1. Concentrated Polymer Solutions
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2. Application and Importance of Concentrated Polymer Solutions
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3. What does mean? A dilute solution Theta condition Semi-dilute
Concentrated and so On..
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4. What we expect from a concentrated solution?
The phase separation……
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5. What are the requirements for phase separation.
The required conditions in terms of thermodynamical properties of the system.
The effect of temperature.
The effect of molecular weight.
The effect of concentration.
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6. Solubility, The temperature and the Mw.
Tc=The critical temperature is the highest temperature of phase separation.
Phase diagrams for polystyrene fractions in cyclohexane. Circles and solid lines,
experimental. Theoretical curves are shown for two of the fractions. The viscosity-average
molecular weights are PSA, 43,600; PSB, 89,000; PSC, 250,000; PSD, 1,270,000 g/mol
The lower molecular weight species will tend to remain in solution
at a temperature where the higher molecular weights phase-separate
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7. The criteria for concentration based on chain situation in solution
Relationships of polymer chains in solution at different concentration regions. (a) Dilute solution regime, where C < Cov. (b) The transition regions, where C = Cov. (c) Semi-dilute regime, C > Cov. Note overlap of chain portions in space.
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8. A review on chain conformation
n links, each of length
l, joined in a linear sequence with no restrictions on the angles between successive
bonds
 is the bond angle between atoms, and  is the conformation angle
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9. Finally by considering the bond angle =109  and all other constraints such as excluded volumes (lesser total number of conformations) and so on:
The characteristic ratio C∞ = r 2/ l2n varies from about 5 to about 10, depending on the foliage present on the individual chains.
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10. Typical values of C∞
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11. Segmental lengths in Chains
Kuhn Segment Length
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12. The Kuhn Segment, b
The Kuhn segment length, b, depends on the chain’s end-to-end distance under Flory -conditions, or its equivalent in the unoriented, amorphous bulk state, r,
For flexible polymers, the Kuhn segment size varies between 6 and 12 mers , having a value of eight mers for polystyrene, and 6 for poly(methyl methacrylate).
The Kuhn segment also expresses the idea of how far one must travel along a chain until all memory of the starting direction is lost, similar to the axial correlation distance.
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13. one phase
Two phase
composition
temperature
LCST
one phase
Two phase
UCST
composition
temperature
spinodal curves
bimodal curves
upper critical solution temperature (UCST)
lower critical solution temperature (LCST)
Most of the polymer solutions exhibit either UCST or LCST behavior with a few exceptions that exhibit both.

14. Phase Diagram
A= Entropic Part
B= Enthalpic Part
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15. Phase diagram
At Theta Condition
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16. Lever Rule
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17. Lever Rule; Free Energy and Stability
Metastable
Stable
fraction fa of the volume of the material having composition Фα (and fraction fβ = 1 —fα having composition Фβ,
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18. Phase diagram
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19. Phase diagram and chain dimensions
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20. Excluded Volume
Is the difference in repulsion and attraction interactions between the chain segments:
V= the effect of repulsion forces on chain structure attraction forces affects chain structure
So V > 0 if repulsion >attraction => extended chain
V< 0 if attraction > repulsion => collapsed chain
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21. Phase diagram chain dimension and the excluded volume
The thermal blob size is the length scale at which excluded volume becomes important.
1- For v ≈ b3, the thermal blob is the size of a monomer
(ξT≈ b) and the chain is fully swollen in an athermal solvent .
2- For v ≈ — b3, the thermal blob is again the size of a monomer (ξT≈ b) and the chain is fully collapsed in a non-solvent.
3- For |v| < b3N-1/2, the thermal blob is larger than the chain size (ξT > R0) and the chain is nearly ideal.
4- For b3N-1/2 < |v| < b3 the thermal blob is between the
monomer size and the chain size, with either intermediate swelling in a good solvent [v > 0] or intermediate collapse in a poor solvent [v<0].
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22. From Fractal dimension
Chain Dimensions
From Fractal dimension
Exclude volume and the chain dimension
Excluded volume repulsion v>0 (good Solvent)
Excluded volume attraction v<0 (poor solvent)
Dense Package
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23. Good, Poor, Theta and Athermal
ν= 3/5
ν= 1/2
ν= 1/3
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24. At Theta Condition At the θ-temperature, the chains have nearly ideal
conformations at all concentrations:
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25. At Theta Condition
The concentration increases moving from left to right. At T=θ, there is a special concentration that equals the concentration inside the pervaded volume of the coil. This is the overlap concentration for θ –solvent.
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26. At Theta Condition The temperature at which chains begin to
either swell (above θ) or collapse (below θ):
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27. Poor Solvent
The highest point on the binodal line is the critical point with critical composition
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28. Poor Solvent
The highest point on the binodal line is the critical point with critical composition
Phase diagrams for polystyrenes in
cyclohexane, M = gmol-1 (open
circles), M = gmol"1 (filled
circles), M = gmol-1 (open
squares), M= gmoP1 (filled
squares),
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29. Poor Solvent
The size of the globules is proportional to the one-third power of the number of monomers in them:
Supernatant
phase
Precipitant
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30. Poor Solvent
The size of the globules is proportional to the one-third power of the number of monomers in them:
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31. Good Solvent
Overlapping
concentration
in good solvent
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32. What happens with the chain swelling in good solvents (due to the excluded volume) above the overlap concentration?
Important concept is that of the screening of excluded volume interactions in the concentrated solutions (Flory, Edwards):
as the chain concentration increases in the region , the coil swelling gradually diminishes and finally it vanishes in the melt (i.e. coils are ideal in the melt -Flory theorem) .
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33. Polymer coil dimensions in semidilute
solutions: example of scaling arguments
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34. Scaling Theory
Go for it….
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35. Scaling Law And The Polymer Solvent Diagram
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36. The Polymer-Solvent Diagram
dilute
Semidilute
Marginal
Concentrated
Ideal
v= 1-2
= volume fraction,
p = C /6,
N = number of bonds per chain, vc = cross-over from swollen to ideal. 2
p3/2N-1/2
Excluded
Volume=v 1
-N-1/2
Phase
Separation
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cr 

37. Macroscopic Phase Separation
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38. Convex part of the function F(Ф): no macroscopic phase separation.
The typical dependence of the Flory-Huggins free energy on the polymer volume fraction in the solution : F Ф
Binodal
points
Spinodal 1
This dependence contains both convex and concave parts. F Ф
Convex part of the function F(Ф): no macroscopic phase separation.
Free energy of the solution separated into two phases with and
Free energy of homogeneous solution at F
Concave part of the dependence F(Ф): macroscopic phase separation into two phases.
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39. Phase separation by Spinodal mechanisim
Phase separation by Nucleation and Growth
Spinodal decomposition is a mechanism by which a solution of two or more components can separate into distinct regions (or phases) with distinctly different chemical compositions and physical properties. This mechanism differs from classical nucleation in that phase separation due to spinodal decomposition is much more subtle, and occurs uniformly throughout the material—not just at discrete nucleation sites.
Spinodal decomposition is of interest for two primary reasons. In the first place, it is one of the few phase transformations in solids for which there is any plausible quantitative theory. The reason for this is the inherent simplicity of the reaction. Since there is no thermodynamic barrier to the reaction inside of the spinodal region, the decomposition is determined solely by diffusion. Thus, it can be treated purely as a diffusional problem, and many of the characteristics of the decomposition can be described by an approximate analytical solution to the general diffusion equation.
In contrast, theories of nucleation and growth have to invoke the thermodynamics of fluctuations. And the diffusional problem involved in the growth of the nucleus is far more difficult to solve, because it is unrealistic to linearize the diffusion equation.
From a more practical standpoint, spinodal decomposition provides a means of producing a very finely dispersed microstructure that can significantly enhance the physical properties of the material. [1]
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40. Some Review Slides
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41. This dependence is shown in the figure:
Conditions for the phase separation (minimum possible free energy) are determined from common tangent straight line - binodal curve. Conditions for the absolute stability of homogeneous phase at a given concentration are determined from the positions of inflexion points - spinodal curve. or
This dependence is shown in the figure: Ф
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42. Phase diagram with binodal and spinodal
Single
globules
Binodal Ф
Conclusions:
Macroscopic phase separation takes place at the quality of solvent only slightly poorer than the solvent.
The critical point for macroscopic phase separation corresponds to the dilute enough solution.
The region of isolated globules in solution corresponds to very low polymer concentrations, especially at the values of  significantly larger than
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43. The precipitant phase close enough to the - point is very diluted.
For different values of N the binodal curves (boundaries of the phase separation region) have the form: Ф
With the increase of N the critical temperature becomes closer to the point, and the critical concentration becomes lower.
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44. Method of fractional precipitation for polydisperse polymer solution: when the quality of solvent is becoming poorer or polymer concentration increases in the dilute enough range at first the most high-molecular fraction precipitates, then the next fraction, etc…; polymers with lower molecular weights require more significant increase in and to precipitate. In this way polymer fractionation is achieved.
Reverse method is called the method of fractional dissolution: when one moves from the region of insolubility to the region of partial solubility at first the fractions with the lowest values of M are dissolved.
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45. What is the connection of the Flory-Huggins parameter and the temperature T ? Within the framework of the lattice model in the experimental variables T, c the phase diagram has the form shown in the figure, i.e. the poor solvent region corresponds to T c
Such situation is called upper critical solution temperature (UCST) - critical point is “on the top” of the phase separation region.
Examples: poly(styrene) in cyclohexane (around ), poly(isobutylene) in benzene, acetylcellulose in chlorophorm.
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46. increases with the increase of T.
However, due to the complicated renormalization of polymer-polymer interactions due to the solvent, sometimes
increases with the increase of T.
Then the T, c phase diagram has the form shown in the figure below, i.e. the poor solvent region corresponds to .
Such situation is called lower critical solution temperature (LCST)-critical point is “on the bottom” of the phase separation region. T c
Examples: poly(oxyethylene) in water, methylcellulose in water, in general - most of the water-based solutions. The reason: increase of the so-called hydrophobic interactions with the temperature (organic polymers contaminate network of hydrogen in water and water molecules become less mobile (solvated), i.e. they lose entropy - this unfavorable entropic factor for polymer- water contacts is more important at high temperatures).
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47. Suppose that the polymer with UCST is glassy without solvent in this range of temperatures. Then the situation is similar to that shown in the figure below: T c 1
Upon the temperature jump to the region of macroscopic phase separation, the separation begins, but it cannot be completed, because of the formation of the glassy nuclei which “freeze” the system. As a result, microporous system is formed, and this is one of the methods of preparation of microporous chromatographic columns.
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48. The Vitrification Effect
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
Berghmans’ point
Vitrification
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49. Polymer Blends
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50. POLYMER–POLYMER Miscibility
PHASE SEPARATION
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51. Why Phase Separation? BECAUSE OF REDUCED COMBINATORIAL ENTROPY
OF MIXING
Endotherm Enthalpy change Plus small Entropy Change 
Not necessary NEGATIVE FREE ENERGY of MIXING
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52. Temperature Effect 1 Hmix=TSmix T Phase Separation
Lower Critical Solution Temp.
LCST
One Phase 1
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53. PHASE DIAGRAMS
Spinodal Lines
Binodal Line
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54. Triple Mixtures PII PII Solvent MII Enters Phase Separation Region PI
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55. HOWEVER, It is possible to have:
TYPE I
TYPE II
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56. TYPE III TYPE IV TYPE V TYPE VI 12/1/2018 Two Phase One Phase

57. Kinetics of Phase Separation
Major mechanisms by which two components of a mutual solution can phase-separate: nucleation and growth, and spinodal decomposition
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58. Nucleation and Growth
Nucleation and growth are associated with metastability, implying the existence of an energy barrier and the occurrence of large composition
fluctuations.
Domains of a minimum size, the so-called critical nuclei, are a necessary condition.
Nucleation and growth (NG) are the usual mechanisms of phase separation of salts from supersaturated aqueous solutions, for
example.
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59. Spinodal Mechanism:
Spinodal decomposition (SD), on the other hand, refers to phase
separation under conditions in which the energy barrier is negligible, so even small fluctuations in composition grow.
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60. Comparison of the two mechanisms:
Nucleation and growth could be seen as tiny spheres, while spinodal decomposition looked like tiny overlapping worms.
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61. Which kind of phase separation does the picture show?
Investigation under
Microscope
Which kind of phase
separation does the
picture show?
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62. 12/1/2018

63. as observed under a microscope
Phase separation by spinodal decomposition () and nucleation and growth (•),
as observed under a microscope
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64. Thermodynamics of Mixing
The Flory–Huggins theory provides an expression for the
free energy density of mixing of two homopolymers labeled
A and B
Entropy Origin
Enthalpy Origin
Ni is the number of monomers in chain i, and i is the volume of each monomer on chain i, A is the volume fraction of component A in the mixture, v is an arbitrary reference volume,  is the Flory–Huggins interaction parameter, Gm is the free energy change on mixing per unit volume, k is the Boltzmann constant, and T is the absolute temperature.
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65. If we consider:
Then:
Note that Ni depends weakly on temperature (because i depends on temperature) while ^ Ni does not.
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66. At the critical point:
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67. In the case of (NA=NB=N)
The spinodal curve:
The binodal curve:
and critical point is located at:
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68. Temperature Dependence of 
The  parameters obtained from polymer blends are often
linear functions of 1/T.
However, in some cases a distinct nonlinearity is observed when  is plotted versus 1/T. In such cases the data can be fit to a quadratic function in 1/T.
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69. Numerical Values of A, B and C,
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70. Types of Phase Diagram
Type I.  is Positive and Increases Linearly with 1/T (B > 0, C = 0)
Example:SPB(88)/dSPB(78)
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71. UCST= 105 C
in our example
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72. PIB/dHHPP blend as an example
Type II.  is Negative and Decreases Linearly with 1/T (B < 0, C = 0)
PIB/dHHPP blend as an example
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73. Lower Critical Solution Temperature (LCST)
Transition from single phase to
two phase occurs at 1705 C regardless of composition.
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74. Type III.  is Positive and Increases Nonlinearly with 1/T(C  0, d/dT  0)
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75. Type III (cont.) One Phase Two Phase
An example of such behavior is PEB/dSPI(7)
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76. Type IV.  is Positive and Nonmonotonic with Temperature, and C > 0
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77. Type IV (cont).
Example system is HHPP/dSPI(7)
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78. Type V.  is Positive and Nonmonotonic with Temperature, and C < 0
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79. Type V (cont.)
The SPI(7)/dPP blend exhibits such
behavior
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80. Type VI. Athermal Mixing,  is Independent of T (B = 0 and C = 0)
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81. Type VI (cont.)
SPI(50)/SPB(78) blends are the examples
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82. Thermodynamics of phase separation according to Sperling
where V is the volume of the sample (usually taken as 1 cm3), Vr is the volume
of one cell , z is the lattice coordination number (z is usually between 6 and 12), and Nc is the number of molecules in 1 cm3. V/Vr is a count of the number of cells in 1 cm3.
Compare it to our previously studied one:
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83. Thermodynamics of phase separation according to Sperling (Cont.)
An Example Calculation: Molecular Weight Miscibility Limit
V1 is the molar volume of the solvent
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84. Solution An Example Calculation: Molecular Weight Miscibility Limit
V1 is the molar volume of the solvent
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85. 12/1/2018

86. Equation of State Theories: Deficiency of the proposed Eq.
A serious deficiency in the classical theory, is the assumption of incompressibility.
This deficiency can easily be remedied by the addition of free volume in the form of “holes” to the system. These holes will be about the size of a mer and occupy one lattice site. In materials science and engineering, “holes” are frequently called “vacancies.” Imagine that a multicomponent mixture is mixed with N0 holes of volume fraction v0.
Then the entropy of mixing is:
This allows for compressibility, since the number of holes may be varied.
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87. Reduced Quantities:
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88. Thus, an equation of state requires three reducing quantities,
Reduced Temperature:
Reduced Pressure
Reduced Volume
Reduced Density
The starred quantities represent characteristic values for particular
polymers, often referred to as the “hard core” or “close-packed” values, that is, with no free volume.
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89. Hartmann equation of state (An example)
the classical Doolittle equation
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90. Properties of Reduced Density,
When all the sites are occupied,
Fractional free volume (vacant sites):
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91. The Entropy of Mixing, Now!
The entropy of mixing vacant sites with the molecules in equation of
state terminology is given by
When all the sites are occupied, = 1, and the right hand side is zero.
where the quantity * is a van der Waals type of energy of interaction.
Note that ˜ = ˜ (P,T); ˜  1 as T  0; ˜  1 as P  .
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92. where r is the number of sites (mers), in the chains, and
By taking , the equation of state via the lattice fluid theory is obtained
where r is the number of sites (mers), 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,
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93. Criteria for Miscibility:
Basically, T* values must be similar for miscibility.
Polymers
must have similar coefficients of expansion for miscibility; that is, the ratio of the densities must remain similar as the temperature is changed.
If T*1 > T*2, then it is desirable to have P*1 > P*2
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94. Equation of state parameters for some common polymers
From such information, it may be found that the system poly(2,6-dimethyl
phenylene oxide)–blend–polystyrene is miscible but that poly(dimethyl siloxane), which is so different from the others, should be immiscible with all of them.
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95. Very Important Conclusions:
For the Flory–Huggins theory of incompressible polymer mixtures, the small entropy of mixing is dominated by the heat of mixing, leading to the conclusion that the heat of mixing must be zero or negative to induce miscibility.
When two compressible polymers are mixed together, negative heats of mixing cause a negative volume change.
Since reducing the volume of the system reduces the number of holes, the entropy of mixing change is negative.
At high enough temperatures, the unfavorable entropy change associated with the densification of the mixture becomes prohibitive, that is, TS > H, and the mixture phase-separates.
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96. Very Important Conclusions:
The two-phase system has a larger volume as the holes are reintroduced, and hence a larger positive contribution to the entropy.
Significantly differing coefficients of expansion contribute to phase separation.
In virtually all polymer–polymer systems exhibiting critical phenomena, both H and TS are relatively small quantities. Thus relatively modest changes in either the enthalpy or the entropy alter the phase diagram significantly.
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97. Let’s try softwares
Applet by Maye’s group
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98. Radius of Gyration of a Polymer Coil
The radius of gyration Rg is defined as the RMS distance of the collection of atoms from their common centre of gravity. R
For a solid sphere of radius R;
For a polymer coil with rms end-to-end distance R ;
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99. The excluded volume effect
Steric hindrance on short distances limits the number of conformations
At longer distances we have a topological constraint – the self avoiding walk – or the excluded volume effect:
Instead of <R2>1/2=aN1/2 we will have
<R2>1/2=aNν where v>0.5
Experiments tells us that in general: v~0.6
Why?
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100. Excluded volume according to Flory
Consider a cube containing N segments of a polymer
V=r3 where r is the radius of gyration.
The concentration of segments is c~N/r3
Each segment with volume ע “stuffed” into the cube reduces the entropy with –kbעN/V = -kbעN/r3 (for small x; ln(1-x)~lnx)
The result is a positive contribution to F; Frep= kbעTN/r3 (expansion of the coil)
From before; Coiling reduces the entropy; Fel=kbT3R2/2Na
The total free energy F is the sum of the two contributions!
Search for equilibrium!
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101. Scaling Law Theories
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102. Equilibrium and stability
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103. Lever Rule Metastable Stable
fraction fa of the volume of the material having composition Фα (and fraction fβ = 1 —fα having composition Фβ,
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104. with respect to 2 be zero.
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 G in equation :
with respect to 2 be zero.
The critical concentration at which phase separation occurs may be written:
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105. The Critical Values n= number of segments per polymer chain
For large n, 2c =1/n0.5.
For n = 104,2c = 0.01, a very dilute solution.
The critical value of the Flory–Huggins polymer–solvent interaction parameter, 1, is given by
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as n approaches infinity, 1c approaches 1/2

106. The Critical Temp. Where 1 is a constant. Plot of 1/Tc versus
(1/n /2n) should yield the Flory
-temperature at
n = infinity
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107. What do you understand from the following picture?
Critical Concentration
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108. The corresponding volume fraction
Now we consider more systematically the equilibrium properties of concentrated polymer solutions of overlapping coils.
It is to be reminded that the overlap concentration of monomer units is
The corresponding volume fraction
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109. Dilute solution Semidilute Concentrated Polymer melt
Since , the overlap occurs already at very low polymer concentration.
There is a wide concentration region where
coils are overlapping and strongly entangled; and
. Such solutions are called semidilute. 1
~ 0.2
Dilute
solution
Semidilute
Concentrated
Polymer
melt
The existence of the regime of the semi-dilute polymer solutions is a specific polymer feature, for low-molecular solutions such regime does not exist.
The crossover volume fraction between the two regimes is
for solvents (ideal coils)
for good solvents
(swollen coil)
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110. Supernatant phase Precipitant
Behavior of Polymer Solutions in Poor Solvents
In poor solvent (below the point) the attraction between monomer units prevails. Single chains (or chain in dilute enough solutions) collapse and form a globule. However, in concentrated solutions the macroscopic phase separation can take place as well (a kind of intermolecular collapse).
Supernatant
phase
Precipitant
What are the conditions for macroscopic phase separation? To answer this question it is necessary to write down the free energy of polymer solution. This problem was first solved independently by Flory and Huggins ( ) for the lattice model of polymer solution.
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111. Flory and Huggins obtained:
where is the total number of lattice sites and is the so-called Flory para-meter; corresponds to ( very good solvent).
This term describes translational entropy of coils (free energy of ideal gas of coils)
Term responsible for excluded volume interaction
Term responsible for the attraction of monomer units
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112. Binary interactions, second virial coefficient B
With the increase of the quality of solvent becomes poorer. Which value of  corresponds to the point? The expansion of F in the power of :
Ideal gas term
Binary interactions, second
virial coefficient B
Ternary interactions, third
virial coefficient C
At point corresponds to
- good solvent region
- poor solvent region
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113. The Polymer-Solvent Diagram
Semidilute
Marginal
dilute
Excluded
Volume=v
Concentrated 
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114. The Polymer-Solvent Diagram
Semidilute
Marginal
dilute
Concentrated
Excluded
Volume=v
Ideal 
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115. Excluded Volume A chain cannot cross itself in space.
Is a long-range interaction; eliminates conformations in which two widely separated segments would occupy the same space.
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116. <r2>1/2=aNν where v>0.5
Excluded Volume
At longer distances we have a topological constraint – the self avoiding walk – or the excluded volume effect:
Instead of <r2>1/2=aN1/2 we will have
<r2>1/2=aNν where v>0.5
Experiments tells us that in general: v~0.6
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