Rapid meniscus-guided printing of stable semi-solid-state liquid metal microgranular-particle for soft electronics

Liquid metal is being regarded as a promising material for soft electronics owing to its distinct combination of high electrical conductivity comparable to that of metals and exceptional deformability derived from its liquid state. However, the applicability of liquid metal is still limited due to the difficulty in simultaneously achieving its mechanical stability and initial conductivity. Furthermore, reliable and rapid patterning of stable liquid metal directly on various soft substrates at high-resolution remains a formidable challenge. In this work, meniscus-guided printing of ink containing polyelectrolyte-attached liquid metal microgranular-particle in an aqueous solvent to generate semi-solid-state liquid metal is presented. Liquid metal microgranular-particle printed in the evaporative regime is mechanically stable, initially conductive, and patternable down to 50 μm on various substrates. Demonstrations of the ultrastretchable (~500% strain) electrical circuit, customized e-skin, and zero-waste ECG sensor validate the simplicity, versatility, and reliability of this manufacturing strategy, enabling broad utility in the development of advanced soft electronics.


Initial conductivity
In Supplementary Fig. 8, the PaLMP sample with 10 v/v% LM shows Newtonian behavior with viscosity 14 mPa·s as the slope is very close to the unity in the logarithmic graph. When the LM volume fraction is increased to 30%, the viscosity at low shear rate increase more than 10 times. At 50 v/v%, the yield stress and shear thinning behavior appear to be stronger as typical characteristics of non-Newtonian fluids. For delicate For comparison with the meniscus-guided printing (MGP) process, screen printing was utilized to generate the PaLMP film (substrate: polydimethylsiloxane, PDMS). For screen printing, a mask was placed on top of the heated substrate at 70 °C, and liquid layer was formed in the opened areas by blade coating. Subsequent natural drying of the solvent generated the film. This film did not have the closely-packed particle assembled architecture; rather evidence of ruptured particles and cavities within the film were observed ( Supplementary  Fig. 15a). This film exhibited cracking and delamination under strain, as seen in Supplementary Fig. 15b, top. XPS show indium oxide peaks for this film, which are not present in MGP-based films ( Supplementary  Fig. 15b, bottom) (XPS data for other elements are presented in Supplementary Fig. 16). The brittleness of indium oxide may be contributing to the brittleness of the film. Real time optical microscopy show that for MGP, at the drying front near the contact line, the solution dries in 0.4 s. For screen printing, solvent took over

Clogging simulation
Under pressure driven flow, the liquid/liquid interface departs from its spherical shape and the surface tension disturbs the neighboring flow. The deformation is much easier for bigger drops and the size of LMP drop is growing by its low dispersion stability. Here, we calculated the 'printing pressure' varying the size of single drop using the finite element method with COMSOL Multiphysics (COMSOL Inc., v5.6). Because the bottle-neck geometry between nozzle body and nozzle tip is well known as the region showing clogging 10-12 , we focused on the axi-symmetric 2D bottle neck geometry for the calculation efficiency (Supplementary Fig.   6).

Governing equations
Navier-stokes equations are used to describe incompressible Newtonian fluids in geometry shown in Supplementary Fig. 6. Because Re = 2ρUR/µ = 0.3 (R = 50 µm and U = 6 mm/s), the turbulent effect is not included 13 . For all liquids, time ( ) dependent momentum and mass conservation equations are solved as where is the flow velocity; is the pressure. Liquid properties follow Supplementary  (1) is coupled with multiphase model by Fst, the surface tension force.
where is the interfacial tension between aqueous acetic acid and EGaIn; is the identical tensor; is the normal vector; is the Kronecker delta. is determined by the shape of interface while its calculation is described in the next paragraph. Even if the oxide layer is known to reduce interfacial tension with water, it is negligible in acetic acid (pH<3) 14 .
Level-set model is used to describe the transport of the interface between immiscible fluids [15][16][17] . In this model, the parameter is a function presenting the phase is dispersed (liquid metal) or continuous (solvent) phase.
is zero at the continuous phase and unity at the dispersed phase. The flux of phase variable can is solved as where = | | . ε is the thickness of interface and is calculated from the mesh size as hc/2, where hc is the characteristic mesh size in the interface. λ is the reinitialization factor and is determined as the typical flow speed (6 mm/s). One can find the advection of the interface as the second term of LHS, which means that equation (4) is also coupled with equation (1) and (2). In the RHS, the first term is the numerical diffusion of the interface and the second term is compressive flux 15,16,18 . For the smooth transition in liquid properties across the interface, the viscosity and density are calculated from as where 1 and 2 denote for continuous and dispersed phase, respectively. The interface is moving at the nozzle wall while the bulk flow is no-slip. To give such mobility to the interface, the Navier slip stress ( ) is given to the wall as = where is the slip velocity and calculated with the normal vector and velocity field as • ( • ) .
is the slip length and used the size of hc. Furthermore, to keep the wettability between the liquid metal drop and the nozzle wall, the final expression at the wall boundary can be written as where 0 is the contact angle of liquid metal on the nozzle inner surface and is fixed as perfectly non-wetting condition ( 0 =180°). w is the normal vector of the wall boundary wall. in equation (6) is vector form in equation (7) because the slip velocity is calculated in each boundary position for each element of interface.
Finally equation (1) ~ (7) are simultaneously calculated in the time ranging from 0 to 10 ms with interval 0.01 ms.
The entire meshing is done with the 'finer' option of the built-in tool of COMSOL, resulting in 65,058 elements. The mesh size is tested in two steps. First, we tested the convergence with fluidics without liquid/liquid interface and second tested with bubble rising problem 19 . In the Supplementary Fig. 7, the velocity profile converges to finer mesh condition, and the bubble shape excellently agrees with the experimental result. For the bubble rising simulation, liquid properties and geometry follow the experimental conditions.

Printing pressure for different drop sizes
Changing the radius of liquid metal droplet, the printing pressure is calculated as Fig. 2D. The printing pressure is calculated from the pressure difference between inlet and outlet of model nozzle. The simulation results separated into two groups; the drop totally blocks the nozzle tip (red group) or does not (black group).
The maximum printing pressure is displayed for the printing pressure for the red group; whereas the final pressure is displayed for the black group. This is because the black group undergo stress overshoot right before the drop deformation that originated from the surface tension of EGaIn. The local pressure gradient within the simulation geometry (Supplementary Fig. 6) is suggested as the working range of printing pressure.