Download Soft-Condensed Matter Physics

Soft-Condensed Matter Physics
HW 2: Dynamics in Polymer Systems
Due: 27th April., 2005
(1) [ Smoluchowski Equation & Browian Dynamics ] The Brownian dynamics of many particles
at positions {~rn } under a potential U is described by the time-dependent distribution function
Ψ({~rn }, t). The damping/hydrodyanmics effects of the solvent is governed by the mobility
P (3x3)
matrices Lnm and the average velocity of the nth Brownian particle is given by ~vn = − Pm Lnm ·
∂~rm U . Continuity equation relates the total flux to the rate of change of Ψ via ∂Ψ
rm ·
m ∂~
∂t +
(~vm Ψ) = 0. The generalized Fick’s law together with the Einstein’s
relation
give
the
relation
between
P
the current and the chemical potential gradient: ~vn = − m Lnm · ∂~rm (kT ln Ψ + U ). Derive the
Smoluchowki equation
∂Ψ X
=
∂~rn · [Lnm · (kT ∂~rm Ψ + Ψ∂~rm U )].
∂t
n,m
This equation forms the basis of dynamics of polymers and colloidal systems. For the case of a
single Brownian particle with a spatially dependent diffusion coefficient, show that the Smoluchowski
equation is equivalent to the usual Langevin equation
ζ
ζ ~r˙ = −∇U + f~(r) (t) + ∇D
2
where f~(r) is a Gaussian distributed random Browian force with zero mean and hfα (t)fβ (t0 )i =
2ζkT δαβ (t − t0 ).
(r)
(r)
(2) [ Rouse Model ] The Rouse model ignores interactions between the monomers and each monomer
only experiences a damping drag force from the immobile solvent, i.e. in the Smoluchowski equation,
Lnm = δnm
ζ I where I is the 3x3 identity matrix. Show that for a linear polymer chain with N
monomers, the Smoluchowski equation in Rouse model leads to(in the continuum limit) the coupled
Langevin equations
3kT ∂ 2~rn
∂~rn
−ζ
+ f~(r) = 0
2
2
a ∂n
∂t
(r)
(r)
where f~(r) ’s are random forces with zero means and hfnα (t)fmβ (t0 )i = 2ζkT δ(n − m)δαβ (t − t0 ).
By Fourier transforming the polymer index n, solve the above coupled Langevin equations in Rouse
~
~
~ ≡ ~rN − ~r0 ), and the
modes (normal modes). Compute the end-to-end correlation hR(t)
· R(0)i
(R
mean square displacement of the center of mass as functions of time.
(3) [ Hydrodynamic interactions & Zimm Model] The Zimm model takes into account the hydrodynamic interactions between the monomers and the mobility matrix is given by the Oseen tensor
(~
r −~
r )α (~
rn −~
rm )β
Hnn = I/ζ and (Hnm )αβ = 8πηs |~r1n −~rm | [δαβ + n (~m
] for n 6= m. Derive the Zimm
rn −~
rm )2
equation
Z N
3kT ∂ 2~rm
∂~rn
(r)
=
dmHnm · [ 2
+ f~m
].
∂t
a ∂m2
0
Show that by pre-averaging Hnm , the Zimm equation is linear under the pre-averaging approximation and solve the pre-averaged Zimm equation in terms of Rouse modes under the Kirkwood
approximation (i.e. neglect interactions between different Rouse modes) . Then compute the end~
~
~ ≡ ~rN − ~r0 ), and the mean square displacement of the center of
to-end correlation hR(t)
· R(0)i
(R
mass as functions of time.
(4) [ Reptation Model] In the reptation model of a dense polymer melt, the motion of a polymer is
described by a 1-d diffusion along the primitive tube of primitive length L. The position of a point
~ t) where 0 ≤ s ≤ L is the distance parameter along the
on the primitive path is denoted by R(s,
~ t+δt) = R(s+δξ,
~
primitive path. The creeping motion is described by R(s,
t) where ξ(t) is a random
2
0
~
~ 0 , t0 ))2 i denote the
variable with zero mean and h(δξ(t)) i = 2Dc δt. Let Ψ(s, s ; t) ≡ h(R(s, t) − R(s
correlation function, derive the diffusion equation for Ψ and the corresponding initial and boundary
conditions. Hence solve for the mean square displacement of a point on the primitive path Ψ(s, s; t)
and deduce the different scaling behavior of the motion of the chain in different time regimes.
(5) [ Viscoelasticity ]
[a] By considering the entropic forces due to polymer chains crossing a plane, show that the stress tensor
of a polymer melt of concentration c is given by
3kT c
σ=
N a2
Z
N
dnh
0
∂~rn ∂~rn
i.
∂n ∂n
[b] For a polymer solution under a flow field given by ~v (~r, t) = κ(t) · ~r, where κ is the velocity gradient
tensor, write down the corresponding Smoluchowski equation. In the framework of Rouse model,
derive the equations for the Rouse modes.
[c] Compute the stress tensor σ for a melt of Rouse chains, and hence calculate the shear relaxation
modulus G(t) and viscosity η of the system.