Biophysics II // lecture 1 Polymers and Random Walks Julien Varennes January 14, 2015

1

What is a polymer?

A linear polymer is a chain with 2 ends and a bunch of repeating groups in the middle. A co-polymer has different sets of repeats within its chain. block : ... -X-X-X- ... -Y-Y-Y- ... ordered : ... -X-Y-X-Y-X-Y- ... How do we deal with polymers? At short distances, chemistry governs what will happen. At short distances what the chain is comprised of is very important. At long distances, physics (universality) does. At long distances we don’t care exactly what the chain is made out of, we just care that we have a long chain of stuff. In this discussion we will specifically deal with ideal polymer chains. Ideal polymers are made of phantom chains. i.e. the different parts of the chain can cross over themselves and overlap.

1.1

Ideal Polymers

Freely jointed chains compose the set {u~n }N ~n | = b0 . All vectors u~n have the same n=1 with |u magnitude but can have completely different orientation. The probability of any configuration is the following: Ψ({u~N }) =

N Y

ψ(u~n )

n=1

ψ(u~n ) =

1 δ(|u~n | − b0 ) 4πb20

1 ψ(u~n ) is the probability of a configuration of one single chain link. The factor of 4πb 2 is for 0 R normalization (in 3d), i.e. du~n ψ(u~n ) = 1. The following properties follow from these two definitions. Z < ~u >= d~u~uψ(~u) = 0

1

2

Z

< ~u >=

d~u~u2 ψ(~u) = b20

~ = PN u~n , and so < R ~ 2 >= P u~m · u~n . Since We can define the polymer chain as R n=1 m,n ~ >= 0. We can calculate < R ~ 2 > the following way: ~u = 0 for all n, < R X X X X ~2 = R < u~n 2 > + < u~m > · < u~n >= b20 + 0 = N b20 n

n

n6=m

n6=m

~ 2 >= b0 L. Note, b0 is commonly referred to Let L = N b0 be the contour length, and so < R as the Kuhn Length, lk . With phantom chains this work can be easily done in any number of dimensions. The only thing that would need to change is the normalization factor in ψ(u~n ). 1.1.1

In the large N limit

Chain size compared to total length: ~2 > 1 > 1, we see that the chain is small compared to its length. Chain volume compared to total volume (in d dimensions). 1 N bd0 N bd = d/20 d = d−2 ~ 2 >d/2 N b0
If d > 2 and N >> 1, then N

1.2

d−2 2

→ 0. The chain is said to be diffuse.

~ Distribution of R

~ is PN (R). ~ The probability distribution of R ~ = PN (R)

Z

~− du~1 ...du~N Ψ({u~N })δ(R

N X

u~n )

n=1

let δ(~r) =

1 (2π)d

Z

~ d~keik~r

~ = → PN (R) let g(~k) =

Z

Z

Z N d~k i~kR~ i~k~ u e d~ue ψ(~u) (2π)d

~

d~ueik~u ψ(~u)

2

1.2.1

For the case d = 3 g(~k) =

Z

~

d~ue−k~u

1 sin(kb0 ) δ(|u~n | − b0 ) = 2 4πb0 kb0

~ = 1 → PN (R) (2π)3

Z

~ i~kR

d~ke



sin(kb0 ) kb0

N

~ is exact! The above result for PN (R) For the case when N >> 1, (g(~k))N is dominated by small k. Therefore, we can do a Taylor expansion around small k. In doing so we will use the following expansion and approximation:   sin x 1 x3 x2 2 = x− + ... ≈ 1 − ≈ e−x /6 x x 6 6 ~ = → PN (R)

1 (2π)3

Z

2 ~~ d~keikR e−N (kb0 ) /6 ≈



3 2πN b20

3/2 e

−3R3 2N b2 0

3/2 −3R2 3 ~ ≈ PN (R) e 2 ~2 > 2π < R This is a Gaussian distribution. Note, we never mentioned anything about the ends of the chain and as long as N is big our formulations are correct. 

1.2.2

For d dimensions ~ = P (R)

2



d ~2 > 2π < R

3/2

−dR2

e 2

What if the chain is not freely jointed?

What happens when we are no longer dealing with an ideal chain? The chain links {u~n }N n=1 now have some restriction(s) on their orientation. This means that ψ(~u) will change.

2.1

Freely Rotating Chain

In this case, the chain links now have restricted angle between one another. Each link must ~ 2 > since now we have be at a fixed angle β with its neighbors. This will change < R 2 < u~n · u~m >= b0 cos β 6= 0. Fixing ~u1 ...~un−1 implies < ~un >= b0 cos(β)ˆ un−1 . →< ~un · ~um >= cos(β) < ~un−1 · ~um > →< ~un · ~um >= b20 (cos β)|n−m| 3

< uˆn · uˆm >= e−|n−m| ln(cos β) e−b0 |n−m|/ξ b0 0 ξ = ln(cos and is known as the persistence length. For this case, as β → 0, ξ → 2b . The β) β2 persistence length is the length scale over which previous orientations affect the following chain link, after a distance ξ the chain link’s orientation is random.

4