Mechanical Spectroscopy of Colloidal Dispersions BY JAN MEWIS AND GUSTAAF SCHOUKENS Department of Chemical Engineering, Katholieke Universiteit Leuven, de Croylaan 2, B-3030 Heverlee, Belgium Received 19th December, 1977 The mechanical properties of colloidal dispersions are closely related to their stability parameters. Unfortunately, the rheological characteristics which are normally used, such as viscosity and static elasticity, measure the global behaviour. It is extremely difficult to extract from such characteristics detailed information about the structure. Therefore, there is a need for more adequate experimental techniques . The construction and the potential applications of a rheometer, in which an oscillatory movement is superimposed perpendicular to a steady shear flow, are discussed.Measurements on carbon black dispersions are carried out, and the relaxation spectra obtained for the sheared dispersions are compared with theoretical predictions based on a chain-like structural model. In this way, informa- tion concerning interaction forces and structural parameters is obtained. Mechanical measurements have been widely used to collect information on the structure of materials. The technique has been most successful on materials which show a time and frequency dependent response to stress and strain. The resulting viscoelastic behaviour can be represented by a characteristic time function, the relaxation spectrum H(z) which is used in mechanical spectroscopy. Clearly, a spectrum provides more information about structure than do viscosity or static elasticity.However, the available viscoelastic data on colloids have not contributed much to our understanding of structure. In part, experimental difficulties are responsible for the lack of success, the oscillations tend to affect the structure to be measured. In addition the spectra obtained do not have any prominent feature^.^-^ Recently, the viscoelastic behaviour of dilute, thixotropic dispersions under shear has been measured by means of relaxa- tion and oscillatory measurement^.^ The results seemed promising and consequently an experiment was set up to explore the mechanical behaviour of flowing colloids. In general, a given rate of shear entails a corresponding degree of structure in shear sensitive systems. If small oscillations are superimposed on the steady state shear, they do not further affect the structure. The oscillatory velocity can be chosen to be parallel or perpendicular to the steady state flow.In the former case the coupling between both modes of flow is stronger than in the latter. This effect shows up in the calculation of the spectra froin the oscillatory data.5 Therefore, the orthogonal superposition technique was selected. EXPERIMENTAL The general layout of the apparatus is given in fig. 1. The measuring cell (E) consists of two coaxial cylinders (Couette geometry). A variable speed motor (F + G) connected to the external cylinder generates the stationary shear flow. The inner cylinder is kept in position by two membrane springs (D). An electromagnetic vibrator on the same cylinderJ .MEWIS A N D G . SCHOUKENS 59 provides the oscillatory movement, which is detected by means of a linear displacement transducer (A). The geometry of the cell must ensure a nearly constant shear rate through- out the sample because the structure changes with shear rate. In Couette geometry the stresses depend on radius. Owing to the pseudoplastic nature of colloidal dispersions, a difference in stresses enhances the variations in the rates of shear. Hence the gap must be as narrow as possible. In the present instrument a radius ratio of 1.02 has been realized. This value limits the possible differences in shear rate to 10% if the power law index becomes as low as 0.2. With a gap size of 3 x lo4 m, one can expect particle size effects from aggregates and elementary particles (R = 6.5 x loF9 m) to be negligible, with qualifications for the fully developed network structures.As the instrument has only been applied to moderately concentrated, free flowing dispersions, the cylinders were not corrugated. The electromagnetic vibrator contains a permanent magnet (B) of 1 T with an annular gap. The inner cylinder carries an electric coil (C) at its top end, which fits in the gap of the magnet. An alternating current through the coil causes the required oscillation movement. A Solartron Transfer Function Analyser 11 70 is used as the variable frequency generator and for analysis of the complex (stress, strain) relation. In order to ensure a purely sinusoidal response, the oscillation amplitude can be reduced to 1 pm.The data provide the real (G') and imaginary (G") parts of a complex modulus which depends on shear rate (f) and frequency (a). Both parts of the modulus reduce to a single spectrum H(z,2j2), which is the most compact way of representing the data. At the same time any other viscoelastic parameter can be calculated from this spectrum. A shear rate dependent spectrum is used here. There are some fundamental arguments against this procedure7 but they do not interfere with the present analysis. It can be shown that the orthogonal superposition moduli are related to the spectrum by eqn(1): d(ln z). G"(7,P) = I-, H(2,2j2) - COT 03 1 + w2r2 Eqn (1) are identical in form with the normal relations from linear viscoelasticity. The spectra can, therefore, be calculated from the measured moduli in the same manner as in oscillatory experiments without superimposed flow.* RESULTS AND DISCUSSION The accuracy and the sensitivity of the instrument have been verified by means of measurements on homogeneous Newtonian and viscoelastic fluids. With a fluid of viscosity 19 Pa s, consistent results have been obtained down to the cHz frequency range.9 Viscosity and elasticity (first normal stress coefficient) have been computed from the oscillatory data with and without superimposed flow. They coincided with the corresponding values, measured directly on a Weissenberg rheogoniometer within the measuring accuracy of the latter in~trument.~ With the orthogonal superposition rheometer, measurements have been performed on structure-forming dispersions of carbon black (Neo Spectra Mark 11, Cities Services) in mineral oil4 at 20 "C.The samples are thixotropic. At rest they develop into weak, solid-like structures. Under shear the structures break down with time. To a large extent the shear induced changes are reversible. If the changes in structure are relatively slow, as during recovery after shearing, the spectra could be obtained as a function of time.4 In the present work, only equilibrium spectra are discussed. Fig. 2 presents the moduli measured on a 4.7% (volume) concentration at a shear rate of 13.1 s-l. In the terminal zone at low frequencies the G' curve tends to the theoretically required slope of 2, whereas G" attains a slope of unity. The60 MECHANICAL SPECTROSCOPY I FIG.1 .-Layout of the orthogonal superposition rheometer (A : linear displacement transducer; B: permanent magnet; C: electrical coil on inner cylinder; D: membrane springs: E: sample: F: gearbox; G: motor). f / H z FIG. 2.-Superposition moduli on a 4.7% carbon black in mineral oil dispersion (3 = 13.1 s-‘). 0, G”; f, G’.J . MEWIS AND G. SCHOUKENS 61 characteristic high frequency behaviour is also found in the relaxation spectra calcu- lated for the more dilute dispersions (fig. 3). All spectra extend from 1 x s to a variable upper limit which cannot be accurately determined for dilute dispersions. Curves 4 and 5 refer to recovered thixotropic structures. Once flow ceases the systems develop gradually into solid-like materials. The terminal zone then changes into a zone of constant G' (G' = 3.3 x lo2 N mP2 for curve 4, G' = 1.1 x lo4 N m-2 for curve 5).The spectra are wedge shaped and the slope becomes steeper with increas- ing shear rates. The data indicate a limiting high frequency slope of -1/2 for low shear rates at both concentrations. The present data differ from published work on 1 ool I I I I lo-* lo-' loo v FIG. 3.-Relaxation spectra of carbon black dispersions under shear. Curves 1-4: 2.2% carbon black (1, 9 = 26.2 s-'; 2, 9 = 13.1 s-l; 3, 3 = 2.62 s"; 4, after four days standing without shearing; curve 5,4.4% carbon black, after five days standing without shearing. other dispersions mainly through the presence of much more prominent spectral features. In order to analyse the data, we start from a simple chain model as used earlier in the description of static elasticity in colloids.lOvll The chains consist of elementary particles, held together by elastic springs.The spring force is caused by the inter- particle forces considered in stability theory. As such they provide a direct link between the rheology and stability of colloids. The particles in the chain affect the flow of the surrounding fluid and thus cause viscous dissipation of energy. Hence the chains are composed of dissipating spheres alternating with elastic springs. The behaviour of such structures under oscillatory flow can be calculated. As a matter of fact the model is identical to some of the earlier versions suggested for the simula- tion of polymer solutions.8*12 They all reduce to a ladder network of springs and dashpots (fig.4). All these chain models result in a discrete set of relaxation times z, given by: zp = zJp2 ('JI = 1, . . . N). (2)62 MECHANICAL SPECTROSCOPY FIG. 4.-Mechanical ladder network in the simulation of a chain of colloidal dispersions. The smallest relaxation time corresponds to the relaxation of subsections of a chain containing one spring. Larger values of zr, are related to the coordinated movement of increasingly larger sections of the chain. If an equivalent continuous spectrum is derived from eqn (2), or if the high frequency moduli are calculated, one finds the limiting behaviour : l3 ‘ G’ = G” cc m1/2 and H(z) K T ~ / ~ . (3) Hence, if simple chain elements are present, the system should give a spectrum with slope -1/2 at small times.Viscoelastic measurements can, therefore, be used to determine the postulated existence of chains. Fig. 3 provides some evidence that such a structure is present in the carbon black dispersions to hand. A slope of value 1/2 is only encountered if all the chains contribute in the same manner to the different relaxation mechanisms. Since one does not expect the various chains to be identical in size, only a fraction of the chains present can contribute to the longer relaxation times. This effect of polydispersity reduces the spectrum at larger values of z below the expected value, increasing the local slope. If the chain length is systematically decreased the theoretical slope of 1/2 might disappear. This behaviour could be responsible for the steeper slopes found at higher shear rates.Zimm14 has shown that hydrodynamic interaction between the particles leads to slopes of 2/3. This interaction does not require any superimposed stationary flow and, therefore, should also act without superposition. Fig. 3 shows that, in our dispersions, the steeper slopes disappear at zero rate of shear, making the Zimm explanation less likely. Clearly the presence of more complex floc shapes will also alter the spectrum. Except for the qualitative purpose of verifying the structure, the data can possibly be used in a more quantitative manner. The shortest relaxation time corresponds to the smallest relaxing subchain, i.e., a single bead-spring element. Such elements are always present except perhaps at very high shear rates.Fig. 3 gives 1.5 kHz as the highest relaxation frequency. Homogeneous fluids do not show any particular change at that frequency, hence the effect does not seem to be an instrumental artefact. The value of zmin is determined by the viscous dissipation caused by the particle movement and by the spring constant of the elastic interaction forces. Using Stokes’ law for the viscous force one can calculate the spring constant from the value of Zmin. We use the relation obtained for bead-spring models : l5 where yo = viscosity of the medium = 1.2 N s m-2 which leads to Hs = 3.7 x N m-’. The spring constant describes the (force, deformation) relation for particle doublets. Hence it measures the second derivative of the interaction potential at the equilibrium interparticle distance.For the materials under consideration it is difficult to obtain information about the interaction potential in another manner. Accord- ingly, the suggested method could be useful in stability studies of such systems.J. MEWIS AND G . SCHOUKENS 63 In principle the spectrum can also be used to estimate the chain size during flow. Normally, the shape of the terminal zone of the spectrum will depend on the chain size distribution, because not all chains can contribute to all relaxation times. Un- fortunately, the dynamic properties in the terminal zone are too small to be measured with the present instrument. A modified method is suggested here, which is based on the high frequency part of the spectrum. An expression for the chain size distribution under shear has been proposed by Ruckenstein and Mewis.l6 The present authors have shown the resulting distribution to be identical with the most probable distribution of linear polycondensation reac- t i o n ~ .~ At the same time, evidence was presented that the distribution preserved its shape under changes in shear rate. We can then apply a method, suggested by Menefee17 to calculate the average chain size (by weight) N , from the spectrum height H, at zmin: The viscosities of the dispersions have been measured on a Weissenberg rheogonio- meter. The computational results are represented in table 1. The validity of the TABLE 1 .-CHAIN SIZE N , FOR A CARBON BLACK DISPERSION (c = 2.1 %) AS CALCULATED FROM THE RELAXATION SPECTRA [EQN (5)] 26.2 1.6 1400 6.5 13.1 2.3 1700 8.1 26.2 5.5 2530 14 values obtained for N , should be verified by independent measurements.It has been shown that the carbon black dispersions, which are used here, have structure depend- ent dielectric spectra.'' Electric techniques might therefore be suitable to verify the present chain model. Curves 4 and 5 in fig. 3 correspond to dispersions at rest. At the lower frequencies the real modulus reaches an equilibrium value, which indicates a solid-like behaviour, even at these low concentrations. The behaviour can be explained by the presence of a network structure. The high frequency part of the spectrum is not affected by the presence of a network and preserves its characteristic shape. The difference in level of the spectrum between curves 4 and 5 can be understood on the basis of the number of chains that are present.With more chains, the mesh size will decrease, which will reduce the maximum relaxation time. This effect is also shown clearly in fig. 3. The equilibrium moduli could be used in the usual manner to calculate inter- action forces.lo*ll However, doubling the concentration causes a thirty-fold increase in modulus. As it is difficult to model such changes, the same restriction applies to calculation from the data of concentration-independent parameters for interaction forces. This restriction does not apply to the method suggested above. It is concluded that, at least for certain dispersions, superposition measurements provide more detailed information about the colloidal structure than do the earlier viscosity or elasticity measurements.The technique can be applied to flowing dis- persions and hence does not require the presence of a continuous network structure. The high frequency limit can be related to the interaction forces. In addition a64 MECHANICAL SPECTROSCOPY measure of the chain size distribution is obtained. The technique presented here appears to be a potential tool for measuring characteristics of colloids which are difficult to obtain by other means. The authors are indebted to the Nationaal Fonds voor Wetenschappelij k Onder- zoek and to the Fonds Derde Cyclus (K. U. Leuven) for financial support of this project. One of us (G. S.) acknowledges a scholarship from the N.F.W.O. for the period during which this work was performed. C. J. Nederveen, J. Colloid Sci., 1963, 18, 76. R. D. Hoffman and R. R. 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