Mechanical and swelling behaviour of well characterized polybutadiene networks Gregory B. McKenna Polymers Division, National Bureau of Standards, Gaithersburg, MD 20899, USA and Jeffrey A. Hinkley NASA Langley Research Center, Hampton, VA 23665, USA This paper is dedicated to the late Prof. Paul J. Flow who made lasting contributions to polymer physics (Received 22 October 1985) Endlinking of hydroxyl-terminated polybutadiene with the appropriate isocyanate has been used to prepare well characterized networks. Two networks have been studied with molecular weights of the prepolymers being 6100 and 2400 g/mole by g.p.c. Cylindrical specimens were prepared and the derivatives of the stored energy function with respect to the stretch invariants were determined by torque and normal force measurements in torsion. From these data the Valanis-Landel stored energy function derivatives w'(2) were determined for both networks. The stored energy function for the junction constraint model of Flory, which is a special form of the Valanis-Landel function, has been fitted to that determined from the experiments. The contributions, AAphand AAc,to the stored energy function from the phantom network and from the junction constraints respectively do not agree with predictions from the topologies of the networks. In spite of this the form of w'(2) for the junction constraint model gives an excellent 'curve fit' to the data. Comparison is also made with equilibrium swelling. (Keywords: junction constraint model; networks; polybutadiene; rubber elasticity; stored energy function; swelling) INTRODUCTION There has been considerable renewed interest in rubber elasticity theory recently, due to the success of the junction constraint model of Fiery t-6. In this paper we carry out mechanical and swelling measurements on well characterized polybutadiene networks and compare the results with the junction constraint model in a new way. To do this we use the concept from continuum mechanics of the Valanis-Landel 7 (V-L) strain energy density function, i.e. a function in the principal stretches which can be represented as a separable sum of one function evaluated at each of the three principal stretches. We further derive the V-L function for the junction constraint model of Flory, and compare it with the experimental data. There are two salient results from our work: (1) while the Fiery model fits the data extremely well, the parameters related to the network topology are not in particularly good agreement with those calculated from the chemistry involved in making the networks and (2) the assumption that the elastic free energy of deformation is equal to the free energy of mixing at swelling equilibrium can be supported only if one includes a non-zero logarithmic term in the Valanis-Landel strain energy function. In the following sections we describe how one determines the derivatives of the stored energy density function with respect to the stretch invariants for the rubber networks and how the Valanis-Landel function 0032-3861/86/091368503.00 ,~ 1986 Butterworth& Co. (Publishers) Ltd. 1368 POLYMER,1986, Vol 27, September relates to the invariant form of the strain energy density function. We also derive the V-L function relevant to the Flory junction constraint model. Then we describe our experimental procedures and compare our results with the junction constraint model. THEORETICAL CONSIDERATIONS Torsion of a cylinder In this section we are concerned with determining the strain energy density function derivatives DW/dI1 = WI and 0W/012 = W2 for an incompressible elastic material from experiments in which a cylinder is subjected to a twist while the length is held constant. Here 2 2 2 I~=2t+22+2 ~ and I2=1/2~+i/2~+1/2 ~ are the invariants of the deformation tensor, the 2's are the principal stretches and W(I~,I2) is the strain energy density function for the material. The deformation geometry is given in Figure 1. Then the torque, T, and the normal thrust, N, applied to the ends of the cylinder are given by 8: R t~ 47t~b~ (W1 + W2)r3dr (1) T= o/ 0 g = -2n~k2 i (W l + 2W2)r3dr (2) N ~d 0