Springs, entropic

We can take the Rouse term l/ ke 02rm/0m2 (ke = 3kBT//2) entropic spring constant) into consideration formally, if we define the element Tnm of the Oseen tensor as Tnm = E/ . The equation of motion (13) thus becomes... 

The above examples show that we can describe the network modulus of polymer gels by using the concept of entropic springs making up the network. In some cases corrections to the network are required to... 

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

Fig. 3.1 Bead-spring-bead model of a Gaussian chain as assumed in tbe Rouse model. Tbe beads are connected by entropic springs and are subject to a frictional force where v is the bead velocity and fo the bead friction coefficient...

This result states that there is an entropic spring term in the free energy that penalizes deviations of the mean size m from the mean size in the parent distribution. The spring constant is inversely proportional to the variance in the parent distribution thus if the parent is narrower, it is harder to move away from the parental mean size [64]. However, as indicated above, the CLT is not... 

K(R),Kg Entropic spring constant, (7.41), spring constant for a Rouse segment, (7.55). 

Rubber materials are soft, elastic solids, made of mobile, flexible polymer chains (with a glass transition temperature (Tg) typically lower than 0 °C) which are linked together to form a three-dimensional network. They are characterised by a low, frequency independent elastic modulus (of the order 105 to 106 Pa) and usually by a large maximum reversible deformation (up to a few hundred per cent). Rubber elasticity is based on the properties of crosslinked polymer chains at large spatial scales, the presence of crosslinks ensures the reversibility of the deformation, while at short scales, mobile polymer chains behave as molecular, entropic springs. 

Single-Molecule Manipulation of DNA Optical traps and micropipettes are used to hold and to stretch single molecules of DNA, showing how the folded strand behaves as an entropic spring. 

The early molecular theories of rubber elasticity were based on models of networks of long chains in molecules, each acting as an entropic spring. That is, because the configurational entropy of a chain increased as the distance between the atoms decreased, an external force was necessary to prevent its collapse. It was understood that collapse of the network to zero volume in the absence of an externally applied stress was prevented by repulsive excluded volume (EV) interactions. The term nonbonded interactions was applied to those between atom pairs that were not neighboring atoms along a chain and interacting via a covalent bond. 

The resulting physical picture of a rubber-like system as a close-packed collection of mers is radically different from the two-phase image introduced by James and Guth [10]. The latter represents mbber as a network of chains, which act as entropic springs in tension, embedded in a bath of simple liquid. The bath gives rise to an isotropic pressure, whereas the network is responsible for the deviatoric stress. More recent physical pictures consider as well the distribution of network junctions in the liquid and the action of these junctions as constraints on the free motion of a generic chain of the network. The current description is on the mer or atomic level and treats the full stress tensor, both the mean and deviatoric portions, in terms of atomic interactions. 

The concept of a long chain molecule acting as an entropic spring plays a central role in most molecular theories of rubber elasticity. To what extent does this concept remain valid and useful in dense systems of interacting chains This question has been considered by MD simulation in Ref. [12]. 

If we now pull on a chain end to increase the end-to-end distance R, the force we need to exert to overcome the entropic spring force of the chain is... 

The coefficient of proportionality 3kTj Nb ) is the entropic spring constant of an ideal chain. It is easier to stretch polymers with larger numbers of monomers N, larger monomer size b, and at lower temperature T. The fact... 

Thus, the ideal c4iain can be thought of as an entropic spring and obeys Hooke s law for elongations much smaller than the maximum elongation (. R < 7 max = bN). For stronger deformations, the Langevin function [Eq. (2.112)] for freely jointed chains or Eq. (2.119) for worm-like chains can be used to describe the non-linear relation between force and... 

As I have shown, the origin for the finite amorphous fraction formed by the (prevaiUng) loops is due to the balance between the entropic spring force created by the finite separation of the anchoring segments on the one hand side and the effort to remove the loop segments from the thermodynamically preferred crystalline phase on the other side. 

The Alexander-de Gennes approach unrealistically assumes the free ends all to be at the midplane. In order to realize this physically, a fixed force would have to be applied to the chain ends only. In reality, a block is exposed to a force that originates from the thermal motion of all the surrounding blocks. This problem was first analyzed by Semenov who showed that this force originates from a molecular field that compensates the elastic harmonic potential of ideal entropic springs. This field can be calculated self-consistently and is given by... 

Rubber bands are entropic springs. Experiments show that the retractive force / of polymeric elastomers as a function of temperature T and expansion H is approximately given by f(T,C) = aT ( - tn) where a and l ) are constants. 

---