上海交通大学学报, 2025, 59(8): 1145-1155 doi: 10.16183/j.cnki.jsjtu.2023.569

机械与动力工程

过冷水-壁面接触面积对冰成核行为影响实验研究

陆晨律, 王利平, 孟航飞, 刘洪, 王福新,

上海交通大学 航空航天学院, 上海 200240

Ice Nucleation Behavior in Supercooled Water with Varying Wall Contact Area

LU Chenlü, WANG Liping, MENG Hangfei, LIU Hong, WANG Fuxin,

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

通讯作者: 王福新,研究员,博士生导师;E-mail:fuxinwang@sjtu.edu.cn.

责任编辑: 王一凡

收稿日期: 2023-11-10   修回日期: 2023-12-26   接受日期: 2023-12-29  

基金资助: 国家自然科学基金(52202447)
上海市青年科技英才扬帆计划(22YF1419000)
中央高校基本科研业务费专项资金(23X010201110)

Received: 2023-11-10   Revised: 2023-12-26   Accepted: 2023-12-29  

作者简介 About authors

陆晨律(1999—),硕士生,从事飞机结冰研究.

摘要

认知过冷水与固体表面接触对冰成核过程的影响对过冷物质储运以及防冰表面设计等工程应用非常关键,而接触面积变化对冰成核的影响尚不明确,尤其是当接触面积较大时.通过改变硅胶管和聚氯乙烯(PVC)纤维增强软管的长度改变过冷水与壁面接触面积的方式,研究了接触面积对结冰温度的影响.实验结果表明,接触面积变化将显著影响过冷水的结冰温度.根据所得变化规律建立了接触面积-过冷度预测模型,并利用实验结果对经典成核理论中的面积项进行了修正.修正后的成核率预测结果与实验中面积增长对成核的影响基本一致,且接触面积对成核率的影响并非目前认为的线性,而呈非线性.研究方法和所提出的模型为未来工程中过冷水的利用以及接触面积变化影响成核过程的进一步研究奠定了良好的基础.

关键词: 过冷水; 异质冰成核; 接触面积; 成核预测; 修正模型

Abstract

Understanding the influence of the contact area between supercooled water and surface on the nucleation is critical for engineering applications such as anti-icing surface design and the supercooling preservation. However, the effect of contact area between supercooled water and surface on the nucleation still remains unclear, especially when the contact area is large. Therefore, the influence of the contact area on the freezing temperature of supercooled water inside the hose was experimentally studied by changing the length of silicone and polyvinyl chloride (PVC)-reinforced hoses. The results show that variations in contact area significantly affect the freezing temperature of supercooled water. Based on experimentally observed trends, a predictive model for contact area-supercooling relationship was developed. Furthermore, the area term in the classical nucleation theory (CNT) was modified using the experimental data. The modified nucleation rate predictions align well with the experimental findings regarding the influence of increasing contact area on nucleation. Notably, the effect of contact area on nucleation rate was found to be nonlinear, contrary to the commonly assumed linear relationship. The research methodology and the proposed model in this paper provide a solid foundation for future applications of supercooled water in engineering, as well as for further investigation into the role of contact area in nucleation phenomena.

Keywords: supercooled water; heterogenous ice nucleation; contact area; prediction of nucleation; modified model

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本文引用格式

陆晨律, 王利平, 孟航飞, 刘洪, 王福新. 过冷水-壁面接触面积对冰成核行为影响实验研究[J]. 上海交通大学学报, 2025, 59(8): 1145-1155 doi:10.16183/j.cnki.jsjtu.2023.569

LU Chenlü, WANG Liping, MENG Hangfei, LIU Hong, WANG Fuxin. Ice Nucleation Behavior in Supercooled Water with Varying Wall Contact Area[J]. Journal of Shanghai Jiaotong University, 2025, 59(8): 1145-1155 doi:10.16183/j.cnki.jsjtu.2023.569

过冷是指液体温度低于熔点仍保持液态的一种亚稳定状态,其普遍存在于人类生活与自然界中.研究表明在一些温度低至-37.5 ℃的云层中还存在过冷的液态水[1].水在过冷状态发生凝固需首先经历冰成核过程,成核又分为均质成核和异质成核[2].均质成核是在深过冷度下热力学波动的结果,其他情况下则存在壁面及杂质等作为凝结核诱发异质成核,这是由于凝结核的存在降低了成核过程的能垒[2-3].过冷和成核与很多应用领域相关,比如过冷能够延长生物细胞与器官的保存时间[4],过冷保存能维持果蔬与肉制品等食品的品质[5],过冷水能够用于快速制冰[6],输电线、飞机及风力机等表面需要避免成核发生来实现防冰[7],等等.同时有些应用领域需要避免出现过冷甚至主动触发成核,比如应用相变材料进行热能储存[8].过冷导致物质性质的异常,且成核机理非常复杂[9],目前还未被完全认识.因此理解存在不同外界扰动时的成核行为是促进相关应用与技术发展的重要基础.

目前,研究人员能够使用多种技术手段延迟或促使过冷水发生冰成核.Huang等[4]使用油封显著减少在水-空气界面发生的异质成核,实现了大体积水大过冷度的长时间保存.等容条件能够明显降低成核率,增强过冷水的稳定性[10].两壁面间温差在过冷水内形成的温度梯度能够降低成核温度[11-12].另外,研究表明可以通过施加电场、磁场或者超声等方式实现非接触式成核,通过添加成核剂控制成核过程,通过搅拌、机械撞击等机械扰动控制过冷度与成核时间等[13-16].在以上这些研究中,过冷物质都会与壁面接触.接触面积作为异质成核过程中最为基本的参数之一,理解其对成核过程的影响至关重要.

在经典成核理论中,异质成核率与接触面积成正比[17].成核率可以通过改变接触面积来改变,比如超疏水表面接触角越大,水滴与表面间的接触面积就越小,从而可以在不改变成核能垒的条件下降低成核率[18].根据经典成核理论,通常条件下温度每变化1 ℃,成核率会呈现数个量级的变化[3,19-20].因此,试验条件相同时,理论上接触面积应存在量级上的改变才能使结冰温度变化1 ℃.然而,根据Zhang等[21]对铝表面单液滴冰成核特性实验研究结果,等效直径2.12 mm和2.67 mm水滴平均成核温度分别为-17.04 ℃ 和-15.82 ℃,此时水滴与表面的接触面积分别为6.11 mm2和9.66 mm2,比值为1∶1.59,并未出现量级上的变化.另外,Inada等[22]实验研究了玻璃表面水滴的冰成核规律,认为当水与玻璃间的接触面积大于0.01 mm2后,经典成核理论不再适用于冰成核预测.在一些实际应用中,过冷物质与表面的接触面积会远大于单个水滴与壁面的接触面积,比如过冷保存生物样品容积可达数百毫升[4],有些用于储能的相变材料胶囊直径在10 cm左右[23],未来结冰风洞模拟过冷大水滴云雾结冰条件也会需要大量过冷水在管道内进行输运[24].但目前,过冷水体积较大时过冷水-壁面接触面积对结冰的影响规律尚不清晰.

为了理解过冷水-壁面接触面积对过冷水结冰温度的影响,本文开展了过冷水-壁面接触面积较大条件下其对冰成核过程影响的实验研究,建立接触面积与成核温度间的关系,结合实验统计和经典成核理论,对过冷水与不同面积的不同材料表面接触时的异质成核行为进行分析,并根据实验结果对经典成核理论中成核率预测公式进行修正,以实现实际应用系统中过冷水与壁面接触面积较大时壁面诱导冰成核的成核率与成核温度的预测.

1 实验方法

在壁面对冰成核行为影响的相关研究中,采用的实验样本大多数是直接将水滴放置在壁面上[22,25-26].目前,水-空气-壁面间的三相接触线对冰成核的影响尚不明确[27-28].本文使用的实验样本是将超纯水注入圆管内,通过改变圆管的长度l改变水与壁面间的接触面积A,相比改变水滴大小去实现接触面积的改变,此方法能够有效降低三相接触线对成核的影响.设计的实验装置主要包括4个部分:制冷设备、实验测试部分、测温系统以及管道端口控温系统,示意图如图1所示.

图1

图1   实验装置示意图

Fig.1   Schematic diagram of experimental setup


制冷设备使用的是一台高低温交变湿热试验箱(LRHS-504B-LJS, Shanghai Linpin Instrument, Ltd.).其温度控制范围为-40~150 ℃,控温精度±0.5 ℃,箱内温度均匀度小于2 ℃,最大降温速率约为3.0 ℃/min.

实验测试部分由实验样本(注入超纯水的圆管)与相应的支撑架组成.实验所用的管道为工程中常用的硅胶管与聚氯乙烯(PVC)纤维增强软管,内径分别为4 mm和8 mm,壁厚分别为1 mm和2 mm,长度范围分别为0.5~8 m和0.2~4 m.为了尽可能全面地反映过冷水结冰温度随接触面积的变化趋势,硅胶管的长度选取为0.5、0.65、0.8、1、2、5和8 m,PVC管的长度为0.2、0.25、0.5、1、2.5和 4 m.对应的过冷水-壁面接触面积范围分别为6.28×10-3~1.01×10-1 m2和 5.03×10-3~1.01×10-1 m2,过冷水体积范围分别为6.28~100.53 mL(硅胶管)和10.05~201.06 mL(PVC管).为了验证相同接触面积条件下过冷水体积变化对结冰温度的影响,选取内径2、4、6和8 mm的硅胶管进行了实验,对应的管道长度分别为2、1、0.67和0.5 m.受试验箱工作室尺寸(700 mm×800 mm×900 mm)限制,支撑架上每次最多可安装3条实验管道同时进行实验.为了降低冰成核的随机性对实验结果的影响,每个实验条件至少测试20个实验样本.需注意的是在将管路安装在支撑架上时应避免管道弯折.图2展示了实验所用的两种管道,图2(a)为实物图.利用原子力显微镜(atomic force microscopy, AFM)对两种管道内壁面的表面形貌与粗糙度进行了测量,结果如图2(b)2(c)所示,测得硅胶管与PVC管内壁面的表面粗糙度Ra分别为17 nm和62 nm.实验中所用的超纯水由Rephile Bioscience, Ltd.的RS2200QUV制备,其电阻率为18.2 MΩ·cm@25 ℃.

图2

图2   实验所用的硅胶管与PVC管

Fig.2   Silicone hoses and PVC hoses in experiment


测温系统由一台温度测量仪(THM200K, TENGHUI)和PT100温度传感器组成,测温精度为±0.1 ℃,测量频率为1 Hz.为标定管道内外壁面温差,实验前将两个PT100分别粘贴在管道同一位置的内外壁面上,在降温过程中对两种管道的内外壁面温度分别进行了测量,温度曲线如图3所示.降温过程中,管道内壁面温度高于外壁面,管道内外壁面温差Td基本稳定,硅胶管为0.1 ℃,管壁较厚的PVC管为0.8 ℃.结冰温度测量实验中每条管道外壁面至少粘贴两个温度传感器,同一管道上所有传感器温度的平均值记为该实验样本外壁面的温度,再根据实验前标定的内外壁面温差进行修正,得到管道内过冷水的平均温度.过冷水结冰过程中会释放潜热使温度迅速上升至熔点,因此温度上升表明过冷水内已经发生冰成核并开始结冰,温度上升前过冷水的温度则视为结冰温度,亦为冰成核温度TN,如图3所示.图中:Tsi-inTsi-out分别表示硅胶管内壁面与外壁面的温度;TPVC-inTPVC-out分别表示PVC管内壁面与外壁面的温度.

图3

图3   硅胶管和PVC管的降温曲线

Fig.3   Cooling curves of silicone hoses and PVC hoses


为了降低三相界面对过冷水成核的影响,本文通过对管道两端进行加热提高端口处温度来降低此处成核发生的概率,故设计了端口控温系统.该系统由直流电源、缠绕在管道外侧的电阻丝以及外部包裹的保温棉组成.在实验开始降温时对电阻丝施加电流进行加热,使端口处的温度始终高于管道温度至少3 ℃.对两种管道端口加热装置的影响范围分别进行了测量,结果显示:对于硅胶管,距电阻丝 5 cm 处与管路中段处温差小于0.1 ℃;对于PVC软管,这个距离是2 cm.通过这种方法,实验中未在端口处发生成核.

对于每次实验,首先将超纯水注入管内,将端口弯折并用鱼尾夹夹住防止水外流.需要注意的是,在往管内注水过程中要避免出现气泡.之后将实验样本放入高低温试验箱并开始降温,同时开启直流电源控制端口温度.实验样本从室温开始降温直到所有样本发生结冰,为节约单次实验时间,当温度高于5 ℃时,降温速率设定为1.5 ℃/min,当温度降低到5 ℃后,降温速率设定为0.3 ℃/min直到本次实验结束.实验结束后,需等待管路中的冰融化后将水排出,再重新注入超纯水重复上述步骤进行下次实验.

2 结果与讨论

2.1 成核温度分布

过冷水在不同长度硅胶管和PVC管内的结冰温度如图4所示.图中:l为各组实验中管道长度.在实验条件完全相同的情况下,各次实验的结冰温度具有随机性,但仍在一定范围内.一般而言,管道材质相同时,管道长度越短,结冰温度就越低.在本文实验的长度范围内,硅胶管内过冷水的结冰温度分布为-12~-21 ℃,PVC管内的过冷水结冰温度分布为-12~-20 ℃.

图4

图4   过冷水在不同长度管内形成不同接触面积条件下的TN

Fig.4   TN of the supercooled water under different contact areas formed by different lengths


图5给出了硅胶管与PVC管内过冷水平均结冰温度TAVG、中值结冰温度TMEDl之间的变化趋势.图中TAVG的误差线为平均绝对偏差.硅胶管与PVC管的TAVGTMED的变化趋势较为相似,都大致呈现随l增加先迅速增长,随后增长速度放缓并在实验范围内逐步趋向于定值.对于硅胶管,以2 m为分界点,当l<2 m,l的增加会导致TAVGTMED的迅速上升,当l>2 m后,l的进一步增加不再使TAVGTMED发生明显的变化,TAVGTMED在实验所测试的长度范围内逐步趋向于稳定.对于PVC管而言,这个分界点为1 m.平均绝对偏差的变化总体上呈现一个随l增加而略微减小的趋势,这可能是l增大后成核的随机性减弱的结果.

图5

图5   TAVGTMEDl的变化

Fig.5   Variation of TAVG and TMED with l


l发生变化时,A相应改变,同时管道内过冷水的体积V也将发生变化.根据经典成核理论,温度为260 K时,发生在水内部的均质成核率仅有约10-150 cm-3·s-1[29],因此,理论上本文实验中水体积的改变并不会对结冰温度造成影响.为验证,本文进行了保持A不变而改变V的实验.图6展示了实验中TN的散点图,图7展示了不同体积过冷水的TAVG.可以看出,当A相同,V从6.28 mL变化到25.13 mL时,TAVG的变化很小.V相同,A不同(内径din=4 mm、l=0.5 m与din=2 mm、l=2 m)的硅胶管内的过冷水,TAVG相差了4.1 ℃.这表明在本文所进行的实验中,A的变化相比于V的变化对结冰温度有更显著的影响.因此,尽管管道长度的变化会同时导致AV的变化,本文仅考虑A变化对成核带来的影响.

图6

图6   过冷水在A相同(0.012 6 m2) V不同时的TN

Fig.6   TN of the supercooled water with the same contact area of 0.012 6 m2 and different volumes


图7

图7   A相同时TAVGV的变化

Fig.7   Variation of TAVG with V under the same contact area


2.2 过冷度与接触面积间的关系

后续将以A代替l分析成核过程.图5可见,当过冷水A达到一定大小后,A的进一步增加对冰成核的促进作用将不再明显,从而使TAVG趋向于一个定值.另外,管道材质的差异使A的增长对成核的促进程度存在一定的差异,在建立A与平均过冷度ΔTAVG之间关系时需要合理设置参数以体现材料之间的差异.已有研究表明,当撞击物自由下落撞击壁面触发过冷水结冰时,随着下落高度的增加,过冷水结冰温度迅速上升,且上升速度逐渐放缓,结冰温度在下落高度达到一定值后不再发生变化[16,30].本文中A的变化对ΔTAVG的影响与上述研究中结冰温度的变化趋势有一定的相似之处,因此本文参考以上研究中的关系模型建立过冷水-壁面接触面积A与过冷水结冰平均过冷度ΔTAVG两者之间的关系:

TAVG+a)(A+b)=c

式中:abc为与材料性质有关的常数,a反映了随着A增大ΔTAVG逐渐趋向的稳定值,bc则共同反映了过冷水与不同材料表面接触面积变化对结冰温度的影响趋势.ΔTAVG单位为℃,A单位为m2.应用式(1)分别对硅胶管与PVC管的实验结果进行拟合,结果如图8中的实线所示,表1给出了相应的拟合参数.拟合的结果与之前对TAVG的变化趋势的描述相吻合.

图8

图8   ΔTAVGA间的关系

Fig.8   Relationship between ΔTAVG and A


表1   平均成核过冷度与接触面积关系式拟合参数

Tab.1  Fitting parameters in the relationship between supercooling degree and contact area

管道材质abc
硅胶-13.525 11-0.002 820.020 78
PVC-13.368 300.002 380.028 76

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需要指出的是,理想情况下,当过冷水与外界物质接触面积为0时,结冰始于均质成核,根据过冷水的体积,成核温度在-35~-40 ℃[31].因此,当过冷水与壁面接触面积趋近于0时,结冰温度也应当趋近一个有限的值.根据式(1)计算,在A趋向于0时,PVC管的ΔTAVG最终会达到25.0 ℃,硅胶管的拟合结果则出现了奇点,这是由于实验中在硅胶管A接近0的过程中TN迅速下降,进而导致b为负值.由于本文实验方法的限制,未能实现更小接触面积下过冷水结冰温度的测量实验,所以根据实验结果建立的预测模型的适用范围应大于本文实验最小面积,即约0.01 m2.当面积小于0.01 m2时,本文所建立的模型需进一步验证或修正.

2.3 基于经典成核理论考虑接触面积变化的成核行为分析

2.3.1 基于成核温度分布的成核率估算

成核率为单位时间内形成的冰核数量,成核率分析是研究成核行为的主要途径.Seeley等[32]提出了将实验中成核温度的统计数据与经典成核理论相结合来计算成核率的方法,被广泛采用[25-26].通过将温度划分为若干区间,每个区间长度为ΔTi,以每个温度区间的中间值Ti作为特征温度,ni为成核温度处于[TiTi/2, TiTi/2]区间内的成核次数,r为降温速率,则可以根据成核率的物理定义分别计算每个温度区间内的成核率[25-26,32]:

$J\left(T_{i}\right)=\frac{r_{i}}{\Delta T_{i}\left(\frac{n_{i}}{2}+\sum_{j>i} n_{j}\right)}$

不同组实验间接触面积存在差异,因此单位接触面积上的成核率[25-26]

$J_{A}^{*}\left(T_{i}\right)=\frac{J\left(T_{i}\right)}{A}$

温度区间划分及成核次数如图9所示.对于硅胶管与PVC管而言,结冰次数在分布上表现为整个区间中部多、两端相对较少.为了体现成核率计算结果在量级上的变化,对计算结果取以10为底的对数,结果如图10所示.硅胶管与PVC管内过冷水的冰成核率变化都大致呈现随着温度升高而逐渐降低的趋势.

图9

图9   成核次数与成核的累积概率随温度变化关系

Fig.9   Frequency of nucleation and cumulative probability


图10

图10   成核率随T变化关系

Fig.10   Change in nucleation rate with T


根据经典成核理论,异质成核率Jhete[19,33]可表示为

$\lg \left[J_{\text {hete }}(T)\right]=\lg \left(J_{0}\right)-\frac{\Delta G_{\text {hete }}^{*}(T, f)}{2.303 k_{\mathrm{B}} T}$

式中:kB为玻尔兹曼常数;J0是包括了成核率公式指数前各项的指数前因子;f是Fletcher因子,与冰核在凝结核上的杨氏接触角θ以及冰核与凝结核的相对尺寸有关,其数值在0到1之间[2],f=0表示没有成核能垒,比如已经存在冰表面,f=1表示均质成核;Δ${G}_{\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{e}}^{\mathrm{*}}$为形成临界冰核的自由能垒[19,34]:

Δ${G}_{\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{e}}^{\mathrm{*}}$(T, f)=$\frac{16\mathrm{\pi }{\gamma }_{\mathrm{i}\mathrm{w}}^{3}}{3(\mathrm{\Delta }{G}_{\mathrm{V}}{\left(T\right))}^{2}}$f

γiw为冰-水界面能,单位为J/m2.在237.15~273.15 K范围内可以表示为[19]

γiw=[28+0.25(T-Tm)]×10-3

Tm为冰的熔点,即273.15 K;ΔGV为冰与水之间每单位体积的自由能差[19,33-34]:

$\Delta G_{\mathrm{V}}(T)=\frac{\Delta H_{\mathrm{V}}\left(T_{\mathrm{m}}-T\right)}{T_{\mathrm{m}}}$

ΔHV为冰的体积熔解焓,为306 MJ/m3[26].

式(4)中指数前因子J0与Fletcher因子f未知,利用式(4)对由式(3)计算得到的成核率进行拟合,结果如图10所示,拟合参数如表2所示.拟合结果与计算结果所呈现的趋势符合较好,随着温度的升高,成核率会逐渐降低,且降低的速度逐渐增快.

表2   成核率拟合参数

Tab.2  Fitting parameters for nucleation rate

管道材质lg J0f
硅胶0.441 520.014 73
PVC0.771 530.017 85

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2.3.2 基于接触面积修正的成核率预测

在2.3.1节的分析中,可以看到当过冷水与壁面间的接触面积大范围变化时,成核率拟合结果仍具备较好的符合性,但由于在分析时将面积标准化,最终主要体现是管道材质不同对过冷水内冰成核的影响,未能体现过冷水与壁面接触面积变化带来的影响.为了进一步分析接触面积变化对成核的影响,本节将通过具体计算成核率的变化来将其影响定量化,并据此对经典成核理论成核率公式中的面积项做出一定的修正,以使本文实验的结果与其结果一致.

当考虑过冷水与壁面间接触面积时,异质成核率[19,33]可表示为

$J_{\text {hete }}(T)=A K(T) \exp \left(-\frac{\Delta G_{\text {hete }}^{*}(T, f)}{k_{\mathrm{B}} T}\right)$

式中:K为动力学因子,用来表示水分子穿过冰表面的扩散通量[19,33,35],

$K(T)=N_{\mathrm{c}} \frac{k_{\mathrm{B}} T}{h} \exp \left(-\frac{\Delta F_{\mathrm{diff}}(T)}{k_{\mathrm{B}} T}\right)$

Nc为与单位面积冰表面接触的水分子的数量,约为1019 m-2;h为普朗克常数;ΔFdiff为水分子扩散越过冰水界面的活化能[33,35],

$\Delta F_{\mathrm{diff}}(T)=\frac{k_{\mathrm{B}} T^{2} E}{\left(T-T_{\mathrm{R}}\right)^{2}}$

ETR为常数,对于150~273 K温度范围内的液态水,E=892 K,TR=118 K.

异质成核是一个随机事件,可以使用随机过程理论的方法进行研究,当温度为T,持续时间为δt时,样本发生结冰的概率[2]

P=1-exp(-Jhete(Tt)

参考Shardt等[31]的参数设定方法,根据本文实验中的降温速率,取δt=10 s.在结冰概率已知的条件下,结合式(8)和(11)以及Fletcher因子f,可以建立成核温度与接触面积之间的一一对应关系.但现有技术条件测量冰核-水-凝结核间杨氏接触角的值较为困难,目前一般认为f的值与表面材质有关[36].因此,本文分别将硅胶管和PVC软管实验中最小接触面积时的TMEDA代入式(8)和(11),来计算两种管路材质表面的f值.计算得到f值后,可以根据式(8)和(11)分别建立硅胶管与PVC管对应的TMEDA的关系.这里计算得到的面积值与同一温度下实验中的面积大小不一定相等,因此将根据式(8)和(11)计算出的面积定义为虚拟面积AV.表34分别展示了硅胶管和PVC管实验中的真实接触面积Aexp、该面积对应的TMED与该温度下AV的值.

表3   硅胶管的AexpTMEDAV

Tab.3  Aexp, TMED and AV for silicone hose

Aexp/m2TMED/℃AV/m2ln AV
6.28×10-3-19.206.28×10-3-5.07
8.17×10-3-18.433.63×1001.29
1.01×10-2-15.755.30×101331.60
1.26×10-2-15.652.32×101433.08
2.51×10-2-13.901.34×102864.77
6.28×10-2-13.672.36×103069.94
1.01×10-1-13.936.99×102764.11

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表4   PVC管的AexpTMEDAV

Tab.4  Aexp, TMED and AV for PVC hose

Aexp/m2TMED/℃AV/m2ln AV
5.03×10-3-17.505.03×10-3-5.29
6.28×10-3-18.001.02×10-4-9.19
1.26×10-2-15.551.32×10614.10
2.51×10-2-13.951.30×101637.10
6.28×10-2-13.783.00×101740.24
1.01×10-1-13.907.31×101638.83

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表3表4表明,Aexp与通过式(8)和(11)计算得到的AV之间存在着很大的差距,本文希望通过建立虚拟面积与实验真实接触面积之间的关系来对成核率式(8)进行修正.AV的数值范围过大,令Aexp直接与其拟合较为困难,因此先对AV取自然对数后再进行拟合,ln AV的数值在表34中已列出.建立两者间的关系如下:

ln AV=u+v/Aexp

式中:uv为与材料性质相关的拟合参数.拟合结果如图11所示,表5给出了相应的拟合参数.

图11

图11   AVAexp间的关系

Fig.11   Relationship between AV and Aexp


表5   面积修正模型的拟合参数

Tab.5  Fitting parameters for correction model of area

管道材质uv
硅胶77.13-0.53
PVC42.62-0.28

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图11中的点表示根据经典成核理论计算得到的值,实线为拟合曲线.随着实验中Aexp的增大,ln AV迅速增长,且增长速度逐渐放缓.当Aexp趋向0时,ln AV趋向负无穷,即AV趋向0,拟合模型的结果与实际相符.

将式(12)代入式(8),并对其中的指数项进行整理,可以得到修正后的异质成核率预测公式:

$\begin{array}{l}J_{\text {hete }, \bmod }(T, A)= \\\quad K(T) \exp \left[\left(u+\frac{v}{A}\right)-\frac{\Delta G_{\text {hete }}^{*}(T, f)}{k_{\mathrm{B}} T}\right]\end{array}$

为了说明修正面积项对成核率量级的影响,本文选择PVC管在温度为-15 ℃条件下,计算了A变化时式(13)与式(8)计算值的差异,如图12所示.黑色实线与红色虚线分别代表式(8)与式(13)计算得到的结果.A从0.01 m2变化到0.20 m2过程中,根据式(8),A增长对成核率的影响很小,成核率的量级从10-14 s-1增长到10-11 s-1,只改变了3个量级;而根据面积项修正后的计算结果,A的增加会令成核率在量级上迅速增长,导致成核发生并结冰,与本文实验情况相符.

图12

图12   根据经典成核理论应用虚拟接触面积和真实接触面积预测的成核率随A的变化(PVC,-15 ℃)

Fig.12   Variation of nucleation rate with contact area predicted by CNT and modified CNT using virtual and actual contact areas (PVC, -15 ℃)


对成核率分析的结果表明,接触面积变化对成核所造成的影响十分明显.Niedermeier等[37]通过假设凝结核表面存在多个成核区域,各区域内成核性质符合特定的分布函数,利用统计模拟建立了对应的成核模型,初步解释了造成成核行为随机性与奇异性的原因.同样的,实际中与过冷水接触的材料表面上各个区域的性质(如杨氏接触角)并不一定相同,从而会影响不同区域成核发生的概率.根据现有技术水平,还不能准确测量得到接触表面上成核点的分布及数量,另外成核点位的性质是否会随着温度发生改变也无法确定,这些性质的差异最终有可能体现为随着材料表面积增加成核率并非线性增加,从而影响接触面积增加时过冷水可以达到的最低温度.目前,对于与壁面接触的过冷水,基本上只能通过数值模拟或者实际试验积累数据后,再根据拟合或数据分析的方法对过冷水的成核行为进行预测[38-39].未来需要发展纳微尺度成核探测技术,再结合先进的表面表征技术对表面成核开展更深入的研究.

3 结论

本文通过实验研究了不同长度硅胶管和PVC纤维增强软管中过冷水与壁面接触面积对过冷水结冰温度影响,得出以下主要结论:

(1) 在本文实验接触面积范围内,当A较小时,过冷水TAVG会随着A增加迅速上升,且上升速度逐渐放缓,当A增加到一定值后,TAVG基本不再随A增加而变化.尽管管道长度的变化会同时导致过冷水体积的变化,但在本文的实验条件下,实验证明其对冰成核的影响很小.

(2) 通过对实验数据拟合,建立了A与ΔTAVG间的关系模型.基于经典成核理论开展了理论分析,使用经典成核理论中的成核率预测模型对实验获得的成核率进行拟合,发现即使实验样本A变化范围较大,面积均一化后的成核率仍具备较好的拟合结果.

(3) 为理解A变化对实验中过冷水内冰成核的影响,本文定义了过冷水与壁面间虚拟接触面积AV,根据经典成核理论建立了AexpAV间的关系,发现本文实验中冰成核率随接触面积的变化并不是通常认为的线性,而是非线性的,并建立了基于面积修正的成核率预测方法.

(4) 无论是AT还是A-AV关系模型,总体变化趋势与表面材质无关,但关系模型中的拟合参数与表面材质有关.

本文结果能够为表面接触面积变化对成核的影响提供新的见解,同时也能够为大体积过冷相关设施设备的设计提供一定指导.

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The probability of homogeneous ice nucleation under a set of ambient conditions can be described by nucleation rates using the theoretical framework of Classical Nucleation Theory (CNT). This framework consists of kinetic and thermodynamic parameters, of which three are not well-defined (namely the interfacial tension between ice and water, the activation energy and the prefactor), so that any CNT-based parameterization of homogeneous ice formation is less well-constrained than desired for modeling applications. Different approaches to estimate the thermodynamic and kinetic parameters of CNT are reviewed in this paper and the sensitivity of the calculated nucleation rate to the choice of parameters is investigated. We show that nucleation rates are very sensitive to this choice. The sensitivity is governed by one parameter - the interfacial tension between ice and water, which determines the energetic barrier of the nucleation process. The calculated nucleation rate can differ by more than 25 orders of magnitude depending on the choice of parameterization for this parameter. The second most important parameter is the activation energy of the nucleation process. It can lead to a variation of 16 orders of magnitude. By estimating the nucleation rate from a collection of droplet freezing experiments from the literature, the dependence of these two parameters on temperature is narrowed down. It can be seen that the temperature behavior of these two parameters assumed in the literature does not match with the predicted nucleation rates from the fit in most cases. Moreover a comparison of all possible combinations of theoretical parameterizations of the dominant two free parameters shows that one combination fits the fitted nucleation rates best, which is a description of the interfacial tension coming from a molecular model [Reinhardt and Doye, J. Chem. Phys., 2013, 139, 096102] in combination with the activation energy derived from self-diffusion measurements [Zobrist et al., J. Phys. Chem. C, 2007, 111, 2149]. However, some fundamental understanding of the processes is still missing. Further research in future might help to tackle this problem. The most important questions, which need to be answered to constrain CNT, are raised in this study.

ZHANG X, LIU X, WU X, et al.

Experimental investigation and statistical analysis of icing nucleation characteristics of sessile water droplets

[J]. Experimental Thermal and Fluid Science, 2018, 99: 26-34.

[本文引用: 1]

INADA T, TOMITA H, KOYAMA T.

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[本文引用: 2]

ALTOHAMY A A, ELSEMARY I M M, ABDO S, et al.

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王利平. 面向飞机结冰环境模拟的过冷大水滴可控发生原理研究[D]. 上海: 上海交通大学, 2021.

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WANG Liping. The principle of controllable generation of supercooled large water droplets for simulation of aircraft icing conditions[D]. Shanghai: Shanghai Jiao Tong University, 2021.

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LI K, XU S, SHI W, et al.

Investigating the effects of solid surfaces on ice nucleation

[J]. Langmuir, 2012, 28(29): 10749-10754.

DOI:10.1021/la3014915      PMID:22741592      [本文引用: 4]

Understanding the role played by solid surfaces in ice nucleation is a significant step toward designing anti-icing surfaces. However, the uncontrollable impurities in water and surface heterogeneities remain a great challenge for elucidating the effects of surfaces on ice nucleation. Via a designed process of evaporation, condensation, and subsequent ice formation in a closed cell, we investigate the ice nucleation of ensembles of condensed water microdroplets on flat, solid surfaces with completely different wettabilities. The water microdroplets formed on flat, solid surfaces by an evaporation and condensation process exclude the uncontrollable impurities in water, and the effects of surface heterogeneities can be minimized through studying the freezing of ensembles of separate and independent water microdroplets. It is found that the normalized surface ice nucleation rate on a hydrophilic surface is about 1 order of magnitude lower than that on a hydrophobic surface. This is ascribed to the difference in the viscosity of interfacial water and the surface roughness.

FU Q T, LIU E J, WILSON P, et al.

Ice nucleation behaviour on sol-gel coatings with different surface energy and roughness

[J]. Physical Chemistry Chemical Physics, 2015, 17(33): 21492-21500.

DOI:10.1039/c5cp03243a      PMID:26220055      [本文引用: 5]

In this paper, the ice nucleation temperatures of 10 μL water droplets on a series of sol-gel coatings with different roughness and surface energies were obtained using a customized automatic measurement system. Classical nucleation theory was then employed to explain the different icing behaviour on the coatings. It was found that the wetting mode at low temperatures is strongly correlated with the icing behavior of the droplets on the surfaces. Ice-phobic coatings can lower the icing temperature of the droplet on the surface by up to 6.9 °C compared with non-icephobic ones. Using classical nucleation theory, our results support some recent observations that the dominant nucleation sites are along the substrate-water-vapour three-phase contact line rather than at the substrate-water interface.

GURGANUS C, KOSTINSKI A B, SHAW R A.

Fast imaging of freezing drops: No preference for nucleation at the contact line

[J]. The Journal of Physical Chemistry Letters, 2011, 2(12): 1449-1454.

[本文引用: 1]

KAR A, BHATI A, LOKANATHAN M, et al.

Faster nucleation of ice at the three-phase contact line: Influence of interfacial chemistry

[J]. Langmuir, 2021, 37(43): 12673-12680.

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YUDONG L, JIANGQING W, CHUANGJIAN S, et al.

Nucleation rate and supercooling degree of water-based graphene oxide nanofluids

[J]. Applied Thermal Engineering, 2017, 115: 1226-1236.

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YOUNG S W, VAN SICKLEN W J.

The mechanical stimulus to crystallization

[J]. Journal of the American Chemical Society, 1913, 35(9): 1067-1078.

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SHARDT N, ISENRICH F N, WASER B, et al.

Homogeneous freezing of water droplets for different volumes and cooling rates

[J]. Physical Chemistry Chemical Physics, 2022, 24(46): 28213-28221.

DOI:10.1039/d2cp03896j      PMID:36413087      [本文引用: 2]

To understand the crystallization of aqueous solutions in the atmosphere, biological specimens, or pharmaceutical formulations, the rate at which ice nucleates from pure liquid water must be quantified. There is still an orders-of-magnitude spread in the homogeneous nucleation rate of water measured using different instruments, with the most important source of uncertainty being that of the measured temperature. Microfluidic platforms can generate hundreds to thousands of monodisperse water-in-oil droplets, unachievable by most other techniques. However, most microfluidic devices previously used to quantify homogeneous ice nucleation rates have reported high temperature uncertainties, between ±0.3 and ±0.7 K. We use the recently developed Microfluidic Ice Nuclei Counter Zurich (MINCZ) to observe the freezing of spherical water droplets with two diameters (75 and 100 μm) at two cooling rates (1 and 0.1 K min). By varying both droplet volume and cooling rate, we were able to probe a temperature range of 236.5-239.3 K with an accuracy of ±0.2 K, providing reliable data where previously determined nucleation rates suffered from large uncertainties and inconsistencies, especially at temperatures above 238 K. From these data and from Monte Carlo simulations, we demonstrate the importance of obtaining a sufficiently large dataset so that underlying nucleation rates are not overestimated at higher temperatures. Finally, we obtain new parameters for a previous parameterisation by fitting to our newly measured nucleation rates, enabling its use in applications where ice formation needs to be predicted.

SEELEY L H, SEIDLER G T.

Two-dimensional nucleation of ice from supercooled water

[J]. Physical Review Letters, 2001, 87(5): 055702.

[本文引用: 2]

EBERLE P, TIWARI M K, MAITRA T, et al.

Rational nanostructuring of surfaces for extraordinary icephobicity

[J]. Nanoscale, 2014, 6(9): 4874-4881.

DOI:10.1039/c3nr06644d      PMID:24667802      [本文引用: 5]

Icing of surfaces is commonplace in nature, technology and everyday life, bringing with it sometimes catastrophic consequences. A rational methodology for designing materials with extraordinary resistance to ice formation and adhesion remains however elusive. We show that ultrafine roughnesses can be fabricated, so that the ice nucleation-promoting effect of nanopits on surfaces is effectively counteracted in the presence of an interfacial quasiliquid layer. The ensuing interface confinement strongly suppresses the stable formation of ice nuclei. We explain why such nanostructuring leads to the same extremely low, robust nucleation temperature of ∼-24 °C for over three orders of magnitude change in RMS size (∼0.1 to ∼100 nm). Overlaying such roughnesses on pillar-microtextures harvests the additional benefits of liquid repellency and low ice adhesion. When tested at a temperature of -21 °C, such surfaces delayed the freezing of a sessile supercooled water droplet at the same temperature by a remarkable 25 hours.

JUNG S, TIWARI M K, DOAN N V, et al.

Mechanism of supercooled droplet freezing on surfaces

[J]. Nature Communications, 2012, 3(1): 615.

[本文引用: 2]

ZOBRIST B, KOOP T, LUO B P, et al.

Heterogeneous ice nucleation rate coefficient of water droplets coated by a nonadecanol monolayer

[J]. The Journal of Physical Chemistry C, 2007, 111(5): 2149-2155.

[本文引用: 2]

COOPER S J, NICHOLSON C E, LIU J.

A simple classical model for predicting onset crystallization temperatures on curved substrates and its implications for phase transitions in confined volumes

[J]. The Journal of Chemical Physics, 2008, 129(12): 124715.

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NIEDERMEIER D, SHAW R A, HARTMANN S, et al.

Heterogeneous ice nucleation: Exploring the transition from stochastic to singular freezing behavior

[J]. Atmospheric Chemistry and Physics, 2011, 11(16): 8767-8775.

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FITZNER M, PEDEVILLA P, MICHAELIDES A.

Predicting heterogeneous ice nucleation with a data-driven approach

[J]. Nature Communications, 2020, 11(1): 4777.

DOI:10.1038/s41467-020-18605-3      PMID:32963232      [本文引用: 1]

Water in nature predominantly freezes with the help of foreign materials through a process known as heterogeneous ice nucleation. Although this effect was exploited more than seven decades ago in Vonnegut's pioneering cloud seeding experiments, it remains unclear what makes a material a good ice former. Here, we show through a machine learning analysis of nucleation simulations on a database of diverse model substrates that a set of physical descriptors for heterogeneous ice nucleation can be identified. Our results reveal that, beyond Vonnegut's connection with the lattice match to ice, three new microscopic factors help to predict the ice nucleating ability. These are: local ordering induced in liquid water, density reduction of liquid water near the surface and corrugation of the adsorption energy landscape felt by water. With this we take a step towards quantitative understanding of heterogeneous ice nucleation and the in silico design of materials to control ice formation.

SOSSO G C, CHEN J, COX S J, et al.

Crystal nucleation in liquids: Open questions and future challenges in molecular dynamics simulations

[J]. Chemical Reviews, 2016, 116(12): 7078-7116.

DOI:10.1021/acs.chemrev.5b00744      PMID:27228560      [本文引用: 1]

The nucleation of crystals in liquids is one of nature's most ubiquitous phenomena, playing an important role in areas such as climate change and the production of drugs. As the early stages of nucleation involve exceedingly small time and length scales, atomistic computer simulations can provide unique insights into the microscopic aspects of crystallization. In this review, we take stock of the numerous molecular dynamics simulations that, in the past few decades, have unraveled crucial aspects of crystal nucleation in liquids. We put into context the theoretical framework of classical nucleation theory and the state-of-the-art computational methods by reviewing simulations of such processes as ice nucleation and the crystallization of molecules in solutions. We shall see that molecular dynamics simulations have provided key insights into diverse nucleation scenarios, ranging from colloidal particles to natural gas hydrates, and that, as a result, the general applicability of classical nucleation theory has been repeatedly called into question. We have attempted to identify the most pressing open questions in the field. We believe that, by improving (i) existing interatomic potentials and (ii) currently available enhanced sampling methods, the community can move toward accurate investigations of realistic systems of practical interest, thus bringing simulations a step closer to experiments.

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