important points from last lecture: for mixing to occur, the free energy (f) of the system must...
TRANSCRIPT
Important Points from Last Lecture:• For mixing to occur, the free energy (F) of the system must
decrease; Fmix < 0.• At a constant T, 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 of Fmix 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.
PH3-SM (PHY3032)
Soft MatterLecture 6
Polymer Phase Separation, Morphologies and Interfaces
●
An Introduction to Colloids
8 November, 2011
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, Fmixpol
)ln+ln+(= GGRRGRmix kTF
We start with our expression for free energy of mixing per liquid 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 for each of the repeat units.
)ln+ln+(= GGRRGRpol
mix NkTF
The free energy change per polymer molecule is therefore:
Polymer Phase Separation
)lnln( GG
RR
GR
polmixmer
mix NNkT
NF
F
As the polymer consists of N repeat units (mers), we can find the free energy of mixing per mer by dividing through by N:
In general terms, for an asymmetric blend of chains with two different degrees of polymerisation (NA and NB), we can write the Flory-Huggins expression, where is the volume fraction of one of the polymers:
)1ln(
)1(ln)1(
BA
mermix NN
kTF
The points on the coexistence curve are found from the tangent found when the differential is set = 0:
BBAA NNNNkT
F 1)1ln(1ln)21(0
Coexistence Curve in a Symmetric Blend
BBAA NNNNkT
F 1)1ln(1ln)21(0
In a symmetric blend, where the chains have an equal number of repeat units: NA = NB = N
NNkT
F )1ln(ln)21(0
The values of that satisfy this equation define the co-existence curve, which is the line that separates the one-phase and two-phase regions of the phase diagram, are:
NNNbinodal )12(
1ln)1ln(ln
12
1
This equation can be used to plot N as a function of on a phase diagram to define the coexistence curve.
Spinodal Line for a Polymer Blend
BBAA NNNNkT
F 1)1ln(1ln)21(0
The spinodal line is found by equating the second differential to 0:
2
)1(
110
2
2
BA NNkT
F
The curve corresponding to the inflection points defines the boundary between the unstable and metastable regions, called the spinodal line. Solving for gives us:
)1(
11
2
1
BAspinodal NN
For given values of NA and NB, the spinodal line will follow the dependence on given by this equation. For a symmetric blend (NA = NB), we see that the spinodal passes through N = 2 when = ½.
N2
>
Nc2
=Then,
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.
or cN = 2 at the critical point
Spinodal Line for a Polymer Blend
For a symmetric blend, the phase diagram is symmetric, and the critical concentration is at c = 1/2
Polymer Phase Separation in a Symmetric Blend
N
AAs ~ 1/T, we can either increase N or decrease T to cause miscible polymers to phase separate.
NA = NB = N
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 else have a low N.
Entropic contributions in polymers that encourage mixing cannot easily compensate for unfavourable energies of mixing. (Remember that in simple liquids mixing will occur up to when = 2 as a result of entropy.) Therefore, polymer mixing (miscibility) is only favoured when is negative or exceedingly small or when N is very small (such that N<2).
Polymeric interfacial structure and phase separation are often studied by neutron scattering and reflectivity.
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 boundaries.
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
Simulation of spinodal decomposition: http://youtu.be/NSpOX9mfX3g
Structures Resulting from Phase Separation in the Metastable Region
Small fluctuations in composition are not stable.Only1 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:
(N for polymers!)
Poly(ethylene) and poly(styrene) blend
AFM image
10 m x 10 m
Structures Obtained from Two Immiscible Polymers via Nucleation
Phases grow in size to reduce their interfacial area in a process called “coarsening”.
Comparison of “Nucleated” and “Spinodal” Structures
Nucleated structure: islands of one phase in another
Spinodal structure: co-continuous phases
From G. Strobl, Polymer Physics, Springer
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 *
2
3
)2
(43
24
*)(*v
vvnucl
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.
Significance of Surface Tension,Significance of Surface Tension,
droplet
If >0, then the system can lower its free energy by reducing the interfacial area: F = A
But if = 0, then mixing of droplets - or molecules - does not “cost” any energy (no work required!). 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
The interface between immiscible polymers is never atomistically sharp.
If the molecules are forbidden from crossing the boundary, their number of conformations (i.e. arrangements) is reduced. The entropy will then decrease. Entropy increases with interfacial broadening.
Therefore, an interfacial width, w, can be defined for any polymer interface.
A B
w w
Neutron Reflectivity from a Single Interface
sin
=4
Q
Critical angle
Usually a distribution of are used to obtain a broad range of Q values.
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
Deuterium is often used to label polymers to increase their “contrast” with hydrogenated polymers when scattering neutrons.
Comparison of Polymers with Different Parameters and Interfacial Widths
Scattering density profile
w
Width between Two Polymer Phases when Approaching the Critical Point
Data from C. Carelli et al. Phys. Rev. E 72, 031807 (2005)
2 6 10 14 18 22 26 30 34 38
100
200
300
400
500
Inte
rfac
ial w
idth
(A
)
N
Experiments on immiscible polymers confirm that the interface broadens as the critical point is approached.
Also as N decreases toward 2, approaches 0.
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
Colloidal particles are dispersed in a continuous matrix.
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
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.
Phase Equilibria in Colloidal Dispersions
V. Prasad, D. Semwogerere and Eric R. Weeks, J. Phys.: Condens. Matter 19 (2007) 113102 (25pp)
Mono-sized particles can become ordered into crystals at = 0.54 while still in the “wet” state.
RCP = random close-packing;
HCP = hexagonal close-packing
(Volume %)
Equilibrium:
Non-equilibrium (glassy):
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
vaRe V = velocity
a
= viscosity of the continuous medium
= density of the continuous medium
•When Re is low, viscosity dominates over inertia. There is no “coasting”!
•At low Reynolds’ number, when va << , the viscosity of the liquid imposes a significant drag force on the particle’s movement.
•The Stokes’ equation gives this force (Stokes’ or drag force) as: Fs = 6av
•Observe that Fs applies when is large in comparison to a and v.
The Viscous Drag Force on ColloidsThe Reynolds’ number, Re, compares the effects of inertia (momentum) to viscous drag:
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. The same principle applies for separation by size using centrifugation.
Fg
FS
a
Experimental Observation of Brownian Motion
Phenomenon was first reported by a Scottish botanist named Robert Brown (19th cent.)
Brown observed the motion of pollen grains but realised that they were not living.
Brownian motion
Computer Simulation: http://youtu.be/6VdMp46ZIL8
Effect of Molecular Momentum Transfer: Random Brownian Paths
2-D representations of 3-D particle trajectories
Self-similarity: paths appear the same on different size scales
R
0R
Mean displacement:
Mean-Square Displacement
02 R
Displacements of a large number of particles
Mean-square displacement:
Root mean-square displacement:
21
2R
Distance Travelled by Particles
Start
FinishR
Then when observed over n steps at equal 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 the number of steps and hence to the 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.
Einstein 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:
2
2
2
2
dt
RdmF
dt
Rd
dt
Rd
dt
RdmF randrand
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 identity 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 (in 3D) 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 in a fluid. 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.
a
kT
t
RDSE 66
)( 2