Polymer Blends
Good References on Polymer Blends:
•
O. Olasbisi, L.M. Robeson and M. Shaw, Polymer-Polymer Miscibility,
Academic Press (1979).
•
D.R. Paul, S. Newman, Polymer Blends, Vol I, II, Academic Press
(1978).
•
Upper Critical Solution Temperature (UCST) Behavior
Well accounted for by F-H theory with χ = a/T + b
•
Lower Critical Solution Temperature (LCST) Behavior
FH theory cannot predict LCST behavior. Experimentally find that
blend systems displaying hydrogen bonding and/or large thermal
expansion factor difference between the respective homopolymers often
results in LCST formation.
Why?
Phase Diagram for UCST Polymer Blend
Predicted from FH Theory
6
T
A
χ = ; add a constan t B
T
5
Equilibrium
Stability
x1 = x2 = N
binodal
∂2ΔGm
=0
2
∂φ
∂ ΔGm
=0
3
∂φ
4
χN
∂ΔGm
=0
∂φ
low
Two phases
3
A'
B'
2
spinodal
Critical point
A
1
φ'A
3
critical point
0
hot
B
φ"A
One phase
Thigh
0.0
0.2
0.4
A
0.6
φ
0.8
1.0
B
B
Figure by MIT OCW.
A polymer
B polymer
x1 segments
x2 segments
v1 ≈ v2 & x1 ≈ x2
χ=
A
+ B
T
2 Principal Types of Phase Diagrams
T
T
Single-phase region
Two-phase
region
Tc
UCST
LCST
Tc
Two-phase
region
0
Single-phase region
1
φ2
0
1
φ2
PS-PI
2700
180
PS-PVME
160
180
T(oC)
T(oC)
140
140
120
100
P1
80
2100
100
Wps
0.2
0.4
0.6
60
0.8
PS
0
PS
0.2
0.4
0.6
φ
0.8
1.0
PVME
Figure by MIT OCW.
Konigsveld, Klentjens, Schoffeleers
Pure Appl. Chem. 39, 1 (1974)
Nishi, Wang, Kwei
Macromolecules, 8
Today - Measuring Polymer MW& Size
• c2*
• Intrinsic Viscosity – alpha, Mv and more scaling exponents!
• Osmometry – chi and Mn
Spinodal Decomposition of LCST Blend
Bicontinuous Structures
LCST
By up-quenching into the
unstable spinodal regime and
then after a short
hold, rapidly cooling, one
can trap the
bicontinuous spinodal structure.
Over-annealing
results in coarsening
and droplet formation (figure d)
just like in metal alloys.
Image and text removed due to copyright restrictions.
Please see Fig. 28 in Platzer, Norbert A. J.
Copolymers, polyblends, and composites: a
symposium. Advances in Chemistry, no. 142.
Washington, DC: American Chemical Society, 1975.
Overlap Concentration c2*
c2* is the so called overlap concentration; the polymer concentration at
which the coils of molecular weight M just begin to touch.
Note: at and above c2* the solution is reasonably uniform in composition
(i.e. mean field situation, no large regions of pure solvent)
Sphere of pervaded volume
c2* ~
M / NA
4
2
3/2
π<r >
3
Volume of sphere
c2 ≈ c 2
*
c2 > c2*
Viscosity, η
• Basic information on chain dimensions in a particular solvent.
v
Shearing of a fluid
velocity profile
y
x
Boundary Conditions - no slip
τ=
τ= η
If
dγ
dt
N
viscous force
= Pa = 2
area
m
dγ
dv
cm/sec
=
=
= γÝ sec−1
dy
cm
dt
shear stress
η=
= Pa ⋅ sec
rate of shear
η ≠ η (γ tot , γ& ) the fluid is a Newtonian fluid
Polymers are very nonNewtonian!
We are interested in the effect of additives on the viscosity of simple (low
molar mass) fluids.
(1).
suspension of hard spheres -- (Stokes 1850, Einstein 1905)
(2).
suspension of polymer coils
Stokes and Viscosity
η0
v
• sphere of size Rs falling through a fluid of viscosity η0 with
velocity v.
Fviscous= f v f is the friction factor of the solution
f = 6 ρ η0 Rs
Stokes’ Law
• How to treat a polymer molecule?
2 Limiting Cases:
I.
Impenetrable Sphere
II.
Penetrable Coil
x2 segments/chain
Rg2 ≡ s 2 =
r2
6
Polymer Molecule in Shear Flow
• Case I: Impenetrable Sphere
f = 6 π η0 RH
RH ~ <Rg2>1/2
so f ~ x21/2
RH is the hydrodynamic radius
coil / sphere / nondraining
• Case II: Penetrable Coil
Solvent passes through coil
so f ~ ξ x2
where ξ is the monomeric friction factor
coil / open mesh / free draining
The model that is most appropriate depends on the size of the polymer
Scaling Behavior of Friction Factor
f ~ x2
a
a
Condition
1
low MW coil or any rod
1/2
θ condition (Gaussian coil: RW)
3/5
good solvent (coil: SARW)
• For high MW flexible chain molecules correct model is
nondraining, impenetrable units
f = 6πη0R H
2 1/ 2
R H = γ Rg
γ ≈ 0.85
• To develop expression for viscosity use Einstein equation for
viscosity of a solution containing impenetrable spheres of
volume fraction φhs:
η (φhs ) = η0 (1 + 2.5φhs + L)
• choose θ condition and dilute solution ( φ 2 << 1), c 2 << c *2
PS #1
• c2* = overlap concentration, note: φ2 ≠ φhs, indeed φ2 << φhs
2 Important Viscosities
η sp =
• Specific viscosity
η
− 1 = 2.5φhs
ηo
(# sphere) ( vol.of sphere) ⎛ c 2 ⎞ 4 3 2
⎟⎟ πγ R g
φhs =
= ⎜⎜
⎝ M/NAV ⎠ 3
Total Volume
• Intrinsic viscosity
[η] ≡ lim
c 2 →0
[η ]
⎛ 4π 3
⎜ 3 γ N AV Rg
= 2.5⎜
M
⎜
⎝
2
3/2
⎞
⎟
⎟
⎟
⎠
η sp
c2
3/2
2.5φ hs
=
c2
[η]θ
At θ condition and collect constants into prefactor Kθ
For other solvent conditions use α solvent quality factor
R g → α R gθ
Expt’l method for
obtaining alpha
[η] = α 3 [η]θ
3
with K = α Kθ
= Kθ M1/2
in general:
[η]
= KM
a
Intrinsic Viscosity - Key Idea
• A simple method to quickly assess a sample’s molecular
weight is to utilize the dependence of the viscosity of a
polymer solution (in a good solvent) on the molecular
weight of the polymer. The viscosity depends not only on
molecular weight and on solvent quality, but also on
polymer concentration, so one has to extrapolate to zero
concentration to determine the “intrinsic” effect of the
addition of the polymer to the increase in viscosity of the
solution.
Mv Viscosity-average molecular weight
η sp
[η]= kMva
[η ]≡ lim
c2 → 0 c2
1/a
= ⎛ ΣNi Mi1+a ⎞
Mv ⎜
⎟
⎝ ΣNi Mi ⎠ (See following page for derivation)
Viscosity-Average Molecular Weight
ηsp =
η
−1
η0
For a polydisperse solution with ni moles of species i of molecular weight Mi
ηsp,soln = ∑ ηsp,i
[η]i = KM ai
i
Number average MW
∑ ni M i
Mn =
∑ ni
nM
conc of ith species in g/vol. of solution c i = i i
V
[η]soln =
η sp,soln
lim
c2 →0
c2
∑ ηsp,i
[η]soln = lim
i
c 2 →0
∑ ci
∑ ηsp,i
= lim
c 2 →0
i
[η]soln =
i
c 2 →0
∑ n iM i/V
c2 → 0
∑ n iM i
i
Note:
Mv = Mw
For a = 1
But in general 0.5 ≤ a ≤ 0.8 so
∑ n iM i/V
i
∑ ni M1+a
i
i
∑ n iM i/V ⋅ KM ai
= lim i
i
[η]soln = K
∑ n iMi /V
i
∑ ci [η]i
lim
i
≡ KM av
1+a ⎞ 1/a
⎛
⎜ ∑ n iM i ⎟
⎟
or M v = ⎜⎜ i
nM ⎟
⎜ ∑ i i ⎟
⎝ i
⎠
Mn ≤ M v ≤ Mw
Mark-Houwink-Sakurada
[η]
= KM
a
• K and a depend on solvent quality; more scaling exponents!
a < 1/2
poor solvent
θ conditions
a = 1/2
a > 1/2
good solvent
• It turns out α is weakly dependent on coil size (the interior
of the coil expands more than the periphery) so longer chains
expand more in a good solvent than do shorter chains.
α ~ M 1 / 10
which results in a = 8/10 for high MW chains in a good solvent
Determination of Intrinsic Viscosity
ηspec
C2
x
x
x
For a* series of concentrations
C2 < C2
η sp
x
C2
[η]
C2
log[η]
x
x
= [η ] + slope ⋅ C 2
For a series of MWs
[η ] =
K M
a
x
a
x
log K
log M
Atactic Polystyrene
in
(at 25oC)
θ condition
Slopes:
θ solvent
a = 1/2
good solvent 8/10 > a > 1/2; MW dependent
Solvent
K x 102
a
Benzene
Butanone
Chloroform
Cyclohexane
Dichloromethane
THF
Toluene
1.23
3.9
0.7
10.8
2.10
1.10
1.05
0.72
0.58
0.76
0.48
0.66
0.725
0.73
Cyclohexane
θ = 36oC
0.5
Membrane Osmometry ( M n , A2, χ)
•
•
Osmotic pressure, π, is a colligative property which depends only on
the number of solute molecules in the solution.
In a capillary membrane osmometer, solvent flow occurs until π
increases until the chemical potential μ(π) = μ1o
∂μ1
dP
∂P
∂μ1
∂ ⎜⎛ ∂G ⎞⎟
∂ ⎛ ∂G ⎞
=
=
∂P ∂P ⎝ ∂n1 ⎠ ∂n1 ⎝ ∂P ⎠
P0 + π
μ = μ1 + ∫P
p
°
1
0
dG = Vdp − sdT + μdn
Solution
Pure solvent
Semi-permeable membrane
Figure by MIT OCW.
so
T, n fixed
∂G
=V
∂P
∂
(V )≡ V1
∂n1
(partial molar volume
of pure solvent)
Membrane Osmometry cont’d
P0 +π
μ = μ1 + ∫P
°
1
V1 dP = μ1 +πV1
0
μ1 − μ1o = − πV1
Recall F-H expression for a dilute solution (φ2 small):
⎡ −φ 2
μ1 − μ1° = RT⎢
⎣ x2
+( χ −1/ 2) φ22
⎤
⎥⎦
Rearranging:
⎡ ⎛ φ2 ⎞ ⎛ 1
⎛⎜ φ 22 ⎞ ⎤
⎞
⎟ + −χ
π = RT ⎢ ⎜
⎠ ⎝ V 1 ⎠ ⎥⎦
⎣ ⎝ V1 x 2 ⎠ ⎝ 2
Since solution is dilute:
V ≅ n1V1 so V1 ≅ V1
n2 x 2
φ2 ≅
recall
n1
2
⎡⎛ n2 ⎞ ⎛ 1
n
⎛
2⎤
2⎞
⎞
π = RT ⎢⎝ ⎠ +⎝ − χ ⎠ ⎝ ⎠ V1x 2 ⎥
V
2
⎣ V
⎦
The osmotic pressure is
a power series
⎛n ⎞
in ⎜⎝ 2 ⎟⎠ i.e., π depends
V
on the number of
molecules of solute
per unit volume of the
solution.
Osmotic Pressure cont’d
•
For a polydisperse polymer solution, n2 = Σi ni where ni is the number
of moles of polymer of molecular weight Mi in the solution. The
number average molecular weight of the polymer is simply M = ∑ ni Mi = m
n
∑ ni
•
The concentration in grams/cm3 of the polymer in the solution is just
Hence the osmotic pressure can be
written
⎡
m n2 Mn
c
⎛n
c2 =
=
or ⎝ 2 ⎞⎠ = 2
V
V
Mn
V
•
⎞ V1 x 22 ⎤
1 ⎛1
= RT⎢
+⎜ − χ⎟
2 c2 ⎥
⎝
⎠
c2
Mn
⎦
⎣ Mn 2
π
The “reduced osmotic pressure” when plotted as a function of the
solute (i.e. polymer) concentration will give a straight line with:
Key Information:
π
c2
RT
Intercept:
Mn
Slope:
Three types of behavior:
x
x
x
χ < 1/2 (good solvent)
x
x
⎛1
⎞V x
χ
−
⎜
⎟
⎝2
⎠M
2
1 2
2
n
x
x
x
x x
x
x
x
x
χ = 1/2 (θ condition)
χ > 1/2 (poor solvent)
c2
n2
Osmometry cont’d
• At the θ condition, 2nd term disappears (recall χ = 1/2) and solution is
ideal
πV ≅ n 2 RT
• The slope of the reduced osmotic pressure yields a measure of the
polymer segment-solvent F-H interaction parameter,
so we have a way to experimentally determine the important χ parameter
• For gases, the pressure is often developed in a virial expansion as a
function of concentration
P = RT ( A1c+ A2c + A3c +...)
2
3
• We can therefore identify the virial coefficients A1 and A2 from
1
A1 =
Mn
A2 =
⎛1
⎞
⎜ − χ⎟
⎝2
⎠
ρ 22V1
π
c2