超快激光加工的多尺度理论研究及耦合模型构建综述（特邀）特邀综述【增强内容出版】
An ultrashort pulse exhibits an instantaneous high irradiance, which typically induces nonlinear interactions within various materials. In comparison to a short pulse, an ultrashort pulse has an exceedingly brief interaction duration, resulting in a relatively limited energy redistribution induced by thermal transfer. Therefore, ultrashort lasers possess the inherent potential to effectively mitigate the processing constraints encountered in the production of components with superior quality, exceptional precision, elevated hardness, and arduous machinability, surpassing the capabilities of conventional processing methodologies. In the domain of ultrashort laser micro/nano manufacturing, the paramount significance of theoretical inquiries cannot be overemphasized because they establish the fundamental basis for achieving precise control and manipulation. In addition, the intricate nature of the interaction between ultrashort lasers and materials is a subject of profound interest in the field of optical physics.
In stark contrast to the conventional paradigm of laser thermal processing, the interaction between ultrashort lasers and materials manifests a myriad of complex phenomena unfolding across various temporal and spatial scales. When an ultrashort laser interacts with a material, the photons are primarily absorbed by charge carriers. Simultaneously, the excitation and motion of the electrons induce a modification in the potential of the atoms, facilitating the transfer of electron energy from the optical phonon wave to the acoustic phonon wave within a time frame measured in picoseconds (10-12-10-10 s). The duration of the plasma motion, material ablation, and sputtering can range from nanoseconds to microseconds. Therefore, it is imperative to develop an all-encompassing framework that incorporates the intricate dynamics of laser beam propagation, electron ionization and energy transfer, plasma motion, thermal and non-thermal phase transitions, and laser ablation. This holistic model will be indispensable in unraveling the underlying principles governing the intricate interplay between ultrashort laser pulses and materials. Nevertheless, the advancement of such a theoretical framework poses a significant impediment in the ultrashort laser field.
Considering the inherent disparities in the properties of metals, semiconductors, and dielectric materials, this scholarly article commences by elucidating the intricacies of the electron dynamics and intricate interplay between photons, electrons, and ions during ultrashort laser irradiation at the atomic level. This paper first introduces the computation of the electron dynamics of materials under ultrashort pulses. This manuscript initially presents the utilization of time-dependent density function theory (TDDFT) to scrutinize the impact of laser parameters on the rate of electron excitation. Furthermore, it introduces the concept of employing TDDFT to compute the optical properties. Subsequently, taking into account the adherence of a metallic system to the Fermi-Dirac distribution, a streamlined approach that employs density functional theory (DFT) is introduced to derive the electron excitation parameters. Then, the utilization of real-time TDDFT in conjunction with molecular dynamic simulation is introduced to explore the intricate mechanisms underlying the coupling between photons, electrons, and ions. Additionally, a streamlined approach known as ab initio molecular dynamics is presented as a means to investigate the non-thermal phase transition phenomena exhibited by crystalline materials. Finally, the paper highlights that the utilization of an atomic scale model is inherently constrained when investigating phenomena occurring within a few picoseconds or even femtoseconds.
Then, this paper provides a comprehensive overview of the current cross-scale multi-physics coupling models employed in the simulation of ultrashort laser machining. The two-temperature equation is introduced as a means to explore the intricate dynamics of the energy exchange between electrons and ions within the context of metallic systems. Furthermore, we introduce a methodology that combines the two-temperature equation and electron excitation rate equation to analyze the energy transfer dynamics between electrons and ions within semiconductor and dielectric materials. This paper elucidates the two methods capable of manifesting the electron-ion energy in the realm of macroscopic scales.
Based on the principles of energy transfer, this paper presents a comprehensive overview of the contemporary cross-scale multi-physics coupling models utilized in the simulation of ultrashort laser ablation. This manuscript presents a novel approach that synergistically merges the principles of two-temperature equation and molecular dynamics simulations. This combined methodology enables a comprehensive description of material ablation phenomena, encompassing non-equilibrium phase transition thresholds and intricate chemical reactions. It is important to note, however, that the applicability of this method is primarily confined to the realm of nanoscale laser ablation. This paper also presents a novel approach that integrates the lattice temperature with fluid mechanics and heat and mass transfer models, employing the framework of the two-temperature equation (or the two-temperature model-coupled electron excitation rate equation). This methodology facilitates the visualization of laser-induced ablation phenomena at the microscale level.
In addition, it suggests potential avenues of research that could be pursued in the future within this field.
In recent years, researchers have undertaken extensive investigations into the intricate interaction between ultrashort laser pulses and materials. By utilizing DFT and ab initio molecular dynamics, these scholars have simulated the intricate processes of electron excitation, energy transfer, and atomic motion within materials. Furthermore, they have successfully constructed a comprehensive theoretical framework that encompasses the cross-scale coupling of electron excitation, the two-temperature equation, molecular dynamics, and fluid mechanics. The interconnection of these models facilitates the comprehensive characterization of the intricate phenomena occurring in the realm of ultrashort laser processing. This successful integration allows for the accurate simulation of material ablation, nano-ripple generation, and microstructure evolution.
Nevertheless, the intricate examination and formulation of models for ultrashort laser processing at various scales present a series of challenges that will undoubtedly shape the trajectory of future advancements in theoretical modeling. First and foremost, it is imperative to propose a precise and straightforward methodology for the calculation of the photo-induced alteration of material properties. This methodology must take into account the intricate crystal modifications that occur within materials when subjected to multiple ultrashort pulses. Furthermore, it is imperative to develop a novel theoretical framework that can seamlessly integrate the precision of molecular dynamics with the efficiency of fluid dynamics. Finally, novel approaches for coupling and spatial-temporal resolution optimization are being actively pursued in order to maintain the computational precision while enhancing the computational efficacy, enabling the simulation of ultrashort laser processing involving tens of millions of pulses.
1 引言
随着科技不断进步和新材料广泛应用，各零部件正朝着轻量化、小型化和高精细方向发展。在此情况下，对材料高精度、高质量的加工需求愈发迫切。然而，传统的加工方式（如电火花和长脉冲激光加工）由于存在热影响区等缺陷，无法实现设计要求和加工效果的高度统一，已无法满足新领域的需求。相比之下，超快激光可以解决传统加工方式难以解决的加工难题，成为精密制造技术的最佳选择。超快激光具有脉冲宽度极窄、能量密度极高以及与材料作用时间极短的特点。一方面，超快激光具有极高的瞬时激光功率密度，可以将激光非线性吸收变成不可忽略的状态，使得超快激光加工具有普适性，能够加工任何材料；另一方面，超短的脉冲作用时间导致热扩散效应极小，因此可以忽略微裂纹及重铸层［1-2］，同时也可以减少飞溅物的产生［3］，实现高精度加工。上述两种特性的结合赋予了超快激光微细加工特性及三维加工特性。具体为：通过将光束限制在瑞利长度及光斑直径以下的三维有效刻蚀区域，可以突破传统激光加工空间尺寸受限的难题。例如，利用贝塞尔光束可以获得直径在纳米量级且深度在微米量级的孔［4］，通过双光子聚合可以直写三维微纳结构［5］。
表 1. 各种建模方法的性能对比
Table 1. Performance comparison of various modeling methods
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超快激光与材料相互作用的过程是一个极为复杂的过程。与传统激光热加工不一样，超快激光与材料的相互作用涵盖了激光能量的吸收、能量的传递、材料的刻蚀和熔化等一系列时间跨度极大的复杂过程［6-7］，如
图 1. 超快激光照射材料时不同时间尺度下的现象［6-7］以及本文不同内容之间的关系
Fig. 1. Phenomena at different time scales when materials are exposed to ultrashort laser irradiation[6-7] and the relationship of the contents of this article
如
2 原子尺度的超快电子动力学行为研究
在超快激光加工中，光子-电子相互作用，即超快电子动力学行为，是脉冲持续时间内最主要且最重要的过程，对最终的加工结果起到了决定性作用［10］。
半导体及绝缘材料都存在带隙，要使电子从价带跃迁到导带，必须提供足够的能量使其跨越禁带宽度［11］。当禁带宽度小于光子能量时，通常为单光子吸收。当禁带宽度大于光子能量时，只有采用高功率激光照射，才能够将电子从价带激发至导带，同时会引发多光子电离现象。在很强的飞秒激光电场下，库仑场被极大地抑制，束缚在原子上的价电子被强激光撞出，并经历隧穿电离等过程，导带电子吸收单个光子从而获得更多的能量。若导带电子具有足够的能量可以碰撞价带电子并将其激发至导带，则将引起雪崩电离现象［12］。
金属的最高能级电子会占据满带的一部分，电子容易从占据态被激发到空态，此时不存在阈值效应，因此理论上可以吸收任意波长的能量。由于金属内部具有大量电子，超强激光照射过程中材料内部的电子主要发生雪崩电离［13］。此外，金属体内含有高密度（>1020 cm-3）的自由电子，具有非常高的电导率，经过转化后，金属的电导率转变成了介电常数的虚部。研究发现，金属的光趋肤深度在纳米量级。与此同时，金属的等离子体频率高于激光器波长对应的光子能量，介电常数的实部通常为负数。根据电磁波边界条件，可以推断金属表面存在非常强的反射率。
2.1 基于含时密度泛函理论的材料电离过程研究
含时密度泛函理论（TDDFT）方法是目前唯一能够准确且直接地获得超快激光参数对各种材料电子激发特性影响的方法。对于介电材料，超快激光作用引起电子电离过程的研究通常采用TDDFT［14］。在该理论中，含时波函数等价于含时电子密度，激光场则被视为随时间变化的均匀矢量势。通过求解不同时刻电子密度的空间变化，可以获得材料激发电子个数及响应电流等参数［15］。目前，TDDFT已被广泛用于分析不同激光参数（如波长［16-17］、能量［18］、相位差［19］、偏振态［20-21］、波长［22］、脉冲序列［23］等）对电子分布变化的影响规律。该方法可用于分析不同激光参数下的激发电子效率，从而获得激光辐照时材料的非线性光学响应。德国马克斯普朗克物质结构和动力学研究所的Tancogne-Dejean教授及其团队长期从事超快激光参数对材料响应影响的理论研究。他们主要关注超快激光的能量和椭圆度等参数对高次谐波强度及其空间分布的影响规律［20］（如
图 2. 基于 TDDFT 的超快激光作用下介电材料的电子电离过程。（a）激光椭圆度对MgO高次谐波的影响［20］；（b）脉冲时序对石英激发电子个数的影响［23］
Fig. 2. Electron ionization process of dielectric materials when subjected to ultrashort laser action, as described by the time-dependent density functional theory (TDDFT). (a) Influence of laser ellipticity on the generation of higher harmonics in MgO[20]; (b) effect of pulse trains on the population of excited electrons in quartz[23]
北京理工大学姜澜教授［23］领导的研究团队利用TDDFT方法对石英等材料在激光作用下的非线性响应过程进行了研究，旨在揭示激光波长及脉冲时序整形对材料刻蚀的影响机制。结果表明，在超快激光刻蚀过程中，多光子电离起着主导作用，激发电子数为0.69，吸收能量为26.83 eV，平均每个电子吸收的能量为38.99 eV，远高于石英的能带间隙2 eV。此外，该研究团队还发现，当激光输入能量固定时，如
在前文中已经介绍了利用TDDFT方法研究超快激光作用下材料内部电子电离的过程，其重点是构建超快激光作用下的电子电离过程。采用“泵浦-探测”的TDDFT思想可以计算激发态的光学特性。具体方法是：先用泵浦光束作用于材料内部，形成电子激发状态，然后添加阶跃函数的磁矢势，模拟探测过程。其中，对于阶跃函数的磁矢势，其电场强度满足E0δ（t）
2.2 基于有限电子温度的密度泛函理论（DFT）方法的光致材料特性研究
根据近自由电子理论，金属中电子的分布可近似为服从费米-狄拉克分布。在研究金属材料的光致特性变化时，除了可以采用TDDFT以外，还可以使用DFT静态计算方法。目前，研究人员通常通过设定有限的电子温度来获得电子态密度分布的变化以及体系能量，从而推导出电子激发个数及金属的热容［29-32］等参数，如
图 3. 基于有限电子温度的DFT方法的金属光致能量及材料特性的变化。（a）不同金属的态密度（DOS）随电子温度的变化［30］；（b）不同金属的电子热容随电子温度的变化［32］
Fig. 3. Changes in system energy and material properties of metals generated by photons, which is determined using the finite electron-temperature-based density functional theory (DFT) approach. (a) The density of states (DOS) for different metals varying with electronic temperature[30]; (b) electron heat capacity of various metals varying with electronic temperature[32]
目前的研究表明，不同金属材料在电子温度上升时的表现截然不同，尤其是过渡金属［31］：对于碱金属（如铝和镁），它们基本服从费米-狄拉克分布。随着电子温度升高，费米能级向下移动。此外，电子热容会与电子温度成线性关系。然而，对于这类金属，电子-声子耦合系数随着温度升高而增大，但是之后随着温度持续升高，电子-声子耦合系数会逐渐趋于一个常数。对于银、金这样的贵金属以及铜来说，其d轨道的能量一般都低于材料的费米能级（约为2~3 eV）。为此，随着电子温度升高，费米能级会上升。该类金属的电子热容与电子温度的逻辑关系是高于线性的，也就是说，电子热容会随着温度升高而急剧增大。与碱金属一样，这些贵金属和铜的电子-声子耦合系数也会随着温度升高而增大，但增速逐渐变缓。对于镍和铂，其费米能级位于d轨道的高能边缘，随着电子温度升高，费米能级也会随之上升。同时，电子热容会随着电子温度的升高呈现低于线性关系的增长。此外，随着温度升高，其电子-声子耦合系数也会逐渐降低。对于钛和钨，其费米能级位于d轨道中间，随着电子温度升高，费米能级下降。同时，电子热容随着电子温度升高而先增加后减小。其电子-声子耦合系数也会随着温度升高而增大，然而当温度超过5000 K时，钛的电子-声子耦合系数逐渐降低。
合金是指由两种及两种以上金属或非金属材料经过混合熔化、冷却凝固后形成的具有金属特性的物质。因此，在建立原子模型时，必须考虑元素的比例，并进行DFT静态计算［33］。如
图 4. 基于有限电子温度的DFT方法的不锈钢电子参数计算［33］。（a）不锈钢材料的超胞晶体结构；（b）电子-声子耦合系数随电子温度的变化
Fig. 4. Computation of electronic parameters of stainless steel using the finite electron-temperature-based DFT approach[33]. (a) Crystal structure of stainless steel; (b) electron-phonon coupling coefficient varying with electronic temperature
此外，研究人员将基于有限电子温度的DFT方法的静态计算模型应用于介电材料。Sato等［34］进行对比验证，证明了该方法计算激发态硅光学特性的准确性。结果表明，DFT静态计算结果与基于TDDFT“泵浦-探测”物理思想的激发态光学特性相符合。同时，研究人员还将DFT静态计算结果与基于传统Drude模型的光学特性进行了对比研究。结果表明，随着电子温度变化，电子有效质量和碰撞时间发生了变化，揭示了经验公式Drude模型的不足之处。
3 原子尺度的光子‑电子‑离子耦合作用过程研究
为了更深入地理解激光参数对材料加工形貌的影响机制，需要对
3.1 基于实时含时密度泛函理论（rt‑TDDFT）的电子‑ 离子非绝热演变
首先，介绍简化较少的全耦合理论计算方法，即电子-电子、电子-离子及离子-离子之间能量传递全耦合方法，该方法可以用于任何材料。如果系统中原子核的量子效应不明显，就可以使用经典的分子动力学方法处理原子核的运动。经典分子动力学方法的基本思想是结合量子和经典动力学，即：首先求解电子薛定谔方程演化电子波函数，然后通过牛顿力学的形式描述原子核的运动。这样不仅可以准确描述能量传递过程而且降低了计算成本［35］。然而，当系统中原子核的量子效应比较明显时，就需要采用rt-TDDFT和非绝热演化计算相结合的方式求解分子和凝聚态体系的非绝热演化［36］。目前上述方法已被广泛用于小分子在超快激光作用下的分解过程［37-39］以及晶体的非热熔相变过程［40-42］。
北京凝聚态物理国家实验室的孟胜研究团队［43］利用rt-TDDFT研究了VO2材料在超快激光作用下的非热熔相变过程。如
图 5. 原子尺度下激光作用的能量传递以及分子分解过程模拟图。（a）单斜VO2的光致相变机制［43］；（b）铜和镍在超快激光照射下的动力学过程［44］
Fig. 5. Atomic-scale simulation diagram of energy transfer and molecular decomposition process involved in laser irradiation. (a) Mechanism of phototropic phase transitions in monoclinic VO2[43]; (b) dynamic process of copper and nickel when subjected to ultrashort laser irradiation[44]
3.2 基于电子激发的第一性分子动力学模拟
研究人员认为，超快激光与被加工材料作用结束后，电子-电子动力学过程已达到平衡。因此，研究重点转向了电子-原子及原子-原子之间的能量传递过程［45］。这些过程可以被简化为激发态下的电子-离子耦合作用过程，通过设置背景价电子势可以表征电子激发个数［46］，并进行含时计算分析。该方法适用于任何材料。Li等［47］对相变存储合金Ge-Sb-Te进行研究后发现，在9%电子激发量下，该材料可以在低于熔点的条件下于24 ps内完成固态-固态非晶化转变，如
图 6. 原子尺度下激光脉冲结束后Ge-Sb-Te的固态-固态非晶化过程［47］
Fig. 6. Atomic-scale simulation of the solid-state amorphous process of Ge-Sb-Te at the end of laser pulse irradiation[47]
与全耦合方法相比，该方法的计算量较小，仿真时间可达皮秒量级，但仅能描述激光结束后原子的运动。此外，该方法的第一性原理分子动力学仅涉及少量原子，无法准确描述纳米甚至微米级刻蚀。
4 宏观尺度的电子激发与电子‑离子能量传递模拟
由前一节内容可知，原子尺度的电子-离子耦合运动模拟只能描述分子化学键断裂或材料的非热熔相变，无法解释材料在超快激光作用下的刻蚀现象。为了解决这一问题，研究人员提出了一系列简化的宏观及微观模型。
4.1 以电子激发浓度为主的电子激发与能量传递耦合模型
对于半导体和宽禁带半导体材料，目前的主流方法仅着重于研究超快电子动力学过程［50-51］，即仅考虑
图 7. 基于电子激发速率方程的超快激光材料刻蚀仿真［55］。（a）电子密度随激光能量增加的演变图；（b）材料刻蚀临界值确定示意图
Fig. 7. Simulation of ultrashort laser material ablation based on electron excitation rate equation[55]. (a) Evolution of electron density with an increase in laser energy; (b) schematic diagram for determining the critical value for material ablation
目前，这种物理学思想已被广泛应用于纳米波纹刻蚀形貌［54，56］、波纹覆盖区域［55，57］、烧蚀阈值［52，58］及脉冲序列影响机制［59］等理论研究中。然而，脉冲结束与材料最终刻蚀存在时间差，在等离子体共振时原子基本静止；而在脉冲结束后，电子除了发生复合外，还会发生扩散。因此，需要进一步验证“等离子体频率大于或等于激光光子频率的部分材料被刻蚀”这一物理学思想的准确性。
4.2 电子‑离子能量传递的双温方程
超短激光脉冲照射金属时，电子系统会吸收光能并偏离热平衡位置，其中电子热化平衡所需时间远短于电子-声子能量交换所需时间。因此，在皮秒的持续时间内，电子温度维持在高于晶格温度的水平。因此，Kaganov［60］提出的双温方程成为研究超快激光作用下电子-电子、电子-离子以及离子-离子能量传递的主流宏观模型。在双温方程中，每个子方程均为傅里叶传热方程。双温方程中的第一个方程考虑了能量吸收、电子升温、扩散以及电子与声子之间的耦合作用。在此方程中，电子和声子之间的耦合项决定了能量的传递过程。为了拓展双温度方程的应用范围以及提高其精度，北京理工大学的姜澜教授团队［61］对双温方程进行了改进。改进后的方程加入了电子热容、电子弛豫时间、电子热导率、反射率和吸收系数等参数随电子温度的变化规律，可以更加准确地描述电子和晶格升温的过程，如
图 8. 参数未修正与修正双温方程得到的电子温度及晶格温度对比［61］。（a）参数未修正的双温方程；（b）参数修正的双温方程
Fig. 8. Comparison of electronic temperature and lattice temperature obtained by parameter-uncorrected and modified two-temperature equations[61]. (a) Parameter-uncorrected two-temperature equation; (b) parameter-modified two-temperature equation
已有研究表明，双温方程在描述超快激光作用下的电子-离子系统演变过程时，电子热化平衡过程不可忽略［70］。这是因为电子的激发和热化对其参数产生了极大影响，并且热化平衡时间在飞秒时间尺度内［71］。研究人员使用时间依赖密度泛函理论（TDDFT）研究了过渡金属激光激发电子的非平衡动力学，结果表明，超快激光作用下的电子系统可以分为两个子系统［72］。因此，现有基于有限电子温度的DFT方法计算的材料瞬态特性能用于双温方程的可信度受到了质疑，特别是在传统的电子-声子耦合系数不再准确的情况下。为了解决这个问题，人们发展出了一种被称为“三温度模型”的理论框架，该理论主要针对的是铁磁材料［73］。
此外，伦斯勒理工学院的张绳百研究团队采用rt-TDDFT进行模拟后发现，在超快激光照射下硅的熔化温度明显低于其熔点（1680 K）［74］。这种现象可归因于激光引起的化学键软化导致材料的熔化势垒下降。同时，这也揭示了双温方程对物理现象描述的局限性。除此之外，双温方程中没有考虑原子运动的临界点，即相变点，因此无法描述材料的激光刻蚀过程。为此，需要寻找一种合理的方法，以细致地描述材料的刻蚀过程。
5 超快激光作用下宏/微观尺度的整个加工过程模拟
目前的理论认为，激光加工金属材料的去除形式主要包括汽化蒸发、平衡沸腾和液相爆破。Martyniuk等［75］经过不懈的研究，首次提出液相爆破理论可能是对飞秒激光去除金属材料内在机理的最合理解释。同时，现有的超快原位探测技术表明，在超快激光加工下，材料表面经历了相爆、飞溅等热物理过程［76］。因此，为描述超快激光加工下的材料刻蚀过程，大量研究采用了双温方程中晶格温度耦合其他模型的方法。
5.1 双温方程耦合分子动力学
分子动力学模拟是一种典型的非连续介质方法，它是一种微观尺度的模拟手段，适用于任何材料。与其他方法不同的是，它不需要将材料的宏观特性（例如热、机械特性等参数）作为研究的前提条件，也不需要用宏观控制方程解决物理量。此外，该方法能够从分子水平甚至是原子水平对所研究的物理过程进行分析，从而可在极短的时间内精准地跟踪并捕捉到靶材内部各微观物理量的变化过程，然后按照统计物理学的方法计算得到物质的各种宏观特性，例如温度、压力和密度。因此，分子动力学仿真技术在研究飞秒级和数十皮秒级的超快激光与材料相互作用过程方面得到了广泛应用［77-78］。研究所涉及的材料包括金属［79-80］、金刚石［81］等。例如，西安交通大学的梅雪松教授团队［82］采用自主设计的分子动力学软件进行建模仿真，成功获取了超快激光辐照区域内原子温度、相态和微观晶格结构随时间变化的趋势，并认为原子结构观察法和径向分布函数（RDF）可以分别用于晶体硅熔化与再凝固过程中固-液相变接触面的准确定位。同时，分子动力学模拟具备描述材料在超快激光作用下化学反应过程的能力。为了探讨超快激光加工含能材料的安全可行性，北京理工大学爆炸科学技术国家重点实验室的伍俊英等［83］采用分子反应动力学方法研究了1，3，5，7-四硝基-1，3，5，7-四氮杂环辛烷（HMX）的烧蚀机理（
图 9. 基于分子动力学的超快激光刻蚀模拟图［83］。（a）不同激光能量下的粒子扩散图；（b）HMX分子在原子尺度上的演变
Fig. 9. Simulation of ultrashort laser ablation based on molecular dynamics[83]. (a) Particle diffusion graphs at various laser energies; (b) evolution of HMX at atomic scales
考虑到分子动力学计算具有独特的优势，美国弗吉尼亚大学的Zhigilei教授团队［84］采用双温方程耦合分子动力学方法来解决双温方程无法准确描述材料刻蚀情况的问题。如
图 10. 双温方程耦合分子动力学的理论模型［85］。（a）双温方程耦合分子动力学计算方法示意图；（b）多个仿真区域拼接成单脉冲激光刻蚀图
Fig. 10. Theoretical model that integrates molecular dynamics with two-temperature equation[85]. (a) Schematic diagram of calculation method of coupled molecular dynamics with two-temperature equation; (b) amalgamation of many simulation regions into a single pulse laser ablation pattern
为提高双温方程耦合分子动力学模型的准确性，Mo等［92］和Arefev等［93］采用超快X射线技术观察了超快激光作用下原子排布随时间的演变过程，并分析了金薄膜的均匀和非均匀热熔过程，以推断电子-声子耦合系数。研究发现，该参数随着激光能量的增加而变化，如
图 11. 理论推导出的金薄膜的电子-声子耦合系数随电子温度的变化规律［93］
Fig. 11. Electron-phonon coupling coefficient of Au film deduced theoretically varying with electron temperature[93]
通过对原子尺度研究现状的分析可知，原子的运动是由于在激光电场和电子势场的联合作用下，原子周围的势能发生了改变。因此，施加于原子上的力应当是库仑力，而不是温度。因此，需要进一步修正双温方程耦合分子动力学模型。此外，当前分子动力学模拟的尺度仅在纳米级别，计算量相当巨大，急需一种修正方法，以提高分子动力学模拟的尺度和计算效率。
5.2 电子激发、双温方程及流体力学耦合模型
为了对超快激光作用下金属的刻蚀过程进行宏观尺度的描述，许多研究人员采用了双温方程中的晶格温度与流体力学以及传热传质模型耦合的方法，这种方法能够有效地描述超快激光作用下的等离子体运动过程。例如，北京理工大学的姜澜教授团队［94］利用上述耦合模型模拟了不同激光能量下铜的烧蚀过程。研究结果表明：在低激光能量下，材料的热相变过程包括熔化和热塑性变形，此外还包括由电子激发引起的库仑爆炸等非热相变过程；而在高激光能量密度下，气体喷射形成等离子体，同时材料表面密度降低并破碎成微柱结构。此前原子尺度的研究表明，电子的激发会导致离子间产生库仑力。为了解决这一问题，研究人员增加了材料中的电子压强［33］。法国里昂大学的Rudenko等［95］提出了一种考虑电子压强的超快激光刻蚀耦合模型。该模型使用了多物理麦克斯韦方程组、双温方程和可压缩纳维-斯托克斯方程，能够准确地描述材料在低能量情况下产生空化层的过程。该模型的结果与双温方程耦合分子动力学模型的结果极其相似［89-90］。目前，该模型已经被成功应用于多脉冲作用下纳米波纹结构的生成机制［95］，而且该模型能够还原金属表面高频纳米波纹结构的产生过程［96］，如
图 12. 金属表面纳米波纹结构的生成过程模拟。（a）激光能量及脉冲数对不锈钢表面纳米波纹生成过程的影响［95］；（b）镍金属表面高低频纳米波纹的生成过程［96］
Fig. 12. Simulation of the formation process of laser-induced periodic nano-ripple structures on metal surface. (a) Effect of laser energy and pulse number on the formation process of nano-ripples on stainless steel surface[95]; (b) formation process of high and low spatial-frequency nano-ripples on nickel metal surface[96]
为了描述超快激光作用下介电材料的刻蚀过程，需要在上述耦合模型的基础上增加电子激发模型。意大利那不勒斯费德里科二世大学的Allahyari等［97］使用电子激发方程、麦克斯韦方程组、双温方程和不可压缩纳维-斯托克斯方程等耦合模型，模拟了多脉冲作用下硅表面纳米波纹结构演变成微结构的过程，如
图 13. 介电材料表面纳米波纹结构的生成模拟。（a）脉冲数为60时硅表面上纳米波纹结构及微结构的生成［97］；（b）二氧化硅内部结构随脉冲数演变的过程以及纳米波纹结构的生成［99］
Fig. 13. Simulation of the generation of nano‑ripple structures on the surface of dielectric materials. (a) Formation of nano‑ripple structures and microstructures on silicon surface at pulse number of 60[97]; (b) evolution of silica internal structure with pulse number and formation of nano‑ripple structures[99]
从上述可以推知，传统的激光热刻蚀模型（包括流体力学和相变方程等）与双温模型耦合后能够描述与实际加工尺度相当的超快激光多脉冲加工过程，从而将超快激光加工的原子尺度模型推广到了微米级。然而，超快激光光致电离瞬态过程与经典模型中的热物理参数存在差异，化学键软化过程和非常数相变点也无法直观地体现在超快激光加工过程中。
6 结束语
作为激光领域重要的发展方向，超短脉冲激光已经由科研实验室逐渐向工业级应用转化，并已在科研、航天航空、新材料以及精准医疗领域扮演着日益重要的角色。其中，超快激光加工理论研究是微纳可控制造的基础。近年来，国内外学者对超快激光与材料相互机理进行了深入研究，运用DFT和第一性分子动力学模拟了超快激光作用下的材料内部电子激发、能量传递以及原子运动过程，构建了电子激发、双温方程、分子动力学和流体力学之间的跨尺度耦合理论模型。这些模型的耦合使得超快激光加工过程中的复杂现象得到了跨时空尺度的描述，成功模拟了材料刻蚀、纳米波纹结构生成及微观结构演变等过程。目前，超快激光加工的多尺度理论研究以及模型构建依然面临以下问题，这些问题也将成为未来理论建模发展的方向。
1）在材料光致特性理论研究方面，数值模型中的双温方程参数及电子激发参数主要基于经验公式或半经验量子模型获取，精度不高。虽然基于TDDFT及DFT能够高精度地获得材料的瞬态物性，但计算量巨大，不利于建立各种材料的瞬态物性数据库。此外，为获得典型的微纳结构，需要进行超快激光多脉冲照射，这会导致原子布局的多次变化，进而影响了材料瞬态物性的计算。一般情况下，基于DFT获取材料特性所需的计算时间及计算量都远低于TDDFT。因此，未来可以利用TDDFT提取材料的瞬态电子特性，并将瞬态电子特性及晶格分布代入DFT计算中，这样不仅可以提高计算精度，还可以降低计算量。
2）在宏观理论模型研究方面，通过将双温方程与流体力学等经典模型耦合，可以有效描述超快激光的刻蚀过程。然而，现有的流体力学及相变方程中的参数都未考虑超快激光作用下材料非稳态的影响。虽然分子动力学可以相对准确地描述材料内部的空化、相爆、飞溅及熔化等过程，但模型尺度太小。目前涌现出来的粗粒化分子动力学将一些邻近的原子/基团简化成珠子，以此来减少计算量并加速模拟，已被应用于研究高分子及石墨烯等材料的结构及动力学特性，但尚未用于超快激光加工数值模拟中。因此，可以基于粗粒化分子动力学来构造一种新的理论模型，该模型既能保持分子动力学的精度，又能具备流体力学的高效率。此外，超快激光加工过程是一个瞬态的非平衡过程，因此需要考虑修正傅里叶双温方程及其影响区的傅里叶传热模型。
3）在计算精度与模拟时间制约方面，受限于计算量，超快激光加工耦合模型中的微纳结构演变无法实现跨尺度计算。例如，纳米结构无法演变成微米结构，微米结构也无法演变成毫米结构。此外，超快激光加工模型只能在少量脉冲数下模拟结构的演变。然而，超快激光微孔加工、微槽加工及切割等行业需要处理上千万的脉冲数。目前，有限元-离散元混合建模策略被用于大变形且不易收敛的结构数值仿真中，在保证计算精度的同时提高了计算效率。因此，未来的发展方向是将有限元-离散元混合建模策略用于超快激光加工模型中，并寻找新的耦合策略以及时空分辨率调节手段，实现上千万脉冲数的超快激光加工模拟。
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Article Outline
王文君, 潘爱飞, 梅雪松. 超快激光加工的多尺度理论研究及耦合模型构建综述（特邀）[J]. 中国激光, 2024, 51(4): 0402407. Wenjun Wang, Aifei Pan, Xuesong Mei. Review of Multi‑Scale Theoretical Research on Ultrashort Laser Processing and Coupling Model Construction (Invited)[J]. Chinese Journal of Lasers, 2024, 51(4): 0402407.
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