## Reports: ND554804-ND5: Polyhedral Oil Droplets: Nanoscale Elasticity in Emulsions

**Eli Sloutskin**, Bar-Ilan University

**Moshe Deutsch**, Bar-Ilan University

We
have recently discovered that droplets of C_{16} alkane [CH_{3}(CH_{2})_{14}CH_{3}]
suspended in aqueous C_{18}TAB [CH_{3}(CH_{2})_{17}N(CH_{3})_{3}Br]
solutions undergo a temperature-tunable faceting transition. Below the
transition temperature (T<T_{d}), the droplets adopt an icosahedral
shape, while still remaining liquid. These observations contrast with the
behavior of classical free liquid droplets, which are always spherical. Uncovering
the physics of this unique shape transition in liquid droplets is the subject
of our ACS-PRF-ND funded study.

In
the first year of this study, we have demonstrated that the
sphere-to-icosahedron transition occurs due to the buckling of a crystalline
2nm-thick monolayer, forming at the surface of these droplets at T_{s}>T_{d}.
This monolayer is composed of a co-crystallized mixture of alkane and
surfactant molecules, with the interfacially-frozen structure stabilized by
co-crystallization at temperatures far exceeding the bulk melting point, T_{m}.
The crystalline packing for this monolayer is hexagonal, where the molecular coordination
number is Z=6. However, such packing is incompatible with the closed surface
topology, dictating the formation of lattice defects having Z<6
coordinations. Each such defect carries a topological charge Q=6-Z, with the
total charge for the whole droplet being exactly 12, by the classical Euler
condition. To partly relax the tremendous lattice stretching energy associated
with the topological defects, the droplets adopt an icosahedral shape at T=T_{d},
with a topological charge Q=1 situated at each of the vertices[1,2]

During
the second year of this project, we followed more closely the dynamics of shape
transitions in the temperature range T_{m}<T<T_{d}. Within
this temperature range, liquid icosahedra distort along the gravity axis,
forming hexagonal, parallelogram-like, and triangular platelets; more complex shapes
occur occasionally as well. All these shape transitions proceed by merging of adjacent
vertices of the icosahedra. Specifically, when a pair of adjacent vertices of
an icosahedron merges, a hexagon emerges; tetrads of vertices merging together
form a triangle. Since the icosahedra vertices bear a charge of Q=1, each
vertex of the resultant hexagon carries a topological charge of Q=2; similarly,
the vertices of a triangle have a charge Q=4. The parallelograms have a pair of
obtuse Q=2 vertices and a pair of acute vertices of Q=4 [3]. Remarkably, the magnitude
of the vertex angle decreases monotonically with the charge: higher topological
charges form sharper vertices. This scaling is *quantitatively* reproduced
by theoretical models of inextensible closed membranes, as also by simple
macroscopic “kirigami” constructions, where topological defects are modeled by
paper cutting and folding. The observed agreement with the inextensible
membrane models’ predictions indicates that the dimensionless Fӧppl von Kármán
number (Γ_{νK}) of the interfacially-frozen
monolayer, relating its 2D Young modulus to its bending rigidity, is very high,
as indeed anticipated in our previous studies[1]. The observed localization of
high topological charges (Q>1), overcoming charge repulsion, is unexpected in
soft matter. We are currently studying mechanisms underlying this intriguing
and unusual behavior both experimentally and theoretically. Our preliminary
results suggest that the spatial distribution of topological charges across the
surface of a droplet, and consequently the geometry of droplets in their
platelet-like regime, strongly depend on gravity and buoyancy, suggesting that
new exciting shape control methods may be developed.

While
our early droplet faceting experiments involved only the C_{16}:C_{18}TAB
system, recently we have extended this work to several other alkane:surfactant
combinations. In particular, varying the alkyl chain lengths of both the alkane
and the surfactant allows the values of T_{d} and T_{m} to be tuned. The chain length
mismatch between the alkane and the surfactant has an even more dramatic
influence on the faceting phenomena: when the mismatch does not allow for an
efficient co-crystallization, the interfacial crystal is unstable and the
droplets remain spherical down to the alkane’s melting temperature T_{m}.
We employed a combination of several different methods: Wilhelmy plate
measurements, pendant drop technique, and a specially-devised new ‘microdroplet
shape tensiometry’, to measure the interfacial tension of these droplets γ(T) over a range of values spanning several orders of magnitude[3]. From these
measurements, we extracted the entropy loss ΔS_{IF}
at the interfacial freezing transition for a range of alkane:surfactant
combinations (Figure 1). The observed linear dependence of ΔS_{IF}
on the chain length demonstrates that (only) a single monolayer of alkane
freezes in all these systems; moreover, the structure of this monolayer is
largely unchanged for all combinations studied. Finally these ΔS_{IF} values
are similar to the ones detected for crystalline alkane monolayers forming at liquid-air
surfaces of pure alkane melts, an effect known as `surface freezing’. Thus, similar
physical mechanisms possibly play a role in both cases.

Figure 1. γ(T)-derived surface entropy loss ΔS_{IF} at T=T_{s}, for C_{n}:C_{16}TAB
(blue down-triangles) and C_{n}:C_{18}TAB (brown up-triangles),
matching ΔS of surface
freezing in pure alkane melts (red circles); here n is the carbon number. A
similar n-slope is measured for pure alkanes’ bulk freezing (squares), with these
data confirming transition domination by the freezing-out of the internal molecular
degrees of freedom (green dashes). Thus, a similar effect prevails for the
monolayer freezing at the liquid-liquid and liquid-air interfaces.

In conclusion, our research unveiled the topological mechanisms behind the unique sphere-icosahedron and icosahedron-platelet shape transitions of surfactant-stabilized alkane droplets suspended in an aqueous surfactant solution, for several different alkane:surfactant combinations. The ACS-PRF-ND grant enabled obtaining these highly significant results and was, and still is, instrumental in training several graduate students and postdoctoral researchers, as also in developing several collaborations with theoretical teams worldwide.

**References**

[1]
S. Guttman *et al.*,* PNAS* **113**, 493 (2016).

[2]
S. Guttman *et al.*, *Curr. Opin. Colloid Interface Sci.* **22**, 35
(2016).

[3]
S. Guttman* et al.*, *Langmuir* **33**, 1305 (2017).