Equilibrium means the situation of minimum free energy. If a system is left alone for a long time, it settles down to an equilibrium. For normal solids and liquids this is a perfect crystal. For a mixture of two liquids interesting phase behavior may be observed.
If one changes some external parameter, a structural change to a more energetically favorable structure may occur. This structure has the lowest free energy.
Phase transition may be a change from more ordered to less ordered state, e.g. from solid to liquid. In general, at high temperatures disordered phases are more stable than the ordered ones.
Example. Block-copolymers are polymers consisting of covalently bond 'blocks' of two or more polymers with different properties and functional groups.
Example. Gyroid phase: http://en.wikipedia.org/wiki/Gyroid
A reference: Alexandridis et al. Record (Four Cubic, Two Hexagonal, and One Lamellar Lyotropic Liquid Crystalline and Two Micellar Solutions) in a Ternary Isothermal System of an Amphiphilic Block Copolymer and Selective Solvents (Water and Oil) Langmuir 1998; 14(10); 2627-2638
http://pubs.acs.org/doi/abs/10.1021/la971117c
Phase diagrams are used to describe the phase behavior of materials.
Two-dimensional phase diagrams may be e.g.
http://en.wikipedia.org/wiki/File:Phase-diag2.svg
Balance between entropy and energy is reflected in the free energy. For instance, for changes in constant volume the free energy is described as Helmholtz free energy
F = U - TS,
where U is internal energy, S entropy, and T temperature.
Ehrenfest classification of phase transitions:
In the first order transitions the first derivatives of free energy are discontinuous at the phase transition. The discontinuity of entropy, ΔS, leads to an associated exchange of heat, called the enthalpy of transition
ΔHtr = T ΔStr,
where the temperature T is the temperature at the phase transition.
(The enthalpy of transition is also called ’latent heat’, since a small amount of heat exchanged at the transition point does not induce any change in the temperature of the system. )
Due to the intake or release of heat at constant temperature the heat capacity
Cp = T (∂S/∂T)p
of the system diverges at the transition point for first order phase transitions.
http://en.wikipedia.org/wiki/Enthalpy
Order parameters may also show discontinuity at phase transition temperature.
In the second order phase transitions the discontinuities appear only in the second (or higher) derivatives of the free energy. There is no enthalpy of transition associated with these phase transitions.
A second-order phase transition is characterized by a continuous first derivative of chemical potential, but discontinuous second derivative.
Examples of materials which may be called glasses:
When liquid goes through a glass transition, it forms a glass. A material’s glass transition temperature, Tg, is the temperature below which molecules have little relative mobility. Tg is applicable to wholly or partially amorphous phases such as glasses and plastics. For inorganic glasses it is the mid-point of a temperature range in which they gradually become more viscous and change from being liquid to solid. Non-crosslinked polymers have both a melting temperature, Tm, above which their crystalline structure disappears, and lower Tg below which they become rigid and brittle, and can crack.
Glass transition from liquid to glass and glass transition temperature may
depend on cooling rate.
The glass transition shows as a discontinuity in heat capacity. Glass transition is usually detected by using calorimetry. DSC curves show heat flux versus temperature or versus time. http://en.wikipedia.org/wiki/Differential_scanning_calorimetry
There is no wholly satisfactory theory to explain the formation and behavior
of glasses. Physical idea to explain the behavior of glasses is the co-operativity.
At high temperature and low density: particles may move quite freely. Particles
are in uncoordinated random motion. At lower temperatures free space is not
necessarily available. The neighboring particles move less. To change positions,
the particles must move co-operatively.
At high temperature and low density an atom can jump from one position to
another. At low temperature and higher density an atom may move only with the
aid of co-operative movement of neighboring atoms.
The energy barrier for a single molecule is denoted by Δμ and the number
of molecules in the co-operatively rearranging region is denoted by z. The
activation barrier is proportional to the number of molecules that have to move,
z. The relaxation time t may be written as
1/t = v exp(-z Δμ /kT),
where v is the microscopic frequency of the vibration of the particles.
Example. Glass transition of weakly ordered polymer.
Consider a polymer which is amorphous at all temperatures t. However, there may be glass transition at low t. At this temperature, Tg, the material changes from glassy to rubbery. Above Tg, the non-covalent bonds between the polymer chains become weak in comparison to thermal motion, and the polymer becomes rubbery and capable of elastic or plastic deformation without fracture. Cross-linked thermosetting plastics do not show this behavior: they will shatter rather than deform, never becoming plastic again when heated, nor melt.
Order parameters may be useful. Its value is
The behavior of the order parameter as a function of the temperature tells
about the nature of the phase transition.
Assume that there are two types of molecules A and B in the system: A site is
occupied by a molecule A or a molecule B, with probabilities pa and
pb.
Assume neighboring sites to be independent of each other. This is a mean field
assumption.
Thus the entropy is
Smix = -kb (pa ln pa + pb
ln pb).
When entropy of mixing is zero? Either pa or pb is unity –
pure liquid.
The probabilities represent in real experiment volume fractions.
Assume that molecules interact only with their nearest neighbors. Denote energies of interaction between two A neighbors as eaa, between two B neighbors as ebb, and between A and B neighbors as eab. The number of nearest neighbors, coordination number, is denoted by z.
Mean field assumption: a given site has zpa A neighbors and zpb B neighbors.
Energy of unmixed state is
(z/2)(paeaa + pbebb).
Division by two is because all pairs are accounted only once.
Energy of mixing is the interaction energy minus the energy of unmixed state:
z/2 [ pa(pa eaa + pb eab) + pb(pb ebb + pa eab) - (pa eaa + pb ebb)]
Thus the energy of mixing is
Umix = z/2 [(pa2 -pa)eaa+ (pb2-pb) ebb + 2papbeab]
Χ = z / (2kbT) (2eab – eaa – ebb ) .
Using pa + pb = 1, the energy of mixing is in terms of the interaction parameter
Free energy of mixing;
Mixtures with compositions between φ1 and φ2 can lower their free energy by
separating into two phases at these compositions.
Example. Co-existing compositions. The figure presents the free energy of mixing vs composition. The two minima are at points φ1 and φ2.
>> pa = 0:0.01:1; x=2.5; f = pa.*log(pa) + (1-pa).*log(1-pa) + x*pa.*(1-pa)
Landau model is mesoscopic, not an atomistic model. The model has been used for soft matter, magnets, and superconductors.
The basis is the expansion of free energy in a power series of an order parameter.
The model applies to weak phase transitions and small entropy change.
Example. Nematic liquid crystal. An order parameter would parametrize the orientational order of the structural units. In first order phase transition the order parameter shows discontinuity. In second order phase transition it would decrease continuously at the transition.
The starting point is Gibbs free energy, which is defined as:
Landau theory considers changes in free energy density, Gibbs energy per unit volume. The free energy density f is expanded as a power series in the order parameter Z:
f(T,Z) = f(T,0) + A(T)Z + B(T) Z2 + C(T) Z3 ...
The first term is the free energy of the high-temperature phase.
The symmetry of the phases imposes constraints on the terms and puts part of them zero.
Example. Lamellar to isotropic, smectic A to C (tilt of units). A parameter describing the order would be the tilt angle φ. An order parameter would be Z(r) = Z0 exp(i φ(r)). It this is of second order transition, odd powers of Z are zero, since f(t, Z) should be real.
f(T,Z) = f(T,0) + B(T) Z2 + D(T) Z4 ...
How does this look like?
The simplest form for B would be
B(T) = constant (T-T*) = b(T-T*)
where T* is the transition temperature.
If T > T* and D is omitted, there is a single minimum at Z=0.
How to find the equilibrium states?
Assume D constant. Differentiate f(T,Z) with respect to Z and set the result to zero.
2b(T-T*) Z + 4 DZ3 = 0.
Solutions are Z = 0 and
Z = +/- sqrt(b/2D) sqrt(T-T*). = Z2
Above T* the only real solution is Z=0.
When T<T*, Z = 0 corresponds to a (local) maximum in free energy. The solutions Z2 are symmetrically placed minima.
Is this a second order phase transition?
The entropy density would be
s = -(∂f/∂T)v
Above the phase transition s would be constant s0 and below the transition
s = s0 + (b2/2D) (T-T*).
The is thus no discontinuity.
Hamley page 18-19
Spinodal decomposition
Nucleation
http://math.gmu.edu/~sander/movies/spinum.html
(E. Sander and T. Wanner. Monte Carlo simulations for spinodal
decomposition. Journal of Statistical Physics, 95(5-6):925-948, 1999.
E. Sander and T. Wanner. Unexpectedly linear behavior for the
Cahn-Hilliard equation. SIAM Journal on Applied Mathematics,
60(6):2182-2202, 2000.)
A clustering reaction in a homogeneous, supersaturated solution (solid or
liquid) which is unstable against infinitesimal fluctuations in density or
composition. The solution therefore separates spontaneously into two
phases, starting with small fluctuations and proceeding with a decrease in the
Gibbs energy without a nucleation barrier
(IUPAC chemical terminology)
Spinodal decomposition
Spinodal line: any small local change in composition lowers the free energy. The system is unstable and any small fluctuation in composition is amplified.
Uphill diffusion: material may flow from regions of low concentration to regions of high concentration.
Example. Spinodal decomposition by Lattice-Boltzmann simulation
http://sbcb.bioch.ox.ac.uk/oliver/research/Lattice-Boltzmann/spinodal.html
Phenomenological theories:
As mentioned,
Gibbs function or Gibbs free energy is G = E – TS + PV, where P is the pressure and
V the volume of the system.
It is assumed that the system is constrained to be at constant pressure P allowing the
volume V to vary.
Then, for the total entropy of the system and the reservoir to be maximized, the
Gibbs free energy G must be minimized.
The interfaces are assumed not to be sharp. The composition is denoted as φ(x,t).
Furthermore, it is assume that the free energy density of a mixture depends on the local composition φ and its gradient. Free energy may be written as
F = A ∫ [g() + k (dφ/dx)2]dx
where k is the gradient energy coefficient and the function g gives the free energy per unit volume for a uniform mixture of the composition φ.
Is the assumption, that diffusion is driven by gradients in composition,
reasonable?
Statistical mechanics says that it is not the composition that becomes constant
but the chemical potential.
So one should rather assume that the diffusion is driven by the gradient of
chemical potential.
Consider a system that can exchange energy and matter with their
surroundings.
N is the number of the particles.
Chemical potential μ is defined as
μ = -T(∂S/∂N)E,V
According to statistical mechanics the chemical potential should be constant in equilibrium.
Thus for small departures from equilibrium the flux of material should be
proportional to chemical potential μ.
Example. Binary system. Again a system with species A and B is considered. The flux of species A can be written as
Ja = -M (d/dx) (μa-μb),
where M is the transport coefficient.
It is positive: M > 0.
The difference
μ = μa-μb
is called the exchange chemical potential. It represents the change of energy needed to remove a molecule A and replace it by a molecule B.
The exchange chemical potential is written as
μ = d/dφ [g(φ) + k (dφ/dx)2]
It can also be presented in form
μ = dg(φ)/dφ + k d2φ/dx2
Substituting this one obtains for the flux of A-particles:
Ja = -M (d/dx) (μa-μb) = Mg’’ ∂φ/∂x + 2Mk ∂3φ/∂x3
Here it was denoted g’’ = d2g/dx2.
M and k are assumed to be independent on composition. The solution is valid for
only small deviations from some uniform starting concentration.
From the flux
Ja = Mg’’ ∂φ/∂x + 2Mk ∂3φ/∂x3
and the equation of continuity
dφ/dt = -dJ/dx
one obtains the Cahn-Hilliard equation:
∂φ/∂t = Mg’’ ∂2φ/∂x2 + 2Mk ∂4φ/∂x4.
If the last term is missing, this is the diffusion equation with the effective
diffusion coefficient Deff = Mg’’.
Solution of C-H equation is roughly of the form
φ = φ0 + A cos(qx) exp[-Deff q2(1+2kq2/g’’)].
The term -q2(1+2kq2/g’) may be called the amplification factor. The effective diffusion constant can be either positive or negative.
>> q=0:.1:1;
>> kg = 0.5; f1 = -q.^2.*(1+2*kg*q.^2);
>> q=0:.1:1; kg = -0.5; f2 = -q.^2.*(1+2*kg*q.^2);
>> ff = plot(q,f1,q,f2)
>> xlabel('q'), ylabel('amplification factor')
>> legend('k/g = 0.5','k/g = -0.5')
>> figu(ff,[])
Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28, 25.
Example. Phase separation of mixture: 50% 2-propanone, 50% hexadecane at time t = 0
...15 s.
It was found experimentally by following the drop size evaluation that the
amplification factor behaved approximately as R(t) = 0.4t.
Reference: Califano et al. Drop size evolution during phase separation of liquid
mixtures. Ind. Eng. Chem. Res. 43, 349-353, 2004.
Nucleation is the onset of a phase transition in a small but stable region.
The creation of a nucleus implies the formation of an interface at the
boundaries of the new phase.
If a nucleus is too small, nucleation does not proceed. The energy that would be
released by forming its volume is not enough to create its surface.
As the phase transformation becomes more and more favorable, the formation of a
given volume of nucleus frees enough energy to form an increasingly large
surface, allowing progressively smaller nuclei to become viable.
Thermal activation will provide enough energy to form stable nuclei. These can
then grow until thermodynamic equilibrium is restored.
Consider the nucleation of a particle of a size r.
Let the interfacial energy be γ.
Free energy change related to volume change is denoted by ΔFv
Free energy is proportional to the surface area of the nucleating droplet.
Change of free energy on nucleating a droplet of size r:
ΔF(r) = 4/3π r3 ΔFv + 4πr2 γ.
Critical size rc gives the maximum of ΔF(r):
The rate of nuclection is proportional to exp(-ΔF(r)/kT).
This is the probability that the fluctuation increases the free energy by ΔF(r).
(Heterogeneous nucleation due to e.g. dust particles is more common than
homogeneous nucleation.)
Figure. ΔF(r) = 4/3πr3 ΔFv + 4πr2 γ
>> a=25; r = 1:20; fv = -4*pi*r.^3; fa = 4*pi*r.^2; ff = plot(r,fv,r,fa*a,r,fv+fa*a)
>> xlabel('r'), ylabel('\Delta F')
>> figu(ff,[])
Nucleation or spinodal decomposition is followed by growth. Growth is poorly understood. Limiting case is Ostwald ripening where larger particles grow at the expense of smaller ones (Ostwald, W. Z Phys. Chem. 1900, 34, 495). Ostwald ripening is a process where the difference in (local) solubility with particle size leads to a transport of material from small to larger particles, with an accompanying increase in the mean particle size with time.
Ostwald ripening is the process by which larger particles grow at the expense of smaller ones. Small particles tend to be more soluble than large ones and tend to lose their molecules. These molecules diffuse through the continuous phase and re-precipitate onto larger particles. This leads to an increase of average particle size with time.
http://en.wikipedia.org/wiki/Ostwald_ripening
Dynamic scaling hypothesis: The average domain size is the length scale characterizing the structure.
This flux of material J leads to the growth of the domains:
dR(t) / dt ~ J
dR(t) / dt ~ Dγ/R(t)2
Gowth law by integration: R(t) ~ (Dγt)1/3
The local composition near the boundary of the smaller drop is larger than
the bulk coexisting composition. This leads to concentration gradient which
drives diffusion of material from small drops to large drops.
This is a first order phase transition:
At the transition the state of order changes discontinuously.
Thermodynamic quantities that are the derivatives of a free energy with respect
to other thermodynamic variables are discontinuous at the transition.
Freezing is initiated by nucleation.
Homogeneous nucleation produces a uniform distribution of precipitate particles.
Consider crystallization of undercooled melt. It is initiated by spontaneous appearance of a crystal.
It is assumed that the crystal is spherical with radius r.
Free energy change depends on the surface area of the crystal and the volume:
ΔG(r) = 4/3πr3 ΔGb + 4πr2 γ.
1. term: contribution of liquid-solid interface
2. term: change in Gibbs free energy on going from liquid to solid.
When freezing a liquid with only a short range order forms crystals with long range order. The soft material becomes rigid.
Entropy change on melting in terms of the latent heat of melting
ΔSm = ( ∂Gs/∂T)p - ( ∂Gl/∂T)p = ΔHm/Tm,
where s and l denote solid and liquid.
Assume that the undercooling temperature difference ΔT is small enough that the
partial derivatives are approximately constant.
Then free energy change for unit volume when a melt freezes can be approximated
as –(ΔHm/Tm)ΔT for small T
Free energy change is
c= -4/3πr3 ΔHm/Tm ΔT + 4πr2 γ.
Here γ is the interfacial tension between solid and liquid.
Figure. The free energy change as a function of r.
The free energy has maximum at a critical radius
r*= 2 γTm /(ΔHm ΔT)
Crystals with radii less than the critical radius are unstable and can melt.
There is a barrier energy ΔG* which is proportional to 1/ΔT2 .
The probability for a crystal to be nucleated is proportional to exp(-ΔG*/kT).
There is a strong dependence on temperature!
In the case of heterogeneous nucleation, some energy is released by the partial destruction of the previous interface.
Assume that a carbon dioxide bubble forms between water and the inside surface
of a bottle. The energy inherent in the water-bottle interface is released
wherever a layer of gas intervenes, and this energy goes toward the formation of
bubble-water and bubble-bottle interfaces.
The same effect can cause precipitate particles to form at the grain boundaries
of a solid.
A solidification front moving into an undercooled melt is unstable. This can lead to cellular or dendritic growth.
Dendritic means multibranched tree-like form
When liquid crystallises it releases heat which must diffuse away. The rate of growth of the interface is related to the rate at which the heat can diffuse away.
Example. Formation of single calcium carbonate microcrystals as a function of time.
The phase transformation pathway of CaCO3 from the initial supersaturated solution into the final product was followd by SAXS/WAXS.
The mineralization of CaCO3 proceeds in two steps:
At this stage the amorphous particles slowly dissolve again and CaCO3 is turned into the more stable crystalline modifications.
Pontoni et al. Crystallization of Calcium Carbonate Observed In-situ
by Combined Small- and Wide-angle X-ray Scattering. J. Phys.
Chem. B 107, 5123-5125, 2003. http://pubs.acs.org/doi/abs/10.1021/jp0343640
Kinetics of phase transition for a one-component system
Originally for small molecules
Denote time by t, density of crystalline phase ρc, density of liquid phase ρl. V
is the volume of the growing center.
Fraction of transformed in the time t is
1-λ(t) = 1-exp(ρc/ρl ∫0t V(t,s) N(s) ds),
where N is the nucleation frequency of the untransformed volume.
Simplifying assumptions: N constant, rate of crystal growth linear. Analytic
solution
1-λ(t) = 1-exp(-ktn)
where k is constant. The parameter n is called the Avrami exponent.
Homogeneous nucleation, linear growth
3D growth n=4 (steady state)
2D growth n=3
1D growth n=1
Homogeneous nucleation, diffusion controlled growth
3D growth n=5/2 (steady state)
2D growth n=2
1D growth n=3/2
The new phase is nucleated by tiny nuclei in the old phase.
The effective number of nuclei <N> may be altered by temperature and duration of
heating.
The number of nuclei(t) N per unit region decreases from <N> by active growth or
by being swallowed by grains of the new phase.
Let V(t) be the volume of the new phase per unit volume of the new phase. The time dependence may be written as:
N(t) = <N> exp(-nt) [1-V(t)]
Example. Early stages of crystallization in isotactic polypropylene.
Isothermal crystallization of iPP at 130 C (Daplen) was studied by a combined WAXS/SAXS setup. Integrated intensity from SAXS/WAXS data represented the time dependent crystallinity. Both ntegrated WAXS and SAXS intensities obeyed Avrami model.
Avrami model for the time dependence of the volume fraction of the (here) crystalline material
1-X = exp(ktn),
where k is the rate constant of the crystallization process and n
is the Avrami exponent which, under certain assumptions, provides information
about the dimension of the growing crystalline material.
For i-PP the Avrami exponent was n = 3.4.
Example. SAXS and spinodal decomposition
Scattering intensity is proportional to the autocorrelation function of the electron density. A strong peak in the intensity suggests that the phase separation occurs via spinodal decomposition. The theory of Cahn and Hilliard predicts that the compositional fluctuations, and thus the scattered intensity, have a maximum for a given wavelength. Then peak position q* should remain constant in the early stages, but the peak intensity should exhibit an exponential increase with time:
I(q,t) = I(0) exp(2R(q)t)
Growth rate
R(q) = -q2 Deff[1-1/2 (q/qm)2]
is the amplification rate of the composition fluctuation. The inverse of the wave vector q gives the typical length scale of the phase separating structure.
In conventional binary fluid phase separation at fixed average composition, a typically bicontinuous structure emerges at early times with a constant characteristic length scale until nonexponential (nonlinear) growth is reached, at which point the structure coarsens, the kinetics slows, and eventually a macroscopically phase-separated state is attained.
In a polymer melt there is a single species, so any spinodal-like behavior
must necessarily be of different origin. Possibilities include the following:
A demixing instability induced purely by polydispersity in one of the
quantities, such as branching, stereoregularity, or molecular weight, which is
coupled to density. A liquid-liquid instability due to the coupling of the
density to other "hidden" internal degrees of freedom, with the most likely
candidate being the polymer conformation. An instability to an orientationally
ordered state, such as a nematic or smectic precursor phase, within which
crystallites nucleate quite quickly.
W. Li and AJ Ryan. Morphology development via reaction induced phase
separation in flexible polyurethane foam. Macromolecules 35(13),
5034-5042, 2002. http://pubs.acs.org/doi/full/10.1021/ma020035e
