Asymptotic Behavior of Random-Flight Model Elasticity Asymptotic Behavior of Random-Flight Model Elasticity

Low force (f < kB T/b):

< x
L
 > = 1
3
 f
kB T/b
 + O(f2)

The low-force result just comes from the leading linear term in the power-series expansion of cothq - 1/q in powers of q = b f/kB T. The inverse of this (force as a function of extension) is also useful:

f = 3 kB T
N b2
 x + O(x2)

Note that the total chain length L = N b.

High force (f > kB T/b):

< x
L
> 1 - kB T/b
f
 + O(exp[-2 b f/kB T)

The high-force result just comes from ignoring the cothq which is 1 plus exponentially small contributions for large q. The inverse of this is:

f = kB T
b
 1
1-x/L
 + ¼

Conclusion - for small forces, flexible polymers behave like linear springs; for large forces, they show a diverging nonlinear force vs. extension.

Work Done Stretching the Random Flight Polymer

When a polymer is stretched to some end-to-end extension x, force must be applied, and therefore work is done. Thus energy can be stored in the deformation of a polymer.

In regular old thermodynamics we recall that dW = p dV; for one-dimensional extension we have dW = f dx. Therefore the total work done during extension of a polymer is just

DW = ó
õ 		x

0 
 dx f(x)

Low forces and small extensions: (f < kB T/b, x < 0.5 L)

In this regime the force comes up linearly with extension x, so the stored energy is just µ x2:

DW = 3 kB T
2 N b2
 x2
This is a fundamental equation of polymer physics. Note that it indicates that the `force constant' of a flexible polymer is k = 3 kB T / (N b2), i.e. that longer and longer polymers have lower and lower force constants and are thus easier and easier to stretch out.

This formula can be rewritten as

DW = 3 kB T
2
 æ
ç
è 		x
R0
ö
÷
ø 		2

 
where R0 = N b2 is the unstretched random coil mean-square end-to-end distance. As you can see, the free energy cost reaches kB T when the end-to-end extension hits about R0, which makes good sense considering typical end-to-end distance fluctuations must be of amplitude » R0.

High forces and large extensions: (f > kB T / b, x > 0.5 L)

The exact formula for all extensions can be found without much trouble. But it is really simple to figure out how the work done must diverge as you force x ® 1:

DW » constant + kB T
b
 ó
õ 		x

0.5 
 dx 1
1 - x/L
= constant - kB T L
b
 ln( 1 - x/L)
= constant - N kB T ln(1-x/L)
To force x ® 1 requires large forces, to overcome the progressively larger Brownian forces encountered as the chain is forced to become more and more extended. As before, this diverging force is purely of entropic origin (it goes to zero if T ® 0).

The work done DW represents real work that could be recovered by slowly allowing the polymer to relax against some load. Since all of this is being done at constant temperature, we see that DW must also represent the free energy (call it Helmholtz F or Gibbs G depending on your taste) relative to a random coil, of the polymer itself. We will tend to refer to this free energy as `stretching free energy' Fstretch.

File translated from TEX by TTH, version 2.53.
On 25 Feb 2001, 09:22.