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.