Chain elastic forces

Nonionic surfactants stabilize colloidal systems not by electrostatics but basically by osmotic forces. If two sterically stabilized particles approach each other, the soluble parts of the adsorbed chains causes a higher concentration in the interstitials when compared to the average continuous phase. This will cause a flux of continuous phase into the interstitials, which subsequently leads again to drop separation. As nonionic stabilizers are mainly polymeric in nature (for instance, poly(ethylene glycol) chains), elastic forces may contribute to the stability as well. The elastic force per... 

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

The value should be that of single polymer chain elasticity caused by entropic contribution. At first glance, the force data fluctuated a great deal. However, this fluctuation was due to the thermal noise imposed on the cantilever. A simple estimation told us that the root-mean-square (RMS) noise in the force signal (AF-lS-b pN) for an extension length from 300 to 350 nm was almost comparable with the thermal noise, AF= -21.6 pN. 

Figure 4. Schematic description of the swelling process. The molecules of the swelling liquid start to penetrate inside the polymer framework from its surface (a) and to solvate the polymer chains. The polymer chain start to stretch out and to move away from one another the apparent volume of the polymer increases and the first nanopores are formed (b). Swelling stops when increasing elastic forces set up by the unfolding of the polymer chains counterbalance the forces which drive the molecules of the swelling agent into the polymer framework (c).

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

Key words intertwining chains, SARW statistics, conformation, polymer chain, random walks, lattice, thermodynamics, modules of elasticity, forces, work.. 

In Eq. (III-9) the deformation ratios are defined with respect to a reference state in which the chain dimensions are such that they do not exert any elastic forces on the crosslinks (state of normal coiling). In general, the chains in a network may not actually be in this state at the beginning of a deformation experiment, because the ciosslinking process may quite well exert a, largely unknown, influence on the chain dimensions. 

There are few results in the literature on the evolution of transformation diagrams during chain reactions of thermosets. From a thermodynamic point of view, before the gel point the behavior is similar to the well-studied synthesis of high-impact polystyrene (Bucknall, 1989). But after the gel point, which arrives at low conversions, the contribution of elastic forces to the free energy of mixing has to be added in Eq. (8.1) (De Gennes, 1979). 

A homemade SMFS with a silicon nitride cantilever (Park, Sunnyvale, CA) was used. Each tip was calibrated by using a standard sample. The spring constants of these cantilevers were in the range 0.010-0.012 N/m. By moving the piezo tube, one could bring the sample into contact with the AFM tip so that some polymer chains were physically adsorbed onto the tip, resulting in a number of bridges . As the distance between the tip and the substrate increased, the chains were stretched and the elastic force deflected... 

In other words, it is assumed here that the particles are surrounded by a isotropic viscous (not viscoelastic) liquid, and is a friction coefficient of the particle in viscous liquid. The second term represents the elastic force due to the nearest Brownian particles along the chain, and the third term is the direct short-ranged interaction (excluded volume effects, see Section 1.5) between all the Brownian particles. The last term represents the random thermal force defined through multiple interparticle interactions. The hydrodynamic interaction and intramolecular friction forces (internal viscosity or kinetic stiffness), which arise when the macromolecular coil is deformed (see Sections 2.2 and 2.4), are omitted here. 

The fourth term on the right hand side of (3.4) represents the elastic forces on each Brownian particle due to its neighbours along the chain the forces ensure the integrity of the macromolecule. Note that this term in equation (3.4) can be taken to be identical to the similar term in equation for dynamic of a single macromolecule due to a remarkable phenomenon - screening of intramolecular interactions, which was already discussed in Section 1.6.2. The last term on the right hand side of (3.4) represents a stochastic thermal force. The correlation function of the stochastic forces is connected... 

Elastomers. Elastomers are polymeric materials with irregular structure and weak intermolecular attractive forces. Elastomers are capable of high extension (up to 1000%) under ambient conditions. That is, they have the particular kind of elasticity characteristic of rubber. The elasticity is attributed to the presence of chemical and/or physical crosslinks in these materials. In their normal state, elastomers are amorphous, and as the material is stretched, the random chains are forced to occupy more ordered positions. Releasing the applied force allows the elongated chains to return to a more random state. Thus, the restoring force after elongation is largely because of entropy. (Fig. 14.3)... 

In Fig. 3-5a, the polymer coil is modeled as a series of beads equally spaced along the polymer backbone and connected to each other by springs. The beads account for the viscous forces and the springs the elastic forces in the molecule the portion of the chain represented by a single spring is called a submolecule. The bead-spring model is... 

Figure 3-21 Distribution of bead mass as a function of position downstream of the tether point of a DNA molecule of length L -67.2 tm for various velocities measured in experiments similar to those described in the caption to Fig. 3-1. The lines are the predictions of Monte Carlo molecular simulations using the elastic force from the worm-like chain model, Eq. (3-57), and conformation-dependent drag, as described in the text. The value of the parameter fcoii/ ksT — 4.8 sec(/im) is obtained from the diffusivity measurements of Smith et al. (1995) Crod/ a = 9.1 sec(/Ltm) 2 is obtained from Eq. (3-62) for a fully stretched filament (From Larson et al. 1997, reprinted with permission from the American Physical Society.)...

This section seeks to make a quantitative evaluation of the relation between the elastic force and elongation. The calculation requires determining the total entropy of the elastomer network as a function of strain. The procedure is divided into two stages first, the calculation of the entropy of a single chain, and second, the change in entropy of a network as a function of strain. 

The aim of this section is to find the relation between the elastic force and the deformation for a polymer network. For that purpose the change in entropy associated with deformation of the chains in the network must be evaluated. Figure 3.7 shows the distribution of the chain end-to-end vectors in the deformed (stretched) and undeformed (unstrained) states. The distribution has spherical symmetry in the undeformed state, and when the... 

Let us first consider the phantom chain. The sharper is C(q) (see Figure 2), the larger is the number of neighboring atoms that contribute to the intramolecular force on any given atom. In particular, from Eqn. (2.1.44) we may prove that each atom exerts an elastic force on its kth neighbor with an elastic constant 12 8 cos(qk) sin (q12)/C(q), which decreases slowly with... 

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