The Lubricity theory

According to the lubricity theory, the function of a plasticizer is to reduce intermo-lecular friction between the polymer molecules. This theory has been presented in different ways, but the idea persisting in all of them is that when a plastic part is flexed, the polymer molecirles have to slip over each other. The plasticizer acts by lubricating the movement of the molecules and reducing their internal resistance to sliding. 

The lubricity theory is attributed to Kirkpatrick, Clark, and Houwink among others but, as mentioned above, some concepts common to the lubricity and to the gel theories appear in their works. 

Kirkpatrick established that plasticization can result from  

presence of groups that afford points of mutiral attraction in both, plasticizer and 

proper location of these groups in relation to each other so as to permit the attractive 

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One of the apparent results of introducing couple stress is the size-dependent effect. If the problem scale approaches molecular dimension, this effect is obvious and can be characterized by the characteristic length 1. The size effect is a distinctive property while the film thickness of EHL is down to the nanometre scale, where the exponent index of the film thickness to the velocity does not remain constant, i.e., the film thickness, if plotted as a function of velocity in logarithmic scale, will not follow the straight line proposed by Ham-rock and Dowson. This bridges the gap between the lubrication theory and the experimental results. 

The effect of plasticizers has been explained by the lubricity, gel, and free volume theories. The lubricity theory states that the plasticizer acts as an internal lubricant and permits the... 

The relationship between volume, maximum thickness and radius gives a measure of the flatness of the blob away from its leading edge. That is, the volume is a constant times ttR Aq. Tfie constant is unity for a flat blob with vertical sides, 0.79 for the inviscid blob, and 0.67 for the lubrication theory blob (Huppert 1982). The central region of the inviscid blob is flatter than that of the lubrication theory blob as expected, but the difference is not large. 

The preceding description is appropriate for the case of parallel boundary motion. However, if the motion in the lubrication layer results from the relative motion(s) of the boundaries toward (or away from) one another with velocity V, we see from (5-59) that we may simply choosenc = V/s. Finally, although the characteristic pressure could also be adapted directly from the analysis of the preceding section, its scaling is a key result of the lubrication theory, and we thus initially adopt the symbolic notation... 

The most straightforward comparison is between the solution (5-167) for the pressure and the solution (5-151) from the lubrication theory with the latter evaluated for small a. Because... 

Apart from the trivial inclusion of the gravitational body-force terms in (6-2) and (6-3), the governing equations, and the analysis leading to them, are identical to the governing equations for the lubrication theory of the previous chapter. The primary difference in the formulation is in the boundary conditions, and the related changes in the physics of the thin-film flows, that arise because the upper surface is now a fluid interface rather than solid surface of known shape. The boundary conditions at the lower bounding surface are ... 

By 1950 two theories had evolved to account for the major "flexibilizing" effect of plasticization the "lubricity" theory and the "gel" theory ( ). In addition, the "free volume" theory devised to explain fluid flow was also being adapted to plasticization (10). 

The lubricity theory explains the resistance of a polymer to deformation. Stiffness and rigidity are explained as the resistance of intermolecular friction. The plasticizer acts as a lubricant to facilitate movement of macromolecules over each other, thus giving the resin an internal lubricity. The gel theory is applied to predominantly amorphous polymers. It proposes that their rigidity and resistance to flex are due to an internal three-dimensional honeycomb structure or gel. The spatial dimensions of the cell in a brittle resin are small because their centers of attraction are closely spaced and deformation cannot be accommodated by internal movement in the cell-locked mass. Thus, the elasticity limit is low. Conversely, a thermoplastic or thermosetting polymer with widely separated points of attachment between its raacroraolecules is flexible without plasticization. 

When the fiber suspension is concentrated, Christensen (101) used the lubrication theory to derive the limiting behavior... 

Mixing rheology — lubrication theory. When considering only the nip region, it is possible to develop a rheological approach based on the lubrication theory (11). Analyses by Bernhardt (3), Bolen and... 

Mixing rheology - elongational flow. The above considerations are obviously oversimplified and neglect the particular behaviour of viscoelastic materials. On one hand, we have already pointed out that it is difficult to introduce the more realistic viscosity function n = f ( y> T), and,on the other hand, it is quite clear that the mixing process is not isothermal. Furthermore, Tokita and White (13, 14) demonstrated that the lubrication theory can be applied to a nip of length L and depth H providing that... 

A priori, this condition is satisfied only with intermeshing rotors. Moreover, with polymer melts, normal stresses in shear flow are very large, even at relatively low shear rate for example, in the range of 10 dyn/cm at Y = 1 sec for polystyrene at 180 C (15) (and probably higher with elastomers). Consequently the N./ o ratio is larger for viscoelastic fluids than for Newtonian fluids and therefore the validity condition for using the lubrication theory at the nip is surely not fulfilled. 

The first two theories which try to explain the plasticization of polymers were the gel theory and the lubricity theory. Both were developed simultaneorrsly and sometimes the authors to whom the theories are attributed were not aware of the differences between both... 

Some results, which were considered supporting the lubricity theory, were also found Verhoek, who measured vapor pressures of plasticizers in polymers, found that plasticizers reduce solvent retention, which was considered to support these theories (plasticizers fill voids and so they exclude solvents). Rententivity and permeability of cellulose acetate plasticized with different sulfonamides, phthalates and phosphates, were measured by Kirkpatrick who concluded that these properties depended on the attractive forces resulting between the plasticizer and the polymer. Houwink measured swelling of some butadiene-acrylonitrile copolymers with different acrylonitrile content when these were mixed with solvents of different polarities, and proved the importance of the polarity of the plasticizer and the polymer. Moreover, Houwink measured the brittle point of ethyl... 

Consequently analytical methods are mostly confined to creeping flows. Roughly, there are two types of problems that can be solved. The first of these deals with interfaces that show small deviations from simple geometric forms, as for instance the case of a slightly deformed sphere settling in an infinite fluid. The second type constitutes cases where interfacial position changes, but only very slowly. Then its variation can be neglected to the first approximation and the lubrication theory approximation or the slender body approximation applied. It should be noted that both the above methods yield approximate solutions. 

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

The lubrication theory approximation is used to formulate this problran. In this scheme it is assumed that the film is almost flat. Consequently, if x = h(y) is the profile shape, then dhldy 1. Thus the boundary eonditions apphcable on h(y) are used by treating h to be almost a constant. This procedure also allows the use of the assumption that, to a good approximation, the flow is fully developed, with 0 and a function of x alone. Then the equations of motion become... 

The lubrication theory approximation and the closely related slend body approximation (see Problem 7.5) have been applied successfully in many cases. The critical assumption is that the variation of the velocity profile in the direction of flow is ignored in the first approximation. The flow under this approximation becomes one-dimensional in most cases, or at least retains a simple form. 

A thin liquid film lies on the solid surface which forms the floor of a narrow horizontal slit. Through the slit, air is blown at a steady rate. The air is seen to exert a constant shear stress on the liquid surface, thus the film thickness varies linearly with the distance from the leading edge, which is also the contact line (Derjaguin et al., 1944 Levich, 1962). Very close to the contact line the profile changes to retain the equilibrium contact angle at the contact line. The equations of motion and continuity under the lubrication theory approximation reduce to (Neogi, 1982) (see Problem 7.13)... 

Consider the flow of a thin film down a wall, as shown in the accompanying figure. Write the relevant equations of motion under the lubrication theory approximation, including the effect of surface tension. Obtain the thickness h far from the entrance. If h is the thickness at any distance from the entrance which can be approximated as... 

A drop is spreading slowly on a solid surfaee. At large times of spreading, the drop appears to be thin and flat and the lubrication theory approximation can be used. Obtain the appropriate equations of motion (do not neglect gravity). 

The Lubrication Theory. The lubrication theory is based on the assiunp-tion that the rigidity of the resin arises from intermolecular friction binding the chains together in a rigid network. On heating, these firictional forces are weakened to allow the plasticizer molecules to lubricate the chains. Once incorporated into the polymer, the plasticizer molecules shield the chains from each other, thus preventing the reformation of the rigid network (1). 

Th0 GgI Theory. This theory extends the lubrication theory by having the plasticizer break the resin-resin attachments of a three-dimensional honeycomb or gel structure and by masking these centers of attachment from each other, preventing their reformation. This gel is formed by loose attachments occinring at intervals along the polymer chain. This facilitates the movement of plasticizer molecules, thus imparting flexibihty. 

The results suggest that the friction and lubrication conditions within the sealing zone of lip-type oil seals cannot probably be explained by one of the lubrication theories alone. 

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