Random Flight Polymer

A simple model - and one which will tell us a lot about the physics of polymers - is the random flight model . This model of a N-monomer flexible polymer is made up of N line segments each of length b.

Total (contour) length is L = N b

This polymer model supposes total flexibility between each pair of successive segments, so that in thermal equilibrium each segment can point independently in any direction.

So, each segment can be described as a unit vector of arbitrary direction [^(n)]i, |_i| = 1, times the segment length b. Here the index i runs from 1 to N.

I'll drop the hats on these unit vectors, so I'll ask you to remember that all vectors named n have unit length.

Now we can easily write - for any configuration - the distance between the ends, as

rN -r0 =  N
å
i = 1		 b ni
 
This quantity is called the end-to-end vector.

Now we need to do averaging as a start on doing the statistical mechanics of polymers. We will want to sum (integrate) over all orientations of the unit vectors ni. Unfortunately we need to remember how to integrate on the unit sphere (spherical polar angular coordinates):

ó
õ 

unit sphere

 d2 ni =  ó
õ 		 2p

0

 dfi ó
õ 		 p

0

 dqi sinqi =  ó
õ 		 2p

0

 dfi ó
õ 		 1

-1

 d(cosqi) 
 
Here the `polar' angle (angle between ni and the z-axis) is qi; the `azimutha' angle is fi. This is the physics-chemistry convention, and will be used exclusively here - engineers and mathematicians sometimes use different conventions.

Note that if we actually do these integrals (i.e. with integrand 1), we get the value 4p. This is referred to as the total solid angle on a sphere. The unit of solid angle is sometimes called the steradian, analogous to the idea of a radian on the circle.

Finally, recall the (x,y,z) components of n, which are (cosfsinq, sinfsinq, cosq).

So, now we are ready to compute average quantities, by just integrating over all the possible directions to the ni, independently.

Average value of ni

< ni > =  ó
õ 		 d2 ni
4 p		 ni = 0 
 
If we integrate ni over all directions of ni, we get zero. Nonbelievers can explicitly work out the integrals of components of ni.

Average of Any Function of the Segment Directions on a N-segment polymer

< g(n1,n2, ¼,nN) > =  ó
õ 		 d2 n1
4 p		 ó
õ 		 d2 n2
4 p		 ¼ ó
õ 		 d2 nN
4 p		 g(n1,n2, ¼,nN)
 
This average corresponds to statistical mechanics, if we suppose that all configurations of the random-flight polymer have the same energy, i.e. the probability distribution is just constant.

Average Value of ni ·nj
There are two cases, i = j, and i ¹ j.
For i = j,

< ni ·ni > = < 1 > = 1 
 
For i ¹ j,
< ni ·nj > = < ni > · < nj > = 0
 
The decoupling of the average of the product into a product of averages simply corresponds to a reordering of the integrals (write it out if necessary).

So we can summarize the general case with a handy formula:

< ni ·nj > = di,j
 
where di,j = 0 for i ¹ j and 1 for i = j (this is sometimes called the Kronecker or discrete delta function). Now we are ready to compute.

Average End-To-End Vector for Random Flight Model

< rN -r0 > = <  N
å
i = 1		 b ni > 
 
= b  N
å
i = 1		 < ni > = b  N
å
i = 1		 0 = 0
 
The average is over all configurations and therefore over all orientations of the end-to-end vector, and therefore must be zero.

This does not mean that the two ends of the random-flight model are always at the same place! Next we'll figure out the rough average distance between the ends.

Mean-Square End-to-End DIstance for Random Flight Model

< (rN -r0)2 > = < ( N
å
i = 1		 b ni)2 > 
 
< ( N
å
i = 1		 b ni) ·( N
å
j = 1		 b nj) > = b2 N
å
i = 1		 N
å
j = 1		 < ni ·nj > 
 
but we are prepared for the average now,
b2 N
å
i = 1		 N
å
j = 1		 di,j = b2 N
å
i = 1		 1 = N b2
 
This is a fundamental result - the mean-squared end-to-end distance of a long flexible polymer is Nb2, and therefore the overall size is » N1/2 b.

If this sounds familiar from our discussion of diffusion, it should. It is exactly the same arithmetic responsible for < r2 > = Dt for a diffusing particle.

Elastic Response of the Random-Flight Polymer

We will directly compute the average end-to-end displacement resulting from a force (or tension) applied to the ends of the polymer. To do this we will need to use the Boltzmann distribution including the contribution from force:

exp[ f z / kB T ] 
 
where z is the end-to-end distance in the (z) direction of the applied force f. This factor must now be included in the averaging.

What is important is that z can be written as a sum over contributions from the N segments:

z = (rN -r0) · ^
z		 = b  N
å
i = 1		 ni · ^
z		 = b  N
å
i = 1		 cosqi
 
which means that
exp[ f z / kB T] = exp[b f cosq1 / kB T]exp[b f cosq2 / kB T] ¼exp[b f cosqN / kB T] 
 
Now we can write down the average of z, including the Boltzmann factor, including the proper normalization constant,
< z > = 
ó
õ 		 d2 n1 ó
õ 		 d2 n2 ¼ ó
õ 		 d2 nN z exp[f z / kB T]
ó
õ 		 d2 n1 ó
õ 		 d2 n2 ¼ ó
õ 		 d2 nN exp[f z / kB T] 
 
 
After plugging in the sum for z, you will note that a lot of integrals cancel on the top and bottom. You will be left with a sum of N terms:
< z > = Nb 
ó
õ 		 d2 n cosqexp[bcosq/ kB T]
ó
õ 		 d2 n exp[ b f cosq/ kB T] 
 
 
The form of this is not so surprising. It is just the normalized average value of b cosq added up for the N independent segment directions, which is just the projection of each segment onto the z direction, where the average is done with respect to a probability distribution µ exp[ (b f/kB T) cosq].

You will note that everything is going to be a function of the single dimensionless variable a = b f / kB T. So you see there is a characteristic force of kB T/b associated with the polymer's stretchiness.

Just to get an idea of the scale for this force, note that for a b = 1 nm, we have
kB T/b = 4×10-21 J/10-9 m = 4×10-12 N = 4 piconewtons (pN)
We can already see that the force associated with stretching out thermal bending of a random-flight polymer with nm-long segments will be on the piconewton scale.

Now we just have to do two integrals. The one in the denominator of the above is easier:

ó
õ 		 d2 n exp[acosq] =  ó
õ 		 2p

0

 df ó
õ 		 1

-1

 d(cosq) exp[acosq] = 2p ea - e-a
a		 = 4p sinha
a
 
We can use this integral to figure out the numerator integral, using the trick of differentiation through the intergral by the parameter a:
ó
õ 		 d2 n cosqexp[acosq] =  ó
õ 		 d2 n  d
d a		 exp[acosq]
 
=  d
d a		 ó
õ 		 d2 n exp[acosq] =  d
d a		 4p sinha
a
 
= 4p cosha- a-1 sinha
a
 
So, the ratio of the two integrals is
< z > = Nb [ cotha- 1/a] = Nb L(a) 
 
where the special function L(x) = cothx - 1/x is sometimes called the Langevin function (it played an important role in the theory of magnets).

In elementary terms, we have shown that

< z > 
Nb		 =  ebf/kB T + e-bf/kB T
ebf/kB T - e-bf/kB T		 - kB T
b f
 
for the random-flight model.

This result is plotted below, and for us represents the extension of a polymer in units of its total contour length z/L, versus applied force in units of the special force kB T/b.

We'll move on to dissect this result in the next lecture.

Review of hyperbolic functions

You should all remember that

sinhx =  ex - e-x
2		             coshx =  ex + e-x
2
 
tanhx =  sinhx
coshx		             cothx =  1
tanhx
 
and
d
dx		 sinhx = coshx d
dx		 coshx = sinhx 

Additional Reading

For more background on flexible polymers, see the excellent textbooks by

de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (especially the first few chapters)

Doi and Edwards, Theory of Polymer Dynamics, Oxford  University Press (for graduate students)


File translated from TEX by TTH, version 2.53.
On 15 Feb 2001, 12:27.