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water droplet stabilized by the particles. Part I
377
CoUoid8 and Surfaces, 59 (1991? 377-336 Elsevier Science Publishers B.V., Amsterdam
Capillary interaction of spherical particles adsorbed on the surface of an oil/water droplet stabilized by the particles. Part I S. Levine and B.D. Bl;wen Deprtmcnt of Chemical Engineering, I W5, Canada
Unioersity
of British Columbia, Vanct,L
‘Jet;
B.C. V6T
iReceived 2 Janunry 1991; accq,ted 5 Apri; 1991)
Abstract The capillary interaction energy M.ween ideaticd spherical particles adsorbed as a monolayer on the surface cf an oiI/watcr emulsion dmple: stabilized by the particles is determired. Use is made of the throretical work by Menon, Nagarajan and Wasan on the free enemy change due to the adsorption of the particles on the droplet. from the continuou phase. It is assum that the oil/water interfeces joining neighboring adsorbed particles are portions of spheres wil II ticwmrnnn radius and a common center. The capillary intesction between tha particles is ~I!c~;~vv’I:’ial;~n vnolayer of into account through the change in radius of the droplet on the adsorptioil of t 1.
particles. The capillary interaction is obtained as the difference between two Jarge energim of the order of lo* kT per particle which are almost equal. Hence that part of the adsorption energy attributed to a particle which excludes its interactionswith neighboring a&orbed particles must be clearly ideatile& The capillary interaction is attractive and is comparable with a multiple of kT, diminishing as the inverse square of the separa&- - ! ~~!tween the particles and proportional to the fourth power of the particle radius.
INTRODUCTION
Pickering oil-in-water (O/W) or water-in-oil (W/O) emulsion droplets are stabilized by a film of fine particles adsorbed in a monolayer at the oil/water interface. These particles ar8 partially wettable by both oil andwater (e.g. by treatment with suitable organic molecules), so that the three-phase contact angle is neither close to 0” nor close to 18OO.The interactions between the interfacial particles are due to elt~tric double layer, van der Waals, capillary and, for deneely-packed pa&cl-, solvation forces, and in this paper we are concerned with the capillary interaction. In a recent paper [l] we assumed that the radius of a droplet is so large compared with that of an adsorbed particle that the oil/watir interface of the droplet can be regarded as planar. On
378
applying a two-din;en:;inn::l cylindrical cell mot!:,! to determine llltl cupiihtr~ I’r~rccsclcting hctwecn the interl\witll pnrticle!r we obtained a small repulsion. which we nnw believe is incorrect. In ordrx t.o :~llow fm u poxdde (:! - ,I !I!‘(! effect a conical cell model was suh::erluentlyused and this gave a small capil!ary aLtraction, the details of which will be reported later in Part II. As explained in the previous paper [ 1 ] the capillary interaction between the interfacial pnrtitles is expressed as a difference between two Jnr~r p ,y’. ;- .. 1 : !! ue almost ! .*.,’ , i,i ‘r approximations. An equal and conseqclently an error I. : ’ example of such an approximutiull cuuu Lt: Lhe assumption that the oil/water interface cbfthe droplet is Flanar. FurthermorlX ;I i: * 1 dwious how to define that part of the interfacial energy of the droplet with its film of particles which excludes the capillary 1,l beraction betw:+en the particles. A solution to the last problem mentioned above is suggested from a study of a paper by Menon, Nagarajan and Wasan [2], which we proceed to analyze and develop krthrr. ‘i I:. it t.horswere concerned with W/O emulsions, but here we sh~rll!a!z;rl~y.rjtl;.: ‘. /JqT fl,.f,i i:a11 .I I !.’ ‘r’ need only mention the W/O emulsion Lie:i:i’the em!. ‘I’hey cU?lsider the free energy change due to the adsorptio-.. c!L’11llluslolayer of spherical particles on the surface of an oil/water drople ,. They mention deformation of the oil/water interface as a result of perticle adsorption and apparently take this deformation into account by inC-d~~ci:~g ti new (increased) radi-usof the droplet. However, they do not exLlicitly sI.ate that the oil/water interfaces joining neighboring adsorbed particles remain spherical. An unambiguous expression for the capillary interaction between the narticles requires the shape of the oil/water interface tobe known. 1 ,A ;i~lrilyXill g their work we shall assume that this interSace remains spherical with a common radius and a common center. As a consequence the capillary interaction 1;L;;veex the particic:; i., 1, ‘Irectly taken into account through the change in radius of the droplet resulting from the adsorption of the particles, and this interaction will be found to be small and attrectivp A.ltimu~hwe expect that their model of sn oil/water interface with a common radius is ;approximate for a densely packed monolayer of particles, we shall show that it does suggest an expression for the interfacial energy which vmits the capillary interactionbetween the adsorbed particles. In the conical cell model which will be described in Part II, we attempt to account for the departure from a spherical shape of the oil/water interface by applying the Young-Laplace equation, particularly when the monolayer of adsorbed particles is close-packed. THEORETICAL
Geometricalconsiderations Consider one particle of radius a adsorbed on an oil-in-water emulsion droplet. Gravity is ignored and it is assumed that the droplet remains spheric&
379
Fig. 1. Geometry of one particle having center 0’ and radius a, adsc;rbud uli indroplet having center 0 and radius rd. I, II, III and IV ate the voiumes of sections of two cones with apexes 0 and 0’, with a common base. The relative eize of *theparticle is greatty uxaggzrated.
which ib justified in a recant paper by di Meglio and Raphael [3]. These authors have shown that for a single spherical yarticle adsorbed on a curvedliquid-liquid interface, the Young-Laplace equation is obeyed for a sphericalinterface when the free energy of adsorption is minimized. Let r, be the radius of the droplet in the absence of the particle and F, the “adjusted” radiusdue to the adsorption of the particle. We wish to find the relation between r, and r,. Consider the geometrical configuration in Fig. 1.0 and 0’ are the centers of the droplet and particle respectively.A and cyare the angles of the two cones having as apexes 0 and 0’ and aa a common base the circular arec whose perimeter is the oil/water/solid contact line. The sum of the volumes of the two cones equals (see Fig. 1)
i fYTt IV= (43)
3 - rtsin" ( t&in2 a COBCL
A cos A )
01
Also the correspondingsegments of the two tiphereshave volumes I+II+III= II+III+IV=
(2na3/3) (1-cos a) (2&f/3) [l-cos A)
(2)
The p& of the volume of the droplet of radius r, occupied by the sphere is therefore (4x/3) cr: -r:,
=11+111
(3)
380
by volume balance in an incompressible system. It may be verified from Eqns (1) and (2) that II+III=
(ncz3/3)
(2tcos
Lx!)(l-cos
+ (nr3,/3) (2-kcos A)
a!)2
(1-cosA)2
3)
The relation between r, and r, obtained from Eqns (3) and (4) is identical with the formula quoted by di Meglio and Raphael 13 1. Suppose now that NP particles are adsorbed on the droplet and assume that the oil/water interfaces joining the particles remain spherical, with common radius I’d and common center. Then (47V3) cr: -r~)=(7dVp/3)[d(2+cosa)(l-ccosilc)2 tri(2tcosA)
(1-c0sA)~]
(5)
We imagine that the NP particles are uniformly distributed on the surface of the droplet. That some small deviation from uniformity may be required does not matter because of the assumption that the oil/water interfaces joining the particles are portions of spheres with a common radius. The depth of immersion of the particles in the droplet and the angles a! and A are also assumed to be the same for all particles. It should be noted that when the droplet radius changes from r, to rd, corresponding changes will occur in angles cy and A, which should strictly be relabelled. This is clarified below. Menon et al. [2] have only the first term on the right-hand side of Eqn (5). We wish to estimate
Fig. 2. A typical spherical particle with center 0’ and radius a, adsorbed on a droplet. Portions of the oil/water interface joining particles are assumed to be spherical and have a common center 0 and common radius rd. The relative size of the particle is greatly exaggerated.
381
the error in neglecting the second term, assuming a common radius of the oil/ water interface. Consider a typical member of the Np adsorbed particles immersed in the droplet to a depth !, which is so defined that the distance between the centers of the droplet and particle equals at rd- 1. Since we assume that the oil/water interface of the droplet remains spherical, the contact angle 8 is that shown in Fig. 2. Then we have the trigonometric relations sin a=
(rd/a)sinA=rdsin
(o+rd -E)2=
@/(a+rd-l)
(6)
8
(7)
a2+fi+2ardcos
Expanding in powers of a/r& Eqn (7) becomes ~=a(l-cos8)[1-((a/2~d)(~+cos8)+(1/2)(a/~~)2(~+cos8)+..]
(8)
and therefore sin
A=(a/r,) sin8 [I-(a/r,)cos e-(1/2)(a/rd)2(X-3cOS2~)+..]
(9)
The corresponding series for sin a follows from Eqn (6). The expansions of -. cos A and cos a! are cosA=l-i
,_,(asin o/rd)2[1-2(a/rd)cos
8-
(3/4)
(c/rd)2X
wo
(i-5c092e)+..j COB c”=
cos B-t- (c/rd)sin2 e-
(3/2)
(a/rd)“sif?
8 cos
et ..
(11)
It is observed that cos a> cos 8 and therefore cy< 8 where for an O/W emulsion droplet 8< 90”. For a single adsorbed particle rd is replaced by r, in Eqns (6)- (11) and therefore I, A and a! will be different. If we denote cuby a0 for a single particle, then since rd> r,, it follows from Eqn (11) that cy,< GY.This means that the presence of surrounding adsorbed particles on a given adsorbed particle produces an increase in the depth of immersion of the particle in the oil droplet and this increase may be attributed to the capillary attraction between the particles. We also obtain cf!+=z 0 with our conical cell model (Part II) but for the cylindrical cell model [1] a > 8, predicting incorrectly a capillary repulsion between the: particles. We readily derive from Eqn (11) the relation sin (he)= -a sin t?/rd to order a/r,-& so that a! is very slightly Smaller than 0, suggesting that the capillary interaction between the adsorbed particles will also be very small. On substituting Eqns (10) and (11) into Eqn (5), it is readily verified that Eqn (5) may be written as (4x/3)
(r$ -rz)
= (7t.a3~~/3)1 -(%/4rd)sin4
~(~+cos~)(~-ccos~)~ e+..]
(12)
382
The formula used by Menon et al. [ 2 ] is (4n/3)(&-r5)=(7ra3Np/3)(2+cosa)(1-cosa)*
(13)
which, on using Eqn (11) yields (47r/3)(r$-J$r?,)=(
za3NP/3) [ (2tcos (l-cos
0) x
e)* -9(a/rd)sin4
so that, bearing in mind that a/r dz IO-*, slightly.
(14) et..] they have underestimated rd very
Free energy calculation
We now examine the calculation by Menon et al. [2] of the free energy change due to the adsorption of the NP particles onto a droplet from the continuous water phase. The free energy before adsorption is N, y,,
FI =4m$, yaw+4na*
2
(15)
and the corresponding energy after adsorption is F11=27ra2NP[(1-coscr)y,,+(1+cos~)y,,] + 47rr:Yaw-2nr$
Np (l-cosA)y,,
(16)
where yaw, ‘~0~and yws are the interfacial tensions of oil/water, oil/solid and water/solid systems respectively. In the formulation of Menon et al. [2] the last term on the right-hand side of Eqn (16) is replaced by - za2iV,, sin’ a! yaw. By making use of Eqns (6), (9) and (10) we calculate that 26(1-cos
A) -a*
sin’ (x= (a2/4) (a/rd)2sin4 8
(17)
so that the difference between the Eqn (16) for FII and the corresponding expression of Menon et al. [ 2 3 is proportional to the very small factor ( a/rd)2. On eliminating yoSby using Young’s equation ‘yaw cos 0 = Ycls- Yws
(18)
substituting for cos A from Eqn (10) and for cos a! from Eqn (11>,we collect separate powers of a/rd in Eqn (16) to obtain F I~=~~E~Y,,+~~~N~~~;,,,--~,,[(~-cos~)*
- (3/4) (a/rd)“sin”
tit . . I>
IIence the free energy of adsorption of NP particles on to an O/W original radius r, is
09)
droplet of
383
61 -Fi =4n(r:
-r:)‘~o~
-7raWpyow[ (l-cos
e)2
- (3/4) (a/rd)2sin4 et . . ]
(20)
Equation (12) expresses the relation between the two radii rd and write this equation in the form (r&-,)3=1+
(a/4r,)N,(&,)2[ -
(9a/4r,)sin4
(2+c0!3
6) (l--cos
F,. We
can
@I2
8+ . . ]
(21)
For a hexagonal close-packed array of spherical particles on the droplet, we obtain N,(a/r,)“=
%/J3
(22)
On substituting Eqn (22) into, Eqn (21) it is seen that rd/rO differs from 1 by a term of order a/r,. This justifies our replacing rdby r, inside the square brackets in Eqn (21). If the particles are not closely packed so that ND is less than the value in Eqn (22 ) , then rd/rOwould be even closer to 1. On neglecting order (a/r,)” and using Eqn (22)) we can obtain from Eqn (21) rs-rz=
(a/6r,)NPa2[
(2-tcos
- (7ra/12J3r,)(2+cos
8) (l-cos
@)“-
6,)2(1-cos
(9rz/4?,)sin4 8
@)“-t-..]
(23)
Substituting into Eqn (20)) we obtain F,, - FI = -7dVpa2y0,(1-cos + (3/4) [ (2+cos
[a(l+cos 0) [l-cos
8)2{l-Za(2+cos @)/FO]12+
t9)]“+..)
9)/(3r,)
(NJ36)(a/r,)“X (24)
On substituting Eqn (22), the last term inside the large brackets in Eqn (29) becomes (z/18,/3)
[U(~+COS
19) (I-cos
O)/F,,]~
(251
should be noted t3at the expressions above for rd and the free energy only depend on the five variabYes M,,, 8, a, F, and ‘yoW,where NP can vary from I to the close-packed value. This is a consequence of the assumptions that the immediate surroundings of any particle is a portion of an oil/water spherical surface with a common radius rdand thaf, all the particles are equally immersed in the droplet. One would expect the interfacial free energy to depend on a conceivable local variation in the distribution of particles on the oil/water interface of the droplet. For example, this energy should change if the particles were to form clusters. The result of di Meglio and Raphael [ 31 suggests that the model of an oil/water with a common radius is a good approximation when the adsorbed particles are far apart and uniformly distributed on the interface. It
384
However, when the particles are densely packed, the assumption of a spherical oil/water interfaco with a common radius rd must be an approximation. For a single parl-icle on the droplet, NP= 1, and Eqn (22) and therefore Eqn (25) no longer apply. The last term inside the braces in Eqn (24) is of order (c~/r,)~ and is therefore negligible. Thus the change in free energy of a single particle when adsorbed on the droplet is defined by
FYI-F+
-71CZ2&.,(~_-CcOS
+ (3/4)
[U(l+COS
e)2(1-226L(2+COS
e)/(3F,)
W)
e)/F,]“+..)
If we retain the leading term only in IEqn (26) then we obtain
Ff;, --Fy = -7m2yo,(I-cos
e)”
(27)
which is the free energy change when a single particle is adsorbed on a planar oil/water interface, since we have put &/ro--0. The difference between Eqns (26) and (27) represents the effect of curvature of the oil/water interface on which a single particle is adsorbed. Suppose that the last term inside the large brackets and higher order terms in Eqn (24), as well as higher order terms in Lqn (26), are omitted. Then
(FK -FW’&
=Frr-FI
Equation (28) can be interpreted as an approximation which neglects the c f.pillary interaction between the adsorbed particles to order (a/~,)~. It follows that the last term on the right-hand side of Eqn (24) may be regarded as due to capillary interaction between the adsorbed particles because it is proportional to NE. Making use of Eqn (25) this interaction energy is cos f3)2(l-cos
A WI, -FI)/~~~ = -na2yw,
e)4
which is negative indicating an attraction between the particles. An interesting feature is the magnitude of the difference between Eqns (24 ) and (27) constituting a correction to Eqn (27) which is of order a/r,,. Nevq rtheless according to Eqn (21), when NP = 1 a good approximation to the ratio, ro/ro is given by G/G -I=
(u/+~(~-~-cos
8) (I-cos
e)2/r_2
wx
and we see that the right-hand side of Eqn (30) is of order (a/roj3 which is very small. Pn the above analysis we need only interchange JJ_with yWeand write X-- 8 in place of 0 throughout in order to treat adsorption of particles from a continuous oil phase onto a water-in-oil droplet.
3R5
Dependence on particle separation The dependence of the interaction energy in Eqn (29) on the separation between adsorbed particles can be estimated approximately as follows. We ignore the curvature of the droplet surface and assume that the particles retain their hexagonal array as they separa&. If the area allotted. per particle is Z+& a2 i, vl;ltiie i> 1, then the dis$nce between the centers of nearest neighboring particles becomes R= 2ad’f. The total number of particles NP diminishes as l/j’and in place of Eqn (22) we can write (30 which is substituted for NP into the last term inside the large brackets in Eqn (24 ) . Equation (29) is replaced by J (F,, -FL) /lV, = - na2y0, 2’ 9Jz
a4 (2+~0~3j~(l+c0st3)~ rZR2
(32)
which shoulti be a reasonable approxim &ion p;_ovided R <( r,. It seems significant that this energy decreases much more siowly with separation R than either van der Waals or electric double layer interactions. Also, for the given ratio a/R, the capillary interaction energy is proportional to a4, If a= 0.286 pm and r, = 50 pm, (values qu,oted in Ref. : 1 ] ) then at contact b fztween two nearest neighbors R = 2a and (2n:/9fi) (a2/ (4rz ) ) = 3.3 10B6. Assuming yOH. = 30 mN m- I, 7m2yo,= -2 - lo6 kT where k is Boltzmann’s constant and T is the absolute temperature. Thus at t?= n/2 and R=2a the interaction energy in Eqn (31) equals about - 26 kT, bjecreasing to -6 kT when the distance R is doubled. It is possible that this accounts for the experimental observation that at low concentrations of particla2s,the adsorption seems to occur in clusters. (Areas of close-packed coverag are observed along with uncovered patches [ 1 ] ) . However, Eqn (31) is interpreted as the additional energ; per particle due to the presence of surrounding particles. If only nearest neighbor interactions se considered, then i:i a hexagonal array the capillary interaction between two particles will be 1/6th of the quantity in Eqn (31) and a larger value is perhaps needed to account for clustering. l
l
DISCUSSION
It might be srgued that if the oil/water inter.G:ocesjoining particles remtin spherical, as it is assumed here, then there should be no capillary interaction between the particles, whatever the variation of the droplet radius. Such a possibility suggests that m.ore explanation bc given of the physical origin of the newly found attraction force in Eqn (29). The basis of our prediction of
386
this force is the difference between Eqns (24) and (28). To order (~/r,)~, the left-hand side of Eqn (28) is the sum of t.he changcls in fryp energies attributed to Np individual Particles, each of which is separately adsorbed on the droplet. This sum is the change in interfacial energy due to the adsorption of the ZVp particles when the capillary interaction between the pai-ticles is ignored. The corresponding total interfacial free ent rgy change due to the particle adsorption is given in Eqn (24) which differs f-Tornthe energy in Eqn (28) by a term proportional to N z. It is this difference which is identified with the capillary interaction. The model of spherical oil/water interfaces with a common radius Permi.ts -Y- UP O_PT2ximat.ebut elegant evaluation of the capillary interaction between the -1. adsorbed particles. A.n essential requirement is the correct value of the comrnorl radius of i;iie oil/water interfaces. This is achieved by satisfyir?.gthe necessary condition that for the incompressible system considered here the liquid volume of the droplet be conserved on adsorption of the particles. Having established this common radius there remained to deter-mine the changes in the areas of the three types of interfaces, oil/water, oil/se/lid an& water/solid and the corresponding changes in interfacial free energies. A feature of these calculations of practical importance is the expansion in powers of a/l;. We find that postulating a common radius for the spherical interfaces is not exactly equivalent to mere117forming the sum of the interfacial energy changes of the individual adsorbed particles. There is a small difference which is the capillary interaction. Setting up a common radius creates an interactive correlation between the separate particle energies. A similar capillary attraction is obtained with the conical cell model in which the oil/water interfaces are not spherical. ACKNOWLEDGMENT
The authors are grateful to the Alberta Oil Sands Technology and Research Authority (AOSTR.4) for supporting the present project (Agreement No. 451B).
REFERENCES 1 2 :1
S. Levine, B.D. Bowen and S-J. Partridge, Colloids Surfaces, 38 (1989) 345. V.B. Menon, R. Nagarajan and D.T. Wsan, Sep. Sci. Technol., 22 (1987) 2295. J-M. di Meglio and E. Raphael, J. Colloid Interface Sci., 136 (1990) 581.
CoUoid8 and Surfaces, 59 (1991? 377-336 Elsevier Science Publishers B.V., Amsterdam
Capillary interaction of spherical particles adsorbed on the surface of an oil/water droplet stabilized by the particles. Part I S. Levine and B.D. Bl;wen Deprtmcnt of Chemical Engineering, I W5, Canada
Unioersity
of British Columbia, Vanct,L
‘Jet;
B.C. V6T
iReceived 2 Janunry 1991; accq,ted 5 Apri; 1991)
Abstract The capillary interaction energy M.ween ideaticd spherical particles adsorbed as a monolayer on the surface cf an oiI/watcr emulsion dmple: stabilized by the particles is determired. Use is made of the throretical work by Menon, Nagarajan and Wasan on the free enemy change due to the adsorption of the particles on the droplet. from the continuou phase. It is assum that the oil/water interfeces joining neighboring adsorbed particles are portions of spheres wil II ticwmrnnn radius and a common center. The capillary intesction between tha particles is ~I!c~;~vv’I:’ial;~n vnolayer of into account through the change in radius of the droplet on the adsorptioil of t 1.
particles. The capillary interaction is obtained as the difference between two Jarge energim of the order of lo* kT per particle which are almost equal. Hence that part of the adsorption energy attributed to a particle which excludes its interactionswith neighboring a&orbed particles must be clearly ideatile& The capillary interaction is attractive and is comparable with a multiple of kT, diminishing as the inverse square of the separa&- - ! ~~!tween the particles and proportional to the fourth power of the particle radius.
INTRODUCTION
Pickering oil-in-water (O/W) or water-in-oil (W/O) emulsion droplets are stabilized by a film of fine particles adsorbed in a monolayer at the oil/water interface. These particles ar8 partially wettable by both oil andwater (e.g. by treatment with suitable organic molecules), so that the three-phase contact angle is neither close to 0” nor close to 18OO.The interactions between the interfacial particles are due to elt~tric double layer, van der Waals, capillary and, for deneely-packed pa&cl-, solvation forces, and in this paper we are concerned with the capillary interaction. In a recent paper [l] we assumed that the radius of a droplet is so large compared with that of an adsorbed particle that the oil/watir interface of the droplet can be regarded as planar. On
378
applying a two-din;en:;inn::l cylindrical cell mot!:,! to determine llltl cupiihtr~ I’r~rccsclcting hctwecn the interl\witll pnrticle!r we obtained a small repulsion. which we nnw believe is incorrect. In ordrx t.o :~llow fm u poxdde (:! - ,I !I!‘(! effect a conical cell model was suh::erluentlyused and this gave a small capil!ary aLtraction, the details of which will be reported later in Part II. As explained in the previous paper [ 1 ] the capillary interaction between the interfacial pnrtitles is expressed as a difference between two Jnr~r p ,y’. ;- .. 1 : !! ue almost ! .*.,’ , i,i ‘r approximations. An equal and conseqclently an error I. : ’ example of such an approximutiull cuuu Lt: Lhe assumption that the oil/water interface cbfthe droplet is Flanar. FurthermorlX ;I i: * 1 dwious how to define that part of the interfacial energy of the droplet with its film of particles which excludes the capillary 1,l beraction betw:+en the particles. A solution to the last problem mentioned above is suggested from a study of a paper by Menon, Nagarajan and Wasan [2], which we proceed to analyze and develop krthrr. ‘i I:. it t.horswere concerned with W/O emulsions, but here we sh~rll!a!z;rl~y.rjtl;.: ‘. /JqT fl,.f,i i:a11 .I I !.’ ‘r’ need only mention the W/O emulsion Lie:i:i’the em!. ‘I’hey cU?lsider the free energy change due to the adsorptio-.. c!L’11llluslolayer of spherical particles on the surface of an oil/water drople ,. They mention deformation of the oil/water interface as a result of perticle adsorption and apparently take this deformation into account by inC-d~~ci:~g ti new (increased) radi-usof the droplet. However, they do not exLlicitly sI.ate that the oil/water interfaces joining neighboring adsorbed particles remain spherical. An unambiguous expression for the capillary interaction between the narticles requires the shape of the oil/water interface tobe known. 1 ,A ;i~lrilyXill g their work we shall assume that this interSace remains spherical with a common radius and a common center. As a consequence the capillary interaction 1;L;;veex the particic:; i., 1, ‘Irectly taken into account through the change in radius of the droplet resulting from the adsorption of the particles, and this interaction will be found to be small and attrectivp A.ltimu~hwe expect that their model of sn oil/water interface with a common radius is ;approximate for a densely packed monolayer of particles, we shall show that it does suggest an expression for the interfacial energy which vmits the capillary interactionbetween the adsorbed particles. In the conical cell model which will be described in Part II, we attempt to account for the departure from a spherical shape of the oil/water interface by applying the Young-Laplace equation, particularly when the monolayer of adsorbed particles is close-packed. THEORETICAL
Geometricalconsiderations Consider one particle of radius a adsorbed on an oil-in-water emulsion droplet. Gravity is ignored and it is assumed that the droplet remains spheric&
379
Fig. 1. Geometry of one particle having center 0’ and radius a, adsc;rbud uli indroplet having center 0 and radius rd. I, II, III and IV ate the voiumes of sections of two cones with apexes 0 and 0’, with a common base. The relative eize of *theparticle is greatty uxaggzrated.
which ib justified in a recant paper by di Meglio and Raphael [3]. These authors have shown that for a single spherical yarticle adsorbed on a curvedliquid-liquid interface, the Young-Laplace equation is obeyed for a sphericalinterface when the free energy of adsorption is minimized. Let r, be the radius of the droplet in the absence of the particle and F, the “adjusted” radiusdue to the adsorption of the particle. We wish to find the relation between r, and r,. Consider the geometrical configuration in Fig. 1.0 and 0’ are the centers of the droplet and particle respectively.A and cyare the angles of the two cones having as apexes 0 and 0’ and aa a common base the circular arec whose perimeter is the oil/water/solid contact line. The sum of the volumes of the two cones equals (see Fig. 1)
i fYTt IV= (43)
3 - rtsin" ( t&in2 a COBCL
A cos A )
01
Also the correspondingsegments of the two tiphereshave volumes I+II+III= II+III+IV=
(2na3/3) (1-cos a) (2&f/3) [l-cos A)
(2)
The p& of the volume of the droplet of radius r, occupied by the sphere is therefore (4x/3) cr: -r:,
=11+111
(3)
380
by volume balance in an incompressible system. It may be verified from Eqns (1) and (2) that II+III=
(ncz3/3)
(2tcos
Lx!)(l-cos
+ (nr3,/3) (2-kcos A)
a!)2
(1-cosA)2
3)
The relation between r, and r, obtained from Eqns (3) and (4) is identical with the formula quoted by di Meglio and Raphael 13 1. Suppose now that NP particles are adsorbed on the droplet and assume that the oil/water interfaces joining the particles remain spherical, with common radius I’d and common center. Then (47V3) cr: -r~)=(7dVp/3)[d(2+cosa)(l-ccosilc)2 tri(2tcosA)
(1-c0sA)~]
(5)
We imagine that the NP particles are uniformly distributed on the surface of the droplet. That some small deviation from uniformity may be required does not matter because of the assumption that the oil/water interfaces joining the particles are portions of spheres with a common radius. The depth of immersion of the particles in the droplet and the angles a! and A are also assumed to be the same for all particles. It should be noted that when the droplet radius changes from r, to rd, corresponding changes will occur in angles cy and A, which should strictly be relabelled. This is clarified below. Menon et al. [2] have only the first term on the right-hand side of Eqn (5). We wish to estimate
Fig. 2. A typical spherical particle with center 0’ and radius a, adsorbed on a droplet. Portions of the oil/water interface joining particles are assumed to be spherical and have a common center 0 and common radius rd. The relative size of the particle is greatly exaggerated.
381
the error in neglecting the second term, assuming a common radius of the oil/ water interface. Consider a typical member of the Np adsorbed particles immersed in the droplet to a depth !, which is so defined that the distance between the centers of the droplet and particle equals at rd- 1. Since we assume that the oil/water interface of the droplet remains spherical, the contact angle 8 is that shown in Fig. 2. Then we have the trigonometric relations sin a=
(rd/a)sinA=rdsin
(o+rd -E)2=
@/(a+rd-l)
(6)
8
(7)
a2+fi+2ardcos
Expanding in powers of a/r& Eqn (7) becomes ~=a(l-cos8)[1-((a/2~d)(~+cos8)+(1/2)(a/~~)2(~+cos8)+..]
(8)
and therefore sin
A=(a/r,) sin8 [I-(a/r,)cos e-(1/2)(a/rd)2(X-3cOS2~)+..]
(9)
The corresponding series for sin a follows from Eqn (6). The expansions of -. cos A and cos a! are cosA=l-i
,_,(asin o/rd)2[1-2(a/rd)cos
8-
(3/4)
(c/rd)2X
wo
(i-5c092e)+..j COB c”=
cos B-t- (c/rd)sin2 e-
(3/2)
(a/rd)“sif?
8 cos
et ..
(11)
It is observed that cos a> cos 8 and therefore cy< 8 where for an O/W emulsion droplet 8< 90”. For a single adsorbed particle rd is replaced by r, in Eqns (6)- (11) and therefore I, A and a! will be different. If we denote cuby a0 for a single particle, then since rd> r,, it follows from Eqn (11) that cy,< GY.This means that the presence of surrounding adsorbed particles on a given adsorbed particle produces an increase in the depth of immersion of the particle in the oil droplet and this increase may be attributed to the capillary attraction between the particles. We also obtain cf!+=z 0 with our conical cell model (Part II) but for the cylindrical cell model [1] a > 8, predicting incorrectly a capillary repulsion between the: particles. We readily derive from Eqn (11) the relation sin (he)= -a sin t?/rd to order a/r,-& so that a! is very slightly Smaller than 0, suggesting that the capillary interaction between the adsorbed particles will also be very small. On substituting Eqns (10) and (11) into Eqn (5), it is readily verified that Eqn (5) may be written as (4x/3)
(r$ -rz)
= (7t.a3~~/3)1 -(%/4rd)sin4
~(~+cos~)(~-ccos~)~ e+..]
(12)
382
The formula used by Menon et al. [ 2 ] is (4n/3)(&-r5)=(7ra3Np/3)(2+cosa)(1-cosa)*
(13)
which, on using Eqn (11) yields (47r/3)(r$-J$r?,)=(
za3NP/3) [ (2tcos (l-cos
0) x
e)* -9(a/rd)sin4
so that, bearing in mind that a/r dz IO-*, slightly.
(14) et..] they have underestimated rd very
Free energy calculation
We now examine the calculation by Menon et al. [2] of the free energy change due to the adsorption of the NP particles onto a droplet from the continuous water phase. The free energy before adsorption is N, y,,
FI =4m$, yaw+4na*
2
(15)
and the corresponding energy after adsorption is F11=27ra2NP[(1-coscr)y,,+(1+cos~)y,,] + 47rr:Yaw-2nr$
Np (l-cosA)y,,
(16)
where yaw, ‘~0~and yws are the interfacial tensions of oil/water, oil/solid and water/solid systems respectively. In the formulation of Menon et al. [2] the last term on the right-hand side of Eqn (16) is replaced by - za2iV,, sin’ a! yaw. By making use of Eqns (6), (9) and (10) we calculate that 26(1-cos
A) -a*
sin’ (x= (a2/4) (a/rd)2sin4 8
(17)
so that the difference between the Eqn (16) for FII and the corresponding expression of Menon et al. [ 2 3 is proportional to the very small factor ( a/rd)2. On eliminating yoSby using Young’s equation ‘yaw cos 0 = Ycls- Yws
(18)
substituting for cos A from Eqn (10) and for cos a! from Eqn (11>,we collect separate powers of a/rd in Eqn (16) to obtain F I~=~~E~Y,,+~~~N~~~;,,,--~,,[(~-cos~)*
- (3/4) (a/rd)“sin”
tit . . I>
IIence the free energy of adsorption of NP particles on to an O/W original radius r, is
09)
droplet of
383
61 -Fi =4n(r:
-r:)‘~o~
-7raWpyow[ (l-cos
e)2
- (3/4) (a/rd)2sin4 et . . ]
(20)
Equation (12) expresses the relation between the two radii rd and write this equation in the form (r&-,)3=1+
(a/4r,)N,(&,)2[ -
(9a/4r,)sin4
(2+c0!3
6) (l--cos
F,. We
can
@I2
8+ . . ]
(21)
For a hexagonal close-packed array of spherical particles on the droplet, we obtain N,(a/r,)“=
%/J3
(22)
On substituting Eqn (22) into, Eqn (21) it is seen that rd/rO differs from 1 by a term of order a/r,. This justifies our replacing rdby r, inside the square brackets in Eqn (21). If the particles are not closely packed so that ND is less than the value in Eqn (22 ) , then rd/rOwould be even closer to 1. On neglecting order (a/r,)” and using Eqn (22)) we can obtain from Eqn (21) rs-rz=
(a/6r,)NPa2[
(2-tcos
- (7ra/12J3r,)(2+cos
8) (l-cos
@)“-
6,)2(1-cos
(9rz/4?,)sin4 8
@)“-t-..]
(23)
Substituting into Eqn (20)) we obtain F,, - FI = -7dVpa2y0,(1-cos + (3/4) [ (2+cos
[a(l+cos 0) [l-cos
8)2{l-Za(2+cos @)/FO]12+
t9)]“+..)
9)/(3r,)
(NJ36)(a/r,)“X (24)
On substituting Eqn (22), the last term inside the large brackets in Eqn (29) becomes (z/18,/3)
[U(~+COS
19) (I-cos
O)/F,,]~
(251
should be noted t3at the expressions above for rd and the free energy only depend on the five variabYes M,,, 8, a, F, and ‘yoW,where NP can vary from I to the close-packed value. This is a consequence of the assumptions that the immediate surroundings of any particle is a portion of an oil/water spherical surface with a common radius rdand thaf, all the particles are equally immersed in the droplet. One would expect the interfacial free energy to depend on a conceivable local variation in the distribution of particles on the oil/water interface of the droplet. For example, this energy should change if the particles were to form clusters. The result of di Meglio and Raphael [ 31 suggests that the model of an oil/water with a common radius is a good approximation when the adsorbed particles are far apart and uniformly distributed on the interface. It
384
However, when the particles are densely packed, the assumption of a spherical oil/water interfaco with a common radius rd must be an approximation. For a single parl-icle on the droplet, NP= 1, and Eqn (22) and therefore Eqn (25) no longer apply. The last term inside the braces in Eqn (24) is of order (c~/r,)~ and is therefore negligible. Thus the change in free energy of a single particle when adsorbed on the droplet is defined by
FYI-F+
-71CZ2&.,(~_-CcOS
+ (3/4)
[U(l+COS
e)2(1-226L(2+COS
e)/(3F,)
W)
e)/F,]“+..)
If we retain the leading term only in IEqn (26) then we obtain
Ff;, --Fy = -7m2yo,(I-cos
e)”
(27)
which is the free energy change when a single particle is adsorbed on a planar oil/water interface, since we have put &/ro--0. The difference between Eqns (26) and (27) represents the effect of curvature of the oil/water interface on which a single particle is adsorbed. Suppose that the last term inside the large brackets and higher order terms in Eqn (24), as well as higher order terms in Lqn (26), are omitted. Then
(FK -FW’&
=Frr-FI
Equation (28) can be interpreted as an approximation which neglects the c f.pillary interaction between the adsorbed particles to order (a/~,)~. It follows that the last term on the right-hand side of Eqn (24) may be regarded as due to capillary interaction between the adsorbed particles because it is proportional to NE. Making use of Eqn (25) this interaction energy is cos f3)2(l-cos
A WI, -FI)/~~~ = -na2yw,
e)4
which is negative indicating an attraction between the particles. An interesting feature is the magnitude of the difference between Eqns (24 ) and (27) constituting a correction to Eqn (27) which is of order a/r,,. Nevq rtheless according to Eqn (21), when NP = 1 a good approximation to the ratio, ro/ro is given by G/G -I=
(u/+~(~-~-cos
8) (I-cos
e)2/r_2
wx
and we see that the right-hand side of Eqn (30) is of order (a/roj3 which is very small. Pn the above analysis we need only interchange JJ_with yWeand write X-- 8 in place of 0 throughout in order to treat adsorption of particles from a continuous oil phase onto a water-in-oil droplet.
3R5
Dependence on particle separation The dependence of the interaction energy in Eqn (29) on the separation between adsorbed particles can be estimated approximately as follows. We ignore the curvature of the droplet surface and assume that the particles retain their hexagonal array as they separa&. If the area allotted. per particle is Z+& a2 i, vl;ltiie i> 1, then the dis$nce between the centers of nearest neighboring particles becomes R= 2ad’f. The total number of particles NP diminishes as l/j’and in place of Eqn (22) we can write (30 which is substituted for NP into the last term inside the large brackets in Eqn (24 ) . Equation (29) is replaced by J (F,, -FL) /lV, = - na2y0, 2’ 9Jz
a4 (2+~0~3j~(l+c0st3)~ rZR2
(32)
which shoulti be a reasonable approxim &ion p;_ovided R <( r,. It seems significant that this energy decreases much more siowly with separation R than either van der Waals or electric double layer interactions. Also, for the given ratio a/R, the capillary interaction energy is proportional to a4, If a= 0.286 pm and r, = 50 pm, (values qu,oted in Ref. : 1 ] ) then at contact b fztween two nearest neighbors R = 2a and (2n:/9fi) (a2/ (4rz ) ) = 3.3 10B6. Assuming yOH. = 30 mN m- I, 7m2yo,= -2 - lo6 kT where k is Boltzmann’s constant and T is the absolute temperature. Thus at t?= n/2 and R=2a the interaction energy in Eqn (31) equals about - 26 kT, bjecreasing to -6 kT when the distance R is doubled. It is possible that this accounts for the experimental observation that at low concentrations of particla2s,the adsorption seems to occur in clusters. (Areas of close-packed coverag are observed along with uncovered patches [ 1 ] ) . However, Eqn (31) is interpreted as the additional energ; per particle due to the presence of surrounding particles. If only nearest neighbor interactions se considered, then i:i a hexagonal array the capillary interaction between two particles will be 1/6th of the quantity in Eqn (31) and a larger value is perhaps needed to account for clustering. l
l
DISCUSSION
It might be srgued that if the oil/water inter.G:ocesjoining particles remtin spherical, as it is assumed here, then there should be no capillary interaction between the particles, whatever the variation of the droplet radius. Such a possibility suggests that m.ore explanation bc given of the physical origin of the newly found attraction force in Eqn (29). The basis of our prediction of
386
this force is the difference between Eqns (24) and (28). To order (~/r,)~, the left-hand side of Eqn (28) is the sum of t.he changcls in fryp energies attributed to Np individual Particles, each of which is separately adsorbed on the droplet. This sum is the change in interfacial energy due to the adsorption of the ZVp particles when the capillary interaction between the pai-ticles is ignored. The corresponding total interfacial free ent rgy change due to the particle adsorption is given in Eqn (24) which differs f-Tornthe energy in Eqn (28) by a term proportional to N z. It is this difference which is identified with the capillary interaction. The model of spherical oil/water interfaces with a common radius Permi.ts -Y- UP O_PT2ximat.ebut elegant evaluation of the capillary interaction between the -1. adsorbed particles. A.n essential requirement is the correct value of the comrnorl radius of i;iie oil/water interfaces. This is achieved by satisfyir?.gthe necessary condition that for the incompressible system considered here the liquid volume of the droplet be conserved on adsorption of the particles. Having established this common radius there remained to deter-mine the changes in the areas of the three types of interfaces, oil/water, oil/se/lid an& water/solid and the corresponding changes in interfacial free energies. A feature of these calculations of practical importance is the expansion in powers of a/l;. We find that postulating a common radius for the spherical interfaces is not exactly equivalent to mere117forming the sum of the interfacial energy changes of the individual adsorbed particles. There is a small difference which is the capillary interaction. Setting up a common radius creates an interactive correlation between the separate particle energies. A similar capillary attraction is obtained with the conical cell model in which the oil/water interfaces are not spherical. ACKNOWLEDGMENT
The authors are grateful to the Alberta Oil Sands Technology and Research Authority (AOSTR.4) for supporting the present project (Agreement No. 451B).
REFERENCES 1 2 :1
S. Levine, B.D. Bowen and S-J. Partridge, Colloids Surfaces, 38 (1989) 345. V.B. Menon, R. Nagarajan and D.T. Wsan, Sep. Sci. Technol., 22 (1987) 2295. J-M. di Meglio and E. Raphael, J. Colloid Interface Sci., 136 (1990) 581.