Faraday Discuss. Chem. SOC., 1990, 90, 41-55 Use of Viscoelastic Measurements for Investigating the Stability/ Flocculation of Concentrated Dispersions Tharwat F. Tadros” and Andrew Hopkinson? ICI Agrochemicals, Jealotts Hill Research Station, Bracknell, Berkshire RG12 6E Y, UK The use of viscoelastic measurements for studying stability/ flocculation of concentrated dispersions is discussed. With electrostatically stabilised dis- persions, the system becomes predominantly elastic when significant double- layer overlap occurs. This was demonstrated using polystyrene latex disper- sions in and mol dm-3 NaC1. Plots of the elastic modulus G’ versus surface-to-surface separation distance h showed a rapid increase when h was less than twice the double-layer thickness. The experimental G’ values were compared with theoretical values calculated from the second differential of the interaction energy us.distance relationship. With sterically stabilised latex dispersion [containing grafted poly(ethy1ene oxide) chains], predominantly elastic response was also obtained when h became less than twice the adsorbed layer thickness (26). The G’ vs. volume fraction curves were converted to G’ vs. h and this was compared with the values of G’ calculated from direct force, F, vs. distance curves. In both cases there was a rapid increase in G’ with decrease in h when h < 26. Viscoelastic measurements could also be applied for flocculated disper- sions. With weakly flocculated systems such as those obtained by addition of a free non-adsorbing polymer (depletion flocculation), the systems showed pronounced non-Newtonian behaviour above a critical volume fraction of free polymer (4;).The occurrence of this behaviour decreased with increas- ing molecular weight of the free polymer. The extrapolated yield stress was used to calculate the energy of separation between particles, Esep, in the flocculated dispersion. Esep was compared with the theoretical value of the free energy of depletion, Gdep, that was calculated using available theories. With strongly flocculated dispersions, scaling concepts could be applied and the power exponent in # (G’- # n ) could be used as a measure of the strength of flocculation. From G’ and the critical strain y,, above which non-linear response is obtained, the cohesive energy of the flocculated structure, E,, was calculated.log-log plots of E, versus # were used to cbtain the power exponent in # and this should give a measure of the strength of flocculation. Concentrated dispersions, both of the aqueous and non-aqueous types, find applications in many industrial systems of which the following are worth mentioning: paints, dyestuffs, pigments, paper coatings, printing inks, detergents, ceramics, cosmetics, pharmaceutical and agrochemical formulations. In all these systems, it is essential to control the stability/flocculation of the dispersion during its preparation, subsequent storage and on application. This requires control of the interparticle interaction forces between the particles. Four different types of such interactions may be distinguished: ’ hard-sphere, electrostatic (soft), steric and van der Waals.In practice it is possible to obtain dispersions with various combinations of these interactions. By controlling the nature t Present address: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW. 4142 Stability/ Flocculation of Concentrated Dispersions and magnitude of these interaction forces one can control the bulk properties of these dispersions. Investigations of the stability/ flocculation of concentrated dispersions are by no means simple, since measurements need to be carried out without diluting the system. With model systems it is possible to apply scattering methods for such investigations, of which small-angle neutron scattering SANS is by far the most appropriate.2 Informa- tion on the microstructure of the particles and their arrangement in space can be obtained.Clearly these studies form the basis of any quantitative relation between the particle interaction potential and the bulk properties of the system. However, such measurements of SANS are accessible in a small number of laboratories and they can be applied only to model systems. Viscoelastic measurements offer an alternative procedure for investigation of interpar- ticle interactions3 and hence they give direct information on the stability/flocculation of the concentrated dispersion. Although less rigorous and less quantitative than SANS, such measurements are available in many laboratories and they do not require any special preparation of the dispersion.It is the objective of this paper to illustrate the usefulness of these measurements in studying the properties of concentrated dispersions. Various systems will be discussed to illustrate the general applicability of these measure- ments. These systems will range from electrostatically and sterically stabilised disper- sions to flocculated systems. The latter will include both weakly and strongly flocculated (coagulated) dispersions. To simplify the analysis, model latex dispersions that can be prepared fairly monodisperse were used. Two different types of rheological measure- ments were carried out, mainly dynamic (oscillatory) measurements, and steady-state (shear stress-shear rate) measurements. Experimental Latex Preparation Two main polystyrene latex dispersions were prepared. The first was an electrostatically stabilised latex that was prepared using the surfactant-free emulsion polymerisation technique described by Goodwin et aZ.4 The latex was extensively dialysed and then concentrated.The particles were fairly monodisperse, with an average radius of 700 nm as determined by photon correlation spectroscopy (Malvern PCS 47000) and the Coulter Counter (standard deviation <5%). This latex was also used for studying steric interac- tion by coating the particles with a physically adsorbed poly(viny1 alcohol) layer (PVA). The weight-average molecular weight of the polymer, M,, was 45 000 with 12% acetate groups. The adsorption isotherm of the polymer on the latex was determined at 21-23 "C and this gave a plateau value of 2.8 * 0.4 mg m-2, in good agreement with the results obtained b e f ~ r e .~ The adsorbed layer thickness of the PVA layer was determined using PCS to be 46 nm.6 The concentrated sterically stabilised latex was prepared by adding PVA solution to the dilute latex which was stirred overnight to ensure complete adsorption. NaCl solution was then added to the latex to give lo-* mol dmP3 electrolyte in order to suppress any contribution from double-layer effects. The resulting latex was then concentrated by centrifugation. The supernatant solution was used to dilute the concentrated sediment to give suspensions of various volume fractions. The second polystyrene latex dispersion was a sterically stabilised system containing grafted poly( ethylene oxide) (PEO).It was prepared using a dispersion polymerisation technique in water-alcohol mixtures.' The basic recipe consisted of a styrene monomer, a macromonomer of PEO ( Mw = 200) methacrylate, 2,2'-azo-bis( 2-methylpropion- onitrile) and peroxide inhibitors and a water-alcohol mixture (2: 3 ratio). Details of the preparation of this latex were given elsewhere.' The z-average particle size of theTh. F. Tadros and A. Hopkinson 43 latex was determined by PCS. This showed that the latex was fairly monodisperse with an average radius of 175 nm. The adsorbed layer thickness 8 h was obtained from the relative viscosity, as described in detail before.8 8 h was found to decrease with increasing volume fraction, 4, of the suspension, owing to the possible interpenetration and compressure of the layers.A summary of the 8 h values as a function of 4 is given below. 4 0.33 0.38 0.43 0.46 0.50 0.52 0.54 0.57 S h 20.5 19.3 17.0 15.1 15.0 13.3 11.1 9.2 Free Polymer for Depletion Flocculation Two types of polymer were used for investigation of the effect of addition of dissolved polymer on the rheology of the dispersions. These were poly(ethy1ene oxide) (PEO) with molecular weights of 20 000, 35 000, 90 000 and hydroxylethyl cellulose (HEC) with molecular weight of 70 000, 124 000, 223 000. Both polymers were commercial materials that were used as received. Rheological Measurements Dynamic ( Oscillatory) Measurements A Bohlin VOR rheometer (Bohlin Reologie, Lund, Sweden) interfaced with facit DTC2 or IBM microcomputer was used for such measurements.The instrument can operate in the frequency range 10-3-20 Hz and has interchangeable torsion bars covering a wide range of sensitivities. A coaxial cylinder (C25) with a moving cup of radius 27.5 nm and a fixed bob of radius 25.0 nm was used. The Bohlin VOR rheometer is a constant- strain instrument which performs oscillatory tests by turning the cup back and forth in a sinusoidal manner. The shear stress in the sample is measured by measuring the deflection in the bob which is connected to the interchangeable torsion bars. The phase-angle shift is computed from the time displacent between the sine waves of stress and strain ( A t ) , i.e. S = wAt, where w is the frequency in rad s-’ ( w = 27~v where v is the frequency in Hz).The complex modulus G*, storage modulus G’ and a loss modulus G” are calculated from the stress and strain amplitudes ( T ~ and yo, respectively) and phase-angle shift S : G* = To/ Yo GI= G” cos S In oscillatory measurements, one initially fixes the frequency and measures the rheological parameters as a function of strain amplitude. This enables one to obtain the linear viscoelastic region, where G*, G’ and G” are independent of strain, at any given frequency. Above a critical strain y c , G* and G’ start to decrease with increasing applied strain (non-linear response) whilst G” begins to increase. Once the linear region is indicated, then measurements are made as a function of frequency at a fixed amplitude. As we will see later, many systems show only a linear region at very low amplitudes, and therefore measurements need to be made at low deformations.Shear Modulus Measurements These were performed by using the Rank pulse shearometer (Rank Brothers, Bottisham, Cambridge). This is based on the model originally described by van Olphen’ and later44 Stability/ Flocculation of Concentrated Dispersions developed by Goodwin and Smith.” The suspension is placed in a cell that is fitted with two parallel plates or Perspex discs, whose distance of separation, d, can be changed by using a micrometer attached to one of the plates. Each plate is connected to a piezo-electric crystal (LiCl, transducer). An electric generator is connected to the bottom crystal and is used to initiate a small amplitude (ca. rad) strair, with high frequency (ca.200 Hz). This produces a shear wave that propagates through the dispersion and is detected at the upper crystal. The latter is connected to a microcomputer, enabling one to display the damping of the shear wave. By measuring the time t , for shear wave maximum to go to a minimum, at a given distance d between the plates, one can calculate the shear wave the shear wave velocity u. t, is usually measured as a function of d, and u is obtained from the slope of the linear plot of d us. t,. The shear modulus G, (which is the elastic modulus at a frequency of ca. 200 Hz) is given by where p is the density of the suspension. G,= u2p ( 6 ) Steady -s ta te Measu rem en ts Shear stress ( r ) vs. shear rate curves (Y) curves were obtained by using a Haake- Rotovisco rotational viscometer fitted with concentric cyclinder platens.For Newtonian systems, the viscosity 37 was obtained from the slope of the linear curve between r and 7. For non-Newtonian systems, which showed pseudoplastic flow, the data were ana- lysed by using the Bingham,” or Casson’s,12 equation: where rp is the extrapolated (to j = O ) yield stress and qPl the apparent or plastic viscosity, that is the slope of the linear portion of the r vs. y curve, r, is the Casson yield value, and 37, is the Casson viscosity. Results and Discussion Electrostatically Stabilised Polystyrene Latex Dispersions Strain-sweep measurements at a frequency of 1 Hz showed a continuous reduction in the values of the moduli with the applied strain and it was not possible to identify the linear region, as this was at strains lower than 0.004 for lo-’ mol dmP3 NaCl and 0.01 for lov3 mol dm-3 NaCl.Measurements were made at fixed low strains that are as near as possible to the linear region. Within this strain range plots of G”, G’ and G” us. v showed some dependence on the applied frequency, particularly at low # values. The results will be published in detail e1~ewhere.l~ The variation of G”, G’ and G” with # (at v = 1 Hz and yo 0.004 or 0.01) at the two NaCl concentrations (loP5 and mol dm-3) showed the expected behaviour of increase of modulus with increasing #, and there were significant differences between the values obtained in and mol dmP3 NaC1. These plots have been published before14 and the results showed the following trends.At any given # value, the values of the moduli are orders of magnitude lower in 10P3mol dm-3 NaCl when compared with the results in lop5 mol dmP3 NaCl. Within the volume fraction range studied in lop5 mol dm-3 (0.465- 0.525), G’> GI’ over the whole frequency range studied) (1OP5-5 Hz). In contrast, in mol dm-3, G’ were either close to or lower than G”, within the 4 range 0.253-0.566. The above results can be qualitatively explained in terms of the double-layer interac- tion between the particles. This is best illustrated from plots of G’ us. h, the surface to surface distance between the particles in the dispersion, i.e. h = 2a[(4,/+)i/3 - 11 (9)Th. F. Tadros and A. Hopkinson 0 0 10 45 50 100 150 200 h/nm Fig. 1. Variation of G' with h at two NaCl concentrations ( triangles are calculated values in and mol dm-3 NaCl).rnol dm-3 NaC1) closed mol dm-3 NaCl using data from ref. (15) acd (16). 0, Gk,, (lop5 mol dm-3 NaC1); A, G:heor; 0, GLxp where 4m is a constant that is charactristic of the type of array, e.g. 0.74 for hexagonal or face-centred cubic array, 0.68 for body-centred cubic, 0.64 for random arrangements of particles. For the present calculation of h a value of &, = 0.68 was used. The plots of G' vs. h are shown in fig. 1. G' increases rapidly with decreasing h, as expected from double-layer interaction. With the dispersion in mol dm-3, the double-layer extension ( 1 / ~ ) is 100 nm and hence one would expect a rapid increase in the modulus at h < 200, as found experimentally. Under these conditions, the dispersions are highly elastic as a result of the strong repulsive force between the particles.With the dispersions in lo-* mol dmP3, ( 1 / ~ ) is only 10 nm and hence a high modulus will be reached only at h separation distances comparable to 20 nm. This separation distance was not reached at the highest volume fraction studied, namely 0.566. Indeed at this 4 value the dispersion is less elastic than viscous and the G' value is still low (176 Pa). To obtain highly elastic dispersions in mol dm-3, one needs to go to much higher volume fractions (>0.6). It is perhaps useful to compare the experimental G' values with theoretical values that may be obtained using the analysis of Goodwin and co-worker~.'~*'~ These authors have theoretically related the high-frequency limit of the storage modulus (the shear46 Stability1 Flocculation of Concentrated Dispersions modulus) to the interparticle force as a function of volume fraction, and thereby of interparticle distance.The following expression was derived for the theoretical shear modulus: where cy = (3/32)4,n (n being the coordination number), R is the centre-to-centre separation for the particles (R = 2a + h ) and VT is the total energy of interaction between the particle. V - is given by the expression 4.rr~~,a’rl/; exp[-K(R-2a)] R VT = where E is the permittivity of the medium, E, that of the free space and K is the Debye-Huckel parameter. By differentiating eqn (1 1 ) twice, Goodwin and co- w o r k e r ~ ~ ~ , ~ ~ obtained the following expression for Gth (where KU < 10): G:,, = 4 m a ~ s , a ~ r l / ~ exp-[K(R-2a)] G:, were calculated for the dispersions in lo-’ mol dmP3 ( K a < 10) since these were highly elastic and the modulus showed little dependence on frequency.In these calcula- tions a was taken to be 0.833. The results of calculation are shown in fig. 1 (closed triangles). It can be seen that the theoretical values increases less rapidly with decrease of h than the experimental values. This implies that the simple model suggested by Goodwin and c o - w o r k e r ~ ’ ~ ~ ~ ~ is probably only applicable within a limited range of separation distances. There is, as yet, no theory that can accurately predict the elastic modulus under these conditions of significant overlap between the double layers.Another useful way of describing the interactions in concentrated dispersions is to apply scaling concepts. These have been successfully applied for the aggregation behaviour of concentrated dispersions. *’”’ Generally speaking, the modulus of a disper- sion scales with the volume fraction 4 with an exponent n, G‘= k4“. (13) The power n can be obtained from a log-log plot of G’ vs. 4. These plots are usually linear above a critical 4 value that denotes the onset of gel formation. For the above latex dispersions in mol dmP3 NaCl, the value of n, obtained from these linear plots, was in the region 20-30. These values are significantly higher than those obtained with flocculated suspension, as will be discussed later. Again, no theories are available for predicting the value of the exponent and as such it can be used only in a qualitative manner to describe the strong repulsive nature of the latex dispersions. and Sterically Stabilised Dispersions With polystyrene latex dispersions containing grafted PEO chains, the relative viscosity vs.volume fraction relationship was used to obtain the adsorbed layer thickness (6) as a function of 4. Details of the results obtained were published elsewhere’ and these showed a reduction of 6 from a value of 20.5 nm at 4 = 0.33 ( +eR = 0.460) to a value of 9.2 nm at 4 = 0.57 ( +efi = 0.665) (see Experimental). These results indicate that by increasing 4, compression of the PEO chains occurs as the particles approach each other very closely in a concentrated dispersion. This compression is reflected in the elastic properties of the dispersions as discussed below.Th. F.Tadros and A. Hopkinson 47 Plots of G”, G‘ and G” vs. 4 (published elsewhere*) showed an interesting transition as the volume fraction of the dispersion was increased. At 4 < 0.5, G”> G’, i.e. the dispersion showed a predominantly viscous response within the frequency range 1 O-*- 1 Hz. This is not surprising since at 4 < 0.5 the surface-to-surface distance of separation between the particles, h, is less than 28 and hence the interactions between these terminally anchored PEO chains are weak. These chains are only slightly compressed (8 decreases from 20.5 to 15.0 nm as 4 increases from 0.33 to 0.48). Such compression probably occurs with the larger PEO chains (note that grafted PEO chains are polydis- perse) and such a small effect does not lead to a predominantly elastic response within the frequency range studied.However, when 4 > 0.5, G’> G” and the magnitude of the moduli starts to rise strongly with further increase in 4. For example, when 4 is increased from 0.5 to 0.575, G’ at v = 1 Hz increases by an order of magnitude, and on further increase of 4 to 0.585, G’ increases by another two orders of magnitude. Moreover, at such high 4 values, the moduli tend to show little dependence on frequency within the range studied. In other words, the dispersion behaves as a near-elastic gel, as a result of the strong steric interaction between the PEO chains. The latter may undergo interpenetration with the peripheries of the larger PEO chain and further compression of the whole grafted polymer layers may occur. Indeed at 4 = 0.585, G” == GI= 4.8 x lo3 Pa and q’= 8.82 x lo3 Pa s, whereas at 4 = 0.65, G” = G’ = 1.12 x lo5 Pa and q‘ = 1.6 x lo5 Pa s, i.e.the latex behaves as a near-elastic solid. The above viscoelastic results clearly show their value in studying steric interaction in concentrated dispersions. It is possible to convert the modulus vs. 4 curves into G’ vs. h curves using eqn (9) and compare these with the F vs. h curves obtained from direct force meas~rements.~~ In these calculations a value of 4m = 0.68 was used. These F vs. h results can be converted to GYh results using a similar analysis to that of Goodwin and c o - ~ o r k e r s . ~ ~ ~ ’ ~ These calculations were recently obtained by Costello2’ and will be published in detail in the near future.21 A summary of the results obtained is shown in fig.2 which gives the measured G’ value and the theoretical values as a function of h. Although the values of G’ and Gbh vary considerably, the general trend is the same, namely a rapid increase in the elastic modulus with decrease in h, when the latter is less than 30 nm. As discussed above, when h < 30 nm, interpenetration and/or compress- ion of the PEO chains occurs. Thus, viscoelastic measurements can give a quantitative measure of the steric interaction between particles in a concentrated dispersion. It is also possible to apply scaling concepts to concentrated sterically stabilised dispersions. Recent results obtained in our laboratory” using similar latex with grafted PEO showed that GI- 430.The exponent in 4 is high and of the same order as that obtained for electrostatically stabilised dispersions. This high value of n is common in dispersions, where the net force is repulsive, but as mentioned before there are no theories that could predict such scaling factors. Similar results were obtained for physically adsorbed polymers on polystyrene latex.23 For example, results using poly(viny1 alcohol) (with M , = 45 000, S = 46 nm) on 700 nm polystyrene latex showed a rapid increase in G* and G’ at 4 > 0.53, whereas GI’ remained fairly At 4 > 0.53, h < 28, when one assumes a value of 4m = 0.64, i.e. random packing. The latter is more probable with dispersions containing long dangling tails.Under these conditions, elastic interaction between the long, dangling PVA chains occurs resulting in a predominantly elastic response for the dispersions. Weakly Flocculated Dispersions These are best exemplified by sterically stabilised dispersions to which a free (non- adsorbing) polymer is added in the continuous phase. Two such systems were investi- gated, namely, addition of poly( ethylene oxide) ( PEO)24 and hydroxyethyl cellulose (HEC)25 to a polystyrene latex dispersion with grafted PEO (the latex volume fraction48 Stability1 Flocculation of Concentrated Dispersions I I I 10 20 30 h/nm Fig. 2. Variation of G' with h for a sterically stabilised latex dispersion: e, calculated from F / h relationship; 0, experimental data for G', M , (PEO) = 2000.was kept constant at 0.3). Fig. 3 shows the variation of T~ and G, with free-polymer volume fraction &, for the three PEO polymer studies, whereas fig. 4 shows the variation of T~ with & for the three HEC sample studies. Above a critical &, value the rheological parameters increase with increase of &,. This critical value is denoted by @;, i.e. the free-polymer volume fraction above which depletion flocculation results in a structured system with measurable yield value and modulus. It is useful to compare ++ with the semidilute polymer volume fraction, which can be calculated for the equation where R, is the radius of gyration of the polymer that could be calculated from the intrinsic viscosity, using the Stockmayer-Fixman relationship,'6 6 is a constant that is equal to 5.63 for hexagonal close packing of the polymer coil, N,, is the Avogadro constant and p is the density of the polymer.A summary of the values of 4; and 4; and R, is given in table 1. 4; values are significantly higher than q5;, particularly with HEC. However, calculation of 4; using eqn (14) is approximate and based on a crude model of hexagonally close-packed polymer coils behaving as hard spheres. As discussed before27728 it is possible to relate the Bingham yield stress rP to the interparticle interaction. The latter may be equated to the amount of energy required to separate the flocs into single i.e. Tp = NEsep (15) where N is the total number of contacts between particles in flocs and Esep is the energy required to break each contact.The total number of contacts, N, may be related to theTh. F. Tadros and A. Hopkinson 49 2000 - ( a ) 1600 - a“ 1200 1 0” 1 800 - 400 - I . . > I . . . I . . . 0 0.02 0.04 0.06 0 0 0 0.02 0.04 0.06 0.08 4 P Fig. 3. Variation of G, and rp with volume fraction of free polymer (PEO) for a latex dispersion (4 = 0.3). M,(PEO): 0, 20 000; A, 35 000; 0, 90 000. volume fraction, 4, and the average number of contacts per particle (the coordination number), n, by Combining eqn (15) and (16) one obtains 3 4nEsep rp =-. 8 r a 3 Thus, Esep can be calculated from 4, provided a value can be assigned for n. The maximum value of n is probably 8, which is the average number of contacts in a floc for random close packing. However, recent work on the structure of aggregates indicates that quite open structures often arise in which n would be lower than 8 .It is also likely that n may decrease with increase in free-polymer concentration as more open flocs are produced at higher values of &. However, for the sake of comparison, two values of n were assumed, namely 8 and 4, and these were kept constant at all 4,, values. Another assumption that has to be made for calculating Esep from T~ is that above the yield point all contacts are broken. This assumption is probably justified with weakly flocculated systems, whereby the floc can be reversibly broken under high shear, resulting50 Stability/ Flocculation of Concentrated Dispersions l o t P P U 0 0.002 0.004 0.006 0.008 0.01 0.012 &J Fig. 4.ra us. #, for hydroxy ethyl cellulose with various molecular weights for a polystyrene latex dispersion (# = 0.3). Mw: 0, 70 000; A, 124 000; 0, 223 000. Table 1. #i7 #: and R, for the polymers used in depletion flocculation ( a ) poly( ethylene oxide) (PEO) 20 000 0.02 0.03 5 5.52 35 000 0.01 0.024 7.59 90 000 0.005 0.0 12 12.9 ( b ) hydroxy(ethy1 cellulose) (HEC) 70 000 0.0035 0.012 11.9 124 000 0.002 0.007 17.1 223 000 0.001 0.004 24.8 in the formation of primary units. Evidence that this is the case is obtained from the relatively small dependence of plastic viscosity on the free polymer concentration. The results of the calculation of Esep from T~ on the basis of the above assumptions are given in table 2 for the PEO system and in table 3 for the HEC system. These values of Esep may be equated to the free energy of flocculation due to depletion Gdep.The latter can be calculated using Asakura’s and 00sawa’s~~ or Fleer et aZ.’s3* theory. In the first case, Gdep/ kB T = - ( 3 / 2 ) 42pX2; 0 < x < 1 (14) where #2 is the volume concentration of the polymer, p = a / A , where A is the depletion thickness (equal to R,) and x = [A - (h/2)/A].Th. F. Tadros and A. Hopkinson 51 Table 2. Summary of the results for PEO of Esep calculated from the experimental T~ values and Gdep calculated using Asakura and 00sawa’s*~ (AO) and Fleer et aL’s ( FSV)30 models Esepl k0 T Gdep/ k0 c,bp T ~ / N ~ - ~ n = 8 n = 4 A 0 FSV 0.025 0.03 0.04 0.06 0.08 0.015 0.02 0.03 0.04 0.01 0.01 5 0.02 0.025 2.0 2.8 3.8 5.8 13.1 2.3 4.4 7.0 11.7 1.2 2.8 4.4 5.9 ( a ) PEO, M, = 20 000 9.1 18.2 12.7 25.4 17.3 34.6 26.4 52.8 59.6 111.2 ( b ) PEO, M, = 35 000 10.5 21 .o 20.0 40.0 31.9 63.8 53.2 106.0 (c) PEO, M , = 90 000 5.5 11.0 12.7 24.5 20.0 40.0 25.9 53.8 25.3 30.3 40.5 60.7 80.9 16.4 21.8 32.7 43.6 12.3 18.4 24.5 30.6 81.9 102.9 149.5 261.1 397.4 54.1 78.0 135.0 203.0 48.4 85.6 131.3 185.7 Fleer et aZ.-’” gave the following expression for Gdep: where v, is the molecular volume of the solvent, pl is the chemical potential of the solvent at a volume fraction c$p of the free polymer and py the corresponding value at &, = 0.(pl - /LO) can be calculated from c$p using the expression31 where n2 is the number of polymer segments and x is the Flory-Higgins interaction parameter which for PEO is 0.473 and for HEC is 0.47.Results of calculation of Gdep based on the above two theories are given in tables 2 and 3. For the PEO system the calculated values based on Fleer et aZ.’s3’ model deviate from the experimental Esep values, wherever reasonable agreement is obtained using Asakura and Oosawa’s In contrast with HEC, better agreement is obtained using Fleer et d ’ s model,-’” particularly at high M , values. Strongly Flocculated Systems Two main systems were investigated. In the first case, a sterically stabilised dispersion (polystyrene latex with physically adsorbed PVA) was flocculated by making the disper- sion medium a poor solvent for the chains. This was obtained by addition of electrolyte (KC1 or Na2S0,) or heating the dispersion at constant electrolyte concentration.In the second case, an electrostatically stabilised dispersion was flocculated by addition of 0.2 mol drn--’ NaCl. Details of the results obtained will be published elsewhere.32 As an illustration, fig. 5 shows the variation of G*, G’ and G” for a PVA-coated latex52 Stability1 Flocculation of Concentrated Dispersions Table 3. Summary of the results for HEC of Esep calculated from the experimental rs values and Gdep calculated from theory ~~ ~~ Esep/ kB Gdep/ kB c#+, rs/Nm-’ n = 8 n = 4 A 0 FSV 0.002 0.003 0.004 0.006 0.008 0.010 0.0 12 0.001 0.002 0.003 0.004 0.005 0.007 0.010 0.001 0.002 0.003 0.004 0.005 0.007 0.010 0.05 0.6 0.7 2.2 3.4 5.1 8.3 0.5 0.9 1.6 2.7 4.0 5.7 10.0 0.4 1.9 4.9 7.9 10.7 20.0 25.4 (a) M,., = 70 000 2.3 4.6 2.8 5.6 3.2 6.4 10.0 20.0 15.5 30.9 23.2 46.4 37.7 75.4 (b) Mw= 124000 2.3 4.6 4.1 8.2 7.3 14.5 12.3 24.5 18.2 36.4 25.9 51.8 45.5 90.9 ( c ) Mw = 223 000 1.8 3.6 8.7 17.3 22.3 44.5 35.9 71.9 48.7 97.3 90.9 181.8 115.5 23 1 .O 2.7 4.0 5.4 8.0 10.7 13.4 16.0 1.6 3.1 4.7 6.2 7.8 10.9 15.6 1.8 3.7 5.5 7.3 9.2 12.8 18.3 28.6 43.4 58.5 89.7 122.2 156.0 191.0 17.0 34.6 53.0 72.0 91.8 133.3 200.7 20.7 42.9 66.7 91.8 118.5 176.3 274.2 Fig.5. Variation of G* ( x), G’ (0) and G” (A) with CNu2S04 for a PVA-coated latex dispersion. Cp = 0.5, o = 1 Hz, y = 0.01.771. F. Tadros and A. Hopkinson 53 G " 1 10 100 2' 50 0 a" I- \ 1 1 1 G & * " J 0.1 0.2 0.5 1 2 5 10 20 50 a" ... 0 0.5 1 2 5 10 20 50 Yo/ 1 Fig. 6. Strain-sweep results for a flocculated latex ( CNaCl = 0.2 mol dm-3) at various volume fractions.( a ) 4 = 0.346, (b) 4 = 0.205, ( c ) 4 = 0.121, jd) 4 = 0.065. dispersion with 4 = 0.5, as a function of Na,SO, Concentration. The moduli (particularly G* and G') initially decrease with increasing CNaZSo4, reach a minimum and then suddenly increase above a critical electrolyte concentration. The initial reduction is due to the reduction in the adsorbed layer thickness, as a result of a reduction of solvency with an increase in the electrolyte concentration. However, above the critical flocculation concentration of Na2S04 (0.15 mol dm-3) all moduli show a rapid increase. This critical flocculation concentration marks the onset of incipient flocculation, i. e. the 8-point for the chain. Incipient flocculation of sterically stabilised dispersions can also be obtained by increasing the temperature at constant electrolyte concentration.This was demon- strated by using a dispersion in 0.15 mol dm-3 Na2S04 and changing the temperature from 15 to 30 "C. At 15 "C, the dispersion was stable and gave low modulus values, which decreased slightly with increase in temperature (due to reduction of solvency and collapse of the PVA chain), but above 25 "C there was a rapid increase in the modulus with increase in temperature. This temperature denotes the critical flocculation tem- perature (CFT) of the dispersion. One important finding from this work was that on cooling the flocculated dispersion, i.e. from 30 to 15 "C, the modulus values did not reach their initial low values, i.e. restabilisation did not occur in full.In other words, flocculation of such sterically stabilised dispersion is not completely reversible. The second flocculated system studied was that based on an electrostatically stabilised latex to which 0.2 mol dm-3 NaCl was added. Measurements were carried out as a function of strain amplitude at various 4 values, as illustrated in fig. 6. The structure of such coagulated suspensions becomes partially broken down above a critical strain (deformation) value that depends on 4. It can be seen that above a critical strain, ycr, the response is non-linear. A frequency sweep in the linear region, shows that G* = G', and G"= 0, and there was hardly any dependence on frequency in the range 1OW2-1 Hz.54 A log-log plot of G’ vs. 4 gave the following scaling equation: Stability/ Flocculation of Concentrated Dispersions G’ = 1.98 x 10746.0.(17) The above exponent is higher than most of the reported values in the literature. Many author^^^-^^ have reported that the exponent for flocculated suspension is in the range 2.0-3.5. However, the value of the exponent depends to some extent on the treatment to which a coagulated suspension has been subjected before the measurements were made. Recently, Ball36 used a novel method for calculating the elasticity of individual fractal clusters, and an exponent of 4.50*0.2 can be predicted for chemically limited aggregation in three dimensions and of 3.5 f 0.2 for diffusion-limited aggregation. The power produced by Ball’s theory is lower than that found experimentally in our investiga- tion.The higher experimental power in n is indicative of a strong flocculated structure as recently found by Sonntag and Several other flocculated systems were studied in our laboratory to obtain the power exponents. This first was a PVA-coated latex that was flocculated using KCl(2 mol dm-3) at 25 “C. This system gave the following power-law relation: G’= 1.2 x 10645.5. (18) The exponent in 4 is lower than that for the coagulated latex suspension using NaCl. The second system was a PVA-coated latex which was flocculated by heating to 50 “C at constant KCl concentration (0.2 mol dmP3). The power-law relationship was: G’= 5.95 x 10’4~.0. (19) The exponent is significantly less than that obtained for NaCl-flocculated bare latex or KC1-flocculated PVA-coated latex.This indicates that flocculation by heating a sterically stabilised latex produces structures which may be different from those obtained using high concentration of electrolyte. These structures are also different from those obtained by coagulation of an electrostatically stabilised latex dispersion. From a knowledge of y, and G’, one can calculate the cohesive energy, E,, of the flocculated structure. E, is related to the stress in the flocculated structure a, by the following equation:38 Since (T,= ycG’, then E, = joy‘ ycG’ d y = (1/2)yfG’. log-log plots of E, vs. 4 are linear, and the exponent n may be used as an index for strength of flocculation. For example, for the coagulated base latex suspension, n was found to be 9.1, which is indicative of a deep primary minimum coagulation process that involves hundreds of k , T units.Conclusions Viscoelastic measurements offer a powerful tool for investigating the stability/floccula- tion of concentrated dispersions. With stable systems, repulsion can be studied by measurements of the elastic modulus G’ as a function of volume fraction 4 while fixing other parameters of the system, e.g. particle size, electrolyte concentration and adsorbed layer thickness. The G’ vs. 4 curves can be compared with G’ vs. 6 results obtained using direct force measurements. These rheological measurements can also be applied to study interactions in flocculated system. With a weakly flocculated dispersion it isTh. F. Tadros and A. Hopkinson 5 5 possible to obtain the energy of interaction between the particles and compare this with the free energy of attraction, e.g.with depletion flocculation. With strongly flocculated systems, scaling concepts may be applied to obtain the power exponent for the depen- dence of G' on #. This exponent may be used as a qualitative index for the strength of flocculation. References 1 R. H. Ottewill, in Concentrated Dispersions, ed. J. W. Goodwin, Royal Society of Chemistry Publication, 2 R. H. Ottewill, in Future Directions in Polymer Colloids. ed. M. S . El-Aasser and R. M. Fitch, Nato 3 Th. F. Tadros, in Flocculation and Dewatering, ed. B. M. Mougdil and B. J. Scheiner (United Engineering 4 J. W. Goodwin, J. Hearn, C. C. Ho and R. H. Ottewill, Colloid Polym. Sci., 1974, 252, 464. 5 M. J. Gamey, Th. F.Tadros and B. Vincent, J. Colloid Interface Sci., 1974, 49, 57. 6 Th. van den Boomgaard, T. A. King, Th. F. Tadros, H. Tang and B. Vincent, J. Colloid Interface Sci., 7 C. Bromley, Colloids Surf, 1985, 17, 1. 8 C. Prestidge and Th. F. Tadros, J. Colloid Interface Sci., 1988, 124, 660. 9 H. 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