The aggregation of large polystyrene latex particles

Colloids and Surfaces,

23 (1987)

Elsevier Science Publishers

273-299

B.V., Amsterdam

273 -

Printed

in The Netherlands

The Aggregation of Large Polystyrene Latex Particles PAUL A. REYNOLDS* International Kingdom)

Paint PLC, Stoneygate

Lane, Gateshead,

Tyne and Wear NE10 OJY (United

JAMES W. GOODWIN School of Chemistry,

(Received

University

22 November

of Bristol, Bristol BS8 ITS (United

1985; accepted in final form 19 September

Kindom)

1986)

ABSTRACT The rate of aggregation of monodisperse polystyrene latices in various electrolyte concentrations has been studied. A charge stabilised polystyrene latex dispersion was used and comprised 3.47 pm spheres in water. In perikinetic conditions a finite amount of aggregation of the latex dispersion occurred until a plateau in the single particle number concentration was reached. It was found that a critical flocculation concentration, or at least a narrow range of concentrations existed. Below this region, slow aggregation took place and above, the aggregation was more rapid. Orthokinetic aggregation was studied as a function of shear rate and electrolyte concentration. The initial rate of aggregation increased with increasing electrolyte concentration and shear rate. The aggregation kinetics of the dispersion in prolonged periods of Couette flow at low shear rates was studied. It was shown that the plateau in the single particle number concentration could be altered to higher or lower values depending on the conditions used. The phenomena observed were explained in terms of the secondary minimum flocculation.

INTRODUCTION

For large polystyrene latex particles stabilised by forces of electrostatic origin, the DLVO theory [ 1,2] predicts the existence of a deep attractive energy minimum next to the particle surface, a repulsive energy maximum a short distance from the surface and a shallow attractive energy minimum beyond the energy maximum. Thus particles may be aggregated (a term used for an undeclared particle association) in either the primary minimum state or the secondary minimum state. The two types of aggregation are distinguished by convention [ 31. Primary minimum association is termed coagulation whilst secondary minimum association is termed flocculation. The primary minimum *To whom all correspondence

0166-6622/87/$03.50

should be addressed.

0 1987 Elsevier Science Publishers

B.V.

274

state has attracted most attention experimentally in colloid science and so it is intended to concentrate on the secondary minimum in this paper. The secondary minimum can only be studied at electrolyte concentrations below the critical coagulation concentration (CCC ) , that is below the electrolyte concentration at which the primary energy maximum is approximately zero. Cornell et al. [ 41 have studied flocculation, in the secondary minimum regime, using a method of direct observation of large monodisperse electrocratic polystyrene latex particles. The method used observed the rates of flocculation and, furthermore, provided evidence of the existence of a plateau in the single particle population with time curves for flocculation. The work reported here was designed to study the secondary minimum flocculation phenomena further in both perikinetic and orthokinetic conditions [ 51. Perikinetic conditions obtain when the movement within a dispersion originates from both translational and rotational Brownian movement of particles. Orthokinetic conditions obtain when the particle movement is caused by flow of the fluid medium. Colloidal stability The DLVO theory of colloid stability [ 1,2] assumes that the electrostatic repulsive potential energy, V,, and the van der Waals’ attractive potential energy, V,, are additive and produce the total potential energy VT. Derjaguin [ 6,7] obtained an approximate solution for the repulsive potential energy around the interacting spheres, each of radius a, and surface separation H, such that for Ka > 10

vR=2g ln[l+exp(

-KH)]

where (u, is the surface potential of a particle and ICthe Debye-Hiickel reciprocal length. Schenkel and Kitchener [8] have given a convenient approximate form of the attractive potential energy of interaction for large particles v A

= -Aa H

2.45

2.17

0.59 +[ 6Op - 180~1~ 420~1~

1

(2)

where p=-

2nH 1

(3)

A is the dispersion wavelength (taken as 100 nm) and A the combined Hamaker constant. This expression includes a correction for the retardation effect. Assuming that it is reasonable to equate the zeta potential, [, with the surface

potential yy, and that the Hamaker constant for polystyrene in water is of the order of 6 x 10pzl J, then the pair potentials of interaction can be calculated. Primary

minimum

coagulation

The kinetics of an aggregation process have two determining factors; firstly the rate at which the particles collide and secondly their interaction on collision. In perikinetic conditions the collision rate is determined by the Brownian movement of the interacting bodies. Smoluchowski [ 91 considered the case of no repulsion between colliding spheres. This is in the region above the CCC and is therefore rapid coagulation. If a spherical particle of radius a has an infinitely deep potential well at a distance da from its surface, then two particles whose centres are separated by 2(a+da)

=R

(4)

or less will coagulate irreversibly. For large particles a first approximation is that R = 2a. The number of collisions per unit time is obtained by solving the diffusion equations yielding the flux of particles through a sphere of radius a + da. If n, is the initial concentration of monodisperse particles and Do their diffusion coefficients from Ficks first and second laws of diffusion, the particle collision rate, J, is given by J=hR

Do no

(5)

The initial rate of disappearance of single particles with time is given by -

=8n

RD,

n:

(6)

where n, is the concentration of single particles and n2,= no at t = 0. Fuchs [lo] considered the case where particle interactions are important and found the total flux JT

JT=

8xRDo

w

no

(7)

Thus the rate of coagulation has been slowed down by an amount W, where W is the stability ratio W= 2a

m ew s 2a

(VW”) r2

dr

(8)

where V is the potential energy of interaction and kT the thermal energy. The stability ratio is therefore the ratio of the rate of fast coagulation to slow coagulation. The initial rate of disappearance of singlets is

216

8zRDOng

9dn

4 > dt

t_,,=

W

Overbeek [ 111 and Reerink and Overbeek [ 12 ] have given approximate solutions for W and based their solution upon the height of the primary maximum since this forms the major energy barrier to coagulation. Secondary minimum flocculation The flocculation of particles into the secondary minimum state can be considered in the same way as coagulation into the primary minimum. Here the primary maximum is assumed to be insurmountable and thus the distance of closest approach of two particle centres is given by R = 2a + HSmin

(10)

where HSminis the distance from the particle surface at which the secondary minimum occurs. Because the secondary minimum is relatively shallow and comparable with M’, clearly there will be particles with sufficient energy to escape from the flocculation state. If E is the fraction of total particles arriving at the secondary minimum forming aggregates then 8;rrRD,nEE W

(11)

Assuming a Boltzmann distribution of energies, those particles possessing sufficient energy to escape the secondary minimum are exp ( V,,i,/kT) and (Vs,i,/kT)

E=l-exp

(12)

The secondary minimum stability ratio is therefore given by R

oo exp ( V/W

WSmin

=

[

1 -exp

dr

2

s

( V~,i,,kT)

]

(13)

which is exactly analogous to the case given by Frens [ 131 when considering the peptization process generally. Since V is negative for all values of r greater than R the numerator is less than unity, consequently the stability with respect to flocculation arises when the denominator is small. Therefore an approximation in many cases can be given by 1 WSmin

=

1+ exp ( VsminlkT)

04)

271 EXPERIMENTAL

Materials

All water used was doubly distilled, the second distillation being carried out in an all Pyrex apparatus. The sodium chloride and sodium hydroxide used was BDH Analar grade reagent. The deuterium oxide used was Fluorochem 99.9% spectroscopic reagent. Polymer

latex characterization

The polystyrene latex used in this work was prepared by the seeded growth mechanism [ 141 in the absence of surface-active agents. The particle size of the latex was determined by transmissian electron microscopy [ 151. The number average diameter of the particles was found to be 3.47 ,umwith a coefficient of variation on the mean diameter of 5.6%, showing the particles to be monodisperse. The electrophoretic mobility of the latex particles was measured as a function of sodium chloride concentration at pH > 8. Determinations were made using a Rank Brothers Mark II microelectrophoresis apparatus in conjunction with a four-electrode van Gils and Kruyt type electrophoresis cell [ 171. Particle mobilities were obtained in several electrolyte concentrations from which the zeta potentials were calculated using the Smoluchowski equation (Ica>>lOO) U=$$

(15)

and applying corrections for relaxation and retardation effects when Ku < 100 using the tables of Ottewill and Shaw [ 161. The standard deviation of the measurements made was found to be _’ 5%. The results are plotted in Fig. 1. Clearly the values of the zeta potentials are high for this electrocratic polystyrene latex and go through a maximum in the region 1O93-1O-2 mol dme3 sodium chloride. The critical coagulation concentration was determined by pipetting a known volume of latex dispersion into each tube of a series of tubes, and to each of these an equal volume of salt solution was added. The concentration range of electrolyte was chosen to be sufficiently broad to include the value of the CCC, typically 0.3 mol dme3 for symmetrical monovalent electrolytes [ 111. The dispersions were shaken and allowed to stand for 2 h, then shaken again. The volume of the sediment was measured after one day (but was stable for periods in excess of one year). The sedimentation volume was taken to be indicative of coagulation since,

100 ZETA

T POTENTIAL /mV

80

60

!

1

4ot I

; -6

-5

Fig. 1. Zeta potential chloride).

-2

-3

-4

LOG (ELECTROLYTE

of polystyrene

latex particles

-1

/ mol

CONCENTRATION

)

dme3

versus log electrolyte

concentration

(sodium

in rapid coagulation, the particles appeared to form a network which had a greater volume than the sedimented particles in a more dilute electrolyte solution. The increase in sedimentation volume with increasing electrolyte concentration is shown in Fig. 2. The CCC was found at 0.33 mol dme3 sodium chloride, this being the point where the maximum sedimentation volume was first obtained. The experiment could only be performed with higher concen-

I

-05 LOG (ELECTROLYTE

O0 CONCENTRATION

/

mol

dm9

1

Fig. 2. Sedimentation volume of polystyrene latex particles versus log electrolyte concentration (sodium chloride) to determine the CCC. Particle number concentration: 4 x 10’ cm-‘.

279

trations of latex particles, typically of the order of 4 X 103 cm 3, since at lower particle concentrations the sedimentation volumes were too low for reasonable measurement. The sedimentation volume that each dispersion occupied was assumed to be indicative of the degree of coagulation. A smooth curve was obtained for the sedimentation volume-electrolyte concentration curve below the CCC which again indicates that a steady state or equilibrium process occurs at electrolyte values below the CCC. If the tubes were shaken, and the rate of sedimentation studied, a rapid initial build up of sediment was observed followed by a turbid region which settled most quickly for the highest electrolyte concentration and became progressively slower for lower electrolyte concentrations. Again, this indicates that the degree of aggregation increased with increasing electrolyte concetration.

Apparatus for studying aggregation kinetics The microtube apparatus used in this study has been described by Vadas et al. [ 18] and Cornell et al. [ 4 ] and was used here to study the kinetics of particle aggregation in quiescent, Poiseuille flow and stopped-flow experiments. This apparatus provided a method for observing large colloidal particles directly and distinguishing the number and size of aggregates. Direct observations of the particles was achieved by focussing a microscope on the particles contained within a microtube, Fig. 3 shows the apparatus schematically. The microtube itself was a precision bore glass tube of circular cross section with a radius of 200 Itm or less. Alternatively the microtube could be replaced by a capillary tube. In either case the tube which contained the particle dispersion was constructed as part of a cell which was held in place on the cell holder assembly. This part of the apparatus consisted of a moveable stage so that a particle flowing in a microtube could be held stationary in the field of view. The particles could be made to flow through the microtube by controlling the hydraulically driven gas tight syringes placed at either end of the microtube. A specific particle or group of particles could be continuously observed in flow by moving the cell holder assembly at the same rate but in the opposite vertical sense. A detailed description of the apparatus and method operation is given elsewhere [ 19 ]. All data were collected from film taken of the median plane of the microtubes. Prior to data collection the film was headed with exposures of a 0.01 m m spacing micrometer stage taken at the same magnification as that used for filming in microtubes. This enabled accurate measurements to be made regarding grid sizes. The temperature of each experiment was measured using a thermistor bead immersed in a water jacket surrounding the microtube. During experimental runs the wall of the microtube was kept within the field of view

280

-. . qtlcal . axIs

Fig. 3. Schematic diagram of the microtube apparatus. (A) Support plate; (B) observation cell; (C) microtube; (D) movable stage (movable in the vertical direction) ; (E) 50.~1 gas tight syringe with luer hub; (F) hydraulic drives for the movable stage and syringes; (G) Fluon bearings; (H) balance weight for syringe drive.

to enable measurements to be taken velocities to be calculated. Aggregation

and hence shear rates and streamline

kinetics

Methodology

for perikinetic

conditions

Two methods were used to study the rates of aggregation in quiescent conditions. The first method employed an external mixing procedure for mixing the dispersion and electrolyte. This was followed by rapid filling and filming of a rectangular section of the capillary tube. A stopwatch was started simultaneously with the mixing of the latex and electrolyte. An area of the tube was selected and filmed at a time as close to t = 0 as possible. Subsequently, other areas of the tube were randomly chosen and filmed at known times, t, after mixing. The second method employed was a stopped-flow technique in which the mixing of electrolyte and latex dispersion was carried out immediately before entry into the microtube. Electrolyte and latex were contained separately in 50 ~1 syringes and made to flow into a mixing hub build on top of a microtube. In this case the 50 ,ul gas tight syringes were being driven at the same rate in the same sense, i.e. ejecting the fluids into the mixing hub and microtubes. The mixing hub was built from a plastic luer hub into which 0.63 mm cannula tub-

281

ing connecting the syringes to the hub were placed. The fluids exiting from the outlets of the cannula tubing impinged onto a baffle where they were mixed and flowed down the microtube. The mixing hub was kept as small as possible to minimise the retention time of fluid in the hub. The mixed dispersion then flowed down the microtube into the field of observation. The flow was stopped and simultaneously a stopwatch started as filming was begun. The experiment was continued in the same way as the external mixing method. The stopped-flow technique had two advantages; the first was that the mixing process remained constant for all experiments, and secondly experimental times closer to the time t= 0 for the experiment were obtained. Methodology

for orthokinetic

conditions

The aggregation of polystyrene latex spheres in shear flow was observed directly using the microtube apparatus. The experiments were conducted in the following manner. Latex and electrolyte solutions were introduced into a 5 cm3 flask which was covered and inverted two or three times to mix the components. Immediately a 50 ,~l syringe was filled with a sample taken from the flask and placed in the microtube apparatus. The suspension was then made to flow through the microtube at a known constant flow rate using the hydraulically driven syringe. High speed tine photography (500 frames per second) was used for observing the particles in flow, in a particular radial regime. This regime was chosen so that it was small enough to produce reasonable particle numbers to maximise the statistical validity. All experiments were repeated five times for the same reason. The aggregation process was followed as a function of time by filming at different distances down the microtube from the syringe. Since the flow was laminar, the shear history of the radial volume element being studied was known. Several values of shear rate were studied and this was achieved by varying the volume flow rate. The experiments were also conducted as a function of electrolyte concentration, using four electrolyte concentrations below the CCC. Aggregation

in couette flow

A method was devised in order to study the flow induced aggregation-disaggregation of a dilute suspension of aggregated polystyrene latex. It was intended that the stability of the doublets (or aggregates), formed in shear flow, should be tested in order to ascertain whether or not they remain as doublets (or aggregates), or break up after the cessation of flow. The experiment entailed the continuous use of a Couette for several hours whilst sampling the dispersion within periodically. A Couette was built with an inner wall radius (r) of 0.965 cm, an outer wall radius (a) of 1.04 cm and a cylinder height (h) of 5 cm. The base of the inner

282

cylinder was machined to a conical shape with an angle (0) of 5’) in order to retain the constant shear rate regime throughout the sample [ 201. The base of the Couette was made using optically flat glass. The inner cylinder was rotated at a constant speed of 25.4 rpm giving a shear rate of 30.2 s ~ ‘. A second outer cylinder was built with radius 1.385 cm and when used with the original inner cylinder rotated at a constant speed of 6.32 rpm gave a shear rate of 1.5 SK’. The experimental design of this apparatus was such that it required a large run time. Because of this the latex dispersion was prepared in a water/deuterium oxide mixture [ 211 such that the latex particles were neutrally buoyant and eliminated sedimentation effects. The latex was used at a number concentration of 1.49 x 10’ cmp3 in an electrolyte concentration of lo-’ mol dm-” at a pH of 8. Prior to commencing the flow experiments the dispersion was left for a minimum of two hours in order to attain the steady state aggregation-disaggregation state. A control sample was taken and analysed at the beginning and termination of the flow experiment. In both shear field cases the sample was made to flow for several hours and the dispersion was periodically sampled by drawing a portion into a capillary tube. The sealed tube was then placed in the microtube apparatus and filmed as described previously. Samples were taken halfway down the Couette to prevent measurement of surface effects at either end of the Couette. Samples were followed as a function of time after the discontinuation of flow and the particle numbers and aggregate distributions were recorded. RESULTS Perikinetic External

aggregation

kinetics

mixing method

Using the external mixing method, the aggregation kinetics of the polystyrene latex dispersion were determined. The number concentration of the latex used was 1.49 x 10’ cmp3 and was studied in five different electrolyte concentrations (sodium chloride). The single particle number concentration, n,, and the time of experiment, t, were determined for each experiment. Figure 4 shows a typical curve obtained for an experiment. The results show that in all cases a plateau in the single particle concentration was reached. In the case of low electrolyte concentrations the plateau was quickly achieved but for higher electrolyte concentrations the curve extended further in time before reaching a plateau value. With increasing electrolyte concentrations the singlet concentration plateau value became successively lower. Clearly, the two parameters that characterize a curve are firstly, the initial rate of decay of single particle population - (dn,/dt ) t-, 0 and, secondly the plateau value of the single particle concentration, nspl.

283

7

INSET 100

0

120

60 TIME

180

I mrn

Fig. 4. Percentage of the total number of polystyrene latex particles existing as single particles, n,, with time for a flocculating system. Electrolyte concentration (sodium chloride): 0.1 mol dm-“; particle number concentration: 1.49 x 10’ cmm3. The inset shows the behaviour over the initial 10 min after the onset of flocculation. Errors are indicated by the error bars drawn at 30 s, 10 min, and 180 min.

Stopped-flow method The initial rate of decay of the single particle population and the plateau values of singlet concentration were determined using the stopped-flow technique for several electrolyte concentrations. The experimental curves all show behaviour similar to that shown in Fig. 4 and similar trends to the external mixing method. The fastest possible rate of coagulation can be achieved only when no potential energy barrier exists to prevent coagulation. Since this barrier does not exist above the CCC, which was determined at 0.33 mol dmP3 sodium chloride, the results using an electrolyte concentration of 1.0 mol dmP3 must represent the fastest possible coagulation rate. The curve obtained using 1.0 mol dmP3 sodium chloride is reproduced in Fig. 5 and represents this case when the potential energy of interaction is wholly attractive. It was apparent that the method by which an experiment was conducted had a large effect on the results at higher electrolyte concentrations. The rates measured by the stopped-flow method at higher electrolyte concentrations were slower than the corresponding rate determined by the external mixing method. This effect was due to the tendency of the rapid mixing process to break up aggregates.

284

100

c

200 TIME

I mln

Fig. 5. Coagulation of polystyrene latex particles in 1 mol dmm3 sodium chloride with time. The single particle concentration, n2,,is expressed as the percentage of total particle numbers. Particle number concentration; 1.49 x 10’ cme3.

Orthokinetic

aggregation

kinetics

The orthokinetic aggregation behaviour of the latex was studied at a particle number concentration of 1.23 x lo8 cmm3 in four different electrolyte concentrations ranging from 1 x 10e3 mol dmP3 to 8x lop2 mol dmP3 sodium chloride. The aggregation at each electrolyte concentration was studied for five different shear rate regimes from 7.1 s-l to 35.5 s-l. Figure 6 shows the results for the single particle concentration-time curves in an electrolyte concentration of 0.02 mol dmP3 sodium chloride. Unfortunately, this method was found to have quite large errors, of the order of +-20%, because of the low particle number counts and the difficulty in measuring the time t. For this reason the smoothed curves of n, versus t are drawn and are only intended to reflect the general trend of behaviour rather than fit the data accurately. The experiments could not be continued for long times and hence no plateau values were observed. However, the value of the initial rate of aggregation were estimated and are plotted in Fig. 7. The results show that for a given electrolyte concentration the increase in shear rate produces an increase in the initial rate of aggregation. At a given shear rate increasing electrolyte concentration produces an increase in the initial rate of aggregation. Aggregation

in Couette flow

The shear flow induced aggregation of the polystyrene latex particles at a particle number concentration of 1.49x 10’ cmP3 in an electrolyte concentration of 0.01 mol dm-3 sodium chloride was studied in a Couette for long periods

285 100

80

-? 0 E 60

1

21 3

7.1

14.2 shear

rate

I s

-1

35-5 20.4

40

I

I

100

50 TIME

1

I

150

200

,

I s

Fig. 6. Orthokinetic aggregation of polystyrene concentration: 0.02 mol drnm3 sodium chloride. imental points, which have been omitted for n,, is expressed as the percentage of total particle cmm3.

latex particles with varying shear rates. Electrolyte The smoothed curves are drawn through the experclarity. The single particle number concentration, numbers. Particle number concentration: 1.49 x 10’

”

INITIAL RATE AGGREGATION I 108

OF

Q

/ /

c m3 s-’

/ /

//p

/

4-

/ / / / /

/

/ / 2-

_ /

/

LOG (ELECTROLYTE shear

rate

I<’

0 =71,

o-142,

/

-2 CONCENTRATION @:21 3, 0.284,

I

mol

dmm3)

@ 535.5

Fig. 7. The initial rate of orthokinetic aggregation of polystyrene latex particles with varying electrolyte concentrations (sodium chloride) and shear rate. Particle number concentration: 1.49 X lOa cmm3.

286

I

2 SHEARING

I 3 TIME

i 4

1

1

5

6

I hours

Fig. 8. Percentage of total number of particles existing as singlets in Couette flow as a function of time. The dotted line was the plateau concentration obtained by 2 h perikinetic controlled conditions, and was constant throughout the experiment. The full line was obtained in Couette flow at 30.2 s-’ after 2 h perikinetic aggregation. The dashed-dotted lines were the behaviours after the shear flow was stopped. Electrolyte concentration (sodium chloride): 0.01 mol drn-“.

of flow. The suspending medium used was 48% H,0/52% D,O and prevented sedimentation. The results are as shown in Fig. 8 for the dispersion in a shear field of 30.2 s’, having been left to aggregate under perikinetic conditions for 2 h to achieved the plateau region of aggregation. The sample was then sheared at 30.2 s’ for 4 h. The heavy continuous line shows the single particle concentration whilst in flow. It can be seen that the degree of aggregation achieved initially was the same as that for purely perikinetic conditions. After an hour the flow began to break up the aggregates, thus increasing the single particle number concentration. After 2 and 4 h samples were taken and the shear flow stopped in these samples. The degree of aggregation again increased in the samples (dashed-dotted line) to give plateau values found in the unsheared control sample (dashed line). A similar experiment in a shear field of 1.5 s-l (Fig. 9) showed that continuous shear increased the number of aggregates (heavy continuous line) compared with the unsheared control sample (dashed line). Again upon cessation of flow in a sample taken from the sheared suspension the aggregates reformed to return to the plateau value of the unsheared sample. DISCUSSION

Calculation of pair potentials. Evaluation of the total potential energy of interaction as a function of distance from a particles surface for the latex system studied here yields a curve as shown in Fig. 10. This has been drawn for the latex in 10-l mol dmP3 sodium

I

0

1

I

I

1

2

3 TIME

SHEARING

/

I

I

I

4

5

6

hours

Fig. 9. Percentage of total number of particles existing as singlets in Couette flow as a function of time. The dotted line was the plateau concentration obtained by 2 h perikinetic controlled conditions, and was constant throughout the experiment. The full line was obtained in Couette flow at 1.54 SKI after 2 h perikinetic aggregation. The dashed-dotted line was the behaviour after the shear flow was stopped. Electrolyte concentration (sodium chloride): 0.01 mol dmm3.

chloride solution. The dotted lines represent the error in zeta potential measurement of 5 5%. Several features are shown in Fig. 10 and can be calculated for the range of electrolyte concentrations used. These features are the size and position of both the primary maximum and secondary minimum potential energies. The size and position of the primary maximum, VP,,, and HP,,, and the size and position of the secondary minimum, VSminand Hsmin,have been calculated using Hamaker constants of 6 x 10P21 J for the range of electrolytes studied. The results are presented in Table 1. The position of the primary maximum alters little with increasing electrolyte concentration, however, the height is altered dramatically to lower values, thus presenting a smaller barrier to coagulation. Increasing electroyte concentration also has a large effect on the depth and position of the secondary minimum. These variations are shown graphically in Fig. 11. As the electrolyte concentration is increased so the double layer is compressed and the primary maximum is suppressed and the depth of the secondary minimum increases. The secondary minimum also moves closer to the particle surface. At the point at which the primary maximum just disappears, i.e. when the total potential energy is zero, there exists no longer a barrier to coagulation and the interaction energy becomes wholly attractive. This point is regarded to be the CCC and is the point above which rapid coagulation is observed. Critical coagulation concentration The CCC was measured and found at 0.33 mol dme3 sodium chloride for the latex system used. From the determination of the CCC (Fig. 2) it can be seen

288 TOTAL POTENTIAL ENERGY VT I kT

400 --

200 --

o--

’

DISTANCE

-_ o-005

H

0 010

/

).rrn 0 015

-100

t

/

Fig. 10. The total potential energy versus distance from the particle surface, calculated from DLVO theory, for polystyrene latex particles in a 1:l electrolyte concentration of 0.1 mol dmm3. The zeta potential was that measured experimentally in sodium chloride solution (Fig. 1) . The dotted lines shows the variation of + 5% in the zeta potential. Hamaker constant: 6 x lo-*’ J, particle size: 3.47 pm.

TABLE 1 DLVO calculation of primary maximum and secondary minimum potential energy conditions in varying electrolyte concentrations (sodium chloride). Combined Hamaker constant: 6 x 10mzl J Electrolyte concentration (mol dmm3)

VP,,, (kT)

HPmax (10m7 cm)

1x10-4 1 x lo-” 5x10-” 1x10-2

2.6

2x10-2

5669 8436 6917 5570 3877

5x10-2 8x lo-* 1x10-’

1735 822 511

VSmin (kT) -

H Srnl” (1O-6 cm)

-

2.0

- 0.092

13.20

1.8 1.8 1.8

- 0.505 - 1.09 -2.38

5.08 3.28 2.13

1.8 1.95 2.0

- 6.44

1.20

- 10.41 - 12.88

0.87 0.76

289

0.1 5 DISTANCE

Hsmln I Qm

0 --

- 5-DEPTH OF SECONDARY MINIMUM ‘& I kT

-,o__

I

-3

I

-2

LOG (ELECTROLYTE /mot dni3 )

-1 CONCENTRATION

Fig. 11. The variation of the position and depth of the secondary minimum with respect to a particles surface, with electrolyte concentration (1:1) for polystyrene latex spheres. The curve was calculated using DLVO theory. Hamaker constant: 6~ lO-*l J; particle size: 3.47 pm.

that the sedimentation volume was a smooth but increasing function of electrolyte concentration. It would normally be expected that below the CCC, and after the sample had completely settled, the sedimentation volume would be at a minimum. This being due to the relatively shallow secondary minimum allowing particle mobility within an aggregate. Coagulation, having a deep primary minimum, would result in strong open aggregates, and thus a sharp cut off point might have been expected. The smooth curve obtained was probably due to the high particle number concentration in the sedimented bed. Firstly, the particles in the bed would be in close proximity to each other and each particle would be interacting with several neighbours. A steady state condition would be biased to weak aggregation at these concentrations. Secondly, there would be insufficient room for the particles to move and progressively larger forces would be required to cause cooperative motion of large groups of particles as the well depth increased. Stability ratios From the results obtained for the initial rate of aggregation the experimental stability ratios can be obtained. The fastest rate possible is given by the value

290

LOG STABILITY RATIO

I

l5

1-o

(> cl

.

I I I

0

0.5 A

9

!:

-1

LOG (ELECTROLYTE

l =perlklnetlc

I ,

0

,

CONCENTRATION

()=stopped

A.,

LLL

/ mol

0 drf3

)

flow

Fig. 12. Log stability ratio versus log electrolyte concentration (sodium chloride) for the polystyrene latex dispersions. The full line represents the calculated stability ratios for secondary minimum flocculation [ Eqn (14) ] and the dotted line represents the stability ratios for primary minimum coagulation [ Eqn (19) ] . Polystyrene latex particle number concentration: 1.49 X 10’ cm-‘.

obtained in 1.0 mol dmP3 sodium chloride, since at this level of electrolyte there is no energy barrier to coagulation and only an extremely small probability of peptization. This concentration corresponds to the condition where W= 1, and the stability ratios for all other experiments can be found by comparing with this value. The experimentally determined rate was found to be higher than the absolute rate calculated from Eqn (6). This was most probably due to collisions induced by flow. The values for the series of experiments carried out at a latex number concentration of 1.49 x lo8 cme3 are plotted in Fig. 12 as a function of electrolyte concentration. Also plotted are the theoretical values obtained using Eqn (19 ) for coagulation and (14) for flocculation. It is clear that aggregation was proceeding at a very much faster rate than the stability ratio would predict for a coagulation process, i.e. the experimental data points lie to the left of the coagulation line. Thus the aggregation process was unlikely to be coagulation. Furthermore, the experimental points lie to the right of the theoretically derived secondary minimum flocculation curve at higher electrolyte values, showing that the flocculation process is slower that

291

would be predicted. It is tempting to suggest that this is due to a hydrodynamic resistance retarding the rate of flocculation. The results show that there is a small range of electrolyte concentrations at about 1 x 10-l mol dme3 sodium chloride, over which the rate of flocculation increases rapidly. This is the analogous case to the CCC which for flocculation is the critical flocculation concentration (CFC). Qualitatively it can be seen that at the electrolyte concentration increases, so the secondary minimum deepens and approaches the particle surface (Fig. 11). At a value of about 0.1 mol dmP3 the secondary minimum deepens significantly over a small range of concentrations but only alters its position with respect to the particle surface slowly. Thus the hydrodynamic resistance to movement in this region remains essentially constant but the attractive potential increases rapidly. Hence it is reasonable to expect a sharp change in the kinetic rates over this region. Initial rate of flocculation The absolute initial rates of flocculation, assessed here as the initial rate of loss of single particles, can be calculated from Eqn (6) for the fastest possible rate, that is above the CCC where W= 1. The absolute initial rates for electrolyte concentrations below the CCC can be calculated using the secondary minimum stability ratio, Eqn (11) . However, because the latex dispersion prior to an experiment did not contain 100% single particles a correction must be made for the fact that aggregates exist in the original dispersion. Thus the single particle population is depleted by collisions of singlets with singlets, doublets, triplets, etc. Hence a biparticulate collision frequency can be defined such that the initial rate is given by

(16) where subscript j refers to the number of particles contained in the aggregate. The combined diffusion coefficients are given by D,j=D,

+Dj

(17)

and are tabulated elsewhere [ 211. The collision diameter R~jcan be calculated from the average dimensions that an aggregate would present in a dispersion. n, and nj are known from direct observational counts using the microtube apparatus. The results of these calculations are presented in Table 2 for the series of experiments carried out at a later number concentration of 1.49 x lOa cm-3. These are compared with the experimental determination of the initial rate of aggregation. Clearly, the theoretical predictions of the rate are reasonably good, however, the region of greatest deviation coincides with the region

292 TABLE 2 Initial rates of aggregation, expressed as a rate of loss of single particles, and an estimation of the hydrodynamic resistance to flocculation at different electrolyte concentrations. Latex particle number concentration: 1.49 X 10s cm-3 Electrolyte concentration (mol dn?)

Experimental initial rate of flocculation (lo5 cm3 9-l)

Theoretical biparticulate collision rate, Eqn 06) (lo5 cm3 SC’)

Hydrodynamic retardation coefficient, Bqn (26) (10C4 cm)

Normalised” hydrodynamic retardation coefficient

Hydrodynamically retarded theoretical biparticulate collision rate (lo” cm3 s-‘)

0.001 0.005 0.02 0.08 0.10

0.4 0.5 0.3 1.4 2.6

0.2 0.9 2.1 2.4 2.4

6.24 7.07 7.82 8.56 8.72

1 1.133 1.253 1.372 1.400

0.2 0.8 1.7 1.7 1.7

“Therefore

the upper limit of the integral

[ Eqn (25) ] was taken as 0.076~.

where changes in the hydrodynamic to be most pronounced.

resistance

to flocculation

would be expected

Steady state secondary minimum aggregation Experimentally it was found that a steady state condition of flocculation and deflocculation exists and so here - (dn,/dt) = 0 at large t. This type of behaviour has been discussed previously by Cornell et al. [ 41 who proposed a kinetic model between the rate of formation of doublets from singlets and the rate of breakup of doublets to form singlets. The experimental results found in this present work can be treated in a similar manner. The results are plotted in Fig. 13 as a plateau concentration against the log electrolyte concentration at a particle number concentration of 1.49x lo8 cm-3. Again it can be seen that there exists a small region of electrolyte concentration over which the degree of aggregation alters dramatically, and this corresponds to the previously observed CFC region of around 0.1 mol dm-3 sodium chloride. Hydrodynamic retardation of flocculation Two mechanisms exist that would explain the descrepancy between calculated and experimental resutls for the initial flocculation rate; firstly the existence of a flocculation to coagulation transition and, secondly a hydrodynamic retardation term. Calculation of the stability ratios has shown that the expected rate of coagulation is extremely slow and therefore offers an unlikely explanation. The alternative explanation is hydrodynamic retardation. Honig et al. [ 221 have given a good approximation for the slowing down of the rates of interacting particles due to the viscous interactions when the par-

293

LOG (ELECTROLYTE

CONCENTRATION

l = external 0 = stopped

I

mol dmJ

)

mixing flow

Fig. 13. Steady state concentration of single particles versus log electrolyte concentration (sodium chloride) for the polystyrene latex dispersions. The experimental points were obtained from the concentration of single particles found in the plateau region of both the stopped-flow experiment and the external mixing experiment.

titles are in close proximity. They showed that the Stokes resistance, f, could be modified to f /?(u) and

P(u) =

6u2 + 13u+ 2

(18)

6u2 + 4u

where u = H/a and H is the interparticle separation. Using this expression the stability ratio can be written [ 3 ] as W=l+

(37rkT Up,,,/A)i

(l/G)

exp( V,,,,/kT)

(19)

where UP,,, is the distance parameter at the energy maximum u Pmax

-

HPmax

la

(20)

and L=

P(u) exp (VAPWdu (u+2)”

(21)

This expression describes the retardation effect of rates of coagulation due to the hydrodynamic interaction of colliding spheres.

294

Spielman [ 231 obtained an expression for the combined diffusion coefficients D, of unequal size particles, a, and oj, as a function of the surface to surface distance of separation and in approximate form is D, =-

kT (l+Ui/a,)2

6rqa

(H/ai)

(22)

Thus for equal size spheres (23) It is clear that D,, in the neighbourhood of Hsmi,, the secondary minimum distance, is very much lower than would be expected from the normal Stokes frictional term. The form of the hydrodynamic retardation to coagulation occurs as a pre-exponential function of diffusion coefficient in the stability ratio such that (24)

where S= (2a+H)/a. An approximate solution can be obtained by assuming that the contribution to retardation occurs from the region of the secondary minimum out until 20, =D1i, i.e. the combined free diffusion coefficient is equal to the combined restricted diffusion coefficient. Thus flocculation will be retarded by an amount

(25) which on substituting for D, and D,, and using the condition that 20, = D,, when H= 100 a (from Spielman [ 231) , the retardation has the form

;1,

[

He SIlllIl 1

However, evaluation and normalisation of the term of the lowest electrolyte concentration, where the hydrodynamic retardation will be small, gives the required effect (Table 2)) such that the greater the electrolyte concentration the greater the retardation. However, the higher electrolyte concentration values are not fitted well. Clearly the theory in the region of the CFC requires modification in order to describe the processes adequately. Orthokinetic

aggregation

The effect of simple shear flow on the aggregation rate is two fold. Firstly, the number of collisions is increased and secondly, the energy of a collision is

295

TABLE 3 A comparison of the orthokinetic aggregation rates measured experimentally and calculated using Smoluchowski kinetics. Particle number concentration: 1.23 x 10’ crnm3 Shear rate (s-l)

Electrolyte concentration (mol dm-“)

Initial rate (cm” s-l)

Calculated initial rate, Eqn (27) (cm” SK’)

7.1

0.001 0.005 0.02 0.08

4.2 x 10” 3.4 x 106 1.0x 10’ 1.1x10’

5.3 x 5.0 x 4.9 x 4.8 x

14.2

0.001 0.005 0.02 0.08

2.0 x 1.7x 4.8 x 1.4x

1.1x1os 1.0x 108 9.7 x 10’ 9.6 x lo7

21.3

0.001 0.005 0.02 0.08

5.2 x 10’ 3.7 x lo7 1.5x108 1.2x108

1.6x lo* 1.5x 108 1.5x108 1.4x 106

28.4

0.001 0.005 0.02 0.08

6.6X lo7 1.1x108 2.7x lOa 5.3x108

2.1 x 2.0x 1.9x 1.9x

lo8 108 108 10”

35.5

0.001 0.005 0.02 0.08

7.0x 10’ 1.0x lo8 5.4 x 10s 5.3x108

2.7 x 2.5 x 2.4x 2.4x

lOa 10s lOa 10’

107 10’ 10’ 10’

10’ 10’ lo7 10’

increased. Smoluchowski [ 241 gave an expression for the collision frequency of particles under the influence of a velocity gradient, (27) Here R, is given by Eqn (10) since we are considering secondary minimum flocculation. Table 3 shows a comparison of the calculated rates from Eqn ( 27) above with the measured rates of aggregation of the latex, with shear rate and electrolyte concentration. For all shear rates the calculated rate of aggregation decreases with increasing electrolyte concentration and this is because the collision diameter is reduced as the secondary minimum approaches the particle surface with increasing electrolyte concentration. This obviously takes no account of the secondary minimum well depth nor of a collision efficiency.

296

It can be seen that experimentally the rate of aggregation increases both with increasing electrolyte concentration, and increasing shear rate. Generally the experimentally determined rates are lower than the calculated rates of aggregation using the Smoluchowski equation, (27)) and are greater than the diffusion controlled rate [ 1.86x lo5 ( cm3 s-l)]. At the higher shear rates and electrolyte concentrations used the experimental values exceed the calculated values by about a factor of 2. This is most probably a reflection of the experimental method, which was developed in order to directly examine secondary minimum flocculation of large particles. However, because very low sample numbers, and a degree of uncertainty regarding the value of the time after the onset of aggregation exist, very large error bars are generated. For this reason only trends are identified from the data and its analysis is limited, hence collision efficiencies are not considered here. Experimental aggregation rates obtained at low shear rates and low electrolyte concentrations are well below the calculated rates [ Eqn (27) 1. Clearly, the low value of the secondary minimum well depth at these electrolyte concentrations would be expected to be insufficient to form stable aggregates in a shear field which would exert a significant hydrodynamic force pulling particles apart. Goren [ 251 obtained an expression for the hydrodynamic force acting on two spheres in shear flow tending to separate them Fdis=6 71rl a2 ~(u,/u,)

(28)

where h ( al/a2) is a tabulated function and h= 2.04 when a, =u2. Thus an increase in shear rate results in an increase in the force pulling particles apart. If this disruptive force, Fdis, is equated with the maximum force of attraction between two spheres in the region of the secondary minimum, FA=-

(i.!$ >

(29)

obtained from calculations of pair potentials, then a shear rate can be calculated which when exceeded will result in aggregate (doublet) breakup and below which the aggregates (doublets) will remain intact. These calculations are presented in Table 4. It is evident from these calculations that all the experiments conducted, with the exception of those carried out in 0.08 mol dme3 sodium chloride are in a regime where aggregate breakup is expected, since all experiments were conducted at 7.1 s-’ and above. The experiments carried out in the highest electrolyte concentration show that the aggregates should be stable at the shear rates used since they do not exceed 85 SK’. Van de Ven and Mason [ 261 claim that the rate of aggregation is proportional to the shear rate (i) to a reduced power, 0.82, because of the capture efficiency of particles upon collision. This is true for fully destabilised parti-

297

TABLE 4 The maximum force of secondary minimum attraction between two latex spheres and the shear rate at which the force of disruption [Eqn (28)] balances this force at various electrolyte concentrations Concentration of electrolyte (mol dm-3)

Maximum force of attraction in the secondary minimum (DLVO) (Nx~O-‘~)

Shear rate at which force of disruption is equal to the attractive force (s-l)

0.001 0.005 0.01 0.02 0.08

5.6 90 170 590 9000

0.054 0.87 1.6 5.7 85

cles, but for fully stabilised particles the capture efficiency is zero. The case where a secondary minimum occurs is a complicated function of both the hydrodynamic and colloidal forces. The results presented here appear to be consistent with the Van de Ven and Mason analysis in that the capture efficiency is reduced on going to a more stable system. (i.e. lower electrolyte concentrations). However, if the j”.82 rate dependence is simply substituted for it’ in the expression for aggregation rate [ Eqn (27) ] then only the lowest shear rate and electrolyte concentration data fall below the calculated aggregation rates. Again, as discussed previously, this is probably more a reflection of the experimental technique than an indication of the absolute rates of aggregation. It is interesting to note that even when the hydrodynamic force of disruption is calculated to exceed the secondary minimum force of attraction, and hence only single particles and transient collision doublets should exist, a finite amount of aggregation is still observed. This may be a reflection of the fact that in shear flow open (separating) and closed (capture) collision trajectories exist, and that it may be possible to cross the limiting trajectory by Brownian movement [ 271. The previously discussed method of evaluating the disruptive hydrodynamic force and comparing this with the secondary minimum attractive force was used in a similar analysis of forces for the Couette flow experiments which had extended flow times of latex in 0.01 mol dmP3 sodium chloride. For the higher shear rate used, 30.2 SK’, the disruptive force was calculated to be 3.13 x lop7 dyn cmP3 and for the lower shear rate used, 1.5 s-l, 1.59x 10e8 dyn cm-3. The attractive force at the secondary minimum is calculated to be 1.58~ lop8 dyn cm -‘. Thus given that the disruptive force was based on touching spheres, which they would not be in the secondary minimum, and that at a finite sep-

298

aration there is a lowering of the required disruptive force, it can be seen that the experimental results are consistent with this interpretation. That is at the lower shear rate aggregation was observed whilst at the higher shear rate aggregate breakup was observed. It is clear from the extended time Couette flow experiments that a plateau in flocculation exists and that this is not the same concentration of singlets that exists in perikinetic conditions but alters according to the flow experienced. Furthermore, it is seen that the perikinetic plateau is returned to upon cessation of flow. CONCLUSION

For the large highly charged electrocratic latex particles used in this work it was found that although the critical coagulation concentration of electrolyte was 0.33 mol dmp3 sodium chloride there was a finite degree of aggregation below this concentration in non-flowing conditions. This effect was explained in terms of secondary minimum flocculation. The rate of flocculation was found to increase with increasing electrolyte concentration and a critical flocculation concentration was found at about 0.1 mol dme3 sodium chloride. This same critical flocculation concentration region was also seen from the rapid decrease of the steady state plateau value in flocculation for large times of experiment. The plateau region of flocculation was studied in shear flow where it was shown that the plateau could be increased or decreased depending on the magnitude of the shear field, but on cessation of flow the original, perikinetic, plateau value was returned. Orthokinetic flocculation was studied and it was demonstrated that an increase in electrolyte concentration or shear rate increased the rate of aggregation. These observations are consistent with the existence of a secondary minimum for this latex system. ACKNOWLEDGEMENTS

We wish to acknowledge, with thanks, the support of this work by the Science and Engineering Research Council. Moreover, we wish to thank Professor R.H. Ottewill for his helpful and constructive discussions during this work.

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