Chapter%204.%20Concentrated%20Solutions%20and%20Phase%20Separation%20Behavior

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Chapter 4. Concentrated Solutions and Phase
Separation Behavior
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4.1 Phase Separation and Fractionation
4.1.1 Motor Oil Viscosity Example
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The viscosity of todays 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.
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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 ?2 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.
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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
?1 is constant
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Plot 1/Tc versus 1/n0.5 1/2n should yield the
Q-temperature at n infinity
fractionation
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The Q-temperature for PS/cyclohexane was 34.5 oC.
The phase separation curve called binodal line.
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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?
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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.
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4.2 Regions of the Polymer-Solvent Phase Diagram

• A polymer dissolves in two stages
• solvent molecules diffuse into the polymers,
  swelling it to a gel state.
• Then the gel gradually disintegrates, the
  molecules diffusing into the solvent-rich
  regions.
• In this discussion, linear amorphous polymers are
  assumed.

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fvs. u (excluded volume parameter)
Daoud and Jannink and others divided
polymer-solvent space into several regions,
plotting the volume fraction of polymer, fvs. 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.
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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
onffollows the scaling law xs f-3/4
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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.
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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
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4.3.1 Phase Diagram
Phase separation and dissolution are controlled
by three variables temperature, pressure, and
concentration.
Lower critical solution temperature (LCST)
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Ex HIPS ABS
Solid line binodal curve Dash line spinodal
curve
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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.
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4.3.3 An Example Calculation Molecular Weight
Miscibility Limit
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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 ?o. 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.
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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
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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
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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
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This information may be used to determine
miscibility criteria
The quantity T and P represent theoretical
values at close packing.
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4.3.5 Kinetics of Phase Separation
Nucleation and growth
Spinodal decomposition
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Spinodal decomposition
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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.
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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,
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Windows of miscibility result when ?GM lt 0.
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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.
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Polymer blends
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photophysics
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4.3.8 Graft Copolymers and IPNs
SBR/PS
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Block Copolymer (microphase separation)
volume fraction
lt 0.20
0.20 0.35
gt 0.35
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order
disorder
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Block Copolymer phase diagram
disorder
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Representative phase diagram of diblock
copolymers(Khandpur et al., Macromolecules 1995,
28, 8796)
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4.3.9 Block Copolymers
K is the experimental constant relating the
unperturbed root-mean-square end-to-end distance
to the molecular weight.
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Idealized triblock copolymer thermoplastic
elastomer morphology
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SBS
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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.
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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

1. Absorption of the permeating species into the
   polymer
2. diffusion of the permeating species through the
   polymer, traveling, on average, along the
   concentration gradient
3. 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.
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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.
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4.4.2 Ficks Laws
Ficks first law governs the steady-state
diffusion circumstance
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Ficks 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.
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The temperature dependence of the solubility
obeys the Clausius-Clapeyron equation
The permeability coefficients depend on the
temperature according to the Arrhenius equation,
Where ?E is the activation energy for permeation
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4.4.3 Permeability Units
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4.4.4 Permeability Data
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4.4.5 Effect of Permeate Size
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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

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4.4.6.2 Gas Separations
Gas selectivity is the ratio of permeability
coefficients of two gases
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4.4.7 Gas Permeability in Polymer Blends
4.4.8 Fickian and Non-Fickian Diffusion
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4.5 Latexes and Suspensions
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4.5.1 Natural Rubber Latex
4.5.2 Colloidal Stability and Film Formation
4.6 Multicomponent and Multiphased Materials