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金属颗粒掺杂型复合涂层是重要的航天器热控材料[1]，吸收发射比是其重要的热控性能指标之一，即对太阳光的平均吸收率与红外平均发射率之比。涂层中的粒子材料、尺寸、体积分数、几何形貌、粒子取向、涂层厚度等是影响涂层辐射特性的重要因素[2]，这些参数的选择是调控涂层辐射特性的依据。研究涂层辐射特性（或者吸收发射比）的有效调控方法对于航天器热控涂层的设计具有重要意义。
近年来，涂层辐射特性对航天器温度场的影响得到了广泛研究[3-4]，但在涂层组织结构对表观辐射特性的影响方面研究较少。以往研究中，通常将颗粒掺杂型涂层视为均匀分布、各向同性的弥散性介质，而将颗粒简化为等效球体或随机取向的棒状、片状的非球形颗粒[5-7]。王博翔等[8]研究了涂层中非球形孔隙及其朝向分布对涂层辐射特性的影响，表明各向异性对涂层辐射热流传输存在影响。在实验中发现含片状粒子涂层的发射率低于含其他形状粒子的涂层，水平方向排列的颗粒比随机取向的颗粒具有更好的红外散射性能[9-10]。目前对于各向异性涂层辐射特性的理论研究较少。Yuan等[11]基于几何光学和Kubelka层级模型研究了含有水平方向排列金属薄片的涂层发射率。Jian等[12]在此基础上增加了粗糙表面模型，但是他们没有考虑其他取向角，也没有考虑粒子的衍射对红外辐射的影响。Chen等[13]采用经典的Kubelka-Munk理论结合改进的几何光学方法，研究了红外光与片状粒子作用时产生的衍射和边缘效应，给出了片状颜料粒子径向尺寸对涂层发射率的影响，但是没有考虑粒子取向的影响。Song等[14]研究了铝片颗粒在丙烯酸树脂中取向随机分布在0~10°、0~30°和0~60°范围内时涂层的光谱发射率和反射率，发现取向角增大后涂层光谱发射率增大。但他们也没有考虑衍射效应，铝片取向也不是单一方向。
在涂层刮涂、刷涂或喷涂过程中，片状颗粒的朝向并不完全随机，而是整体上呈一定的角度分布，不同的工作参数将导致不同的颗粒取向[15-16]。此外，定向剂是涂层中常用的功能助剂，它能使片状颗粒朝向相同方向，从而影响涂层的辐射特性和光泽度[15-16]。此时，粒子在涂层中空间位置上随机分布，但整体取向具有一致性。而目前关于粒子取向角对涂层吸收发射比的影响机理尚不清晰。
在涂层中颗粒尺寸可达几十微米，远大于特征热辐射波长[13-14,16]。由于计算机资源的限制，对于随机分布的非球形大尺寸颗粒的辐射特性，严格的电磁散射理论求解难度很大。同时对于金属颗粒掺杂的吸收散射复合涂层，物理光学计算难以应用于宏观尺度的研究[14]。文中采用几何光学方法结合二流法研究涂层辐射特性，理论模型简单，适用于大尺寸颗粒掺杂型复合涂层的辐射特性快速计算分析[17]。二流法方法简单，在对涂层辐射特性计算中应用广泛，但现有的采用二流法的研究中大多都将散射假定为各向同性。文中以空间位置上随机分布、朝向上具有相同取向角的片状铝颗粒掺杂涂层为研究对象，利用考虑衍射的几何光学方法计算大尺寸铝片粒子在不同取向角下的辐射特性，然后采用考虑各向异性散射的二流法计算涂层的吸收率和发射率，揭示了铝片取向角、体积分数和涂层厚度对热控涂层吸收发射比的调控规律。
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文中研究对象为空间随机分布、取向角一致的铝片掺杂型复合涂层，如图1所示。先计算铝片在不同取向角（图1中倾斜角θ）的辐射特性，再求解涂层整体辐射传输过程。在模型中，假定入射光垂直于涂层表面，掺杂颗粒为半径25 μm、高2 μm的圆片状铝颗粒，涂层基底为铝。铝的光学常数取自固体光学常数手册[18]。涂层介质很多情况下没有吸收或吸收很小[19]，同时也为了凸显粒子取向对涂层辐射特性的影响，文中与文献[20]一样假定涂层介质的折射率为1.5。
图 1 空间随机分布、取向一致的铝片掺杂型涂层示意图
Figure 1. Diagram of a coating containing randomly distributed and identically oriented aluminum flakes
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据几何光学理论，由粒子吸收、折射和反射引起的衰减截面等于粒子的几何投影面积，由衍射造成的衰减截面也与此面积相等，并且衍射能量主要集中在前向[17]。因此可以将辐射传输分成光线追踪（吸收、反射、折射）和前向衍射两个部分。在分界面上，光线满足菲涅耳反射和斯涅耳定律，可用折射矩阵表征。前向衍射部分可根据粒子几何投影面积进行数值计算。文中使用Karri Muinonen等[21]提供的Siris开源程序计算固定取向铝片的辐射特性。其计算原理如下：
入射光Ii=(Ii, Qi, Ui, Vi)T和散射光Is=(Is, Qs, Us, Vs)T之间的关系可以用散射矩阵P描述[21]：
$$ {{{\boldsymbol{I}}}_{\text{s}}} = \frac{{{\sigma _{\text{s}}}}}{{4{\text{π }}{R^2}}}{{\boldsymbol{P}}} \cdot {{{\boldsymbol{I}}}_{\text{i}}} $$ (1) 式中：σs为粒子对非偏振入射光的散射截面；R为粒子与探测器的距离。当粒子粒径与波长相比较大时，σs和P可以分成光线追踪(Ray tracing, RT)和前向衍射(Diffraction, D)两个部分：
$$ {\sigma _{\text{s}}} = \sigma _{\text{s}}^{{\text{RT}}} + \sigma _{\text{s}}^{\text{D}} $$ (2) $$ {{\boldsymbol{P}}} = \frac{1}{{{\sigma _{\text{s}}}}}(\sigma _{\text{s}}^{{\text{RT}}}{{{\boldsymbol{P}}}^{{\text{RT}}}} + \sigma _{\text{s}}^{\text{D}}{{{\boldsymbol{P}}}^{\text{D}}}) $$ (3) 几何光学近似由以下严格定义[21]：
$$ \sigma _{\text{s}}^{\text{D}} = \left\langle A \right\rangle $$ (4) $$ {\sigma _{\text{e}}} = {\sigma _{\text{s}}} + {\sigma _{\text{a}}} = 2\left\langle A \right\rangle $$ (5) 式中：σe和σa分别为衰减截面和吸收截面；〈A〉为粒子投影截面面积；吸收截面只有光线追踪部分σa=σaRT。不对称因子g的定义为：
$$ g = \int_{4{\text{π }}} {\frac{{{\text{d}}\varOmega }}{{4{\text{π }}}}} \cos {\theta _{\text{s}}}{P_{11}} $$ (6) 式中：θs为散射天顶角；P11为散射矩阵P的第一个元素。前向衍射矩阵表达式如下[21]：
$$ {{{\boldsymbol{P}}}^{\text{D}}} \propto \frac{{{k^2}}}{{4{\text{π }}\left\langle A \right\rangle }}\left\langle {{{\left| {u({\theta _{\text{s}}},\phi )} \right|}^2}} \right\rangle {(1 + {\rm{cos}}{\theta _{\text{s}}})^2}{{\boldsymbol{I}}} $$ (7) 式中：
$u({\theta _{\text{s}}},\phi ) = \displaystyle\int_0^{2{\text{π }}} {{\text{d}}\phi ' \int_0^{r(\phi ')} {{\rm{d}}r'} } r'\exp [ - ikr'{\rm{sin}}{\theta _{\text{s}}} {\rm{cos}} (\phi - \phi ' )]$ ；k为波数；$ \phi $ 为散射方位角；I为4×4单位矩阵；r($ \phi ' $ )描述了粒子轮廓。光线追踪法中，每根光线在经历给定最大次数的内部或外部相互作用前都被追踪计算，最终各辐射特性参数是所有光线统计平均值。为验证程序正确性，将Siris程序和Mie理论[22]进行对比。图2为两种方法对不同尺度参数球形颗粒散射截面σs、吸收截面σa和不对称因子g的计算结果，验证了Siris程序的准确性。
图 2 Siris程序和Mie理论计算不同尺度参数球形颗粒辐射特性结果对比（波长为0.6283 μm, 颗粒折射率为1.33+0.5i, 介质折射率为1）
Figure 2. Comparison of radiative properties of spherical particles with varied size parameters calculated by Siris code and Mie theory (wavelength: 0.6283 μm, refractive index of particles: 1.33+0.5i, refractive index of medium: 1)
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二流法方法简单，求解速度快，是目前在对涂层、颜料的反射、吸收和发射特性求解中应用最为经典和广泛的方法[23]。文献中多将二流法中的散射假定为各向同性，文中考虑各向异性散射。如图3所示，二流法将涂层中的能流简化为前向能流I+和后向能流I-。考虑各向异性散射时，两个能流在涂层中的传播满足微分方程[24]：
图 3 二流法求解模型示意图
Figure 3. Sketch of the two-flux theoretical model
$$ \left\{ {\begin{split} &{\dfrac{{{\text{d}}{I_ + }}}{{N{\sigma _{\text{e}}}{\text{d}}z}} = - \left( {1 - \omega f + \dfrac{{(1 - {f_{\text{v}}})4{\text{π }}{k_{\text{m}}}}}{{N{\sigma _{\text{e}}}\lambda }}} \right){I_ + } + \omega b{I_ - }} \\ &{\dfrac{{{\text{d}}{I_ - }}}{{N{\sigma _{\text{e}}}{\text{d}}z}} = \left( {1 - \omega f + \dfrac{{(1 - {f_{\text{v}}})4{\text{π }}{k_{\text{m}}}}}{{N{\sigma _{\text{e}}}\lambda }}} \right){I_ - } - \omega b{I_ + }} \end{split}} \right. $$ (8) 式中：N为粒子数密度；ω为反照率；fv为粒子体积分数；λ为真空中波长；km为介质吸收系数；f, b分别为前向散射和后向散射占总散射能量的份额，由下式计算：
$$ f = \int_{ + 2{\text{π }}} {\frac{{{\text{d}}\varOmega }}{{4{\text{π }}}}} {P_{11}} $$ (9) $$ b = \int_{ - 2{\text{π }}} {\frac{{{\text{d}}\varOmega }}{{4{\text{π }}}}} {P_{11}} $$ (10) 当 f = b = 0.5时，公式(8)退化为各向同性情形，其边界条件为：
$$ \left\{ {\begin{array}{*{20}{l}} {{I_ + }\left( 0 \right) = \left( {1 - {R_{\text{c}}}} \right){I_{}} + {R_{\text{i}}}{I_ - }\left( 0 \right)} \\ {{I_ - }\left( d \right) = {R_{\text{g}}}{I_ + }\left( d \right)} \end{array}} \right. $$ (11) 式中：Rc为涂层前界面的外向镜反射率；Ri为涂层前界面的内向漫反射率；Rg为涂层基底的漫反射率。
界面处的镜反射率由菲涅耳公式[17]得到，漫反射率由镜反射率在半球空间上的积分得到。
至此，可求解得到厚度为d的涂层整体的反射率Rt为：
$$ \begin{gathered} {R_{\text{t}}} = {R_{\text{c}}} + {{\left( {1 - {R_{\text{c}}}} \right)\left( {1 - {R_{\text{i}}}} \right)\left[ {1 - {R_{\text{g}}}\left( {m - n\coth {\text{(}}n\eta d{\text{)}}} \right)} \right]} \mathord{\left/ {\vphantom {{\left( {1 - {R_{\text{c}}}} \right)\left( {1 - {R_{\text{i}}}} \right)\left[ {1 - {R_{\text{g}}}\left( {m - n\coth {\text{(}}n\eta d{\text{)}}} \right)} \right]} {}}} \right. } {}} \\ \left[ {m - {R_{\text{i}}} - {R_{\text{g}}} + n\coth {\text{(}}n\eta d{\text{)}} + {R_{\text{i}}}{R_{\text{g}}}\left( {m - n\coth {\text{(}}n\eta d{\text{)}}} \right)} \right] \\ \end{gathered} $$ (12) 式中：
$m = \dfrac{{1 - \omega f}}{{\omega b}} + \dfrac{{(1 - {f_{\text{v}}})4{\text{π }}k}}{{N{\sigma _{\text{e}}}\lambda \omega b}}$ ；$n = \sqrt {{m^2} - 1}$ ；$\eta = N{\sigma _{\text{e}}}\omega b$ 。由基尔霍夫定律得到涂层发射率ε等于吸收率α，即[17]：
$$ \varepsilon {\text{ = }}\alpha {\text{ = }}1 - {R_{\text{t}}} $$ (13) 使用文献[14]的蒙特卡罗法计算结果来验证文中二流法的正确性。计算对象为铝片掺杂丙烯酸树脂涂层在8~14 μm的光谱发射率，铝片颗粒平铺在涂层中（取向角0°），半径10 μm，厚0.8 μm，体积分数10%，涂层厚度50 μm。结果如图4所示，两者计算结果吻合良好，验证了二流法计算的有效性。
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使用Siris程序计算铝片在不同取向角下波长λ=0.3~16 μm的光谱辐射特性，结果如图5所示。取向角增大，铝片在入射光方向的投影面积减小，所以吸收截面和散射截面减小。不对称因子随取向角增大而增大。吸收截面峰值位于0.83 μm处，对应散射截面的峰谷和不对称因子的峰值。在0.6 μm以下存在零吸收，以及在较大取向角时1.2 μm附近存在吸收低谷。这是由于在对应波段铝的折射率很小，光在粒子表面发生全反射现象，导致吸收截面突变。
图 5 不同取向角θ下铝圆片的光谱辐射特性。(a)吸收截面σa；(b)散射截面σs；(c)不对称因子g
Figure 5. Spectral radiative properties of aluminum flakes at different incident angles θ. (a) Absorption cross-section σa; (b) Scattering cross-section σs; (c) Asymmetry factor g
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文中涂层对太阳辐射的吸收考虑波段λ=0.3~2 μm，涂层热辐射考虑波段λ=2~16 μm，假定涂层厚度0.5 mm。有些文献制备研究了金属含量超过70%的涂层辐射特性[10,25]，当铝片颗粒平行排列时，体积分数也可能达到0.7以上。因此文中体积分数的选取范围较大，达到0.9，以从理论上揭示体积分数对涂层辐射特性的影响规律。实际中应综合考虑涂层力学性能等多方面因素选择合适的体积分数。当体积分数较大时粒子散射为非独立散射，文中计算未予考虑。
图6为铝片体积分数0.1时，颗粒取向角分别为0°、30°、40°、45°、50°、60°和80°时的涂层光谱吸收率。从图中可以看到，初始的低吸收波段随取向角增大先变宽后变窄，45°时最宽，为0.3~0.57 μm。这是因为随着取向角增大，发生全反射的铝的折射率可以更大，所以波段变宽。但取向角超过45°后，圆柱颗粒侧面在波长较小就从全反射转变为吸收光，所以波段变窄，而且由于不对称因子增大，取向角超过45°后，涂层在粒子发生全反射波段的光谱吸收率比45°以下取向角的光谱吸收率更大。此外，当取向角大于50°时，在0.57~2 μm波段也存在几处突变，都是由于圆柱颗粒底面进入或退出全反射造成的，但此时圆柱颗粒侧面吸收光，所以涂层光谱吸收率没有降到很小。波长大于0.57 μm时，涂层光谱吸收率在铝片取向角从0°增大到40°时略微有所增大，在取向角45°和50°时明显增大，之后随取向角继续增大而减小。这是因为铝片反射光较强，当取向角超过45°时，反射光从后向变为前向，大大增强了前向散射，导致吸收显著增强。而随着取向角继续增大，铝片吸收和散射截面的减小使得涂层光谱吸收率减小。
图 6 不同铝片取向角下的涂层光谱吸收率(fv = 0.1)
Figure 6. Spectral absorptivity of the coating under different orientation angles of aluminum flakes (fv = 0.1)
图7为涂层光谱吸收率在铝片颗粒取向角分别为0°、15°、30°、45°、60°、70°和80°时随体积分数和波长的变化。比较不同铝片取向角的光谱吸收率云图，可以看到不同体积分数下涂层光谱吸收率随取向角的变化与图6基本相同。在所选取的参数范围内，涂层光谱吸收率在0.01~0.91之间，取向角不变时，随体积分数增大，涂层光谱吸收率先增大后减小，最大值位于波长0.83 μm，体积分数0.1~0.2附近。
图 7 涂层光谱吸收率随粒子体积分数fv和波长λ的变化，取向角分别为θ=0、 15°、 30°、45°、 60°、 70°、80°
Figure 7. Dependence of spectral absorptivity of the coating with the particle volume fraction fv and wavelength λ under orientation angles θ of 0, 15°, 30°, 45°, 60°, 70° and 80°
图8是涂层光谱发射率在7个不同铝片取向角下随体积分数的变化。可以看出，涂层光谱发射率随铝片取向角的变化与前述光谱吸收率随取向角的变化类似：涂层光谱发射率在铝片取向角小于45°时较小且变化不大，当取向角大于45°明显增大，并随取向角继续增大而减小。在所选取的参数范围内，涂层光谱发射率在0.33~0.80之间。铝片取向角一定时，随波长增大涂层光谱发射率减小，随体积分数增大涂层光谱发射率先增大后减小。
图 8 涂层光谱发射率随粒子体积分数fv和波长λ的变化，取向角分别为θ=0、 15°、30°、45°、 60°、70°、80°
Figure 8. Dependence of spectral emissivity of the coating with the particle volume fraction fv and wavelength λ under orientation angles θ of 0, 15°, 30°, 45°, 60°, 70° and 80°
按普朗克平均计算涂层平均吸收率α和平均发射率ε，公式如下[6]：
$$ {{i = }}\frac{{\displaystyle\int_{{\lambda _{\text{i}}}}^{} {{E_{\text{b}}}{\text{(}}{T_{\text{i}}},\lambda {\text{)}}\left( {1 - {R_{\text{t}}}(\lambda )} \right){\text{d}}\lambda } }}{{\displaystyle\int_{{\lambda _{\text{i}}}}^{} {{E_{\text{b}}}{\text{(}}{T_{\text{i}}},\lambda {\text{)d}}\lambda } }}{{, i = }}\alpha ,\varepsilon $$ (14) 式中：Eb(Ti,λ)为Ti温度下的黑体辐射光谱，太阳光谱近似认为是Tα = 5770 K的黑体辐射，Tε假定为300 K，积分区间λα=0.3~2 μm、λε=2~16 μm。考察涂层平均吸收率α，平均发射率ε和吸收发射比α/ε随铝片取向角和体积分数的变化，结果如图9所示。由此可以看出，铝片取向角超过45°时，涂层平均吸收率和发射率明显增大，这与涂层光谱吸收率和发射率在45°处的显著变化相关。在所考察的参数范围内，涂层平均吸收率在0.18~0.79之间，涂层平均发射率在0.35~0.79之间。随着铝片体积分数增大，涂层平均吸收率和发射率均先增大后减小。涂层吸收发射比的变化范围是0.48~1.69，其在铝片取向角45°附近最小，且随铝片体积分数增大而减小。图中标出了吸收发射比为0.8, 1, 1.3的等高线。该图表明在不更换掺杂颗粒材料、不增加铝片用量的情况下，通过改变铝片取向也能调控涂层吸收发射比。
图 9 涂层(a)平均吸收率α、(b)平均发射率ε和(c)吸收发射比α/ε铝片取向角θ和体积分数fv的变化（标出了α/ε=0.8, 1.0, 1.3三条等高线）
Figure 9. Dependence of (a) average absorptivity α, (b) average emissivity ε, and (c) ratio of absorptivity to emissivity of the coating α/ε with the flake orientation angle θ and volume fraction fv (contour lines: α/ε = 0.8, 1.0, 1.3)
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当涂层厚度足够大时，公式(12)中coth(βηd)趋近于1，此时涂层反射率基本不受厚度影响。但在涂层厚度较小时，厚度的变化将改变涂层辐射特性。图10是不同铝片取向角下涂层平均吸收率、平均发射率和吸收发射比随涂层厚度的变化。可以看出，不同取向角下，涂层平均发射率随涂层厚度的增大均增大，并且铝片取向角较大时，涂层厚度的影响较大。这是因为涂层越厚，其自身在红外波段的发射越强，取向角较大时铝片颗粒的吸收、散射截面均较小，导致厚度的影响作用更强。不同取向角下涂层的平均吸收率随涂层厚度增大的变化规律并不相同，在铝片取向角小于45°时随涂层厚度增大而减小，在取向角大于45°时先增大后减小。这是因为在0.57 μm以下可能存在由全反射引起的低吸收波段，对于此波段，涂层越厚，吸收率越小，而无全反射的波段的吸收率随厚度增加而增大，不同取向角和厚度下两者比重不同，所以表现出不同的变化趋势。涂层吸收发射比随涂层厚度的增大而减小，并且取向角较大时随厚度变化更大。
图 10 不同铝片取向角θ下涂层(a)平均吸收率α、(b)平均发射率ε和(c)吸收发射比α/ε随厚度d的变化 (fv = 0.2)
Figure 10. Dependence of (a) average absorptivity α, (b) average emissivity ε and (c) ratio of absorptivity to emissivity of the coating α/ε with the thickness d under different flake orientation angle θ (fv = 0.2)
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文中以空间随机分布、取向一致的铝片掺杂型涂层为研究对象，系统研究了粒子取向对热控涂层吸收发射比的调控规律。采用考虑衍射的几何光学方法和考虑各向异性散射的二流法，考察了铝片取向角、铝片体积分数和涂层厚度等因素，为热控涂层的设计和调控提供了新的思路。研究结果表明：
（1）改变铝片取向和体积分数，涂层吸收发射比变化范围是0.48~1.69。涂层吸收发射比在取向角45°附近最小，并随铝片体积分数增大而减小。
（2）当铝片取向角超过45°时，涂层平均吸收率和平均发射率明显增大。随着铝片体积分数增大，涂层平均吸收率和平均发射率均先增大后减小。涂层平均吸收率范围为0.18~0.79，涂层平均发射率范围为0.35~0.79。
（3）涂层光谱吸收率和光谱发射率在铝片取向角小于45°时较小且变化不大，当取向角大于45°明显增大，并随取向角继续增大而减小。
（4）铝片取向角较大时，涂层厚度对涂层平均发射率和吸收发射比影响较大。涂层吸收发射比随涂层厚度的增大而减小。
Regulation of ratio of absorptivity to emissivity of composite thermal control coating via orientation of aluminum flakes
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摘要: 金属颗粒掺杂型复合涂层是重要的航天器热控材料，除了粒子材料、尺寸、体积分数、几何形貌、涂层厚度等常规调控手段，颗粒的取向也是影响涂层辐射特性的重要因素。颗粒取向可以通过改变工艺参数或使用定向剂控制，然而目前关于颗粒整体取向对涂层吸收发射比的影响规律尚不清晰，而且现有的采用二流法的研究中大多都将散射假定为各向同性。以热控涂层中常用的大尺寸铝片粒子掺杂型复合涂层为计算模型，采用考虑衍射的几何光学方法和考虑各向异性散射的二流法研究了铝片取向角对热控涂层吸收发射比的调控规律，同时考虑了铝片体积分数和涂层厚度等因素。结果表明：通过调节粒子取向可实现涂层吸收发射比在0.48~1.69范围内的调控。涂层平均吸收率和平均发射率在铝片取向角超过45°时明显增大。涂层吸收发射比在铝片取向角45°附近最小，并且随铝片体积分数的增大而减小。研究表明：通过控制粒子的整体取向可有效实现吸收发射比的调控，为热控涂层的设计和调控提供了新的思路。Abstract:
Objective Metal particle pigmented coatings play a vital role in the thermal control of spacecraft. The ratio of absorptivity to emissivity of coating is one of the important properties of thermal control performance. Besides the conventional factors of particle material, size, volume fraction, morphology and coating thickness, particle orientation is also an important factor in the regulation of radiative properties of the coating. It is important to study the regulation method of the ratio of absorptivity to emissivity of the coating for the design of thermal control coating. The orientation of the particles can be adjusted by operating parameters or alignment agents. The effect of particle orientation on the ratio of absorptivity to emissivity of coating is not thoroughly explored yet. Moreover, the scattering in two-flux theory is usually assumed to be isotropic in the literature. Therefore, it is necessary to investigate the regulation of the ratio of absorptivity to emissivity of the coating by flake orientation. Methods As the commonly applied heat dissipation coating, the composite coating pigmented with large-size aluminum flakes was studied. The flakes were assumed to be randomly distributed and identically oriented. Due to the limitation of computer resources, it is infeasible to calculate the radiative properties of randomly distributed non-spherical large particles through strict solution of electromagnetic theory. The radiative properties of aluminum flake at different orientation angles were calculated by geometrical optics considering diffraction, and then the radiative transfer of the coating was solved by two-flux theory considering anisotropic scattering. The geometrical optics and the two-flux theory are suitable for the rapid calculation and analysis of the radiative properties of composite coating pigmented with large-size particles. The effects of orientation angle of aluminum flake, volume fraction and coating thickness were investigated. Results and Discussions The spectral radiative properties of aluminum flake at different orientation angles were calculated by geometrical optics considering diffraction (Fig.5), which indicated the variation of absorption cross-section, scattering cross-section, and the asymmetry factor of aluminum flake with orientation angle. The spectral absorptivity (Fig.7) and spectral emissivity (Fig.8) of coatings at different orientation angles were calculated by the two-flux theory considering anisotropic scattering, which revealed the trends at different volume fractions and orientation angles. The regulation of the ratio of absorptivity to emissivity of the coating by flake orientation was investigated. The dependence of average absorptivity, average emissivity, and ratio of absorptivity to emissivity of the coating on the flake orientation angle and volume fraction was illustrated (Fig.9). And the dependence of radiative properties of the coating on the coating thickness was also studied (Fig.10). The ratio of absorptivity to emissivity can be effectively regulated by particle orientation, volume fraction, or coating thickness. Conclusions The regulation of the ratio of absorptivity to emissivity of the thermal coating by flake orientation was systematically studied by modeling the coating pigmented with randomly distributed and identically oriented aluminum flakes. The results show that with the increase of orientation angle, the absorption cross-section and scattering cross-section of aluminum flake decrease, and the asymmetry factor increases. The ratio of absorptivity to emissivity of coating can be regulated in the range of 0.48-1.69 by adjusting the flake orientation. The average absorptivity and emissivity of the coating increase significantly when the orientation angle of aluminum flakes exceeds 45°, and increase first and then decrease with the increase of the volume fraction of aluminum flake. The ratio of absorptivity to emissivity reaches a minimum value at the orientation angle of around 45°. And the ratio decreases with the increase of the volume fraction of aluminum flakes. The coating thickness has greater effects on the average emissivity and the ratio of absorptivity to emissivity when the orientation angle of aluminum flake is larger. The ratio of absorptivity to emissivity decreases with the increase of coating thickness. The spectral absorptivity and spectral emissivity of the coating are small and vary little with orientation angle when the orientation angle of aluminum flake is less than 45°, and increase obviously when the orientation angle is greater than 45°, and then decrease with the continued increase of orientation angle. This work demonstrates that the ratio of absorptivity to emissivity can be effectively regulated by adjusting the particle orientation, providing a new method for the design and preparation of thermal control coating. -
图 1 空间随机分布、取向一致的铝片掺杂型涂层示意图
Figure 1. Diagram of a coating containing randomly distributed and identically oriented aluminum flakes
图 2 Siris程序和Mie理论计算不同尺度参数球形颗粒辐射特性结果对比（波长为0.6283 μm, 颗粒折射率为1.33+0.5i, 介质折射率为1）
Figure 2. Comparison of radiative properties of spherical particles with varied size parameters calculated by Siris code and Mie theory (wavelength: 0.6283 μm, refractive index of particles: 1.33+0.5i, refractive index of medium: 1)
图 5 不同取向角θ下铝圆片的光谱辐射特性。(a)吸收截面σa；(b)散射截面σs；(c)不对称因子g
Figure 5. Spectral radiative properties of aluminum flakes at different incident angles θ. (a) Absorption cross-section σa; (b) Scattering cross-section σs; (c) Asymmetry factor g
图 6 不同铝片取向角下的涂层光谱吸收率(fv = 0.1)
Figure 6. Spectral absorptivity of the coating under different orientation angles of aluminum flakes (fv = 0.1)
图 7 涂层光谱吸收率随粒子体积分数fv和波长λ的变化，取向角分别为θ=0、 15°、 30°、45°、 60°、 70°、80°
Figure 7. Dependence of spectral absorptivity of the coating with the particle volume fraction fv and wavelength λ under orientation angles θ of 0, 15°, 30°, 45°, 60°, 70° and 80°
图 8 涂层光谱发射率随粒子体积分数fv和波长λ的变化，取向角分别为θ=0、 15°、30°、45°、 60°、70°、80°
Figure 8. Dependence of spectral emissivity of the coating with the particle volume fraction fv and wavelength λ under orientation angles θ of 0, 15°, 30°, 45°, 60°, 70° and 80°
图 9 涂层(a)平均吸收率α、(b)平均发射率ε和(c)吸收发射比α/ε铝片取向角θ和体积分数fv的变化（标出了α/ε=0.8, 1.0, 1.3三条等高线）
Figure 9. Dependence of (a) average absorptivity α, (b) average emissivity ε, and (c) ratio of absorptivity to emissivity of the coating α/ε with the flake orientation angle θ and volume fraction fv (contour lines: α/ε = 0.8, 1.0, 1.3)
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