Chapter 4Chapter 4
Polymer SolutionsPolymer Solutions
Ch 4 Slide 2
Thermodynamics of solutionThermodynamics of solutionTypes of solution
ideal soln ΔHm = 0, ΔSm = – R (n1 ln x1 + n2 ln x2)
regular soln ΔHm ≠ 0, ΔSm = – R (n1 ln x1 + n2 ln x2)
athermal soln ΔHm = 0, ΔSm ≠ – R (n1 ln x1 + n2 ln x2)
real soln
for ideal solnΔG1 = μ1 – μ1
o = RT ln x1
ΔG2 = μ2 – μ2o = RT ln x2
ΔGm = n1ΔG1 + n2ΔG2
= RT (n1 ln x1 + n2 ln x2)
ΔHm = 0 ΔSm = – R (n1 ln x1 + n2 ln x2)
for regular solnΔHm ≠ 0, ΔSm = – R (n1 ln x1 + n2 ln x2)
x ~ mol fractionn ~ numberx1 = n1/(n1+n2)
Ch 4 Slide 3
Stat thermo for regular Stat thermo for regular solnsolnΔSm by lattice model
Filling n1 & n2 molecules in n1+n2 = n cellsvolume of 1 ≈ volume of 2 (small molecules)
Boltzmann relation, S = k ln P P ~ number of (distinguishable) ways
Sterling’s approx, ln x! = x ln x – x
n = n1 + n2
x ~ mol fractionn ~ numberx1 = n1/(n1+n2)
Ch 4 Slide 4
applying molar quantities,
heat of mixingfor small molecules
in the absence of specific interaction betw 1 and 2
free energy of mixing
N ~ number of molesN = N1+ N2R = k NA
ΔE ~ heat of vap
Ch 4 Slide 5
ΔΔGGmm vsvs composition plotcomposition plotΔGm vs composition at different B
a, b, cΔGm < 0 and no inflection point
miscible at all compositions
d, e ~ partially misciblebetw the two S’s ~ unstable
S ~ spinodal point ~ d2G/dx2 = 0
local minor composition fluctuation lower the energy
fluctuation stabilized
phase separate (1-rich + 2-rich)
‘spinodal decomposition’co-continuous to discontinuous
Ch 4 Slide 6
betw B and S ~ metastableB ~ binodal point
at B, dG/dx1 = μ1 = μ2 = dG/dx2
local minor composition fluctuation raise energy
back to homogeneous solution
major fluctuation like nucleationphase separate to B comp
‘nucleation and growth’
outside B’s ~ stablesingle phase is stable
and ’ and ” ~ phases
Ch 4 Slide 7
Phase diagramPhase diagramΔGm vs composition at different temp
(temp-comp) phase diagramby connecting B’s and S’s
stable, metastable, unstable
at critical temp
at T > UCST, miscible at all composition
Ch 4 Slide 8
FloryFlory--Huggins theoryHuggins theoryΔSm by lattice model
polymer soln = mixture of solvent/polymervolume of 1 << volume of 2 (by x)
A polymer molecule with x mers (repeat units) takes x cells
volume of 1 mer ≈ volume of 1 solvent molecule
filling n1 solvents & n2 polymers in n1+ xn2 = n cells
number of ways to fill the (i+1)th chain
νi+1 = (n-xi) z(1-fi) (z-1)(1-fi) ----- (z-1)(1-fi)
1st 2nd 3rd xth segment (mer)
z ~ coordination number (# of nearest neighbor)fi ~ probability of a site not available ≈ xi/n
Ch 4 Slide 9
number of ways to fill n2 polymer molecules
1 – fi = (n – xi)/n
n >> x for dilute solution
eqn (4.16)
Ch 4 Slide 10
entropy of solution, S = k ln P2
ln x! = x ln x – x
n = n1 + xn2
Ch 4 Slide 11
solution by(a) disorientation
(b) mixing
ΔSmix = S12 – S1 – S2
S12 from filling n2 polymers into n cells ~ eqn (4.19)
S1 from filling n1 solvents into n1 cells = k ln 1 = 0
S2 from filling n2 polymers into xn2 cells ~ eqn (4.19) with n1 = 0
Ch 4 Slide 12
ΔSmix taking volume fractions
x (mol wt ) ΔSm
for polymer/polymer soln, ΔSm even smaller
v ~ volume fractionN ~ number of molesN = N1+ xN2R = k NA
Ch 4 Slide 13
heat of mixinginteraction energy in solution
interaction energy in pure states
athermal when Δω12 = 0 ΔHm = 0
Ch 4 Slide 14
free energy of mixing
ΔGm = kT [n1 ln v1 + n2 ln v2 + χn1v2]
Drawbacks of F-H theoryno volume change
only for concentrated soln
self-intersection
χ ~ (Flory-Huggins) interaction parameter
~ Flory-Huggins Eqn
~ Flory-Huggins Eqn
Ch 4 Slide 15
Dilute polymer solutionDilute polymer solution
Partial molar free energy of mixing for solvent
ΔG1 = ∂ΔGm/∂N1
from Flory-Huggins eqn
ΔGm = kT [n1 ln v1 + n2 ln v2 + χn1v2]
n1 = NAN1, v1 = N1/(N1+xN2), v2 = xN2/(N1+xN2), kNA = R
ΔG1 = RT [ln (1 – v2) + (1 – 1/x)v2 + χv22]
other form of Flory-Huggins eqn
ΔG1 = μ1 – μ1o = RT ln a1 = RT ln x1γ1
ΔG1 = μ1 – μ1o = (μ1–μ1
o)ideal + (μ1–μ1o)xs
ideal: (μ1–μ1o)ideal = RT ln x1
excess: (μ1–μ1o)xs = RT ln γ1
N: # of moles
a ~ activityγ ~ activity coeff.x ~ mol fraction
~ Eqn (B)
~ Eqn (A)
Ch 4 Slide 16
dilute polymer solnpolymer chains separated by solvent
FH theory does not holdIn F-H theory, chains are placed randomly, and can overlap.
Flory-Krigbaum theory
for dil polym solnx2 = v2/x
v2 = xn2/(n1+xn2) ≈ xn2/n1 (n1 >> xn2)
x2 = n2/(n1+n2) ≈ n2/n1 (n1 >> n2)
ln v1 = ln (1 – v2) = – v2 – v22/2 – v2
3/3 – ---ln x1 = ln (1 – x2) = – x2 – x2
2/2 – x23/3 – ---
= – v2/x – (v2/x)2/2 – ---
Ch 4 Slide 17
from Eqn (A)
ΔG1 = μ1–μ1o = RT [– v2 – v2
2/2 + v2 + v2/x + χv22]
= –RT(v2/x) + RT(χ – ½)v22
from Eqn (B)
ΔG1 = μ1–μ1o = RT ln x1 + (μ1–μ1
o)xs
= –RT(v2/x) + (μ1–μ1o)xs
By Flory-Krigbaum
ΔG1xs = (μ1–μ1
o)xs = ΔHxs – T ΔSxs
= RTκ v22 – T Rψ v2
2 = RT(κ – ψ) v22
ΔG1xs = RTψ [(θ/T) – 1] v2
2 = RT (χ – ½) v22
When T = θ, χ = ½ ΔG1xs = 0 ΔG1= ΔG1
ideal
θ-condition (Flory condition) ~ becomes ideal
When T > θ, χ < ½ ΔG1xs < 0 soluble (good solvent)
κ = ψθ/T
Ch 4 Slide 18
Osmotic pressure exp’t
plotting A vs v22 at various temp χ = f(v2, T)
due to interaction betw solvent and polymer
the 3rd term in F-H eqn is not purely enthalpic
ΔG1xs = RT(κ – ψ) v2
2 = RTψ [(θ/T) – 1] v22
hxs = RTκ v22, sxs = Rψ v2
2
μ1–μ1o = ΔG1 = RT [ln v1 + (1 – 1/x)v2 + χv2
2] ~ F-H eqn
A
Often, χs > χh
Ch 4 Slide 19
Concentration dependence of osmotic pressure
At θ condition, χ = ½, A2 = 0.
ln (1 – v2) = – v2 – v22/2 – v2
3/3 – ---
c ~ conc’n (g/L)V2
o ~ sp vol of polymerMrep ~ mol wt of monomerM ~ mol wt of polymer
2nd virial coeff,
Ch 4 Slide 20
from F-H eqn
at critical point
asymmetric
at M = ∞, v2c = 0, χc = ½, Tc = θ
Phase behavior of polymer Phase behavior of polymer solnsoln
dμ1/dv2 = 0
d2μ1/dv22 = 0
small moleculesvc at 0.5 (sym)
higher mol wt
PS in cx
Ch 4 Slide 21
UCST and LCST in polymer UCST and LCST in polymer solnsolnphase diagram of PS in cx (exp’tal)
shows both UCST and LCST(modified) F-H theory describe UCST onlyfor LCST, ΔHm< 0 and ΔSm< 0
further modification of F-H theory
LCST when ΔCp < 0
molecular interpretation?could be related to volume change LCST at near bp of solventnot by F-H theory
dμ1/dv2 = 0, d2μ1/dv22 = 0
Ch 4 Slide 22
Solubility parameterSolubility parameter
ΔHm = Vm [(ΔE1/V1)½ – (ΔE2/V2)½]2 v1v2
= Vm [δ1 – δ2]2 v1v2
ΔE ~ cohesive energy ~ energy change for vaporization
ΔE = ΔHvap – PΔV ≈ ΔHvap – RT [J]
ΔE/V ~ cohesive energy density [J/cm3 = MPa]
δ ~ solubility parameter [MPa½]
[MPa½] = [(106)(N/m2)½] = [(J/cm3)½] ≈ [(1/2)(cal/cm3)½]
Ch 4 Slide 23
Determination of δfrom ΔHvap data ~ for low mol wt, not for polymers
with solvent of known δswelling
viscosity
group contribution calculation
Ch 4 Slide 24
For solution,ΔHm < T ΔSm
without specific interactionδ1 = δ2 at best ΔHm = 0 ΔGm < 0
Δδ < 20 MPa½ ~ for solvent/solvent solution
Δδ < 2 MPa½ ~ a rough guide for solvent/polymer solution
Δδ < 0.1 MPa½ ~ for polymer/polymer solution
semicrystalline polymers not soluble at RTpositive ΔHf ΔHf + ΔHm > T ΔSm
δ for amorphous state at 25 °C
Ch 4 Slide 25
Polymer blendPolymer blendOnly a few pairs are miscible.
with specific interactionPVAc/PVP (HB)
PS/PPO (π-π)
with similarityin chemical structure?
in physical parameters
Ch 4 Slide 26
Phase behavior of polymer blendPhase behavior of polymer blendclose to symmetrical
polymer/solvent was not
UCST sometimes not observedvitrification
LCST commoncould not be described by F-H theory
F-H does not consider ΔVmix
could be explained by equation of state theory
lattice-fluid theory
gas-lattice theory
Ch 4 Slide 27
FOVE modelFOVE modelan EOS model
employing reduced v, T, pv*, T*, p* ~ ‘hard-core’ or ‘close-packed’
introducing ‘free volume’ or ‘holes’
considers ΔVmix
Volume decreases when mixed ( better miscibility at high P, expt)ΔHm < 0 (ΔV < 0) and ΔSm < 0 (less holes)
at low temp, ΔHm > TΔSm ~ miscible
at high temp, TΔSm > ΔHm ~ phase separates ~ LCST
Ch 4 Slide 28
Miscibility Miscibility Determination of miscibility
turbidity
Tg ~ DSC, DMA, DEA
morphology
spectroscopy
miscibility vs compatibility
For compatibilizationcompatibilizer
IPN
reactive processing
Ch 4 Slide 29
Homework #2 (Due on 23 Oct 2006)Homework #2 (Due on 23 Oct 2006)1. Derive Eqn (A)
ΔG1 = RT [ln (1 – v2) + (1 – 1/x)v2 + χv22]
from ΔG1 = ∂ΔGm/∂N1
and ΔGm = kT [n1 ln v1 + n2 ln v2 + χn1v2]
2. Derive eqn (4.43) from the definition of Π
3. Derive eqn (4.40) and (4.41) from the definition of critical point.