Supplementary materials to
Pinhole formation from liquid metal microdroplets impact on solid surfaces
Hao Yi, Le-hua Qi a), Jun Luo, Yuanyuan Jiang, and Weiwei Deng b)
The heat transfer analysis performed here applies classic model and theory that can be found in many heat
transfer textbooks such as Fundamentals of Heat and Mass Transfer, 7th edition by Incropera and Dewitt, Wiley,
2011, which is referred as (Incropera and Dewitt, 2011) hereafter.
 Convection heat transfer coefficient h
For free falling liquid droplets, the scaling law
for Nusselt (Nu) number is (Incropera and Dewitt,
2011, p465):
Nu=2+0.6Re0.5Pr1/3,
where Re is the Reynolds number and Pr is the
Prandtl number. Typical experimental parameters
(droplet diameter 500 µm, droplet velocity 1m/s,
Pr=0.67) in this work correspond to Re=18 and
Nu=4.2.
The definition of Nusselt number is:
Nu=hD/kg,
where D is the droplet diameter, kg is the thermal
conductivity of the gas. For Argon, kg =0.016
W/(mK), and we obtain h=135 W/(m2K).
 Isothermal approximation
Next we evaluate the Biot number, which is the
ratio of conduction heat transfer resistance within
the droplet to the convective heat transfer
resistance:
Bi=hD/kd,
where kd is the thermal conductivity of the droplet
(kd =200 W/(mK)). Thus Bi is estimated to be about
310-4<<1, which suggests the droplet is
isothermal, i.e. the temperature is uniform within
the droplet.
 Transient heat transfer time constant
For an isothermal object, one can use the
lumped capacitance model to describe the
temperature history. The transient cooling time
constant tc can be estimated using:
tc=mcd/hA,
where m is the mass of the droplet, cd is the specific
heat capacity of molten droplet, A is the surface area
of the droplet. Typical experimental parameters in
this work (cd=1086 J/kg/K, h=135 W/m2/K) yield tc
=1.4 s, which is much longer than the droplet travel
time (0.1 s) prior to impact.
a)
qilehua@nwpu.edu.cn, b) weiwei@vt.edu
 Heat conduction through the gas layer
The typical geometry of the gas layer under the
impacting droplet is ~100 µm in diameter and ~2
µm thick. Assume the temperatures of the two sides
of the gas layer is Td=1023K (droplet) and Ts=303K
(substrate), one can estimate the conduction heat
transfer rate from the droplet to the substrate is
~0.05W.
The latent heat of the droplet is 0.6 J based on
specific latent heat of aluminum (400kJ/kg) and the
droplet mass (1.510-7kg). This suggests that it will
take at least ~ 100 ms to fully solidify the droplet if
the gas layer is preserved.
 Instantaneous contact temperature
To estimate the temperature of the
aluminum/brass interface (Ti) immediately upon
contact, we apply the model of two infinitely large
objects of different temperatures that are suddenly
in contact (Incropera and Dewitt, 2011, p315):
(kc)1d/ 2 Td  (kc)1s/ 2 Ts ,
Ti 
(kc)1d/ 2  (kc)1s/ 2
where k, , and c are the thermal conductivity,
density, and specific heat respectively (subscript d
refers to the droplet; subscript s refers to the
substrate). For typical experimental conditions:
kd = 200 W/(mK), d=2300 kg/m3, cd=1086 J/kg/K,
ks = 109W/(mK), s=8500 kg/m3, cs=380 J/kg/K,
Td=1023K, and Ts=303K:
Ti=694K,
which is below the melting point of Aluminum
(933K). This implies that the thin layer of metal
near the interface will be solidified immediately
upon direct contact without gas cushion layer.
Note that although the droplet may not be
considered infinitely large, the interfacial
temperature can only be overestimated and will not
change the conclusion that Ti is below the melting
point of aluminum.