pubs.acs.org/Langmuir © 2010 American Chemical Society
Formation and Rheology of Viscoelastic “Double Networks” in Wormlike Micelle-Nanoparticle Mixtures Matthew E. Helgeson, Travis K. Hodgdon, Eric W. Kaler, and Norman J. Wagner* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, 150 Academy Street, Newark, Delaware 19716
Martin Vethamuthu and K. P. Ananthapadmanabhan Unilever Research and Development, 40 Merritt Boulevard, Trumbull, Connecticut 06611 Received January 4, 2010. Revised Manuscript Received March 5, 2010 We present a systematic study of thermodynamics, structure, and rheology of mixtures of cationic wormlike micelles and like-charged nanoparticles. Structural and thermodynamic measurements in dilute surfactant-nanoparticle mixtures show the formation of micelle-nanoparticle junctions that act as physical cross-links between micelles. The presence of these junctions is shown to build significant viscosity and viscoelasticity in dilute and semidilute WLMs, even in cases where the fluid is Newtonian in the absence of nanoparticles. Increases in viscosity, shear modulus, and relaxation time, as well as decreases in entanglement concentration, are observed with increasing particle concentration. As such, nanoparticle addition gives rise to a so-called “double network” comprised of micellar entanglements and particle junctions. A simple model for such networks is proposed, where the elasticity can be tuned through two energetic scales, the micellar end-cap energy and micelle-nanoparticle adsorption energy. As a practical application, the results demonstrate that nanoparticle addition provides formulators a unique method to tailor surfactant solution rheology over a wide range of conditions.
1. Introduction Wormlike micelles (WLMs) are long, threadlike aggregates of surfactants or other amphiphiles that form due to the elongation of rodlike micelles under appropriate solution conditions.1 Because of their morphology, WLMs exhibit structure and rheology similar to polymer solutions.2 However, due to their equilibrium self-assembly, they undergo reversible formation and breakage,3 resulting in dynamic or “living” structures exhibiting a hierarchy of length and time scales over which their morphology and dynamics can be widely tuned by manipulating solution conditions, including temperature, pressure, and various additives.4 WLMs will also entangle at sufficient concentration, much like an entangled polymer solution, resulting in significant viscoelasticity,2 as well as shear thinning nonlinear rheology. Because of their detergency, rheology, and easily tunable microstructure, WLMs have become ubiquitous to a wide variety of industrial processes, consumer products, and emerging technologies, including cleaners, soaps, and cosmetics,5 oilfield stimulation6,7 *Corresponding author. (1) Dreiss, C. A. Soft Matter 2007, 3, 956–970. (2) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933–973. (3) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869–6892. (4) Walker, L. M. Curr. Opin. Colloid Interface Sci. 2001, 6, 451–456. (5) Ezrahi, S.; Tuval, E.; Aserin, A.; Garti, N. Daily applications of systems with wormlike micelles. In Giant Micelles; Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; pp 473-492. (6) Armstrong, K.; Card, R.; Navarette, R.; Nelson, E.; Nimerick, K.; Samuelson, M.; Collins, J.; Dumont, G.; Priaro, M.; Wasylycia, N.; Slusher, G. Oilfield Rev. 1994, 7, 34. (7) Bivins, C.; Boney, C.; Fredd, C.; Lassek, J.; Sullivan, P.; Engels, J.; Fielder, E. O.; Gorham, T.; Judd, T.; Sanchez-Mogollon, A. E.; Tabor, L.; Munoz, A. V.; Willberg, D. Oilfield Rev. 2005, 17, 34. (8) Zakin, J. L.; Zhang, Y.; Ge, W. Drag reduction by surfactant giant micelles. In Giant Micelles; Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; pp 473-492. (9) Pileni, M. P. Nat. Mater. 2003, 2, 145–150.
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and production,8 and, more recently, nanoparticle templating.9 Two important commonalities or requirements exist across the broad range of these applications. First, the intimate coupling between the molecular self-assembly of the fluid and the enduse application is critical to the successful formulation of WLMs. A second, often overlooked commonality, is that in many cases the WLM fluid will be mixed with emulsions or colloidal particles, whether in formulation, processing, or end-use application. The addition of colloids to WLMs solutions may influence the self-assembly process or provide additional functionality to the resulting material, leading to new or superior applications. This is particularly true for nanoparticles, where the scales of size, relaxation dynamics, and forces approach the ranges of molecular interactions governing micellar self-assembly. For example, recent studies10,11 have shown that the addition of inorganic nanoparticles to ionic WLMs results in significant changes in the rheology of entangled WLMs. At the same time, the presence of WLMs has been shown to significantly affect the diffusion and colloidal stability of dispersed nanoparticles.10 Nanoparticle addition has also been shown to affect phase behavior, structure and dynamics in other surfactant systems, including microemulsions,12 nonionic surfactants,13 and lamellar phases.14 Previous studies of WLM-nanoparticle mixtures have identified changes in bulk properties, with somewhat contradictory results. Bandyopadhyay and Sood11 studied the affect of adding (10) Nettesheim, F.; Liberatore, M. W.; Hodgdon, T. K.; Wagner, N. J.; Kaler, E. W.; Vethamuthu, M. Langmuir 2008, 24, 7718–7726. (11) Bandyopadhyay, R.; Sood, A. K. J. Colloid Interface Sci. 2005, 283, 585– 591. (12) Kline, S. R.; Kaler, E. W. J. Colloid Interface Sci. 1998, 203, 392–401. (13) Koehler, R. D.; Kaler, E. W. Langmuir 1997, 13, 2463–2470. (14) Salamat, G.; Kaler, E. W. Langmuir 1999, 15, 5414–5421.
Published on Web 03/17/2010
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silica nanoparticles to solutions of cetyltrimethylammonium tosylate (CTAT), where the viscosity increased nonmonotonically with increasing nanoparticle concentration. Furthermore, a transition from non-Maxwellian to Maxwellian linear viscoelasticity was observed with increasing particle concentration. However, Mendoza et al.15 showed that the growth of mesoporous silica in CTAT resulted in the reverse behavior. More recent investigations by Nettesheim et al.10 studied the addition of cationic nanoparticles to WLM solutions of cetyltrimethylammonium bromide (CTAB) in the presence of added sodium nitrate (NaNO3). A monotonic increase in viscosity was observed with increasing particle concentration, and Maxwellian viscoelasticity was observed throughout. Microstructural characterization of the WLM-nanoparticle mixtures showed that this rheological modification is consistent with a measured decrease in osmotic compressibility of the micellar network without fundamentally changing the micellar structure. An important difference among these prior studies is that the nanoparticles varied in surface chemistry. We expect that the interactions between surfactant in solution and the nanoparticles largely determine the effect of nanoparticle addition on WLM properties. However, multiple hypotheses have been proposed in the literature to explain this connection. In some studies it has been proposed that the rheological modification of WLMs by nanoparticles arises simply from contributions of the nanoparticles to the bulk electrostatic properties of the fluid.11 Others have proposed that the nanoparticles materially participate in the viscoelastic network through the formation of effective crosslinks.10 Here, we present a systematic study of the structure, thermodynamics, and rheology of mixtures of CTAB/NaNO3 WLMs and model, cationic nanoparticles to test these hypotheses and develop a microstructural explanation for the observed rheological modification. Previous studies on the CTAB/NaNO3 system in the absence of nanoparticles show the existence of WLMs over a wide range of conditions.16,17 A systematic study of CTAB/ NaNO3 structure, phase behavior, and rheology showed that CTAB/NaNO3 WLMs display dilute, overlapping, and entangled polymer-like rheology and concomitant structural transitions between rodlike and wormlike micelles.17 Building from a previous experimental approach that combines thermodynamic measurements, rheology, and scattering,17 we show that nanoparticle addition results in a so-called “double network” in which the nanoparticles supplement the inherent viscoelasticity of entangled WLMs with additional network junctions. We demonstrate that this behavior can be understood through the measured surfactant-nanoparticle interactions and the microstructure at the surfactant-nanoparticle interface. In the course of these investigations, we also explore the rheological modification of ionic WLMs by nanoparticle addition for surfactant and salt concentrations spanning the dilute and semidilute regimes.
2. Experimental Section 2.1. Materials. The materials used here are the same as in
previous studies.10 Cetyltrimethylammonium bromide (CTAB, Acros Organics, >99%) and sodium nitrate (NaNO3, SigmaAldrich, >99%) were used as received without further treatment (15) Mendoza, L. D.; Rabelero, M.; Escalante, J. I.; Macias, E. R.; GonzalezAlvarez, A.; Bautista, F.; Soltero, J. F. A.; Puig, J. E. J. Colloid Interface Sci. 2008, 320, 290–297. (16) Kuperkar, K.; Abezgauz, L.; Danino, D.; Verma, G.; Hassan, P. A.; Aswal, V. K.; Varade, D.; Bahadur, P. J. Colloid Interface Sci. 2008, 323, 403–409. (17) Helgeson, M. E. Structure, rheology, and thermodynamics of wormlike micelle-nanoparticle mixtures. Ph.D. Dissertation, University of Delaware: Newark, DE, 2009.
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Helgeson et al. or purification. Aqueous stock solutions of these materials were prepared in 18.3 MΩ deionized water (H2O). The model nanoparticles studied in this work are cationically modified silica nanoparticles with a nominal diameter of 30 nm (cal25) provided in a stock suspension containing approximately 30 wt % particles in a mother liquor (30cal25, AZ Electronic Materials). X-ray diffraction measurements suggest that the surface modification performed in order to render the particle surface cationic yields an alumina coating on the particle surface.18 Upon receiving the 30cal25 material from the supplier, the nanoparticle stock was passed through a 0.2 micron syringe filter (Nalgene) prior to sample preparation to eliminate any cal25 aggregates or larger particulate matter. Samples of the filtered cal25 stock were then massed, dried in an oven at 80 C for 24 h in order to evaporate the solvent, then massed again in order to determine the solids content of the filtered stock. This procedure resulted in a nanoparticle stock suspension containing 31.7 wt % solids. Samples of the particle stock suspending medium were also prepared by ultracentrifugation at 235,000 g for 1 h (Beckman Coulter L-100 XP, Rotor 45Ti) to sediment the cal25 particles, followed by decanting of the supernatant. Unless otherwise noted, the presence of dissolved species in the particle stock supernatant had no measurable effect on the data reported. All samples were prepared by mixing or diluting the stock solutions by mass according to the desired final concentration. All samples were equilibrated in a 30 C water bath for at least 24 h prior to measurements. 2.2. Isothermal Titration Calorimetry. Isothermal titration calorimetry (ITC) was performed on a Microcal VP-ITC calorimeter. For all measurements, the sample cell was initially filled with a solution at the same concentration of NaNO3 as the titrant, such that the NaNO3 concentration remains constant throughout the course of the experiment. All experiments were carried out at 25 C using ΔV = 10 μL injections of titrant lasting a period of 1 min, with 5 min between each injection and a total of 28 injections per run. The differential heat input, q(t), was measured as a function of time t over all injections, followed by integration of q(t) over each individual injection to obtain the molar heat of injection, Qinj(T,P,c). The molar heat of injection can then be cumulatively added over all previous injections, yielding the total molar heat, Qtot(T,P,c). Three types of ITC experiments were performed. Dilution enthalpies for both CTAB and cal25 nanoparticles were measured by titrating a dilute solution (0.8 mM) and suspension (φp = 10-4), respectively, into a sample cell initially containing only H2O at 25 C (further details can be found elsewhere17). Measurements of the enthalpy of adsorption of CTAB monomers were carried out by titrating a solution containing 0.8 mM CTAB (below the cmc of CTAB17) into a sample cell initially containing a suspension of cal25 particles with a volume fraction of φp = 10-4. Finally, measurements of the interaction energy between CTAB/ NaNO3 WLMs and cal25 nanoparticles were carried out by titrating a solution containing cal25 nanoparticles with φp = 10-2, 0.8 mM CTAB, and 30 mM NaNO3 into a sample cell initially containing a solution of 10 mM CTAB and 30 mM NaNO3. As such, the titrant consists of nanoparticles with an adsorbed layer of CTAB. During the course of these experiments the concentration of NaNO3 is constant yielding dilute rod-like micelles of nearly constant length,17 such that contributions to the heat of injection from changes in micelle morphology are negligible. The dilution of micelles upon titration, however, is accounted for, as will be discussed later. 2.3. Dynamic Light Scattering. Dynamic light scattering (DLS) measurements were performed on a Brookhaven ZetaPALS (λ = 677 nm) at a temperature of 25 C. Measurements were made for a series of suspensions containing φp = 10-4 cal25 nanoparticles and various amounts of CTAB ranging from 0 to 0.8 mM. The autocorrelation function was analyzed using the (18) Flood, C.; Cosgrove, T.; Espidel, Y. Langmuir 2007, 23, 6191–6197.
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standard cumulant method19 to obtain the effective hydrodynamic radius, Rh, and polydispersity. The ζ-potential of the samples was also measured using a standard gold aqueous electrode geometry at applied AC electric field amplitudes, E0, of 1-2 V/cm. 2.4. Cryo-Transmission Electron Microscopy. Cryotransmission electron microscopy (Cryo-TEM) samples were prepared using an FEI Vitrobot Mark II controlled environment vitrification system. Samples were prepared at 25 C and 100% RH on a Quantifoil holey carbon grid (Electron Microscopy Sciences). Imaging of the samples was performed using an FEI Tecnai G2 12 TEM operating at an accelerating voltage of 120 keV. Digital images were recorded using a Gatan multiscan charge coupled device (CCD) camera and processed with DigitalMicrograph software. 2.5. Rheology. Capillary viscometry measurements were carried out on dilute and semidilute CTAB/NaNO3 solutions with and without cal25 nanoparticles using an AMVn rolling ball viscometer (Anton Paar USA). Measurements were carried out at 25 C unless otherwise specified, and at several tilt angles in order to ensure that the measured value of η0 was independent of measurement angle. Samples in the viscosity range of 1-10 mPa s were measured in a 1.6 mm capillary and those in a range of 10-200 mPa s were measured in a 1.8 mm capillary. Further rheological characterization was performed on a TA Instruments AR-G2 stress-controlled rheometer with a 60 mm, 1 upper cone geometry with a temperature-controlled Peltier lower plate geometry. Linear viscoelasticity was measured using the following protocol. The sample was first given a preshear of 10 s-1 for 60 s then allowed to rest for 100 s to ensure proper loading. An oscillatory strain sweep at 1 rad/s from 1% to 100% strain was first performed in order to identify the linear viscoelastic (LVE) regime for the sample. The storage and loss moduli, G0 and G00 , respectively, were then measured by performing a frequency sweep at 5% strain from 200 to 0.1 rad/s, and then another from 0.1 to 200 rad/s. No hysteresis in LVE measurements was observed for any sample.
3. Results 3.1. Surfactant-Nanoparticle Structure and Interactions. Surfactant Adsorption. Previous approaches for determining the thermodynamics of surfactant adsorption on colloidal particles involve two separate ITC experiments: (1) titration of surfactants above the cmc into a suspension of particles, and (2) titration of the same micellar solution into pure solvent. Subsequently, the results from point 2 are subtracted from the results of point 1, and the result is assigned to be the molar heat of adsorption of the surfactant on the particle surface, ΔHads. However, this procedure has been shown to lead to large uncertainties,20 as the heat released upon adsorption is comparable to the dilution enthalpy and orders of magnitude smaller than the micellization enthalpy, which significantly diminishes the precision of ΔHads. Here, we present a more precise method for the determination of ΔHads, where the surfactant concentration of both the sample and titrant are below the cmc for the entire experiment. Qinj (c) and Qtot(c) are plotted in Figure 1 for injection of 0.8 mM CTAB into a suspension of cal25 silica particles with volume fraction φp = 10-4. In general, the total heat liberated by injection and resulting adsorption of surfactant monomer can be modeled as a sum of contributions of monomer adsorption and rearrangement at the particle surface and dilution of monomers in solution, such as m ðT, PÞ Qinj ðT, PÞ ¼ ΔHads ðθ, T, PÞ þ ΔHdil
ð1Þ
(19) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814–&. (20) Winnik, M. A.; Bystryak, S. M.; Odrobina, E. Langmuir 2000, 16, 6118– 6130.
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Figure 1. Cumulative molar heat of injection for titration of 0.8 mM CTAB into φp = 10-4 cal25 nanoparticles versus total CTAB concentration. Solid and dashed lines give fits to eq 3 and a linear relationship, respectively. Vertical line denotes the experimentally observed critical surface aggregation concentration (CSAC). Inset graph shows the molar heat per injection. Table 1. Adsorption Properties of CTAB on cal25 Silica at 25 C and 1 atm property
value
adsorption equilibrium constant, Kads enthalpy of adsorption, ΔHads critical surface aggregation concentration (CSAC) enthalpy of surface aggregation, ΔHs,agg free energy of surface aggregation, ΔHs,agg enthalpy of surface aggregation, ΔHs,agg
1.07 104 L/mol -17.9 kJ/mol 0.14 mM -5.4 kJ/mol -37.8 kJ/mol 108 J/mol K
where ΔHads is the molar enthalpy of adsorption of a surfactant monomer, which depends on the fractional surface coverage θ. 17 Note that ΔHm dil for CTAB is known from previous experiments. In order to extract useful thermodynamic quantities from the ITC adsorption data, one must assume a model for ΔHads(θ,T,P). In the simplest case, where adsorbed surfactant monomers have no adsorbate-adsorbate interactions, each adsorbate contributes equally to the adsorption enthalpy, and thus: sat ðT, PÞθðcs , T, PÞ ΔHads ðθ, T, PÞ ¼ ΔHads
ð2Þ
where ΔHads is the enthalpy of noninteracting adsorbates at surface saturation (θ = 1), and cs is the total concentration of surfactant in solution. Since eq 2 is restricted to noninteracting adsorbates, an appropriate adsorption isotherm for θ(cs,T,P) is the Langmuir adsorption isotherm, such that: Qtot ðcs , T, PÞ ¼
sat ΔHads ðT, PÞ
Kads ðT, PÞcs 1 þ Kads ðT, PÞcs
ð3Þ
where Kads is an effective equilibrium constant for adsorption. Equation 3 provides a good fit to the data at low overall CTAB concentration, suggesting that monomer adsorption occurs in this range with negligible interactions between adsorbates. We find that Kads(25 C, 1 atm) = 1.07 104 M-1 and ΔHsat ads (25 C, 1 atm) = -17.9 kJ/mol (Table 1). Note that the adsorption is exothermic, suggesting that enthalpic interactions between surfactant and surface significantly influence the adsorption process. Upon increasing the total CTAB concentration above 0.14 mM, significant deviation from the Langmuir behavior is observed (Figure 1), suggesting the onset of surface aggregation of adsorbed species.21 Here, the adsorption enthalpy follows a linear relationship (21) Paria, S.; Khilar, K. C. Adv. Colloid Interface Sci. 2004, 110, 75–95.
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over a large range of CTAB concentration, with a slope in this region of -5.4 kJ/mol. Note that this quantity is approximately one-half the micellization enthalpy for CTAB,17 suggesting that CTAB may form hemispherical surface aggregates similar to what is observed for the adsorption of CTAB on graphite under certain conditions.21 With increasing CTAB concentration, the adsorption enthalpy tends toward an apparent plateau at a solution CTAB concentration of approximately 0.3 mM, suggesting that the surface saturates under these conditions below the cmc. Assuming that the concentration at which the adsorption enthalpy deviates from Langmuir-like behavior is equal to the critical surface aggregation concentration (CSAC), and furthermore that the process of surface aggregation can be treated thermodynamically as a pseudophase separation (similar to the process of bulk micellization), then the free energy of surface aggregation is given by ΔGs, agg ðT, PÞ ¼ RTð2 -R0 Þ ln xCSAC ðT, PÞ
ð4Þ
where xCSAC(T,P) is the mole fraction of surfactant at which the CSAC occurs and R0 is the degree of counterion dissociation. Using eq 5 with the determined value of the CSAC yields ΔGs,agg (25 C, 1 atm) = -37.6 kJ/mol (note: a value of R0 = 0.82 was assumed22). The free energy of surface aggregation is more favorable than that for adsorption, confirming that the process of surface aggregation will be energetically preferred above the CSAC. This value of ΔGs,agg leads to an estimated value of ΔSs,agg (25 C, 1 atm) = 108 J/(mol K). Adsorbed Layer Structure. Measurements of the hydrodynamic radius, Rh, and the ζ-potential were performed on cal25 suspensions in CTAB solutions both below and above the CSAC, and are presented in Figure 2a. The ζ-,potential of the particles increases nearly 2-fold over concentrations ranging from no adsorbed CTAB to surface saturation, and appears to saturate at a CTAB concentration of approximately 0.2 mM, slightly beyond the CSAC. The secondary adsorption evident in ITC measurements (Figure 1) above the CSAC does not lead to a large increase in the ζ-potential (Figure 2a). This indicates an increase in the positive surface charge of the cal25 particles with an increasing amount of adsorbed CTAB, consistent with the measured adsorption of cationic CTAB onto the cationic nanoparticles. Rh increases with CTAB concentration, but limits at the CSAC. Note that the average Rh at saturation is 1.7 nm larger than the bare particle value, which is approximately the length of one surfactant tail group, consistent with tail-on adsorption of CTAB. Although the saturation of Rh at the CSAC indicates the absence of any multilayer adsorption, it does not preclude further adsorption beyond the CSAC, as the formation of surface aggregates does not necessarily require an increase in hydrodynamic size of the adsorbed layer. The ζ-potential measurements enable estimation of the adsorbed amount of CTAB. From the Gouy-Chapman theory for spheres of constant surface potential, the surface charge density σ*is related to the ζ as:23 σ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zeζ 8εkb Tn ( sinh 2kb T
ð5Þ
where ε is the dielectric permittivity of the solvent, kb is the Boltzmann constant, e is the elementary charge, and n( is the (22) Treiner, C.; Mannebach, M. H. Colloid Polym. Sci. 1990, 268, 88–95. (23) Hiemenz, P. C.; Rajagopalan, R. Principles of colloid and surface chemistry, 3rd ed.; Marcel Dekker: New York, 1997; p 650.
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Figure 2. (a) ζ potential (left axis, open symbols) and effective hydrodynamic radius (right axis, closed symbols) of cal25 nanoparticles, as well as (b) number of adsorbed CTAB molecules per nanoparticle estimated using eq 8 versus total CTAB concentration. Vertical lines denote the experimentally observed CSAC. The solid line is drawn to guide the eye.
ionic strength, which includes the free surfactant and any counterions and added salt from the particle stock suspension. The total number of charge groups per particle is Q = 4πRh2σ*. Assuming that the surface charge density in the absence of adsorbed CTAB, σ0*, is independent of the total CTAB concentration, the number of CTAB molecules adsorbed per particle, nads, is given by
nads ¼
4πRh2 ðσ -σ 0 Þ R0
ð6Þ
where R0 is the degree of ionization of CTAB. Figure 2b shows nads for φp = 10-4 cal25 as a function of total CTAB concentration predicted from eq 7. We have assumed that contributions from the cal25 suspension are negligible, such that the ionic strength, n(, is given by n( = NAR0cs. The calculations indicate that nads increases rapidly in the region where Langmuirlike adsorption occurs, reaching a value of nads ∼ 150 at the CSAC. Upon reaching the CSAC, nads increases further, but more mildly with increasing CTAB concentration, and eventually achieves a saturation value of nads ∼ 200. Assuming a headgroup area for CTAB of 0.64 nm2,24 this corresponds to a total projected area for the all adsorbed surfactants that is approximately 38% of the available particle surface area. This suggests that the admicellar structure is not continuous over the entire particle surfaces. Rather, it is contained in discrete “patches” of (24) Zhao, F.; Du, Y. K.; Xu, J. K.; Liu, S. F. Colloid J. 2006, 68, 784–787.
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evolved per injection is given by WLM snt WLM -p ðT, PÞ þ ΔHdil ðT, PÞ þ ΔHassn ðT, PÞ Qj ðT, PÞ ¼ ΔHdil
ð7Þ
Figure 3. Molar heat of injection for addition of the titrants indicated versus real or simulated cal25 volume fraction. Inset graph shows the resulting enthalpy of micelle-nanoparticle association estimated using eq 9, where the line and shaded region show the average value and experimental deviation, respectively.
surfactant. This also explains why the effective Rh of particles does not continue to increase above the CSAC. Given that the increase in the effective particle size is consistent with tail-on adsorption, these patches may be hemimicelles, similar to what is observed for the adsorption of CTAB on graphite.21 This observation is confirmed by ITC measurements of ΔHs,agg, as noted above. Micelle-Nanoparticle Association above the cmc. ITC measurements were carried out in which a titrant comprised of φp = 10-2 cal25 nanoparticles with an equilibrium layer (with both monomers and hemimicelles) of adsorbed CTAB (0.8 mM) were added to CTAB/NaNO3 WLMs. Qinj(ccell,j) is plotted in Figure 3 for injection of this titrant and two reference titrants (cal25 supernatant and DI-water, both with 30 mM NaNO3 and 0.8 mM CTAB) into a sample solution containing 10 mM CTAB and 30 mM NaNO3. These conditions lie below the overlap concentration for CTAB/NaNO3,17 such that the micelles in solution can be considered to be fairly short, rod-like micelles. The data in Figure 3 are plotted versus cal25 volume fraction, where for reference a hypothetical equivalent nanoparticle concentration in the titrant is assumed simply to facilitate comparison of the data and further analysis. The titration of 30 mM NaNO3 and 0.8 mM CTAB ( symbols in Figure 3) measures the dilution enthalpy of micelles at constant salt concentration, and thus serves as a baseline for subsequent measurement (note that the WLM sample solution is diluted to no less than 8.6 mM over the entire course of experiment). The micellar dilution enthalpy is nearly constant with the amount of dilution. The titration of the cal25 supernatant described previously (open squares). measures the dilution enthalpy of any dissolved species in the cal25 suspending medium, such as electrolytes, which can be significant. The final titrant was a suspension of φp = 10-2 cal25 particles to which 0.8 mM CTAB and 30 mM NaNO3 were added (closed squares). The addition of 0.8 mM CTAB is done so that the cal25 particles already have an adsorbed CTAB layer, such that contributions due to monomer adsorption can be neglected. The enthalpy evolved during injection of each titrant used will, in general, contain contributions from four sources: (1) the dilution of surfactant, (2) the dilution of nanoparticles (assumed here to be negligible), (3) the dilution of dissolved species in the cal25 supernatant, and (4) enthalpic interactions between the suspended nanoparticles and WLMs in solution. As such, the molar heat Langmuir 2010, 26(11), 8049–8060
where ΔHWLM (T,P) is the WLM dilution enthalpy (measured with dil the first titrant), ΔHsnt dil (T,P) is the supernatant dilution enthalpy (T,P) is the (measured with the second titrant), and ΔHWLM-p assn enthalpy of association between micelles and nanoparticles. Thus, subtraction of ΔHsnt dil (T,P) measured by injection of the cal25 supernatant solution from Qinj(T,P) for the third titrant (T,P). The resulting (closed squares in Figure 3) yields ΔHWLM-p assn (25 C, 1 atm), normalized by the particle surface area, ΔHWLM-p assn is shown in the inset in Figure 3. It is nearly independent of particle concentration, with an average value of ΔHWLM-p assn (25 C, 1 atm) = -1.05 ( 0.33 μJ/m2. Thus, the association of the cationic nanoparticles with the cationic WLMs is enthalpically favored. (T,P) can be converted to an association energy per ΔHWLM-p assn particle by WLM -p ðT, PÞ Eassn ðT, PÞ ¼ 4πRh 2 ΔHassn
ð8Þ
Using eq 9, the experimentally determined value of ΔHWLM-p assn (25 C, 1 atm) yields a value of Eassn= -9.01 kbT (- 3.71 10-20 J). Note that this energy is of similar magnitude as the scission energy measured previously for CTAB/NaNO3 micelles.17 Cryo-TEM. Figure 4 shows representative micrographs for a solution containing 40 mM CTAB and 120 mM NaNO3 with φp=0.005 cal25 nanoparticles vitrified at 25 C and 100% relative humidity. This sample was chosen because it is below the entanglement concentration in the absence of cal25 particles,17 but upon particle addition, it exhibits a viscosity comparable to an entangled sample containing 55 mM CTAB and 165 mM NaNO3. As shown in the micrographs, the cal25 nanoparticles are dispersed throughout the solution and no aggregates are observed. Micelles are apparent as linear structures, and are not as distinct as the particles due to comparatively lower electron contrast with respect to the solvent. The micelles appear to be of similar radius and rigidity as those without nanoparticles.17 A distinctive feature of the micrographs in Figure 4 is that a majority of the micelles appear to intersect with nanoparticles. Given that the cryo-TEM images are two-dimensional projections of a three-dimensional fluid film, there are two possible explanations; the micelles may pass over or under the particles out of the viewing plane, or they may terminate at the particle surface. Given that nearly all micelles intersect with nanoparticles, the former explanation seems unlikely. Thus, the images further corroborate the hypothesis given by Nettesheim et al.,10 that micelle-nanoparticle association occurs in solution. 3.2. Semidilute Rheology. Viscosity. The zero shear rate viscosity, η0, measured by viscometry at 25 C of CTAB/NaNO3 solutions containing molar ratios of NaNO3 to CTAB of c(/cs = 1, 2, and 3 for samples containing volume fractions of φp = 0, 0.001, 0.005, and 0.01 cal25 nanoparticles was measured using capillary viscometry (measurements were also made for samples with φp = 0.002, but are not shown for visual clarity). The specific viscosity, ηsp = η0/ηs - 1, where ηs is the solvent viscosity (determined to be 0.893 mPa s), was then computed for all measurements, and is plotted versus CTAB concentration for c(/cs = 1, 2, and 3 at various φp in Figure 5. In the absence of nanoparticles (triangles in Figure 5), the data exhibit two different regimes of power law behavior above and below the entanglement concentration, ce, as shown previously.17 DOI: 10.1021/la100026d
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Figure 4. Representative cryo-TEM micrographs of a solution of wormlike micelles (40 mM CTAB, 120 mM NaNO3) containing φp = 0.005 cal25 nanoparticles at 25 C.
At fixed surfactant concentration, the specific viscosity increases monotonically with increasing nanoparticle concentration for all surfactant concentrations. For surfactant concentrations above ce, the viscosity increases by as much as a factor of ∼10 upon addition of only 1 vol % particles, similar to what was observed in previous studies,10 and the effect is more significant at higher saltto surfactant ratio c(/cs. Figure 5 demonstrates that the power law exponents do not change upon addition of nanoparticles. Because these power law exponents are highly sensitive to the effects of electrostatic screening,25,26 this suggests that the presence of nanoparticles does not significantly alter the electrostatic interactions between micelles. This is in direct contrast with the mechanism of viscosity increase proposed by Bandyopadhyay and Sood,11 who suggested that the viscosity increase resulted from additional electrostatic screening through contributions of silica nanoparticles to the bulk ion concentration. If so, one would observe a decrease in power law exponent with increasing particle concentration.17 Figure 5 also shows that the value of ce (given by the intersection of the two power law regimes) decreases with increasing (25) Raghavan, S. R.; Kaler, E. W. Langmuir 2001, 17, 300–306. (26) Raghavan, S. R.; Fritz, G.; Kaler, E. W. Langmuir 2002, 18, 3797–3803.
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Figure 5. Specific viscosity versus CTAB concentration at 25 C for (a) c(/cs = 1, (b) c(/cs = 2, and (c) c(/cs = 3 with the cal25 nanoparticle concentrations indicated. Lines are power law fits to the data below and above ce.
particle concentration. Also note that the value of ηsp at the entanglement concentration decreases with increasing particle concentration. Thus, nanoparticle addition to a WLM solution below ce can result in the formation of an entangled solution in qualitative agreement with previous studies.10 In order to more closely examine the relationship between nanoparticle concentration and micellar entanglement, Figure 6a plots ce versus the salt-to-surfactant ratio c(/cs for various φp. With increasing salt ratio at fixed φp, a scaling of ce ∼ (c(/cs)-1/2 is observed (inset in Figure 6a), consistent with Mackintosh et al.27,28 This scaling is preserved upon the addition of cal25 nanoparticles for all particle volume fractions, providing further evidence that the addition of cal25 nanoparticles do not significantly alter the electrostatic interactions leading to WLM formation. Figure 6b shows that the normalized entanglement concentration decreases monotonically with φp, to as little as 65% of its initial value without particles for the highest cal25 concentration and salt ratio. The same trend occurs for all salt ratios, but the effect increases with increasing salt ratio. This suggests a synergistic (27) Mackintosh, F. C.; Safran, S. A.; Pincus, P. A. J. Phys.: Condens. Matter 1990, 2, SA359–SA364. (28) Mackintosh, F. C.; Safran, S. A.; Pincus, P. A. Europhys. Lett. 1990, 12, 697–702.
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Figure 8. Storage modulus (G0 , closed symbols) and loss modulus (G00 , open symbols) for 50 mM CTAB and 150 mM NaNO3 at 25 C versus applied frequency for the cal25 nanoparticle volume fractions indicated. Lines give Maxwell model fits using the parameters in Table 2.
Figure 6. (a) Entanglement concentration versus salt ratio for the cal25 nanoparticle concentrations indicated. Inset graph shows scaling of ce ∼ (c(/cs)1/2. (b) Fractional decrease in entanglement concentration versus cal25 nanoparticle concentration for c(/cs = 1 (triangles), c(/cs = 2 (circles), and c(/cs = 3 (squares). Solid lines are drawn to guide the eye.
Figure 7. Normalized specific viscosity increment versus cal25 nanoparticle volume fraction both above (closed points) and below (open points) ce for the salt ratios indicated. Dotted line gives prediction from the Einstein equation.
effect between the addition of nanoparticles and the addition of electrolyte that drives the reduction in ce, i.e., the nanoparticles create effective entanglements in the WLM solution. Figure 7 plots the normalized viscosity increment, ηsp/ηsp,φ=0, defined as the ratio of the specific viscosity of a sample containing nanoparticles to that containing no nanoparticles at the same CTAB and NaNO3 concentration, versus φp for all three salt ratios studied. This plot was created by dividing the power law fit (Figure 5) for a given particle volume fraction by that for the WLM solution in the same concentration regime (extrapolated if needed), yielding the closed symbols in Figure 7 for c > ce. and the open symbols for c < ce. Langmuir 2010, 26(11), 8049–8060
The viscosity increment increases with cal25 volume fraction over all conditions. For conditions below ce, the viscosity increment is more than expected by the Einstein prediction for dispersed spheres23 (dotted line), and becomes only weakly dependent on particle concentration at higher concentrations. Interestingly, ηsp/ ηsp,φ=0 also appears to be independent of salt ratio below ce, suggesting that interactions between micelles and nanoparticles leading to the viscosity increase are not effected by electrostatic screening below ce. Contrastingly, the viscosity increment is much larger for conditions above ce. Furthermore, ηsp/ηsp,φ=0 is found to depend more strongly on φp, and increases significantly with increasing salt ratio. This further illustrates the synergistic effects of nanoparticle addition and increasing salt ratio on increasing the viscosity and effective entanglement density in the semidilute regime. Linear Viscoelasticity. LVE measurements on a sample containing 50 mM CTAB and 150 mM NaNO3 with φp= 0, 0.001, 0.005, and 0.01 are shown in Figure 8. The neat WLM solution is just below ce and exhibits Newtonian behavior in the absence of nanoparticles. Addition of φp = 0.001 leads to an increase in G00 , but no measurable elasticity. At φp = 0.005, the solution becomes viscoelastic, as demonstrated by the Maxwelllike behavior of G0 and G00 over the entire frequency range, and further increase in the particle concentration to φp = 0.01 results in a significant increase in viscoelasticity. In this case, the solution exhibits the features typically associated with fully entangled WLMs, namely, Maxwell-like behavior of G0 and G00 at low frequencies, followed by crossover of G0 and G00 and eventually a rubbery plateau in G0 accompanied by a local minimum in G00 at sufficiently high frequencies. The effective breakage time is calculated as λbr = 1/ω*, where ω* is determined from the deviation from Maxwellian behavior via a Cole-Cole plot.29 These results demonstrate that the addition of nanoparticles to an unentangled micellar solution with liquid-like behavior gives rise to significant viscoelasticity and entanglement at sufficient particle concentration. To quantify the viscoelasticity imparted by addition of nanoparticles the data in Figure 8 are fit to a single element Maxwell model with background viscosity, G0 ¼
G0 ðλr ωÞ2 1 þ ðλr ωÞ
2
, G00 ¼
G0 ðλr ωÞ 1 þ ðλr ωÞ2
þ η¥ ω
ð9Þ
(29) Cates, M. E.; Fielding, S. M. Theoretical rheology of giant micelles. In Giant Micelles; Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; pp 109-161.
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Helgeson et al. Table 2. Linear Viscoelastic Properties of CTAB/NaNO3 with cal25 Nanoparticles at 25C
CTAB [mM]
NaNO3 [mM]
φp
G0 [Pa]
λr [s]
λbr [s]
λrep [s]
η¥ [Pa s]
50
150
0 0.001 0.005 0.01 0 0.001 0.005 0.01 0 0.001 0.005 0.01 0 0.001 0.005 0.01
0.61 2.76 4.47 4.69 5.81 7.05 18.98 19.38 20.62 22.39 30.37 31.57 35.44 38.95
0.04 ( 0.02 0.26 ( 0.02 0.20 ( 0.02 0.36 ( 0.01 0.48 ( 0.01 0.53 ( 0.02 0.35 ( 0.01 0.78 ( 0.02 1.13 ( 0.01 1.30 ( 0.01 0.43 ( 0.03 0.75 ( 0.02 0.96 ( 0.03 1.07 ( 0.03
0.04 ( 0.02 0.06 ( 0.02 0.10 ( 0.02 0.11 ( 0.02 0.13 ( 0.02 0.05 ( 0.01 0.13 ( 0.02 0.14 ( 0.02 0.16 ( 0.02 0.10 ( 0.03 0.21 ( 0.04 0.23 ( 0.05 0.24 ( 0.05
1.7 ( 0.7 0.7 ( 0.4 1.3 ( 0.3 2.1 ( 0.4 2.1 ( 0.5 2.5 ( 0.4 4.7 ( 0.7 9.1 ( 0.7 10.6 ( 0.7 2.3 ( 0.7 2.5 ( 0.7 3.7 ( 1.1 4.8 ( 1.0
0.01 ( 0.01 0.03 ( 0.02 0.04 ( 0.02 0.06 ( 0.01 0.07 ( 0.01 0.07 ( 0.02 0.04 ( 0.01 0.31 ( 0.02 0.34 ( 0.02 0.33 ( 0.04 0.03 ( 0.02 1.10 ( 0.02 1.89 ( 0.01 2.10 ( 0.02
100
100
100
200
100
300
Figure 9. G0 (closed symbols) and G00 (open symbols) for 100 mM CTAB with (a) 100 mM NaNO3, (b) 200 mM NaNO3, and (c) 300 mM NaNO3 and the cal25 nanoparticle volume fractions indicated. Lines give Maxwell model fits using the parameters listed in Table 2.
yielding the equilibrium modulus, G0, relaxation time, λr, and high-frequency viscosity, η¥. The resulting model fits are shown as the solid lines in Figure 8, and the best-fit model parameters are listed in Table 2. For the samples exhibiting viscoelasticity, the Maxwell model fits are in fair quantitative agreement with the experimental measurements, except at sufficiently high frequencies, where expected deviations from Maxwellian behavior are observed.17 Both the equilibrium modulus, G0, and the relaxation time, λr, are found to increase with increasing particle concentration. This is consistent with the observed increase in specific viscosity of the solutions noted previously, as η0 = G0λr. 8056 DOI: 10.1021/la100026d
The linear viscoelasticity of solutions containing 100 mM CTAB and c(/cs = 1, 2, and 3 was measured for particle concentrations of φp= 0, 0.001, 0.005, and 0.01 at 25 C, and the results are shown in Figure 9. Note that all of these compositions lie above ce . Both G0 and G00 increase systematically with increasing particle volume fraction for all salt ratios. Interestingly, the effective breakage time increases with increasing salt ratio, but is approximately constant in the presence of cal25 nanoparticles at any particle concentration and fixed salt ratio (Table 2). This is significantly different than what is observed for solutions without nanoparticles, where the breakage time is expected to be inversely proportional to the micellar length29 and therefore decreases systematically with increasing surfactant concentration and salt ratio.17 The LVE data in Figure 9 show significant deviations from Maxwellian behavior at frequencies above ωmin. These deviations become more severe as the particle concentration increases, suggesting a transition to non-Maxwellian behavior with increasing particle concentration. This observation is consistent with the results presented by Bandyopadhyay and Sood for addition of silica nanoparticles to CTAT micelles.11 Table 2 shows that both G0 and λr increase systematically with increasing particle concentration, and that the increase in G0 with particle concentration is more pronounced as the salt ratio is increased. Assuming that the micelles are in the fast-breaking limit,29 the values of λr and λbr were used to estimate the effective reptation time λrep= (λr2/λbr), which also increases monotonically with increasing particle concentration (Table 2). The best-fit values of η¥ show no clear correlation with particle volume fraction. However, the values of η¥ obtained for solutions containing nanoparticles are, in most cases, significantly larger than for the corresponding solutions in the absence of nanoparticles.17 In summary, the characterization of the linear viscoelasticity of CTAB/NaNO3 micelles both above and below ce demonstrates that (1) addition of cal25 nanoparticles to unentangled solutions results in viscoelasticity and entanglement at sufficient particle concentration, and (2) addition of cal25 nanoparticles results in a systematic enhancement of the viscoelasticity of entangled micellar solutions, corresponding to increases in the equilibrium modulus, relaxation time, and reptation time. Furthermore, solutions containing cal25 particles exhibit an apparent breakage time that is independent of particle concentration (but different than that without particles).
4. Discussion The rheological characterization of model WLM-nanoparticle mixtures containing CTAB/NaNO3 micelles and cal25 nanoparticles demonstrate that nanoparticle addition significantly Langmuir 2010, 26(11), 8049–8060
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alters the linear and nonlinear rheology of WLMs both in the semidilute and concentrated regime. Many of the observations made here are in qualitative agreement with previous studies, where nanoparticle addition was found to increase the viscosity and viscoelasticity of WLMs.10,11 In these studies, two very different mechanisms for the rheological modification of WLMs by nanoparticle addition were proposed. Nettesheim et al. proposed that nanoparticles incorporate into micellar solutions by micelle-nanoparticle association, such that the nanoparticles actively participate in the viscoelastic network. This is consistent with the results of our ITC and DLS experiments, where the measured interactions between micelles and nanoparticles indicate the formation of micelle-nanoparticle junctions by association of the micelle with an equilibrium adsorbed surfactant layer at the particle surface. In contrast, Bandyopadhyay and Sood11 proposed that nanoparticles indirectly affect the growth of micelles by contributing to bulk electrostatic screening of charge interactions, essentially acting as “macroions” without specific micelle-particle interactions. However, this hypothesis fails to explain several important phenomena observed in this and previous10 work. For example, Nettesheim et al. showed10 that the diffusion coefficient of cal25 particles in entangled CTAB/NaNO3 WLMs corresponds to an effective medium viscosity that is orders of magnitude greater than the solvent viscosity and greater than the solution viscosity. This would be impossible if the particles simply acted as macroions that diffuse freely through the entangled micellar mesh. Furthermore, the contribution of nanoparticles to bulk electrostatics would suggest that addition of nanoparticles would have a similar effect to increasing the electrolyte concentration of CTAB/NaNO3 micelles. As shown previously, this implies that particle addition should result in (1) a nonlinear viscosity increase with increasing particle concentration,30 and (2) a decrease in power law exponent of the viscosity in the overlap regime.25,26 Here, however, the viscosity increase is linear with particle concentration, and the viscosity exponents are independent of particle concentration. Thus, the macroion hypothesis for nanoparticle addition to WLMs is inconsistent with the results presented here. Therefore, we hypothesize that the rheological modification of wormlike micellar solutions occurs by the formation of micellenanoparticle junctions. The microstructural manifestation of these micelle-nanoparticle junctions is illustrated in Figure 10, which depicts two nanoparticles in an entangled micellar solution. As suggested by the figure, viscoelasticity in the fluid arises from two separate mechanisms. The first is the entanglement of micelles, denoted in the figure as constraints due to overlapping micelles. The second is the micelle-particle junctions themselves, which effectively join two or more micelles, creating additional viscoelasticity. For example, a particle with two junctions will join two micelles, resulting in an effectively longer micelle, which will result in a greater number of entanglements per effective micelle. Particles with three or more junctions will serve as physical crosslinks of a micellar network, much like in a cross-linked polymer gel.31 These two mechanisms of viscoelasticity combine to form a “double network”. Formation of Micelle-Nanoparticle Junctions. Combined results of ITC and DLS measurements indicate that, at surfactant concentrations of interest for WLMs, cal25 nanoparticles have an (30) Nettesheim, F.; Kaler, E. W. Phase behavior of systems with wormlike micelles. In Giant Micelles; Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; pp 223-247. (31) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980; p 672.
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Figure 10. Illustration of the proposed mechanism of double network formation in wormlike-micelle nanoparticle mixtures. Cutout illustrates the process of micelle-nanoparticle association at the solution-solid interface.
adsorbed surfactant layer at the particle surface. The increase in Rh and ζ-potential of the particles, as well as the observed CSAC behavior in ITC adsorption measurements, suggests that this adsorbed layer adopts a hemimicellar morphology that covers a significant portion of the particle surface. Thus, it is reasonable to assume that the interaction of micelles in a bulk fluid will interact with nanoparticles in suspension through specific interactions with this adsorbed surface layer. This situation is similar to that modeled by Jodar-Reyes and Leermakers,32,33 who considered the configuration of rigid rodlike micelles confined between two surfaces decorated with an adsorbed surfactant monolayer. The results showed that bridging of the surfaces by micelles can be achieved by the formation of micellar “stalks” at the solid-solution interface.32 These micellar stalks are thermodynamically stable when the total energy required to form an adsorbed micellar stalk, Eads, is less than the micellar end-cap energy, Ecap. As such, a micelle in solution can achieve a lower energy by sacrificing an end-cap in place of one of these surface stalks. This was found only for moderately hydrophobic surfaces, where rearrangement of the adsorbed surfactant layer to accommodate the micellar stalk is energetically favored. The association energy Eassn(25 C, 1 atm)=-9.01 kbT measured upon the addition of nanoparticles to CTAB/NaNO3 WLMs is of similar magnitude to the end-cap energy measured for CTAB/NaNO3 micelles.17 Given these findings, we hypothesize that the association energy arises directly from the formation of micellar stalks with the cal25 surface. The structure of these stalks, as well as the mechanism by which they occur, is illustrated in Figure 10 (cutout). Here, a micelle in solution adsorbs at an end-cap to form a micelle-nanoparticle stalk, such that the energy of the entire system is lowered by an amount Ecap - Eads. Because the available particle surface area is much larger than the projected cross-section of the micelle, a single particle may accommodate (32) Jodar-Reyes, A. B.; Leermakers, F. A. M. J. Phys. Chem. B 2006, 110, 18415–18423. (33) Jodar-Reyes, A. B.; Leermakers, F. A. M. J. Phys. Chem. B 2006, 110, 6300–6311.
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multiple stalks. Thus, the total association energy between micelles and particles is given by Eassn= f (Ecap - Eads), where f is the average number of micellar stalks per particle. Mechanism of Viscoelastic Enhancement. Addition of cal25 nanoparticles to CTAB/NaNO3 micelles results in the development of viscoelasticity in solutions that are otherwise below the entanglement concentration, as well as an increase in both the equilibrium modulus, G0, and relaxation time, λr, with increasing particle concentration for fluids already exhibiting viscoelasticity. Through the proposed mechanism of micellenanoparticle junction formation, these effects can be tractably explained using rubber elasticity theory.31 For a viscoelastic network of arbitrary contributions (entanglements, cross-links, etc.), the equilibrium modulus is given by G0 ¼ νeff kb T
ð10Þ
where νeff = νent þ νj is the effective number of chains participating in the network per unit volume, where νent is the number density of entanglements, and νj is the number density of micellenanoparticle junctions. Here, we estimate the network junctions due to the presence of nanoparticles via a model that combines classical rubber elasticity31 with the transient network theory of Tanaka and Edwards.34 νeff
φp f -2 G0 ¼ νent þ ¼ kb T Vp f
!
eEassn =fkb T 1 þ eEassn =fkb T
! ð11Þ
where f is the average number of micellar junctions per particle. Thus, the equilibrium modulus is predicted to increase linearly with particle volume fraction, such that a linear plot of G0 versus φp yields the particle functionality, f , if Eassn is known. Note that this analysis neglects network defects such as dangling ends and trapped entanglements. It also assumes that particles with functionality of 2 or fewer do not contribute to the network. Using the best-fit values of G0 listed in Table 2, νeff was computed for 100 mM CTAB at 25 C for all conditions using eq 11, and the results are plotted in Figure 11a versus particle volume fraction for c ( /cs = 1, 2, and 3. The plot confirms that νeff increases linearly with particle volume fraction, as expected from eq 11. Linear fits were made to the data, and the resulting values of νent (intercept) and f (slope) are listed in Table 3, from which the average micellar adsorption energy Eads = Eassn/f is determined (Table 3). The estimated particle functionality ranges between 2 and 4. This magnitude of f agrees qualitatively with cryo-TEM observations, where no more than as many micelles appear to intersect a given particle. The particle functionality increases with increasing salt ratio, which can be rationalized by examining the micellar adsorption energy. At higher salt concentrations, the effective end-cap energy is larger in magnitude,17 such that the reduction in energy realized from the formation of a micelle-nanoparticle junction from a free micellar end-cap will be larger. Therefore, the adsorption energy becomes more negative with increasing salt ratio, which will in turn favor a larger number of micellar junctions per particle. Given the value of f ∼ 3 for all salt ratios, it is tempting to compare micelle-nanoparticle junctions with 3-fold junctions typically observed in branched micellar systems.35 (34) Tanaka, F.; Edwards, S. F. Macromolecules 1992, 25, 1516–1523. (35) Talmon, Y. I. Seeing giant micelles by cryogenic-temperature transmission electron microscopy. In Giant Micelles; Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; pp 163-178.
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Figure 11. (a) Effective elastic contributions per unit volume and (b) apparent reptation time versus cal25 nanoparticle volume fraction for c(/cs = 1 (triangles), c(/cs = 2 (circles), and c(/cs = 3 (squares). Lines are drawn to guide the eye. Table 3. Double Network Model Parameters for cal25 Nanoparticles in CTAB/NaNO3 Micelles with 100 mM CTAB at 25 C c(/cs
νent [μm-3]
f
Eads/kbT
1 2 3
1080 4610 7450
2.4 2.9 3.3
-2.7 -3.1 -3.7
However, micelle-particle junctions are distinct from micellar branching, as branching results in a decrease in viscosity, e.g., for CTAB/NaNO3 micelles,11 due to the presence of sliding junctions.36 The relaxation time, λr, of the WLM-nanoparticle mixtures studied here increases monotonically with cal25 concentration, and this increase is dominated by the contribution of the reptation time. Doi-Edwards theory predicts that the reptation time scales with the effective contour length of the micelles3 as λrep ∼ Leff. Thus, an increase in reptation time upon addition of nanoparticles corresponds to an effective lengthening of WLMs in solution due to the formation of micelle-nanoparticle junctions. Figure 11b shows λrep versus particle volume fraction for 100 mM CTAB at 25 C for each of the salt ratios studied. For all salt ratios, λrep initially increases upon nanoparticle addition. However, after a sufficient increase in nanoparticle concentration, the reptation time appears to saturate. This behavior can be rationalized by considering the number of micellar end-caps available to form micelle-nanoparticle junctions. Assuming cylindrical micelles of radius rc and average contour length Lc, with hemispherical endcaps also of radius rc, the total number concentration of micellar (36) Lequeux, F. Europhys. Lett. 1992, 19, 675–681.
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end-caps, ncap, is given by ncap ¼
2NA ðcs -ccmc ÞV0 4 3 πr þ πrc2 ðLc -2rc Þ 3 c
ð12Þ
where ccmc is the concentration of surfactant at the cmc, and the other quantities are as previously defined. At low particle concentrations, micellar end-caps are in vast excess, such that each nanoparticle added will result in the effective lengthening of micelles, resulting in the consumption of, on average, f micellar end-caps. As more and more nanoparticles are added to the fluid, more end-caps are consumed, until eventually the fluid becomes starved for free micellar end-caps. This will result in a gradual decrease in the effective lengthening of micelles. Eventually, at a particle concentration of φsat p ≈ ncapVp/f , all micellar end-caps will, on average, be consumed in place of particle junctions. Substituting eq 13 for ncap, we find !
φsat p ¼
4a3 6NA ðcs -ccmc ÞV0 3f 4rc3 þ 3rc2 ðLc -2rc Þ
ð13Þ
At this point, any additional particles will compete with other particles for micellar junctions, such that the effective micellar length, and thus the reptation time, will become independent of particle concentration. For 100 mM CTAB and c(/cs = 2 and assuming a surfactant molecular volume of 0.439 nm3,10 eq 13 predicts that this should occur at approximately φsat p ≈ 3 vol %. This approximate value of the saturation concentration of particles is in semiquantitative agreement with the concentration at which the reptation time begins to saturate in Figure 11, suggesting that the effective lengthening of the micelles is indeed limited by the available number of micellar end-caps in solution. Contrastingly, the fact that the effective breakage time is nearly independent of particle concentration for CTAB/NaNO3 solutions containing cal25 nanoparticles suggests a fundamental change in the relaxation processes that give rise to the observed minimum in G00 . Given the discussion above related to effective lengthening of micelles by micelle-nanoparticle junctions, this suggests that the dynamical process leading to the observed minimum in G00 is independent of the overall micellar length. Thus, we hypothesize that the effective “breakage” process for fluids where micelle-nanoparticle junctions dominate the viscoelasticity is dominated by the breakage of those junctions, and not individual micelles. Mechanism of Viscosity Increase. Rationalization of the observed viscosity increase upon nanoparticle addition follows naturally from the previous discussion. First, consider WLMnanoparticle mixtures below ce. The formation of micellenanoparticle junctions leads to the formation of physically cross-linked micellar aggregates in solution. Thus, each nanoparticle will behave as an object with much larger effective hydrodynamic radius. Therefore, the effective volume fraction of suspended particles will be much greater than the solid volume fraction of nanoparticles, which will produce a corresponding increase in viscosity that is much greater than would be predicted by the Einstein equation. This viscosity increase is nearly independent of the salt concentration, as demonstrated in Figure 7. Furthermore, the observed decrease in the entanglement concentration with increasing particle concentration can be reconciled by the effective lengthening of micelles due to the presence of micelle-nanoparticle junctions. Above the entanglement concentration, micellar junctions will produce a more substantial increase in the viscosity due to double Langmuir 2010, 26(11), 8049–8060
network formation. Recalling that η0 = G0λr, the viscosity increase can be trivially explained by the observed increases in the equilibrium modulus and relaxation time. This explains the synergistic effect of increasing both the salt and nanoparticle concentration for solutions above ce, as both lead to nonlinear increases in the effective reptation time. Note that the linear increase in G0 with particle concentration predicted by eq 12 will correspond to a linear increase in viscosity only in the regime where the reptation time is independent of particle concentration. However, this appears to be the case over a significant range of particle volume fraction, as demonstrated in Figure 7. Note that eq 12 predicts that the equilibrium modulus is inversely proportional to the particle size. Thus, the resulting viscosity increase will be lower for larger particles, which may partially explain why no significant viscosity increase is observed for the suspension of non-Brownian particles in WLM solutions.37
5. Conclusions The structuring and rheological modification of model cationic wormlike micelles by nanoparticle addition results from structure and interactions at the nanoparticle-solution interface. Combined ITC and scattering measurements show that, at surfactant concentrations relevant to the formation of WLMs, CTAB forms a hemimicellar adlayer at the particle surface, with which micelles in the bulk can interact. These interactions result in the formation of micellar stalks with energies on the order of the micellar endcap energy, such that reversible micelle-nanoparticle junctions are formed. These results suggest that the effect of nanoparticle addition on micellar self-assembly in a desired system can be controlled by careful selection of the surface chemistry of the added nanoparticles. The presence of micelle-nanoparticle junctions affords unique modification of semidilute WLMs by the formation of a double network in which entanglements and junctions synergistically give rise to viscoelasticity. Below the entanglement concentration, nanoparticles produce mild increases in viscosity by effective lengthening of the micelles. Close to the entanglement concentration, nanoparticle addition can create substantial viscoelasticity in an otherwise Newtonian fluid. Above the entanglement concentration, nanoparticle addition results in more substantial increases in viscosity, rationalized by an increase in modulus as well as the reptation time of the micelles. A simple model for the viscoelastic modification of WLMs based on temporary network theory predicts several features observed in experiments, including linear increases in modulus and viscosity with particle concentration, as well as a number of junctions per particle (2-4) that is consistent with observations made by ITC and cryo-TEM. This double network formation is anticipated to also have consequences for the nonlinear rheology of WLM-nanoparticle mixtures, and is the subject of ongoing investigation. Overall, the results demonstrate that the rheology of WLMnanoparticle mixtures can be controllably tuned independently through two quantities; the micellar end-cap energy, which governs the process of micellar growth and entanglement, and the micellar adsorption energy, which governs the formation of micelle-nanoparticle junctions. As such, nanoparticle addition is distinct from other means of rheological modification, such as the addition of salts or changes in temperature, as it does not change the physicochemical conditions of micellar self-assembly in bulk solution, while at the same time enabling significant increases in viscosity and viscoelasticity. These results have particularly (37) Willenbacher, N.; Oelschlaeger, C.; Schopferer, M.; Fischer, P.; Cardinaux, F.; Scheffold, F. Phys. Rev. Lett. 2007, 99.
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important implications for systems where colloidal species are added for specific function or added benefit, as for such systems the colloids can serve to both add functionality and modify rheology without the need for other additives. Note Added after ASAP Publication. This article was published on the web on March 17, 2010. The axis label of Figure 11 has been corrected. The correct version was published on March 22, 2010. Acknowledgment. Financial support for this work was provided by Unilever, Inc. and the Delaware Center for Neutron Science. Cryo-TEM measurements were performed at the
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W. M. Keck Electron Microscopy Facility, University of Delaware. This manuscript was prepared under cooperative agreement 70NANB7H6178 from the National Institute of Standards and Technology (NIST), U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of NIST or the U.S. Department of Commerce. Some of the rheological measurements were performed on an instrument obtained under ARO Award W911NF-05-1-0234. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Army Research Office.
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