Molecular estimates of the moduli of tough, elastic networks formed through end-linking of poly(dimethyl siloxane) oligomers
Complete enumeration of all conformations possible for short poly(dimethyl siloxane) (PDMS) oligomers, with 9 and 10 repeat units, has permitted the calculation of the dependence of conformational probabilities, energies, entropies, and partition functions on the end-to-end lengths r of these oligomers. From this conformational information, the force required to stretch the PDMS oligomer chains are obtained and used to estimate the moduli of tough, elastic networks formed through their end-linking. The three-chain model of polymer networks is adopted, where it is assumed that in a cube of the PDMS oligomer network, [r] on each side, three chains are contained and lie parallel to opposite cube sides. Upon extension of the cube, one chain is extended from [r] to lambda[r], while the other two chains are compressed from [r] to [r]/lambda(1/2) in keeping with the assumption of constant volume, affine deformation of the three-chain network model. The density of the three-chain cube with sides=[r] is nearly the same as the bulk density of PDMS. The forces of extension, f(e) and compression f(c) on the three chains in the deformed cube are estimated from the dependence of the oligomer conformational energies and entropiesor partition functions on r, and are converted to moduli sigma=(f(e)+2f(c))/A(0)(lambda-1/lambda(2)). Moduli in the range of 10(6) to 10(7) N m(-2) are obtained, which compare favorably to the modulus of 2.2x10(6) N m(-2) observed for the network obtained by tetrafunctional end-linking of the PDMS 9-mer. Note that the observed and calculated moduli are ca. 20 times those normally observed in PDMS rubbers, where the chains between crosslinks typically contain 200 monomer units. This is simply the result of small changes produced in the conformational energy and entropy of long polymer chains as they are extended, because only a small fraction of the chain bonds between crosslinks are required to alter their conformations to accommodate their extension, unlike short oligomeric chains which must change a much larger fraction of their bond conformations when stretched. In uniaxial extension experiments the PDMS 9-mer network shows a maximum extension lambda(max)=1.12, while the PDMS 9-mer is potentially able to stretch from r=[r]=14.82 Angstrom to r=r(max)=21.77 Angstrom, or lambda(max)=1.46. Though we have considered the possibility that the tetrafunctional crosslinks impede those PDMS oligomer repeat units near the chain ends from altering their conformations in response to the macroscopic strain, we are currently unable to reconcile this discrepancy with a molecular explanation.