J. Chem. Soc., Faraday Trans. I, 1984’80, 1673-1688 Interaction Forces in Dispersions Containing Non-ionic Surfactants BY LAURENCE THOMPSON Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW Received 25th April, 1983 Rates and equilibrium positions of aggregation have been measured for dispersions of monodisperse polystyrene latex particles containing the non-ionic surfactant octaoxyethylene glycol dodecyl ether (C12E8). Measurements cover a range of particle sizes and temperatures. Rates are observed which, even in the presence of sufficient electrolyte largely to suppress electrostatic interactions, can be significantly below the diffusion-controlled ‘ rapid ’ rate. Data are interpreted in terms of both primary and secondary minima.The temperature sensitivity of the aggregation appears to arise from variations in the attractive rather than the repulsive contribution, these variations resulting from the effect of temperature on the level of adsorption. The attractive effects are of an essentially short-range nature, and because of this primary- minimum aggregation is more sensitive to temperature than secondary-minimum aggregation. The data have been fitted to potential-energy functions consisting of an attractive van der Waals contribution and a repulsion arising from the surfactant. Making reasonable allowance for the attractive contribution leads to the conclusion that a weak repulsive force is operative at separations considerably in excess of the physical reach of the adsorbed surfactant.Here an arbitrary model for the repulsion has been used which comprises the sum of two exponential decays, a relatively strong short-range component (decay length 0.2 nm) and a weak long-range component (decay length 3.0 nm). Non-ionic surfactants of the alcohol polyethoxylate type are well known as stabilisers of colloidal dispersions. Their effectiveness can decrease with increasing temperature, and flocculation temperatures are often observed below the cloud point of the s~rfactant.l-~ Flocculation is associated with a region of attraction in the plot of potential energy against separation caused by domination of van der Waals forces. It is distinct from the flocculation occurring at the &temperature of a stabilising p ~ l y m e r , ~ - ~ which involves attraction arising from polymer-polymer interaction.The effect is attributable to temperature-induced changes in the adsorbed layer. Reduction of the area per adsorbed molecule, for example, may reflect a reduction of the repulsive forces associated with the surfactant while simultaneously increasing the attractive contribution through its effect on the adsorbed layer’s density.’ The purpose of the work described in this paper is to develop a model for the repulsive force originating within layers of adsorbed non-ionic surfactant so that the relative importance of attraction and repulsion changes in bringing about temperature-induced aggregation can be defined. The temperature dependence of the equilibrium aggregation state of polystyrene latex dispersions containing the surfactant octaoxyethylene glycol n-dodecyl monoether (C,,E,) was described in earlier w0rk.l Conditions were such that the range of the double-layer interactions was very short, being suppressed by electrolyte addition.The aggregation state was related to the effective thickness, 6, of the adsorbed layer 16731674 INTERACTION FORCES IN NON-IONIC SURFACTANTS through an approach based on the second virial coefficient of the particles in the dispersion. A plot of 6 against temperature was constructed which indicated that 6 decreased with increasing temperature from a value of 6.2 nm at 15 "C to 2.2 nm at 43 "C. The lower figure is consistent with the likely dimensions of the C1&8 molecule (random coil length 2.5 nm),8 whereas the larger figure is significantly in excess of even its fully extended length (3.9 nm).The long-range forces implied at the lower temperatures were thought to be of similar origin to the 'solvent-ordering' forces between surfactant or lipid bilayer~,~-ll i.e. they are separate from the osmotic repu1sion12 generated in overlapping adsorbed polymer layers. The main problem with the original interpretation is its foundation on a hard-sphere model for the surfactant's repulsion [fig. l(a)]. This gives a (minimum) estimate of the range of the repulsion but no estimate at all of its dependence on separation. The possibility of obtaining such an estimate from a combination of the equilibrium data and aggregation rate measurements emerged with the realisation that flocculation of the systems can occur at rates below the rapid (Smoluchowski) rate, even in the presence of excess electrolyte. This implies the presence of a maximum in the plot of potential energy against separation [fig.1 (b)], which in itself indicates the limited scope of the hard-sphere approwh. Note that aggregation rates which are slower than the diffusion-controlled (rapid) rate do not necessarily imply the existence of a barrier to aggregation. The rate constant (k,) for aggregation into minima of restricted depth ( Vmin) can be expressedl3 kl=kR 1-exp- Vmin ( kT where kR is the 'rapid' rate constant. An alternative treatment14 suggests an equation of the form k1= kR [1 - ~ X P (Vmin - V*,in)/kTI where Pmin is the value of Vmin below which (using a phase-change model for the aggregation process) aggregation does not occur.Here, however, the observed levels of aggregation imply sufficiently high values of Vmin or (Vmin- Pmi,) that in the absence of a potential-energy barrier k, -+ kR in all cases. Fig. 1 (b) is of similar form to curves typically encountered with charge-stabilised colloids, showing both primary and secondary minima separated by a potential-energy maximum. In the electrostatic case the primary minimum is usually deep and is associated with strong, irreversible and complete aggregation. Here the adsorbed layer restricts the depth of the minimum and weak, reversible and partial aggregation may occur at a rate determined by the magnitude of any potential-energy barrier which may be present. Where a barrier exists, its magnitude is related to the aggregation rate by the expressi on1 9 where the stability constant W is equal to kR/kexptl, the ratio of the rapid rate con- stant (4kT/3q) to the experimental rate constant, q is the viscosity of water, a is the particle radius, and DZ and D,, are the bulk diffusion coefficient of the particles and its value at centre-centre separation, r, respectively:16 V(r) is the potential energy of interaction obtained by summation of attractive and repulsive contributions.Where equilibrium aggregation is observed the depth of the energy well can beL. THOMPSON 1675 v* /” Fig. 1. Schematic plots of potential energy against separation. (a) Hard-sphere repulsion model for particle interactions showing van der Waals attraction V,.(b) Realistic model showing primary and secondary minima separated by a po tential-energy maximum. obtained through the virial-coefficient approach used in the work1? 3* l7 in which the potential-energy function is related to the aggregation state by the expression -= n0 1+2zn, /mr2[exp(F)-l]dr - V(r) - J R where no and n, are the initial and equilibrium particle concentrations, respectively, and R is the shortest distance of centre-centre approach of particles, effectively the inner edge of the energy well. Combination of the ‘energy-barrier ’ and ‘energy-well’ data from a range of particle sizes should in principle yield information on the form of V(r). Since V(r) is the sum of an attractive van der Waals contribution and the repulsive energy that is to be investigated, then provided that a reasonable estimate of the attraction can be made, a model for the repulsion will emerge.To this end, the original equilibrium studies have been extended and the appropriate aggregation rate data have been obtained. EXPERIMENTAL PROCEDURES Aggregation processes were monitored as follows. Test tubes containing the unaggregated dispersions were placed in constant-temperature cabinets. Settling problems were generally avoided by locating the tubes in a rack which was set to rotate end-over-end at ca. 5-10 r.p.m. for 1 min in 15 min. At higher rotation speeds the aggregation state of the dispersion may be affected.” Samples were taken at suitable intervals and the particle concentration was determined using an automatic particle counter which was essentially a laser-illuminated flow ultramicroscope.18~ The samples were withdrawn from the tubes by a 2 cm3 plastic syringe.The shear forces involved in this procedure and in the injection of the samples into the particle counter do not affect the aggregation state of the dispersion (order-of-magnitude variations in injection speed have no effect on the particle count obtained). The errors associated with the counting procedure are close to the statistical prediction nk where n is the number of particles sampled per unit time. Typically the average of ten counts each of ten seconds duration is used in which ca. 1000-2000 particles per 10 s are sampled. This leads to standard deviations of order 2.5%.1676 INTERACTION FORCES IN NON-IONIC SURFACTANTS Rates and equilibria were obtained from plots of reciprocal particle concentration against time.In both types of experiment similar initial particle concentrations were employed throughout [(2-3) x lo8 ~ m - ~ ] . MATERIALS Monodisperse polystyrene latex dispersions were prepared in this laboratory by emulsion polymerisation and were dialysed extensively before use. Particle sizes were determined by electron microscopy or photon correlation spectroscopy. The latex, which in previous work was erroneously assigned a diameter of 1.6 pm, has been reassessed at 1.76 pm (& 10.5%). The other dispersions were more monodisperse, having standard deviations within 5 %. Inorganic reagents were of analytical reagent grade. Octaoxyethylene glycol n-dodecyl monoether (C,,E,) was prepared in this laboratory and was shown by gas-liquid chromatography to be 99% pure.The absence of a minimum in the surface-tension curve indicated the absence of more surface-active impurities. The cloud point of C&, is 78 "C in water and 77 "C in 10-1 mol dm-3 MgSO,, the experimental medium. Water for use with the particle counter was prepared by normal distillation of deionised water from caustic soda/potassium permanganate, followed by a second distillation involving a ' hot spot' to break the water film and minimise particle carryover. This produced water with a very low background particle correction at a counter sensitivity setting appropriate to the smallest particles used (0.25 pm diameter). RESULTS AND DISCUSSION Fig. 2 shows the equilibrium aggregation data of Thompson and Prydel for 0.25, 0.5, 1.0 and 1.76 pm diameter polystyrene latex dispersions stabilised by C,,E,.Excess electrolyte was included to suppress double-layer repulsion. In addition it shows some extra data in the same range of size and temperature together with similar data for 3.8 pm diameter polystyrene particles. The original work indicates that larger particles were less stable to temperature increase than small ones. This was thought to be so because the range of van der Waals attraction increases with size whereas the repulsion due to an adsorbed layer, when viewed as anything like a hard sphere, does not. The residue of attraction at large separations therefore increases with particle size and aggregation is favoured.The new data for dispersions of 3.8 pm particles do not fit this framework in that an increased resistance to complete aggregation was found. Partial aggregation was observed at all temperatures to an extent which bears an unexpected oscillatory temperature dependence (discussed later). The switch to partial aggregation leads to a suspicion that, in terms of fig. 1 (b), the secondary- rather than a primary-minimum mechanism may be involved for these large particles. Further evidence for this emerges from the aggregation-rate data, which are shown in table 1 in the form of the stability ratio W. W is the ratio kR/kexptl, k, is the diffusion-controlled rate constant at the appropriate temperature and kexptl is the experimental rate constant generally determined from the initial slope of the l/n against time curves.Three types of behaviour are apparent. First, the results for the 0.25, 0.5 and 1.0 pm particles indicate the presence of a barrier to flocculation. This implies that aggregation involves a primary minimum. Apart from the 1 .O pm latex, the values of W exhibit little systematic temperature dependence. This appears to be linked to the oscillatory equilibrium behaviour of the 3.8 pm latex and will be dis- cussed later. For the 1.76 and 3.8 pm particles two further modes of behaviour are seen, both of which are typified by an initial rate which is close to the rapid rate. Values for W of up to ca. 2.5 are normally expected for ' rapid ' aggregation because of well known hydrodynamic effects.'* This is in line with the values observed for the 1.76 pm latex.The 3.8 pm latex at low temperature aggregates more quickly, and this may be because it aggregates at relatively greater separations and is therefore free of theseL. THOMPSON 1677 15 35 55 75 TI" C Fig. 2. Plot of equilibrium aggregation state against temperature for C,,E,-stabilised polystyrene latex dispersions in 10-l mol dm-3 MgSO,. 0, 0.25; 0, 0.5; 8, 1.0; V, 1.76 and A, 3.8 pm diameter. no and nco represent initial and equilibrium particle concentrations, respectively. Table 1. Aggregation-rate showing the stability ratio, W, for various temperatures and particle sizes TIT diameter h m 15 20 25 35 40 50 ~~ a a a 46 62 35 0.25 - 40 60 58 55 0.5 1 .o - - 141 118 45 24 1.76 2.1 1 .8b 1.6' 1 .gb 3.2' 1.9 3 . P 0.7 - 0.92 1.8 2.9 2.6 - 31" 64c 406c 72c a Insufficient aggregation to produce reliable data; rates, respectively, where 'type 3' behaviour is observed; 5.5 and 5.2, respectively.and refer to initial and second stage at 60 and 70 "C values of W were effects. In addition there are indications that aggregation of the 3.8 pm latex is slower at high temperature than is allowed for by the simple viscosity correction used here. The observation of diffusion-controlled aggregation indicates that no barrier to flocculation exists, implying that either the secondary minimum becomes important or that the barrier to primary-minimum aggregation has disappeared. The larger particles (3.8 ,urn) exhibit no further complications, and this behaviour will be referred to as 'type 2'.Examination of the details of the aggregation rate curves involving 1.76 pm particles reveals a more complex situation (type 3). After the initial rapid phase (a) in table 1, a period of slower aggregation (b) is observed so that the overall shape of the type 3 curves is quite different to that of type 2. The rate constants quoted under (b) in table 1 were derived from the slope of the appropriate part of the l / n1678 INTERACTION FORCES IN NON-IONIC SURFACTANTS 0 3 1 n E --- 3 v 0. 0.51 I I I I I 0 10 20 30 40 50 tlh Fig. 3. Aggregation-rate data for 1.0 pm diameter polystyrene latex dispersion at 35 "C in 1.86 x lo-' mol dm-3 C,,E, and mol dm-3 MgSO,. (-) Theoretical curve based on initial rate constant, k = 7 x cm3 0, and equilibrium constant [eqn (A 3)], K = 2.31 x cm3.against time curve. For type 3 curves equilibrium may or may not eventually be reached depending on temperature. This suggests concurrent primary (slow) and secondary (rapid) aggregation mechanisms. By extension of the trend from simple primary minimum for the smaller particles through mixed primary/secondary for the 1.76 pm particles, it follows that the largest particles aggregate through a purely secondary-minimum mechanism rather than through a mechanism involving dis- appearance of the barrier to the primary minimum. The likely change of mechanism is supported by the completely different temperature dependence that the 'type 2' data exhibit. Analysis of all of the rate curves was required to confirm this pattern because it was not known what shape the rate curves for reversible aggregation should adopt.(1.e. how quickly does deviation from the initially second-order process occur, and is the apparent difference between types 2 and 3 just a quirk of the kinetics?) For this reason the kinetics of reversible aggregation processes have been analysed (see Appendix) and an expression derived which describes the particle concentration as a function of time. This expression contains the initial rate constant and the equilibrium constant, both of which are determined experimentally. The rate curves predicted by this treatment have been compared with the data. Fig. 3-6 give examples of the three different kinds of behaviour. Fig. 3 shows a 'second-order' plot of reciprocal particle concentration (1 / n ) against time for a dispersion of 1 .O pm diameter particles at 35 "C (type 1).The coincidence of theory and data supports the idea of slow flocculation into a primary minimum of restricted depth, uncomplicated by the presence of a significant secondary minimum. The rate treatment in the Appendix predicts the form of the rate curve remarkably well despite the assumption of non-multibonded, essentially linear aggregation that it contains. This is consistent with the findings of the earlier work' in which the equilibrium treatment [eqn (2)], which contains the same assumption, was found valid to levels of equilibrium aggregation where n,/n, = 2-3, after which it rapidly becomes ineffective. Here n,/n, reaches only ca. 1.5 at equilibrium.L. THOMPSON 1679 Fig.4. Aggregation-rate data for 3.8 pm diameter polystyrene latex at 25 "C in 1.86 x lo-, mol dm-3 C12E8 and 10-1 mol dm-3 MgSO,. (-) Theoretical fit based on initial rate constant, k = 6.7 x 10-l2 cm3 s-l, and equilibrium constant [eqn (A 3)], K = 1.29 x cm3. 0 0 0 0 0 0 100 200 300 400 tlmin Fig. 5. Aggregation of 1.76 pm diameter polystyrene latex in 1.86 x lo-, mol dm-3 C12E, and 10-l mol dm-3 Mg SO, at 40 "C. The line is a theoretical fit for the rapid portion of the curve based on an initial rate constant, k = 2.76 x 10-l2 cm3 s-l, and an estimated equilibrium position, K = 1.31 x cm3. Fig. 4 shows a similar plot for a dispersion of 3.8 pm polystyrene particles at 25 "C (type 2). A good fit is obtained using a rapid initial rate and the observed equilibrium position, supporting the secondary-minimum model.Fig. 5 and 6 show type 3 behaviour, for which it was necessary to propose concurrent primary- and secondary-minimum flocculation. Fig. 5 shows an example in which equilibrium was not achieved and the abrupt transition from the initially rapid aggregation to a clearly defined process about twenty times slower can be seen.1680 INTERACTION FORCES IN NON-IONIC SURFACTANTS 0 5 10 15 ' I 2 5 tlh Fig. 6. Aggregation of 1.76 pm diameter polystyrene latex in 1.86 x lo-, mol dmP3 C,,E, and 10-l mol dm-3 Mg SO, at 20 "C. (---) Theoretical fit based on initial rapid rate, k = 2.0 x 10-l2 cm3 s-l, and observed equilibrium position, K = 1.35 x lo-* cm3. (. . . . .) Theor- etical fit for 'rapid' portion of the curve based on initial rate constant, k = 2.0 x cm3 s-l, and estimated equilibrium position, K = 5.09 x low9 cm3.(-) Theoretical fit for 'slow' portion of the curve based on initial rate constant, k = 1.98 x cm3 s-l, and K = 6.5 x cm3. Initial particle concentration, no, in this case was taken as the intercept of the calculated curve rather than the true value. A theoretical fit for the whole curve cannot be attempted because of the absence of a final equilibrium. If, however, it is assumed that the discontinuity in the curve represents the limit of secondary-minimum aggregation, a theoretical fit for the early time evolution of the process becomes possible and reasonable agreement is obtained. Fig. 6 shows an example in which equilibrium was achieved. The calculated curve based on the initial rapid rate, and the final equilibrium position does not fit the data in that more rapid achievement of equilibrium was predicted than was observed.This confirms that two processes are at work, and (as explained in the Appendix) an approximate fit should be possible if these processes are treated separately. By examination of fig. 6 an estimate can be made of the position of the equilibrium that would have been reached in the absence of the slow primary-minimum process. A combination of the equilibrium constant associated with this position and the initial rapid rate again enables a good fit for the early time evolution of the process to be obtained. An approximate rate constant for the primary-minimum (slow) part of the process was obtained from the slope of the appropriate part of fig.6 and this was used, together with the final position of equilibrium, to model the time evolution of the remainder of the aggregation. Here the fit is poor, and while this fact may contain some message of a fundamental nature it is more likely to be a result of the level of guesswork involved in separating the two processes and will not be pursued. REPULSION MODEL FOR ADSORBED NON-IONIC SURFACTANT In the previous section a qualitative explanation for the data has been established. The aim of this section is to place this interpretation on a more quantitative footing by developing a model for the repulsive interactions between non-ionic stabilised particles. In deriving such a model reasonable allowance must be made for attractiveL.THOMPSON 1681 van der Waals interactions. Here the expressions given by Vincent2O9 21 for particles surrounded by a double sheath of adsorbed material have been used. The Hamaker function of the polystyrene itself was obtained from the Lifshitz computations of Richmond.22 The average value for A,, in the important separation range was 8.2 x J. The double sheath constituting the adsorbed layer contained an inner sheath of pure hydrocarbon (All = 5.8 x J),21 and an outer sheath which was a mixture of polyoxyethylene and water (All = 6.9 x and 3.7 x J, respectively).21 The thickness used for the outer sheath was in general the ‘random- coil’ length of 1.4 nm. The thickness of the inner sheath and the constitution of the outer sheath are determined by the area per molecule.The adsorption data were those obtained in earlier work. An appropriate mathematical form for the repulsion between adsorbed non-ionic surfactant layers is not easily defined because there is no clear understanding of the nature of the repulsion other than the rather nebulous term ‘hydration force’. In a sense this is unimportant because the function of the model is to define the magnitude and separation dependence of the repulsion. This can undoubtedly be achieved by a variety of mathematical forms, but it is useful to apply one which allows ready comparison with the results of other workers. If it is assumed that the forces involved here are those responsible for maintaining a well defined spacing between non-ionic surfactant bilayers in lamellar phase and between other lipid bilayers, then the work of Parsegian et aL9-11 provides a starting point.These workers used osmotic-pressure measurements to obtain the force between lecithin bilayers as a function of their separation. Their results indicated a simple exponential relationship between the pressure, P , and the bilayer separation, d,: P = Po exp (- dw/A). (3) For egg lecithin the value of p0 was 7.05 x lo8 N m-2 and the decay length L was 0.25 nm.ll Note that this type of relationship has been predicted23 through order- parameter considerations. Using Derjaguin’s approximation, eqn (3) can be expressed as an energy between spheres, when the relationship becomes V = zaA2P. (4) Attempts to fit this equation to the aggregation data were unsuccessful because an appropriate combination of the three extrema in fig.1 (b) could not be generated. The model needed to achieve this requires a strong short-range Component to restrict the primary minimum, together with a weaker long-range component to generate a potential-energy barrier. The range of the second component must not, however, be so great that it precludes the presence of a significant secondary minimum with the larger particles. The most obvious way to achieve this is to use the sum of two exponentials so that where h is the separation between the outer edges of the adsorbed layers on adjacent particles. All of the aggregation data, regardless of particle size and temperature, can be fitted to this model using a combination of constants differing only slightly from the following P,, = 5 x lo6 N m-2, PO2 = 2 x lo4 N m-2, A, = 0.2 nm and A2 = 3.0 nm.Despite the longer decay length of 3 nm, these forces are very much weaker than those observed between lecithin bilayer~.~-ll Eqn ( 5 ) is purely empirical, so that it has no theoretical implications concerning the origin of the repulsion. Some observations concerning the origin of the short-range component can, however, be made. They1682 INTERACTION FORCES IN NON-IONIC SURFACTANTS 12 6 Fz, bh * o - 6 -12 0 4 a 12 16 20 separation/nm Fig. 7. Plots of potential energy against separation for different-sized polystyrene particles. The repulsion model [eqn (5)] used the following parameters: Po, = 5 x lo6 N m-2, Al = 0.2 nm and P,, = 2 x lo4 N m-,. A, varied as follows with particle diameter.(a) 3.8 pm, 3.5 nm; (b) 1.76 pm, 3.0 nm; (c) 1.0 pm, 3.0 nm; (d) 0.5 pm, 3.1 nm and (e) 0.25 pm, 3.3 nm. derive from the exclusion from the calculations of an electrostatic repulsion term arising from the native charge of the particle. The original justification for this was that at the high electrolyte concentration used, and in the presence of an adsorbed layer, the range of the electrostatic term was likely to be insignificant. Fitting the aggregation data, however, has produced a model which contains just such a short-range component. It follows that this component contains at least a contribution from the electrostatic term. Indeed, the repulsion arising from a potential of ca. 10 mV (at the outer edge of the adsorbed layer) is approximately equivalent to the above values of Pol and 2,. 10 mV is not an unlikely figure, and the resulting implication that the short-range component may even arise wholly from this source demands an investigation involving electrokinetic measurements of non-ionic stabilised particles.Such an investigation is not trivial in either an experimental or a theoretical sense and it will be considered separately. Fig. 7 shows plots of potential energy against separation calculated for each of the particle sizes used in the experiments. The headgroup area for the adsorbed surfactant (0.65 nm2) refers to 35 "C. The curves have been fitted to within kT [through eqn (1) and (2)] to the kinetic and equilibrium data at this temperature and they are consistent with the mechanisms discussed in the last section.The models used differed only in the values of &, which ranged from 3.0 to 3.5 mm, and in the precise detail of the primary minimum. Fig. 7 does not reflect the restriction placed on the depth of the primary minimum. This is because the balance of forces at the short primary-minimum separations is too model-sensitive for serious curve fitting and we can only establish the feasibility of an appropriately restricted minimum. In fact in all cases shown it is possible to eliminate the primary minimum altogether by a relatively small change in the short-range component through either Pol or L. Both the short-range model sensitivity and the restriction of the primary minimum are better demonstrated when eqn ( 5 ) is used to investigate the temperature sensitivity of the particles (1.76 pm) that exhibit the most complex behaviour.L.THOMPSON 1683 Fig. 8. Plots of potential energy against separation for 1.76 pm diameter polystyrene particles showing the effect of temperature (through area per molecule). (a) 1 nm2 per molecule (15 "C), (b) 0.8 nm2 per molecule (25 "C), (c) 0.55 nm2 per molecule (40 "C). This parameter only affects the attractive component of the potential energy. The repulsion model was that used for fig. 7. Fig. 8 shows the effect on the calculated plot of potential energy against separation for 1.76 pm particles produced when the area/adsorbed molecule is given its experimentally determined temperature dependence, C,,E, adsorption increases significantly between 15 and 35 "C, the area per adsorbed molecule changing from 1 to 0.58 nm2.Above this temperature adsorption remains relatively constant up to the limit of the available data at 50 O C . l This changes the attractive van der Waals part of the interaction potential, because increasing the adsorbed layer's density increases its contribution to the attraction. The repulsion was allowed to remain independent of temperature for the purposes of fig. 8, i.e. the changes in the plot of potential energy against separation shown in fig. 8 are brought about entirely by the effect of temperature on the interparticle attraction. These changes may be summarised as follows. At low temperatures there is no primary minimum, and aggregation is associated with the secondary minimum.As the temperature increases the primary minimum develops and the potential-energy barrier becomes smaller. This is broadly in line with experiment where rapid, weak aggregation at low temperature gives way to slow, weak then slow, strong and finally rapid, strong aggregation at high temperature. This final phase indicates that, in practice, the potential-energy barrier is eliminated completely. Note that although the main features of the transition are predicted, the detailed behaviour is more complex. This is thought to involve changes in the structure of the adsorbed layer which are not dealt with by the simple model used here. This point is discussed later. Fig. 8 contains two implications which are central to an understanding of colloid stabilisation by non-ionic surfactants. First, since it is close to explaining the observed temperature effects solely on the basis of adsorption-induced changes in attraction, it follows that the repulsive contribution is to a first approximation independent of temperature. At first sight it seems contradictory to suggest that the adsorbed layer's repulsion remains constant while the level of adsorption increases, because this apparently involves a decrease in headgroup area which in turn reflects a decrease in1684 INTERACTION FORCES IN NON-IONIC SURFACTANTS headgroup repulsion.In this case, however, it is likely that the change in area per adsorbed molecule is brought about by configurational factors (e.g. flat or vertical orientation of the hydrocarbon chain) rather than by a fundamental change in headgroup repulsion.The phase diagram for C,2E,24 supports this view in that it demonstrates the existence of hexagonal liquid-crystal phase over the whole of temperature range studied here This would not be permitted by the packing requirements of the phase if the headgroup area changed drastically, and indeed X-ray diffraction measurements26 indicate an almost temperature-insensitive headgroup area (0.52 nm2) for this system. Additional evidence for a change in adsorbed-layer configuration is contained in the aggregation data and will be discussed later in this paper. Temperature-independent hydration forces have also been observed between mica ~ h e e t s . ~ ' ~ 28 These forces were attributed to the presence of highly hydrated metal ions, and they followed an exponential decay pattern with a typical decay length of 1 nm compared with the 3 nm found here and the 0.25 nm found in Parsegian's work.When the electrolyte was hydrochloric acid, hydration forces were not observed. It seemed unlikely that the present results could be attributed to ion hydration because there was no apparent mechanism for attaining the required high levels of specific counter-ion adsorption. Nevertheless, aggregation-rate experi- ments have been conducted in 0.4 mol dm-3 HCl, and essentially similar results were obtained to those quoted here for 10-1 mol dm-3 MgSO,. This shows that the long-range component of the repulsion is unaffected. As mentioned earlier the short-range component may be sensitive to zeta potential and hence to pH, but this would significantly affect only the primary minimum equilibrium.Note also that little or no difference was produced by substitution of 0.4 mol dme3 NaCl. The second important implication of fig. 8 is that the temperature effect is basically short-range in nature. This is because adsorbed layers of the kind used here cause only a short-range perturbation of the attraction of the underlying particle. Primary- minimum aggregation equilibria are greatly susceptible, whereas rates are less so because they are controlled by an energy barrier at greater separations. Secondary- minimum aggregation is hardly affected at all. This is why aggregation of the largest (3.8 pm) latex, which according to fig. 7 operates through a secondary-minimum mechanism, is relatively insensitive to temperature in the range in which smaller particles aggregate.Fig. 2 shows that this latex finally aggregates at ca. 70 "C, and this leads to a suspicion that the repulsion forces are not completely independent of temperature. Extension of the calculations to this temperature could not be made in the absence of relevant adsorption data. The temperature sensitivity of the repulsive force may provide an interesting area for future research which may best be tackled using a surfactant which has a less temperature-sensitive area per adsorbed molecule. Finally, attention is drawn to some details of both rate and equilibrium data for which the model resulting in fig. 7 and 8 fails to account despite its success in explaining the main features of the aggregation behaviour.It is not surprising that such details exist, because the attraction expression used employs an idealised view of the adsorbed layer's structure. By examining the difference between the model predictions and the data it is possible to draw qualitative conclusions about adsorbed-layer structure. Fig. 8 predicts that for 1.76 pm particles increasing temperature will cause a progressively decreasing barrier to flocculation together with a slightly increased secondary minimum depth. The predicted effect is similar in direction but smaller in magnitude for the smaller particles. This is because of the different separations relative to particle size at which the potential-energy barriers are situated, Fig. 7 shows that the barriers are at about the same absolute separations, so that those associated with the smaller particles are at greater relative separations and are therefore less sensitiveL.THOMPSON 1685 15 2s 35 4s 55 TI" C Fig. 9. Temperature dependence of the secondary minimum: A, 3.8 pm diameter polystyrene latex (secondary minimum); 0, 1.76 pm polystyrene latex (combined primary and secondary minimum); 0, 1.76 pm polystyrene latex (secondary-minimum contribution). to the short-range temperature effect. The model for 0.25 pm particles predicts a variation in barrier height of only ca. 1.5 W i n the 35-50 "C range. The experimental data in table 1 indicate that the temperature effect on barrier heights is generally less than predicted being negligible for the smaller particles.This and the presence of a maximum in W for the 1.76 pm particles indicates an effective increase in repulsion with temperature relative to the model used. The equilibrium data give rise to a similar conclusion, It is recalled that the equilibrium aggregation state of the 3.8pm latex exhibited an oscillatory temperature dependence with least aggregation at ca. 40 "C. It has been concluded that the mechanism of aggregation involved a secondary minimum whereas the aggregation of 1.76 pm particles involved both primary and secondary minima. When the secondary-minimum part of the latter process is crudely separated out (by inspection of the rate curve) its temperature dependence is seen in fig. 9 to follow the same trend as the larger particles with least aggregation at ca.35 "C. This effect was not immediately apparent for the 1.76 pm particles because it was overshadowed by the highly temperature-sensitive primary-minimum deepening caused by increased area per adsorbed molecule. To explain this observation it is necessary to propose some effect which increases the net repulsion slightly (by 1 or 2 k T ) in the 25-35 "C range. A change in the configuration of the adsorbed layer is one possibility which has the merit of consistency with the discussion earlier in this paper where a difference was noted between the temperature sensitivity of the area per adsorbed molecule and the headgroup area in liquid crystals. The difference implied that a configurational change actually caused the area per molecule variation that gives rise to the temperature sensitivity of the colloid behaviour.The present approach is unable to define changes in the adsorbed layer because a given aggregation effect may arise from a large number of conceivable changes. It can, however, assess the feasibility of likely reorganisations by comparing their calculated consequences with the aggregation data. Of the possible changes a reorientation of the adsorbed hydrocarbon chains from flat to vertical seems the most likely. The1686 INTERACTION FORCES IN NON-IONIC SURFACTANTS consequences of this to the interaction energy are of the right order to explain the experimental results. To illustrate this, the equilibrium level of secondary-minimum aggregation alone has been calculated for 1.76 and 3.8 pm particles, first using the ‘condensed-sheath’ model exactly as defined for fig.7, and secondly substituting a vertically orientated hydrocarbon-chain/water layer of depth I .3 nm.8 In both cases the extended-chain model predicts less aggregation (a larger nln,), so that n/n, = 0.90 and 0.60 for the 1.76 and 3.8 pm dispersions, respectively, compared with values of 0.78 and 0.42 for the condensed-sheath model. APPENDIX KINETICS OF REVERSIBLE AGGREGATION Reversible aggregation can be represented as Pmer +jmw + (P +j)rner* It consists of a series of associations and decompositions involving aggregates of different sizes and morph~logies.~ This situation has recently been discussed by Nir and bent^^^ in terms of the distribution of aggregate types formed and its consequences.A reasonably simple, and for the present systems effective, kinetic model can, however, be extracted if it is assumed that no selective aggregation or dissociation takes place, i.e. that the values of p and j are unimportant. (This assumption is also implicit to the classical Smoluchowski treatment of flocculation kinetics and to the equilibrium treatment used here.) In this circumstance the aggregation rate is given by dn dt -- = k,n2-k-, b where n is the concentration of aggregates of all kinds including singlets, k, and k-, are the forward and reverse rate constants, respectively, and b is the concentration of aggregation ‘bonds’. Where aggregates are not multibonded (as would be the case if they were linear in the ‘string of beads’ sense), b = no-n where no is the original singlet concentration.Eqn (A 1) now becomes dn = Jdt. - I k , n2+ k-, n- k-, no Solution of this integral gives the total aggregate concentration as a function of time, so that where q = kZ,(l +4Kn0) 2k, no + k-,( 1 + 2/ 1 + 4Kn0) 2k, no + k-,( 1 - 2/ 1 + 4Kn0) d = tqt+ln and the aggregation equilibrium constant K = k,/k-,. From eqn (A 2) the time evolution of the aggregation process can be derived if the initial rate constant k, can be determined before aggregate dissociation causes deviation from second-order kinetics and if the equilibrium constant K can be obtained. K can be determined experimentally by measuring initial and equilibrium particle concentration and using the equilibrium form of eqn (A 1): Comparison of eqn (A 3) with eqn (2) shows that the equilibrium constant can be expressed in terms of the second virial coefficient.L. THOMPSON 1687 Where two potential-energy minima separated by an energy barrier are concerned, the situation is more complex in principle but is amenable to some simplification.The aggregation is now represented as (1) Pmer +jmer * (P + A s where s and p refer to secondary and primary bonds, respectively. The rates, R, of processes (1) and (2), respectively, are given by R, = k , n2( 1 - W-l) - k-, b, (A 4) and R, = k , rt2 - k-, 6,. (A 5 ) Step (1) only concerns those collisions not energetic enough to go over the barrier, so that the (1 - W-l) term is necessary to prevent the total forward rate from exceeding the diffusion-controlled rate. The total rate expression analogous to eqn (A 1) now becomes dn -- - k , n2( 1 - W-l) - k-, b, + k , n2 - k-, bp.dt Without some means of expressing b, and b, in terms of the overall aggregate concentration n, a rate equation analogous to eqn (A 2) cannot be derived. Where k, % k,, however, the two processes can be treated independently without too much inaccuracy. 1.e. in the early stages of the aggregation the essentially diffusion-controlled secondary-minimum process dominates since, when t = 0, b = 0, (A 7) dn dt -- - n2 [k,( 1 - W-l) + k,] z k , n2. The primary-minimum aggregation cannot strictly be discussed in terms of a diffusion argument until the secondary-minimum equilibrium has been achieved, because a steady state of diffusion does not exist. After the secondary-minimum process has achieved equilibrium (A 8) and only the slower primary-minimum process need be considered.Even then the application of a simple second-order rate model may appear suspect because of possible effects that the accumulation of particles in the secondary minimum may have on the rate of primary-minimum aggregation. Prieve and Rucken~tein~~* 32 have, however, concluded that accumulation at the secondary minimum does not affect the rate of primary-minimum aggregation. k,( 1 - W-l) n2 = k-, bs L. Thompson and D. N. Pryde, J . Chem. Soc., Faraday Trans. I , 1981,77, 2405. * V. A. Volkov and L. F. Komova, Kolloidn. Zh., 1978, 40, 337. Z. Haq and L. Thompson, Colloid Polym. Sci., 1982, 260, 212. D. H. Napper, J. Colloid Interface Sci., 1970, 32, 106. R. Evans and D. H. Napper, Kolloid Z. Z. Polym., 1973, 251, 329. R. Evans and D. H. Napper, Kolloid Z. Z. Polym., 1973, 251,409. C. Tanford, Y. Nozaki and M. F. Rohde, J. Phys. Chem., 1977,81, 1555. D. M. Le Neveu, R. P. Rand, V. A. Parsegian and D. Gingell, Biophys. J . 1977, 18, 209. lo A. C. Cowley, N. L. Fuller, R. P. Rand and V. A. Parsegian, Biochemistry, 1978, 17, L3163. l 1 V. A. Parsegian, N. L. Fuller and R. P. Rand, Proc. Natl Acad. Sci. USA, 1979, 76, 2750. l 2 B. Vincent, Adv. Colloid Interface Sci., 1974, 4, 193. l 3 P. Richmond and A. L. Smith, J . Chem. Soc., Faraday Trans. 2, 1975,71, 468. l4 C. Cowell and B. Vincent, J . Colloid Interface Sci., 1983, 95, 573. l5 J. Th. G. Overbeek, in Colloid Science, ed. H. R. Kruyt (Elsevier, Amsterdam, 1952), vol. 1, p. 285. l6 L. A. Spielman, J. Colloid Interface Sci., 1970, 33, 562. I’ L. Thompson and A. L. Smith, J. Chem. Soc., Faraday Trans. I , 1981, 77, 557. l8 P. McFadyen and A. L. Smith, J. Colloid Interface Sci., 1973, 45, 573. l9 P. G. Cummins, E. J. Staples, L. Thompson, L. Pope and A. L. Smith, J. Colloid Interface Sci., 1983, ’ D. H. Everett and J. F. Stageman, Faraday Discuss. Chem. Soc., 1978, 65, 230. 92, 189.1688 INTERACTION FORCES IN NON-IONIC SURFACTANTS 2o D. W. J. Osmond, B. Vincent and F. A. Waite, J. Colloid Interface Sci., 1973, 42, 262. 21 B. Vincent, J. Colloid Interface Sci., 1973, 42, 270. 22 P. Richmond, Chem. Ind., 1977, 1, 792. 23 S. Marcelja and N. Radic, Chem. Phys. Lett., 1976, 42, 129. 24 D. J. Mitchell, G. J. T. Tiddy, L. Waring, T. Bostock and M. P. McDonald, J. Chem. SOC., Faraday 25 D. J. Mitchell and B. W. Ninham, J. Chem. SOC., Faraday Trans. 2, 1981, 77, 601. 26 I. G. Lyle, personal communication. 27 R. M. Pashley, J. Colloid Interface Sci., 1981, 80, 153. 28 R. M. Pashley, J. Colloid Interface Sci., 1981, 83, 531. 29 J. Bentz and S. Nir, J. Chem. SOC., Faraday Trans. I , 1981 77, 1249. 30 A. Marmur, J. Colloid Interface Sci., 1979, 72, 41. 31 E. Ruckenstein, J. Colloid Interface Sci., 1978, 66, 531. 32 D. Prieve and E. Ruckenstein, J. Colloid Interface Sci., 1980, 73, 539. Trans. I, 1983, 79, 975. (PAPER 3/653)