Some New Aspects of and Conclusions on Theory of Stability of Colloids and their Experimental Verification BY €3. V. DERJAGUIN Department of Surface Phenomena, Institute of Physical Chemistry, Academy of Sciences of the U.S.S.R., 31 kenin Prospect rCn-1, B-312, MOSCOW, 117312, U.S.S.R. Received 2nd December, 1977 This paper considers three aspects of colloid stability theory: phase stability, the stability of disperse composition and aggregative stability in relation to the merits of a direct operational deter- mination of the disjoining pressure. Four components of the disjoining pressure are treated: the dispersion, ion/electrostatic, adsorption and structural terms. The simplest and most general way to derive the second and third components of disjoining pressure is to use the Gibbs-Duhem equation generalized by inclusion of terms corresponding to the electric work of charging the particle interface.The theory of the adsorption component of disjoining pressure explains the stability of free films of some binary solutions demonstrated experimentally by Sheludko and Ekerova. With regard to other aspects of colloid stability we note that flow-ultramicroscopic measurements of the concentration of colloid particles are free from the shortcomings of other methods. This makes accessible the kinetics of slow coagulation of sols (e.g., gold sols) and reveals the role of disaggregation here and in the establishment of aggregative quasi-equilibrium. The measurements of molecular attraction between crossed metal wires as a function of gap width are presented.Colloids differ from true solutions in that they possess a redundant degree of freedom, dispersity. As a corollary, colloids are able to change their state in three main ways: (1) Like true solutions, through separation into 2 quasi-phases that differ from each other in the concentrations of particles and possibly in their locations (periodic structures). (2) In contrast to true solutions, colloid systems are able to change their dispersity through the dissolution of less stable (usually finer) particles and the growth of coarser particles. (3) Also, in contrast to true solutions, colloid systems are able to change owing to the aggregation of particles. Hence, three kinds of stability have to be distinguished; namely, phase stability, disperse composition stability and aggregative stability.Within the framework of this scheme, a special place is occupied by disperse systems such as micellar solutions and microemulsions, which are quite stable from the thermodynamic point of view and may possess simultaneously all three types of stability. Colloid science has long concentrated on aggregative stability because it is dis- turbed most easily and at the highest rate. Development of the theory led to funda- mental changes, not only in the theory of the colloid state, but in adjacent problems of physics, physical chemistry and biophysics related to surface phenomena and forces. One of the first results was substantiation of the repulsion of colloid particles caused by the overlapping of ionic atmospheres.In this connection, it is interesting to elucidate why in the treatment of Langmuir the atmospheres of counterions lead to a decrease rather than to an increase in the osmotic pressure of a colloid system, whereby the ionic atmospheres cause a loss of stability resulting in phase separation.B . V . DERJAGUIN 307 This apparent paradox is explained in the following way: when considering the pair- wise interactions, the concentration of ions around the two particles under considera- tion is assumed to be constant. However, in such a collective effect, when the recipro- cal distances between all particles, and hence their concentrations, vary simultaneously in a similar way, the concentration of counterions in the dispersion medium must also change simultaneously.This is what causes the opposite effect of contraction arising from the ionic atmospheres. As is well known, the theory of the aggregative stability of lyophobic systems was developed on the assumption that the interaction forces between particles are a sum of the dispersion and ionic/electrostatic terms. In early papers, those forces were deduced from a consideration of the distribution of the forces applied to the surface of particles as they approach each otherm2 The same approach is the basis of Lifshitz’ macroscopic theory of dispersion inter- action~.~ In some subsequent papers, the free energy of the system was adopted as the basis, the disjoining pressure being taken as its derivative with respect to distance. The first approach is mathematically simpler ; it also has the advantage of requiring only directly measurable quantities with physical meaning.In this case, the essential point was the fundamental concept of the disjoining pressure4 of thin interlays (of liquids,”gases or even vacuum), where this concept is construed to be the difference between the normal component of the pressure tensor in an interlayer (including the Maxwell tensor and the pressure tensor of an electromagnetic fluctuation field) and the isotropic pressure in a bulk phase, the interlayer being the continuation (or the offspring) of that p h a ~ e . ~ * * ~ This definition is of an operational character, directly indicating the method of measuring the disjoining pressure. Only if such a definition is available, is it possible to impart physical meaning to the relationship between the disjoining pressure and free energy.Exhibiting as it does this general character, the disjoining pressure may be applied to a number of general problems, not only of the equilibrium, but also of the hydrodynamics of thin interlayers, independent of the specific nature of the effects and the particular formulae defining the disjoining pressure. The concept is limited by the fact that it cannot be applied in a strictly quantitative manner to systems, in which the radii of curvature of particles are either commensurate with or smaller than the thickness of interlayers, at which the overlapping effects arise, and which cause the appearance of the disjoining pressure. From the conceptual standpoint, disjoining pressure presents advantages over the notion of the interaction force of particles or phase boundaries that are separated by thin interlayers.Indeed, the very concept of the force implies the existence of points to which the force has been applied, such points being located in material bodies. Now one asks what points of application of repulsion forces can be imagined for the equilibrium of a free film between two bubbles filled by infinitely rarefied gases (at a low volatility of the film)? As applied to the theory of stability, one recognises various components of the disjoining pressure, such as dispersion, ionic/electrostatic, adsorption and structural terms. In this case, in order to derive the second and the third components, the most general, strict and simple method will be to use the generalized Gibbs-Duhem equation represented in the form : I where G is the thermodynamic potential including terms --a,yl and -o2y2, where (iL and (iz are the charges on the internal surfaces of two plates that are separated by a308 NEW ASPECTS OF COLLOID STABILITY THEORY plane-parallel interlayer having thickness d ; ly, and ‘y, are the potentials at the surfaces of the plates; Ti are the adsorptions of the components dissolved in the interlayer (taking into account the overlapping of diffuse adsorption layers), pi are their chemical potentials, S the entropy, T the temperature and p the pressure.Assuming dT = 0, dp = 0, dp, = 0, dly, = 0, we are able to write: From this equation and the Poisson-Boltzmann equation, it is easy to derive the classi- cal expression ne as a function of tyl and the intensity of electric field El at an interface of the interlayer: where D is the dielectric permittivity, k is the electron charge, k is the Boltzmann constant, 2, and 2, are the charges of the ions, yn, and yn2 are their respective con- centrations.The first term on the right of eqn (3) expresses the ponderomotive force of electric field; the second term, the excess hydrostatic pressure. For binary solutions of nonelectrolytes, the new result may be obtained from eqn (l), assuming7 dT = 0, dp = 0, dy, = 0, dly2 = 0: On integration, an approximate formula can be obtained : I7, + 2kTC, exp - A [(”)” + (d - d)-3] - kTC,{exp ($-3 + I}. (5) kT 3 Here n, is the dispersion component of the disjoining pressure for the case of a pure solvent (Coo = 0), Coo is the concentration of the dissolved substance in the bulk phase whose part is constituted by the interlayer; 6 is the cutting-off parameter which, in order of magnitude, is equal to the diameter of dissolved molecules.A is the constant in the expression V(X) = A[JG-~ + (d - x ) - ~ ] (6) for the effective dispersion interaction of the molecules dissolved in the solvent, with the phases separated by the interlayer, at the distance x from one of them. In accordance with the calculations of Derjaguin et aL8 where el, E,, c3 are the dielectric permittivities as functions of imaginary frequency i t of concentration C, substrates and solvent, respectively. According to Dzyaloshinskij et aL9 Analysing the formula derived for enables one to come to following conclusions.B .V . DERJAGUIN 309 At large values of d (weak overlap of diffuse adsorption layers), the adsorption component ITa, which is proportional to C,A, may have either a negative or a positive sign. In the second case, given certain dielectric properties for the system, that component may overweigh a negative disjoining pressure, IT,. As the value of d decreases, there can occur transitions in the value II, whether from the negative value to the positive one, or vice versa. The signs may also be reversed. For the simpler case of a binary solution interlayer between two gaseous volumes, for large values of d we obtain the film stability condition (I7 > 0) in the form: Thus, the stability of such a free film requires that the dissolved substance reduces the dielectric permittivity of the solvent to a considerable degree.This conclusion provides an explanation of the stability of free films of butyric acid solutions in water, observed by Sheludko and Elcserova.10 Now, passing over to considering the role of the structural component of the dis- joining pressure n$ in the stability of disperse systems, the absence of a general theory will have to be taken into account: experimental findings must form the basis of the argument. The structural component of the disjoining pressure is indicated by the form of the isotherm of the disjoining pressure of multimolecular adsorption (wetting) layers of water on silicate surfaces (e.g., glass, fused quartz).ll At the greater thicknesses that result from the thinning out of aqueous interlayers @-films) under an air bubble, the disjoining pressure isotherm is well represented by the formulae corresponding to the ionic/electrostatic component.However, after considerable thinning out or break down of an interlayer, or when the multimolecular adsorption of water vapour is observed to develop, thinner a-films (d < 100 A) are obtained. The disjoining pressures of these films cannot at all be represented as the sum of the dispersion and the ion/electrostatic components. At a certain thickness of the film (of the order of 50 to 70 9.) the disjoining pressure of cc-films reverse in sign, becoming negative at greater thicknesses.l2 Now, above 65 to 70 "C, a-films become monolayers: the multimolecular bound- ary layers having a structure which is different from that of the bulk phase then disappear.13 In Peschel's measurements l4 of the disjoining pressure of liquid interlayers between quartz surfaces, the influence of the structural component of the disjoining pressure is noticeable, too.There are also studies available, which serve to emphasise in a number of cases that, in order to explain the coagulation phenomena, the structural component of the disjoining pressure must be used as a basis. For example15 with adsorption of some hydrophilizing tensides, the thresholds for coagulation by indif- ferent electrolytes increase dramatically and practically cease to depend on the charges of the counterions. In this case, it has been additionally shown that the coagulation is not caused by a decrease in the adsorption of tensides under the influence of electrolytes.Most extensively used methods for the examination of coagulation kinetics are quite unsuitable for comparisons with theory; for they either yield readings (such as light scattering) which are only indirectly connected with the concentration of colloid particles (including the aggregates) or (e.g., Coulter counters) they are able to measure concentrations of the order of 1O'O to 1011 only after tremendous dilution (by millions of times). We have suggested a flow ultrarnicroscopy scheme16 which is free3 10 NEW ASPECTS OF COLLOID STABILITY THEORY from such disadvantages and enables one to measure the particle concentrations directly, rapidly and without dilution, iiicluding those for highly-dispersed red sols of gold.Application of this method enabled us to observe the slow coagulation kinetics as a function of the inverse concentration of particles plotted against time: the plot is not linear and has a horizontal portion (or portions).10 This indicates that temporary stationary states are established, corresponding to disaggregation processes counter- balancing aggregation processes. At a particular stage, a new increase in the coagu- lation rate begins, attributable to the formation of coarser aggregates (for example, made up of three and more instead of two particles). In such aggregates, the bonding energy per particle is higher owing to the greater number of neighbours present. By analogy with the phenomenon of nuclea- tion in the formation of a new phase (personal communication from G.A. Martinov), one must accept that the critical nucleus of coagulum formation contains only three to four particles even in the cases where the coagulation rate and, hence, super- saturation are very small. Sometimes, even the rapid coagulation may terminate with the establishment of a quasi-equilibrium, at which the number of aggregates ceases to increase. It is then probable that the depth of the potential well is small owing to the structural component of the disjoining pressure. It is possible that such a component can exist for the red sol of gold because of the adsorption of organic substances, attendant on the Zsygmondy preparation method. Aggregate formation may be facilitated by a reduction in the potential barrier, as compared with ordinary estimates, owing to the discreteness effect. It has been shown by Muller and myself,l* that even at the zero charge of two dense layers of ions that have been adsorbed in a nonlocalized manner on two neighbouring surfaces, the effect of discreteness causes the attraction at close distances.There is an important connection between the dispersion component of the dis- joining pressure and the stability (or rather instability) of colloids. With this in view, it is a pity that the Lifshitz theory for the case of two metals" remains still to be verified. It is the opaqueness of metals which hinders the application of optical methods and prevents measurement of the molecular interaction of two metals in air and in liquids.However, a new improved variant of applying negative feedback to the measurement of molecular attraction of macro-objects allows one to dispense with direct measurement of distances. Recall that in the early studies the negative feed- back performed two functions; it stabilized the gap between the two bodies, and thus compensated for the influence of the molecular attraction force to be measured according to the automatic compensation scheme.20 In the studies of Rabinovich and coworkers,21 the feedback performed yet a third function ; it permitted two objects-two crossed threads that are initially in contact-to be drawn apart to a preset distance. The ultimate separation was not measured but preset. This method was applied to the measurement of attraction forces, whether between quartz threads or platinum or gold.Fig. 1 represents the interaction force F(d) calculated in terms of the interaction energy of planparallel surfaces per unit area, U(d), as a function of distance d. The scale on the axes is logarithmic. The formula connecting F(d) and U(d) is as follows: F(d) = 24R1Rz)* U(d) where R1 and R2 are the radii of the threads. * The chromium-quartz case was studied experimentally, and the theory confirmed at an earlier date.lgB . V . DERJAGUIN 311 0 0 0.5 1 10 20 30 50 100 log ( h x 107 crn 1 I I I I I h l nrn FIG, 1 .-Dependence of the energy per cm2 of the molecular attraction of plane-parallel surfaces, U, on distance, h. (a) @, quartz; (b) 0, platinum; (c) e, gold. Thus we observe, in accord with theory, at distances where the interaction is fully retarded, the interaction energy, U(d), obeys a theoretical relationship : B 3d3’ U(h) = - where the constant B is the same for all metals.The experimentally obtained values of B for gold and platinum are close to each other, and equal to B = 10 x erg cm, which agrees, within the error limits, with the theoretical value of B = 13 x erg cm. energy follows the law At lower values of d, corresponding to nonretarded interaction, the interaction A 1 2nd2 U(d) = -. Within this range of distances, in accord with theory, the constants for platinum and gold differ from each other. In the case of gold, the experimental constant A > 2.3 x erg; whereas for platinum, A = 2.0 x 10-l2 erg. For gold, the theoretical value of A x 3.6 x 10-l2; but for platinum the theoretical value of A is unknown.The same graph plots the results of measuring freshly-drawn quartz threads. In this case, because of the near atomic smoothness of the surfaces, we succeeded in obtaining measurements to still smaller distances. Under these circumstances, the transition between the retarded and the nonretarded interaction is distinct. For quartz, the experimental value of A = 0.5 x 10-l2 erg, is close to the theoretical312 NEW ASPECTS OF COLLOID STABILITY THEORY A = 0.8 x erg cm compares well with the theoretical value B = 0.6 x This technique was also applied by Rabinovich21 to the measurement of the dis- joining pressure as a function of thickness for quartz and glass threads in aqueous solutions.As in Israelachvili's the results of measurements made in dilute KN03 solutions proved to be in good agreement with the DLVO theory; certain deviations may be attributed to the influence of the structural component of the dis- joining pressure, l7,. erg; the experimental value of B = 1.0 x erg cm. I. Langmuir, J. Chern. Phys., 1938, 6, 873. B. V. Derjaguin, Izuesl. An S.S.S. 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