Background

Adhesion phenomena in contact problems using elastomers and soft materials play a significant role in design of devices, e.g., microfabricated adhesives [1–3] and wall-climbing robots [4, 5]. Theory of adhesive elastic contact [6–8] considering both of the elastic deformation and adhesion phenomenon in contact interface between elastic bodies is helpful for its applications. Since the adhesive elastic contact theory assumes the total energy equilibrium, contact process in the theory (i.e., consists of loading–unloading or advancing-receding contact) is reversible except for its mechanical hysteresis [9]. However, it has been reported that adhesion hysteresis exists in some contact experiments [10–21]. This adhesion hysteresis shows a completely different force curve (force–displacement or force-contact area) between loading–unloading or advancing-receding in actual contact process. Adhesion hysteresis means that the actual contact process is not in equilibrium as assumed in the theory and also means that the total energy in the contact system is dissipated during the process. Therefore, investigating the energy dissipation is significant for understanding the mechanism and complementing the conventional theory.

The energy dissipation in the adhesive contact is mainly investigated and discussed using the strain energy release rate G (i.e., the energy required to separate unit contact area J/m2) [9–18]. Maugis and Barquins [10] first introduced a concept of linear elastic fracture mechanics into the Johnson–Kendall–Roberts (JKR) contact [6]. They experimentally showed that G has a dependency on the crack speed [10], which is the so-called empirical relationship [11–14]. However, the relationship does not represent how the total energy dissipation changes during the contact process, and the mechanism of adhesion hysteresis is still on discussion assuming capillary condensation or adsorbed layer, etc. [17–19, 22–25]. In this paper, a polydimethylsiloxane (PDMS) block is used as the elastic materials, the contact processes between the PDMS and a glass lens have been investigated to evaluate the energy dissipation. Especially, the change in the energy dissipation during the processes is discussed using the elastic contact theory, assuming non-equilibrium.

Methods

Contact mechanics for evaluating energy dissipation

Figure 1 shows a schematic illustration of the spherical contact model describing contact between the spherical rigid tip and the elastic body considering the equivalent stiffness in the measurement system, e.g., a cantilever-like structure, a strain gauge force sensor, and etc. in an actual measurement system. Total energy U total of the contact model is given by Takahashi et al. [7], who described the contact mechanics considering the equivalent stiffness based on the JKR theory [6]. The model assumes small deformation, linear elasticity, elastic half-space, and frictionless surfaces. Moreover, in this study the external work given by the movement of the gross displacement Z is transferred instantaneously and fully to the contact system. Hence, the total energy U total is spontaneously dissipated toward an equilibrium at a fixed Z. The dissipated energy ΔU dissipation (i.e., same as a negative increment of total energy −ΔU total in the contact system) at a fixed Z is expressed as

$$\Delta U_{\text{dissipation}} = - \Delta U_{\text{total}} = - (\Delta U_{\text{elastic}} + \Delta U_{\text{interface}} + \Delta U_{\text{stiffness}} ),$$
(1)

where ΔU elastic is an increment of the elastic energy stored in the elastic body due to its deformation, ΔU interface is an increment of the interface energy stored in the contact area by the work of adhesion, ΔU stiffness is an increment of the stiffness energy stored in the spring corresponding to the equivalent stiffness k of the measurement system [7]. In the spherical contact, the stress distribution in the contact area can be described by the linear combination using Hertz’s and Boussinesq’s stress distribution although the process is not in equilibrium [26, 27]. The specific derivation is given by Muller et al. [26] in the section of the JKR model interpretation. The relationship between the force F, the penetration depth δ, and the contact radius a [26, 27] is given as

$$F = 2E^{*} a\left( {\delta - \frac{{a^{2} }}{3R}} \right),$$
(2)

where E * = E/(1 − ν 2) is the elastic modulus of the elastic body (ν is the Poisson’s ratio), and R is an effective radius of curvature:

$$\frac{1}{R} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }},$$
(3)

where R 1 and R 2 are the radii of curvature of the spherical rigid tip and the elastic body as shown in Fig. 1. The radius of curvature of the elastic body R 2 is infinite when the surface of the elastic body is flat, i.e., R = R 1. The force F is also applied to the spring k, which can be expressed by Hooke’s law as

$$F = k( - Z - \delta ).$$
(4)
Fig. 1
figure 1

Schematic illustration of the spherical contact model, where Z is the gross displacement controlled by the experimental apparatus, k is the equivalent stiffness of the measurement system, R 1 is the radius of curvature of the spherical rigid tip, R 2 is the radius of curvature of the elastic body surface, F is the applied force between the spherical rigid tip and the elastic body, δ is the penetration depth of the spherical rigid tip into the elastic body, and a is the radius of the contact area

Full size image

Therefore, the relationship between the force F, the gross displacement Z, and the contact radius a is obtained from substituting Eq. (4) into Eq. (2):

$$F = \frac{{2kE^{*} a}}{{2E^{*} a + k}}\left( { - Z - \frac{{a^{2} }}{3R}} \right).$$
(5)

Although the adhesion hysteresis is observed in the measurements (F, Z, a), the measurements must satisfy Eq. (5) if a material behaves as an elastic material in the contact experiments. Therefore, Eq. (5) can be used to confirm the elasticity of the material when k, R are given and F, Z, a are measured in the experiments.

The strain energy release rate G Z at a fixed Z is defined and calculated:

$$\begin{aligned} G_{Z} & = \left[ {\frac{{\partial U_{\text{elastic}} }}{{\partial (\pi a^{2} )}} + \frac{{\partial U_{\text{stiffness}} }}{{\partial (\pi a^{2} )}}} \right]_{Z} \\ & = \frac{{({{4E^{*} a^{3} } \mathord{\left/ {\vphantom {{4E^{*} a^{3} } {3R}}} \right. \kern-0pt} {3R}} - F)^{2} }}{{6\pi R({{4E^{*} a^{3} } \mathord{\left/ {\vphantom {{4E^{*} a^{3} } {3R}}} \right. \kern-0pt} {3R}})}}, \\ \end{aligned}$$
(6)

where U elastic and U stiffness are the components of U total = U elastic + U stiffness + U interface given by Takahashi et al. [7]:

$$U_{\text{elastic}} = \frac{{4E^{*} a^{5} }}{{45R^{2} }} + \frac{{F^{2} }}{{4E^{*} a}},$$
(7)
$$U_{\text{stiffness}} = \frac{{F^{2} }}{2k},$$
(8)
$$U_{\text{interface}} = - \pi a^{2} \Delta \gamma .$$
(9)

U interface is the interface energy stored in the contact area, which is contributed by the thermodynamic reversible work of adhesion Δγ during an entire contact process. The work of adhesion is a material constant of interface defined by Dupré [28]: Δγ = γ 1 + γ 2 − γ 12, where γ 1, γ 2 are the surface free energy, γ 12 is the interface free energy per unit area (J/m2). The strain energy release rate G Z is also expressed using Eqs. (1), (6) and (9) as

$$\begin{aligned} G_{Z} & = \left[ { - \frac{{\partial U_{\text{dissipation}} }}{{\partial (\pi a^{2} )}} - \frac{{\partial U_{\text{interface}} }}{{\partial (\pi a^{2} )}}} \right]_{Z} \\ & = - \frac{{\partial U_{\text{dissipation}} }}{2\pi a\partial a} + \Delta \gamma , \\ \end{aligned}$$
(10)

which shows that G Z consists of a dissipative term (variable; \({{ - \partial U_{\text{dissipation}} } \mathord{\left/ {\vphantom {{ - \partial U_{\text{dissipation}} } {2\pi a\partial a}}} \right. \kern-0pt} {2\pi a\partial a}}\)) and the reversible term (constant; Δγ). It also shows that the energy dissipation is contributed by G Z  − Δγ (the reversible term is excluded from the required energy to change unit contact area) and the equilibrium of total energy is given as G Z  = Δγ at the dissipative term to be zero. From Eq. (10), therefore, the total energy dissipation can be evaluated numerically by using the rectangular rule as

$$U_{\text{dissipation}} = \sum {\left[ { - \left( {G_{Z} (a_{i + 1} ) - \Delta \gamma } \right)\left( {\pi a_{i + 1}^{2} - \pi a_{i}^{2} } \right)} \right]} ,$$
(11)

where a i is the instantaneous contact radius measured during the time-series measurements, G Z (a) is a function of a obtained from substituting Eq. (5) into Eq. (6), and the summation is performed over the range of the time-series measurement in the contact experiment.

Spherical contact measurement system

The measurement system was constructed as shown in Fig. 2; it consisted of the contact between a glass lens (BK7 Plano Convex Lens SLB-30-400P, SIGMAKOKI) and a PDMS block (SYLGARD®184 SILICONE ELASTOMER KIT, Dow Corning). The glass lens (R 1 = 207.6 mm) was attached to a clear acrylic plate that was fixed to the motorized stage (KZL06075-C1-GA, SURUGA SEIKI). The PDMS block (60 × 60 × 10 mm) was placed on the digital balance (strain gauge type TE612-L, Sartorius). The gross displacement Z was manipulated by the motorized stage, and the force F and contact radius a were measured simultaneously by using the digital balance and microscope (SKM-3000B-PC, SAITOH KOUGAKU). The spring constant k of the equivalent stiffness of the measurement system was measured to k = 10.5 kN/m as shown in Fig. 3.

Fig. 2
figure 2

Schematic illustration of the measurement system (left) and loading–unloading processes (right). The gross displacement was controlled by the motorized stage, and the force and contact area were measured by using the digital balance and microscope

Full size image
Fig. 3
figure 3

Measurement of the equivalent stiffness k of the measurement system. The PDMS block in Fig. 2 was replaced to the metal block for the measurement of k. The loading–unloading of the gross displacement Z was tested in the speed of motorized stage at 0.1 μm/s, and both of the loading–unloading are plotted. The result shows that a hysteresis in the measurement system is small enough to be negligible. Therefore, the equivalent stiffness was determined to k = 10.5 kN/m from the gradient as shown

Full size image

The PDMS mixture for the PDMS block was made with a mixing ratio of 10:1 of the base polymer and curing agent for fully cross-linked rubber. The air bubbles in the mixture were removed through degassing in a desiccator under a vacuum of 2 kPa for 1 h. The degassed mixture was poured carefully into a mold (60 × 60 × 20 mm) that had a clean glass bottom for making the PDMS block surface smooth and flat. After the mold was filled with the mixture to about 10 mm high, the air bubbles were removed again for 10 min in the desiccator. The filled mold was cured in an oven at 60 °C for 12 h, and then the PDMS block was removed from the mold. The exposed side of the PDMS block in the heat curing was carefully glued to a glass slide (100 × 100 mm) with the same PDMS mixture. The sample was cured again in the oven at 60 °C for 12 h, and the PDMS block was permanently set on the glass slide. The glass lens and PDMS block were cleaned using an ultrasonic cleaner with ethanol and dried using a nitrogen spray gun. After 24 h from the setting of samples to the measurement system in a clean bench on a vibration isolation table, the experiment was conducted at the ambient conditions of 20 °C with 50% humidity.

Experimental procedure

The gross displacement Z was manipulated in step-by-step movements with a constant dwell time in every step for evaluating the spontaneous energy dissipation at a fixed Z. The amount of movement between steps was set at 1 μm (the speed of the motorized stage was set at 1 μm/s). The dwell time for every step was set at 15 s. The loading process was performed up to the maximum loading displacement (−Z max). After the loading process completed, the unloading process was performed until the lens detached. A dependence of the maximum loading displacement on the adhesion hysteresis has been reported [16, 20]. Hence, three different maximum loading displacement were chosen: −Z max = 10, 20, 30 μm, which is sufficiently smaller than the thickness of the PDMS block 10 mm.

Results and discussions

Experimental results and adhesion hysteresis

The experimental results of −Z max = 10, 20, 30 μm are plotted in Fig. 4. Adhesion hysteresis was observed between the loading–unloading paths in each result, and the larger hysteresis loop was observed in the larger −Z max. Moreover, the calculation results of the force F(Z, a) in Eq. (5) are plotted, which are calculated by substituting the measurements of Z and a into F(Z, a). In the calculation, the effective radius of curvature was given as R = R 1 = 0.2076 m (i.e., the surface of the PDMS block was assumed flat) and the equivalent stiffness was given as k = 10.5 kN/m. The elastic modulus was determined to E * = 2.67 MPa using the method of least squares between the calculated F(Z, a) and the measured F using entire measurements of −Z max = 10, 20, 30 μm; the root mean square error (RMSE) between F(Z, a) and F was 0.6 mN, which is small enough throughout the entire observed range of the force (−120 to 120 mN). Notably, the spring deformation calculated by Eq. (4) was 10 μm (≈−Z −δ) when the maximum force was applied (F ≈ 0.1 N at −Z max = 30 μm in Fig. 3).

Fig. 4
figure 4

The loading–unloading curve of −Z max = 10, 20, 30 μm. The measured force F and calculated force F(Z, a) are plotted as a function of the measured contact radius a. The calculated forces is calculated by substituting the measurements of Z and a into F(Z, a) of Eq. (6); R = 0.2076 m, k = 10.5 kN/m, and E * = 2.67 MPa with an RMSE between F(Z, a) and F was 0.6 mN

Full size image

At each fixed Z (15 s dwell time) the changing of F and a is observed in Fig. 4, and the changing in entire process is fitted well with the calculated force F(Z, a) by Eq. (5) with the constant elastic modulus E *. This result suggests that the PDMS block behaves as an elastic material in the contact process. Also, adhesion hysteresis between the loading–unloading paths represents that the total energy is not in equilibrium state. Therefore, it can be considered that the energy dissipation is induced in the contact interface, not in the PDMS block, from a spontaneous process of the total energy toward an equilibrium at each fixed Z, i.e., the spontaneous energy dissipation.

Strain energy release rate and work of adhesion

The work of adhesion Δγ should be estimated for evaluating the total energy dissipation using Eq. (11). As shown in Eq. (10), G Z consists of the work of adhesion and a dissipative term. Since the equilibrium of the total energy is given as G Z  = Δγ and the total energy is spontaneously stabilized at fixed Z: G Z tends to increase to become Δγ in the loading process; G Z tends to decrease to become Δγ in the unloading process (until the existence of equilibrium) [10]. Figure 5 shows the calculation result of G Z by Eq. (6). In the loading process, an approximately constant value of G Z is observed at the end of each step with the advancing of contact radius a; on the contrary in the unloading process G Z drastically changes with the receding of a. From the observation, we assume that the approximately constant value of G Z observed at the end of each step in the loading process is close to the equilibrium G Z  = Δγ. Therefore, the approximately constant value is estimated to the work of adhesion Δγ = 0.03 J/m2, which is a quite similar value obtained in [20]. Notice that the rapidly changing area marked with (a) in Fig. 5 represents the total energy is quite unstable when the initial contact is formed, thus, (a) is not suitable to the estimation.

Fig. 5
figure 5

Strain energy release rate G Z as a function of the contact radius a. An approximately constant value of G Z at the end of each step is observed in the loading process; on the contrary, in the unloading process, G Z varies with the receding of a. The approximated value of 0.03 J/m2 at the end of each step in the loading process is estimated to the work of adhesion Δγ. The initial contact of the first step is shown as a

Full size image

Evaluation of energy dissipation

The energy dissipation is induced in the contact interface from a spontaneous process of the total energy toward an equilibrium at each fixed Z (15 s dwell time). And this relationship is also shown in Eq. (1) that ΔU total = − ΔU dissipation. Therefore, it can be considered that the total energy dissipation U dissipation calculated by Eq. (11) is a cumulative result of ΔU total at each fixed Z during 15 s in entire contact process.

Figure 6 shows U dissipation as a function of contact radius a. In the loading process, the total energy dissipation U dissipation is little increased. In the unloading process, it is found that U dissipation is observed to be linearly proportional to the contact radius a. The gradient of U dissipation in a is expressed from using Eq. (1) and Eq. (10) as

$$\begin{aligned} - \frac{{\partial U_{\text{dissipation}} }}{\partial a} & = \frac{{\partial U_{\text{total}} }}{\partial a} \\ & = 2\pi a(G_{Z} - \Delta \gamma ), \end{aligned}$$
(12)

which represents the gradient of U total in a determined at each fixed Z. For convenience, we call the gradient using a character f in this paper. Figure 7 shows the gradient f of −U dissipation (or U total) as a function of a calculated by Eq. (12). An approximately constant f is observed for each unloading process, i.e., f = 0.71 mJ/m (−Z max = 10 μm), f = 0.83 mJ/m (−Z max = 20 μm), f = 1.03 mJ/m (−Z max = 30 μm). This result represents that the gradient f is determined as a roughly constant value during the receding contact, and f has a dependency on the maximum loading displacement −Z max.

Fig. 6
figure 6

Total energy dissipation U dissipation as a function of contact radius a. Linearity between U dissipation and a is observed in the unloading process (receding contact)

Full size image
Fig. 7
figure 7

Gradient of total energy dissipation f as a function of contact radius a. An approximately constant value of the gradient f is obtained in the unloading process (receding contact): f = 0.71 mJ/m (−Z max = 10 μm), f = 0.83 mJ/m (−Z max = 20 μm), f = 1.03 mJ/m (−Z max = 30 μm)

Full size image

The total energy dissipation and the gradient show that the contact process is not in equilibrium. Although the occurrence mechanism of the spontaneous energy dissipation is not clear, Fig. 5 shows that the total energy could not be fully stabilized state (G Z  = Δγ) at a fixed Z within 15 s especially in the unloading, i.e., the amount of spontaneous energy dissipation ΔU dissipation at a fixed Z is limited within the dwell time. A not fully stabilized total energy affect a next step as a history by the step-by-step control of Z in every 15 s. In the larger maximum loading displacement −Z max, the more step is required to detach the lens from the PDMS, and it can be considered that the more history might be accumulated. Therefore, the larger value of gradient f is observed in the larger −Z max because of the accumulated history related to the amount of required step in the unloading process.

As expressed in Eq. (12), the gradient of total energy dissipation is calculated using G Z  − Δγ (the dissipative term that the reversible term Δγ is excluded from the required energy to change unit contact area G Z ) and 2πa (the entire length of the crack tip). The dimension of the gradient f is J/m, and is also expressed as the dimension of the force N. In this paper, therefore, we define f as a dissipative force which is applied to 2πa during the receding contact. From this, it can be considered that the dissipative force would be induced by an unknown factor existed at the crack tip 2πa. An unknown factor might be an adsorbate on the surface gathered by −Z max, such as gases, liquids, uncross-linked PDMS fragments or etc. Although the mechanism is not clear, the approximately constant f and the dependency on −Z max observed in Fig. 7 suggests a hint how an unknown factor works at the crack tip 2πa.

Conclusion

The energy dissipation is evaluated in the contact process between the glass lens and the PDMS block. The experiments with the three maximum loading displacement −Z max were conducted. The results (Fig. 4) shows that the adhesion hysteresis would be occurred even using the elastic material (PDMS block). This suggests that the mechanism would be induced by some interfacial phenomena. Furthermore, it is found that the approximately constant gradient f (Fig. 7) of the total energy dissipation (Fig. 6) which has a dependency on −Z max. This fact would suggest that the dissipative force f (the gradient) is possibly induced by an unknown factor existed at the crack tip 2πa gathered by −Z max, e.g., an adsorbate on the material surface.