Viscoelasticity molecular basis

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

In this section we are going to examine such viscoelastic properties in some detail and we will start by examining in turn three important mechanical methods of measurement creep, stress relaxation, and dynamic mechanical analysis. This will lead us to interesting things like time-temperature equivalence and a discussion of the molecular basis of what we have referred to as relaxation behavior. 

Intra-articular hyaluronan (lA-HA) injections are widely used to treat osteoarthritis (OA). This procedure is often referred to as viscosupplementation (22) because it involves the replacement of pathologic synovial fluid with viscoelastic hyaluro-nan-based solutions or gels. In the United States, lA-HA is specifically labeled as an intra-articular analgesic and is indicated to treat pain associated with knee OA when conservative measures and simple analgesics fail (e.g., acetaminophen). In other parts of the world, lA-HA is also approved for treatment of joints other than the knee and in some countries for arthritic conditions other than OA. The molecular basis for this application of hyaluronan, and the history of its development, has been recently reviewed (23). In this chapter, we will update the clinical evidence and describe the different types of hyaluronan formulations available in the United States. 

For further study of DMA see, for example Ferry JD (1980) Viscoelastic Properties in Polymers, 3 edition. J. Wiley, New York Ward IM (1983) Mechanical Properties of Solid Polymers, 2 edn. Wdey, New York Meier DJ (1978) Molecular Basis of Transitions and Relaxations. Gordon and Breach, New York McCrum NG, Read BE, Williams G (1967) Anelastic and Dielectric Effects in Polymeric Solids. Wiley, New York Aklonis JJ, MacKnightWJ(1967) Introduction to Polymer Viscoelasticity. Wiley, New York Matsuoka S (1992) Relaxation Phenomena in Polymers. Hanser Publ, Munich. 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

Since each of the preceding functions can be calculated from any other, it is an arbitrary matter which is chosen to depict the behavior of a system and to correlate with theoretical formulations on a molecular basis. In fact, two other derived functions are sometimes used for the latter purpose—the relaxation and retardation spectra, H and L, which will be defined in Chapter 3. Actually, different aspects of the viscoelastic behavior, and the molecular phenomena which underlie them, have different degrees of prominence in the various functions enumerated above, so it is worthwhile to examine the form of several of these functions even when all are calculated from the same experimental data. A qualitative survey of their appearance will be presented in Chapter 2. 

It turns out that the effects of static confining pressure on the viscoelastic properties can be described by reduced variables and interpreted on a molecular basis in a similar manner. 

Behavior in simple extension is, of course, a relatively elementary aspect of nonlinear viscoelasticity. Other types of deformation, and combinations of deformations, can provide additional information which must be described by appropriate constitutive equations and eventually interpreted on a molecular basis. Some investigations of time-dependent properties in pure-shear - and biaxial extension of thin, flat specimens - of soft rubbery polymers have been reported. 

For the simulation of more complex flows, one needs a constitutive equation or a rheological equation of state. Nearly all of the many equations that have been proposed over the past fifty years are basically empirical in nature, and only in the last twenty-five years have such models been developed on the basis of mean field molecular theories, e.g., tube models. Although the early models were often developed with a molecular viewpoint in mind, it is best to think of them as continuum models or semi-empirical models. The relaxation mechanisms invoked were crude, involving concepts such as network rupture or anisotropic friction without the molecular detail required to predict a priori the dependence of viscoelastic behavior on molecular structure. While these lack a firm molecular basis and thus do not have universal validity or predictive capability, they have been useful in the interpretation of experimental data. In more recent times, constitutive equations have been derived from mean field models of molecular behavior, and these are described in Chapter 11. We describe in this section a few constitutive equations that have proven useful in one or another way. More complete treatments of this subject are given by Larson [7] and by Bird et al. [8]. 

There is considerable evidence that all the hysteresis effects observed in these materials and most of the viscoelastic behavior can be caused by the time dependent failure of the polymer on a molecular basis and are not due to internal viscosity [1,2]. At near equilibrium rates and small strains filled polymers exhibit the same type of hysteresis that many lowly filled, highly cross-linked rubbers demonstrate at large strains [1-8]. This phenomenon is called the "Mullins Effect" and has been attributed to micro-structural failure. Mullins postulated that a breakdown of particle-particle association and possibly also particle-polymer breakdown could account for the effect [3-5]. Later Bueche [7,8] proposed a molecular model for the Mullins Effect based on the assumption that the centers of the filler particles are displaced in an affine manner during deformation of the composite. Such deformations would cause a highly non-uniform strain and stress gradient in the polymer... 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

The space-time resolution of these techniques reveals the molecular motions leading to the viscoelastic and mechanical properties of polymeric systems. This knowledge is of great importance for scientific reasons and is also a basis for the design of tailor-made materials. 

We expect that the modification creates the free volume (Vf) in wood substance from the similarity of the effect of and n on viscoelasticity. The discussion for wood, however, is impossible on the basis of a concept of the free volume, although the flexibility of molecular motion for synthetic amorphous polymers is discussed. Unfortunately, we can not directly know the created free volume because the time-temperature superposition principle is not valid for wood [19]. The principle is related to WLF equation by which the free volume is calculated. The free volume, however, relates to volumetric swelling as follows. 

Ethylene-styrene interpolymers exhibit a novel balance of properties that are uniquely different from polyethylenes and polystyrenes. In contrast to other ethylene-a-olefin copolymers, ESI display a broad range of material response ranging from semicrystalline, through elastomeric to amorphous. The styrenic functionality and unique molecular architecture of ESI are postulated to be the basis of the versatile material attributes such as processability (shear thinning, melt strength and thermal stability), viscoelastic properties, low-temperature toughness and broad compatibility with other polymers, fillers and low molecular weight materials. 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

HA is the basis of the lubricant and "shock absorber" properties of synovial fluid. Osteoarthritis is the most common disease of joints, and correlates with a deterioration of synovial HA. Intra-articular administration of HA is a widely used therapy for OA, providing relief of pain, and other symptoms. The first arthroscopic viscosurgical application of HA was in 1989 [150]. There are several preparations of partially cross-linked HA that are now used in this context. However, only one preparation will be discussed here. Synvisc , also known as hylan G-F 20, is a viscoelastic fluid containing modified HA produced from rooster combs. Hylans are cross-linked derivatives of HA. Synvisc contains hylan A (average molecular size 6x10 Da) and hylan B, a hydrated gel in a buffered salt solution. 

Photophysical and photochemical processes in polymer solids are extremely important in that they relate directly to the functions of photoresists and other molecular functional devices. These processes are influenced significantly by the molecular structure of the polymer matrix and its motion. As already discussed in Section 2.1.3, the reactivity of functional groups in polymer solids changes markedly at the glass transition temperature (Tg) of the matrix. Their reactivity is also affected by the / transition temperature, Tp, which corresponds to the relaxation of local motion modes of the main chain and by Ty, the temperature corresponding to the onset of side chain rotation. These transition temperatures can be detected also by other experimental techniques, such as dynamic viscoelasticity measurements, dielectric dispersion, and NMR spectroscopy. The values obtained depend on the frequency of the measurement. Since photochemical and photophysical parameters are measures of the motion of a polymer chain, they provide means to estimate experimentally the values of Tp and Tr. In homogeneous solids, reactions are related to the free volume distribution. This important theoretical parameter can be discussed on the basis of photophysical processes. 

Ferry suggested on the basis of a molecular theory of viscoelasticity that there should be a small vertical shift factor ro/0o/(T/0),where p is the density at T and is the density at the reference temperature T. These shifts have been made in fig. 7.14, but they are usually small and are often neglected.)... 

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