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Interfacial behaviour in water-oil-amphiphile mixtures
M. Matsen, D. Sullivan
To cite this version:
M. Matsen, D. Sullivan. Interfacial behaviour in water-oil-amphiphile mixtures. Journal de Physique II, EDP Sciences, 1992, 2 (1), pp.93-107. �10.1051/jp2:1992116�. �jpa-00247617�
Classification Physics Abstracts
68.10C 64.60C 64.70M
Interfacial behaviour in water-oil-amphiphile mixtures
M. W. Matsen and D-E- Sullivan
Department of Physics and Guelph-Waterloo Program for Graduate Work in Physics, University
of Guelph, Guelph, Ontario, Canada, NIG 2Wl (Received J8 July J99I, accepted 27 September I99I)
Rksu~nk. Utilisant un moddle en rdseau pour des mdlanges eau-huile-amphiphile, nous
examinons les interfaces entre les phases riches en eau, riches en huile et ddsordonnde, avec les
approximations de Bethe et champ moyen. En calculant les tensions de surface entre ces phases le
long de leur triple ligne de coexistence, on trouve que la phase ddsordonnde mouille l'interface eau-huile lorsque le fluide est un fluide d6sordonnd ordinaire mais non lorsque celui-ci est une micro-Emulsion, la transition de mouillage assoc14e dtant faiblement de premier orate. Nous utilisons ici la ligne de ddsordre comme division entre les rdgions ordinaires et de micro-Emulsion
plutbt que la ligne de Lifshitz. Dans la rdgion de coexistence double-phase entre les phases riches
en eau et riches en huile, on trouve trois diffdrentes phases d'interface eau-huile, ainsi que les transitions entre celles-ci. Une de ces phases, contenant une monocouche d'amphiphile fortement orientde, prdsente une trds faible tension de surface.
Abstract. Using a lattice model for water-oil-amphiphile mixtures, we examine the interfaces between the water-rich, oil-rich and disordered phases in the mean-field and Bethe approxi-
mations. By calculating the surface tensions between these phases along their coexistence triple line, we find that the disordered phase wets the water-oil interface when it is an ordinary disordered fluid but not when it is a microemulsion, where the associated wetting transition is weakly first order. Here, we use the disorder line as the dividing line between the « ordinary » and microemulsion regions, rather than the Lifshitz line. In the region of two-phase coexistence between water-rich and oil-rich phases, we find three distinct water/oil interfacial phases and the transitions between them. One of these interracial phases, which contains a strongly oriented
amphiphile monolayer, typically exhibits ultralow surface tension.
1. Introduction.
There are numerous aspects of water-oil-amphiphile mixtures which are of both commercial and biological interest. One such aspect is the nature of the interfaces between various bulk
phases. Because of their complexity, the molecular models [1-12] used to study such mixtures
often restrict the freedom of the molecules to occupy any position (and orientation) by
employing a lattice. Usually these lattice models are further simplified by treating the water
and oil molecules symmetrically. These simplifications mean that the models will behave
somewhat differently from real mixtures, but there is evidence that they can still account for the basic behaviour of these mixtures [I].
JOURNAL DE PHYSIQUE fl T 2, N'i,JANVWR iW2 6
In this paper, we extend our earlier work [2] on a lattice model for oil-water-amphiphile
mixtures to examine the interfaces between the water-rich (W), oil-rich (O) and disordered (D) phases. In this model, as in several other microscopic models [2-9], these phases can
simultaneously coexist along a triple line extending from a multiphase point up to a tricritical
point beyond which the transition becomes second order. At the multiphase point, the three
uniform phases coexist with an additional lamellar phase, but we will not be directly
concemed with the latter in this work. Of main interest is the interfacial wetting behaviour at three phase W + O + D coexistence. In particular, we examine whether the D phase partially
wets the W/O interface, producing a non-zero contact angle as shown in figure la, or completely wets the interface as shown in figure16. As pictured in figure I, the D phase usually resides between the other two phases due to their relative densities (and hence is often
referred to as the middle phase).
o o
D D
w w
la) (b)
Fig. I. Incomplete (a) and complete (b) wetting of the W/O interface by the D phase.
0 is the contact angle, which vanishes in the case of complete wetting.
Toward the low-temperature end of the triple line near the multiphase point, the D phase is
identified as a microemulsion [13]. In this region, when the water and oil concentrations are
comparable as in the present work, the water and oil are believed to forrn bicontinuous interweaving tube-like structures in which a amphiphile monolayer accumulates on the vast water/oil interface [3]. As one moves up the triple line towards the tricritical point, the disordered phase loses its microemulsion structure and becomes an ordinary disordered fluid.
The « ordinary » and microemulsion regions are not separated by a well-defined line such as a
phase transition where the bulk free energy has a singularity. In several previous works, identification of these two distinct regions has been based on the behaviour of the water-water correlation function and related structure function [1, 2, 4, 5, 10]. Recent theoretical [6, II, 14] and experimental [15-17] work has indicated that a wetting transition, from incomplete to
complete wetting by the D phase, may occur on the triple line. In reference [14], a Landau-
Ginzburg model was used to demonstrate that such a wetting transition is closely related to the change in structural properties of the fluid, and occurs when the microemulsion
transforrns into ordinary fluid. The failure of the microemulsion to wet the W/O interface was attributed to the presence of exponentially-damped oscillatory profiles at the interface
between microemulsion and either W or O phases [14, 18].
Results from the present model do indeed show a wetting transition on the triple line. Our calculations also confirm that the wetting transition occurs in close proximity to the « disorder line » [1, 4, 5, 10], which separates the microemulsion from ordinary fluid based on the
presence or absence of oscillations in the water-water correlation function. With the
exception of reference [14], this aspect has not been examined in earlier studies based on
microscopic models [19]. We find the wetting transition to be first order in agreement with experiment [15]. Earlier studies using microscopic models have either been silent on this point [6] or have found second-order wetting transitions [I Ii, although reference [14] predicts that
the transition can become first order with suitable long-range interactions. The wetting
transition in the present model is difficult to locate precisely, which may explain why its
nature has been elusive.
It is well known that small amounts of amphiphile can greatly reduce the surface tension between water and oil [14, 20, 21]. To investigate the ultralow W/O surface tension of the present model, we also examine the entire W + O coexistence region [7], and in doing so find three distinct interfacial phases. One phase has a low interfacial amphiphile density, and the other two have a concentrated amphiphile layer at the interface. At high temperatures, the amphiphile layer tends to be wide, while at low temperatures it is a monolayer. It is the latter
suriace phase containing a amphiphile monolayer that exhibits the ultralow surface tensions.
The other two phases only exhibit ultralow surface tensions near the second-order line, but
this source of low tension is unrelated to the particular structure of the fluid.
2. Model.
In this paper, we use essentially the same three-dimensional lattice model introduced in
reference [2]. Because the Hamiltonian used in reference [2] does not depend on the
orientations of either the water or oil molecules, we choose not to explicitly include their orientational degrees of freedom. We still retain the amphiphile's orientational degrees of freedom, which allows each amphiphile (surfactant) molecule to orient itself with its polar head-group pointing towards any one of its six nearest-neighbour sites on a cubic lattice. Thus in this paper, there are 8 states for each lattice site as opposed to 18 in reference [2].
As before, the I-th lattice site has a state variable «,, which takes on the values +1, 0 or I if the site is occupied by water, amphiphile or oil, respectively. When the I-th site is occupied by an amphiphile, we associate with it a second state variable s, which is a unit vector pointing towards one of the six nearest neighbours, specifying the
direction of the polar head-group. If the I-th site is occupied by water or oil, then
s, = 0. In terrns of these state variables, the Hamiltonian is
4
li ~
" i I (~a Pa,
<j + Ka Pa, ij) Psi Pi (I)
<"J> « =1 ,
where
p,= I-ml
~l,ij
" ~i~j
~2.ij " ~i (Sj 'rip) + ~j(Si ~ji) (2)
~3,
y " (Si '~ij) (Sj
~j<
P~
,~ =
(s; x r,~) (s~ x r~,).
The symbol ii, j) denotes that the sum is over all distinct pairs of nearest-neighbour sites
I and j. The vector r,~, measured in units of the lattice spacing, is the displacement of lattice
site j relative to lattice site I. The parameters J~ and K~ for a
=
I to 4 deterrnine the bonding
energies. For the explicit results presented in this paper, the ratios of these parameters are
selected such that
Jj =~Kj=~J~=2K~=~J~=K~=J~=~K~wJ~0. (3)
The parameters J~ and K~ are the ones that most strongly control the amphiphilic strength of the amphiphile molecules [2]. This selection of parameters corresponds to amphiphile with an intermediate amphiphilic strength. Finally, in equation (I) p~ is the relative amphiphile
chemical potential. Water and oil are assumed to have equal concentrations, so that chemical-
potential terrns associated with these species have been omitted.
3. Calculations.
A method for calculating the bulk phase diagram by the mean-field and Bethe approximations
is outlined in reference [2]. For the parameters in equation (3), the phase diagram in the temperature and amphiphile-chemical-potential plane obtained using the mean-field and Bethe approximations is shown in figures 2 and 3, respectively [22]. W, O, and D denote the uniforrn water-rich, oil-rich and disordered phases, respectivly. The S phase is a crystalline amphiphile-rich phase. The L~ phases are lamellar phases where the water and oil layers are n lattice sites wide, separated by amphiphile monolayers. From analysis of a low-temperature
series expansion, we conclude that the L~ and L~ phases predicted by the mean-field
approximation are spurious. The corresponding phase diagram in the temperature and amphiphile-density plane is shown for the two approximations in figures 4 and 5.
To obtain the disorder line (DL) and Lifshitz line (LL) defined below [1, 4, 5, 10], we need to examine the water-water correlation and structure functions, respectively, for the D phase.
As in reference [2], we use an exact functional relation for the water-water correlation function, cww(r). The latter is related to the change in the probability of having a water
molecule at a particular site due to an infinitesimal potential acting on water molecules at the site located a distance r away. This exact functional relation produces a matrix equation which is evaluated here using the mean-field approximation. It could also be evaluated using the
6
DL~~ ~~
5 /
~~~(,,o,o) D
_~
4 '-
~
3 j~
uQ
44 ,. L~
2
W+O L S
-4 -3 -2 -1 O 2 3 4
lLs/J
Fig. 2. The phase diagram in the temperature versus amphiphile-chemical-potential plane, calculated
by the mean-field approximation. The solid lines represent first-order phase transitions, and the dashed line is a second-order, transition. The dotted lines in the W + O region represent W/O interracial phase
transitions with
a triple point and critical point marked by dots. The dot on the W + O + D triple line marks the wetting transition. The DL and LL lines in the D phase are shown with dotted lines.
6
5 2.25
D
4 2.00
-n 0.25 0.50 0.75 1.00
~ ~"----
E- 3 ~'
uQ
~ 2
w+o L~ s
i
-4 -3 -2 -1 2 3 4
lLs/J
Fig. 3. The same phase diagram as in figure 2, but obtained in the Bethe approximation. Here the DL and LL lines have not been calculated. The inset shows on an expanded scale the interracial phase
transition line which emerges from the W + O + D triple line, discussed in section 5.
6
, '',
DL~,~o) 'DL~~ ,~
5 ',,~ ",
_~
4 W+O "',
D
~
3 j~
uQ ,
~
2 L S
O
o-o 0.2 0.4 0.6 0.8 1-o
Ps
Fig. 4. The phase diagram in the temperature versus arnphiphile-density plane, calculated by the mean-field approximation. The wetting transition in this case is indicated by the horizontal dotted line in the W + O + D coexistence region.
6
5 ~'~~
_=' 4
"",
2.°( ~~ 0 05 O-lo
Hn~ ',,
1i+O ',
F- 3 D uQ J4
2 S
L~ Li
0
o-o 0.2 0.4 0.6 0.8 1.0
Ps
Fig. 5. The same phase diagram as in figure 4, but calculated in the Bethe approximation.
Bethe approximation, but due to the complexity qf such a calculation we choose not to do so.
A Fourier transforrnation conveniently simplifies the equation for c~v~v(r) allowing one to
straightforwardly calculate the water-water structure function,
Sww (q m £ cww (r) e'~ r (4)
,
Then, in principle, one can perforrn the inverse Fourier transforrnation on Sww(q) to obtain
cww(r).
The structure function S~V~V(q), which is an even function in q, can be expanded for small q m (q( as
sww(q ) = A + Bq ~ + o (q~) (5)
The locus B
= 0 is the LL line shown in figures 2 and 4. Below this line, B ~0 and
Sww(q) has a peak at non-zero q characteristic of a microemulsion. One sees that the LL line
occurs very close to the ordered Li and S phases, and thus identifies only an extremely narrow
region of the D phase as microemulsion. This feature was discussed previously in reference [2].
The DL line, which marks the onset of oscillations in the asymptotic behaviour of
cww(r), is more difficult to obtain than the LL line. In the mean-field approximation, the
structure function has the forrn Sww(q)=Sww,J(q)+Sww,K(q), where Swwj(q) and
Sww, ~(q) depend solely on the J~'s and K~'s, respectively. Long-wavelength oscillations in
cww(r) cannot possibly occur from the inverse Fourier transforrnation of Sww, ~(q) and so we
only need to examine Sww,j(q) for calculating the DL line. The reason for this is that long- wavelength oscillations only occur if the parameter J~ is non-zero. (This is explicitly shown in reference [10] for the one-dimensional lattice but can be generalized to higher dimensions.) Performing the inverse Fourier transformation of Sww,j(q) for large r to obtain the
asymptotic behaviour of cww(r is considerably difficult, and so we resort to an approximation
used in reference [4]. The approximation is based on the small-q representation of
Sww, J(q),
~~~'~~~
Aj + Bj q~ + ~jq~
+ O(q~) ' ~~~
where the coefficient Cj depends on the direction of q. The asymptotic behaviour of
cww(r) is then assumed to be proportional to
l~-
<q rAj + Bj q~ + Cj q~
~~
'
~~~
where the integration is extended over all of q-space, assuming spherical symmetry for
Sww j(q) icy's dependence on the direction of q is ignored). The DL line is the locus
Bj
=
2 ~/Aj Cj. To the right of this line, the above expression for the asymptotic behaviour of cww(r) produces exponentially-damped oscillations. In figures 2 and 4, the curves denoted DL ~j o,~~ and DL~j, i, u are the disorder lines obtained when Cj is evaluated in the (1, 0, 0) and (I, I, I) directions in q-space, respectively. The reasonable agreement between these two lines justifies the assumption of spherical symmetry. The DL lines identify a much larger region of the D phase as microemulsion than the LL line.
To obtain the surface tension between two coexisting phases, we first calculate the effective fields for each bulk phase by either the mean-field or Bethe approximation as outlined in
reference [2]. A slab of the lattice, which is infinite in two dimensions and n lattice sites wide in the third dimension, is used to represent the interface between the two coexisting bulk
phases. This slab consists of n 2 inner layers plus two outer layers in contact with the bulk
phases. Next, the effective-field equations are solved on the inner n 2 layers of the slab
subject to the boundary conditions that the effective fields on each outer layer equals those of the bulk phase in contact with it. The free energy of the slab is always greater than the free
energy for an equal volume of either bulk phase. This excess free energy per unit area of the slab approaches the surface tension as the width of the slab, n, is increased. In most cases, the
interface is thin and the convergence is rapid as the width of the slab is increased.
When calculating the water/oil surface tension, «wjo, there are two solutions to the effective-field equations for the W/O interface. One is obtained when the interface thickness is increased keeping n even and the other while keeping n odd. In general, one must evaluate the interface for both cases and select the one which provides the smaller surface tension. As
one may suppose, this by itself can lead to phase transitions, and in fact does at high
temperatures when the interface is wide. These transitions, which are always extremely weak,
are a consequence of using a lattice model and would not be present in a continuum model.
For this reason, we do not include results in this paper conceming this type of interfacial transition.
4. Interfacial behaviour along the W + O + D triple line.
In figure 6 is a plot of «wjo a~/J and 2 «y~~ a~/J calculated in the mean-field approximation along the W + O + D triple line, where a is the lattice spacing and J is defined in equation (3).
(Note that due to the symmetry of the present model, «o~
= «y~~.) To set the scale,
«y~jo a~/J= 2 at comparable temperatures in the absence of amphiphile. At the multiphase point (kB T/J = 2.6133), «y~jo a~/J
= 4.0771 x IO ~ and not zero as it may appear in figure 6.
However, at the tricritical point (kB T/J
= 3.7732) both surface tensions do in fact approach
zero. In figure 6, 2 «y~~ a~/J is plotted rather than «y~~ a~/J becomes it is the
more relevant
quantity when considering the wetting of the W/O interface by the D phase. Consider a
droplet of D phase at a W/O interface as shown in figure la. Balancing the forces due to surface tension at a vertex one obtains
°
- 2 C°S~ 1~?o~ (8)
o.5
0.4
0.3
0.2
o-i
O-O
2.6 2.8 3.O 3.2 3.4 3.6 3.8
k~T/J
Fig. 6. A plot of the W/O and W/D surface tensions along the triple line, calculated using the mean- field approximation. The dot marks the wetting transition.
for the contact angle R. The dot in figure 6 at k~ T/J
=
3.3009
m k~ Ty~/J marks the wetting
transition where the two curves become equal and o becomes zero. In figure 7 is a plot of the
contact angle from the multiphase point to the wetting transition.
In general, if the surface tensions satisfy the inequality «y~jo~«y~~ + «o~, then D incompletely wets the W/O interface, while if «y~jo = «y~~ + «o~ it completely wets the interface. When the wetting transition is first order, there will exist in the vicinity of the transition a metastable interface with «y~jo ~ «y~~ + «o~. It is by finding such an interface that we conclude the present wetting transition is first order. The first-order nature of the
transition is also indicated in figure 7 by the fact that o intersects the o
= 0 axis with a non-
zero slope.
80
50
20
m go
60
30
2.6 2.8 3.O 3.2 3.4
k~T/J
Fig. 7. A plot of the contact angle along the triple line from the multiphase point to the wetting
transition as calculated by the mean-field approximation.
Figures 8 and 9 show the mean-field distribution functions across the unwetted W/O interface before and at the wetting transition, respectively. The distribution functions, p~, p,, p_, p_ and pi are the respective probabilities that a site is occupied by water, oil,
amphiphile pointing to the bulk W phase, amphiphile pointing to the bulk O phase and
,o
0.8
6p 0.6
0.4 ~~
0.2
o-o
-15 -io -5 o 5 io 15
zla
Fig. 8. The distribution functions across the W/O interface at k~T/J = 3.2 in the mean-field
approximation.
