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1/Mn

Figure 2: A plot of the internal pressure and the CED as a function of the inverse molecular weight.

polymer set, we can use it to compute the functions ni for each of the polymer families. In order to accomplish this task we must find a way to determine the true CED for the molecules in one of these data sets. It is to this task we next turn our attention.

### The CED for a Polyethylene Oligomeric Series.

Of all the polymers studied, polyethylene (PE) and its oligomers have been most extensively studied.10'11 The CED for low molecular weight oligomers exists for molecules up to 20 carbon atom chains (C20) over a wide temperature range. The internal pressure can also be measured for these and higher molecular weight liquids.11How can we use this data to determine the CED's for the higher molecular weight polyethylene's? A simple and well known property of moleculer weight series of this type is the fact that most if not all bulk thermodynamic properties exhibit a linear dependence on the inverse molecular weight, Mn, for higher molecular weight.14 The range of molecular weight over which this linear dependence is observed depends on the temperature, pressure and the nature of the thermodynamic property considered. 11 Figure 2 shows a plot of the CED and the internal pressure for the PE series at 150 C. At this temperature a clear linear dependence is observed and one can use these lines to calculate the CED for the higher molecular weight PE's and to compute the value of ni as a function of molecular weight at this temperature. By repeating this at different temperatures we can construct the function ni (T,Mw) for polyethylene. Figure 3 shows this function for a number of different molecular weights where the calculation either did not involve extrapolation or where that extrapolation was minimal. Because of the information used in this evaluation, in particular the compressibility, a high degree of precision is not possible. However the behavior of the funcion n is consistant with previous observations7 and gives us a path foward to the next step in the calculation. Figure 3 shows an asymptotic approach to a constant value of n close to 1.2 as a function of molecular weight and a weak to nonexistent temperature dependence of n for these oligomer and polymer polyethylenes. Only values of n below the boiling points of the liquids were included in the figure. Above the boiling point the value of n rises rapidly as one approaches the critical point. Since we are only interested in the liquids well below their boiling points, we have excluded these data points.

The Master Curve for the Surface Tension using the CED.

It is now a simple matter to reconstruct a master curve using the following thermodynamic information set ( y, Vsp, a, CED,T, P). Once again we can use the equation of state parameters to fit this data. To insure that the fitting parameters reflect the information set exactly, we do a piece-wise fit to the data as a function of the temperature. We found that most equations of state could not provide a global fit to the data sets with just one set of parameters.10 Figure 4 shows the results of this scaling using the new information set. What is clear from figure 4 is that we appear to have lost the strong scaling behavior we observed in figure 1, i.e. the data shows an increased deviation from a master curve. As we stated above, a complete collapse of the data would be expected if the only contribution to the surface tension came from the local cohesive forces between the molecules. Hong and Noolandi15,16 showed that for long chain molecules we expect an additional entropic contribution to the surface tension. Therefore, for a series of molecules with increasing molecular weight, we expect to see a transition from a universal curve for small molecules to a curve which describes long chain molecules. This is what is described in figure 4. Thus there are two universal curves with

Figure 3: A plot of the function n(T) for polyethylene oligomers as a function of the temperature. The molecular weight of the liquids is indicated in the legend by the number of carbon atoms in the molecule.

100 150 200 Temperature (C)

Figure 3: A plot of the function n(T) for polyethylene oligomers as a function of the temperature. The molecular weight of the liquids is indicated in the legend by the number of carbon atoms in the molecule.

Figure 4: A plot of the scaled surface tension, y/y*, as a function of the scaled temperature T/T* for polyethylene oligomers and some low molecular weight liquids. y* and T* are defined in figure 1.

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