Chapter 4. Concentrated Solutions and
Phase Separation Behavior
4.1 Phase Separation and Fractionation
4.1.1 Motor Oil Viscosity Example
The viscosity of today’s motor oils bears designations such a SAE
5W-30. According to crankcase oil viscosity specification SAE
J300a, the first number refers to the viscosity at -18 oC, and
the second number at 99 oC.
4.1.2 Polymer-Solvent Systems
According to thermodynamic principles, the condition for
equilibrium between two phases requires that the partial molar
free energy of each component be equal in each phase.
This condition requires that the first and second derivatives of
△G1 with respect to ν2 be zero.
The critical concentration at which phase separation occurs
may be written
For large n, the right-hand side of Eq (4.1) reduces to 1/n0.5.
The critical value of the Flory-Huggis polymer-solvent
interaction parameter, c1, is given by
which suggests further that as n approaches infinity, c1c
approaches 1/2
The critical temperature is the highest temperature of phase
separation.
The equation for the critical temperature is given by
Ψ1 is constant
Plot 1/Tc versus 1/n0.5 + 1/2n should
yield the Q-temperature at n = infinity
fractionation
The Q-temperature for
PS/cyclohexane was 34.5 oC.
The phase separation
curve called binodal line.
4.1.3 Vitrification Effects
The effect of the solvent at high polymer volume fraction is to
plasticize the polymer. However, if the polymer is below its glass
transition temperature, the concentrated polymer solution may
vitrify, or become glassy.
The vitrification line generally curves down to lower temperatures
from pure polymer as it becomes more highly plasticized.
Concentration vs. solubility?
The interception of these two curves is known as Berghmans’
point (BP) and defined as the point where the liquid-liquid phase
separation binodal line is intercepted by the vitrification curve.
4.2 Regions of the Polymer-Solvent Phase Diagram
A polymer dissolves in two stages:
1. solvent molecules diffuse into the polymers, swelling it to a
gel state.
2. Then the gel gradually disintegrates, the molecules diffusing
into the solvent-rich regions.
In this discussion, linear amorphous polymers are assumed.
φvs. u (excluded volume parameter)
Daoud and Jannink and others
divided polymer-solvent space
into several regions, plotting the
volume fraction of polymer, φvs. u
(excluded volume parameter)
x : screening length
dilute solution regime, x = Rg.
Semidilute regime, z measures
the distance between chain
contacts.
Cross-linked, x provides a
measure of the net size.
The screening length, x, first introduced by Edwards. This quantity takes
slightly different meanings in different regimes
In the dilute solution regime, x = Rg.
In the semidilute regime, x measures the distance between chain contacts.
If the polymer is crosslinked, x provides a measure of the net size.
For semidilute solutions, the dependence of x onφfollows the scaling law
xs ~ φ-3/4
For semidilute solutions, the dependence of x on j follows the
scaling law
Another quantity of interest in semidilute solutions is called the
blob. It contains a number of mers on the same chain defined by
the mesh volume xs3, inside of which excluded volume effects are
operative.
Some texts define the blob as the number of mers between
adjacent entanglements, distance xs apart.
These blobs are large enough to be self-similar to the whole
polymer chain coiling characteristics; they are coil within coil.
4.3 Polymer-Polymer Phase Separation
When two polymers are mixed, the most frequent result is a system
that exhibits almost total phase separation. Qualitatively, this can be
explained in terms of the reduced combinatorial entropy of mixing.
LCST
UCST
4.3.1 Phase Diagram
Phase separation and dissolution are controlled by three variables:
temperature, pressure, and concentration.
Lower critical solution temperature
(LCST)
Ex: HIPS & ABS
Solid line : binodal curve
Dash line : spinodal curve
4.3.2 Thermodynamics of Phase Separation
The basic equation for mixing of blends reads
V : the volume of the sample
Vr : the volume of one cell
z : the lattice coordination number
Nc : the number of cells in 1 cm3
The first term on the right being the heat of mixing term △HM.
The second term on the right is the statistical entropy of mixing
term, △SM.
4.3.3 An Example Calculation: Molecular Weight
Miscibility Limit
4.3.4 Equation of State Theories
At equilibrium, an equation of state is a constitutive equation that
relates the thermodynamic variable of pressure, volume, and
temperature.
Imagine that a multicomponent mixture is mixed with No holes of
volume fraction υo. Then the entropy of mixing is
Noting that the fractional free volume is given by 1-r, the entropy
of mixing vacant sites with the molecules in equation of state
terminology is given by
Where r is less than unity. When all the sites are occupied, r = 1,
and the right hand side is zero.
The Gibbs free energy of mixing is given by
Where the quantity e* is a van der Waals type of energy of interaction.
Note that r = r(P,T); r  1 as T  0; r  1 as P  ∞
By taking △Gm/r = 0, the equation of state via the lattice fluid
theory is obtain
Where r is the number of sites in the chains, and
For high polymers, r goes substantially to infinity, yielding a general
equation of state for both homopolymers and miscible polymer
blends,
The corresponding equation of state derived by Flory is
Workers in the field prefer to state the equations in terms of
density relations, because for condensed systems, density is easier
to measure than volume.
Again, r = r/r* = V*/V
This information may be used to determine miscibility criteria
The quantity T* and P* represent theoretical values at close packing.
4.3.5 Kinetics of Phase Separation
Nucleation and growth
Spinodal decomposition
Spinodal decomposition
4.3.6 Miscibility in Statistical Copolymer Blends
As stated previously, most homopolymer blends are immiscible due to
the negative entropy of mixing and negative heats of mixing.
Sometimes, however, miscibility can be achieved with the introduction
of comonomers.
Karasz and MacKnight approached the problem through mean field
thermodynamic considerations, arguing that negative net interactions
are necessary to induce miscibility.
Where n1 and n2 are volume fractions, n1 and n2 are degrees of
polymerization, and cblend is a dimensionless interaction parameter
defined as
Where the coefficient cij are functions of the copolymer compositions,
with 0 ≦ cij ≦ 1.
For An/(BxC1-x)n’ blends,
“Windows of miscibility”
result when △GM < 0.
4.3.7 Polymer Blend Characterization
Very small size (20 nm) serve to make good damping compositions,
while domains of the order of 100 nm make better impact-resistant
materials.
Polymer blends
photophysics
4.3.8 Graft Copolymers and IPNs
SBR/PS
Block Copolymer (microphase separation)
volume fraction
< 0.20
0.20 ~ 0.35
> 0.35
disorder
order
Block Copolymer phase diagram
disorder
Representative
phase diagram
of diblock
copolymers
(Khandpur et al.,
Macromolecules
1995, 28, 8796)
4.3.9 Block Copolymers
K is the experimental constant relating the unperturbed rootmean-square end-to-end distance to the molecular weight.
Idealized triblock copolymer thermoplastic elastomer morphology
SBS
4.3.11 Ionomers
Ionomers are polymers that contain 5% to 15% ionic groups.
While these materials are statistical copolymers, the ionic
groups usually phase separate from their hydrocarbon-like
surroundings thus providing properties resembling multiblock
copolymers.
4.4 Diffusion and Permeability in Polymers
Permeation is the rate at which a gas or vapor passes through a polymer.
The mechanism by which permeation takes place involves three steps:
(a) Absorption of the permeating species into the polymer
(b) diffusion of the permeating species through the polymer,
traveling, on average, along the concentration gradient
(c) desorption of the permeating species from the polymer surface
and evaporation or removal by other mechanisms.
Factors affecting permeability include the solubility and diffusivity
of the penetrant into the polymer, polymer packing and side-group
complexity, polarity, crystallinity, orientation, fillers, humidity, and
plasticization.
4.4.1 Swelling Phenomena
If the polymer is glassy, the solvent lowers the Tg by a
plasticizing action. Polymer molecular motion increases.
Diffusion rates above Tg are far higher than below Tg.
4.4.2 Fick’s Laws
Fick’s first law governs the steady-state diffusion circumstance:
Fick’s second law controls the steady state:
The permeability coefficient, P, is defined as the volume of vapor
passing per unit time through unit area of polymer having unit
thickness, with a unit pressure difference across the sample. The
dolubility coefficient, S, determines the concentration. For the
simple case
A study of vapor solubility as a function of temperature allows the
heat of solution △Hs to be evaluated.
The temperature dependence of the solubility obeys the ClausiusClapeyron equation
The permeability coefficients depend on the temperature
according to the Arrhenius equation,
Where △E is the activation energy for permeation
4.4.3 Permeability Units
4.4.4 Permeability Data
4.4.5 Effect of Permeate Size
4.4.6 Permselectivity of Polymeric Membranes and
Separations
4.4.6.1 Types of Membranes
1. Passive transport
2. facilitated transport
3. Active transport
4.4.6.2 Gas Separations
Gas selectivity is the ratio of permeability coefficients of two gases
4.4.7 Gas Permeability in Polymer Blends
4.4.8 Fickian and Non-Fickian Diffusion
4.5 Latexes and Suspensions
4.5.1 Natural Rubber Latex
4.5.2 Colloidal Stability and Film Formation
4.6 Multicomponent and Multiphased Materials