Internal viscosity force

The use of internal viscosity forces permit us to take into account kinetic effects associated with deformation rates which were beyond the scope of most polymer... 

Ohnesorge Number Oh = pL/(pLoD0f5 Compare internal viscosity force to surface tension force Walzel [398]... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

The equation clearly does not satisfy the requirement that the internal viscosity force disappears when the coil is rotated as a whole. By ensuring linearisation of the internal friction force according to Cerf s procedure, equation (2.22) may be modified and written thus... 

The internal viscosity force is defined phenomenologically by equations (2.26) formulated above. Various internal-friction mechanisms, discussed in a number of studies (Adelman and Freed 1977 Dasbach et al. 1992 Gennes 1977 Kuhn and Kuhn 1945 Maclnnes 1977a, 1977b Peterlin 1972 Rabin and Ottinger 1990) are possible. Investigation of various models should lead to the determination of matrices Ca/3 and Ga and the dependence of the internal friction coefficients on the chain length and on the parameters of the macromolecule. 

The internal viscosity of the macromolecule is a consequence of the intramolecular relaxation processes occurring on the deformation of the macromolecule at a finite rate. The very introduction of the internal viscosity is possible only insofar as the deformation times are large, compared with the relaxation times of the intramolecular processes. If the deformation frequencies are of the same order of magnitude as the reciprocal of the relaxation time, these relaxation processes must be taken explicitly into account and the internal viscosity force have to be written, instead of (2.26) as... 

To find the stress tensor, one can use equation (6.7), in which the elastic and internal viscosity forces, according to equations (2.2) and (2.25), have the form... 

They asserted that the force is due to the non-smooth nature of the joint in the molecule or explicitly a finite rate of jumps of the rotation of each chemical bond around the adjacent bond. The coefficient of the internal viscosity (force divided by relative velocity) was derived to be inversely proportional to the molecular weight of the polymer (121). 

In this form the expression for the stresses is valid for any dynamics of the chain. For the macromolecules in an entangled system, the elastic and internal viscosity forces according to equations (26) and (40) - (42) have the form... 

When fluid flow in the reservoir is considered, it is necessary to estimate the viscosity of the fluid, since viscosity represents an internal resistance force to flow given a pressure drop across the fluid. Unlike liquids, when the temperature and pressure of a gas is increased the viscosity increases as the molecules move closer together and collide more frequently. 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

It may be surprising that the effect of the nearest-neighbor bond correlations on the one-dimensional chain depends on the sign of P i.e., that the spectrum broadens when extended conformations are favored and narrows when compact conformations are favored. No simple qualitative explanation of this result has occurred to us. The usual internal viscosity always produces a narrowing of the spectrum. This effect is easily introduced into a one-dimensional Rouse model an internal viscous force is... 

Other modifications to the elastic dumbbell have been considered, such as the concept of internal viscosity, where an additional spring force proportional to the rate of... 

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... 

Expression (2.22) for an internal friction force is non-linear with respect to the co-ordinates. To avoid the non-linearity, some simpler forms for internal friction force were used (Cerf 1958). One can introduce a preliminary-averaged matrix of internal viscosity... 

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. 

We can notice that, apart from the deformation of the coil, the stresses (6.7) are determined by the forces of internal viscosity which satisfy equation (7.3) or, in normal form, it is the last equation from set (7.4). It is convenient to consider quantities... 

In the simplest case, at N = 1, the considered subchain model of a macromolecule reduces to the dumbbell model consisting of two Brownian particles connected with an elastic force. It can be called relaxator as well. The re-laxator is the simplest model of a macromolecule. Moreover, the dynamics of a macromolecule in normal co-ordinates is equivalent to the dynamics of a set of independent relaxators with various coefficients of elasticity and internal viscosity. In this way, one can consider a dilute solution of polymer as a suspension of independent relaxators which can be considered here to be identical for simplicity. The latter model is especially convenient for the qualitative analysis of the effects in polymer solutions under motion. 

When the elastic force and the force of internal viscosity are defined, at N = 1, the expression for the stress tensor directly follows relation (6.7)... 

In a laminar flow or in an external potential field a polymer molecule is subjected to forces that can both make it rotate as a whole and cause a relative shift of its parts leading to a deformation, i.e. changing its conformation. Which of these two mechanisms of motion predominates depends on the ratio of times required for the deformation and rotation of the molecule. If the time of the rotation of the molecule as a whole, tq, is shorter than the time required for its deformation, tkinetically rigid. In the opposite case, when tq > r<, the deformation mechanism of motion will predominate and the molecule will be kinetically flexible. To characterize quantitatively the kinetic rigidity of chain molecules Kuhn has introduced the concept of internal viscosity - a quantity describing the resistance of the molecule to a rapid charge in its shape. Later, the theory of internal viscosity has been developed by Cerf ... 

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). 

Note that the turbulent viscosity parameter has an empirical origin. In connection with a qualitative analysis of pressure drop measurements Boussinesq [19] introduced some apparent internal friction forces, which were assumed to be proportional to the strain rate ([20], p 8), to fit the data. To explain these observations Boussinesq proceeded to derive the same basic equations of motion as had others before him, but he specifically considered the molecular viscosity coefficient to be a function of the state of flow and not only on the system properties [135]. It follows that the turbulent viscosity concept (frequently referred to as the Boussinesq hypothesis in the CFD literature) represents an empirical first attempt to account for turbulence effects by increasing the viscosity coefficient in an empirical manner fitting experimental data. Moreover, at the time Boussinesq [19] [20] was apparently not aware of the Reynolds averaging procedure that was published 18 years after the first report by Boussinesq [19] on the apparent viscosity parameter. 

The Zimm model (Zimm, 1956) extends the spring model by considering intermolecular forces such as hydrodynamic forces (perturbations of the velocity field near beads by other beads), reduced excluded-volume effects (coil expansion and reduced contacts), non-linear spring forces (finitely extendable springs) and internal viscosities (coil sluggishness). One can obtain the following expression for the viscosity from the Zimm model ... 

The three theories summarized in the foregoing have one point in common the mathematics are very involved, whilst the physical assumptions are very crude and necessitate the introduction of undefined and arbitrary friction coefficients and diffusion constants. Under these circumstances one may question the usefulness of refining the theories by considering internal viscosities, friction forces, and other vague parameters. 

Internal Viscosity and Cerf-Peterlin Theory. The concept of the internal viscosity was first employed by Kuhn and Kuhn 120) in an attempt to describe the shear-rate dependent viscosity with a dumbbell model or a bead-spring model with N = 1. They assumed that a force proportional to the relative velocity of the beads is exerted on the bead from the connector (spring) in addition to the spring force which is proportional to the relative position of the beads. This force intrinsic to the polymer molecule is compared with the frictional force from the viscous medium and is associated with the term internal viscosity . 

This concept was applied to the bead-spring model by Cerf (122,123) who tried to describe the non-Newtonian viscosity and the flow birefringence in terms of this model. His idea is essentially to add a force Ff due to the internal viscosity to the left hand side of Eq. (2.1) ... 

---