Simulations of electromechanical shape transformations of Au nanoparticles

Transcription

1 Phy. Statu Solidi B 252, No. 1, (2015) / DOI /b Simulation of electromechanical hae tranformation of Au nanoarticle hyica tatu olidi baic olid tate hyic Vahur Zadin *,1,2, Arkady V. Kraheninnikov 3, Flyura Djurabekova 2, and Kai Nordlund 2 1 Intelligent Material and Sytem Lab, Intitute of Technology, Tartu Univerity, Noorue 1, Tartu, Etonia 2 Helinki Intitute of Phyic and Deartment of Phyic, P.O. Box 43 (Pietari Kalmink. 2), Univerity of Helinki, Finland 3 Deartment of Alied Phyic, Aalto Univerity, P.O. Box 11100, FI Aalto, Finland Received 13 June 2014, revied 29 Setember 2014, acceted 1 October 2014 Publihed online 24 November 2014 Keyword finite element imulation, nanocale contact roblem, nanocale ize effect * Correonding author: vahur.zadin@ut.ee, Phone: þ , Fax: þ Metallic nanocrytal can exhibit intriguing roertie due to large urface ecific for low-dimenional object. Recent exerimental obervation howed an intereting hae tranformation of Au nanoarticle during imultaneou mechanical and electrical load. By alying finite element method extended to include nanoize effect, we tudy the mechanim of uch tranformation. We utilize fully couled electrothermal calculation with the nanocale correction to the electric and thermal conductivitie. The mechanical reone of the material i imulated uing the elatolatic material model while the nanocale mechanical interaction between gold and tungten article i imulated uing adheive contact modelling. We oberve that Joule heating due to high electric current increae the temerature of the nanoarticle to the value cloe to the melting oint. In combination with the mechanical tre, thi caue ignificant latic deformation within the nanoarticle, which can exlain the oberved hae modification of the latter. 1 Introduction Large urface area, characteritic to low-dimenional object can lead to intereting new effect in the nanoytem. Examle are material behaviour deendence from the ize [1, 2], increaed thermal and electrical reitance in nanoytem [3], effect of the urface tree, [4] etc. In a recent exerimental tudy, intereting mechanical behaviour of cobalt and gold nanoarticle being in contact with a tungten STM ti wa oberved during imultaneou electrical and mechanical loading [5]. If nm nanoarticle were in contact, but without electric current aing through it, no hae modification wa noticed. However, if electric current wa alied, the hae of Au nanoarticle wa ignificantly elongated forming a har ti at the lace of the contact. The detail of the exeriment are to be found elewhere [5]. Previouly, imilar exeriment were conducted for Al Cu nanoarticle [6]. In thi ytem, electromigration roce wa found to be redominant deformation mechanim. In cae of Au W, the material behaviour for Au W may not necearily follow the ame trend. To undertand the deformation mechanim, the mechanical, electrical and thermal behaviour of the ytem mut be imulated. Recently a uitable imulation aroach wa imlemented and teted [7]. However, the material behaviour wa imulated uing molecular dynamic, limiting the available time cale to ico- or nanoecond [7, 8] while the exerimental time cale are often ignificantly longer (econd) [1, 5]. To overcome thi limitation and extend the imulation time cale to comarable level with the exerimental tudie, we ue finite element method (FEM). Alication of FEM for nanocale ytem require modification of available model to include finite ize effect. Such modified FEM model wa uccefully ued in Ref. [9] to invetigate the urface effect and etimate the effective mechanical roertie of nanocomoite. Alo, in Ref. [10] it wa hown that inelatic deformation and cree can be tudied by FEM if the nanotructure are larger than 2 nm in radiu. Material deformation ma for cobalt [11] and different metal [12] ugget latic deformation or cree to be reonible for material deformation under the exected mechanical load. The aim of current work i to extend the FEM to include nanocale ize effect and aly it to the Au W contact roblem under couled electro-thermal and mechanical loading to tudy the material deformation mechanim under uch condition.

2 Original Paer Phy. Statu Solidi B 252, No. 1 (2015) Model of material and method 2.1 Simulated ytem The geometry of the imulated ytem conit of Au and W late, forming bulk material, with attached nanoarticle of the ame material a reented in Fig. 1. The hae of the cro-ection of nanoarticle wa choen emi-herical, cloe to emiellitical in the deformed hae due to the urface energy minimization and to be imilar to the hae of exerimental nanoti. The radiu of Au article i 10 nm (ee Fig. 1). Temerature-deendent electric reitivity data for gold are obtained from Ref. [13], for tungten from Ref. [14] and thermal conductivity i calculated uing Wiedemann Franz law. The elatic roertie of tungten and gold available in the Comol Multihyic Material Library [15] were ued for elatic imulation of the nanoarticle. Very oft mechanical roertie of gold caue difficultie in exerimental meaurement of the latic arameter of gold, hence, the etimation from available exerimental data in the literature were ued [16, 17]. The initial yield trength wa et to 30 MPa and iotroic tangent modulu to 200 MPa. 2.2 Electric current and heat tranort Current denity ditribution in the material i calculated uing Ohm law in the differential form: r½ðt; xþrfš ¼ 0; ð1þ where w i the otential (V) and (T, x) conductivity (S m 1 ), T temerature (K) and x rereent atial coordinate (m). In the bottom of W boundary, we ue otential zero boundary condition and in to of the Au boundary, we aly current denity o that it integral over the boundary area reult in the current I 0. Temerature ditribution in the ytem i calculated in teady tate, a thermal rocee in nanocale ytem are very fat comared to the exerimental time cale (econd): r½kðt; xþrtš ¼ Q ð2þ where k (T, x) i the thermal conductivity (W m 1 K 1 ), Q ¼ J 5w volumetric heat generation rate (W m 3 ) due to reitive loe and J current denity (A m 2 ). The material model in Au nanoarticle include the hae change by uing Figure 1 Simulated ytem and the range of nanocale ize effect. the different conductivitie for molten and olid material, available in Comol Multihyic [15]. The hae are earated by oberving the temerature in the model uing moothed Heaviide function with 300 K moothing interval (to enure numerical tability). Finally, if hae change occur, amount of heat, equal to the latent heat of fuion i removed from the material. The finite ize effect in the nanoarticle, are characterized uing Knuden number, defined a Kn ¼ l e /d [18], where l e i the mean free ath of an electron and d i the diameter of the nanoarticle. If Kn 1, the finite ize effect are negligible a the hyical dimenion of the object are much larger than the mean free ath of the electron. The correction factor F ¼ F(Kn), riing from the finite ize effect, i calculated uing reviouly ublihed code [19]: nano ¼ FðKnÞ b and k nano ¼ FðKnÞk b ; ð3þ where ubcrit nano correond to roertie of the nanoarticle and ubcrit b to the roertie of bulk. The influence of the finite ize effect cannot be underetimated. For examle, the conductivity of ub nanometer diameter nanoillar i decreaed by more than 10 time. 2.3 Elatolatic deformation model In thee calculation, we aume iotroic material. The model of a material i exreed a [20]: S ¼ C : e el ; ð4þ where S i the econd Piola Kirchoff tre (Pa), C elatic tiffne tenor and e el elatic train. Since the exerimental ytem goe through large deformation, we ue finite deformation model. To account latic deformation, we decomoe the deformation gradient tenor F multilicatively to elatic and latic art [20]: F ¼ F el F ð5þ and remove the latic deformation from the total deformation gradient F el ¼ FF 1. Finally the elatic and latic Green Lagrange train tenor are e el ¼ 1 2 FT el F el I and e ¼ 1 2 FT F I : ð6þ We ue von Mie yield criterion and the effective tre, von Mie tre, wðþ i defined a [20]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 wðþ ¼ mie ¼ devðþ : devðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 3J 2 ðþ; 2 ð7þ where J 2 i the econd deviatoric tre invariant. If von Mie tre reache yield trength, the material tart to deform latically. Dilocation interaction during the latic deformation lead to the material hardening. The

3 hyica tatu olidi b 146 V. Zadin et al.: Simulation of electromechanical hae tranformation of Au NP yield trength i a function of the latic deformation, y ¼ y (e e ), where e e i effective latic train and y (e e ) ¼ y,0 þ h (e e ) i the latic deformation-deendent yield trength of the material. The hardening behaviour of the material i catured by increaing the initial yield trength a a function of the local latic deformation of the material: h ðe e Þ¼ke e and 1 k ¼ 1 1 E tio E ; ð8þ where E i the Young modulu of the material and E tio i iotroic tangent modulu (loe of the tre train curve in latic deformation art). E tio decribe how the yield trength of the material deend from the latic deformation and lead in current imlementation to contant linear hardening of deformed volume. 2.4 Interface interaction We imulate the Au W interface by adding thin elatic layer boundary condition between the contact urface. It act a adheive contact between the article and ue harmonic ocillator otential to characterize the interaction of the material. Breaking of the contact i imulated uing nonlinear ring coefficient k it maintain contant value, until certain deformation, u max, i reached. During further deformation, the ring coefficient decreae linearly, until critical deformation u break. Then, k become zero and interaction of the material to. Contact model i alied uing Hooke law: F ¼ k u; ð9þ where F i the force, k ring coefficient and u dilacement between the nanoarticle. 2.5 Simulation In the current tudy, we do not imulate the comreion of Au and W nanoarticle, but aume that thi art of the roce ha already taken lace the imulation tarted by auming already deformed Au nanoarticle, reed to the tungten urface. Moreover, to imlify the imlementation of Eq. (9), the contact area between Au and W wa aumed to be flat at tungten ide, not comletely herical. The ytem wa allowed to relax into equilibrium configuration, followed by lifting the Au nanoarticle, by alying contant eed boundary condition (v z ¼ 0.1 nm 1, Fig. 1). At the ame time, fully couled current denity and temerature ditribution were calculated. Fixed current I 0 ¼ 1100 ma wa ued a initial value, with incremental increae (by factor 1, 2 and 3) in following imulation. Currently, the elatolatic roertie of the Au nanoarticle are temerature indeendent. Platic behaviour wa modelled only in gold nanoarticle, and the ret of the ytem wa conidered elatic (a only the rocee in the Au nanoarticle were of interet). The imulation were conducted uing the Comol Multihyic ackage [15]. The material and imulation arameter are ummarized in Table 1. Table 1 Material and imulation arameter. arameter value decrition I ma external current E tio 200 MPa iotroic tangent modulu of Au y,0 30 MPa initial yield trength of Au k 310 W m 1 K 1 thermal conductivity of Au nanoarticle at room temerature (S m 1 ) electric conductivity of Au nanoarticle at 300 K v z 0.1 nm 1 lifting eed of Au nanoarticle 3 Reult and dicuion 3.1 Mechanical deformation of the ytem The von Mie tre ditribution in the Au nanoarticle during the imulation i reented in Fig. 2. A redefined deformation and zero initial tre are aumed, the imulation tart with the initial hae relaxation of the Au naoarticle due interaction of Au and W atom, leading to initial tre ditribution (Fig. 2A). The end of thi relaxation i conidered to be tarting oint of the imulation (t ¼ 0). When the Au nanoarticle i lifted (Fig. 2B and C), it undergoe latic deformation and gradually rehae into a reviouly deformed ti. The contact area between the Au and W decreae during the deformation and neck-like tructure form at the contact due to combined effect of the latic deformation, nanoarticle hae caued tre concentration and ticking effect between Au and W. During the tretching, we regain the comreed hae and achieve elongation beyond original height. The elongation of the amle can be exlained through dilocation-mediated activitie (althoug urface diffuion may alo contribute in exerimental ytem). However, a in thee imulation mechanical roertie of the material are conidered to be temerature indeendent, we underetimate the hae change of the ti. Figure 2 Von Mie tre ditribution in the nanoarticle during the imulation. Snahot A D demontrate the rogreive deformation of the Au nanoarticle during the lifting.

4 Original Paer Phy. Statu Solidi B 252, No. 1 (2015) 147 Figure 3 Temerature ditribution in the nanoarticle during the imulation. Highet temerature i reached firt in the tungten ti due to it lowet conductive roertie. 3.2 Thermal behaviour of the nanoarticle The temerature ditribution in the ytem during the imulation i reented in Fig. 3, where colour rereent the temerature and ioline 500, C and melting temerature of gold. In figure, three nahot of the ytem evolution are reented at t ¼ 6, 6.5 and 6.64, reectively. In the lat nahot, liquid Au ha formed. The increae of the current denity needed for ufficient temerature rie i achieved through the decreae of the contact area of the nanoarticle while lifting the Au nanoti. The tungten ti act a the main ource of the heat a thermal and electric conductivity in W are lower than in Au. However, it i cooled by Au nanoarticle the maximum temerature i achieved in the tungten ti, near the interface. A the temerature of the ytem tart to rie, the electric conductivity become lower, leading to higher reitive heating. Finally, if the combined effect of increaed current denity and higher reitivity of W overcome the cooling effect of the Au nanoarticle at t 6.64, the melting oint of Au i reached and liquid hae form at the Au W interface. Further increae of the current denity lead to the exanion of the molten area into Au a the heat generation move toward gold due to the lower conductivity of the liquid hae. The temerature at the ti of gold nanoarticle during the contact imulation i reented in Fig. 4 for different alied current value. In all cae, at the beginning the temerature rie lowly but increae very fat by the end of the imulation due to combined effect of increae of temerature-deendent reitivity and decreae of contact urface area. While all alied current were ufficient to caue melting of the Au nanoarticle, ignificant difference aeared in enitivity to the contact urface area change. The lowet alied current (1100 ma, blue line in Fig. 4) caued the fat temerature rie and the udden melting of the ytem at the interface between the nanoarticle; the larget current ( ma, green line in Fig. 4) lead to the fat heating of Au already at the beginning of the imulation leading to early formation of molten hae at 2. Thee imulation clearly demontrated that alied methodology can etimate the melting of the Au nanoarticle for alied current in the ame order of magnitude a in exeriment. 4 Concluion In current tudy, a ytem of gold tungten nanoarticle i imulated, uing the FEM, extended to include nanocale ize effect. The imulation include characterization of the Joule heating caued thermal behaviour of the ytem. Joule heating due to high electric current increae the temerature of the nanoarticle cloe to the melting oint. The calculation demontrated enitivity to the temerature-deendent electrical and thermal conductivitie and treed the imortance of imlementation of the finite ize effect to the conductivitie. The mechanical interaction between the nanoarticle wa imulated uing adheive contact model, with ditancedeendent interaction roertie. The imulation demontrated that thi kind of interaction lead to the latic deformation of gold nanoarticle. During the elatolatic imulation of the ytem, a clear hae tranformation wa oberved. During the tretching, the Au nanoarticle i deformed beyond the original hae. Combination of high temerature and alied mechanical tre caue ignificant latic deformation within the nanoarticle, which can exlain the hae modification of the latter. A the latic deformation model manage to cature the deformation of the ytem, the elongation of the amle can be undertood through dilocation-mediated behaviour, even though urface diffuion and electro-migration may alo contribute to the deformation of the article oberved in the exeriment. Acknowledgement Suorted by the Tiger Univerity Program of the Information Technology Foundation for Education, Etonian Reearch Council grant PUT57 and ERDF (Euroean Regional Develoment Fund). A.V.K. acknowledge uort from the Academy of Finland through roject No Reference Figure 4 Temerature at the ti of gold nanoarticle during the imulation. Temerature increae due to decreaing contact urface area between the article. [1] G. Dehm, Prog. Mater. Sci. 54, 664 (2009). [2] Q. Yu, Z.-W. Shan, J. Li, X. Huang, L. Xiao, J. Sun, and E. Ma, Nature 463, 335 (2010). [3] D. Joell, S. H. Brongerma, and Z. T}okei, Annu. Rev. Mater. Re. 39, 231 (2009).

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