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Layered sedimentation in suspensions of monodisperse spherical colloidal particles
Layered Sedimentation in Suspensions of Monodisperse Spherical Colloidal Particles 1 DONALD B. SIANO Exxon Research and Engineering Company, Linden, N e w Jersey 07036
Received February 14, 1978; accepted April 12, 1978 Stratification of the supernatant of suspensions of polydisperse sedimenting colloidal-sized particles, hitherto ascribed to polydispersity, aggregation, convection, or sedimentation potential effects, is also shown to occur in dispersions of highly monodisperse, spherical polystyrene particles in the size range from 2.02 to 0.109 txm. The layers appeared in suspensions with concentrations as low as 0.01 wt% up to at least 0.4 wt%. The qualitative behavior of the layers with their associated interfaces was seen to be highly dependent upon concentration gradients initially imposed on the suspension but to be relatively insensitive to salt (as long as its concentration was below the fluctuation value). Examination of samples from the layers showed that the composition of each layer was the s a m e - - o n l y the concentration varied. Many features of the data suggest that the process is in fact due to the mechanism of spinodal decomposition during the separation of two phases (colloid-rich sediment and colloid-poor supernatant) as the system moves toward its final equilibrium state. INTRODUCTION
It has been known for almost a century that, contrary to expectations, initially homogeneous suspensions of hydrophobic colloidal particles do not sediment in a smooth, continuous fashion (1-15). Instead, bands or layers of differing turbidity are often observed in the column of supernatant liquid after the settling has proceeded for a time. To the eye, each layer appears to be of nearly constant concentration (perhaps with a slight "reverse gradient"), while successive layers increase in concentration when they are farther from the meniscus. The turbidity is such that the boundaries between the layers can be quite marked, and previously published photographs have shown the layers very clearly (6, 13, 15). It has been suggested that this layering of 1 Research Supported by the National Science Foundation under Grant NSF-CHE-76-02414 and NIH-RO1 NS 12714-02 and carried out at the Department of Chemistry, Columbia University.
the supernatant above the denser sediment leads to the production of layers subsequently seen in the compacted sediment, e.g., in clays or sedimentary rock, and is therefore of some geological significance. Another possibility that has been put forward is that the phenomenon might be used to characterize various soils and clays empirically, and this appears to have been the major motivating force behind previous investigations. Sedimentation is, of course, a subject of great importance in many industrial processes such as water clarification, sludge removal, and various separation procedures. It is widely recognized that the theoretical understanding of the process is at present in a very unsatisfactory state except for very dilute suspensions (while they are still homogeneous!) (16). An understanding of the formation of the bands and the mechanism for their preservation should be relevant for the further understanding of this subject. While a very wide range of materials has been reported to exhibit the phenomenon,
111
Journal of Colloid and Interface Science, Vol.68, No. 1, January 1979
0021-9797/79/010111-17502.00/0 Copyright© 1979by AcademicPress, Inc. All rightsof reproductionin any formreserved.
112
DONALD B. SIANO Bentonite ~ 0 . 4 %
25
20 y (cm)
]5
5 -
0 15
-
.
I
I
20
25
,L
I
30 t (days)
-
I
I
i
:55
40
45
FIG. 1. The position of the interfaces between bands, measured from the meniscus, as a function of time for a sample of Bentonite clay with an initial homogeneous concentration of about 0.4 wt%. The bottom of the tube was at y = 28 cm.
e.g., carbon black, oil emulsions, very finely ground quartz (13), glass spheres (15), erythrocytes (9), tripoli (2), and As~S3 (14), most studies have been carried out on various clays. There is agreement in the literature that solutions with higher initial concentrations take longer to form the bands, but more bands are eventually formed (2, 13, 15). When the concentration is initially uniform, a plot o f the positions o f the bands as a function o f time shows that the bands (actually the sharp interface b e t w e e n bands) closer to the top settle more slowly and the plots of the lines o f discontinuity converge more or less toward the origin. Figure 1 shows a record for a dilute suspension of bentonite clay in water which illustrates some o f the salient features of the phenomenon. There is some disagreement in the literature on whether light Journal of Colloid and Interface Science,
Vol. 68, No.
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influences the formation o f the layers (6, 15), but all investigators [except two (3, 4)] agree that the elimination o f temperature gradients in the tube is essential to prevent convection which quickly disrupts the layer. The explanation o f layer formation has been a matter of some speculation. The most obvious explanation is that the material is heterodisperse and that segregation according to particle size, shape, or charge has somehow occurred. This appears to be more or less the consensus, but it is not supported by experimental data. In the single investigation in which the particle sizes in different layers were actually measured, the surprising result was that the composition was virtually identical for different l a y e r s - - o n l y the concentration appeared to differ (15). In spite o f these results, however, the possibility was still
113
LAYERED SEDIMENTATION
raised that the polydispersity of the original material had something to do with the formation of the layers through their production of a sedimentation potential. Clearly, the results of previous investigators are difficult to interpret primarily because of their use of poorly characterized heterodisperse materials. However, given the availability of very monodisperse, stable, hydrophobic colloids in the form of polystyrene spheres which have been widely studied under various conditions (17, 18), the possibility exists that some progress might be made toward elucidating seemingly paradoxical phenomenon. The object of this paper is to present some experimental results on layered sedimentation using these monodisperse colloids and to suggest a possible explanation for the occurence of the stratification. Two sets of experiments were performed. In the first set, uniform diameter spherical colloidal dispersions were introduced into long cylindrical tubes and allowed to settle. After several days, layers appeared, and the spacing of these layers, their settling velocities, their concentration, etc., were measured for a series of different initial solution concentrations, as a function of time. In the second set of experiments, dispersions were introduced into the tubes in such a way that initially there was a linear concentration profile or equivalently a constant concentration gradient. These solutions layered much more rapidly than the uniform solutions and the various parameters were studied as a function of average colloid concentration, concentration gradient, salt concentration, and salt concentration gradient. EXPERIMENTAL
Most experiments were carried out in cylindrical Pyrex test tubes 35 cm long with an inside diameter of 0.6 cm. The tubes were cleaned with sulfuric aciddichromate or alcoholic KOH solution and
rinsed at least 15 times with deionized water which had a specific conductivity of 1 to 2 /zmho cm. The tubes were always uniformly wet by a thin film of water and were not allowed to dry before the solutions were put in. The polystyrene spheres (Dow Chemical Co.) with diameters 0.109, 0.234, and 1.091 /zm (standard deviations of 0.0027, 0.0026, and 0.0082 /zm, respectively) had a density of 1.054 g/cm 8. The polyvinyltoluene particles had a diameter of 2.02/zm with a standard deviation of 0.0135 /zm and had a density of 1.027 g/cm z. Solutions with artificial gradients were made by mixing two solutions, A and B. Solution A was made up to the desired polymer and NaC1 concentration by dilution of the stock with deionized water, In most experiments solution B consisted of either water or a salt solution with a concentration equal to that in A. The linear gradients were made by first adding enough of A to give a depth of about 10 cm in the tube. Then, for example, 2 drops of B were added and then 18 drops of A, 4 drops of B, 16 drops of A, etc., until the last addition was 20 drops of B. Since there was some mixing between the additions, this produced a linear gradient in polymer concentration about 10 cm long, which,after an hour or so was fairly smooth to the eye. The concentration of the suspension thus has the form C = CmY/Ym
[1]
where y = 0 is at the meniscus and Cm is the concentration at the bottom of the inhomogeneous region where y - Ym. A graduated scale was affixed to the side of the tube with the zero at the meniscus. The tubes were corked and placed in a darkened room where the temperature was maintained constant at 22 +__ I°C. Similar tubes were also prepared which were initially uniform in concentration of polymer along the entire length for experiments in which gravity acted to eventually produce a concentration gradient. The interfaces which developed were usually viewed by holding a Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
114
DONALD B. SIANO
1.091 /.Lm polystyrene
0
2
4 E t,) v 6
8
10
5
t=O
6
9
12
15 days
FI~. 2. A schematic representation of the appearance of the layers observed as a function of time in an initially homogeneous suspension of monodisperse polystyrene spherical particles with a diameter of 1.091/~m and a volume fraction of 0.001.
flashlight above and behind the tube. Occasionally, visibility was better when they were diffusely illuminated from behind. Measurements were made of the n u m b e r fraction o f monomers in the various layers by viewing samples that were withdrawn by pipet and placed on a graduated slide under a light microscope. The dimers and trimers, etc., could be clearly distinguished from the monomers in the case o f the 1.091hem particles, and approximate measurements o f the relative concentrations o f monomers in the various layers were made. RESULTS
Figure 2 shows schematically the time development of bands which developed in suspension o f spherical monodisperse polyJournal of Colloid and lnterface Science, Vol. 68, No. 1, January 1979
styrene particles (diameter = D = 2r = 1.091 /zm, density = p = 1.054 g/cm -a) which was initially homogeneous with a weight fraction o f polymers of 0.001 in 0.1% NaC1. At this weight fraction the n u m b e r density is n ~ 1.4 × 10a cm -3 and so the distance between particles is d =-n -1/3 8.9/~m. On the other hand, the D e b y e length for the solution is r -1 ~ 0.0024 ~m, so Kr ~ 230 and Kd ~ 3700, which implies that the electrostatic force between two isolated particles separated by their average distance is very small. The gravitational force acting downward on a single isolated particle is significant, however. The net force acting d o w n w a r d is F o = (47r/3)r 3 gAp = 3.7 × 10-16N, where Ap is the difference in the density o f the particle and the solvent. This results in an infinite dilution velocity
115
LAYERED SEDIMENTATION
14
0.1%
1.091/~m polystyrene
0.1%
NoCI
Homogeneous at t = 0
12
10
8 y (cm) 6
4-
2
0
I
I
I
I
I
I
I
100
200
300
400
500
600
700
t (hr) FiG. 3. T h e position of the interfaces b e t w e e n b a n d s as a function o f time for an initially h o m o g e n e o u s s u s p e n s i o n o f 1.091-p~m diameter p o l y s t y r e n e s p h e r e s with a concentration o f 0.1 wt% in a 0.1% NaCI solution.
of vs = F g / ( 6 ~ r ~ r ) = 0.35 cm/day, where a~ is the viscosity of the solvent. From a plot of the positions of the observed interfaces for this suspension as a function of time, shown in Fig. 3, we can calculate the velocity of the several interfaces. The lower curve (the interface nearest the meniscus) gives a value for the interfacial velocity that is in good agreement with the infinite dilution Stokes velocity, but there is some curvature present. Interfaces farther from the meniscus yield velocities several times as great as this, however. Figure 4 shows the same polymer and salt concentration in a 50 wt% e t ha nol - w a t e r mixture. In this case more bands are produced which also move with increasing velocities toward the bottom of the tube. This figure also illustrates the fact that the
concentration discontinuities do not appear to be propagated from the bottom of the tube toward the t o p - - i n s t e a d they give the impression of diverging from the uppermost " p r i m a r y " discontinuity toward the bottom of the tube. There is some similarity between Figs. 3 and 4 in that some interfaces appear to form by diverging from a previously formed interface, while others appear in an "isolated" fashion. This experiment also shows that, as in numerous other cases we have examined, the bands generally appear only in the space within 10 or 15 cm below the primary concentration change at the top of the tube. Lower down, the tube appears to be homogeneous all the way to the very thin layer of sediment at the bottom of the tube. This observation led us to believe that the Journal of Colloid and Interface Science, Vol, 68, No. 1, January 1979
116
DONALD B. SIANO O.1% ]8
1.09]/zm
50 Ho
16
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10 y(cm) 8
6
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I
0
100
I
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I
200
300
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t
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Fro. 4, The position of the interfacesbetweenbands as a functionof time for an initiallyhomogeneous suspension of 1.091-/xm-diameterparticles in a 50/50 (by weight) water-ethanol mixture. concentration gradient produced after a time by the combined effects of settling and diffusion might have been somehow responsible for the production of the bands and suggested that it would be worthwhile to examine the effects of producing artificial concentration gradients by the method described in the experimental section. An example of the results obtained in this way is shown in Fig. 5. In this case bands appear to be more regularly spaced and appear much sooner. Moreover, the velocities of the bands are now nearly equal and have values close to that expected from Stokes' law. We can also see that the bands Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
all appear at nearly the same time and that the average band spacing slowly increases with time. Another example which adds some confirmation to the idea that the concentration gradient causes the bands is shown in Fig. 6. This experiment shows that even particles with a diameter of 0.234 /zm (which do not form a sharp boundary at the top under the action of gravity even after 60 days) will form bands when an artificial gradient is imposed. Moreover, the interfaces fall with a velocity approximately equal to the Stokes velocity (0.0160 cm/day observed, 0.0162 cm/day expected). Simi-
LAYERED SEDIMENTATION 22'
A: 0.05% +0.10%
20
B : Water
117
1.091Fm polystyrene NaCI
Linear gradient 20cm long at t=O 18
]6
14 y(cm) 12
]0
8
6
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0
100
200
300
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400 t (hr)
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FIG. 5. The position of the interfaces of concentration discontinuities as a function of time for a solution with an initial concentration (both polymer and salt) gradient imposed. l a d y , we h a v e o b s e r v e d that p o l y s t y r e n e spheres with diameters of 0.109 /xm f o r m bands w h e n the weight fraction is 0.001 and a linear concentration gradient is imp o s e d as the initial condition. Particles of this size are considered to be colloidal and are not o b s e r v e d to settle at all f r o m an initially h o m o g e n e o u s suspension e v e n after a very long time. Theoretically, their Stokes velocity is 0.0035 cm/day, which is small
enough to be o v e r w h e l m e d b y other phen o m e n a such as Brownian motion. CONCENTRATION OF POLYMER
The effect of varying the m a x i m u m concentration of p o l y m e r , C m , is s h o w n in Fig. 7 for 1.091/zm particles in 0.1% NaC1. The average distance between bands goes through a minimum at a volume fraction 6 ~ 0.002 Journal of Colloid and Interface Science, Vol. 68, No, 1, January 1979
118
DONALD B. SIANO
"14
A :
0.1%
B :
Water
0 . 2 3 4 / . t m + deionized w a t e r
L i n e a r g r a d i e n t 2 0 c m long at t = 0
"12 •
i
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v
L
'
~
10
8 £
.--
y (cm) 6
a T d l
0
I:
~, "
I
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I
100
200
300
400
500
600
700
t
(hr)
FIG. 6. The position of the concentration discontinuities for a suspension of 0.1 wt% with a diameter of 0.234 /xm. An initial linear concentration gradient of colloid (but not salt) about 20 cm long was imposed at t = 0.
and no bands were o b s e r v e d below ~b 0.0002 or a b o v e ~b-~ 0.005. The latter concentration is, h o w e v e r , so optically opaque that the bands m a y have only been o b s c u r e d f r o m view. It is interesting to note that the volume fraction for which bands are formed for the 0.234- and 0.109/xm particles is also around 0.001 and the bands form about the same distance apart. Therefore the average distance between the bands apparently scales a p p r o x i m a t e l y as d/D [since ~ b - (d/D)S], where d is the average distance between particles and D is their diameter. The observations made during this e x p e r i m e n t also confirmed the earlier observations that the less concentrated solutions formed bands s o o n e r than m o r e concentrated ones, but it is difficult to Journal of Colloid and Interface Science, Vol. 68, No. I, January 1979
put this into precise quantitative terms. We can say, h o w e v e r , that the bands with ~b ~ 0.0005 formed completely in about a day, but the bands at tb = 0.004 took about a w e e k to fully appear. CONCENTRATION OF SALT As long as the salt concentration was kept below the value where the colloid flocculates (about 0.3% NaCI), where the formation of bands is o b s e r v e d to be completely abolished, the effects o f added salt upon the spacing of the bands w e r e surprisingly weak. A c o m p a r i s o n o f the effect o f an initially imposed salt gradient as opposed to a s y s t e m with uniform salt (0.1% NaCI) or no salt upon the average spacing
LAYERED SEDIMENTATION A:
1.091/zm in 0.1% NaCl
B:
0.1% NaCI
119
Linear gradient 20cm long at t = 0
I
\ E
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B
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2
j" R~O
1
0
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I
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.2
.5
.4
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•
c (wt%) Fic. 7. The average band spacing, Ay,, as a function of concentration for 1.091-/~m-diameter polystyrene spheres with an initial linear concentration gradient of polymer, but not salt. The arrow indicates that this concentration was tried, but no bands were observed. is s e e n in T a b l e I. T h e n o n s i g n i f i c a n t diff e r e n c e b e t w e e n t h e u n i f o r m salt a n d z e r o salt c a s e s a p p e a r s t o i n d i c a t e t h a t b a n d f o r m a t i o n is p r o b a b l y n o t d u e to d o u b l e layer interaction or aggregation because both phenomena are highly influenced by salt. T h e e f f e c t o f a salt gradient is p r o nounced, however. It decreases the average s p a c i n g b y a b o u t a f a c t o r o f 3. W e also o b s e r v e d t h a t b a n d s still f o r m in t h e c a s e o f 2.02-/xm d i a m e t e r s p h e r e s w h e n t h e salt c o n c e n t r a t i o n is m a d e a s s m a l l as p o s s i b l e b y u s i n g d e i o n i z e d w a t e r as t h e d i l u e n t , p u t t i n g a f e w milliliters o f m i x e d b e d r e s i n b e a d s into t h e t u b e w i t h t h e p o l y s t y r e n e , thoroughly mixing the suspension, and then l e t t i n g it s t a n d . Some interesting results were obtained by u s i n g a 12% s u c r o s e s o l u t i o n ( c a l c u l a t e d to give neutral buoyancy) with D = 1.091/zm, ~b ~ 0.001, a n d 0 . 1 % NaC1. W h e n t h e g r a d i e n t w a s m a d e w i t h w a t e r as s o l u t i o n B,
a b o u t 20 b a n d s 2 m m o r so a p a r t f o r m e d in a b o u t a w e e k . T h e c o n c e n t r a t i o n profile appeared to be initially of approximately a sinusoidal shape, but later the bands app e a r e d to " s q u a r e o f f . " E v e n t u a l l y s e v e r a l o f t h e s e b a n d s b e c o m e m o r e p r o m i n e n t , so TABLEI Comparison of the Average Spacing of the Bands, Ay,, and the Number of Bands Observed, n, for Initially Nonuniform Suspensions of 1.101-/xm Polystyrene with a Weight Fraction of 0.001 under Conditions of No Salt, Uniform Salt, or a Linear Salt Concentrationa Salt concentration
AY. (cm)
n
0 0.1%, no gradient 0.1%, gradient
1.3 ± 0.3 1.4 ± 0.2 0.5 ± 0.2
4 7 5
The concentration ranged from zero at the top of the tube to a maximum of 0.1% at the bottom of the tube. The length of the column was 24 cm and bands formed in the region from about 2 cm to about 10 cm. Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
120
DONALD B. SIANO TABLE II
The Average Spacing of the Bands, AYn, When the Length of the Gradient, L, is Varied by a Factor of 2 or 4 for Polystyrene Spheres with a Diameter of 1.091/xm A (%)
B (%)
L
~Yn
(era)
C (polymer)
C (NaCI)
C (NaCI)
(cm)
10 40
0.I0 0.10
0.10 0.10
0.10 0.10
1.4 -+ 0.7 1.4 ± 0.6
10 20
0.10 0.10
0.10 0.I0
0.10 0.10
1.9 -+ 0.5 1.7 ± 0.5
that there appeared to be a superposition of two periods. This process continued for several more weeks until only a few bands about 1 cm apart in the lower, more concentrated region were visible. GRADIENT LENGTH
The length of the inhomogeneous region between the regions with constant concentration of polymer (Cm at the base and zero at the top) can be easily varied, and some results are shown in Table II. It can be concluded that the average distance between the bands is not affected significantly when the length of the inhomogeneous region is changed by a factor of 2 or 4. The bands in the tubes with a gradient length of 40 cm, however, had some bands disappear after about a week, so that the average spacing was significantly larger. All of the data in Table II were taken from tubes shortly after the bands were completely formed. Also, some interesting results were obtained when the gradient region was made as short as possible (~2 mm) by carefully layering water or a salt solution on top of a colloidal dispersion in 12% sucrose. In this case bands very close together (AYn ~ 2 mm) appeared to slowly "peel off" the lower solution and move slowly upward through the tube. At the other extreme, when no gradient at all was imposed and the particles (2r Journal of Colloid and Interface Science, Vol. 68, No, 1, January 1979
= 1.091/zm, ~b = 0.001) were suspended in a 12% sucrose solution so that they were at neutral bouyancy, a striking result was obtained. After about 50 days, four or five faint bands were seen to have formed which moved very slowly down the tube but no clear space formed at the top of the tube. Thus it appears that the bands can form even when no concentration gradient is initially imposed and even when the Stokes velocity is very low. The initial concentration gradient apparently serves primarily to shorten the time to first appearance of the bands, but the smaller concentration gradient due to spontaneous fluctuations (or possibly insufficient mixing) was sufficient to induce their formation. STABILITY OF BANDS AND CONVECTION
In view of past assertions that the bands are caused by weak convection currents, it is germane to describe some of our observations in this regard. In agreement with almost all previous observers, we found that it was essential to have the tubes in as constant a temperature environment as possible. If a strong light was directed at them or even if one stood next to them for more than a few minutes (especially for the tubes with 0.8-cm inside diameter), the interfaces between the bands blurred and eventually disappeared. The stability of the bands against slight disturbances in an experiment of this sort is illustrated in Fig. 8. The bands were produced in a suspension with ~b = 0.0012 of 2.02-/xm spheres after about 10 days, and the positions of the bands were noted a number of times during the day. We then stood next to them for about 10 min, causing them to disappear completely, and then the positions were noted again. As can be seen in the figure, the bands reappeared in their former positions. It is certainly true that convection cells can be made to appear in the tubes by imposing a lateral temperature gradient of sufficient magnitude. Experimental studies
LAYERED SEDIMENTATION
121
0.12%
2.02Fro
lO -~
9
.
~
~
y (cm)
-m--~ e
~ _ ~ , _ ~ .
o
,, __.__-.-,---
o
8 j.__A.
0 •
0
5 0
I
I
I
I
i
2
4
6
8
]0
t
(hr)
FIG. 8. The position of the interfaces, y, recorded during a single day for a suspension of 0.12 wt% 2.02-/zm particles. Between 7 and 8 hr the bands were twice destroyed by a small temperature gradient (created by standing near the tube). They were then observed to reappear (open circles) in the expected positions. on slots with transverse temperature gradients (which is easier to do experimentally but is still probably relevant to tubes) have shown that convection cells do not appear unless the Rayleigh n u m b e r is greater than about 3500 (19). For water at 25°C, the Rayleigh n u m b e r is given by Ra = 2 × 104. \-~m/ "\-gC-/ '
[2]
where l is the distance across which the temperature gradient AT is imposed. Under ordinary conditions the temperature difference between the sides of a tube was measured to be less than about 0.01°C, and since l is about 0.6 cm, Ra ~< 43, which is much too small. If the bands were caused by simple convection the particles serve merely as indicators of the layering of the fluid, and it should be possible to observe
the banding by using a dye as indicator in place of the particles. Despite many attempts (using fluoroscein dye and density gradients of various magnitudes) we have never succeeded in producing layers similar to those seen with the particles. The fact that bands form even when the suspension is stabilized against convection by imposing a density gradient in the fluid by the addition of a third c o m p o n e n t such as sucrose also constitutes strong evidence against a convective mechanism. Another argument against convection is given by the appearance of the bands. Convection cells which are made visible by the presence of particulates exhibit swirling lines of flow and appear grossly different from the bands discussed here (20). It seems unlikely that these bands are caused by thermal convection. Journal of Colloid and Interface Science, Vol, 68, No. 1, January 1979
122
DONALD B. SIANO POLYDISPERSITY AND AGGREGATION
10 cm. If some sort of separation process were occurring in the original solution, then particles would have to move during the period of the formation of the bands with a velocity of the order of 5 cm/day. This is larger than the Stokes velocity for these particles by about a factor of 14, and therefore, it appears unlikely that the particles differ only slightly in size and/or shape. Strong evidence against the polydispersity hypothesis was also obtained by observing the velocities of the interfaces which occurred in a system composed of equal weight fractions (0.05%) of 1.091- and 0.234-/xm particles. The average velocity of all the interfaces was found to be 0.17 cm/ day, which is not significantly different from the weight average velocity of the particles of 0.18 cm/day. The observed velocities of the interfaces (12 were observed with an average spacing of about 0.6 cm) were not correlated with their positions in the tube. That is, the interfaces deeper in the tube moved with about the same velocities as those nearer the top. There are several other features of the data which appear to be inconsistent with the polydispersity hypothesis. If some sort of aggregation process were responsible for layered sedimentation, then we would expect the layers to appear sooner in more concentrated suspensions. This is also contradicted by the observations. Aggregation is promoted by salt, but the bands are relatively insensitive to salt as long as the concentration of salt is below the flocculation value. If the original colloid is polydisperse, the existence of a lower and an upper limit on the concentration between which bands form is difficult to understand. The dramatic changes in the appearance of the bands when the gradient is made as short as possible and when sugar is added to solution A are likewise difficult to explain in terms of the polydispersity hypothesis.
It is perhaps natural to suppose that the origin of the bands in the sedimenting colloidal suspensions is due to the separation of classes of different sizes and/or shapes of the particles. To test this hypothesis, samples were withdrawn from various tubes containing layers with 1.091-/zm-diameter particles at different levels from different bands by pipet and examined under a light microscope. The individual particles could be easily seen, and the state of aggregation of the particles was readily ascertained. In every case that we examined, the singlets (single spherical particles) comprised about 95 to 98% of the sample, and this fraction did not vary significantly from one layer to the next, nor between different levels within a single band. Most of the remaining particles were seen to be two monomers that were touching, together with a smaller number of triplets, etc. Unless the hypothesized multiplets were very loosely adhering and were disrupted by the sampling procedure. This would appear to rule out the possibility that the bands consisted of different sizes of aggregates of singlets. There are additional lines of evidence against the polydispersity hypothesis that are more indirect. Consider Fig. 5 again, where as many as 14 bands were seen. It is unlikely that there are 14 distinct classes of particles in this tube. If they differed in size from singlets (presumably at the top) to quatredecamers in the bottom band, the velocities of the bands would not be expected to be so nearly equal as they are observed to be. On the other hand, if the bands consisted of mostly singlets but with slightly different average sizes and/or shapes, then we would certainly have found a very clever, extremely sensitive separation method because the polydispersity of the original colloid is known to be very small ! THE SEDIMENTATIONPOTENTIAL Another independent line of reasoning derives from the fact that the bands form comAs we have noted previously, the sedipletely in less than 50 hr in a length of about mentation potential has been invoked in the Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
LAYERED SEDIMENTATION past to account for the presence of layers in sedimenting suspensions. For a dilute homogeneous suspension of sedimenting particles with number density n, zeta potential ~ in a medium with viscosity ,/, electrical conductivity K, and dielectric constant E, an electric field with a magnitude of
123
mentation of a dilute homogeneous suspension of spheres gives the result that
vo/vs = 1 - 6.55~b.
[6]
If these results apply to the experiments in which an artificial gradient is initially imposed, it would appear that no mechanism is produced for the formation of b a n d s - the particles nearer the bottom would just e~raApgn move a few percent slower than those at the Es [3] 3~)K top, and so the latter would eventually catch up and the uppermost boundary would is produced which tends to retard the sedibe merely sharpened. Density currents (25) mentation (21). The observed velocity, Vo and settling convection (26) also appear to is related to the infinite dilution Stokes be ruled out as explanations of layered velocity, Vs, by the relation (22) sedimentation because of the near equality of the velocities of the interfaces with the ~cr \ 47rr ] " [4] limiting Stokes velocity. In addition, the denser fluid was below the lighter in all of Using the values for ¢, 0, and K for "con- our experiments as opposed to the requireductivity" water, r ~, Ap and n for our 0.1% ments for these phenomena to occur. There 1.101-/zm polystyrene solution and a zeta po- appears at present to be no hydrodynamic tential of 50 mV gives an estimate for E~ theory of layered sedimentation in dilute of about 1 mV/cm and a value for the right- suspensions of spheres. hand side of Eq. [4] of about 4 × 10-4. Thus the electric field is seen to be quite small THEORY (and much smaller when salt is added beThe experiments reported here appear to cause K increases), and the correction to lead to one conclusion--that layered sedithe sedimentation velocity is negligible. Bementation is due to the presence of an initial cause this is the case, it is difficult to see concentration gradient in the region in how the sedimentation potential can acwhich bands later appear. (Strictly speakcount for the rather dramatic layering effect. ing, of course, concentration gradients due to spontaneous fluctuations are always HYDRODYNAMIC EFFECTS present, but bands apparently due to this Measurements of the observed velocity of cause take much longer to form, as we have fall for particles larger than about 50 /zm seen.) Our experiments have shown that (where hydrodynamic effects would pre- layering or composition modulations always sumably dominate double-layer effects) occur in the region or interface between the have been made by a number of observers pure solvent at the top of the tube and the by following the fall of the relatively sharp most concentrated suspension at the bottop of the cloud of particles. Oliver (23) has tom. The case of 0.109-/~m-diameter parsummarized the experimental results, and ticles with volume concentration 0.001 is he finds that for dilute suspensions the data particularly instructive. Particles of this are fit satisfactorily by the expression size have a diffusion coefficient olD0 ~ 2.2 x 10-s cm 2 sec -1, so that in 1 day they Vo/Vs = (1 - 0.754)'/3)(1 - 2.15th). [5] would diffuse a distance of about (Dot) v~ Thus for our typical case with ~b = 0.001 0.04 cm, which is over 10 times as large we find Vo/Vs = 0.93. On the other hand, a as the Stokes law sedimentation distance. hydrodynamic calculation (24) of the sedi- We would therefore expect that these
(vs-
L(
t
Journal of Colloid and Interface Science, VoL 68, No. 1, January 1979
124
D O N A L D B. S I A N O
particles are "diffusion controlled" rather than "sedimentation controlled" so that ordinary Fickian diffusion would tend to make the solution more homogeneous with time. On the contrary, the suspension which initially has a ramp-like concentration profile develops a step-like concentration profile up to the region with constant concentration after a relatively short time. Since the sedimentation distance is small compared to the distance between bands during the initial time interval for the formation of the bands (-0.02 cm compared to 1.5 cm), conservation of particles implies that "uphill diffusion" must have occurred. Thus Fick's law does not appear to be applicable for suspensions of this concentration. However, an alternative diffusion equation has been derived for inhomogeneous systems which does predict many of the features exhibited by our data but has not hitherto been applied to the phenomenon of layered sedimentation. This is the CahnHilliard theory of spinodal decomposition, which takes into account the effect of inhomogeneities on the Helmholtz free energy and yields a modified diffusion equation. When the gradient is neither too small (when Fick's law is approximated) nor too large (when extra terms must be added), the diffusion equation is, neglecting gravity (27, 28),
Oc _ M 02f V2c - 2MKV4c, Ot Oc2
[7]
where f(c) is the free energy density for a uniform system of concentration c, M is a mobility, and K is the gradient energy coefficient which is positive and independent of c. For one dimension Eq. [7] has the solution
c(y,t) = Co + 8c exp[R(/3)t] cos (fly), [8] where
( Ozf + 2K/3~) R(/3) = - M / 3 2 0 c 2
[9]
and fl = 2~r/h is the wavenumber of the Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
sinusoidal composition deviation from co, the average composition. Thus for systems outside the spinodal, where 02f/Oc 2 > 0, and K and M are both greater than zero, R(/3) is negative, so composition modulations die out with a time constant of R - I On the other hand, when 02f/Oc 2 is smaller than zero (i.e., inside the spinodal), there is a critical wavenumber /3¢ for which composition modulations with /3
02f /K11'2 / 2
/3m= [ - - ~ c 2 /
j
/'
[10]
and the rate is ]- 0 2 f ] 2 /
[11] We emphasize that this diffusion equation is only an approximation which holds for the early stages of the development when the composition modulations are small. For example, when R > 0 it predicts that Ac increases without bound, which is not possible. The higher-order terms in the diffusion equation stop the growth and determine the "coarsening" behavior (29, 30). The qualitative aspects of this theory have been verified and successfully applied to an A I - Z n alloy (31), a liquid-liquid mixture (32), a polymer-solvent mixture (33), and a sol-gel transformation (34). In these systems the observed wavelength is found to be of the order of 50 to 1000 lattice spacings. The essential difference in our system is that instead of the two components having approximately the same size, the colloid and solvent particles are very different in size. If we take the lattice spacing to be the average distance between the colloid particles, then the expected wavelength for our typical case of 1.091/zm particles with an average distance apart of about 10 diameters amounts to about 0.05 to 1 cm. This agrees with our ob-
LAYERED SEDIMENTATION
served average band spacing Ayn, which we therefore idz,ntify with the wavelength h. Another order-of-magnitude agreement is obtained by noting that K has the dimensions of force and is the order (35) of k s T / d ~ 4 × 10-16 N for the case of D = 1.091/xm, ~b = 0.001. This is of about the same magnitude as the gravitational force and is therefore large enough to alter the behavior of the suspension significantly. Equation [10] predicts, since K is independent of the gradient, that (within limits) the wavelength is independent of the magnitude of the gradient. This is shown to be the case in Table I. Another prediction that can be made from Eq. [10] is that there is a law of corresponding states for /3, i.e., that/3 = fl(d/D, T/Tc), where D is the diameter of the particles and d is the distance between them. Since d/D ~ ¢b-1/3, this implies that fl is a universal function of ~b, independent of the size of the particle. The qualitative form for this universal function might be predicted on the basis of the well-known Flory-Huggins formulation of the thermodynamics of polymer solutions (36). The spinodal in this case on a T vs ~bphase diagram is concave downward and for high molecular weights is highly asymmetric--its maximum is close to the th = 0 axis and has a much higher slope on the low ~b side than on the high q5 side. Our experiments with varying q5 were made at constant temperature and therefore (Off'/ 0q52) is expected to mirror this behavior. Since K is independent of qS, Eq. [10] predicts that /3 vs ~b will also have a similar general shape. This is in qualitative agreement with the behavior illustrated in Fig. 7. Because of the theory's well-known quantitative limitations, we have not attempted a more detailed comparison with our data. We can note, however, that it is not altogether unreasonable that a spinodal is found in the concentration range of approximately ~b = 0.0002 to 0.004 for polystyrene suspensions. When the salt concentration is very low, suspensions of 0.170-/zm particles undergo transitions to a liquid crystalline
125
structure which have been the subject of many studies (37, 38). The approach taken here is to regard the initial homogeneous mixture of polystyrene particles and water as unstable with respect to a final state in which there is a polystyrene-rich phase at the bottom of the tube and a polystyrene-poor phase above this. This initial state then evolves into the final state via the spinodal mechanism with its well-known attendant band structure. This view is somewhat unorthodox because it assumes that the sedimentation of particles with diameters of a micron (or less) constitute a phase separation so that there is a possibility that the spinodal mechanism can be brought into play. An essential part of the argument is that the final state consists of two well-defined phases. It is certainly true that the two regions (supernatant and sediment) are at least nearly homogeneous with different physical properties, and are separated by a relatively sharp interface. Furthermore, there is no prima facie upper limit for the molecular weight above which a phase separation is not allowed to occur. Without direct observation of a phase transition, we can only offer this explanation as a conjecture--a conjecture that seems to be consistent with the observations made. The fact that the bands are observed to appear much sooner for suspensions which have an initial concentration gradient imposed (5-24 hr) than for similar trials in which the composition is initially uniform (5-10 days) is also neatly explained by this hypothesis. According to this viewpoint, in the latter experiments gravity causes the particles to sediment, which then produces after a certain time a clear space at the top of the tube and then a region of increasing concentration of particles farther down. Thus a concentration gradient or inhomogeneity is produced and the spinodal mechanism generates the observed composition modulations. In the experiments with the artificially imposed concentration gradients, the spinodal mechanism (i.e., the diffusion Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
126
DONALD B. SIANO
equation) can start at once to generate the bands without the preliminary sedimentation step. This hypothesis also clearly explains the crucial observation that bands were produced with the 0.109-/zm particles (which do not sediment at all) when an artificial gradient was imposed but not in its absence. The diffusion equation requires a concentration gradient and the sedimentation is only incidental. In the case of the experiment with 1.091-tzm particles made neutrally bouyant in a 10% sucrose solution, the initial concentration gradient evidently was produced by spontaneous fluctuations in number density, but since the initial composition modulation was small, a long time was required before it grew to observable size. The initial concentration profile often appears to be sinusoidal in shape, but then later becomes more steplike. This, again, is in accord with the predictions of the Cahn-Hilliard t h e o r y - - t h e squaring of the profile is attributed to the higher-order terms in the diffusion equation, which become more important in the later stages of "coarsening." The different qualitative behavior seen when the length of the inhomogeneous region is made shorter than the wavelength seen when the concentration gradient is less steep is also to be expected because these same higher-order terms should not then be neglected in the diffusion equation and a different solution from Eq. [8] is required. SUMMARY Layered sedimentation of colloidal suspensions appears to be caused not by sedimentation but instead by the concentration gradient generated by settling. The evidence is: (1) Bands can be produced by a concentration gradient of particles so small that they do not perceptibly settle, and (2) the layers are produced much more quickly when the suspension starts with an imposed concentration gradient than when the suspension starts in a homogeneous Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
state. The bands are produced even in relatively dilute (volume concentration 0.0002) suspensions in both high (15 mM) and low salt concentrations, and a wide range of sizes (at least 0.1 to 2/~m in diameter) of monodisperse particles. When produced by an initial concentration gradient the average distance between the interfaces increases very slowly with time and the interfaces move downward with a velocity nearly equal to the Stokes velocity. The average spacing between the bands is a function of the volume concentration and is nearly the same for particles of different sizes but constant ~b. The experiments reported here are inconsistent with the hypothesis that the layers are caused by a sorting out process due to polydispersity in size, shape, or charge or due to an ongoing aggregation of the particles. Direct sampling of the layers, parallel movement of the interfaces, their rapidity of formation, and sensitivity to properties of the imposed gradient all militate against the polydispersity-aggregation hypothesis. Similarly, thermal convection and the sedimentation potential seem to be ruled out as successful predictive hypotheses. Many features of the data do appear, however, to be consistent with the CahnHilliard theory of spinodal decomposition. According to this viewpoint, the bands transiently appear in the very diffuse interface between the two separating phases (colloid-rich and colloid-poor) as the system moves toward equilibrium. The dependence of the layers on the imposed gradient, and the average spacing as a function of volume fraction, size, and salt appear to be consistent with this hypothesis. This is not considered to be conclusive evidence, however, and additional evidence is being sought. ACKNOWLEDGMENT I acknowledgestimulatingdiscussions and helpful advise from Dr. B. Berne and Dr. G. Flynn in whose laboratory this work was carried out. The technical assistance of Mr. M. Tederons is greatly appreciated.
LAYERED SEDIMENTATION REFERENCES 1. Brewer, W. H., Mem. Nat. Acad. Sci. USA 2, 165 (1884). 2. Barus, C., Bull. U. S. Geol. Survey 5, 515 (1886). 3. Mendenhall, C. E., and Mason, M., Proc. Nat. Acad. Sci. 9, 199 (1923). 4. Mason, M., and Mendenhall, C. E., Proc. Nat. Acad. Sci. USA 9, 202 (1923). 5. Ehrenberg, P., Hahn, E., and Nolte, O., KoUoid Z. 21, 1 (1917). 6. Morison, C. G. T., Proc. Roy. Soc. London Ser. A , 107, 280 (1925). 7. Ungerer, E., KolloidZ. 14, 63 (1927). 8. Ungerer, E., Kolloid Z. 41, 51 (1927). 9. Ducceschi, V., KoUoid Z. 48, 78 (1929). 10. Krauss, G., Kolloidchem. Beih. 30, 378 (1927). 11. Nutting, P. G., J. Wash. Acad. Sci. 19, 402 (1929). 12. Miller, M. A., J. Phys. Chem. 42, 419 (1938). 13. Tobias, T. J., and Ruotsala, A. P., Clays Clay Miner. 13, 395 (1966). 14. Freundlich, H., in "Colloid and Capillary Chemistry" (H. S. Hatfield, Transl.), p. 370, Methuen., London, 1926. 15. Merchant, S. R., and Rosauer, E. A., Clays Clay Miner. 17, 289 (1969). 16. Davies, L., Dollimore, D., and Sharp, J. H., Powder Technol. 13, 123 (1976). 17. Vanderhoff, J. W., Van Den Hul, H. J., Tausk, R. J. M., and Overebeek, J. Th. G., in "Clean Surfaces: Their Preparation and Characterization for Interfacial Studies" (G. Goldfinger, Ed.), p. 15, Dekker, New York, 1970.
127
18. Papir, Y. S., and Krieger, I. M., J. Colloid Interface Sci. 34, 126 (1970). 19. Elder, J. W., J. Fluid Mech. 23, 77 (1965). 20. Hart, J. E., J. Fluid Mech. 47, 547 (1971). 21. Dorn, E., Ann. Phys. Leipzig 3, 20 (1878). 22. Booth, F., J. Chem. Phys. 22, 1956 (1968). 23. Oliver, D. R., Chem. Eng. Sci. 15, 230 (1961). 24. Batchelor, G. K., J. Fluid Mech. 74, 1 (1976); 52, 245 (1972). 25. Bradley, W. H., Science 150, 1423 (1965). 26. Kuenen, P. H., J. Sediment. Petrology 38, 817 (1968). 27. Cahn, J. W., and Hilliard, J. E., J. Chem. Phys. 31, 688 (1959). 28. Cahn, J. W., J. Chem. Phys. 42, 93 (1965); Trans. A I M E 242, 179 (1968). 29. Cahn, J. W., Acta Met. 19, 151 (1971). 30. Hopper, R. W., and Uhlmann, D. R., Acta Met. 21, 267 (1973). 31. Rudman, K. B., and Hilliard, J. E., Acta Met. 15, 1025 (1967). 32. Schwartz, A. J., Huang, J. S., and Goldberg, W. I., J. Chem. Phys. 62, 1847 (1975). 33. van Aartsen, J. J., and Smolders, C. A., Eur. Polym. J. 6, 1105 (1970). 34. Feke, G. T., and Prins, W., Macromolecules 7, 527 (1974). 35. Cahn, J. W., Acta Met. 9, 795 (1961). 36. Flory, P. J., "Principles of Polymer Chemistry," Cornell Univ. Press, Ithaca, New York, 1953. 37. Hachisu, S., Kobayashi, Y., and Kose, A., J. Colloid Interface Sci. 42, 342 (1973). 38. Kose A., and Hachisu, S., J. Colloid Interface Sci. 46, 460 (1974).
Journal of Colloidand Interface Science, Vol.68, No. 1, January1979
Received February 14, 1978; accepted April 12, 1978 Stratification of the supernatant of suspensions of polydisperse sedimenting colloidal-sized particles, hitherto ascribed to polydispersity, aggregation, convection, or sedimentation potential effects, is also shown to occur in dispersions of highly monodisperse, spherical polystyrene particles in the size range from 2.02 to 0.109 txm. The layers appeared in suspensions with concentrations as low as 0.01 wt% up to at least 0.4 wt%. The qualitative behavior of the layers with their associated interfaces was seen to be highly dependent upon concentration gradients initially imposed on the suspension but to be relatively insensitive to salt (as long as its concentration was below the fluctuation value). Examination of samples from the layers showed that the composition of each layer was the s a m e - - o n l y the concentration varied. Many features of the data suggest that the process is in fact due to the mechanism of spinodal decomposition during the separation of two phases (colloid-rich sediment and colloid-poor supernatant) as the system moves toward its final equilibrium state. INTRODUCTION
It has been known for almost a century that, contrary to expectations, initially homogeneous suspensions of hydrophobic colloidal particles do not sediment in a smooth, continuous fashion (1-15). Instead, bands or layers of differing turbidity are often observed in the column of supernatant liquid after the settling has proceeded for a time. To the eye, each layer appears to be of nearly constant concentration (perhaps with a slight "reverse gradient"), while successive layers increase in concentration when they are farther from the meniscus. The turbidity is such that the boundaries between the layers can be quite marked, and previously published photographs have shown the layers very clearly (6, 13, 15). It has been suggested that this layering of 1 Research Supported by the National Science Foundation under Grant NSF-CHE-76-02414 and NIH-RO1 NS 12714-02 and carried out at the Department of Chemistry, Columbia University.
the supernatant above the denser sediment leads to the production of layers subsequently seen in the compacted sediment, e.g., in clays or sedimentary rock, and is therefore of some geological significance. Another possibility that has been put forward is that the phenomenon might be used to characterize various soils and clays empirically, and this appears to have been the major motivating force behind previous investigations. Sedimentation is, of course, a subject of great importance in many industrial processes such as water clarification, sludge removal, and various separation procedures. It is widely recognized that the theoretical understanding of the process is at present in a very unsatisfactory state except for very dilute suspensions (while they are still homogeneous!) (16). An understanding of the formation of the bands and the mechanism for their preservation should be relevant for the further understanding of this subject. While a very wide range of materials has been reported to exhibit the phenomenon,
111
Journal of Colloid and Interface Science, Vol.68, No. 1, January 1979
0021-9797/79/010111-17502.00/0 Copyright© 1979by AcademicPress, Inc. All rightsof reproductionin any formreserved.
112
DONALD B. SIANO Bentonite ~ 0 . 4 %
25
20 y (cm)
]5
5 -
0 15
-
.
I
I
20
25
,L
I
30 t (days)
-
I
I
i
:55
40
45
FIG. 1. The position of the interfaces between bands, measured from the meniscus, as a function of time for a sample of Bentonite clay with an initial homogeneous concentration of about 0.4 wt%. The bottom of the tube was at y = 28 cm.
e.g., carbon black, oil emulsions, very finely ground quartz (13), glass spheres (15), erythrocytes (9), tripoli (2), and As~S3 (14), most studies have been carried out on various clays. There is agreement in the literature that solutions with higher initial concentrations take longer to form the bands, but more bands are eventually formed (2, 13, 15). When the concentration is initially uniform, a plot o f the positions o f the bands as a function o f time shows that the bands (actually the sharp interface b e t w e e n bands) closer to the top settle more slowly and the plots of the lines o f discontinuity converge more or less toward the origin. Figure 1 shows a record for a dilute suspension of bentonite clay in water which illustrates some o f the salient features of the phenomenon. There is some disagreement in the literature on whether light Journal of Colloid and Interface Science,
Vol. 68, No.
1, J a n u a r y
1979
influences the formation o f the layers (6, 15), but all investigators [except two (3, 4)] agree that the elimination o f temperature gradients in the tube is essential to prevent convection which quickly disrupts the layer. The explanation o f layer formation has been a matter of some speculation. The most obvious explanation is that the material is heterodisperse and that segregation according to particle size, shape, or charge has somehow occurred. This appears to be more or less the consensus, but it is not supported by experimental data. In the single investigation in which the particle sizes in different layers were actually measured, the surprising result was that the composition was virtually identical for different l a y e r s - - o n l y the concentration appeared to differ (15). In spite o f these results, however, the possibility was still
113
LAYERED SEDIMENTATION
raised that the polydispersity of the original material had something to do with the formation of the layers through their production of a sedimentation potential. Clearly, the results of previous investigators are difficult to interpret primarily because of their use of poorly characterized heterodisperse materials. However, given the availability of very monodisperse, stable, hydrophobic colloids in the form of polystyrene spheres which have been widely studied under various conditions (17, 18), the possibility exists that some progress might be made toward elucidating seemingly paradoxical phenomenon. The object of this paper is to present some experimental results on layered sedimentation using these monodisperse colloids and to suggest a possible explanation for the occurence of the stratification. Two sets of experiments were performed. In the first set, uniform diameter spherical colloidal dispersions were introduced into long cylindrical tubes and allowed to settle. After several days, layers appeared, and the spacing of these layers, their settling velocities, their concentration, etc., were measured for a series of different initial solution concentrations, as a function of time. In the second set of experiments, dispersions were introduced into the tubes in such a way that initially there was a linear concentration profile or equivalently a constant concentration gradient. These solutions layered much more rapidly than the uniform solutions and the various parameters were studied as a function of average colloid concentration, concentration gradient, salt concentration, and salt concentration gradient. EXPERIMENTAL
Most experiments were carried out in cylindrical Pyrex test tubes 35 cm long with an inside diameter of 0.6 cm. The tubes were cleaned with sulfuric aciddichromate or alcoholic KOH solution and
rinsed at least 15 times with deionized water which had a specific conductivity of 1 to 2 /zmho cm. The tubes were always uniformly wet by a thin film of water and were not allowed to dry before the solutions were put in. The polystyrene spheres (Dow Chemical Co.) with diameters 0.109, 0.234, and 1.091 /zm (standard deviations of 0.0027, 0.0026, and 0.0082 /zm, respectively) had a density of 1.054 g/cm 8. The polyvinyltoluene particles had a diameter of 2.02/zm with a standard deviation of 0.0135 /zm and had a density of 1.027 g/cm z. Solutions with artificial gradients were made by mixing two solutions, A and B. Solution A was made up to the desired polymer and NaC1 concentration by dilution of the stock with deionized water, In most experiments solution B consisted of either water or a salt solution with a concentration equal to that in A. The linear gradients were made by first adding enough of A to give a depth of about 10 cm in the tube. Then, for example, 2 drops of B were added and then 18 drops of A, 4 drops of B, 16 drops of A, etc., until the last addition was 20 drops of B. Since there was some mixing between the additions, this produced a linear gradient in polymer concentration about 10 cm long, which,after an hour or so was fairly smooth to the eye. The concentration of the suspension thus has the form C = CmY/Ym
[1]
where y = 0 is at the meniscus and Cm is the concentration at the bottom of the inhomogeneous region where y - Ym. A graduated scale was affixed to the side of the tube with the zero at the meniscus. The tubes were corked and placed in a darkened room where the temperature was maintained constant at 22 +__ I°C. Similar tubes were also prepared which were initially uniform in concentration of polymer along the entire length for experiments in which gravity acted to eventually produce a concentration gradient. The interfaces which developed were usually viewed by holding a Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
114
DONALD B. SIANO
1.091 /.Lm polystyrene
0
2
4 E t,) v 6
8
10
5
t=O
6
9
12
15 days
FI~. 2. A schematic representation of the appearance of the layers observed as a function of time in an initially homogeneous suspension of monodisperse polystyrene spherical particles with a diameter of 1.091/~m and a volume fraction of 0.001.
flashlight above and behind the tube. Occasionally, visibility was better when they were diffusely illuminated from behind. Measurements were made of the n u m b e r fraction o f monomers in the various layers by viewing samples that were withdrawn by pipet and placed on a graduated slide under a light microscope. The dimers and trimers, etc., could be clearly distinguished from the monomers in the case o f the 1.091hem particles, and approximate measurements o f the relative concentrations o f monomers in the various layers were made. RESULTS
Figure 2 shows schematically the time development of bands which developed in suspension o f spherical monodisperse polyJournal of Colloid and lnterface Science, Vol. 68, No. 1, January 1979
styrene particles (diameter = D = 2r = 1.091 /zm, density = p = 1.054 g/cm -a) which was initially homogeneous with a weight fraction o f polymers of 0.001 in 0.1% NaC1. At this weight fraction the n u m b e r density is n ~ 1.4 × 10a cm -3 and so the distance between particles is d =-n -1/3 8.9/~m. On the other hand, the D e b y e length for the solution is r -1 ~ 0.0024 ~m, so Kr ~ 230 and Kd ~ 3700, which implies that the electrostatic force between two isolated particles separated by their average distance is very small. The gravitational force acting downward on a single isolated particle is significant, however. The net force acting d o w n w a r d is F o = (47r/3)r 3 gAp = 3.7 × 10-16N, where Ap is the difference in the density o f the particle and the solvent. This results in an infinite dilution velocity
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14
0.1%
1.091/~m polystyrene
0.1%
NoCI
Homogeneous at t = 0
12
10
8 y (cm) 6
4-
2
0
I
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I
I
100
200
300
400
500
600
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t (hr) FiG. 3. T h e position of the interfaces b e t w e e n b a n d s as a function o f time for an initially h o m o g e n e o u s s u s p e n s i o n o f 1.091-p~m diameter p o l y s t y r e n e s p h e r e s with a concentration o f 0.1 wt% in a 0.1% NaCI solution.
of vs = F g / ( 6 ~ r ~ r ) = 0.35 cm/day, where a~ is the viscosity of the solvent. From a plot of the positions of the observed interfaces for this suspension as a function of time, shown in Fig. 3, we can calculate the velocity of the several interfaces. The lower curve (the interface nearest the meniscus) gives a value for the interfacial velocity that is in good agreement with the infinite dilution Stokes velocity, but there is some curvature present. Interfaces farther from the meniscus yield velocities several times as great as this, however. Figure 4 shows the same polymer and salt concentration in a 50 wt% e t ha nol - w a t e r mixture. In this case more bands are produced which also move with increasing velocities toward the bottom of the tube. This figure also illustrates the fact that the
concentration discontinuities do not appear to be propagated from the bottom of the tube toward the t o p - - i n s t e a d they give the impression of diverging from the uppermost " p r i m a r y " discontinuity toward the bottom of the tube. There is some similarity between Figs. 3 and 4 in that some interfaces appear to form by diverging from a previously formed interface, while others appear in an "isolated" fashion. This experiment also shows that, as in numerous other cases we have examined, the bands generally appear only in the space within 10 or 15 cm below the primary concentration change at the top of the tube. Lower down, the tube appears to be homogeneous all the way to the very thin layer of sediment at the bottom of the tube. This observation led us to believe that the Journal of Colloid and Interface Science, Vol, 68, No. 1, January 1979
116
DONALD B. SIANO O.1% ]8
1.09]/zm
50 Ho
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Fro. 4, The position of the interfacesbetweenbands as a functionof time for an initiallyhomogeneous suspension of 1.091-/xm-diameterparticles in a 50/50 (by weight) water-ethanol mixture. concentration gradient produced after a time by the combined effects of settling and diffusion might have been somehow responsible for the production of the bands and suggested that it would be worthwhile to examine the effects of producing artificial concentration gradients by the method described in the experimental section. An example of the results obtained in this way is shown in Fig. 5. In this case bands appear to be more regularly spaced and appear much sooner. Moreover, the velocities of the bands are now nearly equal and have values close to that expected from Stokes' law. We can also see that the bands Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
all appear at nearly the same time and that the average band spacing slowly increases with time. Another example which adds some confirmation to the idea that the concentration gradient causes the bands is shown in Fig. 6. This experiment shows that even particles with a diameter of 0.234 /zm (which do not form a sharp boundary at the top under the action of gravity even after 60 days) will form bands when an artificial gradient is imposed. Moreover, the interfaces fall with a velocity approximately equal to the Stokes velocity (0.0160 cm/day observed, 0.0162 cm/day expected). Simi-
LAYERED SEDIMENTATION 22'
A: 0.05% +0.10%
20
B : Water
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1.091Fm polystyrene NaCI
Linear gradient 20cm long at t=O 18
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700
800
FIG. 5. The position of the interfaces of concentration discontinuities as a function of time for a solution with an initial concentration (both polymer and salt) gradient imposed. l a d y , we h a v e o b s e r v e d that p o l y s t y r e n e spheres with diameters of 0.109 /xm f o r m bands w h e n the weight fraction is 0.001 and a linear concentration gradient is imp o s e d as the initial condition. Particles of this size are considered to be colloidal and are not o b s e r v e d to settle at all f r o m an initially h o m o g e n e o u s suspension e v e n after a very long time. Theoretically, their Stokes velocity is 0.0035 cm/day, which is small
enough to be o v e r w h e l m e d b y other phen o m e n a such as Brownian motion. CONCENTRATION OF POLYMER
The effect of varying the m a x i m u m concentration of p o l y m e r , C m , is s h o w n in Fig. 7 for 1.091/zm particles in 0.1% NaC1. The average distance between bands goes through a minimum at a volume fraction 6 ~ 0.002 Journal of Colloid and Interface Science, Vol. 68, No, 1, January 1979
118
DONALD B. SIANO
"14
A :
0.1%
B :
Water
0 . 2 3 4 / . t m + deionized w a t e r
L i n e a r g r a d i e n t 2 0 c m long at t = 0
"12 •
i
~
v
L
'
~
10
8 £
.--
y (cm) 6
a T d l
0
I:
~, "
I
I
I
I
I
I
I
100
200
300
400
500
600
700
t
(hr)
FIG. 6. The position of the concentration discontinuities for a suspension of 0.1 wt% with a diameter of 0.234 /xm. An initial linear concentration gradient of colloid (but not salt) about 20 cm long was imposed at t = 0.
and no bands were o b s e r v e d below ~b 0.0002 or a b o v e ~b-~ 0.005. The latter concentration is, h o w e v e r , so optically opaque that the bands m a y have only been o b s c u r e d f r o m view. It is interesting to note that the volume fraction for which bands are formed for the 0.234- and 0.109/xm particles is also around 0.001 and the bands form about the same distance apart. Therefore the average distance between the bands apparently scales a p p r o x i m a t e l y as d/D [since ~ b - (d/D)S], where d is the average distance between particles and D is their diameter. The observations made during this e x p e r i m e n t also confirmed the earlier observations that the less concentrated solutions formed bands s o o n e r than m o r e concentrated ones, but it is difficult to Journal of Colloid and Interface Science, Vol. 68, No. I, January 1979
put this into precise quantitative terms. We can say, h o w e v e r , that the bands with ~b ~ 0.0005 formed completely in about a day, but the bands at tb = 0.004 took about a w e e k to fully appear. CONCENTRATION OF SALT As long as the salt concentration was kept below the value where the colloid flocculates (about 0.3% NaCI), where the formation of bands is o b s e r v e d to be completely abolished, the effects o f added salt upon the spacing of the bands w e r e surprisingly weak. A c o m p a r i s o n o f the effect o f an initially imposed salt gradient as opposed to a s y s t e m with uniform salt (0.1% NaCI) or no salt upon the average spacing
LAYERED SEDIMENTATION A:
1.091/zm in 0.1% NaCl
B:
0.1% NaCI
119
Linear gradient 20cm long at t = 0
I
\ E
~3
B
"\
2
j" R~O
1
0
I
I
I
I
I
.I
.2
.5
.4
.5
•
c (wt%) Fic. 7. The average band spacing, Ay,, as a function of concentration for 1.091-/~m-diameter polystyrene spheres with an initial linear concentration gradient of polymer, but not salt. The arrow indicates that this concentration was tried, but no bands were observed. is s e e n in T a b l e I. T h e n o n s i g n i f i c a n t diff e r e n c e b e t w e e n t h e u n i f o r m salt a n d z e r o salt c a s e s a p p e a r s t o i n d i c a t e t h a t b a n d f o r m a t i o n is p r o b a b l y n o t d u e to d o u b l e layer interaction or aggregation because both phenomena are highly influenced by salt. T h e e f f e c t o f a salt gradient is p r o nounced, however. It decreases the average s p a c i n g b y a b o u t a f a c t o r o f 3. W e also o b s e r v e d t h a t b a n d s still f o r m in t h e c a s e o f 2.02-/xm d i a m e t e r s p h e r e s w h e n t h e salt c o n c e n t r a t i o n is m a d e a s s m a l l as p o s s i b l e b y u s i n g d e i o n i z e d w a t e r as t h e d i l u e n t , p u t t i n g a f e w milliliters o f m i x e d b e d r e s i n b e a d s into t h e t u b e w i t h t h e p o l y s t y r e n e , thoroughly mixing the suspension, and then l e t t i n g it s t a n d . Some interesting results were obtained by u s i n g a 12% s u c r o s e s o l u t i o n ( c a l c u l a t e d to give neutral buoyancy) with D = 1.091/zm, ~b ~ 0.001, a n d 0 . 1 % NaC1. W h e n t h e g r a d i e n t w a s m a d e w i t h w a t e r as s o l u t i o n B,
a b o u t 20 b a n d s 2 m m o r so a p a r t f o r m e d in a b o u t a w e e k . T h e c o n c e n t r a t i o n profile appeared to be initially of approximately a sinusoidal shape, but later the bands app e a r e d to " s q u a r e o f f . " E v e n t u a l l y s e v e r a l o f t h e s e b a n d s b e c o m e m o r e p r o m i n e n t , so TABLEI Comparison of the Average Spacing of the Bands, Ay,, and the Number of Bands Observed, n, for Initially Nonuniform Suspensions of 1.101-/xm Polystyrene with a Weight Fraction of 0.001 under Conditions of No Salt, Uniform Salt, or a Linear Salt Concentrationa Salt concentration
AY. (cm)
n
0 0.1%, no gradient 0.1%, gradient
1.3 ± 0.3 1.4 ± 0.2 0.5 ± 0.2
4 7 5
The concentration ranged from zero at the top of the tube to a maximum of 0.1% at the bottom of the tube. The length of the column was 24 cm and bands formed in the region from about 2 cm to about 10 cm. Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
120
DONALD B. SIANO TABLE II
The Average Spacing of the Bands, AYn, When the Length of the Gradient, L, is Varied by a Factor of 2 or 4 for Polystyrene Spheres with a Diameter of 1.091/xm A (%)
B (%)
L
~Yn
(era)
C (polymer)
C (NaCI)
C (NaCI)
(cm)
10 40
0.I0 0.10
0.10 0.10
0.10 0.10
1.4 -+ 0.7 1.4 ± 0.6
10 20
0.10 0.10
0.10 0.I0
0.10 0.10
1.9 -+ 0.5 1.7 ± 0.5
that there appeared to be a superposition of two periods. This process continued for several more weeks until only a few bands about 1 cm apart in the lower, more concentrated region were visible. GRADIENT LENGTH
The length of the inhomogeneous region between the regions with constant concentration of polymer (Cm at the base and zero at the top) can be easily varied, and some results are shown in Table II. It can be concluded that the average distance between the bands is not affected significantly when the length of the inhomogeneous region is changed by a factor of 2 or 4. The bands in the tubes with a gradient length of 40 cm, however, had some bands disappear after about a week, so that the average spacing was significantly larger. All of the data in Table II were taken from tubes shortly after the bands were completely formed. Also, some interesting results were obtained when the gradient region was made as short as possible (~2 mm) by carefully layering water or a salt solution on top of a colloidal dispersion in 12% sucrose. In this case bands very close together (AYn ~ 2 mm) appeared to slowly "peel off" the lower solution and move slowly upward through the tube. At the other extreme, when no gradient at all was imposed and the particles (2r Journal of Colloid and Interface Science, Vol. 68, No, 1, January 1979
= 1.091/zm, ~b = 0.001) were suspended in a 12% sucrose solution so that they were at neutral bouyancy, a striking result was obtained. After about 50 days, four or five faint bands were seen to have formed which moved very slowly down the tube but no clear space formed at the top of the tube. Thus it appears that the bands can form even when no concentration gradient is initially imposed and even when the Stokes velocity is very low. The initial concentration gradient apparently serves primarily to shorten the time to first appearance of the bands, but the smaller concentration gradient due to spontaneous fluctuations (or possibly insufficient mixing) was sufficient to induce their formation. STABILITY OF BANDS AND CONVECTION
In view of past assertions that the bands are caused by weak convection currents, it is germane to describe some of our observations in this regard. In agreement with almost all previous observers, we found that it was essential to have the tubes in as constant a temperature environment as possible. If a strong light was directed at them or even if one stood next to them for more than a few minutes (especially for the tubes with 0.8-cm inside diameter), the interfaces between the bands blurred and eventually disappeared. The stability of the bands against slight disturbances in an experiment of this sort is illustrated in Fig. 8. The bands were produced in a suspension with ~b = 0.0012 of 2.02-/xm spheres after about 10 days, and the positions of the bands were noted a number of times during the day. We then stood next to them for about 10 min, causing them to disappear completely, and then the positions were noted again. As can be seen in the figure, the bands reappeared in their former positions. It is certainly true that convection cells can be made to appear in the tubes by imposing a lateral temperature gradient of sufficient magnitude. Experimental studies
LAYERED SEDIMENTATION
121
0.12%
2.02Fro
lO -~
9
.
~
~
y (cm)
-m--~ e
~ _ ~ , _ ~ .
o
,, __.__-.-,---
o
8 j.__A.
0 •
0
5 0
I
I
I
I
i
2
4
6
8
]0
t
(hr)
FIG. 8. The position of the interfaces, y, recorded during a single day for a suspension of 0.12 wt% 2.02-/zm particles. Between 7 and 8 hr the bands were twice destroyed by a small temperature gradient (created by standing near the tube). They were then observed to reappear (open circles) in the expected positions. on slots with transverse temperature gradients (which is easier to do experimentally but is still probably relevant to tubes) have shown that convection cells do not appear unless the Rayleigh n u m b e r is greater than about 3500 (19). For water at 25°C, the Rayleigh n u m b e r is given by Ra = 2 × 104. \-~m/ "\-gC-/ '
[2]
where l is the distance across which the temperature gradient AT is imposed. Under ordinary conditions the temperature difference between the sides of a tube was measured to be less than about 0.01°C, and since l is about 0.6 cm, Ra ~< 43, which is much too small. If the bands were caused by simple convection the particles serve merely as indicators of the layering of the fluid, and it should be possible to observe
the banding by using a dye as indicator in place of the particles. Despite many attempts (using fluoroscein dye and density gradients of various magnitudes) we have never succeeded in producing layers similar to those seen with the particles. The fact that bands form even when the suspension is stabilized against convection by imposing a density gradient in the fluid by the addition of a third c o m p o n e n t such as sucrose also constitutes strong evidence against a convective mechanism. Another argument against convection is given by the appearance of the bands. Convection cells which are made visible by the presence of particulates exhibit swirling lines of flow and appear grossly different from the bands discussed here (20). It seems unlikely that these bands are caused by thermal convection. Journal of Colloid and Interface Science, Vol, 68, No. 1, January 1979
122
DONALD B. SIANO POLYDISPERSITY AND AGGREGATION
10 cm. If some sort of separation process were occurring in the original solution, then particles would have to move during the period of the formation of the bands with a velocity of the order of 5 cm/day. This is larger than the Stokes velocity for these particles by about a factor of 14, and therefore, it appears unlikely that the particles differ only slightly in size and/or shape. Strong evidence against the polydispersity hypothesis was also obtained by observing the velocities of the interfaces which occurred in a system composed of equal weight fractions (0.05%) of 1.091- and 0.234-/xm particles. The average velocity of all the interfaces was found to be 0.17 cm/ day, which is not significantly different from the weight average velocity of the particles of 0.18 cm/day. The observed velocities of the interfaces (12 were observed with an average spacing of about 0.6 cm) were not correlated with their positions in the tube. That is, the interfaces deeper in the tube moved with about the same velocities as those nearer the top. There are several other features of the data which appear to be inconsistent with the polydispersity hypothesis. If some sort of aggregation process were responsible for layered sedimentation, then we would expect the layers to appear sooner in more concentrated suspensions. This is also contradicted by the observations. Aggregation is promoted by salt, but the bands are relatively insensitive to salt as long as the concentration of salt is below the flocculation value. If the original colloid is polydisperse, the existence of a lower and an upper limit on the concentration between which bands form is difficult to understand. The dramatic changes in the appearance of the bands when the gradient is made as short as possible and when sugar is added to solution A are likewise difficult to explain in terms of the polydispersity hypothesis.
It is perhaps natural to suppose that the origin of the bands in the sedimenting colloidal suspensions is due to the separation of classes of different sizes and/or shapes of the particles. To test this hypothesis, samples were withdrawn from various tubes containing layers with 1.091-/zm-diameter particles at different levels from different bands by pipet and examined under a light microscope. The individual particles could be easily seen, and the state of aggregation of the particles was readily ascertained. In every case that we examined, the singlets (single spherical particles) comprised about 95 to 98% of the sample, and this fraction did not vary significantly from one layer to the next, nor between different levels within a single band. Most of the remaining particles were seen to be two monomers that were touching, together with a smaller number of triplets, etc. Unless the hypothesized multiplets were very loosely adhering and were disrupted by the sampling procedure. This would appear to rule out the possibility that the bands consisted of different sizes of aggregates of singlets. There are additional lines of evidence against the polydispersity hypothesis that are more indirect. Consider Fig. 5 again, where as many as 14 bands were seen. It is unlikely that there are 14 distinct classes of particles in this tube. If they differed in size from singlets (presumably at the top) to quatredecamers in the bottom band, the velocities of the bands would not be expected to be so nearly equal as they are observed to be. On the other hand, if the bands consisted of mostly singlets but with slightly different average sizes and/or shapes, then we would certainly have found a very clever, extremely sensitive separation method because the polydispersity of the original colloid is known to be very small ! THE SEDIMENTATIONPOTENTIAL Another independent line of reasoning derives from the fact that the bands form comAs we have noted previously, the sedipletely in less than 50 hr in a length of about mentation potential has been invoked in the Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
LAYERED SEDIMENTATION past to account for the presence of layers in sedimenting suspensions. For a dilute homogeneous suspension of sedimenting particles with number density n, zeta potential ~ in a medium with viscosity ,/, electrical conductivity K, and dielectric constant E, an electric field with a magnitude of
123
mentation of a dilute homogeneous suspension of spheres gives the result that
vo/vs = 1 - 6.55~b.
[6]
If these results apply to the experiments in which an artificial gradient is initially imposed, it would appear that no mechanism is produced for the formation of b a n d s - the particles nearer the bottom would just e~raApgn move a few percent slower than those at the Es [3] 3~)K top, and so the latter would eventually catch up and the uppermost boundary would is produced which tends to retard the sedibe merely sharpened. Density currents (25) mentation (21). The observed velocity, Vo and settling convection (26) also appear to is related to the infinite dilution Stokes be ruled out as explanations of layered velocity, Vs, by the relation (22) sedimentation because of the near equality of the velocities of the interfaces with the ~cr \ 47rr ] " [4] limiting Stokes velocity. In addition, the denser fluid was below the lighter in all of Using the values for ¢, 0, and K for "con- our experiments as opposed to the requireductivity" water, r ~, Ap and n for our 0.1% ments for these phenomena to occur. There 1.101-/zm polystyrene solution and a zeta po- appears at present to be no hydrodynamic tential of 50 mV gives an estimate for E~ theory of layered sedimentation in dilute of about 1 mV/cm and a value for the right- suspensions of spheres. hand side of Eq. [4] of about 4 × 10-4. Thus the electric field is seen to be quite small THEORY (and much smaller when salt is added beThe experiments reported here appear to cause K increases), and the correction to lead to one conclusion--that layered sedithe sedimentation velocity is negligible. Bementation is due to the presence of an initial cause this is the case, it is difficult to see concentration gradient in the region in how the sedimentation potential can acwhich bands later appear. (Strictly speakcount for the rather dramatic layering effect. ing, of course, concentration gradients due to spontaneous fluctuations are always HYDRODYNAMIC EFFECTS present, but bands apparently due to this Measurements of the observed velocity of cause take much longer to form, as we have fall for particles larger than about 50 /zm seen.) Our experiments have shown that (where hydrodynamic effects would pre- layering or composition modulations always sumably dominate double-layer effects) occur in the region or interface between the have been made by a number of observers pure solvent at the top of the tube and the by following the fall of the relatively sharp most concentrated suspension at the bottop of the cloud of particles. Oliver (23) has tom. The case of 0.109-/~m-diameter parsummarized the experimental results, and ticles with volume concentration 0.001 is he finds that for dilute suspensions the data particularly instructive. Particles of this are fit satisfactorily by the expression size have a diffusion coefficient olD0 ~ 2.2 x 10-s cm 2 sec -1, so that in 1 day they Vo/Vs = (1 - 0.754)'/3)(1 - 2.15th). [5] would diffuse a distance of about (Dot) v~ Thus for our typical case with ~b = 0.001 0.04 cm, which is over 10 times as large we find Vo/Vs = 0.93. On the other hand, a as the Stokes law sedimentation distance. hydrodynamic calculation (24) of the sedi- We would therefore expect that these
(vs-
L(
t
Journal of Colloid and Interface Science, VoL 68, No. 1, January 1979
124
D O N A L D B. S I A N O
particles are "diffusion controlled" rather than "sedimentation controlled" so that ordinary Fickian diffusion would tend to make the solution more homogeneous with time. On the contrary, the suspension which initially has a ramp-like concentration profile develops a step-like concentration profile up to the region with constant concentration after a relatively short time. Since the sedimentation distance is small compared to the distance between bands during the initial time interval for the formation of the bands (-0.02 cm compared to 1.5 cm), conservation of particles implies that "uphill diffusion" must have occurred. Thus Fick's law does not appear to be applicable for suspensions of this concentration. However, an alternative diffusion equation has been derived for inhomogeneous systems which does predict many of the features exhibited by our data but has not hitherto been applied to the phenomenon of layered sedimentation. This is the CahnHilliard theory of spinodal decomposition, which takes into account the effect of inhomogeneities on the Helmholtz free energy and yields a modified diffusion equation. When the gradient is neither too small (when Fick's law is approximated) nor too large (when extra terms must be added), the diffusion equation is, neglecting gravity (27, 28),
Oc _ M 02f V2c - 2MKV4c, Ot Oc2
[7]
where f(c) is the free energy density for a uniform system of concentration c, M is a mobility, and K is the gradient energy coefficient which is positive and independent of c. For one dimension Eq. [7] has the solution
c(y,t) = Co + 8c exp[R(/3)t] cos (fly), [8] where
( Ozf + 2K/3~) R(/3) = - M / 3 2 0 c 2
[9]
and fl = 2~r/h is the wavenumber of the Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
sinusoidal composition deviation from co, the average composition. Thus for systems outside the spinodal, where 02f/Oc 2 > 0, and K and M are both greater than zero, R(/3) is negative, so composition modulations die out with a time constant of R - I On the other hand, when 02f/Oc 2 is smaller than zero (i.e., inside the spinodal), there is a critical wavenumber /3¢ for which composition modulations with /3
02f /K11'2 / 2
/3m= [ - - ~ c 2 /
j
/'
[10]
and the rate is ]- 0 2 f ] 2 /
[11] We emphasize that this diffusion equation is only an approximation which holds for the early stages of the development when the composition modulations are small. For example, when R > 0 it predicts that Ac increases without bound, which is not possible. The higher-order terms in the diffusion equation stop the growth and determine the "coarsening" behavior (29, 30). The qualitative aspects of this theory have been verified and successfully applied to an A I - Z n alloy (31), a liquid-liquid mixture (32), a polymer-solvent mixture (33), and a sol-gel transformation (34). In these systems the observed wavelength is found to be of the order of 50 to 1000 lattice spacings. The essential difference in our system is that instead of the two components having approximately the same size, the colloid and solvent particles are very different in size. If we take the lattice spacing to be the average distance between the colloid particles, then the expected wavelength for our typical case of 1.091/zm particles with an average distance apart of about 10 diameters amounts to about 0.05 to 1 cm. This agrees with our ob-
LAYERED SEDIMENTATION
served average band spacing Ayn, which we therefore idz,ntify with the wavelength h. Another order-of-magnitude agreement is obtained by noting that K has the dimensions of force and is the order (35) of k s T / d ~ 4 × 10-16 N for the case of D = 1.091/xm, ~b = 0.001. This is of about the same magnitude as the gravitational force and is therefore large enough to alter the behavior of the suspension significantly. Equation [10] predicts, since K is independent of the gradient, that (within limits) the wavelength is independent of the magnitude of the gradient. This is shown to be the case in Table I. Another prediction that can be made from Eq. [10] is that there is a law of corresponding states for /3, i.e., that/3 = fl(d/D, T/Tc), where D is the diameter of the particles and d is the distance between them. Since d/D ~ ¢b-1/3, this implies that fl is a universal function of ~b, independent of the size of the particle. The qualitative form for this universal function might be predicted on the basis of the well-known Flory-Huggins formulation of the thermodynamics of polymer solutions (36). The spinodal in this case on a T vs ~bphase diagram is concave downward and for high molecular weights is highly asymmetric--its maximum is close to the th = 0 axis and has a much higher slope on the low ~b side than on the high q5 side. Our experiments with varying q5 were made at constant temperature and therefore (Off'/ 0q52) is expected to mirror this behavior. Since K is independent of qS, Eq. [10] predicts that /3 vs ~b will also have a similar general shape. This is in qualitative agreement with the behavior illustrated in Fig. 7. Because of the theory's well-known quantitative limitations, we have not attempted a more detailed comparison with our data. We can note, however, that it is not altogether unreasonable that a spinodal is found in the concentration range of approximately ~b = 0.0002 to 0.004 for polystyrene suspensions. When the salt concentration is very low, suspensions of 0.170-/zm particles undergo transitions to a liquid crystalline
125
structure which have been the subject of many studies (37, 38). The approach taken here is to regard the initial homogeneous mixture of polystyrene particles and water as unstable with respect to a final state in which there is a polystyrene-rich phase at the bottom of the tube and a polystyrene-poor phase above this. This initial state then evolves into the final state via the spinodal mechanism with its well-known attendant band structure. This view is somewhat unorthodox because it assumes that the sedimentation of particles with diameters of a micron (or less) constitute a phase separation so that there is a possibility that the spinodal mechanism can be brought into play. An essential part of the argument is that the final state consists of two well-defined phases. It is certainly true that the two regions (supernatant and sediment) are at least nearly homogeneous with different physical properties, and are separated by a relatively sharp interface. Furthermore, there is no prima facie upper limit for the molecular weight above which a phase separation is not allowed to occur. Without direct observation of a phase transition, we can only offer this explanation as a conjecture--a conjecture that seems to be consistent with the observations made. The fact that the bands are observed to appear much sooner for suspensions which have an initial concentration gradient imposed (5-24 hr) than for similar trials in which the composition is initially uniform (5-10 days) is also neatly explained by this hypothesis. According to this viewpoint, in the latter experiments gravity causes the particles to sediment, which then produces after a certain time a clear space at the top of the tube and then a region of increasing concentration of particles farther down. Thus a concentration gradient or inhomogeneity is produced and the spinodal mechanism generates the observed composition modulations. In the experiments with the artificially imposed concentration gradients, the spinodal mechanism (i.e., the diffusion Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
126
DONALD B. SIANO
equation) can start at once to generate the bands without the preliminary sedimentation step. This hypothesis also clearly explains the crucial observation that bands were produced with the 0.109-/zm particles (which do not sediment at all) when an artificial gradient was imposed but not in its absence. The diffusion equation requires a concentration gradient and the sedimentation is only incidental. In the case of the experiment with 1.091-tzm particles made neutrally bouyant in a 10% sucrose solution, the initial concentration gradient evidently was produced by spontaneous fluctuations in number density, but since the initial composition modulation was small, a long time was required before it grew to observable size. The initial concentration profile often appears to be sinusoidal in shape, but then later becomes more steplike. This, again, is in accord with the predictions of the Cahn-Hilliard t h e o r y - - t h e squaring of the profile is attributed to the higher-order terms in the diffusion equation, which become more important in the later stages of "coarsening." The different qualitative behavior seen when the length of the inhomogeneous region is made shorter than the wavelength seen when the concentration gradient is less steep is also to be expected because these same higher-order terms should not then be neglected in the diffusion equation and a different solution from Eq. [8] is required. SUMMARY Layered sedimentation of colloidal suspensions appears to be caused not by sedimentation but instead by the concentration gradient generated by settling. The evidence is: (1) Bands can be produced by a concentration gradient of particles so small that they do not perceptibly settle, and (2) the layers are produced much more quickly when the suspension starts with an imposed concentration gradient than when the suspension starts in a homogeneous Journal of Colloid and Interface Science, Vol. 68, No. 1, January 1979
state. The bands are produced even in relatively dilute (volume concentration 0.0002) suspensions in both high (15 mM) and low salt concentrations, and a wide range of sizes (at least 0.1 to 2/~m in diameter) of monodisperse particles. When produced by an initial concentration gradient the average distance between the interfaces increases very slowly with time and the interfaces move downward with a velocity nearly equal to the Stokes velocity. The average spacing between the bands is a function of the volume concentration and is nearly the same for particles of different sizes but constant ~b. The experiments reported here are inconsistent with the hypothesis that the layers are caused by a sorting out process due to polydispersity in size, shape, or charge or due to an ongoing aggregation of the particles. Direct sampling of the layers, parallel movement of the interfaces, their rapidity of formation, and sensitivity to properties of the imposed gradient all militate against the polydispersity-aggregation hypothesis. Similarly, thermal convection and the sedimentation potential seem to be ruled out as successful predictive hypotheses. Many features of the data do appear, however, to be consistent with the CahnHilliard theory of spinodal decomposition. According to this viewpoint, the bands transiently appear in the very diffuse interface between the two separating phases (colloid-rich and colloid-poor) as the system moves toward equilibrium. The dependence of the layers on the imposed gradient, and the average spacing as a function of volume fraction, size, and salt appear to be consistent with this hypothesis. This is not considered to be conclusive evidence, however, and additional evidence is being sought. ACKNOWLEDGMENT I acknowledgestimulatingdiscussions and helpful advise from Dr. B. Berne and Dr. G. Flynn in whose laboratory this work was carried out. The technical assistance of Mr. M. Tederons is greatly appreciated.
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Journal of Colloidand Interface Science, Vol.68, No. 1, January1979