Microlens array (MLA) has become an important element in various applications such as flat panel display[1], micro-scanning system[2], fiber-optic probe imaging[3], optical data storage and optical communication etc., for its multi-functional capability. Recently, many manufacturing method of MLA have been proposed and studied, like hot embossing[4], femtosecond and CO2 lasers[5], reflow techniques, UV proximity printing[6], injection molding[7], and other molding processes[8]. Among these, injection forming process is regarded as the best method for replicating MLA mass-production[9], and the mold accuracy determines the forming accuracy. Conventional MLA mold are fabricated by micro-turning, micro-milling and other machining methods, but they are complicated and time-consuming[10]. Indentation process, which is widely used for measuring material properties such as hardness, elastic modulus etc.[11-12], has been studied recently as a novel method for MLA mold manufacturing. By pressing the mold surface with a shaped indenter, a single dimple is formed, which is corresponding to a microlens unit in injection forming.By repeating the indentation, MLA mold can be obtained.

In a single indentation, springback of the workpiece is the main factor affecting the dimple dimensional accuracy. Jiwang Yan et al. investigated the dimensional error caused by springback and observed that the depth of the dimple was remarkably smaller than the nominal indentation depth as elastic deformation played a significant role in the deformation[13]. Repetitive indentation was proposed by them to reduce the form error, but the form error is reduced by only 50%. In multi-indentation, smaller pitch may lead to compressional deformation on the former indented dimples. Yasunori Kobayashi proposed a method to avoid compressional deformation in the manufacture of MLA mold which combined ball-end milling process and sphere indentation[14]. In their method, U-shape grooves intersected at right angle were machined by a ball-end milling, and then indentation was utilized to generate lens at the intersection between U-shape grooves. With this method the deformation of neighboring lens forms caused by following indentation was effectively mitigated. Nevertheless, when the pitch is above a critical value, U-shape grooves won’t be necessary when no compressional deformation between two adjacent dimples.

In this paper, single indentation and multi-indentation were studied to produce a precise MLA mold. Indenter compensation method, by compensating the indenter shape in single indentation, was introduced to increase the single dimple form accuracy. Then, multi dimples at seven different pitches were indented with simulations and experiment to find the critical pitch for MLA manufacturing.

1 Experimental Setup and Simulation

Fig.1 shows the schematic of the indentation process to manufacture MLA mold, which was conducted on a forming process platform as shown in Fig.2, which is a self-developed equipment in Beijing Institute of Technology. An effective displacement of the indenter is controlled by the grating encoder with a resolution of 0.1 μm, and the servo motor is controlled by a force sensor feedback at a resolution of 0.1 N. The working platform is driven by an XY-stage to move the workpiece in the horizontal plane.

Fig.1 Schematic of indentation process manufacturing MLA mold

Fig.2 Forming process platform conducting indentation process

Metal block made of oxygen-free copper was prepared as a work piece for MLA mold manufacturing. The size of workpiece was 5 mm×5 mm×2 mm, and working surface was polished to Ra 0.08 μm. Johnson-Cook (J-C) constitutive model was utilized for indentation simulation shown as

(1)

(2)

所以该题其实是以“产品检验”这一实际问题作为载体，赋予问题以真实的情境，从中考查学生的数学建模素养，并对数学运算、逻辑推理等素养提出一定的要求，是一道综合性较高的题目，有较好的区分度，彰显高考命题育人导向．

TAT载流子产生率GTAT可由场效应增强的SRH模型计算[10-11]，因此GSRH +GTAT可以由场效应增强型的SRH模型给出：

The mechanical property of the material and its parameters are listed in Tab.1.

As shown in Fig.3, a tungsten steel ball with a diameter D of 1.0 mm is glued on the end of Aluminium alloy rod as the indenter, which is used to generate spherical dimples at a specified depth h of 0.1 mm. Diameter of the indented dimple d, D and h take a geometrical relationship as described with

本研究采用亚铁氰化钾为绿色氰源与邻甲基苯甲酰氯通过一锅两步反应来合成邻甲基苯甲酸-1,1-二氰基邻甲基苯甲酯。该反应合理的条件：当 n(邻甲基苯甲酰氯)∶n(亚铁氰化钾) =4∶1，反应介质为二氯甲烷，吡啶为催化剂，反应温度为25 ℃，反应时间为40 min时，产率达到84.2%，该法具有环境友好、操作简单、产率高、后处理方便等优点。

Tab.1 Characteristics of oxygen-free copper and J-C model parameters

Young’smodulus/GPaPoisson’sratioJ-CmodelA/MPaB/MPanCε*pmTRTM1150.33902920.310.02511201356

(3)

Fig.3 Photograph of the indenter

2 Results and Discussion

2.1 Indenter compensation in single indentation

Therefore, multi-indentation processes at seven pitches when k equals 1.00, 1.20, 1.53, 1.67, 2.00, and 2.36 were simulated under the same condition to ensure all the values of k mentioned above included.

Fig.4 Cross-sectional profile generated by sphere indenter

Fig.14 shows the microscopic photograph of dimples when k=1.00. Pile-ups can be observed around the dimples. The far right dimple was indented at last and has a better shape. A cross-sectional profile of the indented dimples is shown in Fig.15. It can be seen that the profile of the last formed dimple agrees very well with an ideal one and the compressional deformation is 0.13d (80.0 μm).

Fig.5 Cross-sectional form error

Fig.6 Cross-sectional profile generated by sphere indenter

2.2 Indentation interaction in multi-indentation

Multi-indentation was simulated to investigate indentation interaction in adjacent areas at different pitches.

Fig.7 shows a schematic of dimples array formed by multi-indentation. As shown in this figure, the distances between two adjacent dimples are the pitch t and the hypotenuse pitch In this study, pitch is defined by a coefficient k multiple the dimple diameter (t=kd), and k should be larger than or equal 1. In this study, as D and d has been calculated (D=1.0 mm, d=0.6 mm), k is 1.67 when the pitch t=D.

Fig.7 Schematic of dimples array generated by multi-indentation and the pitch

Single indentation on the oxygen-free copper was simulated using MSC.Marc code. Loading and unloading velocity was 0.1 mm/s, and holding time was 0.1 s. The cross-sectional profile of indented dimple was measured and shown in Fig.4. Due to the springback, a profile shift occurs comparing to the profile of the indenter.

where σy is the equivalent plastic stress; εp is the equivalent plastic strain; A, B, C, n and m are material parameters, exponent n describing strain hardening, and exponent m is strain rate hardening.

Fig.8a and Fig.8b show the form of dimples array generated by multi-indentation and the distribution of residual Von Mises stress and strain in top view and cross section when k=1.00. Former indented dimple was seriously distorted by the indentation of its two neighboring dimples, where is termed compressional deformation. Maximum residual stress concentrates on the center of the last formed dimple. But,within former indented dimple, area along new neighbor shows higher stress than anywhere else. Residual strain has the largest value around the interaction area between two neighboring dimples.

Fig.8 Multi-dimples forms and distribution of residual Von Mises stress and strain from top view andcross section when k=1.00

Compressional deformation of the former indented dimple along X-axis and Z-axis are not the same. Here we defined c to quantify compressional deformation which is the value of the difference between the ideal diameter of the dimple and the diameter of the distorted dimple. Fig.9a and Fig.9b display the cross-sectional profiles in YZ plane and XY plane of the first indented dimple before and after the neighboring dimple indented respectively, where compressional deformation c is marked. In YZ plane, dimple profile is coincident with the shape of indenter, which means that there is no compressional deformation occurring along Z-axis. On the contrary, cross-sectional profile in XY plane is seriously distorted. Effective maximum compressional deformation c reaches 0.14d (84.0 μm) distinctly, and only occurs on the right half. Far away from neighbor dimple, the other half is not affected.

Fig.9 Cross-sectional profiles when k=1.00

Fig.11 illustrates multi-dimple contours at seven different pitches in simulation, and when k is larger than 2.00, no compressional deformation can be observed, while when k is smaller than 1.41, profile of neighboring dimple is severly distorted. Fig.12 displays the cross-sectional profiles of the distorted dimples at seven different pitches compared with the dimples’ original cross-sectional profile. It can be obviously observed that compressional deformation decreases with the increase of k.

Fig.10 Load-displacement curves

Fig.11 Top-view of contours at different pitches in simulation

2.3 Indentation pitch

On account of the indentation interaction, a critical pitch should be calculated and utilized to ensure the dimension accuracy of the dimples array.

In the simulation, indentation load is also affected by the formation of its neighbor dimples. Load-displacement curves during the forming of three dimples when k=1.00 are depicted in Fig.10. The curves are all near-linear, and the afterward formed dimple needs larger indentation load resulting in larger curve slope. The maximum indentation force to form the three dimples are approximately 210 N, 230 N and 240 N in sequece. Residual stress and strain and pile-ups around the dimple are the mere differences in the three indentations regardless of the variation of material inherent structure.

The maximum compressional deformation c varying with k is recorded in Fig.13. When k is smaller than 1.41, compressional deformation decreases rapidly as k increases, while k is larger than 1.41, the decrease becomes slower. By curve fitting method, it can be calculated that when k is larger than 1.47, c is smaller than 0.01d. Hence, k more than 1.47 can be utilized in the manufacture of MLA mold whose precision is larger than 0.01. Under this condition, critical pitch should be 1.47d, then accurate MLA mold can be obtained. Therefore, critical pitch is a variable related to the processing precision. When k is larger than 2.36, there is no compressional deformation.

Fig.12 Cross-sectional profiles influenced by various pitches

Fig.13 Maximum compressional deformation c vs. k

3 Experiment Verification

Multi-indentation experiments were performed to validate the numerical results and reveal the deformation mechanism.

Fig.17 Top-viewed shape of dimplesat different pitches in experiment

By subtracting the ideal round profile from the dimple profile, form error was calculated in Fig.5. As it can be seen in the figure, the maximum form error is about 2.6%h (2.6 μm). By compensating the form error to indenter shape,the dimple with a cross-sectional profile approximating to the ideal spherical curve can be obtained, and the maximum form error was reduced to 0.007 5%h(7.5 nm)as shown in Fig.6.The following simulations of MLA indentation were all performed using this compensated indenter.

Indenter compensation in single indentation and compressional deformation in a multi-indentation process to manufacture MLA mold were studied in this paper. The following conclusions can be drawn.

The indentation force during the three indentations is recorded in Fig.16 and a linear relation between indentation load and displacement with a maximum load of 250 N can be observed.

Fig.14 Microscopic photograph of dimples when k=1.00

Fig.15 Experimental cross-sectional profile of dimples when k=1.00

Fig.16 Load-displacement curves of three sequent indentations

Micro-grinding process was performed after indentation to remove pile-ups. The effective contours of dimple arrays at seven different pitches are shown in Fig.17. When k was smaller than 1.41, the former indented dimples were severly distorted, which agrees well with the simulation result.

4 Conclusions

叶澜教授曾说过:“课堂应是向未知方向挺进的旅行，随时都有可能发现意外的通道和美丽的图景，而不是一切都必须遵循固定线路而没有激情的行程。《新课程标准》也强调：教学是教师与学生的交往、积极互动、共同发展的过程。学生的学习不是预约的，而是学生与教师、同伴“思维碰撞、心灵沟通、情感融合”的“动态”过程。 作为课堂组织者的教师必须独具慧眼，善于捕捉课堂上的生成资源，努力促进学生的学习在“动态生成”中发展，让课堂焕发出生命的活力。

① Indenter compensation by compensating form deviation to the shape of an indenter is an effective way to eliminate form error in single indentation.

② In multi-indentation process, a small pitch affects the shape accuracy significantly with compression deformation between adjacent dimples. When the pitch is larger than 1.47d, compressional deformation is less than 0.01d. While the pitch is larger than 2.36d, there will be no compressional deformation.

2.2症状消退时间 研究组各症状消退时间均明显短于对照组,差异具有统计学意义(P<0.05)。见表2。

③ Simulation results agree well with the experiment results, which can be utilized for further study on manufacturing MLA molds by an indentation process.

References:

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[2] Pang K, Song L, Fang F Z, et al. An imaging system with a large depth of field based on an overlappen micro-lens array[C]∥66th General Assembly of the International Academy for Production Engineering (CIRP),Guimaraes, Portugal, 2016.

[3] Shinoj V K, Murukeshan V, Tor S B, et al. Design fabrication and characterization of thermoplastic microlenses for fiber-optic probe imaging[J]. Applied Optics, 2014, 53(6):1083-1088.

[4] Xi Qiufan. Realization of mass production of microlens array by hot embossing on silicon substrate[J]. Advanced Materials Research, 2010, 1037(139):1562-1565.

[5] Choi H K, Ryu J, Kim C, et al. Formation of micro-lens array using femtosecond and CO2 lasers[J]. Journal of Laser Micro Nanoengineering, 2016, 11(3):341-345.

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[7] Kim J S, Ko Y B, Hwang C J, et al. A study on the fabrication method of middle size LGP using continuous microlens array made by LIGA reflow[J]. Korea-Australia Rheology Journal, 2007, 19(3):171-176.

[8] Moore S, Gomez J, Lek D, et al. Experimental study of polymer micro-lens fabrication using partial-filling hot embossing technique[J]. Microelectronic Engineering, 2016, 162(16):57-62.

[9] Chang C Y, Yang S Y, Chu M H. Rapid fabrication of ultraviolet-cured polymer micro-lens arrays by soft roller stamping process[J]. Microelectroinc Engineering, 2007,84(2): 355-361.

[10] Davies M A, Evans C J,Patterson S R, et al. Application of precision diamond machining to the manufacture of micro-photonics components [J]. Proc SPIE 5183, Lithographic and Micromachining Techniques for Optical Component Fabrication II, 2003, 12(4): 94-108.

[11] Moussa C, Hernot X, Bartier O, et al. Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[J]. Journal of Material Science, 2014, 49(2): 592-603.

[12] Oliver W C, Pharr G M. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refineness to methodology[J]. Journal of Materials Research, 2004, 19(1): 3-20.

[13] Yan J W, Akihiro H, Tsunemoto K, et al. Manufacturing structured surface by combining microindentation and ultraprecision cutting[J]. CIRP Journal of Manufacturing Science and Technology, 2012, 5(1): 478-487.

[14] Yasunori K, Ryo I, Haruhisa S. Proposal of finishing method of MLA mold applied sphere indentation[J]. Advanced Materials Research,2013, 2631(797): 475-480.

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选择2017年12月份之前入职的低年资手术室护士为研究对象。入选标准：（1）手术室护士。（2）工作年限＜5年。（3）自愿参加本次调查。

生物学试题中计算题较多，有些计算题无法直接求解（无法通过已知量直接推算结果），必须通过巧妙设元，利用方程思想来解答。那么，中学生物学中哪类的试题必须利用方程思想来解答呢？