DIVISION S-9—SOIL MINERALOGY

Particle Shape and Sodium Self-diffusion Coefficient in Mixed Sodium-calcium Montmorillonite1.

JOSEPH E. DuFEY2, AMOS BANIN," HENRI G. LAUDELOUT,2 AND YONA CHEN3

ABSTRACT

Various parameters related to the shape of clay particles and self-diffusion coefficients of sodium were measured in Camp-Berteaumontmorillonite suspensions equilibrated with NaCl-CaCl2 solutionsof 0.01 total normality, and variable ratios of Na to Ca. As the equiva-lent ionic fraction of exchangeable sodium decreased from 1.0 to 0.0,the anion exclusion volume decreased from 2.9 to 0.5 ml g ', thehydration of clay sediment decreased from 8.6 to 4.4 ml g~', theaverage number of plates per tactoid increased from 1.4 to 3.9, andthe axial ratio of the clay particles decreased from 190 to 90. At claycontent of about 27 g/liter, the self-diffusion coefficient of Na increasedfrom about 0.40 x 10~5 to 0.76 x 10~5 cm2 sec~' at 16°C and from 0.55x Ifl-5 to 1.16 x 10~5 cm2 sec"1 at 33°C, as the ionic fraction ofexchangeable Na decreased from 1.0 to 0.15.

The calculated surface diffusion coefficient of sodium was found tobe about 30% of its value in the free solution for a pure sodium clayand thereafter increased up to about 50% of that value for a clayloaded by 15% Na. The activation energy involved in the surface dif-fusion process was calculated to be in the range of 3 to 6 kcal/mole, anda minimum was found for the clay containing about 70% Na. As evalu-ated from the data on the number of platelets per tactoid the internalsurfaces of which were assumed to be preferentially occupied by Ca,this minimum could perhaps correspond to a completely demixed sys-tem. A less refined but nevertheless rather satisfactory approach wasalso attempted for interpreting the variation of the observed self-dif-fusion coefficient of Na as a simple result of the change in the distribu-tion of that ion between the surface phase and the free solution.

Additional Index Words: tactoid formation, clay minerals, exclusionvolume.

THE SELF-DIFFUSION of cations in clay gels has beenrather extensively studied, especially in systems with

high clay contents (Lai and Mortland, 1961; Low, 1968;Mokady and Low, 1966; Calvet, 1967). In such concen-trated clay systems, cation mobility is reduced due to thecombined effects of two factors: electrical forces operatingnear the surfaces where almost all the cations reside and thetortuosity of the diffusion path, which is mainly dependenton the mutual arrangement of the clay particles. The cationdiffusion in dilute clay gels and suspensions received lessattention, and almost all studies were performed on mono-ionic clays (Cremers and Thomas, 1966). Beyond the twofactors mentioned above, the relative distribution of the cat-ions between the surface phase and the free solution has to

'Contribution to a joint project of the Soil Science Dep. of the HebrewUniv. of Jerusalem and of the Univ. of Louvain. Received 7 July 1975.Approved 22 Dec. 1975.

^Research-fellow (Aspirant) of the "Fonds National de la RechercheScientifique" and Professor, respectively; Univ. of Louvain, Place Croixdu Sud, 2; B-1348 Louvain-la-Neuve, Belgium.

3Associate Professor and Research-assistant, respectively; HebrewUniv. of Jerusalem, PO Box 12, Rehovot, Israel.

4The Na-Ca exchange isotherms determined for this work at 16°C and33°C were found to fall very accurately on the isotherms previouslyreported by Van Bladel, et al. (1972.)

be taken into account in dilute gels. Moreover, the obstruc-tion of the clay matrix to the free movement of ions is morerelated to the actual size and shape of the clay particles. Thepurpose of this work was to examine how data on the shapeof the clay particles, on the anion exclusion volume, and onthe Na-Ca exchange isotherms can be combined to give aninterpretation of the self-diffusion coefficients of Na inmixed Na-Ca clay gels.

MATERIALS AND METHODSPreparation of Monoionic Clays and Determination of Ex-

change Isotherms—The clay used was the Camp-Berteau(Morocco) Montmorillonite. The 0.35 /am (equivalent sphericaldiameter) fraction was separated from the crude clay by repeatedcentrifugation after treatment with \N NaCl and washing with dis-tilled water. For further light scattering measurements a sample ofthat clay was converted to the Li form by repeated centrifugalleachings with \N LiCl followed by removal of excess electrolytewith distilled water.

Exchange isotherms between sodium and calcium ions were de-termined at 16°C and 33°C by adding a Na-clay suspension pre-viously equilibrated by batch dialysis with a O.OW NaCl solutionand containing about 10 g dry clay/liter, to a solution of NaCl andCaCl2 of 0.01 total normality. This electrolyte solution was pre-viously labeled with either 22Na or 45Ca. The resulting mixture wasshaken during 4 days and then centrifuged. The distribution of Naand Ca ions between the supernatant and the colloid phase wascalculated by measuring the specific radioactivity of the superna-tant and of the initial solution.

Preparation of Sodium-calcium Clay Suspensions—Na-claysuspensions were first dialysed for one week against mixed NaCl-CaCl2 solutions of O.OW total concentration. The relative com-position of the solutions, calculated from the exchange isotherms,4was such that after equilibration the desired exchangeable Ca per-centage in the surface phase was obtained. In order to ensure thatthe desired equilibrium point had actually and accurately beenreached, 10 successive dialysis periods of half a day were thenmade with the corresponding equilibrium solutions. After collect-ing the suspensions, the dry clay contents were measured byevaporating at 105°C.

Exclusion Volume and Hydration of Clay Sediments—The anionexclusion volume was measured as follows: 10 ml of suspensioncontaining about 8 g of clay/liter were shaken in polyethylenetubes with 2 ml of the corresponding last dialyzate labeled with36C1. After reaching isotope equilibrium (2 days) and centrifuging,the anion exclusion volume was calculated from the specific radio-activity of the supernatant and of the initial solution measured witha Packard 3380 Tricarb liquid scintillation spectrometer.

The solution retained in the clay sediment was measured by cen-trifuging about 5 ml of the same suspensions as above at 40,000 gfor 20 min. The clay-free supernatants were decanted and the tubeswere then lightly centrifuged (about 30 sec, 80 g) in a reversedposition in order to collect the remaining free solutions at the upperpart of the tubes. After removing the drops of solution and dryingwith paper, the sediments were weighed.

Light Scattering—Suspensions containing about 250 mgclay/liter were used for light scattering determinations. The coeffi-cient of absorbance of these suspensions was measured at thewavelength of 700 nm using a Gary-15 spectrophotometer. Theabsorbance of Li-clay at the same concentration was also mea-

310

DUFEY ET AL.: PARTICLE SHAPE AND SODIUM SELF-DIFFUSION COEFFICIENT IN Na-Ca MONTMORILLONITE 311

sured. According to the relationship developed by Banin andLahav (1968a and 1968b) the ratio between the coefficient of ab-sorbance of Na-Ca clay suspension and the coefficient of absorb-ance of Li-clay suspension allowed a calculation of the number ofplatelets per tactoid relative to the Li-clay which is usually consid-ered as existing in single platelets form.

Viscosity—The viscosity of the suspensions was determinedusing an Ubbelohde type capillary viscometer, automaticallyoperated by a Viscometer Reader (Rehovot Instruments) whichallows a precision of ± 0.1 sec in the readings of flow times. Thecapillary and reservoir volumes in the viscometer were such thatflow times were in the range of 100 to 150 sec. After at least threerepetitive readings were taken of the original clay concentration(3-5g/liter) the suspension was diluted, inside the viscometer,with the equilibrium electrolyte solution and again a sequence ofreadings of the flow times was performed. This procedure of dilu-tion and readings was repeated four or five times. The viscosityrelative to solvent, T)rel, was calculated for each dilution. Specificviscosity (T)sp = 7)rd- 1) was then plotted for each Na-Ca suspen-sion, versus the volume fraction p of clay. A straight line wasusually obtained, its slope equal to K in the extended Einsteinequation,

= Kp. [1]

The factor K was then interpreted following Kahn (1959) to ob-tain the average axial ratio of the clay particles from the rela-tionship

a/b=\.1K. [2]

Self-diffusion—Na-Ca clay suspensions containing about 27 g

Fig. 1—Effect of the surface cation composition on the anion-exclusionvolume, on the volume of solution retained in the clay sediment, onthe average axial ratio of the clay particles and on the tactoid sizeexpressed by the number of single platelets per particle (Li-claytaken as 1), from top to bottom, respectively.

dry clay/liter were labeled with 22Na by placing them in dry testtubes in which an appropriate amount of carrier-free 22NaCl hadbeen evaporated. The method developed by Thomas (1956) andAlien et al. (1963) was used for measuring the self-diffusion coef-ficient of sodium ions in the suspension. It involves measuring therate at which y radioactivity is removed from the suspension whena nonlabeled equilibrium solution is circulated against a membranewith which the suspension is in contact. The cell containing thelabeled clay system was placed in the well of a solid scintillationcounter which allowed a measure of the decrease of radioactivitywith time. The circulating solution as well as the diffusion cell andthe y-ray detector were maintained at the desired constant temper-ature. All measurements were made at least in triplicate.

RESULTS AND DISCUSSION

Hydration and Shape of Clay Particles

Figure 1 shows the values obtained with respect to anionexclusion volume, hydration of sediment, average axialratio, and size of the clay particles as expressed by thenumber of single platelets per tactoid, at various Na+/Ca2+

fractions on the clay surface. It is obvious from the datapresented in this figure that different properties of Na-Camontmorillonite clays showed a rather abrupt change atequivalent fractions of adsorbed Na, NNa, ranging from 0.0to about 0.4. Similar variation has been reported before(Banin, 1968; Lahav and Banin, 1968) and was also foundto affect other properties of the clay gel, such as yield val-ues and rheological properties (Williams, et al., 1953), ionexchange selectivity (Banin, 1968) and hydraulic conduc-tivity (Shainberg and Caiserman, 1971).

Anion Exclusion Volume and Hydration of Clay Sedi-ment—The values found for the exclusion volumes of Na-Ca-clays are quite consistent with those that can be calcu-lated using depth of exclusion data given by Schofield(1974), taking into account the number of platelets per tac-toid and assuming a 10A distance between platelets insidetactoids. Based on these assumptions, values of 3.37 and0.88 ml g"1 have been calculated for the exclusion volumeof the pure Na- and Ca-clays, respectively. These are closeto the experimental values of 2.90 and 0.50 ml g"1, respec-tively.

As expected, the volume of solutions retained in the sedi-ment is related to the exclusion volume; it decreases with in-creasing calcium loading due to the contraction of doublelayers on the one hand, and to the increase of tactoid size onthe other hand.

Tactoid Thickness—The number of single platelets pertactoid was found to be about 1.4 for the pure Na-clay andabout 3.9 for the pure Ca-clay. This last value is muchsmaller than the value of about 10 found by Banin andLahav (1968b) for the Wyoming montmorillonite that theystudied. The difference probably is due mainly to the dif-ferent methods used for preparing the clays. A dialysis tech-nique was used here, whereas Banin and Lahav used re-peated centrifugations, which lead to the formation of largertactoids in Ca-montmorillonite. In addition, the type ofclay, especially as related to its charge density, may affectthe prevailing tactoid size. Since the surface charge densityof the Wyoming montmorillonite is about 20% smaller thanthat of the Camp-Berteau montmorillonite (exchange ca-pacities of 82 to 90 and 102 to 110 meq/100 g were mea-

312 SOIL SCI. SOC. AM. J., VOL. 40, 1976

Table 1—Observed self-diffusion coefficient of Na+, D0Na, in mixed

Na-Ca clay suspensions as function of the equivalent ionicfractions of Na+ in the 0.01 N bulk solution, NNa,

and the surface phase, NNa.

0.5

Temperature

°c16

33

NNa

1.00.950.850.750.650.550.450.350.250.151.00.950.850.750.650.550.450.350.250.162

NNa

1.00.999630.998630.997250.995100.991850.98650.9790.9610.9001.00.999630.998670.997350.99510.99170.98650.9760.9540.898

Clay content

g/ioo g,3.832.712.752.642.562.802.682.762.912.722.912.712.792.792.812.852.822.762.762.79

%N10 ~5 cm

0.3070.3960.4150.4430.4770.4860.5470.5820.6460.7610.5530.5980.5750.6250.6600.7290.8290.9241.0411.155

a

2, secH

0.004f0.0050.0070.0030.0080.0030.0030.0040.0060.0020.0020.0080.0140.0190.0160.0040.0020.0040.0070.002

t Standard deviation for 3 to 5 experimental measurements.

sured for these clays), higher repulsion between adjacentplatelets may cause a decrease in the equilibrium size of thetactoids.

Axial Ratio of Clay Particles—Conditions prevailing dur-ing the viscosity measurements were such that almost com-plete orientation of the clay particles parallel to the stream-ing lines could be assumed since the velocity gradient wasmuch larger than the rotational diffusion coefficient of theparticles (Alexander and Johnson, 1949). The shearingplane between the solid particles and their surrounding solu-tion at these conditions is rather close to the surface of theparticles. Therefore the axial ratio measured by this methodis dependent on the true dimensions of the solid particlesand has to be well correlated with the axial ratio calculatedfrom the average tactoid size described above. The axialratio of the tactoids was found to decrease from about 190for pure Na-clay to about 90 for Ca-clay. This is understoodsince the increasing number of platelets per tactoid results ina decrease of the axial ratio.

Self-diffusion of SodiumThe values of the self-diffusion coefficient of sodium

ions, D9Na, measured at two temperatures and at various

equivalent fractions of adsorbed Na, NNa, in equilibriumwith the equivalent Na fractions N N a in solution are reportedin Table 1.

Surface Self-diffusion of Sodium Ions—An attempt wasfirst made to interpret these results in terms of surface dif-fusion and tortuosity effects. If the tortuosity of the systemis expressed by a formation factor (F), the product of themeasured self-diffusion coefficient by the formation factorgives the self-diffusion coefficient in a hypothetical gel inwhich the plates would be arranged parallel to the diffusionflux. According to the relationship developed by Cremersand Thomas (1966), this fictitious self-diffusion coefficientF-Z)9

Na may be apportioned between the sodium ions in thefree solution and those in the surface phase by

0° 3'

H———I———I———h-——I———I———I———t———r-

0.5

Fig. 2—(a) Surface self-diffusion coefficient of sodium (D« ), and (b)Arrhenius activation energy for the surface diffusion process, asfunctions of the equivalent ionic fractions of adsorbed calcium (NCa)and sodium (NNa).

where DsNa and D^" are the self-diffusion coefficients of Na

in the free solution and in the surface phase, respectively;XCT

Na is the ratio between the amount of adsorbed Na ionsand the total amount of Na ions in the system.

The self-diffusion coefficient, DsNa, in the mixed equilib-

rium electrolyte solution (the solution phase) was obtainedby the use of Onsager (1945) equations as corrected byMills and Godbole (1960) for the effect of the ionicstrength. The fraction X^1" was calculated from experi-mentally measured quantities (ionic fractions of Na in sur-face and solution phases, cation exchange capacity, totalnormality, dry clay content, and anion exclusion volume)according to the relationship developed by Dufey and Lau-delout (1975b).

The relationship between F and the clay content(W,g/100 g) has been previously determined by anion self-diffusion for pure Na systems (Dufey and Laudelout,1975a). At a bulk NaCl concentration of 10 meq/liter, thefollowing empirical relationship was obtained for theCamp-Berteau monttnorillonite

N^ = 1 - 0.075(±0.013)W. [4]

For mixed Na-Ca systems, this formation factor was cor-rected on the basis of the number of platelets per tactoid,again assuming a 10A thickness for the inside water layers.If the large dimension of the particles ("a" axis) remainsconstant during the Na+-Ca2+ exchange, at least for Caloadings used in diffusion experiments, the relative de-crease of the axial ratio (a/b) with increasing Ca content canbe calculated. The way this affects the formation factor canbe derived from a Fricke (1924) relationship

-'-'(4*) [5]

= DsNa(l - [3]

where </> is the porosity, k is a shape factor of the particleswhich are assumed to be oblate ellipoids (axis a = c > b).

DUFEY ET AL.: PARTICLE SHAPE AND SODIUM SELF-DIFFUSION COEFFICIENT IN Na-Ca MONTMORILLONITE 313

Fig. 3—Sodium distribution in the surface phase of the clay tactoids asfunction of the overall surface cation composition: (1) Na on internalsurfaces, (2) Na on external surfaces, (3) percentage of external ad-sorption sites occupied by Na, (4) Na on external surfaces expressedin percentage of the total exchangeable Na.

For large values of the axial ratio, alb, the shape factor issimply given by

b [6]

Assuming that the effective overall porosity for sodium at agiven clay content does not change much with tactoid for-mation, and that the axial ratio is large enough to justify theassumption (2/3Tr)(a/b)» 1, the corrected formation fac-tor, Fc, for mixed systems can be approximated by

Fc - 1 - [7]

when FNa is the formation factor for the pure Na-clay (fromEq. [4]) and R is the ratio between alb of a mixed Na-Caclay and alb of the pure Na-clay. The variation of the sur-face self-diffusion coefficient of sodium ions, •D0

Na> calcu-lated as outlined above, is presented in Fig. 2a.

Before discussing the obvious trend of this coefficient toincrease with Ca loading, it seems worthwhile calculatingthe Arrhenius activation energy involved in the surface dif-fusion of Na. As shown in Fig. 2b, the activation energytends to decrease as the equivalent fraction of adsorbed Caincreases from 0 to about 0.30, followed by an obviousincrease at larger Ca fractions. This could be explained bythe combination of two effects on the surface diffusion coef-ficient of Na as follows: According to the works of Shain-berg and Kemper (1966) and Shainberg and Otoh (1968) itcan be assumed that in the first stage Ca ions preferentiallyoccupy only the internal surfaces of the tactoids (to the ex-tent allowed by the number of platelets per tactoid) displac-ing the Na ions to the external surfaces where the activationenergy for diffusion is certainly smaller than inside the tac-toids. When all the available internal surfaces are filled withCa, additional amounts of divalent cations will be distrib-uted on the external surfaces by mixing with Na ions. Thiscould result in increased activation energy because of an in-creasing "hopping" distance for Na ions. Figure 3 depicts

12

10

S s

8 6I?

0 2 4 6 8 10 12Experimental Dg (10'5cm! sec-')

Fig. 4—Statistical test of the agreement between the calculated andexperimental values of the self-diffusion coefficient of sodium inmixed Na-Ca clay gels (see text).

schematically the change in distribution of Na between in-ternal and external surfaces of the tactoids and also the frac-tion of Na, out of the total exchangeable ions (Na+ + Ca2+)on the external surfaces, and the fraction of external sodiumout of the total exchangeable Na. It can be seen that theminimum activation energy coincides with a completelydemixed system in which all external surfaces would be sat-urated by Na ions and all internal surfaces by Ca ions. Thecontribution of the edges to the cation exchange capacityhas been neglected in this simple model. But, assuming alsoa preferential Ca loading on them, because of their highcharge density, the possible shift of the position of theminimum activation energy would certainly be small com-pared to the accuracy of the experimental minimum. Thus,the increase in the surface diffusion coefficient of Na withincreasing Ca fraction may then be explained at low Ca con-tents by the decrease of the activation energy. Thereafter, inspite of an increasing activation energy, the surface dif-fusion coefficient increase could be due to an increasinghopping distance and possibly to an increasing jumpfrequency. Another explanation could perhaps be found in amodification of the distribution of the cations within the ex-ternal double layers.

Theoretical Calculation of Gel Diffusion Coefficients—Amore practical, but maybe less refined, approach to cationmobility in mixed-ion clay suspensions consists of evalu-ating how the change of the Na distribution between the sur-face phase and the bulk solution accounts for the observedchanges in the self-diffusion coefficient in the whole gel as-suming a constant surface diffusion coefficient. This, in es-sence, is an extension of the Boyd et al. (1947) approachwhich, for ion exchange resins, assumes that most of thecounter ions are localized on the solid surface and only fewions diffuse in macropores containing solution with com-position and diffusion rates similar to that of the equilibriumsolution. Since, for the relatively dilute clay gels studiedhere, it is unlikely that the contribution of the ions in thesurface phase to the overall diffusion process in the gel can

be completely neglected, a nonzero but constant surface dif-fusion coefficient was taken into account. The justificationfor assuming a constant surface diffusion coefficient for Naions can be found in the reasoning given above for the"demixing" tendency of the ions in the Na-Ca systemwhich leads to Na being always concentrated on outer sur-faces of tactoids. By using constant values of 0.30 x 10~5

and 0.45 x 10~5 cm2 sec'1 forDffNa at 16 and 33°C, respec-

tively, gel diffusion coefficients were backcalculated by theuse of Eq. [3], [4], and [7]. In Fig. 4, the calculated valuesare plotted against the observed ones. The best-fit straightline for both temperatures, forced through the origin has aslope of 0.928 indicating an average deviation of about 7%between observed and calculated values. The slopes calcu-lated separately for the values at 16 and 33°C are 0.932 and0.926, respectively. This relatively good agreement in-dicates that the major factor affecting the mobility of Naions in the clay gels is their distribution between the surfacephase and the bulk solution as modified, however, by theformation of larger tactoids in the Ca-rich systems.