The effect of cationic polymers on colloidal stability

8/17/2019 The effect of cationic polymers on colloidal stability

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The Effect of Cationic Polymers on the Colloidal

Stability of Latex Particles

J O H N G R E G O R Y

Department of Civil and Municipal Engineering University College

Gower Street London WC 1E 6BT England

Received June 20, 1975; accepted December 12, 1975

Using a simple turbidimetri c procedure, relative flocculation rates of a polyst yrene latex suspension

with three cationic polym ers and a cationic surfacta nt have been determined. All four materials

showed abo ut the same optim um flocculation concentration when expressed in equivalen t amounts,

indicating the predominance of charge effects. In agreem ent with previous findings, a considerable

enhancement of flocculation rate was observed with cationic polymers of moderately high molecular

weight, which is consistent with the electro static pat ch model of polyel ectrolyt e adsorption. A t

opti mum poly mer concentration, it was found th at the rate of flocculation after a two-stage add ition

of suspension was just the same as th at after th e conventional one-stage addition. This, at first sight,

surprising result also can be explained in terms of a simple patchwise adsorption model. However,

the rate enhancement is considerably underestimated by this simple model, probably because the

mutual orientation of colliding particles is not taken into account.

INTRODUCTION

The de s ta b i l i z a t ion a nd re s ta b i l i z a t ion o f

ne ga t ive pa r t i c le s by c a t ion ic po lyme rs oc c urs

pr ima r i ly by c ha rge ne u t ra l i z a t ion a nd c ha rge

re ve r sa l , a s de mons t ra te d by the f a c t tha t

the e le c t rophore t ic m obi l i ty o f pa r t i c le s a t the

op t imu m f loc c u la n t c onc e n t ra t ion i s c lose to

zero (1-4) . S l ight ly less direc t , t houg h n o less

s ignif icant, evid ence is the f inding th a t ca t ionic

po lyme rs , d i f f e r ing on ly in mole c u la r we igh t ,

h a v e t h e s a m e o p t i m u m c o n c e n t r a t i o n f o r t h e

f loccula t ion of nega t ive la tex par t ic les (5 , 6) .

In o the r words , op t imum f loc c u la t ion oc c urs

whe n a c er ta in a m oun t o f pos i t ive c ha rge ha s

be e n a dsorbe d by the pa r t i c le s , i r r e spe c t ive o f

the l e ng th o f the po lyme r c ha ins c a r ry ing the

charge . Fur ther evidence a long these l ines wil l

be pre se n te d in th i s pa pe r .

The impl ic a t ion o f the se r e su l t s i s tha t

b r id g ing o f ne ga t ive pa r t i c le s by ca t ion ic

1 Presented at the 49th Nati onal Colloid Symposium,

Potsdam, New York, June 16-18, 1975.

po lyme rs ma y no t be o f g re a t p ra c t ic a l s ign i f i -

cance . Needless to say, th is conc lus ion has no

be a r ing on the be ha v io r o f

anionic

p o l y m e r s

with nega t ive par t ic les , where br idging f loccu-

la t ion re ma ins by fa r the mos t p la us ib le

me c ha n ism (7 ) .

I t s e e ms l ike ly tha t , whe n a dsorbe d on

ne ga t ive pa r t i c le s , po lyc a t ions f a i r ly r a p id ly

a dop t a r a the r f l a t c onf igura t ion be c a use o f the

s t rong ion ic in te ra c t ion be twe e n oppos i te ly

c ha rge d g roups on the pa r t i c le su r fa c e a nd

a long the po lyme r c ha in . Ka spe r (8 ) ha s

sugge s te d tha t suc h a c onf igura t ion would

ver y much reduce the l ike l ihood of br idging

f loccula t ion. There a re , however , two respec ts

in whic h the s imple c ha rge ne u t ra l i z a t ion

p ic tu re ma y ne e d re f ine me nt .

F i r s t ly , the e qu i l ib r ium c onf igura t ion o f a n

a dsorbe d po lyme r c ha in i s a c h ie ve d a t some

fini te t ime (perhaps severa l seconds) a f te r

in i t i a l c on ta c t , a nd dur ing th i s t ime , the

d is t r ibu t ion o f po ly me r s e gme nts f rom the

par t ic le surface is more extens ive than in the

Copyright ~ 1976 by Academic Press Inc.

All rights of reproduction in any form reserved.

35

Journal of Colloid and Interface Science Vol. 55 No. 1 Apr il 1976


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36

JOHN GREGORY

equil ibrium state, so that bridging may be

possible. Fro m the stan dp oin t of flocculation,

the crucial factor is the particle collision

frequency (and hence, the part icle number

concentrat ion). In a concen trated sol a part icle

may experience many collisions during the

t ime in which the adsorbed polym er achieves i ts

equil ibrium configurat ion and bridging could

then be a significant destabilizing mechanism.

In more dilute sols the collision frequency

would be too low for this effect to be appreci-

able. Recent results (6) with latex particles

and high molecular weight cat ionic polymers

indicate tha t this nonequ il ibrium floccula-

t ion becomes imp ortan t with part icle concen-

tratio ns of a bou t 10 n cm 3 or greater.

The second effect arises when particles of

fair ly low negative surface charge densi ty are

neutral ized by highly charged cat ionic poly-

mers. In such cases, elementary geometric

considerat ions show that the part icle charge

cannot be neutral ized

uniformly

by the ad-

sorbed polycat ions but rathe r t hat the part icle

surface will have a mo saic pat te rn of

posi t ively and negatively charged areas. The

importance of this electrostat ic pa tch effect

in f locculat ion was f irst recognized by Kaspe r

(8) and i t was also invoked by Gregory (5)

to account for the enhanced rate of floccu-

lat ion observed with cat ionic polymers of

moderately high molecular weight . In this

paper further experimental evidence is pre-

sented on the electrostat ic pa tch effect ,

and by varying the mixing procedure, i t is

shown th at there is a close analogy with

the heteroflocculat ion of opposi tely charged

particles.

I t is worth point ing ou t tha t the concepts of

patch wise adsorption and the f lat configura-

t ion of adsorbed polycat ions imply that the

part icles are larger than the polymer mole-

cules. Clearly, with v ery high mo lecular weig ht

polymers and very small colloidal particles,

this assump tion would no longer be val id and i t

is not difficult to envisage a situation like

that depicted by I ler (9) in which several

part icles are at ta ched to one polym er chain.

E XPE RI ME NT AL ME T HODS

Polystyrene suspension This was prepared

by emulsion polymerizat ion, using sodium

dodecyl sulfate (2.5 raM) as emulsifier and

sodium persulfate (2.5 mM) as ini t iator . The

styrene was distilled prior to use and repre-

sented about 10% by weight of the react ion

mixture (45 g total) . Poly merizat ion was

carried out overnight in a sealed tube rotat ing

in a wate r b ath, at 70 ° . The result ing latex

was extensively dialyzed in a Dow hollow-fiber

beaker dialyzer , to remove residual ini t iator

and react ion products and to reduce the level

of emulsifier (it is extrem ely difficult to rem ove

all

adsorbed surfacta nt by dialysis (10)) . After

dilution of the purified latex to 100 cm ~, the

polys tyrene content was found to be 3 .92%

(w/v), by evaporat ing a small port ion to

dryness, heat in g at 110 ° a nd weighing the

residue.

The p art icle s ize of the latex was determ ined

by electron microscopy. A part icularly effective

method of specimen preparat ion was to

immerse a coated copper grid in a di lute

solut ion of cat ionic polymer (0.02% Polymer

2, see below), rinse in water and then immerse

for a few seconds in a diluted (1/100) latex

sample. The negatively charged latex part icles

adhere strongly to the adsorbed cat ionic

polymer layer on the grid and appear to

deposi t fair ly uniformly. Samples prepared

in this way are ready for examination in the

electron microscope after a few minutes

drying on f i l ter paper.

The mean part icle diameter was found to be

296 nm with a s tandard deviat ion of 34 nm.

Cationic polymers

Three polymers were

used in this work:

1. Poly (d imethylaminoethyl methacryl -

ate) of nom inal molecu lar weig ht 5 X 10~,

ful ly quaternized with dimethyl sulfate.

2. As polymer 1 but with a molecular

weight of about 1.5 >( 105.

3. Poly (1-ethyl 2-methyl 5-vinyl py-

ridinium bromide) with a molecular weight of

about 106

Journal of Colloid and Interface Science Vol. 55 No. 1 April 1976


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CATIONIC POLYMERS AND LATEX STABILITY 37

Polymers 1 and 2 were described in (5)

(there designated I and III, respectively).

The preparation of polymer 3 was given in

(6), where it was labeled B.

Stock solutions of 0.1 polymer were pre-

pared and these were diluted to working

solutions of 10 or 20 ;~g cm 3.

Reagents. Cetyl trimethylammonium bro-

mide (CTAB) was BDH laboratory reagent

grade and sodium nitrate was of Analar

quality. Both were used without further

purification.

All solutions and suspensions were prepared

with water that had been passed through a

mixed-bed demineralizing column and then

distilled from alkaline permanganate solution.

From the very short persistence of bubbles

after shaking it was assumed that the water

was substantially free from organic impurities.

When working with very dilute solutions of

cationic polymers (in the present case down

to a few nanograms per cubic centimeter) it is

essential to eliminate interference from traces

of organic matter, especially surfactants.

Otherwise, reproducible results cannot be

obtained.

Flocculation procedure. Suspensions of poly-

styrene particles at a concentration of 13 ug

cm 3 (9.19 X 108 partic les cm 3) were floccu-

lated with a cationic surfactant (CTAB) and

the three polymers mentioned above, in the

presence of 10 4 M NaNO3. The flocculation

was carried out in a 4-cm spectrophotometer

cell and the increase in turbidity was followed

as a function of time. The output from the

spectrophotometer (Unicam SP500 Mk2) was

fed via a Quaylab log-lin converter to a chart

recorder. The slope of the trace was then

proportional to the rate of turbidity increase,

and will also be assumed proportional to the

rate of flocculation.

The spectrophotometer cell was first charged

with 8 cnl ~ of a 1.5 10-4 M NaNO3 solution,

containing the appropriate amount of floccu-

lant. Four cubic centimeters of a 39 ug cm ~

polystyrene suspension was then rapidly added

to the cell from a dispensing pipet te (E-Mil)

and the turbidity record was commenced

within a few seconds of mixing. For each

flocculant, the range of concentrations was

chosen so that the onset of flocculation and

restabilization could be clearly established.

When the optimum concentration of each

flocculant had been established (i.e., that

giving the maximum rate of flocculation), a

two-stage mixing procedure was carried out

as follows : To 8 cm3 of a 1.5 10 4 M NaNO3

solution containing the optinmm amount of

flocculant, 2 cm 3 of the stock latex suspension

was added, followed, after a measured time,

by a second 2-cm~portion. The rate of turbidi ty

increase was measured after the second

addition.

RESULTS AND DISCUSSION

One Stage Addition

The results are presented in Fig. 1, where

the ordinate is the slope of the recorded trace

divided by the initial reading, thus giving the

0 . 1 C

O O e

~'~ 06

E

rl,~

.04

0 . 0 2

0 5

- - O C T A B

- - - - x - - P O L Y M E R 1

+ 2

D~S, -H-- 3

\

x

/ _ l\

10 15 20 25 30

F L O C C U L A N T C O N C E N T R A T I O N ( g e q /g l

F i C . 1 . R a t e o f t u r b i d i t y i n c r e a s e f o r l a t e x s u s p e n s i o n

flocculated with a cationic surfactant (CTAB) and

three cationic polymers, plotted as a function of

flocculant concentration. All results obtained in 10 4 M

NaNO3 solution. Dashed horizontal line indicates the

corresponding rate for the same suspension flocculated

with 0.3 M Ca(NOa)2.

Journal of Colloid and Interface Science Vol. 55, No. 1 , Apri l 1976


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38 JOHN GREGORY

rate of turbidi ty increase relat ive to the

original value r0, or 1/ro)dr/dt, where r is

the turbidi ty at t ime t af ter mixing. As dis-

cussed previously (5, 6), the initial slope (i.e.,

at t = 0) is not a reliable indicato r of the

optimum flocculant concentrat ion, s ince ad-

sorpt ion and rearrangement of polymer chains

may st i l l be taking place during the f irs t few

seconds after mixing, and this can lead to

anomalous values of the rate. This effect is

most pronounced wi th polymers of h igh

molecular weight at concentrat ions above

optimu m. Consequently, the slopes of the l ines

have been taken over the interval between one

and two minutes after mixing. The abscissa in

Fig. 1 is the amount of flocculant, expressed

in microequivalents , per gram of polystyrene

part icles. The equivalent weight of the poly-

mers is taken as the formula weight per

quaterna ry n it rogen a tom. For polymers 1

and 2 this is 283 and for polymer 3 the value

is 228 . Th e dash ed horizo ntal line in Fig. 1

represents the rate of turbidi ty increase when

the particles are flocculated with 0.3 M

calcium nitrate solut ion.

I t is clear from Fig. 1 that the f locculat ion

effects of a s imple cat ionic surfactant CTAB,

and a low molecular weight cat ionic polymer

1, are not great ly different . The maximum

rate is achieved with essential ly the same

amount of adsorbed posi t ive charge, 14

/~equiv g-l, or a bou t 7 ~C cm 2. It is reasonable

to assume from the work of Connor and

Ottewil l (11) that quanti tat ive adsorption of

C T A+ ions occurs on a polysty rene surface, at

least up to the point of charge neutral izat ion.

This wil l certainly be the case for polycat ions.

Although, there is some scat ter in the resul ts ,

the breadth of the f locculat ion region appears

to be about the same in the two cases. I t

seems very like ly tha t for C TAB and Polymer

1, flocculation and restabilization are simply

due to charge neutral izat ion and charge

reversal by strongly adsorbed cat ionic ma-

terials. Since the ionic strength of the solution

is low, very small amounts of negative or

positive charge are sufficient to stabilize the

particles.

With the higher molecular weight polymers

2 and'3 , f locculat ion occurs over a som ewha t

broader range , a l though the maximum ra te i s

achieved with about the same equivalent

amounts, so that the primary effect is s t i l l

one of charge neutralization. It is difficult to

accoun t for the broader f locculat ion regions by

a bridging effect, since the considerable differ-

ence in molecular weight between polymers 2

and 3 appears to have no fur ther infuence .

I t is just conceivable that , because of the

strong interact ion, loop dimensions of the

adsorbed polymer chains are independent of

molecular weight above a certain value, but

are still sufficiently large to prom ote b ridging

between part icles. I t is more l ikely that the

observed effects with increasing molecular

weight can be explained by the electrostat ic

patch model discussed earl ier . If the adsorbed

positive patches are of sufficient size, en-

counters with negative areas on othe r part icles

will be sufficiently favorab le to cause atta ch-

ment, even though the overal l charge on the

part icles, i f uniformly distr ibuted, would st i l l

be high enough to prevent f locculat ion.

Fu rth er increase in the size of the pa tche s (i.e.,

increasing molecular weight of adsorbed

polycat ion) would necessari ly reduce the

number of pa tches per par t ic le and tend to

keep the chance of favorable contacts with

other part icles about constant . Similar argu-

ments apply to both the destabil izat ion and

restabilization regions and so explain the sym-

metrical broadening of the flocculation zone.

Turning to the maximum rates of f loccu-

lat ion observed, i t is appar ent tha t appreciably

faster f locculat ion can be achieved with

polymers of mod erately high molecular weight

over that found by the addit ion of s imple

salts. In the presence of 0.3 M Ca(NO3)2,

double lay er repulsion is complete ly suppressed

and the f locculat ion rate observed should be

the rate of Brownian collisions between

par tic les (modif ied by the hydr odyn am ic

effect, see below). Th e results in Fig. 1 show

that the ra te obta ined wi th polymers 2 and 3

can be about twice this value, as reported

previously for other suspensions_ (5, 6), Again,

Jour nal of Colloid and Interface Science Vol. 55 No. 1 April 1976


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polymers 2 and 3 behave a lmost i dent i ca l ly

in this respec t , in spite of sixfold differenc e in

molecular weight . A patchwise adsorpt ion of

polymer could probably expla in th i s l imi t ed

rate enhancement , as shown in a later sect ion.

The maxim um ra t e of f l occula tion found

wi th polymer 1 i s about 50% higher t han tha t

with salt and this is more difficult to explain

by a patch model . I t i s not easy to see why

posi t ive patches large enough to enhance the

rate of f locculat ion appreciably, should not

also cause a more substant ial broadening of

the f locculat ion zone. However, even adsorbed

CT A + ions app ear to enh ance the f locculat ion

rate sl ight ly and in this case there should be

no possibi l i ty of patchwise adsorpt ion, so that

o ther fac tors , as ye t uncer t a in , m ay hav e to be

t aken in to account .

Two-Stage Addition

By choos ing the opt imum polymer concen-

t rat ion, b ut only adding hal f of the suspension,

each part icle receives, on average,

twice

t he

opt imum amount of f l oeculan t . I t can be seen

from Fig. 1 that , for al l of the polymers

employed, this would be sufficient to com-

pletely restabi l ize the part icles (because of

thei r excess posi t ive charge), and so essent ial ly

no flocculat ion should occur after the addi t ion

of the f i rst hal f of the suspension. When the

second hal f of the

will be l i t t le or no

solut ion and i t i s

suspension i s added there

polymer remain ing in f ree

safe to assume tha t t he

adsorbed polymer on the f i rst part icles wi l l

no t desorb to any apprec i ab le ex ten t . Thus ,

the part icles in the second port ion wi l l remain

negat ive ly charged and the two-s t age addi t ion

procedure should produce a suspens ion con-

t a in ing equal numbe rs of pos i ti ve and n egat ive

part icles. In fact , this cond i t ion can be veri fied

by di rect observat ion of such a suspension in

microelect rophoresis cel l (Sheiham, unpub-

l i shed observat ions, 1973). In the mixed

suspension, the only coll isions result ing in the

format ion of aggregates should be those

between opposi tely charged part icles, which

represent hal f the total number of col l i s ions.

From th is s imple p i c ture i t might be pred ic t ed

tha t t he ra t e of f locculat ion found af t e r a

two-stage addi t ion of suspension would be

just hal f that observed fol lowing the one-stage

addi t ion procedure . Resul t s f ro m the two-s tage

addi t ion exper iment s do not bear ou t t h i s

predict ion, bu t do lead to a possible explana t ion

of the effect of patchwise adsorpt ion on

flocculat ion rates general ly.

The fo rm of the tu rb id i ty versus t ime t races

for one-stage and two-stage addi t ions i s

shown schemat ical ly in Fig. 2 for the case

of op t imum polymer concent ra t ion . Af t er t he

one-stage addi t ion of the suspension at point

A, the turbidi ty at tains i t s ini t ial value r0,

and then ri ses at a rate that depends on the

rate of f locculat ion under opt imum condi t ions.

In a two-stage addi t ion, the f i rst hal f of the

suspension i s added at point A and the turbid-

i ty becomes

ro/2.

Apar t f rom a s l igh t non-

equi l ibrium flocculat ion in the case of fai r ly

high molecu lar weight polym ers (e.g. , 2 and 3),

no change in turbidi ty occurs, as expected.

The second por t ion i s added a t po in t B ,

whereupo n the turb id i ty r ises to abou t to and

then begins to increase at a measurable rate.

Al though, the reproducibi l i ty of the resul ts

i s not ent i rely sat i sfactory, sufficient experi -

ment s have been conducted , us ing var ious

>~

£ ~o

1

STAGE

/

¢

,,

i

. _ . . . . . . . . . . . J

i

S t

t' ¢

A B

TIME

2 S T A G E

/

/

/ /

/ /

/

/

/

FIG. 2. Schema tic llustration of the turbidity change

with time after a one-stage (at A) or a two-stage (at

A and B) addition of latex suspension to a solution

containing the op timu m concentration of cationic

polymer.

Journal of Colloid and Interface ~¢ien¢¢ V o l . 5 5 , N o . 1 , A p r i l 1 9 7 6


6/10

40 JOHN GREGORY

latex part icles and polymers, under di fferent

condi t ions and by two di fferent experimental

techniques, to make the fol lowing conclusions

with some confidence:

a) The t ime interval between the addi-

t ions of the two port ions of suspension At has

no infuence on the f loccula t ion ra t e a f t e r t he

second addi t ion. V alues of At hav e bee n

varied from a few seconds to many hours, wi th

no change in the f locculat ion rate.

b) Th e rate of f locculation after the

addi t ion of the second hal f of the suspension

is the

same

as tha t observed af t e r a one-s t age

addi t ion of suspension, rather than hal f of

the o ne-stage rate as exp ected on the basis of a

simple heteroflocculat ion picture.

Some typical resul ts for polymers 1 and 3

are shown in Fig. 3, in which the po ints shown

for At = 0 are those corresponding to a one-

stage addi t ion of suspension.

In the next sec t ion an a t t e mp t wi ll be made

to account for these f indings in terms of the

elect rostat ic patch model .

Patchwise Adsorption and Flocculation Kinetic s

In the case of one-stage addi t ion of suspen-

sion, assume tha t a frac t ion f of the to tal

surface area of the part icles i s covered by

adsorbed ca tionic polymer , and hence , 1 - - f )

i s the fract ion of negat ively charged surface.

I t i s then possible to envisage three dist inct

types of col l i sion, each characterized by a

certain stabi l i ty rat io W as in the Fuchs

theo ry of slow flocculat ion 12):

a ) be tween negat ive a reas Wn ;

b) between po si t ive areas W~2;

c) between negat ive and posi t ive areas

}V12.

If i t i s further assumed that the relat ive

frequency of each type of col l i sion is propor-

t iona l t o the product of the appropr i a t e a rea

fract ions, then the overal l s tabi l i ty rat io for

the suspension Wt, can be wri t ten, fol lowing

Hogg, Hea ly and Fuers t enau 13) :

1 I 1 - - f ) 2 f f 2 f 1 - - f ) 1

w , . E13

0 1 0

0 0 ~

d

¢

I ¢

0 0 ~

0 02

[]

m

POLYMER 3

POLYMER 1

X

J

At rnin)

Fro. 3. Rate of turbidity increase after a two-stage

addition of suspension, as a function of

At

the time

interval between the additions of the two portions.

The values at At = 0 are those for a one-stage addition

of suspension.

Journal of Colloid and Interface Science Vol. 55. No. 1. April 1976

In the case of two-stage addi t ion, hal f of the

part icles receive al l of the polymer so that a

fract ion 2f of thei r surface consists of posi t ive

patches. I t i s conv enient to assume f < ,

i .e . , that there i s enough part icle surface to

accom m oda t e t w i ce t he op t i m um am oun t o f

polymer, al though exact ly the same resul t i s

obtaine d i f f > ). A fract ion 1 -- 2f) of the

surface of the f i rst part icles remains n egat iv ely

charged. When the second hal f of the suspen-

sion is added, al l of i t s part icle surface area

remains negat ively charged since no cat ionic

polymer i s avai lable for adsorpt ion. Assuming

the same values of stabi l i ty rat ios

Wn W22

an d W12, and considerin g all six possible kind s

of coll ision, the following expression for the

overal l s tabi l i ty rat io, Wt , can be derive d:

4 f 1 2 f ) q - - - + . [ 2 ]

+ W12 W n

1 _ 1V 1- 2/ ) 2+ 4ff 1

+ - -

W Wn


7/10


Rearrangement of this expression gives:

1 F 1 _ f ) 2 f 2

2f(1 -- f)-] 1

j - E3-]

i .e . , exact ly the same stabi l i ty rat io as in the

case of one-stage addi t ion. Hence, from the

patch model , there i s good reason to expect

tha t f l occula tion ra t e should be ind ependent

of the mixing procedure.

In solut ions of low ionic st rength, the inter-

act ion between areas of l ike charge wi l l be

high ly repulsive, i .e. , Wn and W22 will be

very large. Conversely, because of the at t rac-

t ion betw een are as of unlike charge, W12 < 1,

i .e. , such coll isions should occur more fre-

quen t l y t han expec t ed f rom t he B row n i an

col l i s ion rate. Consequent ly, the overal l s ta-

bi l i ty rat io

Wt,

should be de t ermined en t i re ly

by the th i rd t e rm on the r igh t -hand s ide of

Eq. [ -17 and hence , by the va lues o f f and W12

Unfor tunate ly , ne i ther of t hese va lues can be

est imated wi th much confidence.

The f rac t iona l pa t ch area wi l l depend on

the nature of the cat ionic polymer, especial ly

on the charge den si ty and the f lexibi li ty of

the cha in . However , t he va lue of t he t e rm

2f (1 - f ) is no t very sens it ive to the va lue

of f , e .g., over the rang e 0.2 < f < 0.8 this

term varies between 0.32 and 0.5, wi th the

ma ximu m value at f = 0.5. For a rou gh

estimate of Wt i t wil l be sufficient to assume

f= 0 .3 .

The value of W12 requires rather more

careful considerat ion.

Calculation of [/V12

In principle the stabi l i ty rat io W, for a

suspension of spherical part icles can be calcu-

l a t ed i f t he to t a l i n t e rac t ion energy be tween

two spheres Vt , i s known as a funct ion of the

distance of separat ion d.

If the spheres are of equal radius a, the

t reatment of Fuchs (12) leads to:

f0 ~ exp

V~/kT)

W = 2 2 + u 2 du, [-4-1

where k i s Bol tzmann's constant , T i s the

absolu t e t empera ture , and

u = d/a.

Usual ly,

the Fuchs method is appl ied to slowly floccu-

lat ing systems, where

Vt

i s posi t ive at most

separat ion distances and W > 1. If there i s

no interact ion between the part icles

Vt = 0),

Eq . [4-] gives W = 1 and then , by definit ion,

the f locculat ion rate i s governed ent i rely by

the Brownian col l i s ion frequency of the

particles.

It is now recognized (14, 15) that Eq. [-47

needs to be modified to take into account a

hyd rody nam ic, or viscous, effect which tends

to reduce the rate of f locculat ion. The effect

arises essent ial ly from the need to squeez e

ou t the l iquid between appro aching spheres,

and this becomes increasingly di ff icul t as the

gap narrows.

A useful empirical approximat ion was given

by Honig , Roebersen and Wiersema (15) ,

which adequa te ly represent s t he hydrod yna mic

effect and leads to the fol lowing, expression for

the stabil i ty rat io, replacing Eq. [-4-]

~ (6u 2 + 13u + 2) exp

V t / k T )

W = 2 du.

0 (6u 2 + 4u)( 2 + u) 2

ES-]

If Vt = 0, Eq . [-5-] gives W = ~ , i.e., no

flocculat ion. Only i f there i s some at t ract ion

betw een the pa rt icles (i .e. , Vt < 0 at close

approach) can flocculat ion take place.

Ignoring the possibi l i ty of ster ic effects ,

t here a re two impor t an t cont r ibu t ions to the

in t erac t ion energy , t he van der Waal s and

electrical te rms :

V, = V~dw + VE. [-6-]

The v an der Waal s a t t rac t ion be tween equal

spheres i s given by the Hamaker expression

(16) :

V v d w ~ - - - -

2 2

6 2 + 4u (u + 2) 2

u 2 + 4u

+ In u + , [-7~

where A i s t he Hamaker cons t an t , which , for

Jour na l o f Co llo id and In terface S¢ ienc¢ Vol. 55 No. 1 A pril 1976


8/10

42 JOHN GREGORY

the present case of polystyrene in water , wil l

be as sum ed to be 5 X 10 21 J (17).

Equat ion [ -7] does not take re ta rda t ion

effects into account, nor have possible short-

comings of the simple Ham aker approach (18)

been considered. However, for the present

case where the electrical term in Eq. [-6] is

also at tract ive, the van der Waals at t ract ion

is quite a minor contr ibution to Vt, so that no

serious errors will arise by the use of Eq. [7].

The VE term, presents some difficulties, for

instance in the choice of surface potentials for

the positive and negative regions. There is also

the quest ion of whether the interact ion occurs

at constant potential or constant charge (19),

which is of crucial importance in the case of

unlike surfaces, even the sign of VE can be

different depending on the condit ion assumed

(20). A nother problem is whethe r the posi t ive

patch es are su fficiently large to justify the use

of expressions based on the interaction of

spheres, but s ince such an assumption is

implici t in the previous derivat ion of the

stability ratio Wt, it will not be discussed

further .

I t has been shown (21 ) tha t expressions

based on the linear superposition approxi-

mation (LSA) give resul ts intermediate be-

tween those for the constant charge and

consta nt potential condit ions and, s ince nei ther

of these extreme conditions is likely to be

appropriate in pract ical s i tuat ions, the LSA

approach might be a reasonable one to adopt .

A convenient expression for spherical particles

of equal s ize but unequal potentials in a

symmetrical (z-z) electrolyte solution is (22):

128rrnk T a

v~

3 ,3 2 - - e ~p ( -- Kd ), l- S]

K2 u q- 2)

where n is the number of cations (or anions)

per cubic meter , K is the Deby e-Hiicke l re-

ciprocal length parameter (~2= 2e~nz2/~kT,

where e is the electron charge and ~ the per-

m itt iv ity of the med ium). 3'1 and 3 2 are fu nc-

tions of the surface po ten tial s ~bl an d ~k2 of th e

negative and posi t ive patches, respect ively:

3 1 = tanh(ze~bl/4kT) etc.

Str ict ly, Eq. I-8] only applies whe n the

diffuse pa rt of the electrical double layer is

fair ly thin compared to the radius of the

particles (Ka > 10). For the latex particles used

in the present work, a~---150 nm and, in

10 4 M NaNO3, Ka is onl y abou t 5, but the

errors involved in using Eq. [-8] are still no t

serious.

The appropriate values of ~bl and ~b2 ~tre

not known, but real is t ic est imates can be

made as follows. From electrokinetic studies

on dialyzed latex suspensions (23), zeta po-

tent ials in di lute electrolyte solut ions are

often about --60 inV. After charge reversal

by cat ionic surfactants (11) or polymers (2)

the zeta potential reaches a maxim um value in

the region +30 to 60 mV, al though local ized

areas of higher potential probably exist under

these conditions. Bearing these figures in mind ,

the reduced potential ,

y l (= ze~ l / kT) ,

of the

nega tive areas has been take n as --2 (i.e.,

~bl = -- 51 mV ) an d y2, for the positiv e pat ch es,

has been given va lues of q- l , +2 , and + 4

(i.e., ~b2 va ry ing betw een 26 and 102 mV ).

After computing values of Vvaw and VE

from Eqs. [7] and [.8], W12 has been obtai ned

from Eq. 1-5] by a num erical integrat ion

procedure. Results are shown in Fig. 4 for

concentrat ions of 1- 1 electrolyte between

10 5 an d 10 2 M, assum ing the radius of the

part icles is 150 nm and the Hamaker constant

is 5 X 10 21 J.

Using Eq. [1] and assumingf = 0.3, values

of W t may be calculated and these can be

compared with the stabi l i ty rat io expected for

the particles in the absence of electrical

repulsion (e.g., in high salt concentration),

W t°. By pu t t ing V~ = 0 and car ry ing out

the numerical integrat ion procedure, W? is

found to be 1.784, i .e., the flocculation rate

should be about 56% of the Brownian coll is ion

rate, in agreement with previous computat ions

(15) of the h ydro dyn am ic effect .

The ra t io W t ° / W , should then give the factor

by which the flocculation rate is increased as a

result of patchwise polymer adsorption. This

ratio is plotted in Fig. 5 as a function of

electrolyte concen trat ion and for various values

Journal of Colloid and Interface Science Vol . 55 No. 1 Apri l 1976


9/10


of y2 . I t i s a ppa re n t tha t the r e su l t de pe nds

ve ry s t rong ly on the ion ic s t r e ng th bu t ve ry

little on the value of y2.

A s t rong de pe nde nc e o f ma ximum f loc c u-

la t ion ra te on the ion ic s t r e ng th o f the so lu t ion

ha s be e n re por te d p re v ious ly (5 ) a nd the

resul ts in Fig. 5 shows a s imila r t rend. How-

e ve r , the c om pute d re su lt s d i f fe r ma r ke d l y

f rom e xpe r ime nta l f ind ings . Ac c ord ing to the

resul ts in Fig. 5 pa tchwise adsorpt ion of

c a t ion ic po lyme r shou ld

decrease

the ra te of

f loccula t ion i f the sa l t concentra t ion is grea te r

tha n a bou t 10 4 M, wh e re a s e xpe r im e nta l

ra te s ha ve be e n shown to be e nha nc e d up to

abo ut 10 2 M sa l t (5). The pres ent resu l ts in

10 4 M NaN O3 show tha t th e f loccula t ion ra te

c a n be ne a r ly doub le d wi th c a t ion ic po lyme rs ,

whereas Fig. 5 indica tes a s l ight decrease in

ra te a t th i s s a l t c onc e n t ra t ion .

The c h ie f d i f f i c u l ty wi th the p re se n t mode l

i s tha t a l thoug h the f a vora b le in te ra c t ion

be twe e n pos i t ive a nd ne ga t ive a re a s l e a ds to a

decrease in th e ca l cula ted va lue of W12, as

20

1

11

(35

\\ \

\ \ \ \

- 4 - 3 - 2

LOGloSALT CONCN ( IM)

1 1o.

5 . Th e r a t i o

W?/Wt

see t ex t ) as a func t ion o

e l e c t r o l y te c o n c e n t r a t i o n . Co n d i t i o n s a s f or F i g . 4 .

1 0

W12

O.

/ / /

/ / /

/

/ /

- 5 - 4 - 3 - 2

L O G l o S A LT C O N C N ( M )

FIG. 4. Calculated values of the stab ility rat io W ~,

for the interacti on between o ppositel y charged spherical

surfaces, as a function of the concentration of 1-1

electrolyte. The reduced potential of the negative

surface yt, is --2 and the numbers on the curves denote

the values of y~, for t he po sitive surface.

required, th is is usua l ly outweighed by th~

small frac tio n of such collisions. I f f = 0.3

on ly 42 o f c o ll i sions a re be twe e n pos i t iw

and nega t ive a reas , according to Eq. 1-1]

Thu s , the e ffec t ive s tabi l i ty ra t io W12, woulc

have to be reduced by a fac tor of about 2 . . :

to show a n ove ra l l r a te e nha nc e me nt . I t i ,

very l ike ly tha t col l is ions a re not ent i re l)

rand om, as assum ed in Eq. I -1] , but tha t ther~

is a mu tua l o r ienta t ion e ffec t be tw een ap-

proa c h ing pa r t i c le s , so tha t pos i t ive a nd ne ga -

t ive a reas a re more l ike ly to col l ide than th~

s imple f a c to r 2 f (1 - f ) would sugge s t . How.

ever , a qua nt i ta t i ve t re a tm ent of this e ffec l

would present cons iderable dif f icul t ies .

REFERENCES

1.

TEOT,

A. S.,

Ann. N. Y. Acad. Sc i .

155, 593 (1969)

2. GREGORY, ., Trans. Faraday Soc. 65, 2260 (1969)

3. DIx oN, J. K. AND ZIELYK, M. W.,

Environ. Sci

Technol.

3, 209 (1969).

4. SANDELL,L. S. AND LUNER,

P. J . Appl . Polym. Sc i

18, 2075 (1974).

5. GREGORY, J. J. Colloid Interfa ce Sci. 42, 441

(1973).

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4 4 J O H N G RE G O RY

6. GREGORY, J. , AND SHEIHAM, I. Brit. Polym. Y. 6

47 (1974).

7. MICHAELS,A. S.,

Ind. Eng. Chem. 46

1485 (1954).

8. RASPER, D. R., P h.D. Thesis, California Institu te

of Technolog y, 1971.

9. ILER,

R. K. J. Colloid Interface Sc i.

37, 364 (1971).

10. VAN DEN HU L, H. J. AND VANDERHOFF J. W.,

J. Colloid Interface Sci. 28, 336 (1968).

11. CONNOR, P. AND OTTEWILL, R. H. J. Colloid

Interface Sci.

37, 642 (1972).

12. OVERBEEK, J. TH. G.

in

Colloid Science, Vol. 1,

p. 283. (H. R. K ruyt , Ed.), Elsevier, Amsterdam ,

1952.

13. HOG G, R. , HEA LY, T . W. , AND FUERSTENAU,

D. W. Trans. Faraday Soc.

62, 1638 (1966).

14. SPIELMAN,L. A. J. Colloid Interface Sc i. 33, 562

(1970).

15. HON IG, E. P. , ROEBERSEN, G. J., AND WIERSEMA,

P. H. J. Colloid Interface Sci.

36, 97 (1971).

16. HAMAKER,H. C., Physica 4 1058 (1937).

17. GREGORY, J.,

Adv. Collo id Interface Sci.

2, 396

(1969).

18. PARSEGIAN,V. A. AND NINHA~I,

B. W. ]. Colloid

Interface Sci.

37, 332 (1971).

19. FR ENS, G. ANn OVERBEEK J. Th. G., ] .

Colloid

Interface Sci.

38, 376 (1972).

20. BELL , G. M. AND PETERSON,

G. C. J. Colloid

Interface Sci. 41, 542 (1972).

21. GREGORY,

. J. C olloid Interface Sci.

51, 44 (1975),

22. BELL , G. M. , L EVm E, S., AND MCCARTNEY,L. N..

J. Colloid Interface Sci.

33, 335 (1970).

23. OT'rEWILL,R. H. AND SHAW,J. N. J. Electroanal.

Chem. 37, 133 (1972).

Journa l of Colloid and Interface Science. Vol. 55. No. 1, Apr il 1976

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The effect of cationic polymers on colloidal stability

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