Elastomer swelling in mixed associative solvents.
One of the most important performance parameters for elastomers in equilibrium with mixed solvents is the degree of solvent absorption or swell. As little as 20% volume swell can reduce properties such as hardness, stiffness, and strength by 60% or more . Correlation of elastomer swell in regular solutions may be accomplished via solubility parameter approaches. However, correlation of elastomer swell in more complex nonideal, associative solvent blends continues to be of theoretical as well as practical interest.
From a theoretical perspective, predictive methods based upon the well known three-dimensional Solubility Parameter (SP) concept  are only applicable over a limited concentration range. Either endothermic or exothermic excess Gibbs free energy of mixing negates the regular solution assumption. The utility of a recently proposed method , based upon partitioning of solvent species, depends upon the relative concentration and swelling power of monomer versus associated chemical species in solution. Complex solutions containing mixed associative species are more commonly encountered than ideal solutions. Indeed many pure solvents, such as chloroform (CH[Cl.sub.3]), methanol (MeOH) among others, may be considered blends of monomer and associated species. When combined with the intention of conducting a chemical reaction, mixed solvents often exhibit substantial mutual attraction . Either positive or negative excess swell may be observed for elastomers in equilibrium with complex solutions.
From a practical perspective, application of modern three-dimensional SP analysis to multicomponent solutions requires a considerable volume of data. Three constants are required for the elastomer and three more for each constituent in solution. Furthermore, estimation of SP by group contribution schemes requires either prior knowledge of the chemical structure of the elastomer or extensive characterization of elastomer swelling behavior in model solvents. Often, swelling estimates are quantitative over a limited range of solution compositions. At best, only qualitative pass-fail estimates can be considered reliable. Thus, a practitioner must often rely upon prior experience or engineering judgment to select elastomers for these services.
In the present study, a previously disclosed linear partition swelling model  is generalized to account for solutions containing associative chemical species. This generalization is accomplished by incorporating concepts taught by Chemical Theory for nonideal solutions . In Chemical Theory, nonideal solutions of mixed associative solvents are considered to behave as ideal solutions containing both monomer and associated chemical species. Individual partition coefficients are then assigned to each chemical species. An advantage of the present method is that it requires thermodynamic data for the solution and only one elastomer swell observation for each chemical species present. No information about the elastomer's chemical structure is required to correlate swelling data in either ideal or nonideal solutions.
In a previous review , the evolution and limitations of theoretical methods commonly used to predict swelling of elastomers in mixed solvents were summarized. Selections from that review are included in Appendix A. Historically, much attention has been given to Hildebrand's concept of cohesive energy density or SP . These well-known methods are apparently adequate when the solution is "regular" and there are no interactions among the solvents in the solution or in the polymer. When an opportunity exists for hydrogen-bonding, three-dimensional SP analysis is predictive of elastomer swell for only a limited range of solution compositions. Alternative approaches to model elastomer swell in mixtures of polar solvents have been proposed. These methods attempt to characterize polymer-solvent interactions by using activity coefficients , acid-base parameters , interaction parameters , polarity indices , and others. The authors' effort, briefly summarized below, generalizes upon Flory's original concept of polymer solvent equilibrium .
A prior study by the present authors showed that an elastomer in equilibrium with an ideal solution swells in direct proportion to the volume fraction of solvents in solution . A unique partition coefficient for a given elastomer-solvent pair is defined from observed swelling in pure solvents. Key to this formulation is representing the observed volume swell in terms of the volume fraction of solvents within the elastomer phase ([v.sub.m,e]). When [v.sub.m,e] is plotted against volume fraction of solvent in an ideal solution ([v.sub.i,s]), a straight line is observed.
When an elastomer is in equilibrium with a nonideal solution, activity coefficients based upon the constituent volume fractions are introduced. More generally, elastomers swell in proportion to the above partition coefficients multiplied by the activity of solvents in solution. As long as the character of the solution is adequately represented by physical forces and spatial considerations, the linear partition model applies. Thus, a more general equation for elastomer swell in mixed solvents may be written as:
[v.sub.m,e] = [summation over i] [a.sub.i,s] [v.sub.i,e.sup.o]. (1)
Here, [v.sub.i,e.sup.o] are the partition coefficients and [a.sub.i,s] are the volume fraction based activities of each solvent in solution. Activity coefficients for solvent blends may be obtained from vapor pressure observations or from computational schemes such as NRTL , UNIFAC , and others. Equation 1 simplifies all contributions to the Gibbs free energy balance in the swollen elastomer due to entropy, enthalpy and elastic retraction energies, with a partition coefficient. One partition coefficient is required for each polymer-solvent pair. Knowledge of an elastomer's chemical structure is not required to evaluate Eq. 1.
As a general expression for elastomer-liquid equilibrium, Eq. 1 is analogous to the well-known expression for vapor-liquid equilibrium (VLE), below.
[P.sub.tot] = [summation over i] [a.sub.i] [P.sub.i]. (2)
[P.sub.i] is the vapor pressure of component "i" and [P.sub.tot] is the total pressure in a closed system. [P.sub.i] may be considered a partition coefficient which describes the tendency of a solvent molecule to leave the liquid phase and enter the vapor phase. In VLE, the value of [P.sub.i] is unique for any given temperature whenever the vapor phase behaves as an ideal gas and molecular interaction is reasonably neglected. In elastomer-liquid equilibrium [v.sub.i,e.sup.o] is a partition coefficient which describes the tendency of a solvent molecule to leave the liquid phase and enter an elastomer. In contrast, an elastomer behaves as a phase separated liquid wherein polymer-solvent interactions cannot be neglected. An elastomer's inherent affinity for a given chemical species affects the chemical partitioning behavior. Thus, at any given temperature, numerous values of [v.sub.i,e.sup.o] will be observed, one for each elastomer-solvent pair.
For example, consider a moderately nonideal solution of reformulated gasoline containing methyl tertiary-butyl ether (MTBE). As Lewis base, MTBE may interact with acidic hydrogen atoms of other constituents. Since no other constituent in gasoline contains acidic hydrogen atoms, the solution is not associative. However, MTBE does interact with polar moieties in poly(vinylidenefluoride cohexafluoropropylene) (FKM-66) , a common elastomer used in gasoline service. Because of the sequential vinylidenefluoride -(C[H.sub.2]-C[F.sub.2])[.sub.n]- groups in FKM-66 and electro-negativity of the fluorine atom, this elastomer contains many acidic hydrogen atoms. As such, FKM-66 selectively and preferentially absorbs MTBE from solution. Absorption is driven by the activity of MTBE in gasoline and by the exothermic interaction between the ether linkage and the acidic hydrogen atoms of the elastomer. Equation 1 quantitatively predicts swelling of FKM-66 over the entire composition range of MTBE in gasoline .
Now consider highly nonideal solutions exhibiting chemical association. The elastomer-liquid equilibrium problem is now complicated by an increased number of distinct chemical species available to partition into the elastomer phase. This mechanism of excess swelling by associative species was proposed by Myers and Abu-Isa  to explain the large volumetric swell (100%) observed for FKM-66 elastomer in equilibrium with pure MeOH. Recognizing that pure MeOH may exist as a self-associated hydrogen-bonded tetramer, Abu-Isa postulated that the tetramer presents a more aggressive swelling agent to FKM-66 than other forms of MeOH. To demonstrate this effect, water and other associative diluents were added to MeOH and the swelling power of the solution decreases dramatically because, it was argued, MeOH tetramers dissociate into linear or branched chains in the presence of water. As a further demonstration, the swelling power of pure MeOH was shown to be inversely related to temperature. Since the heat of formation of the associated tetramer is exothermic, increasing thermal energy decreases the mole fraction of associated species in solution and therefore decreases their resulting swelling power.
Accounting for positive and, more significantly, negative excess swelling of elastomers in equilibrium with associative solvent mixtures is the subject of the present study. This accounting is accomplished by identifying the relevant chemical species, their activity in solution and their relative partitioning power in elastomers. When strong molecular associations exist among solvent molecules in solution, either positive or negative excess elastomer swelling behavior may be observed. Negative excess swelling occurs when an associated chemical species is less aggressive toward an elastomer than the parent monomer species. Positive excess swelling occurs when the opposite is true. A detailed examination of this method is presented below using examples of elastomer swell in equilibrium with associative solvent mixtures. First, an accounting method for associative chemical species in solution is reviewed.
Theoretical treatments of activity coefficients in nonideal solutions have been discussed for a century. An excellent review of solution theories of fugacity can be found in the text by Prausnitz et al. . Many theories attempt to describe solution thermodynamics in terms of molecular size and also physical intermolecular forces, primarily London dispersion forces, which operate among them. However, apparent activity coefficients for monomer species in strongly associative solvent blends are typically much greater or lesser than unity and are difficult to analyze mathematically.
When associative forces are sufficiently strong, Chemical Theory better characterizes the solution thermodynamics. Chemical Theory postulates the existence of distinct self-associated or cross-associated species which are assumed to be in chemical equilibrium with the monomer species. The theory further assumes that the blend of monomer and associated species forms an ideal solution. Chemical Theory has a distinct advantage of being able to account for both positive and negative deviations from ideal mixing and it is applicable to mixtures containing polar and hydrogen-bonded species. Chemical Theory allows one to fit experimental data for any solvent mixture, provided there are a sufficient number of adjustable parameters.
The fitting power of Chemical Theory also leads to arbitrariness and the disadvantage of having to decide which chemical species are present in solution. Independent experimental evidence of molecular association is required before Chemical Theory can be prudently applied. Fortunately, much experimental evidence exists for associations among hydrogen-bonded species. Infrared and NMR techniques are especially useful in characterizing the existence of these species. An excellent review of hydrogen-bonding phenomena is given by Joesten and Schaad . A summary of equilibrium data for many hydrogen-bonded systems is included in that reference.
Coleman, Painter and coworkers  have applied Chemical Theory to model the phase behavior of polymer blends in which polymers can cross-associate and one of the polymers can self-associate. Their expression for the excess Gibbs energy of the polymer blend resembles the traditional Flory-Huggins expression with a hydrogen-bonding term added. Physical interactions among polymer functional groups are represented with traditional [chi] parameters, while Chemical interactions are represented with equilibrium constants.
There is an important distinction between the representation of physical and chemical interactions. In its simplest form, the [chi] parameter is a composition-independent binary parameter resulting in a term for physical interactions which are symmetric in composition. In contrast, the contribution from chemical interactions is often highly asymmetric with respect to composition, reflecting significant differences in the equilibrium constants describing self-association and cross-association. The [chi] parameter could become a complex function of composition but this introduces additional empirical parameters which must be fit to mixture data. Thus, it is generally more convenient and efficient to represent mixtures where chemical interactions dominate with Chemical Theory.
For strongly self-associating solvents, such as alcohols, consider the pure solvent to be composed of at least two chemically distinct forms: the monomer, the dimer and possibly other higher oligomers. These species are in chemical equilibrium according to the formula:
nM [left arrow] [right arrow] [M.sub.n] [k.sup.1] = [k.sup.2] = ... = [k.sup.n]. (3)
Here "[k.sup.n]" is an equilibrium constant whose value depends upon the heat of formation of the associated species and the temperature. For associations more complex than dimerization, "[k.sup.n]" may be considered as a lumped constant where "n" is the order of the multiple association. It is customary to adopt the simplifying assumption that all equilibrium constants in this multiple association are equal to each other. As a result, the apparent activity coefficient of the monomer species will be greater than one when the pure solvent is diluted into another nonpolar solvent.
Now consider CH[Cl.sub.3]. As a Lewis acid, it may cross-associate with oxygenated hydrocarbons such as ethers, esters, and ketones acting as Lewis bases. In strongly cross-associated two component solutions, consider the mixed solvent to be composed of at least three chemically distinct forms: two monomers and a complex according to the formula:
A + B [left arrow] [right arrow] AB k = [a.sub.(AB)]/([a.sub.(A)]*[a.sub.(B)]). (4)
Here, A and B represent the Lewis acid and Lewis base molecules, respectively. Pure CH[Cl.sub.3] has also been shown to exist in equilibrium with its dimer [19, 20], as in Eq. 1 with n = 2. The latter reference reports an equilibrium constant of 0.16 for the dimerization of CH[Cl.sub.3] at 25[degrees]C. Thus, for solutions including CH[Cl.sub.3], relationships similar to Eqs. 3 and 4 may be considered simultaneously to provide an adequate description of the equilibrium condition. The dominant association will depend on the relative equilibrium constants and the apparent concentrations of the interacting monomer species in the solution of interest.
The objective of Chemical Theory is to identify a rational set of equilibrium associations that creates an ideal solution among all chemical species. Identifying appropriate association equilibrium is exactly analogous to identifying chemical reaction equilibrium. Schemes for accomplishing this are discussed in textbooks  and are briefly summarized as follows. First, assume a set of equilibrium associations that account for all associated and monomer species in solution. Next, write expressions for the mole fractions of each component in solution. Fit these functions with thermodynamic data and impose the requirement of ideal behavior among all species in solution. In other words, the mole fractions of all chemical species, or their activities, must be between zero and one and the sum of all activities must also be equal to one.
0 [less than or equal to] [a.sub.i] [less than or equal to] 1 (5a)
[SIGMA][a.sub.i] = 1 (5b)
Of course, the simplest association model simultaneously satisfying Eqs. 5a and 5b and also fitting the activity data is desired.
For the purpose of modeling elastomer swell in mixed associative solvents, it is very convenient to adopt the principles of Chemical Theory. In this way, the applicability of Eq. 1 is extended by considering the partitioning power of all chemical species present in solution and a unique partition coefficient is assigned to each one. This fit procedure requires one experimental observation of elastomer swell for each chemical species. In the simpler case of nonassociative solutions, one observation of elastomer swell in each pure solvent is sufficient. For the case of associating solvents, one additional observation of elastomer swell in the mixed solvent is required for each associated chemical species present.
The requirement of one partition coefficient for each chemical species does not grant as much fitting power as might be expected. Constraints arise because the function describing the activity of each component in solution is not arbitrary; rather, it is restricted by the equilibrium model chosen for the solution. Even the available equilibrium models are constrained by the requirements imposed by Eqs. 5a and 5b. Therefore, it is not generally true that a given number of partition coefficients can always be selected to fit the same number of observations of elastomer swell.
In many cases, Eqs. 5a and 5b may be satisfied without any further simplifying assumptions. However, a simplifying assumption may be required when a solution is composed of a nonideal blend of associating and nonassociating solvents. An example is alcohol diluted into gasoline. Because of their [pi]-electrons, aromatic hydrocarbons exhibit some nonideal mixing with aliphatic hydrocarbons as well as alcohols. In this case, Chemical Theory applies better in alcohol-rich solutions while physical activity models apply better in gasoline-rich solutions. Therefore an approximate hybridized scheme may be adopted.
To demonstrate the utility of the linear partition model for associative solvent blends, several elastomer-solvent systems are examined in the following section. Some systems exhibit negative excess swelling behavior while others exhibit positive excess swelling behavior.
Solutions of CH[Cl.sub.3] and oxygenated hydrocarbons are convenient systems to demonstrate the competing effects of self-association versus cross-association and the resulting unusual swelling behavior of elastomers exposed to these solutions. Alcohols diluted into hydrocarbons are convenient systems to demonstrate the effects of multiple self-associations. Both literature and original data will be used to demonstrate the modeling technique for natural rubber (NR), nitrile rubber (NBR), and two fluorinated elastomers (FKM-66 and FKM-70).
Consider the solvent absorption behavior of NR and NBR in equilibrium with CH[Cl.sub.3] and oxygenated solvents such as: diethylether (DEE), n-butyl acetate (NBA) and dimethyl ketone (DMK). These oxygenated hydrocarbons at as Lewis bases while CH[Cl.sub.3] exhibits amphoteric proton donor and proton acceptor characteristics.
NR, composed of cis-1,4-polyisoprene, is a nonpolar elastomer which swells considerably in both DEE and NBA. NBR, a more polar elastomer than NR, is a copolymer of acrylonitrile and butadiene that also swells considerably in these solvents. Using an SP approach, it seems anomalous to observe that NR swells to a greater extent in CH[Cl.sub.3] than does NBR despite the fact that CH[Cl.sub.3] is a strong polar solvent. However, as shown below, it is the less polar self-associated dimer of CH[Cl.sub.3] that is the more aggressive swelling agent for NR compared to monomer form.
Consider a two-component solution containing self-associating and cross-associating chemical components in equilibrium with an elastomer. Bristow and Watson  investigated the swelling behavior of NR in solutions of CH[Cl.sub.3] and DEE and observed positive excess swelling of the elastomer. Table 1 summarizes data from their study. An equilibrium condition consisting of four distinct chemical species is proposed, according to the following two equilibrium relationships.
2(CH[Cl.sub.3]) [left arrow][right arrow] (CH[Cl.sub.3])[.sub.2] [k.sub.1] = 0.5 (6)
CH[Cl.sub.3] + DEE [left arrow][right arrow] CH[Cl.sub.3]*DEE [k.sub.2] = 1.1. (7)
Apparent activity coefficients of the monomer species in solution as a function of concentration are computed from van Laar coefficients of [A.sub.12] = -0.3494 and [A.sub.21] = -0.5647 . Using these data and the above association model, activities for all associated and nonassociated chemical species are computed as a function of apparent concentration of the monomer species. These activities are summarized in Table 1 and plotted in Fig. 1. The equilibrium constants [k.sub.1] and [k.sub.2] are computed where [k.sub.1] = 0.5, for the dimerization of CH[Cl.sub.3], compares favorably with a value previously reported for this system . Notice that all component activities fall between zero and one inclusive and the sum of all activities in solution is equal to one. Note also that no adjustable parameters are required to fit all the activity data to this solution model. In essence, two van Laar coefficients were used to generate two equilibrium constants. Therefore a rational chemical model for this solution has been proposed.
The four partition coefficients are determined from four experimental observations of elastomer swell. Absorption data in both pure solvents and two more in solvent blends at ~25% by volume of each component in the other were selected. Now the four partition coefficients may be computed by elimination. The partition coefficients are also summarized in Table I and they are considered unique values for these solvent species in this elastomer at ambient temperature.
Using the activity data and the partition coefficients as per Eq. 1, the predicted swelling behavior of NR in solutions of DEE and CH[Cl.sub.3] is shown in Fig. 2 along with the data of Bristow and Watson. The predicted absorption curve is smoothly varying and the resulting fit successfully predicts positive excess swelling of NR over all solution compositions.
Chloroform-n-Butyl Acetate Solutions
Now consider a more complex example of chemical cross-association. NR exhibits negative excess swelling in solutions of CH[Cl.sub.3] and NBA. Gee  investigated this system and Table 2 summarizes data from his study. An equilibrium condition consisting of five distinct chemical species is proposed, according to the following three equilibrium relationships.
2(CH[Cl.sub.3]) [left arrow][right arrow] (CH[Cl.sub.3])[.sub.2] [k.sub.1] = 0.5 (assumed) (6)
CH[Cl.sub.3] + NBA [left arrow][right arrow] CH[Cl.sub.3]*NBA [k.sub.3] = 1.1 (8)
CH[Cl.sub.3] + CH[Cl.sub.3]*NBA [left arrow][right arrow] (CH[Cl.sub.3])[.sub.2]*NBA [k.sub.4] = 1.5. (9)
[FIGURE 1 OMITTED]
Three equilibrium associations are proposed for this system because the ester moiety (R-O-C=O-R'0) of NBA is dibasic with respect to CH[Cl.sub.3]. NBA accepts two CH[Cl.sub.3] molecules, each acting as a Lewis acid. Since only two van Laar coefficients are available, only two of the three equilibrium reactions can be fit independently. Thus we assume the dimerization of CH[Cl.sub.3] has the same equilibrium constant as was observed in the prior example, namely [k.sub.1] = 0.5.
Apparent activity coefficients for the monomer species, as a function of their concentration in solution, are computed from the van Laar coefficients: [A.sub.12] = -0.4229 and [A.sub.21] = -1.0839 . Using these data and the above association model, activities of all chemical species in equilibrium are computed, summarized in Table 2 and plotted in Fig. 3 as a function of the apparent concentration of CH[Cl.sub.3]. Values for [k.sub.3] and [k.sub.4] are computed as 1.1 and 1.5, respectively. Perhaps it is significant that the first molecule of CH[Cl.sub.3] attaches to NBA with the same equilibrium constant as was observed for the CH[Cl.sub.3] association with DEE. Notice that Eqs. 5a and 5b are again satisfied and a rational equilibrium model has been proposed.
Five experimental observations are required to compute the five partition coefficients. In addition to the two observations of elastomer swell in the pure solvents, three additional observations in the mixed solvents are required. The computed partition coefficients are also summarized in Table 2.
[FIGURE 2 OMITTED]
Using the activity data and the partition coefficients as per Eq. 1, the predicted swelling curve for NR in solutions of NBA and CH[Cl.sub.3] is shown in Fig. 4 along with the data of Gee. The predicted absorption curve is smoothly varying and the fit successfully predicts negative excess swelling of NR over all solution compositions. Note that negative excess swelling is more pronounced in dilute CH[Cl.sub.3].
[FIGURE 3 OMITTED]
The swelling behavior of NBR in equilibrium with solutions of CH[Cl.sub.3] and DMK is of interest because a slight negative excess swell is observed for DMK-rich solutions while a slight positive excess swell is observed for CH[Cl.sub.3]-rich solutions. It was previously proposed that an association of the CH[Cl.sub.3] and DMK species presents a tertiary swelling agent to the elastomer .
In this study, the same activity data will be treated using Chemical Theory. NBR swelling data in this solvent system  are summarized in Table 3. An equilibrium condition consisting of four distinct chemical species is proposed for this system as follows.
2(CH[Cl.sub.3]) [left arrow][right arrow] (CH[Cl.sub.3])[.sub.2] [k.sub.1] = 0.5 (assumed) (6)
CH[Cl.sub.3] + DMK [left arrow][right arrow] CH[Cl.sub.3]*DMK [k.sub.5] = 1.75. (10)
[FIGURE 4 OMITTED]
Again, assume the dimerization of CH[Cl.sub.3] has the same equilibrium constant as observed in the first example, namely [k.sub.1] = 0.5. Apparent activity coefficients for the pure components were computed using NRTL . Using this data and the above equilibrium model, activities for all four chemical species are computed, summarized in Table 3 and plotted in Fig. 5 as a function of apparent concentration of CH[Cl.sub.3]. No adjustable fit parameters are required to model chemical associations in this solution. A value for [k.sub.5] is found to be 1.75. Notice that Eqs. 5a and 5b are satisfied and a rational equilibrium scheme for this solution has been proposed.
Using the activity data and the partition coefficients from Table 3 in Eq. 1, the predicted swell of NBR in solutions of DMK and CH[Cl.sub.3] is shown in Fig. 6 along with the data of Westbrook and French . The predicted absorption curve is smoothly varying and the fit successfully predicts a negative excess swelling response of NBR in dilute CH[Cl.sub.3] and a positive excess swell in dilute DMK compositions.
Alcohol solutions are intrinsically more difficult to model than CH[Cl.sub.3] solutions. Lower alcohols tend to form complex associations that can be affected by trace amounts of water or other polar species. A summary of efforts to characterize the structure of alcohol solutions has been given by Wolff . Many authors have studied these solutions and there is no general agreement regarding their specific nature. Cyclic, linear or branched structures, dimers, trimers, tetramers, and higher polymers have all been reported. As such, apparent activity coefficients for lower alcohols diluted into hydrocarbon solutions are often very much greater than one.
As alcohols are diluted into hydrocarbon blends containing aromatic hydrocarbons, the equilibrium condition becomes more complex. There is a considerable body of spectroscopic evidence that the hydroxyl moiety of the alcohol molecule interacts with the conjugated [pi]-electrons of an aromatic ring . An equilibrium constant for the specific MeOH-toluene ([C.sub.7][H.sub.8]) interaction has been estimated to be 0.44 at 28[degrees]C .
Complicating the equilibrium condition further is the fact that aromatic and aliphatic hydrocarbons form slightly nonideal solutions themselves. In the absence of alcohols, physical interaction theories for nonideal behavior adequately model the activity coefficients. In the presence of alcohols, Chemical Theory is perhaps better suited to model the multiple associations which occur as physical interactions become insignificant. No single thermodynamic theory is quite adequate for alcohol blends with gasoline over the entire composition range.
Therefore a hybridized scheme is required to adequately model the swelling power of alcohol solutions with aromatic hydrocarbons over the entire composition range. The present authors accomplish this by first assuming the nonpolar hydrocarbon components behave ideally such that the principles of Chemical Theory may be applied. Then the minor excess activity contribution from the nonpolar hydrocarbons is added back to the total activity of the solution.
Given these complications, the reader may note that simple association models chosen to fit activity data in alcohol solutions are not as quantitative as those chosen for the CH[Cl.sub.3] solutions. They are, however, adequate to qualitatively show the effect of the associated species on the swelling response of elastomers. Elastomers in equilibrium with alcohol solutions tend to exhibit substantial positive excess swelling.
Ethanol-Toluene-Isooctane Ternary Solutions
In a previous article, absorption of ethanol (EtOH)-gasoline blends by NBR were fit to a linear partition swelling model considering only the activity of the monomer species . An adequate correlation of the data was observed; however, some additional positive excess swelling was evident, especially in the lower EtOH blends. In this study, the same data are also treated using Chemical Theory to describe the activity of associated species.
For modeling purposes, gasoline is considered to be an equal volume percent blend of toluene ([C.sub.7][H.sub.8]) and isooctane (i[C.sub.8][H.sub.18]). This blend is known as ASTM Reference Fuel C . Table 4 summarizes data for NBR swell as a function of EtOH concentration in gasoline, according to Abu-Isa . NBR exhibits a broad swelling maximum with EtOH concentration. The following equilibrium associations are proposed to account for the associative behavior of this solvent system.
n(EtOH) [left arrow][right arrow] (EtOH)[.sub.n] [k.sub.6.sup.n] = large (11)
EtOH + [C.sub.7][H.sub.8] [left arrow][right arrow] EtOH*[C.sub.7][H.sub.8] [k.sub.7] [approximately equal to] 1. (12)
Here, i[C.sub.8][H.sub.18] is considered a noninteractive diluent and "n" is an adjustable fit parameter. The value of "n" is relatively unimportant because activity coefficients are somewhat insensitive to "n" as it becomes large. Apparent activity coefficients for EtOH in Reference Fuel C, as computed by NRTL , are much greater than one. Using these data and the above association model, activities for all chemical species are estimated in Table 4 and plotted in Fig. 7. Partition coefficients are computed for each chemical species and they are also summarized in Table 4 for the associated and nonassociated partition models.
Predicted swell, using different models for associated and also nonassociated species is summarized in Table 4 and plotted in Fig. 8 for comparison to the data. Corresponding primarily to the activity of EtOH, NBR exhibits positive excess swell in these solutions. In this case a nonassociated partition coefficient model, utilizing component activities based only on the physical interaction of monomer species, is capable of describing most of the swelling power EtOH-gasoline blends in NBR. Neither of the associated species are strong swelling agents for NBR. Therefore, incorporation of Chemical Theory to account for all species present in these solutions contributes only a secondary correction to the predicted behavior. As will be shown in the following section, a minor contribution by associated species is not always observed for elastomers in equilibrium alcohol solutions.
Methanol-Toluene-Isooctane Ternary Solutions
Attention is now turned to MeOH diluted into Fuel C. Here interest is focused on FKM-66 copolymer and also FKM-70 terpolymer. FKM-66 contains about a 3.5 to one mole ratio of vinylidenefluoride to hexafluoropropylene in the backbone. FKM-70 achieves higher fluorine content by replacing about one mole of the vinylidenefluoride monomer with tetrafluoroethylene monomer. As such, FKM-70 has many fewer acidic hydrogen atoms and is a much less polar elastomer. Accordingly, FKM-66 is known to swell almost 100% by volume in pure, dry MeOH whereas FKM-70 swells an order of magnitude less.
[FIGURE 5 OMITTED]
Table 5 summarizes observed swelling data of FKM-66 and FKM-70 as a function of MeOH concentration in Reference Fuel C. FKM-66 exhibits a complex swelling response generally increasing with MeOH content. However, FKM-70 exhibits a broad swelling maximum similar to the behavior of NBR in solutions of EtOH and Reference Fuel C. The following solvent interactions are proposed to account for the nonideal behavior of this system.
n(MeOH) [left arrow][right arrow] (MeOH)[.sub.n] [k.sub.8.sup.n] = large (13)
MeOH + [C.sub.7][H.sub.8] [left arrow][right arrow] MeOH*[C.sub.7][H.sub.8] [k.sub.9] [approximately equal to] 1. (14)
As before, i[C.sub.8][H.sub.18] is a noninteractive diluent in these solutions. Apparent activity coefficients for MeOH in Reference Fuel C, as computed by NRTL , are much greater than one. Using these data and the equilibrium model, activities for all chemical species are estimated, summarized in Table 5 and plotted in Fig. 9. Partition coefficients are assigned for each component in solution and for each elastomer. Absorption data are summarized in Table 5 and plotted in Fig. 10 for both elastomers.
The behavior of FKM-66 is unusual because it tends to absorb a significant quantity of both the associated MeOH oligomer and the MeOH*[C.sub.7][H.sub.8] species. In contrast, FKM-70 elastomer shows no affinity for the MeOH oligomer. Because of its very low activity, the MeOH monomer is not a significant swelling agent for either elastomer.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
To further investigate the behavior of MeOH toward FKM-66, absorption measurements were conducted with pure MeOH vapor at ambient temperature with a fugacity of 0.75 . A superheated vapor was chosen to avoid the possibility of condensing liquid onto the sample and also solvent-solvent association may be neglected in an ideal gas. The total equilibrium weight gain in the vapor phase was ~1%. Compared to 100% absorption of pure methanol liquid, reported by Myers and Abu-Isa , FKM-66 is almost inert to methanol vapor.
Direct evidence for the role of associated chemical species in the swelling behavior of elastomers is obtained by comparing the absorption of pure methanol liquid and vapor in FKM-66 elastomer. A single component vapor at equilibrium with a single component liquid is equally effective at swelling a given elastomer because their respective activities are equivalent. However, MeOH vapor at ambient temperature and pressure is clearly not as aggressive toward FKM-66 as the MeOH liquid. This apparent anomaly may be resolved only if a self-associated chemical species is present in the liquid phase and it is the dominant swelling agent for the elastomer. Absent an associated form, MeOH vapor is not aggressive toward FKM-66.
[FIGURE 8 OMITTED]
The tendency of any chemical species to swell a given elastomer is characterized by its activity in solution and its partition coefficient in the elastomer. A linear partition coefficient model is a useful simplification of the Flory-Rehner equation because, for aggressive solvents, partition coefficients adequately account for the entropic, enthalpic, and elastic retraction energy balance in the elastomer. A single partition coefficient may be assigned for each chemical species in solution. Once determined, partition coefficients are expected to be readily applicable in other solvent blends provided the same elastomer compound is being considered at the same temperature. A partition coefficient model can be applied generally to ideal solutions and also to various forms of nonideal solutions exhibiting both positive and negative deviations.
[FIGURE 9 OMITTED]
If nonideal solution behavior originates only from spatial and/or physical considerations among the solvents, nonideal solution activity coefficient models will adequately account for excess swelling observed in elastomers. If nonideal solution behavior originates from the presence of associated chemical species, it is convenient to assume that the solvent blend is an ideal mixture of monomer and associated forms. It is observed that associated chemical species exhibit unique partitioning behavior in elastomers creating either positive or negative excess swelling. If nonideal solution behavior originates from both physical and chemical effects, one must apply the thermodynamic theory which best describes the nonideal behavior in the solution composition of interest.
Recall that three to five partition coefficients are required to fit the complex absorption behavior observed in the examples. This lower number of fit parameters compares favorably with the nine constants required by three-dimensional SP analysis for an elastomer in equilibrium with a binary solvent blend. However, even when using twice the number of fitting parameters, SP analysis is incapable of describing swelling behavior over the entire solvent composition range and, more importantly, any negative excess swelling behavior of the type observed in Figs. 5 and 6.
Another advantage of the current approach is that prior knowledge of the chemical structure of the elastomer is not required to model its swelling behavior in a mixed solvent. The resultant chemical energy of the elastomer governs its interaction with solvents and as various chemical species interact with an elastomer, they reveal vital information in the form of partition coefficients.
However, if the structure of an elastomer is known, one can rationalize its swelling behavior in equilibrium with solutions of interest. For example, it is interesting to compare the results of Gee to those of Bristow and Watson on the swelling behavior of NR in CH[Cl.sub.3] solutions . First, it is the CH[Cl.sub.3] dimer, not the monomer, aggressively swells NR. By molecular association, the hydrogen-bonded dimer is considerably less polar and the resulting swelling power in nonpolar NR is correspondingly high. Second, association of dibasic NBA with a single CH[Cl.sub.3] molecule creates a chemical species with a single remaining basic site. As such, it is chemically similar to the DEE monomer. Both these somewhat polar species exhibit similar, relatively low swelling power in NR. However, the associated DEE species is similar to the secondary-associated NBA species because all basic sites are occupied. As a result, both of these nonpolar associated species are powerful swelling agents for NR.
Further examination of Gee's data reveals more insight. Recall that substantial negative excess swell is observed in all solution compositions but it is more pronounced in dilute CH[Cl.sub.3] solutions. Gee wrote, "The swelling power of NBA is slightly reduced by adding a small proportion of the much more powerful swelling agent, CH[Cl.sub.3]." Although Gee provides no explanation for this interesting behavior, the present authors suggest that the dibasic character of the ester moiety in NBA is very effective at removing the more aggressive dimer form of CH[Cl.sub.3] from solution by their chemical association. Thus, the resulting cross-associated chemical species is less aggressive toward NR than either of the neat solvents.
Notice that the partition coefficient for CH[Cl.sub.3] monomer is much greater in NBR relative to NR. Accordingly, the partition coefficient for CH[Cl.sub.3] dimer is much less in NBR. These observations are consistent because NBR is a more polar elastomer than NR. In fact, the [pi]-electrons in the nitrile moiety probably function as weak Lewis bases [35, 36] which are therefore attractive to the CH[Cl.sub.3] monomer acting as a Lewis acid. NBR should naturally exhibit more affinity than NR for the polar CH[Cl.sub.3] monomer and less affinity for the nonpolar CH[Cl.sub.3] dimer. Despite differences in polarity, both elastomers absorb a significant quantity of pure CH[Cl.sub.3], because CH[Cl.sub.3] itself exists as two distinct different chemical species at ambient temperature.
The applicability of a linear partition swelling model has been extended for the case of elastomers in equilibrium with mixed associative solvents. This generalization is accomplished by considering the associative solvent blend to behave as an ideal solution containing both monomer and associated chemical species in equilibrium. Individual elastomer-solvent partition coefficients are then assigned for each chemical species. The required number of partition coefficients compares favorably to the larger number of fitting parameters required for three-dimensional SP analysis. Once known, these partition coefficients and equilibrium constants are expected to be valid for other solutions provided the same elastomer compound is considered at the same temperature.
[FIGURE 10 OMITTED]
The current swelling model requires a rational accounting of the mole fraction of all chemical species present in solution to compute the activity of each species. This accounting can be accomplished through the use of Chemical Theory of nonideal solutions, often without adjustable parameters.
The methodology proposed herein accounts for both positive and, more significantly, negative excess swelling behavior of elastomers in equilibrium with certain solutions over the entire composition range. The method also accounts for the fact that certain self-associating solvents, such as CH[Cl.sub.3], are aggressive swelling agents for both polar and nonpolar elastomers alike. Prior attempts to model elastomer swelling behavior in strongly nonideal solutions fail to adequately describe these complex behaviors.
The authors express their appreciation to Jerry B. Wilt and George Koplos for their capable assistance in the laboratory and with computations and also to Shell Oil Company for permission to publish this work.
APPENDIX A: AN HISTORICAL PERSPECTIVE ON THEORETICAL APPROACHES TO MODEL ELASTOMER SWELLING
By considering the statistical thermodynamics of macro-molecules in solution, Flory  showed that swelling of a cross-linked polymer network by a pure solute may be computed from the dilution entropy, the heat of dilution and the elastic retraction energy. Use of volume-fraction based activity coefficients are required to properly account for the dilution entropy. It is recognized that whenever a swollen network and a solvent are in phase equilibrium, the activity of a solute molecule in the network phase may be equated to the activity of a similar molecule in the solvent phase.
The regular solution theory of Hildebrand estimates the excess chemical potential of two miscible solutes in solution from their heat of dilution effect alone. The entropy of mixing is ignored because the solutes are assumed to have the same size and shape (regular solution assumption). Cohesive energy density of a solute was defined as its vaporization energy divided by its molar volume and the square root of this property was called "Solubility Parameter" (SP), or [delta]. Since the mixing enthalpy is estimated to be proportional to the square of the difference between [[delta].sub.i] of the solutes, SP was said to be very important in determining the "mixing range" of a component.
By considering a cross-linked polymer network to be a fluid, Gee was the first to apply the new SP concept to correlate the swelling of natural rubber by pure hydrocarbon solvents  and solvent mixtures . However, when he applied the method to synthetic copolymers and/or mixed polar solvents, quantitative agreement was not obtained.
Scott  stated that regular solutions generally exhibit "homogeneous liquid" properties whereby the SP of the mixture may be estimated by the volume-fraction average of the SP's of the components.
[[delta].sub.m] = [v.sub.1][[delta].sub.1] + [v.sub.2][[delta].sub.2]
Within the context of Flory-Huggins theory, Scott  further addressed ternary mixtures comprising solvent 1, solvent 2, and polymer 3 where the interactions among the solvents and polymer become important. He treated the binary solvent mixture as a single liquid, essentially a "pseudo-solvent." An interaction parameter for the solution was defined as the sum of the 1,3 and 2,3 interaction parameters weighted by their volume-fraction in solution. A 1,2 interaction term was added which allows the pseudo-solvent interaction parameter to also depend upon interactions between the solvents.
[[chi].sub.m,3] = [v.sub.1][[chi].sub.1,3] + [v.sub.2][[chi].sub.2,3] - [v.sub.1][v.sub.2][[chi].sub.12]
Systematic estimation of SP for a cross-linked polymer network was first addressed by Small . He proposed that the molar attraction constants should be additive. Then he derived values for molar attraction constants for structural groups from vapor pressure and heat of vaporization data and from molecules of known SP. Thus the SP of any solvent or polymer can be estimated by functional group contribution analysis. Hence, the concept that molecules with like structure tend to be good solvents for each other was born. Much later, Fedors  expanded and improved upon the group contribution method of Small for estimating both SP and molar volume of liquids.
During the 1960's, Beerbower and coworkers published a series of papers on elastomer swelling. The first paper  was a comprehensive review on the use of SP for predicting elastomer-fluid compatibility and later , the SP unit was named "Hildebrand." SP values of many polymers and solvents were tabulated. Although a general correlation between SP differential and swelling power emerged, several problems with this method became clear. First, some elastomers, such as poly(butadiene-acrylonitrile) copolymer (BuNa-N) seemed to exhibit two solubility parameter values. The authors concluded that BuNa-N forms a block copolymer but presented only anecdotal evidence to substantiate this claim. Second, the extent of volume swell could not be reliably computed. For every solvent that swells an elastomer to a great extent there are others in the same SP range that do not.
From a study of paint resin solubility in mixed solvents, Burrell  already had concluded that each polymer should be listed with three SP values. One value is for strong, moderate, and non hydrogen-bonding solvents, each. Ten years later, Hansen [47, 48] quantified this concept by expressing cohesive energy as the sum of contributions from hydrogen-bonding, permanent dipole-dipole interaction and nonpolar dispersion forces. Dividing by the molar volume of the solution, a three-dimensional SP was subsequently defined as:
[[delta].sup.2] = [[delta].sub.d.sup.2] + [[delta].sub.p.sup.2] + [[delta].sub.h.sup.2]
The three-dimensional SP was conveniently incorporated into the single-liquid approximation and also the group contribution method for estimation of SP was expanded to include the three-dimensional analysis technique.
Using the new three-dimensional SP approach, Beer-bower  systematized selection of elastomers for use with various fluids. However, despite tripling the amount if information required for this correlation, anomalous swelling behavior was still observed.
By the 1970's, it was evident that the paint industry would enjoy wider success utilizing solubility parameters to predict the miscibility of resins in complex solutions than the rubber industry would to predict elastomer swelling in solutions. This success may be a result of two factors: uncross-linked solvent-borne paint resins exist in a single phase where they exhibit no network retraction energy and also paint resins typically utilize theta solvents whereby a low dilution entropy is manifest. Although Hansen and Beerbower  claimed a 97% correlation for elastomer swelling behavior in pure solvents, questions remained regarding the ability of SP based methods to account for electron donor-acceptor interactions.
Despite these questions, SP's have become widely used in industry to characterize polymer-solvent interactions for estimating solubility and compatibility, viscosity, diffusion, surface energy, etc . Continuing utilization of the SP concept to predict elastomer swelling is probably related to the large amount of historical data available on SP of various liquids and polymers and, as shown below, their apparent utility when applied to mixed solvents with limited composition range. More recent examples of SP application for estimating elastomer compatibility come from the early 1980s when oxygenated hydrocarbons were being considered as alternative octane enhancing additives for gasoline.
Nersasian [51-53] published swelling contour plots for predicting volume increase of fuel handling elastomers exposed to oxygenated fuel blends. These plots were based on the three-dimensional SP method of Hansen and Beerbower. Interestingly, although excellent correlation for elastomer swelling in ternary fuel blends was obtained, the contour plots could not be used to predict elastomer swelling by the pure components. He attributed this behavior to strong binary interactions among the alcohol and gasoline constituents.
Petrovic et al.  studied the swelling of polyurethane copolymer networks in toluene, methanol, and their mixtures. An effective interaction parameter was estimated from the elastomer swelling data in pure solvents. Similar to the work of Nersasian, a swelling maximum was observed for mixtures containing 15-30% methanol. This swelling maximum was qualitatively correlated using a volume-fraction averaged solubility parameter approach.
Indeed, there continue to be many contributions to the SP literature since the definitive work of Hansen and Beerbower. Barton's  handbook summarizes the state-of-the-art on SP up to the early 1980's. Shortly after its publication, alternatives to the use of SP for predicting network swelling were already being considered.
For example, Holten-Anderson and Eng  promoted the use of weight-fraction based infinite dilution activity coefficients to estimate weight gain of elastomers exposed to solvents. These activity coefficients may be obtained from experimental observation; however, absorption data of very low activity solutes into elastomers are not as widely observed as absorption data at unit activity. Therefore group contribution methods were recommended for computing these parameters.
Jensen  expanded upon the electron donor-acceptor concept where, in addition to the dispersion forces, the Lewis acid-base sensitivity of the elastomer and fluids are also considered. For a binary polymer-solvent system, this method requires as many constants (six) to be evaluated as the conventional three-dimensional SP approach. Hertz  reviewed the work of Jensen and others and supported the view that the acid-base sensitivity hypothesis can explain the maximum swell observed in alcohol blended fuels as well as other apparently anomalous behavior.
Starmer  reviewed alternative parameters that describe certain aspects of polar solvency such as: dielectric constant, aniline point, miscibility number, molar transition energy, and polarity index. A "Polarity Index" parameter (useful in liquid-liquid chromotography) was found to describe the influence of dispersive and polar forces in elastomers more accurately than does solubility parameter. In a companion paper , he proposed an emperical three parameter fit to correlate the swelling of nitrile vulcanizates as a function of acrylonitrile content. For example, swelling of nitrile rubbers in isooctane, toluene, and their blends with alcohols were adequately described over the entire composition range. An additional parameter is still required to adequately describe acid-base interactions.
Theoretical Limitations of SP Methods to Correlate Solute Absorption. Although the SP approach for estimation of solvent compatibility of elastomers is widely applied in industry, it embodies several assumptions that impose certain theoretical limitations to the method.
All SP models approximate solvent activity in the elastomer from the heat of mixing effect alone. They ignore contributions to the free energy of mixing from dilution entropy and from the elastic retraction force of the network. However, whenever the polymer-solvent interaction parameter is close to zero (a good solvent) these latter effects may constitute the only significant contributions to the free energy of mixing. Therefore any successful quantitative description of elastomer swelling must account for these contributions to the free energy.
Another well-known limitation of SP models is their inherent inability to represent negative excess heat of mixing. Fowkes  Piccarolo and Totomanilo , Jensen, and Hertz have all discussed this subject extensively. The SP concept is based upon the assumption that compounds that have similar intermolecular forces will be miscible. This force-matching concept works well for dispersion and dipole forces. However, the ability of two compounds to form acid-base interactions is derived from dissimilar but complementary chemical functionality. Ignoring exothermic interactions among solute mixtures implies that a material cannot absorb a mixture of solutes to a lesser extent than it absorbs any of the pure solvents. This negative deviation from ideal swelling behavior for polymers in mixed solvents has been reported for several systems. For example, poly(butadiene co-styrene) (Buna-S) in chloroform/n-butyl acetate solutions, natural rubber in chloroform/diethyl ether solutions , copolymers of styrene, tetrahydrofuran and methyl-methycrylate in tetrahydrofuran/methanol solutions , poly (ethylene co-terephalate) in nitrobenzene/chloroform and nitrobenzene/dichloroethane solutions . These solutions exhibit an exothermic heat of mixing and the component activity coefficients are less than unity [24, 64]. Therefore SP methods are not expected to yield quantitative swelling predictions in these systems.
Finally, the single-liquid approximation can not account for partitioning of species between an elastomer and a mixed solvent. It has generally been assumed that the pseudo-solvent which enters the elastomer has the same composition as the bulk solvent or, alternatively, has the best SP match with the elastomer. Such reasoning discounts the fact that permeation is driven by individual fugacity gradients for each component in solution. Ironically, Beerbower was aware of this deficiency as early as 1963. Citing experimental data on absorption of solvent mixtures he writes, "It is clear that elastomers are selective. In fact the process is basically one of liquid-liquid extraction and an additive will have a definite partition coefficient between the solvent and elastomer." Although many workers have observed preferential solute absorption, there are surprisingly few attempts to model such behavior. Beerbower gives little guidance on this subject and he scarcely mentions it in later papers.
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P.A. Westbrook, (1) R.N. French (2)
(1) PPG Industries, 940 Washburn Switch Rd, Shelby, NC
(2) Equilon Enterprises LLC, Westhollow Technology Center, Houston, Texas 77251-1380
Correspondence to: P.A. Westbrook; e-mail: firstname.lastname@example.org
TABLE 1. Activity of monomer and associated species in chloroform (C) and diethylether (D) solutions. xC vC,S a(C) a(D) a(CD) a(C2) atot NR data 1.000 1.000 0.740 0.000 0.000 0.260 1.000 0.855 0.950 0.938 0.703 0.036 0.027 0.236 1.002 0.900 0.878 0.664 0.074 0.053 0.212 1.003 0.850 0.819 0.626 0.113 0.076 0.189 1.004 0.800 0.762 0.586 0.154 0.098 0.166 1.004 0.750 0.706 0.546 0.197 0.118 0.144 1.004 0.825 0.700 0.651 0.505 0.241 0.135 0.123 1.004 0.650 0.598 0.464 0.288 0.149 0.103 1.004 0.600 0.545 0.423 0.336 0.161 0.084 1.003 0.550 0.494 0.382 0.386 0.169 0.066 1.003 0.500 0.444 0.341 0.438 0.173 0.050 1.002 0.789 0.450 0.395 0.300 0.491 0.174 0.036 1.001 0.400 0.348 0.260 0.546 0.170 0.025 1.001 0.350 0.301 0.220 0.603 0.162 0.015 1.000 0.300 0.255 0.182 0.660 0.150 0.008 1.000 0.250 0.210 0.146 0.719 0.133 0.003 1.000 0.750 0.200 0.167 0.111 0.777 0.112 0.000 1.000 0.150 0.124 0.079 0.835 0.087 -0.002 1.000 0.100 0.082 0.049 0.893 0.059 -0.001 1.000 0.050 0.040 0.023 0.948 0.030 -0.001 1.000 0.000 0.000 0.000 1.000 0.000 0.000 1.000 0.708 [k.sub.2], [k.sub.1] 1.1 0.5 [v.sub.i,NR.sup.o] 0.25 0.71 1.5 2.6 xC NR fit 1.000 0.855 0.950 0.850 0.900 0.844 0.850 0.838 0.800 0.831 0.750 0.824 0.700 0.816 0.650 0.809 0.600 0.801 0.550 0.793 0.500 0.785 0.450 0.778 0.400 0.771 0.350 0.765 0.300 0.758 0.250 0.752 0.200 0.745 0.150 0.738 0.100 0.730 0.050 0.720 0.000 0.708 [k.sub.2], [k.sub.1] [v.sub.i,NR.sup.o] Equilibrium constants for the formation of each associated species are shown along with partition coefficients and computed swelling data for natural rubber. TABLE 2. Activity of monomer and associated species in chloroform (C) and n-butyl acetate (B) solutions. x(C,S) v(C,S) a(C) a(B) a(BC) a(C2) a(BC2) 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.050 0.031 0.025 0.950 0.025 -0.001 0.002 0.100 0.064 0.050 0.898 0.047 -0.001 0.005 0.150 0.097 0.077 0.846 0.068 -0.002 0.011 0.200 0.133 0.104 0.793 0.085 0.000 0.018 0.250 0.169 0.133 0.739 0.100 0.003 0.025 0.300 0.208 0.163 0.685 0.112 0.008 0.032 0.350 0.248 0.194 0.629 0.122 0.015 0.040 0.400 0.289 0.227 0.573 0.128 0.025 0.047 0.450 0.333 0.261 0.516 0.132 0.037 0.053 0.500 0.379 0.297 0.459 0.133 0.052 0.059 0.550 0.428 0.335 0.402 0.132 0.068 0.062 0.600 0.478 0.375 0.345 0.128 0.087 0.064 0.650 0.532 0.417 0.289 0.122 0.108 0.064 0.700 0.588 0.461 0.235 0.115 0.130 0.060 0.750 0.647 0.507 0.182 0.105 0.152 0.054 0.800 0.710 0.555 0.134 0.094 0.174 0.043 0.850 0.776 0.604 0.090 0.082 0.196 0.029 0.900 0.846 0.652 0.052 0.067 0.217 0.012 0.950 0.921 0.699 0.022 0.046 0.237 -0.003 1.000 1.000 0.740 0.000 0.000 0.260 0.000 [k.sub.3], [k.sub.1], 1.1 0.5 1.5 [k.sub.4] [v.sub.i,NR.sup.o] 0.35 0.52 0.46 1.7 0.85 x(C,S) atot NR data [v.sub.m,NR] 0.000 1.000 0.519 0.519 0.050 1.000 0.514 0.100 1.000 0.505 0.509 0.150 1.000 0.504 0.200 1.000 0.503 0.501 0.250 1.000 0.500 0.300 1.000 0.507 0.501 0.350 1.000 0.506 0.400 1.000 0.517 0.513 0.450 1.000 0.523 0.500 1.000 0.542 0.536 0.550 1.000 0.551 0.600 1.000 0.573 0.568 0.650 1.000 0.587 0.700 1.000 0.605 0.606 0.750 1.000 0.624 0.800 1.000 0.640 0.642 0.850 1.000 0.657 0.900 1.000 0.671 0.670 0.950 1.000 0.682 1.000 1.000 0.701 0.699 [k.sub.3], [k.sub.1], [k.sub.4] [v.sub.i,NR.sup.o] Equilibrium constants for the formation of each associated species are shown along with partition coefficients and computed swelling data for natural rubber. TABLE 3. Activity of monomer and associated species in chloroform (C) and dimethylketone (A) solutions. x(C,S) v(C,S) a(C) a(A) a(CA) a(C2) atot 1.000 1.000 0.734 0.000 0.000 0.266 1.000 0.975 0.954 0.694 0.029 0.003 0.274 1.000 0.950 0.910 0.657 0.056 0.008 0.278 1.000 0.900 0.827 0.655 0.058 0.070 0.217 1.000 0.800 0.680 0.572 0.129 0.131 0.168 1.000 0.700 0.553 0.484 0.213 0.178 0.125 1.000 0.600 0.443 0.394 0.310 0.209 0.087 1.000 0.500 0.347 0.311 0.417 0.219 0.053 1.000 0.400 0.261 0.230 0.533 0.210 0.028 1.000 0.300 0.185 0.157 0.655 0.177 0.011 1.000 0.200 0.117 0.093 0.776 0.130 0.002 1.000 0.100 0.056 0.041 0.893 0.067 -0.001 1.000 0.000 0.000 0.000 1.000 0.000 0.000 1.000 [k.sub.5], [k.sub.1] 1.75 0.5 [v.sub.i,NBR.sup.o] 0.602 0.439 0.449 0.689 x(C,S) NBRdata [v.sub.m,NBR] 1.000 0.625 0.625 0.975 0.620 0.621 0.950 0.613 0.616 0.900 0.603 0.601 0.800 0.576 0.576 0.700 0.553 0.551 0.600 0.528 0.527 0.500 0.506 0.505 0.400 0.487 0.485 0.300 0.474 0.469 0.200 0.460 0.456 0.100 0.445 0.446 0.000 0.439 0.439 [k.sub.5], [k.sub.1] [v.sub.i,NBR.sup.o] Equilibrium constants for the formation of each associated species are shown along with partition coefficients and the computed swelling data for NBR elastomer. TABLE 4. Activity of monomer and associated species in ethanol (E), toluene (T) and isooctane (I) solutions. x(E,S) v(E,S) a(E) a(T) a(I) a(En) 0.000 0.000 0.000 0.454 0.627 0.000 0.105 0.050 0.033 0.403 0.604 0.005 0.199 0.100 0.037 0.388 0.600 0.009 0.283 0.150 0.038 0.381 0.601 0.009 0.359 0.200 0.038 0.376 0.602 0.008 0.427 0.250 0.038 0.370 0.603 0.007 0.490 0.300 0.038 0.362 0.603 0.008 0.547 0.350 0.039 0.351 0.602 0.012 0.599 0.400 0.039 0.339 0.599 0.017 0.647 0.450 0.040 0.324 0.594 0.027 0.691 0.500 0.040 0.307 0.585 0.041 0.732 0.550 0.041 0.288 0.573 0.061 0.771 0.600 0.042 0.267 0.556 0.088 0.806 0.650 0.042 0.243 0.533 0.125 0.839 0.700 0.043 0.217 0.502 0.174 0.870 0.750 0.044 0.189 0.461 0.238 0.900 0.800 0.044 0.158 0.407 0.322 0.927 0.850 0.045 0.124 0.339 0.428 0.953 0.900 0.046 0.087 0.251 0.565 0.977 0.950 0.047 0.045 0.140 0.736 0.000 1.000 0.048 0.000 0.000 0.953 [k.sub.6], [k.sub.7] large [v.sub.i,NBR.sup.o] 2.28 0.63 0.00 0.00 x(E,S) a(ET) atot NBR data 0.000 0.000 1.081 0.265 0.105 0.031 1.076 0.329 0.199 0.041 1.075 0.351 0.283 0.046 1.074 0.355 0.359 0.050 1.074 0.427 0.055 1.073 0.342 0.490 0.061 1.073 0.547 0.068 1.072 0.599 0.076 1.071 0.647 0.085 1.069 0.691 0.094 1.067 0.338 0.732 0.102 1.065 0.771 0.110 1.062 0.806 0.115 1.059 0.839 0.118 1.055 0.870 0.117 1.049 0.265 0.900 0.111 1.043 0.927 0.099 1.035 0.953 0.078 1.026 0.977 0.046 1.014 0.000 0.000 1.001 0.107 [k.sub.6], [k.sub.7] [approximately equal to]1 [v.sub.i,NBR.sup.o] 0.74 x(E,S) [v.sub.m,NBR] 0.000 0.265 0.105 0.332 0.199 0.340 0.283 0.341 0.359 0.342 0.427 0.342 0.490 0.343 0.547 0.343 0.599 0.343 0.647 0.342 0.691 0.340 0.732 0.337 0.771 0.331 0.806 0.323 0.839 0.312 0.870 0.297 0.900 0.276 0.927 0.248 0.953 0.212 0.977 0.166 0.000 0.107 [k.sub.6], [k.sub.7] [v.sub.i,NBR.sup.o] Partition coefficients for the swelling of NBR elastomer and computed swelling data are also given. TABLE 5. Activity of monomer and associated species in methanol (M), toluene (T) and isooctane (I) solutions. x(M,S) v(M,S) a(M) a(T) a(I) A(Mn) 0.000 0.000 0.000 0.668 0.479 0.000 0.144 0.050 0.008 0.572 0.461 0.062 0.262 0.100 0.008 0.542 0.463 0.066 0.361 0.150 0.008 0.525 0.468 0.060 0.444 0.200 0.009 0.511 0.473 0.051 0.516 0.250 0.009 0.498 0.477 0.040 0.578 0.300 0.009 0.484 0.481 0.028 0.632 0.350 0.009 0.471 0.485 0.014 0.681 0.400 0.009 0.458 0.489 -0.002 0.723 0.450 0.009 0.444 0.492 -0.018 0.762 0.500 0.009 0.429 0.493 -0.032 0.796 0.550 0.009 0.413 0.492 -0.044 0.827 0.600 0.009 0.393 0.488 -0.049 0.856 0.650 0.009 0.371 0.478 -0.044 0.882 0.700 0.009 0.343 0.460 -0.024 0.906 0.750 0.009 0.310 0.432 0.018 0.927 0.800 0.009 0.269 0.392 0.092 0.948 0.850 0.010 0.220 0.334 0.208 0.966 0.900 0.010 0.160 0.253 0.383 0.984 0.950 0.010 0.088 0.145 0.635 1.000 1.000 0.010 0.000 0.000 0.990 [k.sub.8], [k.sub.9] large [v.sub.i,FKM-66.sup.o] 4.4 0.18 0.00 0.46 [v.sub.i,FKM-70.sup.o] 6.3 0.09 0.00 0.00 x(M,S) a(MT) atot FKM-66 Data 0.000 0.000 1.147 0.120 0.144 0.036 1.138 0.181 0.262 0.057 1.137 0.228 0.361 0.075 1.136 0.444 0.093 1.136 0.279 0.516 0.111 1.135 0.578 0.131 1.133 0.316 0.632 0.151 1.130 0.681 0.172 1.126 0.353 0.723 0.194 1.121 0.762 0.216 1.115 0.398 0.796 0.237 1.108 0.827 0.258 1.098 0.429 0.856 0.275 1.088 0.882 0.287 1.075 0.475 0.906 0.291 1.061 0.927 0.283 1.046 0.482 0.948 0.258 1.030 0.966 0.210 1.016 0.488 0.984 0.128 1.005 1.000 0.000 1.000 0.496 [k.sub.8], [k.sub.9] [approximately equal to]1 [v.sub.i,FKM-66.sup.o] 1.35 [v.sub.i,FKM-70.sup.o] 0.05 x(M,S) [v.sub.m,FKM-66] FKM-70 Data [v.sub.m,FKM-70] 0.000 0.120 0.060 0.000 0.144 0.214 0.091 0.002 0.262 0.242 0.104 0.005 0.361 0.261 0.007 0.444 0.278 0.105 0.009 0.516 0.296 0.011 0.578 0.314 0.106 0.014 0.632 0.333 0.016 0.681 0.353 0.106 0.018 0.723 0.373 0.021 0.762 0.393 0.102 0.023 0.796 0.414 0.025 0.827 0.436 0.102 0.028 0.856 0.457 0.030 0.882 0.478 0.092 0.032 0.906 0.498 0.034 0.927 0.515 0.075 0.037 0.948 0.526 0.039 0.966 0.531 0.057 0.041 0.984 0.524 0.044 1.000 0.499 0.048 0.046 [k.sub.8], [k.sub.9] [v.sub.i,FKM-66.sup.o] [v.sub.i,FKM-70.sup.o] Partition coefficients for the swelling of FKM-66 and FKM-70 elastomers and computed swelling data are also given.
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|Author:||Westbrook, P.A.; French, R.N.|
|Publication:||Polymer Engineering and Science|
|Date:||Oct 1, 2007|
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