The Role of Extracted Free Chains in Adhesion on Soft Gel Surfaces

Abstract

Adhesion on soft polymeric materials is crucial for many engineering applications. However, the substantial surface deformations of compliant substrates present challenges for classical contact theory in describing the adhesion behavior. Furthermore, surface contact can drive the migration of free solvent within polymer networks, further complicating the governing physics of soft adhesion. In this work, we used dual-wavelength confocal microscopy to characterize how contact-induced extraction of free chains affects the adhesion profiles of microspheres on soft silicone gels with varying cross-linking densities. Locally, a phase separation region emerged near the contact points while the balance of surface stresses was preserved. Globally, the indentation depth and contact radius of the spheres were described by incorporating the competition between adhesion energy and surface stress into classical contact models. Over extended measurement periods, we imaged in situ a slow increase in the indentation depth of attached spheres in all experiments. This finding indicates a gradual rise in adhesion energy, occurring on a time scale consistent with poroelastic diffusion in soft silicone gels.

1. Introduction

Adhesion between a rigid object and a soft substrate plays a critical role in various emerging engineering systems, such as bioinspired interfaces, soft robotics, , and flexible electronics. The high compliance of soft materials results in unique adhesion behaviors, distinct from those observed at stiff interfaces. , For instance, the Johnson–Kendall–Roberts (JKR) framework in classical contact mechanics breaks down within the elastocapillary length, where solid capillarity dominates. As a consequence, an effective description of soft adhesion mechanics requires precise experimental characterizations and a tailored theoretical analysis of interfacial profiles.

Furthermore, adhesion to soft polymeric materials can extract free solvent near the contact area. When a rigid object is placed on a soft substrate, the adhesion energy leads to a highly deformed contact ridge that induces pronounced strain and stress. The enhancement in local elastic energy may drive the migration of solvents within the gel toward the contact line. This phase separation process not only prevents stress divergence at the contact line but also alters the overall adhesion profiles. , However, while this extraction is widely expected under soft contacts, − direct observation of the solvent migration remains largely elusive due to experimental difficulties. Consequently, conventional microindentation characterizations on soft gels have primarily focused on overall force responses, , overlooking the dynamic formation of a phase-separated region and its evolving influence on adhesion. Additionally, theoretical modeling of soft adhesion often neglects impacts on adhesion profiles caused by the emergence of solvent extraction. ,

In this work, we used dual-wavelength confocal microscopy to image contact deformations induced by microspheres adhered to soft silicone gels with varying cross-linking densities. These measurements enabled in situ characterization of extracted free chains, revealing how solvent migration influences adhesion profiles on multiple length scales and modulates adhesion energy over time. Given the coupling between adhesion energy and surface stresses of soft gels, their adhesive responses can be quantitatively captured by a modification of classical contact models.

2. Experimental Methods

2.1. Materials

To synthesize the substrates of soft gels, we first prepared a base polydimethylsiloxane (PDMS) polymers (Gelest, DMS-V31) mixed with copolymer cross-linkers (Gelest, HMS-301) and catalysts (Gelest, SIP6831.2) on a glass coverslip (Marienfeld, 22 mm × 22 mm). The weight fraction of cross-linker (k) was varied to quantitatively tune the stiffness of the resulting gels. After performing spin-coating at 1000 rpm for 1 min, a thin layer of the uncured mixture formed on glass. The sample was then placed in a 40 °C oven for 24 h, resulting in a film of cured silicone gels with a thickness of 45 μm. Glass beads with nominal radii of 10–30 μm (Polysciences) were placed at the soft gel surfaces to induce adhesion contacts.

2.2. Confocal Microscopy

To visualize the adhesion-induced surface deformation near the contact line, both the surface and bulk of the soft gels were fluorescently labeled. The soft gel surfaces were coated with a layer of fluorescent 100 nm nanobeads (FluoSpheres, Thermo Fisher). Since the fluorescent beads preferentially attach to the cross-linked networks through physical adsorption on the silicone gel surfaces and remain immobile during gel deformations, , they were used as the tracers to quantify the network deformations. Due to a low surface coverage density (<0.5%), the presence of these nanobeads does not affect the adhesion profiles. Simultaneously, the bulk of soft gels was stained with a 0.1 wt % UV-sensitive fluorescent dye (Tracer Product, TP3400-1P6), which was soluble in the base PDMS polymers. The nanobeads and UV dye were excited at 555 and 488 nm, respectively, using laser beams on a confocal microscope (ZEISS, LSM 980 with Airyscan 2). By combining the dual-channel fluorescent signals, we were able to reconstruct the deformed surfaces along with the extracted free chains.

3. Results and Discussion

3.1. Adhesion Profiles

We first examined how the adhesion profiles depend on the stiffness of soft PDMS gel substrates. By adjusting the weight concentration of cross-linkers in the gel layer from k = 0.77% to k = 2.0%, the resulting shear modulus was varied between G = 0.26 kPa and G = 14.1 kPa. As shown in Figure , the relationship between G and k can be quantitatively captured by an empirical relationship $G(k)=G_0{(k-k_0)}^n$ , with the fitting parameters G ₀ = 10.5 kPa, k ₀ = 0.72%, and n = 1.47. Notably, the power-law index n represents an empirical result rather than a critical-gel scaling exponent. For k < k ₀, the polymer mixture can not properly cure to form a gel. Figure b,c show the side-view images of a microsphere of radius R = 14 μm adhered to soft gel substrates with k = 1.25% and k = 2.0%, respectively. The indentation depth (d) decreases from 8.7 to 3.8 μm as k increases, demonstrating that the adhesion-induced deformation becomes progressively weaker with increasing gel stiffness.

1.

Adhesion profiles on soft gels. (a) Plot of shear modulus (G) against the cross-linking density (k). The solid dots indicate the measured G(k), with error bars indicating the standard deviations from at least three individual measurements. The solid black line indicates $G(k)=G_0{(k-k_0)}^n$ with G ₀ = 10.5 kPa, k ₀ = 0.72%, and n = 1.47. Inset: schematic illustration of soft silicone gels. The blue and yellow chains represent cross-linked chains and un-cross-linked free chains, respectively, and the red dots indicate the cross-linkers. (b,c) Side-view microscope snapshots of a glass particle adhered to soft silicone gels with k = 1.25% and 2.0%, respectively. Scale bars: 20 μm. (d,e) Confocal images of adhesion profiles near contact areas on soft gels with k = 1.25% and 2.00%, respectively. Scale bars: 5 μm.

Using dual-wavelength confocal microscopy, the deformation of cross-linked networks was determined by locating the positions of nanobeads in three dimensions, while the free solvent distribution was mapped via the intensity profile of the UV-excited dye. Figure d,e show the resulting images of the deformed gel surfaces near the contact regions for k = 1.25% and 2.0%, where the red dots mark the locations of fluorescent nanobeads and the yellow domains indicate the dyed polymers. In both cases, a microscale triangular region is clearly visible near the contact line. This region is filled with dyed polymers and is bounded below by a line of fluorescent nanobeads, indicating the formation of a distinct free-chain-rich phase adjacent to the deformed gel surface. This local feature represents the extraction of free chains due to adhesion, similar to the interfacial structures previously observed for liquid droplet wetting on soft silicone gels. ,,

We further validated the conservation of the total material volume under adhesion. As shown in the inset of Figure , we define V _ind as the volume occupied by the bead below the plane of the undeformed gel surface. Meanwhile, the volume of adhesion ridges includes both the elastic network displaced above the undeformed surface (V _gel) and the extracted free chains (V _liquid). Using glass beads with radii ranging from 8 to 20 μm, we systematically measured V _ind, V _gel, and V _liquid on soft gel substrates with different shear moduli of G = 1.36, 3.87, 10.9, and 14.1 kPa. Figure plots V _gel + V _liquid versus V _ind. All data points align closely with the volume conservation relationship, V _gel + V _liquid = V _ind, confirming the overall incompressibility of silicone polymers.

2.

Conservation of total material volume. Plots of V _gel + V _liquid against V _ind for adhesions on soft gels with different shear moduli. The dashed line indicates volume conservation, V _gel + V _liquid = V _ind. The inset illustrates how V _ind, V _gel, and V _liquid were characterized based on confocal images.

3.2. Local Balance of Surface Stresses

Unlike the surface profiles induced by liquid droplets wetting soft silicone gels, the contact ridges in soft adhesion are inherently asymmetric due to the presence of a rigid particle. A comprehensive description of the adhesion profiles (e.g., Figure d,e) must therefore account for both the adhesion energy (W) at solid interfaces and the surface stresses (ϒ) at fluid interfaces. Below, we first examine the role of surface stresses in determining local contact geometry.

As illustrated by the schematics in Figure a, the region with extracted solvent is constrained by three distinct contact points (A, B and C). While A and C correspond to two contact points located at the rigid surface of microspheres, B denotes a contact point involving three deformable phases: extracted free chains, cross-linked soft-gel regions, and air. On a scale smaller than the elastocapillary length of gels (∼10 μm), the contact profiles at B are determined by the surface stresses at three interfaces. We define γ_lv as the surface tension of extracted free chains in air, ϒ_gv as the surface stress of soft gels in air, and ϒ_gl as the surface stress at the phase boundary (AB) between extracted solvent and cross-linked gels. Considering the stress balance at point B (Figure b), γ_lv, ϒ_gl, and ϒ_gv construct the three sides of the Neumann triangle. − Thus, the law of sines gives rise to

$$\frac{{\gamma }_{\mathrm{l}\mathrm{v}}}{\sin \varphi }=\frac{{\Upsilon }_{\mathrm{g}\mathrm{l}}}{\sin \beta }=\frac{{\Upsilon }_{\mathrm{g}\mathrm{v}}}{\sin (\varphi -\beta )}$$ 1

where ϕ is the opening angle between ϒ_gl and ϒ_gv, and β represents the supplemental angle between γ_lv and ϒ_gv. Since γ_lv = 23.3 mN/m is a constant material parameter at the solvent–air interface, ϒ_gv and ϒ_gl are determined by the relationship between β and ϕ.

3.

Stress balance near the adhesion contact point. (a) Schematic of the region containing polymer chains extracted by soft adhesion. Near the contact line, a region (ABC) filled with free chains forms under the balance of elastic and osmotic stresses. The interface AB marks the phase boundary between the extracted polymers and gels. (b) Neumann’s triangle is formed by ϒ_gv, ϒ_gl, and γ_lv. The opening angles β and ϕ determine the interfacial profiles near the contact point B. (c) Plots of ϕ and β against k, where the error bars represent the standard deviations from at least three independent measurements. (d) Plots of ϒ_gv and ϒ_gl against k, obtained from eq . The error bars represent the standard deviations from at least three independent measurements.

Figure c shows the variation of these opening angles with the cross-linking density (k) of gel substrates. As k increases from 1.0% to 2.0%, β rises sharply from 2° to 16°, while ϕ remains constant at approximately 133°. Using the relationships

$${\Upsilon }_{\mathrm{g}\mathrm{l}}={\gamma }_{\mathrm{l}\mathrm{v}}\frac{\sin \beta }{\sin \varphi },{\Upsilon }_{\mathrm{g}\mathrm{v}}={\gamma }_{\mathrm{l}\mathrm{v}}\frac{\sin (\varphi -\beta )}{\sin \varphi }$$ 2

we obtained ϒ_gl and ϒ_gv versus k in Figure d. For k = 1.0% and 1.1%, ϒ_gv ≈ γ_lv ≈ 24 mN/m, indicating that the surface stress of cross-linked gels is close to the surface tension of un-cross-linked free chains at low cross-linking densities. In this regime, the surface stress at the phase boundary (AB) is minimal, with ϒ_gl ≈ 1.2 mN/m. In contrast, for k ≥ 1.25%, ϒ_gv increases from 24 to 28 mN/m, remaining higher than γ_lv. Meanwhile, ϒ_gl rises rapidly from 1.2 to 8.7 mN/m. Consistent with previous soft wetting measurements, , these findings suggest that a higher cross-linking density increases the surface stresses of cross-linked gels, thereby amplifying the material contrast between cross-linked gels and un-cross-linked free chains.

3.3. Adhesion Energy

We further characterized the adhesion energy (W) of gel substrates with different cross-linking densities. Without external load, classical JKR theory predicts the relationship between the indentation depth (d) and the particle radius (R)

$$d_{\mathrm{J}\mathrm{K}\mathrm{R}}={\left[\frac{\sqrt{3}\pi W(1-\nu )}{4G}\right]}^{2/3}{R}^{1/3}$$ 3

where ν represents the Poisson ratio of substrates. With pronounced elasto-capillary effects, the governing force balance has been previously derived based on energy minimization

$$\frac{8G{R}^{1/2}{d}^{3/2}}{\sqrt{3}(1-\nu )}-2\pi WR+2\pi {\Upsilon }_{\mathrm{g}\mathrm{v}}d=0$$ 4

Equation indicates that the deformation of a soft gel substrate is initiated by adhesion, and balanced by the bulk elasticity, and solid capillarity. Using the elastocapillary length (l _e = ϒ_gv/G) as a normalization parameter, eq can be expressed in a dimensionless form

$$\frac{4{(\mathrm{R̅})}^{1/2}{(\mathrm{d̅})}^{3/2}}{\sqrt{3}\pi (1-\nu )}-\left[\frac{W}{{\Upsilon }_{\mathrm{g}\mathrm{v}}}\right]\mathrm{R̅}+\mathrm{d̅}=0$$ 5

with $\mathrm{R̅}=R/l_{\mathrm{e}}$ and $\mathrm{d̅}=d/l_{\mathrm{e}}$ . When solid capillarity is neglected $(\mathrm{d̅}\ll 1)$ , eqs and are both reduced to the classical JKR prediction given by eq . A concise derivation of eqs and , based on the competition among bulk elasticity, adhesion energy, and solid surface stress, is provided in Section S2 of the Supporting Information. Notably, the first term in eq is derived from the linear elasticity of a semi-infinite compliant substrate. This contribution of bulk elasticity is significant only for relatively stiff gels (G = 10.9 and 14.1 kPa) when R ≫ l _e. In this regime, the linear elastic approximation holds since the indentation depth stays below 5 μm, which is significantly smaller than the substrate thickness. Conversely, for soft gels with G < 3 kPa, the indentation depth can reach up to 20 μm and finite-size effects can arise. Nevertheless, the specific formulation of the bulk elastic term is noncritical when R < l _e, as the profiles are solely governed by the balance between solid surface stress (ϒ_gv) and adhesion energy (W). Consistent with this argument, we further demonstrate that adhesion profiles on soft gels with different moduli remain independent of the substrate thickness (see Figure S3 in the Supporting Information).

Figure a shows the equilibrium indentation depth (d) as a function of the particle radius (R) for soft gels with shear moduli ranging from G = 0.46 to 14.1 kPa. Over this stiffness range, the corresponding elastocapillary length (l _e = ϒ_gv/G) varies between 52 and 2 μm. Thus, solid capillarity is expected to be critical for adhesion on weakly cross-linked gels.

4.

Indentation depth (d) in soft adhesion. (a) Plot of d against R for gels with different shear moduli. The dotted lines represent the best fits of R(d) to eq with W/ϒ_gv = 1.9. (b) Plot of $\mathrm{d̅}=Gd/{\Upsilon }_{\mathrm{g}\mathrm{v}}$ against $\mathrm{R̅}=GR/{\Upsilon }_{\mathrm{g}\mathrm{v}}$ . The solid line represents the numerical results obtained from eq with W/ϒ_gv = 1.9, while the dashed line indicates the prediction of classical JKR theory (eq ) with W = 48 mN/m.

By normalizing the indentation depth as $\mathrm{d̅}=d/l_{\mathrm{e}}=Gd/{\Upsilon }_{\mathrm{g}\mathrm{v}}$ , and the sphere radius as $\mathrm{R̅}=R/l_{\mathrm{e}}=GR/{\Upsilon }_{\mathrm{g}\mathrm{v}}$ , the adhesion results from different gel substrates collapse onto a single master curve when plotting d̅ against R̅ (Figure b). The master curve is well-described by eq using a fitting parameter W/ϒ_gv = 1.9. This constant normalized adhesion energy indicates that both ϒ_gv and W increase slightly with the cross-linking density of soft gels. For large spheres (R̅ ≫ 1 or R ≫ l _e), the indentation depth follows the d ∼ R ^1/3 scaling, predicted by the classical JKR theory (eq ). In contrast, d remains significantly lower than the JKR predictions for small spheres (R̅ ≪ 1 or R ≪ l _e), indicating the significant contribution from solid capillarity of the soft substrates.

3.4. Time-Dependent Adhesion Profiles

Due to a gradual relaxation of silicone gel substrates, the adhesion profiles slowly evolved over time. For example, Figure a shows the overlaid profiles of a gel surface (k = 0.83% and G = 0.46 kPa), measured at t = 0 h (red circle), 2 h (magenta square) and 24 h (blue triangle) after depositing a glass microsphere (R = 21 μm). Over this 24 h measurement period, we observed moderate sinking of the sphere (∼2 μm), accompanied by a gradual growth of the region containing extracted free chains near the contact line.

5.

Time-dependent adhesion profiles. (a) A glass microsphere (R = 21 μm) was placed on a soft silicone gel (G = 0.46 kPa and k = 0.83%) at t = 0. The surface profiles measured at t = 0 (red circles), t = 2 h (magenta square) and 24 h (blue triangle) are overlaid. (b) Plots of W/ϒ_gv against t for G = 0.46 kPa, 1.36 kPa, 3.87 kPa. The solid black line represents the best fit to an exponential growth function with a characteristic time scale τ _d = 10⁴ s.

The slow settling of the microspheres upon adhesion at soft interfaces implies a gradual, temporal rise of the adhesion energy (W) between the particle and substrates. Considering the near-equilibrium nature of this relaxation process, we obtained the time-dependent ratio of W(t)/ϒ_gv from the indentation depth (d(t)) using eq . Figure b shows the plots of W(t)/ϒ_gv for soft gels with different shear moduli, G = 0.46, 1.36, and 3.87 kPa. The growth of W(t)/ϒ_gv for different moduli are well collapsed together, which are quantitatively fitted to an exponential function, W/ϒ_gv = A _∞ – A ₀ exp­(−t/τ _d ). By maintaining A _∞ equal to W/ϒ_gv = 1.9 that was obtained from the equilibrium adhesion profiles (Figure ), the best fit gives A ₀ = 0.13 and τ _d = 1.0 × 10⁴ s.

While contact line aging has been reported on material surfaces with microscale heterogeneities, its influence on compliant gel interfaces is expected to be negligible. Also, the long growth time scale τ _d cannot be attributed to the viscoelastic nature of the gel substrates. Using standard oscillatory rheological characterizations, we found that the characteristic viscoelastic relaxation time scales of soft silicone gels are in the order of 10 and 100 ms across all different shear moduli, while remaining substantially lower than τ _d (see Figure S1). Since the increase in indentation depth (d) is associated with the migration of free chains to the contact region, we consider the role of poroelastic diffusion driven by the competition between elastic stresses and osmotic pressure. ,, Considering the thermo-equilibrium states of cross-linked silicone gels with a porosity (κ), the effective diffusivity of free polymer chains through the networks can be approximated as

$$D_{\mathrm{f}}\approx \frac{G\kappa }{{\eta }_s}\frac{1-2\nu }{1-\nu }\sim \frac{G^{1/3}{(k_{\mathrm{B}}T)}^{2/3}}{{\eta }_s}$$ 6

with η_s ∼ 10³ Pa·s being the zero-rate viscosity of migrating extracted chains, as obtained through shear rheological characterization (see Figure S1b in Supporting Information). Because the viscosity of the un-cross-linked base polymers (∼1 Pa·s) is substantially smaller than η _s , we conclude that the extracted solvent primarily consists of partially cross-linked chains that are unattached to the main network. For a soft silicone gel with a shear modulus G ∼ 1 kPa at room temperature T = 300 K, we obtained D _f ∼ 10^–15 m²/s. As the extractions of free chains were concentrated near contact points, we use the size of phase separation domains to estimate the characteristic diffusion length, L _D ∼ 3 μm. Consequently, the diffusion time scale of free chains is given by τ_p ∼ L _D /D _f ∼ 10⁴ s, of the same order of magnitude of τ _d . We conjecture that, during soft adhesion, the extracted free chains reduce the divergence of elastic energy within the soft substrates and thus improves solid–solid contacts, leading to a time-dependent adhesion energy.

3.5. Contact Area

We further investigated the effective contact area under soft adhesion with the presence of extracted free chains. The classical JKR approximation, a _JKR = [9πW(1 – ν)/4G]^1/3 R ^2/3, fails to quantitatively capture the contact area under large surface deformations. , For instance, Figure a,b show the adhesion profiles on a soft gel (k = 1.25%), measured at t = 5 min and t = 125 min, respectively. For both cases, the value of a _JKR calculated with W = 48 mN/m substantially overestimates the contact radius.

6.

Contact area in soft adhesion. (a,b) Adhesion profiles of a glass bead on a soft gel substrate with k = 1.25% measured at t = 5 min and t = 125 min, respectively (scale bars: 10 μm). Inset: zoomed plots near the contact points (scale bars (white): 3 μm). The dashed lines are best fits of the measured surface profiles to eq using an optimal contact radius a ₀. The dotted black lines indicate the predictions from classical JKR theory. (c) Plot of the normalized contact radius $(\mathrm{a̅}=a_{0}G/{\Upsilon }_{\mathrm{g}\mathrm{v}})$ against the normalized bead radius $(\mathrm{R̅}=RG/{\Upsilon }_{\mathrm{g}\mathrm{v}})$ on soft gels with different cross-linking densities (k). The solid line represents the numerical results obtained from eq with W/ϒ_gv = 1.9. The gray dashed line indicates the prediction of the classical JKR theory. Inset: the ratio a ₀/R measured at different time points (t = 5 min, 125 min, and 24 h) for different bead radii on soft gels with G = 1.36 kPa.

To address this issue, we considered the relationship between the adhesion energy (W) and true contact radius (a ₀) through an energy release rate $({\mathcal{G}}_c)$ , following the extended adhesion model by Maugis ,

$$W-\frac{{\Upsilon }_{\mathrm{g}\mathrm{v}}d}{2R}={\mathcal{G}}_c=\frac{G}{\pi a_0(1-\nu )}{(d-\frac{a_0}{2}\ln \frac{R+a_0}{R-a_0})}^2$$ 7

where the term ϒ_gv d/(2R) accounts for the contribution from solid capillarity. Using the normalized parameters, $\mathrm{R̅}=R/l_{\mathrm{e}}$ , $\mathrm{d̅}=d/l_{\mathrm{e}}$ , and $\mathrm{a̅}=a_0/l_{\mathrm{e}}$ , eq becomes

$$\frac{W}{{\Upsilon }_{\mathrm{g}\mathrm{v}}}=\frac{\mathrm{d̅}}{2\mathrm{R̅}}+\frac{1}{\pi \mathrm{a̅}(1-\nu )}{(\mathrm{d̅}-\frac{\mathrm{a̅}}{2}\ln \frac{\mathrm{R̅}+\mathrm{a̅}}{\mathrm{R̅}-\mathrm{a̅}})}^2$$ 8

Accordingly, considering the Sneddon’s solution to the axisymmetric Boussinesq problem and the spherical indentation profile, , the surface profile can be numerically evaluated by

$$z(r)=\frac{2}{\pi }{\int }_0^1(d-\frac{a_{0}t}{2}\ln \frac{R+a_{0}t}{R-a_{0}t})\frac{a_0\mathrm{d}t}{\sqrt{r^2-{(a_{0}t)}^2}},(r>a_0)$$ 9

A brief derivation of eqs –, based on the Griffith criterion and the corresponding energy release rate, is provided in Section S4 of the Supporting Information. While maintaining W/ϒ_gv = 1.9, we quantitatively determined the contact radius (a ₀) by fitting the experimental adhesion profiles to eqs –. As shown by the dashed black line in Figure a,b, we obtained a ₀ = 10.5 μm at t = 5 min, which is nearly identical to the a ₀ = 10.7 μm measured at t = 125 min.

We systematically varied the radius of adhered particles from R = 8 to 24 μm on soft gels with different shear moduli, G = 0.46, 1.36, 3.87, 5.95, 10.9, and 14.1 kPa. For each adhesion profile, we characterized the contact radius a ₀ at t = 125 min using eqs –. Figure c represents the measured $\mathrm{a̅}=a_0/l_{\mathrm{e}}$ plotted against $\mathrm{R̅}=R/l_{\mathrm{e}}$ for all different bead radii (R) and substrate stiffnesses (G). The experimental data a̅ versus R̅ collapse onto a single master curve, which is well described by eq . For large glass beads (R̅ ≫ 1 or R ≫ l _e), the contact radius follows the classical JKR theory, a ₀ ∼ R ^2/3. In contrast, for small glass beads (R̅ ≪ 1 or R ≪ l _e), the JKR theory significantly overestimates a ₀. In this regime, the true contact radius remains close to the bead radius, a ₀ ≈ R, independent of the gradual extraction of free chains.

Given the relationship between W and a ₀, eq quantitatively explains why the adhesion-induced phase separation plays an insignificant role in varying a ₀. With the logarithmic term ln­[(R + a ₀)/(R – a ₀)] and the approximate relationship a ₀ ≈ R in eq , a moderate variation in W only generates a negligible change in a ₀. Thus, the adhesion contact area in our experiments was primarily determined by the bead sizes, even as the indentation depth (d) gradually increased over time.

Finally, to examine the generality of the surface profiles and phase separation observed during soft adhesion, we further characterized the adhesion induced by polystyrene (PS) particles on soft PDMS surfaces. Their equilibrium adhesion profiles can also be quantitatively captured by eqs and (see Figure S2a,b in Supporting Information). As the free chains were extracted from the gels at the contact lines, the PS particles gradually settled, resulting in a time-dependent increase in adhesion energy (Figure S2c in Supporting Information).

4. Conclusions

In contrast to the previous studies of wetting-induced phase separation that focus on liquid droplets on soft surfaces, , this work characterized the distinct geometry and mechanics of gel-particle adhesion. The axisymmetric nature of soft adhesion, combined with the absence of liquid interfaces, leads to unique deformation responses on large scales and extraction dynamics near contact lines.

Specifically, we measured in situ the extraction of free polymers during the adhesion process on soft silicone gels with different cross-linking densities (Figure ), while the phase separation process conserves the total volume of silicone polymers (Figure ). By measuring the adhesion profiles on gels with different cross-linking densities, we demonstrated that the extracted free chains near contact points preserved the local balance of surface stresses (Figure ). The adhesion energy (W) can be characterized from the indentation depth (d) by considering the contribution from solid surface stress (Figure ).

During the extraction of free chains, the adhered particles slowly sank into soft gels, indicating a time-dependent adhesion energy (W) caused by poroelastic diffusion (Figure ). In contrast, the effective contact radius (a ₀) remains approximately constant, independent of the emergence of the phase separation region. Using an extended JKR-Maugis framework, the entire adhesion profiles can be quantitatively described for varying bead radii (R) on soft gels with different cross-linking densities (Figure ). These findings will benefit the future design of functional soft interfaces, such as lubricant-infused substrates, for controlling the multiscale adhesion processes.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.langmuir.6c01851.

• The Supporting Information includes (1) Oscillatory Shear Rheology of Silicone Gels, (2) Rheological Characterization of Free Chains, (3) Adhesion measurements of polystyrene microspheres on soft PDMS gels, (4) Theoretical Predictions for Adhesion Profiles, and (5) Adhesion profiles on soft substrates with different thicknesses (PDF)

†.

W.Q. and K.Z. contributed equally to this work. W. Q. performed the adhesion experiments. W. Q. and K. Z. analyzed the experimental results. K. Z. and Q. X. wrote the manuscript.

The authors declare no competing financial interest.