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M a g n e t i c E f fec ts in Par t i c le A d h e s i o n

I. Kinetics of HematRe Particle Deposition on Stainless Steel 1

M. F. HAQUE, 2 N. KALLAY, 3 V. PRIVMAN, AND E. MATIJEVIC 4

Departments of Chemistry and Physics, Clarkson University, Potsdam, New York 13699

Received March 1, 1989; accepted September 29, 1989

The effect of the magnetic field on the deposition of uniform spherical hematite particles of submicron size on stainless steel beads was studied by the packed column technique. The kinetics was interpreted on the basis of the theoretical models for convection diffusion in terms of the excluded area and collision efficiency. The application of high intensity magnetic field reduced the effective excluded area to 0 and increased significantly the collision efficiency. The interaction energy was calculated by considering magnetic, electrostatic, and dispersion contributions. The enhanced deposition was explained in terms of particle accumulation in the secondary minimum created by the magnetic field. This finding was supported by detachment measurements. © 1990 Academic Press, Inc.

INTRODUCTION

The effect of a magnetic field on particle deposition and release is of interest in appli- cations, such as filtration of solid/liquid suspensions or separation of magnetic particles from nonmagnetic materials, as well as in the general theory of adhesion phenomena ( 1-5 ). I f the system is of suitable magnetic charac- teristics, the imposition of the field promotes the attachment.

The adhesion of colloidal particles (e.g., hematite or magnetite) was investigated using wire filters (1, 5) and packed column techniques (2-4) . The latter method is convenient as the process can be described by the convection diffusion model (6 -8 ) . The mechanism of at tachment depends on the particle size; if the latter is in the submicron range, diffusion prevails.

This work deals with the kinetics and mechanism of the deposition of uniform col-

Supported by Air Force Contract F49620-85-C-0142. 2 Part of the Ph.D. thesis of M.F.H. 3 On leave of absence from the University of Zagreb,

Zagreb, Yugoslavia. 4 To whom correspondence should be addressed.

0021-9797/90 $3.00 Copyright © 1990 by Academic Press, Inc. All fights of reproduction in any form reserved.

loidal spherical hematite particles from an aqueous suspension on steel beads in a packed column under the influence of a superimposed magnetic field. Extensive studies on the adhesion phenomena of the same system in the absence of an external force were carried out previously (9-11 ). When the particles and the substrate bear potentials of the same sign, the energy barrier greatly reduces the rate of deposition. Once this barrier is sufficiently lowered by the addition of electrolytes, free diffusion governs the process ( 11 ).

Deposition of particles on surfaces of op- posite charge is enhanced at low ionic strengths due to the electrostatic attraction at long separations (12). I f the ionic strength is >1110 -3

mol dm -3, the double layer thickness is reduced, resulting in a steep potential drop in the diffuse layer. As a result, attraction acts only over close separations and does not ac- celerate the deposition process (13) above the rate expected in the absence of electrostatic interactions.

To examine the effect of another kind of force on the adhesion phenomena, it is convenient to apply a magnetic field to a known system. In such a case, the kinetics of depo-

36

Journal of Colloid and Interface Science, Vol. 137, No. 1, June 1990

http://dx.doi.org/10.1016/0021-9797(90)90041-L

http://dx.doi.org/10.1016/0021-9797(90)90041-L

M A G N E T I C E F F E C T S I N P A R T I C L E A D H E S I O N 37

sition can be investigated by varying the magnitude of the force over a broad range. As a rule, the rate of particle attachment decreases with their accumulation at the surface, due to surface charge reversal (14, 15) and /o r excluded area ( 13, 16, 17).

In this work, the deposition of colloidal hematite particles on steel beads was followed as a function of time and of the flow rate of aqueous dispersions passing through a column packed with steel beads in the presence of a magnetic field, the strength of which was var- ied. The data were used to evaluate the rate constant of attachment and the effective excluded area. The results were interpreted by considering the influence of the magnetic field on the interaction energy function between the particle and the substrate.

In the separation processes, it is also of interest to release adhered particles from the collector surface. For this reason, detachment measurements were carried out, which offer some insight into the state of the attached particles under the influence of the magnetic field.

E X P E R I M E N T A L

Materials

Spherical hematite particles of radius rp = 0.041 + 0.007 #m (density pp = 5.24 g cm -3) were prepared by hydrothermal aging of an acidified aqueous solution of FeC13 and purified as described earlier (18). The isoelectric point (IEP) of such particles was at pH 7.1 (19), and their magnetic properties were extensively evaluated (20).

The stainless steel spheres (C 1018, Nuclear Metals) had a radius rb = 33.4 + 4 .4 / ,m and a density pb = 8.3 g cm -3.

In principle, electrokinetic potential of the steel surface cannot be determined by con- ventional techniques due to the electric con- ductivity of the metal. An adhesion method (21) was instead employed to obtain the IEP of the heads used in this work. The procedure is based on independent measurements of the deposition of positively and negatively charged latex particles on metal beads as a function of

the pH at a low ionic strength. No surfactant should be present in the suspension and the sign of charge of the latex must be independent of the pH. As the magnitude of the charge of metal beads is altered with the addition of an acid or a base, the amount of adhered latex changes accordingly; i.e., the attachment of the positively charged particles decreases with the lowering of the pH and vice versa. The IEP is then obtained as a cross-section point of two adhesion curves of oppositely charged latex on the same substrate.

Figure 1 shows the results obtained with the steel beads used in this work. The IEP is at pH 5.7. This value is in good agreement with the results obtained earlier, in which the micro- electrophoresis was used on tiny scrapings separated from the steel beads (9).

Methods

The rate of particle deposition was measured at 22°C using the packed column described in detail earlier (9-11 ). The glass column of the inner cross-section area S = 0.79 cm e was packed with 3 g of beads, with the void volume fraction q~ = 0.40. The flow rate was regulated by a peristaltic pump.

The uniform transverse magnetic field was applied by placing the column between the poles of an electromagnet, the field strength of which ranged from 0 to 5000 Oe.

The number concentration of inlet and outlet suspensions was determined by light scattering from an appropriate calibration curve on the basis of the density and the size of hematite particles. The'inlet solutions con- tained 4.4 × 1015 particles/m 3, except for measurements at high ionic strengths, where more dilute dispersions were used. The concentration of particles in the effluent was determined as a function of time or flow rate under the influence of different strengths of the imposed magnetic field.

The dependence of the deposition rate on the surface coverage was measured at a constant flow rate for extended periods of time, while the flow rate effects were examined at

Journal of ColloM and Interface Science, Vol. 137, No. 1, June 1990

38 HAQUEETAL.

1.5

"" l.O :::I

c ° ~

cO

0.5

f ~'TEEL -LATEX l-I positive

0 negot ive pH (iep) : 5.7

0 I I L i I t 2 4 6 8 l0 12

pH FIG. 1. Rate of adhesion expressed as lg(CiJCou~) for positively ((2) and negatively (O) charged IDC

surfactant-free latex particles on steel beads (C 1018) as a function of pH. The crossover gives the isoelectric point at pH 5.7. Conditions: mass of steel, 10 g; flow rate, I? = 3 cm 3 min-1; initial concentrations of acid (HNO3) or base (NaOH), I × 10 -3 tool dm-3; temperature, 22°C.

the early stage of the particle at tachment process. The latter data can be used to discrimi- nate between different possible reaction mechanisms (22).

In experiments requiring high ionic strength (NaNO3), the electrolyte solution and the hematite dispersion were mixed in a specially designed chamber just before entering the col- u m n (9-1 t, 13). In doing so, the coagulation effect was minimized. Calculations based on the Smoluchowski equation for rapid coagulation (23) showed that, with the experimental setup employed, less than 5% of the particles underwent aggregation.

All experiments were carried out at pH 11 (NaOH) , which resulted in the negative charge on both surfaces.

The effect o f the magnetic field strength, under which the particles were deposited, on the detachment of particles in the absence of the field was examined as follows. In the first step (sample preparation), the hematite suspension (4.4 × 10 ~5 par t ic les /m 3, p H 11, I


= 0.002 mol dm -3) was passed for 6 h through the column filled with 3 g of steel beads at a flow rate of 1.9 c m 3 / m i n under the influence of an applied magnetic force of different strengths. The particle release was then measured once the field was removed by rinsing the column with a solution of the same p H and ionic strength. The results were expressed in terms of the fraction of detached particles as a function of time.

INTERPRETATION OF DATA

In the particle deposition experiments, the following quantities were controlled: particle number concentration of inlet suspension (Cin), pH, electrolyte concentration, and volume flow rate (1?) of suspension, which passes through the column of the cross-section area S and the void volume fraction ~b. The column was packed with spherical steel beads of radius rb and total mass rob. The particle number concentration in the outlet suspension (C out) was measured as a function of time, t.

M A G N E T I C EFFECTS IN PARTICLE ADHES ION 39

The kinetics of deposition (9-11) is described by the rate constant of attachment or mass transfer coefficient (k) defined as

- d C / d t = A ~ k C / V , [1]

where As is the available surface area exposed to a volume element of suspension V. Particle number concentration C corresponds to a fluid element moving through the packed bed. As may be calculated from the total collector surface area by subtracting the area occupied by deposited colloid particles. The latter is equal to the product of the surface area per deposited particle, a, and their number. The maximum value of particle surface concentration, Fmax, is equal to

rmax = 1 / a . [21

By taking into account the excluded area and the properties of the column, and by neglecting any detachment, one obtains (13)

G i n - G u t Q n a k h In - In b + - - Cinakt, [3]

Gout /)

where

b = exp[3kh(1 - ~ ) / ( r b O v ) ] -- 1 [4]

and the velocity of the suspension flow, v, is given by

v = ~ / ( s ~ ) . [5]

The height of the packed column, h, is related to the mass of beads mb as

h = m b / [ P b S ( 1 -- ~b)]. [6]

The slope and the intercept of the linear plot of ln[(C~, - Cout)/Cout] vs t (Eq. [3]) yields the rate constant of attachment, k, and the excluded area per particle, a.

For the initial state of zero coverage, Eqs. [ 3 ] - [ 5 ] yield

3h(1 - ~b) In (C~n/Co~t) - k

rbq~ V

- 3ink k. [7] p b r b V

The rate constant of attachment, k, may be expressed as the product of the collision efficiency, a, and the rate constant of rapid deposition (kdie), the latter being the rate constant of attachment in the absence of a repulsion barrier:

k = O~kdiff. [ 8 ]

In a column packed with spherical beads kdie is given for a convection diffusion process by Pfeffer, Happel, and Ruckenstein (6 -8) ,

kaif f = 0 .624D2/3rbz /3 ( f l v ) I/3, [9]

where/3 is the porosity term defined by

/3 = [1 - - (1 - - q ~ ) 5 / 3 ] / [ 1 - - 1.5(1 - q~)1/3

+ 1 . 5 ( 1 - 4 ) 5/3 - ( 1 - 4)21 [10]

(for 4 = 0.4,/3 = 38).

The diffusion coefficient D of spherical colloidal particles of radius r o depends on the vis- cosity of the medium n:

O = kBT/(67rTlrp). [t 1]

Equations [ 7 ] - [ 9 ] yield

In(Gin/Gout) = 1.870~D2/3rgS/3 mbPb 1

× f l l / 3 0 - l / 3 s - l / 3 v - 2 / 3 . [ 12]

Equation [ 12] is based on the assumption that convective Brownian diffusion is the dominant mechanism in the deposition process. Thus, a linear plot oflg[ln(Cin/Cout)] as a function of lgl? with a slope o f - 2 / 3 woul d suggest that the condition of convective diffusion prevails. Other mechanisms, such as interception and sedimentation, which are dominant for deposition of larger particles ( rp > 1 #m), would be characterized by a different slope (22), since the rate constants for these processes are not proportional to v 1/3 (Eq. [9]) .

The kinetics of deposition could also be described by the relative rate with respect to fast attachment in the absence of a repulsion bar- tier. Since a = 0.5 was obtained for the deposition at high ionic strength, the relative rate


40 HAQUE ET AL.

is simply twice the value of a as defined by Eq. [121.

RESULTS

Figure 2 shows deposition data as a function of the flow rate, plotted according to Eq. [ 12], at two different applied transverse magnetic field strengths. Solid lines that best fit the data are drawn with a slope o f - 2 / 3 , as required for convective diffusion, for a equals 1, 0.47, and 0.26, respectively. Similar data were obtained at lower values of H, including H = 0.

The relative number of deposited particles (O) is shown as a function of time in the absence and in the presence of weak (Fig. 3) and strong (Fig. 4) magnetic fields. The average surface coverage, O, is related to a saturated monolayer in a cubic packing by

0 = N4rao/A, [13]

where A is the total surface area of the collector and N is the total number of deposited particles. The slope of the lines in these plots is

proportional to the rate of particle deposition. At H = 0 the rate obviously decreases as the coverage increases, while hardly any change is detectable when the magnetic field is sufficiently strong.

The influence of the surface coverage on the rate of adhesion can be expressed in terms of Eq. [3 ] and the corresponding plots are shown in the top panels of Figs. 3 and 4. The corresponding calculated values of a and a are given in the legends. It is noteworthy that at the high magnetic field strength (Fig. 4) the average value of O exceeds unity, which in- dicates a multilayer of particles on the surface. This finding is corroborated by the zero value of a. Electron micrographs in Fig. 5 illustrate these systems in the absence and in the presence of the magnetic field ( H = 2000 Oe); the crowding of particles is clearly shown in the latter case.

Measurements of the type displayed in Figs. 3 and 4 were carried out at additional magnetic field strengths and the results are summarized in Fig. 6 in terms of the collision efficiency,

0.4

•r,•° 0,2 w-

,m

(n w"

t ~ 0

-%

HEMATITE - STEEL

pH : 11

/ : 2 - 16 3 mot dm 3

-0.2 0 0.2

o ( : 1

o 2000, Oe

, :0.20 o °N

J 0,4 0.6 0.8 1.0

Ig ( V ' / c m 3 rain -1 )

FIG. 2. The effect of the flow rate on the deposition of hematite particles on steel beads plotted according to Eq. [12]. Conditions: hematite, rp = 0.041 t~m, Cin = 4.4 × 1015 m-3; steel beads, rb ---- 33.4 #m, mb = 3 g; pH 11; ionic strength, I = 2 × 10 -3 tool din-3; temperature, 22°C; magnetic field strengths, 1000 Oe (O) and 2000 Oe (©). The straight lines were calculated assuming collision efficiencies of a ~ 1, 0.47, and 0.26.


MAGNETIC EFFECTS IN PARTICLE ADHESION 41

-1

i

[ i

HEMATITE - STEEL

pH = 11 " ' , , , , ~ Oe I = 2 ,, 16 3 mot drn 3

(: 0 0e -3

I I I I I~""~

0.15 ~ 0 0e

0.10 r ' 4 " " " " -

0.05

0 0 20O 40O 60O 80O

t / r a in

FIG. 3. The effect of surface coverage on the rate of particle deposition. Lower: average surface coverage, (9, as a function of time in the absence (O) and in the presence (H = 250 Oe, [3) of a magnetic field for the same system as in Fig. 2, at a constant flow rate of 1.9 cm 3 min -~. Dashed line at zero magnetic field represents the initial rate of deposition. Upper: the plot of the same data in terms of Eq. [3]. Evaluated quantities--H = 0: c~ = 0.015, a = 9.6 r~r; H = 250 Oe: a = 0.06, a = 2.9 rpZTr.

a , and the relat ive exc luded area. The fo rmer increases wi th H while the la t ter (a/rZp~r) decreases rap id ly to zero as H approaches 500 Oe.

I t should be no ted tha t par t ic le depos i t ion exper imen t s with the same h e m a t i t e / s t e e l system at high ionic s trength in the absence o f the magne t i c field also y ie lded a ~ 0.5.

F igure 7 gives the results o f de t achmen t ex- pe r imen t s at H = 0 for samples p repared by depos i t ion at 0, 250, and 2000 Oe. On ly 30% o f the par t ic les were released, i f they were depos i ted wi thou t the inf luence o f the magne t i c field. Once the magne t ic field exceeds 250 Oe in the course o f deposi t ion , the d e t a c h m e n t efficiency is 100%. Exper imen t s carr ied ou t wi th samples loaded unde r the inf luence o f H

= 375, 500, 750, 1000, and 1500 Oe showed the same results as those p lo t ted for 2000 Oe.

TOTAL INTERACTION ENERGY

In order to in te rpre t the effect o f the magnetic field on the kinet ics o f par t ic le deposi- t ion, it is necessary to cons ider the in te rac t ion energy func t ion for the system, E t o t , which in- cludes the magne t ic (Em), electrostat ic (Ee) , and dispers ion (Ed) contr ibut ions , together with the shor t - range repuls ion (Er) :

Etot =Em + Ee + Ea + Er. [14]

The shor t - range repuls ion m a y be approx-

ima ted by a hard wall at Xo; i.e., Er ( x < Xo) = oo and E~ ( x > xo) = 0. Since x0 ~ 5 to 10

for systems o f interest (23) , while all rele- vant p h e n o m e n a in this s tudy occur at dis- tances o f the o rder o f at least 100 A, Er is neglected in the present calculat ions.

i ¢- .m

e "

2

pH =11 J= 2 x 10 -3 tool dr~ 3

1 I 1

1 I

HEMATITE - STEEL

H-- 2000 Oe

I I

J J

J J

0 , J ~ " - t t t t 1 0 20o ,~0o 600 800 1000

t/rain

FIG. 4. The same plots for the system described in Fig. 3 at H = 2000 Oe. Evaluated quantities: a = 0.47, a n 0 .


42 H

AQ

UE

E

T A

L.

O

,.m

¢)

o~ o= t~

< off o,o

L~

e-~

O

O

'-6 ¢.2.

OO

O

• =- ¢2

H

Journal of Colloid and Interface Science, V

oI. ! 37, No.

1, Jun

e 1990


I0(~" HEMATITE-S TEE---~

L I pH:11 I: F I I: 2,, ~3 ~l dm ~

0 ~ o, - 0.6 1

8

0 ' 2 / / I ,,i

0 I000 2000 5013o HlOe

FIG. 6. The effect of the magnetic field strength on the collision efficiency, a (lower part), and the relative excluded area, a/r~Tr (upper part), for the same system as illustrated in Figs. 3 and 4.

The electrostatic contr ibution was evaluated in the Hogg, Healy, and Fuers tenau ( H H F ) app rox ima t ion (24) , which in CGS reads

~rbrp I 1 + e x p ( - K x ) Ee = 4( rb + %) L2~b~vln 1 -- exp( - -Kx)

--+(~bb 2 + ~ b 2 ) l n ( 1 - e - 2 " x ) ] , [151

where e is the dielectric constant ( o f water) , ~bb and ¢/p are surface potentials, x is the distance o f separat ion, and K is the inverse Debye length.

The _+ sign in Eq. [15] applies to the constant potent ial ( + ) and constant charge ( - ) models. The fo rmer case was found to bet ter describe adhesion p h e n o m e n a (25) . This expression is valid for low surface potentials ( < 2 5 m V ) and for Kr > 10 (relatively high ionic strength a n d / o r relatively large particles).

The dispersion contr ibut ion, due to Lon- d o n - v a n der Waals interactions, is given by (26)

AH E d -

6

[ 4rbrp[X 2 + 2X(rb + to) + 2rbrp]

x[x + 2(rb + rp)] } [16] + In (x + 2rb)(X + 2rp) '

where the overall H a m a k e r constant AH is the " m e a n " of individual constants for particles (p ) , water (w) , and beads (b ) ,

AH = ( f ~ v - - f Y £ ) ( V ~ b - - ~ ) . [17]

!

HEMATITE - STEEL

O1 LU d n,

c3 I.U

=, hl n~

U_ 0

Z 0

o.

/ = 0.002 rnol dm "3

[~ DEPOSITION at H : 0.6 O 0

A 250 0e a 2000 0e

0.4 ~DETACHMENT a t H =0

0.2

0 o

| : I

20 40 t / min

FIG. 7. Fraction of released hematite particles from steel beads as a function of time in the absence of magnetic field at pH 11 and I = 0.002 tool dm -~. Flow rate, 1.9 cm 3/min. The sample was prepared by passing the suspension of hematite particles (C = 4.4 × 1015 m -3) atthe same ionic strength for 6 h in the absence of and under the influence of the magnetic field of different strengths-- H/Oe: 0 (©); 250 (A); 2000 ([~).

Journal of Colloid and Interface Science, VoL 137, No, 1, June 1990

44 HAQUE ET AL.

In the absence of the magnetic field, any discrepancy between the calculated and measured quantities can be compensated for by adjusting the Hamaker constant. It was also shown that the interaction energy at the maximum is a linear function of the particle size for the plate / sphere system (27), which would imply no barrier for extremely small particles and an infinite barrier for very large particles. As a consequence, small particles should deposit at a rapid rate, whereas the large ones should not adhere at all, which is obviously not the case. Any compensation made by al- tering the Hamaker constant for systems of different particle size would require that An vary with size, which is physically meaningless.

The magnetic energy, Era, due to the interaction of a particle and a bead of magnetic moments t~p and t~b, is given by

E~ = /Zb/3/tP 3(/tb l)(/~p'[ 5 l) , [18]

where CGS units are used. Here I is the center- to-center separation vector. The magnetic interaction is strongly orientation dependent. For particles (and beads) magnetized along the applied field (see below), the magnetic interaction is most attractive for an approach along the bead axis in the direction of the field. In our case, the latter is transverse to the flow of the fluid. For such geometry

E m = - 32"lr2r3 r3ppbPp~bfrP , [19] 9(rb + rp + x) 3

where p b and P p are the densities, and ~b and ~rp are the corresponding mass-specific magnetic moments (i.e., magnetic moments per unit mass), for beads and particles, respectively. The values of ab and ~p are not constant. For beads, the main source of variation of ab is the magnetization due to the external field, H. The directly measured function ab(H) is presented elsewhere (28). For field strengths of interest in the present calculation, one can use the approximate (to within several per- cent) expression

Journal of Colloid and Interface Science, VoL 137, No. 1, June 1990

~u/(Oe cm 3 g - l )

= 0.252 + 0.0314H/Oe. [20]

The evaluation of ap in Eq. [ 19] is more complicated. There are two important effects to consider. First, the effective field acting on a particle consists not only of the applied field, H, but also of the field due to the magnetized bead, which cannot be neglected. In the uniaxial geometry considered in connection with Eq. [19 ], the bead field is given by

87rCrb0b r3 Hb -- 3(rb + X) 3 -- 87r°'bPb/3' [21]

where the x-dependence can be neglected be- cause calculations in this study are for x ~ rb. For example, for applied fields of the order of 2000 Oe, one estimates Hb ~ 3600 Oe. For steel beads, Hb is comparable to H in order of magnitude, for all fields of interest. Hence, we define

H~fr = H + Hb. [221

The direct measurement of a for the hematite powder of particle size rp --- 0.041 tzm underestimates the effective value of a in solution (20), due to uniaxial magnetic properties of this iron oxide. Recent experiments in solution (29) indicate that a better approach is to regard hematite particles in this size re- gime as single domain, with superparamagne- tism due to particle rotation, but not due to demagnetization [ see (29) for details ]. Thus, we take

_ [4~rr3anemppHefc'~ ~p = ~ner.L 1 ~ J , [23]

where L(y) = cotanh(y) - y-1 is the Langevin function and ffhem is the magnetization of the bulk hematite which in turn can be repre- sented for fields of interest as (20)

O_nem/(O e cm 3 g - l )

----- 0.264 + 0.196 × 10-4aeff/Oe. [24]

M A G N E T I C EFFECTS IN P A R T I C L E A D H E S I O N 4 5

Note that Heff/kBT in Eq. [23] is multiplied by the "bulk" magnetic moment in the particle volume.

Thus, ~v has a nontrivial variation with the applied field and also with particle size. Nu- merical results suggest that for r v = 0.041 urn, with system parameters considered here, the superparamagnetic reduction described by Eq. [23 ] is important only for very small applied fields, i.e., H ~ 100 Oe.

D I S C U S S I O N

This study deals with a system in which the collector and the particles bear potentials of the same sign. Consequently, the existence of a repulsion barrier inherently inhibits the deposition.

Figure 8 illustrates calculations of the total interaction energy as a function of distance in the absence and in the presence of the magnetic field of four different strengths, using the Hamaker constant A n ~ 7 X 10 -2o j (25, 26 ).

4O

r, I . - .

• '~ 20

-40

-60

I 0

LLI

H E M A T I T E - S T E E L

H / 0 e :

0

500

I I I 500 1000

FIG. 8. Total interact ion energy as a function of distance at different magne t i c field s t rengths for the bead /pa r t i c l e

system in an aqueous electrolyte system using the following parameters : Ionic strength, I = 2 X 10-s mo l d in-3; 22oC;

A = 7 X 10-z° J; rp = 0.041 urn; r b = 33.4 #m; pp = 5.24 g cm 3; Pb = 8.3 g cm-S; ~pp = --40 mV; ~bb = --30 inV.

These data show that the absolute value of the magnetic contribution to the total interaction energy is essentially constant over the separation distance at which the electrostatic effect normally dominates (x ~< 500 A). Since Em does not change much with the distance near the surface, the value of E~ot at the maximum is lowered, but the barrier remains practically the same. Furthermore, a secondary minimum develops and extends far from the surface, the depth of which depends on H. At larger separations Ee and E~ tend to zero so that E, ot "~ Em.

The magnetic force (F = [dEm/dx[ ) decreases slightly with separation and increases with the strength of the magnetic field. For example, at 2000 Oe at the surface F = 1.3 X 1 0 -14 Nwhich is ~ 1000 times greater than the weight of the particle. According to Stokes, this force would result in a particle velocity of 17 ~tm s -x, which is small relative to the velocity of the liquid flow employed in this work (1000 u m s -1), but comparable to the rate constant for rapid deposition as calculated by Eq. [91 (3 um s - l ) .

The particles accumulate in the secondary minimum by convective diffusion. In the presence of a sufficiently strong magnetic field, the secondary minimum can be sufficiently deep to prevent their escape into the bulk (see Fig. 8). This conclusion is strongly supported by detachment experiments (Fig. 7). In the absence of the magnetic field, particles slowly deposit only in the primary minimum. This process is partially irreversible; on washing some of the attached particles may be released, and the process is characterized by a distri- bution of rate constants (13, 27). If the magnetic field is applied in the course of deposition, all attached particles are released instantly once the field is turned off. Such a mechanism is possible only if adhesion takes place in the secondary minimum, as indicated by the na- ture of the energy functions (Fig. 8).

Additional support for this conclusion is the direct observation of the aggregation of dis- persed hematite under the influence of a weak magnetic field (30); the particles from ordered


46 HAQUE ET AL.

agglomerates. If mutual electrostatic repulsion (low ionic strength) prevails, the process is re- versible; i.e., the chains are broken up by the application of a perpendicular magnetic field of low strength (a few Oe). Furthermore, particles within the ordered clusters show oscil- latory thermal motion.

The particle accumulation in the secondary minimum can also explain the observed changes in the effective excluded area (a) with the application of the magnetic field. The large value of a at H = 0 is caused by a higher energy barrier between a particle approaching already adhered particles, as compared to those ad- vancing toward the bare surface. When a sufficiently strong magnetic field is applied, caus- ing a significant accumulation of particles in the secondary minimum, the energy barrier does not affect the rate of deposition in the "polar cap" regions of the beads along the applied field axis. As a result, the rate of attachment is equal for the bare and occupied surfaces, which brings about a formation ofmul- tilayers. This effect is signified by the disappearance of the excluded area, i.e., a = 0, as is clearly shown in Fig. 6.

Figure 9 shows that the interaction energy well does not differ much for the 1 st and the I 0th particle, which offers an explanation for their agglomeration at the surface. The calculations for the case N -- 10 were performed for a single particle interacting with a chain of nine particles attached in an array perpendicular to the surface along the direction of the magnetic field. The total energy was obtained by the summation of the bead and single particle contributions and is plotted as a function of the surface-to-surface separation of the particle from the end of the tail (x). It is apparent that the energy barrier prevents deposition in the primary minimum, but the deep secondary minimum enables the formation of multilayers.

The calculation given in Fig. 9 explains, in principle, the experimental results shown in Fig. 6. For example, at H = 50 Oe essentially no secondary minimum develops. Under these conditions, the excluded area is large, while

Journal of Colloid and Interface Science, Vol, t 37, No. 1, June 1990

60 HEMATITE- STEEL

A 40 . ~ ~

Ill --- N=! N =10

20

HlOe : 50

0

- 4 0 " " - . . . . . .

I I i i

500 1000 x/~

FIo. 9. The same plot as in Fig. 8 for the deposition of one particle to the bead surface (N = 1, dashed lines) and for the 10th particle attached to a chain of nine particles (N = 10, solid lines).

the rate of deposition (a) is small. In contrast, at H ~ 1000 Oe the deep minimum persists even when multilayers are formed, corresponding to zero excluded area and accelerated deposition.

This work has offered a viable explanation of the adhesion effects as influenced by a magnetic force. The particles are collected in a secondary minimum, the depth of which can be manipulated by varying the applied magnetic field. Since the repulsion barrier persists under these conditions, the transfer of particles into the primary minimum is rather infrequent, relative to the flux of incoming particles. As a consequence, the experimentally measured rate of adhesion is determined by the retention efficiency caused by the existence of the secondary minimum. At a sufficiently high strength of the magnetic field, this efficiency, which corresponds to a, tends to 100%.

ACKNOWLEDGMENTS

The authors are indebted to Professor S. Arajs for the use of his facilities and to Professor M. Ozaki for helpful


discussions. We are also grateful to Miss Z. Torbi6 for carrying out the determination of the isoelectric point of steel.

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Journal of Colloid andlnterface Science, Vol. 137, No. 1, June 1990

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