The subject of polymer blends is vast and has been studied for decades. Blending together different polymers has enabled many new materials with improved physical, optical, and chemical properties. One way to promote or modify polymer–polymer (P–P) miscibility is to incorporate hydrogen-bond interactions into blend components.1, 2 Site-specific H-bonding interactions have been shown to improve the miscibility of linear polymer with traditionally immiscible components including rigid-rod polymers,3 liquid crystals,4, 5 and thermosetting polymer precursors.6, 7 Several factors influence the ability of sidegroups to participate in hydrogen bonding: H-bond strength, backbone tacticity, backbone rigidity, and side-group bulkiness, and rotational freedom of participating functional groups. Studies reveal that far fewer H-bonds form in low temperature, semicrystalline8 and glassy states,9 where molecular motion is restricted, compared with higher temperature melt states. In addition, adequate spacing of functional groups along a polymer backbone has been shown to promote H-bonding due to improved molecular rotational freedom of H-bonding groups.4, 10 Terminal H-bonding groups possess a high degree of rotational freedom and can be systematically incorporated into a polymer blend. Massa et al.11 have demonstrated this concept through their investigation of the miscibility of hyberbranched polymers with linear polycarbonates, polyesters, and polyamides. Miscibility was significantly enhanced when the hyperbranched component contained hydroxy-terminated groups compared with those containing acetoxy-terminated groups. To further explore this issue we are exploring the use of end-group functionalized polymers to influence P–P phase behavior.
Linear telechelic polymers bearing associating endgroups, in particular halato-telechelic polymers, have been studied for over two decades.12, 13 Over the last decade, stronger and more directional multiple H-bonding groups have been developed.14–23 These functional groups typically employ an array of complementary or self-complementary hydrogen bonds. For example, the ureidopyrimidinone group contains four hydrogen-bond sites and reversibly dimerizes with an association constant Ka exceeding 106 M−1 in chloroform.15 Such strongly associating functional groups can guide the self-assembly of small molecules, oligomers, and polymeric components into linear, polymer-like chains, gels, and other ordered phases. At elevated temperatures, noncovalent bonds dissociate, and the material behaves similar to a liquid or a low-viscosity melt. The ability to reversibly transform a noncovalent polymer, gel, or crosslinked elastomer into a low-molecular weight liquid offers an approach to reducing the energy costs associated with polymers and materials processing and could foster technologies including thermoplastic elastomers24, 25 and shape memory polymers.26, 27 An improved understanding of phase behavior of blends containing associating terminal groups will aid in developing such applications. Associating endgroups may also enable new polymer blends containing traditionally immiscible polymers and will further development the concept of supramolecular block copolymers.28–33
Much of our understanding of solution phase behavior of telechelic associating polymers derives from the classical models of reversible polycondensation developed by Jacobson and Stockmayer34 and from the lattice theory proposed by Flory and Huggins.35 More recently, Semenov and Rubinstein36 developed a mean-field model for thermoreversible association of polymers in solution. Their model suggests that association between chains can lead to phase separation even under mild solvent conditions. Similar mean-field models can describe thermoreversible association of dissolved chains in the case where associating groups can only form pairs and are located at the end of chains.37, 38
Despite the progress in understanding the behavior of associating polymers in solution, few theoretical studies have been carried out on bulk P–P blends involving reversible association. Coleman and Painter2 investigated the effect of side-group H-bonding interactions on bulk P–P miscibility. They developed an association model where the free energy of H-bond formation is accounted for by incorporating an additional, term, ΔGH, into the classical Flory–Huggins mixing equation. The magnitude of this additional term is shown to depend on the number of new hydrogen bonds (dimers) that form upon mixing. The number of newly formed hydrogen bonds upon mixing can then be expressed as a function of the side-group association constant, and the number of “free” sidegroups—a quantity that can be experimentally measured using infrared spectroscopy. Through an extensive experimental campaign, this model was successfully applied to study and predict the phase behavior of several polymer pairs.
In this article, Painter and Coleman's association model is modified to predict how P–P miscibility changes if polymers are functionalized with noncovalent, reversibly binding endgroups. The analysis considers both complementary and hetero-complementary [donor–acceptor (DA)] binding groups as well as monofunctional versus telechelic systems. The predicted phase diagrams display the delicate balance between three factors: (1) the combinatorial entropy of mixing polymer chains, (2) the repulsive interactions between dissimilar polymers, and (3) additional enthalpic and entropic changes due to end-group association of chain ends.
In the following section, the scope of the study will be defined and a blend classification scheme will be introduced. This will be followed by a description of the model, its assumptions, and its limitations. In the Results and Discussion section, selected examples of predicted phase diagrams will be presented and their critical point behavior will be discussed.
The scope is limited to the case where associating groups are not within the chain interior, but terminate one or both ends of polymer chains. Self-complementary interactions [Fig. 1(a)] and hetero-complementary interactions [Fig. 1(b)] are considered. Examples of self-complementary endgroups include carboxylic acids (DA), diaminopurines (DADA),39 ureidopyrimidones (DDAA),15 and ureido-naphthyridine (DDAADDAA).40 Quadruple and higher H-bonding functional groups have been shown to exhibit self-sorting with high specificity.41 For that reason blends are considered that involve up to two different site-specific interactions, and each associating group may only bond with associating groups of the same type, that is, cross-association of different endgroups is prohibited. Filled and hollow circles in Figure 1(a) denote different types of endgroups. There are a total of 12 unique combinations of binary blends involving two or fewer different types of self-complementary interactions. Polymer blend components are classified into three types: polymers without associating groups (P), monofunctional polymers (M) that can form dimers, and telechelic polymers (T) that can form chain-like structures. Figure 1 uses the notation that will be adopted throughout the article: upper-case letters denote the polymer type, and subscripts denote the interaction type. For example P–T11 denotes a blend containing a nonfunctionalized polymer and a telechelic functionalized polymer with each end capable of reversibly bonding to other ends. For hetero-complementary binding, the seven unique combinations of binary polymer blends are shown in Figure 1(b) where associating groups are denoted as either donors (D) or acceptors (A). Here, donor endgroups bond exclusively with acceptor end-groups and are prohibited to bond with other donor endgroups. Experimental analogs include adenine- and thymine-derived nucleobase interactions,42 diamido-naphthyridine complexes with urea (ADDA:DAAD),43 and oligoamide duplexes (ADAADA:DADDAD).18
Painter and Coleman have modified the classic Flory–Huggins expression for free energy of mixing (ΔGM) by including an additional term (ΔGH) to account for enthalpic and entropic free energy changes due to hydrogen bonding:
where Φi is the volume fraction of the ith component, Ni is the number of segments the ith polymer chain, and χ is the Flory–Huggins interaction parameter. Equation 1 describes the free energy of mixing on a per-site basis using a reference volume of a single chemical repeat unit. The final term is an excess free energy term and is determined by the number of associated and free H-bonding species. This term can be analytically evaluated by considering the stoichiometry of H-bonding and by defining and specifying an association constant. This procedure, developed by Coleman and Painter,2 will be briefly outlined for a simple blend pair in our study: T11–P. The stoichiometry and resulting equations are similar to those derived to explain phase behavior of polymers carboxylic acid sidegroups that can form intermolecular dimers.44
For the blend P–T11, component A in eq 1 is a linear polymer, and component B is a linear telechelic polymer that has self-complementary endgroups of the same type. If the molecules' molar volumes are VA and VB respectively, and the molar volume of one associating endgroup is Vβ, then the total number of segments (including endgroups) within a B molecule sB is
and the volume fraction of end groups is
Endgroups can dimerize according to
where the subscripts 1 and 2 denote free and associated endgroups. The volume of a dimer is twice that of a free end and, therefore, the equilibrium constant is
The average degree of association h is the ratio of the total number of B molecules to half the number of free endgroups (since there are two free endgroups per associated chain):
To apply eq 1, an expression for the excess free energy of hydrogen bonding is needed. Treating free endgroups and associated endgroups as separate and distinguishable species this term can be written in the form:
The first term on the right accounts for the entropy of dilution of unbounded endgroups in going from the pure reference state to the mixed state. The second term is equal to the change in the number of dimers in going from pure B to the mixture. The final term is subtracted to make ΔGH a true excess function to describe only the redistribution of H-bonds. The functional form of eq 9 depends on the architecture of the blends and on the stoichiometry of associating groups. Analogous expressions to eq 9 were derived for each of the 20 unique cases listed in Figure 1.
Together, eqs 1, 3, 7, and 9 define the ΔGm for the P–T11 blend which can be used to address the question of miscibility. By examining how ΔGm depends on composition, regions of phase miscibility, and stability can be constructed. Spinodal decomposition leading to the formation of two thermodynamically stable phases occurs if
The limits of stability are determined when the second derivative term is zero. The metastable region is delineated by the points of double-tangency on a plot of ΔGm versus Φ forming a line of coexistence, that is, when the chemical potential of one phase equals that of another.
In real systems, both the interaction parameter χ and the intermolecular associating constant Ki depend on temperature. We assume that χ varies simply as 1/T, and that the association constant can be modeled using the Van't Hoff equation:
where K0,i is the value of the association constant at an absolute temperature T0, R is the gas constant, and hi is the molar enthalpy of end-group association.
The calculations that follow are used to illustrate the broad applicability of association models to blends involving end-associating groups. To proceed, χ, representing nonassociative interactions between chains, will be assigned a value of 0.1 at 300 K. This assignment, was selected for the following reasons: it is an experimentally reasonable value, it enables repulsive interactions to play an important role at experimentally relevant temperatures (300–500 K), and the same choice was made in developing other association models.44 The molar enthalpy of end-group association hi is taken as 40 kJ/mol. This value is about the same as the enthalpy of dimerization for a typical carboxylic acid dimer.
Mathematica (v 5.2) was used to symbolically derive expressions for the excess free energy of hydrogen bonding ΔGH for each blending scenario shown in Figure 1. The results of these derivations are available in the Appendix section. Using derived expressions for ΔGH, Igor Pro (v 5.0.1) was employed to locate spinodal and binodal phase boundaries. An additional root-finding routine was written, using Igor Pro, to locate critical points.
RESULTS AND DISCUSSION
The organization of this section will be briefly outlined. The predicted phase behavior of the 12 unique cases involving self-complementary blends [Fig. 1(a)] will be discussed first. These blends are classified into three categories: (i) combinations involving associating polymers that are blended with nonassociating polymers (e.g. T11–P), (ii) combinations where both components associate, but only in their pure phases (e.g. M1–M2), and (iii) combinations where both components bear associating groups and association is possible between the two blend components as well as within the pure phases (e.g. T12–M1). The phase behavior of blends with DA type interactions will be discussed as a separate topic.
To simplify the discussion and to enable direct comparisons between blend combinations, the following constraints were imposed: (i) as expressed in eq 11, hi is taken as 40 kJ/mol and T0 is 300 K; (ii) χ varies as 1/T and is assigned 0.1 at 300 K; (iii) the length of both blend components is always held equal. The last constraint ensures that any asymmetry present in phase diagrams is a result of reversible association, and is not due to differences in chain lengths between components.
Blends of Associating Polymers with Nonassociating Polymers (P–T11, P–M1, P–T12)
Figure 2 shows predicted upper critical solution temperature (UCST) spinodal and binodal phase boundaries for P–T11. Curves were generated for different polymer chain lengths using a fixed value of the association constant (K0,1 = 1000). In the phase-space below the spinodal (dashed) line a single phase is unstable, and concentration fluctuations can lower the system's free energy resulting in spinodal decomposition into two thermodynamically stable phases. The region adjacent to this space, bounded by the binodal (solid) and the spinodal lines, forms a region of single phase metastability. The spinodal and binodal curves meet at the UCST critical point. When the T11 chains are short (e.g. N = 50), the model predicts asymmetric phase diagrams, and UCST critical points are skewed toward the left. For these short T11 chains, a greater number of associating endgroups are present, which can self-associate into equilibrium polymers that exhibit long-chain characteristics, explaining the observed asymmetry. For longer-chain mixtures, phase boundaries are located at higher temperatures because less entropy is gained upon mixing long-chain molecules. Also, at high temperatures end-group association is negligible, and phase boundaries mimic the ideal-mixture phase boundaries predicted by classical Flory–Huggins theory.
Another set of phase boundaries for P–T11, generated using different values of K0,1 for a fixed chain length (NT-11 = NP = 100), is shown in Figure 3. The two-phase region grows with increasing values of K0,1. The figure demonstrates how end-group association results in asymmetry. Also, predicted phase diagrams show a larger metastable region on the left side of the unstable, two-phase region (rich in T11) compared with that on the right (rich in P).
A revealing picture of the phase behavior can be deduced by considering how the UCST critical point coordinate depends on model input parameters (K0,1 and N). Figure 4 shows this type of analysis for P–T11, P–M1, and P–T12 blends. In Figure 4(a–c), temperature–composition plots are shown, and each curve represents a collection of calculated UCST critical points for different values of N holding K0,1 constant. For example, the bold curve for P–T11 in Figure 4(a) is for K0,1 = 1000 and consists of the critical points taken from UCST curves shown in Figure 2. For each curve, at high enough N or low enough K0,1, critical points converge along the common vertical line of ϕ = 1/2. This indicates that the critical point occurs at high enough temperatures—where the equilibrium constant is small—such that end-group association does not affect the phase diagram symmetry and location of the critical point. The same effect is seen for the P–M1 and P–T12 blends in Figure 4(b,c) in the limits of large N or small K0,1.
Figure 4(a,b) shows that, in the limit of strong association, P–M1 blends exhibit very different critical point behavior compared with P–T11. With increasing K0,1, the critical point of P–T11 approaches ϕ = 0. In this limit, telechelic polymers assemble end-to-end forming infinitely long associated chains. On the other hand, association of M1 chains can only lead to dimers, and for strong association, the critical point approaches ϕ = 0.414—the same asymmetry that is observed in a binary mixture of two polymers when one is half the length of the other.
The phase behavior of P–T12 blends requires two association constants, one for each type of self-complementary interaction. When these association constants are identical, the resulting temperature–composition plot of critical points resembles that of P–T11. To illustrate this, Figure 4(c) includes the case when K0,1 = K0,2 = 105 for P–T12 blends (square symbols) as well as the case when K0,1 = 105 for the P–T11 blend (solid line). The P–T12 critical points lie slightly beneath, and to the right of the P–T11 curve. The model accounts for the fact that the concentration of each type of functional groups is a factor of two lower in the P–T12 than the concentration of Type-1 end groups in the P–T11 case. Consequently, the degree of association is predicted to be slightly less for P–T12 blends.
Another interesting feature can be seen in Figure 4(c) when K0,1 and K0,2 are several orders of magnitude different. The “+” symbols show a case when K0,1 is a factor of 104 larger than K0,2. At low values of N, the critical point is low enough in temperature such that association is important at both ends of T12 chains and the temperature–composition curve resembles the P–T11 case. At higher intermediate values of N, the critical point is high enough in temperature (∼0 °C) where Type-2 association can be neglected, and chains can only form dimers resulting in critical point behavior that resembles the P–M1 case.
Figure 5 complements Figure 4, and provides another way to understand the critical point behavior. In Figures 5(a–c) the temperature of the UCST critical point is plotted versus the length of unassociated chains for P–T11, P–M1, and P–T12 blends. In each case, a reference line is included for P–P blends. End-group association leads to predicted phase diagrams with critical points that are higher than the P–P reference line. In the limit of large N, critical points are at high enough temperature that association becomes negligible, and all curves collapse onto the P–P reference line. Several features that were discussed for Figure 4 can also be observed in Figure 5. For example, in Figure 5(b), P–M1 blends again transition from the strong association limit (dashed line) to the P–P reference curve. In Figure 5(c), P–T12 blends are shown to deviate most from the P–P reference when K0,1 = K0,2. However, P–T12 blends can never exhibit higher UCST temperatures than analogous P–T11 blends.
Blends of Self-Associating Polymers That Can Only Associate with Polymers of the Same Type (M1–M2, T11–M2, T11–T22)
Predicted T–ϕ and T–N critical point diagrams are shown in Figures 6 and 7. All blends in this class involve two types of self-complementary associating groups characterized by K0,1 and K0,2. Just as observed before, the UCST critical points plotted in T–ϕ diagrams all approach the vertical line ϕ = 1/2 in the limit of large N, or, equivalently, when end-association becomes negligible. Likewise, at this limit in T–N diagrams the critical points fall onto the reference line calculated for nonassociating P–P mixtures. Away from the limit of large N, much more interesting behavior is predicted.
For M1–M2 blends, both components can form dimers. In the limit of small N, the critical point occurs at low temperatures where both components form dimers and mimic a blend of linear polymers that are twice their original length (2P–2P). Such an M1–M2 blend is symmetric, and its critical point approaches ϕ = 1/2 as seen in Figure 6(a) (in the limit of low N). On the T–N diagram, the same critical point lies on the 2P–2P reference line. If both association constants are identical (K0,1 = K0,2), then, upon increasing N, the blend remains symmetric and the critical points move upward on the ϕ = 1/2 line. Likewise, on the T–N diagram, if K0,1 = K0,2, the critical point moves from the 2P–2P reference line at low N to the P–P reference line for larger N. If, on the other hand, K0,1 ≠ K0,2, then the T–ϕ diagram is no longer symmetric. For example, consider the curve where K0,1:K0,2 = 105:10 [the darkened curve Figs. 6(a) and 7(a)]. At low values of N, the critical point occurs near ϕ = 1/2. At intermediate values of N, M1 dimers form near the critical point, however M2 dimers do not. Consequently, the phase diagram becomes asymmetric, and the critical point moves left toward the limit of ϕ = 0.414 for a 2P–P blend. Finally, for high values of N, both components are dissociated near their critical point and, hence, ϕc approaches 1/2. The same physical picture is obtained from Figure 7(a). Upon increasing N, the critical temperature moves from the 2P–2P curve to the 2P–P curve, and finally to the P–P curve.
T11–T22 and T11–M2 blends behave phenomenologically similar to M1–M2 blends. However, end-to-end association of telechelic chains can lead to much more asymmetric phase diagrams. Figure 6(b) shows that, upon decreasing N, critical points converge onto a constant-ϕ line whose positions depend on the ratio K0,1:K0,2. Also, it should be noted that association in T11–T22 blends can raise the critical temperature much higher than is observed in equivalent M1–M2 blends. This is especially apparent by comparing T–N diagrams in Figure 7.
Blends of Self-Associating Polymers That Can Associate Both with Polymers of the Same Type and/or Polymers of the Opposite Type (M1–M1, T11–T11, T12–T12, T11–M1, T12–M2, T12–T11)
Blends M1–M1, T11–T11, and T12–T12 are always symmetric, even in their highly associated states, and are predicted to exhibit identical phase behavior to the binary polymer mixture P–P. This is not surprising because, for these cases, the number of “new” dimers that form upon mixing is exactly zero, and it can be shown that ΔGH is a constant, independent of composition.
Figures 8 and 9 show calculated T–ϕ and T–N critical point diagrams for blends T11–M1, T12–M2, T12–T11. For T11–M1, predicted phase diagrams are only slightly asymmetric and closely mimic the P–P blend. For K0,1 = 105, the critical points converge on a vertical line ϕ = 0.492 in the limit of low N. This asymmetry is due to difference between the end-group concentration of the components' pure state and their blended state. The T11 chains have double the concentration as M1 chains do in their respective pure states. Upon mixing both concentrations change, and hence the density of dimerized endgroups also changes. For the T12–M2 blend, the asymmetry of phase diagrams is more pronounced, and the phase behavior matches the M1–P blend [Figs. 4(b) and 5(b)]. The presence of Type-2 interactions does not influence the phase behavior because the number density of free or associated Type-2 endgroups does not change upon mixing. Finally, the blend T12–T11 exhibits aspects from both the M1–P and T11–M1 blend. To appreciate the subtleties of this blend, consider the case in Figure 8(c) where K0,1:K0,2 = 10:105. At high values of N, the critical point lies at high temperatures where association is not important and the critical point is near ϕ = 1/2. Upon lowering N, the critical point moves to a region where only Type-2 interactions are important, and asymmetry is predicted that mimics the M1–P blend. Here, critical points are observed at about ϕ ∼ 0.42. At lower values of N, the critical temperature decreases, and both types of interactions become important, and the critical point moves to slightly higher values of ϕ.
Blends of DA Associating Polymers with Nonassociating Polymers
Critical point diagrams for TDA–P are shown in Figure 10. Predicted asymmetry in the T–ϕDA plot is limited by the P–T11 reference case. The model accounts for the fact that the concentration of both donor and acceptor functional groups is a factor of two lower than the concentration of Type-1 end groups in the analogous P–T11 case. Consequently, the degree of association is predicted to be slightly less than the P–T11 blend. Note that the critical point behavior predicted for P–TDA is identical to that predicted for P–T12 when K0,1 = K0,2 for the T12 chains.
Blends of DA Associating Polymers with Other Polymers Containing Complementary (D or A) Interactions
All remaining blends in this section involve DA interactions that can act between opposite blend components. As expected, this use of DA interactions is predicted to promote polymer mixing, and offers an approach to developing new, stable blends of dissimilar polymers. Remaining blends are divided into blends that only involve inter-component association (e.g. MA–MD) and those that involve inter-component and intra-component association (e.g. TDA–MD).
Blends involving only inter-component association (MA–MD, TAA–MD, TAA–TDD) will be discussed first. Three predicted phase diagrams of the MA–MD blend are shown in Figure 11 for different values of chain length N. At relatively low values of N = 350 a two-phase region appears that has a UCST critical point and an LCST critical point. Above the UCST point phase mixing occurs whereby the entropy of mixing outweighs the enthalpic penalty of interacting chains. At these high temperatures, association effects are negligible. On the other hand, below the LCST critical point, association of endgroups is important and drives the formation of a one-phase region. Similar low-temperature one-phase regions have predicted for polymers containing hydrogen-bonding groups present as sidegroups, along the chain.44 A two-phase region lies between the UCST and LCST critical points. At these temperatures dissimilarity between chains overcomes both entropy of mixing and the free energy gain due to end-group association. Upon increasing the chain length to N = 450 [Fig. 11(b)] two UCSTs appear at lower temperatures, introducing two additional two-phase regions. Increasing the chain length leads fewer endgroups, and the inter-chain association that stabilizes the one-phase region becomes less dominant. This phase diagram also demonstrates the ability of the model to account for stoichiometry of endgroups. The low-temperature one phase region is largest around ϕMA = 1/2 because each donor endgroup is matched with an acceptor endgroup, leading to a greater degree of association. Upon increasing N, the low-temperature one phase region continues to shrink until it becomes a completely inverted one-phase region [Fig. 11(c)].
Similar predicted phase diagrams for the TAA–MD blend are shown in Figure 12. The main difference is that the diagrams are no longer symmetric and a low temperature two-phase region only appears for compositions rich in TAA. This feature is easily explained on the basis of stoichiometry. An equal number of acceptor and donor groups is present at ϕ = 1/3, and this promotes association which stabilizes the one-phase region. Phase diagrams for the TAA–TDD appear very similar to the MA–MD case, and for this reason are not shown.
To summarize the critical point behavior for these three blends, Figure 13 displays how UCST and LCST critical temperatures depend on chain length. Note that, in every case, the critical point curve lies below the P–P reference curve. This highlights the ability of DA association groups to form stable one-phase regions. Comparing these three cases, the blend TAA–TDD demonstrates the greatest ability to stabilize the one-phase region (note the difference in the x-axis scale). The hollow circle at low N marks the location (T, N) where a new two-phase region forms and the hollow circle at high N denotes the location (T, N) where the LCST and the UCST critical points annihilate, forming a continuous two-phase region.
The last three blends considered include TDA–MD, TDA–TDD, and TDA–TDA. These blends have the possibility for inter-component and intra-component association. Figure 14 shows T–N critical point diagrams for TDA–MD, and TDA–TDD. In general, these blends deviate less from the P–P reference curve than the blends with only inter-component association. The phase diagram for TDA–TDA is perfectly symmetric, and is not shown.
A mean field model was developed to predict P–P phase behavior for blends involving polymers functionalized with noncovalent, reversibly binding endgroups. Critical point diagrams (T vs. ϕ, and T vs. N) were generated for several unique blend combinations and provide a useful way to assess differences in phase behavior. The analysis reveals the extent to which self-complementary interactions can stabilize two-phase regions, resulting in asymmetric phase boundary curves. The two-phase region is stabilized most for blends that involve telechelic molecules and that self-associate in their pure states (T11–T22). The model predicts how polymers with only one functional end have a much smaller impact on phase boundaries since they can only form dimers. The model also suggests that polymers with DA interactions can stabilize the single-phase region of the phase diagram, and this is most effectively accomplished using the TAA–TDD blend. For DA systems, unusual phase behavior is predicted, including LCST behavior and inverted phase diagrams (e.g. MA–MD). The model has a few adjustable parameters and will be used to guide future experiments on blends with associating endgroups.
Acknowledgement is made to the donors of The American Chemical Society Petroleum Research Fund for support of this research. Acknowledgement is also made to Paul Painter at Pennsylvania State University for several discussions and assistance in deriving free energy expressions.
Table Appendix. Table of Derived Expressions for the Excess Free Energy of Hydrogen Bonding for Several Different Binary Blends
ϕA, ϕB, volume fraction of A and B components; ϕα, volume fraction of type-α self-complementary associating groups; ϕ, volume fraction of type-α dissociated self-complementary associating groups; ϕα1,pA, volume fraction of type-α dissociated self-complementary associating groups in pure A; ϕβ, volume fraction of type-β self-complementary associating groups; ϕδ, volume fraction of donor type-δ hetero-complementary associating groups; ϕπ, volume fraction of acceptor type-π hetero-complementary associating groups.