The Rubberlike Liquid

We now proceed to examine the most important ways in which Eq. 10.3 fails to describe the behavior of melts when the deformation is large and fast. Each nonlinear phenomenon is introduced by showing how Eq. 10.3 might be modified in an ad hoc way to deal with it, and we then provide an interpretation of the phenomenon in terms of the tube concept. 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

Lodge [10] constructed a very simple model of nonlinear behavior that can be represented as Eq. 10.3 with the Finger tensor B t,t ) replacing He called the material described by 

An equation like 10.5, obtained from the Boltzmann principle by replacing the infinitesimal strain tensor by one that can describe a large deformation, is sometimes called a model of finite linear viscoelasticity . If the memory function in the rubberlike liquid is taken to be the relaxation modulus of a single Maxwell element [G(f) = Gq exp(f/T)], we obtain the special case of the rubber like liquid that we will call Lodge s equation this is shown as Eq. 10.6. 

This is also the integral form of the differential constitutive equation called the upper convected Maxwell model , which is given in the next section. 

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The relaxation of the primary normal stress difference after cessation of steady-state flow at strain rate 7 can also be expressed in terms of linear viscoelastic properties by these models. For example, in terms of the relaxation spectrum, the rubberlike liquid theory of Lodge ° provides ... 

The rubberlike liquid model is able to predict, qualitatively, certain nonlinear viscoelastic phenomena. In particular, some effects arising from the finite orientation of chain segments are predicted, for example a nonzero first normal stress difference. However, it fails to describe many other nonlinear effects. For example, it predicts that the viscosity is constant with shear rate and the second normal stress difference is zero. In fact, all its predictions for the shear stress in simple shear are the same as those of the Boltzmann superposition principle. We can gain some insight into the origins of nonlinearity by examining the features of the rubberlike liquid model that limit its predictive ability. 

For the relaxation of the first normal stress difference following a step strain, the rubberlike liquid model (Eq. 10.6) predicts that [35] ... 

For start-up of steady-simple shear the rubberlike liquid model predicts that the shear stress is given Eq. 4.8. While h/2 is predicted to be zero, the first normal stress difference is ... 

Another symbol sometimes used in place of Pj is 6. The rubberlike liquid model (Eq. 10.5), predicts that Pj is independent of shear rate and related to the linear relaxation modulus ... 

Figure 10.14 Recoverable shear for steady simple shear for an LDPE. Also shown is the shear-rate dependent recoverable shear 2 aiy), the predictions of the rubberlike liquid model (straight line.fromEq. 10.5) and Wagner s equation (solid curve,from Eq. 10.10).FromLaun...

Turning to the behavior of typical melts, it is found that the damping function is not nearly as sensitive to molecular structure as are the linear viscoelastic properties, e.g. the storage and loss moduli. The rubberlike liquid, as well as the tube model, predict that the ratio of the first normal stress difference to the shear stress in step shear should be equal to the strain at all strains, and this is in fact observed. The other quantity measured in simple shear experiments is the second normal stress difference, but this is difficult to measure and few data are available. Of the shear histories other than step strain than have been used to study nonlinear viscoelasticity, start-up of steady simple shear has been the most used. If the shear rate is sufficiently large, some degree of chain stretch can be generated in the early stages. 

Mason77 developed ideas about the distribution of the free-volume to explain the existence of the broad transition region from glassy to rubberlike state. He believed that there is some localization of that part of the free-volume that distinguishes the rubberlike state from the true liquid state in which the free-volume is not localized. In the non-crosslinked state of some rubbers there may be an arbitrary distribution of the free-volume Vf connected with the free-volume of each monomeric... 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

The DSC is widely used to measure the glass-rubber transition temperature (Tg-value), which is an important parameter for polymer characterisation. The Tg-value represents the temperature region at which the (amorphous phase) of a polymer is transformed from a brittle, glassy material into a tough rubberlike liquid. This effect is accompanied by a step-wise increase of the DSC heat flow/temperature or specific heat/ temperature curve. Enthalpy relaxation effects can hamper the... 

To start with, 3.4 liters of sodium silicate solution (sodium waterglass, d ° 1.37) is diluted with one liter of water (mechanical stirring). Then, ION HCl is added at a rate of 10 ml./min. until thymol blue shows an acid reaction (pH 2-2.8). (After addition of 400 ml. of the acid the mixture becomes viscous and rubberlike. The acid addition is interrupted and the mass is broken up. It is then manually stirred while acid is added in drops. The mixing is continued until a thin suspension is obtained. The remainder of the acid is then added at the original rate until the desired pH is reached.) The mixture is then stirred for two additional hours at room temperature, suction-filtered and washed until the wash liquid is no longer acid. The gel is dried at 200°C for 12 hours, groimd to the desired particle size, and finally washed free of Cl . 

For finite strains, however, several measures of strain are available, and each of these reduces to the same quantity in the limit of infinitesimal strains. The situation is therefore similar to the one encountered previously in connection with the multiplicity of time derivatives for the stress. The simplest molecular network theories o) suggest the use of the so-called Finger measure of strain, and the resulting equation is called the Lodge rubberlike liquid. Not surprisingly, one finds(9,8i) that, with the use of the Finger strain measure, Eq. (31) is mathematically the same as Eq. (26). 

Generally, it is found that while the upper convected Maxwell fluid, Eq. (26), and the Lodge rubberlike liquid, Eq. (31), predict the correct qualitative features of polymeric fluid behavior, the representation is not quantitative. In particular, in a stress-relaxation experiment, the relaxation takes place over too broad a range of time to be described by a single exponential. One therefore uses a spectrum of relaxation times, and modifies Eq. (30) to... 

Here y[o] is shorthand for YiojitXl- This can be regarded as an expansion about the Lodge rubberlike liquid, which in turn includes the general linear viscoelastic model. By expanding the strain tensors in equation (49) about time t, the retarded-motion expansion of equation (38) is obtained, with the... 

At temperatures well above the glass transition of the polymers, the molecular segments are highly flexible and slip past each other almost without restriction. They behave like the molecules of a liquid except for the fact that their ends are linked with each other. Just the existence of crosslinks distinguishes rubberlike materials from ordinary liquids. The bulk moduli K of liquids and of rubberlike materials are of similar magnitude, e.g. K = 1 to 2 GPa [26]. 

Techniques and procedures of such thermoeleastic measurements under unidirectional or uniform (hydrostatic) deformation of solid and rubberlike polymers are described in 1 64 66). Similar methods have been used more often for recording the temperature changes resulting from the plastic deformation of solid polymers. Besides thermocouples, fluorescent substances, liquid crystals and IR-bolometers are used for such measurements. 

It was found that Afi Tg and Aa Tg are not constant and therefore the SB equation has limited applicability. Hie results indicate an increase in Aa Te with increasing Tg. Therefore it is inadmissible to use the product A a Tg as a universal value in any theoretical discussion of the glass-transition phenomenon. At the same time, this conclusion in no way excludes the free-volume theory and the role of free-volume in the transition from the glassy to the liquid or rubberlike state. 

The limiting value of the creep is equal to 00 = a0/E = Voigt-Kelvin element is only able to describe qualitatively the creep behaviour of rubberlike materials with a limited creep and not the creep of an elastic liquid. In general the creep compliance may be expressed as... 

In 1839, Eduard Simon, an apothecary in Berlin, distilled storax resin obtained from the Tree of Turkey , (liquid ambar orientalis) with a sodium carbonate solution and obtained an oil which he analysed and named styrol (what we now call styrene) [1]. He recorded the following observation that with old oil the residue which cannot be vaporised without decomposition is greater than with fresh oil, undoubtedly due to a steady conversion of the oil by air, light and heat to a rubberlike substance . Simon believed he had oxidised the material and called the product styrol oxide. Later, when he realised that it contained no oxygen, the product became known as metastyrol. This puzzled the early chemists as there was no change in empirical formula despite the very pronounced alteration in chemical and physical properties. Unknowingly, this was the first recorded instance of polymerization. 

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