Analogy Between a Polymer Chain and

Imagine you are in a thick forest. You have picked enough mushrooms and berries (or whatever you were gathering there), the weather has become bad, and all you want now is to get out of the wretched place. But how?

The trees and bushes hinder your view and make it hard to walk. You cannot see the sun behind the clouds. . . It seems quite certain that you are facing a hard time — unless you have got a compass. (Well, in theory they say that an experienced person can tell directions by looking at how moss and lichen grow on tree trunks, and where ant hills are, etc. — but that is not really our subject.) But with a compass — would it be of any use without a map? You would not know which direction you need to take. . . Well, it appears a compass would still be extremely useful. Soon we shall see why.

We are telling you this story for a good reason. It will help us to comprehend a deep mathematical concept which has been very fruitful when explaining the behavior of polymers as well as many other things, in the

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fields ranging from biology to economics. Historically, this concept was first developed when Brownian motion was studied.

Brownian motion itself, as its name reveals, was discovered in 1827 by the English botanist Robert Brown (1773–1858). Looking through a mi-croscope at little particles of pollen suspended in water, he was fascinated by their random “dances”. The particles were moving by themselves, ap-parently with no external encouragement. So a lot of people decided that there must have been some “living power” causing the motion (because the flowers were animate!) They reckoned this had proved that there was some mysterious “substance” which made the animate di↵erent from the inanimate.

The question was debated for a long time. Everyone was free to think what they wanted. Then there was a dramatic boom in interest at the end of the 19th century. Brownian motion was regarded as a kind of perpetual motion, so it tantalized those who were puzzled by the general problems of science. These included the nature of irreversibility (i.e. the distinction between past and future) as well as the di↵erence between Darwinian bio-logical evolution leading to perfection of species, and the thermodynamic evolution described by Clausius, Thompson, and Boltzmann, which leads to dissipation, or, as it was then called, “thermal death.”

Eventually, the answer was found by Albert Einstein² and the Polish physicist Marian Smoluchowski (1872–1917), then a professor at the Uni-versity of Lviv. The title of one of Einstein’s papers on the theory of Brownian motion is rather telling: “On the motion of particles suspended in resting water which is required by the molecular-kinetic theory of heat”.

Einstein and Smoluchowski considered chaotic thermal motion of molecules and showed that it explains it all: a Brownian particle is “fidgeting” because it is pushed by a crowd of molecules in random directions. In other words, you can say that Brownian particles are themselves engaged in chaotic ther-mal motion. Nowadays, science does not make much distinction between the phrases “Brownian motion” and “thermal motion” — the only di↵er-ence lies back in history. The Einstein–Smoluchowski theory was confirmed by beautiful and subtle experiments by Jean Perrin (1870–1942)³. This was a long awaited, clear and straightforward proof that all substances are made of atoms and molecules⁴.

2By the way, Einstein presented his theory of relativity and the concept that light consists of photons in exactly the same year, 1905.

3You can read about Perrin’s experiments in a very interesting book [55].

4The atomic hypothesis was suggested long ago by the ancient Greeks, but it had to wait for more than two thousand years to be proved!

We will skip further details of this adventure story. We just need to emphasize one more thing before we get back to polymers. Since a Brow-nian particle moves due to collisions with molecules, its path breaks into a sequence of many very short flights and turns. In this sense, a Brownian trajectory is pretty similar to the shape of the polymer chains which we saw in Section 2.4 (Figure 2.6). Another obvious example of this sort is of a man who is lost in a forest, with no compass, and has no choice but to wander at random.

Certainly, no microscope would let you see the twists and turns of an individual molecule’s path. However, the Einstein–Smoluchowski theory tells us how to spot the di↵erence between a “fuzzy” line which consists of a great number of tiny random kinks, and an ordinary smooth curve, even though we cannot discern the individual kinks. (We do not always need to see everything, e.g. we can happily tell water from alcohol even though the individual molecules are invisible!) In the same way, a polymer chain looks nothing like a shape stretched in a certain direction. And the path of a man in a forest would depend quite noticeably on whether he is equipped with a compass or not!

So what is the di↵erence between a smooth and a “kinky” path?

For motion in a straight line: R = v (t2 t1) (6.1) For a Brownian particle: R = `^1/2[v(t2 t1)]^1/2 (6.2) The notation here is as follows: in formula (6.1), R is the displacement, i.e. the distance R =|R¹ R2| between the initial (R¹at time t1) and final (R2at time t2) points of the motion (t1< t2), v is the average velocity of the motion. In formula (6.2), R also characterizes the distance between initial and final positions, but since the motion is random, R should be understood as an average; more specifically, it is the root-mean-square displacement:

R =⌦

(R2 R1)²↵1/2

, where the angle brackets indicate that the average is taken over a number of di↵erent Brownian paths. Apart from this technical detail of the definition of R, Equation (6.2) is fundamentally di↵erent from (6.1) because R is proportional to the square root of elapsed time instead of time itself. The price for that is the appearance of a new parameter ` which has the dimension of length and whose physical meaning we will have to discuss and explain⁵).

5The Einstein–Smoluchowski theory leads to the value ` = (mkBT )^1/2/(3⇡⌘r) for spherical Brownian particles of radius r and mass m moving in a liquid of viscosity ⌘ at a temperature T .

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What is the polymer analogue of the Einstein–Smoluchowski equation (6.2)? Let L be the contour length of a polymer chain. It is bound to be proportional to the number of monomers in the chain, given that the chemical structure does not change. The chain length L plays the same role for a polymer as the value v(t2 t1) for a Brownian particle, that is the total distance traveled by the particle along the path. Since the chain wiggles around a lot, the root-mean-square distance between its ends, R =⌦

(R2 R1)²↵^1/2

, is totally di↵erent and not even proportional to the contour length L. You can easily find R from Equation (6.2) if you replace v(t2 t1) by L:

R = `<sup>1/2</sup>L<sup>1/2</sup>= (`L)^1/2. (6.3)