Langmuir 1995,11, 1870-1876
1870
Articles Effect of Flow on the Distribution of Colloidal Particles near Surfaces Studied by Evanescent Wave Light Scattering Marco Polverari and The0 G. M. van de Ven* Paprican and Department of Chemistry, Pulp and Paper Research Centre, McGill University, Montreal, Quebec, Canada H3A 2A7 Received August 16, 1994. I n Final Form: December 22, 1994@ The intensity of light scattered from a suspension of colloidal latex particles near an interface and subjected to flow was measured as a function ofthe penetration depth of an evanescent wave and compared to the results predicted from the mass transport theory, in which the colloidal forces are described by the DLVO theory. From the scattered light intensity profiles, the particle concentration profiles near the interface were calculated. The experiments were done as a function of salt concentration and flow rate. The results were found to be in excellent agreement to predicted results, confirming, for the first time, that particular concentration profiles at interfaces can be measured in flowing dispersions.
Introduction
particles near surfaces. Using a related method, Schumacher and van de Ven13have used evanescent wave light scattering (EWLS)to study latex-glass interactions. Their results were found to be consistent with results expected from the conventional DLVO theory. The wall-jet technique, combined with EWLS, was shown to be a useful tool to measure the deposition rates of liposomes14 and of bare and polymer-coated particles onto s ~ r f a c e s . l ~In- ~the ~ present paper, we describe a n extension of this technique, to nondepositing systems, that involves the manipulation of colloidal particles by the use of hydrodynamic forces. A comparison of the measured particle concentration profiles to the theoretically expected particle concentration profiles, calculated from DLVO theory, is made.
Since its initial development, the DLVO theory of colloidal interactions has provided a solid reference for the rationalization of interactions present between colloidal particles. In recent years experimental evidence has been found for the existence of additional repulsive and attractive forces not accounted for by the DLVO the0ry.l Past studies have used light scattering and particle-counting techniques to measure colloidal forces. Such studies, however, have only provided indirect evidence for the existence of such force^.^^^ Recently, more direct, macroscopic measurements of the interparticle forces involvedbetween colloidal particles have been used to confirm and describe both the attractive (van der Waals) and repulsive (electrostatic)components ofthe DLVO theory. Two such methods, the surface force Theoretical Background apparatus4 and atomic force m i c r o ~ c o p y have , ~ ~ ~ been Evanescent wave, or total internal reflection spectrossuccessfully used to measure surface forces. copy, is a technique which utilizes the properties of a totally More recently, total internal reflection microscopy internally reflected incident beam of light to probe the (TIRM)has been used to make microscopic measurements of colloidal interaction forces. Work by Bike and P r i e ~ e ~ - ~interface of a n optically denser yet transparent medium. l8 When a beam of light, traveling through the denser has shown that TIRM can be employed to monitor the medium, is totally reflected a t the interface between two position of a particle undergoing diffusion near an interface materials of different refractive indices, a small portion and thereby characterize weak potential energy profiles of the beam penetrates into the rarer medium. The between a particle and a surface. Similarly, Brown et penetrating beam is called the evanescent wave. The aZ.1° have recently combined the TIRM and radiation intensity of the evanescent wave is highest a t the interface pressure techniquel1J2to study weak interaction forces of and decays to zero as one moves away from the interface. Auseful characteristic of evanescent waves is that their Abstract published in Advance ACS Abstracts, May 15, 1995. penetration depth a t the interface is a function of the angle (1)Claesson, P. M. Prog. Colloid Polym. Sci. 1987,74,48. (2) Cahill, J.; Cummins, P. G.; Staples, E. J.; Thompson, L. G. Colloids of incidence.13 It is therefore possible to measure the surf. 1986,18,189. distance from an interface a t which a collection of colloidal @
(3)Lips, A.; Duckworth, R. M. J . Chem. Soc., Faraday Trans. I1988, 84,1223. (4)Israelachvili, J. N.; Adams, G. E. J . Chem. Soc., Faraday Trans. I1978,74,975. (5) Ducker, W. A,; Senden, T. J.; Pashley, R. M. Nature 1991,353, 239. (6) Butt, H.-J.; Biophys. J . Biophys. SOC. 1991,60, 777. (7) Prieve, D. C.; Lanni, F.; Luo, F. J . Chem. SOC.,Faraday Discuss. 1987,83,297. (8) Bike, S. G. Ph.D. Thesis, Carnegie Mellon University, 1988. (9) Prieve, D. C.; Frej, N. A. Langmuir 1990,6, 396. (10) Brown, M. A,; Smith, A. L.; Staples, E. J. Langmuir 1989,5, 1319. (11)Ashkin, A. Sei. Am. 1972,226, 63.
0743-7463/95/2411-1870$09.00/0
(12) Ashkin, A. Phys. Rev. Lett. 1970,24,156. (13)Schumacher, G. A,; van de Ven, T. G. M. Langmuir 1991,7, 2028. (14)Xia, Z.;van de Ven, T. G. M. Langmuir 1992,8,2938. (15) Polverari, M.; van de Ven, T. G. M. J . Colloid Interface Sci., in press. (16) Albery, W. J.; Kneebone, G. R.; Foulds, A. W. J . ColloidInterface Sei. 1986,108 (11, 193. (17)Albery, W. J.; Fredlein, R. A,; Kneebone, G. R.; OShea, G. J.; Smith, A. L. Colloids Surf. 1990,44,337. (18)Harrick, N. J . Internal Reflection Spectroscopy; John Wiley and Sons: New York, 1967.
0 1995 American Chemical Society
Effect of Flow on Colloidal Particles
Langmuir, Vol. 11, No. 6, 1995 1871
where c is a constant dependent on the optical properties of the particles and on the optics of the light scattering instrument and n(z)is the distribution of colloidal particles near the interface given, in the absence of flow, by
-1.
i8 l-
E
Figure 1. Illustration of the phenomenon of total internal reflection and the resulting evanescent wave. nl and n2 are the refractive indexes of the rare and dense medium, respectively. 8i is the incident angle and 8, is the critical angle beyond which the evanescent wave occurs. Here P is the stagnation point. t' = t cos Bi, t being the finite thickness of the laser beam. 6 is the pentration depth of the evanescent wave given by eq 2.
particles resides by measuring the intensity of scattered light as a function of this angle. EWLS can thus be used as a tool that can detect objects located a t various distances from the interface. The incident beam of light is totally internally reflected from an interface when the incident angle, Bi, is larger than the critical angle, 8, = sin-'(nz/nl)), where n2 and nl are the refractive indexes of the denser and rarer media, respectively, as shown in Figure 1. For perpendicularly polarized light with a n electric field amplitudeEo,the intensity of the evanescent wave is given by
where I, = Eo2,with E, = 2Eocos O i / ( l - sin28e)1/2, z is the distance from the interface, and the penetration depth, 5, is given by
where I , is the wavelength of the light used (in vacuum). The light scattered by colloidal particles in a n evanescent wave is proportional to the intensity of the evanescent waves for large z and also when the radius of the particle a 50. Here Y is the kinematic viscosity defined as v = VI@,Q being the density of the medium and 7 being the viscosity of the suspending medium, R is the radius of the inlet tube, and Q is the experimentally measured volume of suspension passing through the cell per unit time. For such a flow the masstransfer equation takes the form
where n is the particle concentration normalized by the bulk concentration, no,H = (z - a)/a,a being the particle radius, fi's (i = 1,2,3)are known correction functions for hydrodynamic interactions between particles and surf a c e ~ ,& ~= ~ 4lkT - ~ ~is the total interaction energy (in kT units) consisting of the dispersive van der Waals and repulsive electrostatic energies, k is Boltzmann constant, and T i s the absolute temperature. The Peclet number, Pe, is defined as 2a3a Pe = -
D
(8)
where D is the diffusion coefficient, given by D = kTl 6x7~. (20) Dabros, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983,261, 694.
(21) Brenner, H. Chem. Eng. Sci. 1961,16,242. (22) Goren, S. L.; O"eil1, M. E. Chem. Eng. Sci. 1971,26,325. (23) Bart, E. Chem. Eng. Sci. 1968,23,193. (24) Goldman, A. J.; Cox,R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22,653. (25) Siffert, B.; Metzger, J.-M. Colloids Surf. 1991,53,79.
Polverari and van de Ven
1872 Langmuir, Vol. 11, No. 6, 1995 Under certain limiting c a s e ~ , ~ ~ eq, ~7Ocan be solved analytically. In many cases, however, a numerical solution is required, which includes the effects of both colloidal forces and hydrodynamic interactions. The boundary conditions for eq 7 are
H-CQ, n - 1 H-8,
n-0
The retarded van der Waals attractive energy term for sphere-wall interaction is given by13
4 l+where A is the Hamaker constant, which for the fused quartz-water-polystyrene system is calculated to be 4 x J.26-28The retardation wavelength, A,is taken to be 200 nm.29 The electric double layer interaction energy was calculated from the equation30
(12) where E , is the permittivity of free space, cr is the dielectric constant of the medium, e is the Columbic charge, K - ~is the Debye length, and Y1 and Yz are given by31
Yl = 4 tanh(lyJ4)
(13)
and
8 tanh(vd4)
++
latex sample L-1 L-2 L-3
comp 60% styrene/ 55% styrene/ 100%styrene
latex samde L- 1 L-2 L-3
Table 2. Solution ProDerties particle diam (nm) particle K a (NaCl) TEM" PCS" concn 108/mL 10-4M 10-5M 6.03 2.19 0.69 134 145 3.61 1.15 245 1.40 220 0.83 3.52 2.62 169 160
(9)
where 6 is usually identified with the distance of the primary energy minimum and is taken to be 1.0 x The total energy actingon the particle, 4, is a summation ofthe electric double-layerforces, &, and the van der Waals dispersion forces, #d:
Yz =
Table 1. Latex Composition
9]"" (14)
l + [ l - (2KU (KU 11, 1) tanh2(
For our study the dimensionless surface potential, \ v i of the particle and surface are assumed to be related to the measured 5 potential by the equation I/+ = e(i/kT.
Experimental Section Latex Dispersions. Three latexes were obtained from BASF corporation (Sarnia, Ontario, Canada). The latexes were cleaned by using a mixed bed of anionic and cationic ion-exchange resins. The details of the procedure are presented elsewhere.32 The compositions ofthe latexes are presentedin Table 1. The physical and chemical properties of these latexes were extensively studied in a previous Colloidal suspensions were prepared for each of the latexes in Table 1 by adding a known amount of latex suspension (10% solids) and a known amount of 0.1 M NaCl solution into a 1-L (26) Visser, J. Adv. Colloid Interface Sci. 1972,3,331. (27) Vincent, B. J . Colloid Interface Sci. 1973,42 (2), 270. (28) Osmond,D. W.; Vincent, B.; Waite, F. A. J . Colloid Interface Sci. 1973,42 (21, 262. (29) Takamura, K. Ph.D. Thesis, McGill University, Canada, 1972. (30) Bell, G. M.; Levine, S.;McCartnev, L. N. J.Colloidlnterface Sci. 1970,33,335. (31)Ohshima, H.; Healy, T. W.; White, L. R. J . Colloid Interface Sci. 1982,90 (11,11. (32)Polverari, M.; van de Ven, T. G. M. Colloids Surf, A , 1994,86, 209.
(I
copolymer acrylic acid itachonic acid none
functional surface acid groups so4-2/cooso4-2/coos04C2
Data from ref 32.
volumetric flask and diluting with distilled, deionized, filtered water. The final particle concentrations and solution properties are shown in Table 2. The differences in particle size obtained from TEM and PCS are ascribed to polydispersity. Details can be found in ref 32. All solutions and suspensions were prepared with freshly distilled, deionized water. The water was filtered three times through a 0.22-pm chromatographic filter. All glassware was cleaned in potassium dichromate acid solution and rinsed with filtered, distilled, deionized water. QuartzHemi-cylinder. The fused quartz hemi-cylinder used for this study was purchased from Harrick Scientific Corporation. It had a smoothness of Ad15 and a refractive index of nl= 1.457 at A. = 632.8 nm. Since the surface of the hemi-cylinder was optically smooth, a two-step cleaning procedure, which would not etch the surface, was developed. Step A. The hemi-cylinder (already glued to the jet cell) is initially subjected, for a 20-min period, to ultrasonic waves in a glass cleaning detergent. Following the ultrasonic cleaning, the hemi-cylinder was thoroughly rinsed with water. A second ultrasonication in acetone, followed by a rinsing in distilled, deionized, filtered water was done. The hemi-cylinder and impinging jet cell are then installed into the sample cell assembly (see Figure 2). After installation, the outer surface of the hemi-cylinder is wiped clean by using lense paper wet with methanol and sparged dry with compressed air. Copious amounts of distilled, deionized, filtered water are then flowed through the cell. The jet cell is then aligned and the reference background scattered intensity is measured as a function of incidence angle, 0,. Step B. After each experimental run, the back of the jet cell is removed and the interior of the cell is first flushed with clean water and then cleaned by the use of a cotton swab immersed in methanol. Once clean, the back of the jet is reinstalled and copious amounts of distilled, deionized, filtered water are flowed through the cell. The reference background scattered intensity is again measured as a function of angle. If the measured background scattered intensity is not found to be the same as initially measured in step A, the jet cell is removed from the cell assembly and step A is repeated. InstrumentationandDataAnalysis. (a)Instrument Setup and Alignment. Schematics of the experimental setup and electronics are shown in Figures 2 and 3. The hemi-cylinder was glued to the impinging jet cell. The impinging jet cell was then placed into a modified Brookhaven Instruments BI-2030 cell assembly housing. The impinging jet cell is held in place in the assembly housing by an XYZ-translation table (see Figure 2b), which also rotates with respect to the Z axis. The cell assembly (with the translation table) is located on a goniometer platform. The laser used was a vertically polarized 50-mW HeNe laser (Spectra Physics) with wavelength of 1, = 632.8 nm. The scattered light was perpendicular to the incident light polarization. The experiments were all conducted a t 25 "C. The laser beam was narrowed down by passing it through two pinholes (0.6 mm and 0.3 mm, respectively, in diameter) separated at a distance of 20 cm. An optical filter and shutter were installed in front ofthe photon counter (located in the light culminatinglense system, LCO), so that only light with the same wavelength as that ofthe laser beam was allowed to pass through. A third pinhole (also located within the LCO system) was set to 0.6 mm. This was purposely done so that the viewing area
Effect of Flow on Colloidal Particles
Langmuir, Vol. 11, No. 6,1995 1873 WBrm
1 0 1
I
I
CPUP
. .
h
El/-(**)
Figure 2. (a) Schematics of the experimental set up: (L) 50mW He-Ne laser, (&I quarter wave plate, (Pl)pinhole 1,(P2) pinhole 2, (GI goniometer table and jet assembly, (LCO) scattered light culminating lenses, (PC) photon counter, (C) correlator, (CPU1) master computer, (CPU2) slave computer, (PR) 115-V power regulator, (GSM) goniometer stepper motor, (E)is the alignement eyepiece, and (HV)high-voltage power supply. (B)Schematics of the light scattering cell assembly: (MS)microstepper, (TI X-Y-2 translation table, (LCL) laser light culminating lense, and (J) impinging jet cell.
Figure 3. (a)Schematics of the impinging jet cell system. The suspensions and/or solutions are pressure driven from reservoir R1 or R2 by using the nitrogen cylinder (C) and a pressure valve (PV). The suspensions flows through the split valve (FV) past the connector (CN) and into the jet inlet tube (IT). The suspension impinges onto the hemi-cylinder (H) and exits through the exit tube (SE).SV is a split joint. The scattered light is detected by the photon counter (PC). (b) Geometry of the stagnation point flow cell. A jet of radius R impinges onto the hemi-cylinder surface and creates a stagnation point flow at P. The plates are a distance h apart. C
of the LCO would be larger than the area illuminated by the evanescent wave, ntt'/4 (cf. Figure 1). Such a procedure guarantees that all the photons from the laser beam which are scattered from the particles near the interface are detected by the photon counter. In order to align the center of the LCO with the center of the impinging jet cell, the geometric center of the orifice of the impingingjet cell are brought to the center of a cross scale located in the alignment eyepiece of the LCO with the aid of the translation table. The alignment is carried out at angles of go", 180", and 270". Once alignment a t these angles has been made, the impingingjet cell is oriented to 0"(the angle facing the laser). If the system is well aligned, the laser beam should strike the center of the jet orifice. As a precautionary step, to verify that the impinging jet cell is well aligned, clean distilled, deionized,filtered water is flowed through the cell. The critical angle, e,, of the system is measured by changing the incident angle until the point a t which the incident laser beam is no longer totally reflected. If the system is well-aligned this angle should correspond to the angle 8, = sin-Yndn1). The impinging jet cell can be rotated, with the use of the microstepper, with an accuracy of 0.15" and the photon counter with an accuracy of 0.5". The critical angle at which total internal reflection was found to disappear was 66.3 f 0.5" for the fused quartz hemi-cylinder. This angle is in good agreement with the theoretical value of 66.2". Figure 3 illustrates the flow schematics for the impinging jet system. The solutions are pressure driven from reservoirs R1 and R2 by using the nitrogen cylinder, C. The solution flows into the impinging jet cell and onto the quartz interface. The scattered light passes through the light-culminatinglense system (Figure 2) and is detected by the photon counter (PC). CPUl is a mastercomputer (XT) which controls both the microstepper and the photon counter tube, as well as CPU2 (286AT). For our cell geometry, h/R = 1.7, h being the distance between the confining plates and R the radius of the inlet tube. (b) Experimental h.ocedures and Data Analysis. Experiments were conducted as a function of solution salt concen-
1 Re 9 715
Re 9 107
time (sec) Figure 4. Schematic of typical experimental results obtained at a fmed scattering angle and salt concentration for various Reynolds numbers. lpis the scattered light intensity and I b is the background scattered light intensity when no particles are present in solution. At points A and C only water is flowing onto the interface, at point B the suspension is being flowed onto the quartz surface.
tration and Reynolds number (flow intensity). The NaCl salt concentrations used were 1 x and 1 x M NaC1. At these low salt concentrations the particles were found to be nondepositing. Somewhat higher salt concentrations resulted in slight deposition, making interpretation of the results more difficult. The Reynolds numbers were varied between 0 and 750. The experiments consisted of, a t first,flowing only clean water through the impinging jet cell at a given flow rate. The background scattered intensity as a function of incident angle, @, was then measured. The suspension is then switched to the reservoir containing the suspension and the suspension is allowed to flow through the impinging jet cell at the same flow rate. The scattered intensity is again measured as a function of the incident angle, Oi. This procedure was repeated for each latex a t both salt concentrations and at Reynolds numbers of 0,107,214,358, and 715. Figure 4 illustrates the typical results obtained for a latex suspension a t a fmed incident angle. At first only water is flowed through the impinging jet cell. The background scattered
Polverari and van de Ven
1874 Langmuir, Vol. 11, No. 6, 1995
l4 12
12
t
10 8 6 4 2
0 4
60
70 80 Bi (degrees)
Figure 5. Intensity of scattered light as a function of incident
and (b) angle, Oi, for latex L-1 at concentrations of (a) M NaC1. The symbols are experimental data points and the solid lines are the fitted data using eqs 3 and 7. Experimental 0.0775, and Peclet numbers: (0)0, (v)0.0127, (I)' 0.0360, (O), (W) 0.219. Solid line with no data points is for n(z) = 1.
70 80 8 , (degrees)
60
90
90
Figure 6. Intensity of scattered light as a function of incident
and (b) angle, Oi,for latex L-2 at concentrations of (a) M NaC1. The symbols are experimental data points and the solid lines are the fitted data using eqs 3 and 7. Experimental Peclet numbers: (0)0, (v)0.104, (I)' 0.294, (0)0.632, and (W) 1.79. Solid line with no data points is for n(z) = 1.
intensity, zb, is measured. At point B, the suspension is flowed through the impinging jet cell. The scattered intensity is seen to increase and to finally reach a constant value, Zp. The time scale to read the plateau is typically in the range of 2-3 min, which is the required time for the latex dispersion to reach the collector. The higher the flow rate, the larger the scattered intensity from the suspension is. The experimental scattered intensity, Is,is calculated by subtracting the measured scattered intensity when the suspension is flowingthrough the impinging jet cell,Zp,from the blank intensitywhen only water flowsthrough the impinging jet cell, Zb. At point C the flow is switched back to water only. The intensity is found to decrease back to its initial value,I b , indicatingno particles are deposited on the glass surface.
14
Results and Discussion
g Potentials. The value of the [potentials for the fused quartz hemi-cylinder, at and M NaC1, was obtained from literature13a3 as being about -100 mV at both salt concentrations. The [potentials of the particles are fitted values. Scattered Light Intensity Profiles. Plots of the scattered light intensity profiles, IS,as a function of the incident angles, Peclet number, and salt concentration are shown in Figures 5-7 for latexes L-1, L-2, and L-3, respectively. The units for the scattered light intensity are s-l, since the measured intensity is in photon counts per second. The solid lines are calculated from integration of eq 3 by using c and the 5 potential of the particles as the only adjustable parameters, with n(z)calculated from numerical integration of eq 7. The values of c and [ were varied so as to minimize the residuals between the theoretical and the experimental data. The same value of c was found when fittingthe experimental intensity profiles for a given latex. The fitted parameters are shown in Table 3. The values of c are related to the scattering intensity per particle. Correcting the values in Table 3 for the differences in particle concentration (cf. Table 2), it follows that the ratio of relative scattering power of the particles is L-l:L-2:L-3 = 1:15:2, while from their sizes and (33) Jednacak, J.;Pravdic, V.; Haller, D. J. J.Colloid Interface Sci. 1974, 49 (11,16.
ut 60
70 80 Bi (degrees)
90
Figure 7. Intensity of scattered light as a function of incident and (b) angle, Oi,for latex L-3 at concentrations of (a) M NaCl. The symbols are experimental data points and the solid lines are the fitted data using eqs 3 and 7. Experimental Peclet numbers: (O),0, (v)0.0235 (v)0.0665, (0)0.143, and (W) 0.404. Solid line with no data points is for n(z) = 1. Table 3. Fitted Parameters
5 potential latex, mV
C
latex sample L- 1 L-2 L-3
10-4~ NaCl
162 554 180
10-5 M
NaCl
10-4 M NaCl
162 554 180
-110 -110 -110
-130 - 135 - 140
i o - 5 ~
NaCl
assuming I , a6, one expects 1:23:2.5. This is in fair agreement with theory, consideringthat the latex particles are too large to act as Rayleigh scatterers. The fitted [-potentials for the latexes at the given salt concentrations are in the range typically expected for polystyrene and carboxylated latexes at low salt concen0~
Effect of Flow on Colloidal Particles 2.5
-
2.0
1
Langmuir, Vol. 11, No. 6, 1995 1875 (a)-
2.5 2.0
-i x
''
1 5
C
4
-.o
10
0
0.5
-.I
L
m
e
0.0
V
2.5
2.0 1.5 1 .o
0.5
1 .o
"...
n n
0.5
0
2
4
6
8
10 1 2
14
16
18 20
normalized d i s t a n c e t o wall
0.0 0
2
4
6
0
10
12
14
18
1 I
20
normalized distance t o wall Figure 8. Normalized particle concentration as a function of distance from the interface, in particle radii, for latex L-1 at and (b) M NaC1. The solid lines concentrations of (a) are calculated by eq 7. Experimental Peclet numbers: (i)0, (ii) 0.0127, (iii) 0.0360, (iv) 0.0775, and (v) 0.219.
Figure 9. Normalized particle concentration as a function of distance from the interface, in particle radii, for latex L-2 at and (b) M NaC1. The solid lines concentrations of (a) are calculated by eq 7. Experimental Peclet numbers: (i)0, (ii)
0.109, (iii) 0.294, (iv) 0.632, and (v) 1.79.
2.0
t r a t i o n ~ . However, ~~ such results may be somewhat fortuitous since the calculated light scattering profiles are found to be not very sensitive to the values of [-potential.13 In contrast, the value of the dimensionless double layer, Ka, was found to profoundly influence the calculated light scattering profiles. Any small change in the value of K a was found to have a large influence on the intensity profiles. To accurately determine the value of Ka, the electrolyte concentration in solution was determined by potentiometic titration of the C1- ion with Ag+.35 Particle Concentration Profiles. Figures 8- 10 show the particle concentration profiles near the wall, n(z), calculated from eq 7 by using the best-fit parameters from EWLS data for latexes L-1, L-2, and L-3, respectively. The particle concentration profiles are plotted as afmction of Peclet number and salt concentration. The particle concentration profiles, for the three latexes, clearly indicate the effect which salt concentration, flow rate, and particle size have on the particle concentration profiles. Inspection of eq 7 shows that for low Peclet numbers the solution can be expressed as n(z) = n,(z) Penl(z), with no(z)given by eq 4 and n(z) the first-order correction due to flow. Hence one expects the deivations from no to be of order Pe. It can be seen from Figures 8- 10 that the actual deviations are much larger. This is, however, a consequence of the length scale used in the definition of Pe (particle radius). The strength ofthe flow is a function of the distance to the wall (cf. eq 5) and thus a more realistic length scale is the average particle-wall distance L , at which deviations in n(z) occur. A more representative Peclet number can be expressed as
+
(34) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1981. (35)Harris, D. C . Quantitative Chemical Analysis; Freeman and Co.: New York, 1982.
x
.d
z
0
pl
0
.d
-.I
L.
m
P
2.5
2.0 1.5 1 .o
0.5 0.0 0
2
4
6
B
10 1 2
14
16 1 8 2 0
normalized distance t o wall
Figure 10. Normalized particle concentration as a function of
distance from the interface, in particle radii, for latex L-3 at and (b) M NaC1. The solid lines concentrationsof (a) are calculated by eq 7. Experimental Peclet numbers: (i)0, (ii) 0.0235, (iii) 0.0665, (iv) 0.143, and (v) 0.404.
Pe, = ~ ~ L ~ I D
(15)
Since L >>a,PeL >> Pe and for low salt concentrations and high flow rates PeL >> 1, and thus deviations from n,(z) are expected to be large. An increase in the strength of the flow or in latex particle size shifts the particle concentration profile nearer to the wall. An increase in salt concentration has the same effect but results in a smaller particle concentration near to the wall. These results a r e in good agreement with both the DLVO theory and the mass transport theoryeZ0 The Peclet number characterizes the convective effects relative to
Polverari and van de Ven
1876 Langmuir, Vol. 11, No. 6, 1995 the diffusion effects in mass transport. Increasing the particle size, a, or the strength of the flow, a,leads to an increase in the Peclet number (eq 8). This implies that mass transport, relative to diffusion, of the particles to the surface also increases. The net effect is that there is a resulting increase of the particle concentration near the surface and a shift of the particle concentration profile nearer to the wall. Similarly, a change in salt concentration has an effect on the colloidal interaction term, &, in eq 7. Decreasing the salt concentration decreases the double-layer thickness (increases K a ) of the electric double layer around the particles and compresses the double layer. Furthermore, as seen in Table 3, an increase in
