[1,3]Jianyu Li

1]Mechanical Engineering, McGill University, Montreal, H3A 0C3, QC, Canada

2]Mechanical Engineering, Colorado University Boulder, Boulder, 80309, CO, USA

3]Biomedical Engineering, McGill University, Montreal, H3A 2B4, QC, Canada

# Programming hydrogel adhesion with engineered polymer network topology

Zhen Yang<EMAIL_ADDRESS>Guangyu Bao<EMAIL_ADDRESS>Shuaibing Jiang<EMAIL_ADDRESS>Xingwei Yang
<EMAIL_ADDRESS>Ran Huo<EMAIL_ADDRESS>Xiang Ni
<EMAIL_ADDRESS>Luc Mongeau<EMAIL_ADDRESS>Rong Long
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [

###### Abstract

Hydrogel adhesion that can be easily modulated in magnitude, space, and time
is desirable in many emerging applications ranging from tissue engineering,
and soft robotics, to wearable devices. In synthetic materials, these complex
adhesion behaviors are often achieved individually with mechanisms and
apparatus that are difficult to integrate. Here, we report a universal
strategy to embody multifaceted adhesion programmability in synthetic
hydrogels. By designing the surface network topology of a hydrogel,
supramolecular linkages that result in contrasting adhesion behaviors are
formed on the hydrogel interface. The incorporation of different topological
linkages leads to dynamically tunable adhesion with high-resolution spatial
programmability without alteration of bulk mechanics and chemistry. Further,
the association of linkages enables stable and tunable adhesion kinetics that
can be tailored to suit different applications. We rationalize the physics of
chain slippage, rupture, and diffusion that underpins emergent programmable
behaviors. We then incorporate the strategy into the designs of various
devices such as smart wound patches, fluidic channels, drug-eluting devices,
and reconfigurable soft robotics. Our study presents a simple and robust
platform in which adhesion controllability in multiple aspects can be easily
integrated into a single design of a hydrogel network.

###### keywords:

controlled adhesion, hydrogel adhesives, tough hydrogels, polymer entanglement

## Introduction

The ability to program hydrogel adhesion has significant implications in
engineering, biology, and medicine. The variables of hydrogel adhesion include
adhesion energy, spatial distribution, and kinetics. Among them, controlling
adhesion energy is needed for bonding reinforcement after placement[1] or for
easy detachment without damaging the adherend surface[2, 3]. Controlling the
spatial distribution enables independent modulation of adhesion at different
locations and can be useful for wound dressings that require strong attachment
to healthy tissues while preventing stickiness to fragile and delicate wound
beds. While most existing research efforts focused on the adhesion magnitude
at the equilibrium stage, controlling adhesion kinetics, i.e., to modulate
transient adhesion temporally, allows one to tune the operating time window
for adhesive placement and is equally important but less explored. Programming
the multifaceted adhesion with high-level control could enable and improve
various applications ranging from tissue repair to soft robotics, yet remains
extremely challenging.

In nature, marine animals such as flatworms control adhesion to substrates
using sophisticated adhesion organs that contain two glands which respectively
release adhesive and de-adhesive agents[4]. Such programmable adhesion is
difficult to achieve for synthetic adhesives because they require the addition
of complex chemistry and apparatus that are potentially difficult to
integrate. For instance, adhesion based on covalent bonds is generally strong
but difficult to modulate [5] unless introducing specific chemistries [6].
Physical interactions offer more flexibility to modulate adhesion energy, but
require specific material properties (e.g., viscoelasticity) or additional
apparatus (light, ultrasound, etc.) [7, 8, 9]. In terms of adhesion kinetics,
the rate of covalent bonding is fundamentally dependent on the specific
chemical reactions involved. Although using different bonds with varying
reaction kinetics can in principle enable tunable adhesion kinetics,
incorporation of multiple reactions into one system is challenging. Physical
interactions often form instantaneously and hence do not offer sufficient
tunability in the kinetics for applications that might desire rather slow
kinetics[10]. Achieving spatial control of adhesion requires forming (or
suppressing) the interactions at selective locations through sophisticated
surface patterning, while the outcomes could be compromised by uncontrolled
diffusion of chemical reagents. A universal design strategy that inherently
allows for robust and multifaceted adhesion programming on diverse surfaces is
still missing.

Figure 1: Engineered network topology and linkages for multifaceted
programming of hydrogel adhesion. (a) Schematics of the stitch linkage (Top)
and slip linkage (Bottom) formed between a bridging polymer and networks
without and with surface dangling chains. The thickness of the dangling chain
layer and the penetration depth of the bridging polymer are denoted as
$h_{\mathrm{dc}}$ and $h_{\mathrm{pen}}$, respectively. (b) Hydrophilic and
hydrophobic molds are used to form a regular network (Top) and a network
carrying surface dangling chains (Bottom), respectively. (c) Rate dependence
and magnitude of the adhesion energy depend on the interfacial linkage types:
stitch linkages with $h_{\mathrm{pen}}/h_{\mathrm{dc}}\rightarrow\infty$, slip
linkages with $h_{\mathrm{pen}}/h_{\mathrm{dc}}\ll 1$, and their hybrid with
$h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 1$. (d) The slip linkage offers
programmable adhesion kinetics through tuning $h_{\mathrm{dc}}$ (Top), which
is also insensitive to processing conditions such as the thickness of bridging
polymer solution $h_{\mathrm{sol}}$ (Bottom). (e) Spatially controllable
adhesion obtained from patterning the topological linkages at the interface.

Here we report that engineering surface network topology provides a facile,
predictive, and robust methodology to program multifaceted hydrogel adhesion.
This approach leverages the principles of polymer entanglement that are
applicable to a wide range of materials. The underlying strategy is through
two adhesion units constructed by interfacial polymer entanglements of
different topologies, referred to as the slip and stitch linkages (Fig 1a).
The slip linkage has the topology of a long polymer chain entangled with
another crosslinked network (chain to network). Such a linkage can dissociate
in a rate-dependent manner so that the adhesion energy can be varied over many
folds without background dissipation by varying loading rate (Fig 1c). The
association kinetics of the slip linkage dominates over other operating
condition-dependent sub-kinetics and is controllable through tuning its
governing length scale, enabling a stable kinetic time that can be tuned in a
wide range from $\sim$50s to $\sim$1000s (Fig 1d). In contrast, the stitch
linkage has the topology of two crosslinked polymer networks entangled
together (network to network), as found in the hydrogel topological adhesion
and offers less tunability in terms of adhesion energy and kinetics[7].
Through a simple fabrication technique, we pattern the slip and stitch
linkages spatially at the same interface (Fig 1b and 1e). Their contrasting
adhesion behaviors enable pre-defined and spatially varying adhesion in a
scalable manner. As such, we can embody adhesion programmability in multiple
aspects in a hydrogel adhesive through a single design of the network
structure, which we refer to as the topologically engineered adhesives (TEA)
in this paper. The robust, facile, and predictive strategy for unprecedented
control over hydrogel adhesion opens up numerous opportunities in engineering
and medicine. We demonstrate the applications of our strategy in a wide range
of devices, including wound patches, drug-eluting depots, fluidic channels,
and soft actuators.

## Design of the interfacial topological linkages

To robustly program the adhesion between hydrogel adhesives and targeted
surfaces, we create a diffusive interface by placing a third species of
diffusive polymer, called bridging polymer, to the interface[11, 7]. Formation
of the chain-to-network topology of the slip linkage demands the following
conditions: (1) the hydrogel network needs to contain dangling chains and (2)
a thermodynamic driving force is needed to facilitate the diffusion of
bridging polymers into the gel network. Meanwhile, the diffusion needs to be
halted once the linkage forms to prevent the over-diffusion of bridging
polymers into the bulk gel, which may reduce the number of linkages at the
interface.

To meet the first condition: we choose polyacrylamide (PAAm) as a model
hydrogel network and polymerize it on mold with low surface tension such as
Poly(methyl methacrylate) (PMMA). The hydrophobicity and other associated
effects inhibit the free-radical polymerization of the gel in the vicinity of
the mold [12, 13, 14]. This results in a surface layer of branched dangling
chains with thickness $h_{\mathrm{dc}}\approx 10\sim 100\mu$m, “protruding”
from the crosslinked bulk network. In contrast, gels polymerized on molds with
high surface tension such as glass are not subject to the hydrophobic mold
effect, and hence contain crosslinked networks instead of branched dangling
chains on their surfaces. The gels with and without engineered surface
dangling chains are hereafter referred to as the TEA and regular gels,
respectively. To meet the second criterion, stimuli-responsive polymers such
as chitosan or gelatin were chosen as bridging polymers. The polarity of the
hydrogel network and chitosan chains and the entropy of mixing promote the
diffusion of chitosan chains into the hydrogel; meanwhile, the chitosan chains
can be triggered to crosslink into a bridging network through a reaction-
diffusion process in responding to pH changes, leading to penetration depths
$h_{\mathrm{pen}}$ on the order of tens of microns[11]. The network formed in-
situ provides a more efficient way to engage the dangling chains as opposed to
a pre-formed network in which the dangling chains have to diffuse slowly
through reptation to form entanglement. Other strategies to form the chain-to-
network topology of slip linkage at soft material interfaces can be found in
Supplementary note 3.

Additionally, to encode adhesion kinetics and ensure the repeatable formation
of slip linkages, the dominating kinetic mechanisms for the association of
slip linkage should intrinsically rely on the gel network rather than being
sensitive to external processing conditions that are difficult to control such
as the thickness of cast solution $h_{\mathrm{sol}}$. The formation of the
slip linkage is associated with the diffusion of the bridging polymers
(kinetic time $t_{\mathrm{d}}$) and their gelation process (kinetic time
$t_{\mathrm{gel}}$), which depend on different governing length scales.
Specifically, $t_{\mathrm{d}}$ and $t_{\mathrm{gel}}$ are associated with the
diffusion of bridging polymers and gelling triggers over the thicknesses of
the dangling chain layer $h_{\mathrm{dc}}$ and that of the cast solution
$h_{\mathrm{sol}}$, respectively. Since $h_{\mathrm{dc}}$ is a well-defined
material property while $h_{\mathrm{sol}}$ is sensitive to various processing
conditions[7, 15], a well-defined and controllable adhesion kinetics ensues if
$t_{\mathrm{slip}}\geq t_{\mathrm{gel}}$. A simple scaling analysis allows us
to determine a rough criterion to fulfill this requirement:
$h_{\mathrm{dc}}^{2}D_{\mathrm{eff,gel}}/h_{\mathrm{sol}}^{2}D_{\mathrm{eff}}\geq
1$, where $D_{\mathrm{eff,gel}}$ and $D_{\mathrm{eff}}$ are the effective
diffusion coefficients of gelling triggers and bridging polymers,
respectively. Taking $D_{\mathrm{eff}}\approx 5\cdot
10^{-12}\mathrm{m}^{2}\mathrm{s}^{-1}$(Supplementary note 3 ) and
$D_{\mathrm{eff,gel}}\approx 10^{-11}\mathrm{m}^{2}\mathrm{s}^{-1}$[15] leads
to $h_{\mathrm{dc}}/h_{\mathrm{sol}}\gtrapprox 1$. $h_{\mathrm{sol}}$ at a
hydrogel interface is often in the range of 10$\sim$100 $\mu$m [15, 7], which
means that $h_{\mathrm{dc}}$ needs to be in the comparable range and is
readily satisfied by our fabrication techniques.

We hypothesize that the condition $h_{\mathrm{dc}}/h_{\mathrm{sol}}\gtrapprox
1$ can lead to well-defined association kinetics of the slip linkage that
encodes the overall adhesion kinetics. Once the slip linkage is formed, the
long PAAm chains entangle with the crosslinked bridging network. This means
that the linkage can dissociate via chain slippage, which is expected to be a
thermally activated process that results in rate-sensitive adhesion as
generally seen at the cell and elastomeric interfaces, and other bonds[16, 17,
18, 19]. Unlike slip linkage, stitch linkages form when the bridging polymer
diffuses into a regular gel and crosslinks into the bridging network, and
their failures must involve the breaking of one of the networks, thereby
leading to strong and rate-insensitive adhesion[7].

### Structural characterization

Based on the above principles, we fabricate a model TEA using single-network
TEA gel made of PAAm and use chitosan as the bridging polymer. To probe the
engagement length between the dangling chains and the bridging polymer chains,
we used confocal microscopy to visualize how fluorescently labelled chitosan
chains penetrate the TEA gel at equilibrium. The fluorescence intensities
exponentially decrease from the outermost surface to the bulk of the TEA gels
with different crosslinker-to-monomer ratios $C$ (colored dash lines, Fig 2a).
For different $C$, we measured similar distances where the intensities meet
the lower plateau (black dash line, Fig 2a), defining the penetration depth of
the bridging polymer $h_{\mathrm{pen}}\approx 70\mu m$. This value depends on
the reaction-diffusion process and thus may vary with the type of bridging
polymers. For instance, $h_{\mathrm{pen}}$ for gelatin is expected to be
temperature-dependent.

As directly imaging the dangling chains is challenging, we made a first-order
estimation of the thickness $h_{\mathrm{dc}}$ of the dangling chain layer from
the experimentally measured elastic moduli. The TEA gel has a total thickness
of $h$ and is idealized with a tri-layer model (Fig 2b): a layer of a regular
network is sandwiched by two layers of branched dangling chains. The elastic
modulus of the sandwiched regular network $E_{\mathrm{reg}}$ can be measured
from a regular hydrogel formed at the same conditions except using a
hydrophilic mold, given their observed structural similarity[13, 14, 20]. The
elastic modulus of the dangling chain layer is assumed to be negligible since
it cannot carry any transverse loads. As such, we can estimate
$h_{\mathrm{dc}}$ from the ratio of measured elastic moduli of the TEA and
regular gels $E_{\mathrm{tea}}/E_{\mathrm{reg}}$ in uniaxial tensile tests
(Eqn. 1 and Fig S1). The estimations of $h_{\mathrm{dc}}$ show a decreasing
trend with the increasing value of $C$ (Fig 2c). The trend
$h_{\mathrm{dc}}\sim C^{-1}$ may be attributed to the competition between bulk
elasticity of the gel network and interface tension during gelation on
hydrophobic mold[12] (Supplementary Note 1), demonstrating a controlled method
for fabricating the dangling chain layer of different sizes.

With the measured length scales, we calculate their ratio
$h_{\mathrm{pen}}/h_{\mathrm{dc}}$ to quantify the extent to which the
bridging polymers engage the dangling chains, which is expected to govern the
formation of different topological linkages at the TEA gel interface (Fig
S2d). When $h_{\mathrm{pen}}/h_{\mathrm{dc}}\ll 1$, the bridging network only
engages a part of the dangling chain layer, so that the interface only
comprises slip linkage. If $h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 1$, a
complete engagement ensues which indicates that part of the bridging polymers
may diffuse across the dangling chain layer to stitch the underlying network
of the TEA gel. In this case, the linkage is expected to behave as the
combination of the slip and stitch linkage and is referred to as the hybrid
linkage (Fig 1c). Lastly, a regular hydrogel interface that only comprises
stitch linkage corresponds to
$h_{\mathrm{pen}}/h_{\mathrm{dc}}\rightarrow\infty$ since
$h_{\mathrm{dc}}\rightarrow 0$. Fig 2c shows
$h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.2$ when $C=0.024\%$ and increases
to unity as $C$ increases to $0.06\%$ for the TEA gel interface. By tuning
$C$, we can vary the degree of engagement and consequently the formation of
different linkages, which will be shown later to modulate the resulting
adhesion energy.

Figure 2: Design and characterization of topology-engineered adhesive (TEA).
(a) Intensities of florescent chitosan chains diffused in TEA gels. The shaded
area represents the standard deviation from 5 measures. The stronger chitosan
intensity for higher $C$ may be due to more chitosan chains trapped by the
denser dangling chains on the interface. (b) An idealized model used to
estimate the thickness of branched dangling chain layer. (c) Left panel:
estimated $h_{\mathrm{dc}}$ and measured $h_{\mathrm{pen}}$ as functions of
the crosslinker-to-monomer ratio $C$. Error bars represent standard deviation.
Right panel: relative engagement length $h_{\mathrm{pen}}/h_{\mathrm{dc}}$ as
a function of $C$. Dissimilar topological linkages lead to contrasting
adhesion behaviors. (d) Slip and stitch linkages-mediated $G^{1/2}$ plotted as
functions of $\ln(V_{\mathrm{crack}})$ for $C=0.024\%$ and $c_{chi}=2\%$ g/mL.
(e) Illustration showing the dissociations of slip and stitch linkages as
thermally activated processes, with reaction rate of dissociation $r_{i}$ ($i$
can be slip or stitch). Upon a separation force $F$, the activation energy of
the linkage is decreased by $-Fx$, where $x$ is the seperation distance. (f)
The formation of topological linkages and the resulting adhesion depend on
$h_{\mathrm{pen}}/h_{\mathrm{dc}}$, which is controlled by $C$. Slip and
hybrid linkages are achieved for TEA gels with $C=0.024\%$ and $0.048\%$,
respectively. The inset shows the same curves as (d) but for $C=0.048\%$.

## Interfaical topological linkages to program rate-dependent adhesion energy

To test our hypothesis, we first focus on two extremities: the interfaces
containing either slip or stitch linkages. To form slip linkage-mediated
adhesion, we adhere two TEA gels using chitosan as the bridging polymer with
$h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.2$ ($C=0.024\%$), followed by a
T-peeling specimen to measure the adhesion energy $G$ as a function of crack
speed $V_{\mathrm{crack}}$ (Methods and Fig S2a). Fig 2d shows that the slip
linkage-mediated adhesion $G^{1/2}$ varies logarithmically with
$V_{\mathrm{crack}}$, the crack speed. We observed a factor of 25 in the
change of $G$ as $V_{\mathrm{crack}}$ varies by two decades. Together plotted
in Fig 2d is the stitch linkage-mediated adhesion formed between two regular
hydrogels for the same $C$ and chitosan concentration $c_{\mathrm{chi}}$,
showing higher magnitude but much weaker rate-dependence. The contrast between
slip and stitch adhesion is the most pronounced at low $V_{\mathrm{crack}}$
but diminishes at high $V_{\mathrm{crack}}$. We also observed adhesive failure
and mixed adhesive-cohesive failure at the slip and stitch linkage-mediated
interfaces, repspectively. Our experiments further confirmed the similar bulk
mechanics between the TEA and the regular gels: they both show minimal
hysteresis in cyclic loadings and weak rate dependences, indicating near-
perfect elasticity (Fig S1(a)-(d)). The data suggest that different
interfacial network topologies regulate hydrogel adhesion independent of the
bulk properties.

These results motivate us to further analyze the data with a kinetic model
proposed by Chaudhary[18]. The model considers the breaking of linkages as
thermally activated processes[16, 18, 17, 19], and treats each linkage as a
linear spring with stiffness $k_{\mathrm{i}}$ and an activation energy of
dissociation $E_{\mathrm{i}}$ (i can be slip or stitch). These parameters
influence the dissociation rates of the linkages (Fig 2e), and consequently
the rate-dependence of the hydrogel adhesion energy. As detailed in
Supplementary note 2, the model states the adhesion energy for linkage $i$
relates to the crack speed via $G^{1/2}\sim\ln V_{\mathrm{crack}}$, which
agrees perfectly with our experimental data for the slip linkage-mediated
adhesion (blue dash lines in Fig 2d). Further, the model shows that the slope
of the linear relation scales inversely to $k_{i}^{1/2}$, while the intercept
depends on $E_{i}$. By fitting this model to our data we were able to
determine $k_{i}$ and $E_{i}$, which are otherwise difficult to characterize
directly. Specifically, we found $k_{\mathrm{slip}}=1.7\times 10^{-7}$ N/m and
$E_{\mathrm{slip}}$ = 75 kJ/mole for the slip linkage with $C=0.024\%$ and
$h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.2$. It is plausible that the
hydrogel dangling chains determine $k_{\mathrm{slip}}$ which is of entropic
type, so $k_{\mathrm{slip}}\sim k_{B}T/R^{2}$ with $k_{B}T$ the energy in
temperature and $R$ the average end-to-end distance of the dangling chains. We
estimated that $R\approx 250$ nm, which is 50 times larger than the mesh size
of the underlying network $\xi\approx 5$ nm (Supplementary note 1). The fitted
value of $E_{\mathrm{slip}}$ is larger than the typical activation energy of
hydrogen bond (4-50 kJ/mole), suggesting potential synergistic contributions
of multiple hydrogen bonds (between chitosan and PAAm) to a single slip
linkage. Besides, the model captures the rate-insensitivity of
$G_{\mathrm{stitch}}^{1/2}$ of the stitch linkage with
$k_{\mathrm{stitch}}\geq\sim 300k_{\mathrm{slip}}$ and
$E_{\mathrm{stitch}}\approx 185$ kJ/mol (red dash line, Fig 2b). The much
larger $k_{\mathrm{stitch}}$ may be due to the full extension of the entangled
networks prior to network rupture, driving the polymer chains far beyond the
entropic limit. The estimation of $E_{\mathrm{stitch}}$ is in the range of the
bond energy of the C-C bond (350 kJ/mole) [21] and the theoretically
estimtated energy stored in each bond prior to rupture using molecular
parameters (60 kJ/mole)[22], in line with the assumption that the stitched
networks must rupture during separation. This model reveals quantitatively
that the slip linkages exhibit much lower stiffness and dissociation energy
compared to those of stitch linkages.

Additionally, the model predicts that the hybrid linkage formed when
$h_{\mathrm{pen}}/h_{\mathrm{dc}}$ is close to 1, would impart tunable
dependence on loading rate through the relation
$G_{\mathrm{hybrid}}=G_{\mathrm{slip}}+G_{\mathrm{stitch}}$. In this case,
$G_{\mathrm{hybrid}}^{1/2}$ is predicted to be a nonlinear function of $\ln
V_{\mathrm{crack}}$ with a finite and constant value of $G_{\mathrm{stitch}}$
(Fig 1c), indicating that the hybrid linkage behaves as slip or stitch linkage
respectively in different ranges of loading rates. To test the hypothesis, we
prepared TEA gels with $h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.6$
($C=0.048\%$, Fig 2c), and the resulting $G^{1/2}$ shows a nonlinear trend as
expected: at high crack speed, the data collapses onto a master curve with
that with $h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.2$ ($C=0.024\%$),
following $G_{\mathrm{slip}}^{1/2}\sim\ln V_{\mathrm{crack}}$ (Fig 2f). Note
that in this regime, the slip linkage-mediated adhesion is higher than that
mediated by stitch linkage for the same $C$ between two regular gels (Fig 2f
inset). Below $V_{\mathrm{crack}}$=0.5mm/s, the data converges to a plateau
corresponding to rate-independent adhesion energy of $\sim 50$ Jm-2. This
baseline adhesion is also close to the value of $G_{\mathrm{stitch}}$ for the
same $C$ ( 60 Jm-2, Fig 2f inset), confirming the co-existence of stitch- and
slip-linkages on the interface. Fixing $G_{\mathrm{stitch}}$ = 50 Jm-2, our
model captures the experimentally measured $G_{\mathrm{hybrid}}^{1/2}$ with
fitting parameters $k_{\mathrm{slip}}=1\times 10^{-7}$ N/m and
$E_{\mathrm{slip}}=71$ kJ/mole (Fig 2f, cyan dot line), closed to the values
of the sample with $h_{\mathrm{pen}}/h_{\mathrm{dc}}\approx 0.2$
($C=0.024\%$). The ability to control the formation of linkage by tuning the
entanglement length between TEA gel and bridging polymers offers a high level
of adhesion programmability: not only can we predictably tune the adhesion
energy by varying loading rates, but also program rate dependence in different
ranges of loading rate. The finite adhesion energy at low loading rates
provided by the hybrid linkage can effectively prevent the adhesive from
failing at static load to ensure good durability.

## Programming adhesion kinetics

In addition to the equilibrium state of adhesion, we next demonstrate that the
association of the topological linkages regulates the transient adhesion,
which can be exploited to encode adhesion kinetics (Fig 1d). When the bridging
polymer solution is placed between the hydrogel and a permeable substrate,
they diffuse into the two networks while crosslinking into a bridging network
in response to a trigger. The reaction-diffusion process comprises two
concurrent sub-processes: the gelation and the diffusion of the bridging
polymer with their respective kinetic time $t_{\mathrm{gel}}$ and
$t_{\mathrm{d}}$, respectively. We assume that the overall adhesion kinetics
is governed by the slower sub-kinetics:
$t\equiv\max\\{t_{\mathrm{d}},t_{\mathrm{gel}}\\}$.

When using chitosan as the bridging polymer, the gelation process is due to
the pH change in the solution, which is associated with the diffusion of
gelling trigger (protons) away from the cast adhesive solution. The thickness
of the solution $h_{\mathrm{sol}}$ sets the critical diffusion length, and
thus its kinetics time follows $t_{\mathrm{gel}}\sim
h_{\mathrm{sol}}^{2}/D_{\mathrm{eff,gel}}$[15] where $D_{\mathrm{eff,gel}}$ is
the effective diffusion coefficient of the gelling trigger. However,
$h_{\mathrm{sol}}$ is sensitive to the applied compression or wettability of
the interface, yielding the gelation kinetics uncertain in practice without
carefully controlled $h_{\mathrm{sol}}$.

Figure 3:  Programmable adhesion kinetics of TEA.(a) Illustrations showing
that the total adhesion kinetics comprises two sub-kinetic processes:
diffusion and gelation. (b) Dimensionless adhesion between two regular
hydrogels $G/G_{\mathrm{eq}}$ as a function of waiting time for different cast
solution thicknesses $h_{\mathrm{sol}}$. The inset shows $t_{1/2}$ as a
function of $h_{\mathrm{sol}}$. Error bars represent 95% confidence intervals
from fitting the exponential function. (c) Similar curves as (b) measured at
the interface between two TEA gels with $h_{\mathrm{dc}}\approx 120\mu m$. (d)
Adhesion kinetics of TEA interfaces with fixed $h_{\mathrm{sol}}$ ($50\mu m$)
and varying values of $h_{\mathrm{dc}}$ ($h_{\mathrm{dc}}\approx 370,120,70\mu
m$, achieved using $C=0.024\%$, $0.048\%$, and $0.06\%$, respectively). The
inset shows $t_{1/2}$ as a function of $h_{\mathrm{dc}}$. The y error bars
represent a 95% confidence interval from fitting an exponential function while
the x error bars represent the standard deviation from 3 measures.

Meanwhile, the diffusion process of bridging polymers depends on the value of
$h_{\mathrm{dc}}$, and hence the type of formed linkages. For a regular gel,
$h_{\mathrm{dc}}\rightarrow 0$, the interface is dominated by stitch linkages
which only require the bridging polymer to diffuse by one mesh size of the gel
network, thus taking negligible kinetic time $t_{\mathrm{d}}\approx 0$ s [15].
Thus, one can expect the adhesion kinetics of the regular hydrogel interface
to be limited by $t_{\mathrm{gel}}$, which is difficult to control in practice
due to the variable $h_{\mathrm{sol}}$. We hypothesize that incorporation of
slip or hybrid linkages with dangling chain layers of finite values of
$h_{\mathrm{dc}}$ can resolve the issue. In this case, the formation of the
linkages requires the bridging polymers to diffuse through the dangling chains
layer (Fig 3a). $h_{\mathrm{dc}}$ thus sets the characteristic diffusion
length such that $t_{\mathrm{d}}\sim h_{\mathrm{dc}}^{2}/D_{\mathrm{eff}}$.
The prolonged diffusion process can bypass the uncertain gelation process to
govern the overall adhesion kinetics. Importantly, since $h_{\mathrm{dc}}$ is
a material property, it can render the overall adhesion kinetics insensitive
to processing or environmental conditions.

To test the hypothesis, we characterized the adhesion kinetics with different
values of $h_{\mathrm{sol}}$ (50 and 120 $\mu$m) controlled by nylon meshes of
different thicknesses[15] (Methods and Fig S3). We define the adhesion
kinetics using the half time $t_{1/2}$ when $G/G_{\mathrm{eq}}$ reaches 1/2,
where $G_{\mathrm{eq}}$ is the adhesion energy in equilibrium. For the regular
gel interface, we observe a strong $h_{\mathrm{sol}}$-dependent adhesion
kinetics, and the associated kinetic time follows $t_{1/2}\sim
h_{\mathrm{sol}}^{2}$ (Fig 3b and inset). On the contrary, we observe that the
adhesion kinetics of the TEA gel interface with $h_{\mathrm{dc}}\approx 120\mu
m$ ($C=0.048\%$) is insensitive to the value of $h_{\mathrm{sol}}$ (Fig 3c and
inset). Our point is further strengthed by applying an initial compression
(15% strain) to the TEA gel interface ($h_{\mathrm{dc}}\approx 370\mu m$)
without controlling $h_{\mathrm{sol}}$, which yields the same adhesion
kinetics as the TEA gel interface with controlled $h_{\mathrm{sol}}$ (Fig
S3c). Thus, incorporation of the engineered dangling chain layer leads to
adhesion kinetics insensitive to processing conditions, validating our
hypothesis.

Importantly, not only is the TEA kinetics insensitive to processing conditions
but also controllable through changing $h_{\mathrm{dc}}$. Fixing
$h_{\mathrm{sol}}$, we observed strong $h_{\mathrm{dc}}$-dependent adhesion
kinetics of the TEA gel interface: the kinetics accelerates as
$h_{\mathrm{dc}}$ decreases, suggesting shorter distance that the bridging
polymers need to diffuse across to form hybrid or slip linkages (Fig 3d). The
half time follows $t_{1/2}\sim h_{\mathrm{dc}}^{2}$ at
$h_{\mathrm{dc}}\approx$ 70 and 120 $\mu$m but deviates from the scaling
relation at $h_{\mathrm{dc}}\approx$ 370 $\mu$m (Fig 3d inset). In the last
case, the kinetics time is presumably bounded by the total diffusion-reaction
time since $h_{\mathrm{pen}}/h_{\mathrm{dc}}\ll 1$, indicating that the
underlying crosslinked network of the TEA gel is beyond the reach of bridging
polymers. The pre-programmable TEA kinetics can be tailored to suit different
applications. For instance, a small $h_{\mathrm{dc}}$ can be used with
compression to achieve fast kinetics for hemostatic applications[23], while a
large $h_{\mathrm{dc}}$ provides a sufficient time window for adhesive
placement.

## Universal applicability

The design and fabrication of TEA are universally applicable to a wide range
of material systems (different bridging polymers, targeted substrates, and TEA
networks) (Fig 4a). Slip linkages formed on the gel-bridging network interface
could be coupled with other interactions such as slip, stitch linkages or
covalent bonds [24] that the bridging network can interact with the targeted
substrates. For instance, the triggered crosslinking and the abundant amino
groups of chitosan provide numerous options to interact with different
substrates through covalent or physical interactions[25]. Based on the
principle, slip-slip, slip-stitch, and slip-bond linkages were achieved
between two TEA gels, between a TEA and a regular gel, and between a TEA gel
and a VHB elastomer, respectively (Fig 4b, Table 1). Remarkably, our data
reasonably collapse for the different mechanisms to engage different targeted
substrates (Fig 4c), suggesting that the overall adhesion behavior is dictated
by the slip linkages while depending less on the types of interactions between
the bridging network and targeted substrates. These results support the
robustness of the adhesion programming through the TEA strategy.

Figure 4:  Universal applicability of the TEA strategy. (a) Schematic showing
that the main constitutes of a TEA interface can be made of a variety of
materials. (b) When the adhesive network is made of SN PAAm TEA gel,
interfacial topological linkages can be engineered to interact with different
targeted substrates. The slip-slip, slip-stitch, and slip-bond linkages are
created on targeted substrates TEA gel, regular gel, and VHB elastomer,
respectively. (c) Different topological linkages lead to a similar trend of
$G^{1/2}$ as functions $V_{\mathrm{crack}}$. The SN TEA gel matrix and the
chitosan solution are prepared with $C=0.024\%$ and $c_{\mathrm{chi}}=2\%$
g/ml. (d) Slip-slip mediated adhesion between two TEA gels using gelatin and
chitosan as bridging polymers. $c_{\mathrm{chi}}=c_{\mathrm{gelatin}}=2\%$
g/ml (d) Adhesion of TEA and regular DN gels on porcine skin at relatively low
$V_{\mathrm{crack}}$.

Besides chitosan, we examine another bridging polymer gelatin, which was
prepared as polymer solution at $37^{\circ}C$ and then applied to the
interface between two TEA gels for $C=0.024\%$ at room temperature. Similar to
chitosan, gelatin diffused into the gel and was crosslinked into a bridging
network in responding to a temperature drop to form slip linkages with the TEA
dangling chains. Our data reveals an identical trend between the data obtained
using gelatin and chitosan as bridging polymers (Fig 4d), highlighting the
dominating role of polymer topology rather than material chemistry in the
formation of slip linkages.

We then explore using double-network (DN) hydrogel as the TEA network.
Compared to single-network (SN) hydrogels, the DN hydrogels exhibit much
higher fracture toughness and adhesion[5, 26, 11, 27] due to background
dissipation. We tested PAAm-alginate and PAAm-chitosan hydrogels as
representative materials. In the two types of DN gels, alginate and chitosan
are physically crosslinked macromolecules and do not covalently interfere with
the PAAm network, we expect that the hydrophobic mold could produce surface
dangling chains in the PAAm network within the DN gels. We confirmed the
presence of the dangling chain layer in the surface of a PAAm-alginate
hydrogel polymerized on hydrophobic substrate by EDTA treatment to remove
calcium-alginate bonds followed by Atomic Force Microscopy (AFM) tests (Fig
S5a, b). We then examined the adhesion of TEA and regular DN gels polymerized
on PMMA and glass molds on porcine skin (for a systematic study on different
gelling molds, see Fig S5c). We use chitosan as the bridging polymer and
EDC/NHS reagent to form covalent bonds between chitosan and tissue surfaces
[11]. Since the energy dissipation in the DN gels is coupled to the
interfacial adhesion energy, we hypothesized that the slip linkage elicits
negligible bulk energy dissipation at low $V_{\mathrm{crack}}$, thereby
resulting in weak adhesion. Here, we only focused on the adhesion behavior of
the DN TEA at low $V_{\mathrm{crack}}$, since its rate-sensitive adhesion is
presumably coupled with the rate-dependent bulk dissipation and is not further
pursued [7]. Both PAAm-alginate and PAAm-chitosan gels show slip linkage-
mediated adhesion 10 times lower than stitch-mediated adhesion at
$V_{\mathrm{crack}}$=0.25mm/s (Fig 4e), demonstrating that our methodology is
applicable to both SN and DN hydrogels as long as the topology of one of the
networks can be engineered.

## Programming spatial adhesion

The contrast between slip and stitch linkages allows us to program the
adhesion spatially. To do so, we patterned a mold substrate with hydrophilic
(Glass) and hydrophobic regions (PTFE films thickness $\sim$0.1mm), followed
by polymerizing a TEA gel on the patterned mold. Given the predefined
geometries (circle, triangle) of the hydrophobic domains, we can design the
dangling chain region where weak adhesion $G_{\mathrm{slip}}$ is formed at low
loading rates; meanwhile, strong adhesion $G_{\mathrm{stitch}}$ is formed in
other areas to sustain tension or twisting applied to the interface without
interface debonding. Fig 5a and S5e show that by shaping the dangling chain
region, we can achieve weak adhesion region of complex shapes between a TEA
gel and a regular gel by slowly injecting liquid dye into the weak interface.
To further characterize the resolution of the spatially programmable adhesion,
we made a series of circular islands of nominal radii $r_{\mathrm{nominal}}$
in which slip linkages are formed. By slowly injecting the liquid dye, we
visualized and measured their radii $r_{\mathrm{measure}}$ of the weak
adhesion region using a digital camera (Fig. 5b and S5c). The excellent
agreement between the nominal and measured radii suggests high spatial
resolution $\sim$0.1 mm achieved with a manual procedure.

As the slip and stitch linkages show different sensitivities to loading rate,
we expect rate-dependent spatially programmable adhesion, characterized by the
adhesion energy contrast $G_{\mathrm{slip}}/G_{\mathrm{stitch}}$. Fig 5c shows
that $G_{\mathrm{slip}}/G_{\mathrm{stitch}}$ predicted by the parameterized
model (Supplementary note 2) approaches unity at high $V_{\mathrm{crack}}$ and
decreases towards zero at low $V_{\mathrm{crack}}$. The prediction is
supported by our experimental observations: A TEA gel with the designed
dangling chain region shows large adhesion contrast to a regular gel at low
$V_{\mathrm{crack}}$ while appearing to be uniformly adhesive at relatively
larger $V_{\mathrm{crack}}$ (Fig. 5d). The rate-dependent spatially-
programmable adhesion can potentially enable applications which desire tunable
adhesion contrast in different regions under different loading rates.
Additionally, not only we can achieve reduced adhesion
($G_{\mathrm{slip}}/G_{\mathrm{stitch}}<1$) but also enhanced adhesion
($G_{\mathrm{hybrid}}/G_{\mathrm{stitch}}>1$) in the region with slip linkages
by leveraging the hybrid linkage at large $V_{\mathrm{crack}}$ (Fig S4). In
this case, the slip linkage acts as a toughening mechanism that
synergistically contributes to the adhesion unit with the stitch linkage.
Moreover, the one-step fabrication allows spatially selective adhesion to be
assembled within a monolithic material, which otherwise requires assembling
different materials at the interface.

Figure 5:  Spatial programming and soft devices enabled with TEA (a) TEA
strategy leads to spatially programmable and deformable adhesion capable of
tracing complex shapes between a PAAm-alginate TEA DN gel and a regular DN
gel. Scale bar: 1cm. (b) Spatial resolution of the spatially programmable
adhesion. (c) Predicated $G_{\mathrm{slip}}/G_{\mathrm{stitch}}$ achieved
using the parameterized model (Eqn.14) using the data of $G_{\mathrm{slip}}$
and $G_{\mathrm{stitch}}$ in Fig 2d. (d) Experimental demonstration of the
rate-dependent $G_{\mathrm{slip}}/G_{\mathrm{stitch}}$ between a TEA gel with
a circular-shaped dangling chain region and a regular gel. Scale bar: 1cm. (e)
Wound patches made of TEA and regular PAAm-alginate DN gel adhered to wounds
on rat skin (top, scale bar: 8mm) and porcine stomach (bot, scale bar: 12mm).
(f) A drug-eluding device enabled by injecting drug into the weakly-adhered
interface between a TEA gel and a regular gel. Grid size of the inset: 10mm.
(g) Deformable hydrogel-based fluidic channels created by adhering a PAAm-
alginate TEA DN gel to a regular DN gel. Scale bars: 2cm. (Bottom) A PAAm-
alginate TEA DN gel with designed adhesion selectivity forms a fluid channel
on the surface of porcine skin. Scale bar: 2.5cm. (h) Reconfigurable soft
actuators. (Top) the fabrication process of the actuator units with connection
surfaces composed of dangling chains. (Bottom) two modes of actuation. The
initial and the actuated stages are indicated by the white and green dash
lines, respectively. Scale bars: 2cm.

## TEA-based devices

The programmable adhesion of TEA enables various applications such as wound
patches, drug depots, fluidic channels, and soft actuators. For the
application of wound patches, TEA allows one to program weak adhesion to wound
beds while maintaining strong adhesion to the surrounding tissue. As such, the
patch could protect the wound without impairing tissue regeneration and wound
closure. Using the one-step fabrication process (Fig 1e and S5d), we prepared
such a TEA gel with its surface composed of a circular region of dangling
chains and the surrounding region of crosslinked network. The dangling chain
region forms slip linkages which attach weakly to the wound site upon slow
removal to minimize the damage to the wound. Meanwhile, the stitch linkages
attach strongly to the surrounding healthy tissue to maintain the stickiness
of the patch. In contrast, a regular hydrogel exerts strong and uniform
adhesion to both wounded and healthy tissues, which ruptures the wound bed
upon removal (Fig 5e).

Besides, the creation of a weak adhesion region between two hydrogels could
serve as a drug depot. Upon slow injection, mock drug solution filled up the
weak interface. Further injection created a bulge of hydrogel to accommodate a
high amount of drug, which can be continuously released through the hydrogel
network when the whole device is immersed in an aqueous environment (Fig. 5f
Top). Our data show that the initial amount of drug injected into the depot
affects the amount of release over time but the relative kinetics of release
remains similar (Fig. 5f Bottom). As well, we can create a drug depot above a
wound site, where drugs can be directly released into wounded tissue. In
contrast, the strong adhesion of a regular hydrogel prevents the injection of
drug solution to interface (Fig. S5f).

We then demonstrate hydrogel-based fluidic devices assembled with TEA. A PAAm-
alginate TEA DN gel with a rectangular-shaped dangling chain region forms a
partially weak interface with a regular DN gel, which subsequently becomes a
fluidic channel upon slow injection of liquids. The resulting device is highly
deformable while no liquid leakage is observed (Fig 5g Top). The one-step
fabrication technique provides a simple approach to fabricate hydrogel fluidic
channels compared with conventional methods that typically involve multiple
molding steps [28, 29]. In addition, the spatially programmable adhesion is
applicable to varying surfaces as it requires no patterning of the targeted
substrate. As such, we can form such a fluid channel directly on tissue
surfaces such as porcine skin (Fig 5g Bottom). This feature could benefit
medical devices that contact with tissue surfaces for sustained drug release
[30], or in-vitro organ-on-chip models to study cellular behaviors [31].

Lastly, we show that the TEA made of SN PAAm hydrogels can be used to
construct reconfigurable soft actuators, featuring minimal bulk dissipation
for efficient actuation and dynamic adhesion for reversible attachment (Fig
5h). Such actuators are formed with hydrogel units that contain surface
dangling chains on each face and are connected to each other with the aid of
bridging polymer (Methods). The slip linkage-mediated adhesion between the
units is strong enough to sustain actuation, and yet can be separated easily
and slowly with a small force. The separated units can then be reconnected
upon reapplying the bridging polymer solution to the interface so that one can
modify configurations of assembly for different actuation. Our data shows that
the slip linkage-mediated adhesion increases and reaches a plateau after
cycles of detachment and reattachment (Fig S2f). This property can be
partially attributed to the fact that the dissociation of the slip linkage
does not rupture the adherend networks, so the slip-mediated TEA interface is
inherently subjected to minimal damage compared with those bonded by stitch
linkages or covalent bonds (Fig S2g).

## Conclusion

In summary, we have demonstrated that designing the interfacial network
topologies of hydrogels provides a facile and robust approach to program
adhesion in multiple aspects including magnitude, space, and kinetics. Our
approach can be potentially extended to different length scales using proper
manufacturing processes. For instance, spatially programmable adhesion with a
spatial resolution on the micro-scale can be achieved with microfabrication of
the hydrogel network topology [32, 33], while that on the metre scale is
expected to be achieved using gelling molds of the same size for applications
such as camouflaging skin [34]. Broadly, our methodology falls into the
emerging paradigm of material intelligence, as the adhesion programming is
directly encoded in the hydrogel network as material properties, similar to
other properties such as elastic modulus. The implementation of adhesion
control requires no external apparatus, making the methodology extremely
facile, robust, and scalable. We hope that the design of TEA can spark
interest in controlling hydrogel adhesion by designing their network
topologies, opening the door to a new design space for intelligent
materials/structures through programmable adhesion.

## References

  *   * Yang et al. [2021] Yang, Z., Yang, X., Long, R., Li, J.: Stimulation modulates adhesion and mechanics of hydrogel adhesives. Langmuir 37(23), 7097–7106 (2021)
  * Cho et al. [2019] Cho, H., Wu, G., Christopher Jolly, J., Fortoul, N., He, Z., Gao, Y., Jagota, A., Yang, S.: Intrinsically reversible superglues via shape adaptation inspired by snail epiphragm. Proceedings of the National Academy of Sciences 116(28), 13774–13779 (2019)
  * Croll et al. [2019] Croll, A.B., Hosseini, N., Bartlett, M.D.: Switchable adhesives for multifunctional interfaces. Advanced Materials Technologies 4(8), 1900193 (2019)
  * Lengerer and Ladurner [2018] Lengerer, B., Ladurner, P.: Properties of temporary adhesion systems of marine and freshwater organisms. Journal of Experimental Biology 221(16), 182717 (2018)
  * Yuk et al. [2016] Yuk, H., Zhang, T., Lin, S., Parada, G.A., Zhao, X.: Tough bonding of hydrogels to diverse non-porous surfaces. Nature Materials 15(2), 190–196 (2016)
  * Chen et al. [2020] Chen, X., Yuk, H., Wu, J., Nabzdyk, C.S., Zhao, X.: Instant tough bioadhesive with triggerable benign detachment. Proceedings of the National Academy of Sciences 117(27), 15497–15503 (2020)
  * Yang et al. [2018] Yang, J., Bai, R., Suo, Z.: Topological adhesion of wet materials. Advanced Materials 30(25), 1800671 (2018)
  * Gao et al. [2019] Gao, Y., Wu, K., Suo, Z.: Photodetachable adhesion. Advanced Materials 31(6), 1806948 (2019)
  * Ma et al. [2022] Ma, Z., Bourquard, C., Gao, Q., Jiang, S., De Iure-Grimmel, T., Huo, R., Li, X., He, Z., Yang, Z., Yang, G., et al.: Controlled tough bioadhesion mediated by ultrasound. Science 377(6607), 751–755 (2022)
  * Wang et al. [2019] Wang, Y., Jia, K., Xiang, C., Yang, J., Yao, X., Suo, Z.: Instant, tough, noncovalent adhesion. ACS Applied Materials & Interfaces 11(43), 40749–40757 (2019)
  * Li et al. [2017] Li, J., Celiz, A.D., Yang, J., Yang, Q., Wamala, I., Whyte, W., Seo, B.R., Vasilyev, N.V., Vlassak, J.J., Suo, Z., Mooney, D.J.: Tough adhesives for diverse wet surfaces. Science 357(6349), 378–381 (2017)
  * Gong et al. [2001] Gong, J.P., Kii, A., Xu, J., Hattori, Y., Osada, Y.: A possible mechanism for the substrate effect on hydrogel formation. The Journal of Physical Chemistry B 105(20), 4572–4576 (2001)
  * Kii et al. [2001] Kii, A., Xu, J., Gong, J.P., Osada, Y., Zhang, X.: Heterogeneous polymerization of hydrogels on hydrophobic substrate. The Journal of Physical Chemistry B 105(20), 4565–4571 (2001)
  * Mandal et al. [2021] Mandal, J., Zhang, K., Spencer, N.D., et al.: Oxygen inhibition of free-radical polymerization is the dominant mechanism behind the “mold effect” on hydrogels. Soft Matter 17(26), 6394–6403 (2021)
  * Steck et al. [2020] Steck, J., Kim, J., Yang, J., Hassan, S., Suo, Z.: Topological adhesion. i. rapid and strong topohesives. Extreme Mechanics Letters 39, 100803 (2020)
  * Evans and Ritchie [1997] Evans, E., Ritchie, K.: Dynamic strength of molecular adhesion bonds. Biophysical Journal 72(4), 1541–1555 (1997)
  * Bell [1978] Bell, G.I.: Models for the specific adhesion of cells to cells: a theoretical framework for adhesion mediated by reversible bonds between cell surface molecules. Science 200(4342), 618–627 (1978)
  * Chaudhury [1999] Chaudhury, M.K.: Rate-dependent fracture at adhesive interface. The Journal of Physical Chemistry B 103(31), 6562–6566 (1999)
  * Pobelov et al. [2017] Pobelov, I.V., Lauritzen, K.P., Yoshida, K., Jensen, A., Mészáros, G., Jacobsen, K.W., Strange, M., Wandlowski, T., Solomon, G.C.: Dynamic breaking of a single gold bond. Nature communications 8(1), 15931 (2017)
  * Zhang et al. [2019] Zhang, K., Simic, R., Yan, W., Spencer, N.D.: Creating an interface: rendering a double-network hydrogel lubricious via spontaneous delamination. ACS Applied Materials & Interfaces 11(28), 25427–25435 (2019)
  * Creton and Ciccotti [2016] Creton, C., Ciccotti, M.: Fracture and adhesion of soft materials: a review. Reports on Progress in Physics 79(4), 046601 (2016)
  * Wang et al. [2019] Wang, S., Panyukov, S., Rubinstein, M., Craig, S.L.: Quantitative adjustment to the molecular energy parameter in the lake–thomas theory of polymer fracture energy. Macromolecules 52(7), 2772–2777 (2019)
  * Bao et al. [2022] Bao, G., Gao, Q., Cau, M., Ali-Mohamad, N., Strong, M., Jiang, S., Yang, Z., Valiei, A., Ma, Z., Amabili, M., et al.: Liquid-infused microstructured bioadhesives halt non-compressible hemorrhage. Nature Communications 13(1), 5035 (2022)
  * Yang et al. [2019a] Yang, J., Bai, R., Chen, B., Suo, Z.: Hydrogel adhesion: a supramolecular synergy of chemistry, topology, and mechanics. Advanced Functional Materials 30 (2019)
  * Yang et al. [2019b] Yang, J., Bai, R., Li, J., Yang, C., Yao, X., Liu, Q., Vlassak, J.J., Mooney, D.J., Suo, Z.: Design molecular topology for wet–dry adhesion. ACS Applied Materials & Interfaces 11(27), 24802–24811 (2019)
  * Sun et al. [2012] Sun, J.-Y., Zhao, X., Illeperuma, W.R.K., Chaudhuri, O., Oh, K.H., Mooney, D.J., Mooney, D.J., Suo, Z.: Highly stretchable and tough hydrogels. Nature 489(7414), 133–136 (2012)
  * Bao et al. [2021] Bao, G., Huo, R., Ma, Z., Strong, M., Valiei, A., Jiang, S., Liu, S., Mongeau, L., Li, J.: Ionotronic tough adhesives with intrinsic multifunctionality. ACS Applied Materials & Interfaces 13(31), 37849–37861 (2021)
  * Gjorevski et al. [2022] Gjorevski, N., Nikolaev, M., Brown, T., Mitrofanova, O., Brandenberg, N., DelRio, F., Yavitt, F., Liberali, P., Anseth, K., Lutolf, M.: Tissue geometry drives deterministic organoid patterning. Science 375(6576), 9021 (2022)
  * Lin et al. [2016] Lin, S., Yuk, H., Zhang, T., Parada, G.A., Koo, H., Yu, C., Zhao, X.: Stretchable hydrogel electronics and devices. Advanced Materials 28(22), 4497–4505 (2016)
  * Whyte et al. [2018] Whyte, W., Roche, E.T., Varela, C.E., Mendez, K., Islam, S., O’Neill, H., Weafer, F., Shirazi, R.N., Weaver, J.C., Vasilyev, N.V., et al.: Sustained release of targeted cardiac therapy with a replenishable implanted epicardial reservoir. Nature Biomedical Engineering 2(6), 416–428 (2018)
  * Vera et al. [2021] Vera, D., Garcia-Diaz, M., Torras, N., Alvarez, M., Villa, R., Martinez, E.: Engineering tissue barrier models on hydrogel microfluidic platforms. ACS Applied Materials & Interfaces 13(12), 13920–13933 (2021)
  * Tudor et al. [2018] Tudor, A., Delaney, C., Zhang, H., Thompson, A.J., Curto, V.F., Yang, G.-Z., Higgins, M.J., Diamond, D., Florea, L.: Fabrication of soft, stimulus-responsive structures with sub-micron resolution via two-photon polymerization of poly (ionic liquid)s. Materials Today 21(8), 807–816 (2018)
  * Carlotti and Mattoli [2019] Carlotti, M., Mattoli, V.: Functional materials for two-photon polymerization in microfabrication. Small 15(40), 1902687 (2019)
  * Pikul et al. [2017] Pikul, J., Li, S., Bai, H., Hanlon, R., Cohen, I., Shepherd, R.F.: Stretchable surfaces with programmable 3d texture morphing for synthetic camouflaging skins. Science 358(6360), 210–214 (2017)
  * Gong et al. [2001] Gong, J.P., Kurokawa, T., Narita, T., Kagata, G., Osada, Y., Nishimura, G., Kinjo, M.: Synthesis of hydrogels with extremely low surface friction. Journal of the American Chemical Society 123(23), 5582–5583 (2001)
  * Chau et al. [2021] Chau, A., Cavanaugh, M., Chen, Y.-T., Pitenis, A.: A simple contact mechanics model for highly strained aqueous surface gels. Experimental Mechanics 61, 699–703 (2021)
  * Hui et al. [2004] Hui, C.-Y., Tang, T., Lin, Y.-Y., Chaudhury, M.K.: Failure of elastomeric polymers due to rate dependent bond rupture. Langmuir 20(14), 6052–6064 (2004)
  * Brown [1993] Brown, H.R.: Effects of chain pull-out on adhesion of elastomers. Macromolecules 26(7), 1666–1670 (1993)

## Acknowledgments

This work is supported by the Natural Sciences and Engineering Research
Council of Canada (grant RGPIN-2018-04146), Fonds de Recherche du Quebec-
Nature et technologies (grants FRQNT PR-281851), and the National Institute on
Deafness and Other Communication Disorders (grant R01-DC018577). Z.Y. and R.H.
acknowledge support from FRQNT and McGill Engineering Doctoral Award. S.J.
acknowledges support from FRQNT Doctoral Award. X.Y. and R.L. acknowledge
support from the National Science Foundation of the United States (NSF
CMMI-1752449). J.L. acknowledges the support from the Canada Research Chair
Program. The authors acknowledge Mr. Tianqin Ning and Mr. Yixun Chen for
taking pictures, Mr. Omar Peza-Chavez for the help with cryosectioning, and
Prof. Reghan Hill for valuable discussion.

## Competing financial interests

The authors declare no competing financial interests.

## Methods

Materials All chemicals were purchased and used without further purification.
Materials for the hydrogel synthesis and the bridging polymer include
acrylamide (AAm, Sigma-Aldrich, A9099), N,N’-methylenebisacrylamide (MBAA;
Sigma-Aldrich, M7279), ammonium persulphate (APS, Sigma-Aldrich, A3678),
N,N,N’,N’-tetramethylethylenediamine (TEMED, Sigma-Aldrich, T7024), Alginate
(I-1G) was purchased from KIMICA Corporation, chitosan (degree of
deacetylation, DDA: 95%, medium and high molecular weight, Lyphar Biotech),
sodium bicarbonate (Fisher Scientific, S233), sodium phosphate monobasic
(NaH2PO4, Sigma, S8282), sodium phosphate dibasic (Na2HPO4, Sigma-Aldrich,
S7907), acetic acid (Sigma-Aldrich, A6283), calcium sulfate (Sigma-Aldrich),
N-hydroxysulfosuccinimide (NHS, Sigma-Aldrich, 130672), and
1-ethyl-3-(3-dimethylaminopropyl) carbodiimide (EDC, Sigma-Aldrich, 03450),
Gelatin (Sigma-Aldrich, G2500). Glass, acrylic sheets (PMMA), PS, silicon, and
PTFE were purchased from McMaster-Carr to make mold substrates for
polymerization. VHB elastomer was purchased from 3M. Porcine skin was
purchased from a local grocery store, then stored in a fridge at -20°C, and
thawed at 4°C before use. Nylon mesh were purchased from McMaster Carr without
further modification (9318T25, 9318T23 for thicknesses of 50 $\mu m$ and 120
$\mu m$, respectively.)

Synthesis of TEA The TEA based on PAAm hydrogels was prepared as follows. AAm
monomers of 6.76 g was first dissolved in 50 mL of deionized water. After
degassing, the AAm solution of 25 ml was mixed with varying amounts of MBAA
aqueous solution (0.02 g mL-1) and 20 $\mu L$ of TEMED in a syringe. The
volumes of MBAA solution added were 90, 120, 150, 180, and 240 $\mu L$ for the
crosslinker$-$to$-$monomer molar ratios $C$ at 0.024$\%$, 0.032$\%$, 0.04$\%$,
0.048$\%$, 0.06$\%$, respectively. Meanwhile, another syringe was added with
565 $\mu L$ of APS solution (0.066 g mL-1) and 478 $\mu L$ deionized water.
The two syringes were connected with a Luer$-$Lock connector, so the two
solutions were syringe$-$mixed to form a homogeneous solution. The mixture was
immediately injected into rectangular acrylic molds of 80 $\times$ 20 $\times$
3 mm3 or 80 $\times$ 15 $\times$ 1.5 mm3, covered with PMMA on both sides, and
then kept at room temperature for 24 hours to complete the reaction. To
prepare the regular PAAm gels, we follow the same procedure except for
injecting the mixed solution into the acrylic molds covered by glass sheets on
the two sides.

The TEA based on double-network gels was prepared with the following protocol.
To prepare PAAm-alginate TEA, 1.5g alginate (I-1g) power and 6.76 g AAM were
dissolved in 50 mL of deionized water. The first syringe was prepared
following the aforementioned protocol, while the second syringe was added with
565 $\mu$L of APS solution and 478 $\mu$L calcium sulfate solution (15% w/v).
The precursor solutions were quickly syringe-mixed and immediately poured into
80 $\times$ 20 $\times$ 3 mm3 rectangular acrylic molds covered with PMMA on
the two sides, and then kept at room temperature for 24 hours to complete the
reaction. To prepare the PAAm-Chitosan TEA, acrylamide and chitosan powders
were first dissolved in 0.2 M acetic acid at 3.3 mol/L and 2.5%, respectively.
MBAA was then added to the AAm-chitosan solution at 0.0006:1 the weight of
acrylamide to complete the polymer precursor solution. To prepare a gelling
solution to crosslink the chitosan, 0.1 M Na2HPO4 and 0.1 M NaH2PO4 were first
mixed with a volume ratio of 50:3. Sodium bicarbonate was then added to the
solution at a concentration of 0.306 M. A mass fraction of 6.6% APS was later
added to the solution as an initiator. Both solutions were degassed, quickly
mixed at 3:2 volume ratio (polymer precursor to gelling solution) using
syringes and injected into a mold with substrates of choice for overnight
gelation. To prepare the regular PAAm-alginate and PAAm chitosan gels, we
follow the same procedure except for injecting the mixed solution into the
acrylic molds covered by glass sheets on the two sides followed by the
gelation process.

Preparation of the bridging polymer solutions. This study tested two types
bridging polymers: chitosan and gelatin. To prepare the chitosan solutions of
2%, 1%, and 0.5% w/v, 50 mL of deionized water was added with chitosan powders
of 1, 0.5, 0.25 g, respectively. 400 $\mu L$ of acetic acid was also added for
a final pH of 4.5. The mixture was stirred overnight to form a homogeneous
solution and then kept at 4∘C before use. To prepare the gelatin solution of
2% w/v, 1g of gelatin powder was dissolved in 50 mL of deionized water. The
solution was stirred in a water bath at 37${}^{\circ}C$ for 30 mins before
use.

Adhesion test. The TEA gels were prepared with a length of 80 mm (or 40 mm),
width of 20 mm, and thickness of 3 mm. To test their adhesion on gels and VHB
elastomers, the surface of TEA was treated with the bridging polymer solution
of 0.25 $\mu$L/mm2, and then immediately covered with the adherend. An instant
compression of 15% strain was applied to remove the excessive solution on the
interface. No prolonged compression was applied. For testing TEA adhesion on
tissue samples, bridging polymer solution with chemical reagents added was
applied to the tissue surface, followed by covering with the TEA gel. A
continuous compression of 15% strain over the whole course of adhesion
establishment was applied to the TEA-tissue sample. To measure the adhesion
energy, standard 180-degree peeling was performed based on on samples with 2
cm width. The tests were conducted using a universal testing machine (Instron
Model 5365) with 10N and 1kN load cells. Before test, PET backing is attached
to the samples using Krazy glue. In a typical test, the peeling force reaches
a plateau $F_{\mathrm{plateau}}$ once reaching the steady state process. The
adhesion energy is calculated as twice the plateau force divided by the sample
width, $G=2F_{\mathrm{plateau}}/w$. The loading rate was varied from 0.2 mm/s
to 40 mm/s. Given the rigid backing, the crack speed for 180-degree peeling is
half the loading rate.

Adhesion kinetics test. To characterize adhesion kinetics, nylon mesh of
different thicknesses (50 and 120 $\mu m$) were used to define the thickness
of the bridging polymer solution $h_{\mathrm{sol}}$ following a previously
reported protocol[15]. We first immersed the nylon mesh in the bridging
polymer solution, and then removed the excessive solution on the surface
before applying it to the interface between two pieces of hydrogels. We waited
for different time $t$ before measuring adhesion energy using T-peeling test
at a relative low crack speed $V_{\mathrm{crack}}=$ 0.25 mm/s. The adhesion
energy increases with $t$ and reaches the equilibrium value $G_{eq}$. The
adhesion energy versus waiting time was fitted by the function
$G=G_{eq}(1-e^{t/t_{1/2}})$, where the half time $t_{1/2}$ can be fitted.

Uniaxial tensile test. Samples with length of 40mm, width of 20mm, thickness
of 3mm were prepared and tested using the Instron machine with 1N and 1kN load
cells. The nominal stress is calculated using $\sigma=F/A$ where $F$ and $A$
are the measured force and cross-sectional area of the sample. The stretch is
calculated as $\lambda=\lambda_{1}/\lambda_{0}$ where $\lambda_{1}$ is the
current length and $\lambda_{0}$ is the initial length.

Atomic force microscopy. An atomic force microscope (JPK NanoWizard@3, Berlin,
Germany) was used to conduct nano-indentation tests. Rectangular silicon
cantilevers with 0.6 $\mu m$-in-diameter spherical beads attached as probes
were used (Novascan, IA, USA). Cantilevers with a nominal spring constant of
0.6 N/m were used for experiment. The cantilever spring constants were
determined using thermal noise method before the experiment. Hydrogel samples
were immersed in PBS for 3 hours before indentation to avoid fast swelling
during test. Then swollen hydrogels were glued to 35-mm Petri dishes and
immersed in PBS during the measurement. Hertzian contact model was used to fit
the indentation data and to calculate the Young’s modulus.

GPC test. Chitosan samples with a concentration of 1 mg/mL were dissolved in
an aqueous acetate buffer (0.25 M acetic acid, 0.25 M Na acetate) overnight,
with stirring. They were filtrated (0.45 $\mu$ m) prior to be transferred to
the autosampler vials. 100 $\mu$L injection was analyzed using a SEC column
OHpak SB-805 HQ (8 mm ID x 300 mm L) at a flow rate of 0.3 mL /min of the
eluting phase (same as the one in which samples were dissolved). The signal of
the RI detector was used for calculation of the molecular weights and PDI
against a calibration curved constructed with pullulan standards of narrow
polydispersity and molecular weight in the 5.9 to 788 kDa range.

Confocal microscopy. Hydrogel samples with diffused bridging polymers were
imaged with a confocal microscope (LSM 710, Zeiss). Hydrogel samples with
diffused FITC-chitosan were cut into thin slices by a blade and transferred to
a 35 mm Petri dish with coverslip bottom (MatTek, P35G-0-10-C). The polymer
network was imaged with 10X and 20X objective lenses. Axiovert A1 inverted
microscope (Zeiss) equipped with a motorized stage was used to obtain
fluorescent signals at multiple locations. Light intensity was maintained the
same for all the samples.

Patterned TEA. A PTFE tape of thickness 0.1mm was designed into different
shapes and then adhered to a glass plate. A rectangular acrylic mold was
placed on top of patterned substrate. After injecting pre-gel solution, the
mold was covered with another glass plate.

TEA-based wound patch. A PTFE tape of thickness 0.1mm was designed into a
circle of diameter 1.5 cm, and then adhered to a glass plate. A rectangular
acrylic mold was placed on top of patterned substrate. After injecting pre-gel
solution, the mold was covered with another glass plate. Biopsy punch was used
to generate circular wounds on rat skin (diameter of 6mm) and porcine stomach
(diameter of 1cm). When de-molded, the TEA gel was applied to wounds on rat
skin and porcine stomach with 2% chitosan and EDC/NHS reagents. A prolonged
compression for 3 mins was then applied by hand. The size of the circular
region can be tailored to suit different sizes of the wounds.

TEA-based fluidic channels. A PTFE tape of rectangular (or ‘S’-shape) was
attached to a glass plate. Pre-gel solution is injected into the mold for
reaction to complete. Once de-molded, we use hole punch of diameter 2 mm to
create an inlet hole and an outlet hole, and then attach it to a targeted
surface using chitosan as bridging polymer. If the targeted surface is tissue,
we also added chemical reagents to form covalent bonds between chitosan and
tissue surfaces. After the adhesion is established, we further insert two soft
tubes into the inlet and outlet, sealed using Crazy glue, and slowly inject
liquid to cleave the weak adhesion by slip linkages. The fluidic channel is
formed once the whole weak interface is separated.

TEA-based Drug eluting devices. A SN PAAm TEA gel was first prepared on a
glass mold substrate patterned with a circular PTFE film of diameter 2 cm. It
was then adhered to a regular gel which was glued onto an acrylic sheet. The
circular low-adhesion region served as local depot to which drugs will be
injected. The albumin-FITC (A9771, Sigma-Aldrich) was used as a model drug and
was dissolved in PBS at 10 mg/mL to get the drug solution. At equilibrium of
hydrogels, different volume of drug solution (0.2 mL and 0.5 mL) was slowly
injected into the weak interface, resulting in local drug punches inside the
hydrogel. After the injection, the whole device was immersed in 200 mL PBS. At
determined time points, the fluorescence intensity of the solutions was
measured using a BioTek Synergy HTX multi-mode microplate reader ($\lambda$ex
= 485/20 nm, $\lambda$em = 528/20 nm). A standard calibration curve was made
in order to calculate the concentration of released drug.

TEA-based reconfigurable soft actuators Individual actuator module was
prepared using a molding process. SN PAAm pregel solution was injected to a 3D
printed mold made of PLA (Fig S6). The mold was then covered by a PMMA sheet.
Once the reaction is completed and demolded, the actuator module readily
carries dangling chains on its surface. Different modules were connected by
applying bridging polymer chitosan to the interface. To separate the connected
modules, a small force with a low separate rate can be applied. The separated
modules can be reconnected following reapplying bridging polymers to the
interface.

##  1 Supplementary Note 1: TEA gel prepared by heterogeneous polymerization.

We consider here the heterogeneous structure of the TEA gel imparted by the
hydrophobic mold substrate. The phenomenon was first discovered by Gong et
al.[12, 13, 35], who proposed that the hydrophobic mold substrate suppresses
the polymerization of the precursor solution near the substrate-solution
interface due to the increased interfacial tension. When the pre-gel solution
starts polymerizing on the low-surface-tension mold such as PMMA, the
interfacial tension between the hydrophobic mold surface and the liquid
solution $\gamma_{sl}$ increases with the increased polymer fraction. To
minimize the free energy of the whole system, the mold surface repulses the
polymers to create a depletion layer of thickness $\xi_{dp}$, where the
polymer fraction is significantly reduced compared to that in the bulk. As the
polymerization continues, the polymer chains in the bulk start to entangle and
from a crosslinked network with the bulk network elasticity $E_{bulk}$. Within
the depletion layer, however, the crosslinking process is influenced due to
the extremely low polymer fraction, resulting in a layer of dangling chains of
thickness $h_{dc}$. We shall consider $\xi_{dp}$ and $h_{dc}$ to be
equivalent. In equilibrium, the increased interfacial energy
$\Delta\gamma_{sl}$ equals the work done against $E_{bulk}$ over the distance
$\xi_{dp}$, leading to the relation $\Delta\gamma_{sl}\sim E_{bulk}\xi_{dp}$
or $\Delta\gamma_{sl}\sim E_{bulk}h_{dc}$. By assuming $\gamma_{sl}$ changes
negligibly with $E_{bulk}$ when $C$ is varied, and all the crosslinkers
effectively contribute to load-bearing chains, it leads to the scaling
relation: $h_{dc}\sim 1/E_{bulk}\sim 1/C$, which is plotted in Fig 2b with the
estimated $h_{dc}$ values.

Recently, several results suggested that the inhibition of free-radical
polymerization by trapped oxygen on hydrophobic substrates may play
significant roles in the heterogeneous polymerization[14, 20]. Despite the
debates on the underlying mechanisms, the substrate effect effectively
produces heterogeneous structure of a TEA gel that comprises a layer of
branched dangling chains spanning a thickness of $h_{dc}$ near the surface,
and a homogeneously crosslinked bulk network that is affected minimally by the
substrate effect of thickness $h-2h_{dc}$.

### Estimation of $h_{\mathrm{dc}}$

We propose a simple model to characterize the value of $h_{\mathrm{dc}}$ of
TEA gels. The model considers the TEA gel has an effective modulus of
$E_{tea}$, composed of the bulk elastic modulus $E_{bulk}$ and the dangling
chain elastic modulus $E_{dc}$. It assumes that the bulk of the TEA gel the
same elasticity $E_{bulk}$ as that of a regular gel polymerized on glass
$E_{reg}$, while the dangling chains has a modulus $E_{dc}\approx 0$,
yielding:

$\frac{2h_{\mathrm{dc}}}{h}=1-\frac{E_{\mathrm{tea}}}{E_{\mathrm{reg}}}$ (1)

where $E_{\mathrm{tea}}$ and $E_{\mathrm{reg}}$ can be measured from uniaxial
tensile tests. We fit the data with $5\%$ of strain to the linear-elastic
model, which is degenerated to from the neo-Hookean model at the small strain
limit:

$\sigma_{\mathrm{linear}}=E(\lambda-1)$ (2)

where $\sigma$ is the nominal stress and $\lambda$ is the stretch. We also fit
the data to another hypereasltic material model, the incompressible neo-
Hookean model:

$W_{\mathrm{neo-Hookean}}=\frac{E}{6}(I_{1}-3)$ (3)

where $I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}$. The Neo-Hookean
model assumes that the polymer chains follow Gaussian distribution and are
free of entanglements. Under uniaxial tensile test, the principle stretches:
$\lambda_{1}=\lambda$, $\lambda_{2}=\lambda_{3}=1/\sqrt{\lambda}$. The nominal
stress-stretch curves derived from the models are:

$\sigma_{\mathrm{neo-Hookean}}=\frac{E}{3}(\lambda-\lambda^{-2})$ (4)

We plot representative stress-stretch curves from uniaxial tensile tests in
Extended Data Fig 1d and along with the fitted models. The Neo-Hookean model
underestimates the slopes of the curves at $\lambda\rightarrow 1^{+}$ for all
samples. The linear model that fit the data within $\lambda$ from 1 to 1.05 is
used to estimate the moduli of TEA and regular gels with varying values of
$C$, plotted in Extended Data Fig 1e. With the values of $E_{\mathrm{tea}}$
and $E_{\mathrm{reg}}$, we estimated the value of $h_{dc}$ using the Eqn. 1
(Fig 2c). Note that our estimation of $h_{\mathrm{dc}}$ is much larger than
those estimated by observing force-displacement curves from nano-indentation
tests[14][36], which may attribute to the different values of $C$ used in the
studies. The projection of our data yields a similar estimation of
$h_{\mathrm{dc}}$ at the same level of $C$ as in Simic et al[14] ($C\approx
1\%$).

### Estimation of bulk mesh size

Assuming the bulk network of the TEA gel and that of a regular gel have affine
structures, so their shear modulus can be expressed as

$\mu=\frac{E}{3}=\nu k_{B}T$ (5)

where the density of the network strands
$\nu=\frac{c_{\mathrm{aam}}}{m_{\mathrm{mono}}N_{c}}N_{A}$. The concentration
of the aam solution is $c_{\mathrm{aam}}=6.76\mathrm{g}/50\times
10^{-6}\mathrm{m}^{3}=1.352\times 10^{5}$ g/m3. The molar mass of aam is
$m_{\mathrm{mono}}\approx 71$ g/mole. Using the shear modulus of the regular
PAAm gels for $C=0.024\%$, the number of monomers between two crosslinkers are
approximated as:

$N_{c}=\frac{c_{\mathrm{aam}}}{m_{\mathrm{mono}}\mu}N_{A}=\frac{c_{\mathrm{aam}}}{m_{\mathrm{mono}}\mu/k_{B}T}N_{A}\approx
2348$ (6)

The mesh size is thus estimated as:

$\xi=aN_{c}^{1/2}=0.1\times N_{c}^{1/2}\approx 5\mathrm{nm}$ (7)

where $a=0.1$ nm is the length of a single bond.

##  2 Supplementary Note 2: Thermally activated processes of chain slippage
and ruptures

In experiments, we observed that $G^{1/2}$ for TEAs varies logarithmically
with $V_{\mathrm{crack}}$ (Figure 2(a)) if $h_{pen}/h_{dc}\ll 1$, reminiscent
of the dynamic adhesion of cell-cell interface[17], elastomer[18], and other
bonds[19] due to thermally activated bond breaking. To rationalize the
results, we adopt a kinetic theory which considers the linkage dissociation as
thermally-activated processes[18, 37]. The activation for the dangling chain
to slip from the bridging network is assumed to decrease by the applied force
(Fig 2f). Using the concept of mechanochemistry, the rate of dissociation of
linkage $i$ can be expressed as:

$r_{i}(t)=-\frac{dN_{i}}{dt}=r_{i}N_{i}$ (8)

where $i$ can be stitch or slip. $N_{i}$ is the areal density of linkage $i$,
and $r_{i}$ is the rate constant for linkage dissociation, and is assumed to
be dependent on the force applied to the chains $F$ via the Arrhenius law:

$r_{i}=r_{0}\exp{\left(\frac{l_{a}F}{k_{B}T}\right)}$ (9)

where $r_{0}=1/\tau_{-}$ is the rate constant of the linkage dissociation
without adding any force, $\tau_{-}$ is the intrinsic relaxation time of the
slip linkage, $k_{B}T$ is the temperature in the unit of energy, $F$ is the
applied force to break a linkage. Note that we have assumed the linkage
dissociation process is irreversible, thus the re-association of the linkage
is not accounted for. We further consider an individual linkage has a spring
constant $k_{i}$ and is stretched at a fixed velocity $V$ over an averaged
bond survival time $\bar{t_{i}}$. Thus, the averaged force that an individual
linkage can bear is given by:$\bar{F_{i}}=k_{i}V\bar{t_{i}}$.

For simplicity, the average linkage survival time $\bar{t_{i}}$ is estimated
by the most probable survival time $t_{i}^{*}$ corresponding to the maximum of
the dissociation rate[16]: $d^{2}N/dt^{2}=0$. Therefore, $\bar{t_{i}}$ can be
expressed as:

$\bar{t_{i}}\approx
t_{i}^{*}=\frac{k_{B}T}{k_{i}Vl_{a}}\ln{\left(\frac{l_{a}k_{i}V\tau_{-}}{k_{B}T}\right)}$
(10)

The energy released upon breaking linkage $i$ can be expressed as:

$e_{i}=\frac{\bar{F_{i}}^{2}}{2k_{i}}=\frac{V^{2}\bar{t_{i}}^{2}k_{i}}{2}$
(11)

Substituting Eqn.10 into 11 and multiplying $e_{i}$ with the number density of
linkage $N_{i}$ across an interface yields the energy released by advancing a
unit area of an interface held by an array of slip linkages, namely, the
adhesion energy:

$G_{i}=\left(\frac{N}{2k_{i}}\right)\left(\frac{k_{B}T}{l_{a}}\right)^{2}\left[\ln{\left(\frac{l_{a}k_{i}V\tau_{-}}{k_{B}T}\right)}\right]^{2}$
(12)

We further assume that the crack geometry remains invariant during the steady
state peeling process, so $V_{\mathrm{crack}}\sim V$. We express
$\tau_{-}=h/(k_{B}T)\exp\left[E_{i}/(k_{B}T)\right]$, where $E_{i}$ and $h$
are the activation energy of linkage $i$ and Planck constant, respectively.

To calibrate $N_{i}$, we deduced that it is limited by the area density of the
chitosan chains in the bridging network $N_{i}\approx N_{\mathrm{chi}}$, since
its size is presumably larger than that of hydrogel network and the spacing
between dangling chains[7]. If true, Eqn. 12 suggests that in the absence of
stitch linkage, $(G/N_{chi})^{1/2}$ should only depend on
$V_{\mathrm{crack}}$. Since we cannot obtain a direct measurement of
$N_{chi}$, we adopt an approximation using the areal density of chitosan
chains homogeneously dispersed in the solution using the chitosan polymer
concentration $c_{\mathrm{chi}}$ (w/v%) through:

$N_{chi}\sim\left(\frac{c_{chi}}{M_{w}}N_{a}\right)^{2/3}$ (13)

Where $M_{w}\approx 300$kDa, estimated from our GPC test (Fig S7)
$N_{a}=6\times 10^{23}$ is the Avogadro number. Although Eqn. 13 provides a
rough estimation of $N_{\mathrm{chi}}$, they lead to a reasonable collapse of
our data following the reformulation $(G/N_{chi})^{1/2}$ using different
values of $c_{chi}$ = 2%, 1%, and 0.5% g/mL (Supplementary Fig 2a and b),
confirming the validity to approximate $N_{i}$ by $N_{chi}$.

Finally, the bridging network may form slip, stitch, or the combination of the
two linkages when engaging with the TEA gel, yielding the expression for the
total adhesion energy[18]:

$G=\sum_{i}G_{i}$ (14)

with

$G_{i}=\left(\frac{N_{i}}{2k_{i}}\right)\left(\frac{k_{B}T}{l_{a}}\right)^{2}\left[\ln\left(\frac{V_{\mathrm{crack}}k_{i}l_{a}h}{(k_{B}T)^{2}}\right)+\frac{E_{i}}{k_{B}T}\right]^{2}$

Given that an inextensible backing film was attached to the gel,
$V_{\mathrm{crack}}$ is determined as the half of the peeling rate in the
T-peel test. We then fit the model to the experimental data to estimate
$E_{i}$ and $k_{i}$.

##  3 Supplementary Note 3: Others

### Estimated and fitted values of the effective diffusion coefficients of
bridging polymers in the dangling chain layer

According to the Rouse model, the diffusion coefficient of the polymer in
water is given by

$D=\frac{k_{B}T}{N\eta b}$ (15)

taking $k_{B}T=4.11\times 10^{-21}$J, $N=1000$ the number of units per polymer
chain, $\eta=10^{-3}$ Pa$\times$S, $b=1$ nm the length of a repeating unit of
chitosan, it gives $D_{\mathrm{eff}}\approx 5\times 10^{-12}$ m2s-1.

### Alternative approaches to create slip-linkage topology

Other approaches exist to create the similar chain-network topology of the
slip linkage, such as by placing uncrosslinked polymer chains between hydrogel
networks[7, 38]. The polymer chains can diffuse into the pre-formed gel
networks to form slip entanglement. However, the highly permeable hydrogel
would promote the diffusion of polymer chains into the gel matrix, greatly
reducing the number of polymer entanglements on the interface.

### Summary of topological linkages

Table 1:  Types of topological linkages, their constituents (adhesive matrix, targeted surface, and bridging polymer), and the associated data.  Linkage types | Adhesive matrix | targeted surface | Bridging polymer | Data
---|---|---|---|---
Slip-slip | SN TEA PAAm gel | SN TEA PAAm gel | Chitosan | Fig 2d,f, Sup Fig 2a,b
$\sim$ | $\sim$ | Gelatin | Fig 4d
Stitch-stitch | SN regular PAAm gel | SN regular PAAm gel | Chitosan | Fig 2d,f
Slip-stitch | SN TEA PAAm gel | SN regular PAAm gel | Chitosan | Fig 4c, Sup Fig 2e
Slip-bond | SN TEA PAAm gel | VHB elastomer | Chitosan | Fig 4c, Sup Fig 2d
SN TEA PAAm gel | Porcine skin | Chitosan+EDC/NHS | Fig 4e
Sitch-bond | SN regular PAAm gel | VHB elastomer | Chitosan | Fig 4c, Sup Fig 2d
SN regular PAAm gel | Porcine skin | Chitosan+EDC/NHS | Fig 4e

Slip or stitch linkages: To form the slip and stitch linkages, bridging
polymer chitosan or gelatin was directly applied to a gel with or without
dangling chain layer, and then covered with a permeable adherend.

Bond linkage: to form bond linkage between bridging chitosan network and
tissue surfaces, we utilize the amine groups on chitosan, which can be
covalently bonded to the carboxylic acid groups on tissue surfaces with EDC
and NHS as coupling reagents[11]: 30 mg of EDC and 30 mg of NHS were added
into 1 mL of the chitosan solution for forming covalent bonds with tissue
surfaces. To form the bond linkage between bridging polymers and VHB elastomer
surfaces, we utilize the Carbonyl bonds on the VHB, which can form imide bonds
with the amine group on the chitosan polymer at pH = 4. Besides, the ionic
bond formed between NH3+ of chitosan of pH $<$ 6.5 and COO- of the VHB of pH
$>$ 4.5 can also contribute to the bond linkage. We prepared chitosan of pH
4.5 and hydrogel of pH 7, thus the chitosan can form an interfacial bridging
network and can form both type of bonds with VHB surfaces[25].

Figure S1:  (a) Uniaxial cyclic tensile tests of a TEA and a regular gel made
of SN PAAm for $C=0.024\%$. (b) the same curves as (a) for $C=0.048\%$.
Uniaxial tensile test results of SN TEA gels with varying strain rates for (c)
$C=0.024\%$ and (d) $C=0.048\%$. (e) representative stress-stretch curves of
TEA (left) and regular (right) gels for different values of $C$ measured in
uniaxial tensile tests. (f) Measured elastic moduli by the linear model for SN
TEA and regular gels, plotted as functions of $C$. Figure S2:  (a) Schematic
showing the T-peeling test and the resulting force-displacement curves. The
adhesion energy is interpret as $G=2F_{plateau}/w$, with $w$ the out-of-plane
thickness of the sample. (b) $G^{1/2}$ by the slip-slip linkage plotted as a
function of $\ln V_{\mathrm{crack}}$ for $C=0.024\%$ and varying $c_{chi}$
($0.5\%,1\%$, and $2\%$ g/mL) (c) reformulated ($G/N_{chi})^{1/2}$) for
$C=0.024\%$, where $N_{chi}$ is estimated using Eqn 13. (d) Illustrations
showing the effects of entanglement length on the formation of slip and stitch
linkages. (e) Stitch- and slip-mediated adhesion on VHB surfaces. (f) Cyclic
peeling test on a TEA-regular interface at $V_{\mathrm{crack}}$=1mm/s. After
every peeling, a new regular gel adheres to the same TEA gel with newly
applied chitosan solution. The adhesion increases and stabilizes after 5
cycles. (g) Peeled regular gel and TEA gel after a peeling test. The interface
was bonded with a slip-stitch linkage using fluorescently labeled chitosan.
After peeling, the chitosan network remains on the regular gel, but remains
minimally in the TEA gel. Scale bars: 1mm. Figure S3:  (a) Illustrations
showing the total kinetics of stitch-mediated (top) and slip-mediated
interface (bottom) as two sub kinetic physical processes. (b) Illustration
showing the experimental procedure to characterize the adhesion kinetics of TA
and TEA. (c) Dimensionless adhesion $G/G_{\mathrm{eq}}$ as functions of
waiting time for TEA with $h_{\mathrm{dc}}=370\mu m$. In one case
$h_{\mathrm{sol}}=50\mu m$ was controlled by using nylon mesh of the same
thickness at the interface, while in the other case an initial compression
strain of $15\%$ was applied to the sample after applying adhesive solution
such that $h_{\mathrm{sol}}$ was not well-controlled. No prolonged compression
was applied. (d) Summarized $t_{1/2}$ of the slip and stitch-linkage mediated
adhesion as functions of $h_{\mathrm{sol}}$ and $h_{\mathrm{dc}}$. Together
plotted is the reported $t_{1/2}$ in Steck et al[15], wherein pH-responsive
polymer cellulose is used as the bridging polymer (closed markers). Error bars
and shaded area represent 95% confidence interval. Figure S4:  Adhesion
contrast between the slip and stitch regions in a patterned TEA for given $C$
and $h_{\mathrm{pen}}/h_{\mathrm{dc}}$. The slip linkage becomes the hybrid
linkage as $h_{\mathrm{pen}}/h_{\mathrm{dc}}$ gets closer to 1. Figure S5:
(a) Uniaxial tensile test results of a DN TEA and a DN regular PAAm-alginate
gels. (b) AFM test on the surface of the DN gels with and without EDTA
treatment on the surface. (c) Adhesion of PAAm-alginate gels made on different
mold surfaces on porcine skin. (d) Schematic illustrating the procedure for
creating spatially programmable adhesion of TEA A needle is used to penetrate
the hydrogel into the interface, or a tube was sandwiched between the two
gels. Next, liquid dye was injected through the needle or the tube. Finally, a
digital camera was used to capture the weak interface through the liquid dye.
(e) Patterned interface formed between a TEA PAAm-alginate gel and a regular
PAAm-alginate gel. The TEA gel is designed with designed dangling chain
regions of different shapes, resulting in weak interfaces of the identical
shape. Scale bar: 10 mm. (f) Mock drug is injected to the weak interface of a
TEA patch adhered to the wound of a porcine stomach. Instead, if a DN PAAm-
alginate TA was adhered to the wounded porcine stomach, strong adhesion to the
wound prevents injection of drug. Figure S6: 3D printed mold for creating soft
actuator modules. The mold parts are 3D-printed using PLA. The cover,
indicated in blue, is made of PMMA. Figure S7: GPC test results.