Langmuir 1997, 13, 2617-2626
2617
Articles Viscosity of Emulsions of Polydisperse Droplets with a Thick Adsorbed Layer Andrew M. Howe* and Andrew Clarke Surface and Colloid Science Group, Research Division, Kodak Limited, Harrow HA1 4TY, England
Thomas H. Whitesides Dispersion Technology Laboratory, Imaging Research and Advanced Development, Eastman Kodak Company, Rochester, New York, 14650-2109 Received September 6, 1996. In Final Form: February 10, 1997X Three oil-in-water dispersions stabilized by the anionic surfactant Alkanol-XC and adsorbed gelatin have been isolated by centrifugation, resuspended in water, and characterized with respect to particle size and gelatin binding. The viscosities of these resuspended dispersions were measured as a function of applied shear rate and dilution with the aim of understanding the rheological role of the adsorbed gelatin shell. The emulsions are polydisperse (weight:number mean size ∼3) with the number-mean diameter ∼50-90 nm. The low-shear behavior (from the Newtonian plateau to the critical shear stress) can be described using a simple hard-sphere model using an estimated length-mean size for the oil droplets with an adsorbed gelatin layer of effective thickness from 25 to 39 nm, increasing with mean oil particle size. At high shear the model breaks down because there is no second Newtonian plateau. Instead, power-law thinning continues to the highest rates (∼105 s-1) and stresses (∼500 Pa) measured. We attribute the excess thinning to the deformation of the gelatin shell. We extract an interparticle pair potential from the flow curves, using an effective hard-sphere model based on that due to Buscall [Buscall, R. Colloids Surf. A 1994, 83, 33]. A self-consistent, physically reasonable picture of the rheology is then apparent for the different dispersions at different concentrations.
Introduction The viscosity of concentrated colloidal dispersions can be of crucial importance to the properties of materials and their efficiency of manufacture. It is well established that the rheological properties of these systems are influenced strongly by the nature and concentration of the dispersed phase. Most of the theoretical and experimental progress in the understanding of dispersion rheology has been made in the study of suspensions of monodisperse hard spheres dispersed in Newtonian liquids,1-5 though, in practice, all colloidal dispersions are polydisperse to some extent. Colloidal systems made using high-energy techniques, such as homogenization, can contain a very wide distribution of particle sizes. More recently, studies have been reported on binary mixtures of “monodisperse” particles,6 but so far little systematic information is available on dispersion systems of broader polydispersity. Most dispersions of practical importance are not stabilized simply by hard-sphere repulsive forces, but rather by steric repulsion resulting from the presence of an adsorbed polymer layer.7-9 These forces are “softer”; that X
Abstract published in Advance ACS Abstracts, April 1, 1997.
(1) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111. (2) Batchelor, K. J. Fluid Mech. 1977, 83, 97. (3) de Kruif, C. G.; van Lersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1985, 4171. (4) van der Werff, J. C.; de Kruif, C. G.; Dhont, J. K. G. Physica 1989, 160A, 205. (5) Mewis, J.; Frith, W. J.; Strivens, T. A.; Russel, W. B. AIChE J. 1989, 35, 415. (6) Rodriguez, E.; Kaler, E.; Wolfe, M. S. Langmuir 1992, 8, 2382. (7) Tadros, Th. F. The Effects of Polymers on Dispersion Properties; Academic Press: London, 1982.
S0743-7463(96)00871-2 CCC: $14.00
is, they change less sharply with surface separation. The effect of softness on the shear viscosity as a function of volume fraction has been studied by varying the size of the hard PMMA core (from 84 to 1220 nm) with a soft adsorbed layer of poly(hydroxystearic acid), 9 nm thick. As the core size decreases, the adsorbed layer occupies a greater proportion of the particle volume and the particles are effectively softer. The dispersions of soft particles showed a smaller viscosity increase at high-volume fractions. This deviation from hard-sphere behavior is because the polymer layers can interpenetrate or compress, leading to a reduction in effective particle size.10 Wolfe and Scopazzi11 found that the shear viscosity of swollen polymer microgels could be approximated well by a hard-sphere treatment of a wide range of conditions. Deviations from hard-sphere behavior decreased as the cross-link density within the microgels increased. In practical formulations of colloidal dispersions and emulsions, it is common for the dispersion medium to contain polymers and/or surfactants, giving the continuous phase significantly non-Newtonian properties. When these dispersions are concentrated, both the continuous and disperse phases make significant contributions to the rheology. Before conventional models can be applied to understand the dispersion rheology in such cases, the properties of the continuous phase must be known. (8) Napper, H. Polymeric Stabilisation of Colloidal Dispersions; Academic Press: London, 1983. (9) Fleer, J.; Cohen-Stuart, J. M. A.; Scheutjens, M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (10) Frith, J.; Russell, W. B.; Strivens, T. A.; Mewis, J. AIChE J. 1989, 35, 415. (11) Wolfe, S.; Scopazzi, C. J. Colloid Interface Sci. 1989, 133, 265.
© 1997 American Chemical Society
2618 Langmuir, Vol. 13, No. 10, 1997
In this paper, we will discuss a set of three oil-in-water dispersions made with photographic couplers in the dispersed phase. [Photographic couplers are the materials in many photographic products that react (couple) with developing agents to yield colored dyes forming the visual image. They generally are very hydrophobic organic compounds with molecular weights around 1000 Da.] Organic materials like couplers can be incorporated into water-based photographic products by emulsification, usually in the presence of an anionic surfactant and gelatin at a pH greater than the isoelectric point. The dispersion droplets, which are often 100 nm in diameter, are stabilized by a relatively thick (tens of nanometers) layer of gelatin adsorbed at the oil-water interface, where the surfactant is also concentrated, and dispersed in a medium containing high concentrations of nonadsorbed surfactant and gelatin. The oil phase consists of a concentrated solution of the coupler (>50% by weight) in a water-insoluble oil such as dibutyl phthalate or tricresyl phosphate and has a high viscosity (>1 Pa s). A critical factor in understanding the rheology of such dispersions is a detailed knowledge of the adsorption of gelatin and surfactant. Adsorbed gelatin increases the effective particle size and hence the total, i.e., effective, volume fraction of the dispersed phase. As this volume fraction approaches the maximum packing fraction for hard spheres, the viscosity increases dramatically. With small particles and with relatively thick adsorbed layers, this can occur at oil volume fractions as low as 0.1-0.2. The viscosity then becomes strongly dependent on oil droplet particle size, concentration, and the adsorbed layer thickness. While the particle size and size distribution of the particles in coupler dispersions can be obtained reasonably accurately by sedimentation field flow fractionation (SFFF) techniques, experimental measurement of the effective thickness of the adsorbed layer in a concentrated dispersion is difficult. Uncertainty in the size and composition of the adsorbed layers in dispersions means that the composition, and hence viscosity, of the continuous phase are also uncertain. In consequence, the deconvolution of the contributions to the dispersion viscosity made by the continuous and disperse phases is not trivial. In photographic systems, the continuous phase usually consists of a mixture of gelatin and anionic surfactant in aqueous solution. Solutions of gelatin and surfactant are known to be Newtonian up to shear stresses of ∼100 Pa,12 after which thinning commences.13 The interaction of gelatin with surfactant14 leads to synergistic increases in the system viscosity. Therefore, the continuous phase contributes not only to the low-shear viscosity of dispersions but also, at high shear stress, to the shear-thinning behavior. In order to understand the rheology due to the particles alone, we have prepared dispersions with very low concentrations of gelatin and surfactant in the continuous phase and studied their viscosity in shear flow. This reduces the complications caused by the uncertainty in the continuous phase composition and allows us to treat the dispersion as a set of polydisperse particles in a Newtonian medium of known, low viscosity. Experimental Section Materials. The materials used in this study consist of highviscosity dispersions of photographic couplers in an aqueous solution of lime-processed gelatin and the anionic surfactant (12) Howe, A. M.; Wilkins, A. G.; Goodwin, J. W. J. Photogr. Sci. 1992, 40, 234. (13) Greener, J.; Contestable, B. A.; Bale, M. D. Macromolecules 1987, 20, 2490. (14) Whitesides, T. H.; Miller, D. D. Langmuir 1994, 10, 2899.
Howe et al. Alkanol-XC (a mixture of isomers of sodium di- and triisopropylnaphthalenesulfonate, supplied by DuPont). The three parent dispersions contain similar (but not identical) formulations of gelatin and surfactant but have different oil phase compositions. The following sections describe the methods used to remove the nonadsorbed gelatin and surfactant from the continuous phase and to characterize the compositions of the resulting “resuspended” dispersions. From a colloidal viewpoint, the major differences among the three resuspended dispersions are the particle sizes. The final materials are given the codes S, M, and L, which correspond to small, medium, and large mean particle size. Isolation and Resuspension of Dispersions. A sample of the dispersion was diluted with 1 mM Alkanol-XC (equiv wt ) 439, by titration) to contain 1% total gelatin. Six 30-g samples of the diluted dispersion were subjected to centrifugation for 2 h in a Model L8-60M Beckman Ultracentrifuge equipped with a SW28 swinging-bucket rotor operating at 28 000 rpm. In order to maintain the samples in as fluid a state as possible, the temperature of the chamber during centrifugation was adjusted to 45 °C and the samples were incubated at 45 °C prior to centrifugation. The pellets could be resuspended in water by warming and agitation with a stirring rod, indicating that the particles of dispersion maintained their identity during the separation, an observation confirmed by particle size measurements. The supernatant solution, containing most of the free gelatin and surfactant, was decanted from the pellet in each tube. The pellets were washed with water, removed from the tube, and stored in the refrigerator at 4 °C under a small amount of water (∼40 mL). Four batches of pellets were collected following this procedure. The combined pellets from all four runs were treated with 40 mL of 1 mM Alkanol-XC and warmed in a constant-temperature bath at 45 °C for 1 h. The dispersion was redispersed by breaking up the pellets with a spatula, followed by stirring in a high-shear mixer for 2 min at 9500 rpm. Finally, the resuspended dispersion was filtered through cheesecloth and stored in the refrigerator. The total gelatin and surfactant and the bound gelatin and surfactant concentrations were determined by the methods described. The particle density and size distribution were determined by density-gradient centrifugation and SFFF, respectively, while the concentration of oil phase components was measured by HPLC. We found the particle size of the original dispersion and that of the resuspended material to be very similar, suggesting that the particles are not qualitatively modified by the isolation procedure. Samples of both the supernatant solution and the diluted dispersion were analyzed for gelatin and surfactant, as described below. Gelatin and surfactant not found in the supernatant were assumed to be bound to the sedimented dispersion, so that bound gelatin and surfactant were determined by the difference between the concentration of these species found in the diluted dispersion and that found in the supernatant solution after centrifugation. The total surface area of the dispersion in a given volume of dispersion was calculated from the expression
A)
6w DsvF
(1)
where A is the area in m2/mL, w is the weight of the oil phase (g/mL), Dsv is the surface-volume mean diameter (determined by sedimentation field flow fractionation (SFFF)), and F is the density of the oil phase (g/mL, as determined by density gradient centrifugation). Γgelatin and Γsa, the binding of gelatin and surfactant per unit surface area, were taken to be the ratio of the concentration of bound gelatin or surfactant to the area A. Bound gelatin is expressed in mg/m2, and bound surfactant, in mmol/m2. Gelatin Analysis. The analysis of gelatin was accomplished by a modification of the biuret method, as described by Mehl.15 This method depends on the spectrophotometric measurement of a colored complex between copper(II) and the amide linkages of a peptide at high pH. Any turbidity in the analyte is interpreted as an anomalously high concentration of gelatin and so must be avoided. We modified Mehl’s procedure by including a substantial (15) Mehl, J. W. J. Biol. Chem. 1944, 157, 173.
Viscosity of Emulsions of Polydisperse Droplets quantity of propanol as cosolvent to dissolve the oil phase of the emulsion and thus obtained clear solutions for analysis. Interference by coupler was minimal in most cases but was corrected for by the subtraction of appropriate controls. Reagent Preparation. A stock solution of biuret reagent was prepared by mixing 85 g of 51% NaOH solution and 100 g of ethylene glycol in a 500 mL volumetric flask and diluting to about 350 mL with water. After the mixture had cooled to room temperature, 50.0 mL of a solution of 4.00 g of CuSO4‚5H2O in water was added, and the dark blue solution was diluted to the mark with water. This solution (“biuret reagent”) appeared to be indefinitely stable (months). Analysis and Calibration. Mixtures for analysis were prepared from 6.00 mL n-propanol, 8.00 mL biuret reagent, and 6.00 mL of sample, containing 0.5-2.0 mg/mL of gelatin. For maximum precision, all components, particularly the sample, were weighed to the nearest milligram and the gelatin concentrations corrected appropriately. After mixing, the optical density of the solution at 542 nm was obtained using a PE552A UV/visible spectrophotometer. A blank (containing no gelatin) and a series of approximately five samples of known, varying gelatin concentrations were run with each set of samples. A calibration curve was constructed using the optical densities of the samples of known concentration using a quadratic model for gelatin concentration as a function of optical density. The correlation coefficient of the resulting fit was always >0.9999. This model was used to calculate the concentration of gelatin in the unknown samples. Interference by the constituents of the oil phase was checked by controls containing only these materials. Surfactant Analysis. Surfactant was determined by Epton titration,16 using an indicator obtained from BDH, Ltd. (England), catalog no. 19189. The Epton method is a two-phase titration depending on the extraction of an intensely colored ion pair into an organic phase in equilibrium with the titrated solution. Alkanol-XC was titrated using standard solutions of Hyamine 1622 [(((Diisobutylphenoxy)ethoxy)ethyl)dimethylbenzylammonium chloride, Rohm and Haas]. Because the method depends on clean separation of the organic and aqueous phases, and since gelatin is an excellent emulsifying agent, it was necessary to digest the gelatin in each sample by treatment with Takamine, a proteolytic enzyme, prior to analysis. Typically, 10 g of sample was treated with 1 g (both solutions weighed accurately) of a solution of 1 mg/mL of Takamine and incubated for several hours at 45 °C. Titrations were performed using a device obtained from Genex Corp. called a Mixxor, which allowed intimate mixing of small quantities of aqueous and organic phases without loss. Typically, 5 mL of methylene chloride and 5 mL of indicator solution (prepared according to the directions supplied by BDH) were placed in the Mixxor, and an aliquot of incubated sample containing a few micromoles of surfactant was weighed into the apparatus. Titrant (4.00 mM Hyamine 1622) was added using a Gilson Micropipet readable to (0.001 mL until, after agitation, the initially pink organic phase had changed to a uniform gray. With practice, precision of (0.005 mL could easily be obtained. From the volume of titrant and the weight of the aliquot, the concentration of surfactant in the incubated sample was obtained, and from this value, together with the weights of sample and Takamine solution, the concentration in the original solution. Rheology. Rheological studies were made on the Bohlin VOR, a computer-controlled, controlled-strain rheometer. At shear rates up to 4000 s-1, measurements were made using a lowvolume, bob-and-cup geometry, the C 2.3/26, which has a cup of diameter 26 mm and a bob of diameter 25 mm with a base of cone angle 2.3°. This geometry uses 1.8 mL of sample and has a torqueto-shear-stress conversion of 38 538 Pa N-1 m-1. At rates of shear above 1000 s-1, measurements were also made using a tapered-plug geometry. The exact value of the gap between bob and cup in this geometry was obtained by matching the data with those obtained at similar shear rates using the C 2.3/26. Thus, these data at very high shear suffer from a higher uncertainty than those obtained with the C 2.3/26 geometry. The resuspended dispersions were used as prepared or diluted to varying concentrations with 1% w/w gelatin and 1 mM Alkanol(16) In Anionic Surfactants-Chemical Analysis; Cross, J., Ed.; Surfactant Science Series, V8; Marcel Dekker: New York, 1977. (17) Krieger, I. M.; Dougherty, T. J. Trans. Soc. Rheol. 1959, 3, 137. (18) Quemada, D. Rheol. Acta 1978, 17, 643.
Langmuir, Vol. 13, No. 10, 1997 2619 XC (this composition is believed to be close to that of the continuous phase of the resuspended materials; see below). After the samples were placed in the rheometer, they were covered with a very thin layer of silicone oil to slow surface drying and gelation. All measurements were carried out at 42 °C. The viscosity of the 1% gelatin, 1 mM Alkanol-XC solution that was used for dilutions, was determined using the Bohlin CS50 constant-stress rheometer. The low viscosity meant that the double-gap geometry DG 40/50 was required.
Theory Low-Shear Viscosity. The low-shear viscosity data were treated according to the Krieger-Dougherty model.17 This model is used to explain the viscosity of a suspension of monodisperse, noninteracting, hard-sphere particles dispersed in a Newtonian medium. The relative viscosity of the particles ηr is given by the equation:
ηr )
(
)
φe η ) 1η0 φm
-[η]φm
(2)
where η is the measured viscosity (we will use η(0), the measured low-shear Newtonian plateau value), η0 is the continuous phase viscosity, [η] is the intrinsic viscosity (2.5 for spheres), and φm is the maximum volume/volume packing fraction, which for polydisperse spheres we will take to be 0.8. (The choice of 0.8 rather than 0.64, which would be appropriate for randomly-packed monodisperse spheres, is an estimate of the correct value for polydisperse samples such as those under consideration. The exact choice of this parameter can be shown to have relatively little effect on our results.) This gives a value of -2.0 for the exponent (as in the expression due to Quemada18), which we use throughout for convenience. The term φe is the effective volume fraction of the dispersed phase. For small particle dispersions stabilized by adsorbed polymer, φe is taken to be the volume fraction of oil φ0 plus the volume fraction of the adsorbed gelatin shell. From simple geometric considerations, the ratio between φ0 and φe is given by
(t +r r)
φe ) φ0
3
(3)
where t + r is the total hydrodynamic radius of the particle (comprising an oil core of radius r and an adsorbed layer of hydrated gelatin of hydrodynamic thickness t). Since η0 can be estimated (see below) and η(0) measured, φe can be determined and the value of t/r calculated from the above equations. Shear Thinning of Hard-Sphere Dispersions. The flow curve of a dispersion of hard spheres can be described using
η(σ) )
η(0) - η(∞) 1 + (σ/σc)p
+ η(∞)
(4)
where η(∞) is the limiting high-shear viscosity and the exponent p describes the width of the shear-thinning transition (this equation was derived by Krieger and Dougherty17 with p ) 1). The critical shear stress σc is defined as the stress at which the viscosity is midway between the high- and low-shear values:
η(σc) ) 0.5[η(0) + η(∞)]
(5)
A minimum value for the limiting high-shear viscosity was estimated by noting that the lower viscosity under shear can be thought of as the result of more efficient packing of the dispersed phase under the ordering
2620 Langmuir, Vol. 13, No. 10, 1997
Howe et al.
influence of the shear field, that is, as an increase in the effective maximum packing fraction. Clearly, the maximum value this quantity can have is 1.0. Therefore, we estimate a lower bound on η(∞) by using this value together with φe in the Krieger-Dougherty/Quemada equation,18 so that
η(∞) ) η0(1 - φe)-2
(6)
In a suspension of hard spheres in a Newtonian medium, the shear-thinning behavior is a reflection of competition between the thermal (Brownian) and the hydrodynamic (shear) forces experienced by the particles. The thermal force is proportional to the thermal energy kBT and the inverse of the particle size. The hydrodynamic force is proportional to the shear stress σ and the square of the particle size. At σc, the Brownian and shear forces are assumed to be in balance. Therefore, we should be able to predict the shear-thinning behavior from the effective particle size determined by the low-shear data. If our particles behave sufficiently like hard spheres, the effective hydrodynamic radius a will be the sum of the core radius and the hydrodynamic thickness of the adsorbed polymer as described above. If, on the other hand, the stresses on the particles are large enough to induce distortions of the shell (or the core of the particle), the effective hydrodynamic size will be reduced, and the viscosity will be lower than for the corresponding hard-sphere dispersion. Krieger1 modeled the shear thinning of hard-sphere dispersions in terms of reduced (dimensionless) variables, namely the relative viscosity ηr (eq 2) and the reduced shear stress σr, given by
σr )
σa3 kBT
σca3 kBT
(8)
If σrc has a constant value for most dispersions, then estimation of a should allow us to calculate σc. In fact, the value of σrc for several hard-sphere dispersions in a Newtonian liquid was found by Krieger1 to be 0.43. More recently, from studies of many hard-sphere dispersions, it has been reported that σrc lies in the range 0.1-0.5.19 We will use the value midway in that range, σrc ) 0.3. We can then predict the viscosity of a dispersion of hard spheres with volume fraction φe from
η(σ) )
η(0) - η(∞) + η(∞) 1 + σr/σrc
( )
φe(σ) ) φ0
ae(σ) r
3
(10)
In this equation, ae(σ) is the shear-dependent equivalenthard-sphere radius of the particle at shear stress σ, r is the radius of the oil core, taken to be independent of applied shear, and φ0 is the volume fraction of oil. At high shear, Buscall shows that, to a good approximation, Re ) 2ae can be related to U(Re) by
U(Re) ) 0.5kBT[1 + σRe3/8kBTK(φe)]
(11)
where K(φe) is an empirical relation for the balance between shear forces and Brownian motion on the dispersion microstructure and is given approximately by
K(φe) ) 0.016 + 0.52φe
(7)
where a ()r + t) is the effective hydrodynamic radius of the particles. He defined a scaling factor for the thermal and hydrodynamic forces, the critical reduced shear stress σrc, the value of σr at which half of the shear thinning has occurred, i.e.,
σrc )
not seem unreasonable in light of the high viscosity of the core, but may not be valid, particularly at very high shear stress. In order to describe the high-shear behavior, we use the “effective hard-sphere” model of Buscall.20,21 In this model, shear forces are assumed to result in compression of the particles. The resulting effective diameter is related to the shear stress by an approximate particleparticle pair interaction potential U(Re), where Re is the average center-to-center distance between particles. The magnitude of this potential can be either assumed (for example, in the case of electrostatic repulsion) or derived from analysis of the flow curves. Buscall assumes the validity of the Krieger-Dougherty equation under all conditions, with an effective, sheardependent, dispersed phase volume fraction φe(σ) given by
(12)
Since we have assumed above that this balance is achieved at σrc ) 0.3, for consistency we will use the same assumption here. Hence, we will use the alternative form
(
U(Re) ) 0.5kBT 1 +
)
σ[ae(σ)]3 σrckBT
(13)
in which K(φe) ) σrc ) 0.3. This model neglects the hydrodynamic ordering effects of shear described in the preceding section. We have included these effects by assuming that the maximum packing fraction is also shear dependent,22 varying from a low shear value (0.8 for our polydisperse dispersions) to a high-shear value (1.0). It was calculated according to the following procedure. We estimate the hard-sphere viscosity at shear stress σ using eq 9, with η(∞) estimated according to eq 6. We then assume (with Buscall) that the Krieger-Dougherty equation (eq 2) applies at any shear, so that the maximum packing fraction as a function of the shear stress can be calculated as the solution to
(9)
-2 ηHS r (σ) ) [1 - φe/φm(σ)]
(14)
0.5 φm(σ) ) φe[1 - (1/ηHS r (σ)) ]
(15)
that is, Shear Thinning of Dispersions of Deformable Particles. We demonstrate below that the hard-sphere model is incapable of describing the flow curves of our dispersions at high shear stress. In the following analysis, we attribute this failure entirely to the deformability of the shell of adsorbed gelatin. In the process, we ignore, for example, possible contributions to the shear thinning due to deformation of the oil core. This assumption does (19) Goodwin, J. W. Contribution 86 (“Rheology of Latexes”), 64th ACS Colloid & Surface Science Symposium, Lehigh, June 1990.
where ηHS r (σ) is the calculated relative viscosity for a hard-sphere suspension (eq 9) at a shear stress σ. The calculation of U(Re) from the flow curve therefore proceeds as follows. The effective dispersion volume fraction φe, (20) Buscall, R. J. Chem. Soc., Faraday Trans. 1991, 87, 1365. (21) Buscall, R. Colloids Surf. A 1994, 83, 33. (22) Wildemuth, R.; Williams, M. C. Rheol. Acta 1984, 23, 627.
Viscosity of Emulsions of Polydisperse Droplets
Langmuir, Vol. 13, No. 10, 1997 2621
hydrodynamic size a, reduced shear stress σr, and calculated hard-sphere relative viscosity for a given shear stress ηHS r (σ) are estimated from the low-shear viscosity. These values are used to estimate the shear-dependent maximum packing fraction from eq 15. The KriegerDougherty equation is then used with this maximum packing fraction to calculate φe, the effective volume fraction. The interaction distance Re is then obtained from
( )
φe(σ) Re ) D2 φ0
Y3(r22/r1r3)} (23) (16)
∑(r + t)3
(17)
where rj is the desired average particle radius, the shell thickness t is assumed to be independent of particle size (but see below), and N is the total number of particles in the sample. This expression is equivalent to stating that what we are trying to find is the radius corresponding to the number average particle volume. The most detailed information about the size distribution of dispersion is obtained through the use of SFFF. The software associated with this technique could in principle be written to yield the desired radius directly, but only certain averages are calculated by the present version of the software. These are
r1 ) r3 ) r4 )
∑(ri)/N ∑(ri)3/∑(ri)2 ∑(ri)4/∑(ri)3
number mean radius ()Dn/2) (18)
If the polydispersity is not too great, r1r2r3 ∼ r23. Further, under the same conditions, r2/r3 and r22/r1r3 will both be close to unity. If these approximations are not too severe, then
X = r23(1 + Y)3
(24)
rj ) X1/3 - t = r2(1 + t/r2) - t ) r2
(25)
Therefore
Unfortunately, even the length mean is not calculated by the SFFF program, so in this paper we use the average of the number of surface means instead. Results Characterization of Resuspended Dispersions. The composition of the dispersions and their particle sizes are given in Table 1. Gelatin and surfactant binding data are included in Table 2. In the worst case (L), the free gelatin concentration is estimated to be only 1.24%; the other two dispersions contain almost no free gelatin. Gelatin concentrations at this level show no appreciable non-Newtonian behavior. The Γgelatin figure of around 4 mg/m2 is typical for gelatin adsorbtion at hydrophobic surfaces.23 Rheological Measurements. The flow curves at 42 °C for the resuspended dispersions and various dilutions in 1% gelatin, 1 mM Alkanol-XC solution are shown in Figures 1 (S), 2 (M), and 3 (L). Note that even for samples with η(0) values as low as 5 mPa s, shear thinning is clearly discernible. The value of η(0) at each oil phase concentration was estimated by fitting the data over the entire range of shear stresses to the Ellis model, eq 4. As we explain later, we believe that the parameters of these fits (other than the zero-shear viscosity), have little physical significance, so only η(0) is recorded in Table 3. We wished to use the Krieger-Dougherty expression to deduce the effective volume fraction occupied by the dispersed phase plus its adsorbed gelatin shell. Consistent Table 1. Bulk Properties of Resuspended Dispersions dimension
surface mean radius ()Dsv/2) (19) volume mean radius ()Dw/2) (20)
As we show below, an approximation to the desired number volume mean diameter is
r2 )
X ) r1r2r3{1 + 3Y(r2/r3) + 3Y2(r2/r3) +
1/3
where D2 is the mean diameter of the oil core (see next section). Finally, U(Re) is calculated from eq 13. Choice of an Appropriate Average Particle Size for Rheological Applications. As pointed out above, most theoretical treatments of rheological data assume monodisperse particle size distributions. The dispersions described in this paper are distinctly polydisperse, and so a question arises as to whether an appropriate average diameter can be chosen that can be used as a single parameter to describe the shear dependence of the viscosity. We describe one approach to this choice, based on a combination of convenience (in particular, what kinds of measurements of particle size were readily available) and plausibility. We start with the observation that the KriegerDougherty model for treating the low-shear behavior of a dispersed system deals with the effective volume fraction of the dispersed phase. This observation suggests that a reasonable guess for an appropriate average size for a polydispersed system of particles is the number volume mean, i.e., the size that gives the same effective dispersed phase volume as the actual polydispersed sample. Thus
N[rj + t]3 )
This expression is not particularly helpful, since it depends on t. However, it involves only the first three of the averages of the distribution; quite generally, the magnitude of the average increases with its degree. Therefore, the best approximation will be to use the length mean, r2. If we define Y ) t/r2, then
∑(ri)2/∑(ri)
length mean radius
2
3
X ) r1r2r3 + 3tr1r2 + 3t r1 + t
S M L a
52 65 86
(22)
92 130 149
154 72.0 198 97.5 197 117.5
9.3 5.3 6.2
% oil
gelatin Alkanol-XC (mg/mL) (mM)
14.30 11.53 15.37
39.8 25.1 33.2
12.3 8.5 11.6
This is the average of Dn and Dsv. See Theory section.
Table 2. Estimated Gelatin and Surfactant Partitioning for Resuspended Dispersions Alkanol-XC
(21)
To show this, we can expand X ) (rj + t)3 in terms of the various kinds of mean radii:
composition
disper- Dn Dsv Dw D2a A sion (nm) (nm) (nm) (nm) (m2/mL)
gelatin
gelatin- oildisper- free bound bound Γsa free bound Γgelatin sion (mM) (mM) (mM) (µmol/m2) (mg/mL) (mg/mL) (mg/m2) S M L
0.82 0.8 1.5
0.2 0.05 1.5
11.3 7.7 8.6
1.2 1.4 1.4
7.6 1.6 12.4
32.2 23.5 20.8
3.5 4.4 3.4
2622 Langmuir, Vol. 13, No. 10, 1997
Figure 1. Flow curves for dispersion S at various volume fractions. Full lines are fits with eq 9 using parameters in Tables 3 and 4. Dotted lines are predictions using Buscall’s effective-hard-sphere model assuming an exponential increase in the pair potential with distance, as described in the Discussion.
Howe et al.
Figure 3. Flow curves for dispersion L at various volume fractions. Full lines are fits with eq 9 using parameters in Tables 3 and 4. Dotted lines are predictions using Buscall’s effective-hard-sphere model assuming an exponential increase in the pair potential with distance, as described in the Discussion. Table 3. Limiting Low-Shear Viscosities and Calculated Values of the Continuous Phase Viscosity, Effective Volume Fraction, Shell/Core Radius Ratio, and Predicted “Hard-Sphere” High-Shear Viscosities for Resuspended Dispersions Diluted with 1% Gelatin, 1 mM Alkanol-XC Solution dispersion S
M
L
Figure 2. Flow curves for dispersion M at various volume fractions. Full lines are fits with eq 9 using parameters in Tables 3 and 4. Dotted lines are predictions using Buscall’s effective-hard-sphere model assuming an exponential increase in the pair potential with distance, as described in the Discussion.
use of this equation requires an estimate of the continuous phase viscosity, η0. We assumed that this viscosity is determined by the amount of gelatin dissolved in the continuous phase of the dispersion. Unfortunately, however, the volume of the continuous phase is a function of the effective volume fraction occupied by the particles. As a result, the concentration of gelatin in the continuous phase, and the resulting viscosity, must be calculated iteratively. The following procedure (expressed as a computer program) was used. First, the concentration of free gelatin in the total volume of dispersion Gt was calculated as the sum of the analyzed amount (from Table 2) and that added with the diluent. Second, an initial guess of the effective volume fraction was made (generally, (23) Kamiyama, Y.; Israelachvili, J. Macromolecules 1992, 25, 5081.
φ0
η(0) (mPa s)
η0 (mPa s)
φe
t/r
η(∞)HS (mPa s)
0.143 0.129 0.114 0.100 0.096 0.115 0.104 0.092 0.069 0.154 0.123 0.092 0.061
127 72.4 30.1 12.8 8.14 12.41 8.3 5.65 3.3 151 22.3 6.2 3.2
1.86 1.73 1.59 1.43 1.34 0.89 0.93 0.97 1.02 2.94 1.96 1.50 1.32
0.70 0.68 0.62 0.53 0.47 0.59 0.53 0.47 0.36 0.69 0.56 0.41 0.29
0.70 0.74 0.76 0.74 0.69 0.72 0.72 0.72 0.73 0.65 0.66 0.65 0.68
20.7 16.9 11.0 6.5 4.8 5.3 3.3 3.5 2.5 30.6 10.1 4.3 2.6
the oil phase volume fraction was used). The following steps were then executed iteratively: (1) The gelatin concentration in the continuous phase was calculated as Gt/(1 - φe). (2) The viscosity corresponding to this concentration was calculated from parameterized empirical measurements of viscosity vs gelatin concentration. The expression utilized was log(η0) ) -0.107 + 0.151Gt - 0.00164Gt2. (3) Using the Krieger-Dougherty expression (eq 2) with the extrapolated value of the low-shear viscosity (η(0), Table 3), a new value for φe was obtained. Steps 1-3 were repeated until the φe no longer changed appreciably. The resulting values for the continuous phase viscosity and the effective volume fraction are recorded in Table 3. Taking the analytically-determined oil concentration as φ0, we can calculate the ratio of the adsorbed layer thickness to the oil phase particle radius from
()
φe t ) r φ0
1/3
-1
(26)
For each dispersion the ratio t/r is fairly constant, independent of oil volume fraction. A lower bound on the
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Langmuir, Vol. 13, No. 10, 1997 2623
Table 4. Calculated Values of Effective Hard-Sphere Particle Radius a, Oil Core Radius r, and Adsorbed Shell Thickness t for the Resuspended Dispersions Diluted with 1% Gelatin, 1 mM Alkanol-XC Solution dispersion
t/r
D2/2 (nm)
a (nm)
t (nm)
σc (Pa)
S M L
0.73 0.72 0.66
36 49 59
62 84 98
26 35 39
5.4 2.2 1.4
limiting values of the high-shear viscosity assuming that the particles are unchanged in the flow field (i.e., hardsphere behavior) may be predicted with φm ) 1.0 (eq 6). From inspection of Table 3 and Figures 1-3, it is clear that the measured values of viscosity at high shear are much lower than the hard-sphere predictions. Assuming D2/2 is a good measure of the radius of the oil core r, the hydrodynamic shell thickness of the adsorbed gelatin t and the effective hydrodynamic radius a of the particles can then be estimated from
(
a) 1+
t D2 r 2
)
(27)
From eq 7, the critical shear stress for each dispersion is then
Figure 4. Interparticle pair potentials for the dispersions S, M, and L calculated from the experimental data in Figures 1-3.
Throughout this paper, we assume a value of 0.8 for the maximum packing fraction φm. For monodisperse systems a value of 0.64 (maximum random close packing) is appropriate, but for a polydisperse system such as ours this value is surely too small. Fortunately, results of the calculations are reasonably insensitive to the exact choice. Trial calculation using data for the concentrated dispersions shows that using a value of 0.64 for φm reduces the calculated shell thickness by less than 10%. Adsorbed Gelatin Layer. From the amount of adsorbed gelatin and the volume of adsorbed layer (φe φ0), the average concentration of gelatin in the adsorbed layer may be calculated. For S, M, and L, the mean concentrations in the hydrodynamic shell are 55, 49, and 39 mg/mL, respectively. Thus, the shell around the large particle dispersion is appreciably more dilute than that around the others. The estimated shell thickness varies from 26 to 39 nm, roughly consistent with the expected dimensions of a gelatin molecule,24,25 and also consistent with the thickness of adsorbed gelatin layers on latex particles measured by
photon correlation spectroscopy.26 There appears to be a systematic increase in the shell thickness with the size of the core. This change may be the result of the high radius of curvature of the interface. Such behavior has been found recently in model studies of adsorbed diblock copolymers.27 For adsorbtion on surfaces with curvature radius of the same order as the adsorbed layer thickness, the maximum adsorbed amount was found to increase and the hydrodynamic thickness to decrease as the particle core radius decreased. This was attributed to a decrease in repulsion between the blocks of the polymer that protrude from the surface into the solvent.27 If we take an average molecular weight for the gelatin to be close to that for a single R chain (about 100 000), we can calculate an approximate value for the area and volume occupied per molecule adsorbed at the surface. Both the area/molecule (ca. 35-45 nm2) and the volume/ molecule (1000-2000 nm3) are very much smaller than the corresponding dimensions of a gelatin molecule in solution (radius of gyration of ca. 30 nm21). Therefore, the adsorbed strands must overlap extensively. Modeling the Flow Curves. From the data in Table 4, we can use eq 9 to calculate flow curves, on the assumption that the particles with adsorbed gelatin behave as effective hard spheres. The predicted hardsphere behavior is shown as solid curves in Figures 1-3. The agreement between these predictions and the data is reasonably good for shear stresses up to about σc. This observation suggests that the gelatin shell is relatively incompressible at low interaction energies. At shear stress substantially greater than σc, large deviations from the behavior expected of hard spheres is observed, in that the dispersions continue to shear thin up to the highest accessible shear stresses instead of reaching a Newtonian plateau. As previously discussed, we attribute this continued shear thinning to the deformability of the adsorbed gelatin layer. We can obtain the interaction potential U(Re) using Buscall’s effective-hard-sphere model20,21 as described in the theory section. The results of this calculation for the three dispersions are shown as a plot of U(Re) vs Re (the center-to-center distance) in linear form in Figure 4 and
(24) Pezroni, K.; Djabourov, M.; Leblond, J. Polymer 1991, 32, 3201. (25) Herning, T.; Djabourov, M.; Leblond, J.; Takerkart, G. Polymer 1991, 32, 3211.
(26) Whitesides, T. H. Unpublished Data. (27) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 1331.
σc )
σrckBT a3
(28)
with σrc ) 0.3. The values derived from the various sets of data are summarized in Table 4. The solid curves in Figures 1-3 show the predicted hard-sphere behavior calculated using eq 9 with the values for η(0) and η(∞) taken from Table 3 and σc as shown in Table 4 (p ) 1). According to these curves, deviations from hard-sphere rheology occur in all of the dispersions at shear stresses of 1-10 Pa. We will attribute the further shear thinning (over the expected hard-sphere behavior) to shear deformation of the particles, particularly of the adsorbed polymer shell. The dotted curves in Figures 1-3 are calculated on the basis of this picture by means of an inversion of the Buscall model20,21 using a procedure and pair potential function described in the next section. Discussion
2624 Langmuir, Vol. 13, No. 10, 1997
Figure 5. Interparticle pair potentials for the dispersions S, M, and L calculated from the experimental data in Figures 1-3. Note how the potential rises steeply (i.e., in a hard-spherelike manner) up to U(Re) ∼ 10 kBT, and then more gradually at higher shear energies.
in semilog form in Figure 5. (Note that although the maximum shear stress experienced by the three samples is similar (200-500 Pa), the energy increases with particle volume U(Re) ∼ σ{ae(σ)}3, hence the difference in the maximum value of U(Re) for the three dispersions.) The data derived from the flow curves at various concentrations give rise to consistent potentials for each dispersion. The calculated potentials have similar shapes for all three dispersions. From Figure 4, the potentials appear consistent with very soft interactions between the particles that increase on decreasing separation. On increasing shear stress, the effective adsorbed layer thickness is reduced by between 15 and 30 nm, dependent on the dispersion. However, presentation of the data in linear form does not show clearly the behavior at low stresses, which turns out to be rather different from that at high shear, as can be seen from inspection of a log-linear plot (Figure 5). Now we can see that the particles behave almost as hard spheres (as expected, given how well eq 4 fits the data to σc) up to interaction potentials of the order of 10 kBT; that is, the potential rises steeply as a function of decreasing Re. More energetic interaction leads to an effective compression of the shell. Since the three dispersions have a shell consisting of a common polymer, we expected that their behavior should show common characteristics and hoped to find a universal scaling that would collapse all of our data to a single curve. Good collapse is, in fact, achieved by appreciating that the surface-to-surface distance (2te) is a more fundamental measure of the interaction than is Re. To the extent that the shell acts like a simple spring, the interaction should scale as the dimensionless distance 2te/t. The result of this scaling is shown in Figure 6, where all of the data from all three dispersions fall nearly on a single curve. The agreement among the three data sets is regarded as quite satisfactory, considering the simple nature of the model. Interactions between gelatin layers adsorbed on mica surfaces have been investigated by Kamiyama and Israelachvili23 using the surface-forces apparatus. This study was carried out over a much wider range of pH and ionic strength than the work reported here. Under conditions comparable to ours (0.01 M NaCl, pH > isoelectric pH), repulsive forces were detected to a range
Howe et al.
Figure 6. Interparticle pair potential U(Re) expressed as a function of the dimensionless surface-to-surface distance, 2te/t.
of about 100 nm surface separation. The inferred layer thickness (ca. 50 nm) is comparable to the values listed for t in Table 4, especially if the observation that the shell thickness increases with the particle core size is not an artifact of our analysis, since the measurements obtained in the surface-forces apparatus are for effectively planar surfaces. It is difficult to compare the shape of the repulsive interactions obtained by Kamiyama and Israelachvili with the data in Figures 4-6, because their data is in terms of force vs distance and ours is potential vs distance. An exponential decay in U(Re) was found to model the interaction between PMMA particles with a grafted deformable layer of poly(12-hydroxystearic acid) dispersed in decalin.28 There are two detailed differences from this work: firstly, we have allowed for the hydrodynamic effects on the effective maximum packing fraction φm (which gives hard-sphere behavior at low applied stresses, to σc); and, secondly, these gelatin-covered particles have a much thicker adsorbed layer relative to the particle size and are therefore much softer. In addition, our particles continue to deform even under the highest stress to which we subject them. It is not necessary to postulate a minimum approach distance within which hard-sphere behavior is found. Witten and Pincus29 have concluded that, for particles with a shell of grafted polymer of a thickness much larger than the core, the pair potential should increase as the logarithm of the surface separation. Figure 7 shows the pair potential derived for our particles plotted as a function of ln(2te/t). The data do not correspond exactly to the prediction, but the deviations from linearity are those that might be expected from an adsorbed polymer (whose local segment density would be expected to increase more rapidly as the surface is approached than would that of a grafted species, because of the presence of populations of loops and trains), and also from particles whose core is not small relative to the shell thickness. In fact, as is seen in Figure 8, an almost linear dependence of U(Re) on the reciprocal of the scaled surfaceto-surface distance is observed. We have used this simple relationship to invert the Buscall procedure and model the high-shear rheology of our dispersions. The pair potential function used is of the form (28) Buscall, R.; D’Haene, P.; Mewis, J. Langmuir 1994, 10, 1439. (29) Witten, T. A.; Pincus, P. A. Macromolecules 1986, 19, 2509.
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Langmuir, Vol. 13, No. 10, 1997 2625
Figure 7. Interparticle pair potential U(Re) plotted as a function of the logarithm of the dimensionless surface-to-surface distance 2te/t.
Figure 8. Interparticle pair potential U(Re) plotted as a function of the reciprocal of the dimensionless surface-to-surface distance, t/2te. Note the approximately linear dependence of the potential on this parameter. The use of a linear potential function allows reasonable prediction of the rheology of all three dispersions, as shown by the dotted curves in Figures 1-3.
(
)
[(r + t) - Re/2] 1 t U(Re) ) m )m 2te 2 (Re - D2)
(29)
The dotted curves in Figures 1-3 show the predicted rheology of the dispersions at high shear based on this equation. There is excellent agreement between the predictions and the observed behavior, particularly for dispersions S and L, where we have the most complete set of data. The collapse of the data shown in Figure 6 is not so good that a single value of m can be used for all three dispersions; in order to get the results shown in the figures, values of m of 880, 540, and 1000 kBT were used for S, M, and L, respectively. Summary In this paper we have described the isolation of three oil-in-water dispersions with particles coated with an
adsorbed layer of gelatin and surfactant. We have characterized the rheological behavior of these dispersions in concentrated suspension in a Newtonian medium of known viscosity. We have shown that the flow curves can be described by a mixture of behaviors. At low shear stress (
