1577
J. Phys. Chem. 1995,99, 1577-1581
Calculation of Equilibrium Sedimentation Profiles of Screened Charged Colloids Jean-Pierre Simonin Laboratoire des Propriktks Physico-Chimiques des Electrolytes (URA CNRS 430), UniversitC Paris VI, Bat. F, Boite no. 51, 8 rue Cuvier, 75005 Paris, France Received: August 4, 1994@
A method for the calculation of equilibrium sedimentation profiles of charged colloids, in the presence of added salt, is presented. It is found that the concentration of the colloidal particles obeys a nonlinear equation, which can be solved by a numerical method. The assumptions used are as follows: each species is subjected to balanced forces; the electrical neutrality condition is verified locally; the colloid particles have a constant charge; they are spherical and their size is much smaller than the gravitational length; the main departure from ideality arises from the hard-sphere contribution, and it may be derived from the Carnahan-Starling equation of state. The results are compared with recent experimental data. It is found that the agreement is excellent in the case of high screening (excess of salt). When the salt concentration is decreased, large discrepancies are observed. The salt-free case is also treated. In each case, tentative interpretations are examined.
Introduction The sedimentation of charged colloidal suspensions has been the subject of interesting recent experimental studies. Sedimentation velocities, showing the importance of the electrical double layer, have been measured.' Equilibrium profiles have been obtained,2 and the hard-sphere equation of state has been shown to describe perfectly the experimental data for strongly screened colloids. However, it was found in this latter work that the data were well fitted by taking an effective mass which was smaller than the expected mass of the particles, as calculated from the known particle parameters. Decreasing the relative amount of added salt led to a mass reduction which could reach ca. 40%. The authors have interpreted this phenomenon by supposing that there was an additional force acting on the particles, likely arising from a residual electric repulsion between the colloid spheres. Theoretical studies have also been reported, dealing for example with the sedimentation equilibrium of a mixture of hard spheres3 or the derivation of the osmotic equation of state of colloidal suspensions? A theoretical treatment of the calculation of density profiles of colloidal particles in sedimentation equilibrium has been developed on the basis of the minimization of the free energy functional of the system. This method has been applied to the case of colloidal particles of size comparable to their gravitational length (1 = kBT/mg,with k~ the Boltzmann constant and T the temperature). The cases of uncharged5 and charged6 colloids have been approached by these authors. In the present work, we give a theoretical treatment, suitable for the case of charged spherical colloidal particles, of size much smaller than the gravitational length. The treatment is applied to the experimental data reported in ref 2. Theoretical Section General Case. We consider here the case where salt, e.g. NaC1, is added to a colloidal suspension, composed of charged colloids and their small counterions. We assume that the size of the colloid spheres is much smaller than the gravitational length (characteristic scale of sedimenta@
Abstract published in Advance ACS Abstracts, November 1, 1994.
0022-365419512099-1577$09.00/0
tion). This assumption allows a macroscopic treatment involving the local concentrations of each species. The electrochemical potential of species i reads
with yi the activity coefficient, Ci the local concentration, Zie the charge, and W the electric sedimentation p ~ t e n t i a l . ~ At equilibrium, we have that
(aJ&
+ mUpg = 0
with 3, the partial derivative with respect to x , the x axis being directed upward, mp the mass of a particle i , and g the gravity. Substituting pi from eq 1 into this equation yields
kBTa, ln(yiCi) - ZieE
+ m,g = 0
(3)
where E is the local electric field and mi the buoyant mass. It is assumed now6 that the departures from ideality arise mainly from the hard-sphere repulsion between the colloid spheres, as calculated from the Camahan-Starling equation of state.* It is found that the activity coefficient of the colloid spheres reads
In y = q(8 - 9q
+ 3q2)/(1 - 1;1)3
(4)
with 3 their local packing fraction. Moreover, the sedimentation length of the small ions is large: for sodium ions for instance, 1 1.1 x lo4 m. Thus, eq 3 can be written for the three unknown functions: C(x) the colloid concentration, C+(x)the concentration of the small positive ions, and C-(x) that of the small negative ions. It is assumed also that the small ions (the counterions and the ions of the salt added) have a charge +e in absolute value. We get then
-
1 eElkBT = 3, In C, = -3, In C- = - -3, ln(yCe4") ( 5 ) Z where q = mg/kBT = 1-' (m is the buoyant mass of the colloid particles) and where the colloid spheres are supposed to bear a negative charge -Z. In eqs 5 it is also assumed that m >> Zmi, with m+ and m- the masses of the small ions. Using eqs 5, we find that 0 1995 American Chemical Society
1578 J. Phys. Chem., Vol. 99, No. 5, 1995
Simonin
and
x+
x-
where and are two constants ensuring the conservation of the small ions
h, thus yielding a and p as functions of yo and Y h . Then, a and being known, the whole profile can be computed numerically from eq 17: at each point x, the value of y was calculated by a Newton-Raphson method. Let LIS mention that the activity coefficient y , as calculated by the Camahan-Starling formula, is a function of r, the packing fraction of the colloid spheres. Also, 17 is proportional to the local concentration given by eq 11. Besides, using eq 11, the conservation of the colloid particles yields
C/C,
and (9) with x = 0 the bottom of the container and h its height, Cs the mean salt concentration, and CN the mean concentration of the positive ions. As a closure relation it will be assumed that the local electrical neutrality condition9-" prevails in the bulk of the solution, that is
= (r(x)e-qx)
(20)
c
with the mean concentration of the colloid particles. The conservation of the salt ions is expressed by eq 15. The conservation of the small positive ions (eq 16) results from eqs 10, 15, and 17. So, the two constraints expressed by eqs 15 and 20 can be written as bs(YCl?Yh)= a ((rY>'lZ)- a =
(21)
and This equation expresses the strong electric attraction between positive and negative species. It does not contradict eq 3, in which the electric field E appears, for this field originates from a gravity-induced charge separation existing at a microscopic scale. When averaged on a larger scale, it may be assumed that the charge density nearly vanishes, yielding eq 10. Combining eqs 6, 7, 8, 9, and 10 and setting C(x) = Ci&> Y ( X )
(11)
b(ro,yh)= (re-"> - c = 0
(22)
with c = CIC~. Solving the basic equation (13) is therefore equivalent to finding the two unknowns yo and Y h which are the roots of eqs 21 and 22. It is worth noting that the repulsive interactions between the colloid particles entail Y O= c'oe-q'xy'(x)
with q' = q/(Z we obtain
+ 1).
-4
(37)
Results The present treatment has been applied to the experiments described in ref 2. In this work, latex spheres of PTFE copolymer were used. Their radius was R = 73 nm and their gravitational length I 0.0235 cm, leading to a ratio 2W1- 6 kg at x lV4. From the value of 1 we find m = 1.784 x 25 "C. Titration experiments showed that the number of ionizable sites was certainly about 1300. From these data, we find also C 10-6q (with C in mom). In the present work, we have studied four among the seven experiments performed on these colloid particles. The experimental conditions are summarized in Table 1. They have been chosen so as to scan the whole salt concentration range. Experiments A and B were performed in conditions of relatively high screening, at the same relative amount of salt but with different values of the container's height h. The relative salt concentration is smaller in experiment C, by a factor of about 11. The suspension used in case E was free of salt. Figures 1-4 illustrate the results obtained from our treatment for experiments A, B, C, and E. On each graph, the decimal logarithm of the packing fraction is plotted against the position x in the cell (x = 0 refers to the bottom of the sedimentation cell). In each figure, the results for the uncharged hard-sphere fluid and for several charges Z are compared to the experimental data. The agreement found in case A between the hard-sphere result and the experimental points is surprisingly good (Figure 1). It extends from the bottom of the cell, where the packing fraction
-
.
':..
t
(36)
with A' a constant. The algorithm in this case is similar to the one used in the former case. We start from an estimate of y'o. From this value, the value of y and I' are computed at the origin x = 0, and the whole profile y'(x) is determined by solving numerically eq 36. So, the conservation of the colloid can be verified. An equation analogous to eq 22 can be written, and it can be considered as a function of y'o. The problem is to find the root of this equation. This was done also by a Newton-Raphson method. Defining the apparent mass and the mass deviation in the same way as in eqs 25 and 26, we find that
4x1 = [Z + e(x>l/[l + z + &)I
...\..:.... ...:. ,
h
(35)
Substituting this expression into eq 34, YYtZfl = 1'
l
0
0.05
0.1
0.15
0.25
0.2
0.3
x (cm)
Figure 1. Experiment A. Decimal logarithm of the packing fraction of the colloid as a function of the position x in the container: squares, experimental data; continuous curve, theoretical result for the fluid of hard spheres; dashed curve, result for Z = 400;dotted curve, result for Z = 700.
; It
G
-3
rl
-4
-5 -6 I
0
I
,
,
I
0.05
0.1
0.15
0.2
0.25
I
I
0.3
0.35
I 0.4
0.45
x Icm)
Figure 2. Experiment B. Same as in Figure 1.
-
reaches 0.2, to the region where In this case the result is slightly sensitive to the charge Z chosen: the agreement is still very good for Z = 400. In experiment B the cell was 4 times higher than in A. The effective experimental mass of the particles, found in this case, was reported to be about 14% lower than the expected value. Here, while the theoretical curves (see Figure 2), obtained for a charge ranging from 0 to 700, are in agreement with the experimental data up to x = 0.15 cm, the latter cannot be described with the use of a single charge at higher altitudes. The theoretical results exhibit a barometric region for x > 0.25 cm, and it is found from the use of eq 28 that the slope yields exactly the expected buoyant mass of the particles. No deviation of the effective mass is obtained for any charge. In experiment C, the screening had been much reduced and the effective mass of the particles was found lowered by about 22%. Here too, the apparent mass (defined by eq 25) coincides with the expected value in the barometric region, for Z ranging
1580 J. Phys. Chem., Vol. 99, No. 5, 1995
Simonin
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joining the experimental points for x > 0.7 cm. The theoretical curve having the same slope is obtained for Z 0.5 (curve 1). However, the packing fraction found with the use of this charge is more than 2 orders of magnitude smaller than the observed value. The strong influence of the charge of the colloid is shown by plotting the results for Z = 2, 3, and 15. This latter value yields a satisfying agreement with the data for x -= 0.6 cm.
-5 -4 -6
1‘
0
Discussion
0.1
0.2
I
I
,
I
I
0.3
0.4
0.5
0.6
0.7
x
(cm)
Figure 3. Experiment C. Same as in Figures 1 and 2. Full line e: experimental barometric line.
0
0.2
0.4
0.6
0.8
1
1.2
x (cml
Figure 4. Experiment E. Decimal logarithm of the packing fraction of the colloid as a function of the position x in the container: squares, experimental data; full line e, experimental barometric line; continuous curve, result for the unchanged hard-sphere fluid; curve 1, result for Z = 0.5; curve 2, Z = 2; curve 3, Z = 3; curve 4,Z = 15.
from 0 to 700. These findings are shown in Figure 3: the slope of the experimental data in the barometric region is lower than that of the theoretical curves. When going from experiment A to C, it is seen that the theoretical results are more and more sensitive to the effect of the particle charge. At the same time, the effective mass, as deduced from the experiment, is found to decrease. Also, the concentration profile of the small negative ions has been studied, using eqs 6 and 15. It is found to be quite flat in all experiments; the maximum relative concentration difference, between the bottom and the top of the container, is less than 1%. The absence of segregation is not a surprising result: for instance, the maximum volume fraction at x = 0, found in experiment C, is ca. 0.5 and corresponds to a concentration (using C = 10-6r)of about 5 x lo-’ m o m ; thus, the quantity ZC is always much smaller than the mean salt concentration, and consequently, as shown by eq 10, the colloid concentration is not sufficient to alter appreciably the homogeneity of the salt profile. As regards experiment E, performed in the absence of salt, the mass reduction was about 37%. Now, it can be seen from eq 37 that, in the region of high dilution, the apparent mass should be mapp= m/(Z l), if Z is not too small. Thus, the charge to be used in the calculation, which may yield the observed barometric slope, should be close to 0.6. In Figure 4 the experimental data are compared with the theoretical results for various charges. The barometric region is shown by a line
+
The excellent agreement obtained in the case of high screening (experiment A) between theory and experiment shows that the repulsion between the colloid spheres is well described by the hard-sphere volume exclusion, at the Camahan-Starling level. A similar result was obtained in ref 2, where the Camahan-Starling equation of state was verified in an experiment with particles of different physical characteristics (made from the same material, but of larger size and mass). However, a 30% lower effective mass was required, not only in the low volume fraction region but also in the intermediate and dense regions. In contrast, taking the buoyant mass of the particles as an input in our calculations, for experiment A, yields a satisfying result. At this stage of the present study, only tentative explanations can be put forward to account for the discrepancies observed between the theoretical results and the experimental data of experiments B, C, and E. Here, three possible ones are presented. First, in the absence of a reliable expression, the electrostatic short-range repulsion between charged colloid particles has not been included in the present treatment, in the departure from ideality expressed by eq 4. Clearly, including this effect would lead to a lower (higher) theoretical packing fraction in the bottom (top) of the container. However, in the barometric region, the experimental colloid concentration is low: for instance, in experiment C (Figure 3), the two points at x 2 0.6 cm correspond to a concentration less than 3 x mom; this result leads to a mean interparticle distance greater than ca. 2 pm, while the Debye screening lengthI3 (the inverse of the Debye screening parameter K ) , arising from the salt at the concentration Cs = 0.02 m o m , is ca. 20 A. As a consequence, it may be expected that including an electrostatic contribution in eq 4 would yield an unchanged barometric slope in comparison with the present “restricted” treatment. Nevertheless, further theoretical work is needed to confirm this conclusion. Next, the fact that the charge of the colloid is assumed constant may be too restrictive. The latex particles used in the experiments bear a negative surface charge, due to the adsorbed anion surfactant and to the end groups of the polymer chains.l4 Owing to counterion condensation effects, the effective charge of the colloidal particles is likely to be different from the charge of the bare particles, and it may be dependent on the local composition. Now, when going from experiment A to B, and then to C, that is to say when the screening is decreased, the experimental points are more and more moved “upward” with increasing x. Qualitatively, everything happens as if the effective charge of the colloids increased with increasing altitude. Moreover, the charge seems to increase more rapidly when the relative mean salt concentration decreases. Thus, it may be thought that salt-dependentcounterion condensation may be involved. However, this hypothesis may be discarded because, as shown in the Results section, the equivalent concentration of the colloid is always much smaller than the salt concentration everywhere. For instance, in experiment C, at the point x = 0.5 cm, the ratio of the equivalent colloid
Sedimentation Profiles of Screened Charged Colloids concentration to the salt concentration is at a maximum (taking Z = 1000) 5 x it is about 2 x at x = 0.6 cm. It is worth noticing that experiment E exhibits an opposite behavior. There, looking at the local slope of the experimental curve, everythmg happens as if the effective charge of the colloid decreased with x , with a tumover in the vicinity of x = 0.7 cm. Below this point the best agreement is obtained for Z 15 and Z 0.5 above. These charges are surprisingly low. Although reduced surface charges in deionized suspensions have been reported,' the interpretation of the present results in terms of position-dependent colloid charge is not clear. The last possible explanation is connected with the time relaxation of the colloid profile. Starting from an initial homogeneous suspension, one may wonder how long one would have to wait before equilibrium is reached. In the experiments of ref 2 the time evolution of the profile was followed, and since, after 100 days, no appreciable changes were observed, it was supposed that the system was equilibrated. As a preliminary study, we have considered the ideal system, without departures from ideality. The characteristic relaxation times of the system can be calculated using the normal-mode technique, extensively used in our group.10-12 This treatment is based on the near-equilibrium linearization of the continuity equations, written for each species, with the fluxes expressed by
-
-
Ji
= -pDici(a#i
+ mPg)
pi being given by eq 1. The diffusion coefficient of the colloid
spheres was estimated from the Stokes' law. The result is a set of three relaxation modes. The inverse of the f i s t one is the Debye time, of the order of 1 ns, related to the local very fast relaxation toward electroneutrality. The other two modes describe the return of the system to equilibrium, which occurs according to a migration process. Calling U Iand uz these two migration velocities, the characteristic times read zi = Uui, with L a characteristic length. Starting from an homogeneous suspension, L must be of the order of the height of the container, h. The result of this simplified treatment is that, in all cases and for any charge Z, the modes are nearly decoupled: the migration velocities correspond respectively to the relaxation of the salt alone on the one hand and of the uncharged colloid on the other. We find uc 1.5 x cm s-l for the colloid and U S 2 x cm s-' for the salt, and then zc and zs are always respectively of the order of lo6 and lo9 s, that is, ca. 10 and lo4 days. Keeping in mind that the
-
-
J. Phys. Chem., Vol. 99, No. 5, 1995 1581 relaxation process of each species is a combination of both relaxation modes, it is found that the relaxation of the present system may be govemed by the relaxation process of the salt. Obviously, this behavior would be owed to the low mass of the salt ions. Though the small ions hardly sediment, the motion of the colloid particles is even so coupled to their motion through the electrical neutrality condition. In conditions of moderate screening, the time to be elapsed before equilibration may be much longer than 100 days, as taken in the experiments of ref 2. From this point of view, a very slow sedimentation process can be consistent with the experiment: the experimental data of Figure 3 for instance (experiment C) are above the theoretical result for a "low-charged" colloid in the top part of the container, which may mean that the colloid has not totally sedimented. However, this conclusion requires further work, including the departures from ideality and a rigorous calculation of the relaxation modes and transient profiles. At the present time, we are working on this subject. The results will be presented in a subsequent paper. Acknowledgment. The author gratefully thanks J. P. Hansen for drawing his attention to this subject, T. Biben for interesting comments, and P. Turq for fruitful discussions and suggestions. References and Notes (1) Okubo, T. J. Phys. Chem. 1994, 98, 1472-74. (2) Piazza, R.; Bellini, T.; Degiorgio, V. Phys. Rev. Left. 1993, 71, 4267-70. (3) Vrij, A. J. Chem. Phys. 1980, 72, 3735-39. (4) Barrat, J. L.; Biben, T.; Hansen, J. P. J. Phys.: Condens. Matter 1992, 4, Lll-L14. (5) Biben, T.; Hansen, J. P.; Barrat, J. L. J . Chem. Phys. 1993, 98, 7330-44. (6) Biben, T.; Hansen, J. P. J . Phys.: Condens. Matter 1994,6 (Suppl. 23A), A345-A349. (7) Levine, S.; Neale, G.; Epstein, N. J . Colloid Inferface Sci. 1976, 57, 424-437. (8) Camahan, N. F.; Starling, K. E. J . Chem. Phys. 1969, 51, 635. (9) Robinson, R. A,; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Buttenvorths: London, 1959. (10) Mills, R.; Perera, A,; Simonin, J. P.; Orcil, L.; Turq, P. J. Phys. Chem. 1985, 89, 2722-25. (1 1) Simonin, J. P.; Gaillard, J. F.; Turq, P.; Soualhia, E. J. Phys. Chem. 1988, 92, 1696-1700. (12) Tura P.; Orcil, L.; Chemla, M.; Mills, R. J . Phvs. Chem. 1982. 86, 4062. Simonin, J. P.; Turq, P.; Soualhia, E.; Michard, G.; Gaillard, J. F. Chem. Geol. 1989, 78, 343. (13) Lowen, H.; Hansen, J. P.; Madden, P. A. J. Chem. Phys. 1993,98, 3275-89. (14) Degiorgio, V.; Piazza, R.; Bellini, T. Adv. Colloid Interface Sci., in press. JP942039U
