I. ELASTICITY OF A SINGLE POLYMER Strands
High molecular weight polymers that exhibit rubber-like behavior are known as elastomers. Above the glass transition temperature, the rubber-like polymers are in a liquid-like state and the mer or repeat units change their position readily and continuously due to Brownian motion. Thus, each polymer chain takes up random conformations in a stress-free state.
When considering conformations of many ideal rubber-like chains or conformations of a single chain at different times, the random distribution of the chain ends can be described by a Gaussian function. The probability of a chain having its ends separated by a distance R(N) is given by
where R0(N) is the average of the squares of the end-to-end distances of the relaxed chain composed of N repeat units.
A tensile or shear deformation induces larger chain dimensions and a reduction in free energy F, which in turn, leads to a retractive force f which is consistent with entropy elasticity:
where Ω is the number of possible chain conformations of a polymer chain with an end-to-end distance R.(1)
In the case of an ideal chain, R0 is independent of temperature because all random arrangements are equally likely. Then the retractive force f in the equation above arises solely from an entropic mechanism, i.e., from the tendency of the chains to adopt conformations of maximum randomness, and not from any energetic preference for certain conformations over others. The tension f is then directly proportional to the absolute temperature T.
The Equations above are only applicable to chains with short end-to-end distances, r2 ≈ Nl2, and in first approximation for distances less than about one-third of the length of fully stretched chains (contour length). For chains with larger end-to-end separation we have to choose a different model. It can be shown that P(N,R) can be estimated with the inverse Langevin function of R/Nl (3,4):
L-1(R / Nl) = L-1(ν) and L(v) = coth(ν) - 1/ν
where L-1 denotes the inverse Langevin function. An expansion of this relation in terms of R/Nl yields
ln P(N,R) = const. - N [3/2·(R/Nl)2 + 9/20·(R/Nl)4 + 99/350·(R/Nl)6 + ...]
whereas the Gaussian distribution leads to the expression ln P(N,R) = const. - N 3R2/2Nl2. With
f = - kT ∂ln P(N,R) / ∂R
the equation above may be written
f = 3kT/l [(R/Nl) + 3/5·(RNl)3 + 99/175·(R/Nl)5 + ...]
This equation predicts a steeper rise in tension when the chain end separation is R/Nl > 0.3 , i.e., when the chain becomes nearly taut (see figure below).
Tension–Displacement Relation for Freely Jointed Chain
The equations above describe the retractive force of a single chain on extension. For small extensions, the retractive force is proportional to R, that is, the chain behaves like a Hookean spring. As will be shown next, this is not the case for a polymer network.
In an entangled or cross-linked elastomer, the chains form a loose three-dimensional molecular network. For polymers of high-molecular-weight, the network could solely consist of entanglements by molecular intertwining (knots) with a spacing between knots characteristic for the particular polymer; or the chains could be chemically crosslinked, that is, the chains form a permanent network, as it is the case for vulcanized rubber. Thus for vulcanized rubber, the effective number n of network chains per unit volume is the sum of two terms, ne and nc, arising from physcial entanglements and chemical crosslinks, respectively:
n = nc + ne
nc = NA ρ / Mc; ne = NA ρ / Me
where ρ is the density of the elastomer, NA is the Avogadro’s number, and Me and Mc denote the average molecular weight between entanglements and between crosslinks, respectively.
To simplify the calculation of the retractive force, we assume that the junction points between the sub-chains move on deformation as they were embedded in an elastic continuum, that is, we assume the individual chains are strained in an identical manner to the entire network. If αi = Li / Li,0 are the extension ratios of the macroscopic rubber cube of volume V = Lx Ly Lz and if Rx,02 = Ry,02 = Rz,02 = Nl2/3 are the mean square end-to-end distances of a strand in the three principal directions, ex, ey, ez, then the average end-to-end distance between two junction points, R, in the strained state in any direction may be written
R2 = 1/3 R02 [αx2 + αy2 + αz2]
where αi ≈ Ri / Ri,0 are the extension ratios, and R is the average size of a rubber strand. Since αx,02 = αy,02 = αz,02 = 1,
R2 - R02 = R02/3 · [(αx2 + αy2 + αz2) - 3]
Then the change in free energy on deformation of N = nV rubber strands is given by:
ΔFel = (Vn 3kT/2) · (R2 - R02 ) / R02 = (VnkT/2) · [αx2 + αy2 + αz2 - 3]
where V = LxLyLz is the volume of the rubber cube and n is the number of network subchains per unit volume. For an incompressible system, αx αy αz = 1, and uniaxial deformation, 1/α1/2 = αy = αz, this expression may be written
ΔFel = (VnkT / 2) · [α2 + 2/α - 3]
and for the tensile force:
fel = ∂Fel / ∂(Lxα) = (VnkT / Lx) · (α - α-2)
Conversion to stress yields
σxx = fel / (LyLzα-1) = nkT · (α2 - α-1)
The Youngs modulus is determined by the slope at the origin, α = 1. One obtaines
E = ∂σxx / ∂α (α = 1) = 3nkT
As these relations demonstrate, the stress-strain relationship of a rubbery material is non-Hookean.
The Young's modulus of a rubber is E = 4.0 MPa and its density 1000 kg/m3 at a temperature of T = 300 K. The molar crosslink density is then
n = E / 3RT = 4 x 106 N/m2 /
(3 x 8.314 Nm/mol-K x 300 K)
= 475 mol/m3,
and the average molecular weight between two crosslinks is
Mc = ρ / n = 1000 kg/m3 / 475 mol/m3 = 2105 g/mol
References and Notes
For an ideal chain, where there are no enthalpic interactions between the repeat units, the total number of states of a chain with fixed ends, R= |r1 - r2|, is given by Ω1 = ZnP(R,N), where Zn is the total number of conformations. Then the total number of conformations of a chain with an end-to-end distance R is Ω2= Zn 4πR2 P(R,N).
For real polymers, consisting of a large number N of primary valence bonds along the chain backbone, each of length l, the true end-to-end distance is noticeably larger than that of a Gaussian chain
R02 = C∞Nl2
where the coefficient C∞ is the so called characeteristic ratio which is the ratio of the unperturbed mean square end-to-end distance to the mean-square end-to-end distance of the freely jointed chain.
C∞ is found to vary from 2 to 10, depending on the chemical structure of the molecule and also on the temperature, because the energetic barrier to bond rotation is more easily overcome at higher temperatures. C∞1/2l can be regarded as the effective bond length of the real chain, which is a measure of the stiffness of the polymer.
- W. Kuhn, Kolloid-Z. 101, 248 (1942), Kolloid-Z. 76, 258 (1936)
- P.J. Flory, Principles of Polymer Chemistry, New York 1953