# Online parameter inference for the simulation of a Bunsen flame using
heteroscedastic Bayesian neural network ensembles

Maximilian L. Croci, Ushnish Sengupta & Matthew P. Juniper
Department of Engineering
University of Cambridge
Cambridge, United Kingdom
<EMAIL_ADDRESS>

###### Abstract

This paper proposes a Bayesian data-driven machine learning method for the
online inference of the parameters of a $G$-equation model of a ducted,
premixed flame. Heteroscedastic Bayesian neural network ensembles are trained
on a library of 1.7 million flame fronts simulated in LSGEN2D, a $G$-equation
solver, to learn the Bayesian posterior distribution of the model parameters
given observations. The ensembles are then used to infer the parameters of
Bunsen flame experiments so that the dynamics of these can be simulated in
LSGEN2D. This allows the surface area variation of the flame edge, a proxy for
the heat release rate, to be calculated. The proposed method provides cheap
and online parameter and uncertainty estimates matching results obtained with
the ensemble Kalman filter, at less computational cost. This enables fast and
reliable simulation of the combustion process.111Code available at:
https://github.com/nailimixaM/iclr-2021-baynne

##  1 Introduction

Simulation-based digital twins of physical systems are becoming cheaper to use
due to improvements in computer processing power and storage (Fuller et al.,
2020). Data-driven machine learning methods for the inference of digital twin
model parameters from experiments are made possible in part due to the
possibility of creating cheap synthetic data sets at scale. These methods can
also address the need to do bring down the computational cost of parameter
inference such that these models may be updated in real-time (online) using
the latest sensor observations.

In the design of modern rocket engines and gas turbines, simulation of the
combustion process (the flame) can be used to design out thermoacoustic
instabilities which would otherwise lead to catastrophic damage (Juniper &
Sujith, 2018). Thermoacoustic instabilities are driven by the coupling of the
heat release rate of the flame and the acoustics in the combustor (Strutt,
1878). It is therefore necessary to model the flame dynamics such that the
heat release rate is quantitatively accurate. The $G$-equation (Williams,
1985) is the kinematic model used in LSGEN2D (Hemchandra, 2009) to simulate a
ducted, premixed flame such as that from a Bunsen burner. By tuning the
parameters of the $G$-equation model to fit a Bunsen flame experiment, a
digital twin of the flame is created and its surface area variation in time,
which is a proxy for the heat release rate, can be calculated. The ensemble
Kalman filter (EnKF) is the current state-of-the-art, which iteratively
performs Bayesian inference of the $G$-equation parameters from $G$-equation
model forecasts generated in LSGEN2D and observations of the flame edge (Yu et
al., 2019). The model forecasting is expensive, however, which makes the EnKF
too expensive to be used online. We propose to infer online the $G$-equation
parameters of Bunsen flame experiments using a heteroscedastic Bayesian neural
network ensemble (Sengupta et al., 2020a; b). In an expensive offline step,
the ensemble learns a surrogate for the Bayesian posterior distribution of the
parameters given observations of the flame front from a library of pairs of
$G$-equation parameters and corresponding flame front shapes generated in
LSGEN2D. The ensemble can then be used to infer online the parameters of the
model for simulation of the Bunsen flames.

##  2 Bunsen flame experiment and simulations

###  2.1 Bunsen flame experiment

A Bunsen burner is placed inside a transparent duct and a high-speed camera is
used to take images of the Bunsen flame at a frame rate of $f_{s}=2500$ frames
per second and a resolution of $1200\times 800$ pixels. Speakers force the
flame at frequencies in the range 250 Hz to 450 Hz. The gas composition
(methane, ethene and air) and flow rate are varied using mass flow
controllers. By varying the forcing frequency and amplitude and gas
composition and flow rate, flames with different aspect ratios, propagation
speeds and degrees of cusping of the flame front are observed. In some cases,
the flame front cusping leads to pinch-off at the flame tip. For each of the
270 different flame operating conditions, 500 images are taken.

The flame images are processed to find a single-valued discretisation of the
flame front, $x=f(y)$. First, the pixel intensities are thresholded and a
position $x$ for every vertical position $y$ is found by weighted
interpolation of the thresholded pixels, where the weights are the pixel
intensities. Next, splines with 28 knots are used to smooth the $(x,y)$
coordinates. Each flame image is therefore converted into a $90\times 1$
vector of flame front $x$ coordinates x (as the $y$ coordinates are the same
for all flames, they are discarded). Observation vectors z are created by
stacking 10 subsequent x vectors. These observation vectors are used for
inference with the neural networks. All 500 images of each Bunsen flame are
processed in this way.

###  2.2 Flame front model

In this paper we use a kinematic model of the flame front as a boundary
between reactants and products (see Figure 1). The flame front is defined to
be the $G=0$ contour of a scalar field $G(x,y,t)$. Regions of negative and
positive $G$ correspond to unburnt and burnt gases respectively (the magnitude
of $G$ does not have a useful meaning). The position of the flame front in
space and time is governed by:

$\frac{\partial G}{\partial t}+\textbf{v}\cdot\nabla G=s_{L}|\nabla G|,$ (1)

where v is a prescribed velocity field and $s_{L}$ is the laminar flame speed:
the speed at which the flame front propagates normal to itself into the
reactants. The flame speed $s_{L}=s_{L}^{0}\left(1-\mathcal{L}\kappa\right)$
is a function of the unstretched (adiabatic) flame speed $s_{L}^{0}$, the
flame curvature $\kappa$ and the Markstein length $\mathcal{L}$, and is
insensitive to pressure variations. The unstretched flame speed $s_{L}^{0}$
depends only on the flame chemistry. The velocity field
$\textbf{v}=u^{\prime}\textbf{i}+\left(V(x)+v^{\prime}\right)\textbf{j}$
comprises a parabolic base flow profile
$V(x)=V(1+\alpha(1-2(\frac{x}{R})^{2}))$ and superimposed continuity-obeying
velocity perturbations
$u^{\prime}(x,y,t)=V\epsilon\sin(\text{St}(\frac{K}{R}y-t))$ and
$v^{\prime}(x,y,t)=-\frac{V\epsilon
K\text{St}}{R}x\cos(\text{St}(\frac{K}{R}y-t))$ where $\alpha$ determines the
shape of the base flow profile ($\alpha=0$ is uniform flow, $\alpha=1$ is
Poiseuille flow), $\epsilon$ is the amplitude of the vertical velocity
perturbation with phase speed $V/K$, $\text{St}=2\pi fR\beta/V$ is the
Strouhal number with forcing frequency $f$ and flame radius $R$, and $\beta$
is the aspect ratio of the unperturbed flame. The parameters
$K,\epsilon,\mathcal{L},\alpha,\text{St}$ and $\beta$ are tuned to fit an
observed flame shape.

Figure 1: (Left) Image of a Bunsen flame. (Middle) $G$-equation model of the
flame front. (Right) Simulated flame in LSGEN2D.

###  2.3 Simulated flame front library

A library of simulated flame fronts with known parameters
$K,\epsilon,\mathcal{L},\alpha,\text{St}$ and $\beta$ is created for neural
network training. The parameter values are sampled using quasi-Monte Carlo
sampling to ensure good coverage of the parameter space. The parameters are
sampled from the following ranges: $0.0<K\leq 1.5$, $0.0<\epsilon\leq 1.0$,
$0.02\leq\mathcal{L}\leq 0.08$, $0.0\leq\alpha\leq 1.0$, $2.0\leq\beta\leq
10.0$ and $0.08\leq f/f_{s}\leq 0.20$. The values of St are calculated by
additionally sampling $0.002\leq R\leq 0.004$ m and $1\leq V\leq 5$ m/s and
calculating $\text{St}=2\pi fR\beta/V$. The parameters are sampled 8500 times,
normalised to between 0 and 1 and recorded in target vectors
$\\{\textbf{t}=[K,\epsilon,\mathcal{L},\alpha,\text{St},\beta]\\}$.

For each of the 8500 unique parameter configurations, LSGEN2D iterates the
$G$-equation model of the flame front until it converges to the corresponding
forced cycle. For each of the 200 forced cycle $G$ field states produced by
LSGEN2D, the flame front is found by interpolating the $G$ values. This
results in a $x=f(y)$ discretisation and the vectors x of $x$ coordinates are
recorded. Observation vectors z are created by stacking 10 subsequent x
vectors. There are 200 observation vectors created from every cycle, resulting
in a library of $1.7\times 10^{6}$ observation-target parameter pairs
$\\{(\textbf{z},\textbf{t})\\}$. This library is split 80%-20% into training
and testing data sets.

##  3 Inference using heteroscedastic Bayesian neural network ensembles

We assume the posterior probability distribution of the parameters given the
observations can be modelled by a neural network:
$p_{\theta}(\textbf{t}|\textbf{z})$ with its own parameters $\theta$. We
assume this posterior distribution has the form:

$p_{\theta}(\textbf{t}|\textbf{z})=N\left(\textbf{t};\bm{\mu}(\textbf{z}),\bm{\Sigma}(\textbf{z})\right),$
(2)

where $\bm{\Sigma}(\textbf{z})=\text{diag}(\bm{\sigma}^{2}(\textbf{z}))$. This
encodes our assumption that the parameters are all mutually independent given
the observations z. The architecture of a single neural network is shown in
Figure 2. Each neural network comprises an input layer, four hidden layers
with ReLU activations and two output layers: one for the mean vector
$\bm{\mu}(\textbf{z})$ and one for the variance vector
$\bm{\sigma}^{2}(\textbf{z})$. The output layer for the mean uses a sigmoid
activation to restrict outputs to the range $(0,1)$. The output layer for the
variance uses an exponential activation to ensure positivity. An ensemble of
$M=20$ such neural networks is initialised, each with unique initial weights
$\bm{\theta}_{j,anc}$ sampled from a Gaussian prior distributions according to
He initialisation (He et al., 2015).

For a single observation z, the $j$-th neural network in the ensemble produces
a sample of the posterior with an associated estimate of the aleatoric noise
in the observations: $\bm{\mu}_{j}(\textbf{z}),\
\bm{\sigma}_{j}^{2}(\textbf{z})$. This is achieved by using the loss function
$\mathfrak{L}_{j}$:

$\mathfrak{L}_{j}=\left(\bm{\mu}_{j}(\textbf{z})-\textbf{t}\right)^{T}\bm{\Sigma}_{j}\left(\textbf{z}\right)^{-1}\left(\bm{\mu}_{j}(\textbf{z})-\textbf{t}\right)+\log\left(|\bm{\Sigma}_{j}\left(\textbf{z}\right)|\right)+\left(\bm{\theta}_{j}-\bm{\theta}_{anc,j}\right)^{T}\bm{\Sigma}_{prior}^{-1}\left(\bm{\theta}_{j}-\bm{\theta}_{anc,j}\right).$
(3)

The loss function comprises the negative log of the Gaussian likelihood
function (probability of the observations given the targets, first two terms)
and a regularising (penalty) term. By regularising about parameter values
drawn from a prior distribution, the NNs produce samples from the posterior
distribution. This is called randomised maximum a-posteriori (MAP) sampling
(Pearce et al., 2020).

Once converged, the prediction from the ensemble for an observation z is
therefore a mixture of $M$ Gaussians each centered at their respective means
$\bm{\mu}_{j}(\textbf{z})$. This mixture is approximated by a single
multivariate Gaussian posterior distribution
$p(\textbf{t}|\textbf{z})\approx\mathcal{N}(\textbf{t};\bm{\mu}(\textbf{z}),\bm{\Sigma}(\textbf{z}))$
with mean $\bm{\mu}(\textbf{z})=\frac{\Sigma_{j}\bm{\mu}_{j}(\textbf{z})}{M}$
and covariance
$\bm{\Sigma}(\textbf{z})=\text{diag}(\bm{\sigma}^{2}(\textbf{z}))$ where
$\bm{\sigma}^{2}(\textbf{z})=\frac{\Sigma_{j}\bm{\sigma}_{j}^{2}(\textbf{z})}{M}+\frac{\Sigma_{j}\bm{\mu}_{j}^{2}(\textbf{z})}{M}-\left(\frac{\Sigma_{j}\bm{\mu}_{j}(\textbf{z})}{M}\right)^{2}$
following similar treatment in Lakshminarayanan et al. (2017). This is done
for every observation vector z. The posterior distribution
$p(\textbf{t}|\textbf{z}_{i})$ with the smallest total variance
$\sigma^{2}_{i,\text{tot}}=||\bm{\sigma}^{2}(\textbf{z}_{i})||_{1}$ is chosen
as the best guess to the true posterior. The $M$ parameter samples from the
chosen posterior are used for re-simulation, which allows us to check the
predicted flame shapes and to calculate the normalised area variation over one
cycle.

Figure 2: Architecture of a single neural network. The input and hidden layers
have 900 nodes each, while the output layers have 6 nodes each. All layers are
fully connected (FC). Rectified Linear Unit (ReLU) activation functions are
used for the hidden layers and sigmoid and exponential (Exp) activation
functions are used for the mean and variance ouput layers respectively.

##  4 Results

The ensemble is trained for 5000 epochs and evaluated on the Bunsen flame
image data. Parameter predictions for an observation vector z take
$O(10^{-4})$ seconds on an Nvidia P100 GPU. Figure 3 show the results of
inference and simulation of two different Bunsen flames. The BayNNE predicts
flame shapes in good agreement with the experiments: the root mean square
distances between the simulated and Bunsen flame shapes of 5 different Bunsen
flames ranges from 0.017 units to 0.024 units, with mean 0.019 units (where
the flame radius is 1 unit). These results show that the BayNNE’s parameter
estimates match those of the EnKF and require $O(10^{8})$ less computing
power.

The uncertainty in the BayNNE’s predictions is greater than that of the EnKF
due to the calculated posterior being the probability of the parameters given
10 flame images, whereas the EnKF considers all 500 flame images of the Bunsen
flame. Unfortunately, it is not possible to combine the BayNNE’s parameter
estimates without knowledge of the probability distribution between the
observation vectors. Any two observation vectors are not independent as
knowledge about the first restricts our expectation of the second to a likely
set of forced cycle states. Future work will address this limitation by using
alternative neural network architectures, such as long-short term memory
networks (Hochreiter & Schmidhuber, 1997), that are better suited to time-
series data.

##  5 Conclusions

This work proposes a method for inferring the parameters of the $G$-equation
model of a Bunsen flame for simulation purposes. Bayesian inference of the
parameters and their uncertainties is performed using heteroscedastic Bayesian
neural network ensembles. The neural networks are trained on a library of
synthetic flame front observations created using LSGEN2D. Once trained, the
Bayesian neural network ensemble accurately predicts the parameters and
uncertainties from just 10 images of the Bunsen flame. The simulated flames
with parameters predicted by the ensemble agree with the experiments, and the
surface area variation, which is a proxy for the heat release rate, is
calculated. A quantitatively accurate digital twin of the Bunsen flame is
therefore created. Future work will focus on improving the parameter and
uncertainty estimates by leveraging alternative neural network architectures.

Figure 3: Results of inference using the BayNNE method compared to the EnKF
method for the simulation of two different Bunsen flames. The flame images are
preprocessed to find the flame front and then used for inference. The green
and orange bands in the EnKF and BayNNE predicted flame shape plots are
regions of high likelihood. The normalised area variations over one period of
the simulated flames show good agreement with the experiments.

## Disclosure of Funding

This project has received funding from the UK Engineering and Physical
Sciences Research Council (EPSRC) award EP/N509620/1 and from the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement number 766264.

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##  Appendix A Supplementary material: Hyperparameter settings

Table 1: Hyperparameter settings. Hyperparameter | Value
---|---
Training |
Train-test split | 80:20
Batch size | 2048
Epochs | 5000
Optimiser | Adam
Learning rate | $10^{-3}$
Architecture |
Input units | 900
Hidden layers | 4
Units per hidden layer | 900
Output layers | 2
Units per output layer | 6
Ensemble size | 20


# Mechanism of pressure sensitive adhesion in nematic elastomers

Hongye Guo<EMAIL_ADDRESS>Mohand O. Saed and Eugene M. Terentjev
<EMAIL_ADDRESS>
###### Abstract

Nematic liquid crystal elastomers (LCEs) have anomalously high vibration
damping, and it has been assumed this is the cause of their anomalously high
pressure-sensitive adhesion (PSA). Here we investigate the mechanism behind
this enhanced PSA by first preparing thin adhesive tapes with LCE of varying
crosslinking density, characterizing their material and surface properties,
and then studying the adhesion characteristics with a standard set of 90-deg
peel, lap shear, and probe tack tests. The study confirms that the enhanced
PSA is only present in (and due to) the nematic phase of the elastomer, and
the strength of bonding takes over 24 hours to fully reach its maximum value.
Such a long saturation time is caused by the slow relaxation of local stress
and director orientation in nematic domains after pressing against the
surface. We confirm this mechanism by showing that a freshly pressed and
annealed tape reaches the same maximum bonding strength on cooling, when the
returning nematic order is forming in its optimal configuration in the pressed
film.

##  1 Introduction

Liquid crystalline elastomers (LCEs) are a class of functional materials that
attract attention in many different applied fields. Originally, the main
interest was in utilizing LCEs as soft actuators with a very high stroke, and
actuation stress capable of reaching the MPa scale.[1, 2] The discovery of
‘soft elasticity’ in nematic LCEs has led to another strand of potential
applications utilizing their anomalously high viscoelastic dissipation and
vibration damping [3, 4], although it has recently been suggested that the
enhanced damping is not actually related to the macroscopic soft elasticity
but is controlled by the high-frequency rotation of nematic director on the
microscopic, sub-domain scale [5]. Recently, there have been several reports
demonstrating a new effect of an enhanced pressure sensitive adhesion (PSA) on
the surface of nematic LCEs [6, 7], alleging a direct link between the
viscoelastic dissipation in the material, measured by the loss factor
$\tan\delta(\omega)$, and the effective energy of surface adhesion $\gamma$.
That is, similar to many other polymer-based PSA systems [8, 9, 10, 11, 12],
the strength of adhesion is dominated by a physical factor additional to the
‘ordinary’ surface energy; this factor is dominated by the integral of
$\tan\delta(\omega)$ over the range of frequencies determined by the pull-off
speed [13, 14]. As the mechanisms of PSA are generally not very well
understood, even in spite of several good theoretical approaches [15, 16, 17],
and since in LCEs we have an additional strong effect of nematic director
relaxation, here we investigate the enhanced LCE adhesion in some detail in
order to ascertain its mechanism.

There are several factors one has to consider in the PSA process, and in
considering them we wish to exclude the chemically-defined surface tension
$\gamma_{0}$ from this list: we will work with a particular fixed chemistry of
an elastomer, so that $\gamma_{0}$ remains constant (which we measure from the
contact angle analysis of water droplets to be $\gamma_{0}=42$ mN/m). The
methods of estimation of surface energy of polymers have evolved since Owens
and Wendth [18], but they all invariably give the range of $\gamma_{0}$
between 20 and 60 mN/m (with water having the high $\gamma_{0}=$72 mN/m). This
is about 4 orders of magnitude lower than the measured adhesion energy $W$
(often called the ‘work of adhesion’) of ‘sticky surfaces’ in many cited
studies, which is in the range of 100’s N/m, to 1000’s N/m [11]. The first key
factor is the roughness of the polymer surface (assuming for simplicity that
the pressing surface is rigid and smooth), which has been extensively studied
over the years [19, 20]. As we will be working with a natural polydomain
nematic LCE, it turns out that its free surface has an unavoidable roughness
on the micron scale, reflecting the equilibrium nematic domain structure that
causes the corresponding local deformation pattern [21, 22, 23]. We just
mentioned the assumption that the pressing surface is smooth, which is of
course incorrect, and the famous Dahlquist criterion has been developed a long
time ago [24] to relate the natural roughness of that surface to the adhesion
strength: it is based on how well the PSA polymer penetrates the rigid rough
surface profile, and shows that the typical rubber modulus needs to be below
0.1 MPa for a typical solid surface with single-micron roghness topography.
Here is another common confusion, in assuming that better penetration
increases the surface contact area which is the cause of better adhesion –
when an easy estimate shows that the change in effective contact area could be
less than 5%. It is the ‘work of adhesion’ [8, 11] that is behind the
empirical PSA effect.

The second important factor that affects PSA is the soft elasticity of
polydomain nematic LCE. Here we send the reader to the extensive original
literature, well summarized in the book [25], to understand what this novel
effect unique to LCE is. Soft elasticity manifests in different ways in the
tensile mode in the peel test and in the lap shear test, but in all cases
leads to an enormous ductility of the LCE material. Unlike in ordinary
isotropic polymer PSA systems, the LCE in a loaded contact with substrate will
be able to gradually adjust and re-align its nematic domains near the surface,
following the local soft trajectory, to significantly lower the overall free
energy of a bonded state. As a result, the thickness of the “affected layer”
(which is about 10-100 nm in ordinary polymer networks [26]) extends to
several microns. The elastic energy stored in this layer is our proposed
mechanism of the enhanced adhesion of polydomain LCE, as discussed towards the
end of this paper, once we get familiar with the measured properties of this
experimental system.

The two factors discussed above are related to the third characteristic
feature of polydomain LCE, having a profound impact on PSA properties. The
relaxation dynamics of polydomain nematic LCE is extremely slow [27].
Therefore, we expect that the contact time during PSA bonding would have a
very strong effect on the strength of adhesion in LCE, allowing the sufficient
domain re-organization and adjustment under pressure against the substrate.
This would be highly unusual for the ordinary isotropic-pgase PSA systems. And
finally, nematic LCEs are renowned for their anomalous viscoelastic
dissipation, measured by the high loss factor $\tan\delta(\omega)$ [3, 28,
29], which arises not because of the ‘usual’ lowering of the storage modulus
(as is the case in lower crosslinking densities or melts), but due to the
additional mechanism of internal dissipation due to local rotational motions
of nematic director, controlled by its rotational viscosity, which is in turn
controlled by the thermally-activated exponential factor of overcoming the
mean-field barrier for mesogen rotation [30, 31]. Both disappear in the
isotropic phase of LCE, returning $\tan\delta$, and the PSA strength, to
ordinary elastomer values.

The first study that has mentioned the PSA in LCE is the theoretical work of
Corbett and Adams [32], much earlier than its first experimental finding [6].
This study applied the equilibrium ‘Trace formula’ for nematic elasticity [25]
and the dynamic dumb-bell rheological model of Mafettoni and Marucci [33] to
explore how the the nematic rotational degree of freedom affects the PSA layer
debonding via cavity nucleation and then fibrillation. They found that in the
director geometry when the rotation would accompany local deformation, the
effective work of adhesion was higher than when there was no underlying
rotation (the director perpendicular to the adhesion surface in the flat probe
tack test). However, they also found that the isotropic-phase adhesion was as
high as the parallel-alignment adhesion, which is strongly contrary to all
observations. In any case, their pioneering work has little to do with out
case, when we deliberately work with the practically relevant non-aligned
polydomain LCE layer, and the local inhomogeneous deformations in it are
behind our proposed mechanism of enhanced adhesion.

In this paper we carry out several technology-standard adhesion tests to
assess how the key factors discussed above affect its strength and other
features. Here we do not aim to ‘optimize’ the polymer system, i.e. find the
best LCE adhesive material or compare with other non-LCE PSA, which should be
a subject of a separate study of the materials chemistry. Instead, we work
with a standard well-characterized LCE material, described and characterized
in dozens of papers, and focus on the adhesion mechanism itself, which is
universal and does not depend on the chemical nature of the material.
Similarly, we do not explore different adhesion surfaces, but stay
consistently with the case of LCE adhesion to glass, in different
configurations and settings.

##  2 Experimental section

### LCE film preparation

The starting materials for our LCE are all commercially available and
relatively cheap. To produce the ‘standard’ reference LCE materials, we
followed the established synthesis [34, 35] using the Michael addition click
reaction between RM257 reacting mesogens (4-(3-acryloyloxypropyloxy)benzoic
acid 2-methyl-1,4-phenylene ester di-acrylate) obtained from Daken Chemical
Ltd. and the EDDT thiol spacers (2,2’-(ethylenedioxy) diethanethiol).
Butylated hydroxytoluene inhibitor (BHT) was used to prevent acrylate
homopolymerization, and Triethylamine (TEA) is one of many that work well with
thiol-acrylate chemistry. We specifically chose the weak TEA catalyst (instead
of a much stronger catalyst such as Dipropylamine) to have a low reaction
rate, in order to obtain a narrower and consistent chain length distribution.

We controlled the crosslinking density by first preparing thiol-terminated
oligomer chains of LC polymer, running the above reaction with an excess
thiol. The di-acrylate mesogen RM257 was dissolved in toluene (70 wt%)
together with BHT (2 wt%) assisted by gentle heating and vigorous mixing by a
vortex mixer. Next, the di-thiol spacer EDDT was added to the same vial. After
obtaining a fully dissolved, homogeneous solution, TEA (1 wt%) was added while
stirring to catalyze the reaction. The vial was then covered with aluminium
foil to avoid light exposure and placed on a roller for 48 hours. There is a
straightforward correspondence between the excess of functional groups during
chain-extension by alternating di-functional monomers, which we have verified
by running gel permeation chromatography (GPC) on these oligomers and finding
a good match with the basic calculation (see [36] for details of GPC
analysis). In this way, we had oligomer chains of the average 20-mesogen
length for the 1.05:1 thiol excess over acrylate, of the average 10-mesogen
length for the 1.1:1 thiol excess, and of the average 5-mesogen length for the
1.2:1 thiol excess. The longer-chain oligomers had the appropriately much
higher viscosity. The vinyl-functional crosslinker was mixed into the
oligomer, together with the photo-initiator Irgacure I-369 (1 wt%), to induce
a rapid UV-stimulated crosslinking of the thin film after its spreading on the
backing plastic substrate. The exact amount of the vinyl crosslinkers was
determined stoichiometrically by balancing the overall thiol and acrylate
groups.

We used 4-functional (2,4,6,8-Tetramethyl-2,4,6,8-tetravinyl
cyclotetrasiloxane)[37] and 3-functional
(1,3,5-triallyl-1,3,5-triazine-2,4,6(1H,3H,5H)-trione)[38] versions, and
verified that the film properties after crosslinking were very similar, both
mechanically and adhesively. This is because the elastomer properties are
determined by the length of network strands, and only very weakly by the
different crosslink functionality. For this reason, we refer to the three
types of LCE material as 5%, 10% and 20% crosslinked, reflecting the
respective 20-, 10- and 5-mesogen long chains in the network. The weight
percentages shown here are all calculated with respect to the total weight of
the elastomer without solvent. All chemicals, apart from RM257, were obtained
from Sigma Merck.

The spreading of the pre-crosslinking oligomer was done using the Tape Casting
Coater (MSK-AFA-III), from MTI Corporation, using the PET backing films of two
thicknesses: the 23 $\mu$m film from Hi-Fi Industrial Film Ltd, and the 200
$\mu$m overhead transparencies (for the lap shear testing when the backing
tape must remain undeformed under tension). The PET films were plasma-treated
using the plasma cleaner from Diener Plasma GmbH & Co. KG to ensure strong
bonding on the adhesive LCE layer. After thickness-controlled spreading (500
$\mu$m), the film was exposed to UV for 1 hour for full crosslinking. The
coated film was placed in a vacuum oven at 70 ${}^{\circ}C$ overnight for
solvent removal. After preparation, the exposed surface of LCE was covered by
silicone paper (we used the backing paper of Avery labels) to protect it from
contamination by atmospheric factors and accidental touch. The final thickness
of the homogeneous polydomain LCE layer on PET was ca. 200 $\mu$m.

### LCE characterization

This class of nematic LCEs has been studied extensively, and its bulk
properties are well established.[34, 35] For instance, the glass transition
$T_{g}$ is ca. 0${}^{\circ}C$ and the nematic-isotropic transition
$T_{\mathrm{NI}}$ is ca. 60${}^{\circ}C$, although the latter could be
adjusted within the range of 30-100${}^{\circ}C$ by minor modifications of
monomers and spacers used.[35]

The equilibrium (chemical) surface energy $\gamma_{0}$ was measured via the
contact angle, following the Owens-Wendth method [18], using the water droplet
on clean glass as a reference. Our LCE is less hydrophobic than typical
polyolefins, and its surface energy was found to be close to Nylon:
$\gamma_{0}\approx 42$ mN/m.

The tensile stress-strain curves were measured using the Tinius-Olsen ST1
tensiometer, with LCE made into the standard dogbone shapes (ASTM D412).
Experiments were conducted at ambient temperature in the nematic phase, and
also at a high temperature of ca. 90${}^{\circ}C$ in the isotropic phase, for
comparison. The enormous ductility of the nematic LCE is in sharp contrast
with the brittle network in the isotropic phase: this finding will be key to
our explaining many observations below.

Figure 1: (a) The contact angle illustration. (b) Tensile test results for the
three nematic LCE materials with different crosslinking densities; the red
curves mark the tests at high temperature, in the isotropic phase. (c) DMA
temperature ramp data for the 10%-crosslinked LCE tape, at constant 1 Hz,
showing the storage modulus $G^{\prime}$ and the loss factor $\tan\delta$ in
three phases. $T_{g}$ and $T_{\mathrm{NI}}$ are labelled in the plot.

Dynamic mechanical (DMA) temperature ramp was conducted using the TA
Instruments DMA850 in tension film mode. For these tests, two PET-based LCE
adhesive tapes were stuck to each other face-to-face, so that the actual test
of the deformable LCE was in the simple shear mode. The temperature was ramped
from -40${}^{\circ}C$ to 120${}^{\circ}C$ with the sample oscillating at 0.1%
strain at 1Hz, and its storage and loss moduli (with
tan$\delta=G^{\prime\prime}/G^{\prime}$) recorded. This test clearly
identifies the low-temperature glass, the high-temperature isotropic, and the
ambient-temperature nematic LCE phases, the latter characterized by the
anomalously high loss factor, see Fig. 1 for details.

### Probe Tack 2 test

The Probe Tack tests are designed to measure the surface adhesion after a load
pressure is applied to the probe. The details of these, and other standard
adhesion tests are well described in the summary monograph by Abbott [26].
Much of the literature describes tests with a flat probe (ASTM D4541) [11,
17], which is useful since the concepts of stress and strain are easily
defined in such a geometry. However, we were unable to build a measuring
device that would ensure the strict parallelity of the adhesive surface and
the probe, and instead opted for the spherical probe: the ‘Probe Tack 2’
method according to the Abbott classification.

Figure 2: The illustration of the Probe Tack 2 test. Above: a photo of the
glass sphere pressing into or pulled off the LCE adhesion tape facing up.
Below: a zoomed-in region of contact during the pull-off.

We used a spherical glass probe of 10 mm diameter mounted on a vertical
dynamometer frame (using the same Tinius Olsen ST1). The test protocol
involved starting with a fresh LCE adhesive surface (the PET tape glued to a
flat rigid base) and a cleaned glass sphere, which was lowered into the LCE
layer until the fixed load of -0.5 N was reached (this load level was
determined empirically, by verifying that no damage was done to the LCE layer
after the test). After that, the pre-determined dwell time (which we called
‘contact time’ in the discussion) was allowed, after which the probe was
pulled up at a fixed rate of 10 mm/min. It is well known that the adhesion
strength depends on the pulling rate very strongly [26, 7], but here we did
not explore this test variable and kept it constant. The adhesion force is
defined as the peak value of the tension force before the adhesive ‘neck’
detaches from the probe, see Fig. 2 for illustration, and the Supplementary
Video 1 for detail.

### 90-degrees Peel test

The Peel Test followed the ASTM D3330 standard parameters and protocol, and we
chose the 90-degrees peel geometry. Our ‘standard’ 25 mm-wide PET tape with
LCE adhesive layer was pressed into the clean glass slide with a free end of
the tape clamped into the vertical dynamometer frame. After a pre-determined
contact time, the tape was pulled up (using the same Tinius-Olsen ST1
tensiometer) with a rate of 100 mm/min, recording the adhesion force. A crude
method of horizontal movement of the flat glass slide base was devised to
ensure the delaminating edge was always below the pulling clamp. See Fig. 3
for illustration, and the Supplementary Video 2 for detail.

Figure 3: The illustration of the 90-degrees Peel test. Above: a photo of the
glass slide with the PET-based LCE adhesive tape being pulled up at 90
degrees. Below: a zoomed-in region of contact during the pull-off.

### Lap shear test

The Lap Shear test followed the ASTM D3163 standard parameters. The PET tape
was pressed into the clean glass slide with a free end of the tape clamped
into the vertical dynamometer frame, while the free end of the glass clamped
into the bottom clamp. In this geometry a shear deformation is imposed on the
adhesive layer, with the shear strain proportional to the clamp displacement:
$\varepsilon=x/d$, where the LCE layer thickness $d\approx$ 0.2 mm. The same
Tinius-Olsen ST1 tensiometer was used in this test.

Here we had to make a variation on our PET tapes, because in this geometry we
could not afford to have the PET backing stretch under the high forces
involved – and our ‘standard’ PET of 23 $\mu$m thickness was stretching
easily. Therefore, for these tests we used PET ‘tape’ of 0.2 mm thickness, to
act as backing for the same LCE layer, which was effectively rigid and did not
provide any measurable deformation to contaminate the shear response of the
LCE. Figure 4 illustrates the test geometry, and the Supplementary Video 3 is
helpful to visualize different modes of delamination in LCE layers of
different crosslinking density.

Figure 4: The illustration of the Lap Shear test. Left: a photo of the glass
slide with the PET-based LCE adhesive tape being pulled up, inflicting the
shear deformation in the LCE adhesive layer. Right: zoomed-in regions of the
PET tape edge during the shear, illustrating different modes of LCE layer
failure.

##  3 Results and Discussion

###  3.1 Probe Tack 2

The test summary in Fig. 5(a) illustrates the meaning of ‘adhesion force’ in
this context, and also of the ‘contact time’, giving examples of two different
adhesive responses: a weaker strength with direct delamination, and a higher
strength with a neck forming on further pullout. We remind that at our chosen
load of -0.5 N there was no visible damage to the LCE layer after detachment.
Figure 5(b), showing the data for the 5%-crosslinked LCE, illustrates the
strong dependence of adhesion on contact time: an almost 6-fold increase in
the adhesion force over 10 hours of contact.

We will discuss the time dependence in detail below, but first show the
summary of this test in Fig. 5(c), for a specific 5-second value of contact
time. It is clear that weaker-crosslinked LCE (with correspondingly longer
network strands) has the higher adhesion force. The same test repeated at
$T=80^{\circ}$C (in the isotropic phase of the LCE) shows a much lower
adhesion. This thermal switching of PSA is fully reversible, with quite a
small error also indicated in the plot.

Figure 5: Probe Tack 2 results for a 10 mm-glass sphere on LCE adhesive tape.
(a) The summary illustration of a typical test output, showing a weak and a
strong adhesion cases. (b) Comparing the pulling force response for the
5%-crosslinked LCE layer after the increasing contact time. (c) The aggregate
data for three different LCE layers, in the nematic and in the isotropic
phase, at a fixed 5-second contact time.

We see a strong dependence on contact time, and Fig. 6 illustrates its
different aspects in greater detail. First of all, we found that while the
glass sphere is pressing into the polydomain LCE layer at constant force, the
depth of indentation slowly increases. This is reminiscent of the studies of
slow stress relaxation in polydomain LCE [27], and certainly relies on the
same internal mechanism. The misaligned nematic domains that find themselves
under stress tend to re-align along the local axis of principal extension.
However, since neighboring domains are initially aligned differently, their
individual shape change must be made mutually compatible. Adjusting this
strongly correlated mechanical system is similar to the relaxation of the
angle of repose in a sandpile, via decreasing local avalanches [39, 27], and
it has been suggested to follow a logarithmic time dependence. We find it
practically very difficult to distinguish the logarithmic time dependence from
the stretched exponential with a low exponent, not without testing at very
long times which our instruments did not permit. Figure 6(a) shows how the
indentation depth increases in three separate experiments with different
contact time, and the overlap of the curves is reassuring. We found the good
fitting of this relaxation to follow the stretched-exponential equation

$d(t)=d_{\mathrm{eq}}+(d_{0}-d_{\mathrm{eq}})\cdot\left\\{1-\exp\left[-\left(\frac{t}{\tau}\right)^{0.4}\right]\right\\}\
,$ (1)

where $d_{0}=0.5$ mm is the initial indentation depth that we measure, the
fitted saturation depth $d_{\mathrm{eq}}\approx 0.535$ mm, and the fitted
relaxation time constant is $\tau\approx$ 43.5 minutes.

Figure 6: The effect of LCE relaxation in the Probe Tack 2 test; an example of
5% crosslinked LCE. (a) Monitoring the indentation depth of the 10 mm glass
sphere held at a contact load of -0.5 N. Three data sets are for the 1 h, 3 h
and 10 h contact time, with the solid green line presenting the fitted curve
(see text). (b) The value of the ultimate (peak) adhesion force as a function
of contact time, again with the fitted curve through the data.

Figure 6(b) shows a different aspect of this relaxation, plotting the set of
peak values of adhesion force, as seen in Fig. 5(b), against contact time.
Again, the logarithmic time axis allows visualization of both the short and
the long time limits, the longest test we were able to carry out had a contact
time of 45 hours. This plot highlights the enormous, 6-fold increase in the
adhesion strength, which cannot be explained merely accounting for the
moderate increase in glass contact area that follows from the indentation
increase. It is clear that the main contributing factor to this increase in
adhesion strength is the slow re-adjustment of deformed nematic domains, which
are able to achieve a much lower free energy state over time, and then
‘resist’ having to lose this deep energy minimum on the sphere pullout.
Importantly, a good fitting of this time dependence is achieved by the same
stretched exponential:

$F(t)=F_{\mathrm{eq}}+(F_{0}-F_{\mathrm{eq}})\cdot\left\\{1-\exp\left[-\left(\frac{t}{\tau}\right)^{0.4}\right]\right\\}\
,$ (2)

where $F_{\mathrm{eq}}=6.23$ N is the saturation force that we measure, the
fitted initial adhesion strength (at zero contact time) is $F_{0}\approx 0.89$
N, and the fitted relaxation time constant is now $\tau\approx$ 23 minutes. We
are not sure whether a small difference in the two time constants $\tau$ has
any relevance, given so different types of data being fitted, but the fact
that both relaxation laws follow the same stretched exponential law suggests a
consistent mechanism of polydomain texture re-adjustment which is dominant in
determining the strength of PSA.

###  3.2 90-degree Peel

The main difference in the peel test is that the adhesive tape is under no
load during the contact time, unlike with the probe that is being pressed at
constant load. The preparation of test samples is described in ASTM D3330
protocol: we use the standard 25 mm wide adhesive tape on the glass slide,
stick it using the prescribed 2 kg roller, leave for a pre-determined contact
time, and then peel at a high rate of 100 mm/min.

Figure 7 shows the essence of the test, with the x-axis representing the
vertical position (which is equivalent to the time, at a constant pulling
rate). After gathering some slack, the pulling force saturates at a constant
level representing the adhesion force (per standard 25 mm of the tape width).
Note, that we follow the standard test notation of ASTM D3330, quoting the
’peel force’ (in N) for the 25 mm tape, but in many cases it may be
instructive to look at the proper intensive parameter in units of N/m, or
J/m2. In this way, the measurement of force =10 N per 25 mm in Fig.7 means 400
N/m for the 20%-crosslinked sample we label ’1.2’, or the force of 20 N means
800 N/m for the 5%-crosslinked sample ’1.05’.

Figure 7: Peel Test results on a 25 mm-wide adhesive tape, comparing the
different LCE crosslinking densities: (a) 5%, (b) 10%, (c) 20%. In each plot,
different curves correspond to the different load-free contact time, as
labelled in the graphs.

As expected, and in agreement with the Probe Tack test (see the discussion
[11]), we find higher adhesion force for lower crosslinking densities, which
in our case directly corresponds to the length of polymer strands in the
network. For each crosslinking density, we again find a strong effect of the
contact time. Interestingly, the saturation is reached much faster, within 24
hours for the highly crosslinked 20% and 10% LCE (samples 1.2 and 1.1), but
takes over 5 days in the weakly crosslinked 5% LCE sample 1.05. At the same
time, the magnitude of the adhesion increase is just about 50% for the 20%
LCE, while reaching over 120% increase for the 5% LCE adhesive film. Clearly,
with longer network chains, the nematic polydomain structure is less
constrained and is able to achieve a much ‘deeper’ free energy state on re-
adjustment near the surface – but it takes a much longer time.

Another important point is also illustrated in Fig. 7. The “annealed” curve in
all plots represents the test carried out on the freshly bonded tape, which
was then annealed to the isotropic phase, and allowed to naturally cool back
to the ambient temperature (all within 10 minutes after the first bonding). It
is clear that such annealing enhances the adhesion strength to its maximum,
which otherwise would be achieved after days of contact time at ambient
temperature. We will return to the discussion of the underlying mechanism
causing this effect later in the paper.

Figure 8: Phase effect on the LCE adhesion, using the example of
10%-crosslinked LCE tape after 1-day bonding. (a) Repeated peel tests at
ambient and at the high temperature show reproducibility of equilibrium
adhesion strength in both phases. (b) The adhesion strength calibrated for
continuously changing temperature on natural cooling of the tape during the
peel test.

Figure 8 explores the switchable aspect of PSA in the LCE adhesive. Plot (a)
combines two cycles of heating and cooling, using as an example the 10% LCE
tape. Starting at the ambient temperature (nematic phase of LCE) after 1-day
contact time adhesion strength is reproducibly ca. 0.64 N/mm, corresponding to
the peel force of ca. 16 N, in Fig. 7(b). When this tape is brought to a high
temperature (isotropic phase), the continuing peel test shows the adhesion
strength dropping to a reproducible ca. 0.08 N/mm (peel force of 2 N). When we
then allow the sample to cool back to ambient temperature, the PSA tape
condition becomes ‘annealed’ (see the discussion around Fig. 7), and the
adhesion strength returns to its equilibrium value. Heating again, returns to
the weak adhesion in the isotropic phase. We could not repeat this test over
many more cycles because the exposed LCE surface becomes contaminated after
repeated adhesion-peel events.

It was very difficult for us to test the accurate temperature dependence of
adhesion strength, which would be desirable since we separately know the
temperature dependence of the nematic order parameter and could make a more
quantitative analysis. The best we could do is to heat the bonded tape, and
calibrate its natural cooling rate (using the thermal camera to read the
surface temperature, accepting the systematic error of the temperature
gradient between the outer PET surface that we can view and the inner adhesive
surface of LCE). Having calibrated the temperature vs. time in this way, we
then started the peel test of the heated tape, allowing it to cool while being
pulled. The result is presented in Fig. 8(b). Acknowledging all the
experimental shortcomings, we nevertheless find a very clear reflection of the
nematic phase transition, and a seemingly close correlation between the
adhesion strength of the annealed LCE tape and the nematic order parameter
that grows on cooling, before reaching the phase saturation.

Figure 9: The residue left in high-temperature isotropic-phase peeling. (a)
The photo of the glass slide left after the test of Fig. 8(a): as the tape was
peeled off, first the residue-free debonding occurred in the ambient nematic
phase, then the strip of residue was left on peeling at 80∘C, then again the
clean debonding after the tape was cooled back to the ambient temperature, and
then again the residue strip left after peeling at 80∘C. (b) The surface
profilometer scans data, showing the characteristic dimensions of the natural
free LCE surface, and of the residue layer left after isotropic peeling.

The peel tests at high temperature have revealed an unexpected phenomenon. We
have to emphasize: in our lab, we have tested LCE adhesive tapes on many
different surfaces (steel, ceramic tile, perspex plastic), and in all those
cases we had no surface residue on peeling the PSA tape – whether in the
ambient nematic phase, or at high temperature in the isotropic phase. On the
clean glass only, and at the high-temperature peeling only, but consistently
across all our LCE materials, the tapes left a residue. Figure 9(a)
illustrates this residue, and the lack of it when peeling the strongly bonded
nematic LCE tape; in fact the photo is exactly of the sample presented in Fig.
8(a), with two segments of hot/isotropic peeling and two segments of
ambient/nematic peeling.

Figure 9(b) shows the surface topography of the natural LCE free surface, and
of the residue layer discussed. The LCE surface profile is characteristic of
the polydomain structure, where the regions aligned perpendicular to the
surface elongate in the nematic phase and create the ‘hills’. However, it is
obvious from comparison, that the residue left on debonding in the isotropic
phase has nothing to do with these domains, as it has a very characteristic
periodic length scale of ca. 80 $\mu$m (much greater than the LCE domain
size). In looking for an explanation for this effect, we have to return to
Fig. 1(b) showing the intrinsic mechanical strength of our LCEs. It is clear
that the isotropic phase of these networks is rather brittle, and breaking
stress not exceeding 0.3 MPa, in great contrast to the nematic phase. We have
to assume that the (low) strength of isotropic adhesion on clean glass is
still strong enough to cause the elastomer network to rupture instead of
debonding (which is apparently not the case for other bonding surfaces we have
tested). Given that we measured the isotropic adhesion strength (which we now
realise is the cohesive failure) at 0.08 N/mm and crudely taking the width of
the stretched region to be 0.5 mm (from the visual observation of peel
boundary in Fig. 3 and Supplementary Video 2), the resulting stress becomes
ca. 0.16 MPa. It is very easy to accept that a bit higher stress at the very
front of the stretched zone reaches twice that value, which necessarily
produces the cohesive crack, according to our tensile data.

We will return to the discussion of residue, and of periodic structures
developing at the debonding edge, in the next section on Lap Shear testing.

###  3.3 Lap Shear

Lap shear test is a very different environment for the adhesive layer, and
although it is an accepted alternative way of assessing the adhesion strength
and features – its results are not directly correlated with the Probe Tack or
Peel tests [26]. For an LCE adhesive layer of 200 $\mu$m thick, the shear
strain of 500% is reached by the 1 mm movement of the top plate. Figure 10(a)
gives the summary of lap shear results, and the comparison of three materials,
in the nematic and isotropic phases. In the other tests, we saw an expected
result that the weaker (5%) crosslinked LCE layer has stronger adhesion.
However, in shear geometry the response is different, and dominated by the
cohesive failure of the bonded LCE layer. Both the 5% and the 10%-crosslinked
LCE tapes reach the stress peak at a similar strain value of 700-800%, similar
to the tensile failure of these materials, see Fig. 1(b). After this LCE
network failure occurs (predictably, at a higher shear stress in the 10% LCE),
the response turns into the plastic flow of the top PET plate over the bottom
glass plate. The fact that stress seems to diminish with further deformation
(as opposed to the ‘constant stress’ in the classical plastic creep) is simply
due to the fact that less contact area is engaged. The ‘apparent’ decrease of
stress is an artefact of us dividing the diminishing force by the initial
overlap area – and when we take the proportionally diminishing area change
into account, the proper plastic creep is obtained (at constant stress). The
Supplementary Video 3 shows the detailed process of cohesive debonding viewed
by the camera from above the overlap edge. Note the continuous thin layer of
residue left on the glass plate.

Figure 10: The Lap Shear testing of 25 mm-wide tape with 10 mm overlap. (a)
Comparing the three crosslinking densities in shear test, in the nematic phase
and in the high-temperature isotropic phase. There are two different modes of
failure: the clean residue-free debonding for the 20% LCE tape, and the
cohesive failure resulting in the tape creep along the surface, and the
residue layer, for the 10% and 5% LCE tapes. (b) A confirmation of the
contact-time effect, for the 20% LCE adhesive with its clean adhesive failure,
but at different strength for different contact time.

In contrast, the strongly crosslinked 20% LCE tape debonds by a clear adhesive
failure without any residue, at a peak shear stress of 0.4 MPa and strain ca.
500%, evidently just before the LCE network failure. All LCE adhesive tapes
debond at a low stress at high temperature, in the isotropic phase. But again,
there is a contrast: the weaker crosslinked LCE tapes break across the LCE
network, leaving the residue layer on glass – while the stronger 20% LCE
continues to debond adhesively, only at a lower stress since we now know the
adhesive strength is lower in the isotropic phase. Finally, we confirm the
finding of other adhesion tests, that increasing the contact time on the glass
surface makes the adhesion stronger, Fig. 10(b). This figure only presents the
data for 20% LCE tape as the only one actually testing the adhesion strength.

It bears repeating the argument made earlier, about why the LCE network
failure occurs in the thin layer, of approximately a domain size, above the
flat adhesive surface with glass. Figure 11 gives the sketch that illustrates
this point. On pressured adhesion, the naturally rough free surface of LCE is
forced to be flat, which distorts the first layer of nematic domains and
forces them to start their slow nematic director re-adjustment to lower the
elastic energy by exploring the local soft elastic trajectory for each domain.
As was argued before, this is a slow process due to many mechanical
constraints (barriers) that elastic domains must overcome to allow their
director rotation, and is the cause of the adhesion strength increase over
many days of contact. However, the local internal stresses that are left in
the polydomain LCE add to the external shear stress imposed in the test, and
the crack nucleation occurs in that thin layer near the contact plane.

Figure 11: The sketch of the lap shear geometry of the polydomain LCE layer,
highlighting the highly deformed zone of domains near the adhesive surface,
which causes high internal stresses at the depth of domain size, and the
resulting crack nucleating in that zone leading to the localized cohesive
failure and the thin residue layer.

##  4 Conclusion

In this paper, we carried out s series of industry-standard adhesion tests to
reveal the mechanism of LCE adhesion, enhanced in the nematic phase and
returning back to the ordinary elastomer levels in the isotropic phase. The
essence of this mechanism is twofold, but in both cases based on the mobility
of nematic director that is the core essence of LCEs [25]. One factor causing
a significant increase in the effective adhesive energy is the anomalous
viscoelastic damping in nematic LCEs. This was alluded to in recent literature
[7], and is based on the theoretical link between the effective surface energy
$\gamma$ and the dissipative loss (measured by $\tan\,\delta$) [14]:

$\frac{\gamma}{\gamma_{0}}\approx\left(1-\frac{2}{\pi}\int_{0}^{2\pi
v/a}\frac{\sqrt{1-\omega a/2\pi
v}}{\omega\tan\,\delta(\omega)}\mathrm{d}\omega\right)^{-1},$ (3)

where $\gamma_{0}$ is the ‘chemical’ surface energy, $v$ is the speed of
pulling (so that $v/a$ is the strain rate). The integral is over the frequency
range determined by the strain rate. The reason why $\tan\,\delta$ is so high
in the nematic phase of LCE is a separate matter, not very well understood but
certainly related to the rotational viscosity of nematic director. However,
the fact of increased viscoelastic dissipation in nematic LCE is not in
question, and is one of the two reasons for the enhanced adhesion. This also
suggests the route for material optimization: the higher the loss factor, the
stronger the adhesion.

The other factor contributing to the high adhesion in the nematic phase is the
significant reduction of the LCE elastic energy when a polydomain texture is
deformed by pressing into the target surface, and the local director in each
deformed domain is allowed to rotate, re-adjusting in correlation with its
neighbors, exploring the soft elastic trajectory of deformation. The full re-
adjustment of director takes a long time, following an unusual (possibly –
logarithmic) relaxation law due to the multiple correlation of local barriers
in the random system. Once the low value of elastic energy is achieved, the
bonded state becomes much preferred compared to the just-debonded state, where
the condition for the energy minimum is removed. This suggests that polydomain
LCE has a higher adhesion compared to an aligned monodomain LCE layer where no
such soft elastic re-adjustment is possible.

Other promising routes for material optimization to further increase the
adhesive strength, while retaining its reversible reduction in the isotropic
phase (which could be achieved by heating above $T_{\mathrm{NI}}$ or by light
stimulation if the LCE is made photo-sensitized), is by reduction of
crosslinking density. However, this is not a simple matter, since an LCE
network with less than ca. 5% crosslinking becomes very weak and mechanically
unstable. Therefore, the valid avenues are either by using the concept of
‘double network’ to strengthen the LCE, or by forming a stronger network with
dangling chain ends to provide the mobility and thus adhesion. Both of these
routes are explored in the classical PSA systems [40, 41, 42, 43], but we
believe they have a great promise in the LCE field as well.

### Supplementary Material

Three videos are supplied as Supporting Information, available on request from
the authors. The following files are available.

  * •

Video S1: the probe tack experiment composition

  * •

Video S2: the 90-deg peel test composition

  * •

Video S3-5x: the lap shear test for the 5% crosslinked LCE

  * •

Video S3-20x: the lap shear test for the 20% crosslinked LCE

### Acknowledgements

We are grateful to Professor Steven Abbott for many useful and encouraging
conversations about PSA and adhesion testing. This work was supported by the
European Research Council (H2020) AdG “Active Polymers for Renewable
Functional Actuators” No. 786659, and PoC “Reversible adhesion damping tapes
of liquid crystalline elastomers” No. 101100553). M.O.S. acknowledges support
of the Royal Society. H.G. acknowledges support from the China Scholarship
Council.

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