J. CHEM. SOC. FARADAY TRANS., 1991, 87(19), 32453250 3245

Transient Behaviour of Polymer-induced Flocculation

Jyh-Ping Hsu* and Der-Po Lin Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10764, R.O.C.

The transient behaviour of polymer-induced flocculation is examined theoretically. The present study, which takes the collision efficiency into account, is an extension of the conventional analysis. Here, the kinetics of the adsorption of polymer molecules onto the surfaces of colloidal particles are incorporated into Smoluchowski's flocculation mechanism. Also, unlike the classical approach, which assumes that the adsorption of polymer molecules onto the surfaces of colloidal particles achieves its steady state instantaneously, the transient nature of the collision efficiency is discussed. The temporal variations of the number of primary particles, the mean and the variance of the floc size distribution, and t h e mean density and porosity of the flocs are derived.

The destabilization of dispersed colloidal particles through introducing a small amount of polymer solution is a tech- nique used extensively in various solid-liquid separation pro- cesses. In this case, the aggregation of colloidal particles is generally ascribed to a bridging As pointed out by H ~ g g , ~ polymer-induced flocculation involves two stages: adsorption of polymer molecules onto the surfaces of dispersed colloidal particles, followed by particle-particle col- lisions leading to the formation of molecular bridges. Pre- vious efforts have almost always assumed that the former is much faster than the latter. In other words, the formation of molecular bridges is negligible until the adsorption stage reaches equilibrium. This assumption, although simplifying the subsequent analysis, is unrealistic for most of the systems that are of practical significance.

Intuitively, the rate of flocculation for a well agitated apparatus is proportional to the product of the frequency of collisions between particles and the fraction of the collisions that yield bridging (collision efficiency). That is,

R a E N i N j (1)

where R is the rate of flocculation, E represents the collision efficiency, and Ni and N j denote the numbers of particles in classes i andj, respectively. Several attempts have been made to determine the dependence of collision efficiency on the fractional surface coverage of a colloidal p a r t i ~ l e . ~ - ~ Among these, the expression suggested by Molski6 appears to be the most general. In his model, it is assumed that three types of effective collision between two colloidal particles are possible : (1) the two sites at the collision point are both bare; the prob- ability of sticking between these particles is a. (2) The two sites at the collision point are both occupied; the probability of sticking between these particles is 8. (3) 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.

The fundamental assumption made in the conventional collision efficiency analysis that particle-particle collision is the rate-determining step implies that E in eqn. (1) is time independent. Since aggregation of primary particles is often appreciable immediately after the introduction of a polymer solution, the adsorption and the flocculation processes occur simultaneously. In the present study, the transient behaviour of polymer-induced flocculation is examined with an attempt to take the kinetics of the adsorption of polymer molecules into account.

Analysis In the following discussion, the temporal variations of the number of primary particles, distribution of floc size, and the mean physical properties of the flocs are derived.

Variation of Mean Fractional Surface Coverage

In the case where the concentration of polymer is low and the adsorption of polymer molecules to the surfaces of colloidal particles is of the Langmuir type, it can be shown that the temporal variation of the mean fractional surface coverage of colloidal particles by polymer molecules (e(t*)) is :7

where X * = X(t ) /S( and t* = k , S ( t . A brief summary of the derivation of this expression is given in Appendix A. Here, X ( t ) denotes the mean number of unoccupied sites on the surface of a colloidal particle that are accessible to polymer molecules at time t , S is the number of active sites on the surface of a colloidal particle, ( represents the fraction of the mean number of unoccupied sites on the surface of a colloidal particle that are accessible to an attached polymer molecule, and k, is the adsorption rate constant. According to their definitions, X* is the fraction of unoccupied sites on the col- loidal surfaces and t* is the ratio (tlr), where T = ( k , S < ) - ' . The characteristic time, r, is the average time required to occupy a site on the colloidal surface. In the derivation of eqn. (2), it is assumed that an attached polymer molecule is capable of rearranging itself on the colloidal surface. X*(t*) is given by

where

a l , a2 = ( - ( r + K - 1) & [ ( r + K - 1)2 + 4K]1'2}/2 (3b)

In this expression, r = rnN/X,S and K = k J ( k , S [ ) , where rn denotes the number of sites on a polymer molecule, X , and N represent the total numbers of colloidal particles and polymer molecules, respectively, and k , is the desorption rate constant. By their definitions, r is the ratio, number of sites on polymer molecules/number of sites on colloidal surfaces, and K is a measure of the relative magnitudes of the rates of desorption and adsorption. The constants a1 and a2 result from solving a quadratic equation which describes the tem- poral variation of X* (see Appendix A).

Variation of Collision Efficiency

Molski6 suggests that

E = 1 - (1 - ax1 - q2 - (1 - f i e2 (4) where 6 is the steady-state fractional surface coverage of col- loidal surface by polymer. If the time-dependent nature of the fractional surface coverage is taken into account, eqn. (2) can

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3246 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

be modified for the case where the number of colloidal par- ticles is large by replacing 0 with (0). Substituting eqn. (2) and (3a) into eqn. (4) gives

E(t*) = [a;(a + /? - 2) + 2a2(1 - D) + fl + 2(a1 - a2X1 - a2)[a2(a + p - 2) + (1 - @]/A

+ [ (a + p - 2Xa1 - - a2)21/A2 ( 5 4 where

A = (1 - a2) - (1 - a,)exp[-(a, - a2)t*] (5b)

Variation of the Number of Primary Particles

Let N, be the number of primary particles, we have, by refer- ring to eqn. (1) for the very early stage of the flocculation process

dNl/dt = -KpEN:/2 ( 6 4 where'

8 k ~ T/3q; perikinetic flocculation Kp = { 2GD,/ 3; orthokinetic flocculation

In this expression, k, and T are Boltzmann's constant and the temperature, respectively, q denotes the viscosity of the fluid, G represents the shear rate and D, is the diameter of the primary particle. For convenience, eqn. (6a) is rewritten as

(6b)

dN:/N:' = -(Xo Kp/2k, S<)E dt* (7)

where N : = Nl/Xo. Substituting eqn. ( 5 4 into this equation and solving the resulting expression subject to the initial con- dition N: = 1 at t* = 0, we obtain

1/N? = 6 ( 8 4 where

In this expression

A = a;(. + p - 2) + 2a2(1 - p) + p

C = (a + p - 2Xa1 - a2)2(1 - ~ 1 ~ ) ~

( 8 4

(84

( 8 4

B = 2(a, - a2X1 - a2)[a2(a + /3 - 2) + (1 - o)]

Distribution of Floc Size

The temporal variation of the distribution of the size of floc is governed by the following set of equations:'

L k = i + j _I k = 1, 2,

where Kij denotes the collision coefficient, and Ii integer. For convenience, this equation is recast as

L k = i + j J k = 1 , 2 , .

Solving this expression directly is non-trivial, if not imposs- ible. However, the mean, (N), and the variance, ((N)), of floc size distribution can be obtained without much difficulty. As an illustration, let us consider three special cases in which the collision coefficient, Kij, is assumed arbitrarily. In the first case, K, = K O , a constant. In other words, the collision coefficient is independent of floc size. It can be shown that (see Appendix B)

( N ) = t~ (1 1)

(124 ((N)) = 40 - 1) where

(a1 - a2)t* + ln[A/(al - (1 - a2Xai - ~ 2 )

x [At* + B(

t* 1 + '((1 - a2)2 A(1 - a2)(al - a2)

--

In the second case, K, = K,(i + j ) , where KO is a constant; that is, the collision coefficient is proportional to the sum of the sizes of the two colliding flocs. Here, we have (see Appen- dix B)

(N) = exp(2a - 2) (13)

(14)

In the third case, K, = KO zj , where K O is a constant. In other words, the collision coefficient is proportional to the product of the sizes of the two colliding flocs. It can be shown that (see Appendix B),

(( N)) = exp(6a - 6) - exp(4a - 4)

( N ) = 1/(2 - a); 0 < 3/2 (15)

((N)) = (a - 1)/[(3 - 2a)(2 - 0 < 3/2 (16)

Temporal Variation of the Mean Physical Properties of the Flocs

The present approach provides a way of estimating the tem- poral variation of the mean physical properties of the flocs. For illustration, the variations of the density and porosity are discussed. Let us denote the average number of primary par- ticles contained in a floc by (Nl), the average density and porosity of a floc by ( p ) and ( E ) , respectively, and the volume of a primary particle by 5. Glasgow and Hsu" suggest that a floc is comprised of a number of primary par- ticles, each of which carries a fixed amount of fluid. If we denote the density of fluid and the amount of fluid carried by a primary particle as p, and 4Vp, respectively, and the density of primary particle as pp, then it can be shown that (see Appendix C)

Results and Discussion In the conventional treatment, the collision efficiency is taken as unity, i.e., E = 1. In this case, it can be shown that"

(19) 1/N: = 1 + 6'

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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 3247

t * / s

Fig. I Temporal variation of the average fractional surface cover- age and the corresponding collision efficiency; the values of the parameters assumed are a = B = K = 0 and r = 0.8

where

6' = X, K , t* /k , S< (20)

Comparison of eqn. (8a) with eqn. (19) shows that neglecting the effect of collision efficiency on the evolution of NT may lead to a significant deviation.

t * / s

Fig. 2 Temporal variation of the number of primary particles: (-) predicted by the present model, (---) E = 1; 0, perikinetic flocculation ; A, orthokinetic flocculation. The values of the param- eters assumed are q = 0.01 g cm-' s-', G = 1.5 s-', D, = 5 pm, k, = 8 x lo-' s-', r = 0.8, S = lo5, X, = 10l2, T = 298.15 K and 5 = 0.01

Table 1 A comparison of the results predicted by the present model, NTp., and those based on the conventional analysis, NTc, at arbi- trarily selected times

10

1000

0.00 1

0.0 10.0 20.0 30.0 40.0 t */s

10

10

0 .

0 0 0

0.1 0 .oo 0.10 0.20 0.30

t * ls

1.5

(N)1.0

-5.0 0.05 0.00 0'O3 t*lS

Fig. 3 Temporal variations of the mean (-) and the variance (---) of floc size distribution. (a) K i j = K O ; (b) K i j = K,(i + j ) ; (c) K i j = K O i j . The values of the parameters assumed are the same as those used in Fig. 2 with K O = 1.098 x s-' . 0, predicted by the present model; A, E = 1

The variations of the mean and the variance of floc size 0.00 1 1.25 1.01 1.16 distribution for the case where the collision eficiency is unity 0.002 2.50 1.03 1.31 have been derived by Ziff." Three types of collision coefi- 0.005 6.25 1.07 1.78 cient have been examined. The results he obtained are sum- 0.010 12.50 1.14 2.55 0.050 62.50 1.66 7.68 0.100 125.00 2.26 10.65 0.250 3 12.50 3.50 9.96 0.500 625.00 4.24 7.37

t* t l S (N7plN:c)peri ":,IN :c)ortho

marized below. If Kij = K O , then

( N ) = 1 + 672 (21)

(22) ( ( N ) ) = (672x1 + 672)

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3248 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

In the case where K i j = Ko(i +j ) ,

( N ) = exp(6’) (23)

(24)

( N ) = 1/(1 - 672); 6’ < 1 (25)

( ( N ) ) = exp(36’) - exp(26’)

If K i j = K O ij, then

( ( N ) ) = 6’/[2(1 - S’Xl - S’/2)’]; 6’ < 1 (26) Again, a comparison of eqn. (11H16) with eqn. (21H26) reveals that neglecting the effect of collision efficiency on the evolutions of ( N ) and ( ( N ) ) may lead to a significant devi- ation.

The temporal variations of the average fractional coverage of colloidal surfaces by polymer molecules and the corre- sponding collision efficiency are shown in Fig. 1. For illustra- tion, the adsorption of polymer molecules on the surfaces of colloidal particles is assumed to be irreversible ( K = 0), and only those collisions in which one of the two sites at the colli- sion point is bare and the other is occupied will yield attach- ment (a = j? = 0). The result shown in Fig. 1 reveals that, while the average fractional surface coverage increases mono- tonically with time, the collision efficiency increases with time, reaches a maximum and then decreases to a steady- state value. This steady-state value is used in the classical col- lision eficiency analysis. Depending on the timescale, the value of E used in the classical collision efficiency analysis can be either underestimated or overestimated. Similar con- clusions can be obtained for reversible adsorption of polymer molecule^.^

Fig. 2 illustrates the simulated variation of the number of primary particles as a function of time. In general, the rate of decrease of the number of primary particles for orthokinetic flocculation is much faster than that for perikinetic floccu- lation. This is because the former is driven by the motion of the fluid element, and the latter mainly occurs via Brownian motion of primary particles. Therefore, the collision fre- quencies are orders of magnitude different. As can be seen from Fig. 2, neglecting the collision efficiency may lead to significant differences for both perikinetic and orthokinetic flocculation. Table 1 illustrates the relative magnitudes of the results calculated by the present model and those obtained by the conventional analysis at arbitrarily selected times. Note that the deviation in N : can be as large as 900% for ortho- kinetic flocculation and 300% for perikinetic flocculation. The rate of decrease of the number of primary particles diminishes if the collision efficiency is taken into account. This is expected since 0 d E d 1.

The evolutions of the mean and the variance of the floc size distribution for different K i j are shown in Fig. 3. This figure reveals that both the mean floc size and the corresponding variance increase monotonically with time. Again, neglecting the collision efficiency may lead to a significant deviation. Both the mean and the variance of the floc size distribution obtained by the conventional analysis are overestimated. In particular, the difference in the variance is of the order of lo4 times. Since the variance of a distribution is a measure of its width, the floc size distribution predicted by the conventional analysis will be much broader than that calculated by the present method. The rate of flocculation follows the sequence (Q) < (b) < (c), see Fig. 3. This is because the collision coeffi- cient is independent of floc size in case (a), proportional to the sum of the sizes of the two colliding flocs in case (b), and proportional to the product of the sizes of the two colliding flocs in case (c).

Fig. 4 presents the evolutions of the average density and average porosity of the flocs. As can be seen from this figure, the average density decreases with time. This means that the

0 . 8 3 I 0.6

0.L ( E )

O e 2 U 0.0

2.0

(P>

1 .o

0.0 0.0 2 : O L : O 0.0 0.5 1.0 1.50.0 0.2 0 .L

t * / S

Fig. 4 Temporal variations of the average density (-) and the average porosity (---) of the flocs for the case of Fig. 3 with p, = 2.54 (g ~ r n - ~ ) , p, = 1 (g crn-j) and 4 = 2.5. (a) K , = K O ; (b) K , = K,(i + j ) ; (c) K i j = K O ij. 0, present model; A, E = 1

density of the floc varies inversely with its size, that is the larger the floc, the lower the density. In contrast, the mean porosity increases with time, in other words, the larger the floc, the greater the porosity. These results are consistent with those observed experimentally. The differences between the results obtained by the conventional analysis and those pre- dicted by the present method are appreciable. In general, the mean density is underestimated and the mean porosity over- estimated by the conventional analysis.

Conclusions In summary, the transient behaviour of polymer-induced flocculation is examined by incorporating the kinetics of the unsteady-state adsorption of polymer molecules onto the sur- faces of colloidal particles into Smoluchowski’s flocculation mechanism. Taking the transient nature of the collision eff- ciency into account is an extension of the conventional analysis, in which the adsorption of polymer molecules to the surfaces of colloidal particles is assumed to achieve its steady state instantaneously. We show that the collision efficiency is a parameter of fundamental significance. Neglecting this parameter may lead to serious deviations in predicting the physical properties of the system under consideration.

This work was supported by the National Science Council of the Republic of China under project number NSC80-0405-E- 002-27.

Appendix A Let the number of polymer molecules with i sites attached to the surface of a colloidal particle at time t be denoted by ni(t). A number balance for the number of attached sites on the polymer side gives

(Al) dno/dt = -&no + 01nl

dn,ldt = Ai - l l t i - 1 - (Ai + 01i)ni + + 1 ni + ;

i = 1, 2, . . . , (m - 1) (A2)

dn,,,/dt = A,,- lnm- - CL), n, (A3) where Ai is the rate of adsorption for the sites on a polymer molecule and oi represents the rate of desorption for the attached sites per polymer molecule. Since the adsorption of polymer molecules on the colloidal surfaces is of the Lang- muir type, we have

I z i = k,(m - i )X(t) (A4)

oi = k,i (A51

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Substituting these expressions into eqn. (AlHA3), and writing in a more compact form, we obtain

dnJdt = k,(m - i + l)X(t)ni-,

- [k,(m - i)X(t) + k, i ]ni

+ kdi + l)ni+ i = 0, 1, 2, . . ., m (A6)

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

m

Then

W(t) = S - C ini(t)/X, i = O

1

Differentiating eqn. (A8) with respect to t yields r m -I/

or

dX*/dt* = K - ( r + K - 1)X* -- X*2 (A10)

The solution to this equation subject to the initial condition X* = 1 at t* = 0 is

If we define the mean fractional surface coverage, ( O ( t ) ) , as

(A 12) (W)> = CS - X(t)/tI/S

then

( O ( t ) ) = 1 - X*(t) (A 13)

or

(e(t*)) = 1 - x*(t*) (A141

This is eqn. (2) in the text.

Appendix B The characteristics of the floc size distribution can be retrieved by resorting to its moments. Let us define the Ith moment about zero of the floc size distribution, p l , as13

R / R R / R

If we define MI as R

MI = 1 i'N* i = 1

then

dM, X o E dt* k , S < -=-

( 4 R R I x {L 1 2 [(i + j)' - i' - j ' ] K i j N: NY} (B3)

2 i = l j = 1

The mean floc size, ( N ) , and the corresponding variance, (( N)), can be evaluated, respectively, by

( N ) = PI = Ml/MO 034)

(B5) ( ( W ) = p-2 - p: = (MOM2 - M : ) / M ;

Case 1. K i j = K O (a constant) In this case, eqn. (B3) gives

dMo/dt* = - (X, K O E/2ka SOM; (B6)

dMl/dt* = 0 (B7)

dM,/dt* = (X, K O Elk , S<)M: (B8)

dM3/dt* = (3x0 K O E/k,S()MlM2 (B9)

and so on. The solutions to the first three of these equations subject to the initial conditions MJO) = 1, i = 0, 1, 2 are

M , = l/a (B10)

Ml = 1 (B11)

M 2 = 2 ~ - 1 (€3121

where a is defined by eqn. (12b) in the text. The value of M i , i = 3, ..., R , can be found by resorting to the method sug- gested by Ziff.12314 In particular, we have

(I3131 M, = M ; + 2(M2M0 - 1)/M;

p l = Ml/Mo = 0

p-2 = a(2o - 1)

p3 = (20 - 1)2a + 2a2(a - 1)

Thus

0314)

(B15)

and

( N ) = pl = 0317)

((N)) = p2 - P: = d a - 1) These are eqn. (1 1) and (12), respectively, in the text.

Case 2. K i j = Ko(i + j ) In this case, eqn. (B3) yields

dMo/dt* = - (Xo KO Elk, S5)Mo M 1 0319)

dMl/dt* = 0 (B20)

dM2/dt* = (2X0 K O E l k , S5)M , M 2 (B21)

dM3/dt* = (3X0 K O Elk, So(MlM3 + M i ) (B22)

and so on. The solutions to the first three of these equations subject to the initial condition Mi(0) = 1, i = 0, 1, 2, . . . , R are

M , = exp(2 - 20) 0323)

M , = 1 0324)

M 2 = exp(4o - 4) (B25)

p1 = Ml/M, = exp(2o - 2) (B26)

p 2 = M 2 / M , = exp(6o - 6) 0327)

Thus

and, therefore

( N ) = p l = exp(2a - 2)

( ( N ) ) = p 2 - p: = exp(6o - 6) - exp(4a - 4) (B29)

Thus, eqn. (13) and (14) in the text are justified.

Case 3. K i j = Koij In this case, eqn. (B3) gives

dM,/dt* = -(Xo K O E/2ka S<)M: (B30)

dMl/dt* = 0 (B3 1)

dMJdt* = ( X , K O E l k , S < ) M i (B32)

dM,/dt* = (3x0 K O Elk, S5)M2 M 3 (B33)

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3250

and so on. The solutions to the first three of these equations subject to the initial condition MLO) = 1, i = 0, 1, 2, . . . , R are

Mo = 2 - 0; B < 3/2 0334)

MI = 1 (B35)

M2 = 1/(3 - 2 ~ ) ; B < 3/2 0336)

= M , / M , = 1/(2 - 6); o < 3/2 (B37)

Thus

~2 = MJMO = 1/[(3 - 2 ~ x 2 - a)]; B < 3/2 (B38)

and therefore

( N ) = ~1 = 1/(2 - 0); B < 3/2 (B39)

( ( N ) ) = p2 - pf = (a - 1)/[(3 - 20x2 - o)'] ;

B < 3/2 (B40)

These are eqn. (15) and (16), respectively, in the text.

Appendix C Let the average mass and volume of a floc be denoted by ( M , ) and ( V , ) , respectively. As discussed in the text, Glasgow and Hsu" suggest that a floc comprises of a number of primary particles, each of which carries a fixed amount of fluid. Thus

(C1)

(C2)

(v,) = W d V p + ((NJ - WVp

( M J = (Nl)Pp yJ + ( ( N J - 1)4VpPw

where the symbols have the same meanings as in the text. Therefore

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

Clearly, (p) and (E) are related by

(P> = (1 - W P p + (+P,

1 - ( E ) = ((P) - P,)/cpp - P,)

(C4)

or

(C5)

Substituting eqn. (C3) into this equation and solving for ( E ) , we obtain

This is eqn. (18) in the text.

References 1 2

3 4

5

6 7

8

9 10 1 1

12

13 14

R. A. Ruehrwein and D. W. Ward, Soil Sci., 1952,73,485. V. K. La Mer and T. W. Healy, Rev. Pure Appl. Chem., 1963, 13, 112. R. Hogg, J. Colloid Interface Sci., 1984, 102, 232. V. K. La Mer and R. H. Smellie Jr., J. Colloid Interface Sci., 1956, 11, 704. B. M. Moudgil, B. D. Shah and H. S. Soto, J. Colloid Interface Sci., 1987, 119, 446. A. Molski, Colloid Polym. Sci., 1989, 267, 371. J. P. Hsu and D. P. Lin, J. Chem. Soc., Faraday Trans., 1991, 87, 1177. K. J. Ives, in Solid-Liquid Separation, ed. J. Gregory Ellis Horwood, Chichester, 1984, p. 196. M. V. Smoluchowski, Z. Phys. Chem., 1917,92, 129. L. A. Glasgow and J. P. Hsu, Part. Sci. Technol., 1984,2, 285. J. Th. G. Overbeek, in Colloid Science, ed. H. R. Kruyt, Elsevier, Amsterdam, 1952, vol. 1 , p. 278. R. M. Ziff, in Kinetics of Aggregation and Gelation, ed. F. Family and D. P. Landau, North-Holland, New York, 1984, p. 191. M. Frenklach, J. Colloid Interface Sci., 1985, 108, 237. M. S. Bowen, M. L. Broide and R. J. Cohen, J. Colloid Znterface Sci., 1985, 105, 617.

Thus, eqn. (17) in the text is justified. Paper 1/00531F; Received 5th February, 1991

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