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Flory Huggins equation for the Gibbs Free Energy of Mixing, DG is given by,

Equation 3.85 and 3.86 pp. 88 Strobl. f_i is the volume fraction of component "i", N_i is the degree of polymerization, c is the Flory Huggins interaction parameter. The first two terms represent the combinatorial entropy of mixing while the latter term represents the enthalpic component of the free energy change on mixing which should reflect only local site-site interactions of neighboring mer units. The interaction parameter in this form is given by,

c has an inverse temperature dependence and since this is the only place in the free energy change expression where temperature is involved the thermal dependence of miscibility can be written in terms of c or T. The first derivative of free energy with respect to composition reflects the condition of equilibrium through equality of chemical potentials between coexisting phases. The second derivative reflects the condition for stability (spinodal) of a system to small perturbations in composition and the third derivative reflects the critical point for miscibility. From these thermodynamic definitions the critical point for a polymer mixture and the spinodal curve in composition can be obtained. The critical point occurs at f_c and at a temperature reflected in c_c,

Figure 3.14 (pp. 91) shows the calculated Free energy change on mixing for a symmetric blend, N_A = N_B, as predicted by the Flory Huggins equation. The above equations show that the critical composition for this situation is 0.5 and the critical c_c is 2/N. Because of this dependence of the critical interaction parameter on molecular weight, N, physicists often plot cN versus composition rather than T versus composition in constructing a phase diagram for a polymer system, lending some degree of universality to the plots.

In Figure 3.14, the free energy change on mixing shows three behaviors which are reflected in the three derivatives discussed above. At low c (high temperatures), there is a single minimum reflecting miscibility. At the critical c, cN = 2, the curve becomes markedly flat associated with a transition to a double minima curve at lower temperatures. At large c (low temperature), two minima in the free energy change are observed. The bottom curve represents a single phase system, the middle curve represents a critical system having a temperature just at which phase separation begins while the upper curve represents a system where phase separation will occur, the two phases which form corresponding in composition to the points where a line is tangent to both curves. The points of tangent are the binodal points at that temperature. Another pair of points, where the free energy change goes from concave down to concave up reflects the stability limit where spontaneous decomposition occurs.

Figure 3.15 on pp. 93 shows these two sets of points for various cN as a function of composition. The critical point is where all of these derived curves meet. In this case the critical point, which defines the limit of phase miscibility, occurs at a low temperature similar to low molecular weight systems, i.e. you heat sugar to dissolve it in water. Often polymer systems display the opposite behavior, phase separation on heating (PVME/PS system). From an empirical perspective such behavior necessarily involves a major modification of the definition of the interaction parameter.

The definition of c, above, is of the form B/T reflecting the enthalpic nature of this parameter. From the Flory-Huggins equation this can lead only to systems which phase separate on cooling (upper critical solution temperature, UCST, systems, figure 3.17 pp. 96) or systems which are always single phase (for negative B). In order to obtain phase separation on heating (LCST systems which are commonly observed in polymers), there must be a negative B and a positive temperature independent term, A, in c which reflects an entropic component, c = A + B/T, figure 3.18 pp. 97. Such an entropic component of c is termed a non-combinatorial entropy reflecting the combinatorial (counting) statistics used in the derivation of the Flory-Huggins equation.

A single phase system such as a blend of two polymers contains fluctuations in concentration due to random thermal motion of the blend components. (Some of these fluctuations are larger than others, particularly near the average coil size in the mixture.) Thermal fluctuations are dampened by the nature of the free energy curve discussed above. For example, consider an average concentration <f> = 0.25 system where small thermal fluctuations exist as shown below,

S(q) = <df_q²>/V_c,

where S(q) is proportional to the scattered intensity, V_c is a lattice volume element from the Flory-Huggins approach for instance, and df_q is the q wave vector component of the difference between a concentration value and the mean concentration in the system,

df_q = f_q - <f>

<df_q> = 0. f_q involves breaking up essentially random thermal concentration fluctuations into Fourier components (a series of cosine waves). Fourier theory is based on the assumption that these Fourier components can be treated independently. For a single Gaussian polymer coil <df_q²>/V_c can be calculated from the Gaussian statistics function leading to the Debye scattering function for a single polymer coil,

where Q = q²Nb²/6 = q²R_g², Rg is the radius of gyration for the polymer coil and q is the wavenumber for concentration fluctuations measured in the scattering experiment as = 4p/l sin(q), where l is the wavelength of the radiation and 2q is the scattering angle. q is also proportional to the inverse of the Bragg spacing, q = 2p/d.

To consider the growth of phases in the spinodal regime consider the free energy of a polymer blend of composition f, which contains small concentration fluctuations, G(f(r)), as described by the "Ginzburg-Landau Functional", Strobl equation 3.136 pp. 111 (and Hashimoto paper Macro 16, 641 equation II-1).

g(f(r)) is the free energy density as a function of concentration and phase size, r, and the second term contains the gradient of the concentration. The term b' reflects the cost associated with the presence of concentration fluctuations associated with df/dr (or del f in 3-d coordinates). This differs from a surface energy term since a "surface" does not exist, see figure 3.24 pp. 104.

The Ginzburg-Landau Functional (function of arbitrary functions) simply states that the average free energy associated with the blend reflects the sum of component free energy densities times volume plus the average free energy associated with fluctuations. There is no linear term in del f because <df> = 0, so the second order term, <(df)²> is the simplest approximation. The expression should also include higher, even-powered terms for large concentration fluctuations. The term b' can be calculated for the specific system of interest, and for polymer blends it is given by,

so is associated with a natural size-scale for fluctuations in the system associated with the coil size (equation 3.163 pp. 114 and RTb"/2 = b', pp. 112 below 3.148). The value of b' is obtained by comparison of the Debye scattering function with the scattering function obtained from the Ginzburg-Landau Functional using Flory-Huggins Theory. (This is done by Strobl on pp. 113-114.)

The equation for concentration growth rate reflects a balance between the change in the chemical potential (second derivative of the free energy density) with respect to concentration, times the derivative of the concentration gradient in space, and a factor which accounts for the energy cost associated with a concentration gradient, b', in terms of an average size parameter for the system related to the radius of gyration for the component coils and the concentrations. The chemical potential is a bulk system parameter, i.e. reflects the largest size scales so that an infinite ensemble is an appropriate approximation. The parameter, b', indicates that the system contains some natural size-constant which you might view as being similar to a time-constant for a temporally oscillating system. i.e. here oscillations are in concentration rather than in time for something like a pendulum. This is basically what Strobl means by and "equation of motion" on the top of pp. 121. Depending on the relative values of the derivative of the chemical potential and the parameter b', the system will select certain "frequencies" in spatial size, r, which are preferred for growth.

f(r, t) - <f> = df(r, t) = S exp[t R(q)] {A(q) cos(q dot r) + B(q) sin(q dot r)}

where R(q) is the linear growth rate for phases at wavenumber "q" associated with wavelength l, given by,

d²g/df² is a measure of the direction of curvature of the free energy versus composition curve, i.e. in the nucleation and growth regime the curve is concave up so d²g/df² is positive and in the spinodal regime the curve is concave down and d²g/df² is negative. For the nucleation and growth regime the growth rate, R(q), is negative (b' is a positive size-related parameter) and no growth occurs spontaneously. For the spinodal regime R(q) can be positive if d²g/df² > 2b'q², implying a critical (maximum) q for phase growth (or a minimum size for phase growth since l = 2p/q). This value of q is termed a critical q, q_c,

q_c = (-(d²g/df²)/2b')^1/2

The maximum growth occurs for dR(q)/dq = 0, or d²g/df² = -4 q_max² b',

q_max = (-(d²g/df²)/4b')^1/2 = q_c/Ã2

R(q_max) = -D_AB (d²g/df²)/8b'

The position of the maximum growth rate and the critical growth rate are governed by thermodynamics and size alone, i.e. there is no transport term, while the rate at the maximum is governed by thermodynamics and transport. The presence of a maximum size of growth reflects a balance between transport and thermodynamics. The expression for R(q) reflects this balance between thermodynamics and kinetics. At small q, q < q_max, or large size, thermodynamics favors rapid decomposition while the rate term's transport prefactor leads to slower growth (q² dependence of the prefactor). At large q, q > q_max, or small sizes, the thermodynamic term (in brackets) lowers the rate due to the -2 b'q² term.

The expression for q_max and R(q_max) can be used to explain the difficulty of kinetic studies of spinodal decomposition in metals and the relative ease of spinodal studies in polymers. Polymers are large molecules so the term b' is typically on the order of close to a micron in size, where as for metals it is of atomic dimensions. This means that spinodal decomposition, for thermally similar systems, will be studied with laser light scattering for polymers and x-ray or neutron scattering for metals. Typically, a light scattering pattern can be easily collected in seconds while x-ray or neutron scattering patterns require times on the order of hours. This size difference, coupled with much higher transport coefficients, D_AB, for metals leads to decomposition rates on the order of seconds for metals and on the order of minutes for polymers. This means that despite, Cahn-Hillard theory's original development for metals, the only real systems where spinodal decomposition can be studied in-depth, without special quenching procedures etc., are polymeric!

Since the terms D_AB and (d²g/df²) are coupled in the Cahn-Hillard approach, these terms are often referred to as the apparent diffusion coefficient, D_app, reflecting both thermodynamic and transport properties,

D_app = D_AB(d²g/df²)

There is a great deal of self-consistency required by Cahn-Hillard theory in the function R(q). A plot of R(q)/q² versus q² yields D_app as the low-q intercept and 2b' as the slope. These can be used to calculate the growth rate at maximum, the q value at maximum and the critical maximum q at which the growth rate goes to 0. This means that any two features of the spinodal growth rate curve will predict the entire curve! Demonstration of self-consistency in a spinodal curve is therefore evidence that the Cahn-Hillard theory is appropriate for a data set.

The Flory-Huggins equation can be substituted for g(f) in the Ginzburg-Landau Functional. For a simple system of matched molecular weights this results in equation II-15 of Hahsimoto,

where D_c is the self-diffusion coefficient for translational diffusion which scales as N^-2. R₀ is the unperturbed coil size, n^1/2a. c_s is the interaction parameter at the spinodal. From comparison with the general, Cahn-Hillard equation given above,

Since c is proportional to 1/T,

Scattering offers a direct interpretation of the development of interfaces through Porod's Law which describes high-q scattering from a system which displays sharp and smooth interfaces (i.e. oil and water type system). Porod's law states that the scattered intensity, S(q) equals 2p r²S_V, where r is a contrast factor such as electron density for x-rays, and S_V is the surface to volume ratio for a two phase system. In a log intensity versus log q plot the Porod Regime can be easily identified by marking off 4 decades in I and one decade in q and drawing a line of slope -4. A Porod regime is observed in Figure 3.36 of Strobl, pp. 127. The decay in the prefactor to the power-law decay in time, shown in the lower graph of Figure 3.36, shows that the surface area of the sample is decreasing as a result of Oswald ripening, i.e. phase growth driven by coalescence of small domains into larger domains.

Much of modern polymer theory is based on an understanding and description of concentration fluctuations. This is because a description of such fluctuations are a natural way to describe random systems with statistical features. Fluctuations in density can be described in terms of a series of wavelengths, that is, any distribution of density in space can be decomposed into a series of sin waves of different magnitude. For each wavelength there is an associated wave number, k = 2p/l, associated with that type of fluctuation. The amplitude of a density fluctuation at wave number k (and wavelength l) which contributes to the density profile of a sample in space, r, can be denoted f_k.

2) Incompressibility is an assumption of almost all theoretical approaches so f_k refers to one component of a mixture but the other component is just - f_k, so fluctuations of component A are matched by opposing fluctuations of B in the opposite direction to yield an average concentration for each wavenumber.

Density fluctuations of f_k follow Boltzman statistics so the probability of a fluctuation of size f_k is given by,

p(f_k) Å exp(-a_k f_k²/2kT)

where a_k is the inverse of the response of the density, f_k, to a potential field, Y_k,

f_k = Y_k/a_k = a_k Y_k

(The increase in free energy, dG associated with a concentration fluctuation, f_k, is given by,

dG = a_k f_k²/2 = Y_k df_k)

The mean square value, < f_k²> = kT/ a_k = kT a_k

S_c(q) = kT a_k/V_c

The calculation of a scattering function, S_c(q=k), is the equivalent of calculation of response functions, a_k.

The RPA approach is an extension of the composition fluctuation ideas presented above. Consider an athermal mixture of two polymer chains, A and B. Consider that the B chains have 0 contrast so are not observable. If a potential, Y_k, is applied to this system that only acts on "A" chains and which leads to excitation of a concentration fluctuation of wave vector "k", f_k, we can write,

f_k = a_k⁰ Y_k

where a_k⁰ is the collective response coefficient for lattice units of chain "A" under athermal conditions (hence the superscript 0) in a mixture with chains "B". Under the discussion above, a_k⁰, is associated with the scattering from the mixture of "A" and "B" chains.

f_k = f_k^A = -f_k^B

This response of "B" units to a fluctuation in "A" units reflects the internal induction of the system. "B" units respond as if they were subjected to a potential Y_k even though "B" units do not respond to the external field which is applied. Y_k is called the "internal field".

A-chains respond to both the internal field, Y_k, as well as to the external field Y_k.

f_k = a_k^AA (Y_k + Y_k)

where a_k^AA is the single-chain response coefficient of "A" units response to forces acting on other "A" units. This is different than a_k⁰ which reflects the response of an "A" unit to an external field. For chains in a melt it is assumed that the Gaussian state is displayed due to screening so analytic forms based on the Debye scattering function apply to describe the single-chain response coefficients. That is, we already have an analytic form which describes scattering associated with a_k^AA.

"B" units respond only to the internal field, Y_k, since the external field, Y_k, acts only on "A" units as introduced. Thus,

f_k^B = -f_k = a_k^BB (Y_k)

The collective response coefficient, a_k⁰, is associated with the scattering from the mixture of "A" and "B" chains. In order to determine a_k⁰ from the known Debye functions for a_k^AA and a_k^BB, the induced field, Y_k, must be expressed as a function of the known functions and the applied field, Y_k,

Y_k = Y_k a_k^AA/( a_k^AA + a_k^BB)

Through addition of the two expressions above. This expression can then be used in the expression for f_k above to yield,

f_k = Y_k { a_k^BB a_k^AA/( a_k^AA + a_k^BB)}

1/ a_k⁰ = 1/ a_k^AA + 1/ a_k^BB

For non-athermal systems, an external potential Y_k leads to a concentration wave f_k which produces a molecular field, c' f_k, where c' = 2ckT/V_c. If c' is positive the field is enhanced and if c' is negative the external field is diminished.

f_k = a_k⁰ (Y_k + c' f_k)

This can be solved for f_k to yield,

f_k = a_k⁰ Y_k/(1 - c' a_k⁰)

The non-athermal collective response coefficient, a_k, is associated with the scattering from a non-athermal polymer blend. This response coefficient can be directly obtained from the above expression by comparison with the definition of the collective response coefficient,

a_k = f_k/ Y_k = a_k⁰/(1 - c' a_k⁰)

1/ a_k = 1/ a_k⁰ - c'

Using the previous expression for 1/ a_k⁰ we have,

1/ a_k = 1/ a_k^AA + 1/ a_k^BB - 2ckT/V_c

(For a polymer solution a_k^BB is a constant in k.)

For an athermal system composed of a block copolymer of A and B segments, the description of concentration fluctuations parallels that for a polymer blends except that response coefficients related to the connectivity of A and B chains must be included. a_k^AB reflects the reaction of A units to a force acting on B units in the same chain (at wave vector k). A response coefficient a_k^BA reflects the reaction of B units to a force acting on A chains (at wave vector k). In parallel with the athermal blend derivation above,

f_k = a_k^AA (Y_k + Y_k) + a_k^AB Y_k

f_k^B = -f_k = a_k^BB (Y_k) + a_k^BA (Y_k + Y_k)

The last term representing a force acting on an A unit and the response of a B unit so includes the internal and external fields. The internal field, Y_k, can be removed by summation of the two equations under the assumption of incompressibility, f_k^B = -f_k,

Y_k(a_k^AA+ a_k^AB + a_k^BB + a_k^BA) = -Y_k (a_k^BA + a_k^AA)

Y_k = -Y_k (a_k^AA + a_k^BA)/(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)

This can be substituted in the expression for f_k yielding,

f_k = Y_k {a_k^AA - [a_k^AA + a_k^AB] (a_k^AA + a_k^BA)/(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)}

a_k⁰ = f_k/Y_k={a_k^AA - [a_k^AA + a_k^AB] [a_k^AA + a_k^BA]/(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)}

= (a_k^AA a_k^AA + a_k^AA a_k^AB + a_k^AA a_k^BB + a_k^AA a_k^BA - a_k^AA a_k^AA - a_k^AAa_k^BA - a_k^ABa_k^AA- a_k^ABa_k^BA)

(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)

a_k⁰ = (a_k^AA a_k^BB - a_k^ABa_k^BA)/(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)

1/ a_k = 1/ a_k⁰ - c' = (a_k^AA+ a_k^AB + a_k^BB + a_k^BA)/(a_k^AA a_k^BB - a_k^ABa_k^BA) - 2kTc/V_c

a_k=q^AA = V_c fN_AS_D(R_Aq)/kT

a_k=q^BB = V_c (1 - f) N_BS_D(R_Bq) /kT

where N_i is the degree of polymerization for chains of type "i", f,is the dilution of chains A in the block copolymer, S_D is the Debye scattering function for a polymer coil and R_i is the coil's Gaussian radius of gyration. The Debye function is the Fourier-transform of the pair distribution function g(r) for the pairs of type AA or BB.

To calculate a_k^AB we need the pair distribution functions for the probability of finding a B or A monomer at a distance r from each other. The latter can be analytically solved as a function of the single chain structures. One result of this is that a_k^AB = a_k^BA. The functional form is,

a_k=q^AB = V_c (N_ABS_D(R₀q) - fN_AS_D(R_Aq) - (1 - f) N_BS_D(R_Bq))/(2kT)

a_k⁰ = (a_k^AA a_k^BB - a_k^ABa_k^BA)/(a_k^AA+ a_k^AB + a_k^BB + a_k^BA)

Variation of c' in the thermal system leads to the determination of a critical point where the intensity becomes infinite as shown below:

For symmetric diblock copolymers c_cN_AB = 10.4. This is the lowest value for c_critical, i.e. other compositions will have higher values. This value should be compared with the critical c for the same two polymers in a symmetric binary blend, c_cN = 2. The higher value for the critical point in a block copolymer reflects the connectivity of the polymer chains which suppresses phase separation to a higher value of c (lower temperature).

The size and separation distance of lamellar domains (for example) can be predicted using thermodynamics. The change in Gibbs free energy, DG, for the transition from a homogeneous system to the micro-phase separated system is composed of an enthalpic term, DH, which has contributions from bulk, DH_bulk, and interface, DH_int, a change in entropy associated with A-B junction points becoming located at the interface, DS_int, and a change in entropy associated with stretching the chains, DS_str. The bulk enthalpy change is given by Flory Huggins theory as,

DH_bulk = - kTcN_ABf_A(1-f_A)

DH_int = + kT c A d_t/V_c

DS_int = k ln(d_t/(d_A + d_B)) = k ln(d_t/d_AB)

DS_str = -kb²(d_AB/R₀)²

DG = DH_bulk + DH_int - T DS_int - T DS_str

DG/kT = - cN_ABf_A(1-f_A) + c A d_t/V_c + ln(d_t/d_AB) + b²(d_AB/R₀)²

(1/kT) dDG/dA = c d_t/V_c - 2b²(V_c N_AB /R₀)²/A³ = 0

A Å V_c (2 N_AB² /( c d_t R₀²))^1/3

A³ Å V_c^7/3 N_AB/(c d_t)

d_AB = N_AB^2/3 V_c^2/9 (c d_t)^1/3

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