| Polymer |
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:
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:
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:
1
and 2 of components
1 and 2 are given by the relationships:
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.
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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:
,
is defined (Orwoll 1977) in terms of the chemical
potential of component 1 as:
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:
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:
to be products
of temperature-dependent, D(T), and concentration-dependent, B(2),
terms (Kamide et al. 1985), such that:
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:
H)
and entropy (S) terms
(Flory 1953) such that:
is the reduced partial
molar heat capacity of solution at constant pressure.
Eq. 263 integrates to:
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):
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:
Spinodal
The spinodal curve defines the boundary between unstable and metastable mixtures. Thermodynamically, the spinodal condition is defined by:
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:
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:
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.
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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:
Quasi-binary systems
In binary systems, the binodal represents three different coinciding curves:
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:
'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:
etc. Explicitly, Eq. 289 is given by:
Cloud point curve
Two phases can exist at equilibrium only if the potentials of the corresponding constituents are equal:
1
and 2. According
to Flory-Huggins theory, the partitioning of each polymer component between two
phases obeys the relations:
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:
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:
''2
becomes a function of '2
and 2:
1
becomes a function of 2
and '2:
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:
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:
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 (j
k). In a quasi-ternary
system in which the third component is polydisperse, the free energy becomes:
µip,
of the p-th constituent of the i-th component (for monodisperse
polymers there is only one constituent) is given by:
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
For ternary systems, this reduces to two conditions:
Points on the boundary of this region satisfy:
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:
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:
In a like manner, when the second polymer is polydisperse
as well, a second separation factor, 2,
replaces '2.
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):
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:
2c,
and the critical value of the interaction parameter,
c, are given by:
)
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:
interaction parameter is, in general,
concentration-dependent in polymer solutions and blends. However, if both
components are copolymers, the same concentration-dependent interaction parameter
implemented in Binary systems
and Quasi-binary systems
would necessitate the use of 12 more coefficients that are unknown for most
polymer systems.
as described
in the section on Thermodynamic
properties.
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:
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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:
. 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:
such that:
is labeled C in Figure
21.
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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:
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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:
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:
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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:
| 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 |
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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.
| 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 | |||
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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.
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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.
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).
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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.
Command summary
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.
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.
inc = (T_high - T_low) / (Ncurves -1)
Copolymers
A phase calculation for a copolymer blend is performed using the following steps:
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:
therm_plots.pdgrf Creates three graphs:
1. Free energy vs. volume fraction.
2. Chemical potentials vs. volume fraction.
3. Activities vs. volume fraction.
Tutorial
Many tutorials are available online for use with the Pilot interface.
Pilot online tutorials
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 |
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.
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.
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.
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.
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.
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.
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.
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.
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.
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