Stability Properties of Hydrophobic Colloidal Solutions. - The Journal

STABILITY OF HYDROPHOBIC SOL8

239

STABILITY PROPERTIES OF HYDROPHOBIC COLLOIDAL

SOLUTIONS’ S. LEVINE2 Department of Physics, University of Toronfo, Toronto, Canada AND

G. P . DUBE’

Department of Physics, Rajput College, Agra, India Received September 93, 1940 I. INTRODUCTION

In a series of papers (7, 23, 25, 26) the authors have developed an expression for the mutual electrical energy of two colloidal particles. A number of applications were given (27),4particularly with regard to the stability properties in hydrophobic sols. In this paper we wish to expand upon some of the previous applications, which were presented in a very condensed form, and also to give additional results arising from our calculations. There are three contributions to the interaction energy of two colloidal particles,-namely, the electrical energy arising from the presence of the double layer,5 the van der Waals attractive energy, and the energy associated with the hydration and electrostrictive effects. A preliminary calculation suggests that the third type is not important, and we shall neglect it here. In addition, we shall neglect the polarization or distortion

* The work on this paper was begun jointly in 1939. One of us (S. L.) is indebted to the Department of Scientific and Industrial Research for an assistantship t o Prof J. E. Lennard-Jones, which was held a t Cambridge University during 1938-1939. The other autho; (G. P. D.) owes thanks to the ,Managing Committee, Rajput College, Agra, India, for granting him leave of absence and also wishes t o express his sincere thanks to Prof. Louis de Broglie for permitting him to work a t the Institut Henri Poincar6. The paper WRS completed by the former author (S. L.) a t the University of Toronto. Formerly a t the University Chemical Laboratory, Cambridge University, Cambridge, England. Formerly a t Institut Hcnri Poincar6, Paris, France. This earlier paper on applications will be referred to a s paper AI in the text. I n a recent paper Hauser and Hirshon (14) raise a number of objections to our theory, the most serious of which is t h a t “the effect of the repulsion forces existing between the ions of like sign in the overlapping ionospheres” is neglected by us. This is certainly not true, since we have calculated the tofa2 electrical energy assoeiated with the two micelles, as has already been explained in the earlier papers. Their criticism of the applicability of the Debye-Huckel equation is, no doubt, partially valid. However, a t present there does not appear t o exist a method essentially different from the one adopted here for calculating the mutual electrical energy of two colloidal particles.

240

8 . LEVINE AND Q. P. DUBE

effect in the diffuse ionic layers and also the correction associated with the complete Debye-Huckel equation, since the approximate equation only has been considered here. The amount of numerical work resulting from the introduction of all these corrections is really quite formidable; consequently, it was considered desirable to investigate first the degree of success in explaining the properties of colloidal solutions with the help of the first approximation only to the energy. It has already been suggested that this should be reasonably accurate for small particles or for low electrolyte concentrations. In paper AI we adopted the very neat method introduced by Hamaker (11, 12) for illustrating graphically the properties of the energy function, and we shall continue to use this approach here. We choose suitable pairs of the quantities that may be varied experimentally,-namely, particle charge, particle radius, electrokinetic potential, or electrolyte concentration, -let them form the axes of a two-dimensional plane, and then draw contours representing constant values of the energy a t contact of the particles and of the energy at the maximum." As explained by Hamaker, these two-dimensional diagrams are very useful in illustrating the stability properties of a sol, various regiom in the plane being assigned to.stable sols, flocculated sols, and so on. It was assumed in paper AI that the energy at the maximum was the controlling factor in the stability properties of a sol. However, this r6le should really be assumed by the rate of coagulation, a preliminary examination of which will be given in this paper. This reveals that it is still useful to draw energy contours, for the following reasons: Firstly, the energy maximum determines the stability properties of a sol fairly accurately in the region of very slow coagulation and roughly in the coagulation zone; secondly, its value is required in computing the rate of coagulation; thirdly, it is much simpler to compute the energy contours than the corresponding contours depicting the rate of coagulation. XI. QENERAL PROPERTIES OF THE MUTUAL ENERGY FUNCTION

Adopting the same notation &s in paper AI, we denote the electrical energy by F and the van der Waals energy by V, so that the resultant energy reads

E =F

+V

= =

+ Ads, a) t2Da(l + 7)2f(s, + (&'/Dalf(s, T )

7)

&(s,

(19 a)

(lii)

6 It has already been explained in the earlier papers that the resultant of the electrical and the van der Waals energy, RS a function of the separation of the particles, exhibits a maximum close to particle contact and a minimum farther out. Recently, Derjaguin (5) has objected to the existence of this minimum, claiming

STABILITY OF HYDROPHOBIC SOLS

241

Here a, Q,and { are the radius, charge, and electrokinetic potential, respectively, of a single particle; s = R/u, where R is the distance between the particles; and D is the dielectric constant of the solution. T

=

KU

= adE/3.04 X

lo-'

a t temperature T = 18' C.

(2)

where p = 1 , 3 , 6 , and 10 for the 1-1,2-1,3-1, and 4-1 types of electrolytes, respectively, and K is the characteristic quantity in the Debye-Hiickel theory of electrolytes, being proportional to the square root of the electrolyte concentration y (in moles per liter). I t is assumed that -tis given by

Q = Da(1 j -

7)

(3)

The forms of J(s, 7) and Ag(s, a),? where A is a constant, have been given in the earlier papers. To illustrate the dependence of the energy function (li) on the particle separation and. electrolyte concentration, we have plotted in figure 1 f(s,7) as a function of s for various 7. Taking a = uo = 10 mp, we assume the electrolyte present in the sol to be a 1-1 type and choose the concentrationsy = 2.31 X 5 X low4, 3.70 X lo-', lo-', 2.31 X lo-*, 5 X lo-', And 9.25 X lo-' moles per liter. Using equation 2, the corresponding values of 7 are 0.329, 0.500,0.736, 1.04, 2.00, 3.29, 5.00, 7.36, and 10.0, respectively. The group of curves in figure 1 illustrates how the electrical energy of the particles near contact diminishes very rapidly with added electrolyte, there being no change in particle charge and how, a t the same time, the range of interaction decreases as the ionic atmosphere is depressed. The minimum in each curve, the position of which will be denoted by skin (the superscript implies that we are dealing only with the electrical energy) the van der Waals energy being neglected), is indicated by an arrow. Its depth reached a maximum a t about T = 1.0, where &in = 3.8 and f(ski,, T) = -0.0043. The dependence of the minimum on T has already been examined in the earlier papers (23, 25). It is to be noticed that, for small 7, the minimum is shallow when compared with the range of positive values that the electrical energy assumes. For larger T( > 5 ) , however, its depth does become comparable with the (maximum) value of the energy at particle contact. The reason for the rapid drop in the electrical energy near contact can be that a correct evaluation of the mutual (free) energy of two particles will not yield the negative term which we obtained in our previous papers. This criticism, which we do not consider t o be valid, has been answered by one of us elsewhere (24). Harnaker (13) haa also pointed out a difficulty with regaid to our minimum, t o which we refer in the last section of this paper. See Hamaker (11) for properties of Ag(s, a ) .

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S. LEVINE AND G . P. DUBE

FIG.1. Form of the electrical energy function f(s, T) as a function of the mutual separation s (in units of the particle radius), for various values of T = Ka; also form of the van der Waals function g(s, a ) for a = 10 mp. Curves I t o I); representf(8, T ) f o r + = 0.329,0.500,0.736,1.04,2.00, 3.29, 5.00,7.36,and 10.0, respectively; curve X, g(s, a)/180;and curves X t o XIV, f(s, T ) g(s, a)/180 for T = 0.736,1.04,2.00, and 3.29,respectively. Magnified diagram in the upper right-hand corner is a contixustion of t h e main diagram for large r . Arrows indicate positions of the minimum.

+

visualized as follows: In the earlier papers (7, 23, 2 5 ) it was shown that the energy associated with a single particle is

provided T >> 1. Here C is the effective capacity of the double layer of the particle. Suppose the outer diffuse ionic layer of the particle be replaced

243

STABILITP OF HYDROPHOBIC SOLS

+

by a sphere of radius a 1/2K and carrying the charge -&. Then we shall have a spherical condenser consisting of two concentric spheres of radii a and a 1 / 2 ~ . The spacing 1 / 2 ~between the two spheres is of the right order, since a t a distance 1 / the ~ charge density in the outer layer falls to l/eth of its value at the particle surface ( e = 2.718). If the capacity of this condenser is denoted by C', the energy required to charge it is

+

provided KU = T >> 1, Le., provided the thickness of the diffuse layer ( 1 / ~ ) is small compared to the particle radius. We see how both C and C' increase with the contraction of the double layer accompanying the addition of electrolyte. This growth of the capacity leads to a drop in the energy, the charge remaining constant. Now this drop is even more marked in the electrical mutual energy a t contact. Thus, when T >> 1 we find that

and this property can again be attributed to the contraction of the outer layer with added electrolyte, a4though a simple picture such as in the case of the single particle has not been found as yet. In order to examine the resultant energy equation li in figure 1, we choose the values Qo = Me, where e = 4.80 X lo-'' E.S.U. is the electronic charge, Do = 80, and A . = kTo = 4 X ergs, say, where k is Boltzmann's constant, just as in paper AI. Then &:/DOCLO = 180 kTo = 7.2 X lo-'' ergs and we may write equation l i as E/180 kTo = E/7.2 X lo-''

= f(s,

T)

+

g(S,

~)/180

(4)

which has been plotted in figure 1. This resultant shows the characteristic maximum already studied in the previous papers. We observe how rapidly the height of this maximum drops with increase of the electrolyte concentration, also to be attributed to the contraction of the double layers. Both the maximum and minimum in the resultant energy function li are given by

Urn = E m / A = 180 vlf(Sm, where

7)

+

g(Sm,

a)

(5)

244

8 . LEVINE AND G. P. DUBE

making use of equation 3; s, denotes the position of either the maximum or the minimum and is defined by

which may be written w (7ii) We have introduced the same units as in paper AI, the charge and the van der Waals constant A being absorbed into one variable y, which may therefore be considered a measure of the charge. We also combine the energy and A into one quantity U = E / A . I n the diagrams y will always be taken positive, although Q, and hence y, can be either positive or negative, depending on whether we are dealing with a positive or a negative sol. Thus, instead of Q we should really write I Q I, its absolute value, throughout this paper; the same will be true of the electrokinetic potential t = I 1 and of the electric mobility u = I u 1. A single prime on the functions f(s, T) and g(s, a) denotes (partial) differentiation with respect to s, and a double prime is used for the second derivative. The same notation will be adopted for the derivatives with respect to s of other functions introduced in this paper. 111. PROPERTIES OF ENERGY CONTOURS

We now consider the properties of the energy maximum E,,, = E,,, (or Urn= Urnax)and of the energy minimum E,,, = Emin(or U , = Urnin)in the ( y - ~ )plane, equivalent to the particle charge-electrolyte concentration plane, for the particular radius a = 15 mp. The method of solving the simultaneous equations 5 and 7ii to obtain the curves representing constant values of U,, and Urnin and also of the positions smaxand sminin the (y-T) plane has already been indicated in paper AI. The results are shown in figures 2a and 3. The properties of the Urnaxcontours have already been studied in paper AI, but we wish to examine the region U , < 0 in figure 2a to show its relation to the Urnincontours in figure 3. I n paper AI we referred to the fact that this region does not extend to y = 0 but is bounded by another curve, which will now be studied in detail. From an examination of figure 2a, it was proved in paper AI that equation 7ii defines the function y = y(sm, T , a), which has the following property. For fixed T and a, as sm increases from its minimum value srn= 2 (particle contact), y is a monotonic decreasing function of sm until sm reaches a certain critical ski,, , which is defined by value 8,

0 and g’(s,, a ) > 0, it follows that equation 7ii has a meaning only when f)(s., , T ) 5 0 which, according to the form of f(s, 7) illustrated in figure 1, implies that sm 5 s&,, . We wish to show that the region 2 5 s, 5 S, refers to the maximum and the region S, 5 sm _< s;,= to the minimum. I t has just been remarked that

urr.(7,

stln

aylas,

20

according as

,S

@

5,

On differentiating equation 7ii, it is readily seen that this implies g’(sm, a)j”(sm

according as sm 3 5,

, 7) a)

0

, and hence, from equations 5 and 7ii, that

= 180yzf”(s,, = (g”(sm

, T ) - g ” ( S m , a)f’(sm , 7) 5 7)

+ g”(s,

, a)f’(sm,

7)

, a) a)f’’(sm

- g’(sm

p 0, according as s, 2

, ~)l/f’(~m

1

7)

(9)

3,

This proves our statement. The lower bound in figure 2a is defined by equation 8 or, according to equation 9, by

C:(sm,

(1Oi)

a) = 0

7,

It is now possible to prove that this lower boundary is the envelope of the two families of curves representing constant values of c‘, and S, in the (y-T) plane. In the following argument, a is assumed fixed, so that it does not enter and hence will be omitted. We may write equation 8 as O = Z 7 L ( s m , y,

7)

= 18Oy2f’(sm, T )

+ g’(sm)

= H(sm

7

Y,

7)

(11)

Kow in the (y-7) plane this represents a family of curves given by the parameter s, . According to the ordinary theory of envelopes, the envelope of this family is given by H’(s, , y, ). = 0, Le., by C:(s,

J

y,

7)

(1Oii)

=0

A comparison of equations 9 and 11 shows that this last equation is identical with 1Oi. Further, if we express equation 5 as L’, = V,(y, S, , r ) , then we can invert this function to obtain sm= s,( G, , y, r ) , i.e., we may consider s,,, as a function of C‘, , y, and 7. Therefore equation 11 becomes

0 = V A ( s m Y J 7) = H { s m ( L r m Y, 9

9

Y,

7))

71

= H*(Crrn9 Y,

T)

246

S. LEVINE AND G. P. DUBE

247

STABILITY O F HYDROPHOBIC SOLS

This relation also represents a family of curves in the (y-7) plane with U , as the parameter, with envelope

aH*/au,

=

H’(s,, y, 7)(as,/acm)

=

c&, , y, T)(as,/ec,)

=

o

provided equation 1Oi is true. We now see how the curves for C‘, in figure 2a pass over into those for Urninin figure 3 a t the point of contact with the common envelope defined by equation 8 and forming the lower bound. A similar connection exists between the smaxand smincurves. These relations are illustrated in figure 3a. It is likely that this behavior is a general one, holding not only for spherical particles but for other shapes. An examination of figures 2a and 3 yields the following properties: (Instead of the coordinates y and 7 we shall speak in terms of the equivalent quantities particle chargc and electrolyte concentration, remembering that the radius of the particles is assumed to be fixed. Also we shall refer simply as the maximum (minimum).) Firstly, to the position smax(smin) a t a given electrolyte concentration, the minimum moves out from ,S to ,,:s as we increase the charge on the particle. The asymptotic value smin+ si,,, assumed for very large particle charge, Le., when the van der Waals energy is negligible, is a function of T only, of course. It is true, however, that for spherical particles this increase in sminis not large.’ Also for fixed charge, as we add electrolyte the minimum moves in, just 8

The range (e,

,

..................... ..................... #,,,(a = 15mp). . . . . . . . . . 1 . .

&in

a s a function of

.I 1 1 3.8 3.4

T

is illustrated in the following table:

2.76 :.52

~

3 2.47 2.31

~

%:;: !& ~

~

9 2.14 2.08

FIG.2a. Energy maximum in the ( y - ~ )plane, i.e., in the particle charge-electrolyte concentration plane, taking 15 mp for the radius. One set of contours represents constant values of Urnax, the other of amax. Broken lines show the paths of the 1-1, 2-1, and 3-1 valence types of electrolytes, the arrows marking critical precipitation points. The paths, C, for the 2-1 and 3-1 types illustrate the influence of an initial amount of electrolyte i n the sol. Scale for y (in millimoles per liter a t T = 18°C.) calculated from equation 2, yielding y = 0.401~2,0.133~*,and 0.0669~2for 1-1, 2-1, and 3-1 types, respectively (see figure 6). FIG.2b. Plot of In f i n the ( U T ) plane, where f is the correction factor to account for slow coagulation in Smoluchowski’s formula for the rate of fast coagulation. The curves for In E = 1, 0, and particularly for -0.2 are subject t o a small error, owing t o difficulties in the calculations. Broken curves A illustrate the case of charge reversal, using the formula { = 4 a q o / D X , nrhere v is the cataphoretic velocity, X is the external field, and q is the viscosity. The long arrows mark equivalent stability conditions. Curves B refer t o the formula = Gaqv/DX. Values a = 15 mp and A = kTo have been assumed and, in addition, D = 81 in calculating the scale for Q / e , where e is the electronic charge.

248

8. LEVINE AND G. P. DUBE

like &in , the minimum for the electrical energy only. On the other hand, if the electrolyte concentration is unchanged and the charge increases, the maximum moves in from S, to 2, so that it occurs when the particles are touching, Le., there is really no maximum. In addition, for fixed charge and increasing electrolyte concentration, the maximum moves out to the position 8,. All these properties hold for any radius of the particles.

FIG. 3. Energy minimum in the (y-r) plane. One set of contours represents constant values of Urnin,the other of amla. Figure 3a illustrates the merging of the U,i. and smincurves with the U,,, and smr. curves, respectively, along the lower bound (envelope).

Secondly, if electrolyte is added and the charge does not change, then the minimum energy increases in magnitude from zero to a maximum and then decreases again, finally reaching the value D,/A, below which the minimum (and also the maximum) ceases to exist, the mutual energy being of a purely attractive character. The corresponding change in the maximum energy is a much more rapid and continual drop to U,,,/A. Also, as we increase the particle charge at fixed electrolyte concentration, the


249

depth of the minimum increases, this also being true of the height of the maximum. We also wish to examine the form of the energy in the electrokinetic potential-electrolyte concentration plane. It is convenient to choose = another radius, namely ai = 50 mp and to put E.S.U. = 30 millivolts, so that

{iDoa; = lOOkTo

FIG.4a. Plot of energy maximum in the (z-T) plan:, i.e. in the r-potential-electrolyte concentration plane or in the electric mobility-electrolyte concentration plane, taking a = 15 mp. To obtain 1‘ = 54.42 (in millivolts), we put D = 81 and A = kTo and the mobility u = 2.452 (in ~rper second) is derived from = 6 n 7 ~ u / D X . Broken curve illustrates the case of the same charge reversal as in figure 2a. FIG.4b. Plot of In 6 in the ( 2 7 ) plane under the same conditions as in figure 2, i.e., taking a = 15 mp and A = kTo . Broken curve same as in figure 4a. Small circles mark equivalent critical points in charge reversal of undialyzed eold sol.

We introduce

Then, making use of equation lii, the energy maximum is given by

where the latter equation simply being the condition 7. Instead of choosing the 1, to z2 = 5.40/r and U c o n = 33.8/r - 3.57, respectively.0 Above the dividing line, equation 15 becomes

+

+

z = {(LTcon 3.57)/f(2, ~ ) ) ~ ” / 1 0 ( 1 r )

(17) Just as in figure 2a, we have a lower limit in figure 4a, defined by equation lOi, below which Urnaxno longer exists. It has already been shown in paper AI that, in figure 2a, the equation of the dividing line is y2 = -1.12f’(2, r ) , which reduces to y2 = 4.5 when r > 1. Above the dividing line we have, in place of equation 17, y = 0.0745( (Uwn

+ 3.57)/f(2,

(18) Assuming D = 81 ( T =’18’C.) and A = kTo and multiplying by 3 X lo5 to change from E.S.U. to millivolts, equation 13 becomes { = 24.4%(in millivolts), from which we obtain the scale for in figure 4a. According to.Henry (15),1° I; (in F.s.u.) is given by r))l”

r

r = 4?rr)fJ/DXf(r)

(19)

where v is the cataphoretic velocity, X is the external field, q is the viscosity of the solution, and f (T ) is a certain function of r increasing from 8 to 1 as r goes from 0 to m . When r 5 10, f ( r ) ranges from Q to 1 and if we neglect the factor 3 / 2 f ( r ) , which lies between 1 and 0.83, the formula developed by Debye and Hiickel (3) is obtained, namely,

-

1 Whenr l , f ( 2 , ~ )= - andf’(2, 169 3 8r

T)

=

‘

Hermaiis (16, see also 18) has considered the relaxation effects in the double layer, neglected by Henry. But his results apply only t o large z (> 25) and even then are not much different from Henry’s formula for 1-1 electrolytes, provided I < 50 millivolts. la

251


Here 7~ is the mobility in microns per second per volt per centimeter. Substituting equation 20 for { into equation 12 and putting D = 81, -4 = k T a , and q = 0.0106, we derive u = 2.452, from which the scale for u in figure 4a is computed. There is no need t o plot the curves representing constant values of the electrical energy a t contact, F,,, , in the (2-T) plane, as has been done in the (y-T) plane in paper -41, since there will not be much difference from the form of the curves in figure 4a. The chief difference is that the contours for F,,, in the ( 2 - 7 ) plane extend to the 7-axis ( 2 = 0), where y,,, = 0. It is of interest to compare the curves for C,,, and smsx a t the given radius 15 mp in figures 2a and 4a with the corresponding curves a t another radius, say 50 mp, as shown in figures 5 and 6. In figure 5 , the upper limit above which C,,, = 17,,, is defined by y2 = -12.7/f’(2,

T)

and

which reduce to y = 7.14 and U,,, = 4.59 X lo3, respectively, when T > 1. Above the limit (expression 21), the form for the constant C,,, curves is given by equation 18 x i t h the number 3.57 being replaced by 13.2. In figure 6 the upper limit is z2 = -22.9/((1

+ 7)2f’(2,

(22)

7))

the corresponding form for C,,, = C,,, being the same as in equation 21. When T > 1 to z2 = 61.1/7. Above the limit the constant energy curves are given by equation 17, with 3.57 again being replaced by 13.2. Making use of equation 12, the relation between in millivolts and z is readily found to be = 29.82, taking the same values for D and A as in figure 4a. From equations 12 and 19, me now obtain u = 1.342. The three equivalent quantities z, {, and u are plotted as the ordinates in figure G.

r

r

IV. DIAGRAMS DEPICTING THE R 6 L E O F THE RADIUS

.

The dependence of the energy on the radius of the particle is now studied, an extension of the results already obtained in paper AI being given in this paper. On comparing the representations of the conditions E,,, = 0 and E,,, = 0 in the (.-a) plane in paper AI (figures 3 and 5), it was seen that there is no qualitative difference. Consequently, it was thought sufficient to examine in detail the properties of E,,, as a function of the radius and only qualitatively to consider E,,, , the computation of which

252

8 . LEVIPI’E AND G . P.

DVBE

is much more tedious. At contact of the particles (s be written as

o = u,,, - 107za(1 +-

=

2), equation 2 may

~a)2f(2,~ a ) g(2, a) =

(P(K,

a)

(23)

FIG.5 . Energy maximum in the (y-r) plane, i.e., in the particle charge-electrolyte concentration plane for a = 50 mr. Broken curves show paths of 1-1, 2-1, and 3-1 valence types of electrolytes such that the electrokinetic potential is decreasing linearly with the square root of the ionic strength, the arrows marking critical precipitation points.

where

z

=

I O - ~ D ~= ~ /IO-%~/U A

(24)

substituting equation 12. At D = 81 and A = kTo , this yields 21.1Z1’*in millivolts. Putting U,, = 1, 2.5, and 5, we plot the relation 23 in the (K-a)plane for different values of 2 (i.e., of the t-poten-

{ =

253


tial), the resulting curves representing constant values of 2 (or f ) being shown in figures 7a, 7b, and 7c, respectively. When K = 0 (along the a-axis), relation 23 becomes

2 sincef(2, 0 )

=

= 2 ( I-,,,

- g(2, a))/lO’a

1/2.

FIG.6. Energy maximum in the (z-r) plane, i.e., in the r-potential-electrolyte concentration plane or i n the electric mobility-electrolyte concentration plane for n = 50 mH. Scales for 5 and u are obtained in a similar fashion to those i n figure 4a. Broken lines illustrate same paths as in figure 6.

The question arises as to when a maximum exists in the curves for constant 2 in the (K-a) plane. The slope - of these curves is given by dK/da =

- (ap/a~)/(ap/a~)

I t is seen from figures 7a, 7b, and 7c that when K = 0 and a is small, d r / d a > 0. For large a and KU >> 1, g(2, a ) = -u/24e, where E = 1.5X em. (radius of an atom)” andf(2, K U ) 2: (1 7 / 2 ~ ~ ) / 1 6 ( ~ a ) * . Substitution of these limiting forms into equation 23 leads to

+

dK -- 2(1 - 2 X 10-’/3aZ) da 11 K 2 11

See footnote 7, on page 00.

2, usually varying from 2 t o 3. We may neglect this small correction in slow coagulation, but not in fast coagulation. Thus, if we assume emax= 3, then Smoluchowski’s formula becomes

eLt

=

1 12aD’na

yielding In E’ = In

(el..t/Ot..t)

= -In 1.5 = - 0.4

According to figure 2, the value of In E in the region of the lower bound (equation 1%) should be of this order (say -0.3 to -0.5), in accordance with experiment.


259

where emlow and efaatare the specific coagulation times for slow and rapid coagulations, respectively, and x = x(Q, T , a,A ) is a certain function of Q, T , a,and A , the explicit form of which will be discussed in a later paper. Here we shall compare the contours for In [ in the (y-T) plane with those of E,,/kTo for the particular values a = 15 mp and A = kTo , so that C,,, = E m a x / k T nthe , results being shown in figures 2a and 2b. I t should be remarked that the form of In (I - x) has only been determined approximately, since its exact computation is very tedious. For a fixed sol concentration ( n ) ,implying a fixed rate of rapid coagulation (efast),In 6 uniquely detgrmines the degree of stability of a sol and not E,,, , as was assumed in paper AI. For example, the so-called critical precipitation condition in a sol should refer to a definite value of the specific (slow) coagulation time or, even more appropriately, perhaps, to a definite value of the ratio $. = Oalaw/0fnat. Ordinarily the critical {-potential refers to the beginning of coagulation, and hence a t the critical point we may assume that In [ > 0, its value likely lying in the range 0 to 3. A comparison of the curves for In [ and E,,,/kTo in figure 2 s h o w that they are quite similar, and in the region where In t = 0 to 3, C,,, lies roughly between 2 and -7, Le.. E,,, ranges from 2 to -7kTo .I8 Thus we were not far wrong when we required that a t the precipitation point the values of E,,, for different electrolytes should be of the same order and equal to a small multiple of liT in magnitude. Furthermore, in the region where >> 1 (Emnx>> k T ) , we find that E,,,/kT >> 1 In (1 - x) I, so that In 5 = E,,/kT, Le., the energy maximum E,,, does determine fairly satisfactorily the stability properties of the sol. If we now draw in figure 2b the same three paths as in figure 2a, depicting the histoiy of three particular electrolytes of 1-1, 2-1, and 3-1 types, it is seen that a t coagulation the three values of In E are 0, 0.5, and 2 roughly, assuming that the initial electrolyte concentration is negligible. If the initial concentration is 10-3M of a 1-1 type, then the values of In 6 become 0, 0.2, and 0.3 roughly (the coagulation point in the (y-T) plane in the 1-1 case is off the figure). We notice that there is good agreement in the second case among the values of In [, although they appear to be rather small, since one would expect a value of In t = 1 to 3, say, a t the critical point. This can be achieved by choosing y larger, Le., either the van der Waals constant A smaller or !he charge &, and hence the {-potential, larger. We shall examine only the second alternative here. For small values of T ( 100 mp, a = 90 m p , and a = 50 mp on these three curves, Le., the larger a the smaller E,,,, for given K , {, and A . Taking Z = 3 and K = 2 as a second example, the three values of a are > 100 mp, = i 5 mp, and =50 mp, leading to the same conclusion. As already explained in paper AI, this property leads to the phenomenon of stepwise coagulation; namely, in a polydisperse sol, the largest particles precipitate first, followed by the smaller ones in order of decreasing size.

+

+

+

+

+

+

22 We first tried the values 50, 25, and 20 millivolts, to agiee with the values found by Powis for arsenic trisulfide sol and quoted by Eilers and Korff (8). However, this leads to a value Emax= lOkTo for the 1-1 electrolyte, which appears to be rather high. We have already remarked elsewhere (22) that the critical potential in the 3-1 case can he expected t o he lower than that in the 2-1 case. 23 If we assume A = 2kTo , then the values of E,,,/kTo a t the coagulation point -8, 1.5, and - 1 for the 1-1, 2-1, and 3-1 types, respectively, which is are rather unsatisfactory.

- -

-

272

S. LEVINE AND G . P . DUBE

The presence of the maximum in the curves representing constant Z is significant. The analysis in the preceding paragraphs refers to the region to thc right of the maximum, i.e. for larger a. A similar examination of the region to the left of the maximum, for smaller a, reveals the following: We compare again the three curves for 2 = 2 in figures 7a, 7b, and 7c, observing now their intersection with the abscissa K = 1 to the left of the maximum. The three values of a are 2 mp, 5 mp and 10 mp roughly, the corresponding values of E,,, being in the ratios 1:2.5:5.0. This illustrates that for small a, E,,, increases with the radius, assuming that the 0 when x = 0. For large a and KU >> 1, an analysis similar to that carried out on equation 23 shows that a$/aK > 0 and that

16 Here 9 = v(e/a) is a monotonic decreasing function of a , varying from 1 to 0 as a varies from e to m . For example, when a = 1 w, 5 w, 10 mp, and 50 mN, the values of q are 0.8, 0.5, 0.2, and 0.05, respectively (to one significant figure). I n paper AI we used the notation g(2, a) = g ( u ) , where u = ( ~ / a ) / (l (r/a) I, so that u = m (and g ( u ) = 0) when a = e. -However, i t simplifies matters when a + 0 to use Hamaker's definition u = €/a, according to which u -9 m and g ( u ) -+ 0 as a + m . This definition is used when we speak of Eco.+ 0 as a --t 0. The difference between the two forms is trivial in the colloidal range where a > e = 0.15 w. For either definition we have utilized the relation

uZdg(u)/du = edg(2, a)/da

in computing the paths of the maximum in figures 7c and 8.


275

< according as K 4.7 X lo6,equivalent to Z $ A$, substituting equation 31 into the limit 30 and putting p = 2 and D = 80. These conditions have the following interpretation: For any fixed K < 4.7 X lo6 (2 > 40/9), as the radius is increased from zero, E,,, increases steadily from zero value and does not pass through a maximum. In other words, when K < 4.7 X lo6, dK/da > 0 for all a. In the region 4.7 X lo6 < K < lo', we have the property that E,,, increases from zero to a maximum value and then decreases again, finally becoming negative. In the region K > lo', E,,, < 0 for all a > 0, increasing in absolute value with the radius, assuming K fixed. Thus the particular example represented in figure 8 serves to illustrate the meaning of the limiting {-potential given by equation 30. The example depicted in figure 2 in paper AI exhibits similar properties, the value 4.7 X 10' for K being replaced by 0.8 X 10'. The general property concerning the maximum in figure 8 can be derived as follows. If me write E,,, in the form

making use of equations lii and 33, then, assuming that K is constant, a ) is a monotonic decreasing function of a, since the functions (1 Ka)'f(2, K U ) and ~ ( e / a have ) this property. Hence for a < a ~ , *(K, a) > * ( K , a ~ )= 0 and therefore E,,, > 0. Since at a = 0 and a = a0 , E,,, = 0, E,, must pass through a t least one maximum in the range (0, ao). This means that for the general condition that { be a function of K only, Le., be independat of the radius, then provided the condition 32 holds, E,,, > 0 in the range (0, Q) and it possesses a maximum in this range. This property has already been discussed (23). The path of the maximum in figure 8 is given by dx/da = 0 or a$/aa = 0, which may be written as *(K,

+

corresponding to equation 11 in paper AI. This maximum, which is shown as the broken curve, is asymptotic to the abscissa K = 4.7 X 10' for large a. Indeed, an examination of equation 31 shows that all the curves in figure 8 are asymptotic to the same abscissa. Thus, from a study of figures 7a, 7b, 7c, and 8 and also of figure 2 in paper 41, E,,, exhibits the following property in the region where it is a small multiple of k T : Assuming the electrolyte concentration to be fixed and the {-potential to be independent of the radius, E,, , considered BS a function of the radius, increases from zero value (when the radius is zero), passes through a maximum, and then decreases again, finally becoming

276

S. LEVINE

AND G. P. DUBE

negative. E,, behaves in a similar fashion. This region may be assumed to comprise the coagulation zone, from the very slow to the fast stages, since E m a x may have values as high as 10kT,say. This behavior is quite consistent with the phenomena of the formation of secondary particles from primary ones (as may occur in aging, for example) and of stepwise coagulation. It is reasonable to assume that this behavior is still observed when the 3.-potential is not strictly independent of the radius, but rather changes to a small degree with it. XII. EFFECT OF SURFACE IRREGULARITIES

There are a number of difficultiesassociated with our expression for the mutual energy of two colloidal particles. Hamaker has pointed out that for large particles the depth of our minimum is roughly proportional to the particle radius, implying that large particles are unstable. This discrepancy is discussed by one of us elsewhere (24) and the conclusion is reached that the irregular shape of the surface of the particle reduces the depth of this minimum to a considerable extent. Indeed, it appears that large particles of truly sphkrical shape cannot be stable, in accord with the fact that emulsions need stabilizing agents. On the other hand, the correction to the electrical energy near contact of the particles does not seem to be so drastic. According to this picture, two particles will touch only a t a few points where the surfaces are projecting and this accounts for the small values of the van der Waals constant A that we obtain in this paper. Another difficulty is illustrated in figures 2, 4, 5, and 6, where we see that smax= 2 very nearly, except in the coagulation region. This means that in slow coagulation the position of the energy barrier over which the particles must pass in order to coalesce is very close to particle contact. Indeed, according to the paths of the 1-1 electrolytes illustrated in this paper, coagulation may take place under conditions where there does not seem to be a maximum, Le., the particles are repelling one another even when they are touching. This suggests that the maximum should be farther out than appears to be the case. We therefore suggest the following modification in the form of equation 1 for the mutual energy of two particles, in order to account for their uneven surfaces. The particles first come in contact at the projecting areas, but with time they adjust their positions in the coagulum until they are closely packed. In the form of the van der Waals energy, say A*g*(s, a), of such particles, where 6 is the average radius, A* > kTa will approach more nearly the high values of A computed by Hamaker (11). Secondly, it is likely that g‘(s, 6) > g*’(8, a), i.e., 1 g*(s, a)\,does not fall so rapidly with distance as in the case of two hypothetical truly spherical particles of radius 6. We shall put g*(2, 6) = g(2, 6 ) . These properties will also be assumed for the new form f*(s, T ) for the electrical energynear contact, Le., 1 f(s, .)I >

277

STABILITT OF HYDROPHOBIC SOLS

T)I

for s = 2, with f ( 2 , 7) = f * ( 2 , T ) ; also we shall assume > f c ( S m i n , 7)l and Smin < i i n . The resulting change in the mutual energy of two particles has been illustrated in figure 9. We have plotted curve VI in figure 1, writing equation 4 as If*'(s,

1

1 f(smin ,

E/kTo = 1 8 0 f ( ~T, )

+ g(s, 6)

(4i)

and also the new form

E* kTc -

+ A*g*(s, a) mo{C~*(S, Doao T)

where 6 = a0 = 10 mp,

-10

T

= 230f*(s, T )

+ 3g*(s, d )

(37)

= 1.04 (y = 10-3M, 1-1 type), A* = 3kTo,

1

I

I

3 5 4 FIG.9. Effect of irregular surface on mutual energy of two particles. Energy curves for truly spherical particles: curve I, 180f(s,r ) ; curve 11, g(s, a ) ; curve 111, E / k T o . Energy curves for irregularly shaped particles: curve IV, 230f*(s, T ) ; curve V, 3g*(s, a ) ; curve VI, E * / k T o .

the curves for g*(s, a) andf*(s, 7) satisfying the conditions specified above, otherwise being arbitrary; the value Q : = 1.1 Qois chosen to satisfy the condition E,,, = E:,, . We notice that, , :s > smax, Le., the maximum is farther out from contact. On combining equations 12 and, 15, we may write the boundary line in figures 2a, 4, 5, and 6 as

1'

r

such that for > equation 37 reads

r:'

= -Ag'(2, d ) / o a ( l f 7)lf'(2,T ) =

lo , C,,,

=

C,,

.

li

The boundary line associate'd with

= -A*g*'(2, a ) / o d ( 1

+ T)lf+'(2,

T)

278


and if {(: > l o , which may be expected, then a maximum will be created in that region above the boundary in the (y, 7) plane where none existed before. In this way we can overcome the difficulty about the absence of a maximum a t the precipitation concentration of 4 to 5 x 10-'M of the 1-1 type, already pointed out in paper AI. This discussion enables one to appreciate the qualitative character of the van der Waals energy as calculated by Hamaker and, to a lesser degree, of our expression for the electrical energy. The small values of A obtained by us are no longer surprising. However, the general agreement with experiment suggests that the results obtained by us will not be changed in a qualitative manner by such corrections and, indeed, that our theory can be considered to be of a semi-quantitative character. This particularly applies to small particles, since it appears that the corrections arising from the protrusions on the surface of the particles become less, the smaller the particles. XIII. GENERAL DISCUSSION

An examination of the literature dealing with stability properties in hydrophobic sols reveals considerable confusion concerning the various factors that need to be considered. The more common factors that are suggested are the reduction in charge, attainment of a critical electrokinetic potential or mobility, compression of the double layer, and recently, according to Ostwald (30), the activity coefficient of the coagulating ion. All these properties are important, but a knowledge of each separate one does not uniquely determine the condition of stability of the sol. The stability properties of a sol are determined uniquely by the (free) energy to be associated with the system, compared to the energy of the corresponding coagulum, For sufficiently dilute sols, the former energy as a first approximation depends only on the form of the mutual energy of two particles and the total number of particles. In practice, it is most convenient to work with the rate of coagulation in describing quantitatively the degree of stability. The dependence of this non-equilibrium property on the energy of the sol can be obtained by applying the kinetic theory of colloidal particles, as developed by Einstein and Smoluchowski. If the rate of coagulation is sufficiently slow, it can be assumed to depend on mutual energy of particle pairs only, since only primary and secondary particles would be present to any appreciable extent, a t least a t the beginning of coagulation. The problem of the stability of a sol can then be divided into two parts. The first is to obtain an expression for the mutual energy of two particles as a function of the various variables upon which it actually depends. We have attempted to solve this problem to a certain extent in the series of earlier papers. The second part is to find the relation between the rate of coagulation and the mutual energy of two particles and the sol concentration. This has been carried out approximately by Fuchs (9) and Der-


279

jaguin (4, G ) , neglecting*multipleencounters of the particles. R e are well aware that our solution of the first problem is only approximate and that a considerable amount of work still remains in the way of obtaining more accurate expressions for the interaction energy of colloidal particles. However, we believe that our results have lead to a first attempt a t a comprehensive and consistent theory of stability?' XIV. SUMMARY

A number of applications of the mutual energy of two colloidal particles are given. The stability properties of a hydrophobic sol can be uniquely described quantitatively by the rate of coagulation. However, it is shown that the energy maximum in our expression for the interaction of two particles does determine the stability properties of a sol fairly satisfactorily in the region of very slow coagulation and roughly in the coagulation zone, assuming the sol concentration to be fixed. The dependence of this energy maximum on the various significant factors,-namely, particle charge, electrokinetic potential, electrolyte concentration, valence of the ions, and particle radius,-are examined. The condition for coagulation is that this energy maximum be a small multiple of kT. The concept of the precipitating ion is shown to be of limited validity. I n the case of electrolytes which coagulate a t high {-potentials and large electrolyte concentrations (e.g., alkali halides in a negative sol), the ions of the same charge as the particle are adsorbed, thereby increasing the particle charge. The drop in the height of the energy maximum is caused by the increase in the effective capacity of double layer? of the two particles, resulting from the contraction which accompanies the addition of electrolyte. I t is shown that even the electrokinetic potential may increase and yet coagulation will occur upon the addition of sufficient electrolyte. On the other hand, the reduction of charge is the important factor for easily adsorbed ions of higher valence. The phenomena of charge reversal by trivalent ions is studied. Our theory predicts that the critical {-potential in the second region of stability must be higher than that in the first. A check on the theory comes from the calculation of the van der Waals energy of attraction of the particles, being given the critical {-potential, precipitation electrolyte concentration, and the particle radius. The van der Waals constant should have the same value for different electrolytes in the same sol, and fair agreement is obtained. The dependence of the energy maximum on the particle radius is dis46 Note added in proof: In a recent issue of this Journal (46, 731, 1941) Hazel has come to conclusions very similar t o our own concerning the factors determining the stability of a sol, particularly those expressed in section VIII. However, we contend t h a t our analysis is more fundamental and thorough. It would be instructive t o construct tables similar to those in this paper from his data.

280


cussed. Assuming that the {-potential is independent of the radius, in the coagulation zone, the energy maximum increases with the radius for very small particles, but reaches a maximum and then decreases with the radius for larger particles. This is consistent with the formation of secondary particles from small primary ones and with the phenomenon of stepwise coagulation. The effect of the irregularities on the particle surface is discussed. It is probable that these will modify our results in a quantitative but not a qualitative manner, the corrections being less the smaller the particles. REFERENCES (1) ABRAMSON:Electrokinetic Phenomena, p. 218. American Chemical Society Monograph, Reinhold Publishing Corporation, New York (1934). (2) BRIGGS: J. Phys. Chem. 34,1326 (1930). (3) DEBYEAND HUCKEL:Physik. Z. 24,49 (1924). Acta Physicochim. U. R. S. S. 10, 333 (1939). (4) DERJAGUIN: Trans. Faraday SOC. 36, 203 (1940). (5) DERJAGUIN: Acta Physicochim. R. s. 8. 10,25, 1% (1939). (6) DERJ.4GUIN AND KUSSAKOV: (7) DUBEA N D LEVINE:Compt. rend. 108, 1812 (1939). (8) EILERSA N D KORFF: Trans. Faraday SOC.36,229 (1940). (9) FUCHS:Z. Physik89,736 (1934). (10) FUCHS A N D PAULI:Kolloid-Beihefte 21, 412 (1925). Physica 4, 1058 (1937). (11) HAMAKER: Rev. trav. chim. W, 1015 (1936); 66, 17, 727 (1937); 67; 61 (1938); (12) HAMAKER: Symposium on Hydrophobic Colloids, Nederland. Chem. Vereen (November, 1937). (13) HAMAKER: Trans. Faraday soc. 36, 186 (1940). A N D HIRSHON: J. Phys. Chem. Is, 1015 (1939). (14) HAWSER (15) HENRY:Proc. Roy. Soo. (London) AlSS, 106 (1931). Phil. Mag. [7]26, 650 (1938); Trans. Faraday SOC. 36, 133 (1940). (16) HERMANS: Physik. : Z. 26, 204 (1924). (17) H ~ C K E L Researches Electrotech. Lab. (Tokyo), No. 387 (1935). (18) KOMAGATA: A N D V A N GILS: Kolloid-Z. 78, 32 (1937). (19) KRUYT A N D V A N DER WILLIGEN: Colloid Symposium Monograph (20) KRUYT,ROODVOETS, 4, 304 (1926). (21) KRUYT AXD VAN DER WILLIGEN: Z. physik. Chem. 190, 170 (1927). (22) LEVINE:J. Chem. Phys. 7,831 (1939). (23) LEVINE:Proc. Roy. SOC.(London) Al70, 145, 165 (1939). (24) LEVINE:Trans. Faraday SOC.98, 725 (1940). 36, 1125, 1141 (1939). (25) LEVINEAPFD DUBE:Trans. Faraday SOC. (26) LEVIXEA N D DWBE:Phil. Mag. 29, 105 (1940). (27) LEVIKEA X D DUBE:Trans. Faraday SOC.36, 215 (1940). Rec. trav. chim. 46,772, 954 (1926). (28) LIMBWRG: (29) MULLER: Kolloid-Beihefte 26, 257 (1928). Kolloid-Z. 73, 301 (1935): J. Phys. Chem. 44, 981 (1938). (30) OSTWALD: J. Chem. SOC. 1930, 1447. (31) PENNYCUICK: Trans. Faraday SOC.36, 69 (1940). (32) RUTGERS: A N D VERLENDE: Proc. bcad. Sci. Amsterdam 42, 71 (1939). (33) RUTGERS Proc. Acad. Sci. Amsterdam U , 763 (34) RUTGERS,VERLENDE,AND MOORKENS: (1938).

u.

Stability Properties of Hydrophobic Colloidal Solutions. - The Journal

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