Intramolecular Friction

If deformation of the system is fast enough (that is, before relaxation of chains can occurs), one expects that macromolecules deform affinely, i.e., for every particle rf = where Vij is the velocity gradient, and r is the position 

This quantity has value of zero for non-entangled systems and increase with increase in the length of macromolecules. As for external force, there is a slight difference in resistance, when the particle moves along the chain or in a perpendicular direction, but, in this subsection, this effect is neglected for simplicity. 

The force of intramolecular resistance appears, when relative motion of the particles exists, so that one can write a general expression (which is identical to expression (2.21) for a chain in a dilute solution) 

One can rewrite the matrix of internal resistance in the following form 

This expression defines the general form of a matrix of internal friction, which allows the force to remain unchanged on the rotation of the macromolecular coil as a whole. The written matrix is symmetrical with respect to the upper and lower indices and, in contrast to matrix Cap, has non-zero diagonal components, which are depicted by the first term in (3.27). In equilibrium situations, after averaging over the orientation, matrix (3.27) can be presented as 

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From the weak dependence of ef on the surrounding medium viscosity, it was proposed that the activation energy for bond scission proceeds from the intramolecular friction between polymer segments rather than from the polymer-solvent interactions. Instead of the bulk viscosity, the rate of chain scission is now related to the internal viscosity of the molecular coil which is strain rate dependent and could reach a much higher value than r s during a fast transient deformation (Eqs. 17 and 18). This representation is similar to the large loops internal viscosity model proposed by de Gennes [38]. It fails, however, to predict the independence of the scission yield on solvent quality (if this proves to be correct). 

On the deformation of the macromolecule, i.e. when the particles constituting the chain are involved in relative motion, an additional dissipation of energy takes place and intramolecular friction forces appear. In the simplest case of a chain with two particles (a dumbbell), the force associated with the internal viscosity depends on the relative velocity of the ends of the dumbbell u1 — u° and is proportional, according to Kuhn and Kuhn (1945) to... 

The characteristics monotonically increasing function of the number of the mode a. This dependence can be fitted by... 

The significance and importance of the internal viscosity can be elucidated by comparing the consequences of the theory with experimental data, which will be discussed further on. However, here one should note that the phenomenological characteristics of the intramolecular friction prove to depend not only on the characteristics of the macromolecule, as might have been expected, but also on the properties of the liquid in which the macromolecule is present (Schrag 1991). 

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. 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

The initial characteristic viscosity defined by equation (6.23) is seen to be independent of the characteristics of intramolecular friction, but this is a consequence of the simplifying assumptions. It has been shown for a dumbbell (Altukhov 1986) that, when account of the internal viscosity and the anisotropy of the hydrodynamic interaction is taken simultaneously, the characteristics of these quantities enter into the expression for a viscosity of type (6.23). This result must be revealed also by the subchain model when account is taken of the anisotropy of the hydrodynamic interaction. 

The characteristic viscosity (6.22) is of special interest in the study of the influence of intramolecular friction on the dynamics of a macromolecule in a viscous liquid. At w —> oo, characteristic viscosity can be written as... 

While the model employed in the present work provides a reasonable picture of a unimolecular reaction involving a large molecule in solution, other ingredients not considered here may play a role in some systems. The possible role played by intramolecular friction (nonlinear coupling between the reaction coordinate and other nonreactive modes near the barrier) has been discussed in Section IV. Also, the dependence of the molecular potential surface, in particular the activation barrier on the molecule-solvent interaction, may dominate in some cases the observed solvent effect on the rate. Such may be the case (see Section VIII) in a polar solvent when the reaction involves a change in the molecular dipole moment (such as a charge transfer reaction). 

The weaker dependence is completely unexpected and contradicts the common view that hydrodynamic friction force drives CST and chain fracture. Nguyen and Kausch [108] ascribed (10) to the intramolecular friction (so-called internal viscosity ) between chain segments in the coiled part in FTF. Unlike the internal viscosity depends on strain rate and could reach much higher values than... 

In MMA and MA no such hydrogen bonds are operative. The distinct difference in the pre-exponential for bulk polymerization of these two monomers (see entries 4 and 7 in Table 1), however also originates from effects on internal rotational mobihty. The lower value of A(fep MMA) is due to enhanced intramolecular friction induced by the a-methyl groups on the polymer backbone. 

In the sections below a brief overview of static solvent influences is given in A3.6.2, while in A3.6.3 the focus is on the effect of transport phenomena on reaction rates, i.e. diflfiision control and the influence of friction on intramolecular motion. In A3.6.4 some special topics are addressed that involve the superposition of static and transport contributions as well as some aspects of dynamic solvent effects that seem relevant to understanding the solvent influence on reaction rate coefficients observed in homologous solvent series and compressed solution. More comprehensive accounts of dynamics of condensed-phase reactions can be found in chapter A3.8. chapter A3.13. chapter B3.3. chapter C3.1. chapter C3.2 and chapter C3.5. 

Straub J E, Borkovec M and Berne B J 1987 On the calculation of dynamical friction on intramolecular degrees of freedom J. Phys. Chem. 91 4995... 

In the present chapter we shall be concerned with quantitative treatment of the swelling action of the solvent on the polymer molecule in infinitely dilute solution, and in particular with the factor a by which the linear dimensions of the molecule are altered as a consequence thereof. The frictional characteristics of polymer molecules in dilute solution, as manifested in solution viscosities, sedimentation velocities, and diffusion rates, depend directly on the size of the molecular domain. Hence these properties are intimately related to the molecular configuration, including the factor a. It is for this reason that treatment of intramolecular thermodynamic interaction has been reserved for the present chapter, where it may be presented in conjunction with the discussion of intrinsic viscosity and related subjects. 

Further examination of the Williams approach seems called for, both to improve the method for estimating parameters such as the relaxation time, and to clarify the relationship between the intramolecular potential form and non-thermodynamic frictional forces. The method might provide a fairly unified description of non-linear flow porperties if a suitable potential function for large scale molecular friction were found. Aside from the Williams work, there have been no theoretical studies dealing with t] vs. y at low to moderate concentrations. The systematic changes in the master curve /(/ ) with coil overlap c[ij] are thus without explanation at the present time. 

With a finite value of necessarily some intramolecular hydrodynamic interaction or shielding must occur. The importance of eq. (3.53) lies at the present time, in the fact that it can be adapted for concentrated, solvent free systems like polymer melts. As Bueche (13) pointed out, in these systems every chain molecule is surrounded by chain molecules of the same sort. As all these molecules are necessarily equivalent, one cannot speak of a hydrodynamic shielding effect. This would imply that certain chains are permanently immobilized within the coils of other chains. The contrary is expected, viz. that the centre of gravity of each chain wiH independently foHow, in the average, the affine deformation of the medium as a continuum. From this reasoning Bueche deduces that the free-draining case should be applicable to polymer melts. Eq. (3.53) is then used (after omission of rj0) for the evaluation of an apparent friction factor . After introduction of this apparent friction factor into eq. (3.50), the set of relaxation times reads ... 

Eisenthal s reports that the solvent dependence of the kinetics of the LE/CT interconversion of DMABN should be attributed predominantly to polarity-induced barrier height changes rather than viscosity (friction) changes, has received a lot of recent attention. Su and Simon have explored the role of intramolecular vibrational fluctuations in the LE/CT kinetics of DMABN [80]. This mechanism is outlined in the following subsection, where the same mechanism has been invoked to understand the photodynamics of bis-(N,N-dimethylaminophenyl)-sulfone. Research on the DMABN class of molecules continues to the very active. It will be interesting to see how the various mechanisms and apparently contradictory arguments are reconciled in the future. 

The reaction of trimethylene biradical was successfully treated by means of dynamics simulations by two groups with different PESs as described above.11 15 The success led one of the groups to extend the study to analyze the collisional and frictional effects in the trimethylene decomposition in an argon bath.16 A mixed QM/MM direct dynamics trajectory method was used with argon as buffer medium. Trimethylene intramolecular potential was treated by AM1-SRP fitted to CASSCF as before, and intermolecular forces were determined from Lennard-Jones 12-6 potential energy functions. 

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