Entropic elasticity of phantom percolation networks

– A new method is used to measure the stress and elastic constants of purely entropic phantom networks, in which a fraction p of neighbors are tethered by inextensible bonds. We find that close to the percolation threshold pc the shear modulus behaves as (p−pc) , where the exponent f ≈ 1.35 in two dimensions, and f ≈ 1.95 in three dimensions, close to the corresponding values of the conductivity exponent in random resistor networks. The components of the stiffness tensor (elastic constants) of the spanning cluster follow a power law ∼ (p− pc), with an exponent g ≈ 2.0 and 2.6 in two and three dimensions, respectively. In the gelation process, monomers or short polymers in a fluid solution are randomly crosslinked. At a certain moment during the reaction, a macroscopically large network, the gel , spans the system. At this point, the system changes from a fluid-like (sol) to a solid-like (gel) phase that has a finite shear modulus. The geometry of gels is frequently described by the percolation model [1]. The percolation geometry is usually defined on a lattice, by randomly occupying a fraction p of the bonds (or sites). The gel point is identified with the percolation threshold pc, the critical bond (site) concentration above which a spanning cluster is formed. Percolation theory predicts that close to pc quantities like the average cluster size or the gel fraction have power laws dependence on (p−pc) with universal exponents, some of which have been measured experimentally for gel systems [2]. Near the sol-gel transition typical polymer clusters are very large, tenuous and floppy. Elastic properties of such systems are primarily determined by the entropy, i.e., distortions of a sample barely modify its energy, but they decrease the available phase space (decrease entropy) and, thus, increase the free energy. Like geometrical quantities near pc, the shear modulus is also expected to follow a power law: μ ∼ (p− pc) . de Gennes [3] used an analogy between gel elasticity and conductivity of random resistor networks (RRN), and conjectured that the exponent f should be equal to the exponent t describing the conductivity Σ of RRN close to pc: Σ ∼ (p− pc). Alternative theories take different approaches and lead to different exponents [4]. An exact calculation of the critical behavior of μ, which takes into account excluded volume (EV) and entanglements effects, is not yet available. Experimental values of