Colloidal Stability: Comparison of Sedimentation with Sedimentation

29 Colloidal Stability: Comparison of Sedimentation with Sedimentation Flocculation 1

D. H. MELIK and H. S. FOGLER

Downloaded by RUTGERS UNIV on May 29, 2018 | https://pubs.acs.org Publication Date: March 27, 1985 | doi: 10.1021/bk-1985-0272.ch029

Department of Chemical Engineering, The University of Michigan, Ann Arbor, MI 48109

We have developed simple analytical equations one can use to estimate the overall stability of a polydisperse colloidal system undergoing simultaneous creaming and gravity-induced flocculation (note: our approach can be easily extended to include other particle loss mechanisms, such as shear-induced flocculation and Brownian flocculation). As an example of this analysis, we have studied the relative particle loss rates of creaming and gravity-induced flocculation under conditions of negligible electrostatic repulsion. From this study we have determined that the total particle concentration, the particle size, and the particle size ratio are the most sensitive operating parameters in controlling the stability of a colloidal system undergoing simultaneous gravity-induced flocculation and creaming. In quiescent media, polydisperse colloidal dispersions are broken by the coupled mechanisms of creaming, Brownian flocculation, and sedimentation (gravity-induced) flocculation (see Figure 1). In creaming, particles either rise or sediment out of the system as a result of the density difference between the particles and the suspending medium. Encounters between particles occur because of random Brownian motion, frequently resulting in particle aggregation if the interparticle interactions are favorable. Gravity-induced flocculation arises from the differential creaming rates of large and small particles. The large particles sweep out the slower moving small particles in their path, often resulting in particle aggregation i f the interparticle interactions are favorable. In the case of emulsions, coalescence of the particles can occur after flocculation. In order to rigorously analyze the behavior of these particle loss mechanisms on colloidal stability, one must solve the 1

Current address: The Procter & Gamble Company, Miami Valley Laboratories, Cincinnati, OH 45247 0097-6156/85/0272-0461$06.00/0 © 1985 American Chemical Society

Shah; Macro- and Microemulsions ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

MACRO- AND MICROEMULSIONS

462


I.

Creaming:

2. Brownian Flocculation:

state

state

Figure 1. C o l l o i d a l breakage mechanisms in quiescent media



29.

MELIK AND FOGLER

Colloidal Stability

463

convective-diffusion equation which describes the d i s t r i b u t i o n of p a r t i c l e s around a central or reference sphere (1). Depending on the p a r t i c l e sizes and the net g r a v i t a t i o n a l force, f i v e important regimes can be i d e n t i f i e d : ( i ) Both Brownian motion and i n t e r p a r t i c l e forces ( i n p a r t i c u l a r , London van der Waals a t t r a c t i v e and e l e c t r i c double layer repulsive) are n e g l i g i b l e , but the effects of d i f f e r e n t i a l creaming are considerable. Under these conditions f l o c c u l a t i o n never occurs due to the hydrodynamic resistance of the f l u i d as the p a r t i c l e s approach each other. However, o r b i t a l pairs do exist (2,3). ( i i ) Brownian motion is n e g l i g i b l e , but the effects of i n t e r p a r t i c l e forces and d i f f e r e n t i a l creaming are s i g n i f i c a n t (4). ( i i i ) Brownian motion is appreciable. Brownian motion, i n t e r p a r t i c l e forces, and d i f f e r e n t i a l creaming must be taken into account. This case requires the complete solution of the convective-diffusion equation; a task which has yet to be accomplished. (iv) Brownian motion and i n t e r p a r t i c l e forces are s i g n i f i c a n t , whereas the effects of d i f f e r e n t i a l creaming are moderate (2). (v) The effects of creaming are negligible, but Brownian motion and i n t e r p a r t i c l e forces are important (5-9). The various f l o c c u l a t i o n models which are v a l i d in the d i f f e r e n t regimes described above allow one to compute the p a r t i c l e / p a r t i c l e c o l l i s i o n rate for any given p a r t i c l e sizes, chemical and physical condition. From the magnitude of this c o l l i s i o n rate, one can estimate a c o l l o i d a l system's s t a b i l i t y in cases ( i v ) and ( v ) . However, in cases ( i i ) and ( i i i ) , both f l o c c u l a t i o n and creaming w i l l be important in the c o l l o i d a l breaking process. Consequently, in order to determine whether a c o l l o i d a l system w i l l be stable in these two cases, we have to determine the net rate of p a r t i c l e loss due to both creaming and f l o c c u l a t i o n . In this paper we propose a simple procedure whereby one can estimate the o v e r a l l s t a b i l i t y of a given c o l l o i d a l system undergoing simultaneous creaming and f l o c c u l a t i o n . Since case ( i i i ) has not been solved yet, we w i l l focus our attention only on case ( i i ) . As an example, we use t h i s procedure to compare the net rates of p a r t i c l e loss due to creaming and gravity-induced f l o c c u l a t i o n when e l e c t r o s t a t i c repulsion is n e g l i g i b l e . Theory For a polydisperse system of p a r t i c l e s undergoing both creaming and gravity-induced f l o c c u l a t i o n , the time evolution of the p a r t i c l e size d i s t r i b u t i o n , and hence the time rate of change of the t o t a l p a r t i c l e concentration, can only be determined from the solution of the governing population balance equations (10,11). For the special case of two d i f f e r e n t sized p a r t i c l e s , t h i s system of balance equations reduces to:

8t


(la)

464

MACRO- AND MICROEMULSIONS

3 N

a

2

W 3 N

= ' 3^

"

)

G

N

of radius a

N

H

+

G

N

H

(

2 12 1 2

are instantaneous

b

)

l

c

)

creaming v e l o c i t i e s for p a r t i c l e s

and a , respectively, u

1

l

ι (u

2

(

" 12 1 2

- âq 12 12>

and u

1

N

2 2

a

i2

at" where u

( U

2

1 2

the instantaneous

creaming

v e l o c i t y for a doublet comprised of one p a r t i c l e of radius a.^ and one of radius a ,

and Ν „ the respective p a r t i c l e concentrations,

2


G

12

t

h

e

c a v i t y - i n d u c e d c o l l i s i o n kernel for p a r t i c l e s of size a

and a , t the time, and x 2

1

the cartesian coordinate axis p a r a l l e l to

1

the d i r e c t i o n of gravity. The creaming terms can be approximated by the following r e l a t i o n s h i p (12,13) : |

(

U

i

„

V

- i - i

(

2

)

(with analogous expressions for the other creaming terms), where h is the height of the suspending medium. Since we are most interested in determining the s t a b i l i t y of a c o l l o i d a l system, an analysis of Equation (1) is only required for i n i t i a l times. This is because i f the system is unstable at t=0, then it w i l l be unstable for t > O. Noting that as t -* 0, N -» N , 1

N

N

2 "* 02'

a

n

d

N

12 "* ° '

a

n

d

m

a

k

i

n

g

u

s

e

o

f

Ration

Q 1

(2), f o r short

times E q u a t i o n (1) reduces t o : d

N

u

i

dt

d N

U

2

dt

dN

12 dt 1

-

N

i oi h

G N N "12 01 02

(3a)

- G. NΝ "12 01 02

(3b)

in

rt1

AO

N

2 02 h

0

Λ

1 2 "12 ± Ο Ν 01Ν 02 ι η

Λ 1

Λ η

(3c)

Therefore, the t o t a l i n i t i a l rate of p a r t i c l e loss due to creaming is given by

Cr=

H

hK 01

+

U

N

2 02>

(

and the i n i t i a l net rate of p a r t i c l e loss due to gravity-induced f l o c c u l a t i o n is given by


4

)

R

465

Colloidal Stability

29. MELIK AND FOGLER

I 12 01 02 G

Gr "

N

N

( 5 )

Equations (4) and (5) provide a basis for not only comparing the net rate of p a r t i c l e loss due to both creaming and f l o c c u l a t i o n , but the sum (R +R ) allows one to determine the i n i t i a l s t a b i l i t y of a cr ur given c o l l o i d a l system. It should be noted that the analysis presented above could e a s i l y be extended to completely polydisperse systems by summing Equation (3) over a l l pairwise combinations of p a r t i c l e sizes a., a~, ... , a . The rest of this paper w i l l be devoted to studying the r e l a t i v e rates of creaming and gravity-induced f l o c c u l a t i o n , that is


.

2 ( U

^ R

'

N

1 0 1 » VW

3 h G

Gr "

N

...

N

12 01 02

for the case of n e g l i g i b l e e l e c t r o s t a t i c repulsion. The gravity-induced c o l l i s i o n kernel for spheres of radius a and a is given by (4):

1

2

G

12 "

where u a

Gr

9

( u

u

02- 01

) < a

l

+ a

2

) 2

< 7 )

is Stokes creaming rate for a p a r t i c l e of radius a^, and

Q i

t h e

o

* Gr

r a v i t

Y~

i n c

*

u c e d

f l o c c u l a t i o n capture e f f i c i e n c y which, for

the case of n e g l i g i b l e repulsion, can be s a t i s f a c t o r i l y computed from (4): «Gr =

«Wf -°'

2 0

(8)

G

with λ being the p a r t i c l e size r a t i o (λ = a / a ) 1

2

f

and f(X) is equal

to O.11 for λ=O.2 and O.17 for λ=O.5. The dimensionless gravity parameter l\l describes the r e l a t i v e importance of g r a v i t a t i o n a l to Q

i n t e r p a r t i c l e a t t r a c t i v e forces and is given by: πgΔpa* #

9 Ζ

=

G

(1+λΓ(1-λ )

(9)

where g is the l o c a l acceleration of gravity, Δρ the density difference between the p a r t i c l e s and the suspending medium, and A the system Hamaker constant. Stokes creaming rate is given by:

u . = 01

where μ

£

2 2gApa. — 9μ £

is the v i s c o s i t y of the suspending f l u i d .

The instantaneous creaming v e l o c i t i e s for moderately concentrated systems can be expressed in the following form (3):


(10)

MACRO- AND

466

U

U

(1

S

U

(1

S

MICROEMULSIONS

S

(lla)

!= 01 " l A - 2lV 2 02 " 12*l " 22*2>

U

=

S

where

is

(

U

the p a r t i c l e volume f r a c t i o n for a p a r t i c l e of radius a^

Colloidal Stability: Comparison of Sedimentation with Sedimentation

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