There are two major models used to describe polymer stretching.  These are the FJC and the WLC models.
 

Excerpt from Jason Bemis's comprehensive exam [PDF](11/19/1998)
 
 

FJC model

The basis of many single polymer theories stem from Flory's Freely Jointed Chain (FJC) model.1,2 This model assumes that chemical bonds are free to rotate and posses a uniform distribution of bond angles. The end-to-end distance, or chain vector (r) is described by Equation 1. This is the summation over all bond vectors (Li), where n+1 is the number of bond in the chain, Figure 1. The uniform distribution of bond angles necessitates that the average chain vector over all conformations is zero. The square of the chain vector averaged over all conformations however, has a finite value. With the assumption that L is the average bond length the average square chain vector simplifies to Equation 2, where the brackets denote the statistical average over all conformations. Flory uses this relation to define the characteristic ratio (Cn) of a polymer, Equation 3.1
 
(1)
(2)
(3) Figure 1  Worm Like chain with fixed bond length and bond angle (109.5), and random dihedral angle

 
 
 
 
 

By definition Cn is unity for freely jointed chains, for other models, such as the worm like chain (WLC), which do not assume that the bond angle is free to rotate, Cn exceeds unity.

Based on viscosity studies conducted by Debye, it is well known that r2 is related to the radius of gyration by Equation 4.3 In the absence of solvent effects the elastic properties
(4)
of such an ideal chain are given by the Kuhn statistical segment length (Ak), Equation 5. The entropic force of uniaxial extension (f) is described by the inverse Langevin function (*), where (R) is defined as coth(R)-1/R. R is the extension ratio, the fraction of the contour length that the polymer is extended. The freely jointed chain is defined as having a Kuhn length equal to the bond length, Equation 6.
 
(5) (6)


 
 

WLC model


A more realistic model for many systems is that of the Freely Rotating Chain, or Worm Like Chain (WLC). This model assumes that the bond angles are fixed at 180-q, but are free to rotate, giving rise to a uniform distribution of dihedral angles. Random conformations were calculated for both models and are shown below (Figure 2). It is difficult to see the three dimensional effect of the
rigid bond angle in the two dimensional figure, so the hypothetical case of restricting the dihedral angle to 0 or 180, to force the molecule into a plane was used to illustrate the two models (Figure 2c and d). The restriction on the bond angle gives rise to Equation 7. In the limit where n® ¥, Cn reduces to C, Equation 8. If n>200 or so Cn is approximately equal to C. The effect of the bond angle and degree of polymerization on Cn is shown in Figure 3. The force of extension for a WLC is given by Equation 9.4 The characteristic length describing the force of extension in the WLC model in is the persistence length (q). This length is defined by Equation 10.5 The persistence length is the sum of the projections of the bonds onto the unit units vector (uo) of the polymer chain.  The unit vector of the polymer chain is the vector of the first bond divided by the bond length.  Essentially q described how far the polymer extends in a given direction before becoming random. Figure 2

A:  Freely jointed chain in three dimensions

B: WLC in three dimensions with a fixed bond angle of 109.5 degrees.

C: FJC confined to two dimensions

D: WLC confined to two dimensions with a fixed bond angle of 109.5 degrees.

 
 
 
 
(7)
(8) Figure 3
Cn as a function of degree of polymerization (N) and bond angle (q)
(9)
(10)

There are many more polymer stretching theories, such as Rotational Isomeric State (RIS) which places restrictions on the available dihedral angles in the WLC model.6 There are also many theories pertaining to volume exclusion and solvent quality.7,8

The tube model, developed by Edwards and Doi.9-11  This model describes the polymer as a tube which is free to move within a network of fixed constraints
 

References
1 Flory, P.J.  In Statistical Mechanics of Chain Molecules, 1969, Interscience Publishers, New York.
2 Also referred to as the random flight, the random coil, or in the limit n®¥, the Gaussian chain.
3 Debye, P. J. Chem. Phys. 1946, 14, 636.
4 Bustamante, C.; Marko, J.; Siggia, E.; Smith, S. Science, 1994, 265, 1599-1600.
5 Yamakawa, H. In Helical Wormlike Chains in Polymer Solutions, 1997, Springer, New York.
6 Volkenstein, M. V. In Configurational Statistics of Polymer Chains, 1963, Interscience, New York.
7 Pincus, P. Macromolecules, 1976, 3, 386-388.
8 Halperin, A.; Zhulina, E. B. Europhysics letters, 1991, 15, 417-421.  Halperin, A.; Zhulina, E. B. Macromolecules, 1991, 24, 5393-5397.
9 Doi, M.; Edwards, S. F. In The Theory of Polymer Dynamics, 1986, Oxford, Oxford.
10 Ball, R. C.; Doi, M.; Edwards, S. F.; Warner, M. Polymer, 1981, 22, 1010-1018.
11 Edwards, S. F.; Vilgis, T. A. Rep. Prog. Phys. 1988, 51, 243-297.