Size, shape and temperature effect on nanomaterials

Idia Joural of Pure & Applied Physics Vol. 53, November 2015, pp. 768-775 Size, shape ad temperature effect o aomaterials G Sharma, S Bhatt, R Kumar & M Kumar* Departmet of Physics, G.B. Pat Uiversity of Agriculture ad Techology, Patagar 263 145, Idia E-mail: muish_dixit@yahoo.com Received 22 March 2014; revised 18 July 2014; accepted 12 August 2015 A simple theoretical model is developed to study the size, shape ad temperature effect o aomaterials. The temperature depedece of thermal expasio ad Youg modulus of differet type (shape) of aomaterials viz. spherical, aowire ad aofilm has bee studied. Size depedece of Youg modulus i two differet regios of temperatures viz. below Debye temperature ad above Debye temperature has bee ivestigated for differet shape of aomaterials. The thermal ad elastic behaviour is related to vibratioal frequecy of aomaterials. Therefore, the size depedece of vibratioal frequecy of differet shape of aomaterials has bee computed. The model predictios are compared with the earlier theoretical ad available experimetal data. A good agreemet supports the validity of the model proposed. Keywords: Youg Modulus, Thermal expasio, Vibratioal frequecy 1 Itroductio Naomaterials are of curret research iterest due to their wide applicatios i sciece ad techology. The stability of aoscale devices is related to the elasticity ad its temperature depedece. For example, the elastic istability iduced by risig the temperature may cause the failure of itegrated circuits as well as other microelectroic devices. I additio to this thermal stability of aomaterial, is of both scietific ad techological iterest because i most of sythesis techiques, the aomaterials are provided i powder form, which have to be cosolidated to obtai dese material for potetial applicatios. The thermal expasio is the fudametal property of aomaterials which directly relates to applicatios. The experimetal measuremets show that the thermal expasio coefficiets of aocrystallie materials are larger tha those of their polycrystallie couterparts which deped o temperature ad iflueced by the effect of size. Lu ad Sui 1 reported that the average liear thermal expasio coefficiets of aocrystallie Ni-P icrease with decrease of grai size. Zhag ad Mitchell 2 studied aocrystallie Se ad foud the similar tred of variatio. Yag et al 3. studied the size depedet root mea square amplitude model. It has bee discussed that model ca predict v (D = 40 m) value accurately i compariso with the experimetal results, while a big divergece betwee them is foud for v (D = 16 m). Moreover, they cocluded that v (D) icreases with decreasig D. Xu et al 4. studied the thermal expasio of as-prepared ad aealed silver aowires embedded i aodic lumia membrae with differet diameters up to 800 K. For both as-prepared ad aealed samples, the coefficiets of thermal expasio have V shape chage as the diameter icreases ad miimum values of the coefficiet of thermal expasio do ot correspod to the same diameters of aowires. Zhao ad Jiag 5 exteded the use of classical thermodyamics to aoscale ad studied the size effect o thermal properties of low dimesioal materials. Kumar et al 6. proposed a simple theoretical model for the size depedece of v of aomaterials. Thus, it seems that some efforts have bee made to study v with varyig size or temperature. Moreover, a simple theoretical model, which icludes the effect of size ad temperature for differet type (shape) of aomaterials, has ot yet bee reported due to a difficult task. I the preset paper, a effort has bee made to achieve this task. Durig thermal expasio, the elastic behaviour plays a importat role. To uderstad elasticity, Youg modulus is oe of the importat elastic costats. It is well kow that whe material dimesios approach aoscale, a sigificat variatio of mechaical properties ca be observed comparig with their bulk couterpart owig to the high surface-to-volume ratio. Cueot et al 7. reported that the Youg modulus of Ag ad Pb aowires

SHARMA et al.: SIZE, SHAPE AND TEMPERATURE EFFECT ON NANOMATERIALS 769 icreases dramatically with decreasig diameters. However, ivestigatios o Cr ad Si aocatilivers showed that the moduli sharply decrease with decreasig diameter 8,9. O the other had, the results reported by Wog et al 10. ad Wu et al 11. showed that the moduli of SiC aorod ad Au aowire are essetially idepedet of the diameter. Besides experimetal ivestigatios, theoretical aalysis has also bee employed to study the mechaical behaviour of aomaterials, which icludes the atomistic simulatio as reviewed by Cao ad Che 12. Koh et al 13. reported that the Youg modulus of Pt ad Au aowires icreases with diameters util they approach to that of the bulk value. The size depedece of elastic properties has bee foud to be relevat to both the film surface crystallographic orietatio ad loadig directio 12,13. The atomic structure aalysis ad eergy calculatios have bee employed to idetify the mechaism of the size depedet elastic properties, uder differet loadig directios ad surface orietatios. Upo small axial deformatio, the relatioship betwee itralayer ad iterlayer bod legth variatio ad film elastic stiffess has bee established. It has bee foud that the atomic layers with larger bod legth variatio have higher elastic stiffess. Bhatt ad Kumar 14 proposed a simple theoretical model to study the size ad orietatio depedece of elasticity of aowires ad aofilms. These authors 14 studied the Youg modulus of aowire ad aofilms. The results obtaied are compared with the available experimetal data ad computer simulatio studies. The theory has bee foud to predict a good agreemet with the experimetal data. Thus, it seems that some theoretical as well as experimetal efforts have bee made to study the size ad orietatio depedece of Youg modulus. Moreover, there is a urget eed to uderstad the effect of temperature ad shape also. I the preset work, we develop simple theoretical formulatio is developed, which icludes the effect of temperature ad shape i additio to the size o differet properties of aomaterials. 2 Theoretical Formulatio To study the temperature depedece of thermal expasio of aomaterials, Kumar et al 15. developed the followig relatio: α [1 α δ 0 T ( T T0 )] α 0 1 (1) where is the coefficiet of volume thermal expasio, T the Aderso-Grüeise parameter, T the temperature ad 0 refers to their iitial value. I Eq. (1), 0 is the iitial value of volume thermal expasio of aomaterials, which we write for simplicity. is the parameter, which depeds o size ad shape of aomaterials ad may be writte as 16 : N α = αb1 2 (2) where b is the coefficiet of volume thermal expasio of bulk material. N is the umber of surface atoms ad is the total umber of atoms of aosolid. The surface atoms refer to the first layer of aosolid. The values of N/2 deped o the size ad shape of the aomaterials. The method to fid N/2 for differet shape of aomaterials has already bee discussed by Qi 17. Accordig to this model N/2 is 2d/D for spherical aosolid with d as the diameter of atom ad D the diameter of spherical aosolid. For aowire, N/2 is 4d/3L where L is the diameter of aowire. For aofilm, N/2 is 2d/3h, where h is the height (size) of aofilm. Thus, we ca write Eq. (1) for differet type (shape) of aomaterials: aosolid: α 2d = 1 αbδt1 ( T T0 ) α0 D : α 4d = 1 αbδt1 ( T T0 ) α0 3L : α 2d = 1 αbδt1 ( T T0 ) α0 3h (3) (4) (5) Eqs (3-5) give the size ad temperature depedece of the coefficiet of volume thermal expasio of differet type (shape) of aomaterials. To study elasticity, the Youg modulus is oe of the importat elastic moduli. For aomaterials, it has bee discussed that thermal ad elastic behaviour is iverse to each other 16. The temperature depedece of Youg modulus is writte as follows:

770 INDIAN J PURE & APPL PHYS, VOL 53, NOVEMBER 2015 Y [1 ( T T )] 0 T 0 Y = α δ (6) 0 where 0 is size ad shape depedet parameter as described above. Therefore, the relatios of temperature depedece of Youg modulus of differet type (shape) of aomaterials is writte as follows:. aosolid: Y 2d = 1 αbδt1 ( T T0 ) Y0 D : Y 4d = 1 αbδt1 ( T T0 ) Y0 3L : Y 2d = 1 αbδt1 ( T T0 ) Y0 3h (7) (8) (9) The elastic ad vibratioal characteristic of aomaterials directly determies the stability ad reliability of the devices. Therefore, i additio to the elasticity, it is very importat to uderstad the vibratioal mechaism of aomaterials. For this purpose, Kumar et al 18. reported the followig relatio for size depedece of frequecy: N v = vb1 2 1/2 (10) where ad b are vibratioal frequecy of ao ad bulk materials. N/2 depeds o size ad shape of aomaterials. Eq. (10) has bee foud to be quite satisfactory for the aomaterials, which follow the Iverse Hall Petch Effect (IHPE). Actually, there are the aomaterials for which bulk modulus decreases by decreasig the size viz. they become soft whe size is decreased. O the other had, there are the aomaterials, which follow the Hall Petch Effect (HPE) viz. they become hard whe size is decreased. Therefore, the size depedet of vibratioal frequecy for materials havig HPE is writte as: N v = vb1 2 /2 (11) Thus, the relatios for differet type (shape) of aomaterials may be writte as: aosolid: v 2d = vb1 D : v 4d = vb1 3 L : v 2d = vb1 3 h /2 /2 /2 (12) (13) (14) I the preset paper, we used Eqs(12-14) to study the size ad shape depedet vibratioal frequecy of aomaterials. 3 Results ad Discussio The temperature depedece of the coefficiet of thermal expasio ad Youg modulus for Cu aofilm (2.2m) usig Eqs (5) ad (9) has bee computed, respectively. The results obtaied are show i Fig. 1 alog with the results reported by Liag et al 19. It should be metioed that Liag et al 19. performed the molecular dyamics (MD) simulatios of the biaxial tesio of Cu thi films at differet temperatures that were carried out to validate the temperature effect o the elastic modulus. The structures of Cu thi film with thickess 12 ad 2.2 m have bee established with x, y ad z beig the [100], [010] ad [001] directios, respectively. 1.3 0.9 0.8 Fig. 1 Temperature depedece of / 0 usig Eq. (5) ad usig Eq. (9) of Cu aofilm (2.2 m), represet the results reported by Liag et al 19. usig M D simulatio

SHARMA et al.: SIZE, SHAPE AND TEMPERATURE EFFECT ON NANOMATERIALS 771 The periodic boudary coditio has bee set i the x ad y directios ad z directio is free ad embedded atom method potetial is used. The simulatios have bee performed at the several temperatures i the rage 300-800 K. The stress-strai curves ad thus the biaxial modulus have bee obtaied at differet temperatures. We have thus selected Cu aofilm (2.2 m) because of the fact that some results are available so that compariso ca be made. The coefficiet of volume thermal expasio is foud to icrease with icreasig the temperature whilst Youg modulus decreases with icreasig temperature. Our computed results are foud to preset to a good agreemet with the available data 19. To iclude the effect of shape we have icluded spherical aosolid ad aowire also ad compared the results. Eqs (3-5) are used to study / 0 ad Eqs (7-9) for to compute their temperature depedece. The results obtaied are show i Fig. 2. It is foud that / 0 icreases with icreasig the temperature for all type of aomaterials cosidered i the preset work viz. spherical, aowire ad aofilm of Cu (2.2 m). Moreover, the value of / 0 is foud to decrease from spherical to aowire ad aofilm. O the other had, Youg modulus is foud to decrease by icreasig the temperature. For differet shape, the reverse effect is foud as observed i the / 0. To cofirm the model predictios we have also icluded the effect of size ad repeated our computatioal work for Cu (12 m). The results obtaied for the temperature depedece of / 0 ad are show i Figs 3 ad 4 alog with the results reported by Liag et al 19., which are available for temperature depedece of of Cu (12 m) aofilm. These are show i Fig. 3. It is foud that model predictios are i good agreemet with the results reported by Liag et al 19. A comparative study is show i Fig. 4. A similar tred as obtaied i Cu (2.2 m), is observed. However, the model predicts that icreasig the size, the shape has less effect. To cofirm the model predictios, the applicatio of our model is exteded by chagig the material. For this purpose, we have selected Si (2.2 ad 10.9 m) because of the availability of MD simulatio results 19 so that model predictios may be judged. The results are show i Figs 5-8. The model is foud to give the similar treds i good agreemet with the available MD simulatio results 19. The origi of size ad temperature depedet of Youg modulus has bee studied by Ao et al 20. by cosiderig the size effects o surface bod cotractio ad meltig temperature variatio. The results show that Youg modulus decreases with a shrikig disparity betwee meltig temperature ad materials applicatio temperature, while surface bod 0.9 0.8 Fig. 3 Temperature depedece of / 0 usig Eq. (5) ad usig Eq. (9) of Cu aofilm (12 m), represet the results reported by Liag et al 19. usig MD simulatio 1.3 1.3 0.9 0.8 0.9 0.7 Fig. 2 Shape ad temperature depedece of / 0 usig Eqs (3-5) ad usig Eqs (7-9) of Cu (2.2 m) 0.8 Fig. 4 Shape ad temperature depedece of / 0 usig Eqs (3-5) ad usig Eqs (7-9) of Cu (12 m)

772 INDIAN J PURE & APPL PHYS, VOL 53, NOVEMBER 2015 cotractio results i icrease i Youg modulus with reductio i size. It has also bee discussed 20 that Youg modulus of aowire ca be cotrolled by maipulatig the size ad applicatio temperature. 6 4 2 0 0.98 0.96 0.94 Fig. 5 Temperature depedece of / 0 usig Eq. (5) ad usig Eq. (9) of Si aofilm (2.2 m), represet the results reported by Liag et al 19. usig MD simulatio 8 5 2 0.99 0.96 0.93 Fig. 6 Shape ad temperature depedece of / 0 usig Eqs (3-5) ad usig Eqs (7-9) of Si (2.2 m) 6 Thus, the effect of size icludig the effect of temperature is a iterestig ad importat property of aomaterials. I the preset paper, the applicatio of our model is exteded to compute the size depedet of Youg modulus at differet temperature. Two differet regios have bee ivestigated viz. whe the applicatio temperature is more tha the Debye temperature (T D ) ad less tha T D. We have selected differet aowires because of the fact that some earlier results 20, 21 are available so that the model predictios may be compared. I additio to this, the effect of shape is also icluded (spherical aosolid ad aofilm). We used Eqs (7-9) is used to study the size ad shape depedet of Cu, Ag, Au ad Al. I Eqs (7-9), T 0 is the referece temperature viz. Debye temperature of the material is cosidered. I the preset paper, the size ad shape depedet of Youg modulus are computed at two temperatures viz. the temperature which is less tha the Debye temperature, ad higher tha the Debye temperature. At the temperature less tha Debye temperature, Youg modulus is foud to icrease by decreasig the size of aomaterial. Moreover, a reverse tred is foud at the temperature greater tha the Debye temperature. The results obtaied for Cu are show i Figs 9 ad 10. At T = 0 K, the theoretical results reported by Ao et al 20. for compariso purpose, are icluded. Our results show the similar tred of variatio as reported by Ao et al 20. At T=300 K, our model predictios are compared with the simulatio results 21. It is observed that our results are foud to be i good agreemet with the computer simulatio studies 21 ad earlier theoretical predictios. It is very iterestig ad importat to metio here that preset model is very simple as compared with earlier ivestigatios 20,21 ad gives good results. To demostrate the 4 2 0 0.98 0.96 0.94 Fig. 7 Temperature depedece of / 0 usig Eq. (5) ad usig Eq. (9) of Si aofilm (10.9 m), represet the results reported by Liag et al 19. usig MD simulatio 6 4 2 0 0.98 0.96 0.94 Fig. 8 Shape ad temperature depedece of / 0 usig Eqs (3-5) ad usig Eqs (7-9) of Si (10.9 m)

SHARMA et al.: SIZE, SHAPE AND TEMPERATURE EFFECT ON NANOMATERIALS 773 0.982 56 53 0.980 50 47 0.978 0.976 44 41 0.974 0.972 38 Fig. 9 Size depedece of usig Eqs (7-9) of Cu at T = 0K, represet the results reported by Ao et al 20. usig theoretical model for aowire 0.976 0.970 Fig. 12 Size depedece of usig Eqs (7-9) of Ag at T = 300 K 60 0.974 55 0.972 Y/Y 0 0.970 0.968 50 45 0.966 0.964 40 35 0.962 Fig. 10 Size depedece of usig Eqs (7-9) of Cu at T = 300 K, represet the simulatio results reported by Liag ad Upmayu 21 for aowire 30 Fig.13 Size depedece of usig Eqs (7-9) of Au at T = 0 K 0.978 75 0.975 70 65 60 55 0.972 0.969 0.966 0.963 50 0.960 45 Fig. 11 Size depedece of usig Eqs (7-9) of Ag at T = 0 K applicability of the model, we have repeated our computatioal work to some other aomaterials viz. Ag. Au ad Al. A similar tred of variatio is foud. The results obtaied are show i Figs 11-16. For the stability ad reliability of aomaterial devices, it is very importat ad iterestig to uderstad Fig. 14 Size depedece of usig Eqs (7-9) of Au at T = 300 K the vibratioal mechaism. The size depedet of vibratioal frequecy of differet type (shape) of aomaterials has bee studied. We have cosidered Si, Cu ad Ag for this purpose because of the fact that the model predictios may be compared. Eqs (12-14) are used to compute the size depedece of vibratioal

774 INDIAN J PURE & APPL PHYS, VOL 53, NOVEMBER 2015 9 0 8 7 7 4 6 5 4 3 2 1 Fig. 15 Size depedece of usig Eqs (7-9) of Al at T = 0 K ν /ν b 1 8 5 2 0.99 0 10 20 30 40 50 60 70 Fig. 18 Size depedet of / b of Cu usig Eqs (12-14), represet MD simulatio results 24 ad * represet experimetal data 25 1.5 0.970 1.4 Y/Y 0 0.965 0.960 0.955 0.950 ν /ν b 1.3 ν /ν b 0.945 Fig.16 Size depedece of usig Eqs (7-9) of Al at T = 500 K 0 6 2 8 4 0 0 5 10 15 20 25 30 Fig. 17 Size depedet of / b of Si usig Eqs (12-14), * represet results based o phoo dispersio relatio 22 ad represet results based o lattice strai ad bidig eergy 23 for aofilm frequecy. The results obtaied are show i Figs 17-19. It is foud that the vibratioal frequecy icreases by decreasig the size for the materials cosidered i the preset paper. I the case of Si aofilm, some results based o phoo dispersio relatio 22, lattice strai ad bidig eergy chage 23 are available. 1 2 3 4 5 6 Fig. 19 Size depedet of / b of Ag usig Eqs (12-14), represet the results reported by Liag et al 23. usig theoretical model These results are icluded i Fig. 17 for compariso purpose. Our model predictios are i good agreemet with these results. I additio to this, the results for spherical aosolid ad aowire are also reported. There are similar tred of variatios. It is observed that frequecy icreases from aofilm to aowire ad spherical. The similar tred of variatio is foud i other solids viz. Cu ad Ag with good agreemet as compared with earlier theoretical ad experimetal results 23-25. The preset model is very simple ad straightforward, which cotaied less iput parameters as compared with earlier models. We have also preseted our results for the materials for which experimetal data are ot available. This may help the researchers egaged i the experimetal study. Due to the simplicity ad applicability, the model may be useful for other aomaterials of curret iterest. Ackowledgemet The authors ackowledge with thaks to the Uiversity for providig the fiacial support i the form of research project (code No. 21A).

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