Eldridge-Ferry is a very important equation. lnc = H/(RT) + constant. c, is polymer concentration, (g/l), R, gas constant, H, gelation/ melting enthalpy. T is absolute temperature (K).
I have already check some literatures, about the temperature, they chose the melting and gelation temperature. (in rheological measurement, Tg at G'>G"). my question is:
Is this equation available if I chose some other point as the Temperature used in this equation, such as critical point, where G' or G" start to increase in rheological measurement?
Thank you very much.
You may choose a different temperature T, but by the nature of the Eldridge-Ferry equation (EFE) this T has to be related to a first order transition with an enthalpic effect. Gel melting can generally be better controlled than gel formation.
I applied the EFE around 40 years ago on aqueous pyrodextrin solutions. Pyrodextrins are starch-derived adhesives which may exhibit unwanted storage instability at high concentrations in water. My aim was to explore the origin of differences in stability between samples. At the time I forced these systems to gel at 4 C in tubes and determined the melting temperature TM by linear programmed heating in a windowed water bath. This was achieved by putting a porcelain spherical bead on top of the gel and monitoring the position of the bead as a function of T visually by means of a scale fixed to the window of the bath. A small amount of decalin was put on top to prevent drying out of samples. TM was estimated by extrapolating the curve to zero displacement. Results were evaluated by plotting ln C vs 1000/TM, revealing that H was similar for all samples. Therefore differences between samples could not be ascribed to the type of cross-linking but rather to a different concentration of physical cross-links in the systems. A much more sophisticated method is to probe the dynamic storage modulus G' at a fixed deformation (or stress) as a function of T in a temperature sweep. In that approach TM is the temperature where a marked decrease in G' is observed or where G'/G" becomes < 1. A feature of my old 'tube' method is that it does not require sophisticated equipment and that several samples can be analysed at a time.
Dear Du.
To understand the equation, it is better to derive it explicitly. Assume that you have a reaction that happens very sharply at a single temperature (like water, which is liquid at 99.99C but a gas at 100C). This is what Peter refers to as a "first-order transition". You are referring to use of the equation in gelation. It is not necessary to be a physicist to appreciate the first-order condition does NOT apply to almost any gelling transition, certainly not due to physical cross-linking, and certainly not in complicated food systems where many steps are involved in the transition from a liquid to a soft solid. Nonetheless, let's go through the derivation. If we write the temperature of the transition T from thermodynamics:
T = ΔH/ΔS,
then we can make some assumptions to relate it to the concentration. Assume that in the liquid state, S is made up of many terms, but the only one of interest is the number of free units that will be immobilized by gelation--> call this number N. All other contributions to the entropy will be assumed to be the same before and after the transition. Finally, assume that after the transition (gelation, in our case), the same term will be reduced to unity (all units are immobilized/linked in a single aggregate). ΔS is then equal to const*R*ln(N) - const*R*ln(1) = const*R*ln(N) = const*R*(ln(C) + another constant). Rearrange, and you find Eldridge-Ferry.
What are the conclusions?
1. The equation should be applied only to first-order transitions (not gelation, as already noted above)
2. The equation assumes that the only change is entropy comes from complete immobilization of units that are completely free to move before the transition, and are completely aggregated into a single unit after the transition. Note that "free to move" doesn't have to refer only to translation. Second, note while the entropy of the units doesn't have to reduce to zero for the derivation to hold, the entropy of the gelling units in the gel state does have to be independent of the concentration. This requirement cannot be expected to hold in real gels. Consider for instance the "elastic entropy", which is zero before the transition and increases with the concentration^2.25 for rubbers.
Final conclusion: there are many inaccurate uses of physical equations in our field. As long as you can explain the approximations involved in their use, and especially when others in your field are making the same approximations, go ahead and use the equations, but be prepared to explain (to both reviewers and readers) that you are dealing with an inaccurate model.
Although Tom may be formally right, the authors themselves applied their equation on the physical cross-linking in gelatin gels. I was motivated by their following publication:
J.E. Eldridge, J.D. Ferry, Studies on the cross-linking process in gelatin gels III. Dependence of melting point on concentration and molecular weight, J. Phys. Chem. 58 (1954) 992-995.
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