ABSTRACT The size-dependent bending modulus of nanotubes, which was widely observed in most existing three-point bending experiments [e.g., J. Phys. Chem. B 117, 4618–4625 (2013)], has been

tacitly assumed to originate from the shear effect. In this paper, taking boron nitride nanotubes as an example, we directly measured the shear effect by molecular dynamics (MD) simulations

and found that the shear effect is not the major factor responsible for the observed size-dependent bending modulus of nanotubes. To further explain the size-dependence phenomenon, we

abandoned the assumption of perfect boundary conditions (BCs) utilized in the aforementioned experiments and studied the influence of the BCs on the bending modulus of nanotubes based on MD

found in the aforementioned experiments. To capture the physics behind the MD simulation results, a beam model with the general BCs is proposed and found to fit the experimental data very

well. SIMILAR CONTENT BEING VIEWED BY OTHERS POLYCRYSTALLINE NI NANOTUBES UNDER COMPRESSION: A MOLECULAR DYNAMICS STUDY Article Open access 03 December 2020 ANALYZING FINE SCALING QUANTUM

EFFECTS ON THE BUCKLING OF AXIALLY-LOADED CARBON NANOTUBES BASED ON THE DENSITY FUNCTIONAL THEORY AND MOLECULAR MECHANICS METHOD Article Open access 28 March 2024 ULTRASMALL SINGLE-LAYERED

NBSE2 NANOTUBES FLATTENED WITHIN A CHEMICAL-DRIVEN SELF-PRESSURIZED CARBON NANOTUBE Article Open access 11 January 2024 INTRODUCTION In the past decades, the discovery of the superior

mechanical and other physical properties in quasi-one-dimensional tubular nanomaterials such as carbon nanotubes (CNTs) and boron nitride nanotubes (BNNTs) have triggered great interest in

their possible engineering applications. For example, owing to their high stiffness and strength, low density and large aspect ratio, CNTs and BNNTs are proposed to be the ultimate material

for the use as nanomechanical resonators for a variety of applications1,2,3. In addition, the extremely high elastic modulus of CNTs and BNNTs reported in the theoretical and experimental

studies suggests that, compared with the conventional nanofibers, CNTs and BNNTs can be regarded as a better reinforcement for the nanocomposites4,5,6. Thus, to make CNTs and BNNTs be

successfully employed in the aforementioned applications, a better understanding of their mechanical properties is required. To characterize the mechanical properties of nanotubes a variety

of experimental approaches have been proposed7,8, among which the three-point bending (TPB) test conducted with the atomic force microscope (AFM) is widely used9,10,11,12,13,14,15,16. In

such AFM-based TPB test nanotubes are deposited onto a stiff substrate with a topographical pattern, such as polished porous aluminium oxide membranes or silica gratings patterned with

trenches (see Fig. 1a). As a result, some nanotube samples can occasionally lie over pores or trenches. During the TPB test the midpoint of the suspended portion of nanotubes is subjected to

a downward force applied by the AFM tip, which will induce the transverse deflection of nanotubes. Force (_F_)-displacement (_δ_) curves are recorded, and the bending modulus _E__b_ thus

can be calculated directly from the slope of the _F_-_δ_ curve together with the second moment of area _I_ and the suspended length _L_ by using following equation11,12,13,16 Theoretically,

the bending modulus of nanotubes obtained from the TPB test is expected to be equivalent to the Young’s modulus measured by the direct tensile test. However, in contrast to the result of the

tensile test17 a unique size-dependent bending modulus was reported in most existing TPB tests of CNTs and BNNTs9,10,11,12,13,14,15. To explain the size-dependent elastic modulus of

nanotubes observed in the TPB tests, a transverse shear theory was proposed by Salvetat _et al_.11 initially in 1999 and then widely adopted by many other researchers12,13,14,15. By tacitly

assuming that the size-dependence phenomenon originates from the transverse shear effect, the “curve fitting technique” was used to explain the size-dependence of the bending modulus.

However, the shear modulus obtained by fitting the shear theory to the experimental results was usually found to be two orders of magnitude lower than the results measured via the torsion

tests18 and the results calculated by the theoretical simulations19. This discrepancy makes us believe that a direct measurement of the transverse shear deformation is desired and some new

theories may need to be formulated to reveal the physics behind the size-dependence of the bending modulus observed in the TPB tests. In addition, it is noticed that Eq. 1 is derived based

on the classical Euler-Bernoulli beam model with the assumption that the ends of the beam are perfectly fixed. But in reality, we can see from Fig. 1a and b that in the TPB experiments a

nanotube is usually deposited onto the surface of a substrate9,10,11,12,13,14,15,16. Under this circumstance, only a few rather than all atoms in the portion attaching to the substrate are

blocked, so the ends of nanotubes may not be perfectly restricted. Therefore, the perfectly fixed ends assumed in the previous studies9,10,11,12,13,14,15 cannot exactly describe the real BCs

of the nanotubes tested in the TPB tests. We need to abandon the assumption of the perfectly fixed ends and study the influence of BCs on the bending behaviours of nanotubes. Motivated by

these ideas, in this paper molecular dynamics (MD) simulations are performed to qualify the influence of the imperfect BCs on the bending behaviours of nanotubes. By changing the number of

the blocked atoms at the ends of nanotubes, we reveal the effect of the imperfect BCs on the bending modulus and the atomic displacements of both single-walled (SW) and multi-walled (MW)

nanotubes. Similar to the influence of the shear effect, when the imperfect BCs are considered the size-dependent bending modulus is also detected for nanotubes. Moreover, by comparing these

two theories quantitatively, we identify the influence of the imperfect BCs as the possible major factor responsible for the strong size-dependence of the bending modulus found in the

experiments for nanotubes10,15. In addition, the Euler-Bernoulli beam model together with the non-ideal BCs was proposed to account for the physics of the observed phenomena. This modified

beam model is also found to fit the size-dependent bending modulus of nanotubes measured in the TPB experiments10,15 very well. SIMULATION METHOD In the present study, we take the BNNT as an

example to investigate the effect of the BCs on the bending properties of the tubular nanostructures. To conduct the computational calculations, the entire nanotube was divided into three

sections, i.e., (1) the boundary layers at the two ends, which correspond to the portion of the nanotubes attaching to the substrate, (2) the moving layer at the middle of the nanotube,

which is used to apply the equivalent transverse displacement loads produced by the AFM tip, and (3) the free layers, which are between the moving and boundary layers. The free layers

together with the moving layer are equivalent to the suspended portion of the nanotubes in the AFM-based TPB test. In this study, the classical MD simulations were employed to investigate

the bending behaviours of BNNTs. In the present MD simulation studies, the interactions between the boron and nitrogen atoms were described by Tersoff potentials20, where the parameters were

adopted from21 and have been successfully employed to evaluate the mechanical properties of SW BNNTs21. Here, the energy-minimized configuration of BNNTs was obtained via the conjugate

gradient method. Then MD simulations were performed with the following procedure. First, the BNNT was completely relaxed for a certain period (20 ps was used in this work) to minimize the

internal energy and reach an equilibrium state. In doing this, the NPT ensemble (constant number of particles, pressure and temperature) was employed to maintain a constant temperature with

the aid of the Nosé-Hoover thermostat algorithm22. In addition, the velocity Verlet algorithm with the time step of 0.5 fs was utilized to integrate the Hamiltonian equations of motion

determined by Newton’s second law. Second, all or a few atoms at the boundary layers were blocked to simulate different BCs. Third, to launch the TPB test a displacement control methodology

was adopted to apply the transverse deflection to the nanotubes, i.e., atoms in the moving layer were moved transversely, while all or a few atoms at the boundary layers were fixed (see Fig.

1c). Last, the moving layer was kept fixed and the system was relaxed for 1 ps so as to allow the BNNT to reach a new equilibrium state. By repeating the above process, the nanotubes can be

bent continuously until the required deflection has been obtained. In the present study, all MD simulations were conducted using a large-scale atomic/molecular massively parallel simulator

(LAMMPS)23 with a periodic BC along the axial direction and a constant temperature of 1 K. Here, we selected such a low temperature to simulate the elastic bending behaviour of nanotubes

because other deformation mechanisms occur rarely at the temperature of 1 K. RESULTS AND DISCUSSION Based on the aforementioned MD simulation technique, we will conduct the TPB test for the

BNNTs. The influence of the shear effect and the imperfect BCs on the bending modulus of BNNTs will be quantified. The phenomena observed in MD simulations will be further analysed by using

the continuum mechanics theories. In addition, based on MD simulations and the continuum mechanics theories we will reveal the correlation between the imperfect BCs and the size-dependent

bending modulus of nanotubes reported in existing experiments10,15. INFLUENCE OF THE LOADING RATE ON THE BENDING MODULUS OF SW BNNTS To simulate the similar quasi-static loading condition

utilized in the experiments9,10,11,12,13,14,15, we need to firstly determine the loading rate utilized in the present study. According to some previous studies24,25,26, the mechanical

properties of nanotubes strongly depend on the loading rate. Specifically, some recent studies show that when the loading rate is relatively high, the external load will make the CNTs and

BNNTs lose their structural integrity25,26. Thus, it is necessary to study the influence of the loading rate on the bending properties of BNNTs. To this end, based on the technique proposed

above we have conducted a series of simulations on a (22, 0) zigzag BNNT (the diameter _D_ is ~17.86 Å) with a length _L_ about 190 Å. In the simulations all atoms at the boundary layers

were fully blocked. Moreover, to study the influence of the loading rate on the bending properties, the atoms at the moving layer of the simulated BNNTs moved transversely with different

velocities ranging from 0.06 Å/ps to 0.48 Å/ps. During the TPB simulations we recorded the relationship between the deflection _δ_ and the stored strain energy _U_ of the BNNTs. In Fig. 2 we

plot the _U_-_δ_ curves of the BNNTs under different loading rates. We can see from Fig. 2 that when the loading rate is smaller than 0.12 Å/ps _U_-_δ_ curves almost overlap with each

other, which means that when the loading rate is smaller than 0.12 Å/ps the influence of the loading rate almost can be ignored. Under this situation the loading condition can be

equivalently treated as the quasi-static loading. Moreover, we can see from Fig. 2 that in the small deformation range _U_ of all simulated nanotubes increases almost quadratically with _δ_.

This _U_-_δ_ relationship offers a means to calculate the bending modulus of the nanotubes. In order to obtain the bending modulus, a continuum model should be introduced. Generally, a

nanotube can be most conveniently modelled as a beam model. Based on the Euler-Bernoulli beam theory the deflection _δ_ is linearly proportional to the load _F_ and is expressed by27 _δ_ = 

_FL_3/192_E__b__I_. Thus, during bending the nanotubes the increase of total energy due to the work done by the applied load is _U_ = _Fδ_/2 = 96_E__b__Iδ_2/_L_3. This beam model thus leads

to the bending modulus of the nanotube as By fitting Eq. 2 to the MD simulation results shown in Fig. 2 we can see that, comparing to the case of the quasi-static loading, a relatively high

loading rate will greatly increase the measured bending modulus of nanotubes. For example, we find that the bending modulus of the nanotubes obtained under the loading rate of 0.48 Å/ps is

over two times larger than that obtained under the loading rate of 0.12 Å/ps. Therefore, to avoid the influence of loading rate and simulate the similar quasi-static loading condition

utilized in the experiments9,10,11,12,13,14,15, in the following discussion we choose the loading rate as 0.12 Å/ps. INFLUENCE OF THE BCS ON THE BENDING MODULUS OF SW BNNTS In this

subsection we will start to study the influence of the BCs on the bending properties of SW BNNTs. A similar zigzag BNNT as illustrated in the above subsection was considered here. However,

to study the influence the BCs three different BCs were considered for the nanotubes, i.e., all, half or a quarter of the atoms at the boundary layers of the nanotubes were blocked (see Fig.

3). _U_-_δ_ curves of the simulated BNNTs were recorded during the TPB test. The recorded _U_-_δ_ curves are plotted in Fig. 3 for SW BNNTs with different BCs. We can see from Fig. 3 that

during the bending process the strain energy stored in the nanotubes with partially fixed ends is smaller than that with fully fixed ends. Accordingly, we can expect from Eq. 2 that the

influence of the imperfect BCs will reduce the bending modulus of nanotubes. Indeed, through fitting Eq. 2 to the MD simulation results depicted in Fig. 3 we find that the bending modulus of

the nanotubes with half and a quarter of atoms at the boundary layers being blocked is respectively 35% and 54% smaller than that of the nanotubes with all atoms at the boundary layers

being blocked. To better understand the influence of the imperfect BCs on the bending properties of the nanotubes, in Fig. 4a we show the atomic displacement of the nanotubes with half fixed

BCs during the TPB simulation. Considering the fact that the displacement of the nanotubes is symmetric to the midpoint, in the present paper we only show the results of the left half part

of the nanotubes. From Fig. 4a we can see that for the nanotubes with half fixed BCs, in the portion attaching to the substrate some atoms also have relative torsional displacements to the

blocked atoms. Moreover, during the simulations the atomic stress of the nanotubes with fully and partially fixed ends is shown in Movie 1 and Movie 2, respectively. In the movies the atoms

are coloured according to the atomic stress along the length direction. We can see from these movies that in the region nearby the boundaries the atomic stress of the nanotubes with

partially fixed ends is much smaller than that with fully fixed ends. Accordingly, in the region nearby the boundaries the moment (proportional to the stress) generated in the nanotubes with

partially fixed ends is much smaller that generated in the nanotubes with fully fixed ends. These results suggest that the perfectly fixed ends widely utilized in the previous

studies9,10,11,12,13,14,15 cannot exactly describe the real BCs of the nanotubes in the TPB test and a general BC should be introduced for nanotubes as far as the beam model is employed.

Thus, as shown in Fig. 4b, we assume that the ends of the nanotubes are restricted by a torsional spring with the coefficient of _k_ rather than perfectly fixed. It is known that when a

transverse force _F_ is applied at the midpoint of a beam as shown in Fig. 4b, its deflection should be symmetric to the midpoint. Then, the deflection of the whole beam can be equivalently

represented by its left (or right) half part with the force and BCs as shown by Fig. 4c. Based on the Euler-Bernoulli beam theory, the differential equation of the static deflection of the

beam illustrated in Fig. 4c is expressed as where _E_ is Young’s modulus, _w_ is the transverse deflection and _x_ is the coordinate. The BCs of the beam in Fig. 4c are hinged with a

rotational spring at _x_ = 0 and free to move laterally with an applied force at _x_ = _L_/2, that is Integrating Eq. 3 four times successively with respect to _x_ yields an algebraic

equation with four constant coefficients, which generally can be written as where _c_1 − _c_4 are the constants of integrations. Using Eq. 4 in Eq. 5 gives Substituting Eq. 6 into Eq. 5

yields the analytical expression of the beam profile The maximum deflection _δ_ of the beam at _x_ = _L_/2 is calculated according to Eq. 7 by Thus, when the influence of the imperfect BCs

is considered, Eq. 1 can be rewritten to give the deflection as Based on Eq. 9, in Fig. 4d we show the influence of the spring coefficient _k_ on the elastic modulus ratio _E__b_/_E_. Here,

for the SW nanotubes their second moment of area _I_ can be written as _I_ = _πD_3_t_/8, where _t_ is the equivalent wall-thickness of the BNNT and usually assumed to be 3.4 Å28. We can see

from Fig. 4d that when _k_ is relatively large the ratio _E__b_/_E_ approaches one, which means that when the torsional spring is relatively large the BCs of the equivalent beam model of the

nanotubes can be equivalently treated as the perfectly fixed ends. In this case the bending modulus obtained from the TPB test is equivalent to the Young’s modulus of the nanotube. On the

other hand, the ratio _E__b_/_E_ is found to decrease as _k_ decreases and approaches 0.25 when _k_ is relatively small. Actually, when _k_ ~ 0 the nanotube should be equivalently regarded

as a simply supported beam and, accordingly, the deflection should be27 _δ_ = _FL_3/(48_EI_). Comparing this equation to Eq. 1 we see that in this case the Young’s modulus is four times

greater than the bending modulus predicted based on Eq. 1. In other words, when the torsional spring is relatively small, Eq. 1 which was widely utilized in the TPB

experiments9,10,11,12,13,14,15 will greatly underestimate the Young’s modulus of nanotubes. Moreover, by fitting the present continuum mechanics solution (Eq. 9) to the MD simulation results

(shown in Fig. 3), we can obtain the equivalent stiffness of the torsional spring utilized in the continuum mechanics model of the SW BNNTs with partially fixed ends. The results are shown

in Fig. 4d, where the equivalent stiffness of the torsional spring is found to decrease as the number of the blocked atoms of the BNNTs decreases. INFLUENCE OF THE BCS ON THE BENDING MODULUS

OF MW BNNTS In the above analysis, we have studied the influence of the BCs on the bending behaviours of SW BNNTs. Subsequently, we will investigate how the BCs affect the corresponding

bending properties of MW nanotubes. To this end, we considered a (6, 0)@(14, 0)@(22, 0) triple-walled BNNT, where the tube layers are stacked inversely (see refs 29 and 30 for details). In

order to model the long-range van der Waals (vdW) interaction for the interlayer interaction, the original Tersoff potential energy proposed above is extended by adding a long-range

Lennard-Jones (LJ) 12–6 potential. The LJ parameters employed in the present simulation were adopted from31. Similar to the above SW BNNT, here the length _L_ of the MW BNNT was taken as 190

 Å. It is noted that because the inner layers of MW BNNTs are wrapped by the outermost layer, when an MW BNNT attaches to the substrate, only the atoms at the outermost layer of the MW BNNT

can be blocked. Similar to the above studies of the SW BNNTs, to study the influence of the BCs on the bending properties of MW BNNTs, at the two edges of the outermost layer of BNNTs half

atoms were blocked (see Fig. 5a). The _U_-_δ_ relationship of such MW BNNT obtained in the TPB simulation is plotted in Fig. 5b, where, for the sake of comparison, the results of the MW BNNT

with all atoms at the boundary layers being blocked are also presented. Through fitting Eq. 2 to the MD simulation results depicted in Fig. 5b we find that the bending modulus of the MW

BNNTs with half fixed BCs is 40% smaller than those with fully fixed BCs. Although the influence of the imperfect BCs on the bending modulus is qualitatively the same for both SW and MW

BNNTs, the influence of the imperfect BCs on the MW BNNTs is stronger than that on the SW BNNTs as the imperfect BCs reduce the bending modulus of the SW BNNTs by only 35%. To shed some

light on the observed difference between the SW and MW BNNTs, in Fig. 5c we show the atomic displacement of the MW BNNTs with half fixed BCs during the TPB test. From Fig. 5 we can see that

the atomic displacement of the outermost layer of the MW BNNTs is similar to that of their SW counterparts, where some atoms in the portion attaching to the substrate have relative torsional

displacements to the blocked atoms. As for the inner layers of the MW BNNTs with half fixed BCs, a rotation displacement is also detected in the boundaries of the second and third layers

since the displacements of the atoms in the inner layers of the MW nanotubes are only restricted by the weak vdW forces. It is worth emphasizing that such rotation displacement in the

boundaries of the inner layers of the MW nanotubes was ignored in the previous studies9,10,11,12,13,14,15, where the boundaries of each layer of the MW nanotubes were tacitly assumed to be

completely fixed. Considering the contribution of the additional rotation displacement in the boundaries of the inner layers, it is reasonable to expect that the influence of the imperfect

BCs on the bending properties of the MW BNNTs should be more significant than that on the SW BNNTs, which is consistent with our MD simulation results. COMPARISON BETWEEN THE INFLUENCE OF

THE BCS AND THE SHEAR DEFORMATION As we mentioned in the introduction, a unique size-dependent bending modulus of nanotube was reported in most existing TPB experiments9,10,11,12,13,14,15,

and the shear effect was widely accepted as a possible explanation for such size-dependent bending modulus. However, those studies were all based on the “curve fitting technique” by

initially assuming that the size-dependence phenomenon originates from the shear effect. Thus, a direct measurement of the influence of the transverse shear deformation is still required.

Moreover, we can see from above discussion that in addition to the shear effect the imperfect BCs can also greatly influence the bending modulus of nanotubes and thus can be regarded as

another possible factor that may induce the size-dependent bending modulus of the nanotubes. To reveal the physics behind this size-dependence phenomenon we need to respectively quantify the

influence of these two factors: the transverse shear deformation and the imperfect BCs. Firstly, we will quantify the influence of the transverse shear deformation. According to the

Timoshenko beam theory27, when the transverse shear deformation is considered Eq. 1 can be rewritten to give the total deflection modulus of a beam with perfectly fixed BCs9,11, which is

shown as follows where _f__s_ is a shape factor and equals to a value of 10/9 for a cylindrical beam9,11, _G_ is the shear modulus and _A_ is the cross-sectional area. After expanding _I_

and _A_, we obtain the equivalent bending modulus of a beam considering the transverse shear deformation as follows where _α_ = _d_/_D_ is the ratio of the inner diameter _d_ to the outer

diameter _D_ of nanotubes. The above Timoshenko beam theory (Eqs 10 and 11) converges to the Euler-Bernoulli beam theory when the beam is rigid in shear, i.e., _G_ → ∞. In this case, the

bending modulus is found to be equal to the Young’s modulus. We can see from Eq. 11 that, in terms of the influence of the geometry of the nanotubes the effect of the transverse shear

deformation on the bending modulus of the nanotubes is mainly determined by their diameter-to-length ratio (_D_/_L_). Inspired by this idea, to quantify the influence of the transverse shear

deformation on the bending behaviours of nanotubes, using the simulation technique proposed above we calculated the bending modulus of five (6, 0)@(14, 0)@(22, 0) triple-walled BNNTs, whose

length is respectively 290 Å, 250 Å, 190 Å, 140 Å and 100 Å (the diameter-to-length ratio is accordingly 0.094, 0.128, 0.179, 0.071 and 0.062). Here, to eliminate the influence of the

imperfect BCs all atoms at the boundary layers of the BNNTs were completely blocked. The obtained elastic modulus ratio _E__b_/_E_ of the simulated BNNTs is plotted in Fig. 6 (triangles) as

a function of _D_/_L_. Here, the Young’s modulus _E_ was obtained by fitting Eq. 11 to the obtained MD simulation results. For the sake of comparison, the experimental results reported by

Tanur _et al_.15 are also presented in Fig. 6. In the experiment conducted by Tanur _et al_.15 the MW BNNT suspension was firstly dropped onto clean silica substrates patterned and was

allowed to dry. After this, the TPB test technique as we described in the introduction was utilized to measure the bending modulus of the BNNTs. In their TPB test a nanotube was ideally

assumed as a homogeneous isotropic beam model15. Additionally, the ends of the beam model of the nanotubes were assumed to have perfectly BCs (completely simply supported or fixed). Based on

these assumptions the bending modulus of the nanotubes measured in the TPB test was found to decrease with increasing diameter-to-length ratio, which was explained by the shear theory. We

can see from Fig. 6 that when atoms at the boundary layers of the BNNTs were completely blocked _E__b_/_E_ obtained by MD simulations (triangles) increases with decreasing _D_/_L_, which is

qualitatively similar to the experimental observations (circles)15. In this case the size-dependent bending modulus obtained in MD simulations is attributed to the shear effect, since

through fitting Eq. 11 to the MD simulation results we obtain the shear modulus _G_ as 245 GPa, which agrees well with 250 GPa that obtained through the torsion test32. On the other hand, in

quantity the gap between the present simulation results and the experimental results is huge. For example, when _D_/_L_ drops to 0.1 _E__b_/_E_ obtained in MD simulations of the BNNTs with

fully fixed BCs is 0.9, which means that when _D_/_L_ < 0.1 the influence of the transverse shear deformation almost can be ignored due to the fact that when _D_/_L_ < 0.1 the shear

effect reduces the bending modulus by no more than 10%. However, in the same range of _D_/_L E__b_/_E_ of the BNNTs that measured in the experiment15 still strongly depends on the geometric

size of the BNNTs. The significant difference in quantity between the MD simulation results only considering the shear effect and the experiment results clearly shows that the transverse

shear deformation may not be the main reason for the size-dependent bending modulus that observed in existing TPB experiments of BNNTs15. Thus, caution must be exercised when the influence

of the transverse shear deformation is considered to explain the experimental data. It is noted here that the current conclusion can be extended from the present BNNTs to CNTs. It is known

that BNNTs and CNTs have comparable Young’s modulus and shear modulus32, thus the influence of the shear effect on the the equivalent bending modulus of CNTs is quantitatively close to that

on the BNNTs (see Eq. 11). Then, to quantify the influence of the imperfect BCs on the size-dependence of the bending modulus, we simulated the same five triple-walled BNNTs with different

lengths (or different diameter-to-length ratios). Here, to take the influence of the imperfect BCs into account half atoms at the boundary layers of the outermost layer were blocked (see

Fig. 5a). The results of _E__b_/_E_ obtained in the simulations are plotted in Fig. 6 (squares) as a function of _D_/_L_. Similar to the influence of the shear effect, from Fig. 6 we see

that the influence of the imperfect BCs also lead to the size-dependent bending modulus of the nanotubes, where the bending modulus declines with increasing _D_/_L_. Specifically, the

imperfect BCs are found to exert more substantial influence on the bending modulus. For example, when _D_/_L_ increases from 0.1 to 0.18 _E__b_/_E_ of the nanotube with partially fixed BCs

decreases from 0.55 to 0.24, whereas in this process the result of the nanotube with fully fixed BCs is found to decrease from 0.9 to 0.72. It is noted in Fig. 6 that both theoretical

predictions are qualitatively similar to the experimental observations. But the theory of the imperfect BCs is found to be even closer to the experiment15. The qualitative agreement between

the proposed theory and the experiments shows the relevance of the imperfect BCs to the size-dependent bending modulus of nanotubes observed in the TPB experiments9,10,11,12,13,14,15. In the

meantime, in Fig. 6 a detectable discrepancy is still observed between the experimental results and the simulation results when the effect of the imperfect BCs is considered. Specifically,

comparing with the results (squares) obtained in the present simulations, the bending modulus measured in the experiment15 drops more significantly with increasing diameter-to-length ratio.

This discrepancy can be possibly attributed to the different geometric sizes of the nanotubes between these two studies. Different to the shear theory (Eq. 11), where the bending modulus of

the nanotubes only depend on the diameter-to-length ratio (_D_/_L_) (see Eq. 11), in the theory of imperfect BCs (Eq. 9) the bending modulus show a more complex relationship with the

geometric size of the nanotubes. For example, as we will illustrate in the following subsection, when the influence of the imperfect BCs is considered, the bending modulus of the nanotubes

with longer length will drop more significantly with increasing _D_/_L_. This result is in coincidence with the results observed in Fig. 6, where the nanotubes tested in the experiment15 is

longer than those simulated in the present study. RELEVANCE TO THE EXPERIMENTAL OBSERVATIONS Finally, we will show the correlation between the beam model considering the imperfect BCs (Eq.

9) and the experimental observations. In Fig. 7a and b (circles) we respectively show the experimental results of the elastic modulus ratio _E__b_/_E_ of BNNTs15 and CNTs10 as a function of

their diameter-to-length ratio _D_/_L. E__b_/_E_ of the BNNTs and the CNTs both apparently decreases with increasing _D_/_L_. In Fig. 7a we give a curve fitting to the experimental data of

MW BNNTs using Eq. 9, which is based on the beam model with the imperfect BCs. Here, the MW BNNT is simply treated as a filled cylinder and thus the second moment of area _I_ = _πD_4/64. In

addition, the length _L_ of the BNNTs is 400 nm and approximately the same in all cases15; the Young’s modulus _E_ of the BNNTs is 1800 GPa as predicted in ref. 15. We can see from Fig. 7a

that the present theory of imperfect BCs can well fit the size-dependent bending modulus of BNNTs observed in the experiment15 when the spring coefficient _k_ is 22 _μ_N · _μ_m/rad.

Moreover, we can see from Fig. 7a that when _D_/_L_ is relatively small _E__b_/_E_ tends to one, which means that the BCs of the nanotubes now can be treated to be completely fixed. On the

other hand, when _D_/_L_ is relatively large _E__b_/_E_ gradually approaches to 0.25, which means that in this case the nanotube tested in the AFM-based TPB test should be considered as a

simply supported beam model rather than a fixed beam model. Similarly, in Fig. 7b we give a curve fitting to the experimental data of MW CNTs10 using Eq. 9. Here, the suspended length _L_ of

the CNTs ranges from 240 nm to 420 nm10 and the Young’s modulus _E_ of the CNTs is 1400 GPa, which was obtained based on the fitting technique proposed by Tanur _et al_.15. We can see from

Fig. 7b that Eq. 9 fits the experimental data10 well when _k_ is 0.18 _μ_N · _μ_m/rad. According to the previous studies33, the binding interaction between the CNTs and the silica substrate

could be weaker than that between the BNNTs and the substrate. Such smaller binding energy of the interface between the CNTs and the substrate is a possible reason for the smaller _k_

detected in the TPB test of CNTs. In addition, we can see from Fig. 7b that different to the shear theory (Eq. 11), where the bending modulus of the nanotubes only depend on the

diameter-to-length ratio (see Eq. 11), in the theory of imperfect BCs (Eq. 9) the bending modulus show a more complex relationship with the geometric size of the nanotubes. For example, for

nanotubes with the same diameter-to-length ratio but different lengths, the bending modulus of the nanotubes will increase as the length decreases (see Fig. 7b). Such complex relationship

between the bending modulus and geometric size of nanotubes is a possible responsible for the large scattering of the measured experimental data (see Fig. 7b). CONCLUSIONS The influence of

the imperfect BCs on the bending properties of BNNTs was investigated based on MD simulations. Our results show that the imperfect BCs will reduce the bending modulus of nanotubes. Moreover,

the influence of the imperfect BCs on the MW nanotubes is more significant than that on the SW nanotubes. At the same time, to capture the physics behind the MD simulation results a beam

model with the general BCs was also proposed. In addition, similar to the influence of the previously proposed transverse shear effect, the influence of the imperfect BCs will also induce

the size-dependence of the bending modulus of nanotubes. However, comparing these two theories quantitatively, we find that the size-dependence phenomenon of the bending modulus induced by

the imperfect BCs is more significant than that induced by the shear effect, which suggests that the imperfect BC can be a possible physical origin that leads to the strong size-dependence

of the bending modulus found in the TPB experiments for nanotubes9,10,11,12,13,14,15. Indeed, the modified beam model proposed in the present study that takes the influence of the imperfect

BCs into account is found to fit the experimental data10,15 very well. ADDITIONAL INFORMATION HOW TO CITE THIS ARTICLE: Zhang, J. Size-dependent bending modulus of nanotubes induced by the

imperfect boundary conditions. _Sci. Rep._ 6, 38974; doi: 10.1038/srep38974 (2016). PUBLISHER'S NOTE: Springer Nature remains neutral with regard to jurisdictional claims in published

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ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (No. 11602074). The author also wishes to acknowledge the financial support from Harbin

Institute of Technology (Shenzhen Graduate School) through the Scientific Research Starting Project for New Faculty. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Shenzhen Graduate School,

Harbin Institute of Technology, Shenzhen, 518055, China Jin Zhang Authors * Jin Zhang View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS J.Z.

designed the project, performed the calculations and wrote the paper. ETHICS DECLARATIONS COMPETING INTERESTS The author declares no competing financial interests. ELECTRONIC SUPPLEMENTARY

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