A. Conditions for Phase Equilibrium

To write the conditions for equilibrium between phases, it is necessary to utilize thermodynamic potentials (the chemical potentials, for example) and constraints which are a function of the variables used to describe the system. They depend on the type of experiments being conducted.

To describe a mixture of several constituents, three, for example, the mole fractions nll n2, n3 and corresponding concentrations X2, X3 are introduced, where n is the total number of moles; we then have:

with:

n = nl + n2 + n3

i.e. also n2 n3

X2 = , X3 = nl + n2 + n3 nl + n2 + n3

(A.I)

(A.2)

The differential of a function f, for example, with respect to nI, is written:

a f a f X2 a f X3 a f anI an n aX2 n aX3

(A.3)

We have similar relations for a/an2 and a/an3. This leads to the sequence of differential equations using (A.2):

a a X2 a X3 a anI an n aX2 n aX3

a a 1- X2 a X3 a -=-+------an2 an n aX2 n aX3

(A.4)

a a X2 a 1- X3 a -=----+---an3 an n aX2 n aX3

Applying these equations to the definition of the chemical potentials Ili (i = 1,2,3) for a ternary mixture, we obtain the following expressions:

ac ac ac III = -- = C m - X2- - X3-

anI aX2 aX3 ac ac ac

112 = - = C m + (1- X2)- - X3- (A.5) an2 . aX2 aX3

380 A. Conditions for Phase Equilibrium

em == ~ = ~~ is the Gibbs molar function.

We easily determine from equations (A.5):

8e 8e /-t2 - /-tl = -, /-t3 - /-tl = - (A.6)

8X2 8X3

As a function of the choice of independent variables (A.I) or (A.2), we can write the conditions of equilibrium between phases either with (A.5), /-tl /-t2 = /-t3, or with equalities (A.6).

B. Percus-Yevick Equation

We should be able to completely determine the equation of state of a fluid if we can calculate the partition function of a system with a very large number of particles (the molecules in the fluid). This is a multiple integral in position and time variables which is a function of the pressure and the temperature. This calculation can only succeed by using an approximation method. The methods used are of three types: series expansions of the density, integral equations, and perturbation methods.

The density expansions have the advantage of giving the exact expres­sions for the virial coefficients which can be compared with the experimental results if the interaction forces are known. On the other hand, they do not allow predicting phase transitions, since truncated series expansions do not have singularity. The so-called Pade approximation nevertheless allows per­forming the calculation for the rest of the series. Kirkwood on one hand and Born-Green-Yvon on the other proposed integral equations for the radial distribution function g( r) which is then used to obtain the equation of state. For this purpose, a hierarchy of functions Cn ( n = 2,3 ... ) which are finite in number is established. A new class of integral equations was subsequently proposed based on a calculation performed by Ornstein and Zernike (1914) to take into account the phenomenon of critical opalescence.

The distribution function for n particles in a system of N particles can be written as:

(n)( ) - N! J J -{3UN d d / PN rl,···,rn - (N-n)! ... e rn+l··· rN ZN (B.1)

In an open system (with a variable number of particles), the probability of observing n molecules in volume drl ... drn at the point (rl, ... rn) is given by:

pen) = L pC;;) PN (B.2) N?n

This function is independent of N. The probability PN is written in the classic manner in statistical mechanics:

P _ ZNZN N- N!E (B.3)

382 B. Percus-Yevick Equation

ZN is the partition function for N particles and S is the grand partition function with

_ (31" (21r"ffikT) 3/2 Z = e h2

Carrying (B.3) into (B.2) for n = 2 and after integration in rl, and r2, we obtain:

j j pC2l(rl' r2)drldr2 = ((N~! 2)!) = (N(N - 1)) = N2 - N (B.4)

and for n = 1, we have:

j j pCl)(rdpCl)(r2)dr1dr2 = (N)2

Subtraction of these two equations gives:

(B.5)

j j[p(2)(Tl,r2) - p(1) (Tl)p(1) (r2)]dr ldr2 = N2 - N 2 - N (B.6)

where (N) = Nand (N2) = N2. We know that:

£J.N2 = ((N - N)2) = kT/'i,T N = N2 _ N 2 V

with p = (NjV) and /'i,T = p-l (g;)T.

(B.7)

Setting p(2)(Tl,r2) = p2g(rl,T2) and p(1) = p(2) = p, i.e. assuming that the fluid is invariant under translation, and utilizing (B.7), (B.6) is rewritten as:

p2V j[g(r) - l]dr = (£J.N2) - N

that is

pkT/'i,T = 1 + p j[g(r) -l]dr

(B.8)

(B.9)

Function h(r) = g(r) -1 is called the pair correlation function and it can be obtained from scattering experiments (light, neutrons). It is a measure of the influence of molecule 1 on molecule 2.

The structure function S(k) is defined using the Fourier transform of g(r) (Fig. B.l) by the equation:

S(k) = 1 + p j exp( -ikr)g(r)dr (B.1O)

Ornstein and Zernike proposed separating the influence exercised by each molecule into two parts. The first term represents the direct interaction of a molecule with its neighbors, and the second is the contribution of the indi­rect influence, which is the force transmitted via an intermediate molecule. The direct correlation corresponding to the first term is the function c(r12),

B. Percus-Yevick Equation 383

(a) (b) g(r) S(k)

r k

Fig. B.l. Pair correlation function (a) and structure function (b) for a liquid in normal conditions.

the second contribution to h(r12) is an average over all positions of the in­termediate molecule r3:

(B.ll)

This equation in fact defines function c(r). It can be solved by taking the Fourier transform and using the properties of the convolutions.

H(k) = c(k) + pH(k) c(k) (B.12)

that is:

H(k) = c(k) 1 - pc(k)

(B.13)

Substituting H(k) in (B.9), we then have:

8p 1 1 kT(8p)T = 1 + pH(O) = 1- pc(O) = 1- pI c(r)dr (B.14)

We must still determine c(r). This can be done by taking an approx­imation that describes short-range interactions. Here c(r) is the difference between the function gtotal (r) corresponding to the effective interaction rep­resented by potential w(r)(g(r) = e-i3w(r)), and the indirect interaction which is the remainder of the potential when the direct interaction is eliminated:

c(r) = gtotal(r) - gindirect(r) = e-i3w(r) - e-i3[w(r)-u(r)] (B.15)

This is the Percus-Yevick approximation for function c(r). Using the Ornstein-Zernicke method, we can then solve the system of equations.

Assuming y(r) = ei3u(r)g(r), we can rewrite (B.ll):

y(r12) = 1 + P ![e-i3U(f'13) -1]y(rh3)h(r32)dr3 (B.16)

This is the Percus-Yevick equation. As h = g -1 = ei3U y -1, this equation determines y(r) and thus g(r). It was solved for hard spheres. The equation for p and thus the equation of state is not the same as a function of whether it

384 B. Percus~Yevick Equation

is determined with the compressibility (E.g) or by direct calculation utilizing the equation for the virial (Problem 4.1):

p p2 {'Xl kT = P - 6kT io ru(r)g(r)471'r2dr (B.17)

The Carnahan-Starling equation is established using the first method.

c. Renormalization Group Theory

Take the Ising Hamiltonian 11.0 for a magnetic system

11.0 = -Jo 2: aiaj - MoHo 2: ai i,j

(C.1)

The free energy F is the sum ofthe singular part near the critical point Fs and the regular part FR corresponding to nonmagnetic lattice contributions. It is written (7.74):

e-F / kT = [2: e-.B1i l Je- FRl / kT

(n1 )

(C.2)

after taking only one out of two spins in the summation E, where (ill) represents the new configurations.

11.1 has the same form as 11.0 but with half the spins and with new values of exchange constant J l and applied field Ho. Introducing the reduced variables K = J/kT and b = MoHo/kT, we have:

Kl = f(Ko) and bl = g(bo) (C.3)

This process is iterated several times, doubling the scale with each op­eration, and a sequence of new variables is obtained: K l , K 2 .•• bl , b2 • The critical point will be a fixed point corresponding to variables Ke and be so that:

Ke = f(Ke) and be = g(be) (C.4)

because no scale change can modify the effective Hamiltonian any further. For finding Te, we operate in the vicinity of this temperature after n

renormalization operations Kn = f(Kn-d and Ke = f(Ke), that is:

oKn = Ke-Kn = f(Ke)-f(Kn- l ) = !'(Ke-Kn- l ) = !'oKn- l (C.5)

At Te, by definition Ke = Jo/kTe (renormalization no longer alters the form of the Hamiltonian). Moreover, bn = g'bn- l and at the critical point, H = be = 0 (C.5) is rewritten:

Kn = Ke - oKn = Ke - !'oKn- l

knowing that

(C.6)

386 C. Renormalization Group Theory

Jo oKo = Ke - Ko = JojkTe - JojkT = kTe c:

Kn = k~e[l- f'nc:]; bn = gln~O~O

(C.7)

After n renormalizations, if d is the dimension of the system and n is the dimension of order parameter a, the total number of remaining spins for N initial sites is Nj(2d)n. The free energy by spins Fsn is a function of reduced variables Kn and bn and thus of (fln)c: and g'n Ho, with c: = (Te - T)jT. As Jo, Te , and Ho are constants, we have:

Fsn2dn = F(f'nc:,g'nH) (C.8)

If we put), = 2dn , f' = (2d)P, g' = (2d)Q, we find an equation similar to (7.69):

(C.9)

We have a scaling law here which allows calculating the critical exponents. A Hamiltonian representing the energy, which has a Landau-type form, is then introduced:

(C.lO)

where t, U4, and a are functions of T, but where t does not necessarily have the form given it in the classic Landau expansion (t can be different from (T - Te)). Application of renormalization represented by the operator R results in an effective Hamiltonian and a fixed critical point lIe defined in the space of parameters (t, U4, a):

lIe = R(IIe) (C.U)

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Index

Acentric factor, 128 Aerogels, 207 Aerosol, 321, 354 Alloy, 9, 33, 67, 117, 213, 232, 242 Amorphous, 105, 115, 117, 166, 242,

323 Antiferroelectric, 227, 229 Antiferromagnetism, 218 Arrhenius, 169, 180 Avrami, 52-54, 107

BCS theory, 235 Bergeron process, 354 Bethe lattice, 137 Bethe model, 221 Binary - alloy, 11, 56 - mixture, 10, 71, 143, 153 - solution, 19 Binodal, 132 Birefringence, 258, 269, 314 Boiling, 359 Born criterion, 91 Bose-Einstein condensation, 288, 290,

296, 345 Bragg-Williams, 20, 216 Broken symmetry, 26 Bubble, 301, 358 - magnetic, 311, 313

Cahn-Hilliard equation, 60 Carnot cycle, 357 Cavitation, 358 Ceramic, 32, 119, 242, 303, 306, 324 Chirality, 249, 251, 254, 284 Clapeyron equation, 80, 133 Climate, 346, 352 Coexistence

- curve, 20 - line, 133 Coherence length, 234, 239 Coil, 197, 201 Colloidal, 29, 68, 70, 104, 185, 299, 318,

320 Composites, 119 Compressibility, 14, 19 Compressibility factor, 134, 135 Cooper pair, 235, 297, 314 Correlation length, 24 Corresponding states, 126, 238 Critical - exponent, 195, 225, 232, 237, 239,

241 - lines, 146, 150 - opalescence, 22, 30, 129 - radius, 43, 44 Crystallisation, 104, 107, 182, 300 Curie-Weiss law, 221

Debye theory, 171 Defect, 92, 303, 316, 336 Dendrites, 52, 68, 71, 116, 300 Density functional, 97 Diamagnetic, 232 - anisotropy, 256 Diamagnetism, 217, 273 Diamond anvil cell, 89, 338 Dielectric constant, 230, 231, 270 Differential thermal analysis, 29, 87 Diffraction - neutrons, 100, 111 Diffusion, 37, 38, 71 Diffusion coefficient, 40 Dislocation, 92 Distillation, 11 DNA, 182, 205, 206, 324

394 Index

Domain, 311 - magnetic, 310 Droplets, 138, 352

Ehrenfest, 14, 15, 33, 134 Einstein model, 90 Elasticity, 271 Electrets, 165, 226 Emulsion, 318, 319 Equation of state, 21, 123, 129 - Berthelot, 136 - Carnahan-Starling, 135, 152, 384 - Dieterici, 136 - mixtures, 146 - Peng-Robinson, 136 - Redlich-Kwong, 136 - Soave-Redlich-K wong, 136 - van der Waals, 22, 33, 125, 134 - virial, 134 Equilibrium - metastable, 5, 56 - stable, 5 - unstable, 6 Ergodic, 174, 176 Eutectic, 52, 363 Extrusion, 118

Ferrimagnetism, 218 Ferroelectric, 226, 228, 242 - domain, 313 - fluid, 283 Ferroelectricity, 226, 254, 283 Ferromagnetic, 28 Ferromagnetism, 28, 217 Fick's law, 38, 49 Film, 243, 285, 300, 319, 327 - Langmuir, 329 - Langmuir-Blodgett, 330, 336 - liquid, 361 - polymer, 336 Flocculation, 29, 320, 321 Fluctuation, 22, 59, 237, 239 Fokker-Planck equation, 49 Fractal, 68, 72, 73, 300 Frenkeltheor~ 39, 92 Frost, 355 Frustration, 224 Fullerene, 3, 236, 324, 342

Gas hydrates, 363 Gel, 71, 73, 185 - chemical, 187 - physical, 187, 199 - silica, 187 Gelatin, 187, 196, 205, 206 Gelation, 37, 68, 73, 115, 185 Geomaterials, 343 Gibbs-Duhem criterion, 5, 17 Gibbs-Thomson effect, 113 Glass, 50, 62, 163 Grain boundaries, 45, 46, 111, 120, 300,

301, 308 Grains, 300 Greenhouse effect, 353, 355

Hall-Petch effect, 323 Hard-sphere, 96, 97, 141, 171 Heat engine, 356 Heisenberg - model, 312 Heisenberg model, 218 Helimagnetism,218 Helium - fountain effect, 290 - four, 157, 286, 288, 291, 293 - mixture, 157 - sound propagation, 294 - three, 157, 286, 296-298 Helix, 196, 201 Holes, 92, 95 Hydrogen bond, 84, 114, 180, 187, 200

Ice, 2, 111, 113, 115, 354 Inertial confinement, 345 Interfacial, 113 Interfacial energy, 303, 311 Irradiation, 3, 93, 117 Ising model, 34, 142, 211, 219 Isotherm, 123, 126, 132

Kadanoff, 34, 130, 238-240 Kauzmann paradox, 94, 168 Kevlar,285 Kohlrausch-Williams-Watts, 171, 179

Landau - expansion, 217 - model, 230, 234, 281

Landau-de Gennes model, 266, 269, 281

Landau-Ginzburg theory, 234, 238 Latent heat, 2, 14, 32, 80, 174, 347,

352, 362 Lattice model, 19 Legendre transformation, 4, 18 Lennard-Jones, 82, 102, 127, 141, 171 Lever rule, 11, 153 Levitation, 118 Lindemann model, 90, 94, 102 Liquid crystal, 2, 26, 208, 247, 248 - cholesteric, 251, 283 - discotic, 255 - nematic, 250, 255, 282 - order parameter, 256, 264, 266 - smectic, 252, 253 Liquidus, 10 Lyotropic, 285

Magnetic susceptibility, 19, 213, 225, 273

Magnetism, 33 Magnetization, 16, 18, 21, 211, 310 Magnetostriction, 242 Magnets, 242 Maier-Saupe theory, 260, 265, 281 Markov process, 49, 142 Martensite, 63, 67, 75, 116, 303, 305 Martensitic transformation, 66, 75, 305 Maxwell's rule, 131, 132 Meissner effect, 232, 314 Melting, 83, 88, 90 Mesomorphic, 26, 247, 285 Metamagnetic, 157 Metastability, 6, 19, 137 Metastable, 58, 84, 166, 265 Micelle, 320 Microcrystalline, 172 Microcrystallites, 41, 56 Microstructures, 37, 51, 76, 105, 180,

299 Mode-coupling theory, 174, 177 Molding, 118 Molecular dynamics, 22, 63, 139 Molecular field, 34, 137, 218, 225, 232,

237 Monte Carlo method, 22, 63, 141, 340 Mutt

Index 395

- model, 92 - transition, 341

Nanomaterials, 76, 299, 324 Non-equilibrium thermodynamics, 188 Nuclear magnetic resonance (NMR),

30, 34, 225, 242, 243, 258 Nuclear reactor, 356, 358, 362 Nucleation, 3, 22, 37, 38, 40, 138 - boiling, 359, 360 - heterogeneous, 44, 45, 352, 358 - rate, 46, 47

Order parameter, 16, 33, 194, 214, 234 Ornstein-Zernike equation, 24, 100, 381 Overheated, 137

Pair correlation function, 382 Paramagnetism, 217 Percolation, 72, 73, 115, 193, 194 Percus-Yevick equation, 101, 383 Peritectic, 11 Perovskite, 26, 227, 243, 344 Phase diagram, 7 Phase transition, 1, 3, 7, 31, 355 Phases, 1 Piezoelectricity, 33, 226 Plasma, 4, 337, 344 Plasma crystals, 105 Plastic crystal, 27, 248 Point - azeotropic, 154 - bubble, 130, 145 - condensation, 351 - critical, 7, 123, 129, 225 - dew, 130, 145, 151, 350 - eutectic, 10, 66, 303 - lambda, 158, 288 - Leidenfrost, 360 - multicritical, 15 - supercritical, 156 - tricritical, 8, 157, 335 - triple, 7, 82, 124, 145 Poisson equation, 351 Polarization, 226, 229, 230, 232, 278 Polycrystalline, 45, 119, 299, 323 Polyelectrolytes, 199, 206 Polymer, 62, 105, 107, 109, 118, 152,

182, 188, 196

396 Index

Polysaccharides, 187, 196, 199 Porcelain, 120, 307, 310 Premelting, 110, 111 Protein, 182, 183, 196, 285 Pyroelectric, 243 Pyroelectricity, 226

Quasicrystal, 305 Quenched, 305 Quenching, 56, 62, 67, 68, 75, 116, 117,

165, 181, 306

Rankine cycle, 357 Renormalization group theory, 34, 129,

239,240 Retrograde condensation, 151 Richard rule, 80

Scaling laws, 34, 129, 238 Scattering - light, 24, 30, 62 - neutrons, 24, 30, 62, 114, 225 - Raman, 66, 114 Shear modulus, 191 Sintering, 120, 306-308 Soft mode, 66, 229 Sol, 185 Sol-gel method, 208 Solidification, 83, 95, 97, 101, 103, 362 Solution - regular, 19 - solid, 10 Spherulites, 51, 106, 107, 188, 300, 303 Spin glass, 224 Spinodal, 6, 20, 86, 132 Spinodal decomposition, 37, 56, 57, 62,

242 States, 1 Steel, 11, 62, 63, 66, 76, 116, 303 Stokes-Einstein law, 39, 174 Structure function, 24, 61, 70, 173, 382 Supercavitation, 359 Superconductivity, 232, 243 Superconductor, 314 - high temperature, 236, 237, 314, 316 Supercooled, 43, 84, 116, 164, 166, 180,

190 Supercooling, 44, 85, 94, 354 Supercritical extraction, 156

Superfiuid, 157, 286 Superfiuidity, 286, 288, 296 Superheated, 6 Superheating, 85, 95, 358, 359 Surface melting, 94, 111 Surfactants, 157, 319 Symmetry, 15,25 Symmetry breaking, 15, 82, 96, 212 Syneresis, 188, 206

Technological applications, 31, 116, 181, 206, 241, 282, 319, 323, 355, 362

Temperature - Bose-Einstein condensation, 291, 292 - critical, 1, 216, 291 - Curie, 218 - gelation, 188 - glass transition, 105, 167, 168 - Neel, 218

superfiuid transition, 292 - transition isotropic liquid-nematic,

263, 268 Thixotropy, 208 Tisza model, 293 Tokamak, 4, 345 Transition

superfiuid A-superfiuid B, 296 - boiling, 360 - coil-helix, 197, 204, 205 - conformational, 200 - displacive, 26, 227 - ferroelectric-paraelectric, 26, 226 - ferromagnetic-paramagnetic, 218,

220 first-order, 14, 16, 17, 200 Frederiks, 276, 278, 280

- glass, 3, 26, 29, 105, 163 helix-coil, 200 insulator-metal, 341, 342

- isotropic liquid-nematic, 260 - Kirkwood-Alder, 104 - Kosterlitz-Thouless, 317 - liquid crystal, 26 - liquid-gas, 125, 356 - liquid-solid, 110 - metal-insulator, 28 - multicritical, 15 - nematic-smectic A, 280

order-disorder, 26, 214, 227, 230 plastic crystal-crystal, 27 second-order, 14, 16, 212, 217, 335 sol-gel, 29, 68, 185 solid-liquid, 102

- solid-solid, 47 - structural, 25, 65 - superconducting, 28, 232 - superfiuid, 28, 288 Turnbull, 85 Two-phase system, 299, 318, 356, 358,

362

Undercooled, 137 Universality, 34, 237 Unmixing, 124, 146, 182 - supercritical, 156

Vacancies, 40

Index 397

van der Waals force, 147, 187, 260, 321 Variance, 7 Virial coefficients, 134 Virial expansion, 21 Viscosity, 191, 195, 287, 289 Vitreous, 163 Vogel-Tammann-Fulcher, 170, 175 Volmer model, 41, 47 Vortex, 295, 314-316

Water, 84, 113, 115, 156, 181, 318, 327, 347

Weiss, 215, 219, 310 Wetting, 113 Wilson theory, 239

Xerogels, 207

Zeldovitch-F'renkel equation, 49