Journal of Colloid and Interface Science 448 (2015) 533–544

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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis

Equilibrium orientations of non-spherical and chemically anisotropic particles at liquid–liquid interfaces and the effect on emulsion stability Nicholas Ballard a,⇑, Stefan A.F. Bon b a POLYMAT and Grupo de Ingeniería Química, Dpto. de Química Aplicada, University of the Basque Country UPV/EHU, Joxe Mari Korta Zentroa, Tolosa Etorbidea 72, 20018 Donostia/San Sebastián, Spain b Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history: Received 5 February 2015 Accepted 26 February 2015 Available online 6 March 2015 Keywords: Pickering Janus particle Interfacial adsorption

a b s t r a c t The effective stabilization of emulsions by solid particles, a phenomenon known as Pickering stabilization, is well known to be highly dependent on the wettability and the adhesion energy of the stabilizer employed at the liquid–liquid interface. We present a user-friendly computational model that can be used to determine equilibrium orientations and the adhesion energy of colloidal particles at interfaces. The model determines the free energy profile of particle adsorption at liquid–liquid interfaces using a triangular tessellation scheme. We demonstrate the use of the model, using a variety of anisotropic particles and demonstrate its ability to predict and explain experimental observations of particle behaviour at interfaces. In particular, we show that the concept of hydrophilic lipophilic balance commonly applied to molecular surfactants is insufficient to explain the complexity of the activity of colloidal particles at interfaces. In addition, we show the importance of the knowledge of the free energy adsorption profile of single particles at interfaces and the impact on overall free energy of emulsification of packed ensembles of particles. The delicate balance between optimization of adhesion energy, adsorption dynamics and particle packing is shown to be of great importance in the formation of thermodynamically stable emulsions. In order to use the model, the code is implemented by freely available software that can be readily deployed on personal computers. Ó 2015 Elsevier Inc. All rights reserved.

⇑ Corresponding author. E-mail addresses: [email protected] (N. Ballard), S.Bon@warwick. ac.uk (S.A.F. Bon). URL: http://www.bonlab.info (S.A.F. Bon). http://dx.doi.org/10.1016/j.jcis.2015.02.069 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.

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1. Introduction The ability of particles to adsorb onto soft, deformable interfaces, for example to adhere to and thus stabilize emulsion droplets, has been studied in great detail since the inception of the topic over 100 years ago by Ramsden and Pickering [1,2]. They found that inorganic particulates were capable of emulsifying oils and serve as surfactants, and noted that particle stabilized emulsions were often superior, in terms of resistance to de-emulsification, to emulsions that use molecular surfactants. A quantitative theoretical description of this effect was given by Pieranski for the case of a sphere of uniform surface tension in the absence of any imposed external force fields [3]. He demonstrated that the main cause of particle adhesion to the interface was due to energy minimization as a result of removal of an area of liquid–liquid interface and that, as a result of this, the energy to remove a particle from the interface is typically much greater than kBT, rendering particle adhesion often effectively irreversible. One important factor in the stability of the emulsions and foam materials listed above is the particle morphology and surface chemistry which govern the overall particle wettability and the strength with which it is adhered to the interface [4]. Whilst spheres are the most studied particulate emulsifier experimentally, there have also been reports on the interfacial activity of discs [5– 7], cylinders [7–10], ellipsoids [11–13], cubes [14–16], dumbbells [17–21] and hemispheres [22–25], aided by advances in the measurement of the adsorption of colloids to liquid–liquid interfaces [26–28]. In addition, new synthetic routes to complex nanoparticles have allowed us to explore the use of chemically anisotropic ‘Janus’ particles as surfactants [29,30]. Thus, Kim et al. used colloidal dumbbells synthesized by seeded emulsion polymerization to stabilize hydrocarbon in water emulsions and illustrated the ability of these particles to outperform spherical particles of the same surface groups [17]. The theoretical description of how such complicated nanoparticles orientate at liquid–liquid interfaces is of the upmost importance and governs the extent to which they are adsorbed, the type of emulsion and ultimately their ability to stabilize an emulsion. While the theoretical case for a sphere has been extensively researched [31–33], more complicated shape and chemically anisotropic particles present difficulties due to the extensive derivations that are required for analytical solutions as the particle shape deviates from a sphere. The case for a chemically anisotropic sphere has been studied by several groups who have shown that amphiphilic spheres orientate to minimize free energy and can potentially give thermodynamically stable emulsion systems [34–36]. Ellipsoidal and cylindrical particles have been shown to orientate according to capillary interactions between particles at interfaces and their orientation is guided by such adhesive bonds [37–39]. In addition, high aspect ratio nanocellulose has been shown to bend due to capillary interactions, inducing a very high surface coverage and highly stable emulsion systems [40]. Recently the effects of both chemical and shape anisotropy have been observed in simulations on amphiphilic cylinders and dumbbells with the orientation being dictated by the balance of shape and chemical nature of the particle surfaces [41]. We present here a program that allows users to predict surface activity of particles of any shape or chemistry in a simple user-friendly manner. The calculation is based on a triangular tessellation scheme originally introduced by de Graaf et al. [42] which allows for the free energy profile of a particle at an interface to be determined from which equilibrium orientations can be predicted. We show the application of the method to a broad spectrum of particle morphologies in the hope that it will aid in the synthesis by design of colloidal surfactants. The surface chemistry and morphologies selected

herein are chosen to exemplify the necessary factors that must be considered when designing particulate emulsifiers and to highlight the complementary nature of the effects of particle surface, shape and packing. In particular, it is shown that there are marked differences between the steric and chemical factors applied to conventional molecular surfactants and their colloidal counterparts. 2. Calculating the free energy surface of shape and chemically anisotropic particles The work of Pieranski demonstrated that the surface activity of spherical colloidal particles could be explained using surface tension arguments. In essence, he demonstrated that the adsorption free energy (Gad) of a colloidal particle is equal to the sum of the three areas shown in Fig. 1 that arise from penetration of a sphere into an interface. While the case for a sphere can be easily solved analytically, upon introducing anisotropy there is often no analytical solution but the general solution (Eq. (1)) still holds.

Gad ¼ AP1 rP1 þ AP2 rP2  A12 r12

ð1Þ

where A corresponds to area and r is the interfacial tension and the subscripts correspond to the particle (P) and liquid phases 1 and 2. Hence for a given particle position the free energy at this orientation can be calculated if the corresponding areas and interfacial tension values are known. In order to calculate the surface areas of a given particle we use a triangular tessellation scheme in which the particle surface is divided into a series of tessellating triangles that define the surface. For most particle morphologies (spheres, cubes, ellipsoids dumbbells etc.) the shape can be approximated by a superellipsoidal model where the location of the x, y z points can be given by Eq. (2) where x, y and z denote the coordinates of the point, rx, ry, and rz are the particles x, y and z radii respectively, and n1 and n2 acts as the ‘‘squareness’’ parameter in the z axis and the x–y plane respectively with 0 < n1, n2 < 1.



x 2=n2 y 2=n2 þ rx ry

n2 =n1 þ

z 2=n1 ¼1 rz

ð2Þ

The details of how this can be implemented for complex particle morphologies are provided in the Supporting information. In general, the model can handle any morphology that can be processed as a series of interconnecting triangles and highly anisotropic particles can be constructed in computer aided design programs and imported easily into the program (see Fig. 2), although the assumptions of the model limit the practical applicability for certain highly complex morphologies with significant capillary interactions.

Fig. 1. Free energy change upon adsorption of colloidal particle of radius, R, at the liquid–liquid interface. The interface energy, Gad, depends on the particle position with respect to the interface given by z0 (where z0 = 1 it is immersed in phase 1 and where z0 = 1 it is immersed in phase 2) and the various interfacial tension, r, values.

N. Ballard, S.A.F. Bon / Journal of Colloid and Interface Science 448 (2015) 533–544

Fig. 2. Examples of the kinds of anisotropic particles for which the free energy profile at interfaces can be computed.

In order to introduce chemical anisotropy into the system the model must have a way of determining the chemical functionality (or interfacial tension) of areas of the particle. This is achieved simply by assigning the triangles that make up the particle surface to a given interfacial tension value and hence Eq. (1) is extended to give Eq. (3) that calculates the adhesion energy for the sum of particle phases P and the corresponding interfacial tension values with the two liquids. This simple method allows for a wide range of chemical anisotropy to be introduced into the model including, striped, biphasic and patchy particles (see Fig. 2).

Gad ¼

X X AP1 rP1 þ AP2 rP2  A12 r12 P

ð3Þ

3 2 31 2 3 2 3 x 0 x B6 7 6 7 6 7C 6 7 6 7 1 0 5  4 0 cosh1 sinh1 5  4 y 5A þ 4 0 5 ¼ 4 y 5 @4 0 z zc z 0 sinh1 cosh1 sinh2 0 cosh2 cosh2 0 sinh2

The triangulation matrix does not change since the relative position of the points is fixed and therefore the tessellation pattern remains the same. After defining a particle position as described above the free energy at this position is calculated. In order to calculate the areas above, below and at the interface, AP1, AP2 and A12 respectively, we determine the surface area of each triangle and its location with respect to the interface (for more details see Supporting information). This calculation accurately takes into account triangles that intersect the interface by subdivision into a series of smaller triangles. The triangles that intersect the interface are used to define the boundary of the interfacial region, A12, whose area is calculated by a constrained Delaunay triangulation, that subdivides the interfacial region into a series of triangles whose individual areas are calculated and summed to give A12. The free energy is thus simply calculated by multiplying the calculated areas by the appropriate interfacial tension values. Therefore the value of Gad(z, h1, h2) can be calculated from Eq. (3) for a series of orientations and the free energy profile can be constructed. In the work presented here the particles are constructed from at least 1000 points and the free energy profile is calculated for a minimum 100 values in the z direction and 90–180 rotations depending on the symmetry of the particles used in the simulations. Using these values the calculated areas have an error of less than 1% compared to the analytical results for a sphere. Greater precision can be obtained by increasing the number of points on the surface of the particle at the expense of computation time.

3. Limitations of the model

P

The free energy profile for a given particle morphology is determined iteratively by scanning through a set of defined rotations and translations. In order to do this we need a consistent model to give the location of the particle. From the initial set of points centred around the origin a minimum bounding sphere with center at the origin which contains all the points is determined (see Fig. 3). The particle is moved in the z direction with a value of z varying from 1 to 1 where z is the ratio of the distance from the center of the bounding sphere to the interface, to the radius of the bounding sphere (z = zˇ/rbounding). The particle can also be rotated about the y axis (h1), or the x axis (h2), to give multiple particle orientations. The set of points for a given rotation is given by the transformation matrix in Eq. (4).

02

535

3 2

1

0

0

ð4Þ

Note that our method falls in line with Pieranski’s model, and does not take into account several factors which may contribute to the overall adsorption energy. We assume a flat interface and ignore effects due to gravity (such as interfacial deformation) because of the low energy involved compared to the effects of interfacial tension for colloidal particles with characteristic length in the region of interest (102–104 nm) [19,42]. The effect of capillarity on deforming the interface is also ignored. This assumption is somewhat validated by the negligible interfacial distortion observed in experimental measurements on anisotropic particles on the order of 100 nm [43] but a more complete description including this effect would be of interest for improving the accuracy of the model. We also ignore the effects of line tension since the magnitude of this force is negligible for smooth particles with a characteristic size exceeding the order of 10 nm [19]. Having said this, due to the calculation method of the model, the inclusion of line tension is a relatively easy task but in this work we do not consider its effects here due to limited literature values of line tension, even for the most common systems.

4. Results

Fig. 3. Example of a bounding sphere that contains all the points of a cube showing the radius of the sphere and the distance of the center of the bounding sphere to the interface zˇ.

In order to demonstrate the necessary considerations for designing particles with optimum surface activity we will use a series of example model systems, building in complexity. These morphologies have been chosen for both their experimental relevance and for their ability to demonstrate specific features of interfacial activity of particulate materials, but the practical relevance of the model is that it can similarly be applied to a huge array of particle morphologies with the caveat that they are in line with the restrictions of the model as detailed above. In order to improve the usability of the model it is freely available from the authors in the form of an interactive software program deployable on standard desktop computers.

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4.1. Effect of particle shape on adsorption to liquid–liquid interfaces To begin, the effect of particle shape on the adhesion energy at liquid–liquid interfaces was determined for a series of ellipsoidal and discoid particles with constant volume by varying the particle aspect ratio (a = width/height). The escape energy, given by the difference in energy between immersion in the oil or water phase and the free energy minima (DGad = Gimmersed  Gad,min(z, h1, h2)) for the series of polystyrene ellipsoidal particles at a hexadecane water interface with varying aspect ratio is shown in Fig. 4 (rP1 = 32 mN m1, rP2 = 14 mN m1 and r12 = 53 mN m1). It can clearly be seen that upon increasing the aspect ratio, the energy required to escape from the interface into either the water or the oil phase is increased. This can be explained by the marked increase in the interfacial area of the ‘removed’ liquid interface, which contributes to the negative energy term in Eq. (3). Similarly, upon decreasing the aspect ratio and forming an elongated, prolate type spheroid the particle orientates itself to maximize the negative energy term. In all cases a single energy minima was observed in the free energy profile as can be seen from the representative contour plots of the free energy profile shown in Fig. S1. For the spherical case the free energy minima is degenerate with respect to rotation due to the symmetry of spherical particles. A similar trend in adsorption free energy is seen for discoidal particles (see Fig. 5). In this case, a sharp transition in the global energy minima orientation between lying upright (flat side parallel to interface) and lying flat (flat side perpendicular to interface) can be observed upon decreasing the aspect ratio below 1 (see contour plots in Fig. S2). Local energy minima were observed in several cases in which both the upright and flat configurations were of similar energy and thus may be expected to coexist as can be observed in the contour plots in Fig. 5. In these cases the predominant orientation is likely to be dictated by the orientation upon approaching the interface, as will be demonstrated later, since interchange between local minima requires a large amount of energy. At extreme high and low aspect ratios only a single minima was observed indicating the importance of particle shape on particle adsorption. The net effect of particle shape can therefore be generalized by saying that the particle will orientate itself to maximize area of the liquid–liquid interface taken up by the particle whilst minimizing

Fig. 4. Escape energy for polystyrene ellipsoids of varying aspect ratio but with constant particle volume (V = 4/3p(100)3 nm3) into the n-hexadecane phase (black square) and the aqueous phase (red circle). The upper images show the particles position at the liquid–liquid interface at the most extreme aspect ratios (0.1, 1, 10). Values for interfacial tension of rHD/water = 53.5 mN m1, rHD/PSt = 14 mN m1, rwater/PSt = 32 mN m1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Escape energy for polystyrene discoids of varying aspect ratio but with constant particle volume (V = p(100)2(10) nm3) into the n-hexadecane phase (black square) and the aqueous phase (red circle). The upper images show the particles position at the liquid–liquid interface at the most extreme aspect ratios (0.1, 0.8, 1, 10). Values for interfacial tension of rHD/water = 53.5 mN m1, rHD/PSt = 14 mN m1, rwater/PSt = 32 mN m1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

unfavourable interactions with either liquid phase. It is also important to note that even fairly basic particle morphologies can exhibit multiple energy minima and thus the adsorption free energy is a rather complicated function with respect to particle shape. 4.2. Synergistic and antagonistic effects of chemical and shape anisotropy on adsorption to liquid–liquid interfaces It has been shown above that by tuning the aspect ratio of colloidal particles their ability to adhere to liquid–liquid interfaces is greatly altered. The effect of adding chemical anisotropy to this and making analogues of molecular surfactants, in which emulsion stability is imparted through differences in wettability throughout the molecular structure, has the potential to greatly increase the efficiency of colloidal surfactants. We simulated the effect of adding a moderately hydrophilic face of 2-hydroxyethyl methacrylate (HEMA) to an ellipsoidal polystyrene particle with aspect ratio 0.4 (rx = ry = 74 nm, rz = 184 nm). We achieved this by cutting along a plane in either the xy or xz axis at a designated value in the plane (vplane) such that Xplane = vplane/rmax (see top of Fig. 6) and measured the escape energy at each value (see Fig. 6) from a hexadecane water interface. It can be seen from Fig. 6 that where the xz plane is used to impart the Janus character then the escape energy is significantly higher. This can be ascribed to the synergistic effects of particle shape and particle chemistry which both encourage the particle to lie flat at the interface. Conversely, where the xy plane is intersected the shape of the particle acts as a driving force to induce the particle to lie flat and maximize the interfacial area at the interface, whereas the chemistry of the particle attempts to make the particle stand upright to encourage favourable interactions between the hydrophilic part of the particle and the aqueous phase and the hydrophobic part and the hexadecane phase. The net effect is a reduction in the amount of energy required to remove these particles from the interface. In fact, in every case the flat orientation is preferred despite the chemistry dictating the opposite. In many cases this results in a secondary minimum, which can be observed in the contour plot in Fig. 6, corresponding to the upright configuration. Upon increasing the lyophilicity of the two faces it would be expected that the chemical interactions would at some point outweigh the tendency to lie flat and an upright configuration would be energetically preferable.

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Fig. 6. Top: Janus ellipsoids with aspect ratio of 0.4 from left to right Xplane = 0, 0.5, 0.5 in the xy plane and Xplane = 0, 0.5, 0.5 in the xz plane. Middle: escape energy for polystyrene-poly(HEMA) Janus ellipsoids (aspect ratio = 0.4) of varying Janus character either intersected in the xy plane (squares) or the xz plane (triangles). The colour of the point represents that the minimum energy for escape is into the oil phase (red) or aqueous phase (black). rHD/water = 53.5 mN m1, rHD/PSt = 14 mN m1, rwater/PSt = 32 mN m1, rHD/PHEMA = 18 mN m1, rwater/PHEMA = 12 mN m1 either calculated from the polymer surface energy or taken from literature [44,45]. Bottom: free energy profile for Xplane = 0 in the xy axis (left) and xz axis (right). The grey circles show minima in the free energy profile. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In select cases for intersection in the xz plane it can be observed that changing the value of Xplane does not induce any change in the escape energy of the particle. The result is initially surprising as it implies that, within a certain relatively broad range, a change in the hydrophilic–lipophilic balance does not result in any change in interfacial activity. In these cases the particle is orientated such that the particle lay flat at the interface to maximize the interfacial area and the hydrophobic part of the particle is completely immersed in the hexadecane phase. In this instance, the polystyrene phase of the particle is not in contact with the aqueous phase and the free energy for the particle at the minimum and in the hexadecane phase is given by

Gmin ¼ APS rPS=HD þ AminHEMA=Water rHEMA=Water þ AminHEMA=HD rHEMA=HD  AminWater=HD rWater=HD

minimum and, by necessity, when the particle lies in the hexadecane phase, and therefore the escape energy can be given by

DGad ¼ GHD  Gmin ¼ AHEMA rHEMA=HD  AminHEMA=Water rHEMA=Water  AminHEMA=HD rHEMA=HD þ AminWater=HD rWater=HD

Since the total area of the HEMA part of the particle (AHEMA) is given by the sum of AminHEMA/Water and AminHEMA/HD then the adsorption escape energy reduces to

DGad ¼ AminHEMA=Water rHEMA=Water þ AminHEMA=HD rHEMA=HD  AminHEMA=Water rHEMA=Water  AminHEMA=HD rHEMA=HD þ AminWater=HD rWater=HD

ð5Þ

DGad ¼ AminHEMA=water GHD ¼ APS rPS=HD þ AHEMA rHEMA=HD

ð6Þ

where Amin corresponds to the areas at the thermodynamic minima in the free energy profile. In these cases all the polystyrene phase (APS) is contained within the hexadecane phase, both at the

ð7Þ



ð8Þ 

rHEMA  rHEMA þ AminWater=HD rWater=HD HD Water

ð9Þ

In cases where the polystyrene particle phase was immersed in the hexadecane phase and where the shape of the particle induces it to sit where the area at the liquid–liquid interface is maximum (see Fig. 7) then the two values of AminHEMA/water and AminWater/HD do not change upon changing the surface chemistry. This results

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Fig. 7. Identical equilibrium orientations of two polystyrene (red)/poly(HEMA) (green) Janus ellipsoids with different intersection planes in the XZ axis (left) Xplane = 0.4 and (right) Xplane = 0.6 at a water (upper phase) hexadecane (lower phase) interface. The minimum escape energy for both is into the hexadecane phase and is identical (6.484  105 kBT). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

in a scenario where changing the amphiphilic balance of the particle does not result in any net gain or loss of interfacial activity and is somewhat contrary to logical thought. This result is in marked contrast to molecular surfactants where the hydrophilic hydrophobic balance is a deciding factor in emulsion type and stabilization efficiency. Thus, we can conclude from these results that in order to maximise the particle adsorption energy the effects of surface chemistry, particle shape and their mutual effects should be considered. When designing colloidal particles to stabilize liquid– liquid interfaces the orientation in which the particle is most stable should be used to determine where the chemical anisotropy should lie to encourage synergistic effects between the two. 4.3. Surface activity of dumbbell like particles This work inspired us to take a look at the surface activity of amphiphilic dumbbell particles. This class of particulate emulsifier is particularly interesting as they have been cited as good potential candidates for emulsion stabilization and are experimentally accessible. Initially we varied the distance between the centres of the two, identically sized, spherical components of the dumbbell, one consisting of polystyrene and the other of poly (2-hydroxyethyl methacrylate) and obtained the free energy profile of the particle at a water/hexadecane interface. Where the distance was zero and the particle was essentially a Janus sphere, the particle orientation is as one would expect, with the hydrophilic lobe residing in the aqueous phase and the hydrophobic polystyrene lobe in the hexadecane phase (see Fig. 8). However as the distance between the spherical components of the dumbbell is increased the particle begins to orientate itself to lie flat with respect to the interface, in effect ignoring the chemical anisotropy. This can be explained by looking at the increasing area of the missing liquid–liquid interface upon tilting the particle. The reduction in energy arising from this tilting outweighs the benefit of maximizing the hydrophilic and hydrophobic interactions of the particle and its medium. The result again indicates that for optimum interfacial adsorption both the chemical and shape anisotropy must promote the same interfacial orientation. If one effect acts antagonistically against the other then the result is an orientation which is a compromise of the two effects and results in a reduction in the adsorption energy. Interestingly, in all cases metastable orientations were observed that correspond to one of the spherical lobes adsorbing to the interface such that the amphiphilic character of the particle is not ‘‘seen’’ by the interface. These orientations are at energy levels such that any motion of the particle results in either: (i) a degenerate energy level; or (ii) a higher energy level, thus particle motion may be expected to be somewhat restricted. Indeed, it has recently been shown that these metastable states exist experimentally and have surprisingly high stability due to contact line pinning hindering rotational diffusion [18]. The degenerate states are not only of significantly different energy level to the

Fig. 8. (Top) Free energy change upon varying orientation angle demonstrating the effect of distance on the equilibrium orientation angle of Janus dumbbells with two spherical lobes of 100 nm radius. Symbols represent centre to centre distances of the two spheres that make up the Janus particle 0 nm(black square), 25 nm (red circle), 50 nm (blue up triangle), 75 nm (pink down triangle), 100 nm (green diamond), 125 nm (dark blue left triangle) and 150 nm (violet right triangle). rHD/water = 53.5 mN m1, rHD/PSt = 14 mN m1, rwater/PSt = 32 mN m1, rHD/PHEMA = 18 mN m1, rwater/PHEMA = 12 mN m1 either calculated from the polymer surface energy or taken from literature [41,42]. (Bottom) Equilibrium orientation of Janus dumbbells (polystyrene (red)/poly(HEMA) (green)) at hexadecane water interface for interlobe distances of (from left to right) 0 nm, 75 nm and 150 nm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

global minimum but also are orientated in a distinct manner, the importance of which is further exemplified in the following section. Examples of the free energy profiles and the potential metastable orientations are given in Fig. 9. It can be observed that both the number of degenerate states and the degree of degeneracy, that is the number of orientations that correspond to an identical energetic state, is increased as aspect ratio increases. One of the advantages to the present technique for calculating the free energy profile is that the propensity of the metastable state with varying particle morphology can be estimated by examination of the gradient at each point in the free energy profile thus obtaining a vector field of adsorption force (see Eq. (10)) [46].

rGad ðz; hÞ ¼

dGad ðz; hÞ dG ðz; hÞ ^ ^z þ ad h dz dh

ð10Þ

At the minimum in the free energy profile, and at metastable orientations, the gradient is 0 and thus the particle is stationary. The particles are initiated from a random angular orientation, completely immersed in the either the oil or water phase. The particles then follow an adsorption trajectory dependent that follows the steepest gradient of the free energy profile and ends up in a given orientation that is dependent upon the initial orientation and bulk phase location (see Fig. 10). It can be seen that for the case of Janus spheres, which have a relatively small number of degenerative states, then the metastable states do not impact greatly on the average orientation and therefore will not be expected to alter drastically the adsorption energy since most particles are directly adsorbed in the global minimum orientation. However, for the highly anisotropic dumbbell particles with largest separation a large number of the particles are expect to pass through the metastable states and only a minority of particles are directly adsorbed in the energetically favourable conformation (see Fig. 11). The

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539

Fig. 9. Contour plots showing the free energy profile of Janus dumbbells with two spherical lobes of 100 nm radius at interlobe separation distances of 0, 100 and 200 nm. The black dashed lines indicate metastable orientations corresponding to the graphical representations in Fig. 11. rHD/water = 53.5 mN m1, rHD/PSt = 14 mN m1, rwater/PSt = 32 mN m1, rHD/PHEMA = 18 mN m1, rwater/PHEMA = 12 mN m1 either calculated from the polymer surface energy or taken from literature [41,42].

indicated that they are long lived states whose population decays only slightly with time and over the course of hours rather than seconds [18]. This result indicates the impact that the combination of shape and chemical anisotropy may have on the surface activity of colloidal particles and further highlights how the model presented herein may help to understand and explain complex relationships between particle morphology and surface activity. 4.4. Thermodynamics of emulsification

Fig. 10. Orientation of Janus dumbbells as a function of the interlobe separation distance depending on initial bulk phase location based on free energy profiles of particles from Fig. 8. The results are based on statistical analysis of 1000 particle initially placed at a randomly distributed orientation in either the water or oil phase. See Fig. 11 for graphical representations of the various metastable states.

impact of this cannot be underestimated since the energy of adsorption will therefore be a function of the orientation of approach to the interface and, critically, the initial bulk phase that the particles are dispersed in. It is also important to note that although the stability of metastable states with respect to time is currently unknown, experimental measurements have clearly

So far, the attachment of a single particle to a flat liquid–liquid interface has been considered. In the formation of particle stabilized emulsions the liquid–liquid interface becomes saturated by the presence of particles and the free energy of emulsification (DGem) in such a case can be determined from the sum of the contribution of the free energy of each droplet DGd. For i droplets this leads to

DGem ¼

X DGd;i

ð11Þ

i

The individual contributions of each droplet can be approximated from the sum of the energy required to create the interfacial area (Aow row ) and the energy of adsorption of a given number of particles at the interface (Aow row )

DGd ¼ Aow row þ np DGad

ð12Þ

where Aow is the area of a bare notionally swollen droplet and np is the number of particles on the surface of the droplet. The droplet is

Fig. 11. Pathways of adsorption of Janus dumbbells (polystyrene (red), poly(HEMA) (green)) from either the bulk oil phase (underneath the interface) or from the aqueous phase (above the interface). Blue arrows represent the initial adsorption of the particle to the interface with the thickness representing the probability of the trajectory occurring. The white arrows represent transitions from metastable states to the global minimum. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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considered to be swollen to take into account that adsorption of np nanoparticles to the interface results in a displacement of volume, thus swelling the droplet. This analysis neglects the entropy change upon demixing of particles from the continuous phase, which can be effectively discounted due to its negligible effect compared to the adsorption free energy. In addition, the use of Aow row , which is calculated on the assumption of a flat interface and ignores the curvature of the droplet, limits the applicability of this equation to scenarios in which the droplet size is much greater than the size of the absorbing particles [19,31]. It should also be noted that in real systems, emulsions are subject to strong external forces, such as collision, which will heavily influence their stability, thus the elasticity of the interface is an important factor in kinetic stability of particle stabilized emulsions. The contributions of lateral interactions (electrostatic, van der Waals and hydration forces) between particles have previously been shown to be negligible, except at very high fractional coverage of particles, with respect to the adsorption energy and can therefore be reasonably ignored [19]. In the context of the current work the anisotropic nature can induce changes in particle–particle interactions but will not drastically affect this assumption as there are several orders of magnitude difference between the adsorption energy and the sum of the lateral interactions [19]. Capillary interactions between particles due to interfacial deformation, which can be significant [38], are not taken into account in the model. This is perhaps the biggest simplification of the model, although it is somewhat validated by the notable absence of the effect in experimental work for a variety of anisotropic colloids [43]. However, great care must be taken when applying the present model to highly complex particle morphologies in which capillary interactions can be strong and may direct particle orientation. A thorough quantitative analysis of these ignored effects has been performed elsewhere and will not be repeated here [19]. It should also be highlighted that although the noted effects make a relatively small contribution to the energy terms, in many cases only a small change in energy is required to drastically alter physical observations (the ability to form thermodynamically stable emulsion is one notable example in the present work) and thus the precise quantitative nature of the results must be treated with caution. Having said this, the demonstration that the thermodynamics of emulsification is heavily influenced by the delicate balance between the effects of particle adsorption and particle packing at liquid–liquid interfaces, which is the main goal of this work, is unaffected by such interactions. Upon adsorption of particles onto the droplet surface, the droplet will swell from radius R0 to radius R. Each adsorbing particle displaces a volume equal to the sum of the volume occupied by the particle underneath the contact line and the volume of the droplet removed that lied above the contact line. For a sphere these volumes can be calculated analytically but for convex/anisotropic particles this is achieved numerically by volumetric integration using the triangular tessellation pattern as the bounding surface. For the simplest case of a spherical particle absorbing onto a droplet of radius R0 we can calculate the number of particles capable of absorbing to the interface. The fractional coverage, g, of a planar interface of randomly packed monodisperse spherical particles is close to 0.84. In this work the fractional coverage is based on adsorption at a planar interface, thus making the assumption that the droplet diameter is considerably larger than that of the adsorbing particles. Where the radius of the droplet is much greater than the characteristic length of the particle, for a given particle orientation the height above the interface, h, is known and the number of particles at the interface can be given by

np ¼ g

Ad Ap

ð13Þ

where Ad is the surface area of a droplet with radius R + h, which equates to the radius of curvature at which particles come into contact, and Ap is the maximum cross-sectional area of the particle (i.e. for a spherical particle Ap = pr2). This is calculated by iteratively scanning through the particle in the z direction and calculating the area of each cross-section and taking the maximum value. The number of particles that can be adsorbed onto an initially bare oil water interface is solved taking into account the swelling of the particle in the following manner. An unswollen droplet of known radius, R, is taken. For an initial guess of the swollen radius of the droplet the number of particles is calculated

np ¼ g

4pðRguess þ hÞ Ap

2

ð14Þ

The volume displaced (Vdisp) by each particle is calculated by volumetric integration using the triangular tessellation pattern and assuming a flat interface as the bounding surface. The radius of the corresponding droplet in the absence of particle can thus be calculated by 1

Runswollen ¼ ðR3guess 

3 3 np V disp Þ 4p

ð15Þ

If |Runswollen  R| < 104 nm then the equation is considered to be solved such that Rswollen is the radius of the swollen droplet with np particles on the surface. Otherwise a new value for the swollen radius is taken as Rguess = Rguess + (R  Runswollen) and the process is repeated until the condition is satisfied. Knowing the radius of the swollen droplet, the number of particles adsorbed and the free energy of adsorption per particle it is possible to calculate the free energy of droplet formation and emulsification using Eqs. (12) and (11) respectively. 4.5. Thermodynamics of emulsification using shape anisotropic particles The present work has shown that it is possible to calculate adsorption energy of any given particle morphology using the triangular tessellation technique. In order to assess how complexity impacts upon the free energy of emulsification we begin, as in the previous section, with the case of a superellipsoidal object of constant volume but varying the squareness parameter. For the series of superellipsoids shown in Fig. 4, varying from ellipsoidal to oblate spheres, the corresponding free energy of droplet formation of a 10 lm droplet of water in hexadecane, assuming maximum coverage is shown in Fig. 12. A w/o emulsion was used because the hydrophobic particles used in this work will tend to create this type of system in accordance with the Bancroft rule [47] and therefore, adsorption to the interface is assumed to take place from the oil phase. As can be seen the formation of droplets becomes less energetically unfavourable when the particle shape becomes less spherical. It must be noted that in all cases the energy of drop formation is positive and therefore energy is required to emulsify the system and the resulting emulsion droplets are thermodynamically unstable. For the case of oblate spheroids, i.e. flattened spheres, the packing density would be expected to be similar to that of spheres, given the similar circular geometrical constraints at the interface, but as in this case the particles used are of constant volume the effective area taken up per particle is increased. As such when the aspect ratio is increased the number of particles that can sit at the interface is lowered. This effect is countered by the large interfacial energy gain for adsorption of a single oblate spheroid compared to a sphere as shown in Fig. 4. The net result is that the large value of DGad is sufficient to counter the reduction in the number of particles and thus formation of droplets is more favourable as aspect ratio is increased.

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Fig. 12. Energy of formation of a 10 lm drop of water in hexadecane using polystyrene superellipsoids of varying aspect ratio similar to Fig. 4. Packing fraction for spherical and oblate spheroids (aspect ratio > 1) is 0.839. For particles of aspect ratio < 1 the packing fraction is given by g ¼ 0:3382a3 þ 0:2438a2 þ 0:141aþ 0:7933 [48].

Similarly for ellipsoids as the ellipsoid becomes longer the reduction in number of particles is countered by the favourable values of DGad thus resulting in emulsification being easier as the particle shape deviates from a sphere. However since the packing fraction of ellipsoids is not identical to spheres and is a function of the aspect ratio this trend is reversed when the shape deviates strongly from a sphere. In this work we used the aspect ratio dependence of the packing fraction taken from Delaney and coworkers on the assumption of a flat interface (i.e. droplet diameter is much greater than diameter of adsorbing particles) [48]. They showed that ellipsoids have a maximum random packing fraction of 0.89 at aspect ratio of 0.7 and could be fit to a 3rd order polynomial. As the ellipsoid is lengthened beyond 0.7 the packing fraction decreases and eventually becomes lower than that of spherical particles. This can be observed for the longest cylinder used in this work where the reduction in packing fraction leads to a greatly lower number of particles at the interface, and resulted in a tendency to increase the emulsification energy where the ellipsoid is very long. This highlights the delicate balance between the particle morphology dependent packing fraction and the adsorption energy of the corresponding particle at an interface. 4.6. Thermodynamics of emulsification using shape and chemically anisotropic particles The additional effect of adding chemical anisotropy to shape anisotropic particles was shown to be critical in the adsorption of single particles to liquid interfaces above but provides additional complications when considering packed ensemble systems. We take the ellipsoids shown in the first section that are cut along the xy plane and were shown to give optimum surface activity when lying flat due to the large interfacial area occupied by this configuration with a secondary minimum in the upright orientation (see Fig. 6). This is beneficial in sparse systems, where there is large unoccupied surface area, but in packed systems, as the number of particles at the interface increases, the flat configuration eventually limits adsorption of additional particles. In this case it can become energetically favourable to orientate in the upright manner, whereby an increased number of particles can be accommodated at the interface. Although this is less energetically favourable on a per particle basis, it allows a significantly larger number of particles to adsorb to the surface and therefore results in an

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energetically more favourable system. This can be seen in Fig. 13 where the flat and upright configurations are compared. As the number of particles adsorbed on the droplet is increased the flat configuration is initially preferred since all the particles can be occupied at the interface and each particle has a highly favourable adsorption energy. However where high numbers of particles are adsorbed the upright configuration is favoured and the possibility of a thermodynamically stable emulsion appears as the increase in number of particles at the interface more than compensates for the slightly lower adsorption energy per particle. This results in a scenario where for single particles the flat configuration is energetically favourable, but for packed ensembles the upright orientation is preferred (see Fig. 14). For ellipsoidal particles that are cut in the xz plane such that the chemical anisotropy and the shape anisotropy both favour the flat position this behaviour is not observed as is shown in Fig. 13. In this case the increase in the number of particles that can adsorb to the interface when in the upright position is insufficient to compensate for the significantly lower adsorption energy per particle. Thus both for single particle and packed ensembles the flat configuration is preferred in this case. In these systems the possibility of thermodynamic emulsion formation (see Fig. 13) has been shown but it must be highlighted that there is a significant energy barrier to reorientation of the particle and thus the system may be kinetically limited from forming a stable emulsion due to freezing in one orientation. This is particularly important for the particle with Janus character imparted in the xy plane since the majority of particles will initially adsorb in the energetically favourable flat orientation and need to be flipped to orientate themselves in the manner that leads to thermodynamically stable emulsions (see Fig. 14). This reorientation energy will result in a sizeable kinetic barrier. Thus, although for both cases a thermodynamically stable emulsion is predicted, the complications arising from the delicate balance between the varying preferential effects of particle shape, chemistry and packing may mean that only in the xz case will the thermodynamically stable emulsion be expected to be formed. 4.7. Thermodynamics of emulsification for anisotropic particle exhibiting multiple energy minima In the examples considered thus far it has been assumed that the particle tends to lie in a configuration that corresponds to a local minimum in the free energy profile. It has also been shown above that dumbbell type particles can possess many metastable intermediates in the free energy adsorption profile which can play an important role in the interfacial activity of these colloids (see Figs. 9–11). While the energy minima is clear in this case, and corresponds to a flat particle at the interface, alternative configurations can lead to increased packing and therefore, as exemplified by the ellipsoidal Janus particles, may be beneficial for emulsion stability. Fig. 15 shows the corresponding free energy for emulsion formation for 3 distinct particle orientations resulting from adsorption from the oil phase. In the calculation it is assumed that the number of particles at the interface in the near flat configuration is given by 2  np,sphere with a contact angle of 90°. This simplification is needed because of the complications with regards to calculating packing fractions when packing anisotropic 3d objects at 2d interfaces but is valid given the similarity in orientation to the global minimum configuration for this colloid. It can be seen that the orientation corresponding to the most favourable configuration for a single adsorbed particle does not yield a thermodynamically stable emulsion but, conversely, one of the two possible metastable orientations can provide thermodynamically stable emulsions. This is a direct result of the large increase in number of particles (approximately two times) that can lie at the interface

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Fig. 13. Energy of droplet formation for a 10 lm water in hexadecane droplet stabilized by ellipsoidal Janus particles with aspect ratio 0.4 and similar to those in Fig. 6. The lines represent the upper limit of the number of particles adsorbed when in the flat and upright configurations. Right figure shows the free energy of emulsification of a 1 m3 of a 20 vol.% water in oil emulsion containing 2 vol.% particles with respect to the oil phase. The particles are initially dispersed in the oil phase. Janus particles are cut either in the xy plane (black squares) or xz plane (red circles). The results of orientating the particle flat (filled symbols) or upright (open symbols) are shown. Packing in the flat orientation is calculated with a packing fraction of 0.867 while in the upright configuration a packing fraction of 0.839 is used [48]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. Difference between preferred orientation when a lone particle is adsorbed at the interface (top phase water, bottom phase hexadecane) and in packed ensemble states for Janus ellipsoids (polystyrene (red)/poly(HEMA) (green)) for a w/o emulsion due to balance between adsorption energy per particle and number of particles that can be adsorbed per unit area. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

in the upright configuration compared to the near flat one. This increase in np is sufficient to overcome the effect of having a smaller adsorption energy per particle compared to the flat orientation. It is also of note that the configuration that is expected to yield a thermodynamically stable emulsion does not involve the second particle phase interacting with the interface at all. Rather the second phase simply renders the particle hydrophobic enough that it tends to reside in the oil phase and makes the escape energy into the aqueous phase very high. The case of the Janus dumbbell can be adequately used to sum up the often conflicting factors contributing to the interfacial activity of anisotropic colloidal particles, as shown in Fig. 16. A colloidal particle that can adsorb at a liquid–liquid interface is expected to do so with a unique adsorption trajectory dependent on the initial phase it lies in, its orientation and, crucially, the free energy adsorption profile. From an energetic viewpoint the optimum orientation of the particle will be given by the balance between the shape of the particle, which drives the particle to enhance the surface area removed from the interface, and the chemistry of the particle, which tends to orientate to maximize chemical interactions.

Kinetically, however, due to the nature of the free energy profile metastable or secondary minima can exist that may prevent the global minimum being reached, particularly due to the large energy associated with reorganization. Furthermore, when packed the optimum orientation for a single particle must now compete with the drive to increase the number of particles at the interface. For the colloidal dumbbells shown in Fig. 16, this work has shown that, energetically, the most favourable orientation is one in which the dumbbell lies near flat at the interface, while the most likely initial orientation upon adsorption is the metastable orientation in which the polystyrene lobe is exclusively at the interface. However, to further complicate matters, it is the other metastable minimum that is capable of producing thermodynamically stable emulsions in this case, due to the combination of high packing and relatively high adsorption energy. This complex system demonstrates the importance of understanding the free energy profile in order to comprehend how the subtle interplay between shape, surface

Fig. 15. Free energy for emulsion formation of water in hexadecane, 1 m3 emulsion containing 20 vol.% water and 4 vol% particles compared to the oil phase depending upon particle orientation at the interface for Janus dumbbells shown in Fig. 11. Black squares – metastable state MO-1 (poly(styrene) at interface). Red circles – metastable state MO-2 (poly(HEMA) at interface). Blue triangles – flat particle corresponding to free energy minima. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 16. Potential orientations of Janus dumbbells (polystyrene (red), poly(HEMA) (green)) resulting from adsorption from the hexadecane phase to hexadecane–water interface. Blue arrows denote adsorption to the interface with relative thickness approximating the extent of each orientation. White arrows represent transition from metastable state. Black arrows show transition from single particle to packed states. See Fig. 11 for graphical representations of the various metastable states. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

chemistry and interfacial packing of anisotropic colloids affects the ability to adsorb to, and stabilize, liquid–liquid interfaces. It is hoped by continuing efforts to understand the transitions between orientational states [46,49], the effect of interparticle interactions at liquid interfaces [38,50], and aided by the present model, it should be possible to tailor make particle surfactants for optimum surface activity. Such an advance should allow for the formation of thermodynamically stable emulsions based on relatively low cost solid stabilizers.

5. Conclusions In conclusion, we have developed a computational method to analyse the interfacial activity of highly shape, and chemically anisotropic colloidal particles in order to better understand numerous experimental observations [18,28,51]. Analysis of a series of experimentally accessible particle morphologies has shown that in order to optimize interfacial adsorption, agreement between the shape anisotropy, which will tend to promote the particle to lie in a position in which a large interfacial area is covered, and the chemical anisotropy which will favour the particle lying in an orientation that promotes favourable interactions with the two liquids, should be considered. Crucially, it has been shown that the concept of hydrophilic lipophilic balance which is commonly used in molecular surfactants is not necessarily appropriate for colloidal systems and changes in surface chemistry can have no impact on particle adsorption. This important result should lead to a dramatic reanalysis in the design of colloidal emulsifiers which, to date, have largely been based on the chemical features of molecular surfactants [17]. In addition, the possibility and importance of metastable orientations that have been observed experimentally [15,18] is highlighted, showing that orientations that are metastable for isolated particles can lead to thermodynamically stable emulsions in packed states. The ability to form thermodynamically stable emulsions for a number of additional cases is shown and it is emphasized that the orientation dependant particle packing and adsorption free energy of isolated particles are critical parameters that often work antagonistically with respect to lowering the free energy of emulsification. It is hoped that the current physical description of particles at interfaces should provide the necessary

knowledge to obtain thermodynamically stable emulsions using tailor-made colloidal stabilizers. Acknowledgments Equipment used was supported by the Innovative Uses for Advanced Materials in the Modern World (AM2), with support from Advantage West Midlands (AWM) and part funded by the European Regional Development Fund (ERDF). We thank Unilever for financial support. Appendix A. Supplementary material Electronic Supplementary Information (ESI) available: Full details of the computational method and examples of use of the program are available as ESI. The program is available free to download from the authors website. Supplementary data associated with this article can be found, in the online version, at http:// dx.doi.org/10.1016/j.jcis.2015.02.069. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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