Viscosity of poly(vinyl acetate) and its concentrated solutions

Data on the viscosity of poly(vinyl acetate) (PVOAc) and its concentrated solutions in diethyl phthalate (DEP) and cetyl alcohol (CeOH) are examined over the molecular weight range 8 × 10³ < M < 1.3 × 10⁶, the range 0.15 < φ < 1.0 of the volume fraction φ of polymer, the temperature range 308 to 430°K for PVOAc and its solutions in DEP, and at 396°K in CeOH. The latter is the θ temperature for dilute solutions of PVOAc in CeOH. The data are analyzed with the relation η = KX_c(αφM/M_c)^a exp[1/β(T – T₀)], where a is 1 or 3.4 for αφM less than or greater than a constant M_c, respectively, and X_C, is a constant. The expansion factor α_φ of the chain dimension is found to be essentially unity for φ > ca. 0.25, increasing with decreasing φ for smaller φ. Both β and T₀ depend on φ, and T₀ also depends on M at low M.     

Moderately concentrated solutions of polystyrene. I. Viscosity as a function of concentration,temperature, and molecular weight

Data on the viscosity η of moderately concentrated solutions of polystyrene are reported. Several solvents were investigated, including cyclopentane solutions over a temperature span between θ_U = 19.5°C and θ_L = 154.5°C. The data were analyzed in terms of a relation giving η as a function of αφM, where α_φ is the expansion factor for the chain dimensions in a solution with volume fraction φ of polymer with molecular weight M. It is shown that values of α_φ so determined decrease as ? lnα_φ/? lnφ = (1 ? 2μ)/6μ for φ greater than φ^* = 0.2M/s³ for moderately concentrated solutions, where s is the root-mean-square radius of gyration and μ = ? ln[η]/? lnM with [η] the intrinsic viscosity.     

Structural analysis of aggregates formed by a thermoresponsive homopolymer in dilute aqueous solutions

The aqueous solution of a thermoresponsive polymer, poly[2‐(2‐ethoxy) ethoxyethyl vinyl ether] poly(EOEOVE), contains a tiny amount of large polymer aggregates at low polymer concentrations far below the lower critical solution temperature (～40 °C). The molar mass M_w,slow, radius of gyration 〈S²〉, and hydrodynamic radius R_H,slow of the aggregating component of poly(EOEOVE) were obtained by simultaneous static and dynamic light scattering as functions of the polymer concentration and temperature, while the weight fraction w_slow of the component was estimated by size‐exclusion chromatography. The M_w,slow dependencies of 〈S²〉 and R_H,slow, as well as the ratio 〈S²〉/R_H,slow, indicated that the poly(EOEOVE) aggregate takes a sparsely branched polymer‐like conformation. We have analyzed the structure of the aggregate, using the branched polymer model of random type. The M_w,slow dependence of 〈S²〉 obtained was favorably compared with this model with reasonable structural parameters. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1179–1187, 2006     

Dilute solution properties of a polybenzimidazole

Light scattering and viscometric studies have been carried out on dilute solutions of a polybenzimidazole in N,N-dimethylacetamide. The data, which span the molecular weight range 2.9 ≦ 10^?4M_w ≦ 23.3, and the temperature range 290 ≦ T/K ≦343, yield the dependence of the mean-square radius of gyration 〈s²〉_LS, the second virial coefficient A₂, and the intrinsic viscosity [η] on molecular weight M_w and temperature. The unperturbed mean-square radius 〈s〉_LS was calculated using experimental values of 〈s²〉_LS and A₂. It was found that excluded volume effects on 〈s²〉_LS are very small. The unperturbed hydrodynamic chain dimension 〈s〉_η was estimated by considering draining effects. A small value of the draining parameter was obtained. Analysis of the temperature dependence of A₂ and [eta;] leads to the conclusion that this system approaches a lower theta temperature with increasing temperature. The steric factor σ = 〈s〉/〈s〉f, based on the value of 〈s〉_f calculated for the polymer chain with free rotation, is nearly unity. Most of these properties can be interpreted in terms of long rotational units within the main chain.     

Heterogeneous surface saponification of suspension‐polymerized monodisperse poly(vinyl acetate) microspheres using various ions

Poly(vinyl alcohol) (PVA)/poly(vinyl acetate) (PVAc) microspheres with a skin/core structure were prepared through the heterogeneous surface saponification of PVAc microspheres suspension‐polymerized. The PVA skin formed through the heterogeneous saponification was hydrogel swellable in water. In addition, to obtain monodisperse PVA/PVAc microspheres having various skin/core ratios and morphologies, the ion‐specificities to the heterogeneous saponification were investigated using SO, Cl^?, NO, Br^?, and I^? for anions and Li⁺, Na⁺, and K⁺ for cations, respectively. The ions were not specific significantly to the rate of the heterogeneous saponification, while were related to the degree of saponification (DS). DSs had different values between by weight loss (DS_w) and by proton nuclear magnetic resonance spectroscopy (DS_NMR) measurements. The order of DS_ws was SO < Cl^? < NO < Br^? < I^? for anions and K⁺ < Na⁺ < Li⁺ for cations, and that of DS_NMRs, I^? < Br^? < NO < Cl^? < SO for anions and Li⁺ < Na⁺ < K⁺ for cations. The differences in values between DS_ws and DS_NMRs were caused by the dissolution of PVA skin and were significantly decreased for SO. The peaks at melting temperature of PVA were sharp and their areas were large for ions deswelling PVA skins.     

Relaxation dynamics of salt‐free polyelectrolyte solutions using flow birefringence and rheometry

Relaxation dynamics of salt‐free, aqueous solutions of sodium poly(styrene sulfonate) (NaPSS) were investigated by mechanical rheometry and flow birefringence measurements. Two semidilute concentration regimes were studied in detail for a range of polymer molecular weights. At solution concentrations c < 10 mg mL, limiting shear viscosity η₀ was found to scale with molecular weight and concentration as η₀ ∼ c^0.5M_w over nearly two decades in concentration. At higher solution concentrations, c > 10 mg mL, a change in viscosity scaling was observed η₀ ∼ c^1.5M, consistent with a change from simple Rouse dynamics for unentangled polyions to near‐perfect reptation dynamics for entangled chains. Characteristic relaxation times τ deduced from shear stress and birefringence relaxation measurements following start‐up of steady shearing at high rates reveal very different physics. For c < 10 mg mL, both methods yield τ ∼ c^−0.42M and τ ∼ c⁰M for c > 10 mg mL. Curiously, the concentration scalings seen in both regimes are consistent with theoretical expectations for salt‐free polyelectrolyte solutions undergoing Rouse and reptation dynamics, respectively, but the molecular weight scalings are not. Based on earlier light scattering studies using salt‐free NaPSS solutions, we contend that the unusual relaxation behavior is likely due to aggregation and/or coupled polyion diffusion. Simultaneous stress and birefringence measurements suggest that in concentrated solution, NaPSS aggregates are likely well permeated by solvent, supporting a loose collective of aggregated chains rather than the dense polymer aggregates previously supposed. Nonetheless, polyion aggregates of either variety cannot account for the inverse dependence of relaxation time on polymer molecular weight for c < 10 mg mL. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 825–835, 1999     

Dilute-solution properties of trans-1,4-polyisoprene

Trans-1,4-polyisoprene was fractionated by both fractional precipitation and preparative gel permeation chromatography to obtain possibly sharp fractions of narrow molecular weight distribution. Selected fractions were characterized by light scattering, viscosity, and gel permeation chromatography. Necessary corrections for molecular heterogeneity were applied. Some of the characteristic relations between [η] and M _w are [η] = 1.81 × 10^?4 M in benzene at 30°C, [η] = 1.38 × 10^?4 M in n-hexane at 30°C, which are found to be in good agreement with literature data when corrected for molecular heterogeneity.     

Nonlinear rheology of highly entangled polymer solutions in start‐up and steady shear flow

Orientation angle and stress‐relaxation dynamics of entangled polystyrene (PS)/diethyl phthalate solutions were investigated in steady and step shear flows. Concentrated (19 vol %) solutions of 0.995, 1.81, and 3.84 million molecular weight (MW) PS and a semidilute (6.4 vol %) solution of 20.6 million MW PS were used to study the effects of entanglement loss on dynamics. A phase‐modulated flow birefringence apparatus was developed to facilitate measurements of time‐dependent changes in optical equivalents of shear stress (n₁₂ ≈ Cσ) and first normal stress differences (n₁ = n₁₁ ? n₂₂ ≈ CN₁) in a planar‐Couette shear‐flow geometry. Flow birefringence results were supplemented with cone‐and‐plate mechanical rheometry measurements to extend the range of shear rates over which entangled polymer dynamics are studied. In slow (τ > ) steady shear‐flow experiments using the ultrahigh MW polymer sample (20.6 × 10⁶ MW PS), steady‐state n₁₂ and n₁ results manifest unusual power‐law dependencies on shear rate [n_12,ss ～ ^0.4 and n_1,ss ～ ^0.8]. At shear rates in the range τ < < τ, steady‐state orientation angles χ_SS are found to be nearly independent of shear rate for all but the most weakly entangled materials investigated. For solutions containing the highest MW PS, an approximate plateau orientation angle χ_p in the range 20–24° is observed; χ_p values ranging from 14 to 16° are found for the other materials. In the start‐up of fast steady shear flow (γ˙ ≥ τ), transient undershoots in orientation angle are also reported. The molecular origins of these observations were examined with the help of a tube model theory that accommodates changes in polymer entanglement density during flow. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2275–2289, 2001     

Ionic interaction behavior and facilitated olefin transport in poly(n‐vinyl pyrrolidone):Silver triflate electrolytes; Effect of molecular weight

Interactions of cation/anion and cation/polymer in poly(N‐vinyl pyrrolidone) (PVP):silver triflate (AgCF₃SO₃) electrolytes with different weight‐average molecular weights (M_w's) of 1 × 10⁶ (1 M), 3.6 × 10⁵ (360 K), 4 × 10⁴ (40 K), and 1 × 10⁴ (10 K) have been studied with IR and Raman spectroscopies. According to the change of the C?O peak, coordination of silver ions by C?O in a low M_w (10 or 40 K) PVP matrix tend to be always thermodynamically favorable than high M_w (1 M or 360 K) PVP, demonstrating that the polymer matrix of low M_w dissolves silver salts more effectively. In addition, silver cations interact with both larger SO and smaller CF₃ to form ion pairs, and the former interaction is stronger than the latter in a monomer or low M_w polymer matrix (40 K, 10 K), as demonstrated by theoretical ab initio calculation or experimental spectroscopy, respectively. However, CF₃ interacts more favorably with silver cation than SO in high M_w (1 M and 360 K) PVP, which is ascribed to the steric effect of the bulky SO anion by highly entangled polymer chains. Despite the superior dissolving property of the low M_w polymer matrix, the membranes consisting of low M_w PVP and AgCF₃SO₃ exhibited poor separation performance for propylene/propane mixtures in comparison with those of high M_w, presumably because of the poor mechanical property for membrane formation in low M_w PVP. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1813–1820, 2002     

Influence of free and copolymerized triplet quenchers on the photolysis of poly(vinyl phenyl ketone) in solution

The influence of free, diffusion-control quenchers of triplets (naphthalene, biphenyl, 2,5-dimethyl-2,4-hexadiene) on the photolysis of poly(vinyl phenyl ketone) in benzene solution has been investigated. The Stern-Volmer plots for quenching of main-chain scission were linear, and the quenching constants were independent of the macroviscosity of the solutions. Copolymers of vinyl phenyl ketone with 1-vinylnaphthalene and 2-vinylnaphthalene containing as much as 10% (by weight) vinylnaphthalene were prepared. The photolysis of the copolymers was compared with the photolysis of poly(vinyl phenyl ketone) in the presence of free naphthalene. It was found that the quenching efficiency of found naphthalene units was about 21 times higher. The possibility of migration of the absorbed energy along the polymer chain is discussed. The relation between average-number molecular weight M _n and intrinsic viscosity [η] has been determined osmometrically. For unfractionated poly(vinyl phenyl ketone) in benzene at 30°C, the relation [η] = 2.82 × 10^?5 M _n^0.84 has been found.     

Entanglement friction and dynamics in blends of starlike and linear polymer molecules

We investigate relaxation dynamics in a series of six‐arm star/linear 1,4‐polybutadiene blends with mechanical rheometry measurements. Blend systems are formulated to systematically probe constraint release and arm relaxation dynamics. Zero shear viscosity and terminal relaxation times of star/linear polymer blends with fixed star arm molecular weights (M_a) and compositions (?_S) are found to follow nonmonotonic dependencies on the linear polymer molecular weight (M_L). At low values of ?_S, at least two scaling regimes are apparent from the data (ξ₀ ～ M and ξ₀ ～ M), where ξ₀ refers to the zero shear viscosity or terminal relaxation time of the blend. The two regimes are separated by a critical linear polymer molecular weight M^* that is more than 20 times larger than the critical molecular weight for entanglements. When the linear polymer contribution to blend properties is removed, a clear transition from dilution dynamics, ξ₀ ～ M, to Rouse‐like constraint‐release dynamics, ξ₀ ～ M, is apparent at low values of ?_S. At higher ?_S values, a new activated constraint‐release dynamic regime is evident in which ξ₀ ～ M and ξ₀ ～ ?, where α changes continuously from approximately 2 to 0.5 as ?_S increases and β varies from 2.0 to 1.0 as M_L increases. The experimental results are compared with theoretical predictions based on a drag coupling model for entangled polymer liquids. All features observed experimentally are captured by this model, including the value of M^* for the transition from dilution to Rouse constraint‐release dynamics. Predictions of the drag coupling model are also compared with published data for the zero shear viscosity and terminal relaxation time in bidisperse linear polymer blends and pure entangled starlike molecules. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2501–2518, 2001     

The Influence of the Ligands Cp*(η5‐C5Me5) and Cp(η5‐C5H5) on the Stability and Reactivity of Titanocene and Zirconocene Complexes: Reactions of the Bis(trimethylsilyl)acetylene Permethylmetallocene Complexes (η5‐C5Me5)2M(η2‐Me3SiC2SiMe3), M = Ti,Zr, with H2O and CO2

The reactions of the bis(trimethylsilyl)acetylene permethylmetallocene complexes CpM(η²‐Me₃SiC₂SiMe₃) (M = Ti ( 1 ), M = Zr ( 2 )) with H₂O and CO₂ were studied and compared to those of the corresponding metallocene complexes Cp₂M(L)(η²‐Me₃SiC₂SiMe₃) (M = Ti ( 3 ), L = – ; M = Zr, L = THF ( 4 )) to understand the influence of the ligands Cp(η⁵‐C₅H₅) and Cp*(η⁵‐C₅Me₅) as well as the metals titanium and zirconium on the reaction pathways and the obtained products. In the reaction of the permethyltitanocene complex 1 with water the dihydroxy complex CpTi(OH)₂ ( 5 ) was formed. This product differs from the well‐known titanoxane Cp₂TiOTiCp₂ which was obtained by the reaction of the corresponding titanocene complex 3 with water. The reaction of the permethylzirconocene complex 2 with water gives the mononuclear alkenyl zirconocene hydroxide 6 . An analogous product was assumed as the first step in the reaction of the corresponding zirconocene complex 4 with water which ends up in a dinuclear zirconoxane. In the conversion of the permethylzirconocene complex 2 with carbon dioxide the mononuclear insertion product 7 was formed by coupling of carbon dioxide and the acetylene. In contrast, the corresponding zirconocene complex 4 affords, by an analogous reaction, a dinuclear complex. In additional experiments the known complex CpZr(η²‐PhC₂SiMe₃) ( 8 ) was prepared, starting from CpZrCl₂ and Mg in the presence of PhC≡CSiMe₃. This complex reacts with carbon dioxide resulting in a mixture of the regioisomeric zirconafuranones 9 a and 9 b . From these in the complex 9 a , having the SiMe₃ group in β‐position to the metal, the Zr–C bond was quickly hydrolyzed by water to give the complex CpZr(OH)OC(=O)–C(SiMe₃)=CHPh ( 10 a ) compared to complex ( 9 b ) which gives slowly the complex CpZr(OH)OC(=O)–CPh=CH(SiMe₃) ( 10 b ).     

Intrinsic viscosities of four homopolymers and one copolymer in nonpolar solvents. II. Effect of solvent steric hindrance

Intrinsic viscosities at 25°C of an ethylene-propylene copolymer containing 81% ethylene (81% E) of polypentenamer (PPmer), polyisobutylene (PIB), polypentene-1 (PP-1), and polydimethylsiloxane (PDMS) have been measured in n-C₉ and three branched nonanes and n-C₇ and five branched heptanes. The effect of the solvent steric hindrance on the free energy, i.e., on the χ parameter was investigated. The highly sterically hindered, cruciform molecules 3,3-dimethylpentane and 3,3-diethylpentane are the best solvents for four of the five polymers. The enhancement of solvent quality due to the steric hindrance diminishes when the polymer free volume increases. The difference in [η] between 2,4-dimethylpentane and 2,3-dimethylpentane is ?50%, ?35%, ?2% for PPmer, PIB, PDMS, and can be correlated to a measure of the polymer free volume, i.e., the lower critical solubility temperature. The χ, χH, and χS are calculated from [η] using the Stockmayer-Fixman relation and from h, the heat of mixing at infinite dilution of the polymer, obtained previously. With each polymer, a good correlation is found between h and [η] obtained with the six heptanes and four nonanes. The correlation points to the same effect being at the origin of χH and χS but of different magnitude. In cases showing the steric hindrance effect, a negative contribution occurs in h (or χH) which is larger in magnitude than the corresponding negative entropic contribution leaving a net negative effect in χ itself. Probably due to their very compact shape and fewer degrees of freedom, the cruciform solvents lose less entropy than the chain solvents in mixing.     

Studies on polymerization of vinyl acetate using ylide as an initiator and degradative transfer agent

Polymerization of vinyl acetate initiated by β-picolinium p-chlorophenacylide was carried out at 30, 35, and 40°C, using conventional dilatometric technique. The initiator and the monomer exponent values were 0.80 ± 0.15 and unity, respectively. The polymerization was inhibited in the presence of hydroquinone, but was favored by nonpolar solvent and polymerization temperature. The energy of activation was 90.3 KJ mol^?1. An average value of k/k_t for the present system was found to be 0.37 × 10^?2. The results are explained in terms of a radical mode of polymerization with degradative initiator transfer; the principal mode of termination, however, was bimolecular.     

The DMSO Solvated octahedro-[W6Cl]Cl Cluster Molecule

Octahedro-hexatungsten octadecachloride, W₆Cl₁₈, is soluble in dimethyl sulfoxide (DMSO). Brownish black crystals of W₆Cl₁₈(DMSO)₄ are formed from the brown solution by evaporation of DMSO under dynamic vacuum. The compound crystallizes monoclinically in the space group P2₁/n (no. 14) with a = 10.420 Å, b = 9.271 Å, c = 20.828 Å, β = 91.10° and Z = 2. The crystal structure is formed by isolated cluster molecules [W₆Cl]Cl of the hexameric tungsten trichloride and DMSO molecules. It is the first hierarchical variant of the tetragonal BaAl₄ type of structure where all atoms of the intermetallic phase are substituted by neutral molecules. The mean bond lengths are d(W–W) = 2.878 Å, d(W–Clⁱ) = 2.391 Å and d(W–Cl^a) = 2.447 Å. They will be discussed in relation to analogous clusters. The two crystallographically independent DMSO molecules (d(S–O) = 1.53–1.55 Å, d(S–C) = 1.65–1.78 Å) form a 3 D net of condensed < 4⁸6⁴ > dodecahedra which envelopes the clusters.     

Measurement of rotational diffusivity of rodlike molecules in amorphous polymer matrices by the dynamic kerr effect

Electrooptic (Kerr effect) relaxation experiments, designed to measure the rotational diffusivity of collagen (rodlike) molecules in aqueous poly(ethyleneoxide) (amorphous) semidilute solutions under various conditions have been performed. The experimental results have been compared with the predictions of a previously derived model giving the rotational diffusivity of dilute rods in semidilute amorphous polymer solutions as a function of rod length and amorphous polymer concentration. Excellent agreement is found between the predicted scaling D_r ～ ?L^?7 (D_r = rod rotational diffusivity, pi_p = polymer weight fraction, and L = rod length) and the experiments.     

Visible light induced free radical promoted cationic polymerization using thioxanthone derivatives

The free radical promoted cationic polymerization cyclohexene oxide (CHO), was achieved by visible light irradiation (λ_inc = 430–490 nm) of methylene chloride solutions containing thioxanthone‐fluorene carboxylic acid (TX‐FLCOOH) or thioxanthone‐carbazole (TX‐C) and cationic salts, such as diphenyliodonium hexafluorophosphate (Ph₂I⁺PF) or silver hexafluorophosphate (Ag⁺PF) in the presence of hydrogen donors. A feasible initiation mechanism involves the photogeneration of ketyl radicals by hydrogen abstraction in the first step. Subsequent oxidation of ketyl radicals by the oxidizing salts yields Bronsted acids capable of initiating the polymerization of CHO. In agreement with the proposed mechanism, the polymerization was completely inhibited by 2,2,6,6‐tetramethylpiperidinyl‐1‐oxy and di‐2,6‐di‐tert‐butylpyridine as radical and acid scavengers, respectively. Additionally polymerization efficiency was directly related to the reduction potential of the cationic salts, that is, Ag⁺PF (E = +0.8 V) was found to be more efficient than Ph₂I⁺PF (E = ?0.2 V). In addition to CHO, vinyl monomers such as isobutyl vinyl ether and N‐vinyl carbazole, and a bisepoxide such as 3,4‐epoxycyclohexyl‐3′,4′‐epoxycyclohexene carboxylate, were polymerized in the presence of TX‐FLCOOH or TX‐C and iodonium salt with high efficiency. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Synthesis and characterization of poly(N‐vinyl formamide) hydrogels—A potential alternative to polyacrylamide hydrogels

The synthesis and characterization of solution‐cast, molded gels of N‐vinyl formamide (NVF) has not been previously reported even though NVF is an isomer of acrylamide (AAm) and polyacrylamide (PAAm) hydrogels have many commercial applications. Aqueous NVF solutions were cross‐linked into gels using a novel cross‐linker, 2‐(N‐vinylformamido)ethylether, and the thermally‐activated initiator VA‐044. For a given formulation, PNVF gels swell up to twice that of PAAm gels cross‐linked with N,N′‐methylenebisacrylamide. From swelling and compression measurements, PNVF gels were found to be more hydrophilic than PAAm gels. Flory‐Huggins solubility parameters were χ = 0.38?₂ + 0.48 for PNVF and χ = 0.31?₂ + 0.49 for PAAm, where ?₂ is the polymer volume fraction. The shear moduli for PNVF and PAAm scale with ? and ? respectively, consistent with good solvent behavior, also suggesting PNVF is more hydrophilic than PAAm. Similarity of mechanical properties for both gels as a function of ?₂ suggests that network structures of PNVF and PAAm gels are similar. Fracture strains of both gels declined with ?₂ by the same linear function while fracture stresses were about 500 kPa regardless of formulation. Since NVF is a liquid monomer, less toxic than AAm and can be hydrolyzed to a cationic form, PNVF gels could become technologically significant. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Ultradrawing of high-molecular-weight polyethylene cast from solution. II. Influence of initial polymer concentration

Ultradrawing of films of high-molecular-weight polyethylene (M?_w = 1.5 × 10⁶) produced by gelation crystallization from solution is discussed. The influence of the initial polymer volume fraction (?) on the maximum draw ratio (λ_max) of the dried films is examined in the temperature region from 90–130°C. The results can be described very well by the relation λ_max = λ ?^?1/2 where λ is the (temperature-dependent) maximum draw ratio of the melt-crystallized film. An attempt is made to discuss the marked influence of the initial polymer volume fraction on λ_max in terms of the deformation of a network with entanglements acting as semipermanent crosslinks.     

A Flory–Huggins thermodynamic approach for predicting sorption equilibrium in ternary polymer systems

The Flory–Huggins theory as modified by Pouchlý has been applied to calculate preferential (λ) and total (Y) sorption coefficients for a ternary polymer system. The ternary interaction function (?₁?₂?₃G_T(u₁, ?₃)) is described as the product of three independent binary functions. This expression allows prediction of λ and Y from binary interaction parameters χ, χ, g, g, and g₁₂(?₁₀). Three ternary polymer systems are used to check the validity of the expression. Moreover for polymer systems in which the parameters g and/or g are unknown, a procedure to evaluate them has been developed and verified on systems for which sufficient experimental information is available.