Small-Angle Neutron Scattering from Elastomeric Networks in which the Junctions Alternate Regularly in their Functionality Aris Skliros, James E. Mark, Andrzej Kloczkowski* Introduction The concept of the phantom polymer network in the theory of rubber-like elasticity was first developed over 60 years ago by James and Guth. [1,2] They assumed that polymer chains interact only at junction points (cross-links) and that they may pass freely through one another, that is, they are ‘‘phantoms’’. This phantom approximation neglects inter- (and intra-) chain interactions that lead to entanglements and topological constraints in real polymers. James and Guth also assumed that the distribution of end-to-end vectors of the network chains is Gaussian and is the same before cross-linking and in the cross-linked undeformed network. Additionally, it was assumed that all chains have the same length and that all junctions in the network have the same functionality f (defined as a number of chains connected at each junction or cross-link). They also assumed the network to be composed of two types of junctions: fixed junctions situated on the rubber surface that preserved the volume of the elastomer, and free junctions inside the polymer bulk, fluctuating around their time-averaged mean positions. Additionally, it was assumed that fluctua- tions of junctions are strain independent while all mean (time-averaged) vectors transform affinely with the applied macroscopic strain. The idea of the phantom network is very similar to the concept of the ideal gas in the kinetic theory of gases. Just as the ideal gas is a basis for theories of real gases, the phantom network theory provides a foundation for more advanced theories of real networks. Full Paper A. Skliros Department of Biochemistry, Biophysics and Molecular Biology, and L. H. Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA 50011-0320, USA J. E. Mark Department of Chemistry, University of Cincinnati, Cincinnati, OH 45221-0172, USA A. Kloczkowski Department of Biochemistry, Biophysics, and Molecular Biology and L. H. Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA 50011-0320, USA E-mail: kloczkow@iastate.edu We compute scattering form factors for SANS from labeled paths in Gaussian phantom networks in which junctions alternate regularly in their functionality (the number of chains emanating from a junction). Our calculations are based on the James-Guth model of rubber- like elasticity, which assumes that fluctuations are strain independent, while mean vectors transform affinely with the applied strain. Kratky plots for scattering from isotropic and uniaxially stretched bifunctional networks are computed and compared with corresponding plots for the simpler unifunctional networks. The results show the effects of the length of the labeled path, extent of deformation, direction of scattering with respect to the principal axis of the deformation and the functionalities of the net- work junctions. θ ζ i j Macromol. Theory Simul. 2009, 18, 537–544 ß 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mats.200900041 537