like” (potentially crystallizable) sequences (lowest row moving towards the highest) as the amount of the stereoregular cyclic siloxane trimer in the polysiloxane polymerization mixture is increased.



467

are examples of chains of irregular cross-section sliding alongside one another so that bulges are alongside more slender sequences. The rigid-rod benzobisoxazole polymers provide examples of such an accommodation.47 The diffusion coefficient was also found to increase substantially with increase in the PE bond length. Perhaps this is due to an increase in flexibility somehow giving increased free volume. Introducing the polarity present in PDMS into the PE chain gave an increase in D, but not a significant one. Since this change in D is so small, certainly within the uncertainty of the methods, further discussion of this modification is probably not useful at present.

Unfilled Elastomeric Networks Curing The curing (cross linking) process has been modeled with regard to both peroxide vulcanization48 and end linking of functionally-terminated PDMS chains.49,50

Gelation The formation of network structures necessary for rubberlike elasticity has been extensively simulated by Eichinger and coworkers.51 The basic approach is to randomly end link functionally-terminated precursor chains with a multifunctional reagent, and then to examine the sol fraction with regard to amount and types of molecules present, and the gel fraction with regard to its structure and mechanical properties. The systems most studied in this regard51 involve PDMS chains having end groups X that are either hydroxyl or vinyl groups, with the corresponding Y groups on the end linking agents then being either OR alkoxy groups in an organosilicate, or H atoms in a multifunctional silane.27 The Monte Carlo method for simulating these reactions was used to generate additional information on the vinyl –silane end linking of PDMS.35,52 Results from both the experiments and the simulations are shown in Fig. 3. The simulations gave a very good account of the extent of reaction at the gelation points, and how the gelation points decreased with increase in end-linker functionality. The marked decrease in the molecular weight of the sol fraction after the gelation point results from the larger soluble molecules having more functional groups that can be used to attach themselves to the gel formed in the preceding steps. This is illustrated in Fig. 4.

Stress-Strain Isotherms The present application of these calculated distribution functions is the prediction of elastomeric properties of the chains within the framework of the Mark-Curro theory18,21,53 described below. The distribution P(r) of the end-to-end vectors r is directly related to the Helmholtz free energy A(r) of a chain by A(r) ¼ c 2 kT ln P(r) where c is a constant. The resulting perturbed distributions are then used in the “three-chain” elasticity model54 to obtain the desired stress-strain isotherms in elongation. For the specific case of this model, the


468


Figure 3. Number-average molecular weight of the sol as a function of extent of reaction for the gelation of the vinyl-terminated polysiloxane chains.

general expression for DA takes the form DA ¼

n ½Aðr0 aÞ þ 2Aðr0 a1=2 Þ 3Aðr0 Þ 3

ð1Þ

for elongations that are “affine” (in which the molecular deformations parallel the macroscopic deformation in a linear manner). Here a is the elongation ratio L/Li, n is the number of chains in the network, and r0 is the value of root-mean-square end-to-end distance of the undeformed network chains. One quantity of primary interest here is the nominal or engineering stress f , defined as the elastic force at equilibrium per unit cross-sectional area of the sample in the undeformed state: @DA f ¼ T ð2Þ @a T Substitution of Eq. (2) into Eq. (3) then gives f ¼

nkTr0 0 ½G ðr0 aÞ a3=2 G0 ðr0 a1=2 Þ 3

Figure 4. Origin of the decrease in sol molecular weight beyond the gelation point.

ð3Þ



469

where G(r) ¼ ln P(r), and G0 (r) denotes the derivative dG/dr. The modulus (or “reduced stress”) is defined by [ f ] ; f /(a – a22) and is often fitted to the Mooney-Rivlin semiempirical formula [f ] ; 2C1 þ 2C2a21, where C1 and C2 are constants independent of deformation a.34,54 Some typical results are shown in Fig. 5,21 in which the ordinate is the calculated value of the reduced stress or modulus normalized by the value given by the Gaussian limit. The value of unity is seen to be obtained for long chains, in this case those having n ¼ 250 skeletal bonds, as expected. The shorter chains show upturns in modulus with increasing elongation that are similar to those shown in bimodal networks in which short chains are introduced to give advantageous increases in ultimate strength and modulus.17,55 Because of the improvements in properties exhibited by elastomers having bimodal distributions,27 there have been attempts to prepare and characterize “trimodal” networks.56 The calculations suggest that adding a small amount of very high molecular weight end-linkable polymer could further improve mechanical properties. A sketch of unimodal, bimodal, and trimodal distributions are described in Fig. 6.

Extraction of Reptating Chains The present application addresses questions related to the rate at which diluent absorbs into a network of known and controlled “pore size”, and how rapidly it can subsequently be extracted.57 – 60 Extraction efficiencies can be used to obtain information on the extents of reaction in the end-linking procedure used to form the network, and the degree to which the extractable chains are entangled with the network chains that impede their removal from within the network structure. Of obvious interest is the dependence of these quantities on the molecular weight Mc of the network chains (as a measure of pore size), the

Figure 5. Elongation moduli of PDMS networks having 20, 40, and 250 skeletal bonds, in the Mooney-Rivlin representation. Values of the modulus have been normalized by the number of network chains, the Boltzmann constant, and the absolute temperature, and a is the elongation ratio or relative length of the deformed sample.


470


Figure 6. Network chain-length distributions, in which NM is the number of chains in an infinitesimal interval around the specified value of the molecular weight M. For reference purposes, a unimodal distribution is shown between the two dotted parts of the bimodal distribution. A possible additional component for forming a trimodal network is shown by the solid peak.

molecular weight Md of the diluent, the structure of the diluent (linear, branched, or cyclic), and whether or not the diluent is present during the end-linking process. One way of obtaining a PDMS network swollen with PDMS diluent is to have the diluent mixed into the reactive chains prior to their being end linked into a network structure. Alternatively, the network could be formed in a first step, and then the same unreactive diluent absorbed into it. These two options are illustrated in Fig. 7. In either case, oligomeric and polymeric diluents are of greatest interest, and they must be functionally inactive for them to “reptate” through the network rather than being bonded to it. Both types of networks can then be extracted to determine the ease with which the various diluents can be removed, as a function of Md and Mc.

Figure 7. Solvent extraction of a chain that is reptating through a network structure, for two methods for introducing such a chain.



471

The ease with which a diluent could be removed from a network was found to decrease with increase in Md and with decrease in Mc, as expected. High molecular weight diluents are extremely hard to remove at values of Mc of interest in the preparation of model networks, a circumstance complicating the analysis of soluble polymer fractions in terms of degrees of perfection of the network structure. The diluents added after the end linking were the more easily removed, as is shown schematically in Fig. 8. This is presumably due to the fact that they were less entangled with the network structure, and this could correspond to differences in chain conformations of the diluent. Related results were obtained in the area of single-chain spectroscopy, using experiments which measured the forces required to pull reptating single chains from polymer gels.29 Simulations on single-chain experiments in general24 – 26 should be useful in interpreting these extraction investigations. These results are consistent with results on reptating deuterated chains that were deformed because of the macroscopic elongation imposed on the non-deuterated network in which they resided.61,62 Small-angle neutron scattering measurements of the dimensions of the reptating chains as a function of time while the host network remained elongated demonstrated the extents of the constraints they experience in attempting to return to their more random arrangements. This is illustrated in Fig. 9.

Cyclic Trapping If cyclic molecules are present during the end-linking of chains, some of them will be trapped because of having been threaded by the linear chains prior to the latter being chemically bonded into the network structure.27,63,64 This is illustrated in Fig. 10. The fraction trapped is readily estimated from solvent extraction studies. Some typical results, in terms of the fraction trapped as a function of degree of polymerization of the

Figure 8. Increased ease of extracting reptating chains that had been introduced by absorption into the already end-linked elastomer.


472


Figure 9. Deformation of a reptating chain because of the elongation imposed on the network in which it resides, and restrictions it presumably experiences in attempting to return to its more random state while the network remains elongated.

cyclic are shown schematically in Fig. 11.65 The results were found to be independent of the amount of time given the cyclics and linear chains to intermingle,64 thus demonstrating the very high mobility of the PDMS chains. As expected, very small cyclics do not get trapped at all, but almost all of the largest cyclics do. The trapping process has been simulated using Monte Carlo methods based on a rotational isomeric state model4,8,19 for the cyclic chains.65 The coordinates of each cyclic chain thus generated were stored for detailed examination of the chain’s configurational characteristics, in particular the size of the “hole” it would present to a threading linear chain. Of particular interest was the size of this hole in comparison with the ˚ , of the PDMS chain. known diameter, 7– 8 A The trapping process was simulated using a torus centered around each repeat unit in the cyclic.65 Any torus found to be “empty” was considered to provide a pathway for a chain of specified diameter to pass through, threading it and then “incarcerating” it once the end-linking process has been completed. The simulation results thus obtained gave a good representation of the experimental trapping efficiencies. It is also possible to interpret these experimental results in terms of a power law for the trapping probabilities and fractal cross sections for the PDMS chains.66 It may also be possible to use this technique to prepare networks having no cross links whatsoever.67 Mixing linear chains with large amounts of cyclic are then difunctionally end-linking them could give sufficient cyclic interlinking to yield an “Olympic” or “chain-mail” network.68,69 Such materials would be very interesting topologically, and would be similar in some respects to the catenanes and rotaxanes that have long been of interest to a variety of scientists and mathematicians.70,71 Computer simulations72 could be particularly useful with regard to establishing the conditions most likely to produce these novel structures.

473



Figure 10. Sketch showing the topological trapping of some PDMS cyclics during nework formation by end linking some functionally-terminated chains

Filled Networks Importance One of the most important unsolved problems in the area of elastomers and rubberlike elasticity is the lack of a good molecular understanding of the reinforcement provided by fillers such as carbon black and silica.73 – 76 This issue has a number of challenging aspects with regard to basic research in polymers in general, and is of much practical importance since the improvements in properties fillers provide are critically important with regard to the utilization of elastomers in almost all commercially applications. Some of the work on this problem has involved analytical theory,77 – 79 but most of it is based on a variety of computer simulations.18 In the approach described here, the simulations focus on the ways the filler particles change the distribution of the end-to-end vectors of the polymer chains making up the elastomeric network, from the fact that the filler excludes the chains from the volumes it occupies. The changes in the polymer chain distributions from this filler “excluded volume effect” then cause associated changes in the mechanical properties of the elastomer host matrix.

Distributions In this case, the same Monte Carlo simulations were carried out as was done for the unfilled networks, but now each bond of the chain was tested for overlapping with a


474


Figure 11. Cyclic trapping efficiencies shown as a function of degree of polymerization of the cyclic. The open circles locate the experimental results, and the partly-filled and filled circles the simulated results for two simulation models.

filler particle as the chain was being generated.80 If any bond penetrated a particle surface, the entire chain conformation was rejected and a new chain started. The particle sizes of greatest interest are those used commercially, with small particles giving significantly better reinforcement than larger ones. The primary particles are generally assumed to be spherical. In actual filled elastomers, the particles are dispersed at least relatively randomly, but it is of interest to do simulations on regular particle arrangements as well.80 In the first of these simulations, a filled PDMS network was modeled as a composite of cross-linked polymer chains and spherical filler particles arranged in a regular array on a cubic lattice.81 The filler particles were found to increase the non-Gaussian behavior of the chains and to increase the moduli, as expected. It is interesting to note that composites with such structural regularity have actually been produced,82 and some of their mechanical properties have been reported.83 In a subsequent study,84 the reinforcing particles were randomly distributed. Of greatest interest in these studies was whether the particles cause increases or decreases in the end-to-end distances, with this expected to depend particularly on the size of the filler particles, but presumably on other variables such as their concentration in the elastomeric matrix as well. Some illustrative results for filler particles within a PDMS matrix are shown in Fig. 12.84 One effect of the particles was to increase the dimensions of the chains when the filler particles were small relative to the dimensions of the network chains. In contrast, particles that were relatively large tended to decrease the chain dimensions. Since these changes in dimensions arising from the filler excluded volume effects are of critical importance, it is necessary to put them into a larger context. These simulated results on the distributions are in agreement with some subsequent neutron scattering experiments on deuterated and non-deuterated chains of PDMS.85 The polymers contained silica particles that were surface treated to make them inert to the polymer chains, as was implicitly assumed in the simulations. These experimental results also indicated chain extensions when the particles were relatively small, and chain compressions when they were relatively large.



475

Figure 12. Radial distribution functions P(r) at T ¼ 500 K for network chain end-to-end distances obtained from Monte Carlo simulations. The results are shown as a function of the relative extension r/rmax, for PDMS networks having 50 skeletal bonds between cross links.84 The radius of the filler ˚ , and the values of the volume % of filler present are indicated in the inset. particles was 5 A

There have been several reports of simulations, however, that have yielded results in disagreement with the described simulations and the corresponding scattering experiments. The major difference in approach was the use of dense collections of chains instead of single chains sequentially generated in the vicinities of the filler particles. In particular, the simulations by Vacatello86,87 find only compressions of the chains. In a rather different approach, Mattice et al.88 generated particles within a matrix by collapsing some of the chains into domains that would act as reinforcing filler. They found that small particles did lead to significant increases in chain dimensions, while large particles led to moderate decreases, in agreement with the single-chain simulations and scattering experiments. Additional simulations eliminated the possibility that the collapse of some of the chains gave rise to an increase in free volume into which the non-collapsed chains expanded.89 Because of these discrepancies, the present simulations were refined in an attempt to understand the differences described.90 This involved i. ii. iii. iv.

relocating the particles periodically during a simulation, starting the chains at different locations, using Euler matrices to change the orientations of the chains being generated, and replacing the “united atom” approach by detailed atom specifications.

None of these modifications significantly changed the results obtained. An additional modification, generating dense collections of chains, is in progress.91 The cases where the filler causes compression of the chain are relevant to another area of rubberlike elasticity, specifically the preparation of networks by cross linking in solution followed by removal of the solvent.27 This is shown schematically in Fig. 13. Such experiments were initially carried out to obtain elastomers that had fewer entanglements and the success of this approach was supported by the observation that such networks came to elastic equilibrium much more rapidly. They also exhibited stress-


476

Figure 13. drying.


Forming a “super-compressed” network by cross linking in solution, followed by

strain isotherms in elongation that were closer in form to those expected from the simplest molecular theories of rubberlike elasticity. In these procedures, the solvent disentangles the chains prior to their cross linking, and its subsequent removal by drying puts the chains into a “supercontracted” state.27 Experiments on strain-induced crystallization carried out on such solution cross-linked elastomers indicated that the decreased entangling was less important than the supercontraction of the chains, in that crystallization required larger values of the elongation than was the case for the usual elastomers cross-linked in the dry state.92,93 The most recent work in this area has focused on the unusually high extensibilities of such elastomers.94 – 96 In any case, the present simulations should help elucidate molecular aspects of phenomena in this research area as well. Also in progress are simulations to determine any effects of having multimodal distributions of particle sizes.97 Looking at this issue was encouraged by the improvements in properties obtained by using bimodal distributions of network chain lengths in elastomers55 and thermosets,98 and bimodal distributions of the diameters of rubbery domains introduced into some thermoplastics.99 – 101

Stress-Strain Isotherms There are two items of primary interest here, specifically increases in modulus in general, and upturns in the modulus with increasing deformation. Results are typically expressed as the reduced nominal or engineering stress as a function of deformation. The area under such curves up to the rupture point of the sample then gives the energy of rupture, which is the standard measure of the toughness of a material.27 Figure 14 shows the stress-strain isotherms in elongation84 corresponding to the distributions shown in Fig. 12. There are substantial increases in modulus that increase with increase in filler loading, as expected. Additional increases would be expected by taking into account other mechanisms for reinforcement such as physisorption, chemisorption, etc., as described below. One additional interesting result is the observation that in some cases, chain compression can also cause increases in modulus. This is being clarified by making some



477

Figure 14. Nominal stress as a function of elongation for the moduli calculated from the distributions shown in Fig. 12.84

arbitrary changes in the distributions obtained and documenting the effects these changes have on the corresponding simulated stress-strain isotherms. Some unexpected results were obtained when the distribution was narrowed at the same most-probable value of the chain dimensions.18 The narrowing caused the peak defining the most-probable value to shift upward to keep the area under the curve the same, as is required. In this case, the change in the shape of the distribution did indeed cause an increase in the modulus. This is consistent with other simulations finding increases in modulus even when the chains are compressed, since it demonstrates that the mechanical properties can depend on subtle features of the distribution, beyond merely some average value of the chain dimensions. Non-spherical filler particles are also of considerable interest. Prolate (needle-shaped) particles can be thought of as a bridge between the roughly spherical particles used to reinforce elastomers102 and the long fibers frequently used for this purpose in thermoplastics and thermosets.103 Oblate (disc-shaped) particles can be considered as analogues of the much-studied clay platelets used to reinforce a variety of materials.104 In one particularly relevant series of experiments, initially spherical particles of polystyrene (PS) were deformed into prolate ellipsoids by i. heating the elastomeric PDMS matrix in which they resided above the glass transition temperature of the PS, ii. stretching the matrix uniaxially, and then iii. cooling it under the imposed deformation.105 The technique is illustrated by the sketch in Fig. 15. It is important to note that this approach also orients the axes of the now-elliptical particles, as shown in the middle section of the figure. If desired, the orientation can be removed by dissolving away the host matrix, and then re-dispersing the particles randomly within another polymer that is subsequently cross-linked. This is illustrated in the bottom part of the sketch. Some relevant simulations106 show that the anisotropy in structure causes the values of the modulus in the longitudinal direction to be significantly higher than those in the


478


Figure 15. The upper sketch represents some originally spherical filler particles being deformed into prolate (needle-shaped) ellipsoids by stretching a polymer matrix in which they reside. This in-situ approach also orients the axes of the deformed particles in the direction of the stretching. The lower sketch shows how this orientation can be removed by dissolving away the host matrix; The ellisoidal particles can then be isotropically dispersed within another polymer, as randomized reinforcing particles.

transverse direction. These simulated results are in at least qualitative agreement with the experimental differences in longitudinal and transverse moduli obtained experimentally.105 Also of interest are randomized arrangements of such prolate ellipsoids. In this case, isotropic behavior is expected, due to the lack of orientation dependence between the nonspherical particles and the deformation axis regardless of the shapes of the particles. The simulated results confirmed this expectation that the reinforcement from randomlyoriented non-spherical filler particles is isotropic regardless of the anisometry of their shapes. In spite of their inherent interest, relatively little has been done on oblate ellipsoidal fillers. In one study, however, oblate particles were again placed on a cubic lattice,107 and were oriented in a way consistent with their orientation in PS-PDMS composites that were the subject of an experimental investigation.108 In general, the network chains tended to adopt more compressed configurations relative to those of prolate particles having equivalent sizes and aspect ratios. The elongation moduli were found to depend on the sizes, number, and axial ratios of the particles, as expected. In particular, the reinforcement from the oblate particles was found to be greatest in the plane of the particles, and the changes were in at least qualitative agreement with the corresponding experimental results.108 In the experimental study, axial ratios were controllable, since they were generally found to be close to the values of the biaxial draw ratio employed in their generation. The moduli of these anisotropic composites were reported, but only in the plane of the biaxial deformation.108 It was not possible to obtain moduli in the perpendicular direction, owing to the thinness of the films that had to be used in the experimental design. With regard to the simulations, it would be of considerable interest to investigate the reinforcing properties of such oblate particles when they are randomly oriented and also randomly dispersed. Such work is in progress.97

Aggregated Particles The silica or carbon black particles used to reinforce commercial materials are seldom completely dispersed,73 – 76,109 as is assumed in the simulations described. As is shown



479

Figure 16. Sketches of (a) primary particles, (b) aggregates, and (c) agglomerates occurring in fillers such as carbon black and silica.

schematically in Fig. 16, the primary particles are generally aggregated into relatively stable aggregates and these aggregates are frequently clustered into less stable arrangements called agglomerates. Simulations should be carried out on such more highly ordered structures.97 It is well known in the industry that such structures are important in maximizing the reinforcement, as evidenced by the fact that being too persistent in removing such aggregates and agglomerates in blending procedures gives materials with less than optimal mechanical properties.73 – 76,109 Friedlander et al. have demonstrated that such aggregates have a remarkable deformability, by carrying out elongation experiments both reversibly, and irreversibly to their rupture points.110 – 113 This is of considerable importance, since when these structures are within elastomeric matrices, their deformations upon deformation of the filled elastomer means that they must contribute to the storage of the elastic deformation energy. This would have to be taken into account both in the interpretation of experimental results and in more refined simulations of filler reinforcement. Potential Refinements The excluded volume effect investigated here is only one aspect of elastomer reinforcement,77 – 79,114 – 117 but some additional effects could be investigated by modeling the adsorption of the elastomer chains onto the filler surface. This could be done by first assuming Lennard-Jones interactions between the particles and chains, in physical adsorption. These aspects could then be extended to include chemical adsorption by assuming that there are randomly-distributed, active particle sites interacting very strongly with the chains (by a Dirac d-function type of potential). If the distance between the chain (generated using the Monte Carlo method) and the active site becomes less than the range of the short-range interactions, then the chain would become chemisorbed. Both types of adsorption would be expected to yield substantial increases in the modulus and related mechanical properties.

Conclusions Although there are obviously unresolved issues, the broad overview presented here should demonstrate the utility of simulations and analytic theory to improve the understanding of


480


structure-property relationships of poly(dimethylsiloxane) in particular, and of polymers in general.

Acknowledgment It is a pleasure to acknowledge the financial support provided to JEM by the National Science Foundation through Grant DMR-0314760 (Polymers Program, Division of Materials Research).

References 1. Oberhammer, H.; Boggs, J. E. “Importance of (p-d)p bonding in the siloxane bond”, J. Am. Chem. Soc. 1980, 102, 7241. 2. Bock, H. “Fundamental of silicon chemistry: molecular states of silicon-containing compounds”, Angew. Chem. Int. Ed. Engl., Adv. Mater. 1989, 28, 1627– 1650. 3. Brook, M. A. Silicon in Organic, Organometallic, and Polymer Chemistry; John Wiley & Sons: New York, 2000. 4. Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. 5. Dubchak, I. L.; Pertsin, A. I.; Zhdanov, A. A. “Calculation of helical chains of polyorganosiloxanes”, Polym. Sci., U.S.S.R. 1985, 27, 2340– 2347. 6. Smith, J. S.; Borodin, O.; Smith, G. D. “A quantum chemistry based force field for poly(dimethylsiloxane)”, J. Phys. Chem. B 2004, 108, 20340 – 20350. 7. Grigoras, S.; Lane, T. H. “Ab initio calculations on the effect of polarization functions on disiloxane”, J. Comput. Chem. 1987, 8, 84 –93. 8. Mattice, W. L.; Suter, U. W. Conformational theory of large molecules. In The Rotational Isomeric State Model in Macromolecular Systems; Wiley: New York, 1994. 9. Grigoras, S.; Qian, C.; Crowder, C.; Harkness, B.; Mita, I. “Structure elucidation of crystalline poly(diphenylsiloxane)”, Macromolecules 1995, 28, 7370– 7375. 10. DeBolt, L. C.; Mark, J. E. “Effects of bond-angle inversion on the statistical properties of poly(dimethylsiloxane)”, J. Polym. Sci., Polym. Phys. Ed. 1988, 26, 989– 995. 11. Bahar, I.; Zuniga, I.; Dodge, R.; Mattice, W. L. “PDMS statistics 1”, Macromolecules 1991, 24, 2986– 2992. 12. Bahar, I.; Zuniga, I.; Dodge, R.; Mattice, W. L. “PDMS statistics 2”, Macromolecules 1991, 24, 2993– 2998. 13. Nicolas, J. B.; Winans, R. E.; Harrison, R. J.; Iton, L. E.; Curtiss, L. A.; Hopfinger, A. J. “An ab initio investigation of disiloxane using extended basis sets and electron correlation”, J. Phys. Chem. 1992, 96, 7958–7965. 14. Moore, W. J. Basic Physical Chemistry; Prentice Hall: Englewood Cliffs, NJ, 1983. 15. Kress, J. D.; Leung, P. C.; Tawa, G. J.; Hay, P. J. “Calculation of a reaction path for KOH catalyzed ring-opening polymerization of hexamethylcyclotrisiloxane”, J. Am. Chem. Soc. 1997, 119, 1954– 1960. 16. Chenoweth, K.; Cheung, S.; Duin, A. C. T. v.; Goddard, W. A., III; Kober, E. M. “Simulations on the thermal decomposition of poly(dimethylsiloxane) polymer using the reaxFF reactive force field”, J. Am. Chem. Soc. 2005, 127, 7192– 7202. 17. Mark, J. E. “Some interesting things about polysiloxanes”, Acct. Chem. Res. 2004, 37, 946– 953. 18. Mark, J. E.; Abou-Hussein, R.; Sen, T. Z.; Kloczkowski, A. “Some simulations on filler reinforcement of elastomers”, Polymer 2005, 46, 8894– 8904. 19. Rehahn, M.; Mattice, W. L.; Suter, U. W. “Rotational isomeric state models in macromolecular systems”, Adv. Polym. Sci. 1997, 131/132, 1 – 18. 20. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.



481

21. Mark, J. E.; Curro, J. G. “A non-Gaussian theory of rubberlike elasticity based on rotational isomeric state simulations of network chain configurations. I. polyethylene and polydimethylsiloxane short-chain unimodal networks”, J. Chem. Phys. 1983, 79, 5705– 5709. 22. DeBolt, L. C.; Mark, J. E. “Theoretical stress-strain isotherms for elastin model networks”, J. Polym. Sci., Polym. Phys. Ed. 1988, 26, 865–874. 23. Smith, J. S.; Bedrov, D.; Smith, G. D.; Kober, E. M. “Thermodynamics and conformational changes upon stretching a poly(dimethylsiloxane) chain in the melt”, Macromolecules 2005, 38, 8101– 8107. 24. Jarkova, E.; Vlugt, T. J. H.; Lee, N.-K. “Stretching a heteropolymer”, J. Chem. Phys. 2005, 122, 1149041– 1149047. 25. Zemanova, M.; Bleha, T. “Isometric and isotensional force-length profiles in polymethylene chains”, Macromol. Theor. Sims. 2005, 14, 596–604. 26. Rubi, J. M.; Bedeaux, D.; Kjelstrup, S. “Thermodynamics for single-molecule stretching experiments”, J. Phys. Chem., B 2006, 110, 12733– 12737. 27. Erman, B.; Mark, J. E. Structures and Properties of Rubberlike Networks; Oxford University Press: New York, 1997. 28. Mark, J. E.; Erman, B. Rubberlike Elasticity, In A Molecular Primer, 2nd Edn.; Cambridge University Press: Cambridge, 2007. 29. Okajima, T.; Tao, X.-M.; Azehara, H.; Tokumoto, H. “Force spectroscopy of single polymer incorporataed into polymer gels”, J. Nanosci. Nanotech. 2007, 7, 790– 795. 30. Sides, S. W.; Curro, J.; Grest, G. S.; Stevens, M. J.; Soddemann, T.; Habenschuss, A.; Londono, J. D. “Structure of poly(dimethylsiloxane) melts: Theory, simulation, and experiment”, Macromolecules 2002, 35, 6455– 6465. 31. Frischknecht, A. L.; Curro, J. “Improved united atom force field for poly(dimethylsiloxane)”, Macromolecules 2003, 36, 2122– 2129. 32. Nath, S. K.; Frischknecht, A. L.; Curro, J. G.; McCoy, J. D. “Density functional theory and molecular dynamics simulation of poly(dimethylsiloxane) melts near silica surfaces”. Macromolecules 2005, 38, 8562– 8573. 33. Hanna, S.; Windle, A. H. “Geometrical limits to order in liquid crystalline random copolymers”, Polymer 1988, 29, 207– 223. 34. Mark, J. E. “Some recent theory, experiments, and simulations on rubberlike elasticity”, J. Phys. Chem., Part B 2003, 107, 903– 913. 35. Mark, J. E. “Some simulations on elastomers and rubberlike elasticity”, Makromol. Symp. 2001, 171, 1 – 9. 36. Madkour, T. M.; Mark, J. E. “Mesoscopic modeling of the polymerization, morphology, and crystallization of stereoblock and stereoregular polypropylenes”, J. Polym. Sci., Polym. Phys. Ed. 2002, 40, 840–853. 37. Madkour, T. M.; Mark, J. E. “Sequence distributions and crystallinity in random and semiblocky polysiloxane copolymers. I. Monte Carlo simulations for single chains and comparisons with analytical theory”, Comput. Polym. Sci. 1994, 4, 79 – 85. 38. Madkour, T. M.; Mark, J. E. “Sequence distributions and crystallinity in random and semiblocky polysiloxane copolymers. II. Searches in multi-chain systems for polymer crystallinity”, Comput. Polym. Sci. 1994, 4, 87 – 93. 39. Madkour, T. M.; Kloczkowski, A.; Mark, J. E. “Sequence distributions and crystallinity in random and semi-blocky polysiloxane copolymers. III. Analytical theory and some comparisons with the simulation results”, Comput. Polym. Sci. 1994, 4, 95 – 100. 40. Madkour, T. M.; Mark, J. E. “Simulations on crystallization in polymers”, J. Comput.-Aided Mats. Design 1997, 4, 19 – 28. 41. Madkour, T. M.; Mark, J. E. “Monte Carlo simulations on the stereoregularity, crystallinity, and related physical properties of poly[methyl(3,3,3-trifluoropropyl)siloxaneg”, Macromolecules 1995, 28, 6865– 6870. 42. Mark, J. E.; Allcock, H. R.; West, R. Inorganic Polymers, 2nd Edn.; Oxford University Press: New York, 2005.


482


43. Abou-Hussein, R. Ph. D. Chemistry, Ph. D.Thesis in Chemistry, The University of Cincinnati, 2007. 44. Abou-Hussein, R.; Wu, S.; Zhang, L.; Mark, J. E. “Effects of some structural features of poly(dimethylsiloxane) on its unusually high gas permeabilities”, in preparation 2007. 45. Flory, P. J.; Shih, H. “Thermodynamics of solutions of poly(dimethylsiloxane) in benzene, cyclohexane, and chlorobenzene”, Macromolecules 1972, 5, 761–766. 46. Mark, J. E. “Overview of siloxane polymers”, In Silicones and Silicone-Modified Materials; Clarson, S. J., Fitzgerald, J. J., Owen, M. J., Smith, S. D., Eds.; American Chemical Society: Washington, 2000; Vol. 729, pp 1 – 10. 47. Bhaumik, D.; Mark, J. E. “A theoretical investigation of chain packing and electronic band structure of the rigid-rod polymer trans-poly(p-phenylene benzobisthiazole) in the crystalline state”, J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 2543– 2549. 48. Lopez, L. M.; Cosgrove, A. B.; Hernandez-Ortiz, J. P.; Osswald, T. A. “Modeling the vulcanization reaction of silicone rubber”, Polym. Eng. Sci. 2007, 47, 675– 683. 49. Gilra, N.; Cohen, C.; Panagiotopoulos, A. Z. “A Monte Carlo study of the structural properties of end-linked polymer networks”, J. Chem. Phys. 2000, 112, 6910 –6916. 50. Heine, D. R.; Grest, G. S.; Lorenz, C. D.; Tsige, M.; Stevens, M. J. “Atomistic simulations of end-linked poly(dimethylsiloxane) networks: Structure and relaxations”, Macromolecules 2004, 33, 3857– 3864. 51. Eichinger, B. E.; Akgiray, O. “Computer simulation of polymer network formation”, In Computer Simulation of Polymers; Colbourne, E. A. Ed.; Longman: White Plains, NY, 1994; pp 263– 302. 52. Braun, J. L.; Mark, J. E.; Eichinger, B. E. “Formation of poly(dimethylsiloxane) gels”, Macromolecules 2002, 35, 5273– 5282. 53. Mark, J. E.; Curro, J. G. “A non-Gaussian theory of rubberlike elasticity based on rotational isomeric state simulations of network chain configurations. III. Networks prepared from the extraordinarily flexible chains of polymeric sulfur and polymeric selenium”, J. Chem. Phys. 1984, 80, 5262– 5265. 54. Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd Edn.; Clarendon Press: Oxford, 1975. 55. Mark, J. E. “Elastomers with multimodal distributions of network chain lengths”, Macromol. Symp., St. Petersburg issue 2003, 191, 121– 130. 56. Erman, B.; Mark, J. E. “Calculations on trimodal elastomeric networks. Effects of chain length and composition on ultimate properties”, Macromolecules 1998, 31, 3099– 3103. 57. Mark, J. E.; Zhang, Z.-M. “The use of model networks with reptating chains to study extraction efficiencies”, J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 1971– 1979. 58. Garrido, L.; Mark, J. E. “Extraction and sorption studies using linear polymer chains and model networks”, J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 1933– 1940. 59. Mark, J. E. “Sorption, extraction, and trapping of linear and cyclic molecules in model elastomeric networks”, J. Appl. Polym. Sci., Symp. 1989, 44, 209– 216. 60. Mazan, J.; Leclerc, B.; Galandrin, N.; Couarraze, G. “Sorption of PDMS chains into PDMS gels”, Eur. Polym. J. 1995, 31, 803– 807. 61. Beltzung, M.; Picot, C.; Herz, J. “PDMS orientation neutron scattering”, Macromolecules 1984, 17, 663–669. 62. Wignall, G. D. “Small angleneutron and x-ray scattering”, In Physical Properties of Polymers Handbook, 2nd Edn.; Mark, J. E. Ed.; Springer-Verlag, Inc.: New York, 2007; pp 407– 420. 63. Clarson, S. J.; Mark, J. E.; Semlyen, J. A. “Studies of cyclic and linear poly(dimethylsiloxanes): Effect of ring size on the topological trapping of cyclic polymers into model network structures”, Polym. Comm. 1986, 27, 243– 245. 64. Clarson, S. J.; Mark, J. E.; Semlyen, J. A. “Studies of cyclic and linear poly(dimethylsiloxanes): 24. topological trapping of cyclic polymers into unimodal and bimodal model network structures”, Polym. Comm. 1987, 28, 151– 153. 65. DeBolt, L. C.; Mark, J. E. “Models for the trapping of cyclic poly(dimethylsiloxane) (PDMS) chains in PDMS networks”, Macromolecules 1987, 20, 2369– 2374.



483

66. Galiatsatos, V.; Eichinger, B. E. “An interpretation of the topological trapping of cyclic poly(dimethylsiloxane) in PDMS network structures”, Polym. Commun. 1987, 28, 182 – 184. 67. Rigbi, Z.; Mark, J. E. “Concurrent chain extension and crosslinking of hydroxyl-terminated poly(dimethylsiloxane): Possible formation of catenate structures”, J. Polym. Sci., Polym. Phys. Ed. 1986, 24, 443–449. 68. de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: New York, 1979. 69. Garrido, L.; Mark, J. E.; Clarson, S. J.; Semlyen, J. A. “Studies of cyclic and linear poly(dimethylsiloxanes): 16. Trapping of cyclics present during the end linking of linear chains into network structures”, Polym. Comm. 1985, 26, 53 – 55. 70. Gibson, H. W.; Bheda, M. C.; Engen, P. T. “Rotaxanes, catenanes, polyrotaxanes, polycatenanes and related materials”, Prog. Polym. Sci. 1994, 19, 843– 945. 71. Gong, C.; Gibson, H. W. “Controlling polymeric topology by polymerization conditions: mechanically linked network and branched poly(urethane rotaxane)s with controllable polydispersity”, J. Am. Chem. Soc. 1997, 119, 8585– 8591. 72. Iwata, K.; Ohtsuki, T. “Olympic network problems”, J. Polym. Sci., Polym. Phys. Ed. 1993, 31, 441. 73. Boonstra, B. B. “Role of particulate filers in elsastomer reinforcement: A review”, Polymer 1979, 20, 691– 704. 74. Warrick, E. L.; Pierce, O. R.; Polmanteer, K. E.; Saam, J. C. “Silicone elastomer developments 1967– 1977”, Rubber Chem. Technol. 1979, 52, 437–525. 75. Rigbi, Z. “Reinforement of rubber by carbon black”, Adv. Polym. Sci. 1980, 36, 21 –68. 76. Donnet, J.; Custodero, E. “Reinforcement of elastomers by particulate fillers”, In Science and Technology of Rubber, 3rd Edn.; Mark, J. E., and Erman, B. Eds.; Elsevier: Amsterdam, 2005; pp 367– 400. 77. Witten, T. A.; Rubinstein, M.; Colby, R. H. “Reinforcement of rubber by fractal aggregates”, J. Phys. II France 1993, 3, 367– 383. 78. Heinrich, G.; Kluppel, M. “Recent advances in the theory of filler networking in elastomers”, Adv. Polym. Sci. 2002, 160, 1 – 44. 79. Kluppel, M. “The role of disorder in filler reinforcement of elastomers on various length scales”, Adv. Polym. Sci. 2003, 164, 1 –86. 80. Mark, J. E. “Some simulations on filler reinforcement in elastomers”, Molec. Cryst. Liq. Cryst. 2002, 374, 29 – 38. 81. Sharaf, M. A.; Kloczkowski, A.; Mark, J. E. “Simulations on the reinforcement of elastomeric poly(dimethylsiloxane) by filler particles arranged on a cubic lattice”, Comput. Polym. Sci. 1994, 4, 29 – 39. 82. Sunkara, H. B.; Jethmalani, J. M.; Ford, W. T. “Solidification of colloidal crystals of silica”, In Hybrid Organic-Inorganic Composites; Mark, J. E., Lee, C. Y.-C., and Bianconi, P. A. Eds.; American Chemical Society: Washington, 1995; Vol. 585, pp 181– 191. 83. Pu, Z.; Mark, J. E.; Jethmalani, J. M.; Ford, W. T. “Effects of dispersion and aggregation of silica in the reinforcement of poly(methylacrylate) elastomers”, Chem. Mater. 1997, 9, 2442– 2447. 84. Yuan, Q. W.; Kloczkowski, A.; Mark, J. E.; Sharaf, M. A. “Simulations on the reinforcement of poly(dimethylsiloxane) elastomers by randomly-distributed filler particles”, J. Polym. Sci., Polym. Phys. Ed. 1996, 34, 1647– 1657. 85. Nakatani, A. I.; Chen, W.; Schmidt, R. G.; Gordon, G. V.; Han, C. C. “Chain dimensions in polysilicate-filled poly(dimethylsiloxane)”, Polymer 2001, 42, 3713– 3722. 86. Vacatello, M. “Molecular arrangements in polymer-based nanocomposites”, Macromol. Theor. Sims. 2002, 11, 757– 765. 87. Vacatello, M. “Chain dimensions in filled polymers: An intriguing problem”, Macromolecules 2002, 35, 8191– 8193. 88. Erguney, F. M.; Lin, H.; Mattice, W. L. “Dimensions of matrix chains in polymers filled with energetically neutral nanoparticles”, Polymer 2006, 47, 3689 –3695.


484


89. Erguney, F. M.; Mattice, W. L. “Response of matrix chains tol nanoscale filler particles” Polymer. submitted, 000– 000. 90. Sharaf, M. A.; Mark, J. E. “Monte Carlo simulations on the effects of nanoparticles on chain deformations and reinforcement in amorphous polyethylene networks”, Polymer 2004, 45, 3943– 3952. 91. Sen, T. Z.; Kloczkowski, A. Unpublished results. 92. Premachandra, J.; Mark, J. E. “Effects of dilution during cross linking on strain-induced crystallization in cis-1,4-polyisoprene networks. 1. Experimental results”, J. Macromol. Sci., Pure Appl. Chem. 2002, 39, 287– 300. 93. Premachandra, J.; Kumudinie, C.; Mark, J. E. “Effects of dilution during cross linking on strain-induced crystallization in cis-1,4-polyisoprene networks. 2. Comparison of experimental results with theory”, J. Macromol. Sci., Pure Appl. Chem. 2002, 39, 301– 320. 94. Urayama, K.; Kohjiya, S. “Uniaxial elongation of deswollen polydimethylsiloxane networks with supercoiled structure”, Polymer 1997, 38, 955– 962. 95. Kohjiya, S.; Urayama, K.; Ikeda, Y. “Poly(siloxane) network of ultra-high elongation”, Kautschuk Gummi Kunstoffe 1997, 50, 868– 872. 96. Urayama, K.; Kohjiya, S. “Extensive stretch of polysiloxane network chains with random- and super-coiled conformations”, Eur. Phys. J., B 1998, 2, 75 – 78. 97. Abou-Hussein, R.; Mark, J. E. Unpublished results. 98. Holmes, G. A.; Letton, A. “Bisphenol-A bimodal epoxy resins. Part I: The dynamic mechanical characterization of a 6300 (340/22,500) weight average molecular weight system”, Polym. Eng. Sci. 1994, 34, 1635– 1642. 99. Okamoto, Y.; Miyagi, H.; Kakugo, M.; Takahashi, K. “Impact improvement mechanism of HIPS with bimodal distribution of rubber particle size”, Macromolecules 1991, 24, 5639–5644. 100. Chen, T. K.; Jan, Y. H. “Fracture mechanism of toughened epoxy resin with bimodal rubberparticle size distribution”, J. Mats. Sci. 1992, 27, 111– 121. 101. Takahashi, J.; Watanabe, H.; Nakamoto, J.; Arakawa, K.; Todo, M. “In-situ polymerization and properties of methyl methacrylate-butadiene-styrene resin with bimodal rubber particle size distributions”, Polymer Journal 2006, 38, 835–843. 102. Medalia, A. I.; Kraus, G. “Reinforcement of elastomers by particulate fillers”, In Science and Technology of Rubber, 2nd Edn.; Mark, J. E., Erman, B., and Eirich, F. R., Eds.; Academic: San Diego, 1994; pp 387– 418. 103. Fried, J. R. Polymer Science and Technology; 2nd Edn.; Prentice Hall: Englewood Cliffs, NJ, 2003. 104. Pinnavaia, T. J.; Beall, G. Polymer-Clay Nanocomposites; Wiley: New York, 2001. 105. Wang, S.; Mark, J. E. “Generation of glassy ellipsoidal particles within an elastomer by in-situ polymerization, elongation at an elevated temperature, and finally cooling under strain”, Macromolecules 1990, 23, 4288– 4291. 106. Sharaf, M. A.; Kloczkowski, A.; Mark, J. E. “Monte Carlo simulations on reinforcement of an elastomer by oriented prolate particles”, Comput. Theor. Polym. Sci. 2001, 11, 251– 262. 107. Sharaf, M. A.; Mark, J. E. “Monte Carlo simulations on filler-induced network chain deformations and elastomer reinforcement from oriented oblate particles”, Polymer 2002, 43, 643– 652. 108. Wang, S.; Xu, P.; Mark, J. E. “Method for generating oriented oblate ellipsoidal particles in an elastomer and characterization of the reinforcement they provide”, Macromolecules 1991, 24, 6037– 6039. 109. Donnet, J.-B.; Vidal, A. “Carbon black: Surface properties and iteractions with elastomers”, Adv. Polym. Sci. 1986, 76, 103– 127. 110. Friedlander, S. K.; Jang, H. D.; Ryu, K. H. “Elastic behavior of nanoparticle chain aggregates”, Appl. Phys. Lett. 1998, 72, 11796 –11799. 111. Suh, Y. J.; Prikhodko, S. V.; Friedlander, S. K. “Nanostructure manipulation device for transmission electron microscopy: Application to titania nanoparticle chain aggregates”, Microsc. Microanal. 2002, 8, 497– 501.



485

112. Suh, Y. J.; Friedlander, S. K. “Origin of the elastic behavior of nanoparticle chain aggregates: Measurements using nanostructure manipulation device”, J. Appl. Phys. 2003, 93, 3515– 3523. 113. Rong, W.; Pelling, A. E.; Ryan, A.; Gimzewski, J. K.; Friedlander, S. K. “Complementary TEM and AFM force spectroscopy to characterize the nanomechanical properties of nanoparticle chain aggregates”, Nano Lett. 2004, 4, 2287– 2292. 114. Heinrich, G.; Vilgis, T. A. “Contribution of entanglements to the mechanical properties of carbon black filled polymer networks”, Macromolecules 1993, 26, 1109– 1119. 115. Kluppel, M.; Heinrich, G. “Fractal structures in carbon black reinforced rubbers”, Rubber Chem. Technol. 1995, 68, 623– 651. 116. Kluppel, M.; Schuster, R. H.; Heinrich, G. “Structure and properties of reinforcing fractal filler networks in elastomers”, Rubber Chem. Technol. 1997, 70, 243– 255. 117. Heinrich, G.; Kluppel, M.; Vilgis, T. A. “Reinforcement of elastomers”, Curr. Opinion Solid State Mats. Sci. 2002, 6, 195– 203.