Adsorption from solutions exhibiting consolute temperatures. II. Adsorption of 2,6-dimethyl pyridine and cetyltrimethylammonium bromide

THERMODYNAMICS OF CONTACT ANGLES. II

Evaluating these integrals as before, we obtain

From Fig. 2, we see that Xl

:

Z1 c o t / ~

239

:

--

A11

[21] AGal =

Ly3'Lv (2yLv~]_{2

\-~a~ /

+ cos (~ + 0)}

3

and from Eq. [6], AI' = --cot #

X {1-cos(flq-0)}½-2½1

LV2VLv(I-~c°s4')][ [22-1 Ap--" AGa,~ --

L~v

[29]

{ 2 ~ v [ 1 + cos (# + o)]}~-

Therefore,

3

Apg

AG2~ = --LyyLv cot #

Xcot#[1 + cos (B + 0)1

x [1 + cos (# + 0)-1. [3o] X { 2"YLV[1 q- COS (fl q- 0)-1 }½

[231

Apg

AG2 =

Calculations with the Smooth Tilted Plate Configuration

L~,TLv\ ~ p g ] x [2~ -

(1 - cos (~ + 0))~

- cot#(1 + cos (# + o))q.

[241

AGa Term Work must be done against gravity to raise (or depress) the liquid from the undisturbed level. AGa may be divided into two terms, one for the work done on the liquid that lies between xt and infinity, and the second for the work done on the liquid lying between the origin and xl. AGa = AGal-t- AGa>

[25]

For both terms, we consider a small column of liquid of rectangular cross section, Lflx, which is composed of elements of volume Ludxdz. The work done in lifting each of these elements to its proper position is, ApgLyzdxdz. Integrating with respect to z yields

AGa ooI..... = ApgLuz~dx/2

[26]

and integrating over all columns yields

AGal - LyApg f ~ z%tx 2

zxa~-

L~Apg f xl -~ J0 ~d~.

[27]

To establish the validity of the model, the free energy changes, Eqs. [12], [24-1, [29], and [301, and the free energy change of the entire system, Eq. [-21, were computed for # = 90 °, i.e., a smooth, homogeneous vertical plate, with variation of the contact angle 0 between 0 and 180 °. For these calculations, let L u = 1.0 cm, 9'LV = 50 dynes/cm,

cos-l[('~sv - vsD/vLv] = 40 ° = 0oy, Ap = 1.0 g/cm 3 and g = 981 cm/sec 2. A plot of the change in free energy for the entire system, AG, versus the contact angle 0, is shown in Fig. 3a. The minimum is found at 0e = 0ey = 40 °. The computation method was further checked for various angles of tilt, #, and several equilibrium contact angles 0~. The results for B = 45 ° with 0eY = 15, 40, 65, 90, 115, 140, and 165 ° are shown in Fig. 3b. Again the minimum for the free energy occurred at 0e = 0~y in each case. (The minima for 0eY = 15 and 40 ° are off the graph, in Fig. 3b.) It can be shown, by minimizing the free energy of the entire system, that the following relationship holds for the elevation zl to which the liquid rises on the tilted plate at equilibrium :

Apgz} cos (8 q- 0°) = --1 q- - - . 2"yLv

1-283

[31-1

When # = 90 ° , this reduces to the more

Jo~s,rnal of Colloid and Interface Science, V o l . 53, N o . 2, N o v e m b e r 1975

240

EICK, GOOD AND NEUMANN

*30 +25

+2 +20

+20[ 1150

+1~

+15 ÷IC

,~ +I0 +

90"

0¥ -1,*

. . . . . . . . . 20 40 60

m 100

80

........ 120 140

160

180

~ (DEGREES)

~ (DEfiREES)

FIc. 3. (a) Change in free energy of the system versus contact angle for smooth, homogeneous, rigid, vertical solid surface for which (~/sv -- 7sL)/7~v = cos 40 ° = cos 0oy; 7LV = 50 ergs/cm "°, unit density, and Ly = 1 cm. (b) Change in free energy of the system versus 0 for solid surfaces for which 0eY has the values indicated.

familiar expression (1),

Apgh~ sin 0e = 1 -- - 27Lv

Thermodynamic Theory for a Rough (Sawtooth ) Surface We now develop a theory of capillary rise of a liquid at a rough surface. Fig. 4 shows the profile of the surface. From H = H0 to HD, = ill. At the point H = HD, the angle of tilt changes discontinuously from fll to f12,

and remains at that angle until the point H0, is reached. At that point, it returns to ill, and so on, generating the sawtooth surface. HD gives a measure of the amplitude of the roughness; HD ----ZD/sin ~1. At this point, we define one more angle. If HD is very small, the optical systems normally employed to observe the contact angle will not reveal that the surface is rough. The envelope of the surface is vertical. We m a y treat the envelope as a surface on which we measure a capillary rise ~ which is related (1) to a macroscopic contact angle OM which m a y or m a y not be an equilibrium angle :

\\\

"ZD'

apg~2

HD~

sin 0~ ~ 1 -- - - - .

E32-]

2"yLV

"Zo' HO"~

/

132

H0"~il.

X~

FIn. 4. Construction for derivation of free energy of capillary rise on idealized sawtooth surface. Journal of Colloid and Interface Science,

We can now divide each face of the sawtooth surface into any number of equal parts e.g., by the three points indicated on the section between 0 and HD in Fig. 4. For each value of z, we m a y determine the corresponding 0 by means of Eqs. [6~ and [-11-]. This enables us to calculate the free energy terms,

Vol. 53, No. 2, November 1975

THERMODYNAMICS

OF C O N T A C T A N G L E S . I I

since the height is specified above or below the reference state at 0. We can generate different degrees of roughness by changing H9, fll and ~2. The free energy terms t h a t are significantly changed, from those in the analysis of the tilted, smooth slab, are AG22 and zXG~2. As we progress upward, moving the line of contact over the surface, AG22 will cycle between 0 and some nonzero value at H9, and then back to 0 at H0,. For example, when H9 = 100 t~m and, ~1 = 45 ° and f12 = 135 ° , for Ap = 1.00 g / c m ~, it was found that at Hz), AG2 = 0.353553 ergs. This cycle repeats itself for each sawtooth section. In the present calculations, /31 and ~2 were varied together, so that ~1 = 180 ° -- ~2. Consequently, AG22 was only a function of H9 and ~1. The AG32 term, for the rough surface, corresponds to the energy required to raise elements of liquid from the reference level to their proper locations in the triangular prism volumes between the actual surface and the envelope of the tips of the sawteeth. This was TABLE I VARIATION O~" THE ANGLES Ot" TILT, fll AND /~2; H D = 100~m; O~y = 40 ° Angles of tilt (degrees)

z~/xa

r

r co80eY

0w Absolute (degrees) minimum present

accomplished with the aid of Eq. [303 and a computer program for sections of surface tilted at angles /~1 and B2. See Ref. (11) for details.

Calculations for the Model Rough Surface The internal consistency of this computation was checked by calculating the free energy terms and the total free energy change for the entire system, setting lid -----100 ~m, /~1 ----~2 = 9 0 °, and 0ey = 40% Throughout all the computations, the values, Ly = 1.0 cm, 7LV = 5 0 dynes/cm, Ap = 1.00 g / c m 3 and g = 981 cm/sec ~ were assumed. The curve shown in Fig. 3a was recovered. Variations of the parameters of this model were then explored. Three conditions were considered: (a) varying the angles of tilt, ~1 and ~2 while keeping 0,y and HD constant; (b) varying 0ey while keeping ~1 and 32 and HD constant; and (c) varying lid while keeping 0ey and ~t and f12 constant. ¢~1 and ¢~2 Varied t31 and j32 were varied in steps of 5 °, from 5 ° to 90 ° for ~1, and 175 to 90 ° for ~2, with HD =100 v m and 0ey = 40 ° • Table I lists the values of 31 and ~32, and in addition, the corresponding Wenzel [-21 roughness factor r, r cos 0oy, and 0w. The Wenzel equation is r('Ysv -- 7SL) = ~LV COS 0W

fll

f12

5 10 15 20

175 170 165 160

2.858 1.418 0.933 0.687

25

155

0.536

30 35 40 45 50 55 60 65 70 75 80 85 90

150 145 140 135 130 125 120 115 110 105 100 95 90

0.433 0.357 0.298 0.250 0.210 0.175 0.144 0.117 0.091 0.067 0.044 0.022 0,000

11.473 5.759 3.864 2.924 2.366 2.000 1.743 1.556 1.414 1.305 1.221

1.155 1.103 1.064 1.035 1.015 1.004 1.000

8.784 4.411 2.960 2.240

-----

No No No No

1.813

--

No

1.532 1.336 1.192 1.083 1.000 0.935 0.885 0.845 0.815 0.793 0.778 0.769 0.766

----0.0 20.7 27.8 32.3 35.4 37.5 38.9 39.7 40.0

No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes

241

[-33~

where r =- a/A = da/dA _-> 1, and a = actual area of interface, A = apparent area of the "geometrical" interface, i.e., area projected on a plane defined by the envelope of true surface, and 0w -- Wenzel contact angle, defined by Eq. [-331. For comparison, the values of roughness Zo/Xo as defined by Johnson and Dettre (6), are given in Table I. (2Z0 -- height of ridge on surface and x0 = distance between ridges. This ratio was employed in Ref. (6) with respect to a sinusoidal surface profile.) The last column indicates whether the computed free energy curves (e.g., Fig. 5) exhibit an absolute minimum or not. A curve of free energy vs the macroscopic or measurable contact angle, 0M, for ~t = 45 °

Journal of Colloid and Interface Science, Vol. 53, No. 2, November 1975

242

EICK, GOOD AND NEUMANN +35 +30

/

+25 +20 +15

O

+lO

+5 0

-5 -10

-%

" io " ~o " ~o " ~

" 1o0 " 1io " l;o " 1~ ' l;o

0M(OEGRrEs)

FIG. 5. Free energy versus macroscopic contact angle

0z¢ for sawtooth surface with fit = 45°, fir = 135°.

and 82 --- 135 °, and HD = 100 ~m, is shown in Fig. 5. I t is evident that metastable states, or positions separated by free energy barriers, are present. These states are similar to those suggested b y Good (5). The metastable states are present between 5 and 85 °, or over a range of 80 ° . I t also can be noted that this particular free energy curve does not have an absolute minimum. This observation is in agreement with Johnson and Dettre (6), who also found that for surfaces with a certain degree of roughness, there is no minimum of free energy, or equilibrium contact angle 0e. The model predicts that 85 ° is the highest contact angle OM that could actually exist, with this system; it would correspond to the largest possible advancing contact angle, OA. . . . Since the free energy curve determined with the above parameters does not reach an absolute minimum above 0 °, the measured receding contact angle, OR mln, would be 0 °. In other words, observable contact angles OM would be possible between 0 and 85 °. The free energy curves for a variety of 8's are given in Fig. 6. The curves for ~1 = 60 ° andS2 = 120 °, and for~l = 75° andfl2 = 105 °,

have absolute minima; and the height of the energy barriers is less than for those that do not exhibit minima. As 81 and 82 approach 90 °, the energy barriers decrease, and the range of the metastable states becomes smaller, until a smooth curve is obtained for 81 --/~2 -- 90 °. Table I I lists results as to the metastable states, and the equilibrium contact angle, if one existed. The range of metastable states is 80 ° , as long as there is no minimum. When an absolute minimum in the free energy exists, i.e., when an equilibrium contact angle, 0e is found, the range of the metastable states decreases as fll and 82 approach 90 °. In the case where 8 1 = 55 ° and 82 = 125 °, the metastable states begin at OM = 5 ° and end at Ou = 75 °, and the calculated equilibrium contact angle is 0e = 20.7 °. Therefore, the lower limit for the receding contact angle OR mir~ will be 5 ° (in contrast to OR mh~ being 0 ° when no absolute minimum exists) and the upper limit for the advancing contact angle 0a ..... will be 75°; and experimentally, any angle OM between 5 and 75 ° should be observable. Thus, the observed angles, and

+35 +30

/

+25 +20 +15

+5

-10 -15

20

40

60

SO

lO0

~ M(DEGRKS)

120

140

160

180

FIO. 6. Curves of free energy versus OM for rough surfaces w i t h a v a r i e t y of values of ill,/~2, w i t h 0ey = 40 °.

Journal of Colloid and Inlerface Science, Vol. 53, No. 2, November 1975

243

THERMODYNAMICS OF CONTACT ANGLES. II TABLE II

METASTABLE STATES AND MINIMA I~OR DIFFERENT ANGLES OF TILT, ~1 AND ~2; ~]~ = 100#m; 0ey = 40 °

Angles of tilt /31and ¢2 (degrees)

Macroscopiccontact angle 0Mfor beginning and end of metastable states (degrees)

Rangeof metastable states (degrees)

Calculated equilibrium contact angle degrees 0o

(degrees)

(degrees)

5 175

45 125

80

None

125

0

10 170

40 120

80

None

120

0

15 165

35 115

80

None

115

0

20 160

30 110

80

None

110

0

25 155

25 105

80

None

105

0

30 150

20 100

80

None

100

0

35 145

15 95

80

None

95

0

40 140

10 90

80

None

90

0

45 115

5 85

80

None

85

0

50 130

0 80

80

None

80

0

55 125

5 75

70

20.7

75

5

60 120

10 70

60

27.8

70

10

65 115

15 65

50

32.2

65

15

70 110

20 60

40

35.4

60

20

75 105

25 55

30

37,5

55

25

80 100

30 50

20

38.9

50

30

85 95

35 45

10

39.7

45

35

90 90

40 40

0

40.0

40

40

Oa

OR

t h e h y s t e r e s i s , will c o n s t i t u t e a m e a s u r e of roughness.

a b s o l u t e m i n i m u m free e n e r g y is p r e s e n t , t h e

A n i n t e r e s t i n g r e s u l t is e v i d e n t w h e n t h e

e q u i l i b r i u m c o n t a c t angle, 0o, is e q u a l to t h e

s i x t h c o l u m n , 0w of T a b l e I, is c o m p a r e d w i t h

c o n t a c t angle p r e d i c t e d b y t h e W e n z e l e q u a -

t h e f o u r t h c o l u m n , 0o of T a b l e I I . W h e n a n

Journal of Colloid and Interface Science, Vol. 53, No. 2, November 1975

244

EICK, GOOD AND NEUMANN

tion [2], i.e., 0~ = 0w. (When no minimum is found in o u r c o m p u t a t i o n s , the Wenzel equation predicts a value of cos0w above 1.0 or below - 1 . 0 , i.e., a value for 0w which has no meaning.) This agreement has an interesting consequence: The suggestion by Johnson and Dettre (6) that Wenzel's equation m a y not be valid in a gravitational field would appear to be incorrect.

O~v Varied In the second set of calculations, 0oy w a s varied while fl~ and C/s, together with HD, were held constant. A composite graph for ¢h = 45 °, f12 = 135 ° and -~D 100 Urn, with 0eY set equal to 15, 40, 65, 90, 115, 140, and 165 °, is shown in Fig. 7. Again, metastable states are present for all curves, with the range of the metastable states depending upon 0eY. Also some curves have absolute minima (i.e., the curves for 0oy = 65, 90, and 115°), while the others do not. Table I I I contains results as to the metastable states for this series. Additional results for ~1 = 300, ~2 = 150° and lid = 100 pm are listed in Table IV, and

for/31 = 60 °,/35 = 120 ° and lid = 100 vm in Table V. In all cases, the equilibrium contact angle (0e) was equal to the values predicted by the Wenzel equation. For any particular value of ~, there is a OM, see Eq. [31], and a local microscopic angle 0, see Eq. F32], such that there is a purely geometrical relation

OM = 0 + /3 -- 90 °.

For the macroscopic contact angles corresponding to the beginning and end of metastable states (when there is a 0e) it is found that, for the beginning of metastable states,

=

+3a L165°

/

÷2t

+1.t

20

4o

60

zo 1oo ro eMtoe,reesj

m

16o

OM= 0Rmi~ = 0 ~ y + / 3 1 - - 9 0 ° . [35] For the macroscopic contact angle at which metastable states end, 0~t = 0a .... = 0~y + [35 - 90 °.

[36-]

When 0oy < (132--/31)/2, there is no absolute minimum free energy. Likewise, when 0ey >___180 -- (¢/2 --/31)/2, there is no 0o. Hence, in either of these cases, OR mi~ will be 0 ° and 0n .... will be 180 °. This last conclusion is notable because it enables us, for the first time, to identify the conditions under which a macroscopic contact angle of 180 ° will be observed. Equations [35] and [36] are of similar form to equations suggested by Shuttleworth and Bailey (3) and Johnson and Dettre (6). Those authors, however, assumed local angles 0 -- 0ey for all configurations of their liquids. In our model, we have not assumed in advance that the liquid drop always meets the surface with a constant local equilibrium contact angle 0.y. Moreover, in our model we could make our calculations of free energy changes as a function of O, the instantaneous microscopic con= tact angle, whereas Johnson and Dettre (6) could not. Hence, our results have considerably greater generality than those previously reported.

m

FIG. 7. Curves of free energy versus OM, for a variety of values of O~y, (noted on the curves) for B~ = 45°, ~2 = 135 °.

[34]

HD Varied In the third set of calculations, HD w a s varied while flz and/32, together with 0eY, were held constant. An example of the type of

Journal of Colloid and Interface Science, Vol. 53, No. 2, November 1975,

THERMODYNAMICS

245

OF C O N T A C T A N G LES . I I

TABLE III METASTABLE STATES AND MINI?aA FOR VARIOUS 0ey'S, fll = 45 ° AND f12 = 0eY (degrees)

Macroscopic contact angle 0,u for beginning and end of metastable states Beginning End

Range of metastable state (degrees)

Calculated equilibrium contact angle 0~ (degrees)

30 80 90 90 90 80 30

None None 53.3 90.0 126.7 None None

0~ Corre0M Corre(degrees) sponding (degrees) sponding angle angle tilt ~ tilt (degrees) (degrees) 15 40 65 90 115 140 165

30 5 20 45 70 95 120

135 135 45 45 45 45 45

60 85 110 135 160 175 150

135 135 135 135 135 45 45

135°; HD

=

100vm

Maximum Minimum advancing receding contact contact angle 0A m~x angle 0~ ~i~ (degrees) (degrees)

60 85 110 135 160 180 180

0 0 20 45 70 95 120

T A B L E IV METASTABLE STATES AND MINIHA FOR VARIOUS 0eY'S, ~1 = 30 ° AN])~2 = 150; /]D = 100~m 0eY (degrees)

Macroscopic contact angle 0.1¢ for beginning and end of metastable states Beginning End

Range of metastable state (degrees)

Calculated equilibrium contact angle 0k (degrees)

30 80 120 120 120 80 30

None None 32.3 90.0 147.7 None None

0;~¢ CorreO~v Corre(degrees) sponding (degrees) sponding angle angle tilt ~ tilt (degrees) (degrees)

15 40 65 90 115 140 165

45 20 5 30 55 80 105

150 150 30 30 30 30 30

75 100 125 150 175 160 135

150 150 150 150 150 30 30

Maximum Minimum advancing receding contact contact angle 0A n~a* angle ORml. (degrees) (degrees)

75 100 125 150 175 180 180

0 0 5 30 55 80 105

TABLE V METASTABLE STATES AND MINIMA FOR VARIOUS 0eY'S; fll = 60 °, AND f12 ~ 120°; Hi) = 100#m GY (degrees)

Macroscopic contact angle 0.~f for beginning and end of metastable states Beginning End

Range of metastable state (degrees)

Calculated equilibrium contact angle 0~ (degrees)

30 60 60 60 60 60 30

None 27.8 60.8 90.0 119.2 152.2 None

O.~r CorreO_u Corre(degrees) sponding (degrees) sponding angle angle tilt 13 tilt (degrees) (degrees)

15 40 65 90 115 140 165

15 10 35 60 85 110 135

120 60 60 60 60 60 60

45 70 95 120 145 170 165

120 120 120 120 120 120 60

Maximum Minimum advancing receding contact contact angle Oa max angle Onmi~ (degrees) (degrees)

45 70 95 120 145 170 t80

0 10 35 60 80 110 135

Journal of Colloid and Interface Science, Vol. 53, No, 2, November t975

246

EICK, GOOD AND NEUMANN

results obtained is shown in Fig. 8. Here, #: = 30 °, #3 = 150 ° and 0or -- 90 °, and HD = 100, 50, 25, 12.5, and 6.25 pro. As can be seen from the figure, the general shape of the curve remains the same with varying HD. However, as might be expected a priori, the height of each energy barrier decreases with decreasing HD. The point, 0~, at which the minimum free energy occurs, and the range of the metastable states, are not affected by an alteration of the height, liD; this is likewise in agreement with intuitive expectations. Figure 8 is a special case, in that 0~v = 90°; under this condition, r cos 0~y = 0, so that the Wenzel angle, 0w, is independent of r, and cos 0w = 0 = cos 0o. When 0~y = 90 ° , all the free energy curves for different HD values have, as their envelope, the smooth free energy curve that is obtained when t3: =/52 = 90 °. However, when O~y ~ 90 °, the free energy curves do not touch the curve for a smooth surface at any point, nor do they have a similar shape. This fact is illustrated in Fig. 9, for #: = 30 °, ~2 = 150 °, 0~y = 65 ° and HD = 100, 50, and 25 ~m. The existence of the minima of the envelopes at the respective Wenzel angles, Figs. 9 and 10, and the fact that the computed ranges of metastable states are independent of HD, give rise to an apparent paradox. When the linear dimension parameter, HD, is allowed to reach zero, according to the mathematical model

8MIDEGREES) FIO. 8. C u r v e s of free e n e r g y v e r s u s ~M, for a s s o r t e d

values of linear dimension parameter H9. 6eY = 90°, ill=30 °,f12= 150°.

-2

-3

•

x

7\7\7

-530

\o/ \o/

3'S

4'0

,

4'S

.

2S ----s~h

5'0 ' S'5

~

~'S

F l a . 9. Free e n e r g y versus ~M for a r a n g e of v a l u e s of HD. ~e¥ = 65 ° a n d ~1 = 30 °, #2 = 150 °.

there should be a discontinuous change, from the existence of an absolute minimum of free energy at the Wenzel angle, to one at the Young angle. Therefore, there must be some smallest dimension (or set or region of dimensions) below which the present model should not be used. The considerations, which we will now develop, all apply with increasing force as the dimensions of the roughness decrease. The first is the indefiniteness of the liquidvapor surface, which is due to microripples and to the intrinsic diffuseness of the interface at any temperatures above 0°K. The latter is of the order of 10 to 50 A for organic liquids; and regarding the former, ripples due to random vibrations in any laboratory probably have a mean height of at least 100 A. In interactions with the asperities of the solid, it would be possible to neglect this indefiniteness (and treat the liquid, mechanically, as being represented by a surface of tension) only for asperities that are at least an order of magninitude larger (cf. (1)). Hence, the predictions from the model can be expected to break down when the characteristic dimensions of the

Journal of Colloid and Interface Science, VoL 53, No. 2, November 1975

THERMODYNAMICS OF CONTACT ANGLES. II

-38[

30

135

140

145

150

155

150

I65

170

L75

~0

~ M (oEcms)

FIG. 10. Free energy versus 0~ for different value of

HD. 0oy = 115%¢~ = 30% ¢~2= 150°• roughness fall below about 1000A. See Ref. (13) for an experimental report that is consistent with this prediction. The second criterion for breakdown of the model can be seen directly from the mathematics of the model itself. The model assumes the radii of curvature of the regions of intersection of planes to be small compared to the other relevant dimension of the system. This automatically limits the applicable range of the model to a range of values of HD which, we can guess, would start well above 100 A. The third component of breakdown of the model is a physical one. It is that with decreasing dimensions of asperities, the sawtooth picture of the rough surface becomes an increasingly poor approximation to any real surface. When the roughness is on the scale of, say, 500 A or less, and certainly below 50 A, we should use the detailed contour of the real surface, since this may be the ultimate that is operationally attainable. In other words, there are a great many systems for which a molecularly flat surface is not attainable; even if the surface starts as a crystallographic plane of a single crystal, it may (for certain systems) achieve a lower free energy by breaking up into steps, ledges or facets. The fourth component is that, on the microscale just noted, the assumption of energetic homogeneity breaks down. Sharp tops of ridges and bottoms of valleys are, on an atomic

247

scale, regions where adsorption will be stronger than on uniform, tilted faces. The importance of these regions will depend on the angles/31 and ~32, and on the material that constitutes the solid. Also, on this scale, the natural angles of cleavage and the growth faces of crystals would have to be considered, with real solids. Thus the discontinuity that would be expected on the basis of Figs. 8-10 is not to be anticipated in real systems. There must be a transition, from the existence of a free energy minimum at the Wenzel angle (see Figs. 9 and 10) to the minimum occurring at the Young angle (see Fig. 8). The details of this transition are too complicated to predict exactly, at this time. However, the changeover must occur in a region of magnitudes of HD well above the atomic level. As indicated above, we estimate that the transition probably occurs when HD is somewhere in the general neighborhood of 103 A. 3. SUMMARY A thermodynamic model has been developed, which predicts the macroscopic contact angle of a liquid on a tilted plate and on a rough, sawtooth surface, under the influence of gravity. I t is shown that metastable states will exist, the range of which is influenced by the contact angle 0e,z for the liquid on the smooth solid, and by the roughness as defined by the angles of inclination ~1 and t32. As ~1 and /S2 approach 90 °, the hysteresis due to roughness decreases to zero. When an equilibrium contact angle, 0e, exists, the influence of 0eY on contact angle hysteresis is predicted by OR 1,1in = 0o-~d- fll - 90 ° (Eq. [-351) and OA.... = 0ey-4- ~2 -- 90 ° (Eq. 1-361). The calculated equilibrium contact angle, 0~, obtained in a gravitational field, is the same as that predicted by the Wenzel equation. I t is to be noted, however, that the Wenzel contact angle, 0w is not amenable to experimental measurement, because of the existence of metastable states. The second parameter that measures roughness is HD, the height of the physical ridges. As HD is reduced, the height of the energy

Journal of Colloid and Interface Science,

Vol. 53, No. 2, November 1975

248

EICK, GOOD AND NEUMANN

barriers is reduced, too. However, the range of the metastable states and the occurrence (or nonoccurrence) of a minimum free energy is not altered. The minimum of free energy occurs at the Wenzel angle, irrespective of decrease in the height of the ridges, until HD reaches a magnitude that is probably of the order of l0 s A. As liD decreases below this range, there must be a transition in the location of the free energy minimum, from the Wenzel angle 0w to the Young angle 0y. There appear to be several advantages of this model over previously developed ones, such as that of Johnson and Dettre (6). First, this model explicitly includes the effects of gravity. Second, a clearer understanding of the influence of the parameters that define roughness, (i.e., the angle of the tilt /3, the contact angle 0ey, and the height He) on the existence of metastable states and hysteresis of contact angles, is obtained. The influence of these parameters on the hysteresis, and on observation of Wenzel versus Young angles, can be separated, one from the other. Third, this model has the potential of being extended, for the purpose of examining a variety of rough surfaces. Fourth, it should be possible to check the theoretical results experimentally, for example, by measuring contact angles on

a diffraction grating, using the vertical plate method (9, 12). REFERENCES 1. NEVMANN, A. W., AND GOOU, R. J., ]. Colloid Interface Sci. 38, 341 (1972). 2. WENZEL, R. N., Ind. Eng. Chem. 28, 988 (1936); J. Phys. Colloid Chem. 53, 1466 (1949). 3. SEUTTLEWORTH,R., AND BAILEY, G. L. J., Disc. Faraday Soc. No. 3, 16 (1948). 4. BIKERMAN,J. J., J. Phys. Chem. 54, 653 (1950). 5. GooD, R. J., J. Amer. Chem. Soc. 74, 5041 (1952). 6. JOHNSON, R. E., AND DETTRE, R. H., in "Contact Angle, Wettability and Adhesion," Advances in Chemistry Series, No. 43, pp. 112, 136. Amer. Chem. Soc., Washington, D.C., 1964. 7. GOOD, R. J., "Aspects of Adhesion" (D. J. Alner and K. W. Allen, Eds.), Vol 7, p. 182. Transcripta Press, London, 1973. 8. YOUNG, T., Trans. Roy. Soc. (London) 95, 65

(~805). 9. NEUMANN, A. W., Advances in Colloid Interface Sci. 4, 105 (1974). 10. NEIJ~ANN, A. W., AND GOOD R..[., in "Surface Techniques," R. J. Good and R. R. Stromberg, eds., Vol. I, Plenum, Press, N.Y. (in press). 11. EICI~,J. D., Thesis, State University of New York at Buffalo, 1971. 12. NEUMANN,A. W., Z. Phys. Chem. (Frankfurt) 41, 339 (1964). 13. NECMANN, A. W., RENZOW, D., REt;~IJT~I, H., AND RICHTER, I. E., Fortsckr. Koll. Polym. 55, 49 (1971).

Journal of Colloid and Interface Science, Vol. 53, No. 2, November 1975

Adsorption from Solutions Exhibiting Consolute Temperatures II. Adsorption of 2,6-Dimethyl Pyridine and Cetyltrimethylammonium Bromide G. P. G L A D D E N

ANn

M. M. B R E U E R

Unilever Research Isleworth Laboratory, 455 London Road, Isleworlh, Middlesex, TW7 5AB, England

Received March 10, 1975; accepted July 31, 1975 The paper describes the measurement of adsorption for a three-component solution system onto silica. The aqueous solution of 2,6-dimethyl pyridine (DMP) and cetyltrimethylammonium bromide (CTAB) shows a lower consolute temperature, the value of which increases with increasing CTAB concentration. The isotherms generally show competition between the CTAB and DMP for the available surface. However, at low DMP concentration (less than 0.5 M), both CTAB and DMP show enhanced adsorption at the expense of water. The effect of temperature is much less marked in the three-component system than in the corresponding two-component (DMP/water) system, where large increases in adsorption occur near to the consolute point. INTRODUCTION In a previous paper (1) we described the adsorption of 2,6-dimethyl pyridine (DMP) onto silica from aqueous solution. This system showed a lower consolute temperature (To) at 31.5°C, and adsorption isotherms were measured to within 2.5°C of this temperature. The composite isotherms were S-shaped, showing preferential D M P adsorption at low concentrations. Since the apparent adsorption increased with increasing temperature, the solution properties were thought to have a dominant effect on the adsorption process; similar marked increases in adsorption near the consolute point have been observed for nonionic detergents (2). The introduction of a third component can have marked effects on the value of Te, depending on its solubility (3), and for a system in which the third component is soluble in both of the other components, To would be expected to increase. I t would also be expected that such changes in solution parameters affect the amount of any given solute adsorbed from solution. The literature on multicom-

ponent adsorption is sparse. Gryazev et al. (4) have investigated the acetic acid/lauric acid/ decane system and showed that the lauric acid displaced the acetic from the substrate (in their case, a silica gel). This paper describes a method of measuring adsorption from threecomponent systems and discusses the results obtained for the cetyltrimethylammonium bromide (CTAB)/dimethyl pyridine/water/ silica system. METHODS AND MATERIALS 1. Materials

The following materials were used. 2,6dimethyl pyridine (ex Hopkins & Williams), redistilled once; cetyltrimethylammonium bromide (Cambrian Chemicals Ltd), used as received. 14CTAB (Radiochemical Centre, Amersham), used as received; this material had a specific activity of 9 Ci/mole. Aerosil 200 was used as received except for drying at 140°C to remove physisorbed water. I t was stored in a vacuum desiccator. Its surface area was 220 mS/g and an a~ plot [-after Sing et al.

249 Copyright © 1975by AcademicPress, Inc. All rights of reproductionin any form reserved.

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GLADDEN AND BREUER

(5)-] failed to show any significant microporosity. Water was double distilled from alkaline permanganate, and pH adjustments were made with sodium hydroxide.

2. Methods of Analysis To define the adsorption isotherms for the three-component system uniquely it is necessary to determine the changes in concentration of both D M P and CTAB. In the previous paper (I), an interferometric method was used to measure concentration changes by refractive index differences. Providing refractive index increments for CTAB and DMP are additive, it would be possible to adapt this method to the three-component system in the following manner. It was assumed that the following equation held : RToT = RcTA~ + RDMP,

[11

where RTOT is the experimentally determined reading on the interferometer and Rcr,~ and Rnm, are the respective contributions from the two adsorbing species. Having calibrated the interferometer for both individual solutes, solutions containing both CTAB and DMP at known concentrations were made up. It was found that the total interferometer reading obtained for the mixed solutions agreed with those predicted by adding the appropriate values from the calibration graphs. It was decided to evaluate RCTABindependently using a radiotracer technique. For this to be successful it was necessary to obtain consistent results between the radiotracer method and the interferometer. Again, solutions of known concentration were prepared and the experimentally determined interferometer readings were in good agreement with the values obtained by calculating the concentration differences from the radiotracer measurements and using the calibration to obtain equivalent interferometer readings. Thus, RToT could be measured directly, RCT~tB could be obtained from radiotracer measurements, and hence, RDMP could be calculated.

3. Phase Diagrams Solutions of known composition were stirred vigorously while being heated on a water bath. The temperature differential between solution and bath was never greater than 2°C and the rate of temperature rise was kept as low as practically possible. The temperatures of phase separation were noted.

4. Adsorption Isotherms Aliquots (10 ml) of solution were run into tubes contalning weighed quantities of adsorbent (0.2 g). The tubes were sealed and rotated in a thermostat bath until equilibrium had been reached. The bath was maintained to better than 0.01°C, and measurements were carried out at 25°C and (T, -- 2.5). The silica was centrifuged off and the change in refractive index was measured in a stoppered 1-cm cell using a Rayleigh interferometer (Hilger-Watts Ltd). Measurements were carried out at the natural pH of the DMP/CTAB/silica system ( ~ p H 9). The CTAB concentrations were measured using a liquid scintillation counting technique, it having been found that 10 or 20 rain counts gave readings with a standard deviation of 0.1 or 0.2~Vowhen 0.1 ~Ci of active CTAB was added to each of the solutions used for the adsorption measurements. A dioxan/napthalene based scintillant was used, lg of active solution being added to 14 ml of scintillant. RESULTS

1. Phase Diagrams The results are given as Fig. 1; these curves can be regarded as sections through a threedimensional composition/temperature space and it is apparent that the addition of CTAB results in a rise in the phase separation temperature across the whole composition range studied. The minimum in the curves also moves to higher DMP concentrations with increasing CTAB concentration. The curves shown are those for the onset of phase separation; this occurred very suddenly over approximately

Journal of Colloid and Interface Science, Vol. 53, No. 2, N o v e m b e r I975

ADSORPTION FROM SOLUTION

251

\\

Temp,°C

•

Q

•

\

\

"<.-.J-a I "------i

~

i

~

i I

I

Concentration of DMP (molar)

FIG. 1. Phase separation curves; O, 0.000 M CTAB; I , 0.088 M CTAB; A, 0.016 M CTAB; O, 0.027 M CTAB.

0.1°C. Before this point slight haze formation was sometimes observed but it was difficult to determine the exact temperature at which this first occurred.

2. Adsorption Isotherms Figures 2 and 3 give the adsorption of DMP and CTAB at 25°C. The base plane of the three-dimensional diagram represents the bulk concentrations in equilibrium with the adsorbed phase. The surface excess at a given concentration is shown by a vertical dotted line connecting the point with the appropriate concentration. Both the surface excesses of CTAB and DMP show a maximum, the one for CTAB adsorption occurring at lower DMP concentrations. The general tunnel shape for the DMP adsorption is shown more clearly in Fig. 4, which gives sections cut through the three-dimensional diagram at 0.00, 0.01, and 0.02 M CTAB. Increasing the amount of CTAB is seen to reduce the amount of DMP adsorbed, irrespective of the DMP concentration. On the other hand, CTAB adsorption is independent of CTAB concentration at low DMP concentrations (Fig. 5). Beyond Point

A, the sections diverge with increasing amounts of DMP and a strong dependence on CTAB concentration develops. The effect of temperature on the adsorption of both DMP and CTAB can be seen by comparing Figs. 2 and 3 with Figs. 6 and 7. The latter give isotherms at ( T o - 2.5)°C. Because of the haze formation, particularly at lower CTAB concentrations, it was not possible to obtain any measurements nearer to To, and sometimes at DAmP concentrations greater than 2.0 M, phase separation was found to have occurred during an adsorption experiment. In the two-component DMP/water system there is seen to be a considerable increase in the amount of DMP adsorbed as the temperature is increased. The effect of CTAB is now much more dramatic, producing a rapid fall in the DMP adsorption with increasing CTAB concentration to a value slightly lower than that at 25°C (Fig. 8). Figure 9 gives the adsorption of CTAB at these higher temperatures. There is no value for the adsorption in the absence of DMP since there is no appropriate value of T~ for the CTAB/water system. However, the adsorption would not be expected to decrease

Journal of Colloid and Interface Science, Vol. 53, No. 2, N o v e m b e r 1975

252

GLADDEN AND BREUER

)-

I

t

7

E

W Ill

o II

?MP(molar)

I

iI I I I i

II I i I I I I

I

Concentration of CTAB (molar)

FIG. 2. Adsorption of DMP at 25°C. very much from the 25°C value for a small increase in temperature. The change in adsorption of CTAB due to increased temperatures is very slight, but there appears to be a slight enhancement. As in the results for adsorption at 25°C, the isotherms diverge with increasing D M P concentration.

DISCUSSION

1, Phase Diagrams The results given as Fig. 1 and described above indicate a steady rise in the phase separation temperature. I t is not the main purpose of this paper to discuss the phase

Journal of Colloid and Interface Science, Vol. 53, No. 2, N o y e m b e r 1975

ADSORPTION FROM SOLUTION

253

?

¢ £

0"3

I 1 I

>?1 ~c~E 0"2 Zi I

I

I

,

,

'

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I

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I

"qo005\ \t

Concentration (Molar)

I

?

I 2Jo

Ti, I

of

?

II II jl

J

lit I I

IOonc. DMP 3!0 I (Mo la r)

4

;,

I

i

h I

I

X ;

[

"L

',

o.o2-~

Fzc,. 3. Adsorption of CTAB at 25°C. separation behavior though some comments should be made about these results. In a two-component system, the minimum in the phase separation curve corresponds to To, but this does not necessarily hold for a threecomponent system (6). However, since no points of inflexion were observed for the

system under study, the critical point was taken as the lowest point of phase separation (7). The addition of CTAB also moves the value of Tc to higher CTAB concentrations and Figs. 10 and 1l show lhat both Tc and the mole fraction of CTAB at the critical point vary

Journal of Colloid and Interface Science, VoL 53, No. 2, N o v e m b e r 1975

254

GLADDEN AND BREUER

"6 E

-8 <

11o

2!o

31o

Concentration of DMP (molar)

FIG. 4. Adsorption of DMP at 25°C; O, 0.00 M CTAB; A, 0.01 M CTAB; I , 0.02 M CTAB.

linearly with the mole fraction of DMP. This suggests that the interaction energies between the components are independent of concentration over the range investigated. Very little work has been reported on the phase separation behavior of mixed surfactant systems but it has been shown that the addition of anionic detergent increases the T~ of a nonionic detergent solution (8). The authors concluded that the anionic detergent is capable of solubilizing solute that is on the point of precipitation and it is probable that the CTAB

0.5[

acts in a similar way in the D M P / w a t e r system. Two theories are available for discussing the behavior of ternary systems that exhibit a consolute point: the regular solution theory (3) and the Flory-Huggins model (9). The former only predicts an upper consolute temperature and is therefore not applicable in our case. Both these theories describe the behavior of ternary solutions in terms of three parameters and these can be evaluated from the variation of the consolute point as a function

A

0-4

~1

o.'

•

.~_ g ~

(?'1

1!o

2!o

' 3!d

Concentration of DMP (molar)

FIo. 5. Adsorption of CTAB at 25°C; O, 0.005 M CTAB; I 0.01 M CTAB; j,, 0.02 M CTAB. Journal of Colloid and Interface Science, VoI. 53, No. 2, N o v e m b e r 1975

o

o

+

&

s:

B+

9

b,

b-3

°1

o

o

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---41

b,5 ¢J1 tj1

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o

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>.

256

GLADDEN AND BREUER

9

0.3

?

l I I I I

E z!

iz II

Conc,DMP "X. (molar) " ~

o,oo~

II II 11 I

].o I

I I I I

t I

I I

2.0

II I,

II II

I

[I

I

Ooncentratio£ m°~o.m 5 of CTAB (molar)

?

I

I 1

FIG. 7. Adsorption of CTAB at (To -- 2.5)°C. of composition and from the Gibbs-Duhem equation. The adsorption of any component is governed by the difference in chemical potential between the solution and the surface and, provided thaL the solution behavior agrees with the appropriate theory, it is possible in principle to

predict the adsorption characteristics of solutions exhibiting consolute points.

2. Adsorption Isotherms Figure 4 shows sections through the adsorption isotherm of DMP at 25°C. In the absence of CTAB the maximum ~rises from the

Journal of Colloid and Interface Science, V o l . 53, N o . 2, N o v e m b e r 1975

ADSORPTION FRON[ SOLUTION

J

257

~o

f

C& 0J -5 E


•

n~~

"~,,

,!0

2!o

3'

Concentration of DMP(molar)

Fro. 8. Adsorption of DMP at (T~ - 2.5)°C; e, 0.00 M CTAB; A, 0.01 M; II, 0.02 3I CTAB.

competition between D M P and water molecules for the available surface. If the D M P molecules are assumed to lie flat, then monolayer coverage is approximately 0.57 mmole/g. The amount of D M P adsorbed is considerably more than this and in an earlier paper (1) a model was developed from which the thickness of the adsorbed phase could be estimated. The experimental isotherm is described by the equation : N o A x A / m = : V j x c - NcSxA,

where i is the number of adsorbed layers. Initially i is set at unity and the equations are solved. The number of layers is then increased either when the layer is filled with D M P or when the individual isotherm passes through a maximum since such an occurrence is not thermodynamically allowable for a two-corn-

[-2-]

where m is the weight of solid, No is the total number of moles present, AXA is the change in mole fraction, and XA, Xc are the equilibrium mole fractions in solution. When adsorption is limited to a monolayer then Nx s and N e s (the number of moles adsorbed per gram) can be calculated by solving Eq. [2~ with

.,1 E

o Ol

A = NasaA + NeSac,

E3~

where A is the available surface area and aA, ac are the respective molar areas of D M P and water. I n the case of multilayer adsorption, Eq. [ 3 ] can be replaced by A / i = NASaA + NcSac,

¢o

2!o

Concentration of DMP (molar)

FIo. 9. Adsorption of CTAB at ( % - 2.5)°C; O. 0.005 M CTAB ; NL0.01 M CTAB ; A, 0.02 M CTAB.

Journal of Colloid and Interface Science, Voi. 53, No. 2, N o v e m b e r 1975

258

GLADDEN AND BREUER

10

CTA B E mole froction

(*'°~)

4

2

•

•

0 0.04

006

008

010

DI'4P mole fraction

Fro. 10. Effect of CTAB on the critical point: effect on concentration.

ponent system. Using this model, the twocomponent D M P / w a t e r system is found to form a surface phase six molecular layers thick. The introduction of CTAB reduces the amount of D M P adsorbed, though it is difficult to describe this in any quantitative manner since the simple two-component model described above cannot be extended to the three-component system. This is for two reasons. First, a monolayer of CTAB will be "thicker" than a monolayer of D M P or water,

whereas for the two-component system, D M P adsorbed flat onto the surface is very similar in "thickness" to a water molecule. The second reason is that in a three-component system it is possible to have maxima in the individual isotherms; in a two-component system, these are not allowable. Such a maximum in a threecomponent system is demonstrated clearly by the CTAB adsorption. The curves shown in Figs. 3 and 5 are, strictly speaking, surface excesses (NoAx~/m.), but it can be shown

Tc (°C)

3,i

0.04

I

01.06

s

0108

I

O!lO

I

DMP mole fraction

FIG. l 1. Effect of C T A B on the critical point: effect on temperature; O, 0.000 M C T A B ; I , 0.008 M, C T A B ; A , 0.016 M C T A B ; V , 0.027 M CTAB. Journal of Colloid and Interface Science, Vol. 53, No. 2, November 1975

ADSORPTION FROM SOLUTION

excess of water increases in magnitude (Fig. 12). At higher DMP concentrations, the competition between CTAB and DMP tends to result in a leveling of the surface excess of water. The divergence of the isotherms at higher DMP concentrations (Fig. 5) probably reflects changes in the state of the CTAB in the mixed solvent system. In the absence of DMP, the CTAB concentration is above the critical micelle concentration (9 X 10.4 M) and the adsorption isotherm has reached the plateau region. As the amount of DMP in the mixed solvent is increased, the cmc of the CTAB moves to higher values. Consequently, the concentration range studied tends to represent the approach to the plateau region and the adsorption becomes dependent on the bulk CTAB concentration. Adsorption of both DMP and CTAB at higher temperatures show similar effects to the adsorption at 25°C. The general shapes and trends of the isotherms have already been described. To obtain a more detailed explanation of the effect of temperature on the adsorption process would also require information about micellization and solubilities in the bulk solutions. For example, the rise in T~ with CTAB concentrations indicates an increased solubility that could lead to a fall in adsorption if the use of (T~ -- 2.5) is not an adequate ap-

readily that Axe is related to the number of moles adsorbed per gram, NA s (for DMP), NB s (for CTAB), and N c s (for water) by the equation /XxB

:¥BS(xA + XC) -- ~CB(2¢As + NO' s) -

.

m

[4-1

~¥o

Since x~ is very small, Nof~x~/m is a close approximation to N~ ~ and the maximum in (NozXx~/rn) reflects a real maximum in the adsorption of CTAB. In the absence of DMP, the amount of CTAB adsorbed is 2.25 X 10-4 moles g-l; this figure is much less than a close packed monolayer (approx 7.3 X 10.4 moles g-l, assuming an area of 0.5 mn2/molecule). Adsorption of less than a monolayer at high pH's has been reported previously (10). Suspensions prepared at the solution composition corresponding to the maximum CTAB adsorption were found to separate much more rapidly on standing than any others, presumably because of the greater hydrophobic nature of the silica surface, though even at this point the amount of CTAB adsorbed is still less than a monolayer. The maximum in the adsorption of CTAB is at the expense of water rather than DMP. Figure 4 shows a steady increase in the surface excess of DMP in the concentration region around 0.5 M DMP while the negative surface

Concentration o,

259

of D M P ( M o l a r )

1.0

2.0

i

c~ E

Q

g

-o ~c

Q Q

Fro. 12. Adsorption of water at 25°C. Journal of Colloid and Interface Science, Vol. 53, No. 2, N o v e m b e r 1975

260

GLADDEN AND BREUER

proximation for a reduced temperature scale. This is specially true since the lower consolute point moves to both higher concentrations of D M P as well as higher temperatures. The chemical potential of CTAB is a function of the concentrations and temperatures and can be described by / sue\ - -

concentration is increased a rise in the cmc results in the development of a marked CTAB concentration dependence. Increasing the temperature to (T~ -- 2.5) has little effect on the amounts adsorbed in the three-component system, whereas a marked increase in adsorption with temperature occurs in the twocomponent (DMP/water) system.

AT

ACKNOWLEDGMENTS

If the chemical potential of the CTAB on the surface is considered to be constant over the temperature range under study, then the adsorption profiles represent changes in the bulk chemical potentials. Comparison of Figs. 5 and 9 show that the curves are very similar and consequently (6Ue/6T)CAis very small. Consequently, the divergence of CTAB isotherms at (Te -- 2.5) is caused for the same reason as at 25°C rather than any use of concentrations instead of activities as the abscissae. SUMMARY

The CTAB has a considerable effect on the lower consolute temperature of the D M P / water system. Competitive adsorption occurs for this three-component system and the effects are most marked at high concentrations of dimethyt pyridine. The adsorption of CTAB is unaffected by the bulk CTAB concentrations at low D M P concentrations, but as the D M P

The authors acknowledge the technical assistance of Miss R. Drozdek and wish to thank Unilever Limited for permission to publish this paper. REFERENCES 1. BREUER, M. M., AND GLADDEN, G. P., in "ACS Colloid Symposium, Austin, Texas," 1974. 2. CORKILL, J-. M., GOODMAN,J. F., AND SYMONS, P. C., Trans. Faraday Soe. 65, 287 (1968). 3. PRIGOGENE,I., ANDDEFAY, R., "Chemical Thermodynamics," Longmans, London, 1954. 4. GRYAZEV, N. N.~ et al., Russ. J. Phys. Chem. 44, 1198 (1970). 5. BHA~BHANI,M. R., CU~TINO,P. A., SIN% K. S. W., AND TURK, D. H., J. Colloid Interface Sci. 38, 109 (1972). 6. TOMPA, H., "Polymer Solutions." Butterworths, London, 1956. 7. COWlE, J. M. G., AND McEwAN, I. J., Trans. Faraday Soc. 70, 171 (1974). 8. MACLAY,W. N., f . Colloid Sci. 11, 272 (1956). 9. FLORY, P. J., "Principles of Polymer Chemistry." Cornell University Press, New York, 1953. 10. BLACKMAN,L. C. F., AZCDHARROP, R., J. Appl. Chem. 18, 43 (1968).

Journal of Colloidand Interface Science, Vol. 53, No. 2, November 197.5