ADSORPTION AND
PLOCCULATlON
3539
BEHAVIOR OF SILICA SUSPEN8IONS
provide immediate repair to sections of the monolayer damaged by wave action. A final observation, which supports elther the collapse or the submergence mechanism, 1s of interest. At all pressures below the region of the plateau, when the disturbance is removed, the static pressure is recovered. This rate of pressure recovery is an exponential curve, and successive application and removal of the disturbance show negligible hysteresis in the plot of recovery at, pressure us. time. In the plateau
reglon, final approach to the static pressure after the lnittal rapid recovery of pressure is slow, so that successive application and removal of the disturbance 1s difficult to analyze. Acknowledgment. We thank Dr. W. A. Zisman and Dr. W. D. Garrett of U. S. Naval Research Laboratories and Drs. Sylvester and Wiegel of the Civil Engineering Department of the University of California, Berkeley, Calif., for helpful discussions, and JIr. l f . Jefferis for assistance in some of the experiniental work.
The Effect of Solid Content on the Adsorption and Flocculation
Behavior of Silica Suspensions
by Jacqueline C. Kane, Victor K. La Mer, and Henry B. Linford Departments of Chemical and Mineral Engineering, Columbia Unzversity, New York, New Yorlc 10027 (Receitied February 19, 1964)
This research analyzes the authors’ previously published results on flocculation in the silicapolymer system by considering both the theory of Smellie and La Mer and the modification of Healy and La Mer, which treats the adsorption phenomenon more fundamentally. The variation of P , a t decreasing solid content is explained in ternis of secondary adsorption resulting from the increased time intervals between unit floc-unit floc collisions. A method for calculating b , the ratio of rate constants for the adsorption and desorption processes, and e, the fraction of the solid surface covered, as a function of added polymer concentration, 1s outlined. An analysis of the deviations from the eighth power filtration law in terms of b is also presented. The basic assuniption that the adsorption mechanism obeys a Langmuir-type equation has been justified. Preliminary calculations are included to indicate the geometrical dimensions that exist when flocculation occurs in the silica-polymer system.
Introduction I n 1958, Smellie and La Mer1 published a quantitative theory of filtration of suspensions flocculated by high polymers. This theory was later modified by Healy and La Rler12who interpreted the mechanism of the adsorption of high polymers by the substrate more fundamentally. They introduced the concept that a fraction of the total number of polymer segments interacted with a certain fraction of the active sites
on the surface of the solid substrate and thus improved the picture of simple Langinuirian adsorpt,ion. Since this approach does not affect the original theory in respect to either the kinetics of flocculation or the filtration mechanism, we coinpare the equations in the two theories for the purpose of interpreting the ~~
(1) R. H . Smellie, Jr., and V. K. La Mer, J . Colloid Sei., 13, 589 (1958). (2) T. W. Healy and V. K. La Mer, J . Phys. Chem., 66, 1835 (1962).
Volume 68, Number 18 December. 136.4
3540
J. C. KANE,V. K. LA MER,AND H. B. LINFORD
previously published data of Kane, La Mer, and Linford. 3
Theoretical
or in a logarithmic form
I n their treatment of the adsorption process, Smellie and La Mer’ proposed that the equilibrium polymer concentration could be represented by
P
=
Po - ksLWB
(1)
where P was the equilibrium polymer concentration, Po the initial polymer concentration, W the solid content, e the fraction of surface covered, and k S L the constant used by Smellie and La Mer dependent upon the fineness of grinding or specific surface area of the solid. I n our previously published research on silica,a the pertinent quantities were expressed as
P and Po = W = kSL
=
g. of polymer g. of water
x
g. of solid g. of water
10-2
x
104
g. of polymer adsorbed g. of solid
x
10-4
(e is dimensionless extending from 0 to
1)
Smellie and La Mcr assumed that the adsorption process could be represented by an isotherm of the Langmuir type
e=---- bP 1
+ bP
(2)
where b, as a first approximation and by analogy with the original derivation, is proportional to the ratio of the forward to reverse reaction rate constants for adsorption. Combining eq. 1 and 2 yields
By employing the geometry involved in the limiting form of the Kozeny-Carman filtration equation, together with basic area and volume relationships, Smellie and La Mer derived the expression Q - Q~ =
rzoz K2W404(1Qo
e14
(4)
Here Q and Qo are the refiltration rates with and without polymer, respectively, Ro is the radius of a primary particle, K is a constant, and W and e are defined above. Rearranging and combining eq. 3 and 4 yields
- Qo) = 8 In B In ( P o 4 ) / ( &
+ 8 In
(Pm
+ Po) (5A)
Differentiating and setting the derivative equal to zero to correspond to an optimum gives
Po (optlmum)
=
Pm
=
(1
+ bksLW)’ b
1 b --
+ 2ks~W+ bksL2W2 (6)
For details of this derivation, the reader should consult ref. 1. Healy and La Mer2 proposed that when a polymer molecule concentrated a t a solid-liquid interface, only @-segments per average polymer molecule adsorbed and covered surface sites, with the remainder of the molecule protruding into the surrounding medium as .extended segments. It should be emphasized that these j3-segments need not occupy B consecutive sites on the surface. I n fact, it is most probable that loops of extended segments separate the adsorbed segments from each other, such that they can be treated as separate “small molecules,” independent of other adsorbed segments along the polymer molecule.4 By introducing the parameter @ in the original derivation of Smellie and La Mer, Healy and La Mer obtained
p(p0 - P ) N / S S , = e
(1A)
P = Po - k=O/@
(1B)
where
I n these equations, s is the number of sites per unit area of surface, Sois the total surface area of solid, N is Avogadro’s number, P and P o are expressed in moles of polymer, and e is again dimensionless. The subscript H refers to the constant of the Healy modification. Reconciliation of the Theories. Since the two approaches are not mutually exclusive, an expression for ksL in terms of k H can be obtained by equating eq. 1 and 1 B and dividing by 0. (3) J . C. Kane. V. K. La Mer, and H. B. Linford, J . Phys. Chem.. 67, 1977 (1963). (4) T. W. Healy, Ph.D. Dissertation, Columbia University, 1963.
ADSORPTIONA N D FLOCCULATION BEHAVIOR OF SILICASUSPENSIONS
ksLW - __ - (Po - P ) - IC_.
e
M
P
354 1
(7)
Since ksL was expressed in weight units while kH was in molar units, the molecular weight of polymer, M , has been included. Substituting for k H from eq. 1C
If the suspended particles are assumed to be spherical, and the density of the solid is p s , then the surface area of the solid is So
=
3W/Rops
.003
.O6
.002
.04
.oo I
-02
(9)
and eq. 8 becomes
For dimensional consistency, it must be assumed that a surface sit,e is equal in area to a polymer segment.6 Calculations of Values of and b. By definition, the optimum polymer concentration, P,, is that concentration of polymer which maximizes the refiltration rate. From eq. 4, however, it is obvious that the refiltration rate can be a maximum only when the expression - e ) 4 has its largest value. This occurs a t e = 0.5. Figure 1 illustrates the functional dependence of e( l - e), the modifier of the Smoluchowski coagulation equation necessary to convert it into the Smellie and La Mer flocculation equation, upon the fraction of the surface covered. P(1 a quantity proportional to the floc size, and e4(l - e)l, a quantity proportional to the rate of filtration, are also included. The importance of these quantities in interpreting flocculation and filtration has been emphasized recently by La Substituting 8 = 0.5 into eq. 4 produces (& -
&o)mex
=
C/256
(11)
where
8 Figure 1. Variation of e(1 - e), e y l - e)$, and e4(l - e)4 with e, the fraction of solid surface covered.
persions flocculated by various high polymers, the relationship is linear. These findings indicate that, for silica, the third term of eq. 6 is negligible in comparison to the terms l / b and 2 k s ~ W . Hence, for tjhis system, the optimum polymer concentration can be expressed simply as
P,
+ 2ks~W
(13)
where the slope of a plot of P , vs. W is 2 k s ~ . For more quantitative information about these plots, the reader should consult ref. 3. Once both 8 and ksr, are known, and since W and P are arbitrarily predetermined quantities, eq. 3 can be solved directly forb, i.e. b =
Obviously, C becomes a known quantity once the maximum filtration improvement is determined. Subsequently, C can be used to evaluate the fraction of surface covered, e, a t refiltration rate improvement values, (& - Qo),other than the maximum. Equation 6 predicts a parabolic dependence of the optimum polymer concentration on the solid content; this has been verified experimentally for many flocculated clay-polymer systems. However, our recent experimental results3 have shown that, for silica dis-
= l/b
e
(14)
( p 0 - lcsLwe)(i- e)
which combined with eq. 1 recovers the familiar form of the Langmuir equation bP
e
(15)
= -
1-e
( 6 ) R. Sirnha, H. Frisch, and F. Eirich, J. Phys. Chem., 57, 684
(1953). (6) V. K. La Mer and T. W . Healy, ibid., 67,2417 (1963).
Volume 68. Number 1.9
December, 1064
3542
J. C. KANE,V. K. LA h l ~ ~AiN,D 1-1.13. LINFORD
Experimental A detailed explanation of the experimental procedure, equipnient, and materials employed in obtaining the data used in this paper has been published previously.3 As in the earlier work the p1-f was held constant between values of 5.6 and 6.0.
20
2.00
2.0 3.0
1.0
I.6 CI
0
Results and Discussion The values of b calculated by eq. 14 appeared to be constant, within experimental limits, over a centrally located range of Po values. In all cases, this range corresponded to the linear portion of the s-shaped curves which typified the Sniellie and La Mer plots of eq. 5 and 5A. Decreasing b values accompanied deviations which were concave downward, while increasing values of b produced positive deviations from linearity. See Fig. 4 of ref. 3. I t was also noted that for flocculation of systems of different solid contents by a given polymer, the product bW, based on an average of the constant b values, was reasonably constant. Furthermore, the magnitude of b increased with increasing molecular weight. Healy4 demonstrated that for the calcium phosphate-polyacrylamide system, b has an optimum with respect to molecular weight,. If these findings apply to other solid-polyacrylamide systems, then the optimum molecular weight of polyacrylamide for the silica systeni must be greater than 10 million. On the average, the values of fractional surface coverage, 8, obtained by the calculations outlined above, ranged from 0.30 to 0.80. For example, see Fig. 2. Figure 3 demonstrates graphically that the assumption of a Langmuirian-type adsorption is valid throughout a central range of surface coverage. The circled points are the result of plotting e against the equilibrium polymer concentration calculated from the Sniellie- La iller theory, P = Po - ksr,WB. T h i s method of proof i s valid since Po, k S I , , and Vr were all determined experimentally, and reference to the theoretical discussion will reassure the reader that the calculations of 0 were accoiuplished without recourse to the assumption of Langiiiuirian adsorption. The solid curve of Fig. 3 is a plot of the equilibrium polynier concentration obtained directly from a Langmuir-type equation, P = e/b(l - e), as a function of the fractional surfacc coverage, e. The value of b used corresponded to an average of the b values of those points which comprised the linear portion of the eighth power Sniellie-La bIer plots. According to eq. 13, if P,, is linear with respect to W , t hen a plot of ksr, us. W should be a horizontal straight The Journal of Phynical Chemintry
al
v)
2
1.2
EV Y
&‘8 CY
.4
0 0
.2
.4
.8
.6
1.0
9 Figure 2. Typical plot of refiltration rate improvement as a function of the fraction of solid surface covered, e, and the added polymer, PO. Results are for ET494 with 0.5 g. of silica/100 g. of water.
r
.8
t
.69
.2 .4
V I
0
.2
I
I
.4
.6
I
I
I
I
I
I
.8 1.0 1.2 1.4 1.6 g polymer a t equilibrium
to6 g
1.8
water
Figure 3. Langmuirian-type plot. Data of ET494 with 0.5 g. of silica/100 g. of water.
line, independent of dilution. This has been verified experinientally for Jaguar and ET-494, for Superfloc 16 above 2.5 g. of siliea/100 g. of water, and for PAM 3 above 4 g. of silica/100 g. of water. See Fig. 4. As expected from the results of Healy4 for the polyacrylamides, p,,, decreased with increasing niolecular weight at a given solid content. Equation 10 can thus be written as
+
~ S = L
J/P
(16)
where
J
c
3sMIRop.N
(17)
A D S O R P T IAND ~~
n
1
1
0
3543
FL~CCUIA~TIO BEIIAVIOR ?~ O F SILICA SUSPEPU’SIONS
1
I
2
4
6
8
8
1
1 0 1 g silica Solid Content 100 g water
1
2
0
3. p Increases. This is the behavior of high Iriolecular weight polymers and in this case proceeds vzu n different mechanism. The long extended segnients of the polynicr inolecule are favorably situated for a “wrap-around” type of adsorption which increases the number of adsorbed segments per average molecule. This interpretation implies that at high dilution ( < 3 g. of solidA00 g. of water) the tinic interval between successive particle-particle collisions is greater than that necessary for secondary (continued) adsorption. In dilute susperisioiis then, adsorption will continue and flocculation will be slow with large flocs resulting. In concentrated suspensions, on the other hand, the rate of the flocculation reaction will be fast, continued adsorption will be slow, and the average floc size will be smaller. These two cases are illustrated schematically below where P and S are the polymer and solid, respectively.
P
+ s =+=
*PS2 0
PSI
Figure 4. Variation of ksr. with solid content. Suspensions of dilrerent solid contents were prepared by mixing 5 g. of silica with various volumes of water.
As before, p is the number of segments of a polynit~ niolecule adsorbed at the solid surface (the remainder, the greater part of each adsorbed molecule, extends into the surroundings) and represent s an average value which can change by either molecular or segment adsorption. Otwiously, J should be constant for any determination inadc using the saint’ solid, suspcmion iiiediuni, polymer, and total weight of solids. Hence, for any system which obeys eq. 13, variations in ksr, must be attributed to a variation in p with decreasing solid content. The authors postulate the following types of behavior as the dilution is increasrd, keeping in mind that at high solid contents, and especially for high niolecular weight polyniers, the crowded environment may slow the adsorption of polymer molecules and consequent production of extended segnients. 1 . p I s Constant. This is the behavior of low molecular wt.ight polymers which are completely adsorbed even a t high dilution or which lack the chain flexibility necessary to change p by the adsorption of extended segments. 2. p Decreases. This is the behavior exhibited by polynicrs of intermediate molecular weight. Since, in this caw, the ex1 mded segments of the polynicr nioleculr are fairly short, adsorption will be continued by new molecules attaching a t only a few segnients to give a p decrease based on the iiuniber of adsorbed segniciits per average molecule.
Here I’Sl and PS2 are, in the terminology of Healy,’ unit flocs (single colloidal particles with molecules of polymer adsorbed), I’S2 rcsulting from the secondary adsorption of polynier on PS1 via one of the mechanisms listed above. (l’S)nR, and (l’S)nRz are each niacroflocs resulting from the bridging action of the polymer niolecules between adjacent unit flocs of PSI or adjacent I’S2 flocs. Reaction 1 occurs if the solid content is high producing a floc of radius R1, while reactions 2 and 3 proceed in dilute suspensions producing a floc of radius R2, wherc R1 is ltss than Rz. Qualitatively, we theorize that there will be a definite solid content at which the time necessary for reaction 2 to proceed will be less than that for reaction 1. That the floc radius is indeed a function of the solid content can best be demonstrated by reference to eq. 4. Experimentally, it has been found that (Q - Q0)/‘Qo varies by a factor of less than 2 when W4 changes by 1.6 X lo5 for a constant value of 8. Ilence, K 2 niust vary inversely with TV4 when Ro2and 8 are constant. Sirice K = ZklIk2 where Z is the proportionality constant between the number of particles and thr solid content, kl is the rate constant of the floc forniation, and k 2 is the deflocculation rate constant, and since (Q - Qo)/Qo varies only slightly with W4, then K , and hence the ratio k l l k 2 , must decrease with incrmsing u’. Furt hermore, kl,/kz is proportional to (7) T. W. IIealy arid V K . La Mer, J . Colloid Sci., 19, 323 (1964)
V o l ~ m 68, e Turnher 12 December, 19Gg
3544
J. C. KANE,V. K. LA MER,AND H. B. LINFORD
Table I : Physical Characteristics of the Various Silica Suspensions Solid content, of silica/ 100 g . of water g.
particles/
Particulate length, 1,
hfean free path, L,
P
Ir
crn.8'
10 9.1 48 5 4.6 60 2.5 2.3 76 1 0.91 103 0.5 0.46 130 Extended length of polymer*
88 175 350 878 1750
Superfloc 16 mol. w t . 4,000,000
2.66 X lo6 2.66 X 105 2.66 X lo6 3.15 X lob 3 . 4 9 x 106 11.2 /l
-Polymer molecules available/silica particle at P m---PAM 3 ET-494 mol. wt. = mol. wt. = >10,000,000 50,000
8.60 x 104 9.27 x 104 6.60 x 104 4.97 x 104 3.97 x 104 28.2 p
2.69 x 107 2.60 X 10' 2.60 x 107 2 . 6 0 X loT 2.60 x 107 0.35 ,u
-
JAG+ mol. w t . = 200,000
6.31 X lo6 6.98 X lo6 6.65 X IO6 6.65 X lob 6 . 6 5 X IOB 0.25 p
* Extended length calculations were based on the nominal molecular weights a The specific gravity of silica was taken as 2.65 g./cm.a. listed in the headings of the table. Recent viscosity measurements b y Healy on PAM 3, however, indicate that the actual molecular weight and hence the extended length of this polymer may be as much as four times as great as the nominal. the radius of the floc, Le., R = (kl/k2)no2e2(1-e ) 2 ; thus the decrease in the extent of flocculation, kl/k2, with increasing solid content is reflected in a smaller floc radius. We are aware that we have not presented a quantitative explanation of the variation of @ below solid contents of 3 g. of silica/100 g. of water. However, elementary calculations based on an average particle diameter of 20 1.1 do shed some light on the actual geometrical dimensions of the suspension. The followiiig assumptions have been employed. (1) Due to coiling, the C-C, C-0, and C-N bond lengths can be taken as 1 A. (2) A particle occupies a cubic particu-
The Journal of Physical Chemistry
late volume of particulate length 1. (3) Mean free path length equations from kinetic theory can be applied. Table I summarizes the calculations.*
Acknowledgment. The authors wish to acknowledge the assistance of Dr. T. W. Healy, whose suggestions and ideas were most helpful in the preparation of this paper. The research has been supported by U. S. Public Health Service Grants RG-8389 and WP00240-02. (8) NOTEADDEDI N PROOF.-It is worthy of note that the values of in Fig. 4 are, a t least, roughly proportional to the calculated values of the extended length of polymers in harmony with the basic concept of the bridging mechanism of flocculation by high polymers. k8L
