A droplet robotic system enabled by electret-induced polarization on droplet

Robotics for scientific research are evolving from grasping macro-scale solid materials to directly actuating micro-scale liquid samples. However, current liquid actuation mechanisms often restrict operable liquid types or compromise the activity of biochemical samples by introducing interfering mediums. Here, we propose a robotic liquid handling system enabled by a novel droplet actuation mechanism, termed electret-induced polarization on droplet (EPD). EPD enables all-liquid actuation in principle and experimentally exhibits generality for actuating various inorganic/organic liquids with relative permittivity ranging from 2.25 to 84.2 and volume from 500 nL to 1 mL. Moreover, EPD is capable of actuating various biochemical samples without compromising their activities, including various body fluids, living cells, and proteins. A robotic system is also coupled with the EPD mechanism to enable full automation. EPD’s high adaptability with liquid types and biochemical samples thus promotes the automation of liquid-based scientific experiments across multiple disciplines.


Supplementary Note 1. Comparison between EPD mechanism and the existing AC/DC-induced liquid polarization.
The most fundamental physical principle of EPD can be attributed to polarization, a principle that is indeed already present and discussed in the literature about micro-object actuation 1,2 :  = 4  ( ⋅ ) where  is the electric field strength,  is the radius of the particle,  is the permittivity of the surrounding medium and  is the Clausius-Mossotti factor related to the effective polarizability of the particle.The traditional method to polarize liquid is to apply high-frequency and high-voltage AC/DC electric fields in a microfluidic device [3][4][5][6] .Despite originating from the same governing principle of polarization as AC/DC-induced polarization, the polarization induced by electrostatic charges deserves to be discussed and explored separately.The equivalent circuit models are different for the AC/DC-induced and electrostatic charges-induced polarization, where the former model is externally connected by a power supply, while the latter model is an isolated system.In the model of AC/DC electric field 4,7 , an external power supply connects the upper and lower electrodes and maintain the voltage  constant.In this closed circuit, both conduction and displacement currents exist and compete frequency-dependently.Thus,  in the equation above should be expressed as 1,2 : in which  is the frequency of the applied electric field,  and  are the permittivity and electrical conductivity, and the subscripts d and m represent the droplet and the surrounding media, respectively.Therefore, the AC/DC induced polarization is related to both electrical conductivity and permittivity of the operated droplet, and the frequency setup needs to be customized according to the electrical property of the operated droplet 5,6 .
As for the isolated system model of the electrostatic charges, there is no conduction current. is thus independent of conductivity and frequency, only related to the permittivity: Besides the expression of ,  also differs for two kinds of polarization.In the model of AC/DC electric field, the voltage  is maintained as constant.Meanwhile, the charge  accumulated on the electrode ( =  • , supplied by the power source) changes with the overall capacitance  of the system, leading to a variable the electric field strength  4,7,8 .For example, the equivalent capacitance of the electric double layer at the electrode interface could become very small in some cases 9 (e.g.,  below 100 mS/m and field frequency below about 15 kHz 10 ).At this time,  within the surrounding medium tends to be zero, leading to an undesired shielding effect 10,11 .In contrast, in the equivalent model for polarization induced by electrostatic charges, the charge accumulated on the electret  is a constant, leading to a stable :  =   4  in which  is the surface charge density of the electret and  is the distance from electret.Based on the analysis above, the polarization induced by AC/DC and electrostatic charges has different impact parameters and limitations due to the difference in their equivalent circuit models.

Supplementary Note 2. Theoretical relationship between maximum velocity and charge density of the electret for EPD-based droplet actuation.
The droplet actuated by EPD should be subjected to two forces, one is the driving force generated by EPD effect,  , and the other is the drag force on the droplet as it moves through the oil layer, .According to Equation (1) and Stokes Law, we have:  ∝  and  = 6 where  is the radius of the droplet and is assumed much smaller than the scale of the field nonuniformity 1 ,  is the viscosity of the oil layer,  is the speed of the droplet.
Therefore, when the droplet accelerates to the maximum velocity, we should have: which lead to:  =  As a result, we can have: ∝

Supplementary Note 3. Theoretical relationship between maximum velocity and droplet volume for EPD-based droplet actuation.
The droplet actuated by EPD should be subjected to two forces, one is the driving force generated by EPD effect,  , and the other is the drag force on the droplet as it moves through the oil layer, .
According to Equation ( 1) and Stokes Law, we have:  ∝  ∝  And  = 6 ∝  where  is the volume of the droplet,  is the radius of the droplet and is assumed much smaller than the scale of the field nonuniformity 1 ,  is the viscosity of the oil layer,  is the speed of the droplet.Therefore, the acceleration of the droplet  can be described as: where  is the mass of the droplet.Therefore, when the droplet accelerates to the maximum velocity, we should have:  = 0 which leads to:  =  As a result, we can have:

Supplementary Note 4. Comparison of droplet actuation resistance on hydrophilic and hydrophobic surface.
The contact angle hysteresis of droplets on hydrophilic surfaces significantly grows larger than those on hydrophobic surface 12 , while the contact area of the droplets also increases 13 , thus leading to a substantially larger relative friction force 14,15 : where  is width of the droplet bottom perpendicular to the direct of the motion,  is surface tension at liquid-air interface, cos and cos are advancing and receding contact angles of the liquid-solid interface.Compared to a superhydrophobic surface (in the case of  = 160° and  = 150°) 16 , the resistance of a hydrophilic surface (in the case of  = 58° and  = 12°) 12 can theoretically increase more than 40 times.

Supplementary Note 5. Theoretical analysis of the attraction between two droplets floating on the oil-air interface.
Two droplets of water, which has a lower density than the oil medium, floats at the oil-air interface, will distort the interface.When the deformation of adjacent interfaces of two droplets overlaps, lateral capillary forces are triggered to minimize the net curvature of the interfaces, which leads to the relative motion of the droplets 17 .The lateral capillary forces between two particles of radius , separated by a distance between their centers  can be roughly described as 18,19 :  = −2   () where  =  sin , ( = 1,2) and  = / Here  is the surface tension,  is the meniscus inclination angle,  and  are the radii of the interfacial contact lines,  () is the first-order modified Bessel function of the second kind,  is the difference in the mass densities of the upper and lower fluids forming the fluid interface and  is the gravity acceleration. > 0 for light droplets, while the same sign of  will lead to the attractive force.Under the force of attraction, the two droplets will gradually come closer and drain the trapped oil between them gradually 20,21 .Below a critical value (typically < 100 Å), the surfaces spontaneously confluence at one or several points and form a thin liquid bridge between two droplets, finally resulting in the merging of them [21][22][23] .