Self-calibration performance analysis of sheet micro-components based on droplet array

The droplet array-based micro-component self-calibration method features simple operation, easy batch control implementation, and holds great promise in the field of micro-assembly. The paper investigates the self-calibration process, analyzes the factors influencing self-calibration performance, discusses self-calibration errors, self-calibration range, and their relationship with micro-component parameters, droplet array parameters, and fluid parameters, and proposes a construction method suitable for droplet arrays suitable for micro-components. The research results indicate that the self-calibration error and calibration range of micro-components are related to micro-component parameters, substrate array parameters, and droplet parameters. When the micro-component size is greater than or equal to the droplet array size, the self-calibration error is relatively small. Constructing an appropriate droplet array based on the size and shape of the micro-components can reduce self-calibration errors and meet the self-calibration accuracy requirements for micro-components of varying sizes and shapes.


Introduction
The self-calibration of micro-components position attitude is an important part of micro-assembly.With the diversification of mechanical and electronic products, components are also gradually moving toward the development trend of miniaturization and thinness, which puts forward higher requirements for the position attitude adjustment of micro-components of various shapes.The position attitude deviation of micro-component directly affects the quality and performance of subsequent assembly operations.Therefore, it is important to study the calibration performance of micro-components to achieve high precision, high efficiency and high adaptability assembly of micro-components.
There are two main ideas in self-calibration methods, one is to perform self-calibration of micro-component based on the surface tension and energy minimization principle of liquid bridge.The other is to complete the self-calibration of micro-component and substrate through the coupling effect of magnetic field or electric field and vibration.
Literature [1] proposed the idea of accomplishing low temperature fluidic self-assembly and self-alignment of chip through core-shell transformation-imprinted solder bumps, which can realize the self-calibration and self-assembly of large flux chips, and analyzed the effect of Sn diffusion from core to BiInSn shell and BiIn shell on the self-calibration performance.Bo Chang and Ali Shah et al. [2] achieved self-transport and self-alignment of millimeter-scale, submillimeter-scale chips by spraying microscopic rain to form a liquid bridge in the assembly region and inducing the microchips to move toward the substrate.Srinivasan et al. [3] used hydrophobic SAM(self-assembly monolayer) to construct a variety of shapes of substrate microstructure, which can be applied to the self-assembly of different shapes of parts in the nano-to milliscale with an alignment precision of less than 0.2 μm and rotational misalignment within 0.3°.
Ramadan et al. [4] proposed large scale microcomponents assembly method using an external magnetic array, assembly of 2500 microcomponents has been accomplished in 5 min with assembly yield of 97% by combining magnetic force, shape recognition, and vibration.Bashir's group [5] proposed a dielectrophoresis and electrohydrodynamics-based fluidic assembly method for silicon resistors that enables submicron precision alignment and assembly of silicon resistors and silicon blocks on microelectrode structures, it can be used for heterogeneous integration of micro-and nanoelectronic devices.
The above self-calibration methods generally require the fabrication of different substrates for different micro-components, and suffer from the problem of fixed alignment shapes and weak adaptability.To solve this problem, literature [6] proposes a droplet manipulator-based micro-components self-calibration method, which changes the shape of the droplet manipulator operating surface according to the shape of the micro-component and forms a droplet of the corresponding shape on the manipulator operating surface.When the droplet manipulator operating surface is in contact with the micro-component, the micro-component will be quickly and automatically aligned with the manipulator end surface position attitude driven by the recovery force of the liquid bridge.The literature [7] further proposes a self-calibration method for micro-components based on droplet array, through which droplet array can be constructed with different shape droplet combinations to meet the self-calibration requirements of different shapes and sizes of sheet micro-components.Building upon the previous research, this paper further analyzes the self-calibration process and the impact of droplet array parameters and micro-component parameters on calibration performance.

Analysis of droplet array self-calibration process
The self-calibration platform consists of a calibration substrate and a droplet generation mechanism.The calibration substrate is uniformly embedded with microtubes, spaced equidistantly to form a microtube array.The lower ends of these microtubes are connected to the droplet generation mechanism, supplying the required droplets to the microtube array.Select a set of microtubes that corresponds to the shape and size of the micro-components to be calibrated.Then liquid is injected into the chosen microtubes using the droplet generation mechanism, forming droplets on the calibration substrate's surface, as depicted in Figure 1.When a micro-component placed on the substrate exists a position and attitude deviation with the envelope of the droplet array, the asymmetric capillary forces between the micro-component and the array quickly drive the micro-component to align its position and attitude with the micro droplet array, thus enabling self-calibration of the micro-component.Taking the calibration of a square micro-component as an example, let's analyze the calibration process.Depending on the geometric dimensions and shape of the micro-component, an appropriate microtube array is chosen.Assuming the coordinate system of the micro-component is denoted as ∑P, and the coordinate system of the droplet array envelope is represented as ∑O, as shown in Figure 2. To facilitate the explanation, the marked as 1-4 correspond to the contact surfaces between the respective microtubes and the array substrate, and the corresponding liquid bridges are assigned the same numbers.When studying a square micro-component, each liquid bridge within the droplet array exerts a liquid bridge force along the axis direction on the contact surface of the micro-component, where = 1~4.This liquid bridge force consists of two components.One part is the effect of the liquid negative pressure inside along the axis of the liquid bridge within the micro-component wetted region, denoted as .The other part is the surface tension on the liquid bridge lateral surface acting along the liquid bridge's axis direction at the three-phase contact line with the micro-component.For ease of description, the surface tension acting on the liquid-air interface at the three-phase contact line of the micro-component is defined as .can be decomposed into the force components along the axis direction ( ) and the horizontal direction ( and ), where the resultant of and is referred to ℎ .We have Σ = , Σ = , Σ = , and Σ = . The resultant of and represents the total liquid bridge force exerted by all the liquid bridges on the micro-component.The positional deviation in the horizontal direction between coordinate systems ∑P and ∑O is represented as (, ), and the attitude deviation is represented as .and their direction is in the negative direction of , hindering the calibration movement of the micro-component.On the other hand, the asymmetry of liquid bridges 2 and 3 decreases, causing 2 and 3 to gradually become smaller as the micro-component moves.When the resultant force = 2 + 3 − 1 − 4 > 0 , the micro-component continues to move in the direction of , approaching the position of the micro droplet array.When the resultant force = 0 , the micro-component reaches an equilibrium state.At this point, the error between the position of the micro-component coordinate system ∑P and the micro array coordinate system ∑O is referred to the position error, denoted as , .During the calibration movement process, liquid bridges 1 and 4 transition from initially symmetric liquid bridges to asymmetric ones.The magnitudes of 1 and 4 are influenced by the viscosity forces between the droplets and the micro-component.When the contact angle of liquid bridges 1 and 4 increases to the advancing contact angle or decreases to the receding contact angle , the viscous resistance on the micro-component's contact surface increases to a maximum.Beyond this angle, the liquid bridges will slide on the micro-component, forming new symmetric liquid bridges, change back to 1 = 4 = 0 .As the calibration process progresses, it exhibits periodic variations.= , produced by the ℎ forces at the geometric center of contact surface on the micro-component, drives the micro-component to rotate in the direction opposite to , aligning it with the attitude of the micro droplet array.When the resultant torque = 0 , the micro-component reaches an equilibrium state, and the attitude error between the micro-component coordinate system ∑P and the micro droplet array coordinate system ∑O is referred to the attitude error .

Factors affecting self-calibration position accuracy
When the center-to-center distance between the microtubes in the array changes, the size of the droplet array will vary accordingly.Taking a 4-microtubes array as an example, with the center-to-center distance between adjacent microtubes denoted as and the radius of the freely spreading circular droplets on the micro-component surface as , the envelope of the array is a square with side length = + 2 .Assuming the micro-component has a side length of , when the droplet array size matches the micro-component size, i.e., = , the initial positional deviation of the micro-component is represented as (, ), as shown in Figure 4.The force from asymmetric liquid bridges drives the micro-component rapidly to align with the position of the droplet array.In the equilibrium state, = 0, = 0, and = .The error , between the coordinates ∑P and ∑O represents the self-calibration error.In an ideal state, neglecting the viscous forces between the micro-component's surface and the droplets, the post-calibration errors and are 0. The calibration range is , ∈± 0, .(a) Initial positional deviation of the micro-component (b) Micro-component position after calibration From the above analysis, it is evident that when the droplet array size is larger than or equal to the micro-component size, the theoretically calculated self-calibration errors and are 0. As the array size increases, the calibration range of the micro-component decreases.When the droplet array size is smaller than the micro-component size, there exists an inherent error that cannot be calibrated in the theoretically calculated self-calibration errors and .With a smaller array size, the self-calibration error increases.This is because when the array size is smaller than the micro-component, each liquid bridge in the droplet array spreads into a symmetric liquid bridge before the micro-component is calibrated to its ideal position.The micro-component is no longer subjected to forces in the horizontal direction, and the calibration stops.Therefore, the guiding principle for selecting a droplet array is that the envelope size of the array should not be smaller than the micro-component size.

The self-calibration attitude deviation adjustment range
On the premise of ensuring the accuracy of position error, select a droplet array that is not smaller than the size of the micro-component for attitude calibration.As shown in Figure 7, when the attitude deviation of the micro-component is 45°, there is a certain geometric relationship ℎ +  radius can influence the surface tension coefficient , the liquid contact angle , and the size of the droplet spreading radius , thus affecting the magnitudes of liquid bridge forces and tension horizontal forces ℎ .Changes in the droplet spreading radius , result in variations in the envelope size of the array, which may introduce increased errors.

Conclusion
This paper has analyzed the self-calibration process and its underlying mechanisms, studied the factors influencing self-calibration performance, discussed the relationship between self-calibration errors and micro-component parameters and droplet array parameters.We proposed a method for constructing droplet arrays suitable for micro-components.The research results indicate that, by selecting an appropriate droplet array that matches the size and shape of the micro-component, self-calibration errors can be minimized, meeting the self-calibration accuracy requirements for micro-components of different sizes and shapes.
The performance analysis method presented in this paper, which focuses on symmetric micro-components, can be equally applied to the analysis of asymmetric micro-components.Further refinement of the model and validation of the proposed method through extensive simulations and experiments will be our future work, ensuring the reliability and applicability of the approach.

Figure 1 .
Figure 1.Micro droplet array corresponding to micro-component

Figure 2
Figure 2 Coordinate systems of the droplet array and micro-component

Figure 3
Figure 3 Schematic representation of the self-calibration process for rectangular micro-component position deviations When there is an initial attitude deviation of the micro-component from the droplet array about the z-axis, as shown in Figure 2(b), all 4 liquid bridges become asymmetric.Each liquid bridge generates a vertical outward component force ℎ along the edge of the micro-component.The resultant torque= , produced by the ℎ forces at the geometric center of contact surface on the micro-component, drives the micro-component to rotate in the direction opposite to , aligning it with the attitude of the micro droplet array.When the resultant torque = 0 , the micro-component reaches an equilibrium state, and the attitude error between the micro-component coordinate system ∑P and the micro droplet array coordinate system ∑O is referred to the attitude error .

Figure 4
Figure 4 Overhead view of the droplet array calibration which matching the micro-component size (a) Initial positional deviation of the micro-component (b) Micro-component position after calibration When the droplet array size is larger than the micro-component size, i.e., > , as shown in Figure 5, theoretically, the calculated post-calibration errors and are 0. The calibration range is , ∈± 0, − 2 .

Figure 5 2 .Figure 6
Figure 5 Overhead view of the droplet array calibration which larger than the micro-component size (a) Initial positional deviation of the micro-component (b) Micro-component position after calibration When the droplet array size is smaller than the micro-component size, i.e., < , as shown in Figure 6, the theoretically calculated post-calibration errors and are − 2 − .The calibration range is , ∈± 0, − 2 .
0, there is the feasibility of attitude deviation calibration.At this point, the 4 liquid bridges provide equal magnitudes of outward perpendicular forces ℎ along the micro-component's edges, and their directions are centrally symmetric.Therefore, the resultant force is ℎ = =1 4 ℎ = 0, the torque around the z-axis is 0. The micro-component remains in an equilibrium state without any motion.If is less than 45°, the micro-component is subjected to the resultant torque and tends towards the attitude direction of the droplet array.If is greater than 45°, due to the micro-component's symmetric shape and the principle of proximity, the micro-component will tends towards the 90°direction for calibration.Therefore, the attitude calibration range for a square micro-component is ∈± 0,45°.

Figure 7
Figure 7 Top view of the droplet array calibrating the micro-component attitude deviation