Polymer



8       Phase_Diagram Module

The Phase_Diagram module allows you to calculate and display thermodynamic properties such as free energy, chemical potential, and activity in binary, quasi-binary, ternary, quasi-ternary, and copolymer systems. The module also computes liquid-liquid phase diagrams: the spinodal and binodal curves (replaced in quasi-binary and quasi-ternary cases by cloud-point and shadow curves) with their critical points, and in the copolymer cases, the so-called miscibility window showing the relationship between copolymer composition and miscibility.

The module does not work from first principles, but requires as input the system-specific coefficients that describe the temperature and composition dependence of the or g interaction parameters. These may be obtained from the literature, or extracted from liquid-liquid phase data on a well characterized system. This data can be experimental, calculated using the PRISM module, or derived from atomistic modelling using the Amorphous_Cell module. An automatic fitting procedure is used within the Phase_Diagram module to calculate the interaction parameter coefficients, which are then used to predict phase behavior for cases for which experimental data is sparse or nonexistent.


Theory

Introduction

The design and synthesis of new polymers for specific applications is a time consuming and costly process. A highly successful alternative to this approach is to blend existing polymers to obtain a balance of the desired properties that are exhibited individually by the components. There are many examples of commercially successful blends, such as rubber-toughened plastics; however, a common goal in blend formulation is to optimize the cost-to-benefit ratio. The identification and implementation of a successful polymer blend requires sophisticated knowledge and technology, albeit of a different nature from the more chemistry-oriented technology required to introduce a new polymer. The first step in the development of a new polymer blend is the determination of its liquid-liquid phase diagram.

Undoubtedly the best-known theory of the thermodynamics of mixing and phase separation in polymer systems is the Flory-Huggins lattice theory (Flory 1953). However, the original Flory-Huggins theory interaction parameter (FH) is concentration-independent and linear in the reciprocal of the temperature. Experimentally, it is known that this is an oversimplification and that the interaction parameter is actually a much more complicated function of both concentration and temperature for most real polymer systems. In addition, the Flory-Huggins theory describes the phase separation qualitatively rather than quantitatively.

Theories such as corresponding states, equation of state, and others were introduced to take account of pressure and volume effects on the interaction parameters (Patterson 1967, Eichinger and Flory 1968, Sanchez and Lacombe 1976, Yamakawa 1971). While these newly developed theories successfully explain both the Lower Critical Solution Temperature (LCST) and Upper Critical Solution Temperature (UCST) phenomena in some polymer systems, most of the characteristic properties of the pure components required as input by these theories are unknown, especially for polymer systems.

Ideally, one would like to predict accurately and from first principles, the interaction parameter (or the phase behavior), but this is not yet possible for polymer systems of commercial importance. An alternative approach is to treat as an empirical function of both temperature and concentration. This approach has the advantage that a considerable amount of data has already been published on the temperature and concentration dependence of for polymer solutions, and some of this data has been tabulated (Koningsveld and Staverman 1968, Riedl and Prud'homme 1988, Orwoll 1977). Once the parameter has been determined from limited experimental data on a simple model system, it can be applied to predict the phase behavior of more complex systems of practical interest.

Binary systems

Liquid-liquid phase diagrams here typically consist of two curves:

The latter distinction is important for the mechanism and outcome of the phase separation process; a metastable mixture separates slowly by nucleation, leading typically to a sphere-in-matrix morphology. On the other hand, spinodal decomposition in an unstable mixture starts instantaneously everywhere throughout its entire volume, and results in a characteristic bicontinuous morphology often preferred for various commercial applications.

Free energy of mixing

The Flory-Huggins expression for the Gibbs free energy density of mixing of component 1 with component 2 is given by:

Eq. 241     

where is the free energy of mixing per mole of lattice sites, R is the gas constant, T is the absolute temperature, 1 and 2, and r1 and r2 are the volume fractions and relative molar volumes of component 1 and component 2, respectively, and FH is the Flory-Huggins interaction parameter. In this expression FH is concentration-independent and is linear in the reciprocal of temperature. However, FH may be replaced by a more general interaction parameter, g(T,2), that is dependent on both temperature and concentration such that:

Eq. 242     

The volume fractions 1 and 2 of components 1 and 2 are given by the relationships:

Eq. 243     

and:

Eq. 244     

where n1 and n2 are the number of moles of components 1 and 2. Multiplication of Eq. 242 by n1r1 + n2r2 yields the expression for the Gibbs free energy of mixing:

Eq. 245     

Figure 18 shows G as a function of 2 at a temperature T', along with the corresponding phase diagram. At temperature T', the system is only partially miscible. The phase diagram is of the LCST type. On the phase diagram, the outer bold curve is the binodal and is defined by the points of the common tangent to G (i.e., a and e at T'). At these compositions the chemical potentials of the two components are equal and two phases can coexist. The inner curve on the phase diagram in Figure 18 is the spinodal, which is defined by the points of inflection on the graph of G (i.e., b and d at T'). For compositions between b and d, the system is unstable for even the smallest concentration fluctuations, and the phase separation process is called spinodal decomposition. Between a and b, and d and e, phase separation proceeds by a nucleation and growth process. The point where the binodal and the spinodal curves meet is the critical point.

Figure 18

In order to construct a phase diagram for a binary system, it is necessary to utilize Eq. 245. However, the interaction parameter g cannot be determined directly from experiment, and it is more convenient to relate g to an experimentally derivable interaction parameter (T,2) defined in terms of chemical potentials. The relationship between g and is developed in the following section.

Partial molar quantities

The chemical potential µi of the component i in the solution is defined as:

Eq. 246     

From Eq. 245 and Eq. 246 it follows that:

Eq. 247     

and:

Eq. 248     

where:

Eq. 249     

The interaction parameter, , is defined (Orwoll 1977) in terms of the chemical potential of component 1 as:

Eq. 250     

Comparing Eq. 247 with Eq. 250, we have:

Eq. 251     

and upon integration:

Eq. 252     

As can be seen, g and are equal only if g is independent of concentration. The concentration dependence of the interaction parameter can be determined from data on osmotic pressure, vapor pressure, gas-liquid chromatography, freezing point depression, swelling equilibria, intrinsic viscosity, light scattering, critical points, and other methods (Orwoll 1977).

It follows from Eq. 251 that:

Eq. 253     

and:

Eq. 254     

where prime, double prime, and triple prime denote the first, second, and third derivatives with respect to the volume fraction of component 2.

An analysis of Eq. 242, Eq. 247, Eq. 248, and Eq. 251 and Eq. 252, gives the free energy and chemical potentials in terms of the interaction parameter as:

Eq. 255     

Eq. 256     

and:

Eq. 257     

Functional form of chi

The exact analytical form of the expression for the interaction parameter need not be known. If it is not known, Accelrys supplies its own form. On the other hand, if you wish, you can supply an alternative form. (See the Chi_Input section.) For definiteness, we will assume throughout the Theory section that the interaction parameter takes the Accelrys-supplied form. Thus, in what follows, we regard both g and to be products of temperature-dependent, D(T), and concentration-dependent, B(2), terms (Kamide et al. 1985), such that:

Eq. 258     

The form of the concentration-dependent term is taken to be (Kamide et al. 1985):

Eq. 259     

where the bi (i=1, ..., n) are constants. These are the most general and realistic functional forms available in the literature (Eichinger 1970, Koningsveld et al. 1970, Siow et al. 1972, Kamide et al. 1982, Kamide and Matsuda 1984, Kamide et al. 1985, Matsuda 1986, French 1988). In addition, the original concentration-independent Flory-Huggins interaction parameter may be readily recovered from these expressions. The concentration dependence of has been reported for certain polymer solutions, and although less information is available for blends, the approach is equally applicable. Furthermore, b1 and b2 can be estimated from experimental data of critical temperatures and concentrations. In most cases, the concentration-dependent interaction parameter function is adequately described by a second order function, so the following form:

Eq. 260     

is used in our work. In order to determine an appropriate functional form for D(T), we start by separating D(T) into enthalpy (H) and entropy (S) terms (Flory 1953) such that:

Eq. 261     

In addition, we have the two standard thermodynamic functions:

Eq. 262     

and:

Eq. 263     

where is the reduced partial molar heat capacity of solution at constant pressure.

Eq. 263 integrates to:

Eq. 264     

on the assumption that is an affine function of T. This expression for S may be substituted into Eq. 262 and integrated to get (Eichinger 1970, Koningsveld et al. 1970):

Eq. 265     

where d0, d1, d2, and d3 are constants.

Binodal

Under certain thermodynamic conditions, a homogeneous polymer solution or mixture may separate into two or more phases. The conditions for equilibrium between two phases in a binary system may be expressed by:

Eq. 266     

and:

Eq. 267     

where the primed and double primed symbols denote the potentials in the two phases. Using Eq. 256, Eq. 257, Eq. 266, and Eq. 267, we obtain the two expressions:

Eq. 268     

and:

Eq. 269     

Combining Eq. 258 with Eq. 268 and Eq. 269, we have:

Eq. 270     

and:

Eq. 271     

The above two equations contain variables for temperature and concentration (in the two phases). The set of temperature-concentration-concentration triples which solve these equations constitutes the binodal curve.

Spinodal

The spinodal curve defines the boundary between unstable and metastable mixtures. Thermodynamically, the spinodal condition is defined by:

Eq. 272     

When this expression is negative, the system is always unstable. Explicitly, the spinodal curve is described by:

Eq. 273     

By substituting g¢ and g¢¢ from Eq. 252 and Eq. 253 into the above equation, it can be shown that:

Eq. 274     

Upon substituting Eq. 258 into Eq. 274, we find

Eq. 275     

This spinodal equation can now be utilized to set conditions for various types of phase diagrams (Solc et al. 1989). For simplicity, assume here that is independent of concentration (i.e., b1 = b2 = 0), D(T) and that d3 = 0 (see Eq. 258, Eq. 259 and Eq. 265). The first and second derivatives of temperature along the spinodal at the critical point are:

Eq. 276     

The second derivative has been simplified by using the condition (S/2) = 0, which is always satisfied at the critical point. Since its numerator is always positive, the curvature of the spinodal is determined by the sign of the denominator: dD/dT > 0 should produce a temperature minimum (LCST) while the opposite sign should result in a maximum (UCST). If the temperature function D(T) is given by Eq. 265 (and d3 = 0), its derivative is:

Eq. 277     

Evidently, an important role will be played by the temperature:

Eq. 278     

where dD(T) / dT becomes zero, UCST behavior changes to the LCST type, and the spinodal slope becomes indeterminate of 0/0 type (i.e., the spinodal switches its pattern).

If the signs of d1 and d2 are opposite, then T* is negative, and all miscibility gaps are of one type; for d1 > 0, d2 < 0 the derivative in Eq. 277 is always negative, yielding exclusively the UCST type. In the opposite case of d1 < 0 and d2 > 0, the derivative is always positive, leading exclusively to the LCST type (strictly speaking, the above classification remains valid even if one of the coefficients becomes zero). However, in the remaining cases where both coefficients possess the same sign, the temperature T* is physically important -- with d1 > 0 and d2 > 0, the derivative in Eq. 277 is positive (LCST type) for T > T* but changes to negative (UCST type) for T < T*, resulting in an hourglass type of phase diagram for lower molecular weights. Finally, for d1 < 0 and d2 < 0, the pattern is switched and the diagram has the form of a closed loop. These results are summarized in Figure 19.

Figure 19

Evidently, the above rules are only valid if the spinodal and critical points exist. This is decided by d0, which must guarantee that the total D(T) is positive, as required by Eq. 275.

Critical point

The critical point is the point on the binodal and spinodal curves where the two phases become identical. Thermodynamically, it is expressed as:

Eq. 279     

Explicitly, these conditions are:

Eq. 280     

and:

Eq. 281     

After substituting Eq. 252 throughEq. 254 and Eq. 258 into Eq. 280 and Eq. 281, we obtain:

Eq. 282     

and:

Eq. 283     

The above two equations can be solved simultaneously to get the critical temperature and critical volume fraction.

Quasi-binary systems

In binary systems, the binodal represents three different coinciding curves:

In ternary and higher systems, there still exists a single binodal hypersurface in multidimensional T-composition hyperspace, which fulfills simultaneously all of the three above functions; however, it is not within our capability to perceive it and make use of it. Typical phase diagrams employing only two coordinates represent merely a section of the binodal hypersurface, and various loci mentioned above are projected as three different curves: cloud point curves, shadow curves, and coexistence curves. Obviously, all of them are sensitive to the molecular weight distribution of the dissolved polymer (Solc 1970, Koningsveld and Staverman 1968).

Thermodynamic properties

In a quasi-binary polymer blend, one or both of the components can be polydisperse. In the general case of two polydisperse components, the free energy of mixing is given by:

Eq. 284     

where the subscripted n's refer to numbers of moles, 's to volume fractions, and r's to relative molar volumes. The interaction parameters g(T, 2) and (T, 2) are related by Eq. 251. The effect of molecular weight on the interaction parameters is typically small (Koningsveld and Staverman 1968), and is therefore neglected in Eq. 284.

The chemical potential, µ1i, of the i-th constituent of the first component is obtained by differentiating:

Eq. 285     

where rn1 and rn2 are the number average chain lengths of the two components. Likewise, the chemical potential of the j-th constituent of the second component is:

Eq. 286     

Spinodal

The spinodal curve defines the boundary between unstable and metastable mixtures. Thermodynamically, it is expressed by the determinant (Koningsveld and Staverman 1968):

Eq. 287     

with ij and kl ranging over constituents of the two polydisperse components. The spinodal curve is explicitly given by:

Eq. 288     

Critical point

The critical point is the point on the cloud point curve and spinodal curve where the two phases become identical. Thermodynamically, it is determined by simultaneously solving Eq. 287 and the equation (Koningsveld and Staverman 1968):

Eq. 289     

where Y´ is the determinant derived from Eq. 287 by replacing the elements of any

horizontal line by

, ,

etc. Explicitly, Eq. 289 is given by:

Eq. 290     

Here, rw2 and rz2 are the weight-average and z-average chain lengths of component 2, respectively.

Cloud point curve

Two phases can exist at equilibrium only if the potentials of the corresponding constituents are equal:

Eq. 291     

Thus, calculation of equilibrium concentrations involves the solution of a system of equations; the number of equations to be solved is equal to the total number of constituents in the two components. If, however, the distribution of chain lengths of both polymers are known before phase separation, only two nonlinear equations need to be solved. This dramatic reduction in complexity is facilitated by the introduction of two separation factors, 1 and 2. According to Flory-Huggins theory, the partitioning of each polymer component between two phases obeys the relations:

Eq. 292     

with ik ranging over all the constituents of component i, where i=1,2. At the cloud point, where phase separation begins, one of the phases is identical with the phase existing before phase separation. In this phase, known as the principal phase, the chain length distribution is known. Once the i are known, the distribution of chain lengths in the second phase (the conjugate phase) can be calculated as well with the help of Eq. 292. This extra detail enables us to reduce the many equations making up Eq. 291 to two equations involving temperature, the separation factor 2, and the total volume fraction in the principal phase. The solution of this system of equations is known as the cloud point curve.

The actual derivation of the two equations that define the cloud point curve is immediate. By Eq. 285, Eq. 286, and Eq. 292:

Eq. 293     

and:


Eq. 294     

where the primed and double-primed symbols denote quantities in the principal and conjugate phases respectively. Eq. 293 and Eq. 294 are equations in 1, 2, '1, '2, ''1, ''2, the temperature T, and the different number average chain lengths.

Obviously, '1 is a function of '2 and''1 is a function of ''2 since:

Eq. 295     

Once the normalized weight distributions, w'1i and w'2j, of the polymer constituents at cloud point are given, ''2 becomes a function of '2 and 2:

Eq. 296     

and 1 becomes a function of 2 and '2:

Eq. 297     

In addition, the normalized weight distributions on the shadow curve (see the following section) can be determined from Eq. 292, so that all the number averages can be calculated once T, 2, and '2 are known. Thus, Eq. 293 and Eq. 294 reduce to equations in T, 2, and '2 alone.

If we select as the interaction parameter, Eq. 293 and Eq. 294 take the form:

Eq. 298     

and:

Eq. 299     

When the first component is monodisperse, Eq. 292 becomes:

Eq. 300     

Substitution of Eq. 300 into Eq. 293 and or Eq. 298 and Eq. 299, yields the equations for the monodisperse-polydisperse case.

Shadow curve

The shadow curve describes the total concentration of component 2 in the incipient phase isothermally separated at the cloud point. Concentrations in the cloud point and shadow curves are related by Eq. 296.

Ternary systems

As in binary systems, polymer solutions and blends containing three chemically different components may separate into several phases. Likewise, it is possible to define the spinodal boundary between unstable and metastable ternary mixtures. Formally, the equilibrium and spinodal conditions for ternary systems are analogous to those of binary systems. However, the addition of a third component increases the dimension of the configuration space by one, so that surfaces, not curves, now constitute boundaries within the space.

Thermodynamic properties

In a ternary system, the free energy of mixing is given by:

Eq. 301     

where ri, ni, and i are the relative molar volume, number of moles, and volume fraction of the i-th component (i=1,2,3), and gjk are interaction parameters for interactions between components j and k (jk). In a quasi-ternary system in which the third component is polydisperse, the free energy becomes:

Eq. 302     

For both ternary and quasi-ternary systems, the chemical potential, µip, of the p-th constituent of the i-th component (for monodisperse polymers there is only one constituent) is given by:

Eq. 303     

Note the presence of the number average chain length of the third component. Note too that the last two terms vanish when the interaction parameters are not dependent on concentration. Moreover, when the interaction parameter is dependent solely on temperature and component concentrations, the r's and n's in the last two terms cancel and they become functions of component concentration and temperature alone. In the Ternary pulldown, the Accelrys form of g is:

Eq. 304     

Alternatively, the user can supply his own form for each of the three g parameters.

Polydispersity in the second component as well requires an addition (and analogous) generalization of the formulas for the free energy of mixing and the potentials.

Spinodal

As in the binary case, the spinodal equations define the boundary between the unstable and metastable regions. A mixture is in stable (or metastable) equilibrium when the matrix of second derivatives of the free energy is positive definite, that is, when the matrix

is positive definite.

For ternary systems, this reduces to two conditions:

and

Points on the boundary of this region satisfy:

Eq. 305     

Critical points

The definition of a critical point in binary systems carries over, mutatis mutandis, to ternary systems. Thus it is a point on the binodal and spinodal surfaces where two phases become identical. The formal conditions remain unchanged: Eq. 305 must be satisfied, along with Y' = 0, where Y' is the determinant derived from Y in Eq. 305 by substituting partial derivatives of Y for partial derivatives of G in a row of the original matrix.

In a ternary system, the set of critical points is a one-dimensional curve, since it is formed from the intersection of two surfaces.

Binodal and cloud point curves

The conditions for equilibrium between two phases in a ternary system are:

Eq. 306     

where i runs over all the polymer constituents. To determine the binodal surface in a pure ternary system, we substitute the potentials for the three components in one phase (Eq. 303) into the left side of Eq. 306 and the potentials for the second phase into its right side. In this way, we obtain a set of three equations in five variables:

T, '2, '3, ''2, ''3

The set of 5-tuples satisfying these equations constitutes the binodal surface. For systems in which the third polymer is polydisperse, the situation is slightly more complicated -- preliminary versions of the first two equations are obtained by substituting the potentials for the two monodisperse components into Eq. 306. A third equation is derived by substituting the potentials for a constituent (say the 3p-th constituent) of the third component into Eq. 306, isolating

on one side of the equation, and replacing it with 3. Then Eq. 296 enables us to replace ''3 in the three equations with an expression in 3, '3, and the molecular weight distribution of the third component. In this way we again obtain three cloud point curve equations in the variables:

and 3.

In a like manner, when the second polymer is polydisperse as well, a second separation factor, 2, replaces '2.

It is worth noting that if

is a solution to the binodal equations, so is

.

This symmetry suggests that we represent the binodal surface by projecting it onto R3 (3-dimensional Euclidean space):

The information lost in this representation - namely, the correspondence between ('2, '3) and (''2, ''3) in the conjugate phases can be recovered with tie lines connecting the corresponding points in R3.

Copolymer systems

This section describes the calculation of phase diagrams for a blend of two monodisperse random copolymers, each consisting of two monomer units. One of them (component 1) may be in fact a homopolymer or even a solvent. The theory described below is based on published work (Kambour, et al. 1983, ten Brinke et al. 1983, Paul and Barlow, 1984).

Thermodynamic properties

The thermodynamics of such a copolymer system is the same as of any other binary/quasi-binary system (see the Binary systems section). The only remaining task is the evaluation of the effective interaction parameter, , acting between two copolymer molecules, in terms of parameters for interactions between up to four types of monomer units. If one copolymer, AxB1-x, consists of monomers A and B, and the other, CyD1-y, of monomers C and D, then (assuming random mixing) the effective is (Roe and Rigby 1987):

Eq. 307     

where x is the volume fraction of A in the AB copolymer and y is the volume fraction of C in the CD copolymer. The AB copolymer is selected as component 1, and the CD copolymer as component 2. If component 1 is a homopolymer or a solvent, x is equal to 1.0.

The Flory type interaction parameters between monomers A, B, C, and D are considered to be a function of temperature only, and are expressed as:

where d0_ij and d1_ij are the interaction parameter coefficients between monomers i and j (i, j = A, B, C, and D, i j), T is the temperature in Kelvin. If component 1 is a homopolymer or a solvent, d0_AB, d0_BC, d0_BD, d1_AB, d1_BC, d1_BD are all equal to 0.0. Due to the functional form of the interaction parameter, only LCST or UCST types of phase diagrams can be generated.

Critical point, and binodal and spinodal curves

Since in this case is independent of concentration 2, the equations for the spinodal curve and critical point become particularly simple. For instance, the spinodal (Eq. 275) becomes:

Eq. 308     

The critical concentration, 2c, and the critical value of the interaction parameter, c, are given by:

Eq. 309     

and:

Eq. 310     

Miscibility diagram

Since the composition y of the CyD1-y copolymer necessarily effects the phase behavior of the mixture, it is customary in the case of copolymer systems to plot data also in another way, as a so-called miscibility window. Here, the cloud point T (or ) is plotted as a function of the copolymer composition y, while keeping the blend concentration, 2, and the other copolymer's compositions, x, constant. In the case that component 1 is a homopolymer or a solvent, the volume fraction of A in the AB copolymer is equal to 1.0.

Assumptions made with respect to copolymer systems

There are several assumptions involved in the theoretical model described above. Of them the most restrictive are:


Application

Binary systems

Types of binary phase diagrams

In polymer solutions and polymer blends, LCST, UCST, combined UCST and LCST, hourglass, and closed-loop shaped phase diagrams have been found experimentally. These five types of phase diagrams are the most commonly observed phase diagrams in polymer systems. Polystyrene in cyclohexane exhibits a UCST phase diagram whereas polystyrene in benzene exhibits an LCST phase diagram (Kamide et al. 1985). However, polystyrene in acetone may exhibit either a combined UCST and LCST or an hourglass shaped phase diagram, depending on the molecular weight of the polystyrene (Siow 1972). The closed loop phase diagram is found in the polyvinyl alcohol-water system (Elias 1983). Recently, the closed-loop shaped phase diagram was also found in polycarbonate and poly(methyl methacrylate) blends (Kyu and Lin, 1989). The majority of known compatible polymer blends exhibit LCST phase diagrams, but some examples of UCST behavior are known, such as the polybutadiene and polymethylstyrene system (Lin and Roe 1987). Polybutadiene and poly(styrene-co-butadiene) blends give a combined UCST and LCST phase diagram (Ougizawa et al. 1988). According to Sanchez (1983), 95% of binary blends have phase diagrams of the hourglass shape and are incompatible over much of the range of composition. A detailed description of phase diagrams and the compatibility of polymer blends and solutions can be found in the literature (Paul and Newman 1978, Krause and Li 1989).

Fitting to experimental data

The curve fitting facility in the Phase_Diagram module can be used to calculate the interaction parameter for binary homopolymer blends and solutions from experimental data. Typically, this data is derived from systems which though amenable to experimentation, are not representative of commercial systems. The model of polymer phase behavior in the Phase_Diagram module may then be used to extrapolate from the limited experimental data available to predict all the experimentally observed phase diagrams mentioned in Types of binary phase diagrams above, and to predict the behavior of systems with different molecular weights and weight distributions for which experimental data are not available.

As described in the Theory section, the functional form of the interaction parameter or g, required to model all experimentally observed phase diagrams, contains six coefficients defining its temperature and concentration dependence. However, not all may be required to describe a particular system. Since the number of data points is usually quite small, it is wise to start by attempting to fit with few coefficients and increasing the complexity as required to get a good fit.

Examples

Walsh et al. (1989) reported that a blend of polystyrene (PS) and poly(vinyl methyl ether) (PVME) shows an LCST phase diagram. The phase diagram determinations were performed on samples of narrow molecular weight distributions in order to reduce polydispersity effects.

The experimental phase diagrams of PVME (Mn = 95000, Mw/Mn = 1.27) with four different fractions of PS (Mn = 35700, 67000, 106000, and 233000, Mw/Mn < 1.08) are replotted in Figure 20. The theoretical binodal curves of the different molecular weight combinations have been calculated using a single interaction parameter and are also plotted in this figure. The x-axis represents the PS volume fraction.

The interaction parameter used is:

Eq. 311     

Figure 20 Experimental and Theoretical (Curves) Phase Diagrams of Poly(styrene) and Poly(vinyl methyl ether) Blends

Mw of Poly(styrene) = 35700 (x), 67000 (o), 10600 (), and 23300 (+)  

As you can see, the calculated curve in Figure 20 is a good fit to the experimental data corresponding to PS samples of two molecular weights (Mn = 67000 and 233000). However, it does not fit as well with data from the other two samples (Mn = 35700 and 106000). This indicates that other parameters such as the molecular weight, polydispersity, and density change may also affect the interaction parameter in this system. Of course, different interaction functions may be used to fit each curve if desired.

A blend of deuterated polybutadiene (DPBD) (Mn = 134000, Mw/Mn = 2.0) and protonated polybutadiene (PBD) (Mn = 135000, Mw/Mn = 1.8) was investigated by Sakurai et al. (1990) using small angle neutron scattering and was found to exhibit a UCST-type spinodal phase diagram. This example provides a particularly clear illustration of the advantages of determining the temperature and concentration response of the parameter using the method described in this publication, compared with other methods of data analysis such as those used by Sakurai et al. (1990). These authors used a Flory-Huggins type of temperature-dependent interaction parameter, such that Eq. 259 and Eq. 265 become:

Eq. 312     

and:

Eq. 313     

and fitted to data at each concentration. This resulted in values of d0 and d1 corresponding to the different concentrations that differed in some non-systematic way by a factor of two to three. The variation in the values of these coefficients is presumably due to experimental uncertainty and, indeed, the authors (Sakurai et al. 1990) state that they detected no significant composition dependence of the interaction parameter within experimental error and the range of the data. However, if one now wishes to describe the spinodal curve, it is unclear what temperature dependence to use for . To illustrate this point, the experimental spinodal data (Sakurai et al. 1990) for the DPBD/PBD blend are plotted in Figure 21 together with three theoretical curves. The DPBD fraction is component 2. Curve A is obtained by using average values for d0 (= 0.000877) and d1 (= 0.295), calculated from the data in Sakurai et al. (1990). This curve has a poor correspondence to the experimental data. However, a much better fit to the data is obtained by considering all the points simultaneously using the approach described here. This results in curve B by using:

Eq. 314     

as the temperature-dependent interaction parameter. The critical volume fraction of DPBD determined using the above concentration-independent, temperature-dependent interaction parameter is 0.501, which is identical to that determined experimentally (Sakurai et al. 1990). A slight improvement in the fit may be produced by including a concentration dependence of such that:

Eq. 315     

The spinodal curve that results from this concentration dependent is labeled C in Figure 21.

Figure 21 Experimental (l) and Theoretical Spinodal of a Deuterated Polybutadiene and Protonated Polybutadiene Blend

Curve A is calculated according to the average of the experimentally determined interaction parameters. Curve B results from a temperature-dependent interaction parameter derived here. Curve C has a concentration-dependent component.  

A closed loop phase diagram is observed in the polycarbonate (Mw = 64000, Mw/Mn = 2.1) and poly(methyl methacrylate) (Mw = 30000, Mw/Mn = 2.4) blend (Kyu and Lin 1989). This phase diagram is replotted in Figure 22. The theoretically fitted binodal curve is also plotted in the figure. The polycarbonate volume fraction is expressed as 2. The interaction parameter used for this plot is:

Eq. 316     

As we can see in Figure 22, the theoretically fitted curve is in good agreement with the experimental data.

Figure 22 Experimental (o) and Theoretical (Curve) Phase Diagrams of a Polycarbonate and Poly (Methyl Methacrylate) Blend

A combined LCST- and UCST-type of phase diagram is observed for PS (Mw = 4800 and 10300, Mw/Mn = 1.06) in acetone (Siow et al. 1972). The experimentally determined phase diagram of PS with molecular weight 4800 is replotted in Figure 23. The theoretically fitted binodal curve is also plotted in the same figure. PS is component 2. The interaction parameter is:

Eq. 317     

As we can see from Figure 23, the theoretically fitted UCST curve agrees well with the experimentally measured UCST curve. However, the theoretically fitted LCST part does not fit with the experimentally measured part as well as the UCST.

The hourglass type of phase diagram is observed for PS (Mw = 19800, Mw/Mn = 1.06) in acetone (Siow et al. 1972). The experimentally determined phase diagram is replotted in Figure 24 together with the theoretically fitted binodal curve. PS is again component 2. The interaction parameter that best fits the data is:

Eq. 318     

As you can see from Figure 24, the theoretically fitted curve agrees very well with the experimentally measured curve.

Figure 23 Experimental (l) and Theoretical (Curve) Phase Diagrams of Polystyrene (Mw=4800) in Acetone

Figure 24 Experimental (l) and Theoretical (Curve) Phase Diagrams of Polystyrene (Mw=19800) in Acetone

Quasi-binary systems

Types of quasi-binary phase diagrams

The different types of phase diagrams exhibited in the true binary systems (LCST, UCST, combined UCST and LCST, closed loop, and hourglass), are also exhibited by quasi-binary systems. However, molecular weight distribution can have a significant effect on phase diagrams of quasi-binary polymer solutions and blends. This is discussed in the Comparison with published experimental data section below. The focus is on quasi-binary systems of the monodisperse-polydisperse type because of the limited published experimental data on well characterized polydisperse-polydisperse systems.

Comparison with published experimental data

Tong et al. (1985) performed measurements on three model polymer mixtures prepared by mixing four narrow molecular weight distribution PS samples in different ratios. The weight-average molecular weight and degree of polydispersity of these four fractions (F1, F2, F3, F4) are listed in Table 5, and the composition by weight of the three mixtures (M1, M2, M3) of these fractions is given in Table 6. The experimental Cloud Point Curves (CPC) of the three mixtures in cyclohexane are replotted in Figure 25. Using the distribution of molecular weight given in Table 6, theoretical CPCs were fitted to the experimental data, and these theoretical curves are also plotted in Figure 25. The interaction parameter used to generate the theoretical curves is:

Eq. 319     

Table 5      Weight Average Molecular Weight and Polydispersity of the Polystyrene Fractions*

Fraction Mw Mw/Mn
F1   11000   1.02  
F2   44500   1.01  
F3   195000   1.07  
F4   807000   1.01  
* From Tong et al. 1985  

Unfortunately, it was not possible to obtain a good fit to all three CPCs simultaneously. Although the theoretical CPC produced using the parameter above is in good agreement with the experimental data for the solution containing the polystyrene blend with the lowest Mw, the quality of the agreement decreases as the Mw of the system increases. This implies an apparent molecular weight dependence of for this system (Qian et al. 1991(a), and the enhanced Flory-Huggins theory described here assumes to be independent of molecular weight. Virial theories would give rise to a molecular weight dependence through the second virial coefficient. However, no such theory is yet available that can produce quantitative agreement with experimental data. Fortunately, for most polymer solutions and blends is not strongly dependent upon molecular weight). For the polystyrene-cyclohexane system, good fits could be readily achieved to each of the CPCs individually. However, this would result in several parameters.

Table 6      Composition by Weight of the Polystyrene Mixtures*

Fractions M1 M2 M3
F1   0.4   0.25   0.1  
F2   0.3   0.25   0.2  
F3   0.2   0.25   0.3  
F4   0.1   0.25   0.4  
Mw   137500   264400   391300  
Mw/Mn   6.08   7.92   6.11  
* From Tong et al. 1985  

Figure 25 Experimental* (l) and Theoretically Fitted Cloud Point Curves for the Polystyrene Solutions in Cyclohexane

*From Tong et al. 1985  

Copolymer systems

Types of phase diagrams

The types of phase diagrams that may be generated for copolymer systems are the LCST and the UCST.

Miscibility diagram

The miscibility diagram can be used to illustrate the range of copolymer composition over which a copolymer solution or blend is miscible. Figure 26 illustrates the concept of a miscibility diagram. The diagram on the left shows a regular binodal for a blend of a copolymer CD with another polymer (homo or copolymer) or a solvent showing LCST behavior. In the diagram on the right, the volume fraction of the copolymer (component 2) in the blend is fixed while the volume fraction of the monomer C in the copolymer CD is varied. The miscibility diagram shows that for the LCST system the miscibility of the copolymer blend is increased within a certain range of concentrations of the C monomer. There are many cases of miscibility involving copolymers when their corresponding homopolymers are not similarly miscible. For example, neither poly(o-chlorostyrene) nor poly(p-chlorostyrene) is miscible with poly(phenylene oxide) (PPO) (ten Brinke et al. 1983), but over a certain composition range, random copolymers formed from these monomers are miscible with PPO.

Figure 26 Illustration of an LCST Binodal and a Miscibility Diagram for a Solution of a 30/70 CD Copolymer

Shaded areas are two-phase regions.  

Comparison with published experimental data

There are up to 12 interaction parameter coefficients allowed in the Binary_Cop/BC_Run command. If you want to study just the spinodal or binodal curves of the copolymer blends, you can use the Binary_Homop/BH_Run command, although this does not utilize information on the interaction parameters between monomeric units.

Quantitative study
For most copolymer systems, it requires a tremendous amount of phase information to obtain the twelve interaction parameter coefficients between the monomeric units in order to study the system quantitatively. At this point, no published examples have been found in the literature. However, the phase and miscibility diagrams of copolymer blends have been discussed quantitatively by assuming that the interaction parameter is a constant, even independent of temperature (ten Brinke et al. 1983, Paul and Barlow, 1984). A brief qualitative discussion is presented here.

Qualitative study
Figure 27 presents the miscibility diagrams obtained for the hypothetical copolymer blends in Table 7.

Table 7     

Parameter Value Parameter Value
M1   10000.0   d0_AC   0.06  
M2   100000.0   d1_AC   -6.0  
Rho1   1.0   d0_AD   0.015  
Rho2   1.0   d1_AD   -4.0  
Uniform_Seg_Vol   on   d0_BC   0.015  
Segment_Volume   82.0   d1_BC   -3.0  
T_Thermo   300.00   d0_BD   0.030  
VF_A_AB   0.12   d1_BD   -5.0  
VF_C_CD   0.4   d0_CD   0.045  
d0_AB   0.01   d1_CD   -12.0  
d1_AB   -2.0      

Curve 1 in Figure 27 is qualitatively the same as those reported in the literature (ten Brinke et al. 1983, Paul and Barlow, 1984). However, the miscibility diagrams of the two copolymers generated by calculation of curves 2 and 3 in the same figure are more complicated than those reported in the literature (ten Brinke et al. 1983, Paul and Barlow, 1984, Brannock et al. 1990).

Figure 27 Hypothetical Binodal Miscibility Diagram Generated Using the Phase_Diagram Module


Command summary

In addition to the core pulldowns in the top menu bar, the Phase_Diagram module adds six pulldowns to the lower menu bar: Mol_Wt_Distrib, Binary_Homop, Binary_Cop, Ternary, Textfile, Graph, and Background_Job.

Mol_Wt_Distrib pulldown

The Mol_Wt_Distrib pulldown allows you to create discrete molecular weight distributions. There are four commands in this pulldown: Simple, Combined, Fine_Tune, and Examine.

Simple

The Mol_Wt_Distrib/Simple command is used to create discrete approximations to the Flory, Gaussian, logarithmic-normal, Schulz-Zimm, and Poisson distributions.

Combined

The Mol_Wt_Distrib/Combined command is used to create combined distributions, that is, convex combinations of the five basic distribution types.

Fine_Tune

The Mol_Wt_Distrib/Fine_Tune command allows you to choose the method of discretization, fine tune discretization parameters, and modify the domain of the output distribution.

Examine

Using the Mol_Wt_Distrib/Examine command, you can create graph files of distributions created by the Simple and Combined commands. These graph files can be displayed with the Graph pulldown. In addition, the Examine command displays several parameters which measure the adequacy of your discretizations.

Binary_Homop pulldown

The Binary_Homop pulldown is used to start Phase_Diagram computation jobs from within Insight II. These jobs compute phase diagrams for binary and quasi-binary polymer systems and fit interaction parameters to experimental data. Insight II notifies you upon job completion, and allows you to run other commands while a job is running. The four commands within the Binary_Homop pulldown are Chi_Fit_Setup, Chi_Fit_Run, Chi_Input and BH_Run.

Chi_Fit_Setup

The Binary_Homop/Chi_Fit_Setup command creates files which contain descriptions of binary and/or quasi-binary systems and specify the location of related (experimental) data. These files can then be used to backfit the phase diagram interaction parameter.

Chi_Fit_Run

The Binary_Homop/Chi_Fit_Run command invokes the background job that fits the interaction parameter to phase_diagram data. As input, you must specify initial guesses for the parameter coefficients and the name of a file (created by the Chi_Fit_Setup command) describing the data.

Chi_Input

The Binary_Homop/Chi_Input command allows you to specify the coefficients of an interaction parameter. These coefficients are then stored in a file which can be read by the BH_Run background job.

BH_Run

The Binary_Homop/BH_Run command allows you to calculate and display thermodynamic properties of binary and quasi-binary polymer solutions and blends; these properties include free energy, chemical potential, and activity. It also allows you to calculate and display critical points and spinodal, binodal, cloud point and shadow curves. Depending on the input parameters, one or more of these curves may not exist.

Binary_Cop pulldown

The Binary_Cop pulldown is used to start the Phase_Diagram background job that computes properties of binary copolymer systems. Insight II notifies you upon job completion, and allows you to run other commands while the job is running. The Binary_Cop pulldown contains one command: BC_Run.

BC_Run

The Binary_Cop/BC_Run command allows you to calculate and display thermodynamic properties of binary solutions or blends of monodisperse copolymers. Component 1 may be a solvent, a monodisperse homopolymer, or a monodisperse copolymer (AB). Component 2 is a monodisperse copolymer (CD). The properties calculated include free energy, chemical potential, and activity. It also allows you to calculate and display the critical point, the spinodal and the binodal curves, and the miscibility window for the systems. Depending on the input parameters, one or more of these curves may not exist.

Ternary pulldown

The Ternary pulldown provides an interface for running Phase_Diagram ternary background jobs and examining their results. Insight II notifies you upon job completion, and also allows you to run other commands while a job is running. The four commands in the Ternary pulldown are Ternary_gs, Ternary_Run, Ternary_Examine, and Ternary_Defaults.

Ternary_gs

The Ternary_gs command lets you input coefficients for the ternary interaction parameters. These interaction parameters serve as input to the ternary background job.

Ternary_Run

The Ternary_Run command invokes the ternary background job, which computes spinodal, binodal, and cloud point curve phase diagrams for ternary systems and traces lines of critical points.

Ternary_Examine

The Ternary_Examine command allows you to display the phase diagrams computed by the ternary background job.

Ternary_Defaults

The Ternary_Defaults command allows you to set several parameters which affect the calculation of ternary binodals.

Equations pulldown

The Equations pulldown will provide a set of general mathematical utilities. At present, the Equations pulldown contains one command: Estimate_Parameters.

Estimate_Parameters

The Estimate_Parameters command lets you fit a (nonlinear) equation to experimental data. The equation is of the form f(p1,...,pn,x1,..,xk) = y0 where p1, ..., pn are parameters to be determined, and x1, .., xk, and y0 are variables representing experimental data. The function f is supplied by the user.

Textfile pulldown

The Textfile pulldown allows you to view the contents of any file containing only text, such as a .log or .outfile. This pulldown contains one command: Get.

Get

The Textfile/Get command presents a list of files that have file extensions matching the search criteria specified by the module in which you are operating. You can select one of the files or type in the name of any other text file. Get automatically pops the textport, then the UNIX more command is used to display the file. When you finish examining the file, a final <Enter> pushes the textport behind the Insight II screen.

Graph pulldown

The Graph pulldown includes the following commands: Boolean, CharSize, Color, Contour, Correlate, Differentiation, Equation, FFT_Real, Get, Info, Interpolation, Integration, Label, Line_Fit, Modify_Display, Move_Axis, Put, Scale_Axis, Smoothing, Threshold, and Tick_Mark.

See the Graph Pulldown chapter for more information.

Background_Job pulldown

The Background_Job pulldown allows you to set up background jobs to run concurrently or interactively with Insight II. You are given the choice of whether to send background jobs to a local or remote host. This pulldown is generic and is found in many Insight II modules that run background jobs. The Background_Job pulldown contains the following commands: Setup_Bkgd_Job, Control_Bkgd_Job, Completion_Status, and Kill_Bkgd_Job.

Please see Chapter 15, Background_Job Pulldown for more information.


Methodology

Phase diagram calculations

Note: Before you can calculate binary, ternary, or copolymer phase diagram, you must supply one or more interaction parameters. These interaction parameters scale linearly with the volume of a lattice site. The choice of a value for this Segment_Volume is arbitrary; in the Phase_Diagram module it is defined as the molar volume of a repeat unit. To obtain correct phase diagrams, it is imperative that the interaction parameter be consistent with the Segment_Volume of the binary or ternary system in question. The potential for inconsistency is greatest in ternary systems, in which three interaction parameters are required. Each of these must be normalized to the molar volume of the repeat unit of the first component. In the copolymer user interface, the relation between each interaction parameter and the associated segment volume has been made explicit in order to reduce the likelihood of confusion.

Binary

A phase calculation on a binary homopolymer solution or blend is performed using the following steps.

1. Select the Phase_Diagram module.

2. Select the Binary_Homop/BH_Run command.

3. Set the kind of system you will be working with: binary, monodisperse-polydisperse quasi-binary, or polydisperse-polydisperse quasi-binary. If a component is monodisperse, enter its molecular weight; otherwise enter the name of a file containing its molecular weight distribution.

4. Enter the name of the file containing the interaction parameter type and the interaction parameter coefficients. You can create this file using the Binary_Homop/Chi_Input command or with the Binary_Homop/Chi_Fit_Run background job (see the Chi input section).

5. Enter the density of both components, Rho1 and Rho2, in units of g cm-3, and the number-average degree of polymerization of the first component, DP1. The number-average degree of polymerization of the second component is calculated as:

rn2 = (Rho1 / Rho2) ·(Mn2 / Mn1) · (DP1)

where Mn1 and Mn2 are the number-average molecular weights of the two components. Likewise, the weight-average and z-average chain lengths, rw2 and rz2, are related to the weight-average and z-average molecular weights, Mw2 and Mz2, by:

rw2 = (Rho1 / Rho2) ·(Mw2 / Mn1) · (DP1)

and

rz2 = (Rho1 / Rho2) ·(Mz2 / Mn1) · (DP1).

6. If you wish to calculate thermal properties (free energy, chemical potentials, activities), toggle Calculate to On and enter the temperature at which the properties are to be calculated.

Ternary

A phase calculation on a ternary solution or blend is performed in the following steps.

1. Select the Phase_Diagram module.

2. Select the Ternary/Ternary_gs command.

Enter the interaction parameters. In ternary systems there are three interaction parameters. Each interaction parameter can take either the standard Accelrys form or a customized form supplied by the user. The Accelrys form is:

where gij is the interaction parameter associated with components i and j and T is the temperature.

To select the Accelrys form for gij, toggle User_gij off and enter the coefficients d0_ij, d1_ij, and d2_ij.

To supply your own form, toggle User_gij on and enter the desired formula in gij_Formula. The following elements can appear in gij_Formula.

After you have entered the formula for gij, Insight II builds a list of the parameters in it (other than t0, v1, v2, and v3). With the first parameter in the list highlighted, enter its value in the gij_Parameter_Value parameter and press <Enter>. When the next parameter is highlighted, enter its value as well. Continue in this way until all the parameters appearing in the formula have been assigned values. If you wish to change the value assigned to a parameter, simply highlight the parameter in the parameter list and enter its new value.

When you run the ternary background job (in the Ternary/Ternary_Run command), an input file is created (containing among other things the current interaction parameters). You can retrieve coefficients from this file by toggling Input from file to on and entering the name of the appropriate Input_File (or selecting the appropriate filename from the Input_File_List).

3. Select the Ternary/Ternary_Run command.

Enter the molecular weight (Mw1) of the first component along with either the molecular weight (Mwi) or the name of the file containing the MWD of the second and third components (MWD_File_i). Also enter the densities (Rho1, Rho2, Rho3) of the three components along with the chain length of the first component (DP1). The chain lengths of the remaining components can then be calculated, since:

rwi = (Rho1/Rhoi) · (Mwi/Mw1) · DP1 (i=2,3)

When you run the ternary background job, an input file is created. You can retrieve these parameters from this file by toggling Input from file to on and entering the name of the appropriate Input_File (or selecting the appropriate filename from the Input_File_List).

The ternary background job determines cross sections of the spinodal and binodal surfaces over a range of temperatures. To determine this range, enter its lower and upper limits (T_low and T_high) and the number of cross sections to be calculated (Ncurves). The cross sections are evenly spaced over the temperature range, so that the temperature increment is:

inc = (T_high - T_low) / (Ncurves -1)

To examine the results of a ternary background job:

4. Select the Ternary/Ternary_Examine command.

The ternary background job creates three different TBL files. One contains spinodal curves (hence, its file extension is .sptbl), another contains binodal curves (with file extension .bitbl), and the third contains curves of critical points (with file extension .crtbl). You can plot the various curves by selecting the Plot_Curve Examine_Option and then choosing the appropriate Phase_Diagram_Type. All curves in the Ternary_Examine command are plotted on triangular graphs. By modifying the values of T_low and T_high, you can plot a limited set of the spinodal or binodal curves calculated by the background job.

If you wish to determine the conjugate point corresponding to a given point on a binodal curve, select the Show_Tie_Line Examine_Option, and when the Plot_Spec parameter is in focus, click on the point in question. A tie line connecting it to the corresponding point in the conjugate phase will then be displayed.

Copolymers

A phase calculation for a copolymer blend is performed using the following steps:

1. Select the Phase_Diagram module.

2. Select the Binary_Cop/BC_Run command.

3. Set the kind of system you will be working with: binary, monodisperse-polydisperse quasi-binary, or polydisperse-polydisperse quasi-binary. If a component is monodisperse, enter its molecular weight; otherwise enter the name of a file containing its molecular weight distribution.

4. Enter the densities of both components, Rho1 and Rho2, in units of g cm-3.

5. If all chi parameters corresponds to a uniform segment volume, toggle Uniform_Seg_Vol on and enter the segment volume in Segment_Volume. Otherwise, toggle Uniform_Seg_Vol off and enter values for SegVol_AB, SegVol_AC, SegVol_AD, SegVol_BC, SegVol_BD, and SegVol_CD.

6. Select the temperature at which the free energy, chemical potentials, and activities are to be calculated.

7. Enter the volume fraction of monomer A in component 1, VF_A_AB. If component 1 is a homopolymer, VF_A_AB should be 1.0.

8. Select the coefficients of the chi parameters, d0_AB, d1_AB, d0_AC, d1_AC, d0_AD, d1_AD, d0_BC, d1_BC, d0_BD, d1_BD, d0_CD, and d1_CD. If component 1 is a homopolymer, d0_AB, d1_AB, d0_BC, d1_BC, d0_BD, and d1_BD should all be zero.

9. Enter the volume fraction of the CD polymer, Mis_VF2_CD. The miscibility diagram is calculated at this volume fraction.

Chi input

When running a homopolymer background job, the definition of the interaction parameter is retrieved from a file. If the interaction parameter has the Accelrys-supplied form, it is stored in a coefficient file (with file extension ".coef"). Coefficient files can be generated either by the Binary_Homop/Chi_Fit_Run background job or, interactively, with the Binary_Homop/Chi_Input command. If the interaction parameter is in a form supplied by the user, it is sorted in a g file (with file extension ".g"). At present, only the g parameter (not the parameter) can be user-defined. Typically, g files are generated interactively with the Binary_Homop/Chi_Input command.

Interactive creation of the coefficient file

To store coefficients of the chi parameter interactively, perform the following steps:

1. Select the Binary_Homop/Chi_Input command.

2. Enter the interaction parameter type. You can select either the chi parameter or the g parameter. Both have the form:

but the coefficients d0 - b2 take on different values for the two parameter types. Parameters g and chi are related by Eq. 251 in the Partial molar quantities section.

3. Enter the coefficients d0 through b2. To save them, the Calculation_Mode/Save_Parameters (or Both) option must be selected. Enter the name of the coefficient file in the Coeff_File_Out parameter, and select Execute.

4. You can plot the chi (or g) function against temperature and volume fraction by selecting the Calculation_Mode/Plot_Chi_Surface option and then selecting Execute. To see the effect of a change in one or more of the coefficients, plot the interaction parameter for the original set of coefficients, toggle the New_Chi_Graph boolean to on, modify the coefficients, and replot the interaction parameter.

5. Once coefficients have been saved, you can view them by selecting the Input_from_File option and entering the name of the coefficient file in which they are stored in the Coeff_File_In parameter. If you wish to make changes, you can modify the coefficients, view the resulting chi (of g) surface, and/or save the new set in a coefficient file.

6. To see which chi coefficients correspond to given g coefficients, select the Chi_to_G Calculation Mode, set the Parameter_Type to g, enter the g coefficients, and select Execute. The Parameter_type changes to chi, and the corresponding chi coefficients are displayed. To retrieve the g coefficients, select Execute again.

Interactive creation of the g file

To supply your own form of the g parameter:

1. Select the Binary_Homop/Chi_Input command

2. Toggle the User_Defined boolean on

3. Enter the form of g in the g_Formula parameter. The following elements can appear in formula.

Examples of valid formulas:

d0 + d1/t0 + b1*v2^1.2.

(d0 + d1/t0 + d2*ln(t0)) * (1 + b1*v2 + b2*v2^2).

4. After you have entered the formula for g, Insight II builds a list of the parameters in it (other than t0 and v2). With the first parameter in the list highlighted, enter its value in the Parameter_Value parameter and press <Enter>. When the next parameter is highlighted, enter its value as well. Continue in this way until all the parameters appearing in the formula have been assigned values. If you wish to change the value assigned to a parameter, simply highlight the parameter in the parameter list and enter its new value.

5. To save the formula for g and the values you have assigned to its constant parameters, enter the name of the g file in g_File_Out and select Execute.

Fitting the interaction parameter

Fitting the interaction parameter to experimental data is accomplished in two steps. In the first step you describe the experimental data:

1. Select the Binary_Homop/Chi_Fit_Setup command. The data may come from one or more phase diagrams. In the Chi_Fit_Setup command you build a list of data descriptions with each member in the list corresponding to a set of data points.

2. For each phase diagram (data set), you must provide information about the system in question. Thus, you must specify the type of curve (whether spinodal, binodal, or cloud point), the dispersity of the components (whether monodisperse or polydisperse), the average molecular weight (in the case of a monodisperse polymer) or the name of a MWD file (for a polydisperse polymer), the densities of the two components and the number average degree of polymerization of the first component.

3. You must also provide the location of the data points. The coordinates of these points are assumed to be stored in graph (or .tbl) files. (For a description of the graph file format, see Appendix C, File Formats.) Enter the name of the graph file containing the data and specify which columns contain the temperature and concentration coordinates.

4. To create a list of data descriptions, select the Add Setup_Action_Type. For each data set, enter the appropriate information. When you select Execute the parameter block clears and you can enter information pertaining to the next data set. In order to save the list to a file, select the Save_List Setup_Action_Type, enter the name of the Instruction_File in which the list will be stored, and click on Execute.

5. Depending on the Setup_Action_Type you select, you can restore saved lists from instruction files (select Restore_List) or clear the current list (select Clear_list). If you wish to modify a data set description, select the Replace Setup_Action_Type. You can then travel backward or forward in the list by selecting the Move_Backward or Move_Forward parameter. If you wish to modify the currently displayed data set description, make the desired changes and select Execute. To save your changes, select the Save_List_Setup_Action_Type and select on Execute.

Once you have set up your description of experimental data, the second step is to fit the data:

1. Select the Binary_Homop/Chi_Fit_Run command.

2. Enter the name of the Instruction_File containing the data set descriptions.

3. Specify whether you want to fit the Accelrys chi (or g) by toggling User_Defined off or on.

If you wish to fit a Accelrys interaction parameters, enter the type of interaction parameter you wish to fit. You can select either the chi parameter or the g parameter. Both have the form:

but the coefficients d0 through b2 take on different values for the two parameter types. Parameters g and chi are related by Eq. 251 (see the Partial molar quantities section).

When fitting the interaction parameter coefficients, the background job uses the Nelder-Mead simplex algorithm, which requires initial estimates of the coefficients. You can enter your own initial estimates for d0, d1, d2, d3, b1, and b2 or you can let the program calculate them. To supply your own initial estimates, toggle Linear_Estimates to off and enter the initial values. When Linear_Estimates is toggled on, the background job arrives at its estimates by fixing the concentration-related coefficients, b1 and b2, and performing a multiple linear regression on the (variable) temperature-related coefficients. You can fix the value of any coefficient xx (where xx can be d0, d1, d2, d3, b1 or b2) by toggling Fix_xx to on and selecting the desired value for xx.

If you wish to fit a g parameter with a form different from that of the Accelrys g, enter its formula in g_Formula. (For a description of the syntax required for g_Formula, see the section on the Interactive creation of the g file.) Insight II then builds a list of the terms appearing in g_Formula (other than t0 and v2). When the first term is highlighted, specify whether it is a Constant, a Parameter (Est), or a Parameter (Range): A Constant is a fixed real number which must be assigned a Value; A Parameter (Est) is an unknown parameter for which the user supplies an Initial_Estimate; A Parameter (Range) is unknown parameter which is likely to appear in an interval bounded by LBound and Ubound. When the second term is highlighted, enter its type and associated value(s) as well. Continue in this way with the remaining terms in g_Formula. If you wish to change the type or value of a term, highlight the term in the term list and make the desired changes.

4. Enter the name of the Coefficient_File (or g_File_Out for user-supplied g's) in which the final estimates will be stored (along with statistics of the fit) and select Execute to run the curve-fitting background job.

Parameter estimation

In the Chi_Input and Ternary_gs commands, you input a function representing an interaction parameter and this function is then "interpreted", evaluated, and differentiated by the BH_Run or the Ternary_Run background job. Needless to say, the ability to perform operations like addition or differentiation on arbitrary functions has utility beyond the realm of phase diagrams. The Estimate_Parameters command illustrates this utility. It allows the user to estimate the value of unknown parameters appearing in a user-supplied equation.

1. Select the Equations/Estimate_Parameters command and enter a Run_Name.

2. It is assumed that the equation to be fitted is of the form f(p1,...,pn,x1,...,xk) = y0 where p1, ..., pn are unknown parameters to be determined and x1, ..., xk, and y0 are variables representing the data. Enter the formula for f in the Function parameter. The following elements can appear in the formula:

Note that there is no restriction on the first letter in a term. However, as a rule, the term y0 should not appear as a term in Function; it is always treated as the "dependent" variable representing the value taken by the function f(p1,...,pn,x1,...,xk) = y0.

3. After you have specified the Function, Insight II displays the terms appearing in it (along with y0). With the first term in the List_of_Terms highlighted, select its type with the Type_of_Term enum. A term can be a Parameter, a Variable, or a Constant.

When the second term is highlighted, enter its type and the appropriate value(s). Continue in this way until you have assigned a type and value(s) to each term appearing in Function. If you wish to change the type or value assigned to a term, simply highlight the term in the List_of_Terms and make your changes.

4. When you select Execute, the Estimate_Parameters background job performs a least squares minimization (using the Marquardt-Levenberg algorithm). The results of the fit are reported in a file named <Run_Name>.fres.

Standard graphs

With the Graph/Get command, you can create a number of phase diagram-specific standard graphs. Below is a list of the standard graphs available from the Phase_Diagram module.

Table 8     

Standard graph Description
spinodal.pdgrf   Creates a plot of the spinodal curve.  
binodal.pdgrf   Creates a plot of the binodal curve, but only if the system in question contains two monodisperse polymers.  
cpc.pdgrf   Creates a plot of the cloud point curve, but only if the system in question contains at least one polydisperse polymer.  
shdw.pdgrf   Creates a plot of the shadow curve, but only if the system in question contains at least one polydisperse polymer.  
sp_bi_cp.pdgrf   Creates a plot of the shadow curve, but only if the system in question contains at least one polydisperse polymer.  
sp_bi_cp.pdgrf   Creates a graph containing plots of the spinodal curve (cyan), the binodal curve (red), and the critical points (yellow). It creates a plot of the binodal curve only if the system in question contains two monodisperse polymers.  
sp_cpc_cp.pdgrf   Creates a graph containing plots of the spinodal curve (cyan), the cloud point curve (red), and the critical points (yellow). It creates a plot of the cloud point curve only if the system in question contains at least one polydisperse polymer.  
therm_plots.pdgrf   Creates three graphs:   1. Free energy vs. volume fraction.   2. Chemical potentials vs. volume fraction.   3. Activities vs. volume fraction.   The potential and activity of the first components are plotted in red, while those of the second are plotted in green. The potentials of each component are divided by the degree of polymerization.  
copd.pdgrf   Creates two graphs:   1. The spinodal curve (cyan), the cloud point curve (red), and the critical points (yellow).   2. The spinodal miscibility curve (cyan) and the binodal miscibility curve (red).    
copd_misc.pdgrf   Creates a graph containing spinodal (cyan) and binodal (red) miscibility curves.  

therm_plots.pdgrf Creates three graphs:

1. Free energy vs. volume fraction.

2. Chemical potentials vs. volume fraction.

3. Activities vs. volume fraction.


Tutorial


Pilot online tutorials

Many tutorials are available online for use with the Pilot interface.

To access the online tutorials for Phase_Diagram, click the biplane or mortarboard icon in the Insight interface, or select Online_Tutorials from the Help pulldown.

Then, from the Open Tutorial window, select Polymer Modeling and Property Prediction tutorials, and then select Phase_Diagram Module Tutorials from the list of modules. Choose from the list of available lessons:

Lesson 1: LCST Phase Diagram: Blend of polystyrene and Poly(Vinyl Methyl Ether)

Lesson 2: UCST and Closed Loop Phase Diagrams

Lesson 3: Hourglass and UCST_LCST Phase Diagrams

Lesson 4: Example of Using the Chi Fitting Utility

Lesson 5: Example Calculations of Quasi-Binary Phase Diagrams

Lesson 6: Copolymer Phase Diagrams

You can access the Open Tutorial window at any time by clicking the Open File button in the lower left corner of the Pilot window.

For a more complete description of Pilot and its use, click the on-screen help button in the Pilot interface or refer to the Introduction in the Insight II manual.

A ternary example

The following example illustrates the use of the Phase_Diagram module to calculate phase diagrams of ternary systems.

The calculations are performed on a blend of three polymers with molecular weights of 1000, 2000, and 2300. The densities of the three components are unity, and the degree of polymerization of the first component is ten.

We assume throughout that the interaction parameters g13 and g23 are independent of concentration, and we examine the effect of increasing the dependence of g12 on the concentration of the first component.

1.   Enter the coefficients of the three interaction parameters

Select the Ternary/Ternary_gs command.

At first we assume that all three interaction parameters are independent of concentration.

With User_g12, User_g13, and User_g23 toggled off, enter the following interaction parameter coefficients: d0_12 0.1

d0_13 0.1

d0_23 0.1

d1_12 1.0

d1_13 2.0

d1_23 3.0

d2_12 0.0

d2_13 0.0

d2_23 0.0

and select Execute. The resulting interaction parameters are:

g12 = 0.1+1.0/T, g13 = 0.1+2.0/T, g23 = 0.1+3.0/T.

2.   Setup and run the Ternary background job

Select the Ternary/Ternary_Run command. Before running the ternary background job, you must enter the molecular weights and densities of the three components along with the degree of polymerization of the first component. The degrees of polymerization of the second and third components are computed:

DPi = (Rho1/Rhoi) (Mwi/Mw1) DP1 i = 2,3.

Enter the following values for the component parameters: Mw1: 1000.

Rho1: 1.0

DP1: 10.0

Is_Mono_2: on

Mw2: 2000.

Rho2: 1.0

Is_Mono_3: on

Mw3: 2300.

Rho3: 1.0

You must also select a range of temperatures at which the phase diagrams are calculated. In this example we determine phase diagrams at T=100 K, 200 K, 300 K, and 400 K; set T_low to 100, T_high to 400, and Ncurves to 4. Enter conc_00_ as the Run Name and select Execute.

A message appears at the bottom of the screen confirming that the Ternary background job has started. After several minutes, when the job has completed, a notifier box appears. Status 0 means that the job completed successfully.

3.   Examine the Phase Diagrams

The Ternary background job creates three graph files as output. One (with file extension .sptbl) contains spinodal curves, a second (file extension .bitbl) contains binodal curves, and a third (file extension .crtbl) contains curves of critical points.

To view the phase diagrams, select the Ternary/Ternary_Examine command. Select the Plot_Curve Examine_Option and the Spinodal Phase_Diagram_Type. A list of files with extension .sptbl will appear when the File Name parameter comes into focus. Select conc_00_0.sptbl from the list. When the Tlow parameter comes into focus, a list appears containing temperatures at which spinodal curves were found. Select 100 from the list. Likewise set Thigh to 400, and select Execute.

After several seconds, a triangular graph containing four curves will appear. These four curves are the spinodals calculated at the different temperatures selected. The points on these curves should be interpreted as follows: the perpendicular distance of a point from the left axis corresponds to the volume fraction of the first component, the distance from the right axis corresponds to the volume fraction of the second component, and the distance from the bottom axis corresponds to the volume fraction of the third component.

In addition, a window containing the data in tabular form appears in the lower right hand side of the Insight II screen. This window contains the input parameters along with the coordinates of the spinodal points. At the top of each column of spinodal data is the corresponding temperature. The window can be manipulated (resized, pushed, popped, etc.) like any X window. Similar windows appear when binodal curves and curves of critical points are plotted.

4.   Modify g12 and compare phase diagrams

We now examine the effect of introducing concentration dependence into g12.

Enter the Ternary/Ternary_gs command and set User_g12 on. Enter
(0.1 + 1.0/t0)*(1+b1*(v1/(v1+v2))) as the g12_Formula, and when the g12_Parameter_List appears with b1 highlighted, enter .5 as the
g12_Parameter_Value. Press <Enter>. With these changes, g12 takes the form:
g12 = (0.1+1.0/T) (1+.51/(1+2))). Select Execute. Then enter the Ternary/Ternary_Run command, change Run Name to conc_05_, and run a new ternary background job by selecting Execute again.

When the background job finishes, reenter the Ternary/Ternary_Examine command. Once again, select the Plot_Curve Examine_Option along with the Spinodal Phase_Diagram_Type. Select conc_05_0.sptbl as the File Name and set Tlow and Thigh to 100 and 400, respectively. Then select Execute.
The graph that now appears differs significantly from the original graph. When the chi parameters were concentration independent, each of the spinodals was composed of one branch. Now, with the change to g12, each spinodal is composed of two disconnected branches.

5.   Plot some binodal curves

Unlike the binary case, it is impossible to tell which of the regions bounded by the spinodal curves is unstable. We can resolve this uncertainty if we plot the binodal curves as well.

Select Binodal as the Phase_Diagram_Type and choose conc_05_0.bitbl from the list of File Names. Leave Tlow and Thigh unchanged, toggle New_Graph to off, click on the Graph Name parameter (so that it is highlighted), and pick on the graph created in the previous step.

After several moments the binodals appear. Like the spinodals, each of the binodals is composed of two branches. With their appearance, the phase behavior of the blend becomes clear. The center region of the triangle, exterior to the two binodals, is miscible; the metastable region is bounded by the spinodals and the corresponding binodals; the unstable region is restricted to the area bounded by the spinodals and the left and bottom axes of the triangle.

6.   More concentration dependence for g12

Let us see what happens if we further increase the concentration dependence of g12.

Repeat the steps in 4 above, but this time set g12_Parameter_Value to 1.0 (with b1 highlighted) in the Ternary/Ternary_gs command, enter conc_10_ as the Run Name in the Ternary/Ternary_Run command, and plot the spinodal curves in conc_10_0.sptbl. Likewise, on the same graph, plot the binodal curves in conc_10_0.bitbl, following the steps of 5 above.

The set of solutions to the spinodal equation is a surface if the temperature is allowed to vary. So too is the set of solutions to the binodal equations (when temperature varies). The spinodal (binodal) curves we have been plotting are cross sections of spinodal (binodal) surfaces at constant temperature. By examining these cross sections, we can gain insight into the geometry of the surface under consideration.

As an example, let us examine the curves we have just plotted. At each temperature, the spinodals contain two branches. But the orientations of the outer curve differ from those of the three interior curves. The same holds true of the binodal curves. The exterior curve was calculated at T=100.0 K. To ascertain this, you could examine the corresponding tables. But, for binodal curves, you have another option.

Select the Show_Tie_Line Examine_Option in the Ternary/Ternary_Examine command. The Plot Spec parameter is immediately highlighted. Pick on a point on the exterior binodal curve and, when a point specification is displayed in the parameter box, select Execute. A tie line is then drawn, connecting the point you specified to the corresponding point in the conjugate phase. At the same time, the coordinates of the two points (along with the temperature) are displayed on the information line below the Insight II graphics window.

It is worth noting that the tie line connects the two branches of the binodal curve at 100 K; each of the binodal branches corresponds to a separate, distinct phase. Since the two branches do not meet, we should not expect to find a critical point at T=100 K. On the other hand, tie lines drawn at higher temperatures reveal conjugate points occupying the same binodal branch. As a result, we can expect to find critical points at these higher temperatures.

Reselect the Plot_Curve Examine_Option of the Ternary/Ternary_Examine command, and select Critical_Point as the Phase_Diagram_Type. Choose conc_10_0.crtbl from the list of File Names, set New_Graph to off, and, with Graph Name highlighted, pick on the graph you have been examining. When the graph name appears in the parameter box, select Execute.

A line of critical points is now plotted. This line joins the separate spinodal branches calculated at 200 K, 300 K and 400 K. But, as expected, it does not intersect the spinodal branches calculated at 100 K. The geometry of the spinodal and binodal surfaces has become clear. Both are saddle-shaped; they meet along the line of critical points, which runs along the ridge of the upper part of the saddle. This line attains a minimum temperature of just above 100 K; the exact minimum can be ascertained from the table which was created when the line of critical points was plotted.

If you wish to continue in Insight II, it is advisable to close the windows created in this tutorial. This can be done with the Windows/Close command. (The Windows pulldown is located on the top Insight II menu bar.) In the Windows/Close command you can use an asterisk to specify that you wish to close all non-essential Insight II windows). To delete all graphs, type delete *. If you wish to exit Insight II, you can simply enter quit at the command prompt and press <Enter>. To confirm, press <Enter> again.




Last updated April 20, 1998 at 09:57AM PDT.
Copyright © 1997, Accelrys Inc. All rights reserved.