The Lodge Liquid

Treatments of flows of polymer melts require a rheological equation of state that allows a calculation of the stress tensor for all points r of a given flow field. We have a complete description of the flow kinetics if we specify for each material point, located at r at time t, the full trajectory in the past. The 

This general expression first accounts for the principal of causality by stating that the state of stress at a time t is dependent on the strains in the past only. Second, by using the time dependent Finger tensor B, one extracts from the flow fields only those properties that produce stress and eliminates motions like translations or rotations of the whole body that leave the stress invariant. Equation (9.163) thus provides us with a suitable and sound basis for further considerations. 

The problem is to find expressions that can describe the experimental observations. We have a perfect solution if we succeed in constructing an equation of state that correctly formulates the stresses for an arbitrary strain history. To reach this final goal is certainly very difficult. It might appear, however, that we have a suitable starting point. Poljrmer fluids have much in common with rubbery materials. In a simplified view, we find, as the only difference, that cross-links are permanent in rubbers whereas, in fluids, they are only temporary with lifetimes that are stiU large compared to all the internal equilibration processes. Hence, it makes sense to search for a formula that includes, from the very beginning, the properties of rubbers as expressed by the equation of state of ideal rubbers, Eq. (9.74). 

Lodge was the first to formulate such an equation by a combination of Eq. (9.74) with the Boltzmann superposition principle as expressed by Eq. (6.111). ExpUcitly, the Lodge equation of state of rubber-like liquids, when written for homogeneous deformations, has the following form 

The Boltzmann superposition principle represents the stress as a result of changes in the state of strain at previous times. In the linear theory that is valid for small strains, these can be represented by the linear strain tensor. In Lodge s equation the changes in the latter are substituted by changes in the 

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The model of appearing and disappearing load-bearing chain sequences thus provides us with a possible interpretation of the memory function of the Lodge liquid. 

Develop expressions for the elongational viscosities for the Lodge rubber-liquid in steady shearfree flow. 

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 constitutive equations benefiting from the specific representations of reptation theory have the general form of the Lodge rubber-like liquid equation, since they are all... 

A careful pressure drop survey indicated that there was an inexplicably high pressure drop in the liquid inlet line. When the tower was opened for inspection, the carcass of a dead rat was discovered lodged in the reboiler liquid inlet nozzle. Inadvertently, this rat caused the tower to flood as the reboiler rate was increased. If the rat had expired in the vapor outlet line, the effect would have been the same. 

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... 

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 difference between the pressure measured flush with the surface (P ) and at the bottom of a hole (P2) during steady-state flow through a channel (slit-die apparatus) can also provide information on normal stress differences, as shown by Lodge and associates." " The relations have been derived by Higashitani and Pritchard subject to certain assumptions concerning the nature of the viscoelastic liquid and the flow pattern they can be expressed in terms of P — P2 as a function of ffi2 for steady-state flows with different shear rates. If the hole is a slot perpendicular to the flow direction. 

Representing a combination of the equation of state of ideal rubbers and Boltzmann s superposition principle, Lodge s equation provides an interpolation between the properties of rubbers and viscous liquids. The limiting cases of an elastic rubber and the Newtonian liquid are represented by... 

As we can see, both are independent of the strain rate 7. Hence, as a first conclusion. Lodge s equation of state cannot describe the shear thinning phenomenon. Equation (7.147) is in fact identical with Eq. (5.107) derived in the framework of linear response theory. The new result contributed by Lodge s formula is the expression Eq. (7.148) for the primary normal stress difference. It is interesting to note that the right-hand side of this equation has already appeared in Eq. (5.108) of the linear theory, formulating the relationship between G t) and the recoverable shear compliance. If we take the latter equation, we realize that the three basic parameters of the Lodge s rubber-like liquid, rjo, and 1,0 are indeed related, by... 

M(t — / ) is called the memory function. Eq. 101 is the constitutive relation for the Lodge rubber-like liquid (LRL). 

The strain is given by A. and the function h X) is a damping function which corrects for the too large strains which are predicted by the Lodge [21] elastic liquid constitutive equation which is identical to the equation above with h(X) = 1. 

In the work of Piau and colleagues [23,24], the Lodge elastic liquid constitutive model was used directly. These authors found that the Lodge model provided reasonable agreement with their uniaxial extension results, showing slightly more strain hardening than was observed in the data. 

Validity of the linear stress-optical rule points at the dominant role of the network forces in pol3mier melts. The Lodge equation of state can be interpreted on this basis. We introduced the equation empirically, as an ap>-propriate combination of properties of rubbers with those of viscous liquids. It is possible to associate the equation with a microscopic model. Since the entanglement network, although temporary in its microscopic structure, leads under steady state conditions to stationary viscoelastic properties, we have to assume a continuous destruction and creation of stress-bearing chain sequences. This implies that at any time the network will consist of sequences of different ages. As long as a sequence exists, it can follow all imposed deformations. 

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. 

As an example of (b) the Wagner model can be cited this is obtained as an extension of the Lodge elastic liquid... 

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... 

This Lodge network model result is a special case of the Lodge elastic liquid, in that the memory function is a sum of exponentials it is also of the same form as the constitutive equation for the Rouse and Zimm models, except that here the constants Aj and f]j are free parameters to be determined from the experimental data. If these quantities are both taken to be proportional to then the zero-shear-rate viscosity is proportional to and the first normal-stress... 

Other possibilities exist to solve the frame invariant problem Cauchy-Maxwell equation uses the Cauchy tensor, C, which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor but contrarily to the Lodge model, it uses a generalized memory function ... 

Rubber elasticity theory can also be used to derive a general class of singleintegral equations, as opposed to a particular equation such as that for the Lodge rabberlike liquid. To proceed, consider a cube of rabber, initially of unit edge, in an extensional deformation as shown in Figure 14.23. In the deformed state, the block of rabber has dimensions Aj, X2, and /I3, which also happen to be the extension ratios. 

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