Macromolecular deformation

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

Lindahl, E., Azuara, C., Koehl, P. and Delarue, M. (2006) NOMAD-Ref visuahzation, deformation and refinement of macromolecular structures based on all-atom normal mode analysis. Nucleic Acid Res. 34, W52-56. 

Jancar J (1987) Deformation behavior of filled polypropylene. PhD thesis. Institute of Macromolecular Science, Czech Academy of Sciences, Prague... 

Elastomer A macromolecular (polymeric) material that, at room temperature, is capable of recovering substantially in shape and size after removal of a deforming force. 

Takserman-Krozer.R., Ziabicki.A. General dynamic theory of macromolecular networks. IL Dynamics of network deformation. J. Polymer Sci. Part A-2 8, 321-332... 

The possibility for the existence of mesophase in a rubbery state 36,46), typical only for macromolecular compounds with their natural ability to display big reversible deformations, reveals interesting prospects from the viewpoint of creation of new types of liquid-crystalline materials in the form of elastic films, as well as for development of the theory of viscoelastic behaviour of such unusual elastomers. 

The viscoelastic response of polymer melts, that is, Eq. 3.1-19 or 3.1-20, become nonlinear beyond a level of strain y0, specific to their macromolecular structure and the temperature used. Beyond this strain limit of linear viscoelastic response, if, if, and rj become functions of the applied strain. In other words, although the applied deformations are cyclic, large amplitudes take the macromolecular, coiled, and entangled structure far away from equilibrium. In the linear viscoelastic range, on the other hand, the frequency (and temperature) dependence of if, rf, and rj is indicative of the specific macromolecular structure, responding to only small perturbations away from equilibrium. Thus, these dynamic rheological properties, as well as the commonly used dynamic moduli... 

The dependence of rf, rf, G, and G" on frequency reflects the ability of macromolecular systems to flow like Newtonian fluids if the experimental time allowed them, feXp = 1 /< , is very large compared to the time that they require to fully respond macromolecularly. This temperature-dependent, material-characteristic time is commonly called the relaxation time, X, although it is actually a relaxation spectrum (7). Conversely, when /exp is very short, that is, co is very high compared to X, the macromolecular system can only respond like an elastic solid, able only to undergo deformation and not flow. In... 

This equation describes only the deformation of the macromolecular coil and therefore r11 is a relaxation time of the deformation process. It can be shown (see Appendix F) that the orientation relaxation process is characterised by the relaxation time tx. 

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. 

To find the influence function rj(s), we shall consider shear deformation of the system at velocity gradient 7y, while two macromolecular coils, separated by a distance dj, move beside each other at velocity 7ijdj. We add to the sum the contributions of every coil, apart from the chosen one, and find the density distribution of the energy dissipation for the chosen coil. The proportionality coefficient depends only on the concentration of the Brownian particles, if an assumption is made that local dissipation is determined by relative velocities of macromolecular coils,... 

A macromolecular coil at equilibrium has a spherical form (Section 1.4). Under deformation of the system, the macromolecular coil change its form that is characterised in this case by the tensor of gyration... 

In a deformed system, the average form of the macromolecular coil can be approximated by an ellipsoid. The effective volume of the macromolecular coil depends on the velocity gradients. The expansion of the effective volume as a series in powers of the velocity gradients does not contain the first-order term, so vu =0. This means that, at low velocity gradients, the coil does not change its volume (one says the coil is orientated by flow). At larger velocity gradients, the volume of the coil is increased. 

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