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The adhesion of colloidal polystyrene particles to cellophane as a function of pH and ionic strength
The Adhesion of Colloidal Polystyrene Particles to Cellophane as a Function of pH and Ionic Strength z J. V I S S E R Unilever Research, Vlaardingen, The Netherlands
Received June 20, 1975; accepted January 12, 1976 A method for the determination of the force of adhesion of colloidal particles adhering to a planar surface in aqueous solutions has been used to study the influence of pH and ionic strength on the adhesion of polystyrene particles to a cellulose film as substrate. It is shown that the adhesion of these particles can be described quantitatively in terms of Van der Waals forces of attraction and electrostatic forces of repulsion. The adhesive force is maximum at the iso-electric point of the adherents where the adhesion is determined by Van der Waals forces only. The electrostatic double-layer interaction forces operating away from the iso-electric point are determined exclusively by the potential of the flat surface (cellophane). 2. MATERIALS AND APPARATUS
1. INTRODUCTION
2.1 Materials
The force of adhesion between solid particles and a fiat surface can be determined directly, both in vacuo and in aqueous environment, by means of a centrifuge (1). The centrifugal method, however, can be applied only to highdensity (e.g., gold) particles of a diameter larger than 1 ~m. Therefore, a new technique has been developed to overcome these problems (2). This technique involves a liquid flow between concentric cylinders and analysis of the liquid flow close to the surface. A calibrating experiment using the centrifuge technique enables the evaluation of removal measurements of colloidal, low-density particles, adhering to the outer surface of the rotating inner cylinder, in terms of forces of adhesion. We applied this technique to a well-defined system, i.e., monodispersed polystyrene particles of a known particle radius distribution (average radius 250 nm) and surface characteristics, adhering to a cellulose film (cellophane) previously used (1, 2).
The polystyrene particles had been prepared by Dr. Nicholls of Unilever Research, Port Sunlight by emulsion polymerization with sodium laurate as emulsifier and hydrogen peroxide as initiator. According to electron micrographs, the particles were perfectly spherical with a modal radius of 260 nm, corresponding to the maximum in the frequency distribution (Fig. la). The log-normal radius distribution (Fig. lb) gives an average radius of 250 nm corresponding to the 50% level, which value was used in the interpretation of the measurements. The particles were dialyzed for 6 weeks at p H 9 and stored in a refrigerator. The cellophane was a regenerated cellulose from AKZO Arnhem, the Netherlands, and was used after rinsing with distilled water, hydrogen chloride (10 m o l e / m 3) and distilled water at 70°C. The rinsing procedure was adopted to remove traces of glycerol and other contaminants. The water used throughout the experiments had been distilled twice, once over permanganate.
1 Presented at the 49th National Colloid Symposium, Potsdam, New York, June 16 18, 1975. The text of this paper is a part of the author's Ph.D. thesis (2).
664 Journal of Colloid and Interface Science, Vol. 55, No. 3, J u n e 1976
C o p y r i g h t ~ 1976 by Academic Press, Inc. All rights of reproduction in a n y form reserved.
ADHESION OF POLYSTYRENE PARTICLES
stainless steel to prevent corrosion by acid or alkaline solutions.
% particles ~< given diameter
/o~O_O~
100 8o
oo
/
/
3. EXPERIMENTAL PROCEDURES
3.1 Removal Experiments
40
20 _-~~5~4~ 0 3~./b°~'
a i I 5 555 615 675 Particle diameter/nm
Particte diameterlnm 800 600 f 400
.-.--o
o'°'°"
5- ' o ' ° ~ °
2OOl-
I I i
I
,
t
,
665
t
,
J
b ,
,
I
I
I
I
1 2 5 10 30 50 70 90 98 % particles ~< given diameter
FIG. 1. Particle radius distribution of polystyrene particles (PS Latex 65251). A. Cumulative radius distribution; B. Log-normal radius distribution.
All chemicals used were of analytical grade or had been purified to the highest possible degree, e.g., by recrystallization.
2.2 Apparatus The apparatus was similar to that used in our previous study (1). It consisted of a cylinder (length 4.5 cm, diameter 4.6 cm) of polymethylmethacrylate (PMMA) or anodized aluminum. A wet cellophane sheet (6.5 X 16 cm) was wrapped on the cylinder surface and tightened by two conical lids (Fig. 2). The previous apparatus used for removal of carbon black particles (1) had a gap width of maximum 0.2 cm. The present apparatus differs in that the inner diameter (8.40 cm) of the outer cylinder results in a gap width of 1.90 cm. Further, the inner cylinder was driven by a more powerful motor allowing of a speed variable between 2250 and 7700 rpm. In combination with the larger gap width, a wall shear stress much larger than that obtained with the previously described machine could be attained. A water jacket of the outer cylinder served to maintain the room temperature during the experiments. The inside of the whole apparatus was of
The polystyrene particles were deposited onto the cellophane by rotating the cylinder at 265 rpm for 30 min in a 250-ml glass beaker containing a 0.005% dispersion of the particles in distilled water. After the cylinder had been removed from the dispersion and dried in air, the number of particles deposited on the surface was determined. Since the size of the particles was comparable with the wavelength of visible light, the particles were detected by their ability to scatter light from a microscope lamp focussed on the area of the cylinder surface under observation. As a result, the particles were visible as light spots on a greyish or black background depending on the type of cylinder used. With the use of a grid fitted in the ocular of the microscope, the number of particles per unit surface area was determined by counting. An overall magnification of 400 was used throughout the measurements. Immediately after the initial particle count, the cylinder was immersed in the apparatus filled with a solution of known pH and ionic strength. After adaptation for 30 min, the cellophane cylinder was rotated at 2250 rpm for 5 min and the number of remaining particles was determined. Then, the rotational speed was repeatedly increased until just more than 50% of the particles were removed. The exact speed at 500-/0particle removal was determined by interpolation.
VIA of aluminium) PMMA or aluminium )
FIG. 2. Inner cylinder with cellophane coating. Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
666
J. VISSER zeta x)tentiatlmV
-60
•
/o :~ o O o" / / ~ o ' ~ ' - ' ~ Io , , ~ p f o
-~ -20C .,,2C
~-
~o / ~ f at
d,o oH FIG. 3. Zeta potential of polystyrene particles as a function of p H and ionic strength (25°C; 1:1 electrolyte). [] = 1 mole/m3; A = 10 mole/m3; O = 100 m o l e / m 3.
For the calculation of the removal percentage, the number of particles adhering to the surface of the inner cylinder before and after a removal experiment was determined. The necessity to examine the same areas before and after the removal was realized by aligning a mark on the rod with markings on the cylinder lid and by using markings on the microscope table to which the cylinder support was fixed. In this way the particles on eight different areas of the cylinder surface, appearing in 9 (out of 25) squares of the ocular grid, were counted (about 600-800 particles per experiment).
3.2 Electrophoretic Mobility Measurements Electrophoretic mobility measurements on polystyrene and cellophane particles were zeta ~otential/mV
D o
o
2(; 1(~ 0
FIG. 4. Zeta potential of cellophane particles as a function of p H and ionic strength (25°C; 1:1 electrolyte). [] = 1 mole/m3; /k = 10 mole/m~; © = 100 m o l e / m 3. Journal of Colloid and Interface Science, Vol. 55, No. 3, J u n e 1976
made at concentrations of 100, 10, and 1 mole/m ~ electrolyte as a function of pH in mixtures consisting of sodium hydroxide or hydrogen chloride at 25°C, that is at the same concentrations and pH's as in the removal experiments. Cellophane particles were obtained after freezing wet cellophane (only available as film) with solid carbon dioxide and grinding in a porcelain mortar. All experiments were performed in a cylindrical cell which was thoroughly flushed with distilled water. Ten readings were made in both directions at the stationary level of the cell. The results are presented in Figs. 3 and 4, after transformation of the data in zeta potentials using the Tables of Loeb, Wiersema, and Overbeek (3) as computerized by Peterson (4) or using, in the case of cellophane, Von Smoluchowski's equation (large KH, see Section 5.3.1), resulting at 25°C, in (5) : zeta potential = 12.83 X[mobility in (~m sec-1)/(V cm-i)]mV [-1] 4. E V A L U A T I O N OF T H E R E M O V A L M E A S U R E M E N T S I N T E R M S OF F O R C E S OF A D H E S I O N
According to Goldman, Cox, and Brenner (6) and O'Neill (7) the tangential force Fn exerted on a spherical particle of radius R in contact with a plane wall in slow linear shear flow (Fig. 5) is given by the modified Stokes' law : Fu = 1.7005 X 6 ~r~Rvx=R [-2] where 1.7005 is a correction factor due to the configuration of a sphere in contact with a plane; ~ is the dynamic viscosity of the liquid, and v,=R the velocity at a distance x = R from the wall. [ I
]
~
!
.>,,
/
Vx=R
vc *F H
~vx FIG. 5. Velocity profile at the surface of the inner cylinder.
ADHESION OF POLYSTYRENE PARTICLES Since by definition, the wall shear stress r is given by r = n(dv/dx)
[33
one obtains, after integration, for a laminar boundary layer, i.e., a linear velocity gradient with v = 0 on the surface: =
667
% Rernova[ 0: ~ 20 -
A rotating outer cylinder v rotating inner cylinder
~ v
8O
[43
Substitution in Eq. [2] gives
RiR9 TM
(Ro_Ri)1,'4
F H = 32R2r.
1-53
In other words, the removal of spherical particles from a flat surface is determined by the magnitude of the wall shear stress r, provided there is a relationship between the force Fzr exerted on the particles by the flow of the liquid and the subsequent removal. For the flow between concentric cylinders, semiempirical relations between flow conditions and wall shear stress exist. These equations are :
FIG. 6. Removal of carbon black particles from cellophane as a function of wall shear stress (r). A = rotating outer cylinder; V = rotating inner cylinder. (3) For a rotating outer cylinder and a stationary inner cylinder (9) : r = lpw~Ro 2 X 0.0013{0.04 + (Ri/Ro)tO}.
[113
Application of Eqs. [9] and [11] to the removal of carbon-black particles from cellophane (2) (Fig. 6) confirms the statement made above that the removal of submicron (1) For a rotating cylinder in a concentriparticles from a flat surface is determined by cal outer cylinder at rest, and a large gap width the wall shear stress r. (8) Since the removal mechanism is unknown, it is not possible to relate Fn to the force of r = 1.6~ X lO-2pv°'2%l'TSRi TM X (R0 -- R0 °'°5 1-6] adhesion FA of the adhering particles on theoretical grounds. As reported before (1), a califor bration experiment is necessary to establish this relationship. I t was found that Fu is 3 X i04 < R e < 107 and numerically identical to FA: 0.1 < ( e o - n 0 / e , < 3 1-7] where the Reynolds number Re is defined by Re = w R ~ ( R o - - R ~ ) / v
[8]
whereas v = kinematic viscosity of the solution, w = angular frequency of the cylinder and p = density of the solution. (2) For the same configuration but with a small gap width (9) : r = 0.23pv½wlRiRol(Ro - - R¢) - l
[-9]
for Reynolds numbers 108 < Re < 104.
[10]
FH = FA.
[-12]
In other words, the force of adhesion of colloidal particles Fa can be directly obtained from a removal experiment using concentric, rotating cylinders. The adhesive force can be related to the angular frequency w by combining Eqs. [5-], I-6], and [-12] for a cylinder rotating in an outer cylinder at a large gap width as used in the measurements reported here. As a result, one obtains after substituting the appropriate constants of the apparatus and the physical quantities at 25°C: Fa = 1.21 X 10-2R% ''Ts.
[13]
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
668
J. VISSER % Removal 100 80
//
60 4O 20 ,
0
2,000
4,6O0 6.0OO Speed of r~ationl(rev.lmin)
FIG. 7. Removal of polystyrene particles (R~ = 260 nm) from cellophane as a function of speed of rotation (25°C; 1:1 electrolyte). A = 1 mole/m~, pH 5.5; C) = 10 mole/m3, pH 6-7. 5. RESULTS AND INTERPRETATION
5.1 The Role of Particle Size Distribution on the Degree of Removal Since colloidal particles are never the same size, but always show a radius distribution, this distribution will also be reflected in the removal results. In studying the removal as a function of speed of rotation, this removal will therefore be a gradual function rather than a step function. The latter situation would be the case for particles of the same size and all adhering with the same force of adhesion. As a consequence, the hydrodynamic force corresponding to 50% removal is a measure of the adhesive force of a particle with a radius equal to the average radius calculated from the particle radius distribution. Because the radius distribution of the carbon black particles used in our previous study (1) was not known very accurately, we decided to
study the removal of the more defined system of polystyrene particles which have a narrow size distribution (Fig. 1) and an average diameter of 500 nm. As a consequence, the removal of these particles should be close to a step function of the speed of rotation. This is indeed the case (Fig. 7). Since the particle radius distribution of these particles is known, it should be possible to relate the removal function to this distribution. An indication of this relation was found by plotting the removal as a function of the speed of rotation on log-probability paper (Fig. 8). A linear relationship was obtained similar to that for the particle radius distribution (Fig. lb). If one assumes that during the removal process the bigger particles come off first followed by the smaller ones at a higher speed of rotation, as was shown in the removal studies of carbon black particles (2), a removal percentage of p % corresponds to the removal of all the particles of a diameter corresponding to (100 -- P)~o of the particles. In this way, it was possible to relate the removal results to the given particle radius distribution. The result is given in Fig. 9, where F n has been calculated with Eqs. [-12] and [-13]. I t appears that Fu and consequently Fa, the force of adhesion of the polystyrene particles, is, under the conditions studied, proportional to the particle radius; the proportionality constant is negative. This is at first sight in contradiction with the theoretically (10) and Particle diameter I nm
6oc
\~
Speed of rotationl(reWmin)
7,00oF 30(
5,~ I
--~--"°'-'-~ ~
2.000[
10(
"°°°'i~ ~,b ~o~0'~o'Tb' ~o' ~ ' (%) removal
FIG. 8. Removal of polystyrene particles as a function of speed of rotation (log-normal paper). /k = 1 mole/ m3, pH 5.5; © = 10 mole/m3, pH 6-7. Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
i
i
2O
4O
I
6O 80 FH I(10-6 dyn)
FIG. 9. Relation between particle diameter and hydrodynamic force (Fn). /k = 1 mole/m3; O = 10 mole/m3.
ADHESION OF POLYSTYRENE PARTICLES experimentally (11) observed proportionality between the force of adhesion of a spherical particle to a flat plate and R with a positive proportionality constant. For this geometry (Fig. 10a), as shown elsewhere (2) Fa
= FVdW
-
-
Frepulsion = ( A R ) / ( 6 H 2) -- ce,
[-14]
where A is the Hamaker constant, H is the shortest distance between the adherents, and c is a constant depending on p H and ionic strength. For the more realistic situation, taking into account surface roughness and nonsphericity of the particles, the adhesion can be described by: FA = FVdW -- FR = ( A R ) / ( 6 H ~) + (~rpc2A)/(6~-H 3) -- cR
[15]
where the second attractive term describes the interaction across the intimate contact area of radius Pe (Fig. 10b). When the contribution of this contact area is dominating, Eq. [15] shows that the adhesion will be proportional to R as observed above, the proportionality constant being negative. As a result the adhesion of polystyrene particles to cellophane can be described by: Fa = (~rm2A)/(67rHO -- cR.
[-16]
The occurrence of an extended contact area in polystyrene particle adhesion to a flat substrate has been demonstrated by Dahneke (12) from electron micrographs. The --cR term in Eq. [-16] describes the influence of the electrostatic double-layer interaction on the adhesion. In view of its particle radius dependency, its magnitude is apparently determined by the double-layer properties outside the contact area, whereas the interaction over the contact area being radius independent, is determined by Van der Waals forces only.
669
ideal
a
41"I
b
ZO¢
Fro. 10. Interaction between a sphere of radius R and a flat surface at a distance H. A. Ideal system; B. system with extended contact area of diameter 2pc. 5.2 The Contribution of Van der Waals Forces to the Adhesion of Polystyrene Particles to Cellophane
As has been shown in the previous section, the contribution of the Van der Waals force to the adhesion (since we are dealing with systems in contact, only nonretarded Van der Waals forces are involved) can be identified with the radius-independent term in the equation describing the adhesion of polystyrene particles to cellophane as obtained from its particle radius contribution and the particle removal as a function of rotational speed. I t can be obtained by extrapolating the graphs in Fig. 9, to R = 0, giving a radius-independent contribution to the adhesion of 78 X 10-6 dyn, being independent of ionic strength. Taking as an estimate for the Hamaker constant of the system 3.5 X 10-14 erg, the correctness of this procedure can be investigated. The condition (A R ) / ( 6 H ~) << (r0 c2A)/(6~r//~)
= 7 8 X 1 0 - 6dyn
[17]
is fulfilled for H = 2.0 nm and pc = 103 nm. Other possibilities do exist but in respect of surface hydration, deformability of the adherents and nonideality of the system such as surface roughness, a value of H = 2.0 nm is quite reasonable. I t is equal to the limiting thickness of the aqueous core of a black film (13) in excess electrolyte, that is to say under conditions where only Van der Waals forces are operating, and somewhat higher than the interlayer spacings of swollen clay platelets as observed by X-ray analysis (14, 15). The value of pc = 103 nm is not unrealistic as can be seen from Fig. 10b, where the corresponding Journal of Colloid and Interface Science, Vol. 55, No, 3, June 1976
670
j. VISSER FH/(IO-6 dyn)
70 60 50 40 30 i
styrene particles indeed show this type of behavior : no longer a gradual increase in removal with increasing speed of rotation was noticed, but a sudden, almost complete removal above a certain speed. Consequently, the accuracy of the measurements in the vicinity of the zero point of charge is low and the results will therefore be omitted in the interpretation (see also Fig. 11).
20
5.3 The Electrostatic Repulsion Forces in the Adhesion of Polystyrene Particles to Cellophane o ~' ~, ~ ~ ~b 1~ pH 5.3.1 Theory. In water, electrostatic doublelayer forces are present in addition to Van der Fla. 11. T h e h y d r o d y n a m i c force (FH) corresponding Waals forces. These forces arise from the to 50070 removal of polystyrene from cellophane as a function of p H (25°C; 1:1 electrolyte). © = 100 charging of materials in the case of water and mole/m3; /~ = 10 mole/m3; [] = 1 m o l e / m 3. they can be influenced by introducing electrolyte into the liquid phase. As the sign of the geometry has been drawn to scale, and is of charge is negative for most materials, mutual the same magnitude (Ripe = 2.5) as Dahneke interpenetration of the corresponding double (12) has shown by electron microscopy for layers usually results in repulsion. polystyrene particles adhering to a chromium From the DLVO-theory of double-layer insurface (R/p~ = 2). Therefore, in the evaluateraction Edeveloped by Deryagin and Landau tion of the adhesion measurements on poly(16) and Verwey and Overbeek (17)-] it is posstyrene in the following sections, the value of sible to calculate the electrostatic double layer an attractive force of 78 X 10-6 dyn and a of repulsion arising from the interpenetration separation distance of H = 2.0 nm will be taken as a basis. The latter value is somewhat of two double layers. There are two conditions higher than that for the system carbon black/ under which this repulsion can be calculated, cellophane (--1.6 nm) (1) which may be due viz: The potential of the interacting systems to polymer chains sticking out from the poly- remains constant during the interaction, or the charge of the interacting systems remains mer surface. constant. A further consequence of the finding that (i) Constant potential interaction. A relathe adhesion of polystyrene particles to cellotion expressing the interaction energy VR ~ bephane can be described by Eq. [16~ is that tween the electrical double layers surrounding for c = 0 when the electrostatic repulsion any two similar or dissimilar colloidal particles forces are zero or can be neglected, the force of adhesion will be independent of the particle has been derived by Hogg, Healy, and radius for the given system. This means that Fuerstenau (18) under the condition of conthe removal will be very close to a step func- stant (low) potential. This equation reads: tion of the speed of rotation, i.e., there will be ~R1R2(~o.12 + ~o,~2) no removal below a certain speed of rotation VR~ = 4 (R1 + R2) and a complete removal of all the adhering particles when exceeding this speed. [ 2~b0,_~b0.2 In 1 -t- exp(--KH) The condition c = 0 occurs when the adX L(~o,x~ --[-~o,J2) 1 -- exp(--KtI) herents are in a situation corresponding to their zero point of charge. In experiments q-In {1 -- exp(--2K//)} 1 E18~ performed under this condition, the polyIC
_.1
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
ADHESION OF POLYSTYRENE PARTICLES where R~, R2 are the radii of the interacting particles 1 and 2 of potential ~//0,1 and ~0,2 the potential ~0.1 at infinite separation. H is the distance of minimum approach, e the dielectric constant of the medium, and r the reciprocal double-layer thickness ~, defined by:
K2= (47re~)/(ekr).Zzfln~o.
[19]
Here, k is the Boltzmann constant, T the absolute temperature, zl the valency of ion i, ni0 the bulk concentration of ion i and e the electronic charge. Equation [18] holds exactly for ~0,1 and/or 60.2 of less than 25 mV and for such solution conditions that the double-layer thickness ~ is small compared with the particle size: a
=
r-' << R1, R2.
~o,~
=
,/'o :
FR+(sphere -- plate) eR~02K•exp (-- KH) [23] 1 + exp(--~H)
(ii) Constant charge interaction. The electrostatic double-layer repulsion energy Va" arising from the interaction of two double layers surrounding a colloidal particle of radius R1 and a colloidal particle of radius R~ under condition of constant charge has been derived by Wiese and Healy (20). Here, ,R,R2@0.~ ~ + ~o/)
VR*= VR + -2 (RI +
[20]
For ~h0,1 and if0.2 of less than 50-60 mV, Eq. [18] is a good approximation. Since Lange has shown (19) that the electrostatic double-layer interaction energy VR between a sphere and a flat plate equals twice the interaction energy between two identical spheres,
VR (sphere - plate) = 2VR(sphere -- sphere)
=
671
R2)
X In {1 -- exp(--2KH)}
where VR¢ is given by Eq. [18-]. For the sphere/plate geometry, the repulsion force due to double-layer interaction at constant charge can be obtained as above. The result is : FR'(sphere -- plate) K
= ½~R(¢o/+
~o2)
[213
the force of repulsion FR due to double-layer interaction for the sphere/plate geometry, the geometry we are interested in, easily can be derived from Eq. [18-]. By putting R1 = R~ = R, the particle radius, and after differentiation with respect to H: FR~(sphere - plate)
1-24]
exp(--KH)
1 -- exp(--2KH)
)
+
.~'o,22
For materials with identical potentials ~o,x = ¢,o.2 = ~o:
FR*(sphere - plate) ~R~dK"exp (--KH) [26]
• 2d VR(sphere - sphere)
dH Kexp(--KH) = ½,R(¢~0/+
x
¢0/)
F 2~o,1~'o,2
1 -- exp(--2KH)
+ +oC+
exp(--KH)l.[22-1
For materials having identical potentials ~0,1
1 -- exp (--KH) If one assumes that ~b0 in Eqs. [22-]-[26] equals the electrokinetic zeta potential of the individual particles, the electrostatic repulsion force FR can be calculated easily, provided H is known. According to Frens (21), there are some indications of a close resemblance to ¢J0 and the zeta potential, but a quantitative theory is not available. Journal of Colloid and Interface Science,
VoL55, No. 3, June 1976
672
j. VISSER
lower potentials is determined by the particle with the larger radius. The latter result sug8 gests that in the case of a sphere/plate inter6 action, the interaction will be controlled by the properties of the plate. On this basis, the adhesion measurements on 2 polystyrene particles adhering to cellophane apparently should be interpreted, considering 0 1.0 2.0 3.0 ~H only the potential of the cellophane. As regards the use of the equations of conFIG. 12. Electro static force of repulsion (FR) as a function of KH. O -~ FR (constant charge) = eR~2~ stant charge or those of constant potential, the •exp(--~H)['l -- exp(--KH)]-l; [] = FR (constant following can be said. For mercury droplets, potential) = eR6*K.exp(-~H)~l + exp(--KH)]-l; ~, the coalescence measurements of Usui and = F~ (experimental) = ½eR62.coth (KH). Yamasaki (22) could be interpreted using the Recently, Usui and Yamasaki (22) have constant potential equations. In the case of oxides and materials (such as shown that their coalescence experiments with oxidized carbon black, polystyrene, and cellomercury droplets could be interpreted quantiphane) of which the charge of the surface is tatively in terms of the DLVO theory, using not based upon the equilibrium of potentialthe potential of the outer Helmholtz plane determining ions but upon the dissociation of instead of the wall potential. Therefore, we groups such as - C O O H and --OH attached will interpret our measurements assuming that to the surface, not the potential but the sur~0 equais the zeta potential. When using Eqs. [18-] and [24], at small face charge should be considered as a constant values of ~H, it can be easily demonstrated that to a first approximation during the interaction for flat plates of dissimilar potential when (25). Together with the suggestions made KH--~ 0, VR~--o -- oo and VR ~--o + o~. This above, one may expect that the adhesion of is an indication that these equations are not polystyrene particles to cellophane is determined by the potential of the cellophane only applicable to colloidal systems in contact. Also Jones and Levine (23) although on and that the influence of double-layer interother grounds, stated that neither the condi- action on the adhesion can be described by the tion of constant ~ nor that of constant ~ is the expression derived under the condition of constant charge, identifying ~0 with the zeta poappropriate one in the case of K t t < 0.5. For equal potentials, however, Eqs. [18] tential of cellophane (Eq. [26]). 5.3.2 Measurements and interpretation. The and [-24] show that VR is always repulsive; Vn ~ still goes to + 0% but less rapidly, VR ~ measurements on carbon black particles re(being smaller than Vn ~) remains finite (Fig. ported previously (1) were thought to be in12). In other words, if the interaction energy sufficiently accurate due to lack of knowledge is determined by one of the two dissimilar of particle radius distribution, geometry, and potentials, Eqs. [-18] and [-24] might be used surface characteristics. Therefore, the meain the interpretation of double-layer interac- surements were repeated for the system polystyrene/cellophane, both of known surface tion at small separations. There are two arguments in favor of this characteristics, as a function of pH and ionic idea: Deryagin (24) has shown that for flat strength, whereas the particle radius distribution was known. The results are presented in plates of dissimilar potential, the height of the potential barrier is determined by the magni- Table I and Fig. 11. tude of the lower potential only, and prelimiAs has been shown above, there is an attracnary calculations by Peterson (4) indicate that tive part for the adhesion of polystyrene parthe interaction between dissimilar particles at ticles to cellophane, which can be described by FR(arbitrary units)
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
673
ADHESION OF POLYSTYRENE PARTICLES TABLE I Speed of Rotation and Corresponding Hydrodynamic Force (FH) a pH
2.23 2.32 3.02 3.12 3.86 4.27 4.93 5.12 5.42 5.65 6.70 7.10 7.30 7.42 7.88 8.82 9.88 10.10 10.75 11.73 12.42
Ionic strength/(mole/m3) 100
10
1
Speed of FH/(IO -6 dyn) rotation (rpm)
Speed of FH/(IO-6 dyn) rotation (rpm)
Speed of FIt/(IO t dyn) rotation (rpm)
7500 --. -7200 -. -6200 . . 4650 . -3900 -3500 . -2000
64.7 --.
. 7100 7000 .
7200 . 6000 .
5500 . . .
. .
.
.
27.6 .
.
-20.2 -16.7 .
.
60.2 .
.
3900 . 2900 . .
37.3 .
.
.
.
.
.
.
.
.
-6.2
1900 .
5.6 .
.
-5100 --
-32.6 --
3900 3900
20.2 20.2
3400 --
15.8 --
--
--
1750 --
4.8 --
.
11.9 .
--48.8 --
.
20.2 .
--6400 -.
43.5
.
-46.1 . .
. 58.7 57.2
.
-60.2 -.
.
.
" Giving 50% removal of polystyrene particles from cellophane as a function of pH and ionic strength (25°C, 1 : 1 electrolyte).
V a n der W a a l s forces, and a repulsive p a r t d e p e n d i n g on p H a n d ionic strength, due to d o u b l e - l a y e r interaction. T h e a t t r a c t i v e p a r t , 78 X 10 -6 d y n in m a g n i t u d e was shown to be i n d e p e n d e n t of p a r t i c l e radius R, w h e r e a s the repulsive p a r t was p r o p o r t i o n a l to R. B y subt r a c t i n g the m e a s u r e d force of adhesion f r o m t h e a t t r a c t i v e p a r t , the repulsive p a r t of the i n t e r a c t i o n is o b t a i n e d : FR = 78 X 10 -e - - F a .
[-27]
T a b l e I I gives t h e results for FR as a function of p H and ionic s t r e n g t h ; for the c o m p u t a tion of FA the F H d a t a of T a b l e I h a v e b e e n used. T a b l e I I includes t h e c o r r e s p o n d i n g z e t a p o t e n t i a l s of cellophane (~bl) and p o l y s t y r e n e ( ~ ) as well as their p r o d u c t s and squares. T h e z e t a p o t e n t i a l s for t h e g i v e n p H v a l u e s h a v e b e e n t a k e n f r o m Figs. 3 and 4.
As a start, t h e finding t h a t the i n t e r a c t i o n b e t w e e n a colloidal p a r t i c l e and a flat p l a t e is d e t e r m i n e d b y the p o t e n t i a l of the flat surface o n l y has been i n v e s t i g a t e d . T h e r e f o r e , Figs. 13, 14, a n d 15 give F R as a f u n c t i o n of ~bl2, ~bl~b2, a n d ~k22, respectively, and for 100, 10, a n d 1 m o l e / m 3 1:1 electrolyte. H e r e , t h e d a t a for p H < 3 h a v e been o m i t t e d because of t h e s c a t t e r in z e t a p o t e n t i a l s in this region and t h e i n a c c u r a c y of the m e a s u r e m e n t s being insensit i v e to the p a r t i c l e radius, as a l r e a d y suggested f r o m the influence of p a r t i c l e radius d i s t r i b u t i o n on r e m o v a l . Figures 13, 14, and 15 show t h a t FR is o n l y p r o p o r t i o n a l to ~bl2 for all the ionic s t r e n g t h v a l u e s studied. I n a c c o r d a n c e w i t h the t h e o r y g i v e n a b o v e a n d t h e results of a dimensional analysis, FR m u s t be p r o p o r t i o n a l to the s q u a r e of a p o t e n t i a l . C o n s e q u e n t l y , in t h e light of Journal of Colloid and Interface S¢i¢n¢¢, Vol, 55, No. 3,
June 1976
.%
3.5
11.0
15.0
19.0
21.0
21.5
23.0
17.8
31.9
50.4
57.8
61.3
71.8
mV
--~1
13.3
(lO-S dyn)
FR/
mV
--~z
529
462
441
361
225
121
43.5
40.0
37.5
33.0
25.0
15.0
12.3 --15.0
(mV) s
~12
100
a As a function of pH and ionic strength.
2.23 2.32 3.02 3.12 3.86 4.27 4.93 5.12 5.42 5.65 6.70 7.10 7.30 7.42 7.88 8.82 9.88 10.10 10.75 11.73 12.42
pH
1892
1600
1406
1089
625
225
225
(mV) 2
~22
TABLE II
1000
860
788
627
375
165
53
(mV) 2
72.4
66.1
32.5
31.5
28.5
23.0
40.7
57.8
22.0
17.0
17.8 34.5
9.0 14.0
--~'1 mV
19.3 20.8
( 1 0 -6 d y n )
Fie/
1056
992
812
529
484
289
81 196
.t~12 (mV) 2
10
69.0
63.0
54.0
35.0
30.0
18.0
--7.5 6.0
---.t~2 mV
Ionic strength/(mole/m
4761
3969
2916
1225
900
324
56 36
,t~22 (mV) 2
s)
2240
1984
1540
805
660
306
68 84
~/1~2 (mV) 2
73.2
62.2
57.8 57.8
45.4
29.2
( l O- O d y n )
FR/
35.0
33.0
32.0 32.5
28.5
18.0
---- ,t/JL mV
1225
1089
1024 1056
812
324
~12 (mV):
1
(FR) and Corresponding Zeta Potentials of Polystyrene (62) and Cellophane (61)~
Wl~2
Electrostatic Force of Repulsion
78.0
75.0
73.0 74.0
64.0
20.0
--.t~2 mV
6084
5625
5329 5476
4096
400
.t~22 (mV) 2
2720
2475
2340 2410
1823
360
~1~2 (mV) 2
(t]
Ox
675
A D H E S I O N OF P O L Y S T Y R E N E P A R T I C L E S FR/(IO-6 dyn) 70
FR/(lO-6dyn) 70
60
60
50
50
40
/-*0
30
30
20
20
10
10
O0
5
i
1,000
I
1,500
i
2,000 O/2/(mV)Z
o? / °
o/o
f
0
1,ooo
2,0oo
3,ooo
4.ooo ~2(rnV)2
FIG. 13. Electrostatic force of repulsion (Fn) as a function of 612, ~1~2, and ~2'; 100 rnole/ma; 25°C.
FIO. 14. Electrostatic force of repulsion (FR) as a function of C/is, ~1~b2, and ~b22; 10 mole/m3; 25°C.
previous findings for carbon black (2), FR appears to be related to ~bl, the potential of the fiat surface. This potential is here the lower of the two, as can be seen from the data given in Figs. 3 and 4. Thus, the interaction is not necessarily determined by the potential of the flat surface, but it also can be regarded as being related to the lower of the two potentials. Quantitatively, the results cannot be described by the theoretically derived Eq. E26~ as suggested (2) from the adhesion of carbon black particles, but all the data given fit the equation :
surface in an aqueous solution, by means of liquid flow between concentric cylinders. The results are reproducible, sensitive to small changes in experimental conditions, and can be interpreted quantitatively in terms of forces of adhesion. It is shown that for polystyrene particles adhering to a cellophane substrate, the adhesion is maximum at the zero point of charge of the adherents, confirming our previous study (1, 2) on the adhesion of carbon black to cellophane. The force of adhesion at this maximum can be described quantitatively by Van der Waals forces. In view of the influence of pH and ionic strength on the adhesion, apart from Van der Waals forces of attraction, the adhesion is determined by electrostatic forces of repulsion due to double-layer interaction. The experimental results concerned could be fitted to theory.
F n = ½ ~ n ~ , 2 coth
(KH).
E28~
A theoretical justification for this equation can be given by comparison with the equations derived under the conditions of constant charge and constant potential respectively (Fig. 12). As can be seen from Fig. 12, Eq. r28-] is close to the relationship derived under the conditions of constant charge. This is what one would expect, since the condition of constant charge will, in practice, never be achieved completely. It is also possible that the deviation is connected with a geometry problem, since we are not dealing with a perfect sphere on a flat surface (Fig. 10b) making point contact, the condition for which this equation has been derived.
?*,,2
FRI(IO-6 dyn) 7O
5G /-,C
2C K i
0
6. CONCLUSIONS A method has been developed for measuring the ~dhesion of colloidal particles to a fiat
/*5
6O
2.000
I
4,000
i
6,000
i
8,000
qJZl(mV)Z FIG. 15. Electrostatic force of repulsion (Fn) as a function of 61~, 61qJ2, and 62'; 1 mole/m3; 25°C. Journal of Colloid and Interface Science, VoL 55, No. 3, June 1976
676
J. VISSER
The adhesion of polystyrene particles to cellophane can be described by:
T V*
F a = (TrpflA)/(67rH
V¢
3)
-- ½~R~b12K.coth (aH)
= absolute temperature (K). = interaction energy at constant electrostatic charge (erg). = interaction energy at constant potential (erg). = elementary charge (4.803 X 10-1°)
[-29-] e
where A is the H a m a k e r constant of the system, / / the minimum separation distance between the adherents, R the particle radius, pc the radius of the contact area, E the dielectric constant of water, K the reciprocal double-layer thickness and ~klthe zeta potential of cellophane. In contrast to the adhesion of carbon black particles to cellophane (1), where the contribution of a possible intimate contact area to the adhesion can be neglected and the adhesion can be described by the sphere/plate model, in the adhesion of polystyrene particles to cellophane, the reverse holds and the adhesion is as regards Van der Waals forces determined predominantly by the contribution of the " i n t i m a t e " contact area. Whereas in the adhesion of carbon black particles to cellophane, the electrostatic repulsion force can be obtained from the DLVO theory under the condition of constant charge during the interaction (2), in the adhesion of polystyrene to cellophane, this force slightly differs from the DLVO theory, probably due to the deviation from the ideal sphere/plate model. In both cases, the potential of the flat surface (cellophane) determines the magnitude of the interaction. APPENDIX: NOMENCLATORE A
FA Fe FH Fa Fvaw H
= = = = = = =
R Re Ri R0
= = = =
H a m a k e r constant (erg). force of adhesion (dyn). centrifugal force (dyn). hydrodynamic force (dyn). force of electrostatic repulsion (dyn). Van der Waals attraction (dyn). distance of minimum approach of adherents (m). radius of spherical particle (m). Reynolds number (1). radius of inner cylinder (m). radius of outer cylinder (m).
Journal of Colloid and Interface ~icn~, Vol. 55, No. 3. June 1976
(esu). k ni.0 v z
K
= Boltzmann constant (1.381 X 10-26) (erg K-l). = bulk concentration of ion i (mole m-3). = velocity of liquid (m sec-1). = valency of ion (1). = double-layer thickness (m). = dielectric constant (1). = dynamic viscosity of solution (P). = reciprocal double-layer thickness
(m-l). O r ~o 1, 2 i 0
= = = = = = = =
kinematic viscosity of solution = ~/p. mass density of solution (kg m - 0 . wall shear stress (dyn). (zeta) potential (V). angular frequency of cylinder (rpm). referring to particle 1, 2. referring to entity i. referring to bulk (concentration). REFERENCES
1. VISSER, J., J. Colloid Interface Sci. 34, 26 (1970). 2. VlSSER, J., in "The adhesion of colloidal particles to a planar surface in aqueous solutions," Ph.D. thesis, Council for National Academic Awards, London, 1973.
3. LOEB,A. L., 0VERBEEK,J. TH. G., AND WIERSEMA, P. H., in "The Electrical Double Layer around
4. 5.
6. 7. 8. 9. 10.
11. 12.
a Spherical Colloid Particle," The M.I.T. Press, Cambridge, Massachusetts, 1961. PETERSON,G. C., personal communication. SMITH,A. L., in "Dispersion of Powders in Liquids" (G. D. Parfitt, Ed.), p. 39. Elsevier, Amsterdam, 1969. GOLDMAN, A. J., Cox, R. G., AND BRENNER, H., Chem. Eng. Sci. 22, 653 (1967). O'NEILL, M. E., Chem. Eng. Sci. 23, 1293 (1968). VAN LOOKERENCAMPAGNE,N., Thesis, Groningen, 1966. WENDT, F., Ing. Arch. 4, 577 (1933). VlSSER, J., in "Surface and Colloid Science," (E. Matijevic, Ed.), Vol. 8, Chap. 1, Wiley Interscience, New York, 1976. KRUPP, H., Advan. Colloid Interface Sci. 1, 111 (1967). DAnNEKE,B., J. Colloid Interface Sci. 40, 1 (1972).
ADHESION OF POLYSTYRENE PARTICLES 13. CLUNm, J. S., GOODMAN,J. F., AND TATE, J. R., Trans. Faraday Soc. 64, 1965 (1968). 14. NORRISH, K., Disc. Faraday Soc. 18, 120 (1954). 15. LAGALY, G. AND W~ISS, A. "25. Hauptversammlung der Kolloid-Gesellschaft," Mtinchen, 1971. 16. DERYAC.IN,B. V. AND LANDAU,L. D., Acta Physicochim. USSR 14, 633 (1941). 17. VERWEY, E. W. J. AND OVERBEEK, J. TH. G., in "Theory of the Stability of Lyophobic Colloids," Elsevier, Amsterdam, 1948. 18. HOGG, R., HEALY, T. W., AND FUERSTENAU, D. W., Trans. Faraday Soc. 62, 1638 (1966).
677
19. LANGE,H., Kollold Z. 154, 103 (1957). 20. WIESE, G. R. AND HEAL¥, T. W., Trans. Faraday Soc. 66, 490 (1970). 21. FRENS, G., Thesis, Utrecht, 1968. 22. Usux, S. AND YAMASAKI,T., J. Phys. Chem. 71, 3195 (1967); J. Colloid Inter/ace Sci. 29, 629 (1969). 23. Jo~qES, J. E. AND LEVINE, S., Y. Colloid Interface Sci. 30, 241 (1969). 24. DERYAGIN,B. V., Disc. Faraday Soc. 18, 85 (1954). 25. OVERBEEK, J. TH. G., in "Colloid Science I" (Kruyt, Ed.), p. 261. Elsevier, Amsterdam, 1951.
Journal of Colloid and Interface Science, Vol. 55. No. 3. June 1976
Received June 20, 1975; accepted January 12, 1976 A method for the determination of the force of adhesion of colloidal particles adhering to a planar surface in aqueous solutions has been used to study the influence of pH and ionic strength on the adhesion of polystyrene particles to a cellulose film as substrate. It is shown that the adhesion of these particles can be described quantitatively in terms of Van der Waals forces of attraction and electrostatic forces of repulsion. The adhesive force is maximum at the iso-electric point of the adherents where the adhesion is determined by Van der Waals forces only. The electrostatic double-layer interaction forces operating away from the iso-electric point are determined exclusively by the potential of the flat surface (cellophane). 2. MATERIALS AND APPARATUS
1. INTRODUCTION
2.1 Materials
The force of adhesion between solid particles and a fiat surface can be determined directly, both in vacuo and in aqueous environment, by means of a centrifuge (1). The centrifugal method, however, can be applied only to highdensity (e.g., gold) particles of a diameter larger than 1 ~m. Therefore, a new technique has been developed to overcome these problems (2). This technique involves a liquid flow between concentric cylinders and analysis of the liquid flow close to the surface. A calibrating experiment using the centrifuge technique enables the evaluation of removal measurements of colloidal, low-density particles, adhering to the outer surface of the rotating inner cylinder, in terms of forces of adhesion. We applied this technique to a well-defined system, i.e., monodispersed polystyrene particles of a known particle radius distribution (average radius 250 nm) and surface characteristics, adhering to a cellulose film (cellophane) previously used (1, 2).
The polystyrene particles had been prepared by Dr. Nicholls of Unilever Research, Port Sunlight by emulsion polymerization with sodium laurate as emulsifier and hydrogen peroxide as initiator. According to electron micrographs, the particles were perfectly spherical with a modal radius of 260 nm, corresponding to the maximum in the frequency distribution (Fig. la). The log-normal radius distribution (Fig. lb) gives an average radius of 250 nm corresponding to the 50% level, which value was used in the interpretation of the measurements. The particles were dialyzed for 6 weeks at p H 9 and stored in a refrigerator. The cellophane was a regenerated cellulose from AKZO Arnhem, the Netherlands, and was used after rinsing with distilled water, hydrogen chloride (10 m o l e / m 3) and distilled water at 70°C. The rinsing procedure was adopted to remove traces of glycerol and other contaminants. The water used throughout the experiments had been distilled twice, once over permanganate.
1 Presented at the 49th National Colloid Symposium, Potsdam, New York, June 16 18, 1975. The text of this paper is a part of the author's Ph.D. thesis (2).
664 Journal of Colloid and Interface Science, Vol. 55, No. 3, J u n e 1976
C o p y r i g h t ~ 1976 by Academic Press, Inc. All rights of reproduction in a n y form reserved.
ADHESION OF POLYSTYRENE PARTICLES
stainless steel to prevent corrosion by acid or alkaline solutions.
% particles ~< given diameter
/o~O_O~
100 8o
oo
/
/
3. EXPERIMENTAL PROCEDURES
3.1 Removal Experiments
40
20 _-~~5~4~ 0 3~./b°~'
a i I 5 555 615 675 Particle diameter/nm
Particte diameterlnm 800 600 f 400
.-.--o
o'°'°"
5- ' o ' ° ~ °
2OOl-
I I i
I
,
t
,
665
t
,
J
b ,
,
I
I
I
I
1 2 5 10 30 50 70 90 98 % particles ~< given diameter
FIG. 1. Particle radius distribution of polystyrene particles (PS Latex 65251). A. Cumulative radius distribution; B. Log-normal radius distribution.
All chemicals used were of analytical grade or had been purified to the highest possible degree, e.g., by recrystallization.
2.2 Apparatus The apparatus was similar to that used in our previous study (1). It consisted of a cylinder (length 4.5 cm, diameter 4.6 cm) of polymethylmethacrylate (PMMA) or anodized aluminum. A wet cellophane sheet (6.5 X 16 cm) was wrapped on the cylinder surface and tightened by two conical lids (Fig. 2). The previous apparatus used for removal of carbon black particles (1) had a gap width of maximum 0.2 cm. The present apparatus differs in that the inner diameter (8.40 cm) of the outer cylinder results in a gap width of 1.90 cm. Further, the inner cylinder was driven by a more powerful motor allowing of a speed variable between 2250 and 7700 rpm. In combination with the larger gap width, a wall shear stress much larger than that obtained with the previously described machine could be attained. A water jacket of the outer cylinder served to maintain the room temperature during the experiments. The inside of the whole apparatus was of
The polystyrene particles were deposited onto the cellophane by rotating the cylinder at 265 rpm for 30 min in a 250-ml glass beaker containing a 0.005% dispersion of the particles in distilled water. After the cylinder had been removed from the dispersion and dried in air, the number of particles deposited on the surface was determined. Since the size of the particles was comparable with the wavelength of visible light, the particles were detected by their ability to scatter light from a microscope lamp focussed on the area of the cylinder surface under observation. As a result, the particles were visible as light spots on a greyish or black background depending on the type of cylinder used. With the use of a grid fitted in the ocular of the microscope, the number of particles per unit surface area was determined by counting. An overall magnification of 400 was used throughout the measurements. Immediately after the initial particle count, the cylinder was immersed in the apparatus filled with a solution of known pH and ionic strength. After adaptation for 30 min, the cellophane cylinder was rotated at 2250 rpm for 5 min and the number of remaining particles was determined. Then, the rotational speed was repeatedly increased until just more than 50% of the particles were removed. The exact speed at 500-/0particle removal was determined by interpolation.
VIA of aluminium) PMMA or aluminium )
FIG. 2. Inner cylinder with cellophane coating. Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
666
J. VISSER zeta x)tentiatlmV
-60
•
/o :~ o O o" / / ~ o ' ~ ' - ' ~ Io , , ~ p f o
-~ -20C .,,2C
~-
~o / ~ f at
d,o oH FIG. 3. Zeta potential of polystyrene particles as a function of p H and ionic strength (25°C; 1:1 electrolyte). [] = 1 mole/m3; A = 10 mole/m3; O = 100 m o l e / m 3.
For the calculation of the removal percentage, the number of particles adhering to the surface of the inner cylinder before and after a removal experiment was determined. The necessity to examine the same areas before and after the removal was realized by aligning a mark on the rod with markings on the cylinder lid and by using markings on the microscope table to which the cylinder support was fixed. In this way the particles on eight different areas of the cylinder surface, appearing in 9 (out of 25) squares of the ocular grid, were counted (about 600-800 particles per experiment).
3.2 Electrophoretic Mobility Measurements Electrophoretic mobility measurements on polystyrene and cellophane particles were zeta ~otential/mV
D o
o
2(; 1(~ 0
FIG. 4. Zeta potential of cellophane particles as a function of p H and ionic strength (25°C; 1:1 electrolyte). [] = 1 mole/m3; /k = 10 mole/m~; © = 100 m o l e / m 3. Journal of Colloid and Interface Science, Vol. 55, No. 3, J u n e 1976
made at concentrations of 100, 10, and 1 mole/m ~ electrolyte as a function of pH in mixtures consisting of sodium hydroxide or hydrogen chloride at 25°C, that is at the same concentrations and pH's as in the removal experiments. Cellophane particles were obtained after freezing wet cellophane (only available as film) with solid carbon dioxide and grinding in a porcelain mortar. All experiments were performed in a cylindrical cell which was thoroughly flushed with distilled water. Ten readings were made in both directions at the stationary level of the cell. The results are presented in Figs. 3 and 4, after transformation of the data in zeta potentials using the Tables of Loeb, Wiersema, and Overbeek (3) as computerized by Peterson (4) or using, in the case of cellophane, Von Smoluchowski's equation (large KH, see Section 5.3.1), resulting at 25°C, in (5) : zeta potential = 12.83 X[mobility in (~m sec-1)/(V cm-i)]mV [-1] 4. E V A L U A T I O N OF T H E R E M O V A L M E A S U R E M E N T S I N T E R M S OF F O R C E S OF A D H E S I O N
According to Goldman, Cox, and Brenner (6) and O'Neill (7) the tangential force Fn exerted on a spherical particle of radius R in contact with a plane wall in slow linear shear flow (Fig. 5) is given by the modified Stokes' law : Fu = 1.7005 X 6 ~r~Rvx=R [-2] where 1.7005 is a correction factor due to the configuration of a sphere in contact with a plane; ~ is the dynamic viscosity of the liquid, and v,=R the velocity at a distance x = R from the wall. [ I
]
~
!
.>,,
/
Vx=R
vc *F H
~vx FIG. 5. Velocity profile at the surface of the inner cylinder.
ADHESION OF POLYSTYRENE PARTICLES Since by definition, the wall shear stress r is given by r = n(dv/dx)
[33
one obtains, after integration, for a laminar boundary layer, i.e., a linear velocity gradient with v = 0 on the surface: =
667
% Rernova[ 0: ~ 20 -
A rotating outer cylinder v rotating inner cylinder
~ v
8O
[43
Substitution in Eq. [2] gives
RiR9 TM
(Ro_Ri)1,'4
F H = 32R2r.
1-53
In other words, the removal of spherical particles from a flat surface is determined by the magnitude of the wall shear stress r, provided there is a relationship between the force Fzr exerted on the particles by the flow of the liquid and the subsequent removal. For the flow between concentric cylinders, semiempirical relations between flow conditions and wall shear stress exist. These equations are :
FIG. 6. Removal of carbon black particles from cellophane as a function of wall shear stress (r). A = rotating outer cylinder; V = rotating inner cylinder. (3) For a rotating outer cylinder and a stationary inner cylinder (9) : r = lpw~Ro 2 X 0.0013{0.04 + (Ri/Ro)tO}.
[113
Application of Eqs. [9] and [11] to the removal of carbon-black particles from cellophane (2) (Fig. 6) confirms the statement made above that the removal of submicron (1) For a rotating cylinder in a concentriparticles from a flat surface is determined by cal outer cylinder at rest, and a large gap width the wall shear stress r. (8) Since the removal mechanism is unknown, it is not possible to relate Fn to the force of r = 1.6~ X lO-2pv°'2%l'TSRi TM X (R0 -- R0 °'°5 1-6] adhesion FA of the adhering particles on theoretical grounds. As reported before (1), a califor bration experiment is necessary to establish this relationship. I t was found that Fu is 3 X i04 < R e < 107 and numerically identical to FA: 0.1 < ( e o - n 0 / e , < 3 1-7] where the Reynolds number Re is defined by Re = w R ~ ( R o - - R ~ ) / v
[8]
whereas v = kinematic viscosity of the solution, w = angular frequency of the cylinder and p = density of the solution. (2) For the same configuration but with a small gap width (9) : r = 0.23pv½wlRiRol(Ro - - R¢) - l
[-9]
for Reynolds numbers 108 < Re < 104.
[10]
FH = FA.
[-12]
In other words, the force of adhesion of colloidal particles Fa can be directly obtained from a removal experiment using concentric, rotating cylinders. The adhesive force can be related to the angular frequency w by combining Eqs. [5-], I-6], and [-12] for a cylinder rotating in an outer cylinder at a large gap width as used in the measurements reported here. As a result, one obtains after substituting the appropriate constants of the apparatus and the physical quantities at 25°C: Fa = 1.21 X 10-2R% ''Ts.
[13]
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
668
J. VISSER % Removal 100 80
//
60 4O 20 ,
0
2,000
4,6O0 6.0OO Speed of r~ationl(rev.lmin)
FIG. 7. Removal of polystyrene particles (R~ = 260 nm) from cellophane as a function of speed of rotation (25°C; 1:1 electrolyte). A = 1 mole/m~, pH 5.5; C) = 10 mole/m3, pH 6-7. 5. RESULTS AND INTERPRETATION
5.1 The Role of Particle Size Distribution on the Degree of Removal Since colloidal particles are never the same size, but always show a radius distribution, this distribution will also be reflected in the removal results. In studying the removal as a function of speed of rotation, this removal will therefore be a gradual function rather than a step function. The latter situation would be the case for particles of the same size and all adhering with the same force of adhesion. As a consequence, the hydrodynamic force corresponding to 50% removal is a measure of the adhesive force of a particle with a radius equal to the average radius calculated from the particle radius distribution. Because the radius distribution of the carbon black particles used in our previous study (1) was not known very accurately, we decided to
study the removal of the more defined system of polystyrene particles which have a narrow size distribution (Fig. 1) and an average diameter of 500 nm. As a consequence, the removal of these particles should be close to a step function of the speed of rotation. This is indeed the case (Fig. 7). Since the particle radius distribution of these particles is known, it should be possible to relate the removal function to this distribution. An indication of this relation was found by plotting the removal as a function of the speed of rotation on log-probability paper (Fig. 8). A linear relationship was obtained similar to that for the particle radius distribution (Fig. lb). If one assumes that during the removal process the bigger particles come off first followed by the smaller ones at a higher speed of rotation, as was shown in the removal studies of carbon black particles (2), a removal percentage of p % corresponds to the removal of all the particles of a diameter corresponding to (100 -- P)~o of the particles. In this way, it was possible to relate the removal results to the given particle radius distribution. The result is given in Fig. 9, where F n has been calculated with Eqs. [-12] and [-13]. I t appears that Fu and consequently Fa, the force of adhesion of the polystyrene particles, is, under the conditions studied, proportional to the particle radius; the proportionality constant is negative. This is at first sight in contradiction with the theoretically (10) and Particle diameter I nm
6oc
\~
Speed of rotationl(reWmin)
7,00oF 30(
5,~ I
--~--"°'-'-~ ~
2.000[
10(
"°°°'i~ ~,b ~o~0'~o'Tb' ~o' ~ ' (%) removal
FIG. 8. Removal of polystyrene particles as a function of speed of rotation (log-normal paper). /k = 1 mole/ m3, pH 5.5; © = 10 mole/m3, pH 6-7. Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
i
i
2O
4O
I
6O 80 FH I(10-6 dyn)
FIG. 9. Relation between particle diameter and hydrodynamic force (Fn). /k = 1 mole/m3; O = 10 mole/m3.
ADHESION OF POLYSTYRENE PARTICLES experimentally (11) observed proportionality between the force of adhesion of a spherical particle to a flat plate and R with a positive proportionality constant. For this geometry (Fig. 10a), as shown elsewhere (2) Fa
= FVdW
-
-
Frepulsion = ( A R ) / ( 6 H 2) -- ce,
[-14]
where A is the Hamaker constant, H is the shortest distance between the adherents, and c is a constant depending on p H and ionic strength. For the more realistic situation, taking into account surface roughness and nonsphericity of the particles, the adhesion can be described by: FA = FVdW -- FR = ( A R ) / ( 6 H ~) + (~rpc2A)/(6~-H 3) -- cR
[15]
where the second attractive term describes the interaction across the intimate contact area of radius Pe (Fig. 10b). When the contribution of this contact area is dominating, Eq. [15] shows that the adhesion will be proportional to R as observed above, the proportionality constant being negative. As a result the adhesion of polystyrene particles to cellophane can be described by: Fa = (~rm2A)/(67rHO -- cR.
[-16]
The occurrence of an extended contact area in polystyrene particle adhesion to a flat substrate has been demonstrated by Dahneke (12) from electron micrographs. The --cR term in Eq. [-16] describes the influence of the electrostatic double-layer interaction on the adhesion. In view of its particle radius dependency, its magnitude is apparently determined by the double-layer properties outside the contact area, whereas the interaction over the contact area being radius independent, is determined by Van der Waals forces only.
669
ideal
a
41"I
b
ZO¢
Fro. 10. Interaction between a sphere of radius R and a flat surface at a distance H. A. Ideal system; B. system with extended contact area of diameter 2pc. 5.2 The Contribution of Van der Waals Forces to the Adhesion of Polystyrene Particles to Cellophane
As has been shown in the previous section, the contribution of the Van der Waals force to the adhesion (since we are dealing with systems in contact, only nonretarded Van der Waals forces are involved) can be identified with the radius-independent term in the equation describing the adhesion of polystyrene particles to cellophane as obtained from its particle radius contribution and the particle removal as a function of rotational speed. I t can be obtained by extrapolating the graphs in Fig. 9, to R = 0, giving a radius-independent contribution to the adhesion of 78 X 10-6 dyn, being independent of ionic strength. Taking as an estimate for the Hamaker constant of the system 3.5 X 10-14 erg, the correctness of this procedure can be investigated. The condition (A R ) / ( 6 H ~) << (r0 c2A)/(6~r//~)
= 7 8 X 1 0 - 6dyn
[17]
is fulfilled for H = 2.0 nm and pc = 103 nm. Other possibilities do exist but in respect of surface hydration, deformability of the adherents and nonideality of the system such as surface roughness, a value of H = 2.0 nm is quite reasonable. I t is equal to the limiting thickness of the aqueous core of a black film (13) in excess electrolyte, that is to say under conditions where only Van der Waals forces are operating, and somewhat higher than the interlayer spacings of swollen clay platelets as observed by X-ray analysis (14, 15). The value of pc = 103 nm is not unrealistic as can be seen from Fig. 10b, where the corresponding Journal of Colloid and Interface Science, Vol. 55, No, 3, June 1976
670
j. VISSER FH/(IO-6 dyn)
70 60 50 40 30 i
styrene particles indeed show this type of behavior : no longer a gradual increase in removal with increasing speed of rotation was noticed, but a sudden, almost complete removal above a certain speed. Consequently, the accuracy of the measurements in the vicinity of the zero point of charge is low and the results will therefore be omitted in the interpretation (see also Fig. 11).
20
5.3 The Electrostatic Repulsion Forces in the Adhesion of Polystyrene Particles to Cellophane o ~' ~, ~ ~ ~b 1~ pH 5.3.1 Theory. In water, electrostatic doublelayer forces are present in addition to Van der Fla. 11. T h e h y d r o d y n a m i c force (FH) corresponding Waals forces. These forces arise from the to 50070 removal of polystyrene from cellophane as a function of p H (25°C; 1:1 electrolyte). © = 100 charging of materials in the case of water and mole/m3; /~ = 10 mole/m3; [] = 1 m o l e / m 3. they can be influenced by introducing electrolyte into the liquid phase. As the sign of the geometry has been drawn to scale, and is of charge is negative for most materials, mutual the same magnitude (Ripe = 2.5) as Dahneke interpenetration of the corresponding double (12) has shown by electron microscopy for layers usually results in repulsion. polystyrene particles adhering to a chromium From the DLVO-theory of double-layer insurface (R/p~ = 2). Therefore, in the evaluateraction Edeveloped by Deryagin and Landau tion of the adhesion measurements on poly(16) and Verwey and Overbeek (17)-] it is posstyrene in the following sections, the value of sible to calculate the electrostatic double layer an attractive force of 78 X 10-6 dyn and a of repulsion arising from the interpenetration separation distance of H = 2.0 nm will be taken as a basis. The latter value is somewhat of two double layers. There are two conditions higher than that for the system carbon black/ under which this repulsion can be calculated, cellophane (--1.6 nm) (1) which may be due viz: The potential of the interacting systems to polymer chains sticking out from the poly- remains constant during the interaction, or the charge of the interacting systems remains mer surface. constant. A further consequence of the finding that (i) Constant potential interaction. A relathe adhesion of polystyrene particles to cellotion expressing the interaction energy VR ~ bephane can be described by Eq. [16~ is that tween the electrical double layers surrounding for c = 0 when the electrostatic repulsion any two similar or dissimilar colloidal particles forces are zero or can be neglected, the force of adhesion will be independent of the particle has been derived by Hogg, Healy, and radius for the given system. This means that Fuerstenau (18) under the condition of conthe removal will be very close to a step func- stant (low) potential. This equation reads: tion of the speed of rotation, i.e., there will be ~R1R2(~o.12 + ~o,~2) no removal below a certain speed of rotation VR~ = 4 (R1 + R2) and a complete removal of all the adhering particles when exceeding this speed. [ 2~b0,_~b0.2 In 1 -t- exp(--KH) The condition c = 0 occurs when the adX L(~o,x~ --[-~o,J2) 1 -- exp(--KtI) herents are in a situation corresponding to their zero point of charge. In experiments q-In {1 -- exp(--2K//)} 1 E18~ performed under this condition, the polyIC
_.1
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
ADHESION OF POLYSTYRENE PARTICLES where R~, R2 are the radii of the interacting particles 1 and 2 of potential ~//0,1 and ~0,2 the potential ~0.1 at infinite separation. H is the distance of minimum approach, e the dielectric constant of the medium, and r the reciprocal double-layer thickness ~, defined by:
K2= (47re~)/(ekr).Zzfln~o.
[19]
Here, k is the Boltzmann constant, T the absolute temperature, zl the valency of ion i, ni0 the bulk concentration of ion i and e the electronic charge. Equation [18] holds exactly for ~0,1 and/or 60.2 of less than 25 mV and for such solution conditions that the double-layer thickness ~ is small compared with the particle size: a
=
r-' << R1, R2.
~o,~
=
,/'o :
FR+(sphere -- plate) eR~02K•exp (-- KH) [23] 1 + exp(--~H)
(ii) Constant charge interaction. The electrostatic double-layer repulsion energy Va" arising from the interaction of two double layers surrounding a colloidal particle of radius R1 and a colloidal particle of radius R~ under condition of constant charge has been derived by Wiese and Healy (20). Here, ,R,R2@0.~ ~ + ~o/)
VR*= VR + -2 (RI +
[20]
For ~h0,1 and if0.2 of less than 50-60 mV, Eq. [18] is a good approximation. Since Lange has shown (19) that the electrostatic double-layer interaction energy VR between a sphere and a flat plate equals twice the interaction energy between two identical spheres,
VR (sphere - plate) = 2VR(sphere -- sphere)
=
671
R2)
X In {1 -- exp(--2KH)}
where VR¢ is given by Eq. [18-]. For the sphere/plate geometry, the repulsion force due to double-layer interaction at constant charge can be obtained as above. The result is : FR'(sphere -- plate) K
= ½~R(¢o/+
~o2)
[213
the force of repulsion FR due to double-layer interaction for the sphere/plate geometry, the geometry we are interested in, easily can be derived from Eq. [18-]. By putting R1 = R~ = R, the particle radius, and after differentiation with respect to H: FR~(sphere - plate)
1-24]
exp(--KH)
1 -- exp(--2KH)
)
+
.~'o,22
For materials with identical potentials ~o,x = ¢,o.2 = ~o:
FR*(sphere - plate) ~R~dK"exp (--KH) [26]
• 2d VR(sphere - sphere)
dH Kexp(--KH) = ½,R(¢~0/+
x
¢0/)
F 2~o,1~'o,2
1 -- exp(--2KH)
+ +oC+
exp(--KH)l.[22-1
For materials having identical potentials ~0,1
1 -- exp (--KH) If one assumes that ~b0 in Eqs. [22-]-[26] equals the electrokinetic zeta potential of the individual particles, the electrostatic repulsion force FR can be calculated easily, provided H is known. According to Frens (21), there are some indications of a close resemblance to ¢J0 and the zeta potential, but a quantitative theory is not available. Journal of Colloid and Interface Science,
VoL55, No. 3, June 1976
672
j. VISSER
lower potentials is determined by the particle with the larger radius. The latter result sug8 gests that in the case of a sphere/plate inter6 action, the interaction will be controlled by the properties of the plate. On this basis, the adhesion measurements on 2 polystyrene particles adhering to cellophane apparently should be interpreted, considering 0 1.0 2.0 3.0 ~H only the potential of the cellophane. As regards the use of the equations of conFIG. 12. Electro static force of repulsion (FR) as a function of KH. O -~ FR (constant charge) = eR~2~ stant charge or those of constant potential, the •exp(--~H)['l -- exp(--KH)]-l; [] = FR (constant following can be said. For mercury droplets, potential) = eR6*K.exp(-~H)~l + exp(--KH)]-l; ~, the coalescence measurements of Usui and = F~ (experimental) = ½eR62.coth (KH). Yamasaki (22) could be interpreted using the Recently, Usui and Yamasaki (22) have constant potential equations. In the case of oxides and materials (such as shown that their coalescence experiments with oxidized carbon black, polystyrene, and cellomercury droplets could be interpreted quantiphane) of which the charge of the surface is tatively in terms of the DLVO theory, using not based upon the equilibrium of potentialthe potential of the outer Helmholtz plane determining ions but upon the dissociation of instead of the wall potential. Therefore, we groups such as - C O O H and --OH attached will interpret our measurements assuming that to the surface, not the potential but the sur~0 equais the zeta potential. When using Eqs. [18-] and [24], at small face charge should be considered as a constant values of ~H, it can be easily demonstrated that to a first approximation during the interaction for flat plates of dissimilar potential when (25). Together with the suggestions made KH--~ 0, VR~--o -- oo and VR ~--o + o~. This above, one may expect that the adhesion of is an indication that these equations are not polystyrene particles to cellophane is determined by the potential of the cellophane only applicable to colloidal systems in contact. Also Jones and Levine (23) although on and that the influence of double-layer interother grounds, stated that neither the condi- action on the adhesion can be described by the tion of constant ~ nor that of constant ~ is the expression derived under the condition of constant charge, identifying ~0 with the zeta poappropriate one in the case of K t t < 0.5. For equal potentials, however, Eqs. [18] tential of cellophane (Eq. [26]). 5.3.2 Measurements and interpretation. The and [-24] show that VR is always repulsive; Vn ~ still goes to + 0% but less rapidly, VR ~ measurements on carbon black particles re(being smaller than Vn ~) remains finite (Fig. ported previously (1) were thought to be in12). In other words, if the interaction energy sufficiently accurate due to lack of knowledge is determined by one of the two dissimilar of particle radius distribution, geometry, and potentials, Eqs. [-18] and [-24] might be used surface characteristics. Therefore, the meain the interpretation of double-layer interac- surements were repeated for the system polystyrene/cellophane, both of known surface tion at small separations. There are two arguments in favor of this characteristics, as a function of pH and ionic idea: Deryagin (24) has shown that for flat strength, whereas the particle radius distribution was known. The results are presented in plates of dissimilar potential, the height of the potential barrier is determined by the magni- Table I and Fig. 11. tude of the lower potential only, and prelimiAs has been shown above, there is an attracnary calculations by Peterson (4) indicate that tive part for the adhesion of polystyrene parthe interaction between dissimilar particles at ticles to cellophane, which can be described by FR(arbitrary units)
Journal of Colloid and Interface Science, Vol. 55, No. 3, June 1976
673
ADHESION OF POLYSTYRENE PARTICLES TABLE I Speed of Rotation and Corresponding Hydrodynamic Force (FH) a pH
2.23 2.32 3.02 3.12 3.86 4.27 4.93 5.12 5.42 5.65 6.70 7.10 7.30 7.42 7.88 8.82 9.88 10.10 10.75 11.73 12.42
Ionic strength/(mole/m3) 100
10
1
Speed of FH/(IO -6 dyn) rotation (rpm)
Speed of FH/(IO-6 dyn) rotation (rpm)
Speed of FIt/(IO t dyn) rotation (rpm)
7500 --. -7200 -. -6200 . . 4650 . -3900 -3500 . -2000
64.7 --.
. 7100 7000 .
7200 . 6000 .
5500 . . .
. .
.
.
27.6 .
.
-20.2 -16.7 .
.
60.2 .
.
3900 . 2900 . .
37.3 .
.
.
.
.
.
.
.
.
-6.2
1900 .
5.6 .
.
-5100 --
-32.6 --
3900 3900
20.2 20.2
3400 --
15.8 --
--
--
1750 --
4.8 --
.
11.9 .
--48.8 --
.
20.2 .
--6400 -.
43.5
.
-46.1 . .
. 58.7 57.2
.
-60.2 -.
.
.
" Giving 50% removal of polystyrene particles from cellophane as a function of pH and ionic strength (25°C, 1 : 1 electrolyte).
V a n der W a a l s forces, and a repulsive p a r t d e p e n d i n g on p H a n d ionic strength, due to d o u b l e - l a y e r interaction. T h e a t t r a c t i v e p a r t , 78 X 10 -6 d y n in m a g n i t u d e was shown to be i n d e p e n d e n t of p a r t i c l e radius R, w h e r e a s the repulsive p a r t was p r o p o r t i o n a l to R. B y subt r a c t i n g the m e a s u r e d force of adhesion f r o m t h e a t t r a c t i v e p a r t , the repulsive p a r t of the i n t e r a c t i o n is o b t a i n e d : FR = 78 X 10 -e - - F a .
[-27]
T a b l e I I gives t h e results for FR as a function of p H and ionic s t r e n g t h ; for the c o m p u t a tion of FA the F H d a t a of T a b l e I h a v e b e e n used. T a b l e I I includes t h e c o r r e s p o n d i n g z e t a p o t e n t i a l s of cellophane (~bl) and p o l y s t y r e n e ( ~ ) as well as their p r o d u c t s and squares. T h e z e t a p o t e n t i a l s for t h e g i v e n p H v a l u e s h a v e b e e n t a k e n f r o m Figs. 3 and 4.
As a start, t h e finding t h a t the i n t e r a c t i o n b e t w e e n a colloidal p a r t i c l e and a flat p l a t e is d e t e r m i n e d b y the p o t e n t i a l of the flat surface o n l y has been i n v e s t i g a t e d . T h e r e f o r e , Figs. 13, 14, a n d 15 give F R as a f u n c t i o n of ~bl2, ~bl~b2, a n d ~k22, respectively, and for 100, 10, a n d 1 m o l e / m 3 1:1 electrolyte. H e r e , t h e d a t a for p H < 3 h a v e been o m i t t e d because of t h e s c a t t e r in z e t a p o t e n t i a l s in this region and t h e i n a c c u r a c y of the m e a s u r e m e n t s being insensit i v e to the p a r t i c l e radius, as a l r e a d y suggested f r o m the influence of p a r t i c l e radius d i s t r i b u t i o n on r e m o v a l . Figures 13, 14, and 15 show t h a t FR is o n l y p r o p o r t i o n a l to ~bl2 for all the ionic s t r e n g t h v a l u e s studied. I n a c c o r d a n c e w i t h the t h e o r y g i v e n a b o v e a n d t h e results of a dimensional analysis, FR m u s t be p r o p o r t i o n a l to the s q u a r e of a p o t e n t i a l . C o n s e q u e n t l y , in t h e light of Journal of Colloid and Interface S¢i¢n¢¢, Vol, 55, No. 3,
June 1976
.%
3.5
11.0
15.0
19.0
21.0
21.5
23.0
17.8
31.9
50.4
57.8
61.3
71.8
mV
--~1
13.3
(lO-S dyn)
FR/
mV
--~z
529
462
441
361
225
121
43.5
40.0
37.5
33.0
25.0
15.0
12.3 --15.0
(mV) s
~12
100
a As a function of pH and ionic strength.
2.23 2.32 3.02 3.12 3.86 4.27 4.93 5.12 5.42 5.65 6.70 7.10 7.30 7.42 7.88 8.82 9.88 10.10 10.75 11.73 12.42
pH
1892
1600
1406
1089
625
225
225
(mV) 2
~22
TABLE II
1000
860
788
627
375
165
53
(mV) 2
72.4
66.1
32.5
31.5
28.5
23.0
40.7
57.8
22.0
17.0
17.8 34.5
9.0 14.0
--~'1 mV
19.3 20.8
( 1 0 -6 d y n )
Fie/
1056
992
812
529
484
289
81 196
.t~12 (mV) 2
10
69.0
63.0
54.0
35.0
30.0
18.0
--7.5 6.0
---.t~2 mV
Ionic strength/(mole/m
4761
3969
2916
1225
900
324
56 36
,t~22 (mV) 2
s)
2240
1984
1540
805
660
306
68 84
~/1~2 (mV) 2
73.2
62.2
57.8 57.8
45.4
29.2
( l O- O d y n )
FR/
35.0
33.0
32.0 32.5
28.5
18.0
---- ,t/JL mV
1225
1089
1024 1056
812
324
~12 (mV):
1
(FR) and Corresponding Zeta Potentials of Polystyrene (62) and Cellophane (61)~
Wl~2
Electrostatic Force of Repulsion
78.0
75.0
73.0 74.0
64.0
20.0
--.t~2 mV
6084
5625
5329 5476
4096
400
.t~22 (mV) 2
2720
2475
2340 2410
1823
360
~1~2 (mV) 2
(t]
Ox
675
A D H E S I O N OF P O L Y S T Y R E N E P A R T I C L E S FR/(IO-6 dyn) 70
FR/(lO-6dyn) 70
60
60
50
50
40
/-*0
30
30
20
20
10
10
O0
5
i
1,000
I
1,500
i
2,000 O/2/(mV)Z
o? / °
o/o
f
0
1,ooo
2,0oo
3,ooo
4.ooo ~2(rnV)2
FIG. 13. Electrostatic force of repulsion (Fn) as a function of 612, ~1~2, and ~2'; 100 rnole/ma; 25°C.
FIO. 14. Electrostatic force of repulsion (FR) as a function of C/is, ~1~b2, and ~b22; 10 mole/m3; 25°C.
previous findings for carbon black (2), FR appears to be related to ~bl, the potential of the fiat surface. This potential is here the lower of the two, as can be seen from the data given in Figs. 3 and 4. Thus, the interaction is not necessarily determined by the potential of the flat surface, but it also can be regarded as being related to the lower of the two potentials. Quantitatively, the results cannot be described by the theoretically derived Eq. E26~ as suggested (2) from the adhesion of carbon black particles, but all the data given fit the equation :
surface in an aqueous solution, by means of liquid flow between concentric cylinders. The results are reproducible, sensitive to small changes in experimental conditions, and can be interpreted quantitatively in terms of forces of adhesion. It is shown that for polystyrene particles adhering to a cellophane substrate, the adhesion is maximum at the zero point of charge of the adherents, confirming our previous study (1, 2) on the adhesion of carbon black to cellophane. The force of adhesion at this maximum can be described quantitatively by Van der Waals forces. In view of the influence of pH and ionic strength on the adhesion, apart from Van der Waals forces of attraction, the adhesion is determined by electrostatic forces of repulsion due to double-layer interaction. The experimental results concerned could be fitted to theory.
F n = ½ ~ n ~ , 2 coth
(KH).
E28~
A theoretical justification for this equation can be given by comparison with the equations derived under the conditions of constant charge and constant potential respectively (Fig. 12). As can be seen from Fig. 12, Eq. r28-] is close to the relationship derived under the conditions of constant charge. This is what one would expect, since the condition of constant charge will, in practice, never be achieved completely. It is also possible that the deviation is connected with a geometry problem, since we are not dealing with a perfect sphere on a flat surface (Fig. 10b) making point contact, the condition for which this equation has been derived.
?*,,2
FRI(IO-6 dyn) 7O
5G /-,C
2C K i
0
6. CONCLUSIONS A method has been developed for measuring the ~dhesion of colloidal particles to a fiat
/*5
6O
2.000
I
4,000
i
6,000
i
8,000
qJZl(mV)Z FIG. 15. Electrostatic force of repulsion (Fn) as a function of 61~, 61qJ2, and 62'; 1 mole/m3; 25°C. Journal of Colloid and Interface Science, VoL 55, No. 3, June 1976
676
J. VISSER
The adhesion of polystyrene particles to cellophane can be described by:
T V*
F a = (TrpflA)/(67rH
V¢
3)
-- ½~R~b12K.coth (aH)
= absolute temperature (K). = interaction energy at constant electrostatic charge (erg). = interaction energy at constant potential (erg). = elementary charge (4.803 X 10-1°)
[-29-] e
where A is the H a m a k e r constant of the system, / / the minimum separation distance between the adherents, R the particle radius, pc the radius of the contact area, E the dielectric constant of water, K the reciprocal double-layer thickness and ~klthe zeta potential of cellophane. In contrast to the adhesion of carbon black particles to cellophane (1), where the contribution of a possible intimate contact area to the adhesion can be neglected and the adhesion can be described by the sphere/plate model, in the adhesion of polystyrene particles to cellophane, the reverse holds and the adhesion is as regards Van der Waals forces determined predominantly by the contribution of the " i n t i m a t e " contact area. Whereas in the adhesion of carbon black particles to cellophane, the electrostatic repulsion force can be obtained from the DLVO theory under the condition of constant charge during the interaction (2), in the adhesion of polystyrene to cellophane, this force slightly differs from the DLVO theory, probably due to the deviation from the ideal sphere/plate model. In both cases, the potential of the flat surface (cellophane) determines the magnitude of the interaction. APPENDIX: NOMENCLATORE A
FA Fe FH Fa Fvaw H
= = = = = = =
R Re Ri R0
= = = =
H a m a k e r constant (erg). force of adhesion (dyn). centrifugal force (dyn). hydrodynamic force (dyn). force of electrostatic repulsion (dyn). Van der Waals attraction (dyn). distance of minimum approach of adherents (m). radius of spherical particle (m). Reynolds number (1). radius of inner cylinder (m). radius of outer cylinder (m).
Journal of Colloid and Interface ~icn~, Vol. 55, No. 3. June 1976
(esu). k ni.0 v z
K
= Boltzmann constant (1.381 X 10-26) (erg K-l). = bulk concentration of ion i (mole m-3). = velocity of liquid (m sec-1). = valency of ion (1). = double-layer thickness (m). = dielectric constant (1). = dynamic viscosity of solution (P). = reciprocal double-layer thickness
(m-l). O r ~o 1, 2 i 0
= = = = = = = =
kinematic viscosity of solution = ~/p. mass density of solution (kg m - 0 . wall shear stress (dyn). (zeta) potential (V). angular frequency of cylinder (rpm). referring to particle 1, 2. referring to entity i. referring to bulk (concentration). REFERENCES
1. VISSER, J., J. Colloid Interface Sci. 34, 26 (1970). 2. VlSSER, J., in "The adhesion of colloidal particles to a planar surface in aqueous solutions," Ph.D. thesis, Council for National Academic Awards, London, 1973.
3. LOEB,A. L., 0VERBEEK,J. TH. G., AND WIERSEMA, P. H., in "The Electrical Double Layer around
4. 5.
6. 7. 8. 9. 10.
11. 12.
a Spherical Colloid Particle," The M.I.T. Press, Cambridge, Massachusetts, 1961. PETERSON,G. C., personal communication. SMITH,A. L., in "Dispersion of Powders in Liquids" (G. D. Parfitt, Ed.), p. 39. Elsevier, Amsterdam, 1969. GOLDMAN, A. J., Cox, R. G., AND BRENNER, H., Chem. Eng. Sci. 22, 653 (1967). O'NEILL, M. E., Chem. Eng. Sci. 23, 1293 (1968). VAN LOOKERENCAMPAGNE,N., Thesis, Groningen, 1966. WENDT, F., Ing. Arch. 4, 577 (1933). VlSSER, J., in "Surface and Colloid Science," (E. Matijevic, Ed.), Vol. 8, Chap. 1, Wiley Interscience, New York, 1976. KRUPP, H., Advan. Colloid Interface Sci. 1, 111 (1967). DAnNEKE,B., J. Colloid Interface Sci. 40, 1 (1972).
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Journal of Colloid and Interface Science, Vol. 55. No. 3. June 1976