of 27
Supplementary Materials for
Imaging-guided bioresorbable acoustic hydrogel microrobots
Hong Han
et al.
Corresponding author: Wei Gao, weigao@caltech.edu; Julia R. Greer, jrgreer@caltech.edu;
Qifa Zhou, qifazhou@usc.edu; Di Wu, dwwu@caltech.edu
Sci. Robot.
9
, eadp3593 (2024)
DOI: 10.1126/scirobotics.adp3593
The PDF file includes:
Methods
Figs. S1 to S21
Tables S1 and S2
Legends for movies S1 to S10
References (
59
65
)
Other Supplementary Material for this manuscript includes the following:
Movies S1 to S10
MDAR Reproducibility Checklist
Supplementary Methods. Numerical simulations
The BAM is modeled as a spherical shell of infinitesimal thickness immersed in a viscous
liquid. A gas bubble is entrapped inside the shell, which possesses one or two circular
openings on its surface. The gas
-liquid interface is assumed to remain flat at
equilibrium
and is pinned to the circular edge of the opening. The interface will vibrate upon actuation
by an ultrasound field, generating acoustic streaming flow and thrust to propel the
microrobot.
To investigate the acoustic streaming phenomenon, we solve numerically the governing
perturbation equations for the thermoacoustic fields
(
59
61
). The temperature
, pressure
, and velocity
fields of the liquid are expanded in perturbation series as
=
!
+
"
+
#
+
.
.
.
,
(1)
=
!
+
"
+
#
+
.
.
.
,
(2)
=
"
+
#
+
,
(3)
where the subscripts 0, 1, and 2 represent the ambient equilibrium state, the linear harmonic
perturbation, and the second-order nonlinear perturbation, respectively. Here, the ambient
flow is absent (i.e.,
!
=
0
). To first order in the amplitude of the imposed ultrasound field,
the continuity equation, the Navier
-Stokes equation, and the thermodynamic heat transfer
equation for the perturbation (
"
,
"
,
"
) are given, respectively, by
$
%
!
$
&
+
!
"
=
0
,
(4)
!
$
!
$
&
=
"
+
#
"
+
훽휇
(
"
)
,
(5)
!
(
$
)
!
$
&
!
(
$
(
!
$
&
=
th
#
"
.
(6)
In the continuity equation,
"
=
!
6
휅훾
(
"
9
, where
is the isentropic compressibility,
is the specific heat ratio, and
(
is the thermal expansion coefficient. In the dynamic and
thermodynamic heat transfer equations,
is the shear viscosity of the liquid,
is the
viscosity ratio,
(
is the specific heat capacity at constant pressure, and
th
is the thermal
conductivity.
As in typical acoustic streaming configurations, the rapid oscillation of the second
-order
perturbation is not resolved. It is sufficient to focus on the mean quantities,
.
.
.
, time-
averaged over one oscillation period. Furthermore, given that the thermal effects in the first
-
order equations are weak in water and other common liquids, the coupling between the
temperature field
#
and the mechanical variables
#
and
#
is neglected in the second-
order equations. The resulting second-order, time-averaged continuity equation and Navier
-
Stokes equation are, respectively, given by
(
61
,
62
)
!
#
=
"
"
,
(7)
!
>
$
"
$
&
?
=
#
+
#
#
+
훽휇
(
#
⟩)
>
"
$
!
$
&
?
!
"
"
.
(8)
The acoustic vibration of the circular liquid
-gas interface,
*
(
*
)
+,&
, in opening
is
expressed in a local coordinate system
*
, where
=
1
. The interface vibration
frequency
is assumed to be the same as the acoustic exciting frequency and the mode
shape
*
is obtained from the semi-analytical model of entrapped microbubbles detailed in
(55)
. An open-source MATLAB implementation of the model is provided in MATLAB
Central File Exchange. The interface motion is coupled to the velocity field in the fluid
through a kinematic condition on the normal component of the velocity on the interface,
-
,
*
=
푗휔
*
+,&
. The no
-
penetration and no
-
slip boundary conditions are applied on all
solid surfaces. The first
-
order harmonic acoustic thermoviscous problem is solved
numerically using a finite element method implemented in COMSOL Multiphysics to
obtain the solution,
"
and
"
. The first
-
order solution is then used for solving the second
-
order, time
-
averaged equations to obtain the time
-
averaged pressure
#
and velocity
#
fields, which are used for visualizing the streaming patterns and calculating the force acting
on the microrobot, following the methodology detailed in
(
61
,
63
)
.
Fig. S1. SEM
characterization of 3D-printed BAMs.
(
A
and
B
) SEM images of
a single-
opening BAM (A) and a dual-opening BAM (B). Scale bars, 10 μm.
Fig
S1.
SEM
Dual-opening BAM
A
B
Single
-opening BAM
Fig.
S2
. Long
-
term propulsion stability of BAMs in PBS.
Error bars represent the SD
from 15 BAMs.
0
2
4
7
0
1000
2000
3000
Time (day)
Speed (
μm
s
-
1
)
Fig
S2.
stability
Fig. S3. Symmetric and asymmetric designs of
dual-opening BAMs.
(
A
and
B
)
Schematic design (A) and propulsion speed (
B) of symmetric and asymmetric
designs of BAMs. O, the geometric center of bubble; O’, the geometric center of
BAM.
9
μm
15
μm
2
μm
4
μm
9
μm
15
μm
6
μm
0
1000
2000
3000
Symmetric
design
Asymmetric
design
Speed (
μm
s
-1
)
B
A
Symmetric
design
Asymmetric
design
O (O’)
O
O’
Fig. S
4
. The schematic of BAMs with varying
angles between the two openings.
F
p
,
propulsion force;
F
SB
, secondary Bjerknes force.
Fig
S4.
Single
-
opening and dual
-
opening structures with different angles
Single
-
opening
Dual
-
opening
(
θ
=
6
0
°
)
F
SB
F
P
θ
F
P1
F
P2
F
SB
θ
F
P2
F
P1
F
SB
x
y
Dual
-
opening
(
θ
=
9
0
°
)
Dual
-
opening
(
θ
=
12
0
°
)
Dual
-
opening
(
θ
=
15
0
°
)
θ
F
P1
F
P2
F
SB
θ
F
P1
F
SB
F
P2
Fig. S5. Locomotion of a BAM near the wall.
Scale bar, 100 μm.