In what follows we present a relatively simple alternative to these approaches and describe some of the successes we have achieved in broadening the range of experimental data which can be used to characterize a system and in being able both to understand and to predict the behaviour of these complex mixtures Lipson et al. More generally, recent theoretical treatments of polymer solutions and blends have been developed using a variety of methods, although lattice descriptions tend to dominate.
An advantage of lattice theories is that they can be compared with lattice simulations, which, for pragmatic reasons, have dominated the field of polymer simulation results. There have also been recent advances in continuum theories, and it is immediately clear that one gains in complexity and realism at the expense of ease and of potential for use by non-theorists. One feature shared by all approaches is that the microscopic parameters inherent in any microscopic theory must be obtained via comparison with continuum experimental data.
The approach featured in the present work is a lattice-based theory for chain molecule fluids. While still relatively simple, it incorporates several important improvements over FH theory including the effects of compressibility and non-random mixing as well as the general ability to model both the mixture properties and the corresponding properties of the pure components rather than simply the mixture properties relative to those of the pure components.
The theory is based on an integral equation approach wherein segment—segment pair distributions are formulated in terms of their corresponding derivatives with respect to separation distance and thus are related to the average force between the segments. We give here, as an example, the equation of state for the pressure of a pure chain fluid,.
Shown in equation 2.
The lattice coordination number, z , is always fixed at a value of 6. Using other values, e. The equation of state has a similar form for binary mixtures i.
It is important to note that for chain molecules which are long enough i. This is done keeping the values of all the other parameters fixed. Examples of the effectiveness of this procedure will be seen in the predicted properties presented in our discussion of the results. A key limitation to the use of any theory, both in analysing experimental data and in making testable predictions, is that a certain amount of data are required in order to characterize the system. Data on polymer mixtures are available in the literature from many different techniques, but, in terms of direct comparison with equation of state models for mixtures, three types have proved particularly useful.
The zero angle scattered intensity of any type of radiation from a mixture in the one-phase region is directly proportional to the second derivative with respect to composition of the free energy of mixing. However, for these neutron scattering experiments the hydrogen in one of the components usually has to be replaced by deuterium in order to provide sufficient scattered intensity. We have been exploring and quantifying the non-negligible effect of deuteration of one component in a mixture and will give a brief preview of our results in the discussion that follows.
Lipson et al. One single datum from the coexistence curve i. In Tambasco et al. The SANS route is particularly appealing, as we have found that even using a small number of data points yields excellent results. However, since the data of choice may not always be available for a system of interest, in the following discussion we explore the use of a variety of kinds of data so as to emphasize the breadth of our approach. It is noted as we now move to a presentation of the results that parameter values and other associated details will be summarized in the captions of the upcoming figures; in cases where they have been made available in other works, we simply cite those sources.
As mentioned above there are serious limitations to the availability of data, so that one of the first goals in applying the lattice integral equation theory to describe pure polymer fluids was to expand upon the kinds of data which could be used in characterizing individual components. The agreement is excellent, providing support for the new characterization route. Having characterized an experimental system of interest using a minimal dataset is useful only if it allows for predictions regarding behaviour which extend beyond what is already known experimentally.
Thus, an important aim of our work has been to exploit the success of the lattice theory in translating our understanding of how a mixture behaves under one set of conditions to new circumstances, going beyond the typical changes in composition or temperature that are usually considered. Examples are shown in the next two figures. The lattice theory predictions are given by the dashed lines for the spinodals and the solid lines for the binodals. Note that we use the scaling of the r parameter described above to make predictions at varied molecular weights.
As is evident from the figure, the lattice theory does an outstanding job both in predicting the coexistence behaviour for a given blend as well as in predicting the effect of changing molecular weight. The theory is even able of capturing the broadening and shifting of the coexistence curve as the molecular weight of the PS is decreased. One point to note is that the transportability of characteristic parameters for a polymeric melt depends on the molecular weight range of interest i.
The fact that a single set of parameters within the oligomeric regime is able to capture the UCST shift for both blends is therefore significant, even given the apparently small difference in molecular weights. Even more notable is the ability of the theory to predict how miscibility changes over such a large pressure range. While the highest pressure value for one blend UCST was chosen for the g determination, similar results are obtained by using another UCST from the combined dataset.
Adapted from Higgins et al. We have demonstrated the success of the lattice integral equation theory in mapping—indeed, predicting—blend miscibility. This presents an opportunity to become more ambitious in terms of what insights to expect from the theory. We have analytic expressions not only for the free energy of mixing, but also for the enthalpy, entropy and volume changes of mixing. Given that such mixing quantities are difficult to obtain experimentally and only infrequently reported, the insights from such predictions are likely to be valuable.
The symbols give experimental data from two separate studies; the error bars give an indication of the difficulty inherent in such experiments and data analysis. The theoretical predictions are shown by the curves; in each case the entire parameter set including g was determined by fitting to the blend PVT surface.
A simple approach to polymer mixture miscibility
Once again, such data are difficult to obtain, as demonstrated by both the very small numbers and the very large error bars. Here the theory is put to another sort of test. The two curves represent lattice theory predictions for this blend using two different routes to the characteristic parameters: the dashed line shows the results applying parameters from fitting blend PVT data, while the solid line uses results from the fits to pure component PVT data along with a g -value obtained via a match to the experimental LCST.
The difference between the curves indicates the range in prediction resulting from use of different parameter sets.
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Interestingly, the two theoretical results appear to provide bounds for the set of experimental values, which neatly fall between the curves. Experimental data symbols are from Shiomi et al.
Two theoretical curves are shown, each representing a different route to the characteristic parameters. The dashed curve is derived from fitting the parameters Tambasco , table 4. The solid curve parameters from Higgins et al. Another advantage which should derive from a successful theoretical model of polymer blends is insight which applies across a range of systems. Once having studied a reasonable number of polymeric mixtures, what larger conclusions can be drawn?
The starting point for each investigation was a fit to pure component PVT data to determine all parameters except for g , which was obtained in each case by a fit to the experimental critical temperature. That is, g exp is the best-fit value yielding a theoretical critical temperature which matches the experimental value T c exp for each system. The very strong sensitivity of the critical temperature to shifts in g is typical of descriptions which use this—or a similar—parameter, and has been remarked upon regularly in the literature.
This can be interpreted as the geometric mean underestimating the strength of the mixed interaction for LCST blends, while overestimating it for UCST blends. Shifts in the modelled critical temperature as a function of the relative strength of mixed interaction parameter, g. T c represents the modelled critical temperature while T c exp is the observed experimental value. Results are shown parameters from Higgins et al.
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The data are adapted from Higgins et al. Despite the fact that there are significant differences not only in the interaction strength but also in the nature of the phase separation, the theory predicts the same shift in critical temperature for a given relative shift in mixed interaction. As noted earlier, one area of current interest is the effect on melt and blend behaviour upon deuteration of one of the components. For blends, this may be manifest by a significant shift in the critical temperature.
The strength and direction of such a shift is currently not well understood; indeed, even the existence of such shifts has not been predicted theoretically. While fitting for g pinned the theoretical curve to the experimental results at the critical temperature, the remainder of the phase diagram comprises a true prediction, and it is evident that the theory does a very good job at characterizing the shape of the phase boundary.
Related Polymer Thermodynamics: Liquid Polymer-Containing Mixtures (Advances in Polymer Science)
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