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\begin{document}

\title{Supporting Information for ``Microengineered materials with self-healing
features for soft robotics''}



\author[1]{Vardhman Kumar}%
\author[1]{Ung Hyun Ko}%
\author[1]{Yilong Zhou}%
\author[1]{Jiaul Hoque}%
\author[1]{Gaurav Arya}%
\author[1]{Shyni Varghese}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 10, 2024}


\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
This supporting information includes details of the computational model,
supplementary figures, interactive 3D CAD model of DraBot and
supplementary movies.~

Corresponding author Email:~{\emph{~shyni.varghese@duke.edu}}%
\end{abstract}%



\sloppy


\par\null

\section*{Details of the computational model for balloon actuators in the
hindwings}

{\label{628372}}

\textbf{Governing equations}

\textbf{Fluid flow:} As the gravitational force is negligible compared
to the pressure generated by the water balloon, the time-dependent flow
field in the microchannel is governed by the conventional continuity and
momentum equations,

\(\nabla\cdot\mathbf{u=}0\),
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)

\(\rho_{f}(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{\ u\cdot}\nabla\mathbf{u)=-}\nabla p+\eta\nabla^{2}\mathbf{u}\),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(2)

where \(\rho_{f},\ p,\selectlanguage{greek}\text{\ η}\selectlanguage{english}\ \)and \(\mathbf{u}\ \ \)are the fluid density,
pressure, viscosity and velocity vector, respectively.

\textbf{Solid mechanics:} Since PDMS and the Ecoflex membrane endure
large deformations, the second Piola-Kirchhoff stress tensor was used in
the governing equations as,

\(\rho_{m}\frac{\partial^{2}\mathbf{x}}{\partial t^{2}}=\ \nabla\cdot\left(\mathbf{\text{FS}}\right)^{T}\),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(3)

\(\mathbf{F=I+}\nabla\mathbf{\text{x\ }}\),
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)

\(\mathbf{S=}2\frac{\partial W_{s}}{\partial\mathbf{C}}\),~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~(5)

where \(\mathbf{x}\) is the displacement vector, \(\rho_{m}\)
is the material density, \(\mathbf{F}\) is the deformation
gradient,\(\mathbf{S}\) is the second Piola-Kirchhoff stress
tensor, \textbf{\(\mathbf{I}\)~}is the identity tensor,
\(W_{s}\) is the strain-energy function,
and\(\mathbf{C}\mathbf{=}\mathbf{F}^{T}\mathbf{F}\) is the right Cauchy-Green deformation tensor. The
Cauchy stress \(\mathbf{\sigma}\)can be reconstructed in terms of
\(\mathbf{S}\).

Since PDMS and Ecoflex membranes are hyperelastic materials, suitable
strain-energy functions are needed to describe the nonlinear
relationship between the stress and strain. For PDMS, the strain-energy
function can be expressed by the Mooney-Rivlin model\hyperref[csl:1]{(Mooney, 1940}; \hyperref[csl:2]{Rivlin, 1948)}
as,

\(W_{s}=\sum_{i,j=0}^{n}C_{i,j}\left(I_{1}-3\right)^{i}\left(I_{2}-3\right)^{j}+\frac{1}{2}K\left(J_{\text{el}}-1\right)^{2}\)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(6)

where \(I_{1}\) and \(I_{2}\) are the first and
second invariant of the left isochoric Cauchy-Green deformation tensor,
\(K\) is the bulk modulus, \(J_{\text{el}}\) is the
elastic Jacobian, and \(C_{i,j}\) are material parameters.

In addition, the strain-energy function for Ecoflex membrane was
represented by the Ogden material model\hyperref[csl:3]{(Ogden, 1972)} as,

\(W_{s}=\sum_{p=1}^{n}\frac{\mu_{p}}{\alpha_{p}}\left(\lambda_{1}^{\alpha_{p}}+\lambda_{2}^{\alpha_{p}}+\lambda_{3}^{\alpha_{p}}\right)+\frac{1}{2}K\left(J_{\text{el}}-1\right)^{2}\)~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
(7)

where \(\lambda_{i}\ \left(i=1,2,3\right)\) are the principal stretches, and
\(\mu_{p}\) and \(\alpha_{p}\) are material constants.

\textbf{Computational domain and boundary conditions:} The out-of-plane
bending motion of the microchannel takes place in the lateral direction.
Therefore, a 3D cuboid domain was chosen for the study set up as shown
in \textbf{Figure S3a} , where only a length of 12 mm in the lateral
direction and 3 mm in the side direction were considered to capture the
main dynamic motions of the channel so as to reduce computational cost.
This model consists of two layers representing the PDMS and Ecoflex
membrane. The fluid domain was embedded into the space between the PDMS
and Ecoflex membrane layers, as shown in \textbf{Figure S3a} , where the
length \emph{L} is an adjustable parameter. Note that the balloon
expansion mainly appears in the rectangular fluid domain and the inlet
tube used in the experimental device only serves as a passage to inject
water into the domain; therefore, the part of tube has not been included
in the computational domain. Thus, the fluid inlet has been directly
assigned to the right end of the rectangular domain. The following
boundary conditions were prescribed:

\textless{}Fluid flow\textgreater{}

Inlet: volume flow rate \(V_{0}\);

All other walls: no slip with \(\mathbf{u\cdot n}=0\);

Initial conditions: \(\mathbf{u}=0\) and \(p=0.\)

\textless{}Solid mechanics (PDMS and membrane)\textgreater{}

Right ends: fixed constraint with \(\mathbf{x=}0\ ;\)

All outer boundaries: free boundaries with zero stress.

Initial conditions: \(\mathbf{x=}0\) and\(\frac{\partial\mathbf{x}}{\partial t}=0.\)

\textbf{Multiphysics:} This fluid-structure interaction was assigned to
the interfaces between the fluid domain and solid mechanics.

\textbf{Numerical approach:} The dynamic bending motion of the cuboid
microchannel in \textbf{Figure S3a} was simulated in COMSOL 5.3a.
Equation 1 and 2 were solved in the Laminar Flow module, and the Moving
Mesh was applied for the fluid domain. The Solid Mechanics module was
used to solve for the structure deformation from Equation 3-7. Within
the module, the five parameters Mooney-Rivlin material model (Equation
6) was used to calculate the strain-energy function for PDMS while Ogden
material model (Equation 7) was used for Ecoflex membrane. The
Mooney-Rivlin parameters for PDMS was used from a previous
work\hyperref[csl:4]{(Kim et al., 2011)}. The Ogden material constants for Ecoflex00-30
membrane determined from the mechanical measurements were used in the
simulation. The Fluid-Structure Interaction was assigned to the coupled
interfaces between fluid flow and solid mechanics. A time-dependent
study was added to solve the model. All the values of the physical and
geometrical parameters are summarized in \textbf{Table 1.}

\par\null\par\null

\section*{Supplementary Figures}

{\label{573369}}\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Picture1/Picture1}

\end{center}
\end{figure}

\textbf{~\textbf{Figure S1.~}} The hydrophobicity of materials was
quantified using contact angle measurement. The contact angles of
Ecoflex, Sylgard 184, and microporous Sylgard 184 were 96.1\selectlanguage{ngerman}° ± 3.07°,
106.3° ± 1.12°, 126.2° ± 5.47° respectively. ~

\textbf{}

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure S2. Mechanical test/analysis of Ecoflex
00-30.}(\textbf{a} ) Tensile test sample dimension according to the ASTM
type V standard. (\textbf{b} ) Curve fitting graph of Ecoflex 00-30
stress-strain data with Ogden model. (\textbf{c} ) Obtained Ogden model
parameters from curve fitting data.

\par\null\par\null\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure S3. Simulation of balloon actuator for optimizing channel
dimensions.} (\textbf{a} ) Isometric view of the 3D cuboid domain of
balloon actuator. The fluid domain is embedded into the space between
the PDMS and membrane layers, as highlighted in the inset. (\textbf{b} )
Comparison of experimental (solid line) and simulation (dash line)
results of displacement angle (\selectlanguage{greek}θ\selectlanguage{english}) with different rectangular channel
lengths: 4.1, 5.6, and 7.1 mm.

\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure S4. Scanning Electron Microscope images of PDMS
microstructure.} (\textbf{a} ) Bare PDMS slab has a clear surface
without any microstructure. (\textbf{b} ) In contrast, microporous PDMS
substrate shows interconnected salt crystal-shaped pores. Scale bar: 250
\selectlanguage{greek}μ\selectlanguage{english}m.

\par\null\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure S5. DraBot velocity in various pneumatic input
pressure.}Velocity increases from \textasciitilde{}5.5 cm/s to
\textasciitilde{} 13.4 cm/s when air pressure is increased from 10 to 40
psi.

\par\null\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure S6. FTIR spectra of bare, plasma treated, TMSPMA modified
PDMS surface.} (\textbf{a} )The total FTIR graph in
4000\textasciitilde{}500 cm\textsuperscript{-1} wavenumber. (\textbf{b}
) \% transmittance change at 3200\textasciitilde{}3400
cm\textsuperscript{-1} following surface oxygen plasma treatment . Peak
at \textasciitilde{}1700 cm\textsuperscript{-1} corresponding to a C=C
bond is observed in TMSPMA-modified PDMS surface. (\textbf{c} ) Control:
bare PDMS surface. Plasma: PDMS modified by oxygen plasma. TMSPMA:
plasma-treated PDMS reacted with TMSPMA.

\par\null

\section*{Table~}

{\label{989971}}\par\null\par\null\selectlanguage{english}
\begin{table*}[H]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCCC}
Component & Symbol & Value  & Description \\
Fluid domain & L & (4.1, 5.6, 7.1) mm & Length \\
 & \selectlanguage{greek}ρ\selectlanguage{english}\textsubscript{f}
 & 1000 kg/m\textsuperscript{3}
 & Water density \\
 & \selectlanguage{greek}η & 8.9 X 10\textsuperscript{-4} kg/ms
 & Dynamic Viscosity of water \\
 & V\textsubscript{0}
 & 0.15 ml/min & Flow rate \\
PDMS & \selectlanguage{greek}ρ\selectlanguage{english}\textsubscript{m}
 & 970 kg/m\textsuperscript{3}
 & Density of PDMS \\
 & K & 9.62 X 10\textsuperscript{8} Pa
 & Bulk modulus \\
 & C\textsubscript{10}
 & 3.08 X 10\textsuperscript{4} Pa
 & Mooney-Rivlin parameters \\
 & C\textsubscript{11}
 & 2.7 X 10\textsuperscript{4} Pa
 &  \\
 & C\textsubscript{01}, C\textsubscript{20}, C\textsubscript{02}
 & 0 &  \\
Ecoflex Membrane & \selectlanguage{greek}ρ\selectlanguage{english}\textsubscript{m}
 & 1070 kg/m\textsuperscript{3}
 & Density of membrane \\
 & K & 5.5 X 10\textsuperscript{4} Pa
 & Bulk Modulus \\
 & m\textsubscript{1}
 & 1241 Pa & Ogden Material Constants \\
 & m\textsubscript{2}
 & 7.88 X 10\textsuperscript{-6} Pa
 &  \\
 & a\textsubscript{1}
 & 3.04 &  \\
 & a\textsubscript{2}
 & 13.02 &  \\
\end{tabulary}
\end{table*}\textbf{Table S1.} Physical and geometric parameters used in the
numerical simulation~

\par\null

\section*{Interactive Figure}

{\label{758771}}\par\null\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Drabot2/static}

\end{center}
\end{figure}

\textbf{Interactive Figure S1}. 3D CAD model of DraBot. Red part
represents Sylgard 184, blue part represents the Ecoflex 00-30 and the
white structures are the microporous Sylgard 184.~

\par\null\par\null\par\null

\section*{Videos}

{\label{471560}}\par\null\par\null

Rich media available at \url{https://youtu.be/TKwdns254c4}

\textbf{Movie S1.~}Ecoflex 00-30 membrane inflates significantly,
causing the entire PDMS structure to bend. The size of the rectangular
channel in the balloon actuator is~7.1 X 1.8 X 0.15 mm (length X width X
height). ~ ~

\par\null\par\null\par\null\par\null

Rich media available at \url{https://youtu.be/-UdIw\_zTxjc}

\textbf{Movie S2.} The maximum bending angle and the increase in rate of
angle~ of the balloon actuator is proportional to channel length. ~

\par\null\par\null

\textbf{}



Rich media available at \url{https://youtu.be/eaXm7\_PYVG4}

\textbf{Movie S3.} Flapping/flexing motion of each hindwing is
controlled separately by two syringes, and the independent flexing of
each hindwing allows the DraBot to change traveling direction.

\par\null\par\null



Rich media available at \url{https://youtu.be/UhOPHEpNMSg}

\textbf{Movie S4.} Both hindwings flexed upward to unblock the air
outlets of both forewings, leading to unidirectional airflow and forward
propulsion. Maintaining the upward flap of the hindwings results in a
straight motion.

\par\null\par\null



Rich media available at \url{https://youtu.be/j\_Bxp68Dfpk}

\textbf{Movie~ S5.} Left hindwing flexing generates net torque,
resulting the bot to turn rightwards. A sustained right-turn signal
leads to continuous clockwise motion.

\par\null\par\null\par\null

Rich media available at \url{https://youtu.be/tje5JFXd8Oc}

\textbf{\textbf{Movie~ S6.}} Net torque, generated by the right hindwing
flexing, makes the bot turn leftwards. A sustained left turn signal
leads to continuous counterclockwise motion.

\par\null\par\null

\textbf{}



Rich media available at \url{https://youtu.be/dOwAy8go2FI}

\textbf{Movie S7.} Complex motion achieved by using multiple input
signal in series.

\par\null\par\null



Rich media available at \url{https://youtu.be/pL81NHX6Qv0}

\textbf{Movie S8.} pH-sensitive self-healing of the A6ACA hydrogel
facilitates welding of fore- and hindwing upon exposure to acid, which
prevents the flapping of hindwing.

\par\null\par\null



Rich media available at \url{https://youtu.be/b8QjHZxDJok}

\textbf{Movie S9.} Welding of fore and hindwings (left side) block the
air outlet on left forewing, which results in DraBot making a continuous
left turn, despite receiving an input signal for forward motion.





Rich media available at \url{https://youtu.be/0XG\_JDVqsEQ}

\textbf{Movie S10.} When exposed to a temperature above
37~\textsuperscript{o}C, the wing color changes from red-to-yellow, and
reverts back when exposed to room temperature.

\par\null\par\null\par\null

Rich media available at \url{https://youtu.be/j1pQ3upYYvg}

~\textbf{Movie S11.~} The Sylgard 184 hydrophobicity along with the
microstructure mediated increase in surface area~allows microporous
structure to imbibe hydrophobic oil while no such absorption is observed
with water.



Rich media available at \url{https://youtu.be/hBDd5QTUAno}

\textbf{Movie S12.} The microporous PDMS structures present at the wings
and abdomen absorb oil from the water when encountered. For better
visualization, colorless Sylgard 184 was used to fabricate DraBot. The
red color shows the absorption of oil.

\par\null

\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\phantomsection
\label{csl:1}{A Theory of Large Elastic Deformation}. (1940). \textit{Journal of Applied Physics}, \textit{11}(9), 582–592. \url{https://doi.org/10.1063/1.1712836}

\phantomsection
\label{csl:2}{Large elastic deformations of isotropic materials {IV}. further developments of the general theory}. (1948). \textit{Philosophical Transactions of the Royal Society of London. Series A Mathematical and Physical Sciences}, \textit{241}(835), 379–397. \url{https://doi.org/10.1098/rsta.1948.0024}

\phantomsection
\label{csl:3}{Large deformation isotropic elasticity {\textendash} on the correlation of theory and experiment for incompressible rubberlike solids}. (1972). \textit{Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences}, \textit{326}(1567), 565–584. \url{https://doi.org/10.1098/rspa.1972.0026}

\phantomsection
\label{csl:4}{Measurement of nonlinear mechanical properties of {PDMS} elastomer}. (2011). \textit{Microelectronic Engineering}, \textit{88}(8), 1982–1985. \url{https://doi.org/10.1016/j.mee.2010.12.108}

\end{document}

