12624
J. Phys. Chem. 1995,99, 12624-12630
Investigation of the Interaction between Emulsions and Suspensions (Suspoemulsions) Using Viscoelastic Measurements Ramon Pons,* Pascale Rossi, and Tharwat F. Tadros ZENECA Agrochemicals, Jealott 's Hill Research Station, Bracknell, Berkshire, RGI2 6EY, U.K.
Received: April 28, 1995; In Final Form: June 20, 1 9 9 9
We studied mixtures of emulsion droplets and solid particles in order to determine the possibility of specific interaction between both types of systems. The solid particles were polystyrene with grafted methylpoly(ethylene oxide) of average 2000 molecular weight. This latex form stable suspensions. This latex was prepared by dispersion polymerization and produced particles with a radius of 3 15 nm and low polydispersity. The emulsion was prepared by shearing a mixture of hexadecane and water and using a polymeric surfactant, Synperonic L92, as emulsifier. The Z-average droplet radius was 280 nm. Latex and emulsion were mixed by using low shear rates, and the rheological properties of the mixtures were measured, as well as rheological properties of the pure systems. The oscillatory behaviors of the emulsion and the suspension are similar at low volume fractions, where the emulsion droplets seem to behave as hard spheres. No specific interaction could be detected, meaning that latex-latex, emulsion-emulsion, and latex-emulsion interactions are of the same type. In steady state measurements some dependence of $+,,ax (maximum volume of packing) with volume fraction and shear rate was detected for the emulsion systems. This dependence shows the influence of the deformability of the droplets. At high volume fractions, the rheological behaviors of the emulsion and the suspension are very different, in both steady state and dynamic measurements. The elastic modulus in the linear region depends on the volume fraction in an exponential fashion for solid particles while the dependence is linear for the deformable particles. The mixtures show intermediate behavior, and their elasticity is well represented by a simple model in which the emulsion and the suspension elastic modulus are used in series with the appropriate weights. A dependence of the critical strain (the strain above which the rheological parameters start to depend on the strain) with volume fraction has been found. The critical strain decreases with volume fraction and goes to a minimum around &,.
Introduction Mixtures of emulsions and suspensions (suspoemulsions) occur in many industrial applications, of which the following are worth mentioning: cosmetics, paints, agrochemicals, and bitumen. Most of the formulations produced in these industries are based on complex compositions which are in most cases arrived at by a trial and error procedure. This is due to the complex nature of the possible interactions that may take place in the system such as particle-particle, droplet-droplet, and particle-droplet. These interactions are influenced by the surfactants and polymers used in their preparation. To understand the nature of these interactions, in particular at high volume fractions, is vital for the preparation and maintenance of the long-term physical stability of these systems. Unfortunately, if one starts with the complex practical systems, it would be extremely difficult to investigate the value of these interactions. As a starting point to study such interactions, one should use well-defined model systems of suspensions and emulsions, where the particles dimension and the surface properties are well established. This is the objective of the present investigation, where we have used model polystyrene latex dispersions with narrow particle size distribution and containing grafted poly(ethy1ene oxide) chains (which produce steric stabilization). The emulsions were made from hexadecane (which does not swell the polystyrene latex particles), and these were prepared
* To whom correspondence should be addressed. Permanent address: Centro de Investigacibny Desarrollo. C.S.I.C. c/ Jordi Girona 18-26. 08034 Barcelona, Spain. Abstract published in Advance ACS Abstracrs, August 1, 1995. @
0022-365419512099-12624$09.0010
using an ABA block copolymer of poly(ethy1ene oxide)-poly(propylene oxide)-poly(ethy1ene oxide). The emulsions prepared were also of narrow droplet size distribution and have radii of the same order as that of the latex. One of the most powerful methods to investigate interactions between particles in a concentrated dispersion is the measure of the viscoelasticity of the system.' As the viscoelasticity is measured as a function of the volume fraction of the dispersion, one can obtain quantitative information on the particle-particle interaction. The same method can be also applied for studying emulsions,2and again the results can be interpreted in terms of interaction or deformation of the droplets. We have adopted this technique to study the interactions between particles and droplets in the mixtures of suspensions and emulsions. These were supplemented by study of the shear rate-shear stress to obtain a value for the maximum packing fraction of the system. Various ratios of volume fractions were used, and our measurements were made as a function of the total volume fraction. Materials and Methods Materials. Synperonic PE L92 was supplied by IC1 surfactants and used as received. This surfactant is an A-B-A block copolymer in which the block A is poly(ethy1ene oxide) and block B poly(propy1ene oxide). Blocks A contain an average of nine ethylene oxide units while block B has an average length of 50 units of propylene oxide. The average molecular weight is 3650, and the molecules contain an average 20% of ethylene oxide. Block B is the hydrophobic part of the surfactant and is insoluble in both water and oil. 0 1995 American Chemical Society
Interaction between Emulsions and Suspensions Hexadecane was Sigma 99% punty and used as received. Styrene was BDH and purified by mixing it with Fullers Earth, followed by filtration. Methoxypoly(ethy1ene oxide) methacrylate of average molecular weight of 2000 was obtained from I.C.I. Paints and used as received. Water was distilled twice in an all-glass apparatus. Methods. Polystyrene latex sterically stabilized was prepared by dispersion polymerization according to the “Aquersemer” m e t h ~ d .The ~ Z-average particle radius was 315 f 10 nm. This value was obtained from PCS (photon correlation spectroscopy) measurements after adequate dilution in water. According to these measurements the polydispersity was 0.1 1, indicating a relatively narrow droplet size distribution. Emulsions were prepared by firstly dissolving the surfactant (54.28 g) in water (245.9 g). The oil phase (542.35 g of hexadecane) was added while shearing the system by means of an Ystral (Ystral GmbH Dottingen) mixer at low speed. Once all the oil was incorporated in the emulsion the speed of the mixer was increased, and this was pursued until an adequate droplet size was achieved. The emulsion Z-average droplet radius, as assessed by PCS measurements of diluted samples, was 280 f 10 nm. The polydispersity of the emulsion (0.15) is slightly higher than that of the latex. PCS (Malvern Instruments 4700) was used to determine the mean particle size of both latex and emulsion. Both systems had to be highly diluted to avoid interaction and multiple scattering. Although the dilution process may change the particle size of the emulsion droplets, it was found that the average droplet size did not show any detectable change after dilution below the threshold of interaction and multiple scattering. Rheology measurements were performed using a Bohlin VOR (Lund Sweden) instrument. Three geometries were used depending on the consistency of the sample: concentric cylinder C14, concentric cylinder C25, and concentric cylinder double gap DG24/27. Some measurements were repeated with two geometries in order to assess the validity of the measured parameters and check for the absence of any wall slip. The samples were put in the cup, and the bob was carefully inserted in the sample. Special care was taken to prevent drying of the samples; either a solvent trap using water or a layer of an insoluble oil was used to prevent this. Drying of the samples is specially troublesome for highly concentrated samples in which a strong dependence of the rheological parameters with the volume fraction is encountered. The stability of some samples was assessed by the repetition of the measurements without changing the sample; this showed absence of drying or coalescence of the samples. The bohlin VOR instrument is a controlled strain apparatus. The cup is moved back and forth in a sinusoidal manner, and the torque is measured on the bob. The torque elements were changed according to the consistency of the sample to obtain the best signal-to-noise ratio. First, a strain sweep experiment was performed to determine the linear viscoelasticity region, and then a frequency sweep was performed at a strain value well within the linear region. Some samples showed a short linear region, and for those the frequency sweep was performed just below the onset of nonlinearity to obtain still some reasonable signal-to-noise ratio. Viscometry measurements were performed with the same arrangements. Discrete shear rates are applied to the sample for 20 s each, and the stress is measured; the shear rates span used was from 0.015 to 925 s-I in 30 steps on a logarithmic scale.
J. Phys. Chem., Vol. 99,No. 33, 1995 12625
m
&
c.
0
200
600
400
800
1000
y I s-’
Figure 1. Shear stress versus shear rate for emulsions with different
volume fractions. Results and Discussion Shear Stress-Shear Rate Measurements. Figure 1 shows plots of shear stress versus shear rate for the emulsions tested. At low volume fraction the plots show a near Newtonian behavior that changes to pseudoplastic behavior with increasing volume fraction. The extrapolated yield stress and plastic viscosity increase with increase of the volume fraction of the oil. Similar trends were obtained for the latex suspension and the emulsion-latex mixtures. From these plots we calculated two values for the viscosity, namely, a low shear viscosity from the slope of the plots between 0.1 and 1 s-I and a high shear viscosity from the slope between 600 and 1000 s-l. These viscosities were used to fit the Eilers equation4
+
(qr”2- l)/# = 0 . 5 [ ~ ] (rr’I2- 1)/#,
(1)
where q r is the reduced viscosity, 4 the dispersed phase volume fraction (oil or latex), 4, the maximum fraction of packing, and [q] the intrinsic viscosity that for hard spheres has a theoretical value of 2.5. Parts a and b of Figure 2 show plots of (qr1l2- 1)/4 as a function of (qrI/* - 1) for the emulsions and latices, respectively. It is apparent from Figure 2a that a straight line can only be fitted for low values of the volume fraction of dispersed phase. At high volume fractions the results deviate from the linear curve; this deviation may be due to the deformation of the emulsion droplets. Moreover, the departure from linearity starts at a volume fraction of approximately 0.6. This corresponds to the maximum packing that can be calculated from the slope at low volume fractions. The maximum packing volume fraction calculated at high and low volume fractions as well as the results for both low-shear rate and high-shear rate are shown in Table 1. The values for 4, and [q]obtained in the high volume fraction region for the emulsion are unreasonably high. In addition, there is a tendency for increase in 4~ with increase in shear rate. These results are consistent with an increase in the oil droplet deformation. At low volume fraction the effect of shear is small as would be expected for small droplets that would behave as almost undeformable. Figure 2b shows the results for the latex suspension. The experimental points can be fitted to a straight line over the whole range of volume fractions measured. The values found for the fit are shown in Table 1. The limiting value of the maximum packing is the same at high volume fractions irrespective of the shear rate at which the viscosity is calculated. This agrees with the picture of the latex system behaving as hard spheres.
Pons et al.
12626 J. Phys. Chem., Vol. 99,No. 33, 1995 100
50
0
150
10000
200
latex emulsion -equation 2 0= ,06.1 0
.'
12
i
0
*
250
200
0
0 140
2
200
6
4
600
400
10
6
800
1400
100
-
-
80
-
- 800
60-
- 600 A
_ _ _ _ _ _ High Shear
1000
- 400 - 200 0
0
20
40
60 (lll'2-
80
0.6
0.5
0.7
4
1000
-
-
0.4
Figure 3. Reduced viscosity as a function of volume fraction for emulsions ( 0 )and latices (m). The lines corresponds to eq 2 with [q] = 2.5 and two values of 4 ~ .
- 1200
F
__.. equation 2 @,,11-0.60
Low shear
-
104
0
1000
100
1)/I$
Figure 2. Reduced viscosity plots for emulsions (a, top) and latices (b, bottom) at several volume fractions. The lines correspond to least linear fits at low volume fraction: (0)low shear viscosity ( y I s-'); (A)high shear viscosity ( y > 600 s-'). TABLE 1: Results of the Fit of Eq 1 for the Emulsion and Latex emulsion low 6 values high 6 values latex" IOW shear &.I = 0.64 5 0.01 4~ = 0.73 f 0.01 4~ = 0.61 f 0.01 [q]= 6.0 [q]= 30.6 [q]= 6.4 high shear & = 0.68 5 0.04 4~ = 0.89 & 0.02 4~ = 0.61 & 0.01 [q]= 2.9 [q]= 8.5 [q]= 2.0 All points could be fitted using a single straight line.
The above results can be also fitted with the Krieger and Dougherty e q ~ a t i o n : ~
Figure 3 shows qr as a function of volume fraction for the latex at high shear rate together with the values calculated using eq 2 in which the theoretical value of the intrinsic viscosity, 2.5, was used. This qr - curves are shown for two values of the maximum packing fraction. Reasonable agreement with the experimental data for the latex is obtained for a maximum packing fraction of 0.61. The good agreement suggests that the assumptions involved in the Krieger and Dougherty equation are valid for this system, Le., a hard-sphere interaction. From the experimental values of maximum packing one can calculate the adsorbed layer thickness if a theoretical maximum
packing is assumed. The adsorbed layer thickness can be calculated from the following equation:
(3) where &ax is the maximum volume fraction obtained from experiment, #Jmax is the maximum volume fraction at which the particles would pack in absence of adsorbed layer, A is the adsorbed layer thickness, and R is the hydrodynamic radius of the particles (Le., core plus adsorbed layer). Using a value for &ax of 0.68 (this value corresponds to the volume fraction of a body-centered cubic arrangement; similar values have been used to account for polydispersity and random packing (6)), the calculated value for the adsorbed layer thickness is around 9 nm. This value is reasonable since the graft polymer has a mean molecular weight of -2000, and this value would suppose a length of 0.20 nm per monomer. This length per monomer is close to the accepted value for a meander conformation of 0.18-0.20 nm.' The above procedure cannot be applied for the emulsions since the droplets are deformable. This can be shown in Figure 3 where the experimental values of qr are also shown for the emulsion. Significant deviation between qr for the emulsion and that of the latex and eq 2 is clearly shown. Summarizing the steady state measurements, we have shown that the rheological behavior of solid particles and deformable particles is similar well below the maximum packing fraction. Systematic differences are found when the dispersed phase volume fraction approaches or exceeds the maximum packing of undeformed spheres or the system is subjected to high shear rates. Under these conditions the liquid particles deform, leading to lower values of viscosity than what is found for solid particles at the same volume fractions. Oscillatory Measurements. In Figure 4 G' and G" at 1 Hz are plotted as a function of volume fraction on a log-log scale. At low volume fraction G" is higher than G', and both quantities increase with increasing volume fraction. However, G' increase faster with volume fraction than G", and a crossover point can be measured with 4 = 0.52 for the latex and 4 = 0.58 for the emulsion. Before the crossover point, the slopes of both curves are little affected by the nature of the particles. For volume fractions above the crossover point the behavior is strongly different. While G' for the latex increases its slope at a volume fraction of 0.60 and continues a strong increase up to the highest volume fraction measured, the same'kind of increase is found for the emulsion, but then the slope is significantly reduced. The variation of G" with volume fraction for the latex follows
J. Phys. Chem., Vol. 99, No. 33, 1995 12627
Interaction between Emulsions and Suspensions 0
1 0
0
100
"1
m $
0
,
G emulsion 0" emulsion Glatex 0" latex
1
0.001
1
0.5
0.4
0.7
0.6
0
Figure 4. A log-log plot of G' (W,0 ) and G" (0,0) as a function of volume fraction for emulsions (closed symbols) and latices (open symbols).
TABLE 2: Crossover Point (G' = G") Volume Fraction for Emulsion, Latex, and Their Mixtures at Three Frequencies fe
0.01 Hz
0.1 Hz
1 Hz
1 0.8 0.6 0.5 0.4 0.2 0
0.59 f 0.01 0.63 f 0.01 0.60 f 0.01 0.60 f 0.01 0.58 f 0.01 0.55 f 0.01 -0.57 f 0.02
0.58 f 0.01 0.59 f 0.01 0.60 f 0.01 0.57 f 0.01 0.55 f 0.01 0.52 f 0.01
0.56 f 0.01 0.61 f 0.02 0.61 f 0.02 0.55 f 0.02 0.54 f 0.02 0.52 i0.0 1 0.52 f 0.02
0.52 f 0.02
the same trend as that obtained for G', except that the slope of the curve is slightly lower. However, this is not the case with the emulsion, where the variation of G" with volume fraction shows significant deviation at higher volume fractions. Indeed, at 4 > 0.63, G" seems to reach a plateau value with further increase of 4. The crossover points for emulsion, latex, and its mixtures are given in Table 2 at three frequencies. The crossover point volume fraction shows a decrease with increase of frequency and increase of the latex content of the mixtures. The crossover point corresponds to the volume fraction at which viscous and elastic forces have the same magnitude at a given frequency. This can be taken as the point at which the form of the interaction between the particles changes its nature from long range to short range. The short-range interaction can be identified, for polymerically stabilized particles, with the onset of strong steric repulsion. This can be taken therefore as the maximum packing fraction of the particles for unperturbed adsorbed layers and undeformed shape. The crossover point volume fraction can also be used to calculate the adsorbed layer thickness. However, it is not clear which value should be used for the maximum packing volume fraction. Random packing of spheres (4max = 0.64) produces unrealistic high values (1 1-20 nm for the latex and 8-13 nm for the emulsion). The use of the maximum packing volume fraction obtained from high shear rate for the latex (&ax = 0.61) or the volume fraction at which the system starts showing high elasticity (values tabulated as a/b in Table 3 obtained from linear plots of G' as a function of volume fraction in Figure 5 ; see below) produces reasonable values for the adsorbed layer thickness. Using these values, we obtain adsorbed layer thickness between 7 and 15 nm for the latex (0.15-0.33 nm per monomer, i.e., reasonable values since the meander conformation for PEO chains corresponds to 0.18-0.20 nm per monomer and the zigzag conformation corresponds to 0.35 nm per monomer (7)). The adsorbed layer thickness calculated for the emulsion is between 5 and 10 nm. Although a zigzag
conformation for nine units of PEO would suppose a length of only 3 nm, the adsorbed layer thickness could be increased due to the polydispersity of the chains and some contribution of the PPO groups due to penetration in the aqueous phase and low penetration in the oil phase. In Figure 6 G' for the pure systems and mixtures is plotted as a function of volume fraction on a log-log scale plot. All the curves look very similar for volume fractions lower than -0.62. This seems to indicate that the interactions inferred from the rheology are of the same kind. This is what one would expect for particles stabilized in the same way, in this case sterically. From the form of the curves it could be deduced that the oil droplets behave as hard spheres up to the point that the interaction energy between two different droplets is high enough to start deforming the droplets to reduce the interaction energy. For concentrated systems the distance between the surfaces of two adjacent particles is small and represents only a fraction of the center-to-center particle distance. In these conditions the interparticle forces are mainly governed by the surface-to-surface distance. This surface-to-surface distance can be calculated similarly to the adsorbed layer thickness:
(4) where R is the mean particle radius, d is half the distance between the surfaces of two neighbor particles, &ax is the volume fraction at which two particles become in contact, Le., the maximum volume fraction for the packing, and 4 is the actual volume fraction. Therefore, the surface-to-surface distance has the form d =R
- 1)
((4ma~4)"~
(5)
As the dependence of the interparticle potential with the distance is usually of the form
~ (a4I/&
(6)
with and exponent of unknown value. Buscall et al.8 showed that the high-frequency modulus of an ordered system is given by the expression
(7) with a = (3/32)&n, where n is the number of nearest neighbors and & the maximum packing for the arrangement. R is the center-to-center particle separation. As d is linear with R, the derivative of the potential is the same except for a factor, and we can write
Functions of this form have been inserted into Figure 6 with exponents /3 = 1 and /? = 2. /3 = 1 gives the best agreement with the experimental results. The behavior of G" with volume fraction is similar below the volume fraction of 0.62 and shows different trends depending on the ratio emulsion-latex above this volume fraction. To interpret this result, we should be able to model the losses of energy in the system. Two possible reasons for this behavior can be proposed. On the one hand, the continuous phase of the emulsion has a higher viscosity than the continuous phase of the latex, and therefore the losses in the combined system are going to be less than in the case of the latex; however, this
Pons et al.
12628 J. Phys. Chem., Vol. 99, No. 33, 1995
TABLE 3: Parameters of the Fit of Eq 9 for the Emulsion, Latex, and Their Mixtures
fe
aylh
b
fe
aylR32
b
1 0.8 0.6 0.5
9896 f 690 9900 f 1700 17500 f 20000 20700 f 2200
0.63 f 0.01 0.63 f 0.02 0.63 f 0.03 0.63 f 0.01
0.04 0.2 0
23700 f 2200 -100000 1oooooo
0.62 f 0.02 0.61 f 0.02 0.62 & 0.02
m
5
(3
0 0 60
0.70
0.65
4
Figure 5. G’ as a function of volume fraction on linear scales for
emulsions, latices, and their mixtures: (m) 100% emulsions, (0)80% emulsions-20% latices, (A)60% emulsions-40% latices, (v)50% emulsions-50% latices, (+) 40% emulsions-60% latices, (+) 20% emulsions-80% latices, ( x ) 100% latices. The lines correspond to the linear fits whose parameters are given in Table 3.
1OOOl 100
A T
-
+
3
,,‘I +
8-2 64
5-5 4-6
2-8
x
0-10 -equation 8 p-1 . equation 8 p-2
10
5 (
jx
10-0
,OW$
1
-
where r is the interfacial tension, R32 is the surface weighted mean droplet radius, and constants a and b have values of 1.76 and 0.71 according to F’rincen. GOis the shear modulus and can be substituted by the elastic modulus of emulsions.* This equation is based on a semiempirical extension of a model system of cylinders arranged in a hexagonal array. When such an arrangement is strained, the total interface is increased; this creates a restoring force that is proportional to the interfacial tension. The origin of constant b is the value of the maximum packing of undistorted cylinders in the hexagonal array. For the samples studied the radius of the emulsion droplets is constant, and due to the excess of surfactant in the emulsion, the interfacial tension is expected to be little dependent on the composition of the system. In Table 3 we report the values that can be obtained from fits of these linear parts. The first constant of the fit corresponds to the factor aT/R3z, and the second corresponds to b in eq 4. Parameter b corresponds to the volume fraction of the onset of elasticity. The value we obtain here for this parameter is well below the value obtained by Princen or P o n ~ ;part ~ . ~of the difference can be accounted for by the role of the adsorbed layer thickness in the present study. However, the adsorbed layer thickness cannot account for the entire difference. As we have seen before, reasonable values of the adsorbed layer thickness increase the effective volume fraction by 0.05 only. It is more reasonable that the packing of undistorted spheres is lower in this system than in the systems studied by Princen or Pons2s9where due to the bigger droplet size the polydispersity may play a role increasing the effective packing of the system. The mixture of latex and emulsion may be regarded as a two elastic element in series with the appropriate weight. The elasticity modulus of a two elastic bodies in series is found to be
. 0.4
0.5
0.6
0.7
4
Figure 6. A log-log plot of G’ as a function of volume fraction for emulsions, latices, and their mixtures. Symbols as in Figure 5. The lines correspond to eq 8 for different values of exponent j3.
factor alone cannot explain a difference in the scaling behavior of the loss modulus, only a reduced value of the emulsion in contrast with the latex. The other reason is due to the deformability of the oil droplets. While the viscous phase in the latex is constrained between two curved surfaces, in the case of the emulsion the droplet surfaces deform to reach the form of parallel surfaces; therefore, the contacting area in a latex suspension must be lower than in the case of the emulsion. In mixtures of emulsions and latices, the higher the ratio of the emulsion, the lower the slope of the G”-# curve. From the linear relationship between G‘ and # that can be shown in Figure 5, it can be concluded that the behavior of the emulsions and mixtures of emulsion and latex above the volume fraction of 0.6 is close to that of a concentrated emulsion. In these systems the elastic modulus is proportional to the volume fraction, following the equation proposed by P r i n ~ e n : ~ Go = a(T/R,,>#”(# - b)
(9)
where Gm,Ge, and G are the elastic moduli of the mixture, emulsion, and latex, respectively, and fe is the fraction of emulsion in the mixture (mass of oil over the total dispersed phase mass). This equation together with the experimental points is plotted in Figure 7. The agreement may be considered good except for fe = 0.2. The general positive deviation, and in particular this for the 0.2 point, could be due to the system behaving in an intermediate way between a series element and a parallel element. A parallel element would imply the latexlatex interaction percolating through the system while a series element would better represent a situation in which the latex particles are surrounded by emulsion droplets that provide a lower elasticity for the whole system. An interesting finding is shown in Figure 8 in which we have represented the critical strain (the strain at which the rheological parameters start to depend on the strain, Le., the onset of nonlinearity) as a function of volume fraction for all the systems studied. yc for all the systems fall on a single master curve. yc is relatively high for diluted systems, and its value goes on decreasing to get to a minimum at a volume fraction around 0.62. For the mixtures containing emulsion and for the emulsion, yc increases again for higher volume fractions. A
Interaction between Emulsions and Suspensions 1000000
h
J, Phys. Chem., Vol. 99, No. 33, 1995 12629 experimental
-eauationlo
1000
0.0
0.2
0.4
1.o
0.8
0.6 fe
Figure 7. Slope (aylR3z) of linear plots of G' as a function of volume fraction (see Figure 5 ) as a function of the fraction of emulsion& The line corresponds to eq 10.
0.1
.:\
+
10-0 8-2 6-4 5-5 4-6 2-8 X 0.10 equation 11, qma,=0.61 X 0.001 :
.
m.' %
...
-equation 11, bm,,=0.62 . . . . . equation 11, om8,=0.63 1
~
1
~
1
'
1
'
1
'
1
'
nonlinearity in the interparticle potential. However, the nonlinearity limit is related not only to a change in the elastic properties but also (and possibly mainly) to an increase in the rate of the relaxation processes. Similar results conceming the critical strain of dispersions were obtained by Jones, Lev,and Boger'* using silica particles coated with stearyl alcohol and dispersed in cyclohexane. They found that the critical strain goes through a minimum at the volume fraction their materials started showing solidlike behavior. These results conceming the critical strain contrast with the findings of Chen and ZukoskiI3 with amphoteric polystyrene particles. Experimentally, they did not find a dependence of the critical strain with volume fraction. They report critical strain values in the range 0.01-0.03. The range of values we obtained expands from 0.1 to 0.001. A possible reason for this different behaviour could be the different kind of interaction potential in this and other systems. Summarizing the oscillatory measurements, we have shown that the dispersion elasticity below the limit for close packing is due to the same kind of interactions regardless of the particles being solid or liquid. Above this critical volume fraction the elastic behavior depends strongly on the nature of the particles. The deformation of the particles produces lower elastic moduli than the obtained for undeformable particles. The dependence of the elastic modulus with volume fraction above the critical volume fraction for mixtures of latex and emulsions is approximately linear; this means that the systems behave as highly concentrated emulsions. The slope of the elastic modulus as a function of volume fraction is dependent on the fraction of emulsion to latex. This slope can be adequately calculated from the values of the pure systems using eq 10 in which the elasticity 1 has been modeled as two elastic elements in series with their weight proportional to the content of latex and emulsion in the mixture. A clear dependence of critical strain with volume fraction has been found. The critical strain seems to be related to the surface-to-surfaceinterparticle distance. This dependence could be related to existing nonlinearities in the interparticle potential. Conclusions We have shown that from the rheological measurements oil droplets can be distinguished from solid particles when the volume fraction is over a critical packing fraction (-0.62). At this volume fraction the viscosity plots for the emulsion departs from the behavior expected for hard spheres. In addition, from viscoelastic measurements the emulsions behave as highly concentrated emulsions, the elastic modulus being proportional to the volume fraction of oil. The elastic modulus of the mixtures of emulsions and latex can be approximated by a simple model in which the elasticity of the mixture arises from the combination in series of two elastic elements that correspond to the emulsion and latex elasticities with the appropriate weights. The critical strain yc has been found to have its minimum at a volume fraction that corresponds to the critical packing fraction of the systems.
This equation has been plotted in the Figure 8 using a value of unity for the ratio d R and three different values for &,,ax. The maximum packing used in this case is the same as that used for the zero-elasticity onset. The good agreement seems to suggest that the resulting nonlinearity is related directly to a
Acknowledgment. The Human Capital and Mobility Program of the E.C. is acknowledged for Contract ERB CHB ITC 920177. We thank Dr. W. Liang for help in the latex preparation, Dr. P. Taylor for help in the rheological measurements and useful discussions, and Mr. K. Kostarelos for useful discussions. References and Notes (1) (a) Tadros, Th. F. Langmuir 1990,6,28. (b) Tadros, Th. F.; Liang, W.; Costello, B.; Luckham, P. F. Colloids Su$ A 1993, 79, 105.
12630 J. Phys. Chem., Vol. 99, No. 33, 1995 (2) Pons, R.; Era, P.; Solans, C.; Ravey, J. C.; StCb6, M. J. J . Phys. Chem. 1993. 97. 12320. (3) Bromley, C. Colloids Surf. 1986, 17, 1. (4) Eilers, H. Kolloid Z. 1941, 97, 113. (5) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 1 1 1. (6) Prestidge, C.; Tadros, Th.F. J . Colloid Interface Sci. 1988, 124, 660. (7) Ribeiro, A. A.; Dennis, E. A. In Nonionic Surfactants: Physical Chemistry; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; p 999. (8) Buscall, R.; Goodwin, J. W. Hawkins, M. W.; Ottewill, R. H. J . Chem. SOC., Faraday Trans. I 1982, 78, 2873.
Pons et al. (9) Princen, H. M.; Kiss, A. D. J. Colloid Interface Sci. 1986, 112, 427. (IO) Princen, H. M. J . Colloid Interface Sci. 1983, 91, 160. ( 1 1) Pons, R. Unpublished results. (12) Jones, D. A. R.; Leary, B.; Boger, D. V. J . Colloid Interface Sci. 1991.. 147.479. . (13) Chen, L. B.; Zukoski, C. F. J . Chem. Soc., Faraday Trans. 1990, 86, 2629,
JP95 1209F
