纳米尺寸效应新应用单键比热强度及应变极限Word文件下载.docx

1、An analytical solution shows that the maximal strain of an impurity-free metallic monatomic chain （MC, or a defect-free nanowire （NW, varies in-apparently with mechanical stress but apparently with the separation between the melting point （Tm （K and the temperature of operation in terms of exp T m （

2、K T-1, where K is the dimension of the NW （for an MC, K = 1.5. Reconciliation of the measured data of Au-MC breaking limit suggests that the discrepancy in measurement arises from thermal and mechanical fluctuations near the Tm of the MC that is 1/4.2 fold of the bulk value. Findings also favour the

3、 mechanism for the high-extensibility of a nanograined NW and further indicate that bond unfolding of the lower-coordinated atoms dominate the grain boundary activities particularly at temperatures approaching to surface melting. E-mail: ; Fax: 65 6792 0415; URL:- 1 -PACS: 61.46.+w; 82.45.Yz; 81.07.

4、Bc- 2 -IntroductionMetallic monatomic chains （MCs and nanowires （NWs have attracted tremendous interest because of the fundamental significance and the fascinating properties in quantum conductance, chemical reactivity, thermal stability, mechanical strength and ductility. These are key issues of co

5、ncern in upcoming technologies such as nano-device. A metallic MC is an ideal prototype for extensibility study, as the MC involves merely bond stretching without bond unfolding or atomic gliding dislocations, as do atoms in a metallic NW consisting of nanograins upon being stretched. 1 Measured usi

6、ng transmission electron microscopy （TEM at room temperature under tension the Au-Au bond breaks at a length that varies from 0.29 nm, 2 0.36 nm （ 30%,3 0.35 0.40 nm4 to even a single event of 0.48 nm,5 while at 4.2 K the breaking limit is reduced to 0.23 0.04 nm as measured using scanning tunnellin

7、g microscopy and mechanically controlled break junction.6 Sophisticated calculations suggest that the Au-Au equilibrium distance （without external stimulus contracts to a range between 0.232 and 0.262 nm7 from the bulk value, 0.2878 nm, whereas the maximal Au-Au distance under tension exceeds no lon

8、ger than 0.31 nm.8 The Au-Au bond in the MC is twice stronger than that in the bulk.9 Discrepancy could not be theoretically solved unless inserting impurity atoms such as H, B, C, N, O, and S into the Au-Au chain in calculations.10 Metallic NWs such as Cu and Al show extensibility that is 1013 high

9、er than the bulk values though the MCs of Cu and Al are hard to form at ambient. The high extensibility was attributed to atomic dislocations or atomic diffusion at grain boundaries that are suggested to be easier.11,12- 3 -Combining the effects of thermal expansion, mechanical stretching, and the a

10、tomic coordination-number （CN imperfection caused bond contraction with the fact that a molten phase is extremely compressible, we have derived a numerical solution to solve the discrepancy in atomic separation of an impurity-free MC with and without thermal and mechanical stimuli. Extension of the

11、solution to a defect-free metallic NW suggests that bond unfolding or atomic sliding of the lower-coordinated atoms at grain boundaries dominates the high extensibility of a nanograined NW at temperatures close to that for surface melting.TheoryThe bond-order-length-strength （BOLS correlation premis

12、e, which has been detailed in ref 13, indicates that the CN imperfection of atoms at sites surrounding defects or near the surface edge causes the remaining bonds of the lower-coordinated atoms to contract spontaneously associated with strengthening of the shortened bond. The bond strengthening cont

13、ributes to the Hamiltonian that dictates the entire bandgap expansion. 14 On the other hand, the structure such as the core-level shift,13 bandatomic CN imperfection lowers the atomic cohesive energy that dominates the thermal stability such as melting20 and phase transition15,16 and determines the

14、activation energy for atomic diffusion, atomic dislocation, and chemical reaction. The competition between energy density increase and the atomic cohesive energy suppression in the relaxed region dominates the mechanical strength and compressibility of a nanosolid.17 The BOLS correlation has also en

15、abled us to determine the identities of a C-C bond in carbon nanotubes,17 the energy levels of an - 4 -isolated atom,13 and the vibration frequency of a Si-Si dimmer bond.18 Matching the BOLS predictions to the measurements reveals that at the lower end of the size limit （one unit cell with atomic C

16、N of 2, the Au 4f-level binding energy increases by 43% （= ci -1 -1 with respect to the bulk value of -2.87 eV and the melting point of the smallest Au nanosolid, or the Au-MC, decreases from 1337.33 K to 320 K, which is 1/4.2 fold the bulk value.19Figure 1 illustrates schematically the BOLS correla

17、tion using the pair-wise inter-atomic potential. When the CN of an atom is reduced, the equilibrium atomic distance will contract from one （unit in d, being the equilibrium bond length in the bulk to ci and the bond energy will increase in magnitude from one （unit in Eb , being the cohesive energy p

18、er bond in the bulk to ci -m . The ci is the bond contraction coefficient. The index m is an adjustable parameter depending on the nature of the bond. For metals, m = 1; for Si and C, m has been optimised to be 4.8813 and 2.67,17 respectively. The solid and the dotted u（r curves correspond to the pa

19、ir-wise inter-atomic potential with and without CN imperfection. The BOLS correlation discussed herewith is consistent with the trend reported in Ref. 7 albeit the extent of bond contraction and energy enhancement that varies from case to case in Ref. 7. The BOLS correlation formulates the bond leng

20、th, di , the bond energy, Ei （T = 0, and the cohesive energy, EBi , per atom in the MC in the following forms:13-15d i （z i d =c i （z i =21+exp （12z i /8z i =0. 69731E i （T =0=c i E b （T =01. 5E b T m , i E Bi =z i E i （z i =2（1- 5 -where z i is the effective atomic CN. Tm,i being the melting point

21、of the MC isi and b denotes proportional to the atomic cohesive energy, EBi = zi E i . 20,21 Subscripta specific ith atom in the MC and an atom in the bulk. The BOLS premise predicts that an Au-MC （zi = 2 bond contracts by 1 - 0.6973 30% from 0.2878 to 0.2007 nm and the bond strength （Ei /di = ci -2

22、 2 becomes two folds the bulk value. The predicted Au-MC equilibrium length is slightly shorter than that measured under tension at 4.2 K, 0.23 0.04 nm,6 and the predicted bond strength agrees with reported values.9 Such consistencies further evidence the validity of the BOLS consideration that attr

23、ibutes the size dependency of a nanosolid to the atomic CN imperfection and the increased portion of the low-CN atoms of the nanosolid.The characteristic energies as indicated in Fig. 1 represent: （i at E = 0, the bond is completely broken with zero interatomic interaction. （ii Separation between E

24、= 0 and Ei （T is the cohesive energy per CN at T, which is the energy required for bond breaking. （iii The spacing between Ei （T and Ei （0 is the energy of thermal vibration. If one wants to melt an atom with zi coordinates by heating the system from T to Tm,i , one needs to provide zi Ei （Tm,i Ei （

25、T = z i 1i （T m , i T energy. When T approaches to Tm,i , the mechanical strength approaches to zero with infinite compressibility.One may apply a tensile stress P to stretch a bond in the MC from its original equilibrium length at T, di （zi , T, 0, to the breaking limit, diM （zi , T, P. Mechanicall

26、y rupturing of the bond at temperature T needs energy 1i （T m , i T +2i that equals the 1/zi fold thermal energy for evaporating an atom in solid state at T:- 6 -d iM （z i , T , P d i （z i , T , 0P （x dx =P d iM （z i , T , P d i （z i , T , 0=P d i=1i （T m , i T +2i（2Ideally, the slope 1i corresponds

27、 to the specific heat per coordinate. The constant 2i represents 1/zi fold energy required for evaporating a molten atom of the MC. 1i and 2i can be determined with the known c i m and the corresponding bulk values of 1b and 2b that have been obtained as shown in Ref.22 in detail.Considering the eff

28、ects of atomic CN-imperfection-induced bond contraction （ci d, thermal expansion （1+T , with being the linear coefficient, and mechanical stretching （1+i （z i , T P , with coefficient i , the distance between two nearest atoms in the interior of an MC can be expressed as:d i （z i , T , P =d c （z i （

29、1+T 1+i （z i , T P ,The maximal strain is then expressed as,d iM （z i , T , P =i （z i , T P d i z i , T , 0（3where d i （z i , T , 0=d c （z i （1+T is the bond length at T without being stretched. One can approximate the mean P to the P （x in eqs （2 and （3, as the diM （zi , T, P represents the breakin

30、g limit and the integration is a constant. Combining eqs. （2 and （3, one has,- 7 -1i （T m , i T +2i P = z , T d z , T , 0i i i 2（4 For tensile stress, P 0, for compressive stress, P 0. The extensibility or compressibility, , of a system is expressed as:23V u （r , T =V V 2V P T d i （z i , T , 0N i E

31、i T m , i E i T 21i （z i , T =Td i （z i , T , 01i T m , i T （5 is the inverse of Youngs modulus or hardness in dimension that equals the sum of bond energy per unit volume.24 The power index = 1, 2, and 3 correspond to the dimensionality of a MC, a rod, and a spherical dot. Ni is the total number of bonds in di volume. One needs to note that the bond number density in the relaxed region does not change upon relaxation. For instance, bond relaxation never changes the bond number between the neighboring atoms in a MC whether the MC is suspended or embedded in the b