液橋內(nèi)熱質(zhì)耦合對(duì)流不穩(wěn)定性及旋轉(zhuǎn)磁場(chǎng)法控制1)

鄒 勇 朱桂平 李 來(lái) 黃護(hù)林,2)

*(南京航空航天大學(xué)能源與動(dòng)力學(xué)院，南京210016)

?(安徽工業(yè)大學(xué)數(shù)理學(xué)院，安徽馬鞍山243032)

液橋內(nèi)熱質(zhì)耦合對(duì)流不穩(wěn)定性及旋轉(zhuǎn)磁場(chǎng)法控制1)

鄒 勇*,?朱桂平*李 來(lái)*黃護(hù)林*,2)

*(南京航空航天大學(xué)能源與動(dòng)力學(xué)院，南京210016)

?(安徽工業(yè)大學(xué)數(shù)理學(xué)院，安徽馬鞍山243032)

浮區(qū)法因具有無(wú)坩堝接觸污染的生長(zhǎng)優(yōu)點(diǎn)而成為生長(zhǎng)高完整性和高均勻性單晶材料的重要技術(shù).但熔體中存在的毛細(xì)對(duì)流會(huì)給浮區(qū)法晶體生長(zhǎng)帶來(lái)極大挑戰(zhàn)，這是由于對(duì)流的不穩(wěn)定會(huì)導(dǎo)致晶體微觀瑕疵的產(chǎn)生和宏觀條紋等缺陷的形成.為了提高浮區(qū)法生長(zhǎng)單晶材料的品質(zhì)，研究浮區(qū)法晶體生長(zhǎng)中毛細(xì)對(duì)流特性及如何控制其不穩(wěn)定性顯得尤為重要.本文采用數(shù)值模擬的方法對(duì)半浮區(qū)液橋內(nèi)SixGe1?x體系中存在的熱質(zhì)毛細(xì)對(duì)流展開研究并施加旋轉(zhuǎn)磁場(chǎng)對(duì)其進(jìn)行控制.結(jié)果表明：純?nèi)苜|(zhì)毛細(xì)對(duì)流表現(xiàn)為二維軸對(duì)稱模式，溫度場(chǎng)主要由熱擴(kuò)散作用決定，而濃度場(chǎng)則由對(duì)流和溶質(zhì)擴(kuò)散共同支配；純熱毛細(xì)對(duì)流呈現(xiàn)三維穩(wěn)態(tài)非軸對(duì)稱流動(dòng)，濃度分布與熔體內(nèi)熱毛細(xì)對(duì)流的流向密切相關(guān)，等溫線在對(duì)流較大的區(qū)域發(fā)生彎曲；耦合溶質(zhì)與熱毛細(xì)對(duì)流則為三維周期性旋轉(zhuǎn)振蕩流.施加旋轉(zhuǎn)磁場(chǎng)后，熔體周向速度沿徑向向外增大，熔體內(nèi)濃度場(chǎng)和流場(chǎng)均呈現(xiàn)二維軸對(duì)稱分布.

浮區(qū)法，毛細(xì)對(duì)流，表面張力，數(shù)值模擬，旋轉(zhuǎn)磁場(chǎng)

引言

浮區(qū)單晶生長(zhǎng)技術(shù)[1-3]作為無(wú)坩堝法，是制備單晶的一種重要方法，廣泛用于高精度硅、高溫合金及其他半導(dǎo)體材料的生長(zhǎng)[4-5].浮區(qū)法晶體生長(zhǎng)是一個(gè)包含固體導(dǎo)熱、熔體對(duì)流、固液相變等多種傳熱流動(dòng)方式的復(fù)雜熱--質(zhì)輸運(yùn)過(guò)程.生長(zhǎng)過(guò)程中的傳熱流動(dòng)特性直接影響功率消耗、溫度分布、界面形狀等宏觀晶體生長(zhǎng)參數(shù)，并最終決定單晶內(nèi)部的微觀結(jié)構(gòu)和缺陷、雜質(zhì)的分布.因此，有必要深入研究熔體中的熱質(zhì)流動(dòng)特性并對(duì)其進(jìn)行有效控制，以改善單晶質(zhì)量[6-9].由于浮區(qū)單晶生長(zhǎng)中存在著自由表面，表面張力驅(qū)動(dòng)的馬蘭哥尼(Marangoni)對(duì)流對(duì)熔體中雜質(zhì)濃度的分布會(huì)產(chǎn)生極大的影響[10].單晶中雜質(zhì)分布的不均勻性是制約晶體質(zhì)量的主要因素之一，它所導(dǎo)致的材料光學(xué)、電學(xué)性質(zhì)的不均勻性將最終損害半導(dǎo)體電子和光電子器件的性能.要想充分利用浮區(qū)結(jié)晶法的優(yōu)點(diǎn)生長(zhǎng)出高品質(zhì)的單晶，首先必須了解雜質(zhì)分布不均勻性的起源和形成，即晶體生長(zhǎng)中的雜質(zhì)分凝現(xiàn)象.晶體生長(zhǎng)過(guò)程中，生長(zhǎng)界面處的雜質(zhì)濃度分布除了受溫度場(chǎng)支配的生長(zhǎng)界面的形狀、穩(wěn)定性的影響之外，還由質(zhì)量輸運(yùn)過(guò)程決定，即通過(guò)擴(kuò)散過(guò)程和伴隨對(duì)流過(guò)程(動(dòng)量輸運(yùn)過(guò)程)實(shí)現(xiàn).擴(kuò)散過(guò)程由熔區(qū)內(nèi)的濃度梯度驅(qū)動(dòng)，而動(dòng)量輸運(yùn)過(guò)程則緊密依賴于熔區(qū)內(nèi)的流場(chǎng).

目前，人們對(duì)于單純的熱毛細(xì)對(duì)流穩(wěn)定性從實(shí)驗(yàn)[11-13]和理論[14-16]上已有較深入的研究.結(jié)果表明：隨著溫差增大，馬蘭哥尼對(duì)流將發(fā)生失穩(wěn)并出現(xiàn)熱流體波不穩(wěn)定性；進(jìn)一步增大溫差，流場(chǎng)將出現(xiàn)更為復(fù)雜的振蕩流動(dòng)，其振蕩特性與馬蘭哥尼數(shù)Ma有關(guān).在地面條件下，熱毛細(xì)對(duì)流還將和重力產(chǎn)生的自然對(duì)流耦合在一起[17-19].吳勇強(qiáng)等[18]采用2cst硅油實(shí)驗(yàn)研究了矮液橋浮力--熱毛細(xì)對(duì)流的起振及轉(zhuǎn)捩到混沌的過(guò)程，并分析了不同液橋的溫度振蕩頻率及相位變化情況.結(jié)果發(fā)現(xiàn)，液橋高徑比、體積比對(duì)起振溫差及振蕩頻率都有很大的影響.隨后，王佳等[19]對(duì)大尺寸液橋內(nèi)部的浮力--熱毛細(xì)對(duì)流流場(chǎng)結(jié)構(gòu)和流動(dòng)模式展開研究.實(shí)驗(yàn)發(fā)現(xiàn)在臨界Ma附近，流場(chǎng)內(nèi)會(huì)出現(xiàn)行波現(xiàn)象，流動(dòng)模式也會(huì)隨高徑比的變化而發(fā)生變化，繼續(xù)增大Ma，流動(dòng)會(huì)進(jìn)入混沌狀態(tài).

實(shí)際上，由于半導(dǎo)體晶體的摻雜需要，尤其是合金材料，在浮區(qū)生長(zhǎng)中，熔體中存在著很大的濃度梯度，這將引起強(qiáng)烈的宏觀分凝.表面處的濃度梯度誘導(dǎo)溶質(zhì)毛細(xì)對(duì)流，這種毛細(xì)流和由溫度梯度產(chǎn)生的熱毛細(xì)對(duì)流耦合在一起，形成復(fù)雜的流動(dòng).Lyubimova等[20-21]利用線性穩(wěn)定性方法分析了浮區(qū)晶體生長(zhǎng)中不同情況下三維熱--質(zhì)耦合對(duì)流的發(fā)展過(guò)程，獲得了非軸對(duì)稱對(duì)流隨時(shí)間演化的特性和結(jié)構(gòu)圖像.從他們得到的不同結(jié)晶速度和高徑比參數(shù)下熱馬蘭哥尼數(shù)MaT--溶質(zhì)馬蘭哥尼數(shù)MaC的穩(wěn)定性平面圖可以看出：即使是弱的溶質(zhì)對(duì)流擾動(dòng)也會(huì)引起熱毛細(xì)流動(dòng)穩(wěn)定性的大幅變化；弱的熱毛細(xì)效應(yīng)的存在同樣會(huì)對(duì)溶質(zhì)毛細(xì)流的穩(wěn)定性造成較大的影響.Minakuchi等[22]以全浮區(qū)液橋模型研究了零重力下溶質(zhì)--熱毛細(xì)耦合對(duì)流，得到結(jié)論是盡管MaC比MaT大，但是自由表面上由濃度梯度引起的軸向?qū)α鲝?qiáng)度反而比熱毛細(xì)對(duì)流小.在國(guó)內(nèi)，游仁然和胡文瑞[23]首先采用了液橋模型開展耦合熱--質(zhì)對(duì)流的研究，他們發(fā)現(xiàn)MaC對(duì)液橋中的流場(chǎng)和濃度場(chǎng)具有重要影響，但對(duì)溫度場(chǎng)分布的影響較弱.Zhou和Huai[24]對(duì)具有變界面的液橋中的熱毛細(xì)對(duì)流和溶質(zhì)毛細(xì)對(duì)流進(jìn)行了數(shù)值分析，研究發(fā)現(xiàn)，當(dāng)MaT與MaC的比值為?1時(shí)，自由表面在冷、熱兩端凸出而中間收縮.目前，這些研究的重點(diǎn)主要集中在毛細(xì)對(duì)流的特征上，在毛細(xì)對(duì)流的控制方面的研究相對(duì)較少.

由于硅、鍺等熔體具有良好的導(dǎo)電性，通過(guò)外加磁場(chǎng)可有效地抑制熔體流動(dòng)并影響熱質(zhì)傳輸，從而減少晶體的生長(zhǎng)條紋，控制晶體生長(zhǎng)的界面、雜質(zhì)的分離等.各種外部磁場(chǎng)[25-29]中，旋轉(zhuǎn)磁場(chǎng)具有低功耗的特點(diǎn)，其在導(dǎo)電熔體中產(chǎn)生洛倫茲力不僅能夠增加傳質(zhì)速率，還能增強(qiáng)熔體流動(dòng)的穩(wěn)定性，因此施加旋轉(zhuǎn)磁場(chǎng)對(duì)導(dǎo)電熔體進(jìn)行主動(dòng)控制是可行的.目前為止，旋轉(zhuǎn)磁場(chǎng)已用于Czochralski法[30-31]、液相擴(kuò)散法[32]、Bridgman法[33]及浮區(qū)法[28-29]等晶體生長(zhǎng)中，相應(yīng)的理論分析及實(shí)驗(yàn)研究已經(jīng)展開，取得了一定的成果.鑒于理論分析和實(shí)驗(yàn)研究在高溫條件下存在的困難，目前僅有少量實(shí)驗(yàn)報(bào)道溶質(zhì)毛細(xì)對(duì)流對(duì)Si-Ge浮區(qū)晶體生長(zhǎng)的不利影響[34]，利用旋轉(zhuǎn)磁場(chǎng)對(duì)Si-Ge體系中毛細(xì)對(duì)流進(jìn)行控制的研究尚未見報(bào)道.本文采用半浮區(qū)液橋模型，通過(guò)數(shù)值模擬研究SixGe1?x體系中熱質(zhì)毛細(xì)力對(duì)熔體對(duì)流的貢獻(xiàn)，對(duì)比分析各種毛細(xì)對(duì)流的特征以及旋轉(zhuǎn)磁場(chǎng)下熔體的傳熱傳質(zhì)流動(dòng)特性.

1 物理數(shù)學(xué)模型

本文計(jì)算采用了柱坐標(biāo)系(r,θ,z)下的半浮區(qū)液橋模型，物理模型如圖1所示.在零重力條件下，不考慮自由面的變形，即認(rèn)為液橋自由面為圓柱面，液橋高為L(zhǎng)，懸浮在具有相同半徑R的圓盤之間，將固液界面簡(jiǎn)化為無(wú)滑移平面，分別位于z=0和z=L處并保持溫差?T(?T=Th?Tc)不變.

假設(shè)熔體為不可壓縮的牛頓型黏性流體，表面張力作用在自由表面上，其大小可表示為溫度及濃度的線性函數(shù)[24]

式中σ0=σ(T0,C0)，γT=?σ/?T，γC=?σ/?C,C為濃度.只考慮表面張力隨溫度升高而減小，隨溶質(zhì)濃度增大而增大這一特殊情況.假設(shè)流體的其他物性不隨溫度變化而僅隨濃度變化，且與濃度呈簡(jiǎn)單的線性混合規(guī)律.施加的外部旋轉(zhuǎn)磁場(chǎng)強(qiáng)度和頻率分別為B0和ω，其分布可表示為[28-29]

式中，eθ和er分別是軸向和徑向單位矢量.

基于以上假設(shè)，外加旋轉(zhuǎn)磁場(chǎng)作用下，熔區(qū)內(nèi)磁流體動(dòng)力學(xué)方程組可表示為

式中，?是在柱坐標(biāo)系下的算符，V是速度矢量，T為熔體溫度，C為熔體中Si的質(zhì)量濃度.ρ為熔體密度，P為壓力，μ為動(dòng)力黏度，κ為導(dǎo)熱系數(shù)，cp為比定壓熱容，D為Si在Ge中的擴(kuò)散系數(shù).假設(shè)混合熔體的物性與Si濃度呈簡(jiǎn)單的線性關(guān)系，即ρ=CρSi+(1 ?C)ρGe，μ=CμSi+(1 ?C)μGe，κ=CκSi+(1?C)κGe，cp=CcpSi+(1?C)cpGe.旋轉(zhuǎn)磁場(chǎng)在熔體內(nèi)部產(chǎn)生的洛倫茲體積力可表示為[35]其中σe為電導(dǎo)率.

邊界條件和初始條件設(shè)定如下：

當(dāng)z=L時(shí)

當(dāng)z=0時(shí)

當(dāng)r=R時(shí)

當(dāng)t=0時(shí)

式中，n是自由表面的法向矢.

表1 SixGe1?x體系物性參數(shù)[34,36]及其他計(jì)算所需參數(shù)Table 1 Physical properties of the SixGe1?xmelt[34,36]and other parameters needed

2 計(jì)算方法

采用非結(jié)構(gòu)化網(wǎng)格的有限體積法對(duì)控制方程進(jìn)行空間離散，并在自由表面及交界面附近進(jìn)行局部加密，網(wǎng)格獨(dú)立性驗(yàn)證曲線如圖2所示，其對(duì)應(yīng)的網(wǎng)格量如表2所示.由圖2可見，當(dāng)網(wǎng)格數(shù)大于20萬(wàn)時(shí)，液橋內(nèi)監(jiān)測(cè)點(diǎn)P(r=R,θ=0,z=L/2)的無(wú)量綱溫度最大值變化較小，綜合考慮計(jì)算準(zhǔn)確性和計(jì)算時(shí)間，選擇40×80×80(r向×θ向×z向)的網(wǎng)格進(jìn)行數(shù)值計(jì)算.計(jì)算中采用時(shí)間步長(zhǎng)為10?3s.對(duì)動(dòng)量方程、能量方程中的對(duì)流項(xiàng)采用QUICK格式離散，擴(kuò)散項(xiàng)都采用二階中心差分離散，時(shí)間項(xiàng)采用二階隱式推進(jìn)法，壓力速度耦合采用PISO算法.算法的正確性同文獻(xiàn)[37]的結(jié)果進(jìn)行了比較.表3顯示的是當(dāng)Pr=0.01,Re=3500時(shí)，相同模型下熔體中最大速度值umax和周向最大速度值wmax同文獻(xiàn)[37]的對(duì)比結(jié)果，可以看出，兩者基本吻合.

圖2 網(wǎng)格獨(dú)立性驗(yàn)證曲線Fig.2 Grid independence veri fi cation curve

表2 網(wǎng)格量Table 2 Grid numbers with di ff erent meshes

表3 本文和文獻(xiàn)[37]的計(jì)算結(jié)果對(duì)比Table 3 Comparison of the present result and the result of Ref.[37]

3 結(jié)果與分析

3.1 無(wú)磁場(chǎng)時(shí)熔區(qū)內(nèi)毛細(xì)對(duì)流特征

零重力環(huán)境下，由重力引起的浮力對(duì)流消失，晶體生長(zhǎng)過(guò)程中熔體質(zhì)量的輸運(yùn)，主要依賴擴(kuò)散，因而可實(shí)現(xiàn)純擴(kuò)散晶體生長(zhǎng)過(guò)程以獲得較大的濃度梯度，同時(shí)又可以避免不穩(wěn)定浮力流對(duì)晶體質(zhì)量的影響.但在空間浮區(qū)法晶體生長(zhǎng)過(guò)程中，雖然克服了浮力對(duì)流的作用，但由表面張力梯度驅(qū)動(dòng)的馬蘭哥尼對(duì)流在熔區(qū)中占據(jù)主導(dǎo)地位.本文中由于所研究的SixGe1?x體系中溶質(zhì)Si濃度很大，自由表面溶質(zhì)毛細(xì)力的影響亦不容忽視.數(shù)值結(jié)果表明，當(dāng)僅考慮溶質(zhì)毛細(xì)力時(shí)，由其驅(qū)動(dòng)的對(duì)流沿自由表面從Si濃度較低的下底面流向濃度較高的上底面，在子午面內(nèi)形成左右兩個(gè)對(duì)稱的對(duì)流渦，并在液橋中心區(qū)域形成逆向回流渦，結(jié)果如圖3(a)所示.在擴(kuò)散作用下，同時(shí)受對(duì)流影響，子午面上半部Si濃度等值線(虛線所示)呈“W”型分布(由對(duì)稱性只顯示右半部)，但是，由于熔體的普朗特?cái)?shù)遠(yuǎn)小于施密特?cái)?shù)，溫度等值線仍呈平行分布(圖中左半部點(diǎn)劃線所示)，這與游仁然和胡文瑞[23]得到的關(guān)于溶質(zhì)毛細(xì)對(duì)流對(duì)溫度場(chǎng)影響較弱的結(jié)論是一致的.中截面z=L/2上速度軸對(duì)稱的分布特點(diǎn)(圖4(a))表明熔體中對(duì)流是二維定常流.

圖3 無(wú)磁場(chǎng)時(shí)子午面上流線(實(shí)線)、無(wú)量綱溫度(T*)(左邊點(diǎn)劃線)和Si濃度(右邊虛線)分布圖Fig.3 Streamline(solid line),dimensionless temperature(T*)(dash-dot line in the left)and Si concentration(dashed line in the right)distribution on the meridian plane without magnetic fi eld

熔體在熱毛細(xì)力的作用下，對(duì)流結(jié)構(gòu)明顯不同于溶質(zhì)毛細(xì)對(duì)流，由圖3(b)不難看出，雖然熱毛細(xì)對(duì)流在子午面內(nèi)也形成左右兩個(gè)對(duì)稱的對(duì)流渦，但并未在液橋中心區(qū)域形成完整的回流渦.兩種對(duì)流渦心位置也有很大的不同，溶質(zhì)毛細(xì)對(duì)流渦心在中截面下方，而熱毛細(xì)對(duì)流的渦心在中截面的上方.由于熱毛細(xì)對(duì)流比溶質(zhì)毛細(xì)對(duì)流強(qiáng)度大，所以回流速度也較大，液橋底部Si濃度較低的熔體被輸運(yùn)到液橋上方區(qū)域，在上底部附近形成較大的濃度梯度，其等值線(虛線所示)也由“W”型變成“π”型.等溫線(點(diǎn)劃線所示)在自由表面附近區(qū)域向上底面彎曲，此時(shí)熔體對(duì)流由二維軸對(duì)稱流動(dòng)轉(zhuǎn)變?yōu)槿S穩(wěn)態(tài)非軸對(duì)稱流動(dòng)，其周向波數(shù)m=4，如圖4(b)所示.

圖4 無(wú)磁場(chǎng)時(shí)中截面z=L/2速度(m/s)分布圖Fig.4 Velocity(m/s)distribution on z=L/2 section without magnetic fi eld

由圖3(c)可見當(dāng)溶質(zhì)毛細(xì)力和熱毛細(xì)力共同作用時(shí)，其液橋內(nèi)對(duì)流結(jié)構(gòu)與圖3(b)中所示熱毛細(xì)力單獨(dú)作用時(shí)的對(duì)流結(jié)構(gòu)相近，此時(shí)對(duì)流渦心較熱毛細(xì)對(duì)流的渦心略有下移，所不同的是在液橋中心部區(qū)域形成了完整的逆向回流渦.此逆向回流渦結(jié)構(gòu)在圖 3(a)所示的僅有溶質(zhì)毛細(xì)力作用下的液橋?qū)α鹘Y(jié)構(gòu)中同樣存在，因此為溶質(zhì)毛細(xì)對(duì)流結(jié)構(gòu)所特有.然而，由監(jiān)測(cè)點(diǎn)P的速度及Si濃度隨時(shí)間變化曲線 (圖 5)可以看出：僅溶質(zhì)毛細(xì)力驅(qū)動(dòng)時(shí)，P點(diǎn)的速度很小，溶質(zhì)毛細(xì)對(duì)流很弱，因此熱毛細(xì)對(duì)流的強(qiáng)度和耦合毛細(xì)對(duì)流強(qiáng)度相當(dāng).但是在溶質(zhì)毛細(xì)力和熱毛細(xì)力共同驅(qū)動(dòng)下，熔體對(duì)流呈現(xiàn)三維周期振蕩模式，由傅里葉頻譜分析可知其振蕩頻率f=0.0292Hz(τ=34.2s).溶質(zhì)毛細(xì)流對(duì)表面處濃度影響很大，從圖5(b)可見，雖然熱毛細(xì)對(duì)流的強(qiáng)度比溶質(zhì)毛細(xì)流的強(qiáng)度大，但是溶質(zhì)毛細(xì)流驅(qū)動(dòng)下P點(diǎn)的Si濃度比熱毛細(xì)流下P點(diǎn)的濃度要小很多，耦合后P點(diǎn)濃度介于兩者之間并隨時(shí)間呈周期性變化.圖6顯示了中截面z=L/2上Si濃度在一個(gè)周期內(nèi)的振蕩圖樣，其周向波數(shù)m=4，與熱毛細(xì)對(duì)流的周向波數(shù)相同.由此可見，即使是弱的溶質(zhì)對(duì)流也會(huì)引起熱毛細(xì)對(duì)流穩(wěn)定性的大幅變化，使穩(wěn)態(tài)的對(duì)流轉(zhuǎn)變?yōu)橹芷谛哉袷幜?，這一點(diǎn)可以從文獻(xiàn)[20-21]的結(jié)論得以驗(yàn)證，但與文獻(xiàn)[38]的結(jié)論不同，他們認(rèn)為當(dāng)熱毛細(xì)對(duì)流與溶質(zhì)毛細(xì)對(duì)流耦合時(shí)，對(duì)流的穩(wěn)定性反而比僅有溶質(zhì)毛細(xì)對(duì)流的穩(wěn)定性要高.

圖5 無(wú)磁場(chǎng)時(shí)監(jiān)測(cè)點(diǎn)P的速度及Si濃度隨時(shí)間變化曲線Fig.5 Curves of velocity and Si concentration of point P with time without magnetic fi eld

圖6 溶質(zhì)與熱毛細(xì)力共同作用時(shí)中截面z=L/2上Si濃度周期性振蕩圖Fig.6 Periodic oscillation of Si concentration on z=L/2 section under the e ff ect of coupled solute and thermo-capillary force without magnetic fi eld

對(duì)文獻(xiàn)[20-21,38]中截然不同的論述，我們更傾向于認(rèn)同文獻(xiàn)[20-21]的結(jié)果.就本文而言，溶質(zhì)毛細(xì)對(duì)流和熱毛細(xì)對(duì)流都是沿自由表面從下底面向上底面流動(dòng)，它們耦合時(shí)流動(dòng)是加強(qiáng)的.分析監(jiān)測(cè)點(diǎn)P的數(shù)據(jù)可知，熔體僅由溶質(zhì)毛細(xì)力驅(qū)動(dòng)時(shí)的速度V1=0.21mm/s，對(duì)應(yīng)溶質(zhì)MaC=7.4；僅由熱毛細(xì)力驅(qū)動(dòng)時(shí)的速度V2=6.75mm/s，對(duì)應(yīng)熱MaT=202.7，兩者速度之比V2/V1=32.這是由于熱毛細(xì)力比溶質(zhì)毛細(xì)力要大得多，兩者馬蘭哥尼數(shù)之比MaT/MaC=27.因而，當(dāng)兩種不同形式的對(duì)流疊加時(shí)，流動(dòng)穩(wěn)定性發(fā)生變化，速度呈現(xiàn)小幅振蕩模式.熔體內(nèi)的濃度則發(fā)生較大幅度的振蕩.前文已經(jīng)分析，溶質(zhì)熱毛細(xì)對(duì)流的渦心在熔體的下半部，所以處在中截面上監(jiān)測(cè)點(diǎn)P的濃度受下半部低濃度的熔體影響較大，C1=0.175；而純熱毛細(xì)對(duì)流的渦心在熔體的上半部，導(dǎo)致監(jiān)測(cè)點(diǎn)P受上半部影響大，C2=0.495.當(dāng)兩種對(duì)流耦合時(shí)，高、低濃度的熔體相混合，不穩(wěn)定性增加，濃度產(chǎn)生較大的振蕩，振蕩頻率和速度振蕩頻率相同，其峰谷值為0.024.

3.2 旋轉(zhuǎn)磁場(chǎng)對(duì)熔區(qū)內(nèi)毛細(xì)對(duì)流的影響

通過(guò)以上分析可知，熔區(qū)內(nèi)流場(chǎng)結(jié)構(gòu)對(duì)于溶質(zhì)Si的輸運(yùn)過(guò)程有直接而強(qiáng)烈的影響，因此均勻合理的流場(chǎng)結(jié)構(gòu)有利于改善Si濃度分布的均勻性.在熔區(qū)內(nèi)施加旋轉(zhuǎn)磁場(chǎng)，可以激發(fā)熔體運(yùn)動(dòng)產(chǎn)生強(qiáng)迫對(duì)流.旋轉(zhuǎn)磁場(chǎng)可以對(duì)熔體產(chǎn)生與磁場(chǎng)旋轉(zhuǎn)方向相同的周向攪拌作用，適當(dāng)調(diào)節(jié)旋轉(zhuǎn)磁場(chǎng)的磁場(chǎng)強(qiáng)度，可以使該強(qiáng)迫對(duì)流成為熔體內(nèi)的主導(dǎo)運(yùn)動(dòng)，從而在晶體生長(zhǎng)過(guò)程中對(duì)傳熱傳質(zhì)有較好的控制.

圖7顯示，選取的不同磁泰勒數(shù)(Ta在7500～17500之間)時(shí)子午面上的流動(dòng)狀態(tài)均呈軸對(duì)稱分布，這是由于旋轉(zhuǎn)磁場(chǎng)產(chǎn)生的洛倫茲力對(duì)熔體攪拌作用的結(jié)果.從計(jì)算結(jié)果看，其溫度等值線為一系列平行線，此時(shí)熔體內(nèi)對(duì)流結(jié)構(gòu)與無(wú)磁場(chǎng)時(shí)各毛細(xì)力驅(qū)動(dòng)的對(duì)流結(jié)構(gòu)均不同.原來(lái)在液橋中心部區(qū)域的逆向回流渦被擠壓到液橋下部，并且隨著磁場(chǎng)的增加，因回流而削弱的表面毛細(xì)對(duì)流的程度更大，這樣在子午面上的4個(gè)頂點(diǎn)附近各形成一個(gè)對(duì)流渦胞，并且上下兩對(duì)對(duì)流渦胞關(guān)于z軸對(duì)稱.在對(duì)流作用下，除表面附近外，熔體中大部分區(qū)域Si濃度的等值線走向基本和流線一致，這樣，液橋上部分Si濃度較大的熔體向中截面方向流動(dòng)，液橋下部分Si濃度較低的熔體也向中截面方向流動(dòng)，因此在中截面附近形成較大的濃度梯度.

圖7 旋轉(zhuǎn)磁場(chǎng)下子午面上流線(實(shí)線)、無(wú)量綱溫度(T*)(左邊點(diǎn)劃線)和Si濃度(右邊虛線)分布圖Fig.7 Streamline(solid line),dimensionless temperature(T*)(dash-dot line in the left)and Si concentration(dashed line in the right)distribution on the meridian plane with rotating magnetic fi eld

圖8 不同磁泰勒數(shù)下中截面z=L/2速度(m/s)等值線圖Fig.8 Velocity(m/s)contour lines on z=L/2 section under di ff erent magnetic Taylor numbers

圖9 不同磁泰勒數(shù)下監(jiān)測(cè)點(diǎn)P的速度及Si濃度隨時(shí)間變化曲線Fig.9 Curves of velocity and Si concentration of point P with time under di ff erent magnetic Taylor numbers

圖8速度分布顯示周向波被完全抑制，在旋轉(zhuǎn)磁場(chǎng)洛倫茲力的攪拌下，周向速度關(guān)于液橋的中心軸對(duì)稱，其大小沿徑向向外增大并在自由表面處達(dá)到最大值.圖9顯示監(jiān)測(cè)點(diǎn)P的速度及濃度振蕩特征已經(jīng)消失，由此可見，施加3mT(對(duì)應(yīng)Ta=7500)的旋轉(zhuǎn)磁場(chǎng)所產(chǎn)生的強(qiáng)迫對(duì)流已經(jīng)成為熔體內(nèi)的主導(dǎo)運(yùn)動(dòng)，從而熔體由三維振蕩流轉(zhuǎn)變?yōu)槎S軸對(duì)稱流動(dòng)；增大磁場(chǎng)強(qiáng)度，監(jiān)測(cè)點(diǎn)P的速度隨磁泰勒數(shù)近似線性增大.當(dāng)B0=7mT(Ta=17500)時(shí)，測(cè)得P的速度為9cm/s，這和Dold等[28]施加7.5mT旋轉(zhuǎn)磁場(chǎng)生長(zhǎng)硅單晶時(shí)得到的實(shí)驗(yàn)數(shù)據(jù)一致，只是由于SixGe1?x熔體的物性參數(shù)不同于Si熔體的物性參數(shù)，這一速度略小于文獻(xiàn)[28]得到的速度值11cm/s.雖然監(jiān)測(cè)點(diǎn)P點(diǎn)的速度變化很大，但是該處的Si濃度基本相同，并且和純熱毛細(xì)流下P點(diǎn)的濃度差別不大.

圖10顯示中截面z=L/2上Si濃度也呈中心對(duì)稱分布，濃度等值線呈環(huán)狀結(jié)構(gòu)，Si濃度周向分布的不均勻性被完全消除，表明旋轉(zhuǎn)磁場(chǎng)的作用可使?jié)舛瘸尸F(xiàn)軸對(duì)稱分布的特點(diǎn).在自由表面附近，由于表面張力流的作用，使得下方的Si濃度低的熔體向上方輸運(yùn)，所以中截面上的邊緣處Si濃度最小.從圖11中截面上Si濃度隨徑向變化曲線可以看出，濃度從中心處沿徑向向外逐漸降低，但在r/R=0.95處出現(xiàn)一個(gè)峰值.結(jié)合圖7不難發(fā)現(xiàn)，由于受到液橋下部逆向回流渦的擠壓，毛細(xì)對(duì)流渦胞的流線在r/R=0.90附近突然向下，然后出現(xiàn)一個(gè)峰值.在峰值兩側(cè)，軸向速度的方向經(jīng)歷了突變，沿半徑增加的方向，軸向速度由原來(lái)的負(fù)向變?yōu)檎?，原本中截面上方Si濃度高的熔體向下輸運(yùn)變?yōu)橄路絊i濃度低的熔體向上輸運(yùn)，因此在中截面上r/R=0.95處Si濃度出現(xiàn)一個(gè)峰值.對(duì)比不同磁泰勒數(shù)下中截面上Si濃度沿徑向變化曲線，我們發(fā)現(xiàn)Ta=17500時(shí)的濃度曲線較為平坦些，因此，針對(duì)文中設(shè)定的晶體生長(zhǎng)條件，施加7mT(Ta=17500)的旋轉(zhuǎn)磁場(chǎng)可以對(duì)傳質(zhì)進(jìn)行更好的控制.

圖10 不同磁泰勒數(shù)下中截面z=L/2上Si濃度分布Fig.10 Si concentration contour lines on z=L/2 section under di ff erent magnetic Taylor numbers

圖11 不同磁泰勒數(shù)下中截面z=L/2上Si濃度隨徑向變化曲線Fig.11 Si concentration on z=L/2 section versus radial direction curves under di ff erent magnetic Taylor numbers

4 結(jié)論

本文采用半浮區(qū)液橋模型，用數(shù)值方法研究了溶質(zhì)毛細(xì)對(duì)流及熱毛細(xì)對(duì)流的流動(dòng)特征，并分析了旋轉(zhuǎn)磁場(chǎng)對(duì)耦合溶質(zhì)--熱毛細(xì)對(duì)流流場(chǎng)及濃度場(chǎng)的影響.得到以下結(jié)論：

(1)僅受溶質(zhì)毛細(xì)力作用時(shí)，熔體在自由表面附近形成低濃度區(qū)域，熔體中部區(qū)域濃度由擴(kuò)散和對(duì)流共同支配，子午面上濃度等值線呈“W”型；溫度場(chǎng)主要由擴(kuò)散作用決定，呈軸對(duì)稱分布；溶質(zhì)毛細(xì)對(duì)流為二維軸對(duì)稱流.

(2)在純熱毛細(xì)力作用下，熔體內(nèi)濃度場(chǎng)受熱毛細(xì)對(duì)流的影響很大，濃度分布與熔體內(nèi)對(duì)流流向相關(guān)，子午面上濃度等值線呈“π”型分布；溫度等值線在自由表面附近區(qū)域向上底面彎曲，熱毛細(xì)對(duì)流呈現(xiàn)三維穩(wěn)態(tài)非軸對(duì)稱流動(dòng)，其周向波數(shù)m=4.

(3)耦合溶質(zhì)--熱毛細(xì)對(duì)流為三維旋轉(zhuǎn)振蕩流，振蕩頻率f=0.0292Hz，其周向波數(shù)與熱毛細(xì)對(duì)流的周向波數(shù)相同，m=4.施加旋轉(zhuǎn)磁場(chǎng)后，在洛倫茲力的攪拌下，熔體周向速度沿徑向向外增大.在選取的3種不同磁泰勒數(shù)(Ta在7500～17500之間)旋轉(zhuǎn)磁場(chǎng)作用下，熔體內(nèi)的周向波均被完全抑制，濃度場(chǎng)和流場(chǎng)均呈現(xiàn)二維軸對(duì)稱分布.因此，施加旋轉(zhuǎn)磁場(chǎng)有助于熔體流動(dòng)的穩(wěn)定性和濃度分布、溫度分布的均勻性，有利于溶質(zhì)具有周向?qū)ΨQ分布特點(diǎn)的合金晶體的生長(zhǎng).

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INSTABILITY OF COUPLED THERMO-SOLUTE CAPILLARY CONVECTION IN LIQUID BRIDGE AND CONTROL BY ROTATING MAGNETIC FIELD1)

Zou Yong*,?Zhu Guiping*Li Lai*Huang Hulin*,2)
*(College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing210016，China)
?(School of Mathematics and Physics，Anhui University of Technology，Ma’anshan243032，Anhui，China)

Floating zone method is an important technology for growth of high-integrity and high-uniformity single crystal materials due to its free of crucible contamination.However,the capillary convection in the melt brings a great challenge to the fl oating zone crystal growth.This is because the instability of convection will cause the formation of some crystal defects such as microscopic imperfections and macroscopic stripes.Therefore,it is very important to investigate the behaviors of the capillary fl ow and control its instability in order to improve the quality of the produced single crystal materials.In this paper,numerical simulations are performed to investigate all kind of the capillary convection in the half floating liquid bridge on the SixGe1?xsystem.And the impact of the external rotating magnetic fi eld is also investigated on the stability of capillary convection.The results show that the purely solute capillary convection is a two-dimensional axisymmetric model，and the temperature fi eld is mainly determined by thermal di ff usion while the concentration fi eld is dominated by convection and solute di ff usion together.On the other hand,the purely thermo-capillary convection presents three-dimensional unsteady axisymmetric fl ow.The concentration distribution is closely related to the fl ow direction of thermo-capillary convection.The isotherms bend in the region with strong convection.The coupled solute and thermocapillary convection is a three-dimensional periodic rotating oscillatory fl ow.When the rotating magnetic fi eld is applied,the circumferential velocity of the melt increases with increasing radius.Both the concentration fi eld and the fl ow fi eld in the melt show a two-dimensional axisymmetric distribution.

fl oating zone，capillary convection，surface tension，numerical simulation，rotating magnetic fi eld

O363.2,O782+.6

A doi：10.6052/0459-1879-17-102

2017–03–27 收稿，2017–09–26 錄用,2017–09–26 網(wǎng)絡(luò)版發(fā)表.

1)國(guó)家自然科學(xué)基金(51276089)、江蘇省“六大人才高峰”(2015-XNY-003)和安徽工業(yè)大學(xué)青年基金(QZ201517)資助項(xiàng)目.

2)黃護(hù)林，教授，博士，主要研究方向：磁流體流動(dòng)與傳熱.E-mail:hlhuang@nuaa.edu.cn

鄒勇,朱桂平,李來(lái),黃護(hù)林.液橋內(nèi)熱質(zhì)耦合對(duì)流不穩(wěn)定性及旋轉(zhuǎn)磁場(chǎng)法控制.力學(xué)學(xué)報(bào),2017,49(6):1280-1289

Zou Yong,Zhu Guiping,Li Lai,Huang Hulin.Instability of coupled thermo-solute capillary convection in liquid bridge and control by rotating magnetic fi eld.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1280-1289