Langmuir 1990,6, 1668-1675
1668
Concentrated Emulsions. Investigation of Polydispersity and Droplet Distortion and Their Effect on Volume Fraction and Interfacial Areal Ashok K. Das and Pushpito K. Ghosh* Alchemie Research Centre, Thane-Belapur Road, Thane-400601, India Received August 28, 1989. In Final Form: May 2, 1990 Concentrated emulsionswith 0.7 < 4 < 0.9, where 4 = volume fraction of internal phase, may be produced, in principle, through polydisperse packing of droplets and/or through distortion of the same. By use of an ideal fcc configuration,it is shown that a volume fraction as high as 0.89 may be achieved with spheres of three sizes. Incorporation of a fourth size leads to virtually no further increase in 6. Emulsions with higher (up to 0.98) may also be obtained through progressive distortion of monodispersedroplets. Plots of normalized total interfacial area vs 4 have been derived for the above cases, keeping the average droplet size constant for the sake of comparison. Experimental studies on emulsion explosive compositionswith paraffin oil as continuous phase, sorbitan monooleate as emulsifier, and aqueous inorganic oxidizer salts as internal phase-indicate that both modes of packing are plausible at 4 0.88. Our results suggest that early stages of emulsification favor formation of polydisperse spheres while the droplets exhibit increasing distortion with droplet comminution. The latter effect is best understood in terms of increasing monodispersity of droplets upon sustained refinement.
-
Concentrated emulsions, dispersions, and foams find numerous applications in industry.2 Consequently, the study of their structure, especially with regard to their packing geometries, is of considerable interest since the latter may have some bearing on the physical properties, e.g. flow behavior, of these systems. Several studies have described packing modes in systems with an internal phase volume fraction (4) close to unity.3 For the most part, such investigations have treated the internal phase as fully distorted monodisperse droplets, although the importance of polydispersity has been recognized? On the other hand, there are few published reports on packing geometries in systems with 0.7 < @ < 0.9. These systems are, nevertheless, intriguing since their packing modes may be complex, involving both polydispersity and partially distorted spheres. As part of our overall program on rheology of emulsion explosives, we report here our studies on the influence of polydispersity and distortion on @ and on the total interfacial area/unit volume in ideal latticelike configurations. Photomicrographic evidence of transformations in packing modes in an emulsion explosive composition is also presented. Note that unlike in the case of hard solids the dispersed phases in emulsions and foams are deformable. Results and Discussions Effect of Polydispersity and Distortion on @. Polydispersity. Figure 1 illustrates the fcc and hcp monodisperse packings for which @ = 0.74.5a This value was earlier believed to be the upper limit of packing density in emulsions, beyond which the system would exhibit phase inversion.5b For loose and close random packing of mono(1) Presented in part at the Seventh IC1 Colloid and Interface Science Symposium, York University, April 11-13, 1989. (2) Emulsions and Emulsion Technology; Lissant, K. J., Ed.; Marcel Dekker: New York, 1974; Vol. 6, Part 1. (3) (a) Princen, H.M.; Aronson, M. P.;Moser, J. C. J . Colloid Interface Sci. 1980, 75,246. (b) Princen, H.M. J. Colloid Interface Sci. 1983,91, 160. (c) Princen, H.M. Colloids Surf. 1984,9,47;1984,9,67. (d) Princen, H. M.; Levinson, P. J. Colloid Interface Sci. 1987, 120, 172. (4) Princen, H.M. Langmuir 1988, 4, 164. (5) (a) Kittel, C. Introduction t o Solid State Physics, 5th ed.; Wiley Eastern Limited: New Delhi, 1985; pp 16,24. (b)Ostwald, W. Kolloid2. 1910, 6 , 103; 1910, 7,64.
0743-7463/90/2406-1668$02.50/0
disperse spheres, the maximum packing fraction has been shown to be in the range 0.60-0.64.6 It has been indicated that @ may be increased-to values greater than 0.9through random packing of polydisperse spheres, although no details have been provided.' Improved packing of powders through polydispersity has been reported for commercially important extenders such as Zeeospheres." While most dispersed systems exhibit random packing, partial ordering in some systems may be possible.8b To our knowledge, there are no published reports on the effect of polydispersity on packing in ordered structures, and the precise requirements for their denser packing remain obscure. Consequently, we have sought to compute the maximum values of 4 for bidisperse, tridisperse, and tetradisperse packings in a few such ordered systems. The unit cell volumes for fcc and hcp packings are given by eqs l a and lb, with b being the edge length of the fcc unit cell
V,,, = b3
(la)
and a and c being the base edge length and vertical height, respectively, of the hcp unit cell (see Appendix). For spherical radius = rl, the cell volumes may be rewritten as shown in eqs 2a and 2b
V,, = [2(2l/')r1l3 = 22.6274r: vhcp
(24
= 6(31/2/4)(2r,)2[1.633(2r,)l = 33.9413r13 (2b)
For monodisperse packing of the fcc and hcp unit cells, with occupancy numbers (n)of 4 and 6, respectively, 4hcp (6) (a) Scott, G. D. Nature 1960,188,908. (b) Cornell, S.W.; Winter, B. Nature 1965,207,835. (c) Bernal, J. D. Nature 1959,183,141; 1960, I85,68. (d) Bernal, J. D.; Mason, J. Nature 1960,188,910. (7) Tadros, Th. F.;Vincent, B. In Encyclopedia of Emulsion Technology; Becher, P.; Ed.;Marcel Dekker: New York, 1983; p 129, Vol. 1.
(8)(a) Technical brochure on Zeeospheres; Zeelan Industries Inc., Minnesota. (b) Davis, K. E.; Russel, W. B.; Glantachnig, W. J. Science 1989,245,507.
0 1990 American Chemical Society
Langmuir, Vol. 6, No. 11, 1990 1669
Concentrated Emulsions
Table 1. Radius Ratio, Occupancy Ratio. and Maximum Attainable Volume Fraction in Spherical Polydisperse Packs radius ratio lattice scc
1:0.41420.2929
bcc
B
Figure I . Three-dimensional projections of the fcc (AI and h (B) monodisperse structures.
= @rcc = 0.7405 (eq 3)9
& = 4(4r/3)r13/Vrc, = 0.7405
(34
6hcp= 6(47r/3)r,3/Vhcp= 0.7405
(3b) The packing fraction may be increased beyond 0.7405 through incorporation of second sized spheres (radius r2. occupancy nz) which may fit into the voids between spheres of radius ri. Similarly, still smaller spheres (radius r3. occupancy n3) can occupy the voids left by rl and rz, while spheres of radius r4 (occupancy n4)can be accommodated in the voids formed by rl, r2, and r3. For polydisperse packing within the fcc and hcp lattices, the 4 values are given by eqs 4a and 4b, respectively @fcc,ply~ =
[(4s/3)Cniri31/(22.6274r,3)
(44
(4b) Table I lists values of radius ratio, occupancy ratio, and corresponding "6 for monodisperse, bidisperse, and tridisperse packing8 in simple cubic (scc),body-centered cubic (bcc), face-centered cubic (fcc), and hexagonal closepacked (bcp) configurations. The maximum permissible &,ep(poly)
= [(4*/3)Cniri31/(33.9413r,3)
(9) For sce and hcc psekings of spheres of radius rI, V,. = (2r# and V,, = (4rl/3112)a,while 4, = [l x (4r/3)rls]/V,. = 0.5236 and & = [z x (4n/3)r,3]/~b,= 0.68113.
occupancy ratio (n1:nz:nd 100 130 1312 200 260 2612 400 430 4324
fcc
Lon 10.41420 1:0.41420.2929
hw
1:OO
eon
1:0.63300 i:n.63300.30m
620 6215
,a" ~
0.5236 0.6358 0.7932 0.6803 0.6878 0.7436 0.7405 0.7799 0.8916 0.7405 0.8031 0.8556
sizes and numbers of the different spheres were based on analyses of the respective unit cells,1° as illustrated by the representative calculationsfor tridisperse fcc and hcp packings (Appendix). While Table I gives the upper limits of occupancy for a given r and the corresponding values of @-, partial occupancy of voids may also be considered to achieve intermediate values of 4. As can be seen from the table, hcp and fcc unit cells lead to facile packing. However, while monodisperse systems of the above have similar &ax, packing efficiencies differ for the corresponding bi- and tridisperse cases. Thus, while bidisperse packing is more efficient in hcp than in fcc (4"(hcp) = 0.803 vs &=(fcc) = 0.780), tridisperse packing in the latter is more dense than in the former ($,,(fee) = 0.892 vs &=(hcp) = 0.855). Incorporation of spheres of a fourth size (r4 = 0.04r1, n4 = 72) in fcc does not appreciably alter 4 (&ba = 0.8923 vs = 0.8916) due to the negligibly small volume of these spheres. Analogous computation of tetradisperse hcp has so far not been possible. We have also analytically computed the spatial coordinates (see Appendix) of all spheres in the tridisperse fcc and hcp unit cells and have used the data to obtain three-dimensional representations of the unit cells by adapting a molecular graphics program." These spacefilling structures (Figure 2) confirm the location and sizes of individual spheres reported in Table I. While only step changes in 4 are possible in the present model, in practice 6 may be varied continuously through non-ideal packing and/or through distortion of polydisperse spheres. Distortion. As stated above, the packing fraction is 0.74 for spherical monodisperse fcc and hcp lattices. This value may be enhanced through distortion of the spheres.'" For simplicity, a monodisperse system is discussed here. Consider the unit cell of hcp shown in Figure 1B. The sphere in the middle of the hexagonal space has six surrounding spheres in the same plane; additionally, it is contacted by three spheres in the plane below and three in the plane above. Thus, this sphere has 12 touching neighbors of equal sizes. Indeed, all spheres in the hcp pack have 12 touching neighbors. Although it is more difficult to visualize, each sphere in the fcc configuration also has 12 touching neighbors.I2b The spheres distort when they are squeezed against one another. Such (IO)Implications of r2 and ra values less than the allowable magnitudes have not been considered in the present work. It is likely, however, that excessive variations in the sizes of the different spheres would not he tolerated hy emulsion systems. (11) Molecular Graphics Program, Version 1.0. Department of Chemistry, University of Washington. (12) (a1 Distortion is controlled not only by 4 hut also by the nature of the emulsifier.lBa (h) Sloane, N. J. A. Sei. Am. 1984, 250, 92.
Das and Ghosh
1670 Langmuir, Vol. 6, No. 11, 1990
h, of the cone decreasing with distortion as illustrated in Figure 3B. We have defined the degree of distortion via a distortion parameter, 6 (eq 5)
P
6 = h / r ; 0.9105 < 6 Q 1 (5) The volume of the composite body-which is the same as that of the original sphere-may be expressed in terms of the volume, V,., of curved (ended) cones having base area, a6,and altitude, h (eq 6)13 Vdmp= (4/3)rr3 = lZV,,
= 12[(1/3)ha,l
(6)
If rl is substituted with r6 in eq 2b, the hcp unit cell volume upon distortion is given by eq 7, while the expression for volume fraction, Q,, can he obtained via eqs 3b and 7 as eq 8, assuming volume conservation of each droplet with equivalent spherical radius, r [V,,,],
-
-
&(h,)
= 33.9413r36’
(7)
= 0.7405/a3
(8)
0.910514and Q 1,as for example in dry foam^,'^ the structure of droplets/voids approaches a regular pentagonal dodecahedron (Figures 3A and 5A). The variations of 46 and h (in units of r ) with 6 are shown in Figure 4. Type I1 Distortion. In type I1 distortion, a flat circular face is assumed to form between two interacting spheres of radius r. Such distortion has been experimentally observed (Figure 5A),’6 and the transformation of a sphere to a polyhedron-like flat-faced structure is schematically depicted by Figure 5B and 5C. Since each drop has 12 touching neighbors, 12 flat circular faces (radius 6) will form per drop. With increase of distortion 13 increases; hence, it is a parameter of distortion. Under volume conservation, the distorted drop may be conceived as a sphere with radius R(R 2 r ) , from which 12 spherical segments (with base radius 8) have been eliminated. (A hypothetical sphere with two eliminated segments is shown in Figure 5D.) The relation among r, @, and R may be given by eq 9 As 6
B
(4r/3)[R3 - r’] = 12((3r/6)P2[R - I(R2 - 82)”211+ (r/6)[R
c:,9,:3-..
Z . , r 2 . .:,1.,
the right-hand side being the sum totalof the volumes of 12 spherical segments.” The corresponding relationship for unit cell volume ([Vhepjp) (eq 10) is obtained by substituting rl = (RE- ,92)1/2 in eq 2b. Finally, the volume fraction $a(h,) is given by eq 11
Figure 2. Three-dimensional projections of the tridisperse fcc (A) and hcp (B) structures. distortion permits better packing of the dispersed phase, leading to a reduction in the unit cell volume with a concomitant increase in 4. The effect of gradual distortion on Qmar has been derived for these configurations, assuming two different geometries of the distorted droplet. Both approaches assume equal distortion of a sphere by its touching neighbors, with conservation of volume. Since the results are identical for hcp and fcc, only the former will be referred to in our subsequent discussion of distortion. Type I Distortion. In type I distortion, compression of deformable spheres (radius r ) against one another leads to 12 equivalent surfaces per drop, each with radius of curvature greater than that of the undistorted sphere (Figure 3A). The distorted body may then be visualized as a composite of 12 curved (ended) cones, the altitude,
- l(R2- Pz)1/2113) (9)
[ V,,,], @8(h,)
= 33.9413(R2- 132)3’2
= 6(4r/3)r3/[33.9413(R2-82)3’21
(11)
Equation 9 has been numerically solved to obtain values of R (in terms of r ) for incremental changes in (0 Q I3 < 0.5627r). (The upper limit of corresponds to the radius of the inscribed circle in the pentagonal face of a regular dodecahedron with volume equal to that of the undistorted sphere.) The plots of R, (R2 - @2)1/2, and Q p vs I3 (13) Kom, G. A,; Korn. T.M.MathematicolHandbookfor Scientists and Engineers; Mffiraw-Hill: New York, 1961; p 748. (14) Perm R.H.. Chilton. C. H.. Eds. Chemical EReineers’ Handbook, 5th ed.; McCraw Hill Kogakwha hi.; Tokyo, IY7:i. pp 2 4
( l j ) l a Aubert,J.H;Krsynik.A.M.;Rand.P.U.Sr,.Am.1986.254. S.H S. I{.:,~ Pandli.d.:Khllar.K.C. J . Colluidlnrerloce ~ , . Sei. 1988, 124; 339. (16) (a) Williams, J. M.; Wrobleski, D.A. Longmuir 1988,4,656. (b) Williams, J. M. Longmuir 1988,4,44. (17) See ref 14, pp 2-7.
_~_
58. 1 h i S m a . D ~~
~~~~~~~~
Concentrated Emulsions
Langmuir, Val. 6, No. 11, 1990 1671
A
,
10pm,
A
B F i g u r e 3. Illustrations of type I distortion. (A) Threedimensional projection of a droplet partially distorted by its 12 touching neighbors. Black circles show the points of interaction between spheres. while the superimposed dodecahedral network depicts the ultimate fate of the droplet. (B) Illustration of the increase in radius of curvature with progress of distortion (1 2 3) of a conical section of a droplet. (Note that each droplet comprises 12 such conical sections in the fcc and hcp lattices.)
-
-
D
-
- .0 90
0 92
09'
096
0 90
1W
Figure 4. Variation of h (in units of the original spherical radius r) and @Awith 6.
are given in Figure 6, with R, (R2 - 82)L/2, a n d 8 being expressed in terms of the original spherical radius, r.I0 Note that even though R increases as 8 increases, ( R 2 p2)'/Lwhich determines the unit cell volume-decreases with 8; as a result, 60 increases with 8. (18) The present ealmlations do not take into account the thickness of the bilayer which, in real syatems, is non-zero and varies depending on the emulsifier/continuous phape blend. Thua, if Figure 5C is considered. a finite bilayer thickness implies flow of mntinuous phase from "cornem" into bilayers. This muld he achieved by distorting the droplets to a greater extent than what is indicated. i.e.. keeping volume fraction fired,the dof distortion ineresws as the bilayer thickness is increased. On the other hand. for B fired hilayer thickness. deviations from the computed (for zero thickness) values of distortion would be increasingly more pronounced as m i n c r e w (Mukesh. D.:Das. A. K.;Ghmh. P. K. Unpublished results).
Figure 5. (A) Freezefracture SEM of an emulsion explosive composition (6 0.90)illustrating type I1 mode of distortion. The arrow at the lower left shows a partially distorted drop with flat circular faces while the upper left arrow shows a completely distorted drop with pentagonal faces. The arrow on the right points to corner voids between drops. (B) Illustration of part of a hcp unit cell indicating a central sphere surrounded by six spheres in the same plane and three spheres in the plane below. Note that this sphere is also contacted by three spheres (not shown) in the plane above. (C) Fate of the central sphere shown in Bas a result of type I1 distortion. (D)Illustration ofa sphere of radius R with two missing segments. In hcp and fcc lattice packings, each distorted drop may be considered as a sphere of radius R from which 12 segments have been eliminated with resultant volume equal to that of the original spherical droplet.
Effect of P o l y d i s p e r s i t y and Distortion on Surface Area This section a t t e m p t s to compare a n d contrast the profiles of normalized surface areas (Abt) for polydisperse and distorted monodisperse packings as a function of 4. For packings with ordered geometries (fcc, hcp), Am =A N I,I,, where A,a = total surface area of dispersed phase per unit cell, N,II = V,,/V,II, and V,II = unit cell volume. With Vbt = unity, Ne" = l/V,u and Abt = A,u/ Vce!l. For the p u r p o s e of comparison, the numberaverage (FN;eq 12) and volume-average (Py, eq 13)droplet
1672 Langmuir, Vol. 6, No. 11, 1990
Das and Ghosh
1.10
3L
I
_/.-
._c
I
0.701
00
01
02
03
04
06
05
P Figure 6. Variation of R, (R2 - p2)1/2, and @JB with 6. Except for @p, the other terms are all expressed in units of the original spherical radius, r.
radii of the polydisperse matrix
are assumed to be of the same size as that of the volume equivalent spherical radius of distorted monodisperse drops; Le., rmono FN or FV. Polydispersity. Computation of Atot involves estimation of Acelland Vcen. The latter is given by eqs 2a and 2b, rl being the radius of spheres of the first size. Acellis given by eq 14, qmax) being the maximum occupancy number of spheres of radius ri (i = 2, 3) in the unit cell (Table I)
2 2' 070
I 075
085
080
090
C95
'00
9
Figure 7. Plots of relative normalized total interfacial area (au) vs volume fraction for type I and type I1 distortions. as 6, we obtain eq 19 for [Ato&. The variation of 4 with 6 and that of [Abt]6 with 4 are shown in Figures 4 and 7, respectively (19) [Atotla = GAdrop(g)/ [vhcpld = 2.2214/(rs4) Type I1 Distortion. Distortion by this mode leads to a droplet structure which can be dissected into 12 right circular cones (with base radius /3 and altitude (R2- P2)1/2) and 20 curved (ended) cones with altitude R and base area ag (see text and Figure 5). In the limit, the curved sections become the 20 vertices of a pentagonal dodecahedron. Assuming conservation of volume upon distortion and a radius r of the undistorted sphere, ag is given by eq 20, A d m a )by eq 21, and [Abt]@by eq 22, where [ vh.& is given by eq 10 above 20(1/3)a$ = (4/3)*r3 - 4aP21(R2- P2)1/21
(20)
Adrop(B)= 12XP2+ 20aB
while Atot is given by [A,,lfCc = 4nxnir?/22.63rl3
= 12*(P2
Equation 15 may be rewritten in terms of PN (eq 16) and Fv (eq 17) by employing the radius ratio data in Table I and the definitions of FN and Fv in eqs 12 and 13 above [Atot(iiN)lfcc= [0.555(n1+ 0.172n2 + 0.086n3)(nl + 0.414n2+ 0.293n,)]/[(n, + n2 + n 3 ) P N ] (16a) [Atot(TN)]hcp [0.370(n1 4- 0.401n2+ 0.093n3)(n1+ 0.633n2 + 0.305n3)]/[(n1 + n2 + n3)PN] (16b)
+
[Atot(Pv)]fcc= [0.555(n1+ 0.172n2 + 0.086n3)(n1
0.071n2 + 0.025n3)"3]/[(nl + n2 + n3)'/3Fv] (17a)
+
+
[Atot(?v)]hcp= [0.370(n1 0.401n2 + 0.093n3)(n1 0.254n2 + 0.028n3)1/3]/[(n,+ n2 + n3)1/3PV](17b)
Type I Distortion. Consider a monodisperse hcp packing with spherical radius r. Upon distortion, the unit cell shrinks (eq 7), with concomitant increase in 4 (eq 8) and Adtop (area of a drop). Assuming the representation of a drop in terms of curved (ended) cones (vide supra), Adrop(6)is given by Ad,op(a)= 12a, = 4 r r 2 / 6
(18)
ag being expressed in terms of r and 6 (vide eqs 5 and 6).
Combining eqs 7 and 18 and taking unit cell occupancy
+ (1/R)[(r3/3) - P21(R2- P2)1/21])
(21)
[Atot](3= 6Adrop(B)/ [ vhcp], = [6.6643(P2+ (1/R)[(r3/3)- P21(R2- P2)'/'1])]/ (R2 - P 2 ) 3 / 2 (22) The plots of Aht vs 4 for type I and type I1 distortions are given in Figure 7. As can be seen from the figure, type I1 distortion is preferred over type I distortion since [Ab+ < [Ato& over the entire range of 4. These results permit rationalization of the actual (type 11)mode of distortion observed in real systems (Figure 5A).lh We note, however, that such a transformation with volume conservation would inevitably involve complex intermediate geometries. It is our conjecture that the intermediate structures resulting from distortion might permit more facile fragmentation of droplets at higher 4.l9 Polydisperse vs Distorted Monodisperse Systems. The normalized Atot of monodisperse (Atot(a)and Atot(b)) and polydisperse lattices (Atot(FN) and Aht(iv)) may be compared with rmono = P N N V) ( ~= ~ r, r being constant for all 4. The data are tabulated in Table I1 while plots for fcc and hcp packings are shown in A and B of Figure 8, respectively. An interesting feature of these plots is the increase (linear for type I and nonlinear for type 11)in the interfacial area of distorted monodisperse systems with increasing 4, with a nonlinear decrease for the corresponding polydisperse matrices. Note' also the difference (19)Das, A. K.; Ghosh, P. K. Unpublished results.
Langmuir, Vol. 6, No. 11, 1990 1673
Concentrated Emulsions
Table 11. Relative Values of Total Interfacial Area per Unit Volume (&t) for Polydisperse and Distorted Monodisperse Systems as a Function of qY polydisperse distorted monodisperse @
AtOdh)
rllh
rlliv
Atot(iv)
Hhlr)
[Atotla
Plr
[At&
1.0000 0.9942 0.9829 0.9677 0.9534 0.9400
2.2214 2.2740 2.3804 2.5334 2.6885 2.8455
O.oo00 0.1100 0.1900 0.2700 0.3300 0.3900
2.2214 2.2607 2.3413 2.4648 2.5836 2.7204
1.oo00 0.9863 0.9733 0.9663 0.9596 0.9530
2.2214 2.3476 2.4754 2.5477 2.6202 2.6934
O.oo00 0.1700 0.2400 0.2700 0.3000 0.3300
2.2214 2.3171 2.4136 2.4648 2.5216 2.5836
Face-Centered Cubic 0.7405 0.7536 0.7799 0.8172 0.8544 0.8916
1.oooO 1.1328 1.3353 1.9775 2.3164 2.5259
2.2214 2.0452 1.8777 1.4607 1.4115 1.4453
1.oooO 1.0709 1.1844 1.5034 1.7082 1.8602
0.7405 0.7718 0.8031 0.8206 0.8381 0.8556
Loo00 1.0553 1.1009 1.4789 1.7449 1.9425
2.2214 2.2456 2.2873 1.8192 1.6405 1.5623
Loo00
2.2214 2.1633 2.1169 1.9212 1.9140 1.9626
Hexagonal Close-Packed 2.2214 2.2824 2.3506 2.1516 2.0685 2.0348
1.0383 1.0713 1.2504 1.3839 1.4914
The corresponding values of rl (expressed in terms of i N and i V ) for polydisperse systems, together with values of 6 and f3lr for distorted monodisperse systems, are also listed. r,,, = i ( N v).
-
A
polydisperse systems is reduced; even so, for 4 0.89, type I1 distortion yields a total area nearly 40 76 larger than the latter. This is mainly due to the fact that even though more spheres are accommodated in a polydisperse unit cell its volume is larger than that of the monodisperse cell, since to fulfil the condition rmono= rwly the radius of the largest sphere (which determines the cell dimension) in the cell is increased as smaller spheres are incorporated to enhance 4. The steeper increase of the cubic function (volume)over the square function (area) leads, therefore, to a decrease in Abt with 4. On the other hand, as 4 increases, the unit cell volume of the distorted monodisperse matrix decreases; therefore, more unit cells are accommodated per unit volume. This, together with the increase in area of a droplet upon distortion, leads to an increase in Abt with
3
8
0
U
.T.'
2
8 A
A
A
A
Atot I
bi
1
tri
4.
0
The notion that Abt decreases with increasing 4 for polydisperse systems is misleading. As pointed out by one of # the referees, Abt depends on the choice of polydispersity employed and the crossover from one form of dispersity B 3 to another. A more appropriate representation of the data in Figure 8 would entail that dispersity be maintained constant over the entire range of 4, i.e., 0 < I$ < 1. Thus a family of curves may be generated for mono-, bi-, and tridisperse systems as illustrated in Figure 9 for fcc packing. The lower volume fractions are obtained by progressively 2 removing spheres from the unit cell; for example, assuming A tot tridisperse fcc packing, a 4 value of 0.25 is obtained with nl, n2,and n3 values of 1 , 3 , and 6, respectively, while for 4 = 0.71, the corresponding values are 3, 3, and 24. The tri dotted line in the f i i e passes through the close pack limits 1 of mono-, bi-, and tridisperse packings. Extension of this curve up to the reported of polydisperse systems with (limiting case)' yields the value of Atot(wly)at F N ( ~= ~ ?"(bi) ~ ) = FN(bi) = rmon0. The dashed h e for Abt@oly) vs 4 is then constructed assuming a linear relationship. 0 I ( '0 0,75 0.80 0.85 0.90 This assumption is not unreasonable since similar behavior is observed with mono-, bi-, and tridisperse systems. The rp above systems would exhibit distortion beyond the close Figure 8. Plots of relative total interfacial area (au) per unit volume vs volume fraction for monodisperse ( ( 0 )type I and (0) pack limits as illustrated qualitatively in Figure 9 (the type 11) and polydisperse ((A)r = i v and (w) r = ?N) fcc (A) and change in Abt with 4 for rmonois more quantitatively shown hcp (B)lattices, with rmmo= f,], = constant for all 6. in Figure 7). Figure 9 may be of interest in the more quantitative estimation of physicochemical properties of in the profiles of Abt vs 4 for polydisperse systems with r = PN and r = Pv. With r = f y , the divergence in the emulsions which depend on the total interfacial area, interfacial areas of distorted monodisperse and spherical typical examples of such properties being rheology,20 (
0
0.75
0.80
0.85
1
(
1674 Langmuir, Vol. 6, No.11, 1990
i
..
0.0
0.1
0.2
0.3
0.L
0.5
0.6
0.7
a8
a9
. -
.
.
.d
1.0
e
F ~ r 9.e Plots of relative At, vs 4 for man+. bi-, tri-, and p l y disperse fcc packings. Note that rN is constantfor all data points and that the curve8 corresponding to distortionare approximate.
electrical capacitance,21 and detonation sensitivity (of emulsion explosives).22 Suffice it to say that average particle/droplet size alone may not be a sufficient criterion for such estimates; information on the degree of distortion/ polydispersity would also be essential. Polydispersity vs Distortion in Emulsion Explosive Formulations
It is anticipated that dilute emulsions will possess spherical droplets while increasing distortion of the same will be observed with increasing @. There are, however, no reports on the possible transformation of packingvis-a-vis spherical polydisperse to distorted monodisperse or vice versa-at a given 6. Figure 10 shows photomicrographsof an explosives type emulsion composition (@ ,.. 0.88)with droplet comminution.23 As is evident from the pictures, the emulsion droplets undergo a gradual transformation from spherical to distorted structure.2' We have rationalized this observation as follows: at early times, a random size distribution of droplets is feasible, permitting (20) (a) Sherman. P. J. Phys. Chcm. I963.67,2531. (b) Sherman,P. J. Coilloid Inferfoee Sei. 1967.24,97. (e) Princen, H.M.J . Colloid Inter/aceSci. 1985,105.150. (d) Pa1.R.; Rhodea,E. J. Colloidlnferface Si.1985. 107. 301. (el Prineen. H. M.LanPmuir 1988.4. 486. Ayrshire. UK,1983. (22) Bampfield. H. A,;
Cooper, J. In Encyclopedia
o/ Emulsion
Technology;Beeher. P.. Ed.; Marcel Dekker: New Yark, 1988, Vol. 3,p
-"..
.?e1
32p m b-----I
Figure 10. Optical micrographs depicting changes in droplet structure on comminution (top to hottom) in an explosive type emulsion composition with 4 0.88 at 85 "C."
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dense packing through aceommodationof smaller droplets into voids created by larger drops. With progress of emulsification, there is an increase in the total interfacial area. Since Am is an inverse function of the radius of the largest sphere, r l (eq 15), the surfactant drives the system to a smaller value of rl to achieve a larger Am. This is illustrated in Figure 9, wherein it is shown that as r l decreases, the system gradually crosses over from polydisperse to monodisperse packing, with concomitant increase in distortion and Amt as mandated by the curves.
(23) Typically. an ea. 800-g emulsion batch contained 284 g of ammonium nitrate. 142gofralcium nitrate. 112gofammoniumrhloride, 44 g of sodium nitrate. IM) g of wavr. 52 g of pardfin oil. and 14.5 g of Conclusion sorbitan monmleatc. The emulaion was prepared by pour~nga hot (85 OC: I ' = 550mL1~olutionofrheo~idirpr~~l~intothemi~rureofpardfin We have demonstrated in the present work the ability oil and aorhitan mondeatc !pII O C ) with rapid tROO rpml mechanical stirring. F d u w i n g initial emulsification for 5 min at high temperature. of moderately concentrated dispersions (4 0.9) to take theemuhion massum furthprrefined thmughstininaand 9imuIlanpOw up either a spherical polydisperse or a distorted monocooling toca. SO T tn M mi". emulsion Wmplp9 being collwtpd st perinlic disperse configuration. Estimates of volume fraction and intervals. The intcmsl phase volume fraction at 85 O C was computed by ".udmg theemuhfierwlwne in therontmuoua phase 'l'ha mclu~glnn interfacial area have also been derived for both models in i*mandated hy s p p r n x x " calculations which S U ~ R P S Ithat only a v"l idealized lattice structures. The present results may be fractim u l th- low H1.H emulsifier occupies the mterfaee of interest in the quantitative interpretation of physical/ t?4iThQsQ f i n d i n ~ ~ a ~ ~ bysimilerobspn.ationsmad~ p p ~ l ~ d in the laboratory of one of the refewes chemical properties of concentrated matrices. Finally, the
