8. The Problem of Excluded Volume 147
8.3 Excluded Volume — Formulating the Problem
Let’s discuss how interactions of the type shown in Figure 8.1 a might influence the shape of an isolated polymer chain in a dilute solution (Figure 4.7 a). First of all, would volume interactions make the coil swell or shrink?
This, it turns out, depends on the temperature of the solution.
Suppose the characteristic energy of attraction " (Figure 8.1 a) is much greater than the thermal energy kBT . Then attraction will dominate. As a result, the macromolecule will shrink to become more compact than an ideal coil. This is a special polymer state, called a polymer globule. We shall come back to it in the next chapter.
It is not the same story if " is smaller than kBT . In this case, attraction is not too important. Repulsion at shorter distances between monomers is the prevailing form of interaction. It makes the coil swell. Such swelling is called the excluded volume e↵ect. (You presumably understand where the name comes from. As we have already said, the repulsion at short distances occurs because the volume of each monomer is excluded for all the others.)
The Problem of Excluded Volume 153
a b
Fig. 8.2 A random walk (a) and a self-avoiding path (b) on the lattice in two dimensions (on the plane). Notice that random walk frequently retraces its own path, while self-avoiding walk never does so.
In this chapter, we are going to tackle the problem of excluded volume, that is, we shall try to picture how a polymer coil swells.
For an isolated polymer chain, the problem is purely geometrical. In-deed, the spatial shape of an ideal chain resembles the path of a randomly wandering Brownian particle (see Chapter 6). What new features will the shape of the chain acquire, if we allow for the excluded volume? Clearly, since the “private space” of each monomer is not available to the rest, the chain cannot possibly cross itself at any stage. This sort of behavior can be described as self-avoiding. For example, if there were an equivalent Brownian particle, it would not be allowed to cross its own track. A two-dimensional version of such a trajectory is sketched in Figure 8.2. Thus, we have made it a purely geometrical problem of self-avoiding random walks.
This problem can be quite successfully approached by computer simu-lation. The simplest way to set it up is to use a random number generator to try out various trajectories of a polymer chain (just as described in Sec-tion 2.4). Then whenever we obtain a trajectory with a self-crossing, we merely ignore it. Thus, we only keep the self-avoiding paths, and when we have enough of them, we can look at some average features. Although more sophisticated (and more efficient) algorithms are normally used these days, in principle they are not that di↵erent from what we have just described.
Typical result is shown in Figure 8.3.
So what has been gathered from the computer simulations of self-avoiding walks? It appears that the conformational properties of a poly-mer coil are quite significantly a↵ected by the excluded volume. The coils become looser, and the fluctuations in the segment concentration become more severe. The mean-square size of the coil increases. Moreover, the mean-square end-to-end distance R2 now depends di↵erently on the num-ber of segments in the chain. Instead of the familiar R2 N (which we derived for an ideal chain in Chapter 6), we now get
R2 N2⌫ , 2⌫ 1.176 6/5 in 3D , (8.2) for self-avoiding walks in three dimensions, and
R2 N3/2 in 2D , (8.3)
b its meaning from the Figure 8.2. A typical collapsed globular conformation is shown in (c);
we will discuss globules later in Chapter 9. The figure is courtesy of S. Buldyrev.
for self-avoiding walks on a plane (i.e. in two dimensions). Of course, the accurate values of indices 1.176 and 3/2 do not appear as such in the computer simulations, particularly on simple minded ones. But within a certain accuracy, the indices produced by a computer are close to these values. Relationships (8.2) and (8.3) confirm that a polymer coil with excluded volume is swollen compared to an ideal coil. You will find it handy to introduce a swelling coefficient ↵, such that
↵2= R2
R2 0 , (8.4)
where R2 0= N `2 is the size of an ideal polymer chain. As N increases,
↵ grows approximately as N1/5 in three dimensions, and as N1/2 in two dimensions.
8.4 The Density of a Coil and Collisions of Monomer Units The problem of excluded volume did not succumb to the e↵orts of theorists for more than 20 years. The way to tackle it, or, to be more precise, the
The Problem of Excluded Volume 155
way to reduce it to some other, better explored problems, was found by P.G. de Gennes in 1972. His solution goes far beyond what we can explain in this book. However, we do not have to jump straightaway to Equations (8.2) and (8.3). To convince you, we o↵er a simple explanation known as Flory’s theory, although the way we are going to present it does not quite follow Flory’s original version.
First of all, the mean spatial size of a coil R is obviously of the same order of magnitude as R2 1/2. This is why we can write that R `N1/2for an ideal polymer (see (6.11)). Meanwhile, for a coil with excluded volume we get R ↵`N1/2 > `N1/2. The volume taken up by such a coil is V `3N3/2(we omit the factor 4⇡/3 as usual). However, a polymer chain never uses up all the space inside the coil. This is clearly seen in Figure 2.6. You can show it is true by the following argument. Let the volume of a single monomer segment be v. Then the total volume of the coil is N v.
Since N 1, we have V > `3N3/2 N v. In other words, the fraction of the volume of the coil taken up by the monomer segments is really very small:
N v
V < N v
`3N3/2 N 1/2 v
`3 1 . (8.5)
(In Section 6.6, we used the same sort of argument for an ideal polymer.) The same thing can be said about the mean concentration of the segments in the coil, n N/V ` 3N 1/2 (cf. (6.14)). At first glance, you may think it implies that a polymer with excluded volume is always ideal. Indeed, if the segment concentration is so low, their encounters are very rare, and one can be tempted to neglect them. On the other hand, we know that the coil is very pliable, and its elastic modulus is small.
This suggests that the question should be treated with more subtlety.
Let’s make a crude estimate of how many encounters (i.e., collisions) be-tween the segments of the coil may occur at the same time. Assume that the coil is a cloud of totally independent particles (segments) spread over the volume V . It would be wonderful if we could take a three-dimensional photo of this cloud. We would then be able to count all the collisions between two, three or more bodies, caught at a moment in time.
Unfortunately, we cannot do this, so we have to use another approach.
There are N particles all together. The probability that each particle has a close “partner” is . This is why the number of pair collisions is of order N . In the same way, the number of three-body collisions is roughly N 2, and so on. In general, the number Ypof p-body collisions can be estimated
as N p 1. From (8.5),
Yp N p 1< N(3 p)/2 v
`3
p 1
. (8.6)
You can see that Yp 1 if p > 3. This indicates that many-body collisions are really rare. Even the number of three-body collisions in a swollen coil is of order 1. So they cannot seriously a↵ect the conformation of the coil.
In contrast, the number of simultaneous pair collisions is about N1/2. This is much less than N (so each particular segment seldom has a collision), yet it is a large number compared to 1.
Besides, as we showed in Chapter 7, a long polymer chain is very pliable, and its elastic constant is small ( 1/N , see (7.21)). Therefore, we have the right to suspect the pair collisions of making the polymer swell in the way implied by (8.2) and (8.3).
What is the free energy of a polymer like, given the excluded volume interactions? (See formula (7.19) for the definition of free energy.) Of course, it has the usual entropy term T S (which would be the only term in the case of an ideal gas or an ideal polymer). In addition, it includes the internal energy U of the segment interactions. This latter term is responsible for the swelling. In other words, it accounts for the excluded volume e↵ect. All we need to know now is the contribution of the binary collisions to the internal energy U of the coil.
Here is how you can find it. The segment density n is very low, as we have seen. So U can be expanded as a series of powers of n:
U = V kBT n2B + n3C + . . . , (8.7) where V is the volume of the coil, and B and C are expansion coefficients, or virial coefficients (i.e., B is the second virial coefficient, C is the third, and so on). These coefficients are fully determined by the form of the interaction potential u(r) and the temperature T . Obviously, the first term in expansion (8.7) stands for the binary interactions. This is because it is proportional to n2, which is just the pair collision probability. Likewise, the second term is related to three-body interactions, and so on1.
1Here is a useful leisure time exercise for a very attentive reader. The purpose is to understand the connection between virial expansion (8.7) and the well known van der Waals equation of state (i.e., the relationship between volume, pressure, and tempera-ture) for an ordinary imperfect gas. You may have studied van der Waals equation in general physics and/or general chemistry class, it reads (p+a/V2)(V b) = N kBT . Say, the volume is V , and the number of molecules in the gas is N . Then n = N/V . You can work out the pressure by di↵erentiation: p = (@F/@V ), where free energy F is defined by formula (7.19), F = U T S = U + Ue↵, the internal energy U is given by (8.7), and
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Notice that B has the units of volume, while C has the units of vol-ume squared. Thus, the energy of all the binary interactions between the segments of a coil is:
U = V kBT n2B , (8.8)
where n is the average segment density in the coil (the number of segments per unit volume inside one coil).
8.5 Good and Bad Solvents, and ✓ Conditions
We have already discussed the potential u(r) in Figure 8.1 a. Repulsion between the segments dominates at higher temperatures (" kT ) (the excluded volume e↵ect), whereas at lower temperatures (" kT ) attraction takes over. Let’s look at the higher temperature region first. The most important values of r are those where u(r) > 0. So the internal energy of the coil (as well as the second virial coefficient) is positive. In contrast, at lower temperatures it is the “attractive” part of u(r), where u(r) < 0, that gives the biggest contribution. So the internal energy of the coil U and B are both negative. In the former case we say that we are dealing with a good solvent, and in the latter case with a bad one. We are not being biased! If in a solvent the segments of polymer chains tend to repel each other, the polymer will dissolve. Conversely, if the segments attract each other, the polymer chains will be rather “sticky”; in other words, they will stick together and precipitate out rather than dissolve.
The quality of a solvent (i.e., whether it is good or bad) may change with its contents or with temperature. Hence, there has to be a special
entropy S or Ue↵ = T S can be thought to be the same as for an ideal gas. Here, we should warn the reader: we never in this book wrote the relation for ideal gas entropy, we only wrote formula for the change of this quantity between two states of di↵erent volumes but the same numbers of particles, this was formula (7.13) — pay attention to on both sides of that formula! In fact, the corresponding formula in general should read Ue↵ = kBT ln(eV /N )N. The extra term N ln N/e arises because particles in the gas are identical; in this book, we do not discuss it, because monomers in the polymer chain are not identical, each of them has its own place along the chain! Besides, this extra term is also not relevant for the determination of pressure, for it cancels away upon di↵erentiation. Thus, using equations above you can derive an equation of state. Check that this leads to the ideal gas equation pV = N kBT if one takes B = 0 and C = 0. For non-zero B and C, compare your answer with the van der Waals equation of state; they are not the same, but prove that your equation, just like van der Waals one, indicates a single value of volume (or density) for any pressure at high enough temperatures (e.g., B > 0 and C > 0), but predicts two possible stable densities (i.e., phase segregation) in a certain interval of pressures at lower temperature. Good luck!
point where the second virial coefficient goes through zero: B = 0. It is usually called the ✓-point (or ✓-temperature — obviously this is the temper-ature when B = 0). At the ✓-point, attraction and repulsion between the segments completely cancel out, and the behavior of the polymer becomes ideal. When T > ✓, repulsion dominates. This is the excluded volume (and good solvent) region. In contrast, when T < ✓ attraction prevails, making the solvent bad. We can now rephrase our initial problem. The swelling of a polymer due to the excluded volume e↵ect is the same as the swelling of a polymer in a good solvent, that is, at T > ✓.
You may wonder why such ✓-conditions are possible in the first place. Is it a mere coincidence that at a certain point repulsion and attraction are so perfectly balanced? For instance, such balancing, or compensation, never quite happens in a real gas. Historically, Boyle found that his law (pV = const for a gas at fixed temperature) is followed at some temperatures more accurately than at others, but never quite perfectly; in modern language, we can say that the gas should be close to ideal at the temperature (called Boyle’s point) when B = 0, but it is not quite ideal because C = 0. By contrast, compensation between attraction and repulsion is indeed nearly perfect for a polymer coil. Why? The answer is that the cancelation only works because three-body interactions (and all the higher ones) are not important. Their contribution to U is always very small. As for the binary collision term (8.8), it is proportional to B, so it falls to zero at the ✓-point. Hence, all that really remains of the free energy F at T = ✓ is the entropy term (see (7.19)). This is why the coil’s behavior becomes ideal.
Thus, the existence of the ✓ point (where the segment interactions have no influence on the shape of the chain) is yet another peculiarity of poly-mers. It is all to do with the very low segment concentration n N 1/2.
8.6 The Swelling of a Polymer Coil in a Good Solvent
Let’s consider an isolated polymer coil in a good solvent (B > 0), and try to find its swelling coefficient ↵. The first calculation of this sort was done by P. Flory in 1949. His approach was as follows. The main cause of the swelling is repulsion between the segments inside the coil (the binary collisions). However, there is also an e↵ect that hinders swelling, arising from the elastic forces whose origin is due to entropy (we discussed them in Chapter 7). These forces emerge because there are fewer di↵erent shapes
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that the chain can take when it is straightened out (or swollen). So Flory’s idea was to obtain the swelling coefficient ↵ from a balance condition be-tween the repulsive and elastic forces.
Both factors contribute to the free energy of a swollen polymer coil (with swelling coefficient ↵), F (↵) = U (↵) T S(↵) (see Section 7.8, Equa-tion 7.19). The potential energy term U (↵) is determined by the repulsive interactions (see (8.8)):
U (↵) = V kBT n2B kBT R3BN2 R6
kBT BN1/2
`3↵3 . (8.9) To write (8.9), we used the following straightforward relationships: V R3, n N/R3, ↵ = R/R0= R/N1/2`, where R0 N1/2` is the size of an ideal polymer coil. As usual, we leave out numerical factors as we are only making estimates. Likewise, the entropy term S(↵) in the free energy of the swollen coil is in one-to-one correspondence with elastic forces. We can work it out from (7.4): Thus, the free energy F (↵) should be obtained by combining (8.9) and (8.10). In doing so, we should be careful about the fact that we left out (and do not know) the numerical coefficient, e.g., in formula (8.9) for internal energy. To make sure that this unknown coefficient does not corrupt all our theory, let’s denote it by some letter, say , and introduce it to the formula (8.9), replacing then the sign with definitive =; then we can write
F (↵) = U (↵) T S(↵) = const + kBT BN1/2
`3↵3 +3
2kT ↵2 , (8.11) where “const”, as before, is independent of ↵.
The function F (↵) is sketched in Figure 8.4. You can see a minimum in the curve at a certain ↵. The minimum of the free energy always gives the equilibrium state. So the equilibrium swelling coefficient is just the value of ↵ at the minimum. Notice that the unknown numerical value of the coefficient does not a↵ect the qualitative shape of F (↵) curve, but does somewhat a↵ect the value of ↵ in the minimum.
Can we find out where exactly the minimum is? The usual way is to di↵erentiate F (↵) with respect to ↵, and to set the derivative equal to zero.
We obtain
@F
@↵ = 3 kBT N1/2B
`3↵4 + 3kBT ↵ = 0 . (8.12)
A/Aeq
Fig. 8.4 The dependence F (↵) given by Equation (8.11). In order to present the most universal plot possible, indepen-dent on the values of parameters, such as N, B, and `, we re-write Equation (8.11) in the following way, using our re-sult (8.13) for the equilibrium value of ↵:
F = 52 B`3N 2/5 25 ↵↵eq 3+35 ↵↵
eq 2
where we also dropped the irrelevant addi-tive constant. In this form, the F /Feqratio becomes a universal function of the ↵/↵eq
ratio; this function is plotted in the figure.
From here, the equilibrium value of ↵ is
↵5eq= BN1/2 In the latter formula we returned to the use of the sign (i.e. “of order of”): we dropped again the numerical factor , because it does not a↵ect the most important feature of the result, namely, the dependence of the equilibrium value of swelling parameter ↵ on the number of monomers N and on the properties of monomers B/`3. We would not be able to find anyway within this simple theory. All that Flory’s theory really gives are the power indices in equations like (8.2). Indeed, from (8.13), the size R of the coil is estimated as:
R ↵R0 ↵N1/2` `N3/5 B
`3
1/5
. (8.14)
The theoretical result R N3/5 agrees reasonably well with the outcome (8.2) of computer simulations.
In the future, we will ignore coefficients similar to when combining various contributions to the free energy.
Using a similar method, we can also explain Equation (8.3) which is the two-dimensional equivalent. Fortunately, the expression (8.10) for S(↵) remains the same. As for U (↵), we need to be careful. The two-dimensional
“volume” is not V R3as usual, but V R2 ↵2N `2. Therefore, instead are given by formulas (8.15) and (8.10), respectively. This time around, we
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don’t have to worry about introducing the coefficient, because we know by experience that it will be dropped out at the end anyway. Therefore, we simply work out where F (↵) reaches a minimum, using the same idea as before. The answer is:
↵ BN
`2
1/4
. (8.16)
Finally,
R ↵R0 ↵N1/2` `N3/4 B
`2
1/4
, (8.17)
in total agreement with (8.3). In fact, it turns out that the result of Flory theory happens to be exact in two dimensions, while in three dimensions it is only approximate, even if reasonably accurate.
Thus, if we allow for the excluded volume e↵ect, the average size of a polymer coil will no longer be proportional to N1/2(as for an ideal chain), but to N3/5 in three dimensions, and to N3/4 in a plane. So, just as we expected, the excluded volume e↵ect is quite significant. This is despite the
Thus, if we allow for the excluded volume e↵ect, the average size of a polymer coil will no longer be proportional to N1/2(as for an ideal chain), but to N3/5 in three dimensions, and to N3/4 in a plane. So, just as we expected, the excluded volume e↵ect is quite significant. This is despite the