807
Langmuir 1992,8, 807-814
Concentrated Emulsions. 2. Effect of the Interdroplet Film Thickness on Droplet Size and Distortion in Mono- and Bidisperse Face Centered Cubic Packings Doble Mukesh, Ashok K. Das, and Pushpito K. Ghosh'yt Alchemie Research Centre, Thane-Belapur Road, Thane 400601, Maharashtra, India Received March 1, 1991. In Final Form: December 5, 1991
Modeling studies of concentrated emulsions have been extended to include the simultaneouseffects of polydispersity and distortion on 4" (maximum attainable volume fraction) and Abt (normalized total interfacial area) in bidisperse and tridisperse face centered cubic (fcc) packings, with smaller droplets occupying the interstices formed by larger drops. The distortion model considered is the one yielding flat circular faces between interacting drops, the radius of the face increasing with increase in distortion. For constant values of 4 and volume average droplet radius, Abt follows the trend mono > bi > tri. The effect of the interdroplet film thickness, t , on emulsion structure has also been considered. It is shown that emulsion droplets get increasingly more distorted as the ratio, a,of this film thickness to droplet radius (r)is increased. This distortion becomes especiallypronounced when 4 exceeds0.9. Although the distortion of all droplets in an emulsion increases with increasing 4 and a,packing considerationsmandate variations in the extent of distortion of different sized droplets in a polydisperse matrix. In proportion to their size, smaller droplets in the bidispersefcc packing would be more distorted than larger drops while the opposite might be expected from energetic considerations. Since a varies inversely with r for constant values of t , smaller droplet sizes and higher volume fractions are likely to promote monodispersity, a prediction partially supported by freeze fracture scanning electron micrographs of emulsion explosive formulations. The minimum possible size of droplets for given values of 4 and t have also been calculated in the present work.
Introduction Concentrated emulsions, foams, and dispersions possess internal phase volume fractions in excess of the monodisperse close packed limit of 0.74. Many such systems find important applications,'+ and there is both theoretical and practical interest in their structure and properties.'J0--'9 Concentrated interfacial systems might also be produced transiently, as during the formation of coatings from + Present address: Research Centre, IC1 Specialties, P.O. Box 42,
Hexagon House, Blackley, Manchester M9 3DA, England. (1) Hoffmann, H.; Ebert, G. Angew. Chem., Int. Ed. Engl. 1988,27, 902. (2) Platz, G.; Ebert, G. In Polymer Reaction Engineering; Reichert, K. H., Geiseler, W., Eds.; Huthig: Heidelberg, 1986. (3) Ruckenstein, E.; Ebert, G.; Platz, G. J.Colloid Interface Sci. 1989, 133, 432. (4) (a) Ruckenstein, E.; Park, J. S. J . Appl. Polym. Sci. 1990,40,213. (b) Park, J. S.; Ruckenstein, E. J. Appl. Polym. Sci. 1989, 38, 453. (5) Ishida, H.; Iwama, A. Combust. Sci. Technol. 1984, 36, 51. (6) (a) Williams, J. M.; Wrobleski, D. A. Langmuir 1988,4,656. (b) Williams, J. M. Langmuir 1988, 4, 44. (7) (a) Bluhm, H. F. US.Patent 3,447,978, June 1969. (b) Bampfield, H. A.; Cooper, J. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1988; Vol. 3, p 281. (8) Lummus, J. L. US.Pat. 2 1953, 661,334. (9) Williams, B. B.; Gidley, J. L.; Schecter, R. S. In Acidiring
Fundamentakr Monograph;S.P.E. Henry L. Doherty Series;SPE-AIME Dalla~,TX, 1979; Vol. 6, pp 58-59,29-37. (10) Clayton, W. In The Theory of Emulsions, 3rd ed.; Blakiston: Philadelphia, PA, 1935. (11) Groves, M. J.; Scarlett, B. Nature (London) 1965,207, 288. (12) Sherman, P. In Emulsion Science; Sherman, P., Ed.;Academic Press: New York, 1968. (13) (a) Lissant, K. J.;Peace,B. W.;Mayham, K. G.J.Colloidlnterface Sci. 1974, 47, 416. (b) Lissant, K. J. In Emulsions and Emulsion Technology;Lissant, K. J., Ed.;Marcel Dekker: New York, 1974;Chapter 8. (c) Lissant, K. J. J. SOC.Cosmet. Chem. 1970, 21, 141. (d) Lissant, K. J. J . Colloid Interface Sci. 1966, 22, 462. (14) (a) Princen, H. M.;Aronson, M. P.;Moser,J. C.J. Colloidlnterface Sci. 1980,75,246. (b) Aronson, M. P.; Princen, H. M . Nature (London) 1980,286,370. (c) Schwartz, L. W.; Princen, H. M. J. Colloid Interface Sci. 1987,118,201. (d) Princen, H. M. J.ColloidInterface Sci. 1983,91, 150. (15) Princen, H. M. J. Colloid Interface Sci. 1990, 134, 188.
polymer latexes.2O As part of our ongoing program of research on emulsion explosives,we have been attempting to identify possible factors which control the structure of concentrated emulsions during their formative stage, with a view toward gaining better control over parameters such as the degree of dispersity and the shape and size of droplets. In a recent paper, we modeled the effects of polydispersity and soft sphere distortion on packing in ordered systems and provided evidence of transformations in packing geometry-from spherical polydisperse to distorted "monodisperse"-of an emulsion prepared at 4 0.88.19 Possible causes underlying such changes were also identified. That effort, however, did not consider the simultaneous effect of polydispersity and distortion. A further limitation was the neglect of the continuous phase film between droplets, i.e., droplets of the dispersed phase were assumed to be in direct contact. This interdroplet film has now been incorporated in the face centered cubic (fcc) packing model along with the simultaneous effects of distortion and polydispersity. However, only a cursory consideration is given to the energetics of such systems. Among other results, consequences of the interdroplet film on the distortion and limiting size of dispersed droplets are discussed.
-
Modeling Mono-, bi-, tri-, and tetradisperse packings of the fcc unit cell yield values of 0.7405, 0.7799, 0.8916, and 0.8923, respectively.21 Increased values of 4 may also be (16) (a) Barby, P.; Haq, 2.European Patent 0060138 (to Unilever), September 1982. (b) Litt, M. L.; Hsieh, B. R.; Krieger, I. M.; Chen, T. T.; Lu, H. L. J . Colloid Interface Sci. 1987, 115, 312. (17) Khan, S.; Armstrong, R. C. J.Non-Newtonian Fluid Mech. 1986, 22, 1. (18) (a) Kraynik, A. M.; Hanson, M. G. J. Rheol. 1987, 31, 175. (b) Aubert, J. H.; Kraynik, A. M.; Rand, P. B. Sci. Am. 1986,254, 88. (19) Das, A. K.; Ghosh, P. K. Langmuir 1990, 6, 1668. (20) Eckersley, S. T.; Rudin, A. J . J. Coating Technol. 1990, 62, 89.
0743-7463/92/2408-0807~03.00J0 0 1992 American Chemical Society
Mukesh et al.
808 Langmuir, Vol. 8, No. 3, 1992 ,-an
eliminated spherical
SIDE VIEW
A
c
B
Figure 1. Effect of increasing distortion on the radii of hypothetical sphere and of circular faces on ita surface.
obtained by distorting droplets. It has been previously shown that the preferred mode of distortion is the one which produces a flat circular face between two interacting droplets, the radius of the face increasing with increasing distortion (Figure 1). Distortion by this mode has been experimentally 0bserved6J~.~2and assumes a finite contact angle between the interdroplet film and the plateau border.14* To conserve its volume during distortion, the noncompressible droplet assumes the shape of a hypothetical sphere whose radius is greater than that of the original sphere and from which segments are eliminated as shown in Figure 2A. For an original sphere of radius r, hypothetical sphere radius, Rr, following distortion, and contact radius Pr, the volume of an eliminated segment, V,, is given by (see Figure 2B)
€=’,( ( 3 r / 6 ) P 2 [ R- (R2- f12)”2]
eliminated segment flat ended or right circular cone
w
+
( a / 6 ) [ R- (R2- p2)1/213)r3 (1) If the hypothetical sphere has nl eliminated segments of radius Plr, n2 eliminated segments of radius 82r, and so on, the total volume of eliminated segments may be given by
( r / 6 ) [ R- (R2- P?)1/213)nir3 (2) (i = 1 , 2 , 3 , ...) while the volume conservation relationship is given by I
In polydisperse systems, the radius, r,, of a droplet may be expressed in terms of the radius, rl, of the largest sized sphere, i.e. r, = mnrl. For contact radii and hypothetical sphere radius Rnrl of a distorted r, droplet, eq 3a may be written in the more generalized form
Note that eq 3b reduces to eq 3a when a system is monodisperse, i.e., m, = 1. The distorted droplet in Figure 2A may be dissected into right circular cones with base area (21) (a)The radius ratio and maximum occupancyratio of r l , r2, r3, and r, spheres in the close packed limit are 1.00:0.41:0.290.04and 43:21:72, respectively.19 (b) For a diacwion on random packing and parking in two- and three-dimensions see Cooper, D. W. J. Colloid Interface Sci. 1987, 119, 442, and references therein. (22) Donaldson, C. C.; McMahon, J.; Stewart, R. F.; Sutton, D. Colloids Surf. 1986, 18, 313.
( B)
Figure 2. (A) Schematic depiction of a distorted r. droplet in
a polydisperse emulsion. All measurements in the figure are expressed in units of r l , the radius of the largest sphere in the emulsion. Also, in the droplet above, nl = 2 and n2 = 1. (B) Figure illustrating the dissection of the hypothetical sphere into right circular and curved ended cones and the relationship between R , 6,and h.
~ ( P i rand ) ~ altitude (Rr - h) = (R2- Pi2W2r,and curved (ended) cones whose collective base area is given by
E A = A,[Vr-
cvR]/vR
= 4 ~ ( R r ) ~ { ( 4 / 3-) xr (r1~/ 3 ) n i * P ; ( R 2 1
where Vrand VRrepresent the volumes of the original and the hypothetical spheres, respectively, AR is the area of the hypoth_eticalsphere; a is the area of a curved (ended) cone, and VR is the volume of a flat (ended) cone in the hypothetical sphere. The total area of a drop, Adrop, is then given by
Langmuir, Vol. 8, No. 3, 1992 809
Concentrated Emulsions
1 * 4r(R2
1 = 4r/2‘I2
-
p2)’12/2112
1 = r[4(Ri2
B
A
-
p i 2 ) 1 1 2+
2=]/21/2
C
Figure 3. Depiction of a face of a monodisperse fcc pack showing (A) undistorted spheres, (B)distorted spheres in direct contact, and (C) distorted spheres separated by a continuous phase film of uniform thickness. Note that volmes of individualspheres have been conserved in A, B, and C and that the unit cell volumes follow the trend A > B = C. All measurements in the figures are expressed in terms of the original spherical radius r. Expressions for the edge length, 1, of the cell are also given below the figures. The more general expression covering polydispersesystems is given by eq 5c [Adrop],,= rrl2{CniPi:
+ (1/Rn)[4m,3-
in parts B and C of Figure 3. However, unlike Figure 3B, wherein droplets of the dispersed phase are in direct contact, droplets in Figure 3C are separated by a film of continuous phase whose thickness, t , is given by ar, a being anumerical parameter. It is apparent that this film alters the degree of distortion of a droplet, i.e., when a # 0, j3 p’ (p’ > j3)and R R’ (R’ > R) when the droplet volume is maintained constant. Equation 6 may now be rewritten as
-
Specific equations for mono- and bidisperse fcc packings, including distortion and bilayer, are derived below. MonodisperseDistorted without Interdroplet Film. Each droplet in the monodisperse fcc lattice (Figure 3A) is surrounded by 12 touching neighbors. Assuming equal distortion by all neighbors (Figure 3B), ni = nl = 12. With j3! = 81 = 6, eq 3a may be rewritten as 2[R3 - 11 = 902[R - (R2- j32)’/21 + 3[R - (R2- j3
2 1/2 3
1
(6) Equation 6 may be numerically solved to obtain R as a function of j3. The unit cell volume, Vfcc(p), volume fraction, 4p,droplet area, Adrop(@), and normalized total interfacial area, Atot(p),may then be calculated as a function of j3 (eqs 7-10). VfCc(@) = 22.6274(R2- j32) 3/2 r3
(7)
4p = [ 4 ( 4 / 3 ) ~ r ~ l / V = ~0.7405/(R2 ~ ~ ( ~ ) - /32)3/2 (8) Adrop(@) = 12rr2(B2+ (l/R)[(l/3) - b2(R2- j32)’/2])
(9)
4
2 [ ~ ’ 3- 11 = gp’2[~’- ( ~ ’ 2 - 8”31/2]
+
3[R’ - (R’2 - p’2)’/213 (11) while the conservation of unit cell volume is satisfied by eq 1 2 (see Figure 3)
+
= l3 = [4(Rt2- p’2)1/2 22.6274(R2- j32)3/2r3 2 a 1 ~ r ~ / (1.2) 2~/~
I being the edge length of the cell. R’ and p’ can be obtained as a function of a by simultaneously solving eqs 11 and 12 for given values of 8, for which the left-hand side in eq 12is constant. As before, Vfccand 4 are given by eqs 7 and 8 while the droplet area is given by Adrop(@,a) = 12rr2(8’2+ (1/R’)[(1/3) F2(R” - p’2)1/23)(13) and normalized total area by Atot(@,a)= 4Adrop(@,a)N*
- (4)12rr2(p2+ (l/R))[(l/3)- 02(R2- /32)’/21 22.6274(R2- j32)3/2r3
(lob)
N* in eq 10a denotes the number of unit cells in a normalized volume (N* = l/Vfcc(p)for volume = unity). This is necessary to relate all data/calculations to the same total volume. For undistorted droplets, Le., when j3 = 0 and R = 1,eq 10b reduces to the simpler relationship, eq 10c.19 Atot(p=O) = 2.22/r (10c) Monodisperse Distorted with Interdroplet Film. The values of Vfccand 4 are the same for packings shown
(14)
These equations are obtained by substituting j3 and R in eqs 9 and 10 with p’ and R‘, respectively. Bidisperse Distorted without Interdroplet Film. There are four spheres of radius r1 and three spheres of radius r2 (r2 = 0.414rl) in the bidisperse fcc unit cell lattice (Figure 4A). The packing fraction may be increased beyond0.78 by distorting the droplets (Figure 4B). I t can be shown that each rl droplet is contacted by 12 rl and 6 r2 droplets, while an r2 droplet is contacted only by 6 rl droplets. Following distortion, rl Rlrl, r2 R2r1, rl-rl contact radius = &lrl (ni = nl = 12), and r1-r2 contact radius = j312rl = j321rl (ni = n2 = 6). The volume conservation expressions for rl and r2 droplets are given via eq 3b as
- -
Mukesh et al.
810 Langmuir, Vol. 8, No. 3, 1992
B
A
L
Figure 4. Depiction of a face of a bidisperse fcc pack showing (A) undistorted spheres, (B)distorted spheres in direct contact, and (C) distorted spheres separated by a continuous phase film of uniform thickness. Note that volumes of individualspheres have been conserved in A, B, and C and that the unit cell volumes follow the trend A > B = C. All measurements in the figure are expressed in terms of the original radius, rl, of the largest sphere.
4[R13 - 11 = (18j3112[R1- (R12- /3112)1/21+ 6[R1- (R12- 8112)1/213) + {98,:[R1 - (R12- 81:)1/21 + 2 1/2 3 3[R1- (R12- 012 1 1 1 (15)
+
(23)
while eqs 15-17 may be rewritten as
+
and 4[R:
+
[ ( R , ~- 81,2)1/2 ( R , ~- p1;)1/213 = [ (Rl’2 - /312/2)1’2 (R ’2 - pl;2)1/2 +
- 0.41431 = 9&:[R2
- /3122)1/21+ 3[R2 - (R22- &22)1/213 (16)
- (R:
respectively, while the Pythagorean relationship between edge length (right-hand side) and face diagonal (left-hand side) of the unit cell is shown [4(R12- &12)1/2]2
+
= 2[2(R12- &:)1/2
2(R: - &22)1/212 (17)
811 may be selected as an independent variable and the dependence of R1, R2, and 8 1 2 on ,811 obtained by simultaneously solvingeqs 15-17. The unit cellvolume, VfcC(pl1), volume fraction, 4pI1, surface areas of rl and r2 droplets, and normalized total interfacial area are given by eqs 1822. Vfcc(B1l) = 22.6274(R12- pll
2 3/2
3
rl
(19)
[Adrop(r,~l~ll = *~1~(6812~ + (1/R2)[4(0.414)3-
- P122)1/21)(21)
Atot(flll) = ~4[Adrop(r1)]f111 + 3[Adrop(rz)]flll)N*
(22)
Bidisperse Distorted with Interdroplet Film. Figure 4C illustrates the effect of the interdroplet film on droplet distortion in bidisperse fcc. Assume that arl is the interdroplet film thickness both of the rl-rl and r1-r~ interfaces. Let R1- Rl‘, 811 fill’, R2 R2‘, and 812 812‘ when a # 0. The conservation of unit cell volume is satisfied by
- -
-
+
9P1i2[Ri - (Ri2- @12’2)1/2] 3[R2’ - (Ri2- P12/2)1/213 (25)
+
[4(R1’2- /311’2)112 + 2aI2 = 2[2(RIr2- /31,12)1/2 2(Ri2- &2/2)1/2 + 2aI2 (26) For constant values of Vf,, and 4, the left-hand side of eq 23 is constant. Rl’, R2’, j311’, and 812’ may then be obtained as a function of a by solving eqs 23-26. For a given 811, the unit cell volume and volume fraction are given by eqs 18 and 19 while the areas of the rl = *r12(12811’2+ 681,” + (1/R1)[4 12811’2(R1’2- Pll’2)1i2- 6P12’2(R1’2- 812‘2 11/2 I) (27)
[Adrop(rl)lflll,a
and
6812’ + (1/R1)[4 [ ~ ~ r o p ~=r l*r12(128112 )l~l, 12Pl12(R12- P112)1/2 - 68122(R12- P122)1/21)(20)
681:(R:
4[Ri3 - 0.4143]
(18)
4B11= [4(4/3)*r13 + 3(4/3)?r(0.414r1)31/Vfcc(B1l) = 0.7798/(RlZ- 8112)3/2
4[R1’3 - 11 = (18811’2[R1’- (Rl’2- &1’2)1/2] 6[R1’ - (Rl’2 - ,811’2)1/213) + (98,i2[R1’ (Rl‘2 - &’2)1/21 + 3[R1’ - (Rl’2 - 81i2)1/213) (24)
r2
[Adrop(r2)lflll,a
= *‘12(6P12’z
+ (1/R2’) [ 4 ( 0 w 36P12’2(R2/2- P1i2)1/21) (28)
droplets for a # 0 are obtained by replacing Rt, R2, 811, and 8 1 2 with Rl’, Rz’, fill’, and 812’, respectively, in eqs 20 and 21. Finally, Abt is given by = (4[Adrop(rl)]flll,a + 3[Adrop(rz)],?ll,rr)N* (29) Distorted Tridisperse Fcc. Equations for the tridisperse case, with and without the interdroplet film, can be obtained in amanner analogous to that described above for bidisperse fcc. In this case, however, rl drops are contacted by 12 rl, 6 r2, and 24 r3 droplets, r2 droplets by 6 rl and 6 r3 droplets, and r3 droplets by 2 r1 and 1 r2 droplet. The volume conservation relationship for each droplet (eq 3b) and the relationship between face diagonal and edge length can be derived as before. In addition, eq 30 [Atot]Bll,a
Concentrated Emulsions
Langmuir, Vol. 8, No. 3, 1992 811
+
21/2[(R12 - p122)1/2 (RC2 - /3122)"2] =
"I
P
2[(RZ2 - /3232)1/2 + 2(R: - &32)1'21 (30)
may be derived. However, since the number of dependent variables (R1, R2, R3, 611, P12, &, P 2 3 ) exceeds the number of equations, only an approximate solution of the tridisperse case is possible, e.g., with the assumption that p13
09-
a 085-
= P23.
Results and Discussion Distorted Fcc without Interdroplet Film. Numerical solution of eq 6 yields R as a function of /3 for monodisperse emulsions. Values of Vf,,, 4, Adrop,and Atot can subsequently be obtained for given values of /3 and R (eqs 7-10). Similarly, solution of the simultaneous equations 15-17 for the bidisperse case yields R1, R2, and P12 as a function of 811. As in the monodisperse case, the unit cell volume,volume fraction, surface areas of r1 and r2 droplets, and normalized total interfacial area can also be computed (eqs 18-22). Relevant data on the two systems are compared in Figure 5. I t can be seen from Figure 5A that 4 increases as /3 or p 1 1 does. The concomitant increase in R is offset by an overall decrease in (R2- p2)1/2,the latter term ultimately controlling the unit cell dimension (eq 7). I t can be seen from Figure 5A that the 4 vs fl (or curves exhibit points of inflection at 611 and /3 values of 0.34 and 0.37, respectively. These numbers tally closely with those reported by P r i n ~ e n 'and ~ ~ Williams.6b The inflection point arises due to the uneven decrease in the (R2- /32)1/2 term with increasing distortion.lg To accommodate a given value of 4, the extent of distortion required at the rl-rl interface is less in bidisperse than in monodisperse emulsions. The distortion felt by r2 dropiets is, however, much greater. Thus, as can be seen from Figure 5A, where /311 = 0.2 at 4 = 0.83, /311:p12 = 2:3 from Figure 5B when the radii at the rl-r-1 and rl-r2 interfaces are expressed in terms o f r l and r2, respectively. The relative distortion of r2 is still more severe at higher values of 4, e.g., P11:P12 = 1:1.7 at 0.94.23 These results based on packing considerations are illustrated by Figure 4B which has been geometrically derived and which also indicates a higher level of distortion of r2 droplets. Although the above arguments mandate a higher degree of distortion of the smaller drops, the opposite is expected based on energetic considerations using the Laplace equation
-
02
(23) In the limit, rz spherescould develop circular faces witha maximum radius = 0.805rz. This value is deduced by transforming the rz sphere into a cube (note that each r2 is contacted by six rl spheres) of e ual volume and estimating the radius of the inscribed circle on a face. %he maximum value of 812 obtained from numerical solution of eqs 15-17 (see Figure 5B)tallies with the geometrically derived limit. However, real systems are unlikely to exhibit such extreme distortions due to energetic considerations.
04
05
(
Po, Pi1
2 826-
2.q
/
20
16
07
where pintand pextare the internal and external pressures acting on a droplet, y is the interfacial tension, and r is the droplet radius; the internal pressure increases with increasing y and decreasing r. For constant y, the "hardness" of a droplet,pint,would increase with decreasing r, making distortion more difficult. As a corollary, a small droplet with a large flattened face would be vulnerable to coalescence, from both kinetic and thermodynamic considerations. The total interfacial areas of mono- and bidisperse emulsions have been computed keeping rmono = PV(bi)
03
0 75
0 80
0 85
090
095
100
Figure 5. Plots showing the effect of (A) increasing distortion on volume fraction in mono- and bidisperse fcc, (B)/311 on ,912 in
bidisperse fcc, and (C)@ on normalized total interfacial area (at constant Pv) in mono-, bi-, and tridisperse fcc. constant for all P, h being defined by (32)
From Figure 5c,Atot(bi) < Atotcmono) at d l 4 (0.78 < 4 < 1). These results confirm the qualitative trends predicted previously on the simultaneous effect of polydispersity and distortion on AtOt.lg Although the solution is inexact, Abt of the tridisperse emulsion conforms to the above trend in the 4 range 0.89-0.98 (Figure 5C). Distorted Fcc with Interdroplet Film. The extent of distortion of a droplet in an emulsion matrix is controlled not only by 4 but also by a. The increase in distortion with CY, at constant 4) is obtained from the solution of eqs 11 and 12 for monodisperse and eqs 23-26 for bidisperse
812 Langmuir, Vol. 8, No. 3, 1992
Mukesh et al.
01
-
-& 03
02-
I
011
0 15 0
0 01
0 03
0 02
d
0'
0 01
0.02
0 01
a
0.03
0.OL
a
B
B
I
25 0 88
a L
0 EL
0.86
23-
2 5r
I
0 EO
23
1
I
'
0
0 02
0 01
0.78
2.1
0 01
0 03
0
a
0.01
0.03
0.02
0.0L
a
..
o.5/
I
E.
.
0.24
/
/
/
0.01
0:02
'
0.03
/
I
0:04
a
Figure 6. Plots showing increase of (A) distortion as measured by j3', (B)normalized total interfacial area, and (c) [Vcp]bl/ [ VcpIwtvs at constant values of 4, in monodisperse emulsions. (Y,
emulsions, respectively. As shown in Figures 6A and 7A, @' > @ (note that @' = @ when a = 0) for all values of 4, but the effect of a becomes especially pronounced beyond 4 = 0.88. Thus from Figure 6A (monodisperse case),"'3j is attained with a = 0.03 a t 4 = 0.92, and with a = 0.008 at 4 = 0.96. The change in Abt with a,at constant 9, is given by eqs 14 and 29 for the mono- and bidisperse systems, respectively, while plots of the same are shown in Figures 6B and 7B. The interdroplet film thickness is presumably controlled by the nature of the continuous phase and, more particularly, by the type of surfactant(s) employed; emulsions of differing compositions, but with comparable 4 and r, might, therefore, exhibit different degrees of distortion as a result of variations in film thickness. This is a likely explanation of the differing degrees of distortion of droplets in the two wto emulsions whose micrographs are shown in Figure 8. Both emulsions were of similar composition and volume fraction (4 0.9) and were prepared under identical conditions, but emulsion A contained sorbitan monooleate as a surfactant and emulsion B a polymeric surfactant, E-476.24 Despite their larger average size (a
-
$,2
I I" ""ill
Of
I,
1
Figure 7. Plots showing the effect of increasing a on (A) rl-rl contact radius, f l l { , and (B)normalized totalinterfacial area, at constant values of 4. A comparison of the effect of a on the rl-rl and r1-r~face radii are also shown in part C for the above range of 4. Note that fill' is expressed in units of rl and 812' in units of r2. The upper limits for 6 = 0.92 and 0.96 correspond to the maximum distortion achievable with the numerical method employed whereaa for the other cases the plots are truncated at (Y
-
0.04.
previous study has shown that distortion increases with increasing commin~tion'~), distortion of droplets in B appears more pronounced, presumably because of the greater thickness of the interdroplet film, although differences in dispersity could also be a contributing factor. In proportion to its size, the distortion of rz spheres in bidisperse fcc would be higher than that of rl spheres, if
-
(24) E-476 (50% active in paraffin oil) was prepared from the reaction of maleinized polybutene (ECA 6132, Exxon; MW 1OOO) and diethanolamine.
Langmuir, Vol. 8, No. 3, 1992 813
Concentrated Emulsions
1 7 microns
34 m i c r o n s
-
Figure 8. Optical micrographs of two w/o emulsion explosive
sampleswith 4 0.90. Both emulsionswere identically prepared and had similar composition except that A contained sorbitan monooleate as emulsifier and B a polymeric emulsifier (E 476, MW 1100).24
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efficient packing of droplets is to be ensured (Figure 7C). This differential distortion becomes increasingly more pronounced as 4 and a! grow larger. The penalty in terms of energy inputamaller droplets would require more energy to deform than would larger droplets (eq 31)-might, however, compelsuch systemsto adopt a more monodisperse configuration. Figure 9 shows the freeze fracture scanning electron micrographs of two emulsions having identical compositions, and which were processed under similar conditions of temperature (80“C)and stirrer speed (300 rpm); the volume fractions during emulsification were, however, different. Thus, whereas A was processed a t 4 = 0.90, B was processed a t 4 0.93 and diluted with balance oil/wax just prior to termination of emulsification. Droplet comminution was more facile in the latter case,25but similar stuff viscosities (25.25 X l o 5 mPa s for A vs 28.06 X lo5 mPa s for B; T = 30 “C;y = 0.16 s-1)26 and droplet sizes (Figure 9) of the final products were achieved when A and B were emulsified for 20 and
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(25) This may be partly due to the higher shear encountered during processing of the more concentrated (4 = 0.93) emulsion (see, for example, Aronson, M. P. Langmuir 1989,5,494) and partly due to the larger initial concentration of emulsifier in the continuous phase. However, an enhancement of the comminutionrate resulting from increased distortion of droplets at higher 4 cannot be ruled 0ut.19 (26) Low-shear viscosity measurements were made on a Haake Model RV-12 viscometer fitted with a SV DIN sensor and M 500 measuring head.
Figure 9. Freeze fracture scanning electron micrographs of two w/o emulsion explosive samples with identical final compositions but prepared via different emulsification methods. Sample A was prepared a t 4 = 0.90 (mixing time 20 min) while B was prepared a t 4 0.93 (mixing time 8 min) and subsequently diluted with residual oil and mixed for an additional minute. The emulsification temperature was 80 “C.
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8.5 min, respectively. The observation which we wish to highlight here is that increased volume fraction during emulsification promotes formationof more uniformly sized droplets (compare parts A and B of Figure9), as speculated by the Notice also from Figure 9 that the greater uniformity in the sizes of droplets in part B compels them to be more distorted as well.19 Effect of Film Thickness on Limiting Droplet Size. For a monodisperse fcc unit cell with droplet radius ( r ) , film thickness ( a r ) ,circular face radius, ( p r ) ,and volume fraction 4, the volume of continuous phase (oil + surfactant), [Vcp]bl, in the interdroplet regions of a unit cell is given by eq 3328a (27) The reliability of the SEM size distribution data was checked through light scattering studies on a Horiba Model LA-500 laser (He-Ne) diffraction particle size analyzer. (See Das,A. K.; Mukesh, D.; Swayambunathan, V.; Kotkar, D.D.; Ghosh, P. K. Submitted for publication in Langmuir. Both emulsions exhibited similar median sizes (2.32 pm for A vs 2.16 pm for B), while the values of u (standard deviation) were 1.36 and 1.09 for A and B, respectively. (28) (a) The right-hand side in eq 33 is derived by estimating the total number of interdroplet films in a unit cell (there are 4 droplets in a fcc unit cell and each droplet is in contact with 12 neighboring droplets, the number of contact points is therefore 24 [(4 X 12)/21) and the volume/ film, 7r(flr)%~r).(b) The total volume of continuous phase in a unit cell is given as (1 - 4)Vfm = (1 - 4)[4(4/3)xP1/4 (see eq 8).
Mukesh et al.
814 Langmuir, Vol. 8, No. 3, 1992 = 24aap”r3
(33) and the fraction of total continuous phase occupying the interdroplet region, Vcpl bd [ Vcplbt ( [ Vcplbt= total volume of continuous phase in the unit cell), by eq 3428b [ vcp]bl
Plots of the above fraction vs a,for given values of 4, are shown in Figure 6C. As is evident from the plots, the function becomes increasingly more steep with increase in 4; indeed, at the highest values of 4, the emulsion is sensitive to even minor fluctuations in a. (Note that the discontinuity in the curves for 4 = 0.92 and 0.96 is not real but arises from the constraints of the numerical technique employed.) An important outcome of the present work is the prediction of the limiting size in an emulsion matrix. In the absence of a film-i.e., when droplets of the internal phase are in direct contact with one another (Figures 3B and 4B)-there is no lower bound on the size of emulsion droplets. However, in real systems, the interdroplet film imposes limits on their size. Clearly, the condition E Vcplbl/ [ Vcpltot< 1 must be obeyed. For an emulsion with constant t, a would increase as r decreases; consequently, 8’ would also increase (Figures 6A and 7A). With 4 constant, the function 9~~/3’~4/[2(14)l (eq 34) would rise steeply with decrease in r, in a manner akin to Figure 6C. FInally, r would have attained its limiting value as the function approaches unity. Figure 10 shows the minimum attainable sizes of the largest droplet [rand rl in mono- (Figure 10A) and bidisperse (Figure 10B)emulsions] as a function of film thickness. It can be seen from these plots that the droplet size is a sensitive function of a,especially at high values of 4. On the basis of a knowledge of actual film thickness (e.g., from TEM measurements), or computation of the surfactant dimension, it may be possible to predict the minimum r for a given 4. Emulsion explosives, for example, would exhibit rmin values of ca. 0.18 pm at 4 0.90 and 0.40 pm at I$ 0.94 if a typical film thickness of 10 nm is chosen.7b It can be stated, in general, that concentrated (4 > 0.9) microemulsions would be difficult to form unless the interdroplet film is made much thinner, for mono- and polydisperse systems.
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Conclusion We have shown in the present work that polydispersity and distortion may be simultaneously treated in the fcc packing model. Relative distortions and normalized interfacial areas have been compared for mono-, bi-, and tridisperse systems. The distortion of emulsion droplets is affected not only by I$ and the level of polydispersity but also by the ratio (a)of interdroplet film thickness ( t ) to droplet radius. Smaller droplets in polydisperse packings must be distorted disproportionately more than larger droplets under geometric modeling. This disparity in distortion levels becomes increasingly more pronounced as 4 and t increase and r decreases. Such conditions would favor monodispersity since, as mandated by eq 31, smaller
0
2
L
6
8
10
12
14
16
18
FILM THICKNESS, nm
FILM THICKNESS, n m
Figure 10. Plots of minimum attainable droplet size vs bilayer thickness, over the 4 range 0.88-0.97,for (A) mono- and (B) bidisperse emulsions. Note that the minimum size in (B)refers to that of rl droplets.
drops are harder than larger drops-other factors remaining constant-and therefore more difficult to distort. Moreover,a small droplet with a large flattened face would be very vulnerable to coalescence. Freeze fracture SEM studies of emulsion explosive samples, both as a function of composition and processing conditions, bear out qualitatively the conclusions of the modeling studies. The limiting sizes of droplets in emulsions have also been estimated for given values of 4 and film thickness. While a more realistic model must await solution of mono- and polydisperse random close packed structures and detailed understanding of the energetics of deformed droplets and of the compressed unit cell, the results described herein should be useful in evolving strategies directed toward specific emulsion requirements.
Acknowledgment. We thankDr. V. Swayambunathan and S. Krishnan and S. V. Chikhale for their assistance with the freeze fracture SEM studies. We also thank one of the reviewers for several valuable suggestions.‘ This work was supported by IC1 India Limited.
