NATIONAL CHENG KUNG UNIVERSITY, TAINAN, TAIWAN
BANYAN
Volume 30 Issue 9 - September 2, 2016
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Commentary
Hsien-Hung Wei
From van der Waals to de Gennes: A journey toward understanding macroscopic and molecular phase transitions
Article Digest
Shoou-Jinn Chang
GaN-Based Power Flip-Chip LEDs With SILAR and Hydrothermal ZnO Nanorods
Chen-Feng You
Evidence for stable Sr isotope fractionation by silicate weathering in a small sedimentary watershed in southwestern Taiwan
Hua-Lin Wu
Thrombomodulin promotes diabetic wound healing by regulating toll-like receptor 4 expression
Yu-Lung Lo
Analysis of optically anisotropic materials by using optical coherence tomography
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Cloud-based health management system iNCKU proven to prevent obesity
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From van der Waals to de Gennes: A journey toward understanding macroscopic and molecular phase transitions
Department of Chemical Engineering, National Cheng Kung University
【Ta-You Wu Memorial Award】Special Issue

Having been teaching and studying thermodynamics for years, it has been a great deal for me to get a deeper understanding on this subject. Among many topics, I was in particular fascinated with phase transition. Especially after listening Stephen Chen’s awesome talk about phase transitions in polymer systems [1], I was eager to know how a phase transition works in detail. But I soon found myself lost in a forest of the literature without getting closer to understanding it. This puzzle continued until I read the works by two great scientists: van der Waals and de Gennes. This brings me an unusual and fascinating journey toward understanding this central topic in thermodynamics.

This journey began when I studied van der Waals's famous equation of state [2]:

(1)

which relates pressure P, volume V, and temperature T with additional parameters a and b. When I first saw Eq. (1), I was amazed by it since this simple single equation is able to describe vapor-liquid transition (VLT). In the meantime, I was always pondering: Why the equation takes such a special form? Why this equation of state, which is designed to describe gas, can also describe liquid as well? (In fact, the latter is a profound question concerning how a singularity of the free energy behaves in the thermodynamic limit, and not resolved until Yang and Lee’s work [3].) Over a period of time I was trying to seek my own version of understanding of VLT by thinking about these questions without looking at the “standard” answers (most of which are not much touchable to me). A revelation spurred when I came across de Gennes’s book and learned that the coil-globule transition (CGT) of a polymer chain—a phase transition at the single molecule level—can be made exactly analogous to VLT [4]! This ingenious connection not only provides me an excellent example for digging out the ideas behind Eq. (1), but also helps me better understand phase transition through molecular interactions from the microscopic point of view.

To answer the above questions, I try to place myself in the position of van der Waals and ask: Is it possible to “guess” Eq. (1) without rigorous derivation? It is not difficult to see that Eq. (1) has to come from a modification of the ideal-gas law. An obvious starting point is treating gas molecules to be finite sized so that the volume that allows these molecule to move freely has to exclude the volume occupied by themselves, which gives the b term in Eq. (1). However, the b term alone cannot describe VLT. This is because the gas molecules, in average, are moving further apart from each other repelled by their excluded volumes. This leads the gas density V-1=(RT/P+b)-1 to be lower than (RT/P)-1 of an ideal gas, making such a gas even more unlikely to condense to liquid.

As VLT cannot happen with the b term alone, a negative attraction term must be added to reduce the undesired pressure mounting due to the b term—liquid can form only when intermolecular attraction exists. Because this intermolecular attraction has to get stronger as the density ρ≡1/V gets higher, the simplest form of this term might be taken as -aρn with n>0. Written in terms of the compressibility factor ZP/ρRT, the resulting equation of state might take a tentative form:

(2)

First of all, n cannot be smaller than 1, otherwise Z won’t be finite as ρ→0. Secondly, n cannot be 1 either since Z=1/(1-)-a/RT won’t get back to 1 in the ideal-gas ρ→0 limit. So we must have n>1. If n is taken to be integer, then its minimum value will be 2, which gives exactly the a term in Eq. (1). In hindsight, n=2 can be understood in the probability sense as follows. The probability for one molecule to perceive attraction from the nearby ones is proportional to ρ.  Because attraction must involve at least two molecules, the joint probability is proportional to ρ2, which is exactly the argument given by van der Waals [2]. Perhaps a more heuristic reason why n must be equal to 2 can be seen by expanding the b term in Eq. (2) as Z=(1++b2ρ2+...)-(a/RT)ρn-1, where each of the ρk terms represent interactions involving (k+1) molecules. Also given that the lower T the stronger attraction and hence the more inclined to form liquid, when VLT occurs by lowering T to some extent, the repulsion due to the b term has to be somehow cancelled out by the attraction due to the a term. This cancellation can be made possible only when both repulsion and attraction appear at the same two-body ρ term [5]. So we end up with the virial expansion of Eq. (1):

(3)


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