鄧世杰趙宇宏?侯華文志勤韓培德

1)(中北大學材料科學與工程學院,太原 030051)

2)(太原理工大學材料科學與工程學院,太原 030024)

高壓下Ti2AlX(X=C,N)?的結構、力學性能及熱力學性質

鄧世杰1)趙宇宏1)?侯華1)文志勤1)韓培德2)

1)(中北大學材料科學與工程學院,太原 030051)

2)(太原理工大學材料科學與工程學院,太原 030024)

(2017年3月26日收到;2017年5月8日收到修改稿)

采用基于密度泛函理論的第一性原理方法,計算研究了壓力對Ti2AlC與Ti2AlN結構、力學性能的影響.研究發(fā)現(xiàn)壓力的增大會使體系的體積比降低,Ti2AlC壓縮性較Ti2AlN好.力學性能研究發(fā)現(xiàn),壓力的增大使材料抵抗變形能力增強,體系的延展性有了很大的提升,當壓力超過40 GPa后,Ti2AlC與Ti2AlN從脆性材料轉變?yōu)檠有圆牧?體模量與剪切模量的比值達到1.75,延展性有了很大的提升.通過準諧德拜模型,分析了壓力與溫度對Ti2AlC與Ti2AlN體模量、熱容及熱膨脹系數(shù)的影響.結果表明,隨著溫度的升高,Ti2AlN與Ti2AlC的體模量下降.定容熱容與定壓熱容的變化趨勢相同,但在高溫下,定容熱容遵循Dulong-Petit極限,溫度對熱容的影響效果較壓力明顯.溫度與壓力對Ti2AlN與Ti2AlC線膨脹系數(shù)的影響主要發(fā)生在低溫區(qū)域.

Ti2AlN與Ti2AlC,力學性能,熱力學性質,第一性原理

1 引 言

Mn+1AXn相材料為三元層狀化合物,其中M代表過渡金屬元素,A代表A組元金屬元素,X代表C或者N[1-4].該材料具有金屬與陶瓷的雙重特性,既有耐高溫與耐腐蝕性能等陶瓷的性能,又具有導電、導熱及可加工性能等金屬的性能,因而受到廣泛關注.Mn+1AXn相材料由于其優(yōu)異的性能而被廣泛用作高溫結構材料、化學防腐材料或高壓發(fā)熱材料,具有廣闊的應用前景和研究價值[5,6].

近年來,對于Mn+1AXn相材料的研究有很多,其對應的材料也有很多種,如Ti2AlC,Ti2AlN和Ti3AlC2等.Xiao等[7]用第一性原理方法預測了Ti2AlC與Ti2AlN等MAX相材料的抗輻射性.Du等[8]通過準諧德拜模型計算分析Ti2AlC, Ti2Al0.5CT0.5與Ti2AlN在0 GPa,0 K條件下的熱力學性質并分析了晶格常數(shù).Manoun等[9]通過X射線測量了Ti2AlC與Ti2AlN的晶格參數(shù)變化.朱佳等[10]借助準諧德拜模型計算了Ti2AlC在零壓下的標準摩爾生成焓、標準熵等.李輝等[11]研究了高壓對Ti2AlC與Ti2AlN結構、彈性及電子性質的影響,發(fā)現(xiàn)Ti2AlC與Ti2AlN為各向異性材料,壓力對電子態(tài)密度影響較小.但是對于Ti2AlC與Ti2AlN材料在高壓下的熱力學性質的研究報道較少,而Ti2AlC與Ti2AlN總處于高溫高壓條件下,所以研究其在高壓下的熱力學性質具有非常重要的意義.本文借助第一性原理的方法,應用準諧德拜模型,研究了高壓下Ti2AlC與Ti2AlN材料的結構、力學性能及熱力學性質,為材料的深入研究提供幫助.

2 計算方法

采用第一性原理方法,使用基于密度泛函理論的CASTEP(Cambridge sequential total energy package)軟件包[12],其波函數(shù)由平面波基組展開.贗勢選取倒易空間表征中的超軟贗勢(ultrasoft)[13].原子之間的交換關聯(lián)函數(shù)選用基于廣義梯度近似(generalized gradient approximation,GGA)的PBE(Perdew-Burke-Ernzerhof)[14]進行計算.平面波截斷能為350 eV,選取布里淵區(qū)K點網(wǎng)格數(shù)為10×10×2.自洽收斂條件設為：總能量收斂值5.0×10-6eV/atom,每個原子的力低于0.01 eV/nm,公差偏移小于0.005 ? (1 ?=0.1 nm),應力偏差小于0.02 GPa.

Ti2AlC與Ti2AlN都屬于六方晶系,其空間群為P63/mmc,每一個晶胞中含有2個Ti2AlX分子,其中Ti原子占據(jù)4f位置(2/3,1/3,z),z為結構參數(shù),Al占據(jù)2d(1/3,2/3,3/4)位置,C或者N占據(jù)2a(0,0,0)位置[11].Ti2AlC與Ti2AlN結構對應的晶胞分別如圖1(a)和圖1(b)所示,其中藍色小球代表Ti原子,粉色小球代表Al原子,灰色小球代表C原子,綠色小球代表N原子.

圖1 (網(wǎng)刊彩色)(a)Ti2AlC與(b)Ti2AlN的晶胞結構Fig.1. (color online)The crystal structures of(a) Ti2AlC and(b)Ti2AlN.

3 結果與分析討論

3.1 結構性質

為了研究壓力對Ti2AlC與Ti2AlN結構的影響,分別計算0—50 GPa下各體系的體積比V/V0(V0表示零壓強下體系的體積).如圖2所示,隨著外加壓力的增大,晶胞體積比逐漸降低,表明壓力增大,原子間的距離減小.Ti2AlC與Ti2AlN體積分別坍縮了20.59%和18.93%,證明兩者都具有較強的可壓縮性.隨著壓強的增大,體積比曲線下降趨勢變得平緩,這是由于原子間排斥力隨原子間距離的減小而逐漸增大.對比曲線可以發(fā)現(xiàn),高壓下Ti2AlC壓縮性較Ti2AlN強,表明壓力對Ti2AlC晶格常數(shù)的影響大于對Ti2AlN晶格常數(shù)的影響.

圖2 Ti2AlC與Ti2AlN體積比隨壓力的變化Fig.2.The volume ratio V/V0as a function of pressure for Ti2AlX(X=C,N).

3.2 力學性質

體模量與彈性模量是表征材料性能的重要參數(shù),可以用來反映材料的力學性能.表1列舉了不同壓力下Ti2AlC與Ti2AlN材料的彈性常數(shù)Cij.可以看出,0 GPa下彈性常數(shù)的計算值與李輝等[11]計算所得的數(shù)值符合,表明計算的準確性與可行性.通過彈性常數(shù)可以判斷力學穩(wěn)定性.Ti2AlC與Ti2AlN均為六方晶系,需要滿足玻恩穩(wěn)定準則[15],其力學穩(wěn)定條件為

計算所得數(shù)據(jù)均滿足力學穩(wěn)定性條件,表明在0—50 GPa下Ti2AlC與Ti2AlN均為力學穩(wěn)定的材料.隨著壓強的增大,其彈性常數(shù)不斷增大,其中,C11與C33增大最為明顯,相同壓力下,Ti2AlN的C11與C33比Ti2AlC高,這是因為Ti—N鍵強于Ti—C鍵的緣故.

通過彈性常數(shù)可以計算體模量B、剪切模量G以及楊氏模量E,用來分析材料抵抗變形的能力[16].根據(jù)Pugh判據(jù)[17],體模量與剪切模量的比值B/G反映了金屬的抗變形能力,臨界值為1.75.當B/G＞1.75時,材料表現(xiàn)為延性,相反,則為脆性,B/G值越大,材料的延展性越好.從表1可以看出,Ti2AlC與Ti2AlN的變化趨勢相同,體模量與剪切模量隨著壓力的增大而增大,表明材料抵抗變形能力增強.體模量反映的是抵抗鍵長的能力,而剪切模量反映的是材料抵抗鍵角的能力,體模量的增加幅度大于剪切模量,所以材料的延性得到提升.兩種材料在0 GPa下都表現(xiàn)為脆性,當壓力增大到40 GPa以后,Ti2AlC與Ti2AlN的B/G值超過了1.75,由脆性材料轉變?yōu)檠有圆牧?楊氏模量變化趨勢與體模量和剪切模量相同,隨壓力的增大不斷增大,材料剛性提升,抵抗變形能力提升.

表1 不同壓力下Ti2AlC與Ti2AlN的彈性常數(shù)Cij、體模量B、剪切模量G與楊氏模量ETable 1.The elastic constants,bulk modulus,shear modulus and Young’s modulus of Ti2AlC and Ti2AlN at various pressures.

圖3 Ti2AlC與Ti2AlN體模量隨壓力和溫度的變化Fig.3.The pressure and temperature dependences of bulk modulus for Ti2AlX(X=C,N).

3.3 熱力學性質

依據(jù)準諧德拜模型[18,19],我們研究分析了Ti2AlC與Ti2AlN的熱力學性質,包括體模量、熱容以及熱膨脹系數(shù).

圖3所示為體模量隨壓力和溫度的變化,0 GPa,0 K下Ti2AlC與Ti2AlN體模量隨溫度的變化與文獻[8]的計算結果相符合.從圖中可以看出,體模量變化趨勢相同,Ti2AlN抗變形能力強于Ti2AlC.在0—100 K間,體模量近似為一個常數(shù),但當溫度超過100 K后,體模量直線下降,表明材料抵抗變形能力逐漸減弱.溫度固定時,體模量隨著壓力的增大而增大.

圖4(網(wǎng)刊彩色)Ti2AlC與Ti2AlN(a)定容熱容CV與(b)定壓熱容CP隨壓力和溫度的變化Fig.4.(color online)The pressure and temperature dependences of(a)CVand(b)CPfor Ti2AlX(X= C,N).

圖4所示為熱容隨溫度與壓力的變化.熱容是熱力學中又一個重要的參數(shù),包括定容熱容CV和定壓熱容CP.定容熱容和定壓熱容變化曲線大致相同,當溫度低于300 K時,曲線直線上升,當溫度超過300 K后,曲線上升趨勢變緩.對于定容熱容,在高溫下,曲線趨近于平行于x軸的直線,這是因為高溫下CV遵循Dulong-Petit極限[20],而CP不受限制,反而繼續(xù)增加.壓力對Ti2AlC與Ti2AlN材料熱容的影響效果與溫度相反,壓力增加,材料的熱容反而下降,且壓力對其影響較溫度小.高溫下Ti2AlC的CV高于Ti2AlN,說明Ti2AlC吸收或者釋放熱量的能力更強.

熱膨脹系數(shù)隨壓力和溫度的變化如圖5所示,可以看出Ti2AlX(X=C,N)熱膨脹系數(shù)的變化趨勢大致相同.隨著溫度的升高,線膨脹系數(shù)增大,當溫度大于400 K后,增長速度下降,曲線變得平緩,表明溫度與壓力對線膨脹系數(shù)的影響主要發(fā)生在低溫區(qū).當壓力超過30 GPa后,壓力對線膨脹系數(shù)的影響較小.

圖5 (a)Ti2AlC與(b)Ti2AlN熱膨脹系數(shù)隨壓力和溫度的變化Fig.5.The pressure and temperature dependences of thermal expansion coefficient for(a)Ti2AlC and(b) Ti2AlN.

4 結 論

利用第一性原理計算方法計算了0—50 GPa下壓力對Ti2AlX(X=C,N)結構及力學性能的影響.隨著壓力的增大,Ti2AlX(X=C,N)體積比下降,壓力對Ti2AlC影響較Ti2AlN大.通過彈性模量的計算,發(fā)現(xiàn)壓力可以增強材料抵抗變形的能力,隨著壓力的增大,材料的延性得到提升,Ti2AlN抵抗變形能力強于Ti2AlC,當壓力超過40 GPa后,Ti2AlX材料由脆性轉變?yōu)檠有?通過準諧德拜模型,分別研究了壓力與溫度對Ti2AlX (X=C,N)體模量、熱容及熱膨脹系數(shù)的影響,發(fā)現(xiàn)體模量隨溫度的升高而減小,但隨壓力的增大而增大.定容熱容與定壓熱容的變化趨勢相同,高溫下定容熱容增加緩慢,且遵循Dulong-Petit極限, Ti2AlC的定容熱容在高溫下高于Ti2AlN.溫度與壓力對線膨脹系數(shù)的影響主要發(fā)生在低溫區(qū)域,壓力超過30 GPa后,壓力對線膨脹系數(shù)的影響較小.

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PACS:61.50.Ah,62.20.fk,62.20.de DOI:10.7498/aps.66.146101

Structural,mechanical,and thermodynamic properties of Ti2AlX(X=C,N)at high pressure?

Deng Shi-Jie1)Zhao Yu-Hong1)?Hou Hua1)Wen Zhi-Qin1)Han Pei-De2)
1)(School of Materials Science and Engineering,North University of China,Taiyuan 030051,China)
2)(School of Materials Science and Engineering,Taiyuan University of Technology,Taiyuan 030024,China)

26 March 2017;revised manuscript

8 May 2017)

The MAX phase has attracted much attention due to its unique properties combined with the merits of both metal and ceramic,including the low density,high electrical conductivity and good oxidation resistance,which makes it signi fi cant for possible applications in various high temperature or other environments.There is a lot of research work on Ti2AlX(X=C,N).However little research about thermodynamic properties at high pressure is carried out.So we study the structural,mechanical and thermodynamic properties of Ti2AlC and Ti2AlN at various pressures and temperatures.

The fi rst-principles calculations based on electronic density-functional theory framework are used to investigate the properties at various pressures.The cut-o ffenergy is 350 eV.Converged results are achieved with 10×10×2 special K-point meshes.The self-consistent convergence of total energy is set to be 5.0×10-6eV/atom.

According to the calculated structural parameters at various pressures,we can fi nd that the ratios V/V0(V0denotes the system volume at 0 GPa)of Ti2AlX are reduced by 20.59%and 18.93%,respectively,so the compressibility of the system is strong.As the internal pressure increases,the curves of V/V0become gentle.Then we calculate elastic constants at pressures ranging from 0 to 50 GPa in steps of 10 GPa.It is obvious that the Ti2AlX is mechanically stable because all of the elastic constants satisfy the Born stability criteria.The bulk modulus,shear modulus and Young’s modulus linearly increase with internal pressure increasing,implying that the pressure can improve the resistance to volume deformation.The ductility and brittleness can be judged according to Pugh’s criterion(ratio of bulk modulus to shear modulus B/G),and the brittle nature turns into ductile nature in a pressure range of 40-50 GPa for the Ti2AlX since the value of B/G exceeds 1.75.Finally,we study the thermodynamic properties at various pressures and temperatures based on the quasi-harmonic Debye approximation theory,including the bulk modulus,heat capacity and thermal expansion coefficient.The bulk modulus decreases with temperature increasing but increases with pressure increasing.The heat capacity at constant volume Cvand the heat capacity at constant pressure Cphave the same variation tendency,while Cvobeys the Dulong-Petit limit.It is easy to see that temperature and pressure have opposite in fl uences on heat capacity and the e ff ect of temperature is more signi fi cant than that of pressure.The e ff ects of temperature and pressure on linear expansion coefficient mainly occur at low temperature and the e ff ect of pressure is not so considerable when the pressure exceeds 30 GPa.Above all,the e ff ects of temperature and pressure on thermodynamic properties are inverse.

Ti2AlN and Ti2AlC,mechanical properties,thermodynamic properties, fi rst-principles

:61.50.Ah,62.20.fk,62.20.de

10.7498/aps.66.146101

?國家自然科學基金(批準號：U1610123,51674226,51574207,51574206,51274175)資助的課題.

?通信作者.E-mail：zhaoyuhong@nuc.edu.cn

?2017中國物理學會Chinese Physical Society

http://wulixb.iphy.ac.cn

*Project supported by the National Natural Science Foundation of China(Grant Nos.U1610123,51674226,51574207, 51574206,51274175).

?Corresponding author.E-mail：zhaoyuhong@nuc.edu.cn