Rolling structure from bilayer nanofilm by mismatch

Jian-Gang Li(李建剛), Xiao-Pi Geng(耿小丕), Qian-Nan Gao(高倩男),Jun Zhu(朱俊), Zhi-Xiang Gao(高志翔), and Hong-Wei Zhu(朱弘偉)

1School of Physics and Electronic Science,Shanxi Datong University&Shanxi Province Key Laboratory of Microstructure Electromagnetic Functional Materials,Shanxi Datong University,Datong 037009,China

2Department of Mathematics and Physics,Hebei Petroleum University of Technology,Chengde 067000,China

3School of Physical Science and Technology,Inner Mongolia University,Hohhot 010021,China

Keywords: nanofilms,nanotubes,surface effects,self rolling

1.Introduction

Over the past decade, nanostructures have received a great deal of attention.[1,2]They were widely applied to nanoelectromechanical systems (NEMS).[3]Nanostructures, such as nanoparticles,[4]nanobeams,[5,6]nanowires,[7]nanofilms,[8]and nanotubes,[9]are significantly different from bulk structures in mechanics.As important components of NEMS, nanostructures were usually used as nanoactuators,[10]nanosensors,[11]nanoresonators,[12]nanogenerators,[13,14]drug delivery devices,[15]etc.But on the other hand, the small size of nanostructures means large surface-to-volume ratio, which induces the size-dependent property in mechanics.[16-19]The size-dependent and surfacemodulated properties in mechanics induce significant difference between nanostructure material and bulk material.[20-22]The applicability of continuous medium mechanics on a nano scale was discussed by Zhao.[23]Cheng[23]improved Thomas-Fermi-Dirac (TFD) model to study the generation mechanism of the residual stress within films.By using Thomas-Fermi-Dirac-Cheng (TFDC) model, the nanotube formation and the corresponding residual stress were studied.[23]

In order to predict the size-dependent elastic properties of nanosized structures theoretically, Gurtin and Murdoch gave a surface elasticity theoretical model.[24]In their theory, the surface slice was considered as an absolutely two-dimensional(2D) slice with elasticity but no thickness.The completely 2D slice with surface elastic coefficient ideally adheres to the bulk-like core of nanofilms.Therefore, their theoretical model is called core-surface model.Miller and Shenoy studied the size-dependent and surface-modulated mechanical behavior of nanofilm by using this core-surface model.[25]In order to develop surface elasticity theory, another theoretical model was established by introducing surface slice thickness,named core-shell model.[26,27]The core-shell model was used to extract nanowire mechanical behavior,[26]nanofilm Young’s modulus,[28]and nanofilm self-equilibrium, and self-bending characteristics, etc.[29]There rises a problem in both coresurface model and core-shell model.These two hybrid models divide a nanofilm into surface area and bulk like area.At interface between these two areas, Young’s modulus changes suddenly.Although this sudden change was delimited by introducing exponential variation Young’s modulus,[30]the Young’s modulus derivative’s sudden change also exists at interface.In order to eliminate the interface,cylindrical theoretical model was introduced to investigate the bending problem,Young’s modulus,Poisson’s ratio,etc.[31,32]

Researchers showed that nanofilms can be rolled by surface stress and interface mismatch to form nanotubes.[33]Zhang and Ionov reported self-rolling polyurethane film.The reversible self-rolling behavior of the bilayer films may allow development of the actuator.[34]Exploiting physical and mechanical properties in rolled-up structures has inspired numerous applications.[35]The research of mechanical self-rolling behavior can promote the development of techniques for manufacturing rolled-up structure.[36]Stoney gave a bending theory about coating.[37]The thinner film thickness was neglected in Stoney theory.Therefore, Stoney formula can work only when film thickness is much less than its substrate thickness.Timoshemko formula introduced film thickness to explain the thermal expansion-induced bending behavior.[38]The bending of bilayer induced by interface mismatch was also explained by Timoshenko formula.[39,40]For a nanofilm with only several nanometers, surface effects are of great importance for film mechanical behavior.[21]Zang and Liu modified Timoshenko formula by considering surface effects.[41]They studied Si-Si nanotube and InAs-GaAs nanotube by using their modified Timoshenko formula.When explaining Si-Si nanotube radius,surface elasticity was chosen as positive.But experimental detection and simulated calculation indicate that Si nanofilm surface elasticity is minus.[19,21]The opposite sign of Si nanofilm surface elastic constant in Ref.[41] needs to be discussed further.Symmetry lowering effect may be a key to solving the surface elastic problem.Cylindrical theoretical model was established to estimate the radius of rolling-up structure.[31]In cylindrical theoretical model, the surface effect and symmetry lowering effect were introduced to modify Timoshenko formula.But the anisotropic rolling structure was missed in Ref.[31].It is not easy to bend a nanofilm into an isotropic bowl-like object.An ultra-thin nanofilm is usually rolled into a nanotube.[42-44]The formed nanotubes are no longer an isotropic structure as explained by Timoshenko’s theory.Cylindrical direction keeps straight and not any bent theoretically.Isotropic Timoshenko formula should be modified anisotropically.Furthermore, cylindrical strain(i.e.the strain in cylindrical direction) affects nanotube radius and tangential elongation strain obviously.Therefore,the theory should be established to explain the formation of nanotubes from nanofilms.Surface elasticity, surface stress,symmetry-lowering effects, and anisotropic structure (nanoscroll structure)should be included in the theoretical scheme.

In this paper,a continuum theoretical model for describing self-rolling of strained bilayers is established.The surface elasticity effect,surface stress effect,symmetry-lowering effect, and the anisotropic structure(nanoscroll structure)are considered.The rolling-up nanotube radius,tangential elongation strain,and cylindrical uniform deformation are discussed.Lattice constant, stress, and strain within nanotube wall are also discussed.

2.Theory and models

Since a nanofilm is rolled into a nanotube, the in-plane directionxcoordinate is replaced by the tangential directiontcoordinate of nanoscroll.The in-plane directionycoordinate is replaced by the cylindrical directionycoordinate of nanoscroll.And the vertical directionzcoordinate is replaced by the radial directionrcoordinate.

Fig.1.Schematic representation of the rolled nanotube,with y-t plane fixed at mid-plane of the bilayer system.

In-plane strains along the tangential direction and cylindrical direction of the nanoscroll are denoted asεtandεy.Radial strain was denoted asεr.Bothεtandεrare dependent onrcoordinate,whileεyis independent ofrcoordinate.The reference plane,i.e.,r=0 is fixed at the mid-plane of the bilayer system,r=(t1+t2)/2 at the top surface,andr=-(t1+t2)/2 at the inner surface,witht1andt2referring to the thickness of nanofilm 1 (inner sublayer) and the thickness of nanofilm 2(top sublayer),respectively.Experiment demonstrates that the rolled nanotube still holds single crystalline structure,and tangential direction lattice parameters vary linearly in their radial direction.[45]Therefore,tangential strain can be given by[42]

wherek=1/Rrefers to the rolling curvature,Rrepresents the radius of nanoscroll,ε1tandε2tare the tangential strains within nanofilm 1 and nanofilm 2,ε1,0andε2,0are the elongation strains of nanofilm 1 and nanofilm 2, respectively,ε0is the elongation of bilayer system, andmis the mismatch between two sub-layers.

The variable quantities to determine nanotube states areε0,k,andεy.And they need to be determined from relaxation conditions.Radial direction strainεrdepends on biaxial relaxation sinceσr=0.According to[32]

wherevbis the effective biaxial Poisson’s ratio,and expressed as

For a film under biaxial loading (here, the film is assumed to maintain plane to determinevbconveniently),σt=σyandσr=0, one can define this parameter asvb=-εr/ε, whereε=εt=εy.Elastic constants under cylindrical symmetry are[31,46]

And then,one can obtain the elastic energy density as follows:

It should be mentioned that the basic principle of elastic energy density is the same as that in Ref.[31],but the expression of Eqs.(10)and(11)are different from those in Ref.[31].This difference is induced by the anisotropic shape of the nanotube.This anisotropy is ignored in Ref.[31].

Total bulk elastic energy can be obtained by integrating bulk elastic energy density in the bilayer system,-(t1+t2)/2≤r≤(t1+t2)/2.Total elastic energy is the sum of bulk elastic energy and surface elastic energy,specifically,

whereVandAare the volume and surface area of the bilayer system (V1andV2correspond to nanofilm 1 and nanofilm 2,respectively).One can obtain the parameterskandε0from the principle of minimum energy:

Stiffness ratioχis dependent on size and surface modulated.Equations (14a) and (14b) describe the rolling curvature and elongation strain in tangential direction,respectively.Elongation strainε0is uniform strain while rolling curvature introduces a gradient of strain.In the cylindrical direction,there is no any rolling(or bending)behavior theoretically.Therefore,cylindrical strainεyremains uniform (independent ofrcoordinate).This anisotropic explanation of nanotubes is different from that of the isotropic Timoshenko formula.[38]Cylindrical strainεyaffects the nanotube radius and tangential elongation deformation.These effects are described in Eqs.(14a)and (14b).The cylindrical strainεycan be taken as a relaxed value (one needs to take the additional condition, i.e.,?Etot/?εy=0).For simplicity,this relaxation can be obtained by uniform deformation (flat condition).Under this condition, there is no difference between tangential direction and cylindrical direction,i.e.,εy=εt=ε.And then,biaxial strain within the flat appears.Then, the elastic energy density reduces into

Biaxial modulus ratioχ′is dependent on size and surface modulated.Owing to mismatch at interface, the relaxation cannot be the same in two sub-layers.Nanofilm 2 cylindrical direction relaxation should beεy+mas shown in Eqs.(18a)and(18b).

The constitutive relation of nanofilms can be obtained by using the derivative of elastic energy density

Owing to straight property, cylindrical strain is identical in each sub-layer and independent ofrcoordinate.But tangential strainεtinduces anr-dependent cylindrical stress via Poisson’s effect as shown in Eq.(20b).

The relationship between lattice parameter and residual strain is

whereais the lattice parameter of nanotube wall,a0is the initial lattice constant without any strain of nanofilms, andεis the residual strain within nanotube wall.

3.Results and discussion

If a compressed (stretched) film grows on another film,lattice atom relaxes outward(inward).On the other hand,lattice constants of these two sub-layers are different from each other.The difference in lattice constant induces mismatch at interface.The relaxation(mismatch)exerts two effects on the bilayer system.One is force(stress)effect,which tends to expand (compress) the bilayer system, and the other is torque(moment) effect, which tends to bend the bilayer system.If one film’s thickness far outweighs the other film’s, the thinner film thickness effect can be neglected,Stoney formula can explain bending behavior well.When the thickness of thinner film is comparable to the that of thicker one,the effect of thinner film thickness influences the bending behavior obviously.For a nanoscaled bilayer system, two sub-layer thickness values are comparable to each other.The bilayer system bending problem induced by interface mismatch cannot be explained by Stoney formula any more.And then, Timoshenko formula, which takes thinner film thickness effect into consideration, came on stage.[38-40]In order to explain the mechanical behavior of nanofilm rolling structures there are still two problems that must be considered.The first problem is surface effect and the corresponding symmetry-lowering effect.The relative number of atoms at surface increases with film thickness decreasing.The atomic environment of surface atoms is different from that of bulk counterpart.Lattice symmetry at surface should also be different from that bulk counterpart.Outside bonding partners are missing, dangling bonds are combined together.The atoms leave from their original equilibrium positions,and then stay in their new equilibrium positions.Lattice structure and interatomic distance are changed.Since elastic constant is very sensitive to the interatomic distance,[26,29]the elastic property is changed.On the other hand,this surface reconstruction expands(or shrinks)the film along in-plane direction and induces reverse relaxation along vertical direction via Poisson’s effect.The symmetry of the film is broken and lowered.Therefore, surface elasticity, surface stress and symmetry-lowering effect should be considered to modify Timoshenko formula.The other problem is the anisotropic structure of nanotubes.For an ultra-thin nanofilm with only several nanometers in thickness, it usually behaves as anisotropic rolling rather than isotropic bending.The isotropic Timoshenko formula should be modified anisotropically.The anisotropic shape of nanotubes should be taken into consideration.

Making use of the theory given in this work, the rolling behavior of bilayer Si-Si,InAs-GaAs,and InGaAs-Cr nanofilms under isotropic lattice mismatch will be discussed.Figure 2 shows the comparison between the data from the current theory and and experimental results.The experimental results in Fig.2 are cited from Ref.[43].The results from our theory (Eq.(14a)) are in good agreement with experimental data as shown in Fig.2.Bulk elastic constants areCα=c11+2c12=219 GPa andCα=c11-c12=102 GPa.Surface parameters are listed in Table 1.The surface elastic constants and surface stresses of Si nanofilm are consistent with those from Ref.[19] The results from the classical theory[43]without surface effect are also shown in Fig.2.Owing to the absence of surface effects,the classical theory does not accord with experiment well.Huanget al.studied Si-Si nanotube radius by considering surface stress and surface elasticity effects,showing that the obtained result is in good agreement with experimental results.[40]The fitted surface elastic constant in Ref.[41] is positive.But at the same time, experimental detection and simulated calculation indicate that Si nanofilm surface elasticity is minus.[19,21]The unreasonable positive surface elastic constant in Zang’s theory needs further discussing.We suggest that there are two main reasons of this trouble,one is the missing of both lowered symmetry and

Other important candidates for rolled-up nanotube are InAs-GaAs and InAGaAs-Cr bilayer systems.The InAs-GaAs and InAGaAs-Cr bilayer systems are underestimated by classical Timoshenko formula.[42,48]The experimental data in Figs.3 and 4 are cited from Refs.[44,47], respectively.Zang and Liu studied the radius of rolled-up nanotube by introducing surface stress effect and surface elasticity effect.[41]The symmetry-lowering effect is introduced to estimate the InAs-GaAs nanotube radius in Ref.[31].In Ref.[31], the anisotropic rolling structure of nanotubes is missed.In the present work, the surface effect, symmetry-lowering effect,and anisotropic rolling structure are considered to explain the rolling behavior of InAs-GaAs bilayer system and InAGaAs-Cr bilayer system.The bulk Young’s moduli and Poisson’s ratios,YInAs=51.42 GPa,vInAs=0.35,andYGaAs=85.06 GPa,vGaAs=0.31,are cited from Refs.[31,41,42,44]and presented in this work.The surface parameters are chosen from Table 1.The InGaAs and Cr nanofilm surface stresses areσInGaAss=1 N/m andσCr

s = 3 N/m, respectively.The bulk Young’s moduli and Poisson’s ratios,YCr=109 GPa,vCr=0.28, andYInGaAs=71 GPa,vInGaAs=0.32,are cited from Ref.[47]and presented from this work.The mismatch strain at interface between InGaAs and Cr is given bym=2.03% in Ref.[47] In computational material science, it is still a challenging topic to investigate the alloy surface elastic constants.Therefore,there is no data of GaAs or InAs, but InGaAs surface elastic constant has been found.The surface stress of GaAs and InAs are consistent with typical values of solid surfaces.[31,41,49]For InAGaAs-Cr bilayer systems,the results of surface stress are in good agreement with experimental results,with the surface elasticity ignored.Using Eq.(14a), we can see from Figs.3 and 4 that the theoretical lines accord well with experimental data.

Tangential elongation deformation is another criterion to explain rolled-up nanotube state.The tangential elongation strain as a thickness function is shown in Fig.5.Interface mismatch exerts two effects on bilayer system, force(stress)and torque (moment) effects.Conventionally, the surface stress difference creates a torque on nanofilm and bends it, while surface stress sum generates a force on the nanofilm and expands(or expresses)it.The difference in mechanical property between nanofilm 1 and nanofilm 2 breaks this convention.The difference in surface stress also expands(or expresses)the nanofilm.And surface stress sum also bends nanofilm.Interface mismatch,as well as surface stress,bends nanofilms only along the tangential direction.On the other hand,they expand(or express) the nanofilm not only in the tangential direction but also in the cylindrical direction.The torque effect does not appear owing to the straight property in the cylindrical direction.The cylindrical uniform deformation is also shown in Fig.5.

Fig.3.Curves of diameter of InAs-GaAs tubes versus GaAs layer thickness,with the thickness of InAs layer fixed at 1.4 ML and experimental data cited from Ref.[44].

Fig.4.Curves of radius of InGaAs-Cr tubes versus Cr layer thickness,with the thickness of InGaAs layer fixed at 9.6 nm and the experimental data cited from Ref.[47].

Fig.5.Tangential elongation strain and cylindrical uniform deformation varying with GaAs thickness.

Cylindrical residual strain affects nanotube diameter and tangential elongation deformation obviously as shown in Figs.6(a)and 6(b).For the sake of discussion,thickness values of the two nanofilms are chosen astGaAs= 4 nm, andtInAs=2 nm, 4 nm, 8 nm, and 16 nm, respectively.Cylindrical direction keeps straight, the uniform residual strain affects nanotube diameter and tangential elongation deformation via Poisson’s effect.According to Eq.(14a), the effect of cylindrical strain on diameter is dependent on the difference in Poisson’s ratio between two nanofilms, (v2-v1).If the two nanofilms have the same Poisson’s ratio,the effect of cylindrical strain on diameter does not exist.But different nanofilms usually have different Poisson’s ratios,especially Poisson’s ratio is influenced by surface effect.[32]No mater whether Poisson’s ratio is equal to each other,the effect of cylindrical strain on tangential direction elongation deformation always exists.Furthermore, if other parameters are fixed, the rolling curvature and tangential elongation deformation will vary linearly with cylindrical strain.This linear dependence can also be found in Eqs.(14a)and(14b).

Fig.6.(a)Nanotube diameter and(b)tangential elongation strain varying with cylindrical strain for different values of thickness tInAs.

The mechanical properties of nanotube are affected by residual strain and stress within nanotube wall.The x-ray microdiffraction analysis demonstrates that the strains within nanotube wall along both tangential direction and radial direction vary linearly with position.[45]In other words, the nanotube wall still holds single crystalline structure.The thickness values of GaAs and InAs sublayers in Fig.7 are the same as those in Fig.6.The lattice parameter can be obtained from Eq.(21a).The initial lattice constants of GaAs and InAs nanofilm areaGaAs=0.5650 nm andaInAs=0.6208 nm,[31,44]respectively.Because of rolling behavior, tangential lattice parameter varies linearly in the radial direction.Cylindrical lattice parameter holds constant at any position and is independent ofrcoordinate owing to its straight state.The lattice parameter in the cylindrical direction is very different from that in the tangential direction.This anisotropic property of lattice parameter is lost in the isotropic Timoshenko formula.Because the coherent lattice parameters,cylindrical lattice parameters, and tangential lattice parameters keep continuous and there is no sudden change at interface.However, the radial lattice parameter behaves sudden change at interface.Initial lattice parameters with no residual strain are different from each other(in the two sub-layers).For different initial lattice parameters, the same lattice parameters in the cylindrical direction and tangential direction, at interface, mean different strains as shown in Figs.8(a) and 8(b).Different in-plane strains induce different radial strains in the two sub-layers.Therefore,radial lattice parameter undergoes a sudden change at the interface.On the other hand,the difference in Poisson’s ratio between two sub-layers also induces the sudden change of radial lattice parameter.Strains along the radial direction and tangential direction are shown in Figs.8(a)and 8(b).The InAs interface means the InAs sub-layer side near the interface, while GaAs interface refers to the GaAs sub-layer side near the interface.The strains at the InAs interface and GaAs interface follow different laws as shown in Figs.8(a)and 8(b),which means the sudden change at interface.The tangential strain is composed of elongation strainεi,0and bending strainrkas shown in Eq.(1a).For a relatively low thickness of GaAs(tGaAs

Fig.7.Lattice parameters within InAs-GaAs nanotube wall versus position at(a)tGaAs=4 nm and tInAs=2 nm,(b)tGaAs=4 nm and tInAs=4 nm,(c)tGaAs=4 nm and tInAs=8 nm,and(d)tGaAs=4 nm and tInAs=16 nm.

Fig.8.Residual strains within InAs-GaAs nanotube wall versus thickness of DaAs in(a)radial direction and(b)tangential direction.

Fig.9.Residual stresses within InAs-GaAs nanotube wall versus GaAs thickness in(a)GaAs layer along tangential direction,(b)InAs layer along tangential direction,(c)GaAs layer along cylindrical direction,and(d)InAs layer along cylindrical direction.

Owing to full relaxation,radial stress does not exist.The tangential stress and cylindrical stress vary linearly in the radial direction owing to rolling behavior.Tangential stresses are shown in Figs.9(a) and 9(b).Various tangential strains fail to induce various cylindrical strains owing to the straight shape.Therefore, various residual stresses emerge as shown in Figs.9(c) and 9(d).The InAs film thickness values in Figs.9(a)-9(d), are the same as those in Fig.3.Stresses at InAs interface and GaAs interface follow different laws (not only in the tangential direction but also in the cylindrical direction), which means the sudden change at interface.Residual stress is dependent on residual strain and film elastic constant.Both different residual strains and different elastic constants induce stress to suddenly change at interface.In other words,the sudden change of strain and the different elastic properties between two sub-layers result in the sudden change of stress.

4.Conclusions

In summary, we present a rigorous method of describing the mechanical behavior of rolled-up nanotubes in the framework of continuum theory.The common phenomenon of isotropic bending caused by isotropic mismatch and isotropic surface stress is usually broken by ultrathin nanofilms.Elastic property of nanofilms is influenced by surface elasticity and surface stress.At the same time,symmetry breaking is also of great importance in studying the elastic property of nanofilms.For the formation of rolled-up nanotube, nanoscroll structure affects nanotube radius and tangential elongation deformation via Poisson’s effect.Timoshenko formula should be modified anisotropically to explain the mechanical behavior of anisotropic rolling structure of nanotubes accurately.Because of the straight shape in the cylindrical direction,the strain and the lattice parameter are uniform along this direction (except the sudden change of strain at the interface).The various tangential strains induce various cylindrical residual stresses via Poisson’s effect,although they are straight along the cylindrical direction.

Acknowledgements

Project supported by the Natural Science Foundation of Shanxi Province, China (Grant No.201901D111316),the National Natural Science Foundation of China (Grant No.11874245), the Teaching Reform and Innovation Pproject of Colleges and Universities in Shanxi Province, China(Grant No.J2021508), and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No.2020MS06007).