Thermodynamics of polymer solutions and blends.
INTRODUCTIONThe Hoch-Arpshofen model for solutions has been applied to metallic and ceramic systems, aqueous solutions of salt mixtures, and now to polymer solutions and blends.
We will show that in polymeric blends the Gibbs energy of mixing can be represented by
Gm/R = T*([z.sub.1]ln [z.sub.1] + [z.sub.2]ln [z.sub.2]) + Wn[z.sub.1][1 - [(1 - [z.sub.2]).sup.{n-1}]]
where z can be mole fraction, volume fraction, or weight fraction. The interaction parameter can be temperature dependent, W = A + [B.sup.*]T but is composition independent. n is an integer (2, 3, 4, . . .), which defines the asymmetricity of the binary phase diagram. In a binary system, n defines the composition where the Gibbs energy of mixing is a maximum or minimum or the composition where the temperature of a miscibility gap is maximum or minimum. In a binary system A-B the maximum effect occurs at [A.sub.n-1]B. This implies that to compensate for the energy effect of one B component, one needs (n - 1) A components. Having two linear polymers containing only carbon atoms, polymer A with a molecular weight of 2800 and polymer B with a molecular weight of 24,000, one needs seven to nine A polymers to counter the repulsive forces of polymer B. To better understand this interaction we "regularize" the phase diagram [forcing it to become symmetrical (n = 2)] by joining some of the smaller polymers together, and have this combination interact with the large polymer. The composition [z.sub.1] becomes now [z.sub.1] = [z.sub.1]/([z.sub.1] + f - f[z.sub.1]), where polymer A is component 1 and f is the number of small polymers that were assembled. We will show that where we could convert the data, all representations are equivalent, obviously with different A, B, and n. Our model permits the evaluation of phase diagrams with highly asymmetric miscibility gaps and calculation of the critical temperatures and compositions.
THEORY
The Hoch-Arpshofen Model
In an earlier paper, Hoch and Arpshofen (1) derived a model for binary solutions. In a subsequent paper, Hoch (2) derived the model for ternary, quaternary, and larger systems. The binary model is merely a special case of the larger model. The model is an extension of Guggenheim's treatment (3) of solutions, combined with an adaptation of Pauling's ideas (4) of the metallic bond.
When treating regular solutions and superlattices, Guggenheim (3) speaks of "treatment of quadruplets of sites, forming regular tetrahedra" and "triplets of sites, forming equilateral triangles." However, Guggenheim always treats the strength of the A-B bond in the same way, regardless of what other atoms are present in the complex. In our model the strength of the A-B bond depends on the number of B atoms to which the A atom bonds, or vice versa. In Pauling's description (4) of a metallic bond, the bond number is defined as the number of bonding electrons divided by the number of neighbors to which the specific atom bonds. In metallic copper, which consists of one bonding electron and 12 neighbors, the bond number is 1/12. This is a one-electron bond, which moves from one neighbor to another. In our model this idea is applied to ionic materials (ceramic) and van der Waals-type forces, in both an attractive and a repulsive mode. This idea is not extravagant because all bonds are caused by the behavior of electrons.
In a multicomponent system with the components A, B, C, D, etc. and their mole-fractions x, y, z, u, etc. the effect of the mixing function Fm (Hm, enthalpy of mixing, [Sm.sub.ex], excess entropy of mixing) of the binary system A-B (mole-fraction x and y) in the multicomponent is
Fm/R = Wnx[1 - [(1 - y).sup.{n-1}]] (1)
W is the interaction parameter and n is an integer (2, 3, 4, etc). The term x is the mole-fraction of the component so that in Eq 1, Fm is maximum (positive or negative) at x [greater than or equal to] 0.5. Atom or molecule A, with the mole-fraction x is the atom or molecule with the lower attractive or repulsive power and in polymers the one with the lower molecular weight. The partial quantities
[F.sub.x]/R = Wn[1 - [(1-y).sup.{n-1}] - xy(n - 1)[(1 - y).sup.{n-2}]] (2)
[F.sub.y]/R = Wnx(n - 1)[(1 - y).sup.{n-1}] (3)
[F.sub.z]/R = [F.sub.u]/R = -Wnxy(n - 1)[(1 - y).sup.{n-2}] (4)
The excess Gibbs energy of mixing, [Gm.sup.ex], is a combination of Hm and [Sm.sup.ex]. Equation 1 can be applied to [Gm.sup.ex] only if [Sm.sup.ex] has the same n and x as Hm, or if [Sm.sup.ex] is zero.
Equations 1 through 4 also applies to binary systems: In a binary system
x + y = 1 (5)
In our model one has to define component 1 or A whose composition is x. In our nomenclature 6, (Ethyl-alcohol) means that n is equal to 6, and Ethyl-alcohol is component 1 or A.
In the multicomponent system (A-B-C-D-etc.) the other binary systems (A-C, A-D, A-etc., B-C, B-D, B-etc., C-D, C-etc., and D-etc.) contribute similarly to the thermodynamic properties of the multicomponent system. The interaction parameter W of the binary system A-B does not change when we go into a ternary or larger system.
The partial quantities, Eqs 2 through 4, do not change sign in a binary system when the composition changes from x = 0 to x = 1 or in a large system when x changes from x = 0 to x = 1 and y changes from y = 0 to y = 1.
One major advantage of our method is that by using regression analysis, we can calculate the binary interactions from the large systems and can compare them with the values calculated from binary data.
It is possible, that in a binary system one side is attractive, and on the other repulsive forces are present (as in Au-Si, CaO-Si[O.sub.2]). In this case two terms of Eq 1 are needed:
Fm/R = [W.sub.1]xn[1 - [(1 - y).sup.{n-1}]] + [W.sub.2]ym[1 - [(1 - x).sup.{m-1}]] (1a)
and
[W.sub.1] [greater than] 0 and [W.sub.2] [less than] 0 or vice versa (6)
and often
n = 2m, or m = 2n (7)
The ideal Gibbs energy of mixing is, as in Guggenheim (3)
[Gm.sup.id] = RT([x.sub.1]ln [x.sub.1] + [x.sub.2]ln [x.sub.2] + [x.sub.3]ln [x.sub.3] + ...) (8)
The model was applied earlier to various binary and ternary systems (5-14); the method of evaluation of binary phase diagrams was also shown (5-7). To describe solid compounds that are very stable and have a wide homogeneity range, we use the Schottky-Wagner model (15) combined with activity coefficients derived by dividing the binary system into two binaries A-AB and AB-B. We refer the reader to the above publications for derivation of the model and the method of application. A great advantage of our model is that we have never needed ternary interaction parameters. More important, we can obtain the binary interaction parameters from ternary or quaternary data by regression analysis; the latter must agree with data obtained from binary data. Enthalpies of mixing and activities in five component systems were also calculated, using only binary interactions. The agreement with experimental data was excellent (16, 17).
As mentioned above, for the case of polymer blends we use the equation:
Gm/R = T*([z.sub.1]ln [z.sub.1] + [z.sub.2]ln [z.sub.2]) + Wn[z.sub.1][1 - [(1 - [z.sub.2]).sup.{n-1}]] (9)
where z can be mole fraction, volume fraction or weight fraction. Component i is the "weaker" component, toward which the miscibility gap leans, and also toward which the extremum (maximum or minimum) of the enthalpy of mixing leans. This component is the "weaker" one because it takes more of it to effect an extremum.
In polymer blends we deal mostly with miscibility gaps, having upper or lower critical point. The miscibility gap leans toward component 1. According to our model in a binary system
W/[T.sub.crit] = [n.sup.(n-2)]/[(n - 1).sup.n] (10)
RESULTS
Table 1 details the description of symbols used.
Enthalpy of Mixing, Organic Materials
Figure 1 shows the enthalpy of mixing in the system Ethylacetate-Chloroform at 25 [degrees] C from Staude (18). Figure 2 shows the enthalpy of mixing in the system i-Amylalcohol-Chloroform at 25 [degrees] C from Staude (18). The original data are given as a function of mole fraction. Chloroform is a round ball, whereas Ethylacetate is a straight molecule and i-Amylalcohol has a side chain. In the i-Amylalcohol-Chloroform system the enthalpy of mixing is positive at high i-Amylalcohol concentrations, and negative at high Chloroform concentrations. We have plotted in Figs. 1 and 2 the data also as a function of volume and weight fractions. The lines are the calculated ones, using the interaction parameters obtained from the measured data. All representations are equivalent. In the system Ethylacetate-Chloroform the volume fraction representation shows a symmetrical enthalpy of mixing (n = 2). This also can be achieved by "joining" Chloroform molecules in the mole fraction (original representation) and using [Mathematical Expression Omitted]. Table 2 gives the properties of the materials used and shows the calculated interaction parameters. The interaction parameters obtained with the volume fraction and "joined" mole fraction representation are equal. Thus there are at least two ways to describe a symmetrical Ethylacetate-Chloroform solution.
Table 1. Description of Symbols Used. Symbol R gas constant component 1 the component close to which the thermal effect is maximum in a binary system. n integer (2, 3, 4 .) defines the asymmetricity of the binary system A - B. (Gibbs energy of mixing maximum or minimum). The composition is [A.sub.(n-1)]B. m If two energy terms are present in a binary system, one with n, then the other with m: A[B.sub.(m-1)] W Interaction parameter (unit depends on the data used) can be temperature dependent W = A + B*T Is composition independent f Number of small or weak polymers assembled to compensate for the strong polymers' attraction or repulsion. Makes the binary system symmetrical (n = 2). r volume ratio r = [V.sub.2]/[V.sub.1] r[prime] molecular weight ratio r[prime] = [Mw.sub.2]/[Mw.sub.1] 3,(silane) In the binary n = 3, and component 1 (A) is silane W 3,(silane) The value of W, together with the above information.
Figure 3 shows the enthalpy of mixing of Cereclor 45 and 52 with PMMA 1 (polymethylmethacrylate), experimental data of Walsh et al. (24, [ILLUSTRATION FOR FIGURE 6 OMITTED]). We had to read the experimental points from a small graph. The properties of the components are given by Walsh et al. (24). In the system Cereclor 52 - PMMA-1 we have 3, (Cereclor 52). In the system Cereclor 45 - PMMA-1 we have 3, (Cereclor 45) and 6, (PMMA-1). The lines are the calculated ones, using the interaction parameters obtained from the experimental data. The scatter of the data is such, that one cannot determine if in the Cereclor 52 - PMMA-1 system there is a repulsive term 6, (PMMA - 1). We treated the binary system Cereclor 52 - PMMA- 1 also with [Mathematical Expression Omitted]: the value of f is extremely small.
[TABULAR DATA FOR TABLE 2 OMITTED]
Polymer Systems
Table 3 summarizes the properties of all materials used in the present analysis. The first column is the figure number, where the polymers appear, the last the reference, from which the data were taken. The columns in between give the properties of the polymers, as described in the references.
In every system we calculate the interaction parameter W from every experimental point. If W is temperature dependent, we vary n, and in the case of properties marked by [Mathematical Expression Omitted], until the line W = A + B*T, has a high regression coefficient.
Binary Systems
Roe and Zin (20) show two complete phase diagrams. In the first diagram (20, [ILLUSTRATION FOR FIGURE 7 OMITTED]), diblock copolymer (Mw 28,000) - polystyrene (Mw 2400) where the molecular weights are very different, the miscibility gap leans toward the material with the low molecular weight material. In the Hoch-Arpshofen (1, 2) description, the polystyrene has weak bonding (or repulsion) capability, because it has fewer carbon atoms. In the other diagram (20, [ILLUSTRATION FOR FIGURE 8 OMITTED]) diblock copolymer (Mw 28,000) - homopolymer polybutadiene (Mw 26,000) where the molecular weights are almost equal, the miscibility gap is symmetric, and the two materials mix much easier. The peritectic reaction in the diagram with the asymmetric miscibility gap (20, [ILLUSTRATION FOR FIGURE 7 OMITTED]) becomes a eutectic one, because the melting point of [M.sub.1] at A does not change. The dotted lines in (20, [ILLUSTRATION FOR FIGURES 7 AND 8 OMITTED]) are required to satisfy the Gibbs phase rule.
Figure 4 shows the miscibility gap from (20, [ILLUSTRATION FOR FIGURE 8 OMITTED]). The line is drawn with the interaction parameter calculated from the experimental points. The system is symmetric, n = 2. The interaction parameter, and the critical temperature are given in Table 4. The first column in Table 4 shows the figure number, the second identifies the curve in the figure, the third n, the fourth f, the fifth r, the sixth and seventh A, the eighth and ninth B, the tenth the regression coefficient, and the eleventh the calculated critical temperature [T.sub.crit].
Figure 5 shows the miscibility gap from (20, [ILLUSTRATION FOR FIGURE 7 OMITTED]). Here the original diagram is strongly asymmetric (n = 8). We have also plotted the diagram as a function of mole-fraction (n = 80), and also as a function of [Mathematical Expression Omitted], (n = 2 by definition) and f = 3.9. All three methods represent the diagram quite well. The [Mathematical Expression Omitted] method, because all points move toward the center of the diagram, best shows the deviation of the points from the calculated curve. All data are summarized in Table 4. In the above two systems, as the components contain only carbon atoms and the component chains are probably linear, the asymmetricity of the diagram depends only on the ratio of the volume fractions.
Figure 6 shows the miscibility gap in the system PMPS poly(methylphenylsiloxane) (Mw 1890) - PDMS poly(dimethylsiloxane) (Mw 1160), both linear molecules, from Kuo and Clarson (21). The data are represented in four fashions, volume-, mole-, weight-, and [Mathematical Expression Omitted]. In this case the volume ratio r = [V.sub.2]/[V.sub.1] = 1.97 is relatively small. However the value of [TABULAR DATA FOR TABLE 3 OMITTED] n is [greater than or equal to] 5 and f = 1.85. The PMPS sample, in which the silicon is bound differently, is more "active" and more PDMS is needed to counteract this effect. All data are summarized again in Table 4.
Goh et al. (25) studied a large number of systems with lower critical point temperature (LCPT). One of the components was always a-MSAN (Poly{a-methyl styrene-co-acrylonitrile}). We picked three systems, two with highly asymmetric miscibility gaps PMMA-2, (Poly{methyl methacrylate}), and PEMA (Poly{ethyl methacrylate}) and the third with a very high molecular weight polymer PMMA-7 (Poly{methyl methacrylate}). Figure 7 contains the three phase diagrams of Goh et al. (25) and Fig. 8 the "regularized" [Mathematical Expression Omitted] (n = 2) diagram. The miscibility gap leans toward the polymer with the lower molecular weight. However, it is surprising that the PEMA-7 polymer, with its very [TABULAR DATA FOR TABLE 4 OMITTED] high molecular weight ([greater than]1,000,000), forms an almost regular solution (n = 3), though r[prime] = 6.25.
The behavior of polymers in organic solvent was studied by Barbarin-Castillo et al. (23). They studied the behavior (especially LCPT) of cyclic (R) and linear (L) poly(dimethylsiloxanes) in TMS (tetramethylsilane) and in NPT (neoptentane) solutions. The various polymers had different molecular weights. Figure 9 shows two binary phase diagrams: the points as given by Barbarin-Castillo et al. (23) and as [Mathematical Expression Omitted] fractions "regularized" by us. The shape of the two [Mathematical Expression Omitted] curves is identical: By adding 7.35 [degrees] C to the lower curve (in NPT) and subtracting the same amount from the upper curve (in TMS) the curves coincide. Figure 10 shows four samples in the two solutions, and one sample calculated. There is some scatter in the data, but the shapes are similar. FInally, Fig. 11 shows the value of A in the equation W = A + B*T, for all samples of Barbarin-Castillo et al. (23) that had four or more points in their Tables 2 and 3. In the regression analysis, B was forced to be the same for all samples, and n = 32 gave the best regression coefficient.
Ternary Systems
Figure 12 shows the ternary diagram of Rigby et al. (26). The critical temperature (cloud temperature) along a section [[Phi].sub.1]/[[Phi].sub.2] = 1, and 0 [less than] [[Phi].sub.3] [less than] 0.5 is plotted. The binary used was equal weights of polystyrene (Mw = 1900) and polybutadiene (Mw = 2650). To this a) a random copolymer R 50/50 (Mw = 25,000). b) a random copolymer RF 50/50 (Mw = 16,300), c) random copolymer SPP 45 (Mw [greater than or equal to] 2.3*[10.sup.5]), d) a diblock copolymer B 50/50 (Mw = 26,000) was added, to obtain the ternary miscibility gaps. In the pure binary system n = 2, and at 120 [degrees] C W = 0.393 kK. The curves drawn in Fig. 12 were obtained by regression analysis and the constants given in Table 4. To see the effects of the ternary additions we assume that these interaction parameters in Table 4 represent binary systems. In Fig. 13 we plotted these data as binary systems, together with the binary diagrams of Roe and Zin (19, [ILLUSTRATION FOR FIGURE 3 OMITTED], curve 1) and (20, [ILLUSTRATION FOR FIGURE 8 OMITTED]; also our [ILLUSTRATION FOR FIGURE 4 OMITTED]). The miscibility gap boundaries of the two true binary diagrams and the cases a) and b) mentioned above are almost parallel (remember that at -273 [degrees] C all diagrams must meet at 0, thus in these cases the ternary effect is negligible. In the case c) the graph shows a much flatter miscibility gap, indicating a strong ternary effect. In the case d) the miscibility gap is reversed: in the ternary a three phase region must be present close to the binary.
Transformation (disorder) Reactions
Roe et al. (27, [ILLUSTRATION FOR FIGURE 3 OMITTED]) show the disordering of a styrene-butadiene diblock (Mw 27,000) and triblock (Mw 57,000) polymer of styrene and butadiene. Both samples contained 25% styrene. Zin and Roe (28, [ILLUSTRATION FOR FIGURE 5 OMITTED]) show the disordering of a styrene-butadiene diblock (Mw 27,000, with 25% styrene) polymer when mixed with various amounts of polystyrene (Mw 2400). The diblock polymer is the same in both studies, and their disordering are similar, thus we treated them together. The original curves look like annealing curves, or the liquid glass transition (29), and were replotted in Fig. 14 according to the equation
ln x/(1 - x) = C + D * T (11)
where x is the amount of material short range ordered (or clustered), and C and D are constants. As expected, all data fall on straight lines, with identical slopes.
All the data from Fig. 14 were analyzed together using regression analysis. It was assumed that D was the same for all systems. Defining [T.sub.0] as the temperature where ln(x/(1 - x)) = 0, where the amount of ordered and disordered material is equal, we plotted in Fig. 15 [T.sub.0] as a function of the styrene content of the mixture, the points fall on a straight line.
REFERENCES
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| Author: | Hoch, Michael |
|---|---|
| Publication: | Polymer Engineering and Science |
| Date: | Oct 1, 1996 |
| Words: | 3689 |
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