PMMA 微注射成型中残余应力的形成机制

PMMA 微注射成型中残余应力的形成机制：分子动力学模拟

中南大学机电工程学院, 长沙 410083

中南大学 高性能复杂制造国家重点实验室, 长沙 410083

新加坡国立大学工程学院机械工程系，9 Engineering Drive 1, Singapore 117575, Singapore

摘要

注射成型是制造具有纳米结构的聚合物零件的一种经济有效的方法，零件中的残余应力是影响成型质量的重要因素。本文以聚甲基丙烯酸甲酯(PMMA)聚合物在长径比为2.0的纳米腔中的注塑成型为例，采用分子动力学模拟方法研究了注塑过程中残余应力的形成机理。对比分析了流动过程中动应力的变化，观察并解释了流动过程中分子链的形态和结构演变。研究了不同纵横比的纳米空腔对纳米结构中应力分布和变形的影响。势能，计算了聚合物的回转半径和弹性恢复百分比。结果表明，应力形成的本质是分子链在流动压力和腔壁的约束作用下发生压缩和缠结。此外，分子链的取向由各向同性变为各向异性，导致应力集中。同时，随着纵横比的增加，脱模后纳米结构的整体应力和变形也随之增加。导致应力集中。同时，随着纵横比的增加，脱模后纳米结构的整体应力和变形也随之增加。导致应力集中。同时，随着纵横比的增加，脱模后纳米结构的整体应力和变形也随之增加。

关键词：

注塑成型；残余应力；亚克力; 分子动力学模拟；纵横比

图形概要

一、简介

具有微/纳米结构的功能表面以其优异的光学、电化学和生物学特性广泛应用于微流控芯片[ 1 , 2 ]、超疏水表面[ 3 ]、光学转换表面[ 4 ]和仿生表面[ 5 ]等领域。因此，微纳制造技术已成为现代科学技术的前沿。目前，微纳米尺寸和高精度要求对加工制造提出了巨大挑战。微注射成型是制造各种形状和尺寸的微/纳米结构件的理想手段，经济、有效，具有大规模生产的潜力[ 6]. 然而，残余应力是影响注塑件最终质量的最重要因素之一。它的存在将直接影响零件的机械、光学和使用性能，甚至可能导致零件出现裂纹[ 7 , 8 ]。因此，许多学者一直渴望通过注射成型来解决表面微/纳米结构零件的残余应力。

众所周知，分子的有序取向表明存在残余应力[ 9 , 10 ]，这可以通过双折射实验来反映[ 11 , 12 ]。迄今为止，在注塑成型过程中，通过实验和有限元模拟已经很好地分析了工艺参数对残余应力的影响，其中温度和压力是最受关注的研究点。翁等。[ 13]使用双折射法检测聚碳酸酯（PC）微透镜阵列中的残余应力，发现模具温度是影响残余应力的最显着因素。通常，在合适的模具温度下，熔体可以顺利地填充模具型腔，而不会形成凝固层 [ 14 ]。林等。[ 15 ] 研究了加工条件对菲涅耳透镜双折射特性的影响，并指出残余应力对熔体温度很敏感。萨拉等人。[ 16 ] 提出保压压力对残余应力也有显着影响。除此之外，通过退火处理可以有效降低残余应力 [ 17 ]。

工艺参数对残余应力和微注射成型质量的影响与传统注射成型不同。当尺度减小到微米/纳米尺寸时，尺度效应和比表面积增加。因此，一些宏观机制很难用微观规律和现象来解释[ 18 ]。这样，分子动力学模拟就具有在分子水平上分析宏观机制的优势。目前，分子动力学模拟被广泛应用于流动行为 [ 19 , 20 , 21 ] 、界面传热 [ 22 , 23 , 24 ] 和界面粘附 [25、26、27 ]。_ _ _

纳米结构在注塑成型中的力学形态和作用位置与纳米压印技术相似。杨等。[ 28 ]利用分子动力学模拟研究了聚甲基丙烯酸甲酯（PMMA）的纳米压印，结果表明压印开始和结束时的势能存在一定差异；它们没有完全释放，导致残余应力。康等。[ 29 , 30 ] 指出模具嵌件的纵横比和形状对结构的应力分布有不同的影响。方等。[ 31 ] 认为温度越高，模具周围原子的剪切应变越大。

在这项研究中，采用分子动力学模拟来研究微注射成型中残余应力的机理。研究了动态应力的变化、分子链的形态和结构演变以及具有不同纵横比的模具嵌件对注塑成型中残余应力分布的影响。引入势能和回转半径探索分子链的流动特性，验证了注塑成型过程中残余应力的形成机理。

2。材料和方法

2.1. 模型构建

考虑到其良好的填充性能和优异的透光性，选择PMMA材料进行模拟。模拟的初始非晶模型如图1所示，包括模具嵌件层、聚合物层和真空层，箱体为60×60×160。模具嵌件层由镍 (Ni) 原子组成，具有 FCC（面心立方）结构，面心为 (1 0 0)，高度为 70 。切割 20 × 40  的凹槽以形成纳米腔，纵横比为 2.0。PMMA 层以聚合度为 20、链数为 50 的配置构建。温度为 298 K、密度为 1.18 g/cm 3 的周期性聚合物电池建立了。然后，在NVT系综下进行500 K高温和298 K低温循环退火处理（粒子数、体系体积和温度保持恒定），形成聚合物体系处于熔融状态。在 PMMA 层上施加高度为 10  的真空层，以防止聚合物与空气接触。此外，x轴、y 轴和z轴分别指定为周期性边界条件、周期性边界条件和自由边界条件。这样，仿真的初始模型就建立起来了。

图 1. 模拟的初始非晶模型。

2.2. 力场

采用一致的价力场 (CVFF) 来表示 PMMA 原子之间的分子间和非键相互作用。如式（1）所示，它由键伸缩势、角弯曲势、扭转势和非键相互作用组成。采用伦纳德-琼斯 12-6 势来描述 PMMA 层中的原子与模具嵌件层中的镍原子之间的非键相互作用。

Utotal=Ubond+Uangle+Utorsion+Unonbond

=kb(l−l0)2+ka(θ−θ02+kt(1+cos(nϕ−ϕ0))+4εij((σijrij)12−(σijrij)6) where kb, ka, and kt were the stiffness constants of bond stretching potential, angular bending potential and torsion potential. l, θ, nϕand l0, θ0, ϕ0 were the bond length, bond angle, torsion angle and the average values of them, respectively. σij and εij were the two Lennard–Jones potential parameters that defined the nonbonding interaction, which represented the well depth and zero-potential distance of the Lennard–Jones potential. Finally, rij was the distance between the atoms i and j. The cut-off distance of the nonbonding interaction was set to be 1.25 nm

2.3. 仿真程序

模拟是使用大规模原子/分子大规模并行模拟器 (LAMMPS) 进行的，它是计算机集群中的开源分子动力学包。在 1.0 ps 初始松弛后，注塑过程的模拟过程包括填充、保压、冷却和脱模，如图2所示. 假设 Ni 模具嵌件是刚性的，因为 PMMA 聚合物的杨氏模量相对于 Ni 可以忽略不计，因此所有 Ni 原子在模拟中都固定在其初始位置。在模拟注塑成型之前，PMMA 层被加热到 533 K，高于熔化温度，并松弛一段时间以达到最佳初始状态。在填充阶段，每个 PMMA 原子被施加 1.0 kcal/mol·ş（等于 0.07 nN）的力，以确保熔体顺利填充纳米腔。当形成清晰的纳米结构轮廓时，模拟程序立即进入堆积阶段，PMMA 原子在堆积压力下变得越来越紧凑。接下来，它进入冷却阶段，因为纳米结构与模具嵌件一起冷却到 353 K。在相反方向的 1.0 kcal/mol  脱模力下，PMMA 纳米结构向上移动并最终移出型腔。至此，一个完整的注塑过程就完成了。在整个仿真过程中，时间步长、总步数和系综分别为0.2 fs、150,000和NVT。

图 2. 注塑成型过程模拟程序的时间和温度设置。

3。结果与讨论

3.1. 注塑成型过程中的动态应力

为了研究注塑成型过程中残余应力的形成机理，有必要对这一过程中的动态应力进行观察和分析。通过计算每个原子的应力和体积，得到应力云图。应力由法向应力和剪应力的组合 von Mises 应力分析，等效应力由式（2）计算。

σ2von=3(σ2xy+σ2yz+σ2xz)+12[(σxx−σyy)2+(σyy−σzz)2+(σxx−σzz)2]     (2)

如图3所示，在注塑过程中有一系列应力变化的快照。动态应力变化主要分为四个注塑阶段。然而，模拟中无法完全划分填充和保压阶段，只能根据PMMA层完全进入模腔形成清晰的纳米结构轮廓来区分这两个阶段。

图 3. 注塑成型过程中应力变化的快照。

可以看出，聚合物的内应力分布基本均匀，初始松弛后的整体应力较小。随着填充阶段的进行，纳米结构的应力逐渐增大，在4.6 ps时纳米结构的肩部出现应力集中，表现出明显的表面压应力。然后，PMMA层的密度在堆积压力下不断增加，形成完整的纳米结构。随着堆积阶段的进行，纳米结构的内应力进一步增加。在 5.8 ps 时，应力集中区开始从肩部转移到底部。然后，在冷却阶段，纳米结构的整体应力保持稳定并有一定程度的波动。最后，在脱模力的作用下，纳米结构逐渐从型腔中分离出来。在脱模阶段，结构的整体应力急剧下降。当脱模阶段进行到18ps时，结构的应力集中区开始扩散，最终应力分布比较均匀。此时，可以看出纳米结构中存在相当大的应力，导致结构产生残余应力和大变形。

势能是反映实际应力变化的重要参数。流动过程中分子结构的变化，如分子链的压缩或拉伸，是应力形成的标志，从而导致相关势能的进一步变化，如键伸缩势、角弯曲势和扭转势潜在的。同时，图4给出了注塑过程中PMMA层的势能和平均应力的变化曲线。可以看出，两者的变化基本相同。因此，我们有充分的证据表明势能与应力的形成密切相关。

图 4. 势能和平均应力的变化。

通过分析势能和平均应力的变化，我们可以大致推断出分子链的变化。结果表明，在注射压力下，势能在 1.8 ps 时减小到零，并沿相反方向增加，平均应力从 0 左右增加。结合应力快照，可以假设松弛后，拉伸的分子链逐渐压缩在纳米结构的肩部，键间距减小，就像弹簧被压缩一样。当PMMA层继续向下移动时，势能和平均应力达到第一个峰值并在3.6 ps处下降，这是由于被压在肩上的分子链开始沿流动方向排列并被迫拉伸所致. 随着注射压力的进一步施加，两者开始快速增加，在6.4ps处达到第二个峰值，这是注射成型过程中的最大值。此时，由于空腔的限制，纳米结构底部的分子链开始再次被压缩和存储。

然后，分子链的势能在堆积和冷却阶段缓慢下降，这可能表明温度是控制该过程的主要因素。随着温度的降低，颗粒的振荡频率降低，平均应力保持在一定范围内。最后，在脱模力、模具嵌件和聚合物之间的粘附力以及纳米结构的弹性恢复力的共同作用下，由于分子链的拉伸和变形，势能和平均应力急剧下降。最终，平均应力略高于初始应力，势能没有完全释放。残余应力仍然存在于结构中。

3.2. 分子形态和结构的演变

研究了注塑过程中 PMMA 层中分子链的迁移和分子取向的变化。讨论了分子形貌和结构的演化规律，有助于从分子水平揭示成型过程中应力的形成机制。

分子链的形态和结构演化通过开放可视化工具（OVITO）输出，如图5所示. 一些有代表性的分子链颜色不同，因此很容易观察到分子链在流动过程中的变化。在注塑成型前，各分子链的拉伸形态和取向是无序的、各向同性的。在 2.0 ps 时，首先到达纳米结构肩部的分子链被压缩，这可能是应力集中区域最先出现在肩部的原因。然后，在 3.0 ps 时，压缩在肩上的分子链展开并伸展，克服空间位阻，并沿流动方向定向。纳米结构肩部的集中区域逐渐消失。随后，分子链以 3.6 ps 的速度不断被压入纳米腔，然后到达纳米结构底部的那些开始被压缩，应力集中区域从肩部转移到底部。此时，我们可以清楚地观察到每条聚合物链的形貌和结构都表现出各向异性的特征，并沿流动方向定向。在脱模阶段，随着应力的不断释放，分子链不断被拉伸，分子链之间在24 ps处出现明显的位移，这是纳米结构变形的原因。我们可以清楚地观察到每条聚合物链的形貌和结构都表现出各向异性的特征，并且沿着流动的方向定向。在脱模阶段，随着应力的不断释放，分子链不断被拉伸，分子链之间在24 ps处出现明显的位移，这是纳米结构变形的原因。我们可以清楚地观察到每条聚合物链的形貌和结构都表现出各向异性的特征，并且沿着流动的方向定向。在脱模阶段，随着应力的不断释放，分子链不断被拉伸，分子链之间在24 ps处出现明显的位移，这是纳米结构变形的原因。

图 5. 分子链的形态和结构演变。

回转半径是表征分子链空间形状分布的重要参数，可用于定量表征分子链的形貌和结构。图 6演示了回转半径随时间的变化。在弛豫期间，各分子链随着温度的升高不断伸长，回转半径增大，在1.4 ps时达到峰值32 。随着纳米结构肩部的分子链被压缩和缠结，回转半径在 3.6 ps 时缓慢减小至 28.2 。此时，聚合物内部的能量不断积累，具有较高内能的分子链在注射压力和型腔壁的限制下开始沿流动方向取向。因此，随着分子链的延长，回转半径再次增加。此外，当填充到 5.4 ps 时，它达到 30.3 Å。

图 6. 回转半径随时间的变化。

然后，到达底部的分子链逐渐压缩并冻结在纳米结构中，此时回转半径减小并稳定在 29.3  左右。该值大于 28.2  的谷值，这可以用分子链的取向行为相对于压缩行为占主导地位这一事实来解释。大多数分子链被迅速拉伸，它们的回转半径相应地急剧增加。

通过对分子形态和结构演变的描述，可以看出应力形成的本质是成型过程中分子链的不平衡构象，如有序取向、压缩和缠结行为。这种不平衡的构象无法立即恢复到适合环境条件的平衡构象，它被冻结在纳米结构中，并以势能的形式储存起来。然而，这种不平衡的构象也是一种部分可逆的变形。因此，在脱模阶段，压缩的不稳定构象会自动转化为自由稳定构象。同时，储存的势能将转化为分子链的伸长和迁移。

3.3. 纳米腔纵横比的影响

不同纵横比的纳米结构在流速、流动阻力等因素上存在差异，因此纵横比影响残余应力的形成和分布。为了进一步研究纵横比对应力形成的影响，在模拟中通过调整腔体深度来控制纵横比。

不同纵横比的纳米结构脱模前的应力分布值对比见图7. 可以看出，三个长宽比不同的结构都被填充在了完整的轮廓中。结果表明，结构的整体应力随着纵横比的增大而增大，三种纵横比的纳米结构的最大应力分别为68.3 GPa、83 GPa和86 GPa。由于纵横比的增加，需要更多的聚合物和更大的变形才能形成完整的纳米结构。因此，需要更大的注射和保压压力，导致整体应力增加。此外，当纵横比为1.0时，应力集中区主要分布在纳米结构的底部和肩部。随着纵横比增加到2.0，应力集中区转移到底部。当纵横比为3.0时，

图 7. 脱模前不同纵横比的纳米结构的应力分布。

势能和平均应力的变化曲线如图8所示。随着纵横比的增加，注塑成型中到达保压阶段​​和完成脱模阶段所需的时间也增加了。考虑到空腔深度的增加，PMMA 层需要更长的时间才能到达底部并从空腔中弹出。随着纵横比的增加，堆积阶段的势能和平均应力增加，表明更多的分子链处于压缩状态。

图 8.不同纵横比下 势能 ( a ) 和平均应力 ( b ) 的变化。

为进一步研究应力分布对不同纵横比纳米结构的影响，成型过程中分子链形貌和结构演化如图9所示. 当纵横比为1.0时，由于腔体尺寸的限制，一些在肩部被压缩的分子链在沿流动方向取向之前已经到达底部。这可能是纳米结构的肩部和底部出现应力集中区的原因。同时可以看出分子链沿流动方向的取向不明确，因此纳米结构中的整体应力较小。当纵横比达到2.0和3.0时，更多的分子链沿流动方向排列，并在纳米结构底部被压缩缠结，导致更大的应力集中区域。在这种情况下，纳米结构的各向异性更加明显，内部的整体应力更大。

图 9. 具有不同纵横比的分子链的形态和结构。

不同纵横比的纳米结构回转半径的变化如图10所示. 充填阶段回转半径不断减小，三条曲线基本重合。当纵横比为 2.0 和 3.0 时，纳米结构的回转半径在 3.4 ps 后增加，但当纵横比为 1.0 时它们仍然减小。这表明大部分分子链处于压缩状态，也验证了上述关于分子链演化的解释。然后，纵横比为 2.0 和 3.0 的纳米结构的回转半径分别在 5.4 ps 和 6.6 ps 处开始缓慢减小。此时，分子链的膨胀和拉伸行为强于压缩和缠结行为，表明达到堆积阶段所需的时间更长。此外，在包装和冷却阶段，回转半径分别稳定在 27.9 埃、29.3 埃和 34.2 埃左右。这表明随着纵横比的增加，更多的分子链沿流动方向取向，从而导致回转半径增加。在脱模阶段，回转半径急剧增加，增加趋势与纵横比成正比。

图 10. 不同纵横比的回转半径。

内应力最直接的影响是脱模后纳米结构的质量。脱模后的纳米结构或多或少的变形，如图11所示，黑色矩形代表脱模前的纳米结构形状，图11中整个纳米结构变宽变长。可以观察到，随着纵横比的增加，纳米结构的变形比脱模前更严重。因此，脱模后的变形程度可以通过计算弹性恢复百分比（等式（3））得到数值。等式（3）中的相关符号在图12中也有说明. 纳米结构脱模前后的形状分别由图12中的深色部分和浅色部分表示。

图 11. 具有不同纵横比的纳米结构的变形。

图 12. 弹性恢复百分比的计算。

当纵横比为 1.0、2.0 和 3.0 时，弹性恢复百分比分别计算为 288%、297% 和 311%。因此，随着纵横比的增加，应力也增加，导致脱模后的弹性恢复增加，变形量变大。

4。结论

在这项研究中，分子动力学模拟用于研究 PMMA 聚合物注射成型过程中残余应力的形成机制。分析了应力、势能和平均应力的变化，以探索成型过程中的动态应力。讨论了迁移和取向以表征分子链的演变。发现势能与应力的形成有关，其变化趋势与平均应力基本一致。应力形成的本质是不平衡的构象，如取向、压缩和纠缠行为，分子链的取向由各向异性变为各向同性。通过比较1.0、2.0不同纵横比的纳米结构的应力分布。0和3.0时，整体应力和变形程度随着纵横比的增加而增加。此外，弹性恢复百分比分别计算为 288%、297% 和 311%。随着纵横比的增加，弹性恢复百分比越大，表明残余应力引起的变形越大。因此，我们将在未来的工作中研究成型过程后的应力松弛行为。

作者贡献

数据管理、TD 和 MZ；形式分析，TD 和 JL；方法论，TD；项目管理，CW；资源，CW；监督、CW 和 HW；写作——初稿，TD；Writing—review and editing, HW 所有作者均已阅读并同意手稿的出版版本。

资金

本研究得到国家自然科学基金资助，资助号51775562；新加坡教育部学术研究基金一级，资助号 R-265-000-686-114；新加坡教育部学术研究基金Tier 2，资助号MOE2018-T2-1-140。

利益冲突

作者声明他们与这项工作没有利益冲突。

原版英文：

Formation Mechanism of Residual Stresses in Micro-Injection Molding of PMMA: A Molecular Dynamics Simulation

by 

Can Weng

 1,2,*,

Tao Ding

 1,2,

Mingyong Zhou

 1,2,

Jiezhen Liu

 1,2 and

Hao Wang

 3

1

College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China

2

State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, China

3

Department of Mechanical Engineering, Faculty of Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117575, Singapore

*

Author to whom correspondence should be addressed.

Polymers 2020, 12(6), 1368; https://doi.org/10.3390/polym12061368

Received: 9 May 2020 / Revised: 10 June 2020 / Accepted: 15 June 2020 / Published: 17 June 2020

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Abstract

Injection molding is an economical and effective method for manufacturing polymer parts with nanostructures and residual stress in the parts is an important factor affecting the quality of molding. In this paper, taking the injection molding of polymethyl methacrylate (PMMA) polymer in a nano-cavity with an aspect ratio of 2.0 as an example, the formation mechanism of residual stresses in the injection molding process was studied, using a molecular dynamics simulation. The changes in dynamic stress in the process were compared and analyzed, and the morphological and structural evolution of molecular chains in the process of flow were observed and explained. The effects of different aspect ratios of nano-cavities on the stress distribution and deformation in the nanostructures were studied. The potential energy, radius of gyration and elastic recovery percentage of the polymer was calculated. The results showed that the essence of stress formation was that the molecular chains compressed and entangled under the flow pressure and the restriction of the cavity wall. In addition, the orientation of molecular chains changed from isotropic to anisotropic, resulting in the stress concentration. At the same time, with the increase in aspect ratio, the overall stress and deformation of the nanostructures after demolding also increased.

Keywords: 

injection molding; residual stress; PMMA; molecular dynamics simulation; aspect ratio

Graphical Abstract

1. Introduction

Functional surfaces with micro/nano-structures are widely used in microfluidic chips [1,2], superhydrophobic surfaces [3], optical conversion surfaces [4] and biomimetic surfaces [5] due to their excellent optical, electrochemical and biological properties. Therefore, micro/nano manufacturing technology has become the forefront of modern science and technology. At present, micro/nano size and high precision requirements pose great challenges to processing and manufacturing. Micro-injection molding is an ideal means for manufacturing various shapes and sizes of micro/nano structured parts, which is economic, effective and has the potential for mass production [6]. However, residual stress is one of the most important factors that influences the final quality of the injection molded parts. Its existence will directly affect the mechanical, optical and service properties of the parts, and it may even lead to cracks in the parts [7,8]. Therefore, many scholars have been eager to resolve the residual stresses of surface micro/nano structured parts via injection molding.

As we all know, the ordered orientation of molecules indicates the existence of residual stresses [9,10], which can be reflected by birefringence experiments [11,12]. So far, in the process of injection molding, the influences of processing parameters on residual stresses have been well analyzed through experiments and finite element simulations, among which temperature and pressure are the research points of most concern. Weng et al. [13] used the birefringence method to detect residual stresses in polycarbonate (PC) micro-lens arrays and found that the mold temperature was the most significant factor affecting residual stress. Generally, the melt could fill the mold cavity smoothly without forming a frozen layer with an appropriate mold temperature [14]. Lin et al. [15] studied the effects of the processing conditions on the birefringence characteristics of Fresnel lenses and stated that the residual stress was sensitive to the melt temperature. Sara et al. [16] proposed that the packing pressure also had a significant effect on residual stresses. Besides this, residual stresses could be effectively reduced by an annealing treatment [17].

The influence of processing parameters on residual stress and the molding quality of micro-injection molding is different from that of traditional injection molding. When the scale is reduced to micro/nano-size, the scale effect and specific surface area are increased. Therefore, some macroscopic mechanisms are difficult to explain using microscopic laws and phenomena [18]. In this way, molecular dynamics simulation has the advantage of analyzing macroscopic mechanisms at the molecular level. At present, molecular dynamics simulation is widely used in the study of flow behavior [19,20,21], interfacial heat transfer [22,23,24] and interfacial adhesion [25,26,27].

The mechanical forms and action positions of nanostructures in injection molding are similar to those of nano-imprint technology. Yang et al. [28] studied the nano-imprint of polymethyl methacrylate (PMMA) using molecular dynamics simulation, and the results showed that the potential energies at the beginning and end of the imprinting were different to some extent; they were not completely released, resulting in residual stress. Kang et al. [29,30] pointed out that the aspect ratio and shape of the mold insert had different effects on the stress distribution of the structure. Fang et al. [31] believed that the higher the temperature, the greater the shear strain of the atoms around the mold.

In this study, molecular dynamics simulation was adopted to study the mechanism of residual stresses in micro-injection molding. The variation of dynamic stress, the morphological and structural evolution of molecular chains and the influence of mold inserts with different aspect ratios on the distribution of residual stresses in injection molding were investigated. The potential energy and radius of gyration were introduced to explore the flow properties of molecular chains, and the formation mechanism of residual stresses in injection molding was verified.

2. Materials and Methods

2.1. Model Constructing

Considering its good filling performance and excellent light transmission, PMMA material was selected for the simulation. The initial amorphous model of simulation is shown in Figure 1, including a mold insert layer, a polymer layer and a vacuum layer, with a box of 60 × 60 × 160 Ȧ. The mold insert layer was composed of nickel (Ni) atoms, which had an FCC (face center cubic) structure with a (1 0 0) plane and a height of 70 Ȧ. A groove of 20 × 40 Ȧ was cut to form the nano-cavity, with an aspect ratio of 2.0. The PMMA layer was constructed in a configuration with a polymerization degree of 20 and chain number of 50. A periodic polymer cell with a temperature of 298 K and a density of 1.18 g/cm3 was established. Then, a cyclic annealing treatment with a high temperature of 500 K and a low temperature of 298 K was carried out under the NVT (the number of particles, the volume and the temperature of the system are kept constant) ensemble to form a polymer system in a molten state. A vacuum layer with a height of 10 Ȧ was applied over the PMMA layer to prevent the polymers from coming into contact with the air. In addition, the x, y and z axes were assigned to the periodic, periodic and free boundary conditions, respectively. Thus, the initial model of the simulation was established.

Figure 1. Initial amorphous model of simulation.

2.2. Force Field

A consistent valence force field (CVFF) was adopted to represent the intermolecular and nonbonding interactions between PMMA atoms. As shown in Equation (1), it was composed of bond stretching potential, angular bending potential, torsion potential and nonbonding interaction. Lennard–Jones 12–6 potentials were adopted to describe the nonbonding interaction between the atoms in the PMMA layer and the nickel atoms in the mold insert layer.

2.3. Simulation Procedure

The simulations were performed using a large-scale atomic/molecular massively parallel simulator (LAMMPS), which was an open-source molecular dynamics package in a computer cluster. After 1.0 ps initial relaxation, the simulation procedure for the injection molding process included filling, packing, cooling and demolding, as shown in Figure 2. It was assumed that the Ni mold insert was rigid because Young’s modulus of the PMMA polymer was negligible relative to that of Ni, so all Ni atoms were fixed in their initial positions in the simulation. Prior to the simulation of injection molding, the PMMA layer was heated to 533 K, above the melting temperature, and relaxed for a period to achieve the optimal initial state. During the filling stage, each PMMA atom was applied with a force of 1.0 kcal/mol·Ȧ (equal to 0.07 nN) to ensure that the melt smoothly filled the nano-cavity. When a clear nanostructure profile was formed, the simulation procedure immediately turned into the packing stage, with the PMMA atoms becoming more and more compact under the packing pressure. Next, it entered the cooling stage, as the nanostructure cooled to 353 K with the mold insert. Under a demolding force of 1.0 kcal/mol Ȧ in the opposite direction, the PMMA nanostructure moved upward and finally out of the cavity. Thus, an entire injection molding process was completed. In the whole simulation process, the time step, the total step number and the ensemble were 0.2 fs, 150,000 and NVT, respectively.

Figure 2. Time and temperature settings of simulation procedure for the injection molding process.

3. Results and Discussion

3.1. Dynamic Stresses during Injection Molding

In order to investigate the formation mechanism of residual stresses during the injection molding process, it was necessary to observe and analyze the dynamic stresses in this process. By calculating the stress and volume of each atom, the stress contour was obtained. The stress was analyzed by the von Mises stress, which was a combination of normal and shear stress, and the equivalent stress was calculated by Equation (2).

Figure 3. Snapshots of stress changes during the injection molding process.

It could be seen that the internal stress distribution of the polymer was nearly uniform, and the overall stresses after initial relaxation were relatively small. With the progress of the filling stage, the stresses of the nanostructure gradually increased, and the stress concentration appeared at the shoulders of the nanostructure at 4.6 ps, showing obvious surface compressive stresses. Then, the density of the PMMA layer increased continuously under the packing pressure, forming a complete nanostructure. With the progress of the packing stage, the internal stresses of the nanostructure further increased. At 5.8 ps, the stress concentration area began to transfer from the shoulders to the bottom. Then, during the cooling stage, the overall stresses of the nanostructure remained stable and fluctuated to a certain extent. Finally, under the demolding force, the nanostructure was gradually separated from the cavity. In the demolding stage, the overall stresses of the structure decreased sharply. When the demolding stage was carried out to 18 ps, the stress concentration area of the structure began to spread, and eventually the stress distribution was relatively uniform. At this point, it could be seen that there were considerable stresses in the nanostructure, resulting in the residual stresses and large deformation of the structure.

Potential energy is an important parameter reflecting the actual stress changes. The changes in the molecular structure in the flow process, such as the compression or stretching of the molecular chain, are signs of stress formation, which lead to further changes in the related potential energy, such as bond stretching potential, angular bending potential and torsion potential. At the same time, Figure 4 shows the change curves of the potential energy and average stress of the PMMA layer during injection molding. It could be seen that the changes in both were basically the same. Therefore, we had sufficient evidence to show that the potential energy was closely related to the formation of stresses.

Figure 4. Variation of potential energy and average stress.

By analyzing the changes in potential energy and average stress, we could roughly infer the changes of the molecular chains. The results showed that the potential energy decreased to zero at 1.8 ps and increased in the opposite direction under the injection pressure, and the average stress increased from around 0. Combined with the stress snapshots, it could be assumed that after relaxation, the stretched molecular chains were gradually compressed on the shoulders of the nanostructure, with the bond spacing decreasing as if a spring were compressed. When the PMMA layer continued to move down, the potential energy and average stress reached the first peak and decreased at 3.6 ps, which was caused by the molecular chains that were compressed on the shoulder beginning to align along the flow direction and being forced to stretch. With the further application of injection pressure, both of them began to increase rapidly, reaching the second peak at 6.4 ps, which was the maximum value in the injection molding process. At this point, the molecular chains at the bottom of the nanostructure started to be compressed and stored again, due to the limitation of the cavity.

Then, the potential energy of the molecular chain decreased slowly in the stages of packing and cooling, which may indicate that the temperature was the main factor controlling this process. With the decrease in the temperature, the oscillation frequency of the particles decreased, and the average stress remained within a certain range. Finally, the potential energy and the average stress decreased sharply due to the stretch and deformation of the molecular chains, under the combination of the release force, the adhesion force between the mold insert and the polymer and the elastic recovery force of the nanostructure. Eventually, the average stress was slightly higher than the initial stress, and the potential energy was not fully released. The residual stresses still existed in the structure.

3.2. Evolution of Molecular Morphology and Structure

The migration of molecular chains and the change in molecular orientation in the PMMA layer during injection molding were studied. The evolution law of molecular morphology and structure were discussed, which helped to reveal the formation mechanism of stresses in the molding process from the molecular level.

The morphology and structure evolution of the molecular chains were output by an open visualization tool (OVITO), as shown in Figure 5. The colors of some representative molecular chains were different, so it was easy to observe the changes in the molecular chains during the flow process. Before the injection molding, the stretching shape and the orientation of each molecular chain were disordered and isotropic. At 2.0 ps, the molecular chains that first reached the shoulders of the nanostructure were compressed, which was possibly the reason that the stress concentration area appeared first in the shoulders. Then, at 3.0 ps, the molecular chains that had compressed on the shoulders expanded and stretched, overcoming the steric hindrance, and oriented along the flow direction. The concentration area of the shoulders of the nanostructure gradually disappeared. Subsequently, the molecular chains were constantly pressed into the nano-cavity at 3.6 ps, and then those that reached the bottom of the nanostructure started to be compressed, with the stress concentration area transferred from the shoulders to the bottom. At this point, we could clearly observe that the morphology and structure of each polymer chain exhibited anisotropic characteristics and were oriented in the direction of flow. During the demolding stage, with the constant release of stresses, the molecular chains were continuously stretched, and there was an obvious displacement between the molecular chains at 24 ps, which was the reason for the deformation of the nanostructure.

Figure 5. Morphology and structure evolution of molecular chains.

The radius of gyration is a crucial parameter to characterize the spatial shape distribution of molecular chains, so it could be used to quantitatively characterize the morphology and structure of molecular chains. Figure 6 demonstrates the variation of the gyration radius with time. In the relaxation period, each molecular chain continuously stretched with the increase in temperature, and its radius of gyration increased, reaching a peak value of 32 Ȧ at 1.4 ps. With the molecular chains on the shoulder of the nanostructure compressed and entangled, the radius of gyration slowly decreased to 28.2 Ȧ at 3.6 ps. At this point, the energy inside the polymer kept accumulating, and the molecular chains with a higher internal energy began to orientate along the flow direction under the injection pressure and the limit of the cavity wall. Therefore, the radius of gyration increased again with the extension of the molecular chains. Moreover, it reached 30.3 Ȧ when filling to 5.4 ps.

Figure 6. Variation of the radius of gyration with time.

Then, the molecular chains reaching the bottom were gradually compressed and frozen in the nanostructure, at which point the radius of gyration decreased and remained stable at around 29.3 Ȧ. This value was greater than the valley value of 28.2 Ȧ, which can be explained by the fact that the orientation behavior of the molecular chains was dominant, relative to the compression behavior. Most molecular chains were rapidly stretched and their radius of gyration correspondingly increased sharply.

Through the description of the evolution of the molecular morphology and structure, it can be seen that the essence of stress formation was the unbalanced conformation of molecular chains during the molding process, such as ordered orientation, compression and entanglement behavior. This unbalanced conformation could not be immediately restored to the equilibrium conformation suitable for environmental conditions, and it was frozen in the nanostructure, where it was stored as potential energy. However, this unbalanced conformation was also a kind of partially reversible deformation. Therefore, during the demolding stage, the compressed unstable conformation would be automatically converted into free stable conformation. At the same time, the stored potential energy would be transformed into the elongation and migration of the molecular chains. At this point, the whole injection molded structure would be deformed.

3.3. Effect of the Aspect Ratio of the Nano-Cavity

Nanostructures with different aspect ratios vary in their flow rate, flow resistance and other factors, so that the aspect ratio affects the formation and distribution of residual stresses. In order to further study the effect of aspect ratio on stress formation, the aspect ratio was controlled by adjusting the cavity depth in the simulation.

The stress distribution values of the nanostructures with different aspect ratios before demolding are compared in Figure 7. It can be seen that three structures with different aspect ratios were all filled in the full outlines. The results show that the overall stress of the structure increased with the increase in aspect ratio, and the maximum stress of the nanostructures with three aspect ratios were 68.3 GPa, 83 GPa and 86 GPa, respectively. Due to the increase in aspect ratio, more polymer and larger deformation were required to form a complete nanostructure. Therefore, greater injection and packing pressure were needed, resulting in an increase in the overall stress. In addition, when the aspect ratio was 1.0, the stress concentration areas were mainly distributed at the bottom and shoulders of the nanostructure. With the increase in the aspect ratio to 2.0, the stress concentration area was transferred to the bottom. When the aspect ratio was 3.0, the spread stress concentration area also appeared at the bottom, and the stress value was obviously larger than the other two.

Figure 7. Stress distribution of nanostructures with different aspect ratios before demolding.

The change curves of potential energy and average stress are shown in Figure 8. As the aspect ratio increased, the time required to reach the packing stage and complete the demolding stage in injection molding also increased. Considering the increase in cavity depth, the PMMA layer needed a longer time to arrive at the bottom and eject from the cavity. With the increase in aspect ratio, the potential energy and average stress were increased during the packing stage, indicating that more molecular chains were in the compression state.

Figure 8. Variation of potential energy (a) and average stress (b) with different aspect ratios.

To further investigate the influence of stress distribution in nanostructures with different aspect ratios, the morphology and structure evolution of molecular chains in the molding process are shown in Figure 9. When the aspect ratio was 1.0, some molecular chains that were compressed at the shoulders had reached the bottom before their orientation along the flow direction due to the limitation of cavity size. This may be the reason for the stress concentration area appearing at the shoulders and bottom of the nanostructure. At the same time, it could be seen that the orientation of the molecular chains along the flow direction was not clear, so the overall stress in the nanostructure was relatively small. When the aspect ratio reached 2.0 and 3.0, more molecular chains were oriented along the flow direction and were compressed and tangled at the bottom of the nanostructure, resulting in a larger stress concentration area. In this case, the anisotropy of the nanostructure was more obvious and the overall stress inside was greater.

Figure 9. Morphology and structure of molecular chains with different aspect ratios.

The changes in the radii of gyration in nanostructures with different aspect ratios are shown in Figure 10. The radii of gyration in the filling stage were constantly decreased, and the three curves basically coincided. With aspect ratios of 2.0 and 3.0, the gyration radii of the nanostructures increased after 3.4 ps, but they still decreased when the aspect ratio was 1.0. This indicated that most molecular chains were in the compression state, and it also verified the above explanation about the evolution of the molecular chains. Then, the gyration radii of the nanostructures with aspect ratios of 2.0 and 3.0 began to decrease slowly at 5.4 ps and 6.6 ps, respectively. At this point, the expansion and stretching behaviors of the molecular chains were stronger than the compression and entanglement behaviors, indicating the longer time needed to reach the packing stage. Additionally, in the stages of packing and cooling, the radii of gyration were stable at around 27.9 Ȧ, 29.3 Ȧ and 34.2 Ȧ, respectively. This showed that with the increase in aspect ratio, more molecular chains were oriented along the flow direction, which led to the increase in the gyration radius. In the demolding stage, the radii of gyration increased sharply, and the increasing trend was proportional to the aspect ratio.

Figure 10. Radius of gyration with different aspect ratios.

The most direct effect of the internal stresses was the quality of the nanostructures after demolding. There were more or less deformations of the nanostructures after demolding, as shown in Figure 11, where the black rectangle represents the shape of the nanostructure before demolding, and the whole nanostructure is widened and elongated in Figure 11. It could be observed that, with the increase in aspect ratio, the deformation of the nanostructures became more serious than that before demolding. Therefore, the degree of deformation after demolding could be obtained numerically by calculating the elastic recovery percentage (Equation (3)). The related symbols in Equation (3) are also demonstrated in Figure 12. The shapes of the nanostructure before and after demolding are represented by the dark and light parts in Figure 12, respectively.

Figure 11. Deformations of nanostructures with different aspect ratios.

Figure 12. Calculation of elastic recovery percentage.

With aspect ratios of 1.0, 2.0 and 3.0, the elastic recovery percentages were calculated as 288%, 297% and 311%, respectively. Therefore, as the aspect ratio increased, the stresses also increased, resulting in an increase in the elastic recovery after demolding and a larger deformation.

4. Conclusions

In this study, molecular dynamics simulation was used to investigate the formation mechanism of residual stresses in the injection molding of the PMMA polymer. The changes in stress, potential energy and average stress were analyzed in order to explore the dynamic stress during the molding process. The migration and orientation were discussed in order to characterize the evolution of molecular chains. It was found that the potential energy was related to the formation of stress, and its variation trend was basically consistent with the average stress. The essence of stress formation was unbalanced conformation, such as orientation, compression and entanglement behaviors, and the orientation of molecular chains changed from anisotropy to isotropy. By comparing the stress distribution of nanostructures with the different aspect ratios of 1.0, 2.0 and 3.0, the overall stress and degree of deformation increased with an increase in aspect ratio. In addition, the elastic recovery percentages were calculated as 288%, 297% and 311%, respectively. With the increase in aspect ratio, a larger elastic recovery percentage indicated a larger deformation caused by the residual stresses. Consequently, the stress relaxation behaviors after the molding process will be investigated in our future work.

Author Contributions

Data curation, T.D. and M.Z.; Formal analysis, T.D. and J.L.; Methodology, T.D.; Projectadministration, C.W.; Resources, C.W.; Supervision, C.W. and H.W.; Writing—original draft, T.D.; Writing—review and editing, H.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant number 51775562; the Singapore Ministry of Education Academic Research Fund Tier 1, grant number R-265-000-686-114; The Singapore Ministry of Education Academic Research Fund Tier 2, grant number MOE2018-T2-1-140.

Conflicts of Interest

The authors declared that they have no conflicts of interest to this work.

Nomenclature