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Depletion Interactions in Model Microemulsions M. Zackrisson,* R. Andersson, and J. Bergenholtz Department of Chemistry, Go¨ teborg University, SE-412 96 Go¨ teborg, Sweden Received November 12, 2003. In Final Form: January 27, 2004 The effects of temperature changes and polymer addition on the behavior of droplet microemulsions of nonionic surfactant, water, and decane are reported and analyzed within polymer depletion theory. Dilution viscometry and dynamic light scattering were used to confirm that these microemulsions behave essentially as hard-sphere dispersions, providing us with an ideal reference system. Addition of poly(ethylene glycol) (PEG) lowers the emulsification failure boundary, where excess oil is expelled, which can be qualitatively understood by an analysis of the available volume for the polymer. Sufficient addition of PEG causes a fluid-fluid phase separation in qualitative accord with experiments on mixtures of rigid colloidal hard spheres and nonadsorbing polymer. Addition of PEG or raising the temperature causes the collective diffusion coefficient DC to decrease. From theory, the initial linear slope of DC versus droplet concentration can be used to discriminate between attractions and repulsions. The measured DC data for the droplets in the presence of PEG are modeled using the Asakura-Oosawa theory of depletion. Fitting the theory to the measured DC data permits for extracting the only unknown parameter, the polymer radius of gyration. Quantitative agreement is found with literature data, demonstrating that polymer depletion occurs in the system and that the Asakura-Oosawa theory provides a faithful description of the phenomenon.
Introduction Mixed aqueous solutions of surfactant and polymer occur in a large number of industrial, environmental, and biological applications. In the mixing of polymer and surfactant, one often seeks to impart to the mixed system some of the attributes possessed by the individual components, yielding systems with improved wetting, detergency, or rheological properties. As is well-known, interactions among the added components complicate matters. Because dissolved polymer is such a widely used additive to surfactant-based formulations, surfactant-polymer interactions have attracted much attention from the research community. Not surprisingly, most of the research has been aimed at strongly interacting systems of anionic surfactants mixed with neutral or charge-bearing polymer.1-3 In contrast, nonionic surfactants interact only weakly with neutral polymer,3 except when hydrophobic interactions are significant,4,5 which explains why such solutions have received far less consideration. While micelle aggregation numbers6 and microemulsion droplet sizes7-9 in nonionic systems appear to be relatively insensitive to added hydrophilic polymers, the phase behavior is significantly altered when polymer is introduced.7,10-14 Attempts have been made to explain the observed (1) Jones, M. N. J. Colloid Interface Sci. 1967, 23, 36. (2) Cabane, B.; Duplessix, R. J. Phys. (Paris) 1982, 43, 1529. (3) Kwak, J. C. T. Polymer-surfactant systems; Marcel Dekker: New York, 1998. (4) Brackman, J. C.; van Os, N. M.; Engberts, J. B. F. Langmuir 1988, 4, 1266. (5) Li, X.; Lin, Z.; Cai, J.; Scriven, L. E.; Davis, H. T. J. Phys. Chem. 1995, 99, 10865. (6) Anthony, O.; Zana, R. Langmuir 1994, 10, 4048. (7) Bagger-Jo¨rgensen, H.; Coppola, L.; Thuresson, K.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 4204. (8) Meier, W. Langmuir 1996, 12, 1188. (9) Filali, M.; Ouazzani, M. J.; Michel, E.; Aznar, R.; Porte, G.; Appell, J. J. Phys. Chem. B 2001, 105, 10528. (10) Wormuth, K. R. Langmuir 1991, 7, 1622. (11) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (12) Clegg, S. M.; Williams, P. A.; Warren, P.; Robb, I. D. Langmuir 1994, 10, 3390. (13) Robb, I. D.; Williams, P. A.; Warren, P.; Tanaka, R. J. Chem. Soc., Faraday Trans. 1995, 91, 3901.
immiscibility by polymer depletion in the immediate neighborhood of the micelles or microemulsion droplets.11-14 As is well-known for nonadsorbing polymers in lipid bilayer systems,15,16 rigid, colloidal-sphere dispersions,17,18 and emulsions,19 systems favor configurations in which polymer-depleted regions overlap; in this way, more volume is made accessible to the dissolved polymer. When viewed as pseudo one-component systems, this effect of polymer depletion appears as an effective attraction, at least at low polymer concentrations. Although added water-soluble polymer causes micellar12-14 and microemulsion7,11 systems to phase separate, leading to polymer- and surfactant-rich phases (segregation or demixing as opposed to association),20 it remains unclear whether a depletion attraction is the driving force. As pointed out by Piculell et al.,21 most cases studied concerned surfactant aggregates that are substantially smaller than the polymer coils. Experiments suggest that in such cases the surfactant aggregates penetrate the polymer coil and cause it to expand, owing to micellepolymer excluded-volume interactions.22 The penetration of small particles in larger polymer coils has indeed recently been addressed theoretically,23,24 a topic that is discussed also in a recent review.25 (14) Pandit, N. K.; Kanjia, J.; Patel, K.; Pontikes, D. G. Int. J. Pharm. 1995, 122, 27. (15) Evans, E.; Needham, D. Macromolecules 1988, 21, 1822. (16) Kuhl, T.; Guo, Y.; Alderfer, J. L.; Berman, A. D.; Leckband, D.; Israelachvili, J.; Hui, S. W. Langmuir 1996, 12, 3003. (17) Lekkerkerker, H. N. W.; Poon, W. C. K.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Europhys. Lett. 1992, 20, 559. (18) Rudhardt, D.; Bechinger, C.; Leiderer, P. Phys. Rev. Lett. 1998, 81, 1330. (19) Bibette, J.; Roux, D.; Nallet, F. Phys. Rev. Lett. 1990, 65, 2470. (20) Piculell, L.; Lindman, B. Adv. Colloid Interface Sci. 1992, 41, 149. (21) Piculell, L.; Bergfeldt, K.; Gerdes, S. J. Phys. Chem. 1996, 100, 3675. (22) Feitosa, E.; Brown, W.; Hansson, P. Macromolecules 1996, 29, 2169. (23) Louis, A. A.; Bolhuis, P. G.; Hansen, J. P.; Meijer, E. J. Phys. Rev. Lett. 2000, 85, 2522. (24) Schmidt, M.; Fuchs, M. J. Chem. Phys. 2002, 117, 6308. (25) Fuchs, M.; Schweizer, K. S. J. Phys.: Condens. Matter 2002, 14, R239.
10.1021/la036132y CCC: $27.50 © 2004 American Chemical Society Published on Web 03/18/2004
Depletion Interactions
The role of depletion in bicontinuous microemulsions has been studied and discussed by Kabalnov and coworkers.26,27 Recently, polymer depletion has been observed in a system containing nonionic microemulsion droplets and hydrophobically modified polymer by smallangle neutron scattering.28 It is prudent to remark also that surfactant aggregates may themselves act as depletion agents, for instance, to cause phase separation in otherwise homogeneous dispersions of solid particles29 or emulsion drops.19,30 In this study, we investigate whether a polymer-induced depletion attraction can be detected among the droplets of a model microemulsion system using low-molecularweight poly(ethylene glycol) (PEG). As shown by the extensive work of Olsson and co-workers, the nonionic surfactant C12E5 together with decane and water forms stable oil-in-water droplet microemulsions. Here, C12E5 is the commonly used nomenclature for penta(ethylene glycol) dodecyl ether, a surfactant with the property that the average area per surfactant at the polar/apolar interface is 45 Å2 and insensitive to changes in composition and temperature.31 In particular, for a surfactant-oil mass ratio of 51.9/48.1, they have shown that the microemulsion phase (L1) is well-modeled as a dispersion of hard-sphere colloids with a hydrocarbon core radius of 75 Å over a wide range of droplet concentrations when the system is near the limit of maximum oil solubilization.32-34 This limit is reached at a particular temperature that is essentially independent of the droplet concentration.32 At this point, the microemulsion droplets are close to expelling a macroscopic excess phase of oil (L1 + O). Self- and collective-diffusion coefficients,32 low-shear viscosity,35 osmotic compressibility,32,34 and osmotic pressure36 are among the properties that can be reconciled within hardsphere theory. In addition, the microemulsion droplets exhibit a size polydispersity of no more than 16%,34 about a factor of 4 or so greater than what can be reached with synthesis of the solid spheres that serve as the best model systems for hard-sphere behavior currently;37,38 hence, we view this particular system as the closest at present to a model, droplet-structured microemulsion system. As with true model systems, the ternary C12E5/water/ decane microemulsion system offers opportunity for systematically studying effects of introducing some added complexity to the system. For example, controlled amounts of charge have been added to the otherwise neutral droplets.39 In this report, we seek to identify attractive depletion interactions by adding neutral, low-molecularweight polymer. We do this by quantitative analysis of the viscosity and collective diffusion coefficient when the system is dilute with respect to the droplets. Part of the (26) Kabalnov, A.; Olsson, U.; Wennerstro¨m, H. Langmuir 1994, 10, 2159. (27) Kabalnov, A.; Olsson, U.; Wennerstro¨m, H. J. Phys. Chem. 1995, 99, 6220. (28) Frielinghaus, H.; Byelov, D.; Allgaier, J.; Richter, D.; Jakobs, B.; Sottmann, T.; Strey, R. Appl. Phys. A 2002, 74, S408. (29) Piazza, R.; di Pietro, G. Europhys. Lett. 1994, 28, 445. (30) Aronson, M. P. Langmuir 1989, 5, 494. (31) Olsson, U.; Wu¨rz, U.; Strey, R. J. Phys. Chem. 1993, 97, 4535. (32) Olsson, U.; Schurtenberger, P. Langmuir 1993, 9, 3389. (33) Olsson, U.; Schurtenberger, P. Prog. Colloid Polym. Sci. 1997, 104, 157. (34) Bagger-Jo¨rgensen, H.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 1413. (35) Leaver, M. S.; Olsson, U. Langmuir 1994, 10, 3449. (36) Bagger-Jo¨rgensen, H.; Olsson, U.; Jo¨nsson, B. J. Phys. Chem. B 1997, 101, 6504. (37) Pusey, P. N.; van Megen, W. Nature 1986, 320, 340. (38) Phan, S.-E.; Russel, W. B.; Chang, Z.; Zhu, J.; Chaikin, P. M.; Dunsmuir, J. H.; Ottewill, R. H. J. Chem. Phys. 1996, 54, 6633. (39) Evilevitch, A.; Lobaskin, V.; Olsson, U.; Linse, P.; Schurtenberger, P. Langmuir 2001, 17, 1043.
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motivation for turning to measurements of the collective diffusion coefficient, aside from the measurements themselves being straightforward to carry out, is to exploit that the collective diffusivity can be used to distinguish to some extent between attractions and repulsions,40 just like the familiar second virial coefficient of the osmotic compressibility. Moreover, we wish to test whether such dynamic measurements of a complex mixture are compatible with theory on the basis of a pseudo one-component model of the system. For static properties, one may expect this on formal grounds,41 but the dynamical response of the system is another matter altogether. In the present case, is the dominant effect of the polymer on the dynamics merely to mediate an effective, isotropic attraction? In what follows, we begin by verifying the hard-spherelike behavior of the one-phase microemulsion system in the absence of polymer, using dilution viscometry and dynamic light scattering. In addition, we assess the range of validity of the effective hard-sphere model and how it fails on displacing the system from the limit of maximum oil solubilization, that is, by increasing the temperature. Addition of nonadsorbing polymer moves the maximum oil solubilization limit to lower temperatures, which we model qualitatively by consideration of the free volume available to the polymer. Next, the collective diffusion coefficient of polymer-containing microemulsions is analyzed using the Asakura-Oosawa (AO) theory of depletion.11,42,43 Fits to the experimental data allow for extracting the only adjustable parameter, the polymer radius of gyration, which is in good quantitative agreement with literature data. 1. Experimental Section The nonionic surfactant C12E5, n-decane, and PEG of molecular weight (Mw) 3350 were obtained from Sigma and were used as received. Microemulsion stock solutions were prepared by weighing in C12E5, Milli-Q water, and decane, giving a mass ratio of surfactant to oil of 51.9/48.1.32 Aqueous PEG stock solutions were prepared by weight as well, and samples were obtained by mixing appropriate weights of the stock solutions. Samples for the phase-diagram studies were enclosed in flamesealed glass ampules together with small magnetic stirring bars, used to provide good mixing. The emulsification failure boundary (EFB; both in the presence and in the absence of polymer), that is, the temperature at which homogeneous microemulsions start to expel oil,44 was determined by visual inspection of samples of varying composition submerged in a constant-temperature water bath. Samples were quenched well into the two-phase (L1 + O) region, and the disappearance of the top oil phase was monitored as a function of incrementally increasing the temperature, allowing for equilibration overnight at each temperature. In other words, the resolubilization of oil was monitored, which provides a more reliable measure of the phase boundary owing to slow kinetics of the reverse process.45,46 However, on approaching the L1 boundary from below, that is, lower temperatures, we found that the emulsification required lengthy equilibration times to prevent premature resolubilization. The density of dilution series of polymer-free and polymercontaining samples was determined using a density meter (Paar DMA5000). The reciprocal of the density behaved as a linear function of the droplet (surfactant plus oil) weight fraction in the (40) van den Broeck, C.; Lostak, F.; Lekkerkerker, H. N. W. J. Chem. Phys. 1981, 74, 2006. (41) Brader, J. M.; Dijkstra, M.; Evans, R. Phys. Rev. E 2001, 63, 041405. (42) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (43) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (44) Leaver, M. S.; Olsson, U.; Wennerstro¨m, H.; Strey, R.; Wu¨rz, U. J. Chem. Soc., Faraday Trans. 1995, 91, 4269. (45) Leaver, M. S.; Olsson, U.; Wennerstro¨m, H.; Strey, R. J. Phys. II (France) 1994, 4, 515. (46) Evilevitch, A.; Olsson, U.; Jo¨nsson, B.; Wennersto¨m, H. Langmuir 2000, 16, 8755.
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range measured (up to 10 wt % droplets), both in the absence and in the presence of PEG (3 wt %). Assuming ideal mixing between droplets and solvent, the droplet density, extracted from the linear least-squares slope, was found to be equal to 0.8450 g/cm3 at 23.00 °C. This value was used to calculate the “gravimetric” volume fractions. In addition, density measurements conducted at varying temperatures were used in converting the kinematic viscosity from the viscometry measurements to the absolute viscosity. Capillary dilution viscometry was conducted at various temperatures using Ubbelohde dilution viscometers (Cannon Instruments). The temperature was controlled by a constanttemperature water bath to within (0.03 °C. The intrinsic viscosity was determined by a linear least-squares extrapolation to infinite dilution of the reduced viscosity, taking into account the variance in the data (obtained from standard deviations from the mean of five consecutive measurements). This analysis is complicated by the need to truncate the dilution series at some droplet concentration to ensure minimal interference from higher-order terms in the concentration expansion (cf. eq 3). This was accomplished by estimating a “goodness-of-fit” (section 15.2 in ref 47) for each data series after successive deletion of highconcentration data points. The maximum number of data points that delivers an acceptable quality of regression is indicated in the legend of Figure 2. Dynamic light scattering measurements were performed on filtered samples using a Malvern Instruments Series 7032 Multi-8 correlator and PCS 100 spectrometer at a wavelength of λ ) 632.8 nm. The scattering angle was maintained at 90°, and the temperature was kept at 23 °C for the polymer-containing microemulsion samples. The reported values are averages of at least five consecutive measurements. In the temperature study, the samples were thermostated at the relevant temperature for 24 h prior to measurement. At each measuring point, the temperature was kept constant within (0.05 °C. The collective diffusion coefficient was extracted from the initial, short-time decay of the auto-correlation function by a second-order cumulant analysis.48
pressure Πp ) n(R) p kBT. As emphasized by Lekkerkerker et al.17 and Ilett et al.,49 the polymer number density n(R) p should be based on the available (free) volume, Vfree; that is, the volume excluded by the ensemble-averaged sphere configuration should be subtracted from the total sample volume. Therefore, the free volume Vfree accessible to the center of the polymer coils is related to the total volume by Vfree ) RV, where the free-volume fraction, R, has been introduced. At very low colloid volume fractions, R ≈ 1 φ(1 + ξ)3, which suffices for our analysis of the collective diffusivity. An approximate expression for R, from scaled particle theory, depending only on φ and ξ, as suggested by Lekkerkerker et al.17 for larger colloid concentration, is referred to in Appendix A. The AO depletion potential has two control parameters: the polymer concentration, which sets the depth of the attraction, and the polymercolloid size ratio, ξ, which controls the range of the attraction. 2.2. Dilute-Limiting Viscosity and the Effective Hard-Sphere Model. The reduced viscosity of a dispersion of particles is given by
ηred )
{
0 < r < 2RHS ∞ -Π V u(r) ) p overlap 2RHS < r < 2(RHS + RG) (1) 2(RHS + RG) < r 0 where r is the center-center separation distance between the spheres, RHS is the radius of the hard spheres, and Πp is the osmotic pressure of a polymer solution in equilibrium with the mixture. In the AO theory, the conformational degrees of freedom of the polymer are neglected by treating polymer coils as spheres with a radius usually taken to be the radius of gyration, RG. The polymer-polymer interaction is not included, but we expect the theory to be reasonably accurate at low polymer concentrations. In eq 1, the so-called overlap volume is given by
Voverlap )
4π 3 R 3(1 + ξ)3 1 - (r/RHS)(1 + ξ)-1 + 3 HS 4 1 (r/RHS)3(1 + ξ)-3 (2) 16
[
ηHS red )
(47) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRAN, 2nd ed.; Cambridge University Press: Cambridge, 1992. (48) Koppel, D. J. Chem. Phys. 1972, 57, 4814.
η/η0 - 1 5 ) + CφHS + O(φHS2) φHS 2
(4)
The coefficient C ) 5.913 for hard spheres,50,51 but it will assume different values for other interaction potentials between the spheres. We follow Olsson and co-workers in modeling the microemulsion droplets as rigid spheres by rescaling the gravimetric volume fraction as φHS ) RHSφ, where RHS will be defined in this work by enforcing the Einstein coefficient to equal 5/2. A comparison between eq 3 and eq 4 leads then to
RHS )
Fp[η] 2 ) lim ηred 5/2 5 φf0
(5)
This procedure is essentially the same as that commonly used in modeling sterically stabilized, rigid spheres with short surface layers,52 except that we also determine Fp in a separate experiment. 2.3. Dynamic Light Scattering and Modeling of the Collective Diffusion Coefficient. Here, we summarize the theory needed for the analysis of the collective diffusion coefficient.40,53 For monodisperse dispersions of colloids, the initial decay of the autocorrelation function is of single exponential form:
]
where ξ ) RG/RHS is the polymer-colloid size ratio. The polymer solution is taken to be ideal with an osmotic
(3)
where η0 is the solvent viscosity, Fp is the mass density of the dispersed particles, and φ is the gravimetric volume fraction, related to the mass concentration via Fpφ ) c. To this order in φ, two coefficients characterize the viscosity, the intrinsic viscosity [η], and the Huggins coefficient kH. For rigid spheres, the corresponding relation is
2. Theory 2.1. Depletion Interaction. For hard spheres immersed in a solution of nonadsorbing, neutral polymer, AO42,43 and Vrij11 have derived an effective sphere-sphere interaction potential, given by
η/η0 - 1 ) Fp[η] + Fp2[η]2kHφ + O(φ2) φ
g(1)(q, t) ) e-q DC(q)t 2
(6)
with (49) Ilett, S. M.; Orrock, A.; Poon, W. C. K.; Pusey, P. N. Phys. Rev. E 1995, 51, 1344. (50) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97. (51) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1988, 89, 3705. (52) van der Werff, J. C.; de Kruif, C. G. J. Rheol. 1989, 33, 421. (53) Russel, W. B.; Glendinning, A. B. J. Chem. Phys. 1981, 74, 948.
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H(q) DC(q) ) D0 S(q)
(7)
H(q) is the so-called hydrodynamic function, and S(q) is the static structure factor, both of which are functions of the magnitude of the scattering vector, q ) (4π/λ)n sin θ/2, where n is the solvent refractive index, λ is the wavelength of light in vacuo, and θ is the scattering angle. Because the droplets we study are small, DC(q) ≈ D0H(0)/ S(0) + O[(qRHS)2], and we require only the q f 0 limit DC/D0 ) H(0)/S(0) ) 1 + λCφHS + O(φHS2), where λC is the dilute-limiting slope on a plot of DC/D0 versus φHS. The coefficient λC depends on the type of interparticle interaction. For hard-sphere particles λC ≈ 1.454.54,55 As realized by van den Broeck et al.,40 the collective diffusion coefficient can be expressed as
DC ≈ 1 + 1.454φHS - 3φHS D0
∫2∞ dx x2[1 - Q(x)] ×
[e
-u(r)/kBT
2
- 1] + O(φHS ) (8)
where
2 1 Q(x) ) [xa11(x) + xa12(x)] + [ya11(x) + ya12] - 1 (9) 3 3 xa11,
is of purely hydrodynamic origin. Here, x ) r/RHS and xa12, ya11, and ya12 are mobility functions, in the notation of Jeffrey and Onishi,56 of the like-sized, two-sphere hydrodynamic mobility problem. From eq 8, λC equals the sum of the last two terms in eq 8 divided by φHS. As recognized by van den Broeck and co-workers, 1 Q(x) is a positive-definite quantity, whereas e-u(x)/kBT - 1 is negative-definite for pure repulsions and positivedefinite for pure attractions beyond the hard core at x ) 2. In other words, for particles with a hard core plus an additional (pure) repulsion, u(x) > 0 for x > 2, the last term in eq 8 containing the integral is positive; hence, λC > 1.454, that is, repulsions lead to an increased λC compared to true hard spheres. On the other hand, for pure attractions beyond the hard-core repulsion, u(x) < 0 for x > 2, we find the opposite result, namely, pure attractions cause λC to decrease below the hard-sphere value. Inserting the depletion potential from eq 1 in eq 8 and integrating numerically, we can determine theoretical estimates for λC. In Table 1, we tabulate λC as a function of the two parameters of the depletion potential, the 3 (R) polymer volume fraction φ(R) p ) (4/3)πRG np , and the size ratio ξ. In these calculations, Q(x) was determined according to the prescription provided by Jeffrey and Onishi.56 3. Results and Discussion 3.1. Polymer-Free Microemulsions. The threecomponent nonionic microemulsions display a rich phase behavior on traversing the phase diagram in terms of composition and temperature, similar to that of binary nonionic surfactant/water systems.57 For a constant surfactant-to-oil mass ratio of 51.9/48.1, Leaver et al.44 determined the microstructure on varying both concentration and temperature, yielding the phase diagram reproduced in Figure 1. At low temperatures, a droplet (54) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1. (55) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1988, 89, 1049. (56) Jeffrey, D. J.; Onishi, Y. J. Fluid Mech. 1984, 139, 261. (57) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253.
Figure 1. Phase diagram of C12E5/decane/H2O microemulsions in terms of the temperature and gravimetric volume fraction of the droplets (surfactant + oil). Table 1. Numerically Calculated Results for the Coefficient λC in the Expansion DC ) D0[1 + λCOHS + O(OH2)] as a Function of the Parameters in the Depletion Potential (cf. Eq 1) ξ ) RG/RHS φ(R) p
0.05
0.10
0.15
0.20
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
1.454 1.283 1.030 0.644 0.045 -0.903 -2.430 -4.925 -9.052
1.454 1.288 1.087 0.841 0.537 0.161 -0.309 -0.899 -1.643
1.454 1.282 1.087 0.863 0.606 0.310 -0.031 -0.428 -0.888
1.454 1.273 1.074 0.854 0.609 0.337 0.034 -0.305 -0.684
phase (L1) is stable up to high droplet concentrations. Upon increasing the temperature, the system transforms to a lamellar phase (LR). At even higher temperatures, a bicontinuous, isotropic sponge phase (L3) appears. The phases encountered on raising the temperature can be rationalized by a preferred curvature of the surfactant that decreases with temperature.58 On crossing the lowertemperature phase boundary of the L1 phase, the surfactant fails to retain the amount of oil present, resulting in an almost pure oil phase coexisting with the droplet microemulsion (L1+O) below this so-called EFB or solubilization limit (Figure 1). Experimental measurements32,35,59,60 have been used to show that near the EFB the microemulsion droplets are well-modeled as hardsphere colloidal particles. Properties such as the osmotic compressibility and collective and self-diffusion can be brought in agreement with hard-sphere theory simply by introducing an effective droplet size. The spherical shape of the microemulsion droplets at the EFB is consistent with theoretical predictions for the free energy over a flexible surface where the spontaneous/preferred radii are close to the actual radii of curvature.61,62 According to (58) Anderson, D.; Wennerstro¨m, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243. (59) Fletcher, P. D. I.; Holzwarth, J. F. J. Phys. Chem. 1991, 95, 2550. (60) Olsson, U.; Bagger-Jo¨rgensen, H.; Leaver, M.; Morris, J.; Mortensen, K.; Strey, R.; Schurtenberger, P.; Wennerstro¨m, H. Prog. Colloid Polym. Sci. 1997, 106, 6. (61) Turkevich, L.; Safran, S. A.; Pincus, P. A. In Surfactants in solution; Mittal, K., Bothorel, B., Eds.; Plenum: New York, 1984. (62) Safran, S. A. Phys. Rev. A 1991, 43, 2903.
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Figure 2. Reduced viscosity as a function of droplet mass concentration and temperature as labeled. The lines are weighted linear least-squares fits to the data. The number within parentheses in the legend is the number of data points (counted from the lowest concentration) used in the regression analysis to yield an acceptable “goodness-of-fit”;47 that is, not all the data shown were used in the regression.
Fletcher and Petsev, the hard-sphere nature of the droplets is connected to the correspondence between the preferred and the actual radii of curvature.63 In Figure 1, we superpose the EFB determined for our microemulsions on the phase diagram of Leaver et al.44 As reported by others and as seen in Figure 1, the EFB is insensitive to changes in droplet concentration. Our microemulsions phase separate at ≈22°, a few degrees below the temperature determined by Leaver and coworkers. We attribute this discrepancy to our using C12E5 from a different supplier. A similar temperature shift of the phase diagram owing to the source of C12E5 has been noted by Kahlweit et al.64 To confirm the hard-sphere-like nature of the droplets in our study as well as to set the range of validity of the effective hard-sphere model, we examine the behavior of the single-phase microemulsions away from the EFB using dilution viscometry and dynamic light scattering. Figure 2 shows that the reduced viscosity behaves as a linear function of φ for sufficiently small values of φ. From a linear least-squares analysis of the data at T ) 23 °C, we obtain Fp[η] ≈ 2.89 and RHS ) 1.15 ( 0.01. In other words, as far as the dilute-limiting viscosity is concerned, a rigid sphere model for the microemulsion droplets requires us to inflate the droplet volume by 15%. Our RHS value is in excellent agreement with that obtained by Olsson and collaborators, who found RHS ) 1.14 by comparing hard-sphere theory with a number of techniques yielding the long-time self-diffusivity, collective diffusivity, isothermal compressibility, and viscosity (measured over a much wider droplet concentration range). Our analysis also demonstrates that it is not necessary to introduce a separate scaling factor for the dilute viscosity as suggested by Leaver and Olsson.35 It is worth noting that more sophisticated models than an effective hard-sphere model can be adopted, for instance, by modeling solvent penetration in the steric layer surrounding the hydrocarbon core. Although some (63) Fletcher, P. D. I.; Petsev, D. N. J. Chem. Soc., Faraday Trans. 1997, 93, 1383. (64) Kahlweit, M.; Strey, R.; Sottmann, T.; Busse, G.; Faulhaber, B.; Jen, J. Langmuir 1997, 13, 2670.
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Figure 3. Relative viscosity of a pure microemulsion with φ ) 0.10 as a function of temperature. The vertical line marks the temperature of emulsification failure, where the L1 microemulsion phase expels excess oil.
models in this vein produce analytical formulas for the single-particle properties such as D0 and [η],65-67 we do not pursue them further here, staying instead with the effective hard-sphere model in keeping with past modeling efforts.32 We have also measured the viscosity of microemulsions as a function of temperature at somewhat higher droplet concentrations. Because the hard-sphere model has been proposed for microemulsions along the EFB, we wish to determine how far from this boundary one can stray before the hard-sphere model fails and in which way it fails. One simple test is to examine the temperature dependence of the viscosity under dilute conditions. Figure 3 shows that the viscosity increases with increasing temperature at a fixed droplet volume fraction. As seen, the deviation is insignificant within 2 °C of the emulsification failure temperature, consistent with the data in Figure 2. Thus, measurements at a fixed temperature of 23 °C instead of closer to 22 °C do not alter the interactions in any significant way. We will exploit this property in the ensuing analysis of the polymer-containing microemulsions. The reason for the increase of the viscosity with temperature away from the L1 + O boundary is apparent from Figure 2. Evidently, the source of the increase is a systematic increase of the O(φHS2) term in the expansion of the relative viscosity (the coefficient C in eq 4) or, equivalently, the Huggins coefficient kH. In passing, we note that an increasing droplet size polydispersity with increasing temperature above the EFB64 should lead to a decreasing kH,68 which we do not observe. In contrast to kH, the intrinsic viscosity remains largely unaffected by temperature. Because the microemulsion droplets are spherical near the EFB34 and the intrinsic viscosity depends on the shape of the dispersed particles, we can say with fair confidence that they remain spherical several degrees away from the boundary. To give an indication of the sensitivity of the intrinsic viscosity to shape deviation, we refer to Scheraga69 to see that prolate or oblate ellipsoids always increase the intrinsic viscosity over that (65) Masliyah, J. H.; Neale, G.; Malysa, K.; van de Ven, T. G. M. Chem. Eng. Sci. 1987, 42, 245. (66) Anderson, J. L.; Kim, J.-O. J. Chem. Phys. 1987, 86, 5163. (67) Zackrisson, M.; Bergenholtz, J. Colloids Surf. 2003, 225, 119. (68) Wagner, N. J.; Woutersen, A. T. J. M. J. Fluid Mech. 1994, 278, 267. (69) Scheraga, H. A. J. Chem. Phys. 1955, 23, 1526.
Depletion Interactions
of spheres in the strong Brownian motion limit, which is fulfilled in our viscometry experiments. Even on the basis of data of high quality, such as those in Figure 2, it is difficult to distinguish the degree to which the particle shape deviates as long as the axial ratios remain less than about 1.5 because it then only produces a j6% increase in the intrinsic viscosity. The increase in intrinsic viscosity grows rapidly, however, with the axial ratio increasing above 1.5. Nevertheless, the data in Figure 2 are consistent with droplets that maintain their shape away from the L1 + O boundary but whose mutual interaction changes, very weakly within the first few degrees but rather dramatically after 4-5 °C from the boundary. These results are not in conflict with previous analyses that attributed deviations away from the L1 + O boundary to droplet growth. The measurements in Figures 2 and 3 are confined to low droplet volume fractions; as seen from Figure 2, the expansion of the viscosity to O(f 2) describes the data well for c j 0.08 (φ j 0.10). Leaver et al.45 used relaxation- and self-diffusion NMR to investigate the system behavior above the EFB at higher φ, where their data were consistent with growth into nonspherical droplets with increasing temperature away from the EFB. Analysis of the initial exponential decay of the autocorrelation function from dynamic light scattering measurements yields a short-time diffusion coefficient DC(q). As is well-known, this technique probes variations in the particle concentration of the spatial extent 2π/q that arise because of Brownian motion. Because the microemulsion droplets in the C12E5/decane/water system are small compared to the wavelength of light, in the present case qRHS ≈ 0.16, the simultaneous motion of many droplets contributes to the relaxation of concentration fluctuations. Hence, we measure the collective diffusion coefficient DC. This is seen also from the leading-order result DC(q) ) DC + O[(qRHS)2], where the correction is small for the small droplets in this study. The collective diffusion coefficient can be used to discriminate between pure attractions and repulsions relative to hard-sphere interactions.40 The effect of droplet-droplet pair interactions enters the slope of DC versus volume fraction in the dilute limit. Pure attractions lower the slope relative to that of hard spheres, and pure repulsions increase it above the hard-sphere value. The collective diffusion coefficient, measured at various temperatures above the solubilization limit, is shown as a function of φHS in Figure 4. As the temperature is raised, the initial slope of DC decreases. The effect is small just above 22 °C, but it becomes pronounced a few degrees away, qualitatively consistent with the viscosity data in Figure 2. As shown in the inset to Figure 4, the hydrodynamic radius remains constant, at least as long as we remain within 4 °C of the solubilization boundary. The value agrees well with that reported by Olsson and Schurtenberger,32 demonstrating that the system studied here behaves the same way as theirs near the solubilization limit despite the observed temperature shift of the EFB (cf. Figure 1). The systematic decrease in the slope in Figure 4 with increasing temperature is consistent with an increasing attraction among droplets. It has been observed previously in similar systems by Fletcher and co-workers,70 who propose the attraction to stem from a growing difference in preferred and actual droplet radii with temperature away from the solubilization limit.63 3.2. Polymer-Containing Microemulsions. Adding nonadsorbing polymer alters the phase behavior of the microemulsion system. As seen in Figure 5, where the (70) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I. Langmuir 1989, 5, 1210.
Langmuir, Vol. 20, No. 8, 2004 3085
Figure 4. Collective diffusion coefficient normalized by the dilute-limiting value D0 versus hard-sphere volume fraction at various temperatures, as labeled. The inset shows the hydrodynamic radius calculated from the Stokes-Einstein relation D0 ) kBT/6πη0RH.
Figure 5. Effect on the emulsification failure of the C12E5/ decane/H2O microemulsion on addition of PEG at a fixed droplet volume fraction, φ ) 0.105.
effect of PEG3350 on microemulsions of fixed droplet concentration, φ ) 0.105, is studied, samples at higher temperature or with enough polymer split into two liquid phases, neither exhibiting optical birefringence between crossed polarizers. This behavior is reminiscent of polymerinduced phase separation in rigid-sphere colloidal systems49 and has been observed in a number of polymermicroemulsion systems.7,12-14 Because we are primarily concerned with the behavior near the EFB at dilute droplet concentrations, we have not pursued an analysis of the composition of the two liquid phases. Bagger-Jo¨rgensen et al.7 observe the same liquid-liquid equilibrium in the same system with added PEG8000 and have determined that the surfactant partitions strongly between the two phases. It remains unknown whether the liquid phases correspond to dilute and concentrated droplet microemulsion phases with the dilute one enriched in polymer, as would be expected from depletion theory.12,13,17,49 The phase behavior at yet higher temperatures can be found in ref 7. Figure 5 shows that the EFB shifts to lower temperatures with increasing polymer concentration, in qualitative agreement with the results of Bagger-Jo¨rgensen et
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al.7 They attribute this trend to a dehydration of the hydrophilic headgroup of the surfactant, leading to a decrease in spontaneous curvature.27 Using the model suggested in ref 27 for this effect, we estimate a maximum swelling (for the 4 wt % PEG samples) of the microemulsion droplets in this study of roughly 7%, compared to the polymer-free microemulsionsa rather small change. Indeed, several studies find little change in droplet dimension on adding nonadsorbing (even hydrophobically modified) ethylene oxide main-chain polymers,7-9 at least at not-too-high polymer concentrations.71 Aside from possible changes in the spontaneous curvature, there is an additional mechanism that can cause the observed shift of the EFB. The “particles” have the freedom to change not only their size and number but also their average spatial arrangement relative each other in response to the added polymer. Making use of expressions from depletion theory,17 which captures the fact that the volume accessible to the polymer is altered by the equilibrium arrangement of the particles, we can qualitatively explain the behavior within a first-order perturbation analysis. As shown in Appendix A, comparing the free volume in the two-phase (L1 + O) system, V2ph free, relative to that in the homogeneous (L1) system, Vfree, within the Lekkerkerker et al. depletion theory,17 gives the following relation: 2 V2ph free/Vfree ) 1 - φ(ξ, φ) + O( )
Figure 6. Collective diffusion coefficient at T ) 23.0 °C as a function of the effective hard-sphere droplet volume fraction and varying PEG concentrations, as labeled. Lines are shown as guides to the eye.
(10)
where ) Vtop/V is the volume of expelled oil in the twophase system divided by the total sample volume and φ is a function of the size ratio ξ and volume fraction φ in the homogeneous system (see Appendix A for details). The function φ is positive for all relevant values of ξ and φ, so it follows that to leading order in the free volume in the two-phase system is smaller, that is, (V2ph free/Vfree) < 1. This implies that there is a gain in translational entropy of the polymer if the system remains homogeneous, providing a driving force toward the fully emulsified system not present in the absence of polymer. Determining whether polymer entropy is indeed the main cause of the EFB shift or merely a contributing factor would require a more complete thermodynamic analysis. We now turn to the measurements of the droplet dynamics. For the purpose of attempting a quantitative analysis of the collective diffusivity on introducing polymer to the system, we have restricted our measurements to low droplet concentrations, φ < 0.12 (φHS < 0.14). As seen from Figure 6, adding increasing amounts of low-molecular-weight PEG (Mw ) 3350) decreases the collective diffusion coefficient, DC, of the microemulsion droplets. We observe a systematic decrease of the intercept, D0, as well as of the dilute-limiting slope, denoted by λC, with increasing PEG concentration. Moreover, we measure the collective diffusion coefficient of the droplets because the contribution to the scattering from the smaller PEG coils is negligible in comparison. Focusing for the moment on the decrease of D0 as a function of increasing polymer content, there are a number of possible rationalizations of this behavior. This coefficient reflects a situation in which microemulsion droplets, isolated from one another, diffuse in a dilute polymer solution because the polymer concentration is below the dilute-semidilute crossover (overlap) concentration (cf. Figure 9). Adding more polymer creates more hindrance to droplet diffusion, and we indeed expect qualitatively (71) Filali, M.; Aznar, R.; Svenson, M.; Porte, G.; Appell, J. J. Phys. Chem. B 1999, 103, 7293.
Figure 7. Representation of the effect of polymer addition on the dilute-limiting diffusion coefficients from Figure 6. The solid line is the model prediction for D(PET) /D0 in eq 11 with ξ ) 0.2 0 and the experimentally determined ηPEG/η0. The chain curve is the same model prediction for D(PEG) /D0 multiplied by ηPEG/η0, 0 ηPEG/D0η0. yielding a prediction for D(PEG) 0
D0 to decrease with increasing polymer concentration. In Figure 7, the dilute-limiting droplet diffusion coefficient is shown together with measured data for the viscosity of PEG3350 solutions at the same temperature. The simplest analysis assumes that the effect of the polymer is merely to increase the solvent viscosity and that the StokesEinstein formula, D(PEG) ) kBT/6πηPEGR(PEG) , remains 0 H valid with the solvent viscosity replaced by the viscosity of the bulk polymer solution. In this case, one obtains D(PEG) /D0 ) (η0/ηPEG)(RH/R(PEG) ) or D(PEG) ηPEG/D0η0 ) RH/ 0 H 0 (PEG) RH , where the superscript (PEG) denotes quantities in the presence of added PEG. If this were the case, one would conclude from the data in Figure 7 that the droplets decrease in size when PEG is added, contrary to the model developed in ref 27 and studies that report little or no size change.7-9 Deviations from the Stokes-Einstein formula have been studied by Gold et al.72 for wide ranges of polymer concentration and varying solvent quality. They report positive deviations under good solvent conditions, which
Depletion Interactions
Langmuir, Vol. 20, No. 8, 2004 3087
Figure 8. Normalized collective diffusion coefficient at T ) 23 °C as a function of the hard-sphere droplet volume fraction and polymer concentration. The lines are single-parameter fits to the data using RG as an adjustable parameter in eqs 1 and 8, except for the top line, which is the hard-sphere prediction DC/D0 ) 1 + 1.454φHS.54,55
prevail in the systems studied here. Moreover, they suggest a model whereby polymer-depleted zones of lower viscosity than that of the bulk polymer solution cause D(PEG) ηPEG/D0η0 to obtain values greater than unity; that 0 is, because polymer is depleted in the immediate neighborhood of the diffusing particles the “solvent” viscosity is somewhat lower there than the bulk polymer solution value. Idealizing the situation considered by Gold et al. rather drastically, we can derive an analytical prediction /D0. We model the droplets as rigid spheres of for D(PEG) 0 , centered in a spherical pocket of radius radius R(PEG) H R(PEG) + R /2 containing water, and surrounded beyond G H R(PEG) + R /2 by a continuum of bulk viscosity ηPEG. As G H shown in more detail in Appendix B, we arrive at the following solution D(PEG) /D0 ) 0
RH R(PEG) H
×
4 + 6κ5 + ηr2(κ - 1)4(4 + 7κ + 4κ2) - ηr(8 - 9κ + 10κ3 - 3κ5 - 6κ6) 2κηr[2 + 3κ5 + 2ηr (κ5 - 1)] (11)
+ RG/2)/R(PEG) ≈ 1 + ξ/2 and ηr ) in terms of κ ) (R(PEG) H H ηPEG/η0. Assuming that the droplets do not change size when polymer is added and that ξ ) 0.24 (cf. Figure 8) is independent of the polymer concentration, we obtain the model predictions shown in Figure 7. Although the improvement over the Stokes-Einstein result is modest, the model correctly predicts a positive deviation from Stokes-Einstein behavior. Considering the overly idealized (step function) viscosity profile used in deriving eq 11 and that no adjustable parameters are used, the agreement with the data is reasonable. A more rigorous analysis should treat the system in this limit as consisting of microemulsion droplets, isolated from one another, but interacting with polymer coils, both by excluded-volume and hydrodynamic interactions. Unfortunately, we cannot pursue this approach simply because of the limited knowledge of such interactions, at least as far as the (72) Gold, D.; Onyenemezu, C.; Miller, W. G. Macromolecules 1996, 29, 5700.
Figure 9. Comparison of the extracted PEG radius of gyration with small-angle neutron scattering measurements of the apparent correlation length ξapp.
hydrodynamic interaction is concerned. Nevertheless, we continue treating the polymer as part of the solvent, bearing in mind that our analysis from here on is subject to error, though hopefully no greater than the ∼10% we find by rescaling the solvent viscosity. The effect of droplet-droplet interactions is contained in the dilute-limiting slope, λc, of the data in Figure 6. The value of the slope relative to that of hard spheres, λc ) 1.454, can be used to distinguish to some extent between attractive and repulsive interactions between droplets. The depletion potential in eq 1, which we expect to be reasonably accurate at low polymer concentrations, predicts an effective attraction among droplets, for which we should expect λC < 1.454. Indeed, as shown in Figure 8, this is the case; an increase in polymer concentration systematically decreases the value of λC. The top line in Figure 8 is the hard-sphere prediction, DC/D0 ) 1 + 1.454φHS,54,55 which is in reasonable agreement with the data without added polymer. The slightly lower DC values obtained experimentally agree with previous observations and point to some weak van der Waals attraction or effects of droplet size polydispersity. Insertion of the depletion potential in the expression for DC permits for extracting the only unknown parameter, the PEG radius of gyration. The lines through the data at a finite polymer concentration are linear least-squares fits using RG as the single adjustable parameter. The results of the single-parameter fits are listed in the legend to Figure 8. We test our analysis by comparing the RG values with independent data obtained from small-angle neutron scattering studies of pure PEG solutions of similar molecular weight.73,74 In the scattering experiments over wider ranges of polymer concentration, a correlation length is extracted, ξapp. In Figure 9, we choose to show the radius of gyration as 31/2ξapp as a function of polymer mass concentration because for dilute polymer solutions ξapp ) RG/31/2. As seen, we find a near-perfect agreement in the concentration range of our measurements. Moreover, our measurements have been done for polymer concentrations significantly below the concentration regime where the correlation length scales as ∼c-3/4, the semidilute scaling prediction. It is also observed that the radius of gyration (73) Branca, C.; Faraone, A.; Magaz, S.; Maisano, G.; Migliardo, P.; Triolo, A.; Triolo, R.; Villari, V. J. Appl. Crystallogr. 2000, 33, 709. (74) Thiyagarajan, P.; Chaiko, D. J.; Hjelm, R. P. Macromolecules 2002, 28, 7730.
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is a decreasing function of polymer concentration, which is caused by excluded-volume interactions under the prevailing good solvent conditions.75 It follows that polymer-polymer interactions are present even at the very low polymer concentrations used in our measurements. Because this is not accounted for in the AO theory and in light of other simplifying assumptions, the quantitative agreement in Figure 9 is somewhat unexpected. Nevertheless, it demonstrates that the analysis captures the main effect and, certainly, that depletion attractions occur in these microemulsions, for which the AO theory provides a faithful description. 4. Conclusions The main focus in this work was to study experimentally the effect nonadsorbing polymer has on a nonionic droplet microemulsion system. Using viscometry and dynamic light scattering, we verify that the droplets behave as a hard-sphere system provided one stays within a few degrees above the EFB. Addition of low-molecular-weight PEG destabilizes the system with respect to fluid-fluid phase separation but stabilizes the system with respect to emulsification failure. Like the former, we suggest that the latter effect can be understood by polymer depletion and free-volume considerations. Making use of results from rigorous colloid theory, we use measurements of the collective droplet diffusion coefficient to quantify the interaction between droplets in the presence of PEG. The results are consistent with the polymer-induced depletion potential based on the AO theory. Acknowledgment. Financial support from the Swedish Foundation for Strategic Research (SSF, Program for Colloid and Interface Technology) and the Swedish Research Council is gratefully acknowledged. We thank Prof. W. Richtering and F. Nettesheim for assistance with some of the dynamic light scattering measurements and Prof. U. Olsson and J. Balogh for numerous useful discussions. Appendix A Free Volume on Crossing the EFB. We start with the free-volume fraction from depletion theory, R, which is a measure of how much of the total volume in the system is accessible to the polymer when the volume occupied by the colloidal particles and the associated depletion zones are subtracted from the total volume, that is,
Vfree R) V
(12)
(ls) of the C12E5 molecules contributes to the size of the hydrocarbon core of the droplets:
R ) 3ls
(
)
Voil 1 + Vs 2
(14)
The two-phase system, consisting of essentially pure oil,32 occupying a volume Vtop coexisting with a droplet phase with a volume V - Vtop, is defined by a droplet radius R2ph and a droplet volume fraction φ2ph. The radius of the polymer is assumed to be unaffected by the emulsification 2ph ) RPEG. Including the entire length failure so that RPEG of the surfactant ls in the definition of the size of the droplets, R, does not alter the analysis. Introducing ) Vtop/V gives the following simple expressions for the volume fractions:
φ)
Voil + (1/2)Vs V
φ2ph )
(15)
φ- 1-
(16)
The polymer-to-droplet size ratio is given by
(
ξ2ph ≈ ξ 1 +
φ
)
(17)
Expansion of φ2ph around ) 0 gives
φ2ph() ) φ + (φ - 1) + O(2)
(18)
We express R2ph as a function of R:
R2ph ) R(1 - φ) + O(2)
(19)
where
φ)
-3ξ2(φ - 1) - (φ - 1)2 + ξ3(2 + φ) (1 - φ)2
(20)
Comparing the available volumes and inserting the expression for φ results in
[
V2ph -3ξ2(φ - 1) - (φ - 1)2 + ξ3(2 + φ) free )1- + Vfree (1 - φ)2
]
1 + O(2) (21) Because the term within parentheses [denoted by φ(ξ, φ) in eq 10] is positive for relevant values of ξ and φ, one can see that
An approximate expression for R that depends only on the volume fraction of the colloidal particles φ and on the ratio between the radius of the polymer and that of the colloids ξ is given by17
V2ph free κR(PEG) H is η0ηr. The equations governing the fluid pressure and velocity are the Stokes equations for steady, low Reynolds number flow of an incompressible liquid,
∇p ) η0ηr∇2u
∇‚u ) 0
(23)
∇p* ) η0∇2u*
∇‚u* ) 0
(24)
where p and u are the pressure and fluid velocity for r > and p* and u* are the corresponding quantities κR(PEG) H < r < κR(PEG) . These equations are defined for R(PEG) H H supplemented by boundary conditions suited to the problem at hand,
u* ) U
r ) R(PEG) H
u ) 0, p - p0 ) 0 u ) u*, n‚σ ) n‚σ*
rf∞ r ) κR(PEG) H
p - p0 ) (A1r-3 - 10A4)r‚U η0ηr
(
u ) -2A4r2 +
(28)
1 1 1 A r-3 + A1r-1 + A2 U + A1r-3 15 3 2 2 1 A r-5 + A4 rr‚U (29) 5 3
)
( )
for r > κR(PEG) and analogous expressions for R(PEG) κR(PEG) H H H 76 respectively. Following Batchelor, the general solution to eqs 23 and 24 can be written as (76) Batchelor, G. K. An introduction to fluid dynamics; Cambridge University Press: Cambridge, 1967.
∫r)R
)2 F ) -(R(PEG) H
(PEG) H
(n‚σ*) dΩ ) 4πη0ηrA1U (30)
where the solution for A1 is A1 ) 3κηr[2 + 3κ5 + 2ηr(κ5 - 1)] R(PEG) H 5
4 + 6κ + ηr (κ - 1)4(4 + 7κ + 4κ2) + ηr(-8 + 9κ - 10κ3 + 3λ5 + 6λ6) (31) 2
Substituting in the solution for A1 results in an expression for the friction coefficient ζ ) 4πη0ηrA1, related to the diffusion coefficient via the Einstein relation kBT/ζ, which is used to obtain eq 11. LA036132Y
