Research data supporting "Roughness spectroscopy of particle monolayer: Implications for spectral analysis of the monolayer image". B-spline representation of radial distribution function.

The files contain knots and coefficients of third order (quadratic) B-spline representation approximating a radial distribution function (RDF). We calculated the function for a hard-disk monolayer generated with event-driven molecular dynamics, of surface coverage 0.85. Specifically, to produce the monolayer, we used the program PackLSD.64.x by Aleksandar Donev, available at https://cims.nyu.edu/~donev/Packing/PackLSD/Instructions.html. We started the simulation of 8.5E7 disks at the initial surface coverage of 0.1 to gradually increase their size. In the nml parameter file, we set the disk expansion rate parameter expansions_=0.001. Once the surface coverage achieved 0.85, we stopped the simulation. We generated 26 replicas of the big system with a constant area of square simulation box. For each replica, we first calculated the discrete RDF g(r) by counting disk pairs in narrow distance intervals of width dr = 1E-3 a, where a is the disk radius. In the narrow interval 3.9900 ≤ r ≤ 4.0020, where the slope of the RDF changes extremely rapidly, we used the ring thickness 1E-4. For each replica of the system, we calculated the mean distance and RDF over the 88108 narrow intervals, averaging over the central particles. We calculated 26 replicas of the function g(r) in the range from r = 2a to r = 90a. Averaging over them, we got 88108 discrete, arithmetic mean values of RDF and standard deviations of the means. We identified the maximum value of the RDF standard deviation to be 0.009. Finally, we fit a third order (quadratic) B-spline representation to the mean RDF. For that, we used the procedure DFC of SLATEC library, with 3786 proper knots. To calculate the RDF with the B-spline, you can use the procedure DBVALU of SLATEC library. The knot vector in the attached file begins and ends with two improper knots, in accordance with requirements of the procedure. For details, see the paper: P. Weroński & K. Pałka, "Roughness spectroscopy of particle monolayer: Implications for spectral analysis of the monolayer image", Measurement 196 (2022) 111263.本数据集文件包含用于近似径向分布函数（radial distribution function, RDF）的三阶（二次）B样条（B-spline）表示的节点与系数。 我们针对通过事件驱动分子动力学生成的表面覆盖率为0.85的硬圆盘单层体系，计算了其径向分布函数。具体而言，我们使用Aleksandar Donev开发的程序PackLSD.64.x构建该单层体系，该程序可从https://cims.nyu.edu/~donev/Packing/PackLSD/Instructions.html获取。我们以初始表面覆盖率0.1启动包含8.5×10^7个圆盘的模拟，逐步增大圆盘尺寸，并在nml参数文件中将圆盘膨胀速率参数expansions_设为0.001。当表面覆盖率达到0.85时，停止模拟。 我们生成了26个拥有固定面积方形模拟盒的大体系复制品。针对每个复制品，我们首先通过在宽度为dr=1×10^-3 a的窄距离区间内统计圆盘对的数量，计算得到离散径向分布函数g(r)，其中a为圆盘半径。在RDF斜率变化极快的3.9900 ≤ r ≤ 4.0020窄区间内，我们将环厚度设为1×10^-4。针对每个体系复制品，我们在88108个窄区间内计算平均距离与径向分布函数，并以中心粒子为基准进行平均。我们在r=2a至r=90a的范围内生成了26组g(r)复制品数据，对其取平均后得到88108个离散的径向分布函数算术平均值与平均值的标准差。经统计，径向分布函数标准差的最大值为0.009。最后，我们使用SLATEC库的DFC子程序对平均径向分布函数进行三阶（二次）B样条拟合，共使用3786个有效节点。 若需通过B样条计算径向分布函数，可使用SLATEC库的DBVALU子程序。附件文件中的节点向量按照该子程序的要求，以两个无效节点作为起始与结束。详细信息请参阅论文：P. Weroński与K. Pałka, "Roughness spectroscopy of particle monolayer: Implications for spectral analysis of the monolayer image", Measurement 196 (2022) 111263。