Predicting elastic modulus of the body centered cubic metallic films

65 HNUE JOURNAL OF SCIENCE DOI 10 18173/2354 1059 2022 0023 Natural Sciences 2022, Volume 67, Issue 2, pp 65 75 This paper is available online at http //stdb hnue edu vn PREDICTING ELASTIC MODULUS OF[.]

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2022-0023 Natural Sciences 2022, Volume 67, Issue 2, pp 65-75 This paper is available online at http://stdb.hnue.edu.vn PREDICTING ELASTIC MODULUS OF THE BODY-CENTERED CUBIC METALLIC FILMS Duong Dai Phuong Military College of Tank Armor Officer Abstract The elastic modulus of the body-centered cubic (BCC) films is studied based on the moment method For Fe, Ta, and W films a clear elastic modulus dependence of the thickness at ambient conditions and under temperatures up to 2000 K The obtained results of the values of elastic modulus metallic films are smaller than the corresponding values of bulk material Calculated results show the effects of temperature and thickness of elastic modulus for Fe, Ta, and W metallic films On the other hand, when the thickness of thin films is about 60nm then the elastic modulus of the BCC films approaches the values of bulk material Our results are compared with the other theoretical results and experimental values of bulk materials Keywords: metallic films, moment method, elastic modulus Introduction To date, metallic thin films have received much attention in material sciences due to their novel applications in technology and industry [1, 2] They exhibit different thermal, mechanical, electrical, and optical properties compared to those of bulk materials [3-5] The knowledge about the elastic quantities of the metallic films as bulk modulus BT , Young’s modulus Y, and shear modulus G enable to determine the stability and reliability of manufactured materials Great efforts with experimental and theoretical studies have been made to estimate the elastic property of metallic and nonmetallic films Using both nano- and microindentation methods, F Seifried et al [6] measured Young’s modulus of Mo, Nb, and Ta thin films on various scales The measured data of thin films showed good agreement with bulk data Progressive scratch tests showed the important role of plastic deformation on the metallic thin films at larger normal forces D Bernoulli et al [7] deposited Ta and TaNi thin films by the Direct current (DC) magnetron sputtering method The effects of the underlying substrate and N2/Ar ratio on hardness and phase of Ta and TaNi thin films were observed Various methods such as X-ray diffraction [8], Brillouin scattering [9] were Received May 11, 2022 Revised June 13, 2022 Accepted June 20, 2022 Contact Duong Dai Phuong, e-mail address: vanha318@yahoo.com 65 Duong Dai Phuong also used to determine the elastic properties of nonmetallic films Literature data showed a significant difference between the mechanical properties of metallic thin films and the values of bulk [10, 11] The measured elastic modulus of Cu, Ag, and Al films were smaller than the values of bulk [10, 12] However, the previous studies estimated the elastic properties of films at low temperatures, and the effects of thickness on the elastic properties have not been studied in detail In this paper, we study the elastic quantities of films (Fe, Ta, W) by the moment method [12, 13] The effects of thickness and temperature and thickness on the elastic quantities of Fe, Ta, and are evaluated in detail The mechanical quantities of Fe, Ta, and W thin films are calculated under temperatures up to 2000 K using the Lennard-Jones potential Content 2.1 Theory n 2.1.1 Expressions of the displacement One separates metallic films has the thickness b into nl layers These layers are as Figure n1 a a b Thickness (n l - 4) La yers t a Figure The metallic films Equation of the displacement  uit  for internal layers atoms of metallic films [12, 13] 66 Predicting elastic modulus of the body-centered cubic metallic films  t d  uit  d  uit   t +    u  +  t  uit 3 + kt  uit  + t ( xt coth xt − )  uit  − p = , t i dp dp kt (1) where xt = t   2 t  ; = k BT , kt =   2io   m0t2 , 2 i  ui eq   4iot   4iot   1t =    ,  2t =   2 48 i  ui eq 48 i  ui ui   4 t    4iot io  t =    +6  2  u u 12 i  ui eq  i i  (2)   , eq (3)     = (  1t +  2t ) , eq  (4) with m0 is the atomic mass of internal layers atoms, t is the vibrational frequency of internal layers atoms; kt ,  1t ,  2t ,  t are the anharmonicity parameters; it0 is the effective interatomic potential The solution to equation (1) is as follows: (5)  uit = uit 0 + A1t p + A2t p when the supplemental force p is at zero then the solution  uit  is given by  uit 0  2 t At , 3kt3 (6) where At = a1t +  t2 kt4 a2t +  t3 kt6 4 t t  a3 + kt a4t +  t5 at 10 kt +  t6 kt12 a6t , with the parameters at ( = 1, , 6) have the same forms as in Ref [12, 13] Similar to the internal layer atoms, the displacement of the next surface layers atoms is the solution of the equation as follows:  1 d  uin1  d  uin1   n1 +    u  +  n1  uin1 3 + kn1  uin1  + n1 ( xn1 coth xn1 − )  uin1  − p = n1 i dp dp kn1 (7) The displacement of the surface layers atoms  uin  can be calculated by moment method formulation as follows: 67 Duong Dai Phuong   uin n kn  u  p + n  u i  p +  p  p n i +   ( xn coth xn − ) − p = 0,  mn2 (8)  where xn = kn = n , = k BT 2 ( ) ( (9) )  n i aix + 0in0  = mn2 ,   i    3 n    3ion io  n =    +   ui ,n uin i , ,  ,   ui , n  eq        eq  (10) (11) when the supplemental force p is at zero then the solution  uit  is given by  uin = uin 0 + A1n p + A2n p (12) here,  uin 0 can be calculated by moment method formulation The expression of  uin 0 is given by  uin 0 = −  n kn2 (13) xn coth xn 2.1.2 The expression of average displacement and free energy The Helmholtz free energies of the internal layers and next surface layers are given by Ref [12] 3N   2  X    t = U 0t + 3N t  xt + ln − e −2 xt  + t2  2t X t2 − 1t 1 + t   + kt    (14)  N t   Xt  Xt   X t − (  1t + 2 1t  2t ) 1 + (1 + X t )   2t + kt4        ) (  ) 3N   2  X    n1 = U 0n1 + 3N n1  xn1 + ln − e −2 xn1  + n21  n1 X n21 − 1n1 1 + n1   + kn1     6N    X   X  + n41   22n1 1 + n1  X n1 − (  12n1 + 2 1n1 n1 ) 1 + n1  (1 + X n1 )  kn1       ( (15) with X t = xt cothxt , X n1 = xn1cothxn1 , and Nt N it0 ( ri ,t ),U 0n1 = n1  in01 ( ri ,n1 ), (16)  2 In the harmonic approximation, the Helmholtz free energy of the surface layers is determined as [13] U 0t =  ( )  n = U 0n + 3N n  xn + ln − e−2 xn  68 (17) Predicting elastic modulus of the body-centered cubic metallic films One considers the system consisting of Nl atoms and the number of atoms in each layer is Nl , then N = nl N l  nl = N Nl (18) The numbers of atoms in the internal, next surface, and surface layers are determined as follows, respectively N  Nt = ( nl − ) Nl =  −  Nl = N − 4Nl ,  Nl  (19) Nn1 = 2Nl = N − ( nl − )Nl and Nn = 2Nl = N − ( nl − )Nl (20) The Helmholtz free energies of the system and an atom, respectively, are given by  = N t  t + N n1 n1 + N n  n = ( N − N l )  t + N l  n1 + N l  n   4 2 = 1 −   t +  n1 +  n N  nl  nl nl (21) (22) The average nearest-neighbor distance (NND) is denoted as atb and btb is the average two-layers thickness, then we have a (23) btb = tb On the other hand, the thickness b can be calculations a (24) b = 2bn + 2bn1 + ( nl − ) bt = ( nl − 1) btb = ( nl − 1) tb From Eq (24), we have nl = + d d = 1+ btb atb The average nearest-neighbor distance is given by 2a + 2an1 + (nl − 5)at atb = n nl − (25) (26) In Eq (25), an , an1 and at have the from an = a0,n +  uin 0 , an1 = a0,n1 +  uin1 0 , at = a0,t +  uit  , (27) From Eqs (22) and (25), Substituting Eq (24) into Eq (23), we determined the expression of the Helmholtz free energy 2atb 2atb Ψ b − 3atb = Ψt + Ψn + Ψ n1 N b + atb b + atb b + atb (28) 69 Duong Dai Phuong 2.1.3 The elastic quantities of the BCC metallic films The elastic quantities of metallic films are derived based on thermodynamic relations The bulk modulus BT is determined  P  BT = −V0    V T  a0tb   atb2 =   P+ 9V  atb    b − 3atb   t 2atb  2 n 2atb   n1   + +    2 b + atb an b + atb an1 T   b + atb at (29) The Helmholtz free energy for surface layers atoms  p ,n is given by Ψ p ,n = Ψ 0,n +  n n = Ψ 0,n + with  n is the elastic strain,  n = Yn n is the stress From Eq (30), one derives Ψ p ,n  n = Yn n2 , Ψ p ,n  n = Yn n  n  n Another face, in the elastic deformation then we have a a − a  n = n = p ,n n , an an (30) (31) (32) where an and a p ,n can be given by   a p ,n = a0 p ,n + yn  On the other hand, from a relationship p =  n Sn =  n an2 an = a0,n + y0n (33) (34) when p is small then we have n = a p ,n − an an  yn − y0n A1n p  , an an (35) where, A1n has form as [12, 13]:  2 n2  xn coth xn   1+ (36) (1 + xn coth xn ) 1 +   kn     From Eqs (34), (35), (36), and (31), for surface layers atoms of metallic thin film, expression of Young's modulus is given by  a 1 (37) Yn = nn n = = n A1 p  an A1  ( a0,n + y0n ) A1n A1n = kn Similar to the next surface layers atoms and internal layers, we can be determined 70 Predicting elastic modulus of the body-centered cubic metallic films Yn1 =  n1an1 n1 A p Yt = =  t at t Ap a A n1 n1 = = ,  ( a0,n1 + y0n1 ) A1n1 (38) 1 = , t  at A1  ( a0,t + y0t ) A1t (39) where A1n1 and A1t have formed as A1n1 =   2 n21  xn1 coth xn1  1+ (1 + xn1 coth xn1 ) 1 +   kn1  kn1    (40)  2 t2  xt coth xt   1+ (41) (1 + xt coth xt ) 1 +   k   t   The Young's modulus and shear modulus of metallic films with body-centered cubic structure is given respectively 2Y + 2Yn1 + ( nl − ) Yt Y= n (42) nl − A1t = kt G= Y (1 +  ) (43) In Eq (25), the Poisson's ratio determined as 1 2  = 1 − Y   3BT  (44) 2.2 Numerical results and discussion To calculate the elastic quantities of Fe, Ta, and W films, one uses the expressions derived in the previous section and the using the Lennard-Jones potential [14] n m D   r0   r0    (r ) = −n m , ( n − m )   r   r   (45) In Eq (45), the values of potential parameters for Fe, Ta, and W films are presented in Table Table Lennard-Jones potential parameters of Fe, Ta, and W metallic thin films [14] Metals n m r0 (A0 ) D / kB (K) Fe 8.26 3.58 2.4775 12576.70 Ta 11.16 2.52 2.8648 21305.51 W 8.58 4.06 2.7365 25608.93 From Eq (26), we obtain the average NND of BCC metallic thin films Based on the expressions obtained in Section 2, the elastic quantities including the analytic 71 ... Figure The metallic films Equation of the displacement  uit  for internal layers atoms of metallic films [12, 13] 66 Predicting elastic modulus of the body- centered cubic metallic films  t d ... modulus of the body- centered cubic metallic films One considers the system consisting of Nl atoms and the number of atoms in each layer is Nl , then N = nl N l  nl = N Nl (18) The numbers of atoms... determine the elastic properties of nonmetallic films Literature data showed a significant difference between the mechanical properties of metallic thin films and the values of bulk [10, 11] The measured