1. Last Lecture:
For mixing to occur, the free energy (F) of the system must decrease; DFmix < 0.
The change in free energy upon mixing is determined by changes in internal energy (U) and entropy (S): DFmix = DU - TDS.
The c interaction parameter is a unitless parameter to compare the interaction energy between dissimilar molecules and their self-interaction energy.
The change of DFmix with c (and T) leads to stable, metastable, and unstable regions of the phase diagram.
For simple liquids, with molecules of the same size, assuming non-compressibility, the critical point occurs when c = 2.
At the critical point, interfacial energy, g = 0.

2. Constructing a Phase Diagram
Spinodal where:
Co-existence where: T1
c >2 T2 T3 T4
c =2 T5 fG
T1<T2<T3….

3. Phase Diagram for Two Liquids Described by the Regular Solution Model
Low T
Immiscible
Miscible
High T
Spinodal and co-existence lines meet at the critical point. fG

4. See Jones’ Soft Condensed Matter, Chapt. 3, 9 and 4
Polymer Interfaces, Phase Morphologies, and an Introduction to Colloids
3SCMP
23/27 February, 2006
Lecture 6
See Jones’ Soft Condensed Matter, Chapt. 3, 9 and 4

5. Significance of Surface Tension
The system will separate into two “bulk” phases; droplets of any size are not favoured.
droplet
If g>0, then the system can lower its free energy by reducing the interfacial area: DF = gdA
But if g = 0, then mixing of droplets - or molecules - does not “cost” any energy. Thus, mixing is favoured at the critical point.

6. Free Energy of Mixing for Polymers
Polymers consist of N repeat units (or “mers”).
The thermodynamic arguments applied to deriving DFmix for simple liquids can likewise be applied to polymers.
The derivation must consider the connectivity of units when putting them on a lattice. N units are mixed all at once rather than individually.

7. Free Energy of Mixing for Polymer Blends, DFmixpol
We start with our expression for free energy of mixing per molecule, DFmix, for simple liquids:
When arranging the repeat units on the lattice, the probability is determined by the volume fractions, f, of the two polymers (assuming equal-sized units). N has no influence on DSmix per polymer molecule.
But the change in U upon mixing polymers must be a function of N times the DUmix of each of the repeat units.
The free energy change per polymer molecule is therefore:

8. Polymer Phase Separation
As the polymer consists of N repeat units (mers), we can find the free energy of mixing per mer by dividing through by N:
The critical point can be found from
Thus,
And cN = 2 at the critical point
Phase separation
Single phase (blend) is stable
For polymers, cN is the key parameter - rather than c as for simple liquids.

9. Polymer Phase SeparationcN

10. Polymer Immiscibility
N typically has a value of 1000 or more, so that cc = 2/N is very small.
Entropic contributions in polymers that encourage mixing cannot easily compensate for unfavourable energies of mixing. (Remember that in liquids mixing will occur up to when c = 2 as a result of entropy.)
Single phases in polymers are only favoured when c is negative or exceedingly small or when N is very small.
This fact explains why most polymers are immiscible - making them difficult to recycle unless they are blended with very similar molecules (low c) or have low N.
Polymeric interfacial structure and phase separation are often studied by neutron scattering and reflectivity.

11. Neutron Reflectivity from a Single Interface
Critical angle q q

12. Sensitivity of Neutron Reflectivity to Interfacial Roughness
Polymer film on a Si substrate with increasing surface roughness, s.
Inversely related to film thickness

13. Reflectivity from a Polymer Multi-Layer
Scattering density profile

14. Scattering density profile
Comparison of Polymers with Different c Parameters and Interfacial Widths w
Scattering density profile

15. Width between Two Polymer Phases when Approaching the Critical Point
Experiments on immiscible polymers confirm that the interface broadens as the critical point is approached.
Also as cN decreases, g approaches 0.
Data from C. Carelli, Surrey

16. Structures Resulting from Phase Separation in the Unstable Region
When moving from the one-phase to the unstable two-phase region of the phase diagram, ALL concentration fluctuations are stable.
(cN if polymers!)
Spinodal points define the unstable region. 1 F fo Fo f1 f2 . c
Leads to “spinodal decomposition” f

17. Two-Phase Structure Obtained from Spinodal Decomposition
The two phases have a characteristic size scale defined by a compromise.
Fourier transform of image
If the sizes of the phases are too small: energy cost of extra interfaces is too high.
If the phases are quite large, it takes too long for the molecules to travel the distances required for phase separation.
Poly(styrene) and poly(butadiene) undergoing spinodal decomposition.

18. Structures Obtained from Two Immiscible Polymers
Phases grow in size to reduce their interfacial area in a process called “coarsening”.
Poly(ethylene) and poly(styrene) blend
AFM image
10 mm x 10 mm

19. Structures Resulting from Phase Separation in the Metastable Region
Free energy change (per unit volume) on de-mixing: c
DFv = Fo - Ff1
Small fluctuations in composition are not stable. F fo Fo f1 f2 .
f2*
Ff1 f
Only f1 and f2* are stable phases! The f2* composition must be nucleated and then it will grow.

20. Nucleation of a Second Phase in the Metastable Region
Growth of the second phase occurs only when a stable nucleus with radius r has been formed.
f2* r f1
g is the interfacial energy between the two phases.
Energy reduction through phase separation with growth of the nucleus
Energy “cost” of creating a new interface

21. Critical Size for a Stable Nuclei+ F*
The free energy change in nucleating a phase, DFnucl, is maximum for a nucleus of a critical size, r*. r*
DFnucl r -
If r < r*, further growth of the nucleus will raise the free energy. The nucleus is unstable.
If r > r*, the nucleus is stable, and its further growth will lower the free energy of the phase-separating system.

22. Calculating the Size of the Critical Nucleus, r*
We can find the maximum of DFnucl from:
Solving for r, we see:
Substituting in our value of r*, we can find the energy barrier to nucleation:
Simplifying, we see:

23. Estimating the Rate of Nucleation during Phase Separation
Nucleation occurs when a fluctuation in DF during the formation of a nucleus is > DF*.
The rate of nucleation is determined by the frequency of the fluctuations and their probability of exceeding DF*.
This probability is given by a Boltzmann factor:
The temperature dependence is complicated by the fact that DF* is a function of g and DFv, which are both temperature dependent.
The barrier DF* can be lowered by the presence of a “nucleant” (a surface on which the phase can grow) in heterogeneous nucleation.

24. Colloids Particles are much larger than the size of molecules. 1 mm
Because the size of colloidal particles is on the order of the wavelength of light, they offer some interesting optical characteristics.

25. Optical Characteristics of Colloidal Films

26. Colloids in Nature: Opals
The effect is a result of the opal structure, which consists of silica spherical particles (typically nm in diameter) about 1/2 the wavelength of light, leading to diffraction of the light by the regular spacing.
Natural opal reflects various colours of light depending on the viewing angle.
Bragg Equation: nl = 2dsinq

27. Using Colloids to Create “Inverse Opal” Structures
The particles are then dissolved to leave a network of air voids.
Colloidal particles are packed into an ordered array.
The space between the particles is filled with a solid through infiltration or deposition from the vapour phase.
• Useful optical and magnetic properties.
• Inverse opals have “optical band gaps”

28. Forces Acting on Colloidal Particles
Gravity: leads to sedimentation or creaming
Coulombic: can be attractive or repulsive; screened by the intervening medium
Drag force from moving through a viscous medium
van der Waals’: attractive for like molecules
Random, “thermal” forces from molecules: lead to Brownian motion
Steric: caused by intervening molecules that prevent close approach

29. Viscous Drag Force v Fs r h
Consider an isolated spherical particle of radius a moving with a velocity of v in a fluid (liquid or gas) with a density of r and a viscosity of h.
In the limit where rva << h, the viscosity of a liquid imposes a significant drag force on the particle’s movement.
The Stokes’ equation gives this force as: Fs = 6phav
Observe that Fs applies when h is large in comparison to a and v. v a
r h Fs

30. Effect of Gravity on Particle Velocity
If the density of a particle is different than that of the surrounding fluid, it will be subject to a gravitational force, Fg, leading to settling (or rising).
If the difference in density is Dr (+ or -) , then Fg = (4/3)pa3Drg, where g is the acceleration due to gravity.
At equilibrium the forces balance: Fs = Fg.
So, 6phav = (4/3)pa3Drg
The velocity at equilibrium, i.e. the terminal velocity, vt, is then found to be (2a2Drg)/9h.
Larger particles will settle out much faster than smaller particles - giving us a means to separate particles by size. Same principle applies for separation by size using centrifugation. FS a Fg

31. Experimental Observation of Brownian Movement
Phenomenon was first reported by a Scottish botanist named Robert Brown (19 cent.)
Brown observed the motion of pollen grains but realised that they were not living.
Brownian motion

32. Effect of Molecular Momentum Transfer: Random Brownian Paths
Self-similarity: appear the same on different size scales
2-D representations of 3-D particle trajectories

33. Distance Travelled by Particles
Start
Finish
If in every unit of time, a particle takes a step of average distance, , in a random direction... 1 2 3 n
Random walk
Then when observed over n time units, the average particle displacement for several “walks” will be 0, but the mean square displacement is non-zero:
Thus the mean-square displacement is proportional to time.

34. Equation of Motion for Brownian Particles
Einstein was unaware of Brown’s observation, but he predicted random particle motion in his work on molecular theory.
He and Smoluchowski wrote an equation for the equation of motion for a Brownian particle in which the net random force exerted by the fluid molecules, Frand, balances the forces of the particle:
where x is a drag coefficient equal to 6pha for an isolated, spherical particle in a viscous fluid.
Writing and in terms of we see:

35. The Mean-Square Displacement
If random, the mean displacements in the x, y and z directions must be equal, so
And we see that
Then, multiplying through by x:
But we recognise that:
Substituting in for the first term, we find:

36. The Mean-Square Displacement
After substituting an identify and taking the average of each term: kT
Because Frand, x and v are uncorrelated, the first two terms on the r.h.s. average to zero.
Finally, the equipartition of energy says that for each d.o.f., (1/2)mv2 = (1/2)kT in thermal energy.
This leaves us with:

37. The Stokes-Einstein Diffusion Coefficient
Integrating and multiplying by three, E and S thus showed that the mean squared displacement of a Brownian particle observed for a time, t, is
A diffusion coefficient, D, which relates the distance to the time of travel, is defined as
So it is apparent that
Recall Stokes’ equation, x = 6pha for a spherical particle. The Stokes-Einstein diffusion coefficient is thus:

38. Applications of the Stokes-Einstein Equation
Observe that the distance travelled (root-mean-square displacement, <R2>1/2) varies as the square root of time, t1/2!
Early work assumed that the distance should be directly proportional to time and made data interpretation impossible.
Experiments, in which the displacement of colloidal particles with a known size was measured, were used by Perrin to determine the first experimental value of k.
Brownian diffusion measurements can be used to determine unknown particle sizes.
The technique of light scattering from colloidal liquids is used to find particle size through a diffusion measurement.