Download NOTES AND PROBLEM SET 3

1
Oleg Krichevsky
Soft Matter Physics
I.
NOTES AND PROBLEM SET 3
due to one week before exam
Problem Set 3
1. Consider a semiflexible polymer of length L and persistence length lp : L can be either
larger or smaller than lp . Consider this polymer to be confined within the long tube
of diameter D. The tube diameter is much smaller than both L and lp .
Estimate the number of collision points between the tube and the polymer. Estimate
the increase in the free energy due to confinement.
2. A great experimental breakthrough in the mid-1990s was a capability to stretch single
DNA polymers (see works of Chu, Bustamante, Chatenay, Bensimon and their coworkers). For example, one end of DNA polymer is attached to an immobile surface and
another to some kind of a handle which is used to pull the end of DNA and measure
the extension (displacement of that end). At about the same time an exact theoretical
expression was derived by Marko&Siggia relating the applied force f~ to the extension
x and to DNA length L and persistence length lp (for L À lp ). Nevertheless, they
suggested to use an approximate expression, which is simpler than the exact one and
is almost as precise:
x
1
1
f lp
= + ¡
¢2 −
x
kB T
L 4 1−
4
L
(1.1)
This expression is in fact a combination of two limiting cases: those of small forces
and of large forces (how small and how large?). Explain these limiting cases: the
case of small forces you can treat exactly and the case of large forces up to a numeric
coefficient so far (you’ll treat it exactly in the next problem). Explain the way they
are combined in a single expression.
3. Semiflexible polymer (e.g. DNA) with length L À lp is stretched along axis x with
rather large force f , such that f lp > kB T .
Oleg Krichevsky
Soft Matter Physics
2
• We define the projected length Lk of a polymer as, well, the length of its projection
onto x axis: say, if the one end of the polymer is hold at x = 0 then the projected
length of the polymer will be just the x-coordinate of the other end. Show that
under strong stretching as defined above, the projected length can be expressed
as:
1
Lk ≈ L −
2
Z
µ
L
dx
0
∂~r⊥ (x)
∂x
¶2
,
(1.2)
where ~r⊥ is defined in the same way as in Problem 3 of Set 2.
• Show that the Hamiltonian of the chain under large force can be expressed as:
κ
H=
2
Z
µ
L
dx
0
∂ 2~r⊥ (x)
∂x2
¶2
f
+
2
Z
µ
L
dx
0
∂~r⊥ (x)
∂x
¶2
,
(1.3)
where as before κ = kB T lp
• Write the Hamiltonian in terms of spatial Fourier modes (you’d better use periodic boundary conditions) and find the mean square values of mode amplitudes.
• Calculate average projected length hLk i as a function of force f . Notice that hLk i
is the same as extension x in the Problem 2 (this set). Verify that the expression
you get is equivalent to Eq.3.1 in the limit of high forces (including the numeric
coefficient.)
4. One of the experimental methods to measure DNA force-extension dependences (developed by Bensimon group) is based on attachment of one end of the DNA molecule
to the substrate (cover glass), the other end to a magnetic bead and placing DNA in
the magnetic field with a gradient in say Z direction. The bead is pulled inside the
magnetic field and the DNA molecule is stretched. One can measure the displacement
of the bead in Z direction (this is the DNA extension) and the thermal fluctuations in
its position in XY plane. Show that the force exerted on DNA by the magnetic field
can be deduced from these two measurements. It is fairly easy to find this relation if
DNA is fully stretched like a pendulum: we did this in the class. You have to show now
that the same relation holds for any amount of stretch and any polymer properties.
5. Consider a semidilute solution of semiflexible polymers of lengths L ¿ lp . The dynamics a semiflexible polymer in semidilute solution is restricted by an effective tube
very much like that of the flexible polymers. However, the properties of the tube in
Oleg Krichevsky
Soft Matter Physics
3
ξ
ξ
Le
Le
D
D
FIG. 1: Schematics of cross-section of semidilute solutions of semiflexible polymers: red line is a
test polymer, solid circles - cross-sections of other polymers, dashed line - different conformations
of the test polymer consistent with topological constrains. These conformations define the effective
tube diameter D. Distance between two collisions along the polymer contour defines the topological
constrains and the entanglement length Le .
solutions of semiflexible polymers are different from those of solutions of flexible polymers. In particular, one has to distinguish three characteristic sizes - the mesh size ξ,
the effective tube diameter D and the entanglement length Le (i.e. the distance along
the contour of a given polymer between consecutive collisions with other polymers,
see Fig.) For the semidilute solutions of flexible polymers, all these three values are
about the same. However, in solutions of semiflexible polymers they can differ a lot.
• Show that one of the relations between the entanglement length, the mesh size
and tube diameter is: Le = ξ 2 /D. Hint: the treatment is very similar to that of
mean-free path in gases.
• Use the results of previous questions in this set to derive the dependence of Le
and D on monomer concentration.
• Estimate the dependence of osmotic pressure of semidilute semiflexible solutions
on concentration. Compare this dependence to the corresponding dependence for
flexible polymers.