Last Lecture:• For mixing to occur, the free energy (F) of the system

must decrease; Fmix < 0.

• The change in free energy upon mixing is determined by changes in internal energy (U) and entropy (S): Fmix = U - TS.

• The interaction parameter is a unitless parameter to compare the interaction energy between dissimilar molecules and their self-interaction energy.

• The change ofFmix with (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 = 2.

• At the critical point, interfacial energy, = 0.

Constructing a Phase Diagram

T1

T2

T3

T4

T5

kTFmix

T1<T2<T3….

Co-existence where:

0=d

dF

Spinodal where:

02

2

=d

Fd

03

3

=d

Fd

G

=2

>2

Phase Diagram for Two Liquids Described by the Regular Solution Model

G

Immiscible

Miscible

T1~

Low T

High T

Spinodal and co-existence lines meet at the critical point.

3SMS

Polymer Interfaces and Phase Morphologies ●

An Introduction to Colloids

19/20 February, 2007

Lecture 6

See Jones’ Soft Condensed Matter, Chapt. 3, 9 and 4

Free Energy of Mixing for Polymers

Polymers consist of N repeat units (or “mers”) each of length a.

The thermodynamic arguments applied to deriving Fmix 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.

a

Free Energy of Mixing for Polymer Blends, Fmix

pol

)ln+ln+(= GGRRGRmix kTF

We start with our expression for free energy of mixing per molecule, Fmix, for simple liquids:

When arranging the repeat units on the lattice, the probability is determined by the volume fractions, , of the two polymers (assuming equal-sized units). N has no influence on Smix per polymer molecule.

But the change in U upon mixing polymers must be a function of N times the Umix of each of the repeat units.

)ln+ln+(= GGRRGRpol

mix NkTF

The free energy change per polymer molecule is therefore:

Polymer Phase Separation

N2

>

)ln+ln+(= GG

RR

GRmer

mix NNkTF

As the polymer consists of N repeat units (mers), we can find the free energy of mixing per mer by dividing through by N:

Nc2

=Thus,

Phase separation

N2

< Single phase (blend) is stable

The critical point can be found from 03

3

=d

Fd

For polymers, N is the key parameter - rather than as for simple liquids.

AndN = 2 at the critical point

Polymer Phase Separation

N

A

Polymer Immiscibility

N typically has a value of 1000 or more, so that c = 2/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 ) or have low N.

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 = 2 as a result of entropy.)

Polymer mixing (miscibility) is only favoured when is negative or exceedingly small or when N is very small.

Polymeric interfacial structure and phase separation are often studied by neutron scattering and reflectivity.

Significance of Surface Tension

droplet

If >0, then the system can lower its free energy by reducing the interfacial area: F = dA

But if = 0, then mixing of droplets - or molecules - does not “cost” any energy. Thus, mixing is favoured at the critical point.

The system will separate into two “bulk” phases; droplets of any size are not favoured.

Polymer Interfacial Width, w

A B

The interface between polymers is never atomistically sharp. If the molecules are forbidden from crossing the boundary, their number of conformations would be reduced. The entropy would decrease.

Therefore, an interfacial width, w, can be defined for any polymer interface.

w

Neutron Reflectivity from a Single Interface

sin

=4

Q

Critical angle

Sensitivity of Neutron Reflectivity to Interfacial Roughness

Polymer film on a Si substrate with increasing surface roughness, .

Inversely related to film thickness

Reflectivity from a Polymer Multi-Layer

Scattering density profile

Comparison of Polymers with Different Parameters and Interfacial Widths

Scattering density profile

w

Width between Two Polymer Phases when Approaching the Critical Point

2 6 10 14 18 22 26 30 34 38

100

200

300

400

500

Inte

rfac

ial w

idth

(A

)

N Data from C. Carelli, Surrey

Experiments on immiscible polymers confirm that the interface broadens as the critical point is approached.

Also as N decreases toward 2, approaches 0.

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.

10

F

o

Fo

1 2

.

Leads to “spinodal decomposition”

N if polymers!) Spinodal points define the unstable region.

Two-Phase Structure Obtained from Spinodal Decomposition

Poly(styrene) and poly(butadiene)

undergoing spinodal decomposition.

The two phases have a characteristic size scale defined by a compromise.

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.

Fourier transform of image

Poly(ethylene) and poly(styrene) blend

AFM image

10 m x 10 m

Structures Obtained from Two Immiscible Polymers

Phases grow in size to reduce their interfacial area in a process called “coarsening”.

Structures Resulting from Phase Separation in the Metastable Region

Small fluctuations in composition are not stable.

Only 1 and 2* are stable phases! The 2* composition must be nucleated and then it will grow.

F

o

Fo

1 2

.2*

F1

Fv = Fo - F1

Free energy change (per unit volume) on de-mixing:

r

2

3

43

4rF

rrF v

nucl +=)(

Nucleation of a Second Phase in the Metastable Region

Energy reduction through phase separation with growth of the nucleus with volume (4/3)r3

Energy “cost” of creating a new interface with an

area of 4r2

1

2*

Growth of the second phase occurs only

when a stable nucleus with radius r

has been formed.

is the interfacial energy between the two phases.

If r > r *, the nucleus is stable, and its further growth will lower the free energy of the phase-separating system.

If r < r *, further growth of the nucleus will raise the free energy. The nucleus is unstable.

Fnucl

+

-

r

The free energy change in nucleating a phase, Fnucl, is maximum for a nucleus of a critical size, r *.

r*

F*

Critical Size for a Stable Nuclei

2

3

43

4rF

rrF v

nucl +=)(

vFr

2

=*Solving for r, we see:

Calculating the Size of the Critical Nucleus, r *

224

3

32

4)(+

)(=*)(=*

vv

vnucl

FF

FrFF

Substituting in our value of r *, we can find the energy barrier to nucleation:

2

3

3

16

vFF

+=*Simplifying, we see:

We can find the maximum of Fnucl from:

rFrdrFd

v

nucl

840 2 +==

Estimating the Rate of Nucleation during Phase Separation

Nucleation occurs when a fluctuation in F during the formation of a nucleus is > F *.

The rate of nucleation is determined by the frequency of the fluctuations and their probability of exceeding F*.

This probability is given by a Boltzmann factor: )*

exp(kTF

The temperature dependence is complicated by the fact that F* is a function of and Fv, which are both temperature dependent.

The barrier F * can be lowered by the presence of a “nucleant” (a surface on which the phase can grow) in heterogeneous nucleation.

Colloids

1 m

Because the size of colloidal particles is on the order of the wavelength of light, they offer some interesting optical characteristics.

Particles are much larger than the size of molecules.

Optical Characteristics of Colloidal Films

Diffraction condition:

sin2dn

d

Natural opal reflects various colours of light depending on the viewing angle.

The effect is a result of the opal structure, which consists of silica spherical particles (typically 250 - 400 nm in diameter) about 1/2 the wavelength of light, leading to diffraction of the light by the regular spacing.

Colloids in Nature: Opals

Bragg Equation: n = 2dsin

Using Colloids to Create “Inverse Opal” Structures

• Useful optical and magnetic properties.

• Inverse opals have “optical band gaps”

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.

The particles are then dissolved to leave a network of air voids.

Forces Acting on Colloidal Particles• Drag force from moving through a viscous medium• Gravity: leads to sedimentation or creaming• Random, “thermal” forces from molecules: lead to

Brownian motion• Coulombic: can be attractive or repulsive; screened by

the intervening medium• van der Waals’: attractive for like molecules• Steric: caused by intervening molecules that prevent

close approach

Viscous Drag Force

• Consider an isolated spherical particle of radius a moving with a velocity of v in a fluid (liquid or gas) with a density of and a viscosity of .

• In the limit where va << , the viscosity of a liquid imposes a significant drag force on the particle’s movement.

• The Stokes’ equation gives this force as: Fs = 6av

• Observe that Fs applies when is large in comparison to a and v.

va

Fs

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 (+ or -), then Fg = (4/3)a3g, where g is the acceleration due to gravity.

• At equilibrium the forces balance: Fs = Fg.

• So, 6av = (4/3)a3g

• The velocity at equilibrium, i.e. the terminal velocity, vt, is then found to be (2a2g)/9.

• 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.

Fg

FS

a

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

Effect of Molecular Momentum Transfer: Random Brownian Paths

2-D representations of 3-D particle trajectories

Self-similarity: appear the same on different size scales

Distance Travelled by Particles

Start

FinishR

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:

22

nR =

If in every unit of time, a particle takes a step of average distance, , in a random direction...

12

3

n

Thus the mean-square displacement is proportional to time.

Random walk

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 is a drag coefficient equal to 6a for an isolated, spherical particle in a viscous fluid.

vAmFFF Spartrand

+=+=

Writing and in terms of we see:

dtRd

dt

RdmF rand

+= 2

2

v

A

R

The Mean-Square Displacement

If random, the mean displacements in the x, y and z directions must be equal, so 22222 3=++= xzyxR

Then, multiplying through by x: 2

2

=)(

dt

xdxmFx

dtxd

x rand

And we see that 21

22/12 3= xR

But we recognise that: dtxd

xdtxd

=

)(21 2

Substituting in for the first term, we find:

2

22

2 dt

xdmxFx

dtxd

rand

=

)(

2

2/1222/12

2

2 33

dt

xdmF

dt

xd

dt

RdmF

dt

Rdrandrand

The Mean-Square Displacement

Finally, the equipartition of energy says that for each d.o.f., (1/2)mv2 = (1/2)kT in thermal energy.

Because Frand, x and v are uncorrelated, the first two terms on the r.h.s. average to zero.

After substituting an identify and taking the average of each term: 22

2 dtxd

mdtxd

xdtd

mFxdt

xdrand

=

)(

kT0

This leaves us with: dtkT

xd

2=2

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 t

kTxR

6

3 22 ==)(

A diffusion coefficient, D, which relates the distance to the time of travel, is defined as

t

RD

6

2)(=

So it is apparent that kTD =

Recall Stokes’ equation, = 6a for a spherical particle. The Stokes-Einstein diffusion coefficient is thus:

akT

DSE 6=

dtkT

xd

2=2

Applications of the Stokes-Einstein Equation

• Observe that the distance travelled, R (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.

akT

t

RDSE 66

2

=)(

=