This work reported that the corneal adhesion possessed a scale effect, affected by the contact area. In this study, the classical JKR model, used to describe the adhesion of solids, was valid to analyze the corneal adhesion with the submillimeter sizes (Fig. 3a). However, the validity would decrease with the increasing of the contact area. While the punch sizes were in the millimeter scales, the corneal Fad obtained by the JKR solutions were obviously smaller than the related experimental data (Fig. 3b-3f).
This contradiction, happened between the JKR solutions and the experimental data, implied that the JKR model partially failed in analyzing the corneal adhesion. To find the reason, we deemed that it should be emphasized on the physiology of the natural cornea, which is fully inflated by IOP, like a membrane. To analyze the corneal adhesion, therefore, it should be considered the contribution of the membranous characteristic. This important factor, however, was not considered in the classical JKR model.
According to the comparison results, it was clear that the scale effect of corneal adhesion was associated with its characteristic varied from solid to membrane. In case of the punch size was sufficiently smaller (i.e., submillimeter scale) than the cornea, the cornea should be considered as a solid, and the classical JKR model could obtained the suitable solution (Fig. 3a). Whereas the punch size was in millimeter scales, the cornea increasingly trended to the characteristic of membrane. Thus, the suitable solution of the corneal adhesion should be obtained by the modified JKR model proposed here.
To simply describe the scale effect of the corneal adhesion, a dimensionless normalization analysis was employed here. The factors, related to the corneal adhesion, contained adhesion force, work of adhesion, elastic modulus, geometry, and IOPs, etc. Taken these factors into account, this work proposed a parameter κ, formulated as Eq. (3), to describe the scale effect.
The parameter of κ, in this study, was described as a function of the ratio between the radii of the punch and the cornea, Rp/Rc. The average value of the corneal radius used here was 9.58 ± 0.22 mm. Compared with the tendencies described in Fig. 3e and 3f, interestingly, the normalization tendencies, functioned with the κ and Rp/Rc, exhibited the opposite (Fig. 4). As shown in Fig. 4, the κ values decreased with the increasing of the ratio Rp/Rc, and the related data were well fitted by an exponential function. For example, in terms of the cornea under the normal IOP of 20 mmHg (Fig. 4a), while the punch was infinitesimal, i.e., the ratio of Rp/Rc boundlessly approaches to zero, the κ value was closer and closer to 1.05, the corneal adhesion increasingly trended to the characteristic of solid. Inversely, while the punch size was infinitely great, it was like that the intact cornea contacted to the ground, the κ value was approximately closer and closer to 0.29, the corneal adhesion increasingly trended to the characteristic of membrane (Fig. 4a). Although the normalization tendencies reversed to the dimensional, the related discrepancies, between the experimental and theoretical data, were the same. Compared with the experimental data, the classical JKR solutions also obtained the lower normalization tendency. And this error also could be offset by the modified JKR solutions (insets in Fig. 4).
Consequently, this work evidenced that the corneal adhesion possessed a scale effect due to its characteristic varied from solid to membrane. While the cornea contacted by a smaller punch in submillimeter, its adhesion could be described by the theory related to solid, i.e., JKR model. While in the greater contact area, however, to analyze the corneal adhesion needed considering the contribution of the surface tension. With the increasing of the contact area, the corneal adhesion increasingly trended to membrane. The related evidences were revealed in the comparison results as shown in Fig. 3, Fig. 4, and Fig. S5. Without wet condition, this study related to the corneal adhesion, also supported the previous discovery of that scale effect was existed in wet adhesion of biological attachment systems.20
Clinically, the corneal adhesion usually appeared with the greater contact area in millimeter scale. In terms of the refractive surgeries, for example, the optical operation diameters were in the range of 5–8 mm.21,22 Additionally, wearing the commercial contact lenses and corneal transplantation, can cover the whole or partial surface of the cornea. To understand the adhesion behavior of cornea itself, this work provided a suitable theory through modifying the classical JKR model.
Compared with the published similar works,3,23 this present work obtained the Fad values of the cornea were smaller than the previous studies in one or two orders of magnitude. This is because the contact areas observed here, between the cornea and the punch, were much smaller than the previous works related to the cornea contacted with the artificial cornea or the contact lenses.3,23 The measured Fad value depends on the contact area (Fig. 2). The Fad of the natural cornea was approximately 20 mN, related to the adhesion interaction between the Boston keratoprosthesis and the corneal disk-samples with the radius of 3 mm.23 However, the maximum critical contact radius ac, related to the 5 mm-Rp punch with the cornea under 20 mmHg-IOP, obtained here was approximately 0.34 mm (Table S1).
In summary, this study evidenced that the corneal adhesion possessed a scale effect. It should be treated as a solid when the cornea contacted in a submillimeter scale, whereas the contaction in a larger size, the characteristic of membrane should be considered in analyzing the corneal adhesion. The modified JKR model proposed here, successfully described the adhesion characteristics of the cornea from solid to membrane.