Entropic Elasticity of a Polymer Network

We have explored what happens when an individual polymer chain is stretched. This was not just an exercise. We have shown that the elasticity of a network is built up from the elasticities of all the subchains (Figure 7.2), so we can make use of what we have found. There is one tricky ques-tion though. Let’s imagine a highly elastic solid body, say, a rubber ball.

The macromolecules are rather closely packed in it and interact strongly with each other. So can we really treat each subchain as an ideal polymer, with no volume interactions at all?

The answer is that we can. Of course, in such a dense structure, the thermal motion of the molecules will be nothing like that of an ideal sin-gle chain. Atomic groups within one monomer will oscillate and rotate in a totally di↵erent fashion. However, the density of the surroundings will make no di↵erence to the entangled shape of the macromolecules (i.e.

the size of a coil will still be proportional to the square root of the chain length). The Gaussian distribution (6.16) will not be a↵ected either. In general, the large-scale properties of chains are the same for both ideal and highly-elastic polymers. This idea was voiced clearly for the first time by P. Flory in 1949; thus it is often called the Flory theorem. You can explain it qualitatively in this way. In a uniform, amorphous substance all the conformations of a certain chain are equally likely (in the sense that they correspond to the same energy of interaction with the other chains). This is because the surroundings of each unit are roughly the same. But this is the only assumption we actually made when deriving the elasticity of an ideal polymer.

Now, as we are convinced we are on the right track, let’s investigate the stretching of a polymer network (see Figure 7.2). We shall treat it as a set of ideal subchains. Suppose each subchain consists of N freely-jointed segments, each of length `. (To make it simpler, we neglect the polydispersity of the polymer.) When the network is stretched, all the subchains are also stretched on average. Their entropy (7.4) decreases (as the end-to-end distance R grows). This causes an “entropic” elastic force. It does not explain the high elasticity yet. The high elasticity is the capability of bearing huge reversible strains at rather moderate stresses.

It occurs because the “elastic modulus” of each chain is fairly small (see (7.20)).

Imagine a polymer network in the shape of rectangular parallelepiped.

Let’s draw the x-, y-, and z-axes along its sides. Suppose we have elongated

130 Giant Molecules: Here, There, and Everywhere

the network by factors x, y, and zalong these axes (respectively). Then, if the initial length of the network along the x-axis was a0x, it will now be

xa0x, etc. Now we need to make some assumption about how the network is deformed. The simplest is to assume what is called affinity (where the cross-links and the whole network deform in the same way). Say, the end-to-end distance of a certain subchain was initially R0, with components R0x, R0y, and R0z. After the deformation, the vector becomes R such that its components are R0x x, R0y y, and R0z z. According to (7.4), the change in entropy of the subchain is

S(R) = S(R) S(R0) To find the total change in the entropy of the whole network, we have to sum contributions like (7.22) for all the subchains. In other words, we can average over R0, and multiply by the number of subchains, ⌫V , in the network. (Here V is the volume of the sample, and ⌫ is the concentration of subchains per unit volume.) Now we can take into account that

⌦R²₀↵

(see (6.11)). We also know that all the three directions (x, y, and z) have equal rights, therefore⌦ It is interesting that the answer does not depend on the parameters N and ` which describe an individual subchain. This indicates that Equation (7.25) is universal. It works whatever the particular structure of the subchains (for instance, regardless of whether they are freely-jointed or wormlike), for whatever contour lengths and Kuhn lengths, and so on. If we glance again at our calculations, we can see that basically all we needed to draw the main conclusion (7.25) was just to regard the subchains as ideal.

We can use (7.25) to find the stress caused by the “entropic” elasticity, for all sorts of deformations. Obviously, one of the most important types of

deformation is the uni-axial elongation (or compression). Let’s see what we can get out of (7.25) in this case. Suppose we have elongated the sample by the factor of along the x-axis, i.e. x= . The size of the network along the y and z coordinates may change freely. Can we find the relative deformations y and z in this case?

Remember that we are talking about a polymer in a highly elastic state.

It seems a sensible assumption that its volume has not changed under the strain. Then, both the y-size and the z-size of the sample ought to have shrunk by a factor of ^1/2, that is, y = z = ^1/2. Thus the total volume after the deformation would not change:

V = xa0x ya0y za0z= x y zV0= V0 . (7.26) How can we justify, physically, that the volume has to be constant? A highly elastic polymer is usually a sort of fluid (a polymer melt). Its chains are linked with chemical bonds. So, if squashed in all directions, such a polymer is bound to behave as an ordinary liquid. In particular, a 1% change in volume can only be achieved with a pressure of roughly 100 atm 10⁷ Pa.

At the same time, the elastic modulus of such polymer is fairly small.

Therefore, the sample can be stretched a few times its length with a much smaller stress ( 10⁵ or 10⁶ Pa). So it is only natural to assume that the volume does not change under such low stresses.

From this point of view, elastic polymers are di↵erent form ordinary solid crystals and glasses, which change their volume just because their length changes. At a molecular level, this di↵erence is not surprising.

When crystals are elongated, their atoms are pulled further apart. Mean-while, polymers increase their length by merely disentangling, uncoiling, and stretching out their wiggly subchains; this way the distances between the atoms are kept unchanged.

There is no problem in finding the elongating force here, using a formula similar to (7.12):

More often we are not interested in the force as such, but rather in the stress, i.e. the force per unit cross-sectional area. There is a little subtlety

132 Giant Molecules: Here, There, and Everywhere

of which we should inform the reader: in defining stress, one can divide force by the actual cross-section at the current state of deformation, or by the initial cross-section of the undeformed sample; the former quantity is called true stress, while the latter is usually dubbed as engineering stress.

Engineering stress is much easier to find in practice, which is why it is most commonly used. We are also using engineering stress throughout this book (see, e.g., Figure 4.4 and its caption). For our present task, engineering stress is computed as follows:

Therefore, let’s rewrite our answer in terms of :

= kBT ⌫

✓ 1

2

◆

. (7.30)

The result (7.30) is a major one in the classical theory for the high elasticity of polymer networks. If the elongation is small (i.e. is close to one), Equation (7.30) can be used to estimate Young’s modulus of a polymer network (see (4.1)). Indeed, in the limit of 1:

1

2 = ( 1)+( + 1)( 1)

2 ( 1)+(2)( 1)

1² = 3( 1) . (7.31) Meanwhile, the value 1 (ax a0x)/a0xis just the relative elongation.

In other words, it plays the same role as the parameter `/` in Equa-tion (4.1). Comparing (4.1), (7.30), and (7.31), we end up with Young’s modulus:

E = 3kBT ⌫ . (7.32)

Thus, E turns out to be the same as the pressure of an ideal gas whose molecular concentration is 3⌫ (i.e. three times the concentration of the cross-links). It means that the more cross-links there are in a highly elastic sample, the less elastic it is. Therefore, the value of E does not indicate a specific polymer. It varies dramatically depending on the density of the cross-links.

However, (7.30) can be used not only to find Young’s modulus. It also describes the nonlinear elasticity, which takes up quite a lot of room on the stress versus strain curve. (In Figure 7.1, it spans from point A where the elasticity ceases being linear up to point B where the reversibility is lost.) What is more, Equation (7.30) is just as good for uni-axial compression.

You only need to bear in mind that will be less than one in this case.

Another warning is that when compressed along the x-axis, the sample will

automatically stretch in both the y and z directions. Even more complex deformations, such as two-dimensional elongation, torsion, shear, and so on are covered by the general relationship (7.25). Although we shall not do it here, you can derive equations similar to (7.30), revealing nonlinear behavior of the stress.

Thus, the result (7.30) ( ) dependence is pretty general — but how accurate is it when compared with experiments? Figure 7.8 brings together both a typical experimental curve and the theory. You can see that up to about = 5 the agreement is far from perfect, but more or less tolera-ble. Then, for > 5, the discrepancy grows more and more. This is not surprising. Expression (6.16) for PN(R) ceases to work for long end-to-end distances R (or, equivalently, large elongations). Why? Because it does not take into account that there is a limit to how much the chains can actually be stretched. Namely, the distance R can never exceed the total contour length N `. This is why Equation (7.27) and all the consequent results will not hold for the case of strong elongation.

Let’s look at the range of moderate elongations: 1.2 < < 5. For most polymer networks, typical discrepancies between the theoretical and experimental ( ) are not that high (about 20% or so), but they tend to be systematic (Figure 7.8). These are explained by the so-called topological constraints to the subchains’ conformations (see Section 2.6).

1 2 3 4 5 6 7 strain for a highly elastic poly-mer network material. Solid line is the theory (7.30); dots show a typical experimental curve (see caption of the Figure 4.4 about the definition of engineering stress). Equilibrium module used to plot the theoretical curve is 3⌫k_BT 3.3 MPa, which cor-responds to ⌫ 0.27 nm ³ — roughly one cross-link per four cubic nanometers. Although the data presented are quite typical in the sense that data first go below theoretical curve and then shoot up above it, the value of ⌫ and the corresponding values of stress can easily change by an order of magnitude either way in di↵erent materials.

Experimental data courtesy of A.A. Askadskii were obtained at room temperature using the sample of polyurethane derived from methylenediphenyl diisocyanate and polyester based on hexanedioic acid and 1,2-ethanediol (the chain extender is 1,4-butanediol). The sample of initial length 25 mm was stretched at the rate 0.16 mm/s, which is very slow. The stress, which has the dimensionality of pressure, is given in the units of megaPascal: 1 MPa = 1_nm^pN₂.

134 Giant Molecules: Here, There, and Everywhere

Despite all the imperfect agreement with experiment, this approach to the description of high elasticity has proved ideologically quite successful, especially because it is universal. All the predictions are satisfactorily accu-rate, whether it is absolute values of Young’s moduli, or their temperature dependencies, or the shape of the nonlinear stress versus strain curve. There are not many other examples in the physics of disordered solids and liquids where such simple arguments have helped to understand so much. The rea-son is rather obvious. The way the chains are entangled and rolled up into coils is not described by the short-scale chemical structure or interactions of individual atomic groups. It is determined by the very fact that the monomers are grouped into chains. So it is a long-scale feature. Soon we shall have a chance to discover that some other most interesting and peculiar properties of polymers have the same sort of origin.

7.11 The Guch–Joule E↵ect and Thermal Aspects