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ACS Award in Colloid or Surface Chemistry Lecture Keeping Pace with Colloids in Motion† Theo G. M. van de Ven Paprican and Department of Chemistry, Pulp and Paper Research Centre, McGill University, Montreal, Quebec H3A 2A7, Canada Received May 15, 1996. In Final Form: August 15, 1996X One of the characteristics of suspended colloidal particles is that they are perpetually in motion. In the absence of any external fields they move around randomly in the fluid, undergoing Brownian motion, while in their presence a deterministic motion is superimposed. These motions lead to collisions with other particles and with surfaces in contact with the fluid. Over the last 20 years considerable progress has been made in understanding the motions and interactions of colloidal particles subjected to gravity and hydrodynamic or electric fields. Theory has advanced significantly and has been verified with new elegant experimental techniques. Examples of our contributions to this progress will be given, varying from the motions of particles, drops, and bubbles to various two-body and particle-wall interactions. Factors that affect these interactions, such as electrolytes and polymers, are being addressed.
Gaining Experience Science is a team effort. New ideas are often generated in discussions and incorporated in our train of thought. Because of this, we sometimes cannot even identify the person who first thought of a new idea, since several people may have contributed to its birth. I have been fortunate to have had around me (and still have) a group of young talented students and numerous colleagues, visiting scientists, and postdoctoral fellows with whom it is a pleasure to discuss colloid science. They all have contributed to the work for which I was awarded the ACS Colloid or Surface Science Award. They did most of the work and deserve most of the credit. I merely channeled their energy in a desired direction. I have been fortunate in many ways. My early interest in colloid science was aroused by Professor Theo Overbeek. I was lucky enough to have him as a first-year teacher at the University of Utrecht, where he taught the course on physical chemistry with lectures based on a book he had written himself (Inleiding tot de fysische chemie1 ). Occasionally he would go far beyond the topics in the book and speak with enthusiasm about current unsolved problems he was interested in. This made quite an impression on me and made me realize that colloid science was a challenging field and full of poorly-explained or unexplained phenomena. It thus became natural to me to do my graduate studies in colloid science, and I joined the group of Professor Agienus Vrij to work on “light scattering on draining soap films.” In all naivety, I had chosen this topic simply because it seemed fun to play with lasers and soap films, little realizing at the time that the topic was at the heart of most colloid and interface science. If one truly understands soap films, one understands colloidal forces, † Award lecture for ACS Award in Colloid or Surface Science, sponsored by Procter and Gamble, presented at the 211th National Meeting of the American Chemical Society, New Orleans, March 26, 1996. X Abstract published in Advance ACS Abstracts, October 1, 1996.
(1) Kruyt, H. R.; Overbeek, J. Th. G. Inleiding tot de fysische chemie, 17th ed.; H. J. Paris: Amsterdam, 1965.
S0743-7463(96)00479-9 CCC: $12.00
surfactant adsorption, micelle formation, surface tension, thin films, stability theory, and hydrodynamics. The experimental techniques exposed me to light scattering and reflection and their underlying theories. It is difficult to think of a better problem with which to train someone in colloid and interface science. After receiving my “doctorandus” degree (M.Sc. Equivalent), I moved to Canada to join Professor Stanley Mason, under whose supervision I worked on “Interactions of colloidal particles in shear flow.” This Ph.D. project exposed me to microhydrodynamics and gave me the first opportunity of working on problems which had both colloidal and hydrodynamic aspects, an interest I have had ever since. Luckily for me, Mason let me more or less work at anything I wanted. When I asked him what I should do, his answer was “Look through the microscope into the traveling microtube and try to observe something interesting.” The traveling microtube is a device, developed in Mason’s group, with which one can observe and film colloidal particles moving in a capillary tube.2 When looking into the traveling microtube one sees particles, and usually aggregates as well, passing by, and one can follow individual particles by moving the tube. My first interesting observation was on triplets (consisting of three latex spheres). They appeared to be flexible, but because they rotate periodically and change their projected angle, it was difficult to tell whether the triplets were flexible or rigid. After observing, filming, and analyzing triplets (with a ruler and protractor), I found that they came in two groups: flexible ones, usually observed at low salt concentrations, and rigid ones, usually seen at high salt concentrations. On one occasion I had the chance of catching the transition from flexible to rigid in the act,3 an observation reproduced in Figure 1. Presumably one of the two bonds was already rigid and the second one became rigid during the observation. This result showed that one could indeed observe interesting phenomena, but only after a thorough analysis of the data. (2) Vadas, E. B.; Goldsmith, H. L.; Mason, S. G. J. Colloid Interface Sci. 1976, 43, 630. (3) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 535.
© 1996 American Chemical Society
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Langmuir, Vol. 12, No. 22, 1996 5255 Table 2. Colloidal Motion and Interactions theory sphere/double layer* sphere/polymer layer spheroid/double layer spheroid/polymer sphere/sphere* sphere/spheroid* spheroid/spheroid sphere/sphere/polymer* sphere/non-sphere/polymer sphere/wall* sphere/wall/polymer*
Figure 1. Variations in the angle R of an aggregate consisting of three latex spheres with time, rotating in a shear flow. Five periods of rotation are shown. In the first three rotations the angle varies periodically while in the last two the angle is constant, implying that a sudden change from a flexible to a rigid triplet has occurred (after ref 3). Table 1. Major Experimental Techniques Used in Our Lab traveling microtube apparatusa microcouette impinging jeta microcollidera transient electric light scattering dynamic light scattering evanescent wave light scattering a
Particles subjected to flow observed through a microscope.
The scarce guidance by Mason was more than compensated for by the presence of Professor Raymond Cox, a brilliant fluid dynamicist and theoretician with profound physical insight. During our daily coffee or tea breaks I learned more about hydrodynamics from him than I ever could have learned from regular lectures or by reading textbooks. He especially taught me how to cast problems in mathematical form. Equally impressive to me were the frequent visits of Professor Howard Brenner to Montreal, at Mason’s invitation. Like Raymond, Howard’s example encouraged us to explain our observations in mathematical terms, and to avoid appearing ignorant, we all had to read from a to z the book “Low Reynolds Number Hydrodynamics”,4 which he coauthored with Happel. The day the ACS phoned to announce that I was this year’s winner of the Colloid or Surface Science Award turned out to be a very sad day for me. I received a second call that same day announcing Raymond Cox’s premature death at the age of 56. His death was a severe loss to his friends and the scientific community at large. My final training consisted of a postdoctoral fellowship with Professor Bob Hunter, who taught me macrorheology and his famous theory of elastic flocs, to which I had the pleasure of making a small contribution.5 Scope of Research Interests A central theme in the study of colloidal dispersions is the motion and interactions of the suspended particles. These colloids in motion can be observed by a variety of techniques. In my lab the major tools are light scattering and microscopic techniques, summarized in Table 1. More details of some of these techniques will be given when discussing specific problems. Most of these techniques (4) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice-Hall: New York, 1965. (5) van de Ven, T. G. M.; Hunter, R. J. Rheol. Acta 1977, 16, 534.
drop/sphere drop/drop bubble/sphere* bubble/bubble bubble/molecule*
experiment latex, gold latex, TiO2/PEI, PEO, PAM clay, nonspherical latex, TMV clay/PEI, PEO, PAM latex, CaCO3 filler/fiber clay, fiber, RBC, platelets latex, CaCO3/PEO, PVA, PEI, PAM filler, fines, fibers/PEO, PEI, PAM latex, TiO2, CaCO3, liposomes, RBC/glass, sapphire, cellophane latex, CaCO3, TiO2, RBC/glass/PEO, PEI, PAM, protein model systems, bitumen/sand bitumen/water air/latex, soap, ink, fines air/air (defoamer) beer bubble/CO2
are either home-made or modifications of existing techniques. My students are encouraged to build their own equipment as part of their training. Over the years we have been looking at increasingly complex systems, starting from the motion of individual particles (subject to Brownian motion, flow, gravity, or an electric field), going to two-body collisions and interactions, and finally going to many-component systems (including polymers and polyelectrolytes) and multibody interactions. We are presently studying five-component systems consisting of fibers, fines, fillers, and a dual-component retention aid system. A summary of the systems we have been studying is given in Table 2. The table shows the model geometries, ideally suited for theoretical interpretations and actual examples of experimentally investigated systems. Reviewing all of this work in a limited amount of time and space is impossible, and I will restrict myself to picking some interesting highlights (indicated in the table by an asterisk). As I hope to show, in studying model systems, one learns a lot about colloid science and, at the same time, has fun doing so. Additional results of earlier studies can be found in my book “Colloidal Hydrodynamics”.6 This monograph was written because I felt that no book was available that adequately treated the hydrodynamics of colloidal systems systematically. Written with the objective of introducing hydrodynamics to colloid scientists (as implied by the limerick I wrote for the preface), I learned that the book was equally helpful for hydrodynamicists who wanted to learn more about colloid science. Resolving Some Classical Problems Diffusion Constant of a Charged Colloidal Particle. Despite the fact that Brownian motion of colloidal particles is well understood since the classical work by Einstein7 and Smoluchowski,8 it was not appreciated by many that the diffusion constant of an electrostatically charged particle is different from that of a neutral, noncharged one. I remember this topic being discussed while I was still at the van’t Hoff Laboratory in Utrecht, and if my recollection is correct, the prevailing opinion of the Overbeek school at the time was that the electrical double layer did not affect the diffusion constant of the particle or that the effect was negligible, an opinion corroborated by the fact that no such effects were taken into account in any of their papers. (6) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press, London, 1989. (7) Einstein, A. Ann. Phys. 1905, 17, 549. (8) Smoluchowski, M. Ann. Phys. 1906, 21, 756.
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Figure 2. Diffusion constants of monodisperse latex particles as a function of salt concentration for various types of monovalent salt (indicated in the figure) (after ref 10).
al.9
In a by now classical paper, Oshima et calculated the drag coefficient, f, of an electrically charged sphere placed in a uniform flow field or, equivalently, sedimenting at a constant speed. Their solution is valid when the Peclet number Pe ) ua/Di , 1, u being the velocity of the sedimenting particle of radius a and Di the (mean) diffusion constant of the ions in solution. We showed10 that, despite the fact that Pe for a single Brownian jump is not small, on average the deviations from the equilibrium ion distribution in the double layer remain small and that the diffusion constant of a charged sphere is given by the Stokes-Einstein equation Df ) kT, with f being the friction coefficient of a charged sphere:
bφ2s ˜f(κa))
D ) D0(1 -
(1)
Here b is the average ion size scaled by the Bjerrum length, φs is the scaled surface potential (φs ) zeζ/kT, z being the valency of the ions, e the unit charge, ζ the ζ-potential, and kT the thermal energy), and ˜f(κa) is a function of the double-layer thickness κ-1 and particle radius a:
1 ˜f(κa) ) [e2κa{3E4(κa) - 5E6(κa)} + 8eκa{E3(κa) 8 E5(κa)} - e2κa{4E3(2κa) + 3E4(2κa) - 7E8(2κa)}] (2)
electric field. Another important consequence of electroviscous drag will be discussed in the next section. Explanation of Lyotropic Series in Coagulation Studies. One of the early successes of the DLVO theory was the explanation of the Schultze-Hardy rule, which predicts that the critical coagulation concentration (CCC) varies as z-6 (z being the valency of the ions). Simple experiments with test tubes containing colloidal dispersions, to which an increasing amount of salt was added, were sufficient to measure the CCC. Kruyt12 found that for AgI sols the CCC varied as predicted with valency, but differences were found between salts of the same valency. A consistently larger CCC was found when the ions in solution were larger. For example, for monovalent salts it was found that the CCC followed the so-called lyotropic series
Li+ > Na+ > K+ > Rb+ (the Li+ ion being the largest ion because it is the most hydrated). The same lyotropic series has been found in many systems, but no satisfactory explanation is provided. Usually it is attributed to specific ion adsorption, which changes the value of the effective ζ-potential and thus the CCC. However the effects follow naturally from the convective diffusion equation which governs the coagulation kinetics. For spherical particles it can be written as6
dn 1 d 2 d 2 r D(r) ) (r D(r)Fcolln) dr dr kT dr
(
)
Equation 3 is, in essence, a mass balance which states that there is no accumulation or depletion of particles in the diffusion boundary layer around a reference particle. The solution of eq 3 yields the distribution of particles n(r) around the reference particle, from which the collision rate per particle can be calculated from Fick’s first law. As indicated in eq 3, D is not a constant but a function of the distance between two particles. The function D(r) is known for hard spheres13 and introduced into the coagulation kinetics by Derjaguin and Muller14 and Spielman.15 For electrostatically charged systems, D(r) needs modification due to the electroviscous drag exerted on the particles16
where
D(r) ) D0(r)(1 - dev)
∫x∞ez
-z
Ei(x) ) xi-1
dz
The trends predicted by eqs 1 and 2 were verified experimentally by photon correlation spectroscopy (PCS) experiments10 on small gold particles (a ) 10 nm) and latex particles (a ) 20 nm); the results for the latex particles are shown in Figure 2. These results show that near κa ) 1 the diffusion constant can be several percent lower than that at κa , 1 or κa . 1. This difference is important when determining the thickness of surfactants or polymer layers adsorbed onto colloidal particles with PCS. The same decrease in diffusion constant was found in rod-shaped tobacco mosaic virus (TMV) particles.11 From measurements of diffusion constants of TMV particles, their ζ-potential could be estimated. It can be seen from eq 1 that ζ-potentials of colloidal particles can be estimated from diffusion experiments, in the absence of any external (9) Oshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans 2 1984, 80, 1299. (10) Schumacher, G. A.; van de Ven, T. G. M. Faraday Discuss. Chem. Soc. 1987, 83, 75. (11) Schumacher, G. A.; van de Ven, T. G. M. J. Chem. Soc., Faraday Trans. 1991, 87, 971.
(3)
(4)
where dev is the electroviscous contribution which depends on b, φs, κa, and the precise form of the colloidal force Fcoll. Inserting eq 4 into eq 3 results in16
CCC ) CCC0(1 + bφ2s h(κa))
(5)
W ) W0(1 + bφ2s g(κa))
(6)
and
Here W is the stability ratio and the subscript “o” refers to predictions of the DLVO-theory with D(r) ) D0(r). Equation 5 explains the lyotropic series, since the CCC increases with the size b of the ions in solution. Equation 6 shows that the stability ratio W (a measure of the flocculation kinetics) is also modified by electroviscous drag forces. The DLVO theory predicts that the slope of (12) Kruyt, H. R. Colloid Science; Elsevier: Amsterdam, 1952; Vol. 1, p 307. (13) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1. (14) Derjaguin, B. V.; Muller, V. M. Dokl. Akad. Nauk. SSSR (Engl. Trans.) 1967, 176, 738. (15) Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562. (16) van de Ven, T. G. M. J. Colloid Interface Sci. 1988, 124, 138.
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Figure 4. Interaction of a sphere with a rod-shaped particle in a simple shear flow (schematic). Figure 3. Comparison of theory (solid lines) of coagulation (with the inclusion of the electroviscous drag force) with experiments (symbols) for two types of sol (after refs 18 and 19). Lower curve, AgI-sol; upper curve, AS2S3-sol.
a log W-log c plot (c being the salt concentration) is proportional to the particle radius a, while experiments show only a weak dependence.17 Equation 6 can provide part of the explanation19 of this classical discrepancy: W0 is an increasing function of a (below the CCC), g(κa) is a decreasing function, and the net effect is a weaker dependence of W on a than predicted by classical theory. Comparisons of experimentally observed CCC’s for various monovalent electrolytes are shown in Figure 3, together with theoretical calculations by Warszynski and van de Ven.18,19 It can be seen that, except for Rb+ and H+, the theory explains the lyotropic series fairly well. Shear-Induced Two-Body Collisions Two-Sphere Collisions. The trajectory equations for two colloidal spheres interacting in a shear flow are well established.20-24 In the absence of colloidal forces, Brownian motion, third-particle interactions, etc., the trajectories are symmetric and reversible (for low Reynolds number Newtonian flows). When colloidal forces are acting, this symmetry is broken. As yet the theory for the electroviscous contribution to the trajectory equations has not been worked out, with the exception of very small particles colliding with a large one.25 When the colloidal forces are attractive, the formation of doublets can occur, the rate of which is given by
4 J ) R0Jsm ) R0nG(a1 + a2)3 3
the absence of electrostatic repulsion it decreases with increasing shear rate and with decreasing radius ratio a2/a1 (a1 being the larger sphere). Sphere-Spheroid Collisions. Many nonspherical particles can be modeled as spheroids: either prolate spheroids (rods, cylinders) or oblate ones (disklike particles). The interaction of a sphere with a rod-shaped particle in a simple shear flow is shown schematically in Figure 4. Since the rod-shaped particle is rotating in a simple shear flow according to Jeffery’s equations,29 the interaction of the sphere with the spheroid depends on the initial orientation of the spheroid. To calculate the collision frequency, one must average over all orientations and orbits, using Jeffery’s equations for the time spent in each orientation. Depending on its orbit, the spheroid can be aligned along the vorticity axis (vertical axis in Figure 4), in which case it simply spins about its long axis, or it can be located in the plane of shear, in which case it rotates periodically according to
dφ 1 ) G(1 + B cos 2φ) dt 2
(8)
φ being the azimuthal angle and B a shape factor. At intermediate orbits, the spheroid also rotates, according to eq 8, but its polar angle θ (with the vorticity axis) is varying periodically as well. Surprisingly, we found that the collision frequency between a sphere and a spheroid is independent of the orbit of the spheroid. For a small sphere the result reduces to30
(7)
Here R0 is the capture efficiency, Jsm is the classical Smoluchowski collision rate,26 and G is the rate of shear. For collisions between unequal-sized spheres of radii a1 and a2, n is the number concentration of particles of type 1 colliding with a sphere of type 2 (or vice versa). For equal-sized spheres a1 ) a2. The capture efficiency R0 has been calculated, without taking electroviscous forces into account, for equal-sized spheres27 and unequal ones.28 In (17) Ottewill, R. H.; Shaw, J. N. Discuss. Faraday Soc. 1966, 42, 154. (18) Warszynski, P.; van de Ven, T. G. M. Faraday Discuss. Chem. Soc. 1990, 90, 313. (19) Warszynski, P.; van de Ven, T. G. M. Adv. Colloid Interface Sci. 1991, 36, 33. (20) Curtis, A. S. G.; Hocking, L. M. Trans. Faraday Soc. 1970, 66, 1381. (21) Batchelor, G. K.; Green, J. T. J. Fluid Mech. 1972, 56, 375. (22) Lin, C. Y.; Lee, K. Y.; Sather, N. F. J. Fluid Mech. 1972, 43, 35. (23) Arp, P. A.; Mason, S. G. J. Colloid Interface Sci. 1977, 61, 21. (24) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 505. (25) Dukhin, A. S.; van de Ven, T. G. M. J. Fluid Mech. 1994, 263, 185. (26) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (27) van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977, 255, 468.
J ) R0GV/π
(9)
V being the volume of the spheroid. This result is identical to the collision frequency between a small sphere and a large one (eq 7 with a2 ) 0). The capture efficiency R0 follows the same trends as collisions between two unequalsized spheres. Equation 9 is an elegant extension of Smoluchowski’s classical theory26 of coagulation. These results can be applied to the deposition of small colloidal particles onto pulp fibers, which can be modeled as slender spheroids. Results are given for the deposition of latex particles on pulp fibers31 in Figure 5a and for clay32 in Figure 5b. Since R0 depends strongly on a/R (R being the radius of the fiber), large latex particles deposit on the fibers with a much larger efficiency R0 (Figure 5a); similarly, large clay aggregates deposit with a larger (28) Adler, P. M. J. Colloid Interface Sci. 1981, 84, 489. (29) Jeffery, G. B. Proc. R. Soc. London, Ser. A 1922, 102, 161. (30) Petlicki, J.; van de Ven, T. G. M. J. Colloid Interface Sci. 1992, 148, 14. (31) van de Ven, T. G. M.; Alince, B. J. Pulp Paper Sci. 1996, 22 (7), J257. (32) Alince, B.; Petlicki, J.; van de Ven, T. G. M. Colloids Surf. 1991, 59, 265.
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semicircle in front of the deposited sphere, various initial positions of mobile spheres are shown. After the collision, the mobile spheres are repelled from the surface by the immobilized sphere and their positions are shown by dots in a semicircle behind the fixed sphere. The trajectories are observed from below through a transparent glass surface. The instrument built to observe these collisions is called a “microcollider.” (Actually, the sphere is stuck to a top plate and observed from above through a microscope.) An example of a recorded collision between two latex particles is shown in Figure 7. The x-coordinate of the mobile sphere can be determined directly from the video image, while the z-coordinate (i.e., the distance from the wall) is determined from the particle’s speed. The trajectory equation of the mobile sphere is
dr ) M‚(Fhydr + Fcoll) dt
Figure 5. (a) Experimentally observed deposition of latex particles of various sizes (indicated in the figure) onto pulp fibers as a function of time. (b) Deposition of individual clay particles onto pulp fibers (circles) and of clay aggregates (other symbols) (after refs 31 and 32). Deposition of the fibers took place in a beaker containing a mixture of fibers and colloidal particles, subjected to gentle paddle stirring.
Figure 6. Principle of colloidal particle scattering (CPS) (see text) (after ref 35).
efficiency on fibers than that for small individual clay particles (Figure 5b). Figure 5a suggests a simple way of separating a mixture of small and large particles. Simply add some (oppositely charged) pulp fibers to the mixture and stir for a short time. Most large particles will have deposited on the fibers, while most small particles remain dispersed in the fluid. The fibers with the large particles will sediment, and the supernatant is rich in small particles. Determining Colloidal Forces from Particle Collisions. As mentioned earlier, collisions between particles in a simple shear flow are symmetrical in the absence of colloidal forces and become asymmetrical in their presence. The degree of asymmetry can be used as a measure of the colloidal forces responsible for the asymmetry. We designed an experimental technique called “colloidal particle scattering” (CPS) with which the forces between colloidal particles can be determined.33 The principles of this technique are shown in Figure 6, which illustrates collisions between suspended particles with a particle stuck to the wall in a simple shear flow. The center of each mobile particle is indicated by a dot. Within a (33) van de Ven, T. G. M.; Warszynski, P.; Wu, X.; Dabros, T. Langmuir 1994, 10, 3046.
(10)
where r is the distance between the spheres, M is the mobility tensor, and Fhydr and Fcoll are the hydrodynamic and colloidal forces acting on the particle. M and Fhydr can be calculated numerically from the solution of the creeping flow equation.34,35 The unknown force Fcoll is found from experiments. Fcoll is parametrized in some appropriate way, and the best fit parameters are determined by comparing experiments and theory. In spirit, this method is analogous to scattering experiments with fundamental particles. A classical example is the experiment by Rutherford (performed at McGill University), who was shooting R-particles onto Au nuclei. The scattering pattern can be used to yield information about the interaction energy of the colliding particles. An example of the location of latex spheres before the collision (open circles) and after the collision (solid circles) is given in Figure 8a, which shows a so-called scattering diagram.36 The best fit force is given in Figure 8b. With this technique forces down to a few femtoNewtons can be detected (which is 4 orders of magnitude more sensitive than forces obtained by a surface force apparatus37 or by atomic force microscopy38 ). When expressing results in terms of interaction energies, an energy minimum which is only a fraction of a kT unit can be detected.36 Besides measuring the colloidal forces between (bare) latex particles, colloidal particle scattering (CPS) has been used to determine the forces between triblock-coated latex spheres39 and poly(ethylene oxide) (PEO)-coated latex spheres.40 These systems are shown schematically in Figures 9 and 10. The triblocks used consisted of PEO-PPO-PEO (PPO being poly(propylene oxide)), with the PPO block adsorbed onto the latex (Figure 9). With CPS one can measure the thickness of the adsorption layer, despite the fact that particles never touch during the experiments. Because of the electrostatic repulsion between the latex spheres, particles never approach closer than about 25 nm, while the thickness is in the range 7-11 nm. However the adsorbed layer affects the force between the particles which can be detected. In essence, the water trapped in (34) Dabros, T.; van de Ven, T. G. M. J. Colloid Interface Sci. 1992, 149, 493. (35) Dabros, T.; van de Ven, T. G. M. Int. J. Multiphase Flow 1992, 18, 751. (36) Wu, X.; van de Ven, T. G. M. Langmuir 1996, 12 (6), 3859. (37) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (38) Binning, G.; Quate, C. F.; Gerber, Ch. Phys. Rev. Lett. 1986, 56, 930. (39) Wu, X.; van de Ven, T. G. M. Langmuir, in press. (40) Wu, X.; van de Ven, T. G. M. J. Colloid Interface Sci., in press.
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Figure 7. Collision betweeen two 4.7 µm latex spheres observed in a microcollider.
Figure 8. (a) Scattering diagram of collisions between two bare latex particles (left); open circles are positions before the collision, and solid circles are after the collision. (b) With the use of eq 10 (arrow) these data can be inverted to obtain the force-distance relationship between the spheres (right).
Figure 9. Two triblock-coated latex particles on a collision trajectory (schematic). Ls is the thickness of the adsorbed layer.
through the central section of the gap between the two particles, and thus we can model the polymer layer as consisting of two parts: a layer in which no flow occurs, of thickness Li, and a layer which is penetrable to water, of thickness Lp. Thus the total thickness Ls ) Li + Lp. From CPS experiments one can determine the values of Li, Lp, and E, the elastic modulus of the polymer layer. These results show that collisions between PEO-coated latex particles are dynamic, and thus, on the time scale of a collision, polymer relaxation is too slow to establish equilibrium forces. Particle-Wall Interactions Impinging Jet. Particle-wall interactions are a limiting case of the interactions between two unequalsized spheres, with the size of one of the spheres approaching infinity. Thus many theoretical results for two-sphere interactions can be readily adapted to spherewall interactions. In our lab we have studied these interactions with an impinging jet apparatus41,42 in which one can control both the hydrodynamic and physicochemical conditions. It consists of impinging a jet with particles onto a surface and observing the surface by microscopy or some other technique. In the area of observation the flow is a stagnation point flow
vr ) Rrz; vz ) -Rz2
Figure 10. Collision between PEO-coated latex particles (schematic). The adsorbed layer thickness Ls consists of two parts: Li, the thickness of the inpenetrable part, and Lp, the thickness of the part penetrable to water.
the adsorption layer reduces the effective van der Waals attraction between the spheres. The situation is different when the latex particles are coated with a thick layer of high molecular weight PEO. Under these conditions the polymer layers come into contact during a collision and are being elastically compressed (Figure 10). Since fluid must flow out of the gap during the collision, water must flow through the adsorbed polymer layer. Most of the water will drain
(11)
where vr and vz are the radial (r) and normal (z) components of the flow and R is a parameter characterizing the strength of the flow which, for a given geometry of the jet, can be found from the volumetric flow rate through the cell.43 In a typical experiment a jet with colloidal particles is impinged upon a glass surface and the number of particles that deposit on the surface is counted (using video analysis). Usually one finds that
N ) N∞(1 - e-t/τ)
(12)
where N is the number of particles per unit area, N∞ is the number at time t ) ∞, and τ is the characteristic time of the process. At short times N varies linearly with time: N ) (N∞/τ)t ) j0t, j0 being the initial deposition rate. (41) Dabros, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983, 261, 694. (42) Dabros, T.; van de Ven, T. G. M. PCH, PhysicoChem. Hydrodyn. 1987, 8, 161. (43) Kamiti, M.; van de Ven, T. G. M. Colloids Surf. A 1995, 100, 117.
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Figure 11. Competition between deposition of polymer-coated particles and adsorption of polymer (1 and 2). After time τads, particle deposition has ceased, since the surface is coated by polymer (3).
Figure 12. Change in rate of deposition of PEI-coated TiO2 particles on glass (after ref 49).
With this technique one can determine j0 and also the blocking and detachment rates (from τ). In the absence of an energy barrier between a particle and the wall, j0 can be accurately predicted from theory.41 We have investigated a number of systems in which the fast deposition rate was observed, such as latex particles above the CDC (critical deposition concentration),44 PEO-coated latex particles,45 bacteria,46 and spherical hardened red blood cells,47 all on glass surfaces. Characterizing Polymers with an Impinging Jet. It is possible to use colloidal particles as probes to study the properties of polymers. One way this can be done is shown schematically in Figure 11. If one impinges a mixture of polymer-coated particles and freely dissolved polymer upon a surface, a competition ensues between particle deposition and polymer adsorption.48 If conditions are chosen correctly, polymer-coated particles will not deposit on a polymer-coated (glass) surface. Thus as soon as the surface is coated by polymer, particle deposition stops. The time at which this happens, τads, can be readily observed.49 τads can be expressed as -1 τads ) kcMp
(13)
where c is the polymer concentration in solution and M is the molecular weight of the polymer. An example of the change in rate for PEI-coated TiO2 particles with excess PEI (polyethylenimine) in solution is given in Figure 12. This result shows that τads can be readily determined. The time τads can yield c, and from this one can determine, for (44) Varennes, S.; van de Ven, T. G. M. PCH, PhysicoChem. Hydrodyn. 1987, 9, 537. (45) Varennes, S.; van de Ven, T. G. M. PCH, PhysicoChem. Hydrodyn. 1988, 10, 229. (46) Xia, Z.; Woo, L.; van de Ven, T. G. M. Biorheology 1989, 26, 359. (47) Xia, Z.; Goldsmith, H.L.; van de Ven, T. G. M. Biophys. J. 1993, 65, 1073. (48) Boluk, Y.; van de Ven, T. G. M. PCH, PhysicoChemical Hydrodyn. 1989, 11, 2, 113. (49) van de Ven, T. G. M.; Kelemen, S. J. Colloid Interface Sci. 1996, 181, 118.
Figure 13. Locations of the center of a sphere bridged to a glass surface by high molecular weight PAM at various times (after ref 50). Notice the symmetry in the locations, indicating that the particle is attached to the surface at a fixed point (0,0) A nontethered particle would execute a regular random walk.
example, adsorption isotherms: if c0 is the concentration initially present and c is the concentration in the jet, the difference is the amount adsorbed onto the particles. Alternatively, eq 13 suggests that it is possible to determine molecular weights this way. This method can be used for very low polymer concentrations, typically in the range 10 µg/L. Higher concentrations lead to very short adsorption times τads, resulting in no particle deposition. Occasionally one observes polymer-coated particles stuck to the surface which move to some extent, especially at high fluid velocities when the jet becomes turbulent. When switching off the flow these particles are observed to undergo translational Brownian motion along the surface but centered around a fixed point. Presumably these particles are tethered to the surface by a macromolecular bridge. By analyzing this motion and modeling the connection as an elastic spring, the spring constant of the polymeric link can be determined. We measured the spring constant of a PAM (polyacrylamide) bridge between a latex particle and glass and between a spherical red cell and glass.50 For displacements of about 0.2 µm we found a spring constant of about 1 µN m-1, in agreement with the prediction of polymer theory. An example of the displacement is shown in Figure 13. These examples show that one can determine polymer concentrations, polymer molecular weights, and spring constants of polymers simply by watching colloidal particles! Liposome-Wall Interactions. Liposomes are hollow particles with a shell consisting of a lipid bilayer. Since these particles are deformable, they behave differently from solid particles. Seifert and Lipowski51 predicted that small liposomes, below a critical size acrit, should not deposit on a surface. The reason is that for deposition to occur the liposome must deform so that a sufficient amount of its surface can come into contact with the wall. This is shown schematically in Figure 14. The critical size is given by51
acrit )
x
κb 2 Wadh
(14)
where κb is the bending modulus of the lipid bilayer, typically of order51 10-19 N m-1, Wadh is the adhesion energy, typically of order52 10-5 J m-2, and thus acrit is expected to be around 100 nm. (50) Kamiti, M.; van de Ven, T. G. M. Macromolecules 1996, 29, 1191. (51) Seifert, U.; Lipowsky, R. Phys. Rev. A 1990, 42, 4768. (52) Evans, E.A. Colloids Surf. 1984, 10, 134.
Keeping Pace with Colloids in Motion
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Figure 14. Deposition of liposomes on a glass surface (schematic). Deformed liposomes scatter light which is observed (evanescent wave light scattering (EWLS)). Figure 16. Deposition of particles on a water-air interface in an impinging jet: (1) an approaching particle; (2) a deposited particle. The stagnation point area in this geometry is similar to that encountered on top of a rising bubble (3).
Figure 15. Rate of deposition of liposomes onto glass as a function of liposome size (after ref 53). Extrapolation shows that liposomes smaller than 60 nm do not deposit (indicated by arrow).
We verified these predictions by depositing monodisperse liposomes of various sizes onto a glass surface.53 The liposomes were prepared by extrusion of solutions of lipids (forming multilamellar versicles) through filters of various pore sizes under pressure. The deposition was observed in an impinging jet with evanescent wave light scattering. In essence, liposomes on the surface find themselves in an evanescent wave created by a reflected laser beam and thus scatter light, the intensity of which is measured with a photomultiplier tube. Results are presented in Figure 15. Although we were unable to prepare liposomes below 80 nm, extrapolation of the data (shown by the arrow) clearly shows that liposomes below about 60 nm in size do not deposit, in fair agreement with the prediction of eq 14. Particles at a Water-Air Interface. The impinging jet can be modified to study the deposition of particles on a water-air interface. One simply replaces the top surface (usually glass) with a surface with a small hole. Capillary forces prevent the impinging jet from flowing through the hole, except at very high flow rates. The geometry of such a jet is shown schematically in Figure 16. The stagnation point flow created in such a way is identical to the stagnation point flow encountered on top of a bubble which rises up through a fluid, an important geometry in flotation processes. Our interest is in the flotation of ink particles in slurries of recycled waste papers. This process can be simulated in an impinging jet, as shown in Figure 16. In a stagnation point flow the (initial) deposition rate for small particles is given by54
j0 ) 0.776R1/3D2/3n
(15)
where R is the strength of the flow (cf. eq 11), D is the diffusion constant of the particle, and n is the number concentration. For large particles this equation needs modification.42 (53) Xia, Z.; van de Ven, T. G. M. Langmuir 1992, 8, 2938. (54) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: New York, 1962.
Figure 17. Photograph of a glass of beer showing rising beer bubbles (after ref 56).
When an electrostatic repulsion is acting between particles and the surface, the right hand side of eq 15 (or a modification thereof) must be multiplied by the deposition efficiency Rd. For latex particles depositing on an oleate-coated water-air interface, we find that Rd ) 0, while for latex particles covered by calcium oleate particles, we find Rd = θ, θ being the fractional coverage of latex by calcium oleate,55 implying that when an oleate patch sitting on a latex particle hits the air-water interface, the particle deposits, but when a bare patch hits the interface, no deposition occurs. Equation 15 applies equally to adsorption of polymers48,49 or even molecules. An interesting example is the accumulation of CO2 molecules on a beer bubble, discussed next. (55) Harwot, P.; van de Ven, T. G. M. To appear.
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Rising Beer Bubbles. When a beer bubble (or move precisely, a CO2 bubble suspended in beer) rises in a glass of beer, it collects dissolved CO2 molecules and thus its size is growing in the process. Shafer and Zare56 measured the increase in bubble size and found that dr/dt = 40 µm s-1, independent of the bubble radius r. A photograph of a rising bubble, reproduced from their paper, is shown in Figure 17. It follows from eq 15, which needs a slight modification to allow for adsorption all around the bubble, not just the stagnation point region,56 that the rate of CO2 accumulation is a function of D and n. It follows from the theory57 that the change in bubble size with time is indeed independent of r (as a first approximation) and that no other parameters, besides known constants of nature, enter into the problem. The concentration of CO2 molecules, n, in solution is related to the supersaturation, S, which depends on the pressure under which the beer was bottled. Typically S = 3 when the bottle is opened and (56) Shafer, N. E.; Zare, R. N. Phys. Today 1991, Oct, 48. (57) van de Ven, T. G. M.; Dukhin, S. S. Phys. Today 1992, April, 15 and 112.
van de Ven
S ) 1 for flat beer. Assuming S ) 2, we found from the relation dr/dt ) f(D,S) = 40 µm s-1 that the diffusion constant D of a CO2 molecule corresponds to a size of a few angstro¨ms.57 This result shows that eq 15 applies down to the atomic scale and that the size of a molecule can be found by simply contemplating rising beer bubbles. Concluding Remarks The examples which have been discussed show that the motion of colloidal particles and their interactions with other particles or surfaces is a fascinating topic. Lots can be learned by studying well-characterized systems under well-defined physicochemical and hydrodynamic conditions. Above all, watching colloids is fun. Acknowledgment. I would like to acknowledge all my students, postdoctoral fellows, visiting scientists, and colleagues with whom I have had the pleasure to collaborate. Also the financial support of various agencies and companies is acknowledged, especially the Pulp and Paper Research Institute of Canada (Paprican) and NSERC. LA9604792
