## 2 The Theoretical Relationship between Pull-Off Forces to Peel Speed and Preload
### The Theoretical Relationship between Pull-Off Forces and Preload  
According to the Hertzian contact theory, the actual contact occurs only on a small part of the apparent area due to the surface roughness when two solid surfaces are in contact. The size and distribution of the zone of contact exert a decisive influence on friction and wear. The shape of the rough peaks on the actual contact surface is usually elliptical. Since the size of the contact area of the ellipsoid is much smaller than its radius of curvature, the rough peak can be approximately regarded as a sphere. The contact of two flat surfaces can be regarded as a series of uneven spheres. The contact between two elastomer can be converted into the contact between an elastic sphere with equivalent radius of curvature \(R\) and equivalent modulus of elasticity \(E\) and a rigid smooth surface. When \(\mu\)LED contacts with PDMS, the Young's modulus of PDMS is much lower than that of \(\mu\)LED, so it can be considered as elastic contact. When the two rough peaks contact each other, the normal deformation \(\delta\) is produced under the action of load \(W\), which makes the shape of the elastic sphere change from dotted line to solid line. The actual contact area is a circle of radius \(a\), as shown in Figure 2. The relationship between load and contact area is given by Equation (2) [22]. \[\left\{ \begin{array}{l}\delta=\left(\frac{9W^{2}}{16E^{*2}R}\right)^{1/3}\\ a=\left(\frac{3 WR}{4E^{*}}\right)^{1/3}\\ W=\frac{4}{3}E^{*}R^{1/2}\delta^{3/2}\end{array} \right. \tag{2}\]  
The ideal rough surface is composed of many orderly rough peaks with the same curvature radius and height, and the load and deformation of each peak are exactly the same and independent from each other. However, the rough peak height of the actual contact surface is randomly distributed in general, so the contact peak should be calculated according to the probability. The contact condition of two rough surfaces is shown in Figure 3.  
Their contact can be converted into the situation where one smooth rigid surface touches another rough elastic surface. Since the surface of \(\mu\)LED is very smooth, while the surface of PDMS is quite the opposite, this assumption is consistent with reality. When the distance between the center lines is \(h\), only the part of the contour height \(z>h\) contacts with. In the probability density distribution curve, the shading area of the \(z>h\) part is the surface contact probability, that is [23]  
\[P(z>h)=\int_{h}^{\infty}\psi(z)\mathrm{d}z \tag{3}\]  
Figure 2: Diagram of single peak elastic contact. If the number of peaks on the rough surface is \(n\), the number of peaks participating in the contact, \(m\), is given by [23]:  
\[m=n\int_{h}^{\infty}\psi(z)\mathrm{d}z \tag{4}\]  
The normal phase deformation of each contact peak is \(z\)-\(h\). From Equation (2), the actual contact area \(A\) is given by [23]:  
\[A=m\pi R(z-h)=n\pi R\int_{h}^{\infty}(z-h)\psi(z)\mathrm{d}z \tag{5}\]  
The total load \(W\) is supported by the contact peak as [23]:  
\[W=\frac{4}{3}mE^{*}R^{1/2}(z-h)^{3/2}=\frac{4}{3}nE^{*}R^{1/2}\int_{h}^{\infty} (z-h)^{3/2}\psi(z)\mathrm{d}z \tag{6}\]  
Usually, the contour height of the actual surface follows a Gaussian distribution [24], in which most of region near the \(z\)-score approximates an exponential distribution. Suppose that \(\psi(z)=\exp(-z/\sigma)\), we get:  
\[m=n\sigma\exp(-h/\sigma) \tag{7}\]  
\[A=\pi nR\sigma^{2}\exp(-h/\sigma) \tag{8}\]  
\[W=\frac{3}{4}nE^{*}R^{1/2}\sigma^{3/2}\exp(-h/\sigma) \tag{9}\]  
From the above equations, it can be derived that \(W\) is proportional to \(A\) and \(W\) is proportional to \(m\). Thus, the actual contact area and the number of contact peaks have a linear relationship with the load in the elastic contact state of the two rough surfaces.