Molecular elastic

Oberhauser et al., 1998] Oberhauser, A. F., Marszalek, P. E., Erickson, H., and Fernandez, J. The molecular elasticity of tenascin, an extracellular matrix protein. Nature. In Press. 

Ea = Arrhenius activation energy Es = excess stress energy AEr = potential barrier for bond rotation Eel = molecular elastic energy F = mean force potential f = average force on the chain fb = bond breaking force H0 = Hookean spring constant kB = Boltzmann constant... 

For the purpose of the following discussions it is helpful to depict the four principal modes of intra- and inter-molecular elastic interaction between neighbouring atoms of a C - C chain (Fig. 1). 

Because nematic liquid-crystalline polymers by definition are both anisotropic and polymeric, they show elastic effects of at least two different kinds. They have director gradient elasticity because they are nematic, and they have molecular elasticity because they are polymeric. As discussed in Section 10.2.2, Frank gradient elastic forces are produced when flow creates inhomogeneities or gradients in the continuum director field. Molecular elasticity, on the other hand, is generated when the flow is strong enough to shift the molecular order parameter S = S2 from its equilibrium value 5 . (Microcrystallites, if present, can produce a third type of elasticity see Section 11.3.6.)... 

Significant shifts in S are not expected to occur in small-molecule nematics unless the shear rate is extraordinarily high. For polymeric nematics, however, molecular relaxation times T are typically 0.001-10 sec, or even higher, and therefore molecular elastic effects are produced at shear rates y r = 0.1-1000 sec L Thus, the order parameter S is significantly distorted away from that of equilibrium when the Deborah number De (discussed in Section 3.6.2.1.1) is of order unity or greater, where... 

When De 1 and molecular elasticity is negligible, the flow properties of polymeric nematics can, in principle, be described by the Leslie-Ericksen equations (see Section 10.2.3). However, at moderate and high De, the Leslie-Ericksen continuum theory fails, and a molecular theory is required to describe the effect of flow on the distribution of molecular orientations. 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

H.E. Gaub and J.M. Fernandez, The Molecular Elasticity of Individual Proteins Studied by AFM-related Techniques. AvH Magazin, 71, 11-18,1998. 

Note that molecular elasticity does not refer to energy actually stored in the molecules i.e. in bond distortions or non-equilibrium conformations. 

The second source. ..is the elasticity stored in the texture, i.e. the micron-scale orientational inhomogeneities created or amplified by the previous shearing flow. Like molecular elasticity, the influence of texture elasticity can be quantified by a dimensionless number, in this case an Ericksen number. 

Based on the observation that molecular elasticity relaxes quickly, in hundredths of a second, whereas band formation and evolution occurs in times of the order of minutes, it seems clear that texture elasticity is the direct energy source. 

However Gleeson e.a. [60] note that molecular elasticity may play an indirect role, since below a threshold value of the Deborah number, bands do not form at all. For example, for a solution of PBG of molecular weight 186000 this critical shear rate was about 40 s , while for a solution of PBG of molecular weight of 86000 bands were formed at a shear rate of 0.25 s . For four solutions of widely varying viscosity and relaxation time, a critical value of De 0.01—0.04 was obtained. Thus molecular elastic effects during shear seem to be necessary to produce the texture that in turn drives band formation after shearing ceases. ... 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

Macro- Rubber molecular elasticity, science solvent swelling... 

New mathematical techniques [22] revealed the structure of the theory and were helpful in several derivations to present the theory in a simple form. The assumption of small transient (elastic) strains and transient relative rotations, employed in the theory, seems to be appropriate for most LCPs, which usually display a small macromolecular flexibility. This assumption has been used in Ref [23] to simplify the theory to symmetric type of anisotropic fluid mechanical constitutive equations for describing the molecular elasticity effects in flows of LCPs. Along with viscoelastic and nematic kinematics, the theory nontrivially combines the de Gennes general form of weakly elastic thermodynamic potential and LEP dissipative type of constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank elasticity in liquid crystalline polymers. It should be mentioned that this theory is suitable only for monodomain molecular nematics. Nevertheless, effects of Frank (orientation) elasticity could also be included in the viscoelastic nematody-namic theory to describe the multidomain effects in flows of LCPs near equilibrium. 

Here the summation convention of repeated indices is used. The Sj are the components of the unit vector in the principal axis directions and co the solid angle. Into this model only one molecular elastic component enters the axial chain modulus. Any interactions by shear or in a direction perpendicular to the chain axes cannot be accounted for. It should only be applied, therefore, if these interactions can be meaningfully neglected. 

Molecular elasticity can be divided into entropic and enthalpic contributions. At low forces, the molecule s extension reduces... 

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