Intramolecular elastic forces

The evolution of the general Fourier BK>de, x ip,t) is therefore determined by the intramolecular elastic forces and a stochastic contribution, due to other chains or solvent, whose statistical properties must be specified Adopting the traditional Langevin specification of the stochastic force we will say that i (p t) is Gaussian of zero mean, white and stationary... 

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... 

Another AFM-based technique is chemical force microscopy (CFM) (Friedsam et al. 2004 Noy et al. 2003 Ortiz and Hadziioaimou 1999), where the AFM tip is functionalized with specific chemicals of interest, such as proteins or other food biopolymers, and can be used to probe the intermolecular interactions between food components. CFM combines chemical discrimination with the high spatial resolution of AFM by exploiting the forces between chemically derivatized AFM tips and the surface. The key interactions involved in food components include fundamental interactions such as van der Waals force, hydrogen bonding, electrostatic force, and elastic force arising from conformation entropy, and so on. (Dther interactions such as chemical bonding, depletion potential, capillary force, hydration force, hydrophobic/ hydrophobic force and osmotic pressure will also participate to affect the physical properties and phase behaviors of multicomponent food systems. Direct measurements of these inter- and intramolecular forces are of great interest because such forces dominate the behavior of different food systems. 

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... 

By comparison of the dynamic viscosity expressions without and with internal viscosity [see Eqs. (3.1.15) and (3.3.16), respectively], we see that in the former case the sum tends to zero for large co, unlike in the latter we conclude that in the presence of internal viscosity the dynamic viscosity deviates from what is commonly regarded as a general law. The reason lies in the fact that with internal viscosity the intramolecular tension contains a contribution depending on x h,t), unlike the other models where it depends on the elastic force only, that is, on x h,t). 

Kawabata44 has panted out on the basis of a simple network model that of the two derivatives, bW/blt and bW/bI2, the former should be related primarily to intramolecular forces such as the entropy force which plays a major role in the kinetic theory of rubber elasticity, while the latter should be a manifestation of intramolecular interactions. He predicted the possibility that bW/bI2 assumes negative values in the region of small defamation. In fact, the prediction was confirmed experimentally by Becker and also by the present authos. 

To understand the mechanical properties of materials, it is important to consider first the microscopic origin of stress. In the usual gas or liquid of small molecules, the stress comes from the momentum transfer due to the intermolecular collision. In polymeric liquids, the stress is mainly due to the intramolecular force, and is directly related to the orientation of the bond vectors of the polymer. This idea, originated from the theory of rubber elasticity, is fundamental in the physics of polymeric materials. ... 

The energetic and entropic contributions to the force in the intramolecular process of stretching the chains can be obtained in experiments where there is no other energetic contribution resulting from changes in (intermolecular) van der Waals forces. Therefore, these experiments must be performed at constant volume. A basic postulate of the elasticity of amorphous pol5uner networks is that the stress exhibited by a strained polymer network is assumed to be entirely intramolecular in origin. That is, intermolecular interactions play no role in deformations at constant volume and composition. 

In another molecular approach Argon (28) has proposed a theory of yielding for glassy polymers based on the concept that deformation at molecular level consists in the formation of a pair of molecular kinks. The resistance to double kink formation is considered to arise from the elastic interactions between chain molecule and its neighbors, ie, from intermolecular forces. This is in contrast to the Robertson theory, where intramolecular forces are of primary consideration. We need to recall that the intramolecular forces are by several orders of magnitude stronger than the intermolecular ones—except for entanglements which operate as if they were primary chemical bonds. 

It was mentioned earlier that rubber elasticity is related to the high levels of entropy present in the undeformed state. Deformation, therefore, involves application of energy, some of which is necessary to overcome inter- and intramolecular forces, the remainder is stored, but is released on recovery from the deformation. The recovery stress-strain curve, therefore, never coincides with the loading curve, with the loss of energy being known as hysteresis. 

Understanding the intermolecular forces responsible for adhesion and cohesion is quite important. The elastic constants, the plastic deformation, the presence of flaws are functions of inter- and intramolecular forces and steric hindrances to rotation in the various molecules. It is important to relate the magnitude of intermolecular forces between molecular species and a particular substrate to the relative adhesive strengths. 

The Helmholtz Free Energy includes both intermolecular forces and intramolecular forces and also entropic contributions. The intramolecular contributions are the same as those required for the single-chain calculation, but there is more difficulty in producing force fields that include intermolecular contributions. The intermolecular contributions are typically Lennard-Jones type interactions and to obtain plausible values that are satisfactory for a range of different chemical compositions is often debatable. It is, however, possible to obtain some confirmation of their validity in a particular instance by verifying that the calculations predict the correct crystal structure and this must be regarded as a the first step to calculating the elastic constants. 

It can be seen that a rather wide range of predicted values is obtained that is partly due to choice of different force constants. The results are also sensitive to the details of the assumed crystal unit cell structure, especially the angle made by the plane of the planar zigzag polyethylene chain with the b-axis of the orthorhombic unit cell. The overall pattern of elastic anisotropy is however clear. The stiffness in the chain axis direction C33 is by far the greatest value, and the shear stiffnesses C44, C55 and Cee are the lowest values. This reflects the major differences between the intramolecular bond stretching and valence bond bending forces and the intermolecular dispersion forces, which determine the shear stiffnesses. The lateral stiffnesses also relate primarily to dispersion forces and are correspondingly low. 

The diffusion of solvating media results in addition of diffused-in media molecules to polymer macromolecules due to intramolecular forces (in polar polymers such as PVC due to dipole-dipole interaction). The macromolecules are then virtually encased by media molecules. This increases the spacing between molecules, thus leading to reduced bond forces and increased macromolecular mobility. In turn, this allows even more molecules to penetrate into the intermediate spaces, resulting in a reduction in strength and an increase in elasticity the plastic becomes soft and under the effect of solvents decomposes completely [243]. 

For the above, detailed knowledge of chain architecture is not necessary. It is well known, however, that the modulus or of each polymer are strongly dependent on the detailed structure, actual and virtual bond lengths, rotational and valence bond angles, and the conformational isomeric states of the individual chains or segments. In a crystalline polymer, the highest elastic modulus is one parallel to the chain axis. The lowest modulus is in the plane transverse to the chain axis. In this plane the interactions are exclusively intermolecular in character and contain no intramolecular, covalent bonds. The intermolecular interactions may be common van der Waals dispersive forces, the somewhat... 

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