Phase Stability

Phase Stability

1

DG = DH – T DS Supercool/Superheat/Supersaturate

At the equilibrium m elting/freezing point or equilibrium concentration for precipitation the driving force for phase separation, DG = DH – T DS = 0. So a phase will only form by lowering the tem perature below the m elting point or increasing the concentration above the solubility lim it. This is related to

“critical slowing down”. At the critical point nothing happens, or anything can happen, there is no therm odynam ic direction for the process.

Cooling below the m elting point produces a supercooled liquid, raising the concentration above the saturation lim it produces a supercritical solution. You can also heat a solid above the equilibrium m elting point and produce a superheated solid.

The rate of phase separation depends on two factors:

DG and kinetics. Kinetics slows exponentially at lower

tem peratures D ~ exp(K/(T-T⁰)). The therm odynam ic driving force, exp(-DG/kT), decays at higher tem peratures, so a m axim um in growth rate results below the equilibrium in the super cooled liquid state.

2

Kauzm ann Paradox,

a therm odynam ic basis for the glass transition

The entropy of the liquid becom es sm aller than the entropy of the solid at the Kauzm ann

tem perature, T^K. This could be the infinite cooling glass transition tem perature.

3

Superheating and M elting

Superheating can occur since m elting occurs at surfaces and if the surfaces are stabilized then superheated solids can be produced

Growth of a liquid phase relies on growth of a m echanical instability

A m echanical instability will not spontaneously grow if it occurs in a m eta-stable region in T and P:

(dG/dx)=0 defines equilibrium or binodal; (d²G/dx²) = 0 defines the m etastable lim it or spinodal (d³G/dx³) = 0 defines the critical point

G = -ST + Vp, dG = -SdT + Vdp

(d²G/dp²)^T = (dV/dp)^T < 0 and (d²G/dT²)^p= -(dS/dT)^p< 0 First requires that the bulk m odulus be positive,

Second requires positive heat capacity, (dS/dT)^p = C^p/T > 0

Shear m odulus goes to 0 at highest possible supercritical solid

4

Therm al (energy) fluctuations are the basis of therm odynam ics

M ap of energy in 3d space at one tim e slice

https://upload.wikim edia.org/wikipedia/com m ons/2/2a/Quant um _Fluctuations.gif

Therm odynam ic Hypothesis:

System s are always probing free energy space (tem perature, com position, pressure, m agnetic field, electrical field, energy, extent of reaction, etc.) through random fluctuations. This enables the attainm ent of the lowest free energy at an equilibrium state.

This hypothesis leads to consideration of the partial derivatives of free energy as defining features in free energy space particularly critical points (d³G/dx³=0; d²G/dx²=0; dG/dx=0), equilibrium (binodal) points (dG/dx=0), and spinodal points (d²G/dx²=0).

5

The book considers first a reversible chem ical reaction A <=> B Cyclohexane from boat to chair conform ation for instance

As tem perature changes you can observer a different m ix of states, E = k^BT ~ 2.5 kJ/m ole at RT But fluctuations allow for 0.1 % boat conform ation. At 1073K 30% boat. Probability is exp(-E/kT).

The percent in boat can be m easured using NM R spectroscopy.

Chair Chair

Boat

Transition State

M etastable Transition

State

The equilibrium point depends on tem perature, k^BT

6

Extent of reaction would be the conversion from chair (A) to boat (B), for instance, at a tem perature T. The m inim um , (dG/dx) = 0 is the equilibrium point. The free energy plot can be concave up (stable equilibrium ) concave down (unstable), flat (critical point/tem perature), and other shapes

A <=> B

Gibbs-Duhem Equation

Affinity

At tem perature T, the reaction is like a spring bouncing between A and B oscillating about the equilibrium point since any deviation from equilibrium increases the free energy.

7

8

a) Stable dG/dx = 0 b) Unstable

c) Spinodal d2G/dx2 = 0

d) M etastable (This can depend on your vantage point)

9

Affinity as a Taylor series in the reaction coordinate

z is a fluctuation in a reaction coordinate, T, p, m, x, etc.

At equilibrium the affinity is 0 so = 0

If then the equilibrium is stable

10

Construction of a Phase Diagram Based on Fluctuations Consider the Hildebrand M odel

Binodal phase equilibria is defined by

x^A = 1-x^B

Spinodal is defined by

Critical Point is defined by

d³G^m/dx^B3 = RT(1/(1-x^B)² - 1/x^B2) = 0 or x^A = x^B= 0.5 Using this com position in d²G^m/dx^B2 = 0 yields

11

Hildebrand Equation

12

If W = -A + BT, and the entropy term is sm all (for polym ers for instance) B is non-com binatorial entropy Phase separation occurs on heating: Lower Critical Solution Tem perature (LCST)

Flory-Huggins Equation

fA is the volum e fraction of A N^A is the num ber of “c” units in A

n^c is the total num ber of c volum e units in the system C is an average of A and B units

Volum e fraction has replace m ole fraction

c is an average interaction energy per c site per RT c ~ W/RT

Sym m etric blend N^A = N^B

Entropy part is sm all since Nⁱ~ 100,000 c ~ a - b/T leads to LCST behavior, a is non-

com binatorial entropy, b is the enthalpy of specific interactions that leads to m iscibilty

2 Phase

1 Phase

13

Since cN depends on 1/T specifying cN specifies the tem perature. Large cN is low tem pearature.

14

15

Joon Roe

16

Spinodal B B

17

Spinodal B B

18

19

H iggins JS, Cabral JT A Thorny Problem ? Spinodal

D ecom position in Polym er Blends M acrom olecules 53 4137−4140 (2020)

20

Series expansion of the free energy density yields,

Positive second derivative leads to increase (Stable) Negative to decrease (Phase Separation)

21

Diffusion in Nucleation and Growth and in Spinodal Decom position

Fickian diffusion down a concentration gradient

Spinodal diffusion up a

concentration gradient driven by free energy

22

Phase grows from fluctuations

H iggins JS, Cabral JT A Thorny Problem ? Spinodal D ecom position in Polym er Blends M acrom olecules 53 4137−4140 (2020)

23

Activation Barrier for Nucleation and Growth

Balance between bulk and surface Free energies

(Next chapter) Different relative bulk

to surface free energies (sm all to large)

Rate of Nucleation

24

Critical Scattering

A polym er chain in a m elt is a random walk like a diffusion path so M ass ~ size²

By Bragg’s law size is related to inverse angle

d = 2/(l sinq)

Reduced angle is q = 4p/l sinq Intensity (S) scales with M ass So S ~ q^-2

Then a plot of 1/S vs q² should be a line that reflects the inverse of contrast at q = 0 At the critical point contrast goes to ∞ so 1/S goes to 0.

Deuteration provides contrast for neutrons, l = 10Å.

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41