# A PDMS/Silicon Adhesion Control Method at Millimeter-Scale Based on Microvibration

Jiawei Yi, Wissem Haouas, Michaël Gauthier, Kanty Rabenorosoa

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Jiawei Yi, Wissem Haouas, Michaël Gauthier, Kanty Rabenorosoa. A PDMS/Silicon Adhesion Control Method at Millimeter-Scale Based on Microvibration. Advanced Intelligent Systems, 2025, 7 (2), pp.2400394. 10.1002/aisy.202400394. hal-05221261

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<figure><img src="image_1.png" /></figure>
# A PDMS/Silicon Adhesion Control Method at Millimeter-Scale Based on Microvibration

Jiawei Yi,* Wissem Haouas, Michaël Gauthier, and Kanty Rabenorosoa

Switchable surface adhesion at a small scale is crucial for robot end-effector design, allowing the manipulation of small objects such as semiconductors, optical lenses, and precision mechanical parts. In this work, a detailed characterization of a millimeter-scale (1–5 mm) adhesion modulation method is performed, demonstrating its effectiveness for switching adhesion on small, lightweight objects with smooth surfaces. This modulation phenomenon arises from the viscoelastic behavior when PDMS interacts with a rigid surface and is controlled via microvibration. A maximum apparent adhesion enhancement of 2400% and a reduction of 50% are achieved with a 1 mm-diameter PDMS hemisphere vibrating at a 30 µm amplitude and a 700 Hz frequency. The effects of different parameters, including size, actuation amplitude/frequency, surface roughness, and material properties, on adhesion performance are carefully measured and analyzed. A monotonic increase in maximum adhesion is observed with increased device size and surface smoothness, while nonlinear relationships of other factors are generalized with a numerical model. A long working lifespan and high endurance are also observed during the characterization. This work serves as a practical reference for the further design of small-scale soft grippers, highlighting its continuous, large modulation range, simple structure, and flexible control.

## 1. Introduction

In the field of robotic manipulation, adhesion control has always been a concern for the scientific community, whether it is to utilize it positively on macroscale to achieve wall climbing,[1,2] object manipulation tasks,[3,4] or to overcome it on microscale as the adhesive contact emerges as a major problem for micromanipulation tasks.[5-8] With a considerable and stable modulation performance, adhesion control could also provide robots with a significant advantage in the application scenario of space manipulation,[9,10] soft gripping,[11,12] and biomedical operation.[13]

J. Yi, W. Haouas, M. Gauthier, K. Rabenorosoa
Department of Automation and Micromechatronic Systems (AS2M)
FEMTO-ST Institute
Besançon 25000, Franche-Comte, France
E-mail: jiawei.yi@femto-st.fr

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/aisy.202400394.

© 2024 The Author(s). Advanced Intelligent Systems published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/aisy.202400394

Over the past decades, the possible merit of adhesion control has inspired scientists to investigate all the possible principles that could be used to modulate adhesion force. For example, in nature, gecko ability to climb on walls and maintain adhesion strength several times its body weight has shed light on the study of microfibrillar structures,[14-16] enabling the strengthening of the adhesion force ranging from 20 to 700 N over large contact surface ranging from 1.5 to 150 cm²[1] in dry media, with its switchability realized by load control in either direction[17-20] or strength.[21-23] In another biomimetic case, the octopus tentacles show another way of improving adhesion strength using both adhesion and suction,[24] which was proven to be effective in both dry and liquid environments.[25] Apart from the mechanical structures, electrostatic force is also reported to provide physical bonding with no preload requirement,[26] showing enormous potential in soft grippers,[27,28] wall climbing,[29,30] and possible hybridization

with other structures such as microfibrillars.[31-33] However, the high voltage demand (usually above 1 kV) and low absolute adhesion strength have placed limitations on this principle. On microscale, the surface tension of liquid has also been developed as an alternative for small-scale gripping.[34] It provides an increasing amount of adhesion strength as the line-to-volume ratio increases. Nonetheless, the liquid residual and its control difficulties have reined up the application of this technology. Over the years, other viable novel methods were explored to refine the performance of adhesion control, including thermal,[35] pneumatic,[36] and magnetic.[37] While possessing their novelty and merits, the adhesion performances have not exceeded the aforementioned ones significantly in the aspects of flexibility, response time, adhesion strength, and so on.

In 2015, Xu et al.[38] introduced a primal design of a rate-dependent pick-and-place device based on a graphene tip and polymer substrate, utilizing the adhesion hysteresis nature of the polymer. Yet it relies on an adhesive substrate to achieve release for it is hard to reduce the adhesive force on small scale. Further developed by Li et al.[39] a polymer hemisphere was directly investigated to show its adhesion strength under different contact velocities. In 2020, Shui et al.[40] displayed a technique of actively modulating the adhesion between polymer-glass contact with acoustically induced microvibration on macroscales, showing a maximum of 76.83 times enhancement
in adhesion strength and a reduction to 0. However, its require-
ment for a smooth contact surface has limited its application on
macroscale. And, to our best knowledge, no available method has
been developed to apply this phenomenon in a precise, flexible,
and repeatable manner.

To address the above dilemma, we explore in this work the
potential of applying the adhesion hysteresis method at small
scales, tailoring it to handle a variety of smooth-surfaced and
relatively clean environments, small objects commonly found
in industries such as semiconductors, ceramics, and photonics.
Our objective is to characterize and model the influence of key
parameters on the method's performance at a small scale to
validate and optimize its practicality. We consider the effects
of surface roughness, Young's modulus, and device size, which
are critical factors arising from fabrication, as well as vibration
amplitude and frequency as active actuation parameters. This
soft adhesion system regulates contact adhesion through micro-
vibration, circumventing traditional load triggers, and enabling
self-attachment and release without additional substrate support.
It offers practical solutions for environments with low air
pressure or vacuum conditions, such as scanning electron micro-
scope (SEM) chambers, where traditional vacuum grippers used
in cleanrooms may not be suitable. This device can enhance
adhesion by up to 2400% for a 1 mm hemisphere and achieve
stable reduction by about 50% for all samples within small ampli-
tude and frequency ranges. We present the evolution of adhesion
modulation ability with dimension scaling, highlighting the
potential for further downscaling. Two separate fabrication meth-
ods were developed to produce small-scale PDMS hemispheres
with varying sizes (1–5 mm in diameter) and different surface
roughnesses. Adhesion behavior measurements of the fabricated
samples informed the development of a new analytical model
based on viscoelastic crack propagation and dynamic modeling
to explain and describe the evolution of adhesion behavior.

## 2. Results

### 2.1. Working Principle and General Performance of Adhesion Modulation

**Figure 1a** shows the basic mechanism of our adhesion modula-
tion device. When a soft hemisphere of radius **R** is in static
contact with a rigid substrate, the apparent adhesion force
**F**<sub>app</sub> is typically defined as the difference between static surface
adhesion **F**<sub>s</sub> caused by intermolecular attraction<sup>[41,42]</sup> and the
mechanical repulsive force **F**<sub>r</sub> caused by elastic deformation,
which is determined by the contact radius **a** and the indentation
**δ** respectively. However, when the hemisphere peels off from the
surface at a speed **V**<sub>cp</sub>, its viscoelastic properties cause crack
propagation to exhibit hysteresis compared to an ideal elastic
hemisphere, increasing the work of adhesion. This work of adhe-
sion increases monotonically with **V**<sub>cp</sub> during crack propagation
and decreases during crack healing within a certain velocity
range, showing as an increase of dynamic adhesive force **F**<sub>d</sub>.
The excess energy generated dissipates into the material after
the crack propagation/healing process, allowing modulation
of the adhesion force with varying **V**<sub>cp</sub>. To achieve continuous
force adjustment, we introduce vertical harmonic vibration,

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Adhesion mechanism. a) Schematics of a vibrating PDMS hemi-sphere in contact with a rigid substrate. b) Hysteresis phenomenon of crack propagation and healing at the crack tip of a dynamic contact under the same indentation. Solid lines and dashed lines represent the profiles of viscoelastic and elastic material movement in the same oscillation condition separately. (I) Direction of crack healing. (II) Direction of crack propagation.</figcaption></figure>

causing the hemisphere to peel and heal repeatedly in a stable
cycle, making $F_{\text{app}}$ a function associated with the vibration ampli-
tude $A$ and frequency $f$ at any fixed indentation.

**Figure 2a** shows a quasistatic approach-retract process (regarded as A–R process) performed between a PDMS hemisphere of 5 mm diameter and a silicon substrate. The retraction speeds in all the experiments are carefully controlled at 1 µm s⁻¹ to simulate a quasistatic condition on our scale. As demonstrated in Movie S1, Supporting Information, the maximum apparent adhesion force ($F_{a\max}$) is achieved when the adhesive force cannot support the elastic deformation and the contact area collapses, inducing a detachment within a short retraction distance $\Delta\delta$ with a typical $F_{d\max}$ of 2.37 mN.

After vibration is introduced, as described previously, the
apparent adhesion would increase along with vibration ampli-
tude and frequency. Figure 2b illustrates the A–R process under
large-amplitude (30 µm), high-frequency (700 Hz) vibration.
These amplitude and frequency values are chosen based on their
earliest enhancement saturation, as shown in **Figure 3**. With a
high-velocity surface peeling, the apparent adhesion force is
enhanced to a typical value of 18.84mN. Concurrently, the verti-
cal vibration causes the pull-off process to become continuous,
extending the retraction distance $\Delta\delta$. The evolution of the
pull-off process is depicted in detail in Movie S2, Supporting
Information. Notably, at the onset of vibration, the average con-
tact radius stabilizes after ≈1.5 vibration periods, providing the
system with a response time of 2.2 ms, as depicted in Movie S3,
Supporting Information.

However, under vibration conditions with a relatively small
amplitude compared to the device scale, such as the typical
6 µm–700 Hz vibration shown in Figure 2c, viscoelastic enhance-
ment is minimal. Instead, the vertical vibration induces instabil-
ity around the contact edge, leading to premature pull-off. This
results in a typical pull-off force of 1.23 mN, demonstrating an
effective reduction in adhesion.

In summary, as shown in Figure 3a,b, a reduction in adhesion
is observed at low amplitudes, which remains nearly insensitive
to frequency changes. In contrast, the enhancement effect occurs
at higher amplitudes and increases almost monotonically with
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** A–R process and adhesion response to vibration. a) Force-displacement plot of an A–R process with no vibration. Numbers in the figure indicate the corresponding steps 1: approach, 2: preload, 3: maintaining preload, 4: slow retraction, 5: pull off. $\Delta\delta$: Retraction distance, $F_{amax}$: maximum apparent adhesion force. b) A–R process with enhanced adhesion. The numbers show a different process as 1: approach, 2: preload, 3: maintaining preload, 4: applying vibration ($A = 30 \mu m$, $f = 700$ Hz), 5: slow retraction, 6: pull-off, 7: adhesion deterioration. c) A–R process with reduced adhesion. The numbers show the same process as (a), but a vibration ($A = 3 \mu m$, $f = 700$ Hz) is applied after 3.</figcaption></figure>

both amplitude and frequency. A clear saturation effect is evident at high amplitudes, attributed to the increase in average repulsive force $F_r$. This effect will be discussed in detail in the modeling section. For a 5 mm PDMS hemisphere contact as shown in this section, an adhesion modulation range of 51–799% is achieved.

## 2.2. Scaling Potential of the Adhesion Modulation

To demonstrate the modulation capability of the phenomena across various sizes, a series of optically smooth PDMS hemispheres with different diameters (1, 2, 3, 4, and 5 mm) were fabricated and tested against a silicon surface. Full scans of the maximum adhesion force under different frequencies and amplitudes were performed. The result, which is chosen as the median value of five repetitions, is demonstrated in **Figure 4a**. In our frequency and amplitude range, the scans are clearly divided into an enhancing zone and a weakening zone.

In the weakening zone, the weakening effect appears to be nearly indifferent to frequency change and maintains a weakening ratio of around 50%. As shown in Figure 4b, the default pull-off force conforms to the Johnson-Kendall-Roberts model[41] $F_0 = 1.5\gamma\pi R$ ($\gamma$ is the surface energy of the contact) with a nearly linear scaling effect while containing a minor offset. The weakened pull-off force also shows a near-linear scaling effect with an offset. Such offset could be induced by fabrication defects, which are observable in Figure 6a.

In the enhancing zone of the 5 mm-diameter scan, $F_{amax}$ nearly monotonically increases as the frequency and amplitude increase, while reaching saturation at around 28 µm amplitude and 700 Hz frequency. Such saturation is observed in multiple hemispheres. In **Figure 5**, a nearly linear scaling effect of the maximum enhancement is observed with negligible offset. The enhancing ratio, however, increased from 7.99 to 24.03 as the diameter decreased from 5 to 1 mm, largely due to the offset of the default force. It should be noted that a full scan consists of 385 A–R experiments with five repetitions in each specific amplitude-frequency combination. Three full scans (each lasts 12 h, 1155 experiments in total) were conducted within a time span of 30 days to examine the endurance of the device and negligible difference in performance is observed. But in order to maintain the consistency of experimental conditions, we only analyze the data in one consecutive scan.

## 2.3. Roughness-Induced Performance Deterioration

Three 5 mm-diameter PDMS hemispheres with different surface roughnesses are fabricated to evaluate the influence of roughness on adhesion modulation performance. **Figure 6a** shows the microscopic image of the fabricated surfaces. The surface roughness is measured with 3D reconstruction using SEM images (see Note S4, Supporting Information) and the detailed measurements are shown in **Table 1**. The smooth surface maintains optical clarity with unidentifiable roughness. The medium surface maintains a coarse wavy surface constructed during fabrication while the rough surface consists of dense, microscopic spikes and valleys.

During vibration, a smooth surface could ensure a wholesome contact and smooth crack propagation/healing process.
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Comparison of maximum adhesion of different vibration conditions. a) Adhesion evolution along amplitude under different frequencies (0, 100, 400, 700 Hz). b) Adhesion evolution along frequency under different amplitudes (0, 6 15, 30 µm). Shadows represent the standard deviation of the acquired data in five consecutive A–R processes.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Full scan of $F_{amax}$ and scaling effect. a) Measured normalized $F_{amax}/F_{default}$ corresponding to different actuation frequencies and amplitudes. The green square indicates the region of reduced force while the red square indicates the region of enhanced force. I: 1 mm diameter, II: 2 mm diameter, III: 3 mm diameter, IV: 4 mm diameter, V: 5 mm diameter. All the values are the median values of five repeated experiments. b) Minimum $F_{amax}$ in the scan corresponding to different hemisphere diameters. Dashed lines show the theoretical data assuming that there is no adhesion when the diameter is 0. The shaded area shows the standard deviation in five repetitions.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Scaling effect of the maximum adhesion force. Maximum $F_{a,max}$ in the scan corresponding to different hemisphere diameters. Dashed lines show the theoretical data assuming that there is no adhesion when the diameter is 0. The shaded area shows the standard deviation in five repetitions.</figcaption></figure>

Nevertheless, a rough surface, as shown in Figure 6b, contains irregularities. During attachment, the protruding spikes would generate excessive deformation, which in turn appears as a repulsive force counteracting the adhesion. The valleys, on the other hand, would perform as cavities during fast vibrations, capturing air in the process (similar to Movie S1, Supporting Information)

<table><caption>**Table 1.** Roughness parameters of different surfaces.</caption><thead><tr><td>Roughness level</td><td>Deviation average [µm]</td><td>Peak-to-valley [µm]</td></tr></thead><tbody><tr><td>Smooth</td><td>&lt;1</td><td>—</td></tr><tr><td>Medium</td><td>2.176</td><td>9.760</td></tr><tr><td>Rough</td><td>2.595</td><td>15.26</td></tr></tbody></table>

and decreasing the effective contact area. These two effects combined could seriously damage the adhesion modulation ability of the device. Figure 6c,d reveals the relationship between surface roughness and modulation capability. In accordance with the adhesion hysteresis analysis, as we increase the vibration amplitude at a certain frequency, the adhesion force should increase until reaching a certain saturation value. This saturation value determines the maximum modulation capability of the device. As we can observe, at 700 Hz, the medium surface would reach enhancement saturation before the smooth surface at a smaller adhesion value. At 100 Hz, the medium surface would reach saturation at a lower level before the smooth surface in both enhancement and weakening effects, demonstrating a generally smaller adhesion modulation range. In both experiments, the rough surface provides only negligible adhesion ($<1 \times 10^{-3}$ mN), rendering it the least capable surface in modulating adhesion.

It is worth mentioning that surface roughness has been observed to contribute to the enhancement of adhesion forces at low retraction speed in previous literature due to elastic instability[43–45] and enhanced hydrostatic pressure,[46] which might partially explain the higher adhesion force of the medium-rough surface at low amplitude and low frequency in

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Measurement and schematics of the roughness effect on adhesion. a) SEM images of the surfaces of different roughness. I: Smooth surface, II: medium surface, III: rough surface. b) Schematic of the roughness influence mechanism. I: schematics of adhesion modulation, II: microscopic behavior of an active contact. c) Amplitude scan of different roughnesses under low-frequency vibration (100 Hz). d) Amplitude scan of different roughnesses under high-frequency vibration (700 Hz). The shaded regions represent the standard deviation within five repetitions.</figcaption></figure>
Figure 6c. However, it is found that the viscoelastic effect could overshadow such an effect at high retraction speed.[47] As our enhancement performance is better at higher amplitude and frequency, it is safe to assume that the smooth surface provides the best adhesion enhancement performance in our context.

## 2.4. Material Influence

A sequence of PDMS hemispheres of 5 mm diameter with different mechanical properties is fabricated by controlling the mix ratio of PDMS polymer and curing agent, baking temperature, and time. The specific fabrication parameters are described in Table 2 with a reference to Young's modulus.[6] A series of A-R experiments are carried out on all the hemispheres against a silicon substrate to find the material that could optimize the performance of the adhesion modulation in our frequency and amplitude range. The most distinguishable experiment results, which are the amplitude scan at high frequency (700 Hz), are shown in Figure 7. The modulation capability increases with Young's modulus until it reaches its maximum (with $E=1.5$ MPa in this experiment). The dominance of the effect of Young's modulus weakens and a nonmonotone

<table><caption>**Table 2.** Different parameters for the PDMS hemisphere fabrication and their corresponding reference Young's modulus.</caption><thead><tr><th>Material number</th><th>Mixing ratio</th><th>Temperature [°C]</th><th>Baking time [min]</th><th>Young's Modulus [MPa]</th></tr></thead><tbody><tr><td>M1</td><td>12:1</td><td>85</td><td>60</td><td>0.75</td></tr><tr><td>M2</td><td>12:1</td><td>85</td><td>120</td><td>1.13</td></tr><tr><td>M3</td><td>10:1</td><td>100</td><td>60</td><td>1.50</td></tr><tr><td>M4</td><td>12:1</td><td>150</td><td>60</td><td>1.65</td></tr><tr><td>M5</td><td>12:1</td><td>150</td><td>90</td><td>1.95</td></tr><tr><td>M6</td><td>8:1</td><td>150</td><td>60</td><td>2.36</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Amplitude-stiffness scan of maximum adhesion at the high-frequency surface plot of the median values of the maximum adhesion force over five repeated experiments. The plot is based on six materials fabricated under different conditions all vibrating under a fixed frequency of 700 Hz.</figcaption></figure>

decrease of enhancement ability appears as Young's modulus further increases. After full performance mapping of the six materials, the material with the optimum modulation ability is material 3, which presents us with an average adhesion enhancement of 799% and weakening of 51% over five repetitions.

With the increase in curing temperature, time, and a decrease in mixing ratio, it is common to find an increase in Young's modulus and a decrease in viscosity in PDMS. With lower viscosity, relatively larger hysteresis caused by longer material relaxation time is observed in dynamic contact. Especially at high-frequency, high-amplitude vibration processes, hysteresis could cause a damping effect in the contact radius oscillation. With decreased viscosity, the damping effect is reduced and the effective adhesion enhancement is elevated. However, when Young's modulus increases higher than a threshold, the mechanical repulsive force takes dominance and it would appear that the apparent maximum adhesion force would decrease (see Note S1, Supporting Information). Yet, as the relationship between viscosity and Young's modulus is not specifically characterized, a nonmonotone decrease is reasonably expected.

# 3. Modeling

## 3.1. Performance Modeling of Adhesion Modulation Systems

A numerical model describing the adhesion response and the corresponding theoretical explanations is provided to understand and analyze the observed phenomena. In a vibration condition as shown in Figure 1, the transient contact load $F_{tra}$ could be expressed as the difference of $F_d$ and $F_r$, while the dynamic adhesion force could be expressed as a function of the strain release rate (see Note S1, Supporting Information).

$$F_{tra} = F_r - F_d \qquad (1)$$

$$F_d = G \frac{d\pi a^2}{d\delta} \qquad (2)$$

where G is the energy release rate of the PDMS/silicon contact and $\delta = \Delta + A \sin(2\pi ft)$ is a sinusoidal vibrating indentation. In viscoelastic contacts, the energy release rate is related to the crack propagation velocity and could be expressed in an empirical function[48,49] (see Note S1, Supporting Information).

$$G = \begin{cases} \Delta\gamma_0(1 + CV_{cp}^n), & \frac{da}{dt} < 0 \\ \Delta\gamma_0(1 + CV_{cp}^n)^{-1}, & \frac{da}{dt} \ge 0 \end{cases} \qquad (3)$$

where $\Delta\gamma_0$ is the adiabatic work of adhesion and C, n are characteristic constants describing the energy dissipation caused by the material's viscoelasticity. According to Movie S2, Supporting Information and Figure 2b, we can assume that at high frequency and high amplitude, the maximum adhesion force takes place when the minimum indentation approaches 0, that is, $\Delta = A$. Yet, as the frequency and amplitude decrease, the weakening effect becomes dominant, and, to the best of our knowledge, no theoretical model has yet been proposed for such phenomena.
Hence here we propose to use a numerical sigmoid function to compensate for the weakening effect as it represents a transition between static adhesion and dynamic adhesion. Thus we could

further derive the expression of the transient adhesion force at its highest adhesion $F_{tra}$ and the highest apparent adhesion force $F_{a max}$ (see Note S1, Supporting Information).

$$F_{tra} = \begin{cases} \Delta\gamma_0\pi R_e \left( 1 + C \left( \frac{\pi f \sqrt{R_e A} |\cos(2\pi f t)|}{1 + \sin 2\pi f t} \right)^n \right) - \frac{4}{3} A^{1.5} E R_e^{0.5} (1 + \sin 2\pi f t)^{1.5}, & \frac{da}{dt} < 0 \\ \Delta\gamma_0\pi R_e \left( 1 + C \left( \frac{\pi f \sqrt{R_e A} |\cos(2\pi f t)|}{1 + \sin 2\pi f t} \right)^n \right)^{-1} - \frac{4}{3} A^{1.5} E R_e^{0.5} (1 + \sin 2\pi f t)^{1.5}, & \frac{da}{dt} \ge 0 \end{cases} \quad (4)$$

$$F_{a max} = \left(1 - \frac{1}{1 + e^{-\frac{a(f-f_0)(A-A_0)}{f_0 A_0}}}\right) F_0 + \left(\frac{1}{1 + e^{-\frac{a(f-f_0)(A-A_0)}{f_0 A_0}}}\right) f \int_0^\frac{1}{f} F_{tra} dt \quad (5)$$

in which E is Young's modulus, $F_0$ is the quasistatic pull-off force, $R_e=R$ is the equivalent radius of the contact sphere, and $a, f_0, A_0$ are characteristic coefficients of the system. However, as both frequency and amplitude decrease, the weakening effect becomes more prominent, and currently, there is no theoretical model available to explain this phenomenon. Therefore, in this context, we employ a numerical sigmoid function to counteract the weakening effect. This function serves as a transitional representation between static adhesion and dynamic adhesion.

In a typical case of $R=2.5$ mm radius contact, we can take $E=1.5$ MPa as a typical value from our fabrication and $\Delta\gamma_0=0.202$ N m$^{-1}$ derived from the quasistatic pull-off force $F_0=2.38$ mN.[41] For simplicity, here we take $f_0=350$ Hz, $A_0=15$ µm, $a=5$, and $n$, $C$ could be taken from the best fitting result, as shown in **Figure 8a**. For different hemisphere radii, the contact geometry difference results in a varied set of $n$ and $C$, and the best-fitting parameters could be obtained by fitting optimization (see Note S1, Supporting Information). The best-fitting coefficients are shown in **Table 3**.

The comparison between the numerical model and the experiment data on different dimensions is demonstrated in Figure 8b. The saturation phenomenon, which is commonly explained as a result of dynamical instability,[40] could be clearly observed here in both experiment data and numerical modeling on all five dimensions. Such a phenomenon could be analytically explained with Equation (1). In a vibrating condition, as the vibration amplitude increases, the dynamic surface adhesion $F_d$ in the process would increase along with crack propagation velocity. However, the mechanical repulsive force $F_r$ also escalates as a result of increased average indentation. As the amplitude reaches a certain level, the increase of $F_r$ takes dominance and the apparent adhesion force starts to decline.

## 4. Discussion

In this work, we have concluded several factors that could affect the adhesion modulation capability of the system and have found

<table><caption>**Table 3.** Best fitting parameters for numerical model.</caption><thead><tr><td>Diameter [mm]</td><td>n</td><td>C [m<sup>n</sup>/s<sup>n</sup>]</td></tr></thead><tbody><tr><td>1</td><td>1.97</td><td>4987.81</td></tr><tr><td>2</td><td>1.85</td><td>1547.83</td></tr><tr><td>3</td><td>1.68</td><td>704.81</td></tr><tr><td>4</td><td>1.37</td><td>347.40</td></tr><tr><td>5</td><td>1.05</td><td>183.22</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** Comparison between the numerical model and experimental data. a) Amplitude scan of the numerical model and experiment data at 700 Hz with sample diameter of 5 mm. $n=1.05$, $C=183.22(m^3/s^3)$. b) Amplitude scan of the numerical model and experiment data at 700 Hz with different diameters. The points represent the experiment data while the solid line represents the numerical model.</figcaption></figure>
the optimum parameters or conditions in each factor. A smooth
hemisphere fabricated with the same process as material 3 could
achieve the best modulation performance in all the comparative
experiments. The scaling study shows a higher enhancement
potential at a smaller scale while the reduction performance
remains practically the same for different sizes. Such a result
shows us the possibility of applying this phenomenon at an even
smaller scale as the maximum and minimum forces show a
near-linear correlation with the dimension. However, more
investigation should be conducted to identify and characterize
the reduction effect at small amplitude, especially in the context

of microgripping, where object release takes priority. In addition,
while our soft elastomeric system offers exceptional repeatability
in controlling adhesion force surfaces by amplifying and reduc-
ing it, its performance should be considered to also conform to
rough surfaces, among other factors.

**5. Experimental Section**

*Structure and Measurement of the Soft Adhesion System: Figure 9a shows the detailed structure of the adhesion system and the experimental setup for time-averaged adhesion force measurement. A smooth, soft PDMS*

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Experiment setup. a) Schematics of the experimental setup for precise adhesion strength measurement in a dynamic process. b) The effective adhesion modulation actuation (microvibration) and the corresponding smooth PDMS/silicon contact. c) Schematics of the control system.</figcaption></figure>
hemispherical fingertip was attached to a piezoelectric displacement actuator (CEDRAT, APA50XS). The actuator could provide microvibration with precise control over amplitude and frequency. Three compact linear stages (PI MIKROMOVE M-111) were attached to the end of the actuator to provide microscale position adjustment while macroscale position adjustment was realized with a three-axis manual linear stage. The soft fingertip was in direct contact with a smooth silicon wafer attached to an aluminum block, as shown in Figure 9b. The heavy aluminum block worked as a devibrator with its overall weight (80 g) significantly higher than the equivalent maximum adhesion weight (2.0 g). The vibration amplitude was verified with the measurement of two laser interferometers (SIOS, NT-04) pointing at both the fingertip cantilever and devibrator surface. The force measurement was performed on an electromagnetic balance (METTLER TOLEDO, MS105DU) with a precision of 0.1 mg. The stabilization effects of the balance and the devibrator ensured the force measurement could receive negligible influence from the mechanical vibration and capture signal with minimum noise (see Note S2, Supporting Information).

As shown in Figure 9c, the control of the system relied on two routes of signal output and three feedback signals. The displacement signal was transmitted to the stage and real measurement feedback was perceived by the PC. Live force measurement was conveyed and recorded simultaneously and paired with the current displacement signal with a sampling frequency of 3 Hz. This sampling frequency proved to be sufficient for capturing the force evolution at our retraction speed, and a faster sampling was deemed unnecessary as the stabilization of the balance prevented further dynamic measurement. Vibration amplitude and frequency were passed on to a real-time control board (DS1104, dSPACE GmbH), which produced a sequence of vibration signals. Passing through a wideband amplifier (KRONE-HITE, Model 7602M), the sinusoidal driving voltage induced precise vibration on the piezoelectric actuator. The vibration displacement was then captured and fed back to the PC for processing and verification. The whole control system was built within the MATLAB SIMULINK framework.

**Hemisphere Fabrication for Different Diameters:** The smooth soft hemispheres used in the experiment were fabricated using a double-molding method; the schematic of the fabrication procedure is shown in **Figure 10a**. The model of the hemisphere is an optically smooth lens (Edmund optics, B270 Half-Ball Lens). The hemisphere was first washed with acetone and then poured with thermally curable polydimethylsiloxane (PDMS, Sylgard 184, Dow Chemical) polymer mixed with a curing agent, typically in a 10:1 ratio by weight. The model and the liquid were then degassed in a vacuum chamber for 30 mins and baked in an oven for 1 h at 130 °C to solidify the PDMS into a negative mold. After demolding, a small piece of silicon wafer was placed upon the cavity as a protective cover while a layer of Dry Film Lubricant (DFL) ELECTROLUBE was applied on the top surface and air dried. Such lubrication could ensure a successful detachment in the following process while the silicon cover protected the cavity from contamination which could result in a deterioration in surface smoothness. Another portion of PDMS was poured into the cavity and degassed for 30 min. A glass substrate was pushed against the top surface to create a flat back support and the whole structure was baked in an oven for 1 h, typically at 100 °C. After demolding, a soft PDMS hemisphere with optical smoothness and the diameter of the model lens was obtained.

**Hemisphere Fabrication for Different Surface Roughnesses:** To fabricate controllable roughness of 10 µm scale on a millimeter-scale hemisphere, it was difficult and time-consuming to use direct patterning fabrication due to the scale difference. Hence a physical approach was applied for the surface roughness control by controlling the curing of the PDMS surface,[50] as is shown in Figure 10b. A mold with a hemispherical cavity was 3D printed with high-temperature resin (Formlab 3) of 25 m layer thickness. The mold was post-treated in UV light for 4 h and heated to 130 °C (0 min for smooth surface, 30 mins for medium surface, and 60 mins for rough surface). After post-treatment, the mold was washed with acetone and air dried. A thin film of DFL was applied on the mold surface for easy detachment. Then, a portion of PDMS was poured into the cavity and degassed for 30 min. A glass substrate was pushed against the top surface to create a flat back support and the whole structure was baked in an oven for 45 min at 100 °C. After demolding, a solid PDMS hemisphere with a semi-fluid surface should be observed. The degree of solidification of the surface

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Schematics of different fabrication processes. a) Fabrication for smooth surface PDMS hemisphere with different diameters. (1) surface cleaning, (2) pouring liquid PDMS, (3) degassing, (4) baking, (5) demolding, (6) spray lubricant, pour liquid PDMS and degassing, (7) baking, and (8) demolding. b) Fabrication of PDMS hemisphere with different surface roughnesses: (1) postcuring, (2) cleaning, (3) spraying lubricant, (4) filling liquid PDMS, degassing, (5) first baking, (6) demolding, (7) second baking, and (8) finish.</figcaption></figure>

depended on the heating time in the post-treatment. The hemisphere was then baked in an oven for another 1 h at 100 °C to produce the end product of a solid PDMS hemisphere. During baking, the semifluid surface would produce a half-smooth surface due to surface tension, and the final smoothness depended on the degree of solidification when demolded.

## Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.
## Acknowledgements

This work was supported by the EIPHI Graduate school (contract "ANR-17-EURE-0002") and by the Bourgogne-Franche-Comté Region. This work was partly supported by the French ROBOTEX network (TIRREX ANR-21-ESRE-0015), RENATECH network, and their FEMTO-ST technological facility CMNR.

## Conflict of Interest

The authors declare no conflict of interest.

## Author Contributions

**Jiawei Yi:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (equal); Validation (lead); Writing—original draft (lead); Writing—review and editing (equal). **Wissem Haouas:** Funding acquisition (equal); Investigation (equal); Project administration (supporting); Supervision (equal); Writing—review and editing (equal). **Kanty Rabenorosoa:** Funding acquisition (equal); Investigation (supporting); Project administration (lead); Supervision (lead); Writing—review and editing (supporting). **Michaël Gauthier:** Investigation (equal); Project administration (supporting); Supervision (supporting).

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Keywords

adhesion hysteresis, fracture mechanics, reversible adhesions, small scales

Received: May 16, 2024
Revised: August 17, 2024
Published online: September 23, 2024

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