# "Magnetic Marshmallows" for Soft Robotics: Magneto-Mechanical Characterization and Application in Switchable Adhesion Structures

Maxime Bès, Samuel Lamont, Azadeh Dufour-Lamartinie, Jean-Claude Mancer, Ludovic Olanier, Jérôme Robert, Franck Vernerey, Artem Kovalenko

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Maxime Bès, Samuel Lamont, Azadeh Dufour-Lamartinie, Jean-Claude Mancer, Ludovic Olanier, et al.. "Magnetic Marshmallows" for Soft Robotics: Magneto-Mechanical Characterization and Application in Switchable Adhesion Structures. Soft Matter, In press, 10.1039/D4SM01503G. hal-05003376

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# "Magnetic Marshmallows" for Soft Robotics: Magneto-Mechanical Characterization and Application in Switchable Adhesion Structures

Maxime Bès¹, Samuel Lamont², Azadeh Dufour-Lamartinie¹,², Jean-Claude Mancer¹, Ludovic Olanier¹, Jérôme Robert³, Franck Vernerey², Artem Kovalenko¹

1. Laboratoire Sciences et Ingénierie de la Matière Molle, ESPCI Paris, PSL University, Sorbonne Université, CNRS, F-75005 Paris, France

2. Department of Mechanical Engineering and Material Science & Engineering Program, University of Colorado at Boulder, Boulder, Colorado, USA.

3. Institut de Physique et Chimie des Matériaux de Strasbourg, F-67200 Strasbourg, France

**Keywords:** actuators; mechanical properties; magnetic materials; polymer foams, soft robotics.

## Abstract

Soft magnetic composites exhibit fast and programmable macroscopic deformations in magnetic fields, making them promising for applications in soft untethered robotics or in designing surfaces with switchable adhesion or wetting properties. However, due to the incompressible nature of soft, non-porous actuators, their compression or elongation leads to important shape change due to lateral expansion or compression, respectively. In practice, bending and folding remain preferred actuation modes. Here, we explore the potential of magnetic elastomer "marshmallows" as compressible actuators with low Poisson's ratio. Using a sacrificial salt pellet template method, we fabricate polydimethylsiloxane foams with open porosity filled with carbonyl iron particles. The obtained foams exhibit strong reversible compression under the influence of a magnetic field gradient. We reveal the significance of stress accumulation in the direction of the field gradient due to magnetic body forces and the key role of the foam thickness in magnetic strain. We propose a simple analytical model based on the action of a magnetic body force that explains these effects. Finally, we demonstrate the development of a novel switchable adhesion structure, in which the magnetic foam covered with a pressure-sensitive adhesive serves as a compressible actuator able to switch between adhesive and non-adhesive states.
# I. Introduction

Actuators made of soft materials find their applications in the field of soft and bio-robotics. Due to their deformable nature, soft robotic grippers are able to adapt their shape, thereby minimizing the risk of damaging complex objects<sup>1,2</sup>, such as biological conducts in surgery or small objects in everyday life. To build more versatile actuators for soft robotics, passive materials are replaced by soft functional composites responsive to light, pH, temperature, magnetic or electric fields. Among existing actuation stimuli, those sensitive to magnetic fields present several advantages<sup>3,4</sup>, including fast, reversible and untethered motion, and the possibility of obtaining complex yet "programmable" deformations.

In general, two types of magnetic forces can be used to trigger deformation<sup>5-7</sup>. On the one hand, an attractive magnetic force can be induced on a magnetic composite by a gradient in the magnetic field. On the other hand, a magnetic torque can be triggered on magnetically anisotropic solids when the field line direction is not collinear to the anisotropy axis. In addition, the mechanical properties of "magnetically soft" composites usually depend on the field because of the interparticle dipole-dipole interactions – this is known as magnetorheological behavior<sup>8,9</sup>. By carefully tailoring the external magnetic field distribution and the magnetic anisotropy of the actuator, a large variety of deformation modes have been demonstrated<sup>4,10-12</sup>, including elongation, compression, bending, rotation, folding and coiling. These deformations were used to drive locomotion and adapt the grasping of soft untethered robots<sup>3,13-19</sup>. Similar magnetic actuation concepts were applied to microstructured surfaces with magnetically-sensitive topography<sup>10,20-25</sup> to switch their adhesion or wetting properties. Recently, a new actuation behavior was demonstrated for magnetic polymer sponges<sup>26,27</sup> and magnetic cellular metamaterials<sup>28</sup>. When placed in a gradient magnetic field, these materials show a strong unidirectional compression with a low Poisson's ratio. While this property showed good potential for applications for drug delivery<sup>26,27</sup> or switching of acoustic properties<sup>28</sup>, its use as a low Poisson's ratio compression-driven actuator was not explored.

The main aim of this work is to rationalize the actuation and stress distribution in magnetic foam actuators and to relate them to material properties, including the Young's modulus and magnetic permittivity, as well as magnetic field properties. Using a robust sacrificial template technique, we fabricate magnetic silicone foams with open porosity, strong magnetic response and large reversible compressibility. Using digital image processing, we characterize the
reversible compression of foams of varying initial height, and illustrate the presence of a strain gradient within the material, which has not been widely considered in other studies on magnetic foams. We propose both a simple analytical model, and more complete numerical analysis that predict the strain and stress distribution in the foam during actuation. Finally, we demonstrate the use of magnetic foams as soft compression actuators, characterized by a low Poisson's ratio, to modulate the adhesion of pressure-sensitive adhesives.

## II. Results

Figure 1a illustrates the fabrication of the magnetic silicone foams. To promote an open-pore foam structure, we adapted a sacrificial template technique²⁹ with porous NaCl pellets. The pellets were obtained by uniaxial compaction of NaCl crystals (Figure 1b) with an average grain size of about 500 µm at compaction pressures varying from 0.4 to 1 GPa. The pellets were impregnated with a suspension of magnetically soft carbonyl iron microparticles (CIP) in a PDMS prepolymer mixture. After thermal curing of PDMS, the salt template was removed by washing with water, leading to a magnetic foam with pores that replicated the NaCl crystals of the pellets. Scanning electron microscopy (SEM) micrographs of a typical foam fracture surface reveal an open porosity structure with cuboid shapes of pores (Figure 1c). The void dimensions are in the 100 µm – 500 µm range and are similar in shape to those of the NaCl crystals used for the pellet fabrication (Figure 1b). The compaction of randomly oriented NaCl crystals leads to a bicontinuous pellet structure (Figure S1.1) with a percolated system of NaCl crystals and open porosity. Indeed, intimate contact between the NaCl crystals under compaction ensures the presence of interconnections between pores in the foam. To better visualize the CIP in the PDMS matrix, we obtained SEM micrographs using a backscattered electron detector. The images (Figure 1d and Figure S1.2) confirm that there is a homogeneous distribution of CIP at the sample scale and fluctuations of CIP concentration at the pore scale. These fluctuations may be explained by clustering of the CIP while passing through narrow channels inside the preform. We found experimentally that the CIP concentration $C_{Fe} = 50\% \text{ wt}$ was the highest concentration that allowed for homogeneous penetration of CIP in all fabricated salt pellets. At higher concentrations (results not shown), the core of the pellet was more transparent, indicating low CIP content, or even completely devoid of any material, indicating complete blockage of the narrow channels by the CIPs.
The foam porosity $\Phi = V_{pores}/V_{sample}$ is governed by the relative pellet density $\rho_{pellet}/\rho_{NaCl}$ where $\rho_{pellet}$ and $\rho_{NaCl} = 2160 \text{ kg m}^{-3}$ are the densities of the pellet and NaCl, respectively. The pellet density was measured from its weight and apparent volume and showed linear variation with the pellet compaction pressure P (Figure 1e) in the range of 400 – 900 MPa. Using this method, we can vary the foam porosity from 82 to 87%.

The compression stress-strain curves of foams with three different porosities are shown in Figure 1f. The foams demonstrate an elastic behavior with a constant tangent modulus at small and intermediate strains and strain-hardening at strains larger than 40% due to densification. The observed curves are different from those typically obtained for monodisperse and low-density polyurethane foams, which usually show a well-defined yielding stress and a characteristic plateau$^{30,31}$. We may explain this difference by the irregular shape of pores walls for which different modes of buckling$^{32,33}$ would take place in a range of stresses. When the foam porosity is increased from 82 to 87%, the initial compression modulus decreases from 10 to 3 kPa, and densification happens at higher compressive strains. This behavior is typical of elastomer foams.

Figure 1g shows the magnetization curves of a magnetic foam sample in comparison with the curve of a non-porous magnetic PDMS and bare carbonyl iron microparticles. The initial magnetic content for all magnetic PDMS samples is equal to 50% wt. The curves show the same shape with no apparent hysteresis and a saturation behavior at a magnetizing field higher than 500 kA m$^{-1}$. This behavior is typical to that reported in the literature$^{34,35}$ for magnetically “soft” carbonyl iron dispersions. De Vicente et al.$^{34}$ have shown that for non-structured CIP embedded randomly in a solid matrix the magnetic susceptibility $\chi (\chi = \mu - 1$, where $\mu$ is the magnetic permeability in the cited publication) is proportional to the content of magnetic particles and independent of the magnetization history for CIP volume fractions lower than 44%. In our study, the volume fraction of CIP in the PDMS matrix is 12% (which corresponds to about 1-2 vol.% in the foams) and the distribution of CIP in the matrix may be considered as random from SEM images. As shown in Figure SI.3, the mass susceptibility and saturation magnetization $M_s$ increase proportionally to the iron content, implying an absence of degradation of CIP during the fabrication process.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** – a) Schematic of magnetic foam fabrication using a sacrificial molding technique with a NaCl template; b-d) scanning electron microscopy (SEM) image of NaCl crystals (b), the final foam structure (sample F82-H6) with a surface contrast (secondary electrons) (c) and elementary contrast (backscattered electrons) (d); e) the relative density of the NaCl templates as a function of compaction pressure; f) compression mechanical curves of foams with different porosities; g) sample weight-normalized magnetization curves of carbonyl iron particles (CIP), a non-porous CIP-filled PDMS and magnetic foams with different porosities.</figcaption></figure>

The fabricated magnetic foams undergo significant deformation when a permanent magnet is introduced and moved underneath its non-magnetic support (see the video SI1). In general, the foam deformation is complex and may include compression, bending and rotation. For clarity, we here specifically focus on the uniaxial compression of the foam when a permanent cylindrical magnet (remanent magnetization *B*<sub>rem</sub> = 1.29 T) approaches the bottom of the support. The compression is due to the attractive force between the sample and the magnet. This force was measured using a setup described in Supporting Information (Section SI.B), and its maximum value was about 0.7 N (Figure SI.4). Using simulations of the magnetic field around the magnet (Figure SI.4), we optimized the geometry of the magnet to produce a magnetic field with strong gradient along the vertical axis *z* which is homogeneously distributed on the horizontal cross-section of the foam which has the radius *R*<sub>foam</sub> = 6.5 mm.
For the selected magnet length L<sub>mag</sub> = 20 mm and radius R<sub>mag</sub> = 10 mm, the border effects may be minimized. The experimentally measured dependence of the vertical component of the applied magnetic field H<sub>z</sub> at the central axis of the magnet was in good agreement with simulations (Figure SI.6). In the closest position to the magnet (fixed by the support thickness equal to 4 mm), the value of H<sub>z</sub> and its gradient dH<sub>z</sub>/dz at the bottom of the foam were equal to 0.46 T and 0.065 T/mm, respectively.

As seen in photographs of the sample in the initial and the magnetically compressed state (Figure 2a and b, respectively), the sample is compressed in the direction of the field, which leads to a total height change on the order of one millimeter. At the same time, almost no lateral expansion is observed, which confirms the low Poisson's ratio hypothesis. By applying digital image processing, we followed the deformation of a rectangular pattern of tracer points represented as red dots in Figure 2a. The vertical distance between dots was chosen to be about 2 mm, which is several times greater than the pores' dimension, thereby homogenizing the strain field<sup>31</sup>. We observe that in the deformed state the density of tracer points is higher in the bottom section of the sample (close to the magnet). From the change of the vertical distance between points, we may estimate the tensile strain field ε<sub>zz</sub>. The typical result for the strain field in sample F87-H6 at the smallest accessible distance from the magnet z = 1.85 mm is shown in Figure 2c. We observe a strong strain gradient from the top to the bottom of the sample, with a value of ε<sub>zz</sub> ≈ 0.6 in the bottom part of the foam and a value close to zero in the upper section. Several factors<sup>7,11,12,36,37</sup> may, in principle, induce non-homogeneities in the strain field of soft magnetic solids, including magnetic field non-homogeneity, demagnetization effects, mechanical instabilities and non-homogeneous boundary conditions. Here, we elucidate another important factor which is the “gravity-like” nature of the magnetic body force inducing a stress distribution in a soft material placed in a magnetic field gradient. In order to support the importance of this factor, we compared the magnetic deformation of three series of samples with similar porosities (82%, 84% and 87%), but different initial heights H<sub>init</sub>, in the similar gradient magnetic field represented on the Figure SI.6. The details of the sample dimensions, porosities, mechanical and magnetic properties are given in the Supporting Information (Table SI.1). As shown in Figure 2e, an increase of H<sub>init</sub> from 4 mm to 8 mm leads to a higher value of ΔH<sub>mag</sub> even though the “added” portion of magnetic foam is in a relatively weak magnetic field.
To explain this effect, we adapted the model of Zrínyi et al.¹² to the case of a magnetic foam by considering the stress equilibrium of the foam (a detailed description of our model is provided in the Supporting information). Briefly, the magnetic body force $f_{mag}$ acting on the sample is determined by the external magnetic field H and the magnetization M as follows:

$$f_{mag} = \mu_0 (M \cdot \nabla) H \quad (1)$$

which may be simplified for a 1D case as

$$|f_{mag}| \approx \mu_0 \chi H_z \cdot \frac{dH_z}{dz} \quad (2)$$

In these equations, $\mu_0$ is the magnetic permeability of free space, M, H and $H_z$ are the magnetization, applied magnetic field and its z component, respectively.

At equilibrium, the magnetic force, which is distributed throughout the sample, is balanced out by the reaction force of the solid support (see Figure 2d), yielding a local equilibrium equation for the stress $\sigma$ of the form:

$$d\sigma = f_{mag} dz \quad (3)$$

The stress distribution through the sample is obtained by integrating this differential equation through the whole sample thickness with boundary condition $\sigma = 0$ on the upper sample surface. The analytical integration of equilibrium (3) may be complicated because of the non-linearity of magnetization curves and the non-homogeneous distribution of the foam density in its compressed state. Here, we limit our discussion to the prediction of a simplified analytical model (detailed in the SI), which neglects these effects and considers the foam as a linearly elastic material. Under these assumptions, and for the particular case of a foam with a small initial height $H_{initial}$ with respect to the magnet dimensions $L_{mag}$ and $R_{mag}$, the height change of the foam in the magnetic is given by

$$\Delta H_{mag} = - \frac{a N_d \chi B_{rem}^2}{8 E \mu_0} H_{initial}^2 \left( \frac{2a}{3} H_{initial} - b \right) . \quad (4)$$

Here, $N_d$ is the demagnetization coefficient and the coefficients $a$ and $b$ are determined only by the geometry of the magnet. They are parameterized by the ratio $z_{limit} = b/a = R_{mag} \left[ 1 + \left( \frac{R_{mag}}{L_{mag}} \right)^2 \right]^{-0.5}$, which is a distance that measures the range of the magnetic field,
such that when $z>Z_{\text{limit}}$, the magnetic field gradient approximately vanishes. Note that in this model, the only adjustable parameter is the value of the demagnetization coefficient $N_d$.

As shown in Figure 2e by the dashed line, equation (4) shows a good quantitative agreement with the obtained values of $\Delta H_{mag}$ in the validity limit of the model, i.e. when the initial height of the sample $H_{initial}$ is lower than the magnetic field range $z_{limit} \sim 8$mm for our magnet. The analytical model predicts that $\Delta H_{mag}$ should attain a maximum value of about 2 mm. This model shows the importance of the vertical stress accumulation for the actuation of the foam in a magnetic field gradient. This phenomenon has different physical nature from the effect of the lateral sample dimensions reported in the literature³⁶ for magnetic compression of soft ferrogels which are mostly due to the magnetic field inhomogeneity.

The accuracy of prediction of our model may be improved numerically by considering the exact equation of the magnetic field, as well as nonlinearities arising from finite deformations. As expected, the numerical model better predicts the experimentally observed local strain in the sample as compared to its analytical counterpart (Figure SI.6). The predicted curve is represented as a continuous line on the Figure 2e, and predicts the increase of $\Delta H_{mag}$ above the $z_{limit}$ to values of about 3 mm. For thicker foams, the “added” section is located in the region of low magnetic field, such that the increase of $\Delta H_{mag}$ with thickness remains relatively small. As a consequence, the total magnetic strain $\epsilon_{mag} = \Delta H_{mag}/H_{initial}$ reaches a maximum value at $H_{initial} \approx z_{limit}$, as represented on Figure 2f.

Based on our results, we may highlight the key parameters that govern the strain behavior of compressible magnetic actuators. To obtain high deformations, the initial height of the foam should be on the order of $z_{limit}$, which can be estimated from the magnet dimensions. A compromise should be found between increasing the magnetic susceptibility $\chi$ and decreasing the Young's modulus $E$. For non-porous composites containing randomly dispersed spherical fillers, such as CIP in PDMS, the increase in modulus may be approximated by the Einstein-Guth's equation³⁸ $E_{composite} = E_{PDMS}(1 + 2.5 \cdot \Phi_{CIP} + 14.1 \cdot \Phi_{CIP}^2)$. At the same time, the susceptibility increases linearly with filler fraction $\Phi_{CIP}$. Thus, an optimized value of $\Phi_{CIP} = 0.26$ would correspond to the highest $\chi/E$ value and to the maximum magnetic strain. However, because of the issues associated with the penetration of the CIP in the NaCl pellet during the preparation process, this high fraction of CIP was not possible to achieve. In
addition, at high CIP fraction in PDMS, our samples became increasingly brittle, making it
difficult to use them as actuators. Another parameter influencing the values of χ and E of the
foams is the porosity. Following the treatment of Gibson and Ashby³⁰, we found empirically
that the modulus of the foams decreased with porosity as an inverse power law, such that
$E_{foam} \approx E_{composite} \cdot (1 - \Phi)^\alpha$ with $\alpha \approx 2.7$. The susceptibility χ, which in equations (2) and
(4) is without units and is a volume-related property, should be proportional to $(1-\Phi)$. As a
consequence, for the same CIP fraction in the matrix composite, we predict a rise in magnetic
deformation with porosity with a scaling of $(1 - \Phi)^{1.7}$. The NaCl pellet template technique we
used in this study allows for obtaining porosities in the range of 75 and 87% and is limited by
the imbibition of the viscous PDMS in the pores between NaCl crystals. In addition to
decreasing the Young's modulus, the introduction of porosity in the magnetic actuators lowers
the Poisson's ratio, which is an advantage for devices working in compression.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** – Deformation of a magnetic foam in a gradient magnetic field. Sample F84-H8: a) the initial state; b) the deformed state; c) the $\varepsilon_{zz}$ map from digital image processing; d) the scheme illustrating the mechanical equilibrium of the foam under action of the magnetic body force; e) the dependence of the normalized maximum sample height change on the sample initial height for all samples; f) the simulated magnetic strain for the sample series F84 as a function of initial sample height.</figcaption></figure>

Next, we used the fabricated foams to demonstrate their use for the design of a magnetic
device with reversible adhesion. In this structure, depicted in Figure 3a, a foam sample is
covered by a layer of a weak pressure sensitive adhesive (PSA) and fixed in a well of depth $H_{well}$
greater than the value of minimal height of the foam in the magnetic field ($H_{init} - \Delta H_{mag}$) but slightly shorter than the initial foam height value $H_{init}$. The details of the fabrication are given in the experimental section. A magnetic field is applied to the device by bringing a permanent magnet close to the structure. In turn, the foam sample contracts, which reversibly switches the sample from sticky to inert. This transition is confirmed by a tack test on the structure as shown on Figure 3b. In the absence of magnet, the structure can sustain a pulling force up to the maximum of 0.2 N, after which it decreases due to adhesive debonding. In the presence of the magnet, no force is detected after contact between the probe and the well surface.

We demonstrate the application of a reversibly sticky structure by triggering the unsticking of a small weight that was initially bonded to the structure. The demonstration is shown in Figure 3c and in the video SI2. A flat non-magnetic PMMA sheet with mass 2.36 g (corresponding to the normal force of 0.023 N) was put in contact by manually applying a small pressure between the sheet and the adhesive surface of the foam. The weight sticks to the foam and, in the absence of a magnetic field, remains attached for at least 15 minutes. The contact surface (underlined in red in the insert) is slightly smaller than the foam's cross-section area and may be approximated by a circle of diameter of about 8 mm. When a magnet approaches from the glass support side, the foam is compressed and puts the PMMA weight into contact with the spacer. In this configuration, the magnetic body force creates a tensile stress at the interface between the adhesive and the PMMA weight. As a result, we observe a detachment of the adhesive by interfacial fracture propagation within a few seconds (debonding time), and the PMMA weight falls under its gravity. Using the same device, the experiment may be repeated several times without changing the behavior. Two immediately consecutive adhesion – debonding cycles are shown on the video SI2.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** – Magnetic foam for actuation of PSA adhesion to a rigid surface: a) Scheme of the switchable structure containing magnetically compressible foam covered with an adhesive; b) the pull-off curve of the structure at different speed with and without magnetic field; c) demonstration of magnetically triggered unsticking of the structure from a rigid weight.</figcaption></figure>

In contrast with traditional modes of PSA debonding, which act at a fixed load, the debonding of our magnetic foams is performed at a fixed overall strain, as illustrated schematically on Figure 4a. In the initial state, the load pulls on the PSA with the force $F_{debond} = F_{load} = mg$ (m is the mass of the load and g the gravity) which is equilibrated by the reaction force of the support of the foam $R_s$. If the load is higher than the critical value $F_{failure}$ (about 0.2 N in for the device shown on Figure 3c), the load will rapidly unstick without an external magnetic field. This remains true in the presence of the magnet, but before the contact between the load and the rigid support, because the magnetic body force $F_{mag}$ decreases the value of $R_s$ without changing the value of $F_{debond}$. If we ignore the compliance of the rigid support, we may consider that the debonding of the PSA takes place at a permanent total foam height equal to the height of the rigid support. In these conditions, the PSA is submitted to $F_{debond} = mg + F_{react}$, where $F_{react}$ is the reaction force of the support. If the rigid support height was chosen as close as possible to the initial height of the foam and for small loads $mg \ll F_{failure}$, the condition of the foam unsticking may be simplified to:

$$F_{react} \approx F_{mag} = \sigma_{mag}A_{adhesive} > F_{failure} \quad (5)$$

where $A_{adhesive}$ is the contact area of the adhesive with the rigid probe and $\sigma_{mag}$ is the magnetic attraction stress acting on the foam. Combining the general relation of Zrínyi et al.¹² with our
approximation at the foam's height lower than the magnetic field range b/a, the magnetic stress may be estimated as:

$$ \sigma_{mag} = \int_{H(z_{max})}^{H(z_{min})} \mu_{0}MdH \approx -\frac{aN_{d}\chi B_{rem}^{2}}{8\mu_{0}}H_{initial}(2b - aH_{initial}) \quad (6) $$

where $H(z_{min})$ and $H(z_{max})$ are the values of the magnetic field at each side of the foam.

To test our hypothesis, we performed measurements of the traction force acting on the adhesive foam in an Instron testing machine equipped with custom-made clamps. This design allows for controlling the magnet motion and the distance between the adhesive surface of the sample and the glass probe surface. A flat glass probe with diameter equal to 10 mm was used to probe the normal force. The sample was similar to those from Figure 3, but without the rigid spacer to avoid contact with the clamp. The test was composed of three stages: 1) the contact stage by indentation of the glass probe in the sample with no magnetic field (shown as negative times in Figure 4b-e); 2) the relaxation stage at a fixed strain corresponding to an external load $F_{load}$, which was varied from 0 to 0.2 N and 3) the magnetic detachment stage, which consists of bringing the magnet below the sample from a distance $z_{magnet}$ = 30 mm (B≈0) to 1.85 mm (B≈0.6 T) at a speed of 0.33 mm/s, as represented on the graphs on top. This stage was only explored in the absence of failure during the relaxation stage (Figure 4b).

The results of the unsticking tests are represented in Figure 4. In the first test (Figure 4b), no external load was applied to the sample during stage 2. When the magnet was brought to $z_{magnet}$ ≈ 9 mm during stage 3, the force started increasing as a result of the foam being drawn by the magnet. At a normal force of about 0.15 N, the sample detached from the probe by interfacial fracture propagation. The value of the normal force corresponding to failure is very close to the failure force $F_{failure}$ in the tack test (Figure 3b), indicating a similar detachment mechanism. Using equation (6) and taking $H_{initial}$ = 8 mm and $A_{adhesive}$ = 78.5 mm², the estimation of $F_{mag}$ gives 0.34 N, which is larger than $F_{failure}$ ≈ 0.2 N obtained from Figure 3b. This validates our criterion of detachment (equation (5)).

To explore the effect of gravity on detachment, the next tests (Figure 4c-e) considered situations where the sample was pulled to a different external loading force $F_{load}$ during stage 2. At a loading force $F_{load}$ = 0.2 N (Figure 4c), we observe an adhesive failure without any applied magnetic field at about 30 s. The use of magnetic switching to bare such a load would
not be efficient. For Fload = 0.1 N (Figure 4d), the normal force decreases to about 0.03 N due
to partial debonding of the adhesive. However, the probe remains in contact with the adhesive
foam. When the magnet is brought in at times t > 150 s, the normal force increases to about
0.05 N, which causes adhesive failure. In the last test with Fload = 0.05 N (Figure 4e), the normal
force slowly relaxes to a low value, although most of the adhesive remains in contact with the
probe. The motion of the magnet towards the foam increases the normal force to about 0.15
N, which leads to the detachment of the adhesive. This experiment shows that magnetic
switching is the most efficient for loads lower than 0.05 N, which is the case of the PMMA
weight (Fload = 0.023 N) used for demonstration in Figure 3c. To increase the range of possible
loads, one may use a foam with a larger surface covered with PSA.

While our results demonstrate the switching of the adhesive, care should be taken to use them
for explication of the working principle of the device on the Figure 3. Under fixed strain (Figure
4b-e), the debonding force Fdebond may strongly decrease at the first stage of detachment due
to the high local foam stretching. This may considerably slower and even stop the interfacial
fracture propagation. At fixed load, the fracture propagates catastrophically due to the
increase of the stress concentration at its edge. The propagation speed should be strongly
influenced³⁹ by the energy dissipation in the PSA layer and the compliance of the foam and of
the attached object. While these factors are out of the scope of the present manuscript, it
should be accounted for the future development design of magneto-switchable adhesive foam
structures for soft robotics.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** – a) Working principle of magnetic unsticking of the device from Figure 3; b-e) Mechanical characterization of adhesive failure conditions for a sample F87-H8 covered with a layer of PSA in absence (b) and presence (a, c, d) of the magnetic field. The loading force $F_{load}$ at zero time was equal to a) 0 N, b) 0.2 N, c) 0.1 N and d) 0.05 N. The curves on top present the imposed distance between the magnet and the sample surfaces; the curves on bottom show the corresponding normal force between the clamp and the foam. The negative times and forces correspond to the bonding contact stage. The curve fragments in light and dark correspond, respectively, to the low field ($B \approx 0$ at constant distance $z_{magnet} = 30$ mm from the magnet) and to the field increasing to its maximum value at $z_{magnet} = 1.85$ mm.</figcaption></figure>

### III. Conclusion

In summary, we showed that magnetic PDMS foams may be used as magnetically compressible actuators with low Poisson's ratio. We studied the magnetic deformation of model PDMS foams filled with carbonyl iron particles in a magnetic field gradient and highlighted the role of both foam height and the magneto-elastic ratio $\chi/E$ on actuation strain. The non-uniform strain field and the presence of an optimum foam thickness to maximize actuation strain is typically not accounted for in the literature. This effect may be generalized for other cases of soft magnetic materials, such as ferrogels, magnetic sponges and magnetic metamaterials.

We proposed an analytical model supported by more advanced simulations that could predict strain and stress distribution within the sample, in good agreement with experimental data. Finally, we demonstrated a proof-of-concept in the form of a switchable adhesive surface
induced by the magnetic compression of a foam sample covered with a layer of pressure sensitive adhesive. The proposed structure allows adhesive bonding of relatively small weights ($F_{load}/A_{adhesive} \sim 0.1$ kPa) for several minutes and on-demand debonding within few seconds as controlled by the controlled motion of a magnet. This concept does not require any complex surface treatment and may be adapted to soft robotics grippers to adjust the grasping ability.
# IV. Experimental section

## IV.1. Materials

Sylgard® 184 and Sylgard® 527 (DOW Corning) silicone elastomer kits were purchased from Neyco. Carbonyl iron particles (CIP) are spherical iron particles with purity of 97.80% and median size of 4.55 µm, and were obtained from Neochimie. NaCl salt was purchased from Sigma Aldrich.

## IV.2. Magnetic foam fabrication and modification

Magnetic foams were synthesized using a sacrificial templating technique adapted from those reported in literature²⁹. The NaCl crystals were compressed in a powder compression mold with internal diameter of 13 mm under pressures ranging from 450 MPa to 750 MPa to obtain porous cylindrical pellets. The height of the pellets depended on the mass of salt used, and was typically between 0.5 and 10 mm. The pellets were infused by the dispersion of CIP in a liquid silicone elastomer. The liquid silicone was composed of a mixture of 25% wt. of Sylgard 184 and 75% wt. of Sylgard 527. The mixing ratio of the curing agent to the base was maintained as 1:10 for Sylgard 184 and 1:1 for Sylgard 527. The mixture was stirred using a Heidolph RZR 2102 control stirrer at 1000 RPM for 5 minutes to ensure homogeneity and degassed under vacuum until the absence of bubbles (around 5 minutes). The infusion of the pellet was performed by placing it in the PDMS-Fe mixture under vacuum. The excess mixture was removed after infusion and the samples were cured at 60°C overnight. The NaCl template was washed by water to give porous cylindrical samples composed of magnetic silicone foam.

In the case of adhesive foams, the cured foam was covered by a thin layer of soft PDMS (10% of Sylgard 184 – 90% of Sylgard 527) that was cured before dissolving the NaCl template, to act as a PSA layer attached to the foam surface. This method avoids PSA infiltration of liquid PDMS inside pores and allows for a strong interface between PSA and the foam. After washing and drying, the magnetic foam covered with an PSA on one side was glued to a glass substrate from the other side and surrounded by a rigid spacer with thickness of 6 mm made of a commercial PDMS composite sheet purchased from Azuc Caoutchouc.
### IV.3. Characterization of pore structure and magnetic properties

The foam structure was observed on cut samples using scanning electron microscopy (SEM, Quattro Fisher-Scientific) using secondary electron detector to visualize pore morphology and using backscattered electron detector to visualize the CIP distribution. Images from both detectors were layered using ImageJ to show both morphology and CIP distribution on a single image.

VSM magnetization measurements were carried out using a Quantum Design Superconducting Quantum Interference Device magnetometer (MPMS SQUID VSM), equipped with a superconducting magnet ($H_{max} \pm 7$ T). Powder samples were encapsulated in standard gellula which were then inserted in magnetometer manufacturer provided plastic straws. Magnetic hysteresis curves ($H_{max} \pm 5$ T) were measured in VSM mode at room temperature to determine the saturation magnetization Ms.

### IV.4. Mechanical and magneto-mechanical characterization

Mechanical curves were obtained using an Instron 6800 testing system machine equipped with a 2530-100N force cell. Three test geometries with home-made duralumin clamps were used. For non-magnetic compression, the sample was placed on the bottom clamp plate and the upper clamp plate was approached at a strain rate $0.01 \text{ s}^{-1}$ until the imposed stress value. To minimize friction, a small amount of paraffin oil was used as lubricant on both plates.

Magnetic field measurements and calculations using the MagPylib[^40] routine are described in the Supporting information. For compression in a magnetic field, the sample was placed on the bottom clamp containing a hole to allow the approach of a magnet. A S20-20-N magnet from Supermagnete which was strongly attached with an adhesive tape on a hollow upper clamp which was fit into the bottom clamp in order to place the magnet axially under the sample. The magnet was approached at a speed 20 mm/min until the contact between clamps, which corresponds to the minimum distance of 1.85 mm between the magnet and the sample. A camera was used to film the deformation of the sample at frequency of 1 Hz. For magnetically-triggered detachment of foams covered with PSA, the sample glued to a glass plate was fixed in the bottom clamp containing a S20-20-N magnet. The position of the magnet may be changed from 30 mm to 1.85 mm below the surface of the foam. The upper clamp contains a 10 mm glass probe parallel to the sample surface. The test consists in approaching
the contact probe at a speed of 0.05 mm/s, up to a given contact force $F_{contact}$. Once this force is reached, the displacement is moved to a positive distance $d_{dwell}$ (the foam is under tension), to act like a constant force as gravity would do. This distance is maintained for a given time $t_{dwell}$. After this time is reached, the magnet is moved manually, by unscrewing the magnet at a speed of 1 turn per second, which corresponds to a height change of 0.33 mm/s.

## IV.5. Digital image correlation

The recorded videos and strain field associated with foam compression under a magnetic field were treated using a custom-built digital image correlation algorithm. A rectangular mesh (i, j) of tracers with positions $(x_i, z_j)^0$ was selected on the initial frame, corresponding to the non-deformed foam. The typical mesh size (typically about 1 mm) was larger to the foam pore sizes. We used the Farneback algorithm available in Python to determine the displacement field $(u_x, u_z)_n^{(i,j)}$ from mesh points between two consecutive frames (n) and (n+1). The new positions of the tracers were found as

$$ (x_i, z_j)^N = (x_i, z_j)^0 + \sum_n^{N} (u_x, u_z)_n^{(i,j)} $$

To represent the local strain $\epsilon$, a mesh whose grid points were located at the centers of the cells (p,q) with new coordinates $(x_p, z_q)^N = \left(\frac{x_i+x_{i+1}}{2}, \frac{z_j+z_{j+1}}{2}\right)^N$ was used. The strain components at the Nth frame were calculated as:

$$ \epsilon_{xx}^{(p,q)} = \frac{(x_{j+1}-x_j)^N}{(x_{j+1}-x_j)^0} $$

$$ \epsilon_{zz}^{(p,q)} = \frac{(z_{j+1}-z_j)^N}{(z_{j+1}-z_j)^0} $$

## Author contributions

M.B. and A.K. conceived and initiated the project; M.B., S.L., A.D.-L., F.V. and A.K. performed most of the experiments and modelling; J.R. performed magnetic characterization; M.B., J.-C. M., L.O. and A.K. designed mechanical and magneto-mechanical setups; M.B. fabricated
materials, characterized their structure and mechano-magnetic properties; S.L. and A.D.-L.
performed COMSOL simulations; M.B. and A.K. drafted the manuscript. All the authors
contributed to the manuscript in its final form.

Conflict of interests

The authors declare no competing financial interest.

Acknowledgments

We thank Jérôme Fresnais, Michel Coret, Guilhem Baeza, Etienne Barthel and Bruno Bresson
for fruitful discussions. We are grateful to Ainhoa Merry, Aubry Martin and Jingyuan Li for their
help on early stages of the project. Behnam Rezaei is acknowledged for his support with
COMSOL. This work was funded by French Agence Nationale de la Recherche, grant MAGELAn
<ANR-19-CE06-00244>. F. J. Vernerey and S. Lamont acknowledge support by the Department
of Energy, National Nuclear Security Administration, Predictive Science Academic Alliance
Program (PSAAP) under Award Number DE-NA0003962.
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