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Thermodynamics

Imperfect fluids and inheritance

The materials that surround us are far from perfect. The polymer molecules that make up the plastics we use to contain our drinks, to mould computer casings and to formulate hair sprays are irregular in both size and shape. Similarly, the latex particles in house paints have a broad distribution of size, surface charge and chemical composition. Even our automotive fuels and lubricants are not simple fluids, but complicated mixtures of linear, branched and aromatic hydrocarbon molecules. When these molecular, chemical and structural irregularities are slight, such ‘complex fluids’ usually have a single phase, which looks homogeneous and is often transparent. Greater irregularity, however, makes a fluid separate into more than one phase, such as the oil and water in salad dressings. Reporting in Physical Review Letters, Evans, Fairhurst and Poon have now derived equations that describe how the disorder in complex fluids partitions among the phases1. Their equations are applicable to fields as diverse as petroleum refining and toiletries, as well as to the design of paints, coatings and plastics.

The fundamental problem can be nicely illustrated in a polymer-science context: ethylene-propylene rubber2 (EPR). This commercial synthetic rubber consists of linear polymer chains that have a more or less random sequence of chemically bonded ethylene (E) and propylene (P) monomer units. Because each position along a chain of N ≈ 1,000 monomer units can be occupied by either an E or a P monomer, there are roughly 2N ≈ 10300 possible molecules that can be made by random linkings of the two monomer types. So a macroscopic sample of EPR, containing small multiples of Avogadro's number of particles (that is, of order 1024), clearly does not contain all of these possible molecules, but it does provide a representative sampling of the composition distribution of the polymer chains.

As shown in Fig. 1, a ‘parent’ EPR sample with a broad composition distribution can phase separate3 into two or more4 ‘daughter’ phases in which chains with more E are spatially separated from chains with more P. Aside from a difference between the mean compositions of the two daughter phases, the full probability distribution functions for composition in these phases can differ considerably from each other, and from the parent distribution.

Figure 1: An ethylene-propylene rubber possessing a broad enough distribution of chain composition can separate into two ‘daughter’ phases.
figure1

ε is the fraction of ethylene units (E) on a given polymer chain in excess of the average ethylene fraction. Typical sequences of E and P (propylene) units in chains from the parent and daughter phases are shown above the distribution functions.

Evans, Fairhurst and Poon have constructed a mathematical description of such situations that is summarized in the box overleaf. Specifically, they derive an equation (equation (2)) that links the various statistical moments of the parent distribution function — mean, variance, skewness and so on — to other moments of the daughter phase distributions. For example, the daughters' mean depends on their parent's variance; their variance on the parent's skewness. These relationships are valid for all types of complex fluid system, provided that the parent phase is characterized by a narrow distribution function — in other words, as long as the disorder is fairly weak. Evans and colleagues have also made an experimental test of this result for a polydisperse system of latex particles.

In the context of EPR, such connections between the composition distributions of parent and daughter phases could have useful applications. When phase separated, the mechanical performance, optical clarity and other properties of EPR are sensitive to the composition distributions of the daughter phases. So Evans, Fairhurst and Poon have created a tool that should aid the design of new EPR rubbers through manipulations of the parent distribution. In principle, this can be accomplished either by blending different grades of EPR, or by modifying the polymerization process used to manufacture it — changing the catalyst, the temperature or the pressure, for example.

Although I have focused on a very specific application of the new formula, it clearly has much broader implications. For example, phase-separation processes have been used to fractionate emulsions consisting of oil drops in water, or aqueous drops in oil5. Equation (2) indicates that it is the skewness (the third moment) of the parent distribution that is responsible for narrowing the drop size distribution of one of the daughter phases. Practically, the consequences are impressive: emulsions with narrow size distributions have already been formulated that actually crystallize into spatially periodic arrays of micrometre-sized drops. These arrays can be fruitfully used in a variety of applications, including the templating of macroporous ceramics for photonic bandgaps, catalyst supports, membranes and filters6. The insights of Evans et al. may help to optimize the tedious fractionation process used to prepare such materials and devices.

Evans, Fairhurst and Poon consider situations in which only two phases coexist, but their approach could be straightforwardly extended to any number of phases. Such extensions may prove important in fields such as petroleum refining, where vast numbers of distinct molecular species are present and can partition between several vapour, liquid and solid phases. It is natural to assume that many other areas will be touched by this development in the physics of disordered materials.

References

  1. 1

    Evans, R. M. L., Fairhurst, D. J. & Poon, W. C. K. Phys. Rev. Lett. 78, 1326–1369 (1998).

  2. 2

    Reisch, M. S. Chem. Eng. News 5 August, 10-14 (1996).

  3. 3

    Scott, R. J. Polym. Sci. 9, 423 (1952).

  4. 4

    Sollich, P. & Cates, M. E. Phys. Rev. Lett. 80, 1365–1368 (1998).

  5. 5

    Bibette, J. J. Colloid Interface Sci. 147, 474–478 (1991).

  6. 6

    Imhof, A. & Pine, D. J. Nature 389, 948–951 (1997).

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