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Note on low-energy ion irradiated silicon nanowires: Anomalous large deformations at lower energy
Vacuum 145 (2017) 308e311
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Note on low-energy ion irradiated silicon nanowires: Anomalous large deformations at lower energy Kwang-Hua R. Chu a, b, * a b
College of Mechanical and Electronic Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China Zhu-Mali Centre, 2/F, No. 24, Lane 260, Section 1, Road Muzha, Taipei, Taiwan 116, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2017 Received in revised form 2 September 2017 Accepted 5 September 2017 Available online 8 September 2017
We adopted the verified absolute-reaction theory which originates from the quantum chemistry approach to explain the anomalous flow for Si nanowires irradiated with 100 keV (at room temperature regime) Arþ ions as well as the observed amorphization along the Si nanowire [Nano Lett. 2015, 15, 3800 e3807]. We also demonstrate some formulations which can help us calculate the temperaturedependent viscosity of flowing Si in nanodomains. © 2017 Published by Elsevier Ltd.
Keywords: Mechanical properties Radiation damage Defects Boundary perturbation
1. Introduction A possible explanation of the influence of intensive plastic deformation (or megaplastic deformation) on the specific behavior of nanomaterials depends on the level of understanding of the essential nature of structural and phase transformations. Structure modifications in low-dimensional Si nanowires can be induced by low-energy ion bombardment and irradiation. Although the literature on ion-matter irradiation is nevertheless huge below we shall focus only on most recent works that address anomalous deformations due to low-energy irradiation on silicon. Recently Johannes et al. [1] reported Si nanowires irradiated at elevated temperatures above 300 C with 300 keV Arþ ions (on a rotatable and heatable stage) retain the geometry of the nanostructure and the corresponding sputtering can be gauged accurately while Si nanowires irradiated at room temperature with 100 keV Arþ ions (on a rotatable and heatable stage), however, amorphize and collapse significantly. The magnitude and direction of the anomalous (plastic) flow is independent of the ion-beam direction and cannot be explained easily. We noticed that Johannes et al. mentioned in Ref. [1]: the irradiation of
* College of Mechanical and Electronic Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.vacuum.2017.09.005 0042-207X/© 2017 Published by Elsevier Ltd.
nanostructured Si reveals a previously unknown (plastic) flow of Si under low energy ion irradiation. As commented in Ref. 1 for a high-energy approach previous researchers [2] developed a convincing viscoelastic model for the flow of amorphous solids under swift heavy ion bombardment seen for ion energies much larger than 1 MeV (that viscoelastic model [2] being based on the assumption of a continuous, intensely excited region of cylindrical shape along the ion path to derive a deformation rate per fluence). Moreover Johannes et al. [1] criticised the classical thermal spike regime is not reached by far. The specific geometrical constraints are much more complex with nanostructured samples and low energy (and thus low-symmetry) collision cascades than in the known swift-heavy ion induced deformation modeled in Ref. [2] and others [1]. To be concise, as also remarked in Ref. 1, the interpretation of what constitutes a thermal-spike is contentious. The total energy loss for 100 keV Arþ in Si is dE=dx ¼ 36 eV/nm of which the electronic energy loss is roughly half. The threshold for the electronic energy loss for the thermal-spike model [2] is, however, given as dE=dx 1 keV/nm. It seems to Johannes et al. [1] that this generalized model [2] based on electronic energy loss derived from MC (Monte Carlo) simulations is not applicable. In fact based on the theory of absolute reaction rate [3e7] we can explain above puzzles or anomalies considering the tuning of activation energy as well as activation volume. As to the viscosity role which Johannes et al. mentioned briefly in Ref. [1] it can be
K.-H.R. Chu / Vacuum 145 (2017) 308e311
demonstrated below after some theoretical derivations. In this short paper we shall adopt verified Eyring's absolute-reaction theory [3e7] to explain Johannes et al.’s puzzles or anomalies [1] considering the tuning of activation energy as well as activation volume. We shall also demonstrate the calculation of viscosity relevant to (plastic) flows using the verified Eyring's approach [3e7].
As developed by Krausz and Eyring [7], the chemical kinetics view of (plastic) flows demonstrated in Ref. [6] leads to the conclusion that because chemical bonds are not always reestablished in a chemical process, broken chemical bonds or misconnected bonds are left in the wake of moving defects (say, dislocations). These faults or defects are potential sources of crack nuclei and stress raisers and are, therefore, the sites at which cracks may develop. When the fault concentration reaches a high density, a new mechanism, the nucleation and growth of cracks, appears [7]. Below we shall investigate the temperature-dependent Si viscosity in nanodomains (presumed to be cylindrical, cf. Fig. 1) by using the verified absolute reaction theory [4e8] which originates from the quantum chemistry approach. In fact according to Eyring's idea: A shearing force is applied across the two layers of particles or atoms, and (plastic) flow takes place when a single particle or atom squeezes past its neighbors and drops into a vacant equilibrium position (a hole or defect) at a distance, k, from its original position [4,5,8]. We remind the readers that our approach for the flow properties of the Si nanowire is essentially a more realistic nonlinear flow law. From the Eyring's absolute reaction model [3e7] (considering the stress-biased thermal activation), structural rearrangement is associated with a single energy barrier (height) EA that is lowered or raised linearly by a (shear) yield stress t. In general we have [4,5,8].
U_ ¼ 2
VA Vm
kB T EA VA t exp sinh ; h kB T 2kB T
VA t VA t exp 2k exp 2kB T BT VA t ¼ sinh : 2kB T 2
VA t VA t The latter becomes sinh 2k zexp 2 T 2k T B B VA t=ð2kB TÞ/∞. Thus above equation (1) becomes
U_ ¼
2. Theoretical formulations
(1)
with VA being the activation volume, Vm being the characteristic _ being the shear strain rate, k being volume under consideration, U B the Boltzmann constant, h the Planck constant, and
309
VA Vm
kB T EA þ VA t=2 exp ; h kB T
once
(2)
once VA t=ð2kB TÞ is rather large. With some manipulations, we then have
2
3
U_
ln4 VA Vm
kB T h
5 ¼
EA þ VA t=2 ; kB T
or
2 3 2 3 _ 2 4 U t¼ k Tln4 5 þ EA 5: VA B VA kB T Vm
(3)
h
If the transition rate is proportional to the shear strain rate (with a constant ratio: L0 z2VA =Vm ), we can calculate the shear stress [4,5].
"
!# _ EA kB T 2U ln t¼2 þ ; VA VA L 0 n0
(4)
where n0 is an attempt frequency [4,5], e.g., for temperatures (T) being Oð1Þ K (kelvin): n0 zkB =h Oð1011 Þ (1/sec). Normally, the value of VA ð≡d2 d3 d) is associated with a typical volume required for a molecular shear rearrangement (d is the distance the patch moves per jump). Here Vm ¼ d2 d3 d1 , d2 d3 : dimensions of unit of flow in the direction of flow and perpendicular to the direction of flow (the directions of d2 and d3 both being perpendicular to that of d1 ) and d1 is the distance between moving layers of particles [3] (cf. Fig. 1). We consider a steady, fully developed transport of the (locally amorphous) Si under high loading in a wavy-rough nanotube of a (in mean-averaged outer radius) with the outer interface being a fixed wavy-rough surface: r ¼ a þ ε sinðkqÞ where ε is the amplitude of the (wavy) roughness and k is the wave number: k ¼ 2p=Lt (Lt is the wave length) (cf. Fig. 2). We can have (via the boundary perturbation series method [8]), after using the forcing parameter
j¼
a 2t0
dp dz
(5)
(t0 ¼ 2kB T=VA and dp=dz being the external forcing along the axis of the cylindrical domain or the transport direction)
U_ ¼ U_ 0 sinhðjÞ þ HOT
(6)
with the small wavy-roughness effect being the first order perturbation which is rather small and thus neglected (HOT means the _ ) is obtained higher order contributions [8]). Here the shear rate (U by Fig. 1. Schematic (plot) of an Eyring's microscopical flow [3]: Flow proceeds by the motion of (composite) particles into holes left open by neighboring ones. Eyring proposed [4] that shear occurs along sets of parallel planes. The distance between these planes is indicated by the symbol d1 . The shear force per unit area is indicated by f. A patch of atoms or molecules whose cross-sectional area is d2 d3 shift or jump as a unit on either side of the shear plane. The distance the patch moves per jump is d.
dv _ ¼ ¼ Vv,n U dn
(7)
with n≡V½r a εsinðkqÞ=jV½r a εsinðkqÞj being the unit normal of the general interface [8] which is between the confined Si
310
K.-H.R. Chu / Vacuum 145 (2017) 308e311
Fig. 2. Schematic of a cross-section (mean radius being a) with wavy-rough wall. ε is the amplitude of the wavy-roughness and the wave number of wavy roughness is k (say, which is 10 here).
and the environment. The temperature-dependent (referenced) shear rate [3,4,6] is
U_ 0 ¼ 2
VA Vm
kB T EA exp ; h kB T
(8)
which is a function of temperature, the activation energy (EA ), the activation volume (the characteristic volume Vm is presumed to be the same as VA here) and the length scale under confinement. L0 n0 in Eq. (4) is temperature dependent and the value could be traced before [3e7] or from equation (3) as well as equation (1). 3. Results and discussion We noticed that above flow (force) law is highly nonlinear. There are difficulties in selecting parameters for experimental condition or process [1], such as dp=dz and VA . Nevertheless, to obtain a qualitative viscosity role only, we can fix all necessary parameters for corresponding temperature (T). The main task is to fix the value of j by prescribing a and dp=dz with different temperatures. The _ . After all these, the remaining shear viscosity is calculated via t=U in equation (4) is the unique relationship between VA and T for a fixed t. Note that most of the mathematical derivations could be found before [3e8]. We start to explain Johannes et al.’s puzzles or anomalies [1] considering the tuning of activation energy as well as activation _ can be related to the volume. To be specific the rate of strain (Y) stress (s) via the application of high-temperature absolute reaction rate theory [3,4,6].
V kB T EA VA s exp sinh ; Y_ ¼ 2 A h Vm kB T 2kB T
(9)
where VA is the activation volume, Vm ¼ d2 d3 d1 , d2 d3 : dimensions of unit of flow in the direction of flow and perpendicular to the direction of flow (the directions of d2 and d3 both being perpendicular to that of d1 ), d1 is the distance between moving layers of particles, EA is the activation energy, T is the absolute temperature with kB being the Boltzmann constant and h the Planck constant. We shall presume VA ¼ Vm here. Using the experimental parameters [1] we have for the high energy ion irradiation case (after 5 1016 cm2 300 keV Arþ irradiations) after taking ion backscattering, surface roughness or sample topography, etc. into account: the temperature for one selected point of the Si nanowire being around (in average) 700 K (which is higher than 300 C or 573.16 K due to absorption of highenergy ion irradiations), the activation energy being around (in average) 4 1019 J, the stress (considering the reported range
(which is of the order of magnitude about 100 GPa) of elastic modulus or effective Youngs modulus for Si nanowire [9]) being around (in average) 6 1010 Pa, and the activation volume being around (in average) 2 1030 m3 (that is reasonable considering Eyring's defect-volume induced flow [3,4,7]). Above selected pa_ rameters give us Yz0:0077 sec1 from equation (6). The small deformation rate agrees with the measurements [1] (cf. Fig. 2 therein). Meanwhile with the experimental parameters [1] we have for the low energy ion irradiation case (after 5 1016 cm2 100 keV Arþ irradiations) after taking ion backscattering, surface roughness or sample topography, etc. into account: the temperature for one selected point (the same as high-energy case) of the Si nanowire being around (in average) 350 K (which is a little higher than the room temperature or 298.16 K due to absorption of low-energy ion irradiations), the activation energy being around (in average) 1 1019 J, the stress (considering the reported range (which is of the order of magnitude about 100 GPa) of elastic modulus or effective Youngs modulus for Si nanowire [9]) being around (in average) 3 1010 Pa, and the activation volume being around (in average) 1030 m3 (that is reasonable considering Eyring's defect-volume induced flow [3,4,7]). Above selected parameters give us _ Yz4:6611 104 sec1 from equation (6). This rather large deformation rate can explain the possible anomalous flows of the Si nanowire (subjected to 5 1016 cm2 100 keV Arþ irradiations) which also agrees with the measurements [1] (cf. Fig. 2 for the clear shrinking and slight broadening of the Si nanowire therein or cf. figure 4 and in the Supporting Information: the Si nanowires look as though they were molten). Furthermore in Ref. [1] electron backscattering diffraction (EBSD) images were made to confirm that the samples irradiated at 300 C remained crystalline and those irradiated at room temperature were amorphized. Nevertheless we remind the readers that in Ref. [10] Pecora et al. irradiated a (Si nanowire) sample with a 300 keV Ge beam at a dose of 1015 cm2. The energy is such that the beam is passing through for the entire length of the Si nanowire. Their results show that the totally amorphous region flows because of the very-large deformation due to the irradiation [10]. Thus the observed amorphization along the Si nanowire in Ref. [1] is relevant to the large (plastic) flow of the Si nanowire as we just calculated or illustrated above. We remind the readers that a comprehensive discussion of the influence of intensive plastic deformation on the physicomechanical properties of nanomaterials can be found in Ref. [11]. Our calculated data (considering the trend) show that generally the larger the activation energy is then the higher is the shear viscosity for the same temperature once the forcing parameter is fixed. To be specific an impinging ion depositing its energy close to the surface can heat the material (then increase the local temperature for the same site) to lower the viscosity and enable normal viscous flow. 4. Conclusion To give a brief summary, we have adopted the verified theory of absolute reaction (originates from the quantum chemistry approach) to explain the anomalous flow for Si nanowires irradiated with 100 keV (at room temperature regime) Arþ ions as well as the observed amorphization along the Si nanowire in Ref. 1 which is relevant to the large (plastic) flow of the Si nanowire as we just calculated or illustrated here. After some theoretical derivations we also can calculate the shear viscosities for flowing Si in the bulk and in nanodomains (our numerical results for a test case were compared with previous experimental data [8] and the good fit inbetween verifies our present approach). Meanwhile we give
K.-H.R. Chu / Vacuum 145 (2017) 308e311
numerical predictions for the shear viscosity of flowing Si in nanodomains. In general our results show that once the activation energy is higher then we have larger shear viscosity for the same temperature once the force parameter is fixed. Confirmation of our prediction for nanoscale shear viscosities of flowing Si must await experiments. We shall investigate other interesting issues [11e17] in the near future.
[10]
Acknowledgements
[11] [12]
The only author thanks Referees for their crucial comments. References [1] A. Johannes, S. Noack, W. Wesch, M. Glaser, A. Lugstein, C. Ronning, Nano Lett. 15 (2015) 3800e3807. [2] H. Trinkaus, A.I. Ryazanov, Phys. Rev. Lett. 74 (1995) 5072e5075.
[3] [4] [5] [6] [7] [8] [9]
[13] [14] [15] [16] [17]
311
H. Eyring, T. Ree, PNAS (USA) 41 (1955) 118e122. H. Eyring, J. Chem. Phys. 4 (1936) 283e301. R.E. Powell, H. Eyring, Nature 154 (1944) 427e428. H. Eyring, J.W. Fredrickson, D. McLachlan Jr., PNAS (USA) 34 (1948) 295e304. A.S. Krausz, H. Eyring, J. Appl. Phys. 42 (1971) 2382e2385. R. Chu, Kwang-Hua, J. Phys. Chem. Solids 104 (2017) 62e67. M.J. Gordon, T. Baron, F. Dhalluin, P. Gentile, P. Ferret, Nano Lett. 9 (2) (2009) 525e529. E.F. Pecora, A. Irrera, S. Boninelli, L. Romano, C. Spinella, F. Priolo, Appl. Phys. A 102 (2011) 13e19. R.A. Andrievski, Physics-Uspekhi 57 (10) (2014) 945e958. G.S. Anjusree, A.M. Asha, K.R.V. Subramanian, N. Sivakumar, A.S. Nair, S.V. Nair, A. Balakrishnan, Int. J. Mater. Res. 104 (6) (2013) 573e577. B.L. Oksengendler, N.N. Turaeva, S.E. Maksimov, F.G. Djurabekova, JETP 111 (2010) 415e420. Zh. Eksp. Teor. Fiz. 138(2010) 469e474. J.-C. Pothier, F. Schiettekatte, L.J. Lewis, Phys. Rev. B 83 (2011) 235206. €sele, Int. J. Mater. Res. 97 P. Werner, N.D. Zakharov, G. Gerth, L. Schubert, U. Go (7) (2006) 1008e1015. C.M. Carlson, H. Eyring, T. Ree, PNAS (USA) 46 (1960) 649e659. G.A. Malygin, Physics-Uspekhi 54 (11) (2011) 1091e1116.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Note on low-energy ion irradiated silicon nanowires: Anomalous large deformations at lower energy Kwang-Hua R. Chu a, b, * a b
College of Mechanical and Electronic Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China Zhu-Mali Centre, 2/F, No. 24, Lane 260, Section 1, Road Muzha, Taipei, Taiwan 116, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2017 Received in revised form 2 September 2017 Accepted 5 September 2017 Available online 8 September 2017
We adopted the verified absolute-reaction theory which originates from the quantum chemistry approach to explain the anomalous flow for Si nanowires irradiated with 100 keV (at room temperature regime) Arþ ions as well as the observed amorphization along the Si nanowire [Nano Lett. 2015, 15, 3800 e3807]. We also demonstrate some formulations which can help us calculate the temperaturedependent viscosity of flowing Si in nanodomains. © 2017 Published by Elsevier Ltd.
Keywords: Mechanical properties Radiation damage Defects Boundary perturbation
1. Introduction A possible explanation of the influence of intensive plastic deformation (or megaplastic deformation) on the specific behavior of nanomaterials depends on the level of understanding of the essential nature of structural and phase transformations. Structure modifications in low-dimensional Si nanowires can be induced by low-energy ion bombardment and irradiation. Although the literature on ion-matter irradiation is nevertheless huge below we shall focus only on most recent works that address anomalous deformations due to low-energy irradiation on silicon. Recently Johannes et al. [1] reported Si nanowires irradiated at elevated temperatures above 300 C with 300 keV Arþ ions (on a rotatable and heatable stage) retain the geometry of the nanostructure and the corresponding sputtering can be gauged accurately while Si nanowires irradiated at room temperature with 100 keV Arþ ions (on a rotatable and heatable stage), however, amorphize and collapse significantly. The magnitude and direction of the anomalous (plastic) flow is independent of the ion-beam direction and cannot be explained easily. We noticed that Johannes et al. mentioned in Ref. [1]: the irradiation of
* College of Mechanical and Electronic Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.vacuum.2017.09.005 0042-207X/© 2017 Published by Elsevier Ltd.
nanostructured Si reveals a previously unknown (plastic) flow of Si under low energy ion irradiation. As commented in Ref. 1 for a high-energy approach previous researchers [2] developed a convincing viscoelastic model for the flow of amorphous solids under swift heavy ion bombardment seen for ion energies much larger than 1 MeV (that viscoelastic model [2] being based on the assumption of a continuous, intensely excited region of cylindrical shape along the ion path to derive a deformation rate per fluence). Moreover Johannes et al. [1] criticised the classical thermal spike regime is not reached by far. The specific geometrical constraints are much more complex with nanostructured samples and low energy (and thus low-symmetry) collision cascades than in the known swift-heavy ion induced deformation modeled in Ref. [2] and others [1]. To be concise, as also remarked in Ref. 1, the interpretation of what constitutes a thermal-spike is contentious. The total energy loss for 100 keV Arþ in Si is dE=dx ¼ 36 eV/nm of which the electronic energy loss is roughly half. The threshold for the electronic energy loss for the thermal-spike model [2] is, however, given as dE=dx 1 keV/nm. It seems to Johannes et al. [1] that this generalized model [2] based on electronic energy loss derived from MC (Monte Carlo) simulations is not applicable. In fact based on the theory of absolute reaction rate [3e7] we can explain above puzzles or anomalies considering the tuning of activation energy as well as activation volume. As to the viscosity role which Johannes et al. mentioned briefly in Ref. [1] it can be
K.-H.R. Chu / Vacuum 145 (2017) 308e311
demonstrated below after some theoretical derivations. In this short paper we shall adopt verified Eyring's absolute-reaction theory [3e7] to explain Johannes et al.’s puzzles or anomalies [1] considering the tuning of activation energy as well as activation volume. We shall also demonstrate the calculation of viscosity relevant to (plastic) flows using the verified Eyring's approach [3e7].
As developed by Krausz and Eyring [7], the chemical kinetics view of (plastic) flows demonstrated in Ref. [6] leads to the conclusion that because chemical bonds are not always reestablished in a chemical process, broken chemical bonds or misconnected bonds are left in the wake of moving defects (say, dislocations). These faults or defects are potential sources of crack nuclei and stress raisers and are, therefore, the sites at which cracks may develop. When the fault concentration reaches a high density, a new mechanism, the nucleation and growth of cracks, appears [7]. Below we shall investigate the temperature-dependent Si viscosity in nanodomains (presumed to be cylindrical, cf. Fig. 1) by using the verified absolute reaction theory [4e8] which originates from the quantum chemistry approach. In fact according to Eyring's idea: A shearing force is applied across the two layers of particles or atoms, and (plastic) flow takes place when a single particle or atom squeezes past its neighbors and drops into a vacant equilibrium position (a hole or defect) at a distance, k, from its original position [4,5,8]. We remind the readers that our approach for the flow properties of the Si nanowire is essentially a more realistic nonlinear flow law. From the Eyring's absolute reaction model [3e7] (considering the stress-biased thermal activation), structural rearrangement is associated with a single energy barrier (height) EA that is lowered or raised linearly by a (shear) yield stress t. In general we have [4,5,8].
U_ ¼ 2
VA Vm
kB T EA VA t exp sinh ; h kB T 2kB T
VA t VA t exp 2k exp 2kB T BT VA t ¼ sinh : 2kB T 2
VA t VA t The latter becomes sinh 2k zexp 2 T 2k T B B VA t=ð2kB TÞ/∞. Thus above equation (1) becomes
U_ ¼
2. Theoretical formulations
(1)
with VA being the activation volume, Vm being the characteristic _ being the shear strain rate, k being volume under consideration, U B the Boltzmann constant, h the Planck constant, and
309
VA Vm
kB T EA þ VA t=2 exp ; h kB T
once
(2)
once VA t=ð2kB TÞ is rather large. With some manipulations, we then have
2
3
U_
ln4 VA Vm
kB T h
5 ¼
EA þ VA t=2 ; kB T
or
2 3 2 3 _ 2 4 U t¼ k Tln4 5 þ EA 5: VA B VA kB T Vm
(3)
h
If the transition rate is proportional to the shear strain rate (with a constant ratio: L0 z2VA =Vm ), we can calculate the shear stress [4,5].
"
!# _ EA kB T 2U ln t¼2 þ ; VA VA L 0 n0
(4)
where n0 is an attempt frequency [4,5], e.g., for temperatures (T) being Oð1Þ K (kelvin): n0 zkB =h Oð1011 Þ (1/sec). Normally, the value of VA ð≡d2 d3 d) is associated with a typical volume required for a molecular shear rearrangement (d is the distance the patch moves per jump). Here Vm ¼ d2 d3 d1 , d2 d3 : dimensions of unit of flow in the direction of flow and perpendicular to the direction of flow (the directions of d2 and d3 both being perpendicular to that of d1 ) and d1 is the distance between moving layers of particles [3] (cf. Fig. 1). We consider a steady, fully developed transport of the (locally amorphous) Si under high loading in a wavy-rough nanotube of a (in mean-averaged outer radius) with the outer interface being a fixed wavy-rough surface: r ¼ a þ ε sinðkqÞ where ε is the amplitude of the (wavy) roughness and k is the wave number: k ¼ 2p=Lt (Lt is the wave length) (cf. Fig. 2). We can have (via the boundary perturbation series method [8]), after using the forcing parameter
j¼
a 2t0
dp dz
(5)
(t0 ¼ 2kB T=VA and dp=dz being the external forcing along the axis of the cylindrical domain or the transport direction)
U_ ¼ U_ 0 sinhðjÞ þ HOT
(6)
with the small wavy-roughness effect being the first order perturbation which is rather small and thus neglected (HOT means the _ ) is obtained higher order contributions [8]). Here the shear rate (U by Fig. 1. Schematic (plot) of an Eyring's microscopical flow [3]: Flow proceeds by the motion of (composite) particles into holes left open by neighboring ones. Eyring proposed [4] that shear occurs along sets of parallel planes. The distance between these planes is indicated by the symbol d1 . The shear force per unit area is indicated by f. A patch of atoms or molecules whose cross-sectional area is d2 d3 shift or jump as a unit on either side of the shear plane. The distance the patch moves per jump is d.
dv _ ¼ ¼ Vv,n U dn
(7)
with n≡V½r a εsinðkqÞ=jV½r a εsinðkqÞj being the unit normal of the general interface [8] which is between the confined Si
310
K.-H.R. Chu / Vacuum 145 (2017) 308e311
Fig. 2. Schematic of a cross-section (mean radius being a) with wavy-rough wall. ε is the amplitude of the wavy-roughness and the wave number of wavy roughness is k (say, which is 10 here).
and the environment. The temperature-dependent (referenced) shear rate [3,4,6] is
U_ 0 ¼ 2
VA Vm
kB T EA exp ; h kB T
(8)
which is a function of temperature, the activation energy (EA ), the activation volume (the characteristic volume Vm is presumed to be the same as VA here) and the length scale under confinement. L0 n0 in Eq. (4) is temperature dependent and the value could be traced before [3e7] or from equation (3) as well as equation (1). 3. Results and discussion We noticed that above flow (force) law is highly nonlinear. There are difficulties in selecting parameters for experimental condition or process [1], such as dp=dz and VA . Nevertheless, to obtain a qualitative viscosity role only, we can fix all necessary parameters for corresponding temperature (T). The main task is to fix the value of j by prescribing a and dp=dz with different temperatures. The _ . After all these, the remaining shear viscosity is calculated via t=U in equation (4) is the unique relationship between VA and T for a fixed t. Note that most of the mathematical derivations could be found before [3e8]. We start to explain Johannes et al.’s puzzles or anomalies [1] considering the tuning of activation energy as well as activation _ can be related to the volume. To be specific the rate of strain (Y) stress (s) via the application of high-temperature absolute reaction rate theory [3,4,6].
V kB T EA VA s exp sinh ; Y_ ¼ 2 A h Vm kB T 2kB T
(9)
where VA is the activation volume, Vm ¼ d2 d3 d1 , d2 d3 : dimensions of unit of flow in the direction of flow and perpendicular to the direction of flow (the directions of d2 and d3 both being perpendicular to that of d1 ), d1 is the distance between moving layers of particles, EA is the activation energy, T is the absolute temperature with kB being the Boltzmann constant and h the Planck constant. We shall presume VA ¼ Vm here. Using the experimental parameters [1] we have for the high energy ion irradiation case (after 5 1016 cm2 300 keV Arþ irradiations) after taking ion backscattering, surface roughness or sample topography, etc. into account: the temperature for one selected point of the Si nanowire being around (in average) 700 K (which is higher than 300 C or 573.16 K due to absorption of highenergy ion irradiations), the activation energy being around (in average) 4 1019 J, the stress (considering the reported range
(which is of the order of magnitude about 100 GPa) of elastic modulus or effective Youngs modulus for Si nanowire [9]) being around (in average) 6 1010 Pa, and the activation volume being around (in average) 2 1030 m3 (that is reasonable considering Eyring's defect-volume induced flow [3,4,7]). Above selected pa_ rameters give us Yz0:0077 sec1 from equation (6). The small deformation rate agrees with the measurements [1] (cf. Fig. 2 therein). Meanwhile with the experimental parameters [1] we have for the low energy ion irradiation case (after 5 1016 cm2 100 keV Arþ irradiations) after taking ion backscattering, surface roughness or sample topography, etc. into account: the temperature for one selected point (the same as high-energy case) of the Si nanowire being around (in average) 350 K (which is a little higher than the room temperature or 298.16 K due to absorption of low-energy ion irradiations), the activation energy being around (in average) 1 1019 J, the stress (considering the reported range (which is of the order of magnitude about 100 GPa) of elastic modulus or effective Youngs modulus for Si nanowire [9]) being around (in average) 3 1010 Pa, and the activation volume being around (in average) 1030 m3 (that is reasonable considering Eyring's defect-volume induced flow [3,4,7]). Above selected parameters give us _ Yz4:6611 104 sec1 from equation (6). This rather large deformation rate can explain the possible anomalous flows of the Si nanowire (subjected to 5 1016 cm2 100 keV Arþ irradiations) which also agrees with the measurements [1] (cf. Fig. 2 for the clear shrinking and slight broadening of the Si nanowire therein or cf. figure 4 and in the Supporting Information: the Si nanowires look as though they were molten). Furthermore in Ref. [1] electron backscattering diffraction (EBSD) images were made to confirm that the samples irradiated at 300 C remained crystalline and those irradiated at room temperature were amorphized. Nevertheless we remind the readers that in Ref. [10] Pecora et al. irradiated a (Si nanowire) sample with a 300 keV Ge beam at a dose of 1015 cm2. The energy is such that the beam is passing through for the entire length of the Si nanowire. Their results show that the totally amorphous region flows because of the very-large deformation due to the irradiation [10]. Thus the observed amorphization along the Si nanowire in Ref. [1] is relevant to the large (plastic) flow of the Si nanowire as we just calculated or illustrated above. We remind the readers that a comprehensive discussion of the influence of intensive plastic deformation on the physicomechanical properties of nanomaterials can be found in Ref. [11]. Our calculated data (considering the trend) show that generally the larger the activation energy is then the higher is the shear viscosity for the same temperature once the forcing parameter is fixed. To be specific an impinging ion depositing its energy close to the surface can heat the material (then increase the local temperature for the same site) to lower the viscosity and enable normal viscous flow. 4. Conclusion To give a brief summary, we have adopted the verified theory of absolute reaction (originates from the quantum chemistry approach) to explain the anomalous flow for Si nanowires irradiated with 100 keV (at room temperature regime) Arþ ions as well as the observed amorphization along the Si nanowire in Ref. 1 which is relevant to the large (plastic) flow of the Si nanowire as we just calculated or illustrated here. After some theoretical derivations we also can calculate the shear viscosities for flowing Si in the bulk and in nanodomains (our numerical results for a test case were compared with previous experimental data [8] and the good fit inbetween verifies our present approach). Meanwhile we give
K.-H.R. Chu / Vacuum 145 (2017) 308e311
numerical predictions for the shear viscosity of flowing Si in nanodomains. In general our results show that once the activation energy is higher then we have larger shear viscosity for the same temperature once the force parameter is fixed. Confirmation of our prediction for nanoscale shear viscosities of flowing Si must await experiments. We shall investigate other interesting issues [11e17] in the near future.
[10]
Acknowledgements
[11] [12]
The only author thanks Referees for their crucial comments. References [1] A. Johannes, S. Noack, W. Wesch, M. Glaser, A. Lugstein, C. Ronning, Nano Lett. 15 (2015) 3800e3807. [2] H. Trinkaus, A.I. Ryazanov, Phys. Rev. Lett. 74 (1995) 5072e5075.
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