Hydrophilic Polymers - American Chemical Society

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16 Associating Polymers Containing Fluorocarbon Hydrophobic Units Eric J. Amis, Ning Hu, Thomas A. P. Seery, Thieo E . Hogen-Esch, Mariam Yassini, and Frank Hwang

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1

2

Department of Chemistry, University of Southern California, Los Angeles, CA 90089-0482

Associating polymers with hydrophilic backbones and with perfluoroalkane comonomers as the hydrophobic units were synthesized, and their properties were investigated. The viscosity of aqueous solutions at low concentration can be increased by more than 4 orders of magnitude. This effect far exceeds that of comparable hydrocarbon hydrophobic copolymers. For random copolymers, optimum comonomer contents for viscosity enhancement reflect the balance of intramolecular and intermolecular associations. Studies of model telechelic and comb-type copolymers with variable lengths and spacing of hydrophobes demonstrate the optimization of intermolecular associations and strong (long-lifetime) associations. For the model systems, Arrhenius activation energies are 16 to 29 kcal/ mol for viscous flow and 5 kcal/mol for unentangled homopolymer chains. The results are interpreted in terms of models for transient networks with modifications to account for the postulate that neither loops nor chain extensions produce active network strands.

R)LYMERS THAT ASSOCIATE VIA PHYSICAL INTERACTIONS i n Solutions have r e c e i v e d m u c h attention as a replacement for high-molecularCurrent address: Polymers Division, National Institute of Standards and Technology, 224/B210, Gaithersburg, MD 20899 Current address: Department of Chemistry, University of Connecticut, Storrs, CT 06269 2

2

0065-2393/96/0248-0279$ 13.00/0 © 1996 American Chemical Society

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weight polymeric viscosifiers. Such associating polymers are now used in a variety of applications because of their unique rheological properties ( 1 , 2 ) . Water-soluble associating polymers are especially interesting because of their roles in food thickeners, coatings, paints, enhanced oil recovery, and water treatment (3, 4). Many of these associating polymers are amphiphilic: They contain a hydrophilic main chain with hydrophobic side chains. For example, polyacrylamide (PAM) and cellulose can be chemically modified by attachment of hydrophobic units along the chain or at the chain ends. Strong associations between these hydrophobic units (sometimes refered to as "stickers") lead to the formation of transient networks that greatly enhance solution viscosity and viscoelasticity. The properties of these polymer solutions depend not only on the structure of the polymer chain, that is, molecular weight of the main chain and structure, number density, size, and distribution of the stickers, but also on external factors such as added surfactants, cosolvents, and salts. Most theoretical efforts to understand the mechanisms of associations have focused on the rheological behavior ( 5 - 9 ) . The classic explanation for the driving force of hydrophobic associations is the entropie gain accompanying the association. The removal of a hydrophobe from aqueous solution into a micellar aggregate is accompanied by the breakdown of a structure of highly ordered water molecules around the hydrophobe, leading to a positive change in entropy. The contribution to free energy from the enthalpy by the transfer of a hydrophobe from water to a micelle is usually much smaller than the entropie term. One difference between polymeric hydrophobic associations and small-molecule micelles is that hydrophobic associations usually have aggregation numbers of less than 10 (10), and smallmolecule micelles have aggregation numbers of several tens to more than 100(11). Associating polymers containing fluorocarbon stickers form much stronger associations than the corresponding hydrocarbons, even when the polymers have much lower hydrophobe contents (12). This phenomenon is consistent with the fact that the micellization of fluorocarbon surfactants occurs at much lower concentrations than that of hydrocarbon surfactants (13). This condition is believed to reflect a more hydrophobic character of the fluorocarbon groups. The entropy increase upon micellization is higher. Furthermore, the solubility of C F is one-seventh of that of C H in water. The solubility of water is almost 25 times less in perfluoroheptane than in heptane. On the basis of critical micelle concentration data, Mathis et al. (13) suggested that fluorocarbon surfactants are about 1.5-1.8 times more effective than their hydrocarbon analogs. The dramatic effect of the greater hydrophobicity of fluorocarbons 4

4

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16.

AMIS E T AL.

Fluorocarbon

1

I—ι

0.2

Hydrophobic

ι

ι

0.4

0.6

Units

ι

0.8

281

I

1.0

log c, wt%

Figure 1. Viscosity (η) versus concentration (c)for telechelic polymers with molecular weight 35,000 and hydrophobic unit -CgFu ( · ) or —C16H33

(M).

is reflected in the viscosity data for hydrocarbon- and fluorocarboncontaining associating polymers, as shown in Figure 1 (14). The vis­ cosity of a model telechelic associating polymer of polyethylene oxide (PEO) with fluorocarbon ( - C F i ) hydrophobic end units is about 10 times higher than that of one with hydrocarbon ( - C 1 6 H 3 3 ) hydropho­ bic units. Even though the hydrocarbon is twice as long, the fluorocar­ bon is more effective. The two polymers have the same mole contents of hydrophobic units. Thus our focus is on the similarities and differences imparted to associating polymers by fluorocarbon hydrophobes. The first part of this chapter reviews experimental results obtained with fluorocarboncontaining, acrylamide-based random copolymers and hydrophobically modified cellulose derivatives. The second section focuses on characterization of polymers with regular structures, telechelic and comb types, and comparisons with theoretical models. 8

7

Random Copolymers Fluorocarbon-containing, water-soluble associating polymers were synthesized by chemical modification of a preformed polymer and by copolymerization of the appropriate monomers. Cellulose derivatives, hydroxyethylcelluloses (HEC), were chemically modified by linking H E C with fluorocarbon-containing modifiers through an ether linkage (15). The degree of substitution is controlled by reaction extent. The

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282

HYDROPHILIC POLYMERS

CeiOCH CH ONa 2

2

+ CH —CHCH OCH (CF ) CF3—2

2

2

2

6

Ο

Η* — -

CeiOCH CH OCH CHCH OCH (CF ) CF 2

2

2

2

2

2

6

3

OH

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Scheme 1 . synthetic route is shown in Scheme 1. PAM-based copolymers were synthesized by copolymerizing acrylamide monomers with fluorocarbon-containing acrylate (e.g., 3M monomer FX13) or methacrylate comonomers (12). Typical fluorocarbon comonomers are shown as structures 1 and 2. The hydrophobe content incorporated into the chain was controlled by the ratio of the comonomers. One difficulty common to both routes is that the fluorocarbon-containing monomers usually have very poor solubility in water because of their extreme hydrophobicity. The hydrophobic comonomer can, however, be dis­ persed by adding a fluorocarbon surfactant or by using a mixed-solvent system such as water-acetone. A major difficulty associated with these materials is ascertaining the degree of hydrophobic comonomer incorporated into the polymer after synthesis. Because the content of the hydrophobic comonomer is exceedingly low, no method has been completely successful in de­ termining the hydrophobe content in the polymer or the hydrophobe distribution in the polymer chain. Schulz et al. (16) addressed this problem in their studies of hydrocarbon associating polymers by incor­ porating a phenyl ring in their hydrophobic unit to allow direct mea­ surement of comonomer content by U V spectroscopy. Fluorescence techniques using pyrene tags have also been used to probe the microstructures of hydrophobic associations, especially at low concentra­ tions (17).

Structure 1. FX13, 2-(N-ethylperfluorooctanesulfoamido)ethyl acryl­ ate.

Structure 2. FX 14, 2-(N-ethylperfluorooctanesulfoamido)ethyl methac­ rylate.

In Hydrophilic Polymers; Glass, J.; Advances in Chemistry; American Chemical Society: Washington, DC, 1996.

16.

AMIS E T AL.

Fluorocarbon Hydrophobic

283

Units

With fluorine-containing polymers, F N M R spectroscopy is a natural possibility for determining the hydrophobe content because of the natural abundance (100%) and relative sensitivity (83.3) of fluo­ rine. However, although F N M R spectroscopy shows spectra of comonomers with well-resolved peaks, F N M R absorption for the copolymer is extremely broadened (18). Although the direct deter­ mination of hydrophobe content is prohibited, monitoring the loss of the sharp monomer peaks during the reaction allows us to measure the incorporation rate and thus the hydrophobe content. The broadening reflects changes in the environment surrounding the fluorine hydro­ phobic units, and thus we anticipate that future work will use N M R spectroscopy to probe the microstructures of associating junctions. 1 9

1 9

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1 9

Viscoelasticity. Shear Rate. A typical feature of associating polymers is a strong shear dependence of viscosity. Figure 2 shows the effect of shear rate on the measured viscosity for an associating copolymer of acrylamide and fluorocarbon-containing acrylate with a molecular weight on the order of 10 . A very strong shear-thinning effect is observed as the viscosity of a 300-ppm solution drops from 2500 to 400 cP over a shear rate range of only one decade. This effect is also polymer concentration dependent. At lower concentrations, shear thinning is much less prominent. Similar results are seen for associat­ ing copolymers of N^V-dimethylacrylamide and fluoroacrylate (J9) and for copolymers with hydrocarbon associating units (20). The strong shear thinning is attributed to the destruction of intermolecular 6

3000 2500 2000 CL ϋ

„ 1500

1000

500 -1.0

-0.5

0.0

0.5

γ, s

1.0

1.5

2.0

1

Figure 2. Effect of shear rate (y) on solution viscosity of copolymer of acrylamide and 0.07 mol% FX 13 at various solution concentrations.

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HYDROPHILIC POLYMERS

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association junctions. The suppression of shear thinning at low con­ centrations may therefore reflect both the overall decrease in associa­ tions and a decrease in the proportion of intermolecular versus intra­ molecular associations. For such a complex system as these random copolymers, it is difficult to separate these effects. Comonomer Content. The comonomer content of associating polymers plays an important role in their viscosifying ability. For all our systems, if solution viscosity at constant polymer concentration is plotted versus the comonomer content for polymers synthesized under conditions varying only in comonomer addition, a maximum is ob­ served, as seen in Figure 3 (21). A complete incorporation of co­ monomers has been confirmed by the F N M R technique, and viscos­ ities are thus plotted against the contents of fluorocarbon acrylate or fluorocarbon methacrylate comonomers. Curves with maxima at inter­ mediate comonomer contents are obtained for these and other fluoro­ carbon copolymers. This maximum is consistent with the formation of an extended network of intermolecular associations favored by the increasing comonomer content. The network is eventually overcome by a preference for intramolecular associations when the comonomer content becomes very high. Chains or clusters of chains collapse under the effect of the association junctions. Several other important points can be illustrated by Figure 3. First, the viscosity varies as much as 2 orders of magnitude, meaning that 1 9

5

4h a. ο 3

ο 2

-3

-2

-1

0

log Comonomer, mol%

Figure 3. Viscosity of PAM copolymers versus content offluorocarboncontaining acrylate (W) or methacrylate (Φ) comonomers at 25 °C. Mea­ surements were made at 0.5 wt% concentration with 0.4-s' shear rate. 1

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AMIS ET AL. Fluorocarbon Hydrophobic Units

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care must be taken to synthesize a copolymer with maximum viscosity. Second, comonomer content at the viscosity maximum is as little as 0.1 mol%, showing the astonishing associative power of fluorocarboncontaining polymers. Third, the peak position changes as different comonomers are used. These changes imply that the interplay of interand intramolecular associations depends on the type of hydrophobic comonomer. Spacer. One structural parameter that alters the association is a variable-length hydrophilic spacer placed between the hydrophobic unit and the hydrophilic main chain. A series of comonomers with nonionic hydrophilic spacers of ethylene oxide units (—CH2CH2O-) placed between the fluorocarbon chain and the acrylate were synthe­ sized. The comonomers C H = C H C O ( O C H C H 2 ) n C H 2 C 7 F (n = 1, 2, or 3) were prepared by reacting C F ( C F ) 6 C H O N a with C l ( C H C H 0 ) - H in a mixture of toluene-diglyme (70/30, vol/vol) to give C F 3 ( C F ) C H ( C H 2 C H 0 ) - H . This compound was then re­ acted with acryloylchloride to give the corresponding acrylates. These were copolymerized with acrylamide (22). Figure 4 shows a plot of viscosity versus comonomer content for several different spacer lengths (one, two, or three). The viscosity increases dramatically as the hydrophilic spacer length increases, especially with the introduc­ tion of the first unit. Similar results were obtained on hydrocarbon associating polymers, and these results have been interpreted as dem2

2

15

3

2

2

2

2

n

2

6

50x10

2

3

0L 10"r3

2

1 I IIIIB|

10"

n

I 1 ΙΙΙΙΜ|

2

10"

I I IIHHj

1

I I I I Hill

10°

10 Κ

1

Comonomer Content, mol% Figure 4. Viscosity versus hydrophobe content for Ρ AM copolymers with one (Π), two (Δ), or three (O) ethylene oxide spacer units. Measure­ ments were made at 0.5 wt% concentration with 0.4-s' shear rate. 1

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HYDROPHILIC POLYMERS

onstrating enhanced formation of micellar structures (23). In that case, a charged moiety in the spacer group was used with the goal of increas­ ing the driving force toward hydrophobic association. Several factors may be responsible for the viscosity enhancement. First, the spacer may increase accessibility of the hydrophobes by decreasing the steric hindrance to association. This increased accessi­ bility would allow denser, more compact, micellelike associations that are stronger and lead to higher viscosities. A second factor could be a decoupling of the motions of associating junctions from the main chain dynamics. To the extent that the solution viscosity of associating polymers is determined by the long lifetime of the junctions, this de­ coupling will reduce the influence of the faster chain dynamics on junction dynamics. There may also be questions regarding the distri­ bution of hydrophobes along the polymer chain. This distribution can also lead to changes in viscosity, and because changing the spacer length may change the kinetics during synthesis, it will be important to characterize the distribution of the hydrophobes. Alternatively, preparation of regular combs with these spaced hydrophobe co­ monomers would address this question. Salt. The addition of salt steadily increases the viscosity of fluorocarbon associating polymer solutions. This same effect is seen with hydrocarbon counterparts. A typical plot is shown in Figure 5 (12). Here the viscosity of a fluorocarbon copolymer is plotted against the concentration of added NaCl for several polymer concentrations. The

3000 2500 2000

Q. ϋ

pf

1500

1000 500

0

1

2

3

4

5

6

7

[NaCl], wt% Figure 5. Viscosity versus salt concentration for PAM copolymers in aqueous NaCl solutions at different polymer concentrations.

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AMIS ET AL.

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Fluorocarbon Hydrophobic Units

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salt effect increases as the p o l y m e r concentration increases, although it also appears that u n t i l the salt concentration reaches about 2%, it has little effect o n the viscosity of the solution. It has b e e n argued that salt enhances h y d r o p h o b i c interactions b y m a k i n g the solvent more polar (11). T h i s action is consistent w i t h the effect of salts on the solu­ b i l i t y of hydrocarbon molecules i n water, where the s o l u b i l i t y de­ creases noticeably w h e n the salt concentration exceeds a certain value (U). Surfactant. U p o n the addition of surfactants, the solution viscos­ ity of fluorocarbon associating polymers i n i t i a l l y increases, passes through a m a x i m u m , and then decreases to l o w levels, as s h o w n i n F i g u r e 6. T h i s particular example shows the effect of fluorocarbon surfactant on the viscosity of a solution of an associating copolymer w i t h fluorocarbon hydrophobes (22). T h e viscosity at a h i g h surfactant concentration is actually l o w e r than the i n i t i a l viscosity w i t h no surfac­ tant present, a n d this observation is consistent w i t h results for hy­ drocarbon hydrophobes (24). T h e effect of surfactants arises from interactions b e t w e e n the h y d r o p h o b i c u n i t of the p o l y m e r a n d the h y d r o p h o b i c part of the surfactant m o l e c u l e . A s a result, the as­ sociating p o l y m e r interactions are altered b y the surfactant (24). A t small doses, surfactant molecules are incorporated into existing p o l y m e r i c m i c e l l e s . W h e t h e r surfactants promote the formation of ad­ ditional associating junctions is not clear. H o w e v e r , the incorporation of surfactants into the junctions stabilizes the associations a n d thus

1000

Q. Ο

0.0

0.1

0.2

0.3

0.4

ΡAM, wt% Figure 6. Viscosity versus added surfactant for 0.25 wt% F AM copoly­ mer solution.

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HYDROPHILIC POLYMERS

increases their lifetimes. With the formation of more stable micelles, an increase in solution viscosity is expected even if the number of associating junctions does not increase (6). As the concentration of surfactant increases to a point at which there are sufficient surfactant molecules to form stable micelle structures with individual hydrophobic units, the hydrophobes separate and exit surrounded by surfactant molecules. As a result, the polymer associations break up, and solution viscosity decreases. In the extreme case when all associating junctions are broken by a high concentration of surfactant, the solution viscosity would be similar to that of polymers without any association. With this as the basis for a technique, single-chain molecular weights have been measured by light scattering or intrinsic viscosity for polymers that would otherwise be aggregated by their strongly associating fluorocarbon or hydrocarbon units (25, 26). Nevertheless, the surfactants change the nature of the system and may even interact with the polymer chain itself (27,28). This method should be applied with caution. Solvent. The addition of an appropriate organic solvent to the aqueous solutions of fluorocarbon associating polymers causes a dramatic decrease in solution viscosity. Figure 7 shows how the viscosity of a fluorocarbon associating polymer solution changes as a function of acetone content in a water-acetone mixed solvent (29). Initially, the viscosity decreases slowly, but it shows a rapid downturn as the acetone content increases further. An increase in acetone content from 4 to 14% decreases the solution viscosity by almost 2 orders of magni-

)2

I

0

I

2

I

4

I

6

I

8

I

10

1

12

L

14

Acetone, wt%

Figure 7. Effect of acetone on solution viscosity of hydrophobically modified associating PAM copolymer.

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Fluorocarbon Hydrophobic Units

hide. Mixed solvents of water and dimethyl sulfoxide can also break up associations of acrylamide-based fluorocarbon copolymers. In the presence of 50% dimethylformamide, the reduced viscosity of the so­ lution drops dramatically, a result indicating that few associations exist at such levels of added solvent (22). The organic solvent probably acts similarly to such hydrophobic bond breakers as urea to break up the association junctions and dramatically decrease the viscosity. Similar experimental results have been obtained on hydrocarbon-containing associating polymers (26) and have been used to determine the molec­ ular weights of associating polymers. As long as the cosolvent is a good solvent for the chain backbone, use of the cosolvent is preferable to use of small-molecule surfactants for measuring isolated chain proper­ ties. In addition to maintaining the chain solvation, the cosolvent does not introduce the complication of small surfactant micelles. Temperature. The effect of temperature on viscosity is not straightforward for fluorocarbon associating polymers, as Figure 8 (22) shows. Similar behavior is seen in other fluorocarbon associating poly­ mer systems (29) and is reproducible. The viscosity first decreases with increasing temperature and then, around 60-80 °C, increases, only to decrease again as the temperature goes higher. A n increase in viscosity with temperature may be viewed as consistent with entropically driven hydrophobic bonding, but a decrease in viscosity with temperature is usually associated with an enthalpic effect. To separate

T, ° C

Figure 8. Viscosity versus temperature for copolymers with one (Ώ), two (Δ), or three (O) ethylene oxide spacer units in the hydrophobic comonomer at 0.5 wt% concentration with 0.1-s" shear rate. 1

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two regions of decrease by a region of increase, another explanation must be invoked. Lundberg and Makowski (30) obtained similar results on the temperature dependence of viscosity for ionomers in a mixed-solvent system. Their explanation was that the first region shows a viscosity decrease due simply to the viscosity of the solvent. The decrease in the second region is primarily due to cosolvent— polymer interactions. In the third region, the ionic interactions weaken as the temperature increases sufficiently. This unusual temperature dependence is seen only for mixed-solvent systems and random fluorocarbon copolymers. Studies of hydrocarbon associating polymers show a consistent decrease in viscosity with increasing temperature (JO, 14). A suitable explanation for the difference is still lacking. The characterization of hydrophobic associations as entropically driven leads to an expectation that as temperature is increased, associations will be favored, and if viscosity is characterized by the number of associations, this favoring will result in an expected increase in viscosity. In this model, viscosity would increase as temperature increases, contrary to the usual result and the Arrhenius description. In fact, the viscosity decrease with temperature increase presents a problem for the hydrophobic model. We can resolve this inconsistency by considering that although the number of associations may increase by entropie forces as temperature increases, the strength of these associations will nevertheless decrease. According to the transient-network theories of Green and Tobolsky (5) extended to associating polymers by Tanaka and Edwards (6), the lifetime of the associations determines viscosity as long as chain dynamics are rapid enough to reequilibrate chains as stickers migrate from one association junction to another. Lifetime is evidently the overriding factor suppressing the effect of the number of associations. One can also imagine these two influences leading to complex temperature dependencies in some cases. The random copolymers are probably not the appropriate model system for testing these ideas. Light Scattering. Light scattering is a natural choice for characterizing polymers in general, but for the associating polymers, this method is quite difficult to employ. The high viscosities make it difficult to clean the solutions either by filtration or by centrifugation. For the fluorocarbon hydrophobes, these problems are compounded by the strength of the interactions and the fact that even for the best cosolvent and the lowest concentrations, significant intermolecular associations persist. In a recent paper, Seery et al. (25) investigated three such copolymers with hydrophobic contents of 0.007, 0.07, and 0.7 mol% by using a combination of static and dynamic light scattering. For the lowest hydrophobic content (0.007%), dynamic light scattering

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is essentially the same as for a homopolymer with no hydrophobes. Static scattering, however, shows a negative osmotic second virial coefficient, A2, indicative of attractive interactions between these chains that have on average only 1.3 hydrophobes per chain. At a hydrophobe level of 0.07%, solutions at typical 0.1% concentrations are extremely viscous and the dynamic scattering at concentrations down to 10 ppm shows large aggregates together with a few small species that are apparently collapsed single chains. The aggregates have 300-nm hydrodynamic radii and do not dissociate upon addition of cosolvents or increase in temperature. Addition of a fluorocarbon surfactant disperses the aggregates and makes it possible to characterize the polymer chains and determine their apparent molecular weight as 6.5 x 10 and their hydrophobe content as 68 per chain. One interesting observation is that the apparent A2 for the aggregated species is positive, a result suggesting a structure of hydrophobes buried within the hydrophilic chains. Increasing the hydrophobe content to 0.7% did not change the scattering behavior much from that of the 0.07% samples. At high concentrations (greater than 1000 ppm), the dynamic scattering collapses to a single mode, and a comparison of the radius of gyration to the hydrodynamic radius indicates that the clusters become increasingly dense. Newly added chains apparently fill in around aggregated ones and thereby cause the volume of the cluster to grow more slowly than its mass. The semidilute overlap concentration is never attained. 6

Polymers with Regular Structures As mentioned, the synthesis of random associating copolymers is straightforward, but the distribution of hydrophobic units and the hydrophobe content in polymer chains remain unresolved. Many studies of associating polymers have avoided the problem and implicitly assumed a random incorporation of hydrophobic units into the polymer chain. Such an assumption, however, is not always valid. For hydrocarbon-associating polymers, McCormick and coworkers (31 ) argued that hydrophobic units form blocky structures and that the length of the blocky structure is related to the surfactant used during the emulsion polymerization. The blocky structures of the hydrophobe that arise by different synthetic routes in turn affect its rheological behavior (20). Because of the uncertainty regarding the distribution and content of hydrophobes in the random copolymers, a systematic study is extremely difficult, and the results may not be consistent between studies. To circumvent this difficulty, we studied associating polymers with two types of well-defined structures: telechelic and comb. Tele-

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chelic polymers have a hydrophobic unit attached at each end of a P E O hydrophilic chain. Combs are composed of fluorocarbon stickers spaced regularly along a chain of coupled P E O units. For telechelic polymers, monodispersed P E O chains are end-capped by a fluorocar­ bon hydrophobic unit, and for comb polymers, the hydrophobe be­ comes part of a linking unit that is used in a condensation reaction of the P E O prepolymer chains. The telechelic polymers simply have two hydrophobes per chain on the ends. The combs have a distribution in overall chain molecular weight, but on every chain the hydrophobe spacing is well defined by the length of the P E O prepolymer. Tele­ chelic polymers have often been used as model associating polymers, as illustrated in the rheological studies of Jenkins (32). Care must always be taken to ensure that both ends are capped with hydrophobic units. The focus of our study is the temperature dependence of viscosity for solutions of each type of polymer structure. We are also interested in how viscosity and network-forming ability differ between telechelic and comb polymers and between polymers with similar structures but different hydrophobe lengths and spacing. T h e o r y . Theoretical studies of the effects of reversible associa­ tions on rheological properties were pioneered by Green and Tobolsky (5). Their theory is based on an extension of classical rubber elas­ ticity theories to transient networks formed by entanglements or breakable physical bonds. It predicts that the steady shear viscosity v(y) equals the zero shear viscosity 17(0)

rj(y) = i?(0) = rGoo where the relaxation time τ is the reciprocal of the bond-breaking and bond-reformation rate, and Goo is the high-frequency storage modulus given by

Goo = vkT where ν is the number density of elastic chains and kT is thermal energy. Tanaka and Edwards (6) expanded the Green-Tobolsky theory. They assumed that the dissociation of an end group from a junction can be approximated as an activation process characterized by a bonding potential with well depth E . In the Green-Tobolsky limit, that is, when the total number of junctions is unaffected by the shear, the association breakage rate β can be expressed by m

0

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293

Fluorocarbon Hydrophobic Units βο =

ω β- ^ 0

Ε

τ

where ω is a characteristic frequency of thermal motion that is esti­ mated to have a typical value on the order of 10 -10 Hz. The lifetime of association r is the reciprocal of βο: 0

8

9

x

τ

χ

= ω