Example V(φ):
Rotational conformations of n-butane
CH3CH2CH2CH3
Potential energy of a n-butane molecule as
a function of the angle φ of bond rotation.
V(φ)
Potential energy/kJ mol-1
20
eclipse
eclipse
15
eclipse
eclipse
gauche
planar
trans
gauche
Planar trans conformer is lowest energy
10
5
0
- 180o - 120o - 60o
60
φ
o
120
o
H3C CH3
H CH3
H
H
H
CH3
0
o
Views along the C2-C3 bond
H CH3
H
H
H
H
180
High energy
states
CH3
H
H
H
V(φ)
eclipsed conformations
CH3
CH3
CH3
H3C
H
H
H
H
H
H
H
H
H
CH3
H
H
CH3
H
Gauche
Planar
Gauche
Low energy
states
Figure by MIT OCW.
Conformers: Rotational Isomeric State Model
•
Rotational Potential
–
•
V(φ)
Probability of rotation angle phi
Rotational Isomeric State (RIS) Model
e.g. Typical 3 state model : g-, t, g+ with weighted probabilities
Again need to evaluate li ⋅ l j +1 taking into account probability of a φ
rotation between adjacent bonds
–
–
•
P(φ) ~ exp(-V(φ)/kT)
This results in a bond angle rotation factor of
combining
r
where
2
1+ cos φ
1− cos φ
⎛⎛ 1− cosθ ⎞⎛ 1+ cos φ ⎞⎞
⎟⎟⎟⎟ = nl 2C∞
= nl ⎜⎜⎜⎜
⎟⎜
⎝⎝ 1+ cosθ ⎠⎝ 1− cos φ ⎠⎠
2
⎛ ⎛ 1 − cosθ ⎞⎛ 1 + cos φ ⎞ ⎞
⎟⎟
⎜
C∞ = ⎜ ⎜
⎟
⎜ ⎜ 1 + cosθ ⎠⎜⎝ 1 − cos φ ⎟ ⎟
⎠⎠
⎝⎝
cos φ =
∫ cos(φ )P(φ )dφ
∫ P(φ )dφ
The so called “Characteristic ratio”
and is compiled for various polymers
with P(φ) = exp(-V(φ)/kT)
The Chemist’s Real Chain
•
Preferred bond angles and rotation angles:
–
•
•
θ, φ. Specific bond angle θ between mainchain atoms (e.g. C-C bonds) with rotation angle
chosen to avoid short range intra-chain interferences. In general, this is called “steric hindrance”
and depends strongly on size/shape of set of pendant atoms to the main backbone (F, CH3, phenyl
etc).
Excluded volume: self-crossing of chain is prohibited (unlike in diffusion or in the mathematician’s
chain model): Such excluded volume contacts tend to occur between more remote segments of the
chain. The set of allowed conformations thus excludes those where the path crosses and this forces
<rl,n2 >1/2 to increase.
Solvent quality: competition between the interactions of chain segments (monomers) with
each other vs. solvent-solvent interactions vs. the interaction between the chain segments
with solvent. Chain can expand or contract.
• monomer - monomer
• solvent - solvent
• monomer - solvent
εM-M vs. εS-S vs. εM-S
Excluded Volume
• The excluded volume of a particle is that volume for which the
center of mass of a 2nd particle is excluded from entering.
• Example: interacting hard spheres of radius a
– volume of region denied to sphere A due to presence of
sphere B
– V= 4/3π(2a)3 = 8 Vsphere
but the excluded volume is shared by 2 spheres so
Vexcluded = 4 Vsphere
Solvent Quality and Chain Dimensions
Theta θ Solvent
Solvent quality factor α
Local Picture
Global Picture
Solvent quality factor
Good solvent
Good Solvent
ent
Solv
Solv
In
ent
θ − Solvent
α2
=
< r2 >
< r2 >θ
20
Out
20
10
10
10
20
10
10
Poor Solvent
10
10
20
10
Poor solvent
Figure by MIT OCW.
• Solvent quality:
– M-M, S-S, M-S interactions: εM-M, εS-S, εM-S
Physicist’s Universal Chain
• Recover random walk relation for real chain at theta
condition (or in melt state!) by redefining a coarse grained
model of N Kuhn steps
r
2 1/ 2
=N
1/ 2
b
Scaling law
b
rn
Fewer, larger steps
r1
N steps of length b,
b = Kuhn step
with
N =
n
,
C∞
b = C∞ ⋅ l
Manipulating {r }
1n
To increase end to end distance
• Excluded Volume (bigger, bulkier monomers)
• Large bond angle and strong steric interactions tend
to favor all trans conformation => C∞ large
To increase or decrease end to end distance
• Solvent quality
• Temperature (affects solvent quality and relative interaction energies)
• Deformation (stretch the coil: rubber elastic behavior)
Characteristic Ratio
Experimentally measure MW and hence chain dimensions
(technique: laser light scattering: Zimm plot)
in a theta solvent for high MW sample
Influence of steric interactions on
characteristic ratio
H3C
n
*
C∞ =
nl 2
H3C
*
O
r2
n
*
O
O
*
θ
H3C
n
*
*
O
O
O
H2C
C∞ = 10
C∞ = 14
C∞ = 20.3
Summary: Flexible Coil Chain Dimensions
Model
<r2>
Mathematician’s Ideal RW
<r2> = nl2
Freely jointed chain, n bonds each of
length 1.
⎛⎛ 1− cosθ ⎞⎛1+ cos φ ⎞⎞
r = nl ⎜⎜⎜
⎟⎟⎟
⎟⎜
⎠
1+
cos
θ
1−
cos
φ
⎝
⎠⎠
⎝⎝
Allow preferred bond angles and preferred
rotation angles about main chain bonds
2
Chemist’s Real Chain
θ fixed ,
2
r 2 = nl 2C∞
θ
V(φ)-RIS
Real Chain θ, V(φ) – and solvent
quality and excluded volume
<r2> = nl2C∞α2
Comments
Characteristic ratio takes into account all
local steric interactions
Factor α takes into account solvent quality
and long range chain self-intersections
(excluded volume)
r2
2
α = 2
r
θ
: α = 1 for two states:
(i) Theta condition - particular solvent and
temperature
(ii) Melt state
Physicist’s Universal Chain
(α = 1 conditions)
<r2> = Nb2
where N = # of statistical segments,
b = statistical segment length, b = C∞l
C∞ is incorporated into Kuhn length b.
Thermodynamics of Polymer Solutions
•
Bragg-Williams Lattice Theory for Phase Behavior of Binary Alloys
W. L. Bragg and E. J. Williams, Proc. Roy. Soc. A145, 699 (1934); ibid A151, 540 (1935).
•
Generalize to polymer-polymer and polymer-solvent phase behavior.
P. J. Flory, J. Chem Phys. 10, 51 (1942), M. L. Huggins, J. Phys. Chem. 46, 151 (1942).
Chapter 3, Young and Lovell
G = H − TS
Δ G = G1 ,2 − (G1 + G 2 )
1
2
MIX
STATE 1
1,2
STATE 2
Binary Component System in State 2
Binary Component System in State 2
solvent-solvent
solvent-polymer
Materials scientists already familiar with B-W Lattice Theory !
Figure by MIT OCW.
polymer-polymer
new
situations
Thermodynamics of Polymer Sol’n cont’d
G= H −TS
Δ G = G1,2 − (G1 + G2 )
1
2
STATE 1
•
1,2
STATE 2
State 1 – 2 pure phases comprised of ni moles of component i
If component is a polymer, it has xi segments and each
segment has volume vi
V1 = n1 x1v1
•
MIX
State 2 – mixed phase,
assume
V2 = n 2 x 2v 2
V = n1 x1 v1 + n2 x 2 v2
ΔV1,2 = 0
Note if system is comprised of a solvent and a polymer, the convention
is to name the solvent component 1 and the polymer component 2
Thermodynamics cont’d – Entropy
1.
2.
Translational - (sometimes called combinatorial or configurational)
due to the number of distinguishable spatial arrangements
Conformational - due to number of distinguishable shapes of a given
molecule keeping center of mass fixed. We will assume that there
are no differences in molecular conformations in the unmixed vs.
mixed state.
3.
4.
ΔS= k ln Ω1,2 − k(ln Ω1 + lnΩ 2 )
Recall S =k lnΩ
Ω1 =1, Ω 2 =1
since there is only 1 way to arrange a pure
•
The increase in translational entropy is due to the increased total volume
available to the molecules in the mixed state. The statistical mechanics is
standard for mixing of 2 gases:
component in its volume
ΔSm = k ln Ω1,2 − 0
ΔSm = k(n1 ln V / V1 + n2 ln V / V2 )
Mean Field Lattice Theory
•
Mean field: the interactions between molecules are assumed to be due to the interaction
of a given molecule and an average field due to all the other molecules in the system.
To aid in modeling, the solution is imagined to be divided into a set of cells within
which molecules or parts of molecules can be placed (lattice theory).
•
The total volume, V, is divided into a lattice of No cells, each cell of volume v. The
molecules occupy the sites randomly according to a probability based on their respective
volume fractions. To model a polymer chain, one occupies xi adjacent cells.
N o = N1 + N 2 = n1 x1 + n 2 x 2
•
V = N ov
Following the standard treatment for small molecules (x1 = x2 = 1)
Ω1,2 =
using Stirling’s approximation:
N 0!
N1!N 2!
lnM ! = M ln M − M
for M >> 1
ΔSm = k(−N1 ln φ1 − N 2 ln φ 2 )
φ1 = ?
Entropy Change on Mixing ΔSm is :
ΔSm
= + k(−φ1 ln φ1 − φ 2 ln φ 2 )
No
Remember this is
for small molecules
x1 = x2 = 1
0.8
0.7
0.6
0.5
0.4
0.3
Note the symmetry
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
φ2
The entropic contribution to ΔGm is thus seen to always
favor mixing if the random mixing approximation is used.
Enthalpy of Mixing ΔHm
• Assume lattice has z nearest-neighbor cells.
• To calculate the enthalpic interactions we consider the
number of pairwise interactions. The probability of
finding adjacent cells filled by component i, and j is given
by assuming the probability that a given cell is occupied by
species i is equal to the volume fraction of that species: φi.
ΔHm cont’d
Mixed state enthalpic
interactions
Pure state enthalpic
interactions
recall
Some math
υij = # of i,j interactions
υ12 = N1 z φ2
υ11 = N1 z φ1 / 2
υ 22 = N2 z φ 2 / 2
H1,2 = υ12 ε12 + υ11 ε11 + υ22 ε 22
z
z
H1,2 = z N1 φ 2 ε12 + N1 φ1 ε11 + N2 φ2 ε 22
2
2
z N1
z N2
H1 =
ε11
H2 =
ε 22
2
2
⎡
⎤
Nε
Nε
ΔH M = z ⎢N1φ 2ε12 + 1 11 (φ1 −1) + 2 22 (φ 2 −1)⎥
⎣
⎦
2
2
N1 + N2 = N 0
1
⎡
⎤
ΔH M = z N 0 ⎢ε12 − (ε11 + ε 22 ) ⎥φ1φ2
2
⎣
⎦
Note the symmetry
χ Parameter
• Define χ:
χ=
z
kT
1
⎡
⎤
(
)
−
+
ε
ε
ε
⎢⎣ 12 2 11 22 ⎥⎦
χ represents the chemical interaction between the components
ΔH M
= k T χ φ1 φ 2
N0
• ΔGM :
ΔGM = ΔH M −T ΔSM
ΔGM
= kT χφ1φ2 − kT [− φ1 ln φ1 − φ2 ln φ2 ]
N0
ΔGM
= kT [χφ1φ2 + φ1 ln φ1 + φ 2 ln φ 2 ]
N0
Note: ΔGM is symmetric in φ1 and φ2.
This is the Bragg-Williams result for the change
in free energy for the mixing of binary metal alloys.
ΔGM(T, φ, χ)
•
•
Flory showed how to pack chains onto a lattice and correctly evaluate Ω1,2 for
polymer-solvent and polymer-polymer systems. Flory made a complex
derivation but got a very simple and intuitive result, namely that ΔSM is
decreased by factor (1/x) due to connectivity of x segments into a single
molecule:
⎡ φ1
⎤
φ2
ΔS M
= − k ⎢ ln φ1 + ln φ2 ⎥
N0
x2
⎣ x1
⎦
Systems of Interest
- solvent – solvent
- solvent – polymer
- polymer – polymer
x1 = x2 = 1
x1 = 1
x1 = large,
For polymers
x2 = large
x2 = large
⎡
⎤
ΔG M
φ1
φ2
ln φ 2 ⎥
= kT ⎢ χφ1φ 2 + ln φ1 +
N0
x1
x2
⎣
⎦
χ=
z
kT
1
⎡
⎤
(
)
ε
ε
ε
−
+
⎢ 12 2 11 22 ⎥
⎣
⎦
Need to examine variation of ΔGM with T, χ, φi,
and xi to determine phase behavior.
Note possible huge
asymmetry in x1, x2