Chain dimensions in polysilicate-filled poly(dimethyl siloxane)

Abstract

We present experimental results on the single chain dimensions of isotopic blends (both mismatched and matched molecular masses) of poly(dimethyl siloxane) (PDMS) containing trimethylsilyl-treated polysilicate particles (fillers) and compare these results with Monte Carlo calculations. For polymer chains which are approximately the same size as the filler particle, a decrease in chain dimensions is observed relative to the unfilled chain dimensions at all filler concentrations. For larger chains, at low filler concentrations, an increase in chain dimensions relative to the unfilled chain dimensions is observed. Both results are in agreement with existing Monte Carlo predictions. However, at even higher filler contents, which are beyond the scope of the Monte Carlo predictions, the chain dimensions reach a maximum value before decreasing to values which are still larger than the unfilled chain dimensions. A simple excluded volume model is proposed which accounts for these observations at higher filler content.

Introduction

Filled polymers constitute a major portion of the commercial polymer market and have been in use since the turn of the century. In most cases, fillers are used as economical additives for altering the mechanical behavior of polymers. In spite of widespread use, a fundamental understanding of how fillers modify mechanical behavior has not been achieved. While some researchers have attempted a rigorous approach toward understanding mechanical behavior in filled polymers [1], [2], [3], empirical relationships have dominated the field. While the partial success of these empiricisms has led to some advances, knowledge of the underlying physical behavior of fillers in polymers is still lacking. One of the greatest needs in the area of filled polymers is a molecular theory of elasticity for filled polymers, analogous to kinetic theories of rubber elasticity. One potential reason for the lack of such a theory is the scarcity of data available for filled polymers on fundamental quantities such as the radius of gyration. Mark and Curro have developed a theory of rubberlike elasticity, which accounts for non-Gaussian probability distributions of chains between crosslinks [4], [5]. Attempts have been made to apply this non-Gaussian theory of rubberlike elasticity to filled elastomers. Mark and coworkers [6], [7], [8], [9] have used Monte Carlo techniques to calculate the distribution of polymer chain dimensions in the presence of filler particles. These theoretical, non-Gaussian distributions have been applied to the Mark and Curro theory to predict the stress–strain behavior and modulus of filled polymers. Experimental confirmation of these predictions are not yet available.

Small angle neutron scattering (SANS) has been used to extract dimensions of polymer chains in multicomponent mixtures. Therefore, the technique is well suited to test the predictions of Mark and coworkers on filled polymer chain dimensions. However, previous studies of polymer chain dimensions have often been restricted to deuterium labeled and unlabeled polymers in the presence of solvent or block copolymers. For filled polymer systems, studies involving a number of different scattering techniques have been reported by other investigators. Among these reports are light scattering studies of soot [10] and carbon black [11]; small angle X-ray scattering studies of filled rubbers [12], [13], [14], and organic–inorganic hybrids [15], [16]; and SANS studies of silica-filled [17] and carbon black-filled [18], [19], [20], [21], [22], [23] polymers. All of these studies have utilized scattering to focus on the structure and nature of the filler particles. To our knowledge, extraction of the single chain dimensions of a polymer in the presence of a filler particle has not been determined experimentally.

In this work, we present experimental results on the single chain dimensions of isotopic blends (both mismatched and matched molecular masses) of poly(dimethyl siloxane) (PDMS) containing trimethylsilyl-treated polysilicate particles (fillers) and compare these results with the Monte Carlo calculations of Mark and coworkers. For polymer chains which are approximately the same size as the filler particle, a decrease in chain dimensions is observed relative to the unfilled chain dimensions. These results are qualitatively in agreement with the Monte Carlo calculations. For larger chains, an increase in chain dimensions at low filler concentrations relative to the unfilled chain dimensions is observed which is also in agreement with Mark and coworkers. However, at even higher filler contents, which are beyond the scope of the Monte Carlo predictions, a maximum in the chain dimensions is observed, followed by a decrease to values which are still larger than the unfilled chain dimensions. A simple excluded volume model is proposed which accounts for these observations at higher filler content.

The Monte Carlo calculations by Mark and coworkers on filled, uncrosslinked polymers have suggested that the radius of gyration of the polymers is a strong function of the filler size and concentration. The Monte Carlo model adopted a rotational isomeric state model for a PDMS chain and assumed no interactions between the filler and the chain. The calculation process was initiated with a chain end located at the center of a sphere with a radius, r_exp, equal to the fully extended chain length (r_exp=nl₀, where n is the number of skeletal bonds and l₀ is the bond length). Filler particles were randomly placed in the interior of the sphere, with the constraint that no filler particles may occupy a volume in the center of the sphere defined by the radius of a single filler particle, r_sph. The filler particles were assumed to be spherical with a uniform size distribution. To obtain reasonable statistics, 30,000–500,000 conformations were generated for a given set of conditions. Any conformations generated for the polymer chain which intersected any spherical particle were rejected in the calculation. A schematic representation of the model calculation is shown in Fig. 1.

Yuan et al. [8], [9] examined PDMS chains with 50 skeletal bonds mixed with filler particles having radii of either 5 or 20 Å, and PDMS chains of 200 skeletal bonds containing filler particles with radii of either 20 or 40 Å. The results for these four cases are plotted in Fig. 2, showing the root-mean-squared end-to-end vector plotted as a function of filler concentration. For the cases where the polymer chains are much larger than the filler particles (n=50, r_sph=5 Å and n=200, r_sph=20 Å), the size of the chains increases with increasing filler concentration. However, for the cases where the polymer dimensions approach the size of the filler particles (n=50, r_sph=20 Å and n=200, r_sph=40 Å), the size of the chains decreases slightly with increasing filler concentration. These results can be understood in terms of the simple excluded volume argument shown schematically in Fig. 3. For the case where the filler particles are smaller than the polymer chain, the chain extends to the extremes of the volume defined by the sphere of radius, nl₀. Since the fillers are randomly placed in the sphere, the fillers near the center of the sphere force the chain to be more extended than if the fillers were not present (Fig. 3a). When the filler particles are nearly as large as the polymer chain (Fig. 3b), the filler particles occupy more of the space toward the periphery of the sphere, and the chain is constrained to the center. Therefore, the chain dimensions are smaller than if the fillers were not present.

Akcasu et al. and Williams et al. provided a framework for extracting single chain structure factors from SANS measurements on multicomponent mixtures by the so-called high concentration method [24], [25]. This method uses fixed compositions of a third component and the total amount of polymer (labeled and unlabeled) and varies the ratio of labeled to unlabeled polymer in the sample. By subtracting the appropriately weighted scattering intensities from samples with different labeling ratios, the single chain structure factor of the polymers can be obtained. This approach has been used extensively on isotopic blends of polymers with matched molecular masses in the presence of a third component.

Based on the formalism of Akcasu et al. and Williams et al., Summerfield and coworkers [26], [27] and King, Ullman, and coworkers [28], [29] derived the following relationship for a three component mixture of labeled and unlabeled polymers of matched molecular mass and solvent:

$$\text{I(q,ρ)=(a}{}_{\mathrm{<mtext>H</mtext>}}\text{−a}{}_{\mathrm{<mtext>D</mtext>}}\text{)}{}^2\text{ρ(1−ρ)S}{}_{\mathrm{<mtext>S</mtext>}}\text{(q)+[a}{}_{\mathrm{<mtext>H</mtext>}}\text{(1−ρ)+a}{}_{\mathrm{<mtext>D</mtext>}}\text{ρ−a′}{}_{\mathrm{<mtext>s</mtext>}}\text{]}{}^2\text{S}{}_{\mathrm{<mtext>T</mtext>}}\text{(q)}$$

where a_H, a_D, and a′_s are the monomer scattering lengths for the protonated polymer, deuterated polymer and solvent, respectively, ρ, is the fraction of total polymer which is deuterated, S_S(q) is the single chain form factor, and S_T(q) is the interchain form factor. Here the solvent molecule is assumed to be much smaller than the other components, hence, the structure factor for the solvent molecule, S_solvent(q)=1. It must be emphasized that Eq. (1) is only valid for pairs of polymers with matched molecular masses. By fixing the total concentration of polymer and varying ρ one obtains a system of linear equations, which can be solved for S_S(q) and S_T(q). For the single chain form factors, S_S(q) may be fit by a non-linear regression fitting routine to the Debye function, g_D(x_i), defined by the following equation:

$$\text{g}{}_{\mathrm{<mtext>D</mtext>}}\text{(x}{}_{\mathrm{i}}\text{)=(2/x}{}_{\mathrm{i}}{}^2\text{)[}\text{exp}\text{(−x}{}_{\mathrm{i}}{}^2\text{)−1+x}{}_{\mathrm{i}}{}^2\text{]}$$

where x_i=q²N_ib²/6, N_i is the degree of polymerization of the polymer, and b is the statistical segment length, yielding R_g of the polymer chains (R_g=N_ib²/6). Alternatively, by plotting 1/S_S(q) against q², a straight line results and R_g may be calculated from the slope and intercept of the line based on the following relation:

$$\text{1/S}{}_{\mathrm{<mtext>S</mtext>}}\text{(q)=}\text{const.}\text{(1+q}{}^2\text{R}{}_{\mathrm{<mtext>g</mtext>}}{}^2\text{/3)}$$

Because of the assumption of matched molecular masses of the polymers, one single chain form factor is obtained which is assumed to apply to both the labeled and unlabeled polymer.

Tangari, Summerfield and coworkers [30], [31], [32] examined the effects of mismatched molecular masses on the high concentration method. However, the presence of a third component was not considered in their treatment. The result is analogous to the results obtained for isotopic blends with matched molecular masses except the single chain form factor is the weighted sum of the individual form factors for the deuterated and protonated polymer chains:

$$\text{S}{}_{\mathrm{<mtext>S</mtext>}}\text{(q)=[ρS}{}_{\mathrm{<mtext>S</mtext>}}{}^{\mathrm{<mtext>H</mtext>}}\text{(q)+(1−ρ)S}{}_{\mathrm{<mtext>S</mtext>}}{}^{\mathrm{<mtext>D</mtext>}}\text{(q)]}$$

where S_S^H(q) is the single chain form factor for the protonated polymer and S_S^D(q) is the single chain form factor for the deuterated polymer. The primary assumption in the approach of Tangari and coworkers is that the system is non-interacting. This assumption helps to eliminate a number of crossterms in the final expression for the total scattering intensity.

In this work, we obtained experimental results on the single chain dimensions of isotopic blends of PDMS with polysilicate fillers by using a treatment similar to Tangari and coworkers. Our principal assumption was that the polysilicate filler may be effectively treated as a solvent molecule. The polysilicate used in this work is smaller than traditional reinforcing fillers and is a liquid which flows at room temperature, but exhibits brittle, glass-like behavior below −70°C. Therefore, the polysilicate is not like most traditional fillers that are rigid, macroscopic particles which come in powder form. However, the polysilicate significantly alters the mechanical behavior of PDMS polymers. The assumption that the polysilicate can be treated as a solvent molecule allows the analysis scheme of Tangari and coworkers to be combined with the results of Summerfield, King, Ullman and coworkers to give the following expression for the scattering intensity:

$$\text{I(q,ρ)=(a}{}_{\mathrm{<mtext>H</mtext>}}\text{−a}{}_{\mathrm{<mtext>D</mtext>}}\text{)}{}^2\text{ρ(1−ρ)[ρS}{}_{\mathrm{<mtext>S</mtext>}}{}^{\mathrm{<mtext>H</mtext>}}\text{(q)+(1−ρ)S}{}_{\mathrm{<mtext>S</mtext>}}{}^{\mathrm{<mtext>D</mtext>}}\text{(q)]+[a}{}_{\mathrm{<mtext>H</mtext>}}\text{(1−ρ)+a}{}_{\mathrm{<mtext>D</mtext>}}\text{ρ−a′}{}_{\mathrm{<mtext>s</mtext>}}\text{]}{}^2\text{S}{}_{\mathrm{<mtext>T</mtext>}}\text{(q)}$$

Eq. (5) will be utilized as the basis for extracting the polymer single chain form factors in this study. The function, S_T(q), which is extracted through this analysis is the form factor which contains the third component. We assume our filler can be treated as a solvent molecule to apply the data reduction scheme of Summerfield et al. This is true as long as the filler particle dimension is sufficiently small to justify the assumption that S_solvent(q)=1. However, it should be noted that the form factor for the third component is contained in S_T(q) regardless of the assumption we make for the physical state of the third component.

For comparison, the radius of gyration values for the polymers were measured in dilute toluene solutions (near theta conditions) and in the bulk isotopic blend without filler. The radii in the bulk isotopic blends were also examined as a function of the ratio of labeled to unlabeled polymer. The results for the isotopic blends were analyzed using the standard two-component random phase approximation (RPA) theory [33]:

$$\text{S(q)=k}{}_{\mathrm{N}}\text{{[φ}{}_{\mathrm{<mtext>A</mtext>}}\text{v}{}_{\mathrm{<mtext>A</mtext>}}\text{N}{}_{\mathrm{<mtext>A</mtext>}}\text{g}{}_{\mathrm{<mtext>D</mtext>}}\text{(x}{}_{\mathrm{<mtext>A</mtext>}}\text{)]}{}^{\mathrm{-1}}\text{+[φ}{}_{\mathrm{<mtext>B</mtext>}}\text{v}{}_{\mathrm{<mtext>B</mtext>}}\text{N}{}_{\mathrm{<mtext>B</mtext>}}\text{g}{}_{\mathrm{<mtext>D</mtext>}}\text{(x}{}_{\mathrm{<mtext>B</mtext>}}\text{)]}{}^{\mathrm{-1}}\text{+(2χ/v}{}_0\text{)}}{}^{\mathrm{-1}}\text{+}\text{Baseline}$$

where N_i is the degree of polymerization index, φ_i the volume fraction, v_i the molar volume of the ith component, v₀ a reference volume and, k_N is the contrast factor given as:

$$\text{k}{}_{\mathrm{N}}\text{=N}{}_0\text{[(a}{}_{\mathrm{<mtext>A</mtext>}}\text{/v}{}_{\mathrm{<mtext>A</mtext>}}\text{)−(a}{}_{\mathrm{<mtext>B</mtext>}}\text{/v}{}_{\mathrm{<mtext>B</mtext>}}\text{)]}{}^2$$

In Eq. (7), N₀, a_i, and v_i are, respectively, Avogadro's number, the scattering length and molar volume of a monomer unit of the ith component. For polydisperse materials, in Eq. (6), the N_i are replaced by 〈N_i〉_n, the number average degree of polymerization and the Debye functions, g_D(x_i) are replaced by the mass average Debye function given by Eq. (8), assuming a Schultz–Zimm distribution for the molecular masses:

$$\text{〈g}{}_{\mathrm{<mtext>D</mtext>}}\text{(x}{}_{\mathrm{i}}\text{)〉}{}_{\mathrm{<mtext>w</mtext>}}\text{=(2/x}{}_{\mathrm{i}}{}^2\text{){x}{}_{\mathrm{i}}\text{−1+[h/(h+x}{}_{\mathrm{i}}\text{)]h}}$$

where, x_i=q²〈N_i〉_nb²/6, h=[〈N_i〉_w/〈Ni〉_n−1]⁻¹ and 〈N_i〉_w is the mass average degree of polymerization. The non-linear regression fitting routine to the RPA equation accounting for the molecular mass distributions of both polymers was performed with b, χ/v₀ and the incoherent baseline as floating parameters. A single value of b is obtained which is an average value for both of the polymers. Based on the values for the degree of polymerization obtained independently by SEC for each polymer, the R_g of each polymer is obtained according to the relation, R_g=(〈N_i〉_wb²/6)^1/2. The radius of gyration values are calculated based on the mass average degree of polymerization.