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将纸币表面的涂层视为单层薄膜,将薄膜细分为若干份,每一份可看作是均匀、透明的薄膜。以其中一份涂层为例,介质1为空气,介质2为涂层,介质3为图纹。光作为入射波首先在介质1与介质2间发生反射、折射,其次在介质2和介质3间发生反射、透射。设空气的折射率为n0,涂层的折射率为n1,图纹的粗糙度为RG,涂层厚度为d1。由于采集涂层光谱数据的傅里叶近红外光谱仪Antaris II的光路是垂直入射的,因此,光垂直入射时在不同介质间发生反射、透射如图1所示。
沿+Z轴方向传播的波用角标“+”,沿−Z方向传播的波用角标“−”,
$E_{0}^{+} $ 表示空气与涂层界面上侧入射涂层中入射电场的振幅,$ E_{0}^{-}$ 表示空气与涂层界面上侧入射涂层中反射和多次透反射电场的总振幅;$E_{11}^{+} $ 表示空气与涂层界面下侧膜层中透射电场及多次反射电场的总振幅,$E_{11}^{-} $ 表示空气与涂层界面下侧膜层中涂层与图纹界面多次反射电场的总振幅;$E_{12}^{+} $ 表示涂层与图纹界面上侧膜层中空气与涂层界面透射和多次反射电场的总振幅,$E_{12}^{-} $ 表示涂层与图纹界面上侧膜层中多次反射电场的总振幅;$E_{2}^{+} $ 表示涂层与图纹界面下侧图纹的多次透射电场总振幅,与之相对应的磁场振幅$H_{0}^{+} $ 、${H}_{0}^{-}$ 等也有相同的意义。应用电场和磁场切向分量在界面两侧连续的边界条件,选取界面的单位法向矢量沿+Z方向,可写出:
$$ \left\{ {\begin{array}{*{20}{c}} {\mathop E\nolimits_0 = \mathop E\nolimits_0^ + + \mathop E\nolimits_0^ - = \mathop E\nolimits_{11}^ + + \mathop E\nolimits_{11}^ - } \\ {\mathop H\nolimits_0 = \mathop H\nolimits_0^ + + \mathop H\nolimits_0^ - = \mathop H\nolimits_{11}^ + - \mathop H\nolimits_{11}^ - = \mathop \xi \nolimits_1 \mathop E\nolimits_{11}^ + - \mathop \xi \nolimits_1 \mathop E\nolimits_{11}^ - } \end{array}} \right. $$ (1) 式中:
$\xi_{1} $ 为涂层光学有效导纳,对于空气与涂层界面和涂层与图纹界面有相同的X、Y坐标的点,光波在两界面间传播,+Z向传播空间相位因子改变${\rm{e}}^{-j \delta_{1}} $ ,−Z向传播空间相位因子改变${\rm{e}}^{j \delta_{1}} $ ,而:$$ \mathop \delta \nolimits_1 = \frac{{2\pi }}{\lambda }\mathop n\nolimits_1 \mathop d\nolimits_1 \cos \mathop \theta \nolimits_{\text{1}} $$ (2) 式中:
$\lambda $ 为真空中的光波长;$n_{1} $ 为涂层折射率;$d_{1} $ 为涂层相对厚度值;$\theta_{1} $ 为入射角。则有:$$ \left\{ {\begin{array}{*{20}{c}} {\mathop E\nolimits_0 = \mathop E\nolimits_{{\text{12}}}^ + \mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } + \mathop E\nolimits_{{\text{12}}}^ - \mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \\ {\mathop H\nolimits_0 = \mathop \xi \nolimits_1 \mathop E\nolimits_{{\text{12}}}^ + \mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } - \mathop \xi \nolimits_1 \mathop E\nolimits_{{\text{12}}}^ - \mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \end{array}} \right. $$ (3) 写成矩阵形式,则:
$$ \left[ {\begin{array}{*{20}{c}} {\mathop E\nolimits_0 } \\ {\mathop H\nolimits_0 } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } }&{\mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \\ {\mathop \xi \nolimits_1 \mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } }&{ - \mathop \xi \nolimits_1 \mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathop E\nolimits_{12}^ + } \\ {\mathop E\nolimits_{12}^ - } \end{array}} \right] $$ (4) 在图纹中仅有+Z向传播的波,涂层与图纹界面应用电场和磁场切向连续的边界条件,有:
$$ \left\{ {\begin{array}{*{20}{c}} {\mathop E\nolimits_2^ + = \mathop E\nolimits_{{\text{12}}}^ + + \mathop E\nolimits_{{\text{12}}}^ - } \\ {\mathop H\nolimits_2^ + = \mathop H\nolimits_{{\text{12}}}^ + - \mathop H\nolimits_{{\text{12}}}^ - = \mathop \xi \nolimits_1 \mathop E\nolimits_{{\text{12}}}^ + - \mathop \xi \nolimits_1 \mathop E\nolimits_{{\text{12}}}^ - } \end{array}} \right. $$ (5) 解出
$E_{12}^{+} $ 和$E_{12}^{-} $ ,因此:$$ \begin{split} \left[ {\begin{array}{*{20}{c}} {\mathop E\nolimits_0 } \\ {\mathop H\nolimits_0 } \end{array}} \right] =& \left[ {\begin{array}{*{20}{c}} {\mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } }&{\mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \\ {\mathop \xi \nolimits_1 \mathop {\rm{e}}\nolimits^{j\mathop \delta \nolimits_1 } }&{ - \mathop \xi \nolimits_1 \mathop {\rm{e}}\nolimits^{ - j\mathop \delta \nolimits_1 } } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathop E\nolimits_{12}^ + } \\ {\mathop E\nolimits_{12}^ - } \end{array}} \right] =\\ &\left[ {\begin{array}{*{20}{c}} {\cos \mathop \delta \nolimits_1 }&{\frac{j}{{\mathop \xi \nolimits_1 }}\sin \mathop \delta \nolimits_1 } \\ {j\mathop \xi \nolimits_1 \sin \mathop \delta \nolimits_1 }&{\cos \mathop \delta \nolimits_1 } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathop E\nolimits_2^ + } \\ {\mathop H\nolimits_2^ + } \end{array}} \right] \end{split} $$ (6) 因为E和H的切向向量在界面两侧是连续的,而且在图纹介质中仅有正向传播的波,所以公式(6)就把空气与涂层界面上侧E和H的切向分量的总振幅
$E_{0} $ 、和$H_{0} $ 涂层与图纹下侧介质中E和H的切向分量的总振幅$E_{2}^{+} $ 、${{{H}}}_{2}^{+}$ 联系起来。根据,
$$ \left\{ {\begin{array}{*{20}{c}} {\mathop H\nolimits_{\text{0}} = Y\mathop E\nolimits_0 } \\ {\mathop H\nolimits_2^ + = \mathop \xi \nolimits_G \mathop E\nolimits_2^ + } \end{array}} \right. $$ (7) 式中:Y为模型的光学有效导纳;
$\xi_{G} $ 为光学有效导纳。但由于复杂的图纹是油墨经印刷转印至纸张形成,所以将$\xi_{G} $ 近似由粗糙度$R_{G} $ 代替,则有:$$ \mathop E\nolimits_0 \left[ {\begin{array}{*{20}{c}} 1 \\ Y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \mathop \delta \nolimits_1 }&{\frac{j}{{\mathop \xi \nolimits_1 }}\sin \mathop \delta \nolimits_1 } \\ {j\mathop \xi \nolimits_1 \sin \mathop \delta \nolimits_1 }&{\cos \mathop \delta \nolimits_1 } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ {\mathop R\nolimits_G } \end{array}} \right]\mathop E\nolimits_2^ + $$ (8) 令
$$ \left[ {\begin{array}{*{20}{c}} B \\ C \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \mathop \delta \nolimits_1 }&{\frac{j}{{\mathop \xi \nolimits_1 }}\sin \mathop \delta \nolimits_1 } \\ {j\mathop \xi \nolimits_1 \sin \mathop \delta \nolimits_1 }&{\cos \mathop \delta \nolimits_1 } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ {\mathop R\nolimits_G } \end{array}} \right] $$ (9) 而矩阵
$$ \left[ {\begin{array}{*{20}{c}} {\cos \mathop \delta \nolimits_1 }&{\frac{j}{{\mathop \xi \nolimits_1 }}\sin \mathop \delta \nolimits_1 } \\ {j\mathop \xi \nolimits_1 \sin \mathop \delta \nolimits_1 }&{\cos \mathop \delta \nolimits_1 } \end{array}} \right] $$ 称为涂层的特征矩阵,它反映了涂层特性的全部物理参数,由此可将列向量
$\left[\begin{array}{ll}B \;\;\;\;\; C\end{array}\right]^{T} $ 称为涂层的组合特征向量。$$ 由于 Y = \frac{C}{B} = \frac{{\mathop R\nolimits_G COS\mathop \delta \nolimits_1 + j\mathop \xi \nolimits_1 \sin \mathop \delta \nolimits_1 }}{{COS\mathop \delta \nolimits_1 + j\left( {\dfrac{{\mathop R\nolimits_G }}{{\mathop \xi \nolimits_1 }}} \right)\sin \mathop \delta \nolimits_1 }} $$ (10) 从而得到涂层的反射系数为:
$$ \tilde r =\dfrac{{\mathop \xi \nolimits_0 - Y}}{{\mathop \xi \nolimits_0 + Y}} = \dfrac{{\mathop \xi \nolimits_0 B - C}}{{\mathop \xi \nolimits_0 B + C}} =\\ \dfrac{{\left( {\mathop \xi \nolimits_0 - \mathop R\nolimits_G } \right)COS\mathop \delta \nolimits_1 + j\left( {\dfrac{{\mathop \xi \nolimits_0 \mathop R\nolimits_G }}{{\mathop \xi \nolimits_1 }} - \mathop \xi \nolimits_1 } \right)\sin \mathop \delta \nolimits_1 }}{{\left( {\mathop \xi \nolimits_0 + \mathop R\nolimits_G } \right)COS\mathop \delta \nolimits_1 + j\left( {\dfrac{{\mathop \xi \nolimits_0 \mathop R\nolimits_G }}{{\mathop \xi \nolimits_1 }} + \mathop \xi \nolimits_1 } \right)\sin \mathop \delta \nolimits_1 }} $$ (11) $\xi_{0} $ 为空气的光学有效导纳,则反射率为:$$ R = \tilde r \mathop {\left( {\tilde r } \right)}\nolimits^* = \left( {\frac{{\mathop \xi \nolimits_0 B - C}}{{\mathop \xi \nolimits_0 B + C}}} \right)\mathop {\left( {\frac{{\mathop \xi \nolimits_0 B - C}}{{\mathop \xi \nolimits_0 B + C}}} \right)}\nolimits^* $$ (12) 由于是垂直入射,将
$\theta_{0} $ 为0o,$\theta_{1} $ 为0o,空气折射率$n_{0} $ 为1,涂层折射率$n_{1} $ 为1.49,代入公式(12)得:$$ R = \frac{{\mathop {\left\{ {\left[ {1 - \mathop {\left( {\mathop R\nolimits_G } \right)}\nolimits^2 } \right] + \left[ {0.671\;1\mathop {\left( {\mathop R\nolimits_G } \right)}\nolimits^2 + \mathop R\nolimits_G - 3.220\;1} \right]x} \right\}}\nolimits^2 + 8.880\;4\mathop {\left( {1 + \mathop R\nolimits_G } \right)}\nolimits^2 x - 8.880\;4\mathop {\left( {1 + \mathop R\nolimits_G } \right)}\nolimits^2 \mathop x\nolimits^2 }}{{\mathop {\left[ {\mathop {\left( {{\text{1 + }}\mathop R\nolimits_G } \right)}\nolimits^2 - \mathop {\left( {{\text{1 + }}\mathop R\nolimits_G } \right)}\nolimits^2 x + \mathop {\left( {0.671\;1\mathop R\nolimits_G + 1.49} \right)}\nolimits^2 x} \right]}\nolimits^{\text{2}} }} $$ (13) $$ \begin{split} \\ \arcsin \sqrt x = \mathop \delta \nolimits_1 \frac{{2\pi }}{\lambda }\mathop n\nolimits_1 \mathop d\nolimits_1 \end{split}$$ (14) 因此,通过将反射率R及同区域的粗糙度
$R_{G} $ 代入公式(13)求得x,再将$\sqrt{x} $ 、$\lambda $ 代入公式(14)求解出涂层相对厚度值$d_{1} $ 。其中,傅里叶近红外光谱仪测得区域的表征是吸收率$A$ ,则:$$ A{\text{ = }}{\log _{10}}\left( {1/T} \right) $$ (15) $$ R = 1 - A - T $$ (16) 式中:R为反射率;A为吸收率;T为透光率;而激光共聚焦显微系统测得同区域的表征是粗糙度。
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首先采用傅里叶近红外光谱仪对样品进行了检测,得到20元、10元、1元已涂布与未涂布纸币多个具有代表性的近红外吸收谱,横坐标是波数,纵坐标是吸收率,如图2(a)~(c)所示。由于面额相同,但衬底图文不同;或衬底图文相同,但已涂布与未涂布存在差别,结果导致每一条谱线的吸收率不同。从图2可知,尤其在红色虚线标记区域4380~4280 cm−1,三种样本的已涂布和未涂布光谱数据交织在一起。为了将已涂布和未涂布的光谱区分,从而确定特征光谱区域,将图2数据做一阶导处理,横坐标是波数,纵坐标是吸收率,如图3(a)~(c)所示。由图3可知,三种样本都在4351~4289 cm−1内,已涂布和未涂布的一阶导数据分别呈现上扬和平缓趋势,由此确定了样本不同但涂层特征光谱相同的区域。
图 2 (a) 20元已涂布与未涂布的近红外谱图表征;(b) 10元已涂布与未涂布的近红外谱图表征;(c) 1元已涂布与未涂布的近红外谱图表征
Figure 2. (a) NIR characteristic absorption spectrum of 20-yuan coated and uncoated detected by Near-IR spectroscopy; (b) NIR characteristic absorption spectrum of 10-yuan coated and uncoated detected by Near-IR spectroscopy; (c) NIR characteristic absorption spectrum of 1-yuan coated and uncoated detected by Near-IR spectroscopy
图 3 (a) 20元已涂布与未涂布的一阶导近红外谱图;(b) 10元已涂布与未涂布的一阶导近红外谱图;(c) 1元已涂布与未涂布的一阶导近红外谱图
Figure 3. (a) First-order derivative NIR characteristic absorption spectrum of 20-yuan coated and uncoated; (b) First-order derivative NIR characteristic absorption spectrum of 10-yuan coated and uncoated; (c) First-order derivative NIR characteristic absorption spectrum of 1-yuan coated and uncoated
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在明确特征光谱区域后,接着确定本区域内的特征波数,从而将对应的吸收射率数据经换算代入模型计算。为此,文中提出了基于多元散射校正(MSC)与二阶导组合的分析方法。MSC是近红外光谱数据预处理的常用算法之一,可以有效消除图纹作为复杂衬底引起的漫反射。首先,计算所有光谱数据的平均光谱作为理想光谱如下:
$$\overline {Data} = \frac{{\displaystyle\sum\limits_{i = 1}^n {\mathop {Data}\nolimits_{ij} } }}{n}$$ (17) 式中:i为样本编号;j为每一个样本内的波数序号;Data为每个样本的光谱数据。
然后将每个样本的光谱与平均光谱做一元线性回归,运用最小二乘法求出每个样本的基线平移量和偏移量如下:
$$ \mathop {Data}\nolimits_i = \mathop k\nolimits_i \overline {Data} + \mathop b\nolimits_i $$ (18) 式中:
$k_{i} $ 、$b_{i} $ 分别为样本的基线平移量和偏移量。最后校正每个样本的光谱如下:
$$ \mathop {Data}\nolimits_{i\left( {MSC} \right)} = \frac{{\left( {\mathop {Data}\nolimits_i - \mathop b\nolimits_i } \right)}}{{\mathop k\nolimits_i }} $$ (19) 结合3.1节中发现的已涂布和未涂布一阶导数据在趋势上的差别,进而提出二阶导。
将3.1节中未经处理的具有代表性的近红外吸收谱数据导入Matlab,经多元散射校正(MSC)与二阶导处理后。以1元某区域为例,相同区域的多个样本在不同波数下,已涂布和未涂布经处理的吸收率。如图4显示,从数据趋势上看,在4346.764 cm−1处,已涂布与未涂布的差别最明显。因此,将4346.764 cm−1定为特征波数,而4346.764 cm−1与涂层中主要成分C-H基团的合频是4347 cm−1[19]基本重合,从而进一步说明确定特征波数方法的可靠性与数据的准确性。
图 4 不同波数下,经MSC和二阶导处理的1元已涂布与未涂布的吸收率
Figure 4. In Different wavenumbers, MSC and Second-order derivative NIR characteristic absorbance of 1-yuan Coated and Uncoated
图 5 (a) 经MSC和二阶导处理的20元已涂布与未涂布的吸收率;(b) 经MSC和二阶导处理的10元已涂布与未涂布的吸收率;(c) 经MSC和二阶导处理的1元已涂布与未涂布的吸收率
Figure 5. (a) MSC and Second-order derivative NIR characteristic absorbance of 20-yuan coated and uncoated; (b) MSC and Second-order derivative NIR characteristic absorbance of 10-yuan coated and uncoated; (c) MSC and Second-order derivative NIR characteristic absorbance of 1-yuan coated and uncoated
由此,对多个相同面额但区域不同的近红外吸收谱数据在特征波数4346.764 cm−1处做多元散射校正(MSC)与二阶导处理,发现已涂布与未涂布在此处的差别都十分明显,如图5(a)~(c)所示,说明特征波数的选取及数据处理方法适用于20元、10元、1元已涂布与未涂布纸币。
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粗糙度是模型求解厚度的重要输入参数之一,使用激光显微系统在其“基本测量”模式下,根据3.2节中的分析结果,测量3.1节中已涂布样本的相同区域。依照国际标准ISO 25178[20],选取
$S_{a} $ 作为图纹粗糙度的表征,$S_{a} $ 如下:$$ \mathop S\nolimits_a = \frac{1}{A}\iint_A {\left| {Z\left( {x,y} \right)} \right|{\rm{d}}x{\rm{d}}y} $$ (20) 式中:
$S_{a} $ 为粗糙度;A为区域面积;x、y分别为区域的横、纵坐标。测量结果如表1所示,纵向比较,同一面额样本的粗糙度与图纹存在直接关系,图纹越复杂,粗糙度测量值越大。横向比较,不同面额但同一工艺制造的样本区域整体趋势一致。表 1 已涂布样本粗糙度测量结果
Table 1. Expression of the coated roughness
Type 1-yuan 10-yuan 20-yuan Region Description Sa/μm Region Description Sa/μm Region Description Sa/μm Plain-Region 1 Coated 3.108 Plain-Region 1 Coated 3.439 Plain-Region 1 Coated 2.472 Plain-Region 2 3.704 Plain-Region 2 3.809 Plain-Region 2 5.353 Plain-Region 3 2.851 Plain-Region 3 3.306 Plain-Region 3 7.192 Plain-Region 4 2.036 Plain-Region 4 8.309 Plain-Region 4 5.227 Complex-Region 5 14.160 Complex-Region 5 13.460 Complex-Region 5 15.938 Plain-Region 6 Complex Region 6 13.624 Complex -Region 6 14.232 Plain-Region 7 Plain-Region 7 6.690 Plain-Region 7 5.777 Plain-Region 8 4.236 Plain-Region 8 5.039 Plain-Region 8 9.440 将粗糙度及3.2节中特征波数4346.764 cm−1对应的反射率导入公式(13),再将结果
$\sqrt{x} $ 和特征波数$\lambda $ 4346.764 cm−1对应的=2300 nm代入公式(14)求解出涂层相对厚度值$d_{1} $ ,最终结果如表2所示。表 2 已涂布样本涂层相对厚度
Table 2. Expression of the coated relative thickness
Type 1-yuan 10-yuan 20-yuan Region Description d1 Region Description d1 Region Description d1 Plain-Region 1 Coated 252.574 Plain-Region 1 Coated 265.339 Plain-Region 1 Coated 241.245 Plain-Region 2 267.249 Plain-Region 2 273.653 Plain-Region 2 332.216 Plain-Region 3 248.799 Plain-Region 3 261.801 Plain-Region 3 171.205 Plain-Region 4 229.470 Plain-Region 4 218.388 Plain-Region 4 321.753 Complex-Region 5 116.244 Complex-Region 5 102.234 Complex-Region 5 124.223 Plain-Region 6 Complex-Region 6 112.302 Complex-Region 6 107.153 Plain-Region 7 Plain-Region 7 195.219 Plain-Region 7 223.175 Plain-Region 8 247.899 Plain-Region 8 239.415 Plain-Region 8 198.442 从表2可以看出,虽然不同面额图纹区域各异,但图纹复杂区域的平均涂层相对厚度值均比简单区域小,相差约132.954、135.035、132.318,符合生产实际情况。
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为了得到直观、真实、可参考的涂层厚度,使用激光显微系统在其“膜厚测量”模式下,测量3.1中已涂布和未涂布样本的相同区域。以20元的图纹复杂和简单区域为例,如表3所示,可直观看出已涂布样本的表面存在一层胶状物质,而未涂布样本的纸张纤维清晰可见。因此,也可以系统采集的微观图像作为判断涂布与否的方法,这种方法更直观,但耗时较长。
表 3 20元具有代表性的复杂与简单区域激光+彩色微观表征
Table 3. Complex and plain characteristic expression of 20-yuan Coated and Uncoated
Region 20-yuan coated 20-yuan uncoated Plain-Region 1 Plain-Region 4 Complex-Region 5 测量涂层厚度时,选择“面膜厚测量”,平均轮廓以有效面为准,测量线条间隔设置0.556 μm。由于纸张的无法完全平整且有图纹十分复杂的区域,因此软件根据有效测量面计算出涂层厚度。在确定有效测量面时,以图像的完整性为主,若图像中存在不完整区域,使用轮廓工具圈定测量面,排除无效面。在计算涂层厚度时,有效面完整的以软件计算值作为测量涂层厚度,但对于无效面较多的区域,测量涂层厚度按公式(21)计算:
$$ \begin{split} \\ \mathop D\nolimits_1 = D\frac{{\mathop S\nolimits_1 }}{{\mathop S\nolimits_2 }} \end{split} $$ (21) 式中:D1为测量涂层厚度;D为软件计算值;S1为圈定的测量面积即有效测量面积;S2为实际测量面积,将测量涂层厚度与表2中d1关联,如表4所示。
表 4 已涂布样本测量涂层厚度
Table 4. Expression of the measurement coated thickness by confocal laser scanning microscopy
Type 1-yuan 10-yuan 20-yuan Region D1/μm Region D1/μm Region D1/μm Plain-Region 1 6.660 Plain-Region 1 8.001 Plain-Region 1 7.332 Plain-Region 2 5.847 Plain-Region 2 9.341 Plain-Region 2 7.267 Plain-Region 3 8.783 Plain-Region 3 5.978 Plain-Region 3 12.738 Plain-Region 4 10.417 Plain-Region 4 6.750 Plain-Region 4 9.565 Complex-Region 5 3.807 Complex-Region 5 4.547 Complex-Region 5 4.015 Plain-Region 6 7.006 Complex-Region 6 4.661 Complex-Region 6 6.223 Plain-Region 7 9.248 Plain-Region 7 7.321 Plain-Region 8 8.779 Plain-Region 8 6.333 从表4可以看出,测量涂层厚度与解耦厚度趋势基本一致,各样本的测量涂层最小厚度为3.807、4.457、4.015 μm,各样本的测量涂层最小厚度为最大10.417、9.341、12.738 μm,图纹复杂的区域涂层厚度较小,图纹简单的区域涂层厚度较大,符合生产实际情况。
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涂层厚度的测量误差包括系统误差和随机误差。对于系统误差,可以通过将测量结果与标准样品经测量结果相比较的方法消除。而对于随机误差,则不能消除。此节主要内容是分析各种随机误差对涂层厚度测量结果的影响。
(1)特征波数定位的不确定度
根据3.2小节中,光谱数据经多元散射校正(MSC)与二阶导组合的处理,有涂层和无涂层数据差异最明显,从而确定涂层的特征波数。通过这一特性,可以获得波数对应的吸收率,进而根据公式(14)求得涂层厚度的解耦值。由于受到近红外光谱仪分辨率的影响,对涂层最有效信号特征的定位存在一定的不确定性,设光谱仪分辨率为
$\Delta \lambda $ ,对应的厚度不确定度是$\Delta {d_{\text{1}}}$ ,由公式(14)可知:$$ \arcsin \sqrt{x}=\delta_{1}=\frac{2 \pi}{\lambda+\Delta \lambda} n_{1}\left(d_{1}+\Delta d_{1}\right)$$ (22) 可以推导出
$\Delta \lambda $ 与$\Delta {d_{\text{1}}}$ 的关系满足公式(23):$$ \Delta {d_1}{\text{ = }}\frac{{\arcsin \sqrt x (\lambda + \Delta \lambda )}}{{2\pi {n_1}}} $$ (23) (2)其他因素导致的测量不确定度
由于纸张在印刷时受到滚筒压印导致变形以及在测量时定位导致的误差,测量结果必然受到影响。因此,将其他因素导致的厚度不确定度归结为
$\Delta {d_{\text{2}}}$ 。(3)合成不确定度
根据本节的分析,涂层厚度测量不确定度主要由特征波数定位和其他因素引起,则涂层厚度测量的不确定度为:
$$ \Delta {d}_{总}\text=\sqrt{{{\displaystyle \left(\Delta {d}_{1}\right)}}^{2}+{{\displaystyle \left(\Delta {d}_{2}\right)}}^{2}} $$ (24) 钞券各区域复杂情况不一,以表4中1元简单区域为例,
$ \Delta {d}_{总}\approx \pm \text{0}\text{.246} $ μm。由于通过模型解耦得到涂层厚度相对值的结果精确到小数点后3位,而且激光共聚焦显微系统测量得到的涂层厚度精度为1 nm,从而文中提出的涂层厚度测量方法的检测重复精度达到nm级。
按照朗伯-比尔定律,对于后期不同区域的涂层厚度定量检测,可根据待测样品在近红外光谱在特征波数4346.764 cm−1处的反射率值和粗糙度导入模型,结合解耦厚度结果,就可以推出待测区域的实际涂层厚度。
Research on quantitative detection method of coating thickness on complex substrates
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摘要: 纸币是国家发行并强制使用的货币符号,2019年中国人民银行发行的2019年版第五套人民币纸币,两面采用了抗脏污保护涂层,使纸币的整洁度明显改善。作为“国家名片”,在纸币生产过程中,对每一道工艺都有严格的质量控制,涂层是通过涂布机将涂布液转移、固化至纸币两面,由此称为涂布工艺。为了更加合理地控制涂布质量,生产中需要检测纸币涂层的厚度。针对该需求,文中建立了纸币图纹作为复杂衬底的涂层厚度光学漫反射模型,采用傅里叶近红外光谱仪和激光共聚焦显微系统对已涂布和未涂布的纸币进行识别并定量检测。文中首先根据涂层物质在近红外光谱可被有效识别的特点,对涂层的近红外吸收光谱数据提出了基于多元散射校正(MSC)与二阶导组合的分析方法,确定4 346.764 cm−1为特征波数。再根据反射率、粗糙度对涂层厚度的模型解耦,最后通过激光共聚焦显微系统检测了已涂布纸币的涂层变化,并将其与模型的厚度解耦结果关联,得出测量涂层厚度最小为3.807 μm,最大为12.738 μm。最终结果表明该检测方法对纸币生产中涂层质量控制具有重要的实践指导意义。Abstract: Banknote which is issued by the government and forced to use. In 2019, the People’s Bank of China issues the fifth set of RMB. The anti-fouling protection coating is used on both sides, which significantly improves the cleanliness of paper currency. In order to control the coating quality more reasonably, it is necessary to detect the coating thickness in the production. The coating thickness of this paper was to investigate, this paper establishes the optical diffuse reflection model of coating thickness with banknote pattern as a complex substrate surface, identifies and quantitatively detects the coated and uncoated banknote by using Fourier near-infrared spectrometer and confocal laser scanning microscopy system. In this paper, one analytical method based on the combination of multivariate scattering correction (MSC) and second-order derivative combination analysis is proposed for the near infrared (NIR) absorption spectrum data of the coating, that could be effectively identified in the NIR spectrum, and 4 346.764 cm−1 is determined as the characteristic wave number. Then, the coating thickness model is decoupled basing on the reflectance and roughness. Finally, the coating thickness changes of coated banknotes are detected by a confocal laser scanning microscopy system, and they are correlated with the decoupling result of the model to obtain the actual coating thickness. The final results show that the minimum is 3.807 μm, the maximum is 12.738 μm. The detection method has an important practical significance for coating quality control in banknote production.
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图 2 (a) 20元已涂布与未涂布的近红外谱图表征;(b) 10元已涂布与未涂布的近红外谱图表征;(c) 1元已涂布与未涂布的近红外谱图表征
Figure 2. (a) NIR characteristic absorption spectrum of 20-yuan coated and uncoated detected by Near-IR spectroscopy; (b) NIR characteristic absorption spectrum of 10-yuan coated and uncoated detected by Near-IR spectroscopy; (c) NIR characteristic absorption spectrum of 1-yuan coated and uncoated detected by Near-IR spectroscopy
图 3 (a) 20元已涂布与未涂布的一阶导近红外谱图;(b) 10元已涂布与未涂布的一阶导近红外谱图;(c) 1元已涂布与未涂布的一阶导近红外谱图
Figure 3. (a) First-order derivative NIR characteristic absorption spectrum of 20-yuan coated and uncoated; (b) First-order derivative NIR characteristic absorption spectrum of 10-yuan coated and uncoated; (c) First-order derivative NIR characteristic absorption spectrum of 1-yuan coated and uncoated
图 5 (a) 经MSC和二阶导处理的20元已涂布与未涂布的吸收率;(b) 经MSC和二阶导处理的10元已涂布与未涂布的吸收率;(c) 经MSC和二阶导处理的1元已涂布与未涂布的吸收率
Figure 5. (a) MSC and Second-order derivative NIR characteristic absorbance of 20-yuan coated and uncoated; (b) MSC and Second-order derivative NIR characteristic absorbance of 10-yuan coated and uncoated; (c) MSC and Second-order derivative NIR characteristic absorbance of 1-yuan coated and uncoated
表 1 已涂布样本粗糙度测量结果
Table 1. Expression of the coated roughness
Type 1-yuan 10-yuan 20-yuan Region Description Sa/μm Region Description Sa/μm Region Description Sa/μm Plain-Region 1 Coated 3.108 Plain-Region 1 Coated 3.439 Plain-Region 1 Coated 2.472 Plain-Region 2 3.704 Plain-Region 2 3.809 Plain-Region 2 5.353 Plain-Region 3 2.851 Plain-Region 3 3.306 Plain-Region 3 7.192 Plain-Region 4 2.036 Plain-Region 4 8.309 Plain-Region 4 5.227 Complex-Region 5 14.160 Complex-Region 5 13.460 Complex-Region 5 15.938 Plain-Region 6 Complex Region 6 13.624 Complex -Region 6 14.232 Plain-Region 7 Plain-Region 7 6.690 Plain-Region 7 5.777 Plain-Region 8 4.236 Plain-Region 8 5.039 Plain-Region 8 9.440 表 2 已涂布样本涂层相对厚度
Table 2. Expression of the coated relative thickness
Type 1-yuan 10-yuan 20-yuan Region Description d1 Region Description d1 Region Description d1 Plain-Region 1 Coated 252.574 Plain-Region 1 Coated 265.339 Plain-Region 1 Coated 241.245 Plain-Region 2 267.249 Plain-Region 2 273.653 Plain-Region 2 332.216 Plain-Region 3 248.799 Plain-Region 3 261.801 Plain-Region 3 171.205 Plain-Region 4 229.470 Plain-Region 4 218.388 Plain-Region 4 321.753 Complex-Region 5 116.244 Complex-Region 5 102.234 Complex-Region 5 124.223 Plain-Region 6 Complex-Region 6 112.302 Complex-Region 6 107.153 Plain-Region 7 Plain-Region 7 195.219 Plain-Region 7 223.175 Plain-Region 8 247.899 Plain-Region 8 239.415 Plain-Region 8 198.442 表 3 20元具有代表性的复杂与简单区域激光+彩色微观表征
Table 3. Complex and plain characteristic expression of 20-yuan Coated and Uncoated
Region 20-yuan coated 20-yuan uncoated Plain-Region 1 Plain-Region 4 Complex-Region 5 表 4 已涂布样本测量涂层厚度
Table 4. Expression of the measurement coated thickness by confocal laser scanning microscopy
Type 1-yuan 10-yuan 20-yuan Region D1/μm Region D1/μm Region D1/μm Plain-Region 1 6.660 Plain-Region 1 8.001 Plain-Region 1 7.332 Plain-Region 2 5.847 Plain-Region 2 9.341 Plain-Region 2 7.267 Plain-Region 3 8.783 Plain-Region 3 5.978 Plain-Region 3 12.738 Plain-Region 4 10.417 Plain-Region 4 6.750 Plain-Region 4 9.565 Complex-Region 5 3.807 Complex-Region 5 4.547 Complex-Region 5 4.015 Plain-Region 6 7.006 Complex-Region 6 4.661 Complex-Region 6 6.223 Plain-Region 7 9.248 Plain-Region 7 7.321 Plain-Region 8 8.779 Plain-Region 8 6.333 -
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