麻省理工-高分子物理4

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?
Sphere of pervaded volume
c2* ~
M / NA 4 2 3/2 π<r > 3
* c2 > c2
Volume of sphere
c2 ≈ c
* 2
Viscosity, η
• Basic information on chain dimensions in a particular solvent. v Shearing of a fluid
[η]θ
= Kθ M1/2
For other solvent conditions use α solvent quality factor
R g → α R gθ
Expt’l method for obtaining alpha
in general:
[η] = α 3 [η]θ
3 with K = α Kθ
2 Important Viscosities
• Specific viscosity
η sp = η − 1 = 2.5φhs ηo
3/2
(# sphere) ( vol.of sphere) ⎛ c 2 ⎞ 4 3 2 φhs = =⎜ ⎜ M/N ⎟ 3 πγ R g ⎟ ⎝ Total Volume AV ⎠
3
binodal spinodal
χN
3
A'
B'
2 Critical point 1 φ'A A One phase B φ"A Thigh 0.0 A 0.2 0.4 0.6 0.8 1.0 B hot
c by MIT OCW.
A polymer B polymer
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
velocity profile
y
Boundary Conditions - no slip
τ=
x
N viscous force = Pa = 2 area m
dγ τ= η dt
dγ dv cm/sec Ý = = = γ sec−1 dy cm dt shear stress η= = Pa ⋅ sec rate of shear
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)
•
•
Phase Diagram for UCST Polymer Blend
Predicted from FH Theory A χ = ; add a constan t B T
x1 = x2 = N
6 T Two phases 5 Equilibrium Stability 4
low
∂ΔGm =0 ∂φ ∂2ΔGm =0 2 ∂φ ∂ ΔGm =0 3 ∂φ
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
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.
2 Limiting Cases: I. Impenetrable Sphere II. Penetrable Coil
x2 segments/chain
2 Rg ≡ 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
(1). (2). suspension of hard spheres -- (Stokes 1850, Einstein 1905) 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?
coil / open mesh / free draining
The model that is most appropriate depends on the size of the polymer
Scaling Behavior of Friction Factor
a Condition
low MW coil or any rod θ condition (Gaussian coil: RW) good solvent (coil: SARW)
Nishi, Wang, Kwei Macromolecules, 8
Today - Measuring Polymer MW& Size
• c2* • Intrinsic Viscosity – alpha, Mv and more scaling exponents!
• Osmometry – chi and Mn
x1 segments x2 segments
v1 ≈ v2 & x1 ≈ x2
A χ= + B T
2 Principal Types of Phase Diagrams
T Single-phase region UCST Tc LCST Two-phase region Tc Single-phase region 1 0 φ2 1 T Two-phase region
[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]= kMa [η ]≡ lim v c2 → 0 c2 ⎛ ΣN M1+a ⎞1/a = Mv ⎜ i i ⎟ ⎝ ΣNi Mi ⎠ (See following page for derivation)