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Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale

Binbo LI Kejing HUANG Wentao WANG Jiaming ZHANG Haizhou XUE Dan MO Jinglai DUAN

李彬博, 黄科京, 王文涛, 张家明, 薛海舟, 莫丹, 段敬来. 快重离子微观尺度二维分布的模拟研究[J]. 原子核物理评论, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
引用本文: 李彬博, 黄科京, 王文涛, 张家明, 薛海舟, 莫丹, 段敬来. 快重离子微观尺度二维分布的模拟研究[J]. 原子核物理评论, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
Binbo LI, Kejing HUANG, Wentao WANG, Jiaming ZHANG, Haizhou XUE, Dan MO, Jinglai DUAN. Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale[J]. Nuclear Physics Review, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
Citation: Binbo LI, Kejing HUANG, Wentao WANG, Jiaming ZHANG, Haizhou XUE, Dan MO, Jinglai DUAN. Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale[J]. Nuclear Physics Review, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037

快重离子微观尺度二维分布的模拟研究

doi: 10.11804/NuclPhysRev.39.2021037
详细信息
  • 中图分类号: O571.53

Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale

Funds: National Natural Science Foundation of China(U1932210)
More Information
  • 摘要: 在快重离子的辐照中,无论通过束流扫描还是束流散焦,在微观尺度上入射离子依然是随机分布的。最近,这种微观上的离子不均匀性对于快重离子的前沿应用,如高密度微孔膜的制备和航空电子器件的单粒子效应的评估等变得至关重要。本工作利用蒙特卡罗方法模拟了垂直于束流方向的二维平面上随机分布的离子。从统计角度来看,蒙特卡罗方法模拟的潜径迹的分布与相同离子注量下微孔膜的电镜观测结果相一致。根据模拟结果,微孔膜的有效孔隙率可以通过辐照注量和孔径来进行有效预测。另外,也提出了计算有效孔隙率的经验公式。利用此公式可以在一定程度上预测最优化的离子辐照和化学蚀刻参数,从而得到理想的孔隙率。同时,为了评估微孔膜的选择性,计算了形成重孔的概率,这有助于在膜的孔隙率和选择性之间达到平衡,从而获得最好的膜性能。通过蒙特卡罗模拟,还研究了微观尺度下离子辐照的均匀性问题,其结果可以为单粒子效应等应用提供重要参考。
  • Figure  1.  (a)~(b) SEM images of microporous membranes etched from heavy ion latent track templates and (c)~(d) the corresponding MC simulations.

    Figure  2.  (color online) The density distribution of $ d_1 $ to $ d_4 $ (from top to bottom) for ion fluence of $1.0 \times 10^8 $$ \text{ions/cm}^2 .$

    Figure  3.  (color online) Simulated effective porosities (a) for membranes with varied pore sizes and different ion fluences; (b) for membranes with varied ion fluences and different pore sizes. (c) The empirical curve versus calculated effective porosities (dots) for an ion fluence of $ 1.0 \times 10^7\ \text{ions/cm}^2 $.

    Figure  4.  (color online) The probability of single pore and multi-pores as a function of effective porosity.

    Figure  5.  (color online) (a) A MC simulation of ion distribution with a nominal ion fluence of $ 1.0 \times 10^8 \ \text{ions/cm}^2 $. Localizated ion fluences are counted for $ 200\; \mu \text{m} \times 200\; \mu \text{m} $ and $ 20\; \mu \text{m} \times 20\; \mu \text{m} $ field of views; (b), (c) Histograms of ion counts for field-of-view sizes of $ 100\; \mu \text{m} \times 100\; \mu \text{m} $ and $ 10\; \mu \text{m} \times10\; \mu \text{m} $. The nominal ion fluence is $ 1.0 \times 10^9 \ \text{ions/cm}^2 $; (d) Calculated RSD for varied area of view and (e) for varied ion fluence from $ 5.0 \times 10^5 $ to $ 1.0 \times 10^{10} \ \text{ions/cm}^2 $.

    Table  1.   Comparison of single and multi-­pores ratios between the PC microporous membranes and MC simulations (shown in parentheses).

    Ion fluence/ (ions·cm-2)Average pore size /nm${P}_1$/%${P}_2$/%${P}_3$/%${P}_{n \geqslant 4}$/%
    $4.76 \times 10^7$ 1 000 23.0 (22.8) 35.60 (32.1) 22.20 (25.4) 19.2 (19.60)
    $1.25 \times 10^7$ 820 76.4 (74.7) 23.00 (21.1) 0.68 (3.84) 0.0 (0.32)
    $8.56 \times 10^5$ 1 420 92.8 (94.2) 7.20 (5.6) 0.00 (0.21) 0.0 (~ 0.00)
    $2.53 \times 10^9$ 20 94.3 (96.8) 5.67 (3.1) 0.00 (0.036) 0.0 (~ 0.00)
    $7.79 \times 10^9$ 24 82.0 (86.3) 15.30 (12.9) 2.67 (0.73) 0.0 (~ 0.00)
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出版历程
  • 收稿日期:  2021-04-28
  • 修回日期:  2021-05-21
  • 刊出日期:  2022-06-29

Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale

doi: 10.11804/NuclPhysRev.39.2021037
    基金项目:  National Natural Science Foundation of China(U1932210)
    作者简介:

    (1995−), Pingliang, Gansu Province, postgraduate student, working on condensed matter physics; E-mail: libinbo@impcas.ac.cn

    通讯作者: E-mail: haizhouxue@impcas.ac.cnE-mail: modan@impcas.ac.cn
  • 中图分类号: O571.53

摘要: 在快重离子的辐照中,无论通过束流扫描还是束流散焦,在微观尺度上入射离子依然是随机分布的。最近,这种微观上的离子不均匀性对于快重离子的前沿应用,如高密度微孔膜的制备和航空电子器件的单粒子效应的评估等变得至关重要。本工作利用蒙特卡罗方法模拟了垂直于束流方向的二维平面上随机分布的离子。从统计角度来看,蒙特卡罗方法模拟的潜径迹的分布与相同离子注量下微孔膜的电镜观测结果相一致。根据模拟结果,微孔膜的有效孔隙率可以通过辐照注量和孔径来进行有效预测。另外,也提出了计算有效孔隙率的经验公式。利用此公式可以在一定程度上预测最优化的离子辐照和化学蚀刻参数,从而得到理想的孔隙率。同时,为了评估微孔膜的选择性,计算了形成重孔的概率,这有助于在膜的孔隙率和选择性之间达到平衡,从而获得最好的膜性能。通过蒙特卡罗模拟,还研究了微观尺度下离子辐照的均匀性问题,其结果可以为单粒子效应等应用提供重要参考。

English Abstract

李彬博, 黄科京, 王文涛, 张家明, 薛海舟, 莫丹, 段敬来. 快重离子微观尺度二维分布的模拟研究[J]. 原子核物理评论, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
引用本文: 李彬博, 黄科京, 王文涛, 张家明, 薛海舟, 莫丹, 段敬来. 快重离子微观尺度二维分布的模拟研究[J]. 原子核物理评论, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
Binbo LI, Kejing HUANG, Wentao WANG, Jiaming ZHANG, Haizhou XUE, Dan MO, Jinglai DUAN. Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale[J]. Nuclear Physics Review, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
Citation: Binbo LI, Kejing HUANG, Wentao WANG, Jiaming ZHANG, Haizhou XUE, Dan MO, Jinglai DUAN. Foreseeing Two Dimensional Distribution of Swift Heavy Ions at Micro-scale[J]. Nuclear Physics Review, 2022, 39(2): 245-251. doi: 10.11804/NuclPhysRev.39.2021037
    • In a solid target, Swift Heavy Ion(SHI) irradiation deposits extremely intense energy, that causes irreversible structural damage and dramatic evolution of target’s properties[1-2]. Factors that determine a SHI irradiation include the ion species, velocity, ion fluence and uniformity. The latest is commonly noticed as a macro-scale uniformity which is achieved by electrical/magnetic scanning, or by magnetic beam defocusing[3]. Noting that during beam scanning and defocusing the entire ion beam is manipulated, at a micro-scale, a number of single ions are still randomly distributed. For ion irradiations, this loss of uniformity at micro-scale has long been neglected. However, it is becoming critical for the emerging applications of SHI, for example, fabrication of high porosity heavy ion microporous membranes, and evaluation of single event effect.

      It is well known that in crystalline insulators and polymers, SHI irradiation may induce latent ion tracks, which are continuous amorphous domains[4] or fragments and rearranged molecular chains with lower mass density[5]. The formed latent ion tracks are micrometers in longitude, and several nanometers to tens of nanometers in diameter, embedded in the intact materials. These cylindrical latent tracks with high aspect ratio can be removed by selective chemical etching[6-7] to obtain heavy ion microporous membranes. The density and size of the etched micropores are modified by ion fluence and chemical etching processes, respectively.

      Recently, high porosity microporous membranes are being employed for seawater desalination, water purification[8-9], and for ion and air molecular separation[10-12]. On the other hand, nuclear ion membranes with uniform pore sizes have been used as templates for the growth of nanostructures with high aspect ratio[13-17]. Ideally, attempts that use microporous membranes as filters and templates desire high porosity as possible. However, because of the randomly distributed incident ions at micro-scale, as the ion fluence increases, the distances between the latent tracks become shorter and uncontrollable. As a result, after chemical etching and expanding of the pores, there are better chances that the micropores are overlapped with their nearest neighbors to form multi-pores[18]. Obviously, the porosity will no longer be linearly improved as functions of the ion fluence and pore size if the overlap starts. Also, the unwanted, uncontrollable and irregular multi-pores could affect the filtrating and selectivity of the membranes. Therefore, a prediction of planar distribution of SHIs is preferred prior to ion irradiation in order to design microporous membranes with better performance.

      On the other hand, for electronics in space, cosmos radiation, especially, highly energetic charged particle is a major threat. In case a charged particle penetrates through or near a logic unit, numerous free charges will be generated by ionization. Sequentially, the generation and diffusion of the charges may cause electrical disturbance that disrupt the normal operation of the circuit, known as single event effect[19]. In order to simulate the cosmos radiation and evaluate the single event effect, low fluence SHI irradiation is employed to test the aerospace electronics[20]. During an ion irradiation, according to the physical size of a circuit, the flux may be as low as several ions per second. As a result, the planar distribution of ions is no longer following a statistical uniformity. Instead, one needs to take into account the randomness of ion distribution, predict the probability of how many ions may hit a logic unit per unit time. By linking this probability with the corresponding single event effect, the radiation tolerance and reliability of the devices can be better evaluated.

      In order to simulate the randomly distributed ions at micro-scale, in this work, Monte Carlo(MC) simulations are performed. The results are presented and discussed in details in the following sections. According to the simulations, for microporous membranes, the porosity and multi-pores formation can be well predicted as functions of both ion fluence and pore size. Meanwhile, the uniformity issues of SHI irradiation are discussed for single event effect.

    • In this study, in order to simulate a single incident ion, a MC code is developed and run in Mathematica. The details of the code are attached in the support information section. During the MC simulation, incident ions are located in a two-dimensional plane one by another. The $ X $ and $ Y $ coordinates of each ion are generated by a random number generator. Because for SHI irradiation on polymer, the threshold of electronic stopping power for the formation of latent track is relatively low[21-22]. We assume that one incident ion exactly produces one latent track, therefore, the distribution of latent tracks can be determined, too. The planar density of latent tracks, and sequentially, etched micropores are as functions of the incident ion fluences and the area of the plane. Therefore, according to the MC prediction, the effective porosity of a simulational plane can be further derived by (1) assuming an average diameter of the micropores; (2) integrating the total area of the pores.

      Experimentally, in order to verify the MC simulation results, microporous membrane samples were fabricated and characterized. The samples are 30$\mu {\rm{m}}$thick polycarbonate (PC) membranes (Makrofol N, Bayer). A serial of 5 cm$ \times $5 cm membranes were irradiated by $ ^{209}{\rm{Bi}} $ (9.5 MeV/u) ions at the Heavy Ion Research Facility in Lanzhou (HIRFL) and at room temperature. The incident ion beam was perpendicular to the surface of membranes. After irradiation, the membranes with latent tracks were firstly sensitized by UV light (MUA­165, Meijiro Genossen, Japan) with wavelength ranging from 280 to 600 nm (peaked at 365 nm) and with a flux density of 4.2 $ {\rm{{mW/cm}}}^2 $. Then the membranes were merged into 5 mol/L NaOH solution at 50 °C to etch the latent tracks and to enlarge the pores. The sizes of pores were controlled by the etching time. As soon as the chemical etching finished, the membranes were immediately thoroughly rinsed with deionized water (18.25 MΩ·cm) to remove the residual etching agent. Finally, the microporous membranes were dried in air at room temperature. The morphologies of microporous membranes were characterized by scanning electron microscope(SEM) (FEI Nova NanoSEM 450). Images were collected with accelerating voltages of 2 and 10 kV.

    • Fig. 1(a) and (b) are the SEM images of PC microporous membranes fabricated by SHI irradiation and chemical etching. The fields of view are $ 29.8\ \mu {\rm{m}}\times 25.7 \ \mu {\rm{m}} $ for Fig. 1(a) and $ 37.0 \ \mu {\rm{m}}\times 32.0 \ \mu {\rm{m}} $ for Fig. 1(b), respectively. The numbers of micropores are counted for both images, which are (a) 365 pores with an average diameter $ \sim 1 \ \mu {\rm{m}} $; (b) 148 pores with an average diameter $ \sim 0.82\ \mu {\rm{m}} $. By assuming that all the latent tracks are etched, the corresponding ion fluences are (a) $ 4.76 \times 10^7 \ {\rm{ions/cm}}^2 $ and (b) $ 1.25 \times 10^7 \ {\rm{ions/cm}}^2 $, respectively.

      Figure 1.  (a)~(b) SEM images of microporous membranes etched from heavy ion latent track templates and (c)~(d) the corresponding MC simulations.

      MC simulations are carried out under the same conditions, which are identical ion fluences and average pore diameters as the SEM results. The predicted planar distributions of micropores are shown in Fig. 1(c) and (d) for nominal ion fluences of $ 4.76 \times 10^7 $ and $ 1.25 \times10^7 \ \text{ions/cm}^2 $, respectively. It is seen that both for the microporous membranes and for the planar simulations, the distributions of micro­pores are random. Further, by following the same rules, the MC simulations are applied to a number of SEM images of microporous membranes. In order to compare between the simulational and experimental results, we counted the numbers of single pore $ {P}_1 $ (free standing single pore), bi-pores $ {P}_2 $ (two pores overlap with each other), triple-pores $ {P}_3 $ (three pores overlap) and quadruple-pores $ {P}_4 $ (four pores overlap). The results are summarized in Table 1. It is found that the ratios of single pore and multi­pores for the MC simulations and for the experimentally prepared microporous membranes are in good coincidence. Therefore, it is confident to employ our MC code to predict the ion distribution of a SHI irradiation.

      Table 1.  Comparison of single and multi-­pores ratios between the PC microporous membranes and MC simulations (shown in parentheses).

      Ion fluence/ (ions·cm-2)Average pore size /nm${P}_1$/%${P}_2$/%${P}_3$/%${P}_{n \geqslant 4}$/%
      $4.76 \times 10^7$ 1 000 23.0 (22.8) 35.60 (32.1) 22.20 (25.4) 19.2 (19.60)
      $1.25 \times 10^7$ 820 76.4 (74.7) 23.00 (21.1) 0.68 (3.84) 0.0 (0.32)
      $8.56 \times 10^5$ 1 420 92.8 (94.2) 7.20 (5.6) 0.00 (0.21) 0.0 (~ 0.00)
      $2.53 \times 10^9$ 20 94.3 (96.8) 5.67 (3.1) 0.00 (0.036) 0.0 (~ 0.00)
      $7.79 \times 10^9$ 24 82.0 (86.3) 15.30 (12.9) 2.67 (0.73) 0.0 (~ 0.00)
    • For microporous membranes, porosity is a vital parameter that is a bottle neck for many applications. For example, the porosity of battery separators used in lithium ion batteries is being improved and currently has reached 40%[23]. The effective porosity $ P_{\text{eff}} $ of a microporous membrane is defined as Eq. (1) shows. Where $ S_{\text{pores}} $ refers to the integrated area of all micropores, and $ S $ refers to the entire area of a membrane.

      $$ P_{\text{eff}} = \frac{S_{\text{pores}}}{S} \times 100 {\text{%}} , $$ (1)

      In order to obtain microporous membranes with higher porosity, one needs to increase the ion fluence during SHI irradiation or/and to enlarge the micropore size. However, because of the randomly distributed ions at micro­scale, if the diameter of the pore becomes larger than the distance between the centers of the nearest latent tracks, overlapping of the pores will occur, that the porosity will no longer linearly increase as functions of ion fluence and pore size. In Fig. 2, the distances (center to center) between the nearest latent tracks are simulated for an ion fluence of $ 1.0 \times 10^8 \ \text{ions/ cm}^2 $. Where $ d_1 $, $ d_2 $, $ d_3 $ and $d_4$ are the distances between the nearest neighbors, the second, the third and the fourth nearest neighbors, respectively. It suggests that for membranes irradiated with relatively high ion fluence, overlapping of the pores starts at very early stage that the average diameter of micropores <100 nm. The impact of pore overlapping is becoming dominating while the $ 3^{\text{rd}} $ and $ 4^{\text{th}} $ nearest neighbors are getting involved. For average pore diameter $ \sim 1 \ \mu \text{m} $, the majorities are multi-pores, and the linearity of porosity as a function of pore size has been diminished.

      Figure 2.  (color online) The density distribution of $ d_1 $ to $ d_4 $ (from top to bottom) for ion fluence of $1.0 \times 10^8 $$ \text{ions/cm}^2 .$

      According to the MC simulations for the nearest neighbors, the non-linearity trend of effective porosity can be well predicted. Examples are shown in Fig. 3, the effective porosities are calculated (a) for continuing increased average pore diameter with different ion fluences from $ 8.0 \times 10^5 $ to $ 6.0 \times 10^6 \ {\rm{ions/cm}}^2 $; (b) for continuing increased ion fluences with varied pore diameters from 1.0 to $ 8.0 \ \mu {\rm{m}} $.

      Figure 3.  (color online) Simulated effective porosities (a) for membranes with varied pore sizes and different ion fluences; (b) for membranes with varied ion fluences and different pore sizes. (c) The empirical curve versus calculated effective porosities (dots) for an ion fluence of $ 1.0 \times 10^7\ \text{ions/cm}^2 $.

      By fitting the curves in Fig. 3, and also by fitting the results for wide ranges of ion fluence and pore diameter, an empirical formula for a quick prediction of the effective porosity is written as

      $$ P_{\text{eff}} = \left(0.991\;7-0.990\;1e^{0.980\;2 N \tfrac{\pi d^2}{4}}\right) \times 100 {\text{%}} . $$ (2)

      Where $ N $ refers to the ion fluences, and $ d $ refers to the average diameter of the pores. To verify the formula, the effective porosities are predicted by the MC simulation for an ion fluence of $ 1.0 \times 10^7 \ \text{ions/cm}^2 $ and for different pore sizes. The results are shown together with the empirical curve in Fig. 3(c), a good coincidence is demonstrated.

      Furthermore, as discussed in the previous sections, if the ion fluence and pore size increases, there are better chances that the nearest micropores overlap with each other. The size and morphology of multi-pores are uncontrollable. As a result, the selectivity of microporous membranes may degrade as the ratio of multi-pores increases. According to Fig. 2, pore overlapping may occur even for small pore sizes (<100 nm). Therefore, it is important to predict the probability of multi-pore formation, and sequentially, evaluate its impact to the selectivity and mechanical strength of the membranes. One example is shown in Fig. 4 that the probability of multipores as a function of effective porosity is predicted by the MC code. As illustrated in Fig. 4, The numbers from 1 to 8 are the orders of multi-pores. For example, if a micropore overlaps with its nearest neighbor only, it is noted as a $ 2^{\text{nd}} $ multi-pore; if a micropore overlaps with its nearest and the second nearest neighbors, the resulting multi­pore is noted as $ 3^{\text{rd}} $, respectively. MC simulation is performed for a wide range of ion fluences from $ 1.0 \times10^6 $ to $ 1.0 \times10^8 \ \text{ions/cm}^2 $ and for varied (average) pore diameters. It is found that multi-pores become dominating in ratio while the effective porosity reaches 20%. Because for most of the applications, the multi-pores are undesired. By employing the predictions, a threshold of effective porosity may be determined. For example, for analytical separations, a low porosity of 5% is suggested to avoid pore overlap as possible[24]. According to Fig. 4, the corresponding bi-pores, triple-pores and quadruple-pores probability are 16.19%, 1.54% and 0.10%, respectively. On the other hand, for air and ion separations, the permeability is important, too[10, 25-26]. The microporous membranes perform best with a reasonable porosity and acceptable loss of selectivity due to the existence of multi-pores.

      Figure 4.  (color online) The probability of single pore and multi-pores as a function of effective porosity.

    • For SHI irradiation, the uniformity is one of the key issues. A macro­scale uniformity can be guaranteed by employing magnetic or electric scanning at appropriate frequencies. Statistically, it means that for relatively large fields of view, for example, $ \sim 200\; \mu \text{m}\times 200\; \mu \text{m} $ as marked in Fig. 5(a), the ion fluences are counted to be very close to the nominal fluence ($ 1.0 \times 10^8 \ \text{ions/cm}^2 $) of the MC simulation. However, for a micro­scale, as demonstrated both in simulations and in fabricated membranes, the planar distribution of ions is random. As a result, for small fields of view [$ i.e.\; <20\; \mu \text{m}\times 20\; \mu \text{m} $, marked in Fig. 5(a)], the local ion fluence may deviate from the nominal fluence significantly. Another evidence of how the field of view, or scale affects the uniformity is shown virtually in Fig. 5(b) and 5(c): For a nominal ion fluence of $ 1.0 \times 10^9 \ \text{ions/cm}^2 $, the planar ion distribution is predicted. Afterwards, the entire plane is divided into squares, which are $ 100 \; \mu \text{m}\times 100 \; \mu \text{m}$ for Fig. 5(b) and 10 $ \mu \text{m} $× 10 $ \mu \text{m} $ for Fig. 5(c), respectively. The numbers of ions which hit each square are counted and plotted in histogram. The results suggest that for smaller field of view, the deviations of ion fluence from nominal fluence are much larger, which indicates the uniformity of ion irradiation is worse.

      Figure 5.  (color online) (a) A MC simulation of ion distribution with a nominal ion fluence of $ 1.0 \times 10^8 \ \text{ions/cm}^2 $. Localizated ion fluences are counted for $ 200\; \mu \text{m} \times 200\; \mu \text{m} $ and $ 20\; \mu \text{m} \times 20\; \mu \text{m} $ field of views; (b), (c) Histograms of ion counts for field-of-view sizes of $ 100\; \mu \text{m} \times 100\; \mu \text{m} $ and $ 10\; \mu \text{m} \times10\; \mu \text{m} $. The nominal ion fluence is $ 1.0 \times 10^9 \ \text{ions/cm}^2 $; (d) Calculated RSD for varied area of view and (e) for varied ion fluence from $ 5.0 \times 10^5 $ to $ 1.0 \times 10^{10} \ \text{ions/cm}^2 $.

      For certain applications, $ i.e $. micro-optics and single event effect, the region of interest is down to several $ \mu \text{m} $ or even less. As a sequence, the loss of irradiation uniformity at micro-scale, and the deviation of real ion fluence from nominal fluence will become critical. In order to quantitatively evaluate this scale effect and its impact to statistic, Relative Standard Deviation(RSD) of ion fluence is calculated. As shown in Fig. 5(d) and 5(e), RSDs are calculated based on the MC predictions for gradually enlarged field of views [Fig. 5(d)], as well as for continuing varied ion fluences from $ 5.0 \times 10^5 \ \text{ions/cm}^2 $ to $1.0 \times 10^{10} \text{ions/cm}^2$ [Fig. 5(d)]. The monotonic trends suggest that RSD is inversely proportional to ion fluence and to the area of field of view. These results and our MC­ based method are important while working with ion irradiations at micro-scale. For example, for the investigation of single event effect, in case that the region of interest, $ i.e $. A logic unit is $\sim (5\; \mu \text{m})^2$, if one wants to optimize the statistic that let the RSD of real ion fluence be <10%, the nominal ion fluence should reach $\sim 4.0 \times 10^8 \ \text{ions/cm}^2$. Meanwhile, if the logic unit is as large as $\sim (20\; \mu \text{m})^2$, less ions will be required for reaching identical RSD, and the threshold of nominal ion fluence is down to $\sim 2.5 \times 10^7 \ \text{ions/cm}^2$. To conclude, the MC method presented in this article serves as a quick approach to provide important references for the investigations and applications which are sensitive to ion beam uniformity.

    • A Monte Carlo method has been developed to simulate the planar distribution of incident SHIs, especially, for a micro-scale situation. This MC method and code provide a quick approach for predicting import features of ion irradiations and sequential micropore distributions. Effective porosity can be derived for varied ion fluences and micropore sizes. Meanwhile, the multi-pores formation and its impact to the performance of microporous membranes can be evaluated. By balancing the porosity and selectivity, ideal irradiation and chemical etching conditions may be determined, according to the MC model. An empirical formula is proposed, too, for a quick estimation of effective porosity. Moreover, for applications such as single event effect, the uniformity issues at micro-scale are investigated. Because of the randomly distributed incident ions, to reach a good statistic result, a threshold of nominal ion fluence is suggested.

      Acknowledgments The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (U1932210). The authors thank the HIRFL operation team for running $ ^{209}{\rm{Bi}} $ SHI beams.

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