Impact of Nanometer-Thin Stiff Layer on Adhesion to Rough Surfaces

Adhesion requires molecular contact, and natural adhesives employ mechanical gradients to achieve complete (conformal) contact to maximize adhesion. Intuitively, one expects that the higher the modulus of the top layer, the lower will be the adhesion strength. However, the relationship between the thickness of the stiff top layer and adhesion is not known. In this work, we quantified the adhesion between a stiff glassy poly (methyl methacrylate) (PMMA) layer of varying thickness on top of a soft poly dimethyl siloxane (PDMS) elastomer with a sapphire lens. We found that only about 90 nm thick PMMA layer on a relatively thicker, softer, and elastic PDMS block is required to drop macroscopic adhesion to almost zero during the loading cycle. This drop in adhesion for bilayers can be explained using a conformal model developed by Persson and Tosatti, where the elastic energy to create conformal contact depends on both the thickness and the mechanical properties of the bilayer. A better understanding of the influence of mechanical gradients on adhesion will have an impact on adhesives, friction, and colloidal and granular physics.


Introduction
Adhesion between two surfaces requires molecular contact, and even a small roughness disrupts this molecular contact; this phenomenon is commonly known as the adhesion paradox [1][2][3].Surfaces with low modulus can deform under pressure to create molecular contact [4,5].However, achieving conformability requires additional energy for elastic deformation, which reduces the work of adhesion during approach [6].When this additional elastic energy exceeds the intermolecular work of adhesion, a complete loss of macroscopic adhesion is observed, and the actual (or nominal) contact area is then dominated by Hertzian mechanics [7].The mathematical formulation for calculating the effective work of adhesion was proposed by Persson and Tosatti in 2001 using height power spectral density (PSD), and this model has been validated for soft elastomeric siloxane polymers in contact with hard diamond surfaces by Dalvi et al. [8,9].
Many natural and man-made surfaces have gradient mechanical properties [10][11][12].These mechanical gradients may result from differences in the composition or topological differences that result in an effective modulus that is a function of thickness [10][11][12][13][14].For example, geckos have setae that branch into finer spatula and produce mechanical gradients controlled by both chemical composition and differences in physical parameters such as the diameter and length of the spatula as compared to setae [13,15].These chemical or mechanical gradients can range from nanometer to micrometer scale and, in some cases, they help or deter mechanical contact.Such gradients offer an exquisite approach for controlling molecular contact and, thus, adhesion and friction.[16] Here we address the fundamental question of how a mechanical gradient can affect adhesion in contact with rough surfaces.To simplify the system, we considered a bilayer system with a high-modulus, glassy polymer poly (methyl methacrylate) (PMMA) on top of a soft elastomer poly dimethyl siloxane (PDMS), where the thickness of PMMA ranged between 10 and 90 nms.We measured the adhesion using the Johnson-Kendall-Roberts (JKR) technique by bringing the bilayer into contact with a sapphire hemispherical lens during approach and retraction.We compared the experimental results with predictions of the Persson-Tosatti model modified for bilayers [17].An understanding of how adhesion is affected by mechanical gradients will shed light on the design principles of biological structures and inform the design of effective adhesives that can improve adhesion by controlling the effective modulus of the topmost layer in many applications including robotics and the biomedical sciences [12,16,18].W app W pull-off using the JKR contact mechanics model.(c) Plot of adhesion values obtained from JKR fits during approach (Wapp; shown as red symbols).Pull-off (Wpull-off) values were determined by retracting the sapphire lens with a speed of 60 nm/s and using the JKR analysis (shown by blue symbols).sapphire lens and a 40-nm bilayer system.The shaded region represents variation in PSD of the sapphire lens within the interquartile range where the green dotted line represents 3 rd quartile and the black dotted line represents the 1 st quartile.The dotted red line represents the extrapolated average (C) of the sapphire lens.These profiles were measured using an atomic force microscope, and PSD was calculated using a previously published procedure [19,20].(c) The experimental adhesion values for the bilayers in contact with sapphire are plotted as a function of thickness and are compared with the theoretical values predicted using Equations 1-5.The blue circles used the average values of C, and the shaded regions are calculated using the interquartile range of C (first and third quartile).(d) The ratio of elastic energy spent by several bilayer systems as a function of qcut.These values are normalized by Uelastic after integrating Equation 2 with q1 = 10 10 (m -1 ).

Figure 2: (a) Schematic showing different stages of a bilayer sample as it conforms to the roughness of the solid sapphire substrate. The center panel shows the intermediate stage to illustrate that the bilayer must deform to match the contours of the rough surface. The final panel shows a conformal contact with the rough surface. The equations provide the total energy of the system in each stage and the apparent work of adhesion is the difference between the final and the initial stage. (b) Results for the PSD (C) of the (a)
To determine Wapp for the bilayers, the original Persson-Tosatti formalism was modified to account for changes in surface/interfacial energies of the PMMA ( ! ) and PMMA-PDMS ( "! ) interfaces.In addition, we needed to account for the effective elastic energy to conform to the roughness of the solid surface [8]. Figure 2A shows the steps involved in the development of the model to calculate Wapp.Here, Aapp is the projected area,  # is the surface energy of the rough solid surface, and  $% is the elastic energy to deform the bilayer to conform to the rough solid surface.Wapp can be calculated based on the energy difference to separate the two surfaces.
To determine the change in surface energy in Equation 1 due to the 0-1 interface, we have assumed that the 0-1 interface has the same PSD as the 1-2 interface after conformal contact, independent of film thickness.This is strictly true only for long wavelength roughness with the wavelength  ≫  (where  is the film thickness).For the bilayers studied here,  "! is small and the accuracy of this assumption does not influence the adhesion values.The elastic energy can be determined using the PSD of the rough surface, C (the definition of C is based on reference [21]).
) is the modulus of layer 1 (top layer) and () is the dimensionless surface responsive function, which depends on .The  in Equation 2 is the wavevector (which is equal to 2/, where  is the wavelength of roughness), and  is the thickness of the top layer.() also depends on the Poisson ratio of two layers ( 3 and  ! ) and the ratio of Young's modulus of the top and bottom layer ( !/ " ) [17] The parameters    are expressed as follows: and where the shear moduli are related to Young's modulus: µ " =  " /(2(1 +  " )) and The value of () as a function of  for PMMA/PDMS in contact with the sapphire substrate is provided in Figure S1 (SI).As  tends to zero, Equation 1 will converge to the value expected for a PDMS (layer 0) in contact with a rough substrate.As  tends to infinity, Equation 1 will converge to the expected value for PMMA (layer 1).
To predict Wapp for the system, we need to measure C for the sapphire lens and the PMMA-PDMS bilayer.Figure 2B shows the PSD for these two surfaces and the effective PSD is a function of the sapphire lens, since the bilayers are much smoother in comparison.The other parameters used to calculate Wapp using Equations 1-5 are provided in Table S2 (SI).The only unknown Wint is the intrinsic work of adhesion between the PMMA layer and the sapphire lens.The value for Wint of 49 mJ/m 2 was determined by minimizing the least squared error.Recent studies suggest that the modulus of PMMA could be a function of thickness.Accounting for this results in slightly higher values of adhesion for lower thickness and those results are summarized in Figure S3 (SI).[22,23] The theoretical predictions of Wapp are shown in Figure 2C as a function of thickness (circles).The contribution of the surface area terms and the elastic energy in Equation 1 are plotted in Figure S2 (SI).The second term is much smaller in magnitude because the area ratio term (Atrue/Aapp) is close to 1 for these almost smooth sapphire lenses.When the elastic contribution exceeds Wint, the adhesion drops to zero.The initial drop in Wapp measured experimentally matches well with the theoretical predictions for smaller thicknesses.However, the Wapp from the experiment dropped off more slowly than predicted by theory.One reason for these discrepancies may be due to the extreme sensitivity of Wapp to roughness.For example, if we take the standard deviation based on different measurements of PSD of the sapphire lens and calculate Wapp, we observe the predictions are spread out in the shaded region predicted by the bilayer model.This finding is intriguing because the results suggest that the sensitivity to roughness is a function of the thickness of the PMMA layer.Another reason for a slower drop in adhesion could be due to plastic deformation of PMMA film.
In Figure 2D we plot the ratio of Uelastic(qcut)/Uelastic (total) as a function of qcut.For this calculation, we had to extrapolate the PSD to higher q values beyond the limit we measured experimentally.We found that for thinner film, this ratio converges to 1.0 at higher q values than that for a thicker film.Therefore, the variation in the roughness of the sapphire lens at lower q values (or larger wavelengths) affects the thicker film more than the thinner film, producing higher standard deviations.However, identifying specific reasons for deviation for thicker PMMA films will require a much more uniform rough surface with very little variation in roughness at lower q values.
The application of Equation 1 to explain the adhesion results assumes that for all these samples, the contacts are conformal.We used surface-sensitive infrared-visible sum frequency generation (SFG) spectroscopy to address this question.Using SFG we measured the shift of the vibrational peak of the OH groups on the sapphire surface after contact with the PMMA surface using a bilayer geometry.This shift is due to hydrogen bonding between surface OH groups and carbonyl groups in PMMA, which is sensitive to variation in separation of less than 0.3-0.4nm [24,25].We observed no differences in the shift of the SFG peak for surface OH between the 10-or 200-nm-thick PMMA bilayers and no differences as a function of the applied load.The SFG results are summarized elsewhere [26].This confirms that in these thickness ranges, the contact is conformaland that Equation 1 is applicable for modeling the adhesion data.
Finally, we explain the insensitivity of Wpull-off to the thickness of the PMMA layer.We observed that for uniform PDMS layers in contact with rough surfaces, Wpull-off does not follow the conformal adhesion model (Equation 1) but is determined by contact line pinning and can be explained using the Griffith model, where the adhesion energy is the product of the real contact area and the Wint.The Wint value for all these thicknesses is dictated by PMMA-sapphire interaction, and they are similar for all these measurements.The long-range van der Waals interaction is almost independent of the thickness of the PMMA layer.(Figure S4, SI) [27,28].Based on our SFG results, we concluded that these samples are conformal and, since both Wint and the real contact area are not a function of the thickness of the PMMA layer, we expect Wpull-off to be a constant, in agreement with our experimental results.
In summary, we show that adhesion is extremely sensitive to the bilayer thickness in contact with solids of modestly low roughness.In the short range in thickness between 10 and 90 nm, the adhesion during approach is lost due to roughness, as the elastic energy to conform to roughness is comparable to the intrinsic thermodynamic work of macroscopic adhesion.The predictions using a conformal bilayer contact model compare well with the experimental results and point towards extreme sensitivity of adhesion to small differences in roughness.These results provide insights on the control of adhesion by controlling the near-surface chemistry and modulus.For example, the model presented here can also predict how adhesion can be improved in contact with rough surfaces by layering a thin film of low-modulus polymers on a rigid polymer underlayer.This finding has a direct impact in many fields including soft robotics as well as adhesives for engineering and biomedical applications.[11,12] For the thickness of the samples used in the experiments, the contribution of  56 is independent of thickness.

Figure 1 :
Figure 1: (a) The geometry showing a sapphire lens bought in contact with a bilayer sample consisting of a top layer of rigid/glassy PMMA on a soft PDMS elastomer.The lens in contact with a flat sheet is used for JKR experiments in which the contact area is measured as a function of applied load.(b) Contact radius cubed vs. applied force data for bilayer samples having PMMA layers with different thicknesses.The solid line is fit

Figure 1
Figure 1 summarizes the adhesion results for bilayers with PMMA thickness in the range of 10-84 nanometers.Figure 1A illustrates the JKR geometry used to measure the bilayer adhesion.A sapphire lens with a radius of curvature of 1.25 mm is brought into contact with the bilayer at a speed of 60 nm/s and the contact area as a function of load is measured in a semi-static way; the results are shown in Figure 1B for 10-, 40-and 84nm-thick PMMA layers.The solid lines in Figure 1B are the fit using the JKR equation to determine the apparent work of adhesion (Wapp), and the effective moduli determined from the fits are summarized in Table S1 (in SI).During retraction, we measured the pulloff force and, using the JKR equation, we related the pull-off forces to the work of adhesion during pull-off (Wpull-off).Both of these values are plotted as a function of film thickness in Figure 1C.The values of Wapp drop rapidly as we increase the thickness (Pearson correlation coefficient of -0.84).In comparison, the values of Wpull-off are not affected by PMMA thickness (Pearson correlation coefficient of 0.27).

Figure S1 :
Figure S1: Dimensionless parameter () vs.  using the geometry of PMMA thin layer on top of a thicker PDMS film.[6].As the thickness of the PMMA film tends to infinity, the () converges to 1.As  converges to zero, the value of () converges to the ratio + !,%-! "

Figure S2 :
Figure S2: Comparison of the magnitude of different components contributing to  !"" in Equation 1 (main text).The first ( ./0 ) and the third term ( 2$!30.4 ) are dominant and are shown using the left y-axis.Since the area (Atrue/Aapp) ratio is close to 1, the second term is much smaller and is shown using the right y-axis.

Figure S3 :
Figure S3:Comparison of the calculated  !"" value using Equation 1 (main text) when assuming a uniform modulus across all thicknesses (represented by the blue symbols) against the scenario where the modulus changes with film thickness (illustrated by the red symbols).

Figure S4 :
Figure S4:Variation in the Lifshitz-van der Waals contributions to the adhesion energy as a function of the thickness of the bilayer with PMMA layer on top of a much thicker PDMS film.[11,12]For the thickness of the samples used in the experiments, the contribution of  56 is independent of thickness.

Table S1 :
Experiment work of adhesion and modulus data obtained by fitting the JKR equation:for the thicknesses of PMMA film on a thick PDMS layer.In the JKR equation,  is the radius of curvature,  is force and a is the contact radius.

Table S2 :
Parameters used to calculate theoretical work of adhesion during the approach cycle.