**2. The Theoretical Relationship between Pull-Off Forces to Peel Speed and Preload** *2.1. The Image of μLED*

The size of μLED is generally smaller than 100 μm [19,20], as shown in Figure 1. A scanning electron microscope (SEM) image of μLED is shown in Figure 1A. The surface of the μLED array obtained by optical microscope is shown in Figure 1B.

**Figure 1.** Image of μLED. (**A**) Scanning electron microscope (SEM) image of μLED. (**B**) The surface of the μLED array obtained by optical microscope.

#### *2.2. The Theoretical Relationship between Pull-Off Forces and Peel Velocity*

The relationship between peel speed and pull-off force has also been extensively studied [21], which can be expressed as:

$$G\_{\varepsilon}(\upsilon) = G\_0[1 + \left(\frac{\upsilon}{\upsilon\_0}\right)^k] \tag{1}$$

where *G*<sup>0</sup> is the critical energy release rate and corresponding detaching speed *v*<sup>0</sup> approaches zero, *v* is the peel speed, and the exponent *k* is a parameter that can be determined from experiments. The power–law relationship (Equation (1)) has been found applicable to low or high peel speed obtained from metal/polymer and polymer/polymer interfaces at various temperatures.

#### *2.3. 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 μLED contacts with PDMS, the Young's modulus of PDMS is much lower than that of μLED, so it can be considered as elastic contact. When the two rough peaks contact each other, the normal deformation *δ* 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].

$$\begin{cases} \delta = \left(\frac{9W^2}{16E^\*R}\right)^{1/3} \\\\ a = \left(\frac{3WR}{4E}\right)^{1/3} \\\\ W = \frac{4}{3}E^\*R^{1/2}\delta^{3/2} \end{cases}$$
 
$$\begin{cases} \sum\_{k} \sum\_{n=1}^{E} \\\\ \sum\_{\substack{R \text{right plane} \\}} \end{cases}$$

**Figure 2.** Diagram of single peak elastic contact.

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 μLED 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]

$$P(z>h) = \int\_{h}^{\infty} \psi(z) dz\tag{3}$$

(2)

**Figure 3.** Contact of rough surfaces. The root mean square values of the roughness of the two surfaces are respectively *σ*<sup>1</sup> and *σ*2, *h* is the distance between the center lines, *z* is the part of the contour height *z* > *h* contacts with, and Ψ(*z*) is the probability of the surface 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]:

$$m = n \int\_{h}^{\infty} \psi(z) dz \tag{4}$$

The normal phase deformation of each contact peak is *z*-*h*. From Equation (2), the actual contact area *A* is given by [23]:

$$A = m\pi\text{R}(z - h) = n\pi\text{R}\int\_{h}^{\infty} (z - h)\psi(z)dz\tag{5}$$

The total load *W* is supported by the contact peak as [23]:

$$\mathcal{W} = \frac{4}{3} m E^\* R^{1/2} (z - h)^{3/2} = \frac{4}{3} n E^\* R^{1/2} \int\_h^\infty (z - h)^{3/2} \Psi(z) dz \tag{6}$$

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 *ψ*(*z*) = exp(−*z*/*σ*), we get:

$$m = n\sigma \exp(-h/\sigma) \tag{7}$$

$$A = \pi n R \sigma^2 \exp(-h/\sigma) \tag{8}$$

$$\mathcal{W} = \frac{3}{4} n E^\* R^{1/2} \sigma^{3/2} \exp(-h/\sigma) \tag{9}$$

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. Separating μLEDs from PDMSs creates two new interfaces, and the force value *F*cr required for this process is obtained as [10]:

$$F\_{cr} = A\gamma \tag{10}$$

where *γ* is the viscosity coefficient of the two surfaces. From Equations (8)–(10), it can be concluded that the adhesive force increases with the increase of preload.

#### **3. The Experimental Results and Discussion**

#### *3.1. Experimental Steps*

To measure the adhesion between a single μLED and the substrate, a cantilever measurement scheme is adopted, and the specific steps are shown in Figure 4A–C.

**Figure 4.** The adhesion measurement results based on cantilever between μLED and substrate. (**A**–**C**) Measurement steps. (**A**) The initial state. (**B**) The loading status. (**C**) The reverse motion. (**D**) Typical single measurement results.

Step 1: Apply glue to the tip of the tipless cantilever with a stiffness of 5.1 N/m.

Step 2: Move the cantilever above a μLED.

Step 3: Lower the cantilever to contact the μLED and wait for the glue (UV photoresist) to solidify.

Step 4: Raise the cantilever to make the μLED separate from the base.

Step 5: Move μLED above the PDMS substrate.

Step 6: Measure the relevant force value at different peel speeds and preload.

A typical adhesion–depth curve on a single μLED with a flexible PDMS substrate (1:10 mixing ratio) measured by AFM is shown in Figure 4D. The tip of the cantilever is controlled at a speed of 10 μm/s. The *x*-axis is the displacement of the μLED. The measurement process is divided into two segments according to the direction of cantilever movement: approach (red line in Figure 4D) and retract (blue line in Figure 4D). The *y*-axis is force between the μLED and PDMS.

The AFM has been well calibrated using thermal method. The relationship between the force acted on cantilever and PSD output has been obtained before measuring the adhesion force.

The μLED on the cantilever was moved above a substrate PDMS, as shown in Figure 4A. Figure 4B is in a loading status. The μLED is pressed on the PDMS and continuously moved through the precision stage. The laser spot moves as the cantilever bends. The cantilever will not stop until the pressure equals the set preload, as shown in section BC in Figure 4D. Figure 4C is in reverse motion. With the reverse movement, the pressure of μLED on the PDMS substrate becomes smaller and smaller until the pressure reaches 0, as shown in the CD section.

Dynamic jumping behavior during approach (such as the BC segment) and measured jumping behavior during return (CF) were measured.

As can be seen from Figure 4D, the measurement process can be divided into four stages according to the contact status: (I) pressure down to contact with PDMS, (II) pressure to the maximum to reach preload, (III) reverse movement, and (IV) separation from PDMS.

I—Initial state: the cantilever is moved by a precision stage and is not in contact with the PDMS, as shown in Section AB.

II—Loading status: μLED contact PDMS. The μLED is pressed on the PDMS and continuously moved through the precision stage. The laser spot moves as the cantilever bends. The cantilever will not stop until the pressure equals the set preload, as shown in section BC.

III—Reverse motion: With the reverse movement, the pressure of μLED on the PDMS substrate becomes smaller and smaller until the pressure reaches 0, as shown in the CD section. As the reverse motion continues, the PDMS deforms due to the tension between the μLED and PDMS. At this point, the elastic force of the cantilever acting on μLED is less than the critical adhesion force of PDMS, as shown in section DE.

IV—Exit stage: The elastic force of the cantilever on μLED is greater than the critical adhesion force. A sudden jump in the position sensitive device (PSD) voltage output can be observed, as shown in the EF section.

The maximum pull-off force can be defined as the minimum force of the force-depth curve, as shown in Figure 4D.

#### *3.2. Measurement of Adhesion under Different Detaching Velocities and Preload*

As shown in Figure 5, the maximum pull-off force was measured at different peel velocities (detaching velocity) varying from 10 μm/s to 300 μm/s, in which high peel speed (300 μm/s) resulted in strong adhesion, while low peel speed (10 μm/s) resulted in weak adhesion. Obviously, there is a strong correlation (proportional relationship) between the maximum pull-off force and the peel velocity.

**Figure 5.** The measured maximum pull-off forces with respect to peel velocity.

We measured the maximum adhesion force at different preload from 0.5 to 3 μN (the peel speed was fixed at 10 μm/s). The proportional relationship between the maximum pull-off force and the preload is shown in Figure 6.

The experimental results show that the preload has a great influence on the adhesion, which is different from the previous research: "Unlike the effects of material property of PDMS, the maximum pull-off force has similar value regardless of the initial indentation force between the tip and the flexible substrate". Our theoretical result is consistent with our experimental result but different from the literature.

It Is hard to compare the experimental results to results from Equations (8)–(10). The equations show a positive proportion relationship between contact area and the maximum pull-off force. However, the real contact area between the μLED and PDMS or other substrates could not be measured. Therefore, it is impossible to directly compare the quantity of theoretical value and experimental value absolutely, but only relatively.

**Figure 6.** Results for the maximum pull-off force at different preload.

## **4. Conclusions**

In this paper, the adhesion force between the μLED and substrate at different peel speeds and preload was measured by AFM. The experimental results show that the separation force between a single μLED and PDMS substrate is not only related to the peel speed, but also related to the preload. Although it is hard to directly compare the absolute quantity of theoretical value and experimental value, the results find a new way to design an apparatus for μLED transfer printing. Future research is required to reversibly change adhesion strength between strong and weak modes by more than two orders of magnitude so that the system can be applied in transfer printing. We will focus on the design of a novel substrate to achieve this target. This system would have broader impacts in transfer printing.

**Author Contributions:** J.B. and P.N. designed the manuscript, wrote and analyzed the data; J.B. designed the figures for the manuscript and performed the experiments, data collection and/or statistical analysis; P.N., S.C. and Q.L. revised the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received financial support by the Scientific Research Program of Tianjin Education Commission (No.2019ZD08).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author, P.J.N., upon reasonable request.

**Conflicts of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

#### **References**

