Langmuir 1997, 13, 2447-2456
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Synthesis, Characterization, and Rheological Behavior of Polyethylene Glycols End-Capped with Fluorocarbon Hydrophobes Bai Xu,1a Lin Li,1b Ahmad Yekta, Zahra Masoumi, Sabesh Kanagalingam, Mitchell A. Winnik,* Kewei Zhang,1c and Peter M. Macdonald* Department of Chemistry and Erindale College, University of Toronto, Toronto M5S 3H6, Canada
Steve Menchen* Perkin Elmer Corporation, Applied Biosystems Division, 850 Lincoln Center Drive, Foster City, California 94404 Received August 12, 1996. In Final Form: January 21, 1997X Two polyethylene glycols (PEG, M ) 35 000) end-capped with short fluorocarbon tails were synthesized and characterized. In aqueous solution, the fluorocarbon portions associate strongly to form micelle-like structures which are bridged by PEG chains to form a three-dimensional network. As a result, these polymers in solution exhibit unusual rheological properties as a function of fluorocarbon length, polymer concentration, and shear rate (frequency). Their zero-shear viscosity increases with concentration, a common behavior of associating polymers. The viscosity is dramatically enhanced by replacing the end hydrophobe C6F13 with C8F17, a consequence of the stronger association interaction of C8F17 in aqueous solution. The polymer with the longer end group exhibits strong shear thinning once a critical shear rate is reached, whereas for the C6F13 end-capped polymer, we cannot with our equipment reach the shearthinning regime. Our data indicate that between 2 and 6 wt %, and perhaps over a wider range of concentrations, both systems can be characterized in terms of identical values of the plateau modulus GN°, implying a similar concentration of chains bridging micelles in each system. The GN° values increase strongly with polymer concentration, consistent with a larger fraction of bridging chains and a smaller fraction of looping chains at elevated concentration. The viscosity difference between the two polymers can be explained in terms of a slower exit rate of the longer fluorocarbon from its micelle.
Introduction Water-soluble polymers containing a single hydrophobic end group associate in water to form micelles. The micelle comprises a core formed from the insoluble hydrophobic groups, surrounded by a corona of long water-soluble chains. When the polymer is a polyethylene glycol (PEG), these substances are referred to as nonionic surfactants. When the PEG chain has hydrophobic substituents at both ends (i.e., telechelic polymers), the end groups associate to form micelle-like structures, but here the PEG chains are in principle able to bridge neighboring micelles. At sufficiently high concentration, one imagines that the polymer can form a three-dimensional network in aqueous solution.2-11 Not all of the PEG chains participate in bridge formation. From the point of view of the simple model just described, a significant fraction of the polymers are looped, with both chain ends located in the same X
Abstract published in Advance ACS Abstracts, March 15, 1997.
(1) Present address: (a) i-STAT Canada Ltd., 436 Hazeldan Rd., Kanata, Ontario K2L 1T9, Canada; (b) Mitsubishi Chemical Corp., Yakkaichi Research Center, 1 Toho-cho, Yokkaichi, Mie 510, Japan; (c) Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290. (2) (a) Water Soluble Polymers; Glass, J. E., Ed.; Advances in Chemistry Series 213; American Chemical Society: Washington, DC, 1986. (b) Polymers in Aqueous Media; Glass, J. E., Ed.; Advances in Chemistry Series 223; American Chemical Society: Washington, DC, 1989. (c) Polymers as Rheology Modifiers; Schulz, D. N., Glass, J. E., Eds.; ACS Symposium Series 462; American Chemical Society: Washington, DC, 1991. (d) Hydrophilic Polymers: Performance with Environmental Acceptance; Glass, J. E., Ed.; ACS Advances in Chemistry Series 248; American Chemical Society: Washington, DC, 1996. (3) (a) Jenkins, R. D. Ph.D. Thesis, Lehigh University, Bethlehem, PA, 1990. (b) Jenkins, R. D.; Silebi, C. A.; El-Aasser, M. S. Polym. Mater. Sci. Eng. 1989, 61, 629. (c) Jenkins, R. D.; Silebi, C. A.; El-Aasser, M. S. In Advances in Emulsion Polymerization and Latex Technology: 21st Annual Short Course; El-Aasser, M. S., Ed.; Lehigh University: Bethlehem, PA, June 1990; Chapter 17.
S0743-7463(96)00799-8 CCC: $14.00
micelle. Recent papers provide a description of the structure of associating polymers in aqueous solution in terms of modern polymer theory.12,13 Polymers of this sort have found important applications in industry as rheology modifiers, particularly for organic coatings formulations. Such polymers are commonly referred to as “associative thickeners” (AT’s).2 Other applications of water-soluble associating polymers include enhanced oil recovery and water treatment.5,6 The rheological properties of AT’s have been examined by a number (4) (a) Maechling-Strasser, C.; Francois, J.; Clouet, F.; Tripette, C. Polymer 1992, 33, 627. (b) Maechling-Strasser, C.; Clouet, F.; Francois, J. Polymer 1992, 33, 1021. (c) Persson, K.; Abramsen, S.; Stilbs, P.; Hansen, F. K.; Walderhaug, H. Colloid Polym. Sci. 1992, 270, 465. (d) Fonnum, G.; Bakke, J.; Hansen, F. K. Colloid Polym. Sci. 1993, 271, 380. (5) (a) Broze, G.; Jerome, R.; Teyssie, P.; Marco, G. Macromolecules 1983, 16, 996; (b) Agarwal, P.; Garner, R. T.; Lundberg, R. D. Macromolecules 1984, 17, 2794. (6) (a) Shaw, K. G.; Leipold, D. P. J. Coatings Technol. 1985, 57, 63. (b) Evali, S.; Rose, G. D. Polym. Mater. Sci. Eng. 1987, 57, 477. (7) (a) Wang, Y.; Winnik, M. A. Langmuir 1990, 1437. (b) Yekta, A.; Duhamel, J.; Brochard, P.; Adiwidjaja, H.; Winnik, M. A. Macromolecules 1993, 26, 1829. (8) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. J. Rheol. 1993, 37, 695. (9) (a) Yekta, A.; Xu, B.; Duhamel, J.; Adiwidjaja, H.; Winnik, M. A. Macromolecules 1995, 28, 956. (b) Yekta, A.; Duhamel, J.; Adiwidjaja, H.; Brochard, P.; Winnik, M. A. Langmuir 1993, 9, 881. (10) (a) Xu, B.; Yekta, A.; Masoumi, Z.; Winnik, M. A. Colloid Surf., A: Physicochem. Eng. Aspects 1996, 112, 239. (b) Yekta, A.; Xu, B.; Winnik, M. A. In Solvents and Self-Organization of Polymers; Webber, S., Ed.; Kluwer: Dordrecht, The Netherlands, 1996. (11) Alami, E.; Almgren, M.; Brown, W.; Francois, J. Macromolecules 1996, 29, 2229. (12) Semenov, A. N.; Joanny, J. F.; Khokhlov, A. R. Macromolecules 1995, 28, 1066. (13) (a) Borisov, O. V.; Halperin, A. Langmuir 1995, 11, 2911; (b) Macromolecules 1996, 29, 2612.
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2448 Langmuir, Vol. 13, No. 9, 1997
of research groups,2,3,8,10,14 with the pioneering mechanistic studies due to Jenkins.3 These fluids have rather complex rheological behavior. In steady shear, beyond a critical shear rate, they undergo a pronounced shear thinning. Over a certain range of shear rates prior to shear thinning, but only over a limited range of concentrations, some of these polymers also exhibit shear thickening. In spite of these complexities, PEG polymers with hydrophobic end groups, under oscillatory shear, commonly have the linear viscoelastic properties of a near perfect Maxwell fluid. More recently, these polymers and their solutions in water have been examined by a variety of scattering and spectroscopic techniques. On this basis, one begins to build up a molecular level picture of the structure of the system and its response to shear. These pictures, inferred from different experiments, and often for different polymers or PEG polymers with different end groups, are not entirely consistent with one another. Systems with long PEG chains (e.g. M ) 20 000) with relatively weakly associating hydrophobic end groups11 appear to exhibit a more complex behavior than polymers of similar or greater length but with larger hydrophobic substituents. In all of these systems, there is a delicate balance between the driving force for micellization (minimization of hydrophobe/water contacts) and the penalty paid through loop formation and chain stretching in starlike micelles.12,13 We have recently become interested in the study of PEG polymers with fluorocarbon substituents at both ends. It is well-known that fluorocarbons are more hydrophobic than hydrocarbons of similar length, and a useful way of thinking about simple surfactants is that a fluorocarbon surfactant with n carbons has the association properties of a hydrocarbon surfactant with 2n carbons. Polyacrylamide polymers with small amounts of fluorocarbon pendant groups have been examined for their association properties.15 It was found that these polymers undergo much stronger associations than those with hydrocarbon substituents, and the association can be strong even at low fluorocarbon content. Our working hypothesis about PEG polymers with fluorocarbon end groups was that these groups would associate to form micelles bridged by PEG chains; and if these chains were all similar in length, the bridges would form pores relatively similar in size which could serve as a sieving medium for electrophoresis.16 We hoped that, with fluorocarbon hydrophobes, the micelles formed would have little tendency to interact with hydrophobic sites on the polyelectrolytes we wished to examine. In this way we might be able to obtain a fluid that could be pumped into a capillary column and serve as a sieving medium for capillary electrophoresis (CE). To be a useful CE fluid, under automated analysis conditions, each column fill should be used for only a single run. After each run, a pump would refill the column with new fluid, expelling the fluid from the previous run along with any chemical byproducts of the analysis. In fact this works very well. A fluid made up of a 1:1 mixture of the two polymers shown below, a PEG of M ) 35 000 with fluorocarbon end groups attached to the (14) Hulden, M. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 82, 263. (15) (a) Zhang, Y.-X.; Da, A.-H.; Hogen-Esch, T. E.; Butler, G. B. J. Polym. Sci., Polym. Lett. Ed. 1990, 27, 213. (b) Hogen-Esch, T. E.; Amis, E. Trends Polym. Sci. 1995, 3, 98. (c) Hu, N. Ph.D. Thesis, University of Southern California, 1994. (d) Cochin, D.; Hendlinger, P.; Laschewsky, A. Colloid Polym. Sci. 1995, 273, 1138. (16) (a) Menchen, S.; Johnson, B.; Winnik, M. A.; Xu, B. Chem. Mater. 1996, 8, 2205. (b) Menchen, S.; Johnson, B.; Winnik, M. A.; Xu, B. Electrophoresis 1996, 17, 1451. (c) Slater, G. W.; Drouin, G. Electrophoresis 1992, 13, 574. (d) Mayer, P.; Slater, G. W.; Brouin, G. Appl. Theor. Electrophor. 1993, 3, 147.
Xu et al. CnF2n+1–CH2CH2O–IPDU–PEG35 000–IPDU–OCH2CH2–CnF2n+1 C6F-35K, n = 6; C8F-35K, n = 8 O
O
NH C
C HN
IPDU
polymer via isophorone diurethane (IPDU) units, serves as an excellent medium for CE sequencing of DNA. The DNA sequencing procedure, which has been described elsewhere,16b involved a sequencing sample derived from a single-stranded template (M13mp18), appropriately primed using dye-labeled dideoxy terminators. The sequencing matrix was a 1:1 mixture of C6F-35K and C8F35K (6 wt % in 40% aqueous urea plus buffer at pH 8.0). The limit of resolution in this formulation was 450 bases in 75 µm capillaries at 200 V/cm. Here we describe the synthesis and characterization of these polymers and their rheological properties in water. We find, for example at 5 wt % concentration, that the zero-shear viscosity of C8F-35K is about 10 times higher than that of a PEO of similar molecular weight with hydrocarbon (OC16H33) end groups also attached via IPDU units and note that Amis et al. have reported similar observations in their recent study of the low-shear viscosity of polymers with similar structures.17 Experimental Section Preparation of C8F-35K. Alcohol Activation. Into a preweighed reaction container (200 mL round-bottom flask with a Teflon stir bar) was placed heptadecafluoro-1-decanol (13.9 g, PCR, Inc., Gainesville, FL). The preweighed container was then used as a vacuum distillation receiver and was placed in a dry ice/acetone bath. Using a short-path distillation setup and a “Firestone splash guard” (Aldrich Chemical, Milwaukee, WI) to protect the receiver from small amounts of material bumping from the distillation pot, a total of ca. 40 mL of isophorone diisocyanate (Aldrich) was distilled onto the alcohol at 0.1 mm/ 85-90 °C. The resulting solution was purged with argon, sealed with a stopcock equipped with a vacuum takeoff, and stirred overnight in a 70-80 °C oil bath. Then the vacuum takeoff was replaced with a short-path distillation setup (a connecting adapter, vacuum adapter, and receiver). The system was evacuated (0.1 mm), and the oil bath temperature was raised to 110 °C, during which most of the excess diisocyanate was collected. The remaining diisocyanate was removed from the pot residue by extraction with dry (distilled from CaH2) hexane: The oily residue was refluxed with 100 mL of hexane. Then the flask was sealed with a rubber septum and cooled to -20 °C for 3 h. To avoid water condensation inside the flask, the supernate was quickly decanted from the waxy residue. The hexane treatment was repeated two more times to insure complete removal of excess diisocyanate from the monoisocyanate product. The intermediate heptadecafluorodecane monoisocyanate was evacuated for several hours, yielding 16.4 g of waxy solid. Heptadecafluorodecane End Capping of PEG 35 000. Polyethylene glycol (PEG) of nominal molecular weight 35 000 (175 g, Fluka, Ronkonkoma, NY) was placed in a 1 L resin kettle. The container was placed under vacuum (0.1 mm) and dried for 12 h in a 110 °C oil bath. The PEG was allowed to cool and to solidify under argon, followed by addition of 200 mL of dry (freshly distilled from LiAlH4) ethylene glycol dimethyl ether (glyme). The total amount of heptadecafluorodecane monoisocyanate from the above reaction was dissolved in 100 mL of dry glyme, and this solution was added to the kettle under argon. An additional 100 mL of dry glyme was used to wash all of the monoisocyanate from the flask into the kettle. The kettle, equipped with a reflux (17) Amis, E. J.; Hu, N.; Seery, T. A. P.; Hogen-Esch, T. E.; Yassini, M.; Hwang, F. In Hydrophilic Polymers: Performance with Environmental Acceptance; Glass, J. E., Ed.; ACS Advances in Chemistry Series 248; American Chemical Society: Washington, DC, 1996.
End-Capped Polyethylene Glycols condenser and overhead stirrer, was slowly heated to reflux with stirring. Dibutyltin laurate (3 drops, Aldrich) was added to the hot solution, and the reaction was refluxed for 8 h under argon. The resulting homogeneous solution was poured into 1 L of hexane, and the liquid was decanted from the precipitated polymer. The crude polymer was triturated two more times with 1 L portions of refluxing hexane and two times with 1 L portions of refluxing methyl tert-butyl ether. The tough rubbery product was dissolved in 600 mL of refluxing ethyl acetate and allowed to crystallize overnight. The finely divided product was collected on a glass frit using positive gas pressure to aid in the filtration. The filter cake was dispersed in hexane and filtered. Vacuum drying yielded 178 g of white powder. Preparation of C6F-35K. Alcohol activation was carried out as described above, using tridecafluoro-1-octanol (9.5 g, PCR Inc.) in place of heptadecafluoro-1-decanol, and isophorone diisocyanate (40 mL). Following purification as described above, tridecafluorooctylisophorone monoisocyanate (8.3 g) was obtained as a mobile, colorless liquid. Attachment of this group to the ends of PEG 35 000 followed the procedure described for the C8F-35K sample. The entire tridecafluorooctylisophorone monoisocyanate reaction product (8.3 g) was reacted with PEG 35 000 (120 g). After purification, 118 g of white powder was isolated. Polymer Characterization. The polymers were characterized by 1H and 19F NMR (Varian 400 MHz) in CDCl3 (3 wt %) containing 1,4-difluorobenzene (7.62 µmol/g) as an internal standard. Data acquisition and data analysis were similar to those described in detail for corresponding polymers with C16H33 end groups.18 The delay time between pulses was adjusted so that the integration values were independent of delay time: for 1H NMR, the delay time was 30 s; and for 19F NMR, it was 4 s. 19F NMR (48 transients were averaged, δ, ppm): C F-35K: -81.1 6 (CF3); -113.8, -122.2, -123.2, -123.9, -126.4 (5 CF2). C8F35K: -81.1 (CF3); -113.9, -122.0 (×2), -122.2, -123.0, -123.9, -126.4 (7 CF2). Internal standard: -120.0. Integration of the spectra and comparison with the internal standard yields 55.9 µmol of CnF2n+1/g of C6F-35K and 54.0 µmol of CnF2n+1/g of C6F35K. On the basis of Mn(PEO) ) 35 000, we calculate 2.02 and 1.96 end groups per polymer, respectively. 1H NMR (2 transients): For IPDU assignments, see ref 18. PEO (3.6 ppm, s). Integration relative to the internal standard (6.98 ppm, t) yields 53.8 µmol of IPDU + 27.1 µmol of PEO (M ) 35 000)/g of C6F-35K and 52.4 µmol of IPDU + 28.7 µmol of PEO (M ) 35 000)/g of C8F-35K. From these values we calculate 1.98 and 1.82 IPDU groups per chain, respectively. Comparing with the 19F NMR results, we obtain CnF2n+1/IPDU ) 1.04 (C6F35K) and 1.03 (C8F-35K). PGSE NMR Diffusion Measurements. Stock solutions consisting of 4.0 wt % polymer in 2H2O were prepared. Samples for PGSE NMR were prepared by serial dilution of the stock solution. Proton self-diffusion studies were performed using an MRI (magnetic resonance imaging) probe with actively shielded gradient coils (Doty Scientific, Columbia, SC) installed in a Chemagnetics CMX 300 NMR spectrometer operating at 300 MHz for protons. A standard Stejskal-Tanner PGSE sequence [90°-τ-180°-τ], with the gradient pulse during τ, was employed.19a The gradient strength (1.04 T/m) was calibrated with a sample of 10 wt % PEO in 2H2O for which the diffusion coefficient is known. The experimental error of the diffusion coefficient was below (5%. All measurements were performed at 25 °C, and the temperature was controlled by an air flow regulator, yielding a temperature stability of (0.5 °C. According to Stejskal and Tanner a spin-echo signal induced by a 90°-τ-180° radio frequency pulse sequence, when a magnetic field gradient, G, is applied during a time, δ, as a twin pulse separated by a time ∆, one before and one after the 180° pulse, has a signal amplitude, I, for a given chemical species given by (18) Yekta, A.; Kanagalingam, S.; Nivaggioli, T.; Winnik, M. A. In Hydrophilic Polymers: Performance with Environmental Acceptance; Glass, J. E., Ed.; ACS Advances in Chemistry Series 248; American Chemical Society: Washington, DC, 1996. (19) (a) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (b) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1.
Langmuir, Vol. 13, No. 9, 1997 2449 I ) I0 exp[-(gGδ)2D(∆ - δ/3)]
(1)
where I0 is a constant, g is the magnetogyric ratio for the nuclei studied, and D is the self-diffusion coefficient. In our experiments, ∆ is kept constant while δ is varied. The self-diffusion coefficient is calculated from eq 1, and at least 12 different values of δ are used. More details of the method may be found elsewhere.19b The self-diffusion measurement directly monitors the random motion of an individual molecule during the time interval (∆δ/3). The relation between the molecular mean square displacement in one dimension, 〈X2〉, and the self-diffusion coefficient is given as
〈X2〉 ) 2D(∆ - δ/3)
(2)
where D is the self-diffusion coefficient, which typically is in the range 10-11 to 10-13 m2 s-1 for polymer systems. In the present study, the ∆ used is in the range 250-400 ms. During this time period the molecules diffuse over a distance which is much larger than the gyration radius of the polymer. Hence, the observed self-diffusion coefficients only reflect purely center-of-mass diffusion of the polymer. Segmental displacement of the polymer exerts no significant influence on the results. Furthermore we note that for the PGSE measurements within the experimental accuracy we always observe monoexponential echo attenuation, and in no case did we observe a dependence of the self-diffusion coefficient on the diffusion time. Therefore, exchange of the polymer molecules between different states is rapid on the time scale (∆-δ/3) of the experiments. The apparent self-diffusion coefficient is the weighted average of the diffusing species at different sites, and furthermore, the distribution of the diffusion coefficients is relatively narrow. Rheology Measurements. Weighed amounts of polymer were dissolved in deionized water (Millipore Milli-Q purification system) by stirring at room temperature (20 °C) until the polymer was completely dissolved. For example, 0.4710 g of C6F-35K was added to 9.0195 g of water to form a 5.0 wt % solution, and 0.5305 g of C8F-35K was mixed with 10.1197 g of water to produce a 5.0 wt % solution. The solutions with lower concentrations could be prepared by diluting a higher concentration of solution with water. To prepare solutions with high concentrations (>5 wt %) of C8F-35K, the mixture of polymer and water was stirred at 60 °C for about 5 h to yield a clear solution. No phase separation was observed during sample preparation or storage. All solutions were kept bubble-free for the rheological measurements. Low-concentration viscosity measurements were made with a standard capillary viscometer at 20 °C, giving intrinsic viscosity values of 2.0 dL/g (C8F-35K) and 1.0 dL/g (C6F-35K). For higher concentrations of polymer, a Rheometrics asphalt analyzer (RAA) with a cone-plate geometry (50 mm diameter, 0.04 rad cone angle) was used to measure the viscoelastic properties in both dynamic and steady-state modes at 20 °C. Depending on the viscoelasticity of a sample, the shear strain was chosen in the range from 20% to 100% to obtain a linear dynamic viscoelasticity.
Results and Discussion Synthesis and Characterization. The reactions used are presented in Scheme 1. End groups were attached to a sample of PEG of M ) 35 000 and narrow molecular weight distribution commercially available from Fluka. The fluorocarbon alcohols (1 and 2) were prereacted with a large excess of isophorone diisocyanate (IPDI, 3) in the absence of catalyst, conditions which lead to selective reaction of the secondary isocyanate, rather than the hindered primary isocyanate.20 Excess diisocyanate could be removed from the monoisocyanate, in part by distillation and in part through washing the sample with hexane, in which the fluorocarbon isocyanates (4-C6, 4-C8) are insoluble. Then excess amounts of these monoisocyanates were reacted with the PEG in the presence of dibutyltin laurate catalyst to yield the desired polymers, (20) Hatada, K.; Ute, K.; Oka, K.-I. J. Polym. Sci., Part A: Polym. Chem. 1990, 28, 3019.
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Xu et al.
Figure 1. Plot of the diffusion coefficient D determined by PGSE NMR for C8F-35K, as a function of concentration. Scheme 1
the hydrodynamic radius of a flower-like micelle,12 formed through hydrophobic association of the fluorocarbon end groups. Analogous, but slightly different, behavior is observed for C6F-35K. Here, at the lowest concentration we can measure (0.06 wt %), there appear to be two populations with different diffusion coefficients, whereas only a single D value is obtained at higher concentrations. This may signal the onset of association in this region, but at this low concentration, the data are not sufficiently good to draw a firmer conclusion. Extrapolating the D values from higher concentrations gives D0 ) 11.3 × 10-12 m2 s-1, corresponding to RH ) 17.6 nm. From the size of the aggregated species one can estimate its aggregation number. For the case of a hard sphere, which is a reasonable approximation for a flower-like micelle in the dilute regime, the mass of the aggregated species, M, is related to its radius, R, through the intrinsic viscosity, [η], via eq 4, where NA is Avogadro’s number.
[η]M ) (10π/3)NAR3
C6F-35K and C8F-35K. These polymers are conveniently purified through recrystallization from warm ethyl acetate. The polymers (in CDCl3 containing a known amount of 1,4-difluorobenzene as an internal standard) were characterized by high-resolution 1H and 19F NMR. Careful data acquisition, with attention paid to the effects of delay time, gave peaks which could be integrated with a precision better than (5%. From analysis of the signals due to the isophorone and CnF2n+1 groups, we learn that the two polymers have a mole ratio of (CnF2n+1)/isophorone ) 1.0 and, based upon Mn(PEG) ) 35 000, contain 1.95-2.05 end groups per chain. PGSE NMR Measurements. In Figure 1, we plot the self-diffusion coefficient (D) of C8F-35K, as determined by PGSE NMR, vs polymer concentration, Cpol. In this lowconcentration regime the system is still above its critical micelle or critical aggregation concentration (cac). The concentration at which the aggregation starts to form is quite low, beyond the detection limits of PGSE NMR with our current instrumental configuration. At the lowest concentrations in Figure 1, the spacing between micelles is larger than the radius of gyration of the system. Thus we can use the Stokes-Einstein equation to calculate the hydrodynamic radius RH of the diffusing units.
RH ) kT/6πηD
(3)
where η is the solvent viscosity, D is the diffusion coefficient, k is the Boltzmann constant, and T is the temperature. Extrapolating D to infinite dilution yields D0 ) 7.74 × 10-12 m2 s-1, corresponding to a hydrodynamic radius of 25.6 nm. This value, we believe, corresponds to
(4)
To calculate M, we take R ) RH. The intrinsic viscosities of C8F-35K and C6F-35K are 2.0 and 1.0 dL/g, respectively. Applying eq 4 and taking the polymer molecular weight as 35 000, we calculate that each micelle of C8F-35K contains an average of 15 polymer chains (30 end groups) and that each micelle of C6F-35K contains an average of 10 polymer chains (20 end groups). These results seem reasonable, but the NMR data are not sufficient to establish whether the associated species have a micelle core with a structure that is independent of concentration in this region. In other systems, an independent measurement of the number of chain ends per micelle (NR) is available from fluorescence quenching (pyrene excimer) experiments.9-11 We have synthesized a pyrene derivative with a short fluorocarbon tail21 that binds to these micelles and have used this probe to obtain apparent NR values. For three different concentrations of C8F-35K in the range 0.5-2 wt %, NR values on the order of 15-20 chain ends per micelle were calculated from the fluorescence decay data. We noticed, however, that the viscosities of the solutions decreased in the presence of near-saturation amounts of the fluorescent probe. Perturbation of micelles by fluorescent probes becomes increasingly likely for micelles of small size composed of relatively short hydrophobic groups. Further experiments are necessary on C6F-35K and C8F-35K before meaningful numbers can be reported. Steady-State Shear Flow. Samples were subjected to steady-state shear flow at shear rates incremented from 0.1 to 1000 s-1 in logarithmically spaced steps (10 points per decade) to determine the steady shear viscosity. For both polymers we observe that the higher the viscosity of a solution, the earlier the appearance of shear thinning. Stable flows persist only to a limited extent in the shear thinning region. Subsequently, the flows become unstable. Air bubbles appear in the sample, and sample material is ejected from the sample holder. This is accompanied by a sudden large drop in the apparent viscosity. We monitored flow stability carefully, and rejected data from unstable flows. Figure 2 shows the steady-state shear viscosity η of C6F-35K solutions, as a function of shear rate γ˘ , at different concentrations. The viscosity is almost independent of shear rate at low shear rates for each concentration. No shear thickening is observed in the high shear rate range, which is different from the behavior of telechelic associat(21) Pham, H. M.Sc. Thesis, University of Toronto, 1995.
End-Capped Polyethylene Glycols
Langmuir, Vol. 13, No. 9, 1997 2451
Figure 2. Steady-state shear viscosity at 20 °C as a function of shear rate for C6F-35K at three different concentrations (3, 5, and 10 wt %).
Figure 4. Effects of polymer concentration on the zero-shear viscosity for aqueous solutions of C6F-35K and C8F-35K at 20 °C.
Figure 3. Steady-state shear viscosity at 20 °C as a function of shear rate for C8F-35K at three different concentrations (1, 2, and 5 wt %).
Figure 5. Shear storage modulus G′ (open points) and shear loss modulus G′′ (filled points) as a function of frequency ω for C6F-35K solutions at room temperature. The polymer concentrations are 3 wt % (circles), 5 wt % (triangles), and 10 wt % (squares).
ing polymers containing hydrocarbon hydrophobes. For the C6F-35K solutions, the viscosity increases with increasing polymer concentration. For example, 3 wt % C6F-35K enhances the viscosity of water (0.001 Pa‚s at 20 °C) by a factor of 1000. Even more significant effects of polymer association on viscosity enhancement can be observed for the C8F-35K solutions. As shown in Figure 3, the zero-shear viscosity (η0, γ˘ f 0) is about 0.1 Pa‚s for the 1 wt % solution and then increases to about 7 Pa‚s for the 2 wt % solution. The 2 wt % solution appears to exhibit a slight shear thickening before the onset of shear thinning, while all other samples show no indication of shear thickening. At 5 wt % polymer concentration, a change in the hydrophobe from C6F13 to C8F17 results in an increase in the zero-shear viscosity of about 50 times, indicating a much stronger association of C8F17 groups than of C6F13 groups. The low shear viscosities observed here are about a factor of 10 higher than those of a PEG polymer of similar molecular weight with C16H33-IPDU end groups,10 in accord with observations reported by Amis et al.17 Figure 4 compares the effect of polymer concentration on the zero-shear viscosity, η0, for solutions of C6F-35K and C8F-35K: η0 increases almost linearly with the polymer concentration up to 10 wt % for C6F-35K solutions. C8F-35K exhibits a much stronger dependence on concentration. On the scale shown in Figure 4, η0 increases dramatically at polymer concentrations above 2 wt %. This behavior is typical of telechelic polymers in water with strongly associating end groups. For example, both Jenkins3 and Annable et al.8 reported that hydrophobic ethoxylated urethane (HEUR) polymers end-capped with C16H33 and other similar hydrophobic end groups show a
Figure 6. Shear storage modulus G′ (open points) and shear loss modulus G′′ (filled points) as a function of frequency for C8F-35K solutions at room temperature. The polymer concentrations are 1 wt % (circles), 2 wt % (triangles), and 5 wt % (squares).
rapid increase in η0 at concentrations beyond 2-3 wt %. Amis et al.17 have reported a strong viscosity increase for a fluorocarbon end-capped PEG with a structure similar to that described here. Dynamic Viscoelasticity. Dynamic measurements were made at frequencies between 10-1 and 5 × 102 rad/s. As shown in Figures 5 and 6, all concentrations of the solutions studied here reach the terminal (flow) region in this frequency range. In the terminal region (ω f 0), at each concentration, G′ and G′′ are nearly exactly proportional to ω2 and ω1, respectively. For the C6F-35K solutions (Figure 5), both G′ and G′′ shift to lower frequency as the
2452 Langmuir, Vol. 13, No. 9, 1997
Xu et al. Table 1. Rheological Characteristics of C6F-35K and C8F-35K Solutions at 23 °C wt %
sample
η0a (Pa‚s)
η0b (Pa‚s)
104Je° 10-3GN° c τc η0Je° d (Pa-1) (Pa) (ms) (ms)
C6F-35K
3 1.25 1.01 15.4 5 3.85 3.83 5.42 10 18.0 16.1 2.11 C8F-35K 1 0.097 0.088 270 2 6.66 7.38 42.8 5 200 190 4.38 6 240 242 4.03
0.30 3.29 3.96
22 51 52
1.6 2.1 3.4 2.4 32 83 98
a Zero-shear viscosity obtained from steady-state shear measurements. b Zero-shear viscosity estimated from dynamic viscoelastic properties. c Obtained by fitting G′ or G′′ to the singleelement Maxwell model. d Calculated from eq 5.
Figure 7. G′(ω) and G′′(ω) for a 5 wt % C6F-35K solution and a 5 wt % C8F-35K solution at room temperature. By a shift factor of 150 rad/s, the G′ and G′′ curves of C6F-35K can be perfectly superposed on those of C8F-35K, as indicated by the solid lines.
polymer concentration, Cpol, is increased, and there is no clear rubbery plateau evident in any of the samples at these concentrations over the frequency range examined in this work. The C8F-35K solutions exhibit more significant variation in their dynamic behavior in this frequency range (Figure 6). As in the case of the C6F-35K polymers, both G′ and G′′ in the terminal region shift to lower frequencies with increasing polymer concentration. Their respective proportionality to ω2 and ω1 is maintained. Here a clear rubbery plateau can be observed for concentrations above 2 wt %. For all concentrations of C8F35K, a crossover of G′ and G′′ can be observed in the frequency range studied, and G′′ exhibits a maximum. In Figure 6 we also see that the plateau modulus increases significantly for C8F-35K solutions as Cpol is increased. While the C6F13-PEG solutions do not exhibit a clear rubbery plateau in the frequency range examined, the G′ data appear to approach a maximum value (cf., Figure 5), and this limiting value increases with polymer concentration. If the G′ data for C8F-35K and C6F-35K solutions at the same concentration are plotted together as in Figure 7, we see that the G′ values for the C6F-35K sample appear to approach a limiting value close to the plateau modulus for C8F-35K. This observation points to an important result which we will elaborate below, that solutions of both polymer samples at the same concentration are characterized by similar values of the plateau modulus. The terminal properties of viscoelastic solutions, such as the zero-shear viscosity, η0, and the steady-state shear compliance, Je°, can be estimated by22
η0 ) limωf0(G′′/ω)
(5a)
Je° ) [limωf0(G′/ω2)]/η02
(5b)
and
η0 is a measure of the fluidity of the solution and is proportional to its longest relaxation time. Je° corresponds to the permanent elasticity. The values of η0 and Je° calculated in this way are given in Table 1. Note that η0 in eq 5 refers to the zero-shear viscosity as the zerofrequency limit of the dynamic viscosity η*. These zero(22) (a) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (b) Mark, J. E.; Eisenberg, A.; Graessley, W. W.; Mandelkern, L.; Koenig, J. L. Physical Properties of Polymers; American Chemical Society: Washington, DC, 1984. (c) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986.
Figure 8. Steady-state shear compliance Je° as a function of polymer concentration for C6F-35K solutions (open circles) and C8F-35K solutions (filled circles) at room temperature. The estimated error for Je° for each concentration is smaller than the points shown.
frequency dynamic viscosity values are consistent with steady-state low-shear viscosity values, as shown in Figures 2 and 3. The product of η0 and Je° is a characteristic time of viscoelastic flow, which is proportional to the longest relaxation of the viscoelastic fluid.22 The elasticity of the polymer solution can also be characterized by the rubbery plateau modulus GN°, which can either be determined from the plateau of the G′ curve or derived from
∫-∞∞[G′′(ω) - Gg′′(ω)] d(ln ω)
GN° ) (2/π)
(6)
where Gg′′(ω) is the loss modulus contribution from the glass transition. GN° is related to Je° by the expression Je° ) (6/5)/GN°.22c Values of Je° calculated from eq 5b are plotted vs polymer concentration in Figure 8. Je° decreases with increasing Cpol. We can draw a common line through all the data points, which suggests very similar values for both polymers at the same concentration. In fact, measurements for C8F-35K were carried out over a set of lower concentrations (cf., Figure 8) than for C6F-35K, so that the inference of similar Je° values is most reliable between 2 and 6 wt %, where both polymers were examined. Because of the relationship between GN° and Je°, GN° should also be independent of end group for these two polymers in this concentration range. One further insight into the properties of the polymer solutions comes from a comparison of dynamic complex viscosity η*(ω) and steady-state viscosity η(γ˘ ) values. Solutions in which these two values are identical at all shear rates satisfy the Cox-Merz rule. Figure 9 presents a comparison of these viscosity values for C8F-35K solutions at 2 and 5 wt %. At 2 wt %, η(γ˘ ) is close in
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Figure 9. Cole-Cole plots of G′′(ω) vs G′(ω) for C8F-35K at 2 wt %, 5 wt %, and 6 wt % polymer concentrations.
Figure 10. Comparison of steady-state shear viscosity η(γ˘ ) and dynamic complex viscosity η*(ω) for C8F-35K at 2 wt % and 5 wt % polymer concentration.
magnitude to η*(ω) over the whole range of shear rates or frequencies. At 5 wt %, however, common values of the two viscosities are found only at low shear rates and frequencies, and the deviation of η(γ˘ ) from η*(ω) becomes more pronounced as shear-thinning develops. The C6F35K solutions also show that the Cox-Merz rule is obeyed at low concentrations. Here also, the deviation becomes more significant with increasing polymer concentration. Network Formation and Relaxation Times. A remarkable finding by Jenkins3 and by Annable et al.8 is that solutions of telechelic associating polymers in water have linear rheological behavior that can be described almost completely in terms of a single-element Maxwell model. This model is characterized by two parameters, GN° and a single relaxation time, τm. A signature of a system that can be described in this way is a Cole-Cole plot, a plot of G′′ vs G′ in the form of a perfect semicircle. Cole-Cole plots for solutions of C8F-35K at three concentrations are presented in Figure 10. Values of GN° and τm are also presented in Table 1. Plateau Modulus. According to the simple theory of rubber elasticity, GN° is related to the number of elastically effective chains per unit volume ν by
GN° ) gνRT
(7)
Here g is a correction factor near unity which we will ignore in the discussion that follows. For polymer melts, ν is related to the molecular weight between entanglements Me by ν ) F/Me, where F is the density of the polymer, and thus GN° ) FRT/Me. In polymer melts, GN° values for a linear and monodisperse polymer are only dependent on Me and are independent of total polymer molecular
weight. A larger molecular weight only contributes to a longer rubbery plateau.23 An interesting example, completely different from the solutions examined here, involves diblock copolymer micelles in the melt. Yoshikawa et al.24 recently reported rheology measurements for polystyrene-b-poly(sodium methacrylate) (PS-PNaMA) diblock copolymers, in which strong association of the ionic blocks leads to stable micelles in the molten state. They found that polymers with similar PS chain lengths, but forming micelles with different aggregation numbers, exhibited nearly constant GN° values. For example, for polymers with 1085 styrene units, the aggregation number Nagg increased from 44 to 112 as the length of the ionic blocks was doubled from 21 to 42 units, but the GN° values were almost unchanged. These results suggests that GN° is related only to Me even in micellar block copolymers. Many of the rheological properties of associating polymer solutions can be described in terms of transient network theory.3,8,25 The connection between entangled polymers and associating polymers is that in both cases the junctions of the network are transient. For entangled systems, these junctions are topological in origin, whereas for associating polymers, exit from the junction is an activated process. This difference becomes striking when one compares the characteristic relaxation times of the two types of systems. The rheology of entangled polymers requires a distribution of relaxation times for a proper description, including the various Rouse modes and the reptation time.23 The finding of a single relaxation time for telechelic associating polymers has been taken as evidence for the absence of entanglements.3,8 While there is a connection at the level of the plateau modulus between polymer melts and associating polymer solutions (an effective chain length between network junction points), the origin of the relaxation time is completely different. Characteristic Lifetimes. We can calculate the characteristic lifetimes of the solutions in two ways, from their terminal properties and from the Maxwell model. Since the terminal region shifts to the lower frequencies as the polymer concentration increases (Figures 4 and 5), the longest relaxation time increases with polymer concentration. In terms of the terminal properties, the longest relaxation time, τm, is approximately equal to the product of the zero-shear viscosity and the steady-state shear compliance (τm ) η0Je°).21 Values calculated in this way are plotted in Figure 11 as a function of the polymer concentration. Both polymers exhibit a linear dependence on Cpol, with a slope of 0.26 ms/wt % for C6F-35K and one of 15.4 ms/wt % for C8F-35K. When lifetimes are calculated by fitting the G′(ω) and G′′(ω) data to the Maxwell model, somewhat smaller values are obtained (Table 1). The two polymers C6F-35K and C8F-35K have strikingly different relaxation times at all concentrations. At 5 wt %, the two values calculated from η0Je° differ by nearly a factor of 40, 2.1 ms for C6F-35K and 83 ms for C8F-35K. Annable et al.8 have ascribed the process dominating this relaxation time to the exit of the hydrophobic group from (23) (a) Roovers, J. (a) Macromolecules 1991, 24, 5895; (b) Polymer 1985, 26, 1091. (c) Onogi, S.; Masuda, T.; Kitagawa, K. Macromolecules 1970, 3, 109. (d) Fetters, L.; Kiss, A. D.; Pearson, D. S. Macromolecules 1993, 26, 647. (24) (a) Yoshikawa, K.; Desjardins, A.; Dealy, J. M.; Eisenberg, A. Macromolecules 1996, 29, 1235. (b) Desjardins, A.; Eisenberg, A. Macromolecules 1991, 24, 5779. (25) (a) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 247, 273, 289. (b) Groot, R. D.; Agterhof, G. M. J. Chem. Phys. 1994, 100, 1649, 1657. (c) Groot, R. D.; Agterhof, G. M. Macromolecules 1995, 28, 6824.
2454 Langmuir, Vol. 13, No. 9, 1997
Figure 11. Longest relaxation time, τm, defined by η0Je°, as a function of polymer concentration for C6F-35K and C8F-35K solutions at room temperature.
the micelle. It is well-known for traditional surfactant micelles that an increase of two CH2 groups can lead to more than a factor of 10 decrease in the exit rates of the surfactant from its micelle.26 Since CF2 groups are more hydrophobic than CH2 groups, this explanation is entirely consistent with our results. The exit rates we calculate are expected to be faster for the network under shear than for a corresponding micelle made up of a PEG chain substituted at only one end, because the forces associated with shear pull on the network. Another way of looking at these results is in terms of the zero-shear viscosity, which can be expressed as the product of GN° and τm. The solutions of C8F-35K are much more viscous than those of C6F-35K. Since both polymers have similar GN° values over the concentration range of interest, the viscosity differences between the solutions of the two polymers at comparable concentrations are due to the difference in the exit rates of the two different hydrophobes from their respective micelles. Annable et al.8 related the activation energy for the exit rate of a chain end of an associating polymer from a micelle to the energy cost of transferring the hydrophobe from the micelle to the aqueous environment. They noted that the activation energies they determined were similar to those for the exit of simple surfactants of comparable chain length from their micelles. While we have not carried out temperature studies, we can draw on the results of Amis et al.,17 who examined the low shear viscosity of comb and telechelic PEG polymers with fluorocarbon substituents. They show that this temperature dependence follows an Arrhenius behavior, and for a C8F17-end-capped PEG of M ) 35 000, they find an activation energy of about 17 kcal/mol, corresponding to 29 kT at 298 K. This is close to the value reported by Annable et al. for PEG polymers capped with C16H33 IPDU groups.8 Model for the Associated Network. Telechelic associative thickeners may be thought of as ABA triblock copolymers with very short end blocks, dissolved in a solvent selective for the middle block. In this kind of selective solvent, the end blocks tend to associate to form the core of micelle-like aggregates. Depending upon the concentration and other factors to be discussed below, some chains will loop back into the same core, while other chain will act as bridges, building up an extended structure. The bridging chains are mechanically active and contribute to the viscosity through the plateau modulus of the system, as described, for example, in eq 7. The key questions about these systems for which one (26) Israelachvili, J. N. Intramolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991.
Xu et al.
would like answers concern the structure of the system over the full range of concentrations: At low concentration, when is the onset of association and what is the nature of the species formed? What kinds of structures are formed at intermediate and high concentrations? In these structures, how many polymer molecules and how many chain ends come together to form each junction, and what is the fraction of bridging and looped chains in the system? In addition, one would like to understand the influence of shear and extensional flow on these structures at all concentrations. While we are far from achieving this goal, significant progress has been made. Two recent publications report experiments that establish that cubic structures can be formed in these kinds of systems at high concentrations. Raspaud et al.27 examined a polystyrene-polyisoprene-polystyrene (PSPI-PS, Mw ) 165 000, weight ratios of each block 15.3%, 69.4%, 15.3%) in heptane, a selective solvent for PI. This polymer undergoes association at concentrations above 1.6 × 10-3 g/mL, referred to as the critical aggregation concentration (cac). At higher concentrations, Bragg peaks appears in the neutron scattering curve. For example, at 16 wt %, five Bragg peaks can be resolved, from which a cubic structure can be deduced. Alami et al.28 examined polymers more closely related to C6F-35K and C8F-35K, a series of PEG samples of narrow molecular weight distribution with C12H25O or C12D25O groups attached directly to the chain ends. As in the case of the triblock copolymer described above, an disorder-to-order transition takes place with increasing concentration, appearing first as a single maximum in the scattering curve, with a second band appearing at higher concentrations. The positions of the peak maxima are consistent with a cubic structure. The extent of ordering was found to decrease as the length of the PEG chain increased (M ) 6000, 10 000, 20 000, 35 000). For the samples of MPEG ) 10 000 and 20 000, the sharp Bragg peak appears at concentrations near 50 wt %. At lower concentrations, the systems are disordered and more difficult to describe. We proposed the model for polymer association shown in Figure 12.9 According to this picture, if the onset of aggregation (the cac or cmc) is sufficiently low, the system initially associates into spherical micelles with looped chains. These are the flower-like micelles that appear in the theory of Semenov et al.12 Our model was based upon experiments involving an HEUR-type associative thickener, with a PEG backbone and C16H33O end groups attached to the PEO chain end IPDU groups.9,10 These polymers are structurally similar to C6F-35K and C8F-35K but with a much broader distribution of PEO chain lengths (Mw/Mn ≈ 1.7). By means of fluorescence and fluorescence decay measurements, using pyrene as probe, we could show that the polymer chains associate into traditional micelle-like aggregates with an average of NR ) 20 chain ends per micelle. The structure of this hydrophobic core was unchanged over a series of concentrations over which the bulk solution viscosity varied by a factor of 104. Association could be detected at concentrations below 10 ppm, and over a range of concentrations on the order of 0.1 wt %, NMR and dynamic light scattering experiments provided evidence for the existence of flower-like micelles. In addition, simultaneous fluorescence and flow experiments (27) (a) Raspaud, E.; Lairez, D.; Adam, M.; Carton, J.-P. Macromolecules 1994, 27, 2956; (b) Macromolecules 1996, 29, 1269. (28) Alami, E.; Rawiso, R. F.; Isel, F.; Beinert, G.; Binana-Limberle, W.; Franc¸ ois, J. In Hydrophilic Polymers: Performance with Environmental Acceptance; Glass, J. E., Ed.; ACS Advances in Chemistry Series 248; American Chemical Society: Washington, DC, 1996.
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Figure 13. Representation of the transition of a flower micelle to a “stripped” micelle with one chain end dangling in solution. The driving force for flower formation represents a balance between the energy of the free chain end in solution and the entropy penalty of the chain looping back into the core of the micelle.
made up of 2p diblocks (NA, NB/2) from the point of view of the blob scaling picture of the corona. The appropriate balance is between the cost in entropy due to backfolding of the middle block and the limited solubility of the end blocks, which drives association to minimize contacts of A-block monomers with the solvent. The free energy of looping takes the form Figure 12. Representation of the association structure of telechelic associating polymers in aqueous solution. At high concentrations, above c* based upon the dimensions of the flower-like micelles, the network fills space to form a weak gel.
suggest that the hydrophobic core retains its structure into the shear thinning region of the viscosity profile. As a consequence, in Figure 12 we depict the influence of shear as being similar to that of dilution, causing bridgeto-loop transitions, leading to smaller sized objects in the solution. This idea is very similar to that developed independently by Annable et al.8 We believe that this model can account for the phenomena we report here for C8F-35K, although for C6F35K the evidence is less compelling. For example, in the low-concentration region, the PGSE NMR results in combination with intrinsic viscosity measurements are consistent with C8F-35K micelles with a hydrodynamic radius of 25 nm containing about 15 polymer chains. For C6F-35K, we find micelles with RH ) 18 nm containing about 10 polymer molecules. According to the model in Figure 12, viscosity increases strongly with polymer concentration because of a rearrangement of the system, with bridging chains replacing looped chains.8-10,12,13 In our view, the micelles serve as the building blocks of the network, and the plateau modulus depends upon the number density of chains bridging micelles. The increase in viscosity and the increase in the plateau modulus with polymer concentration are then related to an increase in the concentration of bridging chains. This type of model has been questioned, both on theoretical grounds and experimentally. For example, Raspaud et al.27 demonstrate very clearly for their PSPI-PS sample in heptane that, at the onset of aggregation, flower-like micelles are not formed; rather, the system aggregates to form branched open structures which the authors refer to as “animals”. In a recent publication, Alami et al.10 report careful fluorescence and light scattering studies on the polymer C12H25O-PEG35 000OC12H25. They show by dynamic light scattering that, above the onset of association, multiple species are present. Their description of the transition with increasing polymer concentration from isolated chains to extended structures is closer to the picture of Raspaud et al. than that of Figure 12 and the theory of Semenov et al. At issue is the energy cost in forming a micelle from a triblock copolymer with poorly soluble end groups. Raspaud et al.27 provide a very nice analysis of this situation. They compare a flower-like micelle made up of p triblock molecules (NA, NB) with a similar star micelle
Floop 3 ) β ln N ˜B kT 2
(8)
where N ˜ B is the number of statistical segments of the middle block. For individual polymers, the looping coefficient β takes the value 1.0 for Gaussian chains and 1.3 for excluded volume chains. In micelles with looped corona chains, the finite radius of the core allows for different loci for reentry of the chain, which tends to lower the value of β. As a kind of minimum stability criterion for a flowerlike micelle, they propose a comparison between a flowerlike micelle and a “stripped” flower obtained by pulling one A-block out into the corona. This situation is depicted in Figure 13. If the energy cost of transferring a chain end into the solvent is less than the entropy of looping, then flower-like micelles will not be stable. The critical micelle concentration (or here, the cac) provides a measure of the energy of a free chain end in the solvent. And thus, if
N ˜B
xp
g
( ) 1 Φcac
2/3β
(9)
the flower conformation will be unstable. In their system, the length of the PI block exceeds N ˜ B, and Raspaud et al. calculate a gain in energy of about 6kT on extruding one PS block from the core into the solvent. They comment that micelles having free chain ends tend to associate and yield extended structures. For the HEUR polymer we studied previously, p ≈ 10 ˜B and φcac < 10 ppm.7a,29 Taking β ) 1, we calculate that N > 6800 from eq 9. For C8F-35K, p is also on the order of 8-10, and we estimate φcac < 100 ppm, so that the (29) The onset of association detected by the pyrene fluorescence probe experiments in this system was very broad and not sharp as one might anticipate for a critical association phenomenon. This is likely due to the broad polydispersity of the sample. Alami et al.28 detect a much sharper transition in their samples of narrow molecular weight distribution, although they comment that the transition in their system is broader than that found for traditional surfactants. It is worth commenting that most fluorescence probe experiments used to detect the onset of aggregation provide only an upper bound to the cmc or cac if these values are sufficiently small (e.g.,
