Analysis of Current-Voltage Characteristics in

Insulating Polymers using a Bipolar Charge Transport Model S. Le Roy, F. Baudoin, C. Laurent, and G. Teyssedre

Abstract-A bipolar charge transport model has first been used to compare the current voltage characteristics (J-E) for different applied voltage protocols and different times under voltage taking low density polyethylene as a case study, as a steady state may take a very long time to be reached experimentally. Simulation results highlight the necessity to spend substantial time under voltage to reach a quasi-stable current. In a second part, evolutions of the model in terms of physical processes are proposed, to observe their impact on the J-E characteristics, and particularly the appearance of electric field thresholds, as proposed by the Space Charge Limited Current (SCLC) theory. To do so, different mathematical expressions are proposed for each physical process related to injection, mobility and trapping. Field dependent mobility equations allow obtaining electric field threshold values comparable to experimental data, while a constant mobility is the only way to observe a trap free limit region. Moreover, all parameters linked to trapping are of most importance in the observation of a SCLC characteristic, but are not enough to observe a trap free limit region when electric field dependent mobility is considered.

Index Terms-bipolar charge transport model, fluid model, J-E characteristics, polymer insulation, SCLC theory

I. INTRODUCTION

URRENT-voltage measurements are necessary to characterize polymers used as electrical insulation under direct current (DC) thermo-electrical stress. They are often analyzed in terms of Space Charge Limited Current (SCLC) theory. This theory has been developed with simple but restrictive physical hypotheses [START_REF] Wintle | Absorption currents and steady currents in polymer dielectrics[END_REF], such as one type of carrier, ohmic contacts, no traps or a unique level of traps, and steady state condition. Figure 1 illustrates a typical current density as a function of voltage curve, where different regions are observed:

-an Ohmic region, where JV. This region is often not observed experimentally for insulating polymers. -A trap limited current region, where JV 2 when traps are not distributed -A trap-filled limited current, i.e. JV  , where theoretically a single trapping level is considered. When traps (deep traps particularly) are distributed, the slope is not infinite. -A trap free limited current. The voltage at which all traps are filled is called VTFL. Even with such restrictive physical hypotheses, the SCLC theory allows describing the specific S-shaped current density-voltage (J-V), or current density-electric field (J-E) characteristic often observed for insulating polymers [START_REF] Bhutta | Steady-State Conduction Current Performance for Multilayer Polyimide/SiO2 Films[END_REF], and particularly polyethylene [START_REF] Hamedi | Impact of the Impurities on the Conductivity of Low-Density Polyethylene[END_REF]. Among the restrictive physical hypotheses of the SCLC theory, obtaining a steady state current is almost impossible for highly insulating polymers [START_REF] Ghorbani | Long-term conductivity decrease of polyethylene and polypropylene insulation materials[END_REF]. In the present paper, a bipolar charge transport (BCT) model [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] has been used to evaluate and understand the differences in results arising from the variation of time under voltage application from seconds to quasi-steady state, and the differences arising from the experimental protocol. The BCT model has then been used with different physical hypotheses related to injection, mobility and trapping, hopefully in the spirit of [START_REF] Dissado | The Role of Theory in Understanding Space Charge Distributions[END_REF], in order to observe the impact on the simulated current densityvoltage characteristics, particularly on the reproduction of a specific S-shape, as predicted by the SCLC theory. Hypothesis of a field dependent mobility seems more appropriate to obtain an electric field threhold in the range of the experimental ones, i.e. 40 kV/mm for LDPE [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF]. However, it goes against obtaining a third region in the J-E characteristic. Although many hypotheses for the mobility rule were tested in the present paper, it was not possible to correctly reproduce the experimental J-E characteristic, for a simple material such as LDPE. C Fig. 1. A typical current density-voltage characteristic of space-charge-limited conduction current. VTR is transition voltage. VTFL is trap-filled limit voltage.

II. MODEL DESCRIPTION

A bipolar charge transport model, already reported in the literature [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF][START_REF] Alison | A model for bipolar charge transport, trapping and recombination in degassed crosslinked polyethene[END_REF][START_REF] Xia | Numerical analysis of packetlike charge behavior in lowdensity polyethylene by a Gunn effectlike model[END_REF][START_REF] Hoang | Charge Transport in LDPE Nanocomposites Part II-Computational Approach[END_REF], has been used for the simulations in a low density polyethylene (LDPE). This charge transport model, also called mesoscopic model in the literature for insulating polymers [START_REF] Doedens | Space Charge Accumulation at Material Interfaces in HVDC Cable Insulation Part I-Experimental Study and Charge Injection Hypothesis[END_REF], is a hydrodynamic model, based on the advection equation. The model is one dimensional, function of the depth in the sample, and features generation of electronic charges (i.e. electrons and holes) through injection at each electrode, charge transport within the bulk, trapping into one level of deep traps, from which charges can detrap, and charge recombination between carriers of opposite sign (Figure 2). The equations to solve are of the form:

𝜕𝑛 𝑎 𝜕𝑡 + ∇. (𝑛 𝑎 µ 𝑎 𝐸 -𝐷 𝑑𝑖𝑓𝑓-𝑎 ∇𝑛 𝑎 ) = 𝑠 𝑎 (1) ∇. (𝜀 0 𝜀 𝑟 𝐸) = (𝑞𝑛 ℎµ + 𝑞𝑛 ℎ𝑡 -𝑞𝑛 𝑒µ -𝑞𝑛 𝑒𝑡 ) = 𝜌 (2)
Where na is the density (m -3 ), and a refers to the charge carrier, being electron (e) or hole (h), mobile (µ) or trapped (t). t is the time, µ the mobility (m 2 /V/s), for each charge carrier, E the electric field (V/m), ε0 is the vacuum permittivity, and εr the relative permittivity of the material (2.3 in the case of LDPE), q is the elementary charge (C), and ρ the net charge density (C/m 3 ).

Ddiff-a refers to the diffusion coefficient, and is of the form, following the Einstein relation:

𝐷 𝑑𝑖𝑓𝑓-𝑎 = 𝑘 𝐵 𝑇 𝑞 µ 𝑎 (3)
With kB the Boltzmann's constant (J/K) and T the temperature (K). This expression of diffusion is adopted since used in organic semi-conductors, even if it is not clear whether or not it is usable when the mobility is field dependent. This particular point has been largely documented in the literature for organic semi-conductors [START_REF] Wetzelaer | Validity of the Einstein relation in disordered organic semiconductors[END_REF][START_REF] Kambili | Electrical transport modelling in organic electroluminescent devices[END_REF]. sa in (1) are the source terms, reflecting all the physical processes taken into account but not linked to transport (trapping, detrapping and recombination in the present paper). An example of source term for mobile electrons is given in:

𝑠 𝑒µ = -𝐵 𝑒 𝑛 𝑒µ (1 - 𝑛 𝑒𝑡 𝑁 0𝑒𝑡 ) + 𝐷 𝑒 𝑛 𝑒𝑡 -𝑆 1 𝑛 𝑒µ 𝑛 ℎ𝑡 -𝑆 3 𝑛 𝑒µ 𝑛 ℎµ (4)
Be is the trapping coefficient (s -1 ), N0et the trap density for electrons (m -3 ), which corresponds to a maximal trapped charge density of 100 C/m 3 . De is the detrapping coefficient for electrons (s -1 ), and S1 and S3 are recombination coefficients (m 3 /s). The same kind of equation holds for trapped electrons and mobile and trapped holes, and can be found in [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF].

All the variables are function of space and time, even if equations (1-4) do not reflect this, for sake of simplicity.

Charge generation is only due to injection of carriers at each electrode, and follows a Schottky law:

𝑗 𝑆𝑐ℎ𝑜𝑡𝑡 (𝑋) = 𝐴𝑇 2 𝑒𝑥𝑝 ( -𝑞𝑤 𝑎𝑖 𝑘 𝐵 𝑇 ) [𝑒𝑥𝑝 ( 𝑞 𝑘 𝐵 𝑇 √ 𝑞𝐸 𝑋 4𝜋𝜀 0 𝜀 𝑟 )] (5)
A is the Richardson's constant, wai is the injection barrier height, for electrons or holes (eV), and the coordinate X refers either to the anode or to the cathode. There is no extraction barrier at the cathode for holes and at the anode for electrons.

The total external current density is calculated based on the following Maxwell equation:

𝐽 𝑡𝑜𝑡 (𝑡) = 1 𝐷 ∫ 𝑗 𝑐𝑜𝑛𝑑 (𝑥, 𝑡) 𝐷 0 𝑑𝑥 + 𝜀 0 𝜀 𝑟 𝜕𝐸(𝑥,𝑡) 𝜕𝑡 (6)
Where jcond(x,t) refers to the conduction current, being the sum of the conduction current of each mobile carrier, and D is the sample thickness.

In the present paper, the sample considered is a LDPE of thickness 150 µm. The model, developed with COMSOL Multiphysics® uses the mathematic module (Partial Differential Equation) to solve the convection-diffusion equation for each kind of carrier (4 kinds). The Poisson equation module is used to couple these equations to the Poisson's equation. The Backward Differentiation Formula solver is used for the time integration (maximum order 2, and minimum order 1).

III. IMPACT OF THE APPLIED VOLTAGE PROTOCOL ON THE J-E RESULTS

A. Impact of the time under voltage application

Simulations have first been performed to observe the impact of the time of voltage application on the current density vs. electric field (J-E) characteristic. The physical hypotheses taken into account are: a constant mobility, a constant trapping coefficient, a detrapping coefficient function of temperature, for each kind of carrier, and constant recombination coefficients. A Schottky injection law ( 5) is used for charge generation at each electrode.

An optimized set of parameters has been previously determined in [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF]. Although no experimental data will be presented in the present paper, the optimized set of parameters chosen is able to reproduce most of the space charge and current measurements available for LDPE, for different experimental protocols. This set of parameters is presented in These time values are based on practices reported in the literature for polyolefins [START_REF] Hashimoto | Space Charge and Conduction Current in Polypropylene Based Material Added Inorganic Filler under DC High Stress[END_REF][START_REF] Kadowaki | Observation of current spikes caused by possible fast carrier transport through a low-density polyethylene sheet under high DC fields[END_REF][START_REF] Teyssedre | Charge distribution and electroluminescence in crosslinked polyethylene under dc field[END_REF][START_REF] Su | Investigation of effects of charge injection and intrinsic ionic carriers in low-density polyethylene and cross-linked polyethylene[END_REF][START_REF] Chen | Space charge dynamics of acetophenone and cumyl alcohol and their synergistic effect in LDPE[END_REF][START_REF] Li | Effect of Crystallinity of Polyethylene with Different Densities on Breakdown Strength and Conductance Property[END_REF][START_REF] Li | Trap Modulated Charge Carrier Transport in Polyethylene/Graphene Nanocomposites[END_REF][START_REF] Kumara | Electrical Characterization of a New Crosslinked Copolymer Blend for DC Cable Insulation[END_REF][START_REF] Yahyaoui | Electrical properties of polytetrafluoroethylene with mineral fillers under high DC electric field[END_REF]. This leads to a total time of experiment varying from 18 min to 54 hours. Figure 3a presents the J-E characteristics for this applied voltage protocol. It is to note that the value taken to plot the J-E characteristic is the mean of the current density over the last minute under the given applied field. In the literature for experimental results, the Authors either take the last value under a given applied voltage [START_REF] Chen | Conduction in linear low density polyethylene nanodielectric materials[END_REF], or they take the mean value over the last minutes of experiment [START_REF] Teyssedre | Charge distribution and electroluminescence in crosslinked polyethylene under dc field[END_REF][START_REF] Su | Investigation of effects of charge injection and intrinsic ionic carriers in low-density polyethylene and cross-linked polyethylene[END_REF][START_REF] Chen | Space charge dynamics of acetophenone and cumyl alcohol and their synergistic effect in LDPE[END_REF][START_REF] Li | Effect of Crystallinity of Polyethylene with Different Densities on Breakdown Strength and Conductance Property[END_REF][START_REF] Li | Trap Modulated Charge Carrier Transport in Polyethylene/Graphene Nanocomposites[END_REF]. For all the simulated time steps apart from 60 s, the J-E characteristic has a specific SCLC shape (or Sshape). The first region, with ohmic regime at low electric field, is not observed, but the other regions, as defined by the SCLC theory (trap-limited conduction, the trap filled limit and the trap free conduction) are observed, though the hypotheses behind the two approaches are clearly different. The impact of the time between two voltage steps on the current density values is important for times up to 600 s. On the contrary, the characteristic is almost unchanged for times above 1800 s, and the current density should have reached a quasi-steady state. This is however not totally true, and simulations results obtained with a stationary BCT model [START_REF] Baudoin | Bipolar charge transport model with trapping and recombination: an analysis of the current versus applied electric field characteristic in steady state conditions[END_REF], with the same optimized set of parameters, presented also on Figure 3a, show a perceptible difference in the current values. The characteristic however presents the same slopes for a steady state and unsteady states. Figure 3b presents the current density as a function of time for an applied electric field of 60 kV/mm, for the different resting times between two steps. The current density at 60 kV/mm has the same characteristic for all resting times, i.e. it increases and reaches a maximum after around 20 s, and then decreases. The differences arise from the value of the current density at the beginning of the step, which is really higher (by up to one decade) for short times under voltage compared to longer ones. In all cases, the current density is still decreasing and has not reached a steady state when the next step in voltage is applied. Charges, being present due to the previously applied voltages or injected at this step, have not achieved an equilibrium, leading to a high current value at the beginning of the next step of applied voltage. This behavior holds for applied electric fields ranging from 10 to 150 kV/mm. Above this field value, there is a slowing down of the differences between each resting time (Fig. 3a). It is to note that the electric field range for which there is a sharp increase of the current density is relatively high (around 200 kV/mm) compared to what is observed experimentally for a LDPE (around 60 kV/mm). This is all the more surprising that the simulation parameters have been optimized for LDPE sample with different experimental protocols, for which the step increase of voltage is one such [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF]. However, during parameters optimization, the objectives were more to quantitatively reproduce most of the experimental data available, letting apart the specific SCLC shape that was not reproduced. One of the possible reasons proposed in [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] was that the mobility of holes was relatively high, making it more difficult for charges to be trapped. In the following section, simulation results are presented with different physical hypotheses, among them electric field dependent mobilities, to see their impact on the J-E characteristics. The present simulated results show that waiting for more than 30 min between two steps is a minimum to obtain a correct value of the current density as regards stationary state, for this specific step increase of voltage protocol and for the current physical parameters values.

B. Impact of the applied voltage protocol

Simulations have then been performed for a polarization/depolarization protocol, for the same applied voltages and different polarization times, in order to investigate the impact of the voltage application protocol on the J-E characteristic. In the previous section, a step increase of voltage has been chosen, which implicitly takes into account the state of charges at the end of the previous step of voltage. In the case of a polarization/depolarization scheme, a new sample is used for each measurement, i.e. the sample is considered free of charges before any voltage application. Figure 4a presents the simulated current density as a function of the applied electric field for different polarization times: a short time of 300 s, and a longer time of 3600 s (i.e. 1 hour). They are directly compared with the previous simulation results of Figure 3a. Simulation results are similar for a step increase of voltage and for a polarization/depolarization protocol, for a time under voltage of 3600 s. In the case of Δt=300 s, the shape of the results is globally the same in both cases, but there are some discrepancies in the simulated results. These differences, mainly in the region 30 to 100 kV/mm, are particularly important in the case of a J-E analysis, where a small error in the slope of the curve leads to a large error in the estimated permittivity, as an example, in the case of a Schottky or Poole-Frenkel (PF) hypothesis, where the permittivity is directly deduced from the J-E slope. Moreover, even if the current values are globally comparable, this does not mean that the current transients are similar during polarization. As an example for medium applied electric fields (60 kV/mm), the global behavior (see Figure 4b) presents a rising current density to reach a maximum, and then a decrease and a stabilization, for both protocols. The amplitude is however different, as for the polarization/depolarization protocol, the increase and decrease of the current is over one decade, while the current density variation in time for the stepincrease of voltage protocol is of the order of 10%. This behavior is due to the charge distribution, and hence electric field distribution in the material prior to field application. The net charge density is null at the beginning of the simulation for a polarization/depolarization protocol, while charges are already present and mostly decrease the electric field distribution next to the electrodes in the case of a step-increase of voltage. Hence, in the first case, there is a high generation rate, a large charge redistribution, leading to a large current transient. The current density converges faster toward a stationary state when a step increase of voltage protocol is applied, i.e. when charges are already present within the material.

According to the model, the magnitude of the current is produced with relatively short resting times between two stepincrease of voltage (>30 min). It is clear that the recommended charging time holds for the particular physical model considered here, and its parameterization. Changing material to a more resistive one would certainly require longer stabilization time.

IV. IMPACT OF THE PHYSICAL HYPOTHESES ON THE J-E RESULTS

It has been noticed before that the range of electric field at which the simulated current density value sharply increases (corresponding to VTFL in Figure 1) is high (around 200 kV/mm) compared to experimental data related to LDPE (40-60 kV/mm). In this section, physical hypotheses related to injection, mobility, trapping, detrapping and recombination have been tested in order to observe their impact on the J-E characteristics. One of the objectives is to determine if this increase of current density could be reproduced for all the proposed physical hypotheses, as the choice of different physical processes can lead to the same simulated result. The second objective is to see if the threshold in electric field could be shifted to lower field values with a change of physical hypotheses (or mathematical expressions).

A. Impact of the mobility equation

Simulations have been performed for different physical hypotheses on the mobility for electrons and holes. The experimental protocol is a step-increase of voltage, for Δt=1800 s, which is not far from a stationary state as concluded in paragraph III.1. As a reference, mobility has been set as constant, with an effective constant value taking into account the possible trapping and detrapping of carriers into shallow traps (values of Frenkel (PF) mobility for electrons and holes have also been tested, in order to see their impact on the J-E characteristics. It is to note that the PF mobility is associated with the necessary presence of ionizable centers localized in the band gap able to release an electron in the conduction band (respectively to accept an electron, i.e. to create a hole in the valence band). These 'donor' or 'acceptor' states (as in the semi-conductor domain) would require too much energy to detrap a carrier in the case of dielectrics, so the PF mobility is not well fitted for our materials. However, the goal here is to have a second fielddependent mobility equation, to evaluate if the current-voltage characteristics could be reproduced. Equations and parameters used for each mobility equation are given in Table II. The other parameters of the model (related to injection, trapping, detrapping, and recombination) have been untouched and are the ones of Table I. Figure 5 presents the J-E characteristics for the different simulations using different mobility laws. The Schottky injection current for holes (having the lowest injection barrier, i.e. the higher injected current) has also been plotted vs. applied field on Figure 5, for comparison purpose. At low applied electric fields, the external currents are clearly lower than the injection current for all mobility hypotheses, i.e. the current density is not driven by the injection process. Of course, the choice of parameters for these mobility equations drives the value of the current at low fields. However, for the cases of field dependent mobility, i.e. hopping and Poole-Frenkel, the current density progressively tends to the Schottky current, whatever the chosen parameters. This means that the injection process rapidly determines the current density, for field dependent mobility cases study. For field dependent mobility equations, the increase in the current density appears at relatively low to middle applied electric fields (30 kV/mm for PF, and around 50 kV/mm for a hopping mobility), which is more consistent with experimental data than for a constant mobility. However, in the cases of field dependent mobility, with the chosen parameters as regards trapping, detrapping and recombination, the third zone of the SCLC characteristic ('trap free region') is not visible, and the current is always following the Schottky current. Other simulations (not presented in the present paper) have been performed with a different trap distribution (i.e. exponential [START_REF] Boufayed | Models of bipolar charge transport in polyethylene[END_REF]), and with a power-law for the field dependence of mobility [START_REF] Zhao | Effect of field-dependent mobility on current density and dynamics of space charge in polyethylene[END_REF], with adapted parameters to have current densities of the same amount as the ones obtained with a constant mobility at 10 kV/mm. These simulations, with field dependent mobility equations, lead to the same Schottky current values in the case of medium to high electric fields. As all other physical processes (injection, trapping, detrapping and recombination) and parameter values are identical, it seems that the value of the mobility in the case of a constant mobility prevents the total current from achieving the Schottky injection current. This has been already described in [START_REF] Baudoin | Bipolar charge transport model with trapping and recombination: an analysis of the current versus applied electric field characteristic in steady state conditions[END_REF], and is directly linked to the balance of charge density between mobile and trapped charges, as predicted by the SCLC theory. In the case of a constant mobility, the mobile charge density for electrons and holes is higher than the trapped charge density, with a trapped charge density being close to its maximal value, i.e. 100 C/m 3 . In the case of a hopping and a PF mobility, the mobile charge density is lower than the trapped one, with values for mobile and trapped charges being very low (lower than 10 - 3 C/m 3 ). In this case, for high electric fields, mobile charges have a high mobility, so their density is low, and the trapped charge density is also low. Hence, the electric field is no more disturbed by the charges. The current is then controlled by the electrodes. It is to note that there is quasi no difference between the simulation results obtained with the PF mobility and the one obtained for a Schottky injection, whatever the applied electric field (Figure 5). It seems that the PF mobility rapidly achieves high values, hence the current remains determined by injection. The same kind of conclusions has been reported in [START_REF] Zhao | Effect of field-dependent mobility on current density and dynamics of space charge in polyethylene[END_REF], i.e. the PF process, even with different parameters, cannot reproduce the space charge behavior at high applied electric fields.

B. Impact of the injection law

Simulations have then been performed for different injection laws, as proposed in Table III: a Schottky injection, an infinite reservoir of charges, and a Fowler-Nordheim (F-N) injection. The validity of an infinite reservoir of charges is questionable: one should account for more detailed physical hypotheses at the interface such as the presence of contact charges at each electrode, the presence of a higher density of traps, on deeper levels, etc. These charges (trapped mainly) should constitute a reservoir of charges. Moreover, one should account for the replenishment of this reservoir of charges, which is hence not constant, being depleted when relatively high fields are applied [START_REF] Dissado | The effect of Contact Charge upon the Injection Current at an Electrode-Insulator Interface[END_REF]. As it is difficult in reality to account for all these physical II.

Other parameters as in Table I. Schottky injection is also presented for comparison.

processes (no information, no parameters) we choose the simplest hypothesis, being the constant reservoir of charges, i.e. injection is not limited. For each hypothesis on injection, a constant mobility and a hopping type mobility have been tested (parameters of Table 2). Other parameters are the ones of Table I. Figure 6 presents the simulated current density as a function of applied electric field for the different cases study. The Schottky and Fowler-Nordheim injection current densities vs. applied electric field values, have also been plotted for a barrier height of w=1.16 eV. In the case of a constant mobility at low to medium applied electric fields (1-100 kV/mm), when a sufficient number of charges are injected inside the dielectric (i.e. case of a reservoir of charge and Schottky injection), the current density is limited by bulk processes. The current is lower than 10 -15 A/m 2 in the case of a F-N injection, as the number of charges injected at low field is quasi null at these applied electric fields. It means that the current for this case, at low applied electric field, is limited by the F-N injection process. At high applied electric fields, and still for a constant mobility, the current remains the same whatever the injection law, i.e. the current is still driven by the 'low' value of the mobility. The case of an electric field dependent mobility is more complex to comprehend. The F-N injection law implies a low current density at low applied electric fields, as in the case of a constant mobility. The current density then increases drastically with the increase of the applied electric field. At high field (>100 kV/mm), the current density value follows the one of the injection current in the case of a Schottky and a Fowler-Nordheim injection. The case of a reservoir of charges associated to a hopping mobility shows a drastic increase of the current density at high applied electric fields compared to other cases, leading to a conclusion different from a constant mobility, as the mobility does not limit the current value. For these cases study for high applied electric fields, the controlling process in the case of a constant mobility is the transport, while it is the injection in the case of a hopping mobility.

C. Impact of recombination

The impact of recombination law on the J-E characteristic has also been investigated, for the different mobility hypotheses proposed in section 4.1. Three cases have been tested, i/ constant recombination parameters, ii/ Langevin recombination, function of the mobility of the carrier, and iii/ no recombination. Figure 7 presents the current density as a function of the applied electric field for the nine different cases. Practically no differences are observed for the cases where the mobility is field dependent, whatever the recombination hypothesis, even no recombination. The only case where recombination plays a non-negligible role is when the mobility is constant. For this hypothesis, considering constant or Langevin recombination almost leads to the same current density values. A difference exists only when no recombination is considered. In that case, the current shape tends to the Schottky injection current. This particular behavior has already been highlighted in [START_REF] Baudoin | Bipolar charge transport model with trapping and recombination: an analysis of the current versus applied electric field characteristic in steady state conditions[END_REF].

D. Impact of the trapping physical hypotheses

One last physical process, the trapping, has been investigated. For these simulations, mobility for electrons and holes are of the hopping type, as it seems more physically sound to have a field dependent mobility. Three parameters are linked to the trapping: the trapping coefficient, the trap density, which is usually kept at 6.25 10 20 m -3 (producing maximal trapped charge of 100 C/m 3 ) in most of the researches linked to BCT models [START_REF] Dissado | The Role of Theory in Understanding Space Charge Distributions[END_REF][START_REF] Xia | Numerical analysis of packetlike charge behavior in lowdensity polyethylene by a Gunn effectlike model[END_REF], and the detrapping barrier height. In the present research the trapping coefficient value has been changed equally for electrons and holes, from 0.1 to 100 s -1 . The trap density for electrons and holes has been increased from 6.25 10 20 m -3 to 6.25 10 22 m -3 . At last, the detrapping coefficient has been varied symmetrically from 0.8 to 1.2 eV. Figure 8 presents the simulation results. Whatever the coefficient, the impact is almost the same. As stated in the SCLC theory, the trapping processes are directly related to the trap-filled limited current as in Fig. 1 (i.e. JV  , when traps are not distributed). An increase of the density of trapped charges (through an increase of the trapping coefficient B, an increase of N0 or an increase of the detrapping barrier height) implies a decrease of the current density at low applied electric fields, a shift of the threshold in electric field to the right, i.e. toward higher values of applied electric fields, and an increase of the slope of the characteristic for medium applied fields. There is hence a clear effect of the trapping parameters on the variation of the current density at low to relatively high fields. At high applied electric fields, all the curves tend to the same current value, which is the Schottky current for holes (i.e. for a barrier height of 1.16 eV), and all the charge densities (not presented here) are low.

E. Discussion

Current density vs. electric field characteristics with a constant mobility and optimized parameters lead to unrealistic thresholds in electric fields regarding the space charge limited and trap free conduction regime (corresponding to VTFL in the SCLC theory) compared to experimental data for LDPE. One of the objectives of this paper was to find how to shift the threshold in electric field to obtain simulation data fitting to experimental ones. To do so, different physical hypotheses and/or mathematical expressions have been tested. Important remarks can be deduced from the simulations in section 4:

-When the mobility is field dependent, the threshold in field approaches the one observed experimentally (around 60 kV/mm in the case of LDPE). However, keeping the optimized parameters as for a constant mobility, the trap free conduction as proposed by the SCLC theory and observed experimentally is not reproduced. Moreover, when adopting electric field dependent mobility equations, the space charge behavior cannot be reproduced, as at high applied electric fields, the amount of charges is quasi null (lower than 10 -5 C/m 3 ), not representative of what is experimentally observed.

-Only a constant mobility 'respects' the trap filled region, as presented in [START_REF] Baudoin | Bipolar charge transport model with trapping and recombination: an analysis of the current versus applied electric field characteristic in steady state conditions[END_REF], i.e. all traps are filled, and the current is driven by bulk transport.

-When the mobility is field dependent, the current limiting step at high applied electric fields is the injection process, for the tested physical hypotheses. Changing the injection hypothesis has then a significant impact on the high field current densities. This is not the case when a constant mobility is accounted for.

-Recombination (constant or mobility dependent) has an impact only for constant mobility; the impact is negligible for field dependent mobility equations.

-Changing the trapping physical hypotheses has been performed only by changing the parameters values. It does not affect the shape of the characteristic to a large extent, but the variations on the current density values are large at low applied electric fields.

Changing the mobility equation from constant to electric field dependent has then a substantial impact on the J-E characteristic. Similar simulations, performed on semiconducting materials [START_REF] Kim | Charge carrier injection and transport in QLED layer with dynamic equilibrium of trapping/de-trapping carriers[END_REF], with an exponential and a Gaussian trap distribution, led to the same kind of conclusions, i.e. the trapping parameters are of most importance, and have to be modulated in order to change the slopes and ranges of the J-E curve. It is however difficult to shift the field threshold without changing the permittivity or the material thickness. Moreover, these simulations [START_REF] Kim | Charge carrier injection and transport in QLED layer with dynamic equilibrium of trapping/de-trapping carriers[END_REF] also led to the conclusion that the injection mechanism determines the current value at high applied electric fields.

From the present results, it seems that changing the mobility law allows obtaining consistent results as regards the current density values, but the shape of the characteristic is not correct. Hence, in its current form, the BCT model is yet not enough to reproduce most of the experimental data available for LDPE, the simplest polymer in its structure. The alternative mathematical or physical hypotheses explored in the present work for such model did not lead to definitive improvement: no hypothesis as regards generation, transport, trapping has been able to reproduce at the same time an electric field threshold at a reasonable applied electric field, and a third region in the J-E characteristic, where most of the traps are filled. A possible reason could be that the real physical processes are a combination of the previously proposed ones: for example, the charge injection rule can be changing from one process to another with increasing the electrode field, or the mobility can switch from a constant to field-dependent one above a threshold. Another possibility is that the physical hypotheses chosen in the present paper are not well adapted to reproduce the experimental behavior. The present model has the advantage of dissociating all relationships between parameters of each physical process. This is however not true in reality, and evolutions of the model should certainly account for that. Other tracks, particularly as regards the trapping/detrapping hypotheses, need also to be investigated, such as for example field dependent trapping/detrapping processes or a modulation of the trapping/detrapping processes as a function of the already trapped density of charges. Fig. 8. Current density as a function of applied electric field for different values associated to trapping processes. Other parameters as in Table I.

III. CONCLUSIONS

A bipolar charge transport model has been used in order to evaluate the impact of each physical process, related to injection, transport and trapping, on the J-E characteristic, for a low density polyethylene. The protocol for voltage application (step-increase of voltage or simple polarization/depolarization protocol) does not seem to be critical as long as the time under a specific voltage is long enough. It has been found that for a polarization time longer than 30 min, the current values do not change to a large amount compared to steady state simulated currents. This time under voltage should then be used as a minimum for any measurement if current-voltage characteristic is under study, for the dynamics of processes considered in the present model. The charge dynamic and the current transient for each protocol (step increase of voltage and polarization/depolarization protocol) is however different, and a richer information can be deduced from polarization/depolarization protocol, compared to a stepincrease of voltage protocol with combining current measurements and space charge distribution measurements. Injection, mobility, trapping and recombination have then been analyzed, to see their impact on the J-E characteristics, and particularly on the electric field threshold linked to the space charge limited current. For the physical hypotheses tested in the present paper, injection is always the dominant process when field dependent mobility is accounted for, at medium to high applied electric fields. Parameters linked to trapped charges (i.e. trapping and detrapping coefficients and trap density) have a significant impact on the shape of the characteristic when field dependent mobility is accounted for, leading to a S-shape characteristic in the case where enough charges can be trapped, but with thresholds in electric field far from the ones observed experimentally for LDPE. More work needs to be done on the mathematical expressions and/or on the physical hypotheses that need to be implemented in such BCT model to have simulation results consistent with experimental J-E characteristics.

  Fig.3. a) Current density as a function of applied electric field for a step increase of voltage protocol and b) current density as a function of time for an applied electric field of 60 kV/mm, for different times between two steps.

Fig. 4 .

 4 Fig. 4. a) Current density as a function of applied electric field for a polarization/depolarization protocol and different times under voltage application; b) Current density as a function of time for a step -increase of voltage and a polarization/depolarization protocol, for an applied electric field of 60 kV/mm.

Fig. 5 .

 5 Fig. 5. Current density as a function of applied electric field for different mobility laws proposed in TableII. Other parameters as in TableI. Schottky injection is also presented for comparison.

Fig. 6 .

 6 Fig. 6. Current density vs. applied electric field for different injection laws and different mobility hypotheses. Parameters as inTable II for mobility and Table III for injection. Other parameters as in Table I.

Table I

 I

	Fig. 2. Schematic representation of the bipolar
	charge transport model. Conduction is by free charges in the transport levels, associated with	and used for the
	trapping and detrapping into one level of deep traps,	simulations in section 3. The experimental protocol is a step
	and recombination between charge of opposite	increase of voltage, from 10 to 500 kV/mm, by steps of 10
	sign.

transport level mobile electrons n em trapped electrons n et trapped holes n ht mobile holes n hm S 3 S 2 2 S 1 S 0 B e e B h h D e e D h h electrons trapping level holes trapping level transport level

	kV/mm from 10 to 100 kV/mm, and by steps of 50 kV/mm from
	100 to 500 kV/mm. The time between each step of voltage has
	been varied from Δt=60 s to 10800 s (i.e. 3 hours).

Table

  Table II for mobility and Table III for injection. Other parameters as in Table I.


Olivier Gazil
email: olivier.gazil@polymtl.ca

Nick Virgilio

Fabienne Gauffre

B Crepec

Synthesis of

come

Introduced by Vitale and Katz, the typical example of the Ouzo effect is when an anis-based alcohol, such as Pastis or Ouzo, is added to water, producing a cloudy solution -the result of trans-anethol nanoprecipitation caused by its low solubility in water. 13 Interestingly, this precipitation yields metastable and nearly monodisperse sub-μm droplets (typically 100-300 nm). 13- 15 In the same way that conventional emulsions can be used as templates for the synthesis of colloidosomes, it has been shown that NPs adsorb at the surface of Ouzo emulsion droplets, and limit the coalescence of droplets. [14][15][16] Thus, Ouzo emulsions stabilized with nanoparticles share similarities with Pickering emulsions, in which particles act as surfactant. 11 However, in contrast with Ouzo emulsions stabilized with particles, Pickering emulsions yield micron-sized droplets or larger, and the formation and stabilization of sub-µm droplets have yet to be generalized. 3,[17][18][19] Furthermore, Pickering emulsions also typically require energy intensive processes complicating their development at the industrial scale. 14,17 These two types of emulsions are different from a fundamental point of view: traditional emulsification is a top-down approach (the oil phase is fractionated from bulk into droplets) while Ouzo emulsification is a bottom-up approach (the oil phase nucleates from the solvated state to droplets), leading to several differences in their outcomes.

In the vast majority of examples, the liquid/liquid interfaces are coated by preformed NPs, initially dispersed in one of the liquid phases. In contrast, reports on the in situ synthesis of NPs directly at the interface in emulsion-based systems remain scarce. López-de-Luzuriaga et al. reported a single-step AuNPs synthesis by reducing a gold complex with triisopropylsilane at the interface of oleic acid droplets, 2 while Sachdev et al. detailed the synthesis of AuNPs at the interface of hexane droplets (formulated in a microfluidic chip) with decamethylferrocene, 20 and Li et Zhang described the formation of Fe3O4 nanoshells on the surface of gas bubbles under hydrothermal conditions (i.e. armored bubbles). 21 In addition, several articles report on the seeded growth of NPs at the interface from an already formed Pickering emulsion. 3,22 This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins Please do not adjust margins Scheme 1. Synthesis of AuNPs at the liquid/liquid interface in Ouzo-type emulsions, and immobilization of AuNPs at the interface with a strongly adsorbing polymer.

Among the different pathways employed to locally synthesize particles at the liquid/liquid interface, the approach developed by Rao et al. is a promising candidate. 23 It involves an aqueous phase comprising a reducing agent, and an organic phase containing a metallic precursor. The reactants can be chosen to yield a variety of metals (e.g. Au, Ag and Pd), oxides (e.g. CuO and ZnO) and chalcogens (e.g. CuS and CdS) 2D thin films. 23 This approach was subsequently used by Gazil et al. to cover complex hydrogel structures with a thin metal film, 24 and by Armstrong et al. to cover emulsified toluene droplets with metal NPs. 25 However, to the best of our knowledge, it was never used in with an Ouzo-type emulsion. We expect that the synthesis of NPs at the surface of Ouzo droplets will stabilize them and allow control over their sizes, leading in fine to the formation of NP-shells (i.e. hollow spheres composed of NPs). Essentially, our results demonstrate that interfacial reactions can be employed in Ouzo and other emulsion-type systems to synthesize ultrasmall metal nanoparticles and thin shells at the interface -a promising pathway to synthesize a wide variety of new materials, owing to the simplicity and reproducibility of the method.

In this work, we take advantage of Ouzo emulsions to achieve an interfacial reduction of metallic precursors and coat submicronic droplets with metal nanostructures. 23 The Ouzo system comprises butylated hydroxytoluene (BHT, a stabilizer added to tetrahydrofuran, THF), and water. The proposed reaction mechanism for the interfacial reaction -hereafter designated as in situ synthesis -is presented in Scheme 1. For the sake of clarity, the various steps are presented in chronological order, though they are in fact happening simultaneously. First, an aqueous phase comprising a reducing agent (sodium borohydride -NaBH4) is mixed with an organic phase (THF) containing a metallic precursor (AuPPh3Cl) and BHT. This leads to the fast nanoprecipitation of BHT with the precursor (they are both hydrophobic), while the reducing agent remains in the aqueous phase. Second, the redox reaction between the reducing agent and the organometallic complex is initiated at the interface of the droplets, resulting in the formation of AuNPs. Finally, when the AuNPs are fully synthesized, a water-soluble polymer is added to stabilize, via adsorption, the AuNP-shells.

In a typical in situ synthesis, 0.75 mL of an aqueous solution containing 10 mM sodium borohydride (NaBH4) is quickly added to 0.25 mL of a THF solution containing BHT (~320 ppm) 26 and 1.5 mM AuPPh3Cl. The high excess of reducing agent versus precursor (20:1 molar ratio) ensures reaction completion in a short period of time. For this reason, sampling times at 1 min (~ at reaction completion) and 30 min were chosen for nanoparticle tracking analysis (NTA) and transmission electron microscopy (TEM) observations. First, TEM analysis revealed the formation of nearly spherical nano-objects formed by closely packed AuNPs, as Figure 1a (1 min) and 1b (30 min) show. High resolution TEM (HRTEM) micrographs are presented in Figure S1, and a micrograph showing individual AuNPs is presented in Figure S2a. No significant differences between the 1 min and 30 min syntheses were observed for the individual AuNPs (Table S1), thus only the 30 min synthesis is presented in Figure S2. The size distribution of the individual AuNPs composing the shells is presented in Figure S2b. The energy-dispersive X-ray spectroscopy (EDX) spectrum (Figure S2c) confirms the formation of AuNPs. AuNPs crystallinity can be observed on the HRTEM micrograph (Figure S2a) with the lattice fringes, which is also confirmed by selected area electron diffraction (SAED -Figure S2d). Hollowness of the NP-shells is shown in Figure S3, where a darker edge can be observed -the core is empty, as the lighter contrast in the center illustrates.

The comparison of Figure 1a and b reveals much larger NPshells after 30 min of synthesis time, as compared to 1 min (note the difference between the two scales). Their respective size distributions, as obtained by NTA, are presented in Figure 1c. After 1 min reaction, the mean size of the NP-shells is 160 ± 50 nm, and the size distribution shows a main mode centered at 140 nm, sharp and narrow. In contrast, a reaction time of 30 min gives a polydisperse distribution with a mean of 300 ± 200 nm and a mode at 150 nm. Size analyses were also obtained based on the TEM micrographs (Figure 1d): after a reaction time of 1 min, the NP-shells average diameter is 160 ± 80 nm (polydispersity index PDI = 0.3), while it is 800 ± 400 nm after 30 min (PDI = 0.2). It is important to mention that NTA gives the Please do not adjust margins Please do not adjust margins hydrodynamic diameter, which is expected to be larger than the geometrical size determined by TEM. 27 However, it is noteworthy to mention that objects larger than 1 μm are excluded from the NTA analysis, mainly because of their limited Brownian motion. This, combined with the time evolution of sample while solution dries on TEM grid, explains why here NP-shells observed by TEM are larger compared to NTA results. It is clear that the average diameter evolves with time between 1 min and 30 min of reaction time -the droplets are not stable and tend to coalesce. This is further supported by NTA measurements after 2 h of reaction time (Figure S4), and a picture of the sample in a vial after 1 day (Figure S5b-d) showing that almost all the NP-shells have coalesced and precipitated at the bottom of the vial. The NPs diameter evolution has an impact on two important factors: coverage and density of the objects. A single layer of 2.7 nm AuNPs around the Ouzo droplets would give the 1 min objects a coverage of 0.8, while it is 1.0 for the 30 min objects (Equation S1) -these numbers are a rough estimate of the coverage based on a few hypotheses (a value of 1 means full coverage). Even though the coverage is around 100%, coalescence is still present -it has been shown in the case of liquid marbles that the formation of holes between contacting spheres is easier for smaller particles. 12,28 Furthermore, the density was measured as an indicator for sedimentation and a value of 2.1 was calculated for the 1 min AuNP-shells, as compared to 1.3 for the 30 min objects (Equation S2). Considering the important coalescence, it is thus required to properly stabilize the AuNP-shells for their storage and later use.

One way to stabilize the shells against aging is by adding a polymer with functional groups that have an affinity for gold (e.g. amine or thiol groups). 29 Here, poly(allylamine hydrochloride) (PAA HCl) was selected as a potential stabilizer due to the affinity of amine groups with Au. 29 Adding 50 µL of a 4.4 mg/mL PAA HCl solution (equivalent to 0.75 mg polymer/mg Au) after 1 min of reaction time yields smaller objects (50 ± 20 nm) and less polydispersed (PDI = 0.1) as compared to the synthesis without polymer (Figure 2a). In addition, the NP-shells suspension remains stable over time against sedimentation. Aged samples stabilized by PAA HCl are displayed in Figure S6. The shells average size slightly increases after 2 weeks, from 50 nm to 70 nm (Figure S6c and Table S1). Furthermore, the individual AuNPs grow from 2.8 nm to 3.4 nm (Figure S6d and Table S1), which results in less homogeneous NP-shells (Figure S6a-b) as compared to the freshly prepared sample (Figure 2a). This increase remains quite limited, but this could indicate that the particles, while being properly stabilized by a polymer as a spherical assembly, can still interact and coalesce with neighboring AuNPs -hence, suggesting the fact that their surfaces are still active and accessible for potential catalysis applications, for example.

In comparison, AuNPs were also synthesized in bulk (by reducing the metal precursor with NaBH4 in THF, designated as "ex situ synthesis"), then added to the Ouzo emulsion. Poly(vinylphosphonic acid) (PVPA) was also tested as a stabilizing polymer (50 µL of a 4.4 mg/mL PVPA), since PAA HCl led to the significant coalescence of AuNPs in that case (Figure S7). This result can be explained by zeta potential measurements. Repulsive forces are deemed strong enough to stabilize droplets when the absolute value of the zeta potential is above 30 mV. The initial Ouzo emulsion is quite stable (-37 ± 4), while the potential of droplets with AuNPs synthesized in situ is close to neutrality (0.43 ± 0.07). The addition of a polymer (PAAH) increases the zeta potential of the AuNP-shells to 30 ± 2 and helps stabilize the droplets against coalescence due to the repulsive forces, which is consistent with previous observations. Please do not adjust margins Please do not adjust margins HCl in grey; c) micrographs of AuNPs synthesized in THF first (ex situ synthesis), followed by the addition of water, and of 50 µL of 4.4 mg/mL PVPA. d) Histograms of NP-shells synthesized in situ without PVPA (grey), compared to NP-shells with AuNPs synthesized ex situ and with the addition of PVPA. In both cases, the stabilizing effect of the polymer (PAA HCl in one case, and PVPA in the other) is significant.

Considering the ex situ synthesis, which is highly positive (46 ± 1), the addition of PVPA gives the droplets a negative charge (-54 ± 3), which is consistent with the observation that PVPA (a negatively charged polymer) stabilizes them, whereas PAA (positively charged) has not effect.

TEM and HRTEM micrographs in Figure 2c show ex situ synthesized AuNPs forming shells around droplets with the addition of 50 µL of 4.4 mg/mL PVPA, with the resulting histogram in Figure 2d. Clearly, this approach, combined with PVPA, also leads to a relatively narrow NP-shells size distribution (100 ± 40 nm and PDI = 0.1). However, shell coverage is not as homogeneous as for the in situ synthesis, since the AuNPs surface density appears to be lower. Unexpectedly, the average sizes of the individual AuNPs are quite similar for the in situ and ex situ approaches (respectively 2.7 ± 0.4 nm and 2.9 ± 0.6 nm, see Table S1). X-ray diffraction (XRD) was also performed for both syntheses (Figure S8) showing that they yield identical crystalline structure. From the Scherrer equation (Equation S3), the size of the crystallites can be calculated (1.6 nm) and compared to the size of the NPs measured with the TEM micrographs (2.7 nm and 2.9 nm, see Table S1), confirming their similarities. This suggests that the reduction of the gold precursor is only weakly sensitive to the local conditions (interfacial versus bulk). This may be due to the high concentration and the strength of the reducing agent used (20:1 ratio of reducing agent per metallic precursor).

Different applications may require different types of metal NPs (e.g. for catalysis, palladium or Au/Pd alloys are also of interest). 5,[30][31][32] For this reason, using a palladium precursor with the same triphenylphosphine ligands (Pd(PPh3)2Cl2) was assessed. Instead of discrete NPs, the reduction of the Pd precursor yielded a continuous and amorphous Pd shell (Figure3a, and Figure S1e for higher magnification). The addition of a polymer (PVPA) was studied as well (Figure S9). histograms of both samples displayed in (b) and (c), obtained by image analysis; e) EDX spectra of (b) and (c) (both spectra are almost identical) highlighting the expected compositions in Au and Pd -i.e., 50/50 mol%. Identified in black are peaks associated to both Au and Pd, with the phosphorous peak. Identified in grey, the elements composing the sample holder and the grid, plus non-relevant elements (e.g. Na of the reducing agent).

While it does reduce the average objects size and narrows the distribution, from 500 ± 200 nm (PDI = 0.1) down to 410 ± 90 nm (PDI = 0.05), its effect is more limited compared to the in situ AuNP-shells synthesis. A complementary in situ synthesis was performed by using exclusively the palladium precursor, without BHT. This synthesis yielded Pd-based objects (with a diameter of 180 ± 30 nm) displaying an increased stability of over several months (Figure S10). In the case of an equimolar combination of Au and Pd, bimetallic shells were obtained. Figure 3 shows bimetallic shells without (b) and with (c) the addition of 50 µL of 4.4 mg/mL PAA HCl. Again, there is a clear stabilizing effect of the polymer, as confirmed by the TEM micrographs and the histogram in Figure 3d. For similar synthesis conditions, the addition of PAA HCl inhibits NP-shells coalescence (800 ± 200 nm without PAA HCl, down to 200 ± 40 nm with the polymer). EDX analysis confirms the bimetallic composition of the shells (Figure 3e). Indeed, the integration of the peaks for Au and Pd gives the expected molar composition -i.e. 50/50 mol% (the initial precursors were added in equimolar amounts). Furthermore, the phosphorus signal is more intense as compared to pure AuNP-shells (Figure S2c), which is caused by the initial Pd complex possessing twice the amount of triphenylphosphine groups per metal atom. HRTEM observations of the Au/Pd shells are shown in Figure 3b and Figure S1f (without PAA HCl): nanoparticles appear to be embedded in a matrix. From the previous observations for the pure Au and pure Pd precursors, it seems that the structure consists of AuNPs embedded in a matrix of Pd. In contrast to gold, the in situ and ex situ (Figure S11) syntheses yield two very different results for the Pd precursor.

The formation of a dense layer of NPs that coat the droplet, as it is the case when Au(PPh3)Cl is reduced at the interface, is reminiscent of Pickering emulsions. However, it should be noted that ultra-small particles are formed here. In fact, these are amongst the smallest particles reported at the fluid/fluid interface in emulsion-type systems (the typical size usually reported is around 0.01-10 μm). 3,4,[17][18][19] In the case of Pickering emulsions, droplet coalescence is limited by the particle layer and stops when the total amount of interface matches the coating ability of the particles. 11,15,33 The same mechanism was also demonstrated recently for spontaneous Ouzo emulsions using particles of ~8 nm. 15 In order to separate the stabilizing effect of the NPs on the droplets (i.e. Pickering emulsion), from the effect of the polymer, the growth of the droplets in the presence of the polymer was monitored by DLS (Figure S12). The results show that with and without polymer, the droplet size stabilizes around the same value (ca 600 nm) over the typical reaction time (< 30min) -i.e. the polymer alone does not particularly inhibit coarsening. Coarsening was also observed for droplets covered only with NPs (Figure S4). Consequently, both the NPs and polymer must be combined to stabilize the Please do not adjust margins Please do not adjust margins NP-shells against coalescence. This significantly improves the lifetime of basic Ouzo emulsions (typically limited to hours, or days at most). 34 It is also noteworthy to underline the structural integrity of the NP-shells even after drying, highlighted by the 3D structure shown by TEM-tomography in the ESI video. Finally, an interesting observation is the presence of a hole on all shells (a few examples are highlighted in Figure S13, and it is illustrated in the ESI tomography video), regardless of their sizes, which may be an artifact due to the drying preparation step prior to TEM observations. This phenomenon was also observed in a previous work investigating the formation of spherical polysaccharide-based nanoparticles, specifically when using THF as the solvent. 35 In conclusion, the in situ interfacial reduction of an organometallic precursor in Ouzo-type emulsions yield metal shells comprised of ultrasmall AuNPs, Pd continuous shells, or a combination of both in bimetallic systems, in a one-pot approach. The elaboration of such ultrasmall nanoparticles assemblies at the liquid/liquid interface is quite challenging since their adsorption is not as energetically favorable as for larger ones, explaining why reports for these types of particles are scarce. 36 Here, the addition of an appropriate polymer stabilizes the NP-shells and reduce their final sizes. To further reduce polydispersity, additional purification steps could be considered, depending on the application. Yet, the simplicity of the process makes it appealing, for example, for applications in therapeutics or catalysis. [2][3][4][5][6] Future work will aim at providing insight into the impact of metallic precursors on the Ouzo droplets size and stability, and at generalizing the approach to other metal-based systems, in order to better understand and control the synthesis of those materials.

Figure 1 .

 1 Figure 1. In situ synthesis of AuNPs at the interface of BHT-rich nanodroplets by the reduction of 0.25 mL of 1.5 mM Au(PPh3)Cl with 0.75 mL of 10 mM aqueous NaBH4. TEM micrographs of the metallic NP-shells after reaction times of a) 1 min and b) 30 min. c) Size distributions of the NP-shells in solution for both samples as acquired by NTA (see methodology in ESI). d) Histogram of the dry samples obtained from the TEM micrographs.

Figure 2 .

 2 Figure 2. a) TEM micrographs of NP-shells (inset: HRTEM) synthesized in situ with 0.25 mL of 1.5 mM Au(PPh3)Cl and 0.75 mL of 10 mM aqueous NaBH4, followed by the addition of 50 µL of 4.4 mg/mL PAA HCl after 1 min reaction time; b) corresponding histogram extracted from TEM micrographs in black, and without the addition of PAA

Figure 3 .

 3 Figure 3. a) TEM micrograph of continuous shells synthesized in situ with 0.25 mL of 1.5 mM Pd(PPh3)2Cl2 and 0.75 mL of 10 mM aqueous NaBH4 (average size of 500 ± 200 nm). b-c) Shells composed of an equimolar Au-Pd ratio, showing the thin layer synthesized in situ (with 0.75 mM of Pd(PPh3)2Cl2 and Au(PPh3)Cl): TEM micrographs without (b) and with (c) the addition of 50 µL of 4.4 mg/mL PAA HCl after 1 min of reaction time; d)

Acknowledgements

This work was supported by the Alexander Graham Bell Canada graduate scholarships program from the Natural Sciences and Engineering Research Council of Canada (NSERC), from the ANR (ANR-20-CE06-0031-01 OUZOFAN), and from the IRN Nanomatériaux Multifonctionnels Contrôlés (IRN NMC). Prof. Virgilio acknowledges the financial support of NSERC via the Discovery Grants program. The authors would like to acknowledge support from Ludivine Rault for the assistance in TEM characterizations performed on THEMIS platform (ScanMAT, UAR 2025 University of Rennes 1-CNRS; CPER-FEDER 2007-2014) and Dr. Mathieu Pasturel for XRD characterizations. We finally thank Dr. Clément Goubault for helping in the laboratory with the methods developed by Dr. Gauffre's research group and Déborah Iglicki for a fruitful collaboration concerning ex situ synthesis.

Conflicts of interest

There are no conflicts to declare.

Notes and references