#### 3.1. Pickering Drop Deformation in an Electric Field

If a drop is less conducting than the surrounding fluid, the drop’s overall dipole moment can be oriented anti-parallel with the electric field direction [

36], and the electric stress distribution can induce oblate drop deformation (compressed in the direction of the electric field). More precisely, drops are oblately deformed if the Maxwell charge relaxation time [

36] of the drop is longer than that of the surrounding fluid, and they become prolate (elongated in the direction of the electric field) in the opposite case, as shown by Taylor [

37]. The observed oblate deformation of a pure silicone oil drop (

Figure 1a) is consistent with the charge relaxation times calculated from the fluid parameters listed in

Table 1 in the Materials and Methods section.

The observed increase (~300%) in steady-state deformation of PE Pickering drops compared to particle-free drops can in general be caused by: (i) reduced resistance to deformation related to reduced surface tension, (ii) added elastic stress from the particle shell, or (iii) an increase in electrical stress.

Because the measured change in surface tension between the PE Pickering drop and the particle-free drop is much smaller (~10%, see

Section 2.1) than the change in deformation (

Figure 1), the increased PE Pickering drop deformation cannot be caused by a reduced surface tension (since the deformation of leaky electric drops is inversely proportional to the surface tension [

37]). Particle-particle capillary interactions can induce resistance to two-dimensional shear deformation of the Pickering layer [

38]. However, such elastic stress will only cause resistance to deformation, and should therefore not be the cause of the increased deformation of the PE Pickering drops. We also performed experiments in an alternating current (AC) electric field (

Figure S3). In this case, there is no free charge build-up at drop interfaces, which means that only dielectric forces are involved in the deformation, and the PE Pickering drop is slightly less deformed than the pure silicone oil drop. In the absence of free charge build-up, the electric stress on the drop interface is expected to be approximately the same for a pure drop and a Pickering drop, because the dielectric constant of PE is very similar to the dielectric constant of silicone oil, and because the volume of the PE particle layer is small compared to the volume of silicone oil. This result supports the assumption that the resistance to deformation (due to Laplace pressure and/or elastic modulus of the Pickering shell) is approximately the same for a pure drop and a PE Pickering drop, provided that the particle layer is below the granular jamming transition, as is the case in

Figure 1 and

Figure 2.

Because both (i) and (ii) are ruled out as the main cause, the increase in deformation of the PE Pickering drop is expected to be caused by increased electric stress that results from larger charge accumulation at the drop interface. The increased charge build-up can be due to both a decrease in electric conductivity of the drop interface and absence of electrohydrodynamic charge convection [

39]. The strength of an electrohydrodynamic charge convection mechanism is quantified by the electric Reynolds number

$R{e}_{\mathrm{E}}$, which is defined as the ratio of the Maxwell-Wagner charge relaxation time

${\tau}_{\mathrm{e}}$ (time for charge to build up at the interface [

36]) to the flow charge convection time

${\tau}_{\mathrm{f}}$ (time for charge transport by convection [

36]). For the silicone oil drop in castor oil system,

${\tau}_{\mathrm{e}}$ is of the same order as

${\tau}_{\mathrm{f}}$. Thus,

$R{e}_{\mathrm{E}}$ ≈ 1, and charge convection may be significant for pure drops [

39]. This is not the case for Pickering drops, because the tangential electric stress is absorbed in the particle layer, and electrohydrodynamic circulation flows (viscous forces) are supressed [

8,

9,

28] (in accordance with our observations) allowing for larger charge accumulation.

When silicone oil drops are covered by clay particles instead of PE particles, the conductivity of the drop interface increases in comparison to that of both the PE Pickering drop and pure silicone oil drop. As a result, the clay Pickering drops behave differently with an oblate-prolate deformation transition (

Figure 2b). In addition, the conductivity of clay particle dispersions is non-ohmic and increases with the applied electric field [

40]. Thus, the conductivity of the clay Pickering film is also expected to increase with the electric field.

Electric field-induced deformation of leaky-dielectric drops has been extensively studied and modelled [

36,

37,

39,

41]. Theoretical models have also been developed for vesicles and membrane-covered drops [

42,

43], and recently also for particle encapsulated drops [

28,

44,

45]. These works consider an elastic particle layer on the capsule, i.e., a particle layer with shear elasticity.

In our present experiments on PE particle-covered drops, the surface concentration is sufficiently low for the particle network to restructure upon deformation, i.e., the particle layer can be considered to be fluid. We have previously shown experimentally that our clay Pickering films are easily deformed and fluid-like [

8]. Therefore, here we describe the Pickering drop layer as thin fluid shell. With this assumption we find that the expression for the electric field induced deformation is:

$D\approx \frac{{E}_{0}^{2}{\epsilon}_{0}}{{\gamma}_{0}{\sigma}_{\mathrm{f}}{\left(2{\sigma}_{\mathrm{ex}}+{\sigma}_{\mathrm{in}}\right)}^{3}}({\alpha}_{0}+{\alpha}_{1}d).$ The

$\alpha $ coefficients are functions of the dielectric constants, conductivities of the materials and drop radius (see

Supplementary Materials for details on the calculation). The fluid shell thickness is

$d$, and its electric conductivity and dielectric constant are

${\sigma}_{\mathrm{f}}$ and

${\epsilon}_{\mathrm{f}}$, respectively. The electric conductivity and dielectric constant of the silicone oil drop are

${\sigma}_{\mathrm{in}}$ and

${\epsilon}_{\mathrm{in}}$, and those of the exterior liquid (castor oil) are

${\sigma}_{\mathrm{ex}}$ and

${\epsilon}_{\mathrm{ex}}$. The electric Maxwell stress working on a leaky-dielectric drop has normal and tangential components. The normal component of the electric stress is balanced by the Laplace pressure with a non-uniform surface tension

$\gamma ={\gamma}_{0}+\delta \gamma \mathrm{cos}2\theta $, where

$\theta $ is the polar angle,

${\gamma}_{0}$ is the uniform component of the surface tension and

$\delta \gamma $ is determined from the tangential stress balance [

46]. In our description, the tangential component of the electric stress is balanced by the gradient of the surface tension of the Pickering film

$\nabla \gamma $, which is due to particle-particle interactions. For PE particles these may be capillary-mediated interactions [

38], whereas clay particles may also be cohesive. The dielectric constants of the fluids and particles, the electrical conductivities of the fluids and the interfacial surface tensions are estimated from independent measurements and are given in the Materials and Methods section. Using the fluid shell description and the data points in

Figure 2a, we estimate the electric conductivity of the PE Pickering film to be ~30% of the electrical conductivity of the silicone oil, which is much larger than the conductivity of PE. The conductivity of PE is approximately seven orders of magnitudes smaller than that of the silicone oil (see Materials and Methods). However, because the PE particle layer on the Pickering drop is porous (we observe that there is silicone oil between the PE particles), the conductivity of the Pickering PE layer is expected to be closer to that of silicone oil than to the conductivity of PE.

Due to the non-ohmic response of clay particles (reported previously in [

40]), the electric field dependence on the conductivity of the Pickering particle layer is unknown. For this reason, the fluid shell description cannot be used to analytically calculate an expression for conductivity of the particle layer from the drop deformation (

Figure 2b). Instead, we first calculate the conductivity of the clay Pickering film numerically from the measured deformation data in

Figure 2b at different electric capillary numbers (proportional to

${E}_{0}^{2}$). The calculated values of conductivities are shown in

Figure 4. We then fit the data points in

Figure 4 with a polynomial function, which gives the particle film conductivity as a function of the applied electric field:

${\sigma}_{\mathrm{f}}\approx 1.98\times {10}^{-10}{\mathrm{Sm}}^{-1}+2.30\times {10}^{-20}{E}_{0}^{2}{\mathrm{SmV}}^{-2}+1.67\times {10}^{-31}{E}_{0}^{4}{\mathrm{Sm}}^{3}{\mathrm{V}}^{-4}.$ The electric permittivity of clay can also change with electric field. However, the change in the electric permittivity of clay is reported to be much smaller than the change in electric conductivity [

40].

#### 3.2. Relaxation of Drop or Pickering Drop Deformation after Turning off the Electric Field

In the present experiments, all Pickering and pure drops relax to a spherical shape when the electric field is turned off.

Figure 5a shows the transient relaxation dynamics of a pure silicone oil drop (blue triangles), a silicone oil drop with PE particle ribbon (red circles), and a PE Pickering drop (green squares). In

Figure 5b it is evident that the silicone oil drop relaxes to a spherical shape (

D = 0) faster than silicone oil drops covered with PE particles, and that in all cases the relaxation is exponential

$D\left(t\right)={D}_{0}{e}^{-\frac{t}{\tau}}$, where

${D}_{0}$ is the initial deformation,

t is the time elapsed since the electric field is removed and

$\tau $ is the relaxation time. The experimental relaxation times of the three cases:

${\tau}_{\mathrm{silicone}}$ = 0.19 s for a pure silicone oil drop,

${\tau}_{\mathrm{ribbon}}$ = 0.48 s for a silicone oil drop covered with a PE particle ribbon, and

${\tau}_{\mathrm{Pickering}}$ = 0.56 s for a PE Pickering drop. The characteristic relaxation time for a silicone oil drop covered by PE particles (estimated particle coverage around 75% and 84%, see Materials and Methods) is significantly longer than that of a pure silicone oil drop.

For the case of a pure drop, balancing capillary and viscous forces [

47,

48] gives the relaxation time

$\tau ={\tau}_{\mathrm{d}}=\left[{\mu}_{\mathrm{ex}}{r}_{0}\left(2\lambda +3\right)\left(19\lambda +16\right)\right]/\left[\gamma \left(\lambda +1\right)\right]$. Here

$\gamma $ is the drop surface tension,

${r}_{0}$ is the radius of the drop,

${\mu}_{\mathrm{ex}}$ is the viscosity of the exterior liquid and

$\lambda $ is the ratio of the drop viscosity and the exterior fluid viscosity. Inserting numbers in the equation above, the theoretical relaxation time for a pure silicone oil drop with a size of 2.4 mm is

${\tau}_{\mathrm{silicone}}$ = 0.25 s, which is close to our experimental observation given above. Even though the electric field is turned off, electric forces can still contribute to the relaxation dynamics of pure drops, because the free charges accumulated at the drop interface discharge with the finite Maxwell-Wagner charge relaxation time, which is

${\tau}_{MW}$ ≈ 1 s for this system [

8]. These free charges interact and can create a force that can influence the shape-relaxation [

49]. The inset in

Figure 5a compares the experimentally measured relaxation of a pure silicone oil drop with the two theoretical models for drop relaxation: (i) only considers capillary forces [

48] and (ii) considers both capillary and electric forces [

49]. The comparison shows that the effect of electric forces after the electric field is turned off can be neglected here, and we will therefore use the model that only consider capillary driven relaxation.

Surface-adsorbed particles can in some cases reduce the Laplace pressure of the interface [

16], which can also slow down the relaxation [

50]. However, we do not observe any significant difference in the surface tension of a particle-free silicone oil drop in castor oil (4.5 mNm

^{−1}) compared with a PE Pickering silicone drop in castor oil (5.0 mNm

^{−1}). The changes in the relaxation time between these drops should therefore be due to viscous dissipation caused by the presence of the particle layer. The PE particle layer imposes constraints on the hydrodynamic flow inside and outside the drop, which can increase the viscous dissipation. In addition, the fluid PE particle layer may in itself possess an effective two-dimensional viscosity due to hydrodynamically mediated friction between particles as they glide along each other [

35].