Elasticity near the vulcanization transition

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 58, Issue 1, 1998, Pages

  Castillo, Horacio E ^a   Goldbart, Paul M ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002728349     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.58.R24     Document Type: Article

References (44)

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34. R. C. Ball, Ph.D. thesis, Cambridge University, 1980 (unpublished).
35. In the vulcanization of polymer melts, the width of the critical region, in which mean-field theory fails, vanishes in the limit of long macromolecules for d>3 2.
36. The complete expression for the free-energy functional for any cross-link density is given in Eq. (5.12b) of Ref. 17.
37. This form for Fn(Ωk̂) can be obtained either from a semimicroscopic model, as envisioned here, or via an argument involving symmetries and the continuity of the transition in the context of a Landau theory 27.
38. This condition reflects the fact that no crystalline order (or any other kind of macroscopic inhomogeneity) appears in the vicinity of the amorphous solidification transition. Consequently, Ωk̂=0 for k̂ of the form 0, … ,0,q,0,...,0.
39. The construction of the free energy for the undeformed case is presented in detail in Ref. 17, Secs. 4 and 5. The only step affected by the displacement of the boundaries is the expansion of the constraints in real space as a sum in replicated Fourier space [c.f. Eq. (5.1b) in Ref. 17]: Πα=0nδ(d)(cα)=∑p̂∈Ru eip̂⋅ĉ/Vn+1→Πα=0nδ(d)(cα)=∑p̂∈Rs eip̂⋅ĉ/Vn+1. The only consequence of this change is the replacement of ∑p̂∈Ru by ∑p̂∈Rs in the final expression for Fns(Ωk̂s).
40. The shear modulus is determined by an expansion of the free energy to quadratic order in the deformation. Thus, only terms to linear order are needed in Ws.
41. To see this, one introduces g(θ)≡θ2e2/θϖ(θ), studies the resulting equation for g(θ), and analyzes the limit θ→∞.
42. It is unclear why the ad hoc strategy for computing the shear modulus used, e.g., by Huthmann [Phys. Rev. E 54, 3943 (1996)] gives a too large value for t.
43. W. Peng, H. E. Castillo, P. M. Goldbart, and A. Zippelius, Phys. Rev. B 57, 839 (1998)
44. W. PengH. E. CastilloP. M. GoldbartA. Zippeliuse-print cond-mat/9709250 (unpublished).