聚合物物理化学课件 lec12

F glob kT
≈N
b d
τ tr ∝ 1 ⇒ τ tr ≈
2
d bN
∝
1
N
Thus poor solvent regime is located below
τ<−
d bN
∝−
1
N
Lecture 12
Confining Polymer Chain
Consider a polymer chain confined inside a pore of size D
D
R
For a chain in a good solvent the number of monomers in a compression blob of size D is
2
This energy can also be written as number of thermal blobs times energy of a blob
F glob kT
≈−
τb d 3 ≈ −N τ ξT d gT
2
N
2 gT
b
2
Surface Energy of a Globule
F conf
2 R2 Nb ≈ kT Nb 2 + R 2

Dependence of Chain Size on Temperature (τ <0)
Total free energy of a chain is
F kT
≈
R
2 2
Nb
+
Nb R
2
2
+τ
b dN R
3
2
≈
d b
2
2
gT
⇒ gT ≈ τ b
d
4
−2
Bulk Energy of a Globule
The free energy of a globule is equal to
F glob kT
≈ − τ db 2
N
3
2
R glob
+ b 3d 3
N
6
3
R glob
≈ −N
b d
τ
The coil-globule transition is the second order transition because the 2 F ∝ − τ free energy of a globule is proportional to glob For chains of finite length the actual transition takes place at when effective temperatures when the globule free energy becomes of the order of the thermal energy kT .
F surf kT
≈
R glob
2
ξT
2
2 τ ≈N b d 3 ξT
2 gT
2/3 d
1/ 3
b
τ
4/3
Total energy of a globule
F glob kT
≈ −N
b d
τ +N
2
2 /3 d
1/ 3
b
τ
4/3
Coil-Globule Transition
D ≈ bg 3 / 5
Solving this equation one can find the number of monomers in a blob 5/3
g ≈ (D / b )
≈D
N
The length of a chain in a tube can be estimated as
R
The polymer globule can be viewed as densely packed aggregate of thermal blobs. The conformation of a chain inside a thermal blob is unperturbed by interactions.
F kT d b
≈ −τ
Nα
−3
+ α −6 b
d
3
The equilibrium chain size is obtained from minimization of a free energy of a chain with respect to α
∂F ∂F ∂α =− =0 − 6 α −7 = 0 ⇔ p = − ∂V ∂α ∂V kT ∂α b b The chain size in a collapsed (globular) state is
ξT
1/ 2 ξT ≈ bg T
The two-body monomer-monomer attractive interactions are of the order of three-body interactions
kTb d τ
2
gT
2
ξ T3
≈ kTb 3 d 3
gT
3
ξ T6
b d ⇒ τ gT ≈ 1 ⇒ gT ≈ τ d b
g ≈ (D / b )
2
The chain size inside a tube is estimated as the random walk 1/ 2 of compression blobs N 1/ 2 R ≈ D ≈ bN g Free energy of confined chain is
F conf
b ≈ kT ≈ kTN g D
N
5/3
R ≈ kT F D
5/3
Confining Ideal Chain
D
R
For an ideal chain the number of monomers in a compression 1/ 2 blob is D ≈ bg and number of monomers in it
2
+
b d N R
6
3
3
3
= α 2 + α −2 + τ
d b
Nα
−3
where α is a swelling ratio α = R / b N Dependence of free energy on α
F kT
(
)
+ α −6 b
d
3
Parameters for plot:
F conf ≈ kT N
1
≈ kTN ≈ kT g D D
b
2
R0
2
Dependence of Chain Size on Temperature
Flory Theory Consider a polymer chain with size R The excluded volume interactions inside chain
b ≈ g D
2 /3
Nb
Free Energy of a Confined Chain
D
R
The free energy of a confined chain is of the order of thermal energy kT per each compression blob
2
4
−2
∝τ
−2
We can also find the blob size and number of monomers inside blob from the requirement of dense packing of blobs
φ≈
Nbd R glob
3 2
≈τ ≈
g T bd
2
ξT 3
1 ∂F
≈ +3 τ
d
Nα
−4
d
3
R glob ≈ b
d b
2 /3
N τ

1/3
N ∝ b τ

1/ 3
The polymer volume fraction inside globule is
φ≈
Nbd R glob
3
2
≈τ
Scaling Picture of Polymer Globule
The globule has an additional contribution to the free energy due to polymer-solvent interface. Origin of surface energy is the different number of neighbors for each blob inside globule and at the globule surface. The surface energy of a globule can be estimated as the number of blobs at the globule surface times the energy of a blob inside a globule.
20
τ =0 τ =-0.05
1
10
d/b=0.5 N=1000
τ =-0.1
2
34α源自-10τ =-0.2
Collapse of Polymer Chain
To analyze a collapse of polymer chain it is sufficient to consider two body and three body interaction terms.
R
F int
vN 2 wN 3 ≈ kT R3 + R6 ≈ kT
two body three body
b 2 dN τ R 3
2
+
3 3 3 b d N
R
6

τ is the effective temperature τ =1-θ /T
Conformational part of the chain free energy is associated with chain confinement and chain stretching