J. CHEM. SOC. FARADAY TRANS., 1991, 87(8), 1177-1181 1177

Transient Collision Efficiency of Polymer-induced Flocculation

Jyh-Ping Hsu* and Der-Po Lin Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10764, Republic of China

The temporal variation of the collision efficiency of primary particles in polymer-induced flocculation is investi- gated. A kinetic model is introduced assuming that the flocculation and the adsorption of polymer molecules are second order, and the desorption of polymer molecules is first order. The adsorption of polymer molecules is assumed to be reversible, and an adsorbed polymer molecule is allowed to rearrange itself. The result of numerical calculations reveals that, depending on the ratio (number of sites on the polymer molecules/number of sites on the surfaces of the colloidal particles) and the relative magnitude of the adsorption rate constant and the desorption rate constant, the collision efficiency calculated by the conventional analysis may be either underestimated or overestimated. Thus, care must be taken in predicting the rate of polymer-induced floccu- lation through the collision efficiency approach.

Among the possible ways of initiating flocculation of a colloi- dal suspension, polymer-induced flocculation is of particular interest. The flocculant thus obtained has some desirable properties from a practical point of view. These include a greater floc size, a more condensed structure, and greater strength to resist hydrodynamic shear force.' Polymer- induced flocculation comprises two processes : adsorption of polymer molecules onto the surfaces of colloidal particles and particle-particle collision leading to the formation of molecu- lar bridges. Intuitively, at a very early stage, the former plays the major role; immediately after some colloidal surfaces are covered by polymer molecules, however, the two processes may occur simultaneously. The rate of flocculation is often estimated by resorting to the number of collisions between particles and the conventional collision efficiency, defined as the fraction of collisions leading to flocculation. The basic assumption of the collision efficiency approach is that the rate of adsorption of polymer molecules onto colloidal sur- faces is much faster than the rate of flocculation. In other words, before the adsorption of polymer reaches equilibrium, the flocculation of particles is negligible. This assumption, although simplifying the procedure of estimating the rate of flocculation, lacks experimental proof. In fact, most of the reported results about the adsorption of polymers on solid surfaces reveal that this phenomenon has been known for some time. Jankovics,* for instance, studied the adsorption of polyacrylamide [ M = (1-6) x lo6] on calcium phosphate. It is known that the adsorption rate constant correlates with the molecular weight of polymer, the time required to achieve an approximate equilibrium ranging from 20 to 2000 min. The time constant for flocculation, although largely depen- dent upon the number of colloidal particles in the suspension medium, is much less than that of polymer adsorption in general.3.4 It appears that the assumption made in the con- ventional collision efficiency analysis is suspicious, and care must be taken in predicting the rate of flocculation by this approach.

The purpose of this study is to provide a quantitative analysis on the effect of the transient behaviour of the adsorption process on the variation of collision efficiency.

Modelling We assume that the adsorption of polymer molecules onto the surface of colloidal particles is reversible. Suppose that the flocculation and the adsorption of polymer molecules are second order, and the desorption of polymer molecules is first

order. An attached polymer molecule is allowed to rearrange itself. In other words, the number of attached sites of a polymer molecule is time dependent. For simplicity, it is assumed that the concentration of polymer is sufficiently low such that two polymer molecules on the surface of a colloidal particle do not interact. The total numbers of colloidal par- ticles and polymer molecules is denoted by X , and N , respec- tively. S is the number of active sites on the surface of a colloidal particle and rn is the number of sites on a polymer molecule. The number of polymer molecules with i sites attached to the surface of a colloidal particle at time t is denoted by ni(t), and the mean number of unoccupied sites on the surface of a colloidal particle, which are accessible to an attached polymer molecule at time t , is X( t ) . A number balance for the number of attached sites on the polymer site gives

dn,/dt = -Lon, + plnl (1)

dnJdt = i i - 1 - (E,i + pi)ni + pi+ ni+ 1 (2)

where

i = 1, 2, ..., (rn - 1)

dnJdt = A,,, - n, - 1 - p,,, n, (3)

where ii represents the rate of adsorption for the sites on a polymer molecule and pi denotes the rate of desorption for the attached sites per polymer molecule. Suppose that the adsorption of polymer molecules on the colloidal surface is of Langmuir type. We thus have

(4)

pi 5 kdi ( 5 )

Ai = k,(m - i )X ( t )

and

where k , and kd are, respectively the adsorption and desorp- tion rate constants. Substituting these expressions into eqn. (1)-(3), and expressing in a more compact form, we obtain

dn,ldt = k,(rn - i + l)X(t)n,-

- [k,(rn - i)X(t) + k d i ] n i

+ kAi + TI,+^; i = 0, 1, 2, .. ., m (6)

and where n- = n,, = 0. The mean number of unoccupied sites on the surface of a colloidal particle at times t , W(t), is

m

W(t) = s - c ini(t) /X, (7) i = O

Publ

ishe

d on

01

Janu

ary

1991

. Dow

nloa

ded

by U

nive

rsity

of

Pitts

burg

h on

30/

10/2

014

14:2

5:57

. View Article Online / Journal Homepage / Table of Contents for this issue

1178

Letting E be the fraction of W(t) that are accessible to an attached polymer molecule, then

(8) ) X(t) = E S - 1 inXt)/Xo

It can be shown that the solution to eqn. (6), subject to the condition that all the colloidal particles are free of polymer molecules initially, is5

P,( t ) = C:[~,(t)]'[l - PT(t)]m-i; i = 0, 1, . . . , rn

( m i = O

(9)

and where the combinatorial symbol C y is defined by

C: = rn ! / ( rn - i)! i ! (94

pi(t) = nXt)/N (9b)

P(t) = [S - X(t)/E]/S (94

= XoSP(t ) / rnN = P(t)/r (94

and

Pdt) = [Xo S - X o X(t)/&]/mN

where r = rnN/XoS. Differentiating eqn. (8) with respect to t yields

dX( t)/di

For convenience,

dX*/c

where

his equation is recast as

!* = K - ( r + K - 1)X* - X*2 (1 1)

x * = X(t)/SE

t* = k,SEt

K = kd/Ek, S

The solution to eqn. ( l l) , subject to the initial condition X* = 1 at t* = 0, is

where

r l = { - ( r + K - 1) + [ ( r + K - 1)2 + 4K]'/2}/2 ( 1 2 ~ )

and

r 2 = { - ( r + K - 1) - [ (r + K - 1)2 + 4K]"2}/2 (12b)

Mean and Variance of Fractional Surface Coverage

The mean fractional surface coverage (8( t ) ) can be defined as

(@(t)> = cs - X(t)/EI/S

= 1 - X*(t) (13)

or

(e(t*)) = 1 - x*(t*) (14)

By definition, the variance of a random variable Y , ( (Y ) , can be evaluated by the following expression:6

= ( Y 2 > - (02 (15)

J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

where ( Y 2 ) and ( Y ) are the mean of Y 2 and the mean of Y , respectively. Since

= (8(t*))( 1 + (e(t*)> (rn - 1))

the variance of the fractional surface coverage is then

Collision Efficiency

Various expressions have been proposed to relate the colli- sion efficiency E to the fractional surface coverage of a colloi- dal particle. By assuming that each particle has the same fractional surface coverage, La Mer and Smellie7 proposed the expression below :

E = 0(l - 8) (18)

where 8 denotes the fraction of a colloidal surface covered by polymer molecules. This model predicts that E achieves a maximum at 8 = 1/2. The experimental evidence, however, reveals that the maximum rate of flocculation does not neces- sary occur at this value.' In an attempt to provide an expla- nation for this discrepancy, Hoggg suggests that the possible reorientation of colliding particles should be taken into account. Moudgil et aL4 assume that there exist two types of sites on the surface of a particle: active site and inactive site. An effective collision between two particles, which yields attachment, occurs if a patch of adsorbed polymer finds a patch of free active sites on another particle, or vice versa. Eqn. (18) is modified by considering the fraction of active sites on the surface of a particle. Molski" assumes that three types of effective collision between two colloidal particles are possible: (i) Two sites at the collision point are both bare. The probability of sticking between these particles is a. (ii) Two sites at the collision point are both occupied. The prob- ability of sticking between these particles is p. (iii) One of the two sites at the collision point is bare and the other is occupied. The probability of sticking between these particles is unity. Molski has shown that

(19) In the following discussion, this expression is adopted with the modification that 0 is replaced by (0). In other words, the mean fractional surface coverage is used in the evaluation of the collision efficiency. It can be shown that if the number of colloidal particles is sufficiently large, using the mean frac- tional surface coverage is appropriate (see Appendix).

E = I - (1 - a)(i - 0)2 - (1 - p)e2

Results and Discussion Fig. 1 shows the temporal variation of the mean fractional surface coverage at different values of r for the case K = 0.01. For a fixed t*, (0) increases with increasing of r. Note that r is the ratio (total number of sites on the polymer molecules/ total number of sites on the colloidal surfaces). Therefore, if r > 1, the total number of sites on polymer molecules is greater than that on the colloidal surfaces. On the other hand, if r < 1, the total number of sites on the colloidal sur- faces is greater than that of the sites on the polymer mol- ecules. The calculated temporal variation of E at different values of a, D and r for the case K = 0.01 is shown in Fig. 2. In Fig. 2(a), since a = p = 0, only those collisions at which one of the two sites at the collision point is occupied and at

Publ

ishe

d on

01

Janu

ary

1991

. Dow

nloa

ded

by U

nive

rsity

of

Pitts

burg

h on

30/

10/2

014

14:2

5:57

. View Article Online

J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 1179

1 0

o a

0 6

( 6 ) 0 4

0.0 0 2 ~ 0 . 0 5.0 10.0 15.0 20.0

t*

Fig. 1 Temporal variation of the mean fractional surface coverage (0) for the case K = 0.01 at different r. t* (= k, Stcwimensionless time; K ( = k d k , ScFrelative magnitude of the desorption rate con- stant and the adsorption rate constant

one of which is empty are effective. In this case, eqn. (19) becomes

As can be seen from Fig. 2(a), for r b 1, E increases with t*, reaches a maximum, and then decreases to a steady-state value. On the other hand, for r < 1, E increases monotoni- cally with t*. Note that if the rate of flocculation is compar- able to the rate of adsorption, the value of E used in the conventional collision efficiency analysis is underestimated if r 3 1, and is overestimated if r < 1.

In the case of Fig. 2(b) since B = 0, the sticking probability is negligible if the two sites at the collision point are both

1 .o 0.8

0.6

0.4

0.2 0.0

E

1 .o 0.8 -

-

0.6

0.4

0.2

E

0.0 / o.ol i 6.b' I ' l 'o!O I '1151.6 ' '201.0

t*

occupied. For r 2 1, E increases with t*, reaches a maximum, and then decreases to a steady-state value. The increase of E in the early stages is due to the increase of the probability of collision between an occupied site and an empty site. In the later stages, the degree of surface coverage is great due to the presence of the large number of sites on the polymer mol- ecules. The effective collisions are mainly due to those between unoccupied sites. This results in a low value of E . On the other hand, for r < 1, E increases monotonically with t*. As in the case of Fig. 2(a), if the rate of flocculation is comparable to the rate of adsorption, the value of E used in the conventional collision efficiency analysis is underesti- mated if r 3 1, and is overestimated if r c 1.

Fig. 2(c) shows the result when a = 0 and p = 0.5, and Fig. 2(d) presents the result when CT = 0.5 and p = 0.1. In Fig. 2(c), the contribution to E due to the collision in which both sites at the collision point are bare is negligible. The same conclusion as that obtained in Fig. 2(a) and 2(b) can be drawn from these figures. In the case of Fig. 3(a), a = 1 and B = 0, eqn. (19) becomes

Since the number of empty sites on the colloidal surfaces decreases monotonically with time, E decreases accordingly. Therefore, if the rate of flocculation is comparable to the rate of adsorption, the value of E used in the conventional analysis is underestimated. In the case of Fig. 3(b), CT = 0.5 and p = 1, and eqn. (19) reduces to

E = 1 - 0 . 5 ~ 1 - (e(t))12 (22)

Since the major contribution to E comes from those occupied sites, E increases monotonically with time. Thus, if the rate of flocculation is comparable to the rate of adsorption, the value of E used in the conventional analysis is overestimated.

1 .o 0.8

0.6

0.4

0.2

0.0

E

1 .o 0.8

0.6

0.4

0.2

0.0

0.5

I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l 1

0.0' 5.0 10'.0 15l.0 20'.0 t*

0.0 4 5.0 10.0 15.0 20.0

t*

Fig. 2 Temporal variation of the collision efficiency E for the case K = 0.01. (a) sl = 0, p = 0 ; (b) a = 0.1, B = 0 ; (c) a = 0, p = 0.5; (6) a = 0.5, B = 0.1. t* ( = k, Stcwimensionless time; K ( = k J k , ScFrelative magnitude of the desorption rate constant and the adsorption rate constant. a and /&probabilities of sticking between two colloidal particles when two sites at the collision point are both bare, and when they are both occupied, respectively

Publ

ishe

d on

01

Janu

ary

1991

. Dow

nloa

ded

by U

nive

rsity

of

Pitts

burg

h on

30/

10/2

014

14:2

5:57

. View Article Online

1180 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

1 .o

0.8

0.6

E

0.4

0.2 I,,,* 0.0

0.0 5.0 10.0 15.0 2( t*

1.5 ‘.O 7 0.8

0.6

E

0.4 4 0.2 I ( b )

0.0 3 0.0 5.0 10.0 15.0 2(

t* 0

Fig. 3 = 1. t* (= k , Stcwimensionless time. K ( = k,/k, ,%+relative magnitude of the desorption rate constant and the adsorption rate constant. a and Fprobabili t ies of sticking between two colloidal particles when two sites at the collision point are both bare, and when they are both occupied, respectively

Temporal variation of the collision efficiency E for the case K = 0.01. (a) a = 1, B = 0 ; (b) a = 0.5,

Steady-state Value of (0) and ((0) The steady-state value of the mean fractional surface cover- age, (0) G o , can be evaluated by

( O ) , = lim ( O ( t * ) ) = lim [l - X*(t*)] t*+ m t*+ Go

= l - q (23)

Similarly, the steady-state value of the variance of the frac- tional surface coverage, ( ( O > > , , is

= (1 - al)[r + (1 - a,Xm - r - l)]/r (24)

The variations of (0) as a function of r is shown in Fig. 4. If the adsorption of polymer is reversible (K > 0), ( O ) , increases with the increase of r . This is because for a fixed value of K ( k a , k , , E , and S fixed), r increases with the total number of sites provided by polymer molecules. On the other hand, if the adsorption is irreversible ( K = 0), { O ) , is unity for r 2 1, and increases with the increase of r for r < 1. For a fixed r , ( O ) , decreases with the increase of K . This is reason- able since K is a measure of the relative magnitude of the desorption rate constant and the adsorption rate constant.

(0) m

0 01

0 5 1 0 1 5 2

/ / /

5 2 -

C O C C 0 5 1 0 1 5

r

Fig. 4 Variation of the steady-state mean fractional surface cover- age ( O ) , as a function of r. K ( = kdk , &)-relative magnitude of the desorption rate constant and the adsorption rate constant. I ( = mN/X, S k r a t i o of number of sites on polymer molecules/number of sites on colloidal surfaces

The variation of ((e>>, as a function of r at different values of m is presented in Fig. 5 . The value of ((e>>, increases with the increase of m. For a fixed value of m, ( ( O > > , increases with r , reaches a maximum, and then decreases.

Conclusion In summary, the temporal variation of the collision efficiency of primary particles in a polymer-induced flocculation is evaluated. The appropriateness of the fundamental assump- tion that the rate of adsorption of polymer molecules on the surface of colloidal particles is much faster than the rate of collision between colloidal particles is examined. The result of the present analysis shows that, depending on the ratio (number of sites on the polymer molecules/number of sites on the surface of the colloidal particles) and the relative magni- tude of the adsorption rate constant and the desorption rate constant, the collision efficiency calculated by the convention- al analysis may be either underestimated or overestimated. Thus, care must be taken in predicting the rate of polymer- induced flocculation through the collision efficiency approach.

8 0 10 1

0 0 0.5 1 .0 1 5 2.0

r

Fig. 5 Variations of the steady-state variance of the fractional surface coverage ( ( O ) , as a function of r for the case K = 0.01 at different m. K ( = k,/k, ScFrelative magnitude of the desorption rate constant and the adsorption rate constant. r (= mN/X, + - ra t io of number of sites on polymer molecules/number of sites on colloidal surfaces. m-number of sites on a polymer molecule

Publ

ishe

d on

01

Janu

ary

1991

. Dow

nloa

ded

by U

nive

rsity

of

Pitts

burg

h on

30/

10/2

014

14:2

5:57

. View Article Online

J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 1181

Appendix Suppose that the fractional surface coverage is different for different colloidal particles. ui denote the number of colloidal particles with a fractional surface coverage ei( = i /S); i = 0, 1, 2, . . . , S. The contribution to E due to all the possible colli- sions at which two sites at the collision point are both bare, E l , is calculated by

U i ( l - S i ) + u,( l - 0,)

S S

x c u ~ i - ei) + . - . + u ~ i - 0,) c ui( i - ei) i = O i = O

s

i = O

Similarly, the contribution to E due to all the possible colli- sions at which two sites at the collision point are both occupied, E , , is

The contribution to E due to all the possible collisions at which one of the two sites at the collision point is bare and the other is occupied, E 3 , is

/ s s

S

(A31 i = O

Thus

+ 2 j ; s: UiUjSi(l - ej) \ i = O j = O

- ; ui Si( 1 - oi))]/2C;0 b44) i = O

In the case where X o is large, since C f o z X 3 2 , eqn. (A4) reduces to

where

and i s

Since 0 < (0) < 1, eqn. (AS) can further be approximated to

E = “1 - (W)>12 + P ( e W 2 + 2(e(t)>C1 - (W)>I = 1 - (1 - a)[l - (0(t))I2 - (1 - P)(S(t))’ (‘48)

Thus, if the number of colloidal particles is large, using the mean fractional surface coverage to represent the exact dis- tribution of the fractional surface coverage is appropriate.

References 1 L. A. Glasgow and J. P. Hsu, AlChE J., 1982, 28,779. 2 L. Jankovics, J. Polym. Sci., 1965, A3, 3519. 3 J. P. Hsu, Ph.D. Dissertation, Kansas State University, 1984. 4 B. M. Moudgil, B. D. Shah and H. S . Soto, J . Colloid Interface

Sci., 1987, 119,446. 5 V. Brendel and A. S . Perelson, Siam J . Appl . Math., 1987, 47,

1306. 6 A. M. Mood, F. A. Graybill and D. C. Boes, Introduction to the

Theory of Statistics, McGraw-Hill, New York, 1974. 7 V. K. La Mer and R. H. Smellie Jr., J. Colloid Interface Sci.,

1956, 11, 704. 8 D. J. Walsh and J. Anderson, Colloid Polym. Sci., 1980, 258, 883. 9 R. Hogg, J. Colloid Interface Sci., 1984, 102, 232.

10 A. Molski, Colloid Polym. Sci., 1989, 267, 371.

Paper 0/05301E; Received 26th November, 1990

Publ

ishe

d on

01

Janu

ary

1991

. Dow

nloa

ded

by U

nive

rsity

of

Pitts

burg

h on

30/

10/2

014

14:2

5:57

. View Article Online