Spatial Configuration of Long Chain Polymers

The study of long chain polymers has been a source of problems within the field of statistical mechanics since about the 1950s. Disordered long chain polymers are too complex to be described using a deterministic method. How-ever statistical approaches can yield results and are often pertinent since large polymers (that is to say, polymers containing a large number of monomers) can be described efficiently as systems at the thermodynamic limit. One of the reasons that scientists were interested in their study is that the equa-tions governing the behavior of a polymer chain are independent of the chain chemistry. The statistical approach to polymer physics is based on an anal-ogy between a polymer and either a Brownian motion, or some other types of random walks.

Random walks in space (or more exactly, random flights since we consider three dimensions) can be thought of as snapshots of the path taken by a random walker in time. The results of random walk analysis have been applied to computer science, physics, ecology, economics and a number of other fields as a fundamental model for random processes in time. For instance, the path traced by a molecule as it travels in a liquid or a gas, the spatial configuration of long chain polymers, and the topic of electronic transport in amorphous photoconductors can all be modeled as random walks. In the present section, we will illustrate such an application for rubber elasticity in polymers.

The freely joined chain is the simplest model of a polymer. In this model, fixed length polymer segments are linearly connected, and all bonds and torsion angles are equiprobable. The polymer can therefore be described by a simple random walk and an ideal chain. The ideal chain model assumes that there are no interactions between chain monomers and polymer segments can overlap each other as if it is a phantom chain. In reality, this occurs when a

2.5 Application of Quantum Statistics 55

single polymer chain is located in bulk polymer materials where the above interactions are effectively canceled out. The ideal chain model provides a good starting point for investigation of more complex systems and is better suited for equations with more parameters. Since two segments cannot occupy the same space at the same time, this interaction between segments can be modeled as excluded volume. This causes a reduction in the conformational possibilities of the chain, and leads to a self-avoiding random walk, which cannot repeat its previous path. This is present when a single polymer chain is in a dilute solution, whose statistics differs from the simple random walk where the idea of the fractal dimension should be introduced.

When the random walk is applied to polymer structures, the walk is rein-terpreted as the physical configuration (static configuration in a glass) of a flexible long-chain molecule, rather than the transitory path of a diffusing particle. Each step of length b (for a uniformly stepping walk) is interpreted as a chemical unit–a monomer segment of the chain. The net displacement magnitude Rb is now the end-to-end length separating the first and last monomers at the two ends of the chain. Above the glass transition tempera-ture Tg, the polymer chain oscillates and Rbchanges over time. The net time averageR^b , or the root mean square (rms) end-to-end length R²b ^1/2(Rrms) is a useful measure of the size of the chain.

By considering an ideal chain, we use N andri to denote the number of steps and the vector position of the i-th link in the chain (|ri| = ri= b). Then the end-to-end vectorRb achieved in walks of N steps isRb = _i=N

i=1 ri and

R²b =

N i=1

ri· ri +

i=j

ri· rj (2.54)

where the angle brackets means an average over all possible walks having exactly N steps, that is configuration average. The second summation is a double sum extending over all values of i and j except for those with i = j. All of the latter diagonal terms, corresponding to the appearance in R_b²=Rb·Rb

of the self-productsri·ri, which represents the square of the length of a given step of the walk, have been separately taken into account in the first sum of Eq. (2.54). Since there are N such self-product terms, and since each contributes b², the first sum is simply N b². On the other hand, the second sum of Eq. (2.54), containing configuration averages over cross termsri· rj, necessarily vanishes because of the assured randomness of the walk. Since two different steps i and j are completely uncorrelated and all orientations of ri andrj occur with equal probability, the averageri· rj of their scalar products, taken over all possible configurations, must equal zero. Hence,

Rrms=R²_b ^1/2= N^1/2· b. (2.55) In addition to Rrms, the full distribution function P (R^b) is also known for random walks. P (R^b) is the probability of finding configuration with R^b.

56 Chapter 2 Statistical Thermodynamics

As no direction is favored over any other, it is isotropic and depends only on the scalar Rb = |R^b|. Thus the frequency of occurrence of end-to-end lengths lying in the range from Rb to Rb+ dRb in configuration space is, in three dimensions, 4πR²bP (Rb)dRb. Assuming that distribution of end-to-end vectors for a very large number of identical polymer chains is Gaussian, the probability distribution has the following form:

P (Rb) =

 3

2πNb²

_3/2

exp−3R^b· R^b

2N b² . (2.56)

Figure 2.1 shows P (Rb) and 4πR²bP (Rb) functions of an assembly of chains assumed to have random-walk conformations and an Rrms of 300 ˚A (a representative value for polymers with N ≈ 10⁵). P (Rb) in Fig. 2.1 provides a linear section, along any radial line, of the spherically symmetric three-dimensional free endpoint distribution P (R^b). The function 4πR²bP (Rb) amounts to the pair correlation function (i.e., the radial density function RDF) for connected chain ends. Its peak occurs at (2/3)^1/2Rrms and its sec-ond moment is R_rms² . Although other linear measures of the region encom-passed by a random coil chain might be adopted (such as the rms distance of the chain segments from its center of gravity, called its “radius of gyration”), they scale each other in the same order.

Fig. 2.1 Distribution functions for the end-to-end distance Rb of an assembly of

“ideal” (i.e., random-walk configuration) chains for which Rrmsis 300 ˚A.

Throughout this discussion, Rb has been employed as a fantastically abridged one-parameter characterization of chain conformation. For each Rb

there exists an astronomical variety of possible configurations. The complete configuration of any particular chain, that is, the full sequence ofri’s of Eq.

(2.54), is virtually unknowable, due to the complete atom-by-atom structure of any amorphous system. Precisely because of the large numbers involved, the statistical approach becomes both necessary and valid.

2.5 Application of Quantum Statistics 57

Upon stretching a material we are doing work on the system to exhibit a limited elastic region while the material regains its original dimensions if the stress is removed. As the resulting strain is related to the extent of movements of atoms from their equilibrium conditions, substances such as crystalline solids and amorphous glasses have elastic limits rarely exceeding 1% because atomic adjustments are localized. The elastic properties of elas-tomers, however, are truly exceptional. Elastomers are polymeric materials, natural or synthetic, that can be stretched to several times its original length without breaking owing to the ability of their constituent polymeric chains to rotate about the chain bonds. By far the most widely studied elastomer is the natural rubber, its deformation is reversible and instantaneous and it shows almost no creep. The reversible character of the deformation is a con-sequence of the fact that rubbers are lightly cross-linked, which prevent the chains from slipping past each other. The chains between adjacent crosslinks contain typically several hundred main chain atoms. The instantaneous de-formation occurring in rubbers is due to the high segmental mobility and thus to the rapid changes in chain conformation of the molecules. The en-ergy barriers between different conformational states must therefore be small compared to the thermal energy. Given our probability distribution function, there is a maximum corresponding to Rb = 0. Physically this amounts to that there are more microstates of chain conformations with an end-to-end vector of zero than any other microstate. Stress acting on the rubber net-work will stretch out and orient the chain between the crosslink joints. This will thus decrease the entropy of the chains and hence give rise to an en-tropic force. The change in chain conformation is expected to change the intramolecular internal energy. The packing of the chains may also change, affecting the intermolecular-related internal energy. Both the intramolecular and intermolecular potentials contribute to the energetic force. The follow-ing thermodynamic treatments yield expression differentiatfollow-ing between the entropic and energetic contributions to the elastic force fe.

According to the first and second laws of thermodynamics, the internal energy change dU consisting of the chain exchanging heat (δQ), deformation and P − V work (δW ) is shown as dU = T dS − P dV + fedRb, where fedRb

is the work done by the deformation. Physically what is more interesting is to consider deformation at constant V in order to view only the direct effects of orientation on entropy and internal intramolecular energy where δW = −P dV = 0. The partial derivative of U with respect to Rbat constant T and V is fe = (∂U/∂Rb)_T,V − T (∂S/∂Rb)_T,V. (∂U/∂Rb)_T,V vanishes for ideal chains, which means that the polymer chains can rotate freely and its U does not change with conformation. We are thus led to

fe=−T

 ∂S

∂Rb



T,V

. (2.57)

Equation (2.57) shows that the elastic force necessarily stems from a purely entropic effect. This entropic force is very similar to P of an ideal gas. U

58 Chapter 2 Statistical Thermodynamics

of an ideal gas depends only on its T , and not on V of its container, so it is not an energy effect that tends to increase V like gas pressure does, or P of an ideal gas has a purely entropic origin. The elasticity of rubbers is predominantly entropy-driven, which brings out a number of spectacular phenomena. The stiffness increases with increasing T and Q is reversibly generated on deformation. What is the microscopic origin of such an entropic force? The most general answer is that the effect of thermal fluctuations tends to bring a thermodynamic system toward a macroscopic state that corresponds to a maximum in the number of microscopic states, which are compatible with this macroscopic state.

Return now to the probability distribution function (Eq. (2.56)) of finding configuration withRb. Recall that according to the principle of equally likely a priori probabilities, Ω at some physical value is directly proportional to the probability distribution at that physical value, viz, Ω(Rb) = cP (Rb), where c is an arbitrary proportionality constant. The entropy associated to a macrostate of an ideal chain is thus equal to

S(Rb) = k lnΩ(Rb) = k ln P (Rb) + C0 (2.58) where C0 is a fixed constant. By combining Eqs. (2.57) and (2.58), it reads

fe=−3kT Rb/(N b²) =−ksRb. (2.59) The above thermodynamic equation is the same as that for the conventional P− V systems where P and V are substituted by −f^eand Rb. The resulting stress-strain relationship is called the equation of state of the ideal chain. It is only exact in the limit of polymers containing a large number of monomers, that is, the thermodynamic limit.

An analogous expression to Eq. (2.57) can be derived: fe= (∂F/∂Rb)_T,V= (∂Ω^∗/∂Rb)_T,V at constant V and T where F = U− T S. Insertion of Eq.

(2.59) into the above equation gives

F =Ω^∗=−3kT Rb²/(2N b²) = ksR²_b/2, ks= 3kT /(N b²). (2.60) Equations (2.59) and (2.60) are known as the entropic spring results. Namely, upon stretching a polymer chain we are doing work on the system to drag it away from its (preferred) equilibrium state and the chain behaves like a conventional spring. Particularly the work can be related entirely to the entropy change of the system.

2.5.2 Statistical Thermodynamics of a Paramagnetic Crystal [3,