Shrinkage and relaxation motions of flexible FePd nanohelices of FePd nanorobots are simulated by a molecular dynamics (MD) model where FePd is a paramagnetic shape memory alloy that can exhibit phase transformation accompanied by softening of the nanohelix under an applied magnetic field (H-field). Two designs of FePd nanorobots are used: (i) a FePd cylindrical head connected to a FePd nanohelix tail and (ii) a FePd nanohelix alone. The geometry and dimensions of the FePd robots are taken after the as-processed FePd nanorobots. In the MD simulation, the FePd head and nanohelix are divided into a number of segmented FePd spheres, each having its magnetic moment. The results of the MD model reveal that upon the applied constant magnetic field, the initial gaps (g = 3nm) between the adjacent turns of the FePd nanohelix are closed, resulting in the total shrinkage (Stot) of 47 nm of the FePd nanorobot. The effects of the applied H-field on Stot are examined by using the MD model and the M-H curve of FePd fitted with Langevin type, resulting in the smaller applied magnetic field leading to the smaller Stot. The results of the MD model provide us with an effective tool in the analysis and design of new nanorobots based on the paramagnetic shape memory alloy of FePd nanohelices that can exert dynamic vibrations on target cells under the oscillating magnetic field.

The use of organic active materials and novel mechanisms1–3 suggests a new direction for designing nanorobots (NRs) based on organic active materials, but the reliability and exact control of such nanorobots for use in the environment are still far from complete. On the other hand, the use of magnetic nanoparticles (NPs) is increasingly popular, particularly the use of Fe2O3 or Fe3O4 nanoparticles with toxic drugs, where the magnetic oxide nanoparticles are used as the carrier of toxic drugs and also as MRI enhancers. To further enhance the MRI image, some researchers are using the magnetic nanoparticles of higher magnetization at saturation (Ms) such as FeCo nanoparticles.4 It is noted that the magnetic navigation of micron-sized robotics has been studied extensively by many researchers.5–13 As to the propulsion of micron-and nano-sized robotics in a body fluid, the use of helical springs under guidance of the applied magnetic field and/or its gradient seems to attract a stronger attention recently5,14–20 where the helical spring shape without its deformation along the spring axis is used as a propulsion unit under the applied rotational magnetic field. The micro/nano-robotics with micro/nano-helical screws used in the past studies are not flexible and they were used only for transporting drugs to target cell sites. If there were flexible nanohelices, which can deform elastically under the applied magnetic field/gradient, such nanorobots (NRs) based on the flexible nanohelices would not only propel under magnetic guidance but also apply mechanical stressing to the cells on which the NRs touch, leading to mechanical stress induced cell death (MSICD).

The Jain group21–24 studied the effects of quasi-static mechanical stress loading on murine mammary carcinoma cell lines. These studies imply that quasi-static compression stress evenly applied externally induces apoptosis and necrosis in cancer cells. Several groups25–28 performed in vivo experiments subjecting cancer cells to dynamic stress loading of very high frequencies (kHz or higher) and were able to use dynamic stress loading to induce cancer cell death. Such high frequency-based stress loading, however, led to heat generation in the cancer cells due to the eddy current effect, and thus, the effects of high frequency dynamic stress loading on cancer cell death are difficult to delineate from those of hyperthermia treatment. Most recently, several groups29–31 reported the use of lower frequency mechanical stress loading on cancer cell death where spinning of magnetic nanoparticles or discs is used to induce the shearing of target cancer cells.

The above literature survey reveals that the application of dynamic normal stress, shear stress, or both on cancer cells may be a new effective approach to induce apoptotic cell death and/or necrotic cell death. If target cells are subjected to narrowly applied mechanical stress loading, then signals of such mechanical stress loading would rapidly propagate through the cytoskeleton network reaching the site of the nucleus, thus damaging DNA and mitochondria structures,32 which is a key process of apoptosis. By an approach similar to this mechanism, Tomasini et al. used the molecular dynamics model to predict the rupture mode of the cell membrane made of lipid bilayers to conclude that the rupture of the cell membrane takes place under both tension and shear loading with the shear mode being more injurious.33 

The above literature survey also reveals that no modeling work on the shrinkage of nanohelices under the applied magnetic field has been attempted yet. Wada and Taya34 demonstrated the rapid shrinkage and relaxation of the macro-sized FePd spring driven by the switching on and off of the applied magnetic gradient using the hybrid mechanism associated with the Fe70Pd30 paramagnetic shape memory alloy (PSMA). The hybrid mechanism is accompanied by sequential events: applied magnetic gradient, magnetic force, stress-induced martensite phase transformation, and reduction of the Young's modulus of Fe70Pd30, resulting in large spring shrinkage. Fig. 1(a) shows the three-dimensional phase transformation diagram of PSMA Fe70Pd30 with temperature (T), stress (σ), and magnetic field (H) as key stimuli of actuation; (b) shows the temperature-dependent Young's modulus, with the stiffer austenite phase being changed to a softer martensite phase at decreasing temperatures.

FIG. 1.

Hybrid mechanism of PSMA Fe70Pd30; (a) three-dimensional phase transformation diagram of the PSMA actuator material under stimuli of stress (σ), temperature (T), and magnetic field (H) where stress-induced martensite (SIM) phase transformation is shown as the dashed line with an arrow sign.35 (b) SIM phase transformation results in softening of Fe70Pd30, and the Young's modulus of Fe70Pd30 is reduced from austenite (80–105 GPa) to martensite (30 GPa) shown as temperature is lowered.35 

FIG. 1.

Hybrid mechanism of PSMA Fe70Pd30; (a) three-dimensional phase transformation diagram of the PSMA actuator material under stimuli of stress (σ), temperature (T), and magnetic field (H) where stress-induced martensite (SIM) phase transformation is shown as the dashed line with an arrow sign.35 (b) SIM phase transformation results in softening of Fe70Pd30, and the Young's modulus of Fe70Pd30 is reduced from austenite (80–105 GPa) to martensite (30 GPa) shown as temperature is lowered.35 

Close modal

The deformation of the macroscopic Fe70Pd30 spring was simulated by using two different finite element methods (FEMs): magnetic and mechanical, suggested by Toi et al.36 resulting in a reasonably good agreement between the FEM and experimental results.34 The PSMA actuators based on the hybrid mechanism and other mechanisms are recently reviewed by Taya et al.35,37,38 If the above macroscopic Fe70Pd30 spring is downsized to a nanohelix made of Fe70Pd30, we may be able to design new nanorobotics based on Fe70Pd30 nanohelices. Allenstein et al.39 demonstrated with an in vitro experiment that single and polycrystalline Fe70Pd30 exhibit good biocompatibility for use in biomedical applications. With such future direction in our mind, we write this paper by focusing on the modeling of the nano-motions of hypothetical Fe70Pd30 nanohelix based nanorobots (i) made of the Fe70Pd30 cylindrical head and Fe70Pd30 tail (nanohelix) and (ii) Fe70Pd30 nanohelix alone under the applied constant magnetic field. PSMA Fe70Pd30 is denoted as FePd hereafter.

We will discuss the geometry, dimensions, and magnetic property of FePd nanorobots in Section I and then the molecular dynamics (MD) model in Section II. The numerical results based on the MD model are given in Section III. Experimental work is stated in Section IV, Discussion is given in Section V, followed by concluding remarks in Section VI.

First, we will discuss the FePd nanorobot made of the cylindrical head and tail (nanohelix). Later on, we will discuss the case of the FePd nanorobot made of only nanohelix. The structure of the FePd nanorobot is schematically shown in Fig. 2 where the spatial parameters which characterize the scale of the nanorobot are also shown: the length LS, outer diameter DO, inner diameter DI, initial pitch p00, and gap g between the nearest neighbor turns in the nanohelix and the diameter Dh and length Lh in the cylindrical head.

FIG. 2.

Nanorobot model made of the FePd cylindrical head and nanohelix with geometrical parameters.

FIG. 2.

Nanorobot model made of the FePd cylindrical head and nanohelix with geometrical parameters.

Close modal

In this model, we fix the spatial parameters as follows: LS 210 nm, DO= 60 nm, DI= 40 nm, and g= 3 nm in the nanohelix and Lh = 200 nm and Dh= 60 nm in the cylindrical head. These dimensions are taken from the as-processed FePd nanohelix and nanorobot made of the FePd cylindrical head and nanohelix (see transmission electron micrographs (TEM) of them in Fig. 3).

FIG. 3.

Transmission electron micrograph (TEM) of (a) as-processed FePd nanohelix and (b) FePd nanorobot made of the cylindrical head and nanohelix.

FIG. 3.

Transmission electron micrograph (TEM) of (a) as-processed FePd nanohelix and (b) FePd nanorobot made of the cylindrical head and nanohelix.

Close modal

In the present model, the FePd cylindrical head and nanohelix are divided into a number of spherical segments, each having magnetic moment. To determine the magnetic moment of a segmented FePd nanoparticle, we will use the magnetization MFePd_VSM of as-annealed FePd nanoparticles (NPs) measured by use of vibrating sample magnetometer (VSM) equipment. The measured magnetization in unit [emu/g] of FePd NPs as a function of the applied magnetic field (H) [Oe] is shown in Fig. 4. From the measured magnetization curve, the magnetic property of the FePd NP is considered to be superparamagnetic or simply paramagnetic where the magnetization of FePd NP reaches its saturation value (Ms) at reasonably low applied H-field. This paramagnetic shape memory alloy of FePd is also observed for bulk FePd.37 

FIG. 4.

The magnetization (M) – magnetic field (H) curves of FePd nano particles; Measured (dashed line), fitted by Langevin function (solid curve) see supplementary material S1.

FIG. 4.

The magnetization (M) – magnetic field (H) curves of FePd nano particles; Measured (dashed line), fitted by Langevin function (solid curve) see supplementary material S1.

Close modal

For simplicity, in this model, we assume first that each FePd NP has reached its magnetization at the saturation (Ms) value upon the applied H-field whose magnitude is as high as 2.24 Tesla [T].

(1.1)

This assumption of the applied H-field of 2.24 T is later changed to a more realistic case; namely, we will consider the case of the lower applied magnetic field, for which we will use the simulated M-H curve of Langevin type (solid curve in Fig. 4) so that the present model can account for the case of the lower applied H-field.

To evaluate the magnetic moments in the FePd nanorobot, we use the density ρ of the FePd material

(1.2)

and the conversion of the CGS unit to the SI unit is

(1.3)

Therefore, magnetization of FePd (MFePd) in unit volume of FePd is given in SI units

(1.4)

Then, the absolute value of the magnetic moment mFePd[Wbm] for the FePd with volume V[m3] is given as follows:

(1.5)

The above data will be used for estimating magnetic moment for a segmented FePd nanosphere which is the key building block of the MD model in Sec. II.

The FePd cylindrical head is discretized into a number of segmented rectangular cuboidal elements, each having its own magnetic moment. The cuboidal elements are further reduced to equivalent segmented spherical NPs which have magnetic moments. Similarly, a nanohelix is also discretized into a number of spherical segments, each having its own magnetic moments. The total number of segmented spheres in the FePd cylindrical head and nanohelix are denoted by NTh and NTsg, respectively (see their formulae in supplementary material S2 and S3). In this section, we will evaluate the following five forces:

  1. Magnetic forces among segmented spheres between the FePd cylindrical head and nanohelix by using the magnetic interaction energy.

  2. Magnetic forces among segmented spheres in the nanohelix only by using the magnetic interaction energy.

  3. Mechanical forces among segmented spheres in the nanohelix by using the mechanical nanohelix energy (harmonic oscillator (HO) potential).

  4. Repulsive forces between segmented spheres across two turns of the nanohelix by using Lennard-Jones (LJ) potential.

  5. Viscous forces arising as a result of the motions of segmented spheres in viscous fluid.

The first four forces are evaluated by taking the derivative of the interaction energy (or potential energy) with respect to position vector r, and then they are incorporated in the Newtonian equation of motion, while the last force is directly incorporated in the Newtonian equation of motion. In the following, we will state the first two forces in Section II A; the segmented spheres in the nanohelix are discussed in Section II B where the use of harmonic oscillator potential and Lennard-Jones potential is also defined to account for the nearest neighboring interactions. Then, the Newtonian equation of motion is stated in Section II C.

In general, the magnetic interaction energy UM(ri,mi;rj,mj) between a magnetic moment mi at position ri and another magnetic moment mj at position rj is given as40 

(2.1)

where the variables rij and rij are the relative position vector and distance between two positions ri and rj, respectively, defined by

(2.2)

Then, the magnetic interaction force FM(ri,mi;rj,mj) between a magnetic moment mi at position ri and a magnetic moment mj at position rj is given by

(2.3)

where the ri is the partial derivative operator defined by

(2.4)

and ex, ey, and ez are the unit base-vectors of the three-dimensional Cartesian coordinate.

By using the general formulae of Eqs. (2.1) and (2.2), the magnetic interaction force FM_head(ri) between the cylindrical head and an i-th segmented sphere in the nanohelix at ri=(xi,yi,zi) is given as

(2.5)

where NTh is the total number of segmented spheres in the cylindrical head and defined by Eq. (S2.9)) and rj is the position vector of the j-th element of the cylindrical head. msg and mgrid are given by Eqs. (S3.10) and (S2.12) respectively in the supplementary material.

Similarly, the magnetic interaction force FM_spring(ri) for the i-th segmented sphere in the nanohelix excluding the i-th segmented element is also calculated as

(2.6)

where NTsg is the total number of segmented spheres in the nanohelix, and it is defined in Eq. (S3.12).

The FePd nanohelix is now subdivided into segmented magnetic moments with total number NTsg, as illustrated in Fig. 5 where two different interactions are given: (i) the interactions among the nearest neighbored segments located at ri1, ri, and ri+1 along the helical coordinate for the nearest neighboring elements are accounted by using harmonic oscillator (HO) potential and (ii) the interactions between the two adjacent segments across the two adjacent turns of the nanohelix, ri and ri + Nsg, is accounted by using Lennard-Jones (LJ) potential. Both HO and LJ potentials will be discussed later.

FIG. 5.

Illustrative interpretation of the positions of the segmented spheres for the interactions of the nearest neighbor HO potential of the three nearest neighbor spheres, ri−1, ri, and ri+1 (shown on the right side) and the repulsive LJ potential between the two nearest neighbor spheres across the adjacent two turns, ri and ri+Nsg (shown on the left side).

FIG. 5.

Illustrative interpretation of the positions of the segmented spheres for the interactions of the nearest neighbor HO potential of the three nearest neighbor spheres, ri−1, ri, and ri+1 (shown on the right side) and the repulsive LJ potential between the two nearest neighbor spheres across the adjacent two turns, ri and ri+Nsg (shown on the left side).

Close modal

1. The mechanical interaction forces between the segmented spheres in the nanohelix

To represent the shrinkage from the initial pitch p00 to the current pitch, p0, the value S of the net shrinkage of one turn of the nanohelix is defined as

(2.7)

where the value of S is constrained by 0Sg and g is the initial gap between the two adjacent turns of the FePd nanohelix. Using the shrinkage S per one turn, the elastic spring mechanical energy UE(S)0 in one turn of the nanohelix is represented by

(2.8)

where k is the spring constant of one turn nanohelix, defined by

(2.9)

where a is the mean radius of the nanohelix defined by Eq. (S3.5), G is shear modulus, and the other parameters are given in Fig. 2. Assuming that the hypothetical FePd nanohelix is in 100% martensite phase and using the Young's modulus of FePd at the martensite phase, E = 30 GPa, Fig. 1(b) and Poisson's ratio (ν) of 0.4, we estimate the shear modulus of FePd as G = 1.07 × 1010 [Pa] in its martensite phase of the FePd material by using the isotropic relation; 2G = E/(1 + ν), the spring constant k of the one turn nanohelix is evaluated as follows:

(2.10)

To represent the elastic spring mechanical energy, we introduce the nearest neighbor harmonic oscillator (HO) potential UHO(ri) for the i-th segmented sphere defined by

(2.11)

where ksg is the spring constant of a single segmented sphere, given by Nsg k, zi is the z-component of position vector ri of the i-th segmented sphere, and it is given by

(2.12)

because we consider the shrinkage only in the axis direction (namely, in this model, z-direction) of the nanohelix.

Then, the force FHO(ri) on the i-th sphere by the nearest neighbor spheres is defined as

(2.13)

For the case of the free nanohelix (p0 = p00), the UHO(r¯i) should satisfy the condition

(2.14)

where r¯i=(x¯i,y¯i,z¯i) is the position of the i-th segmented sphere of the free nanohelix, where θi is given by Eq. (S3.11).

(2.15)

Let us introduce dsg, the distance between the two nearest neighbor spheres in the segmented free nanohelix in the z-direction defined by

(2.16)

The mechanical energy of the nearest neighbor HO potential of Eq. (2.11) for the whole segmented spheres in the one turn nanohelix is defined as

(2.17)

for the case of arbitrary shrinkage of the one turn nanohelix.

Here, we assume that this energy UHO(S) should be equal to the elastic spring mechanical energy UE(S)0 of Eq. (2.8)

(2.18)

Then, we can determine the spring constant (or strength) ksg using Eq. (2.18).

For simplicity, we take the case of the maximum shrinkage S=g to determine the strength ksg as follows:

(2.19)

where g is the initial gap between the two adjacent turns of spring and defined in Section I.

If we take Nsg=25 using Eqs. (2.18) and (2.19), the parameters dsg and ksg are obtained as

(2.20)

We used Nsg = 25 throughout the MD simulation as the use of such a Nsg value makes the MD computational results very stable.

2. Repulsive force between spheres in the nearest neighbor across two turns of spring

To evaluate the shrinkage of the spring by accounting for the magnetic interactions generated by the applied magnetic field H, we take a framework of the molecular dynamics (MD) model, namely, the position ri of the i-th segmented sphere of the segmented nanohelix model is modified to be the time dependent variable ri(t), then the Newtonian equation on the variable ri(t) is solved up to reach the stable configuration of the nanohelix from which the total shrinkage of the nanohelix will be determined.

In the case of strong applied magnetic field H, the shrinkage S becomes very close to the gap g, the i-th sphere collides with the (i±Nsg) spheres of nearest neighbor across the adjacent turns, see the left side of Fig. 5 and these collided spheres would penetrate each other, which is unrealistic.

In order to avoid the unrealistic configuration between the spheres of the nearest neighbor across the adjacent turns, we introduce the reasonable repulsive force between the spheres. To this end, we use the repulsive part of the Lennard-Jones potential which is shown as follows.

The Lennard-Jones potential ULJ(r) between the i-th sphere and (i±Nsg)-th sphere of the nearest neighbor turns is represented as41 

(2.21)

where the variable r is

(2.22)

When the spheres begin to start to collide, then we take the value of dLJ as

(2.23)

where d is the diameter of the nanohelix wire defined by Eq. (S3.3).

The value of strength fLJ in the LJ potential is assumed as

(2.24)

in this model simulation. The reason for the choice of this value is that the MD simulation results are stable with this value.

Then, the repulsive force FLJ(ri) between the i-th sphere and (i±Nsg)-th sphere of the nearest neighbor turns is calculated by

(2.25)

Basically, the shrinkage of the nanohelix is determined by the force balance among four forces: (i) the simulated mechanical spring force FHO(ri) defined by Eq. (2.13) of the nanohelix, (ii) the magnetic forces between the segmented spheres in the head and nanohelix, FM_head(ri) given by Eq. (2.5), (iii) the forces among the segmented spheres of the nanohelix, FM_spring(ri) given by Eq. (2.6), and (iv) the repulsive force FLJ(ri) defined by Eq. (2.25) introduced to avoid unrealistic collapse between the two spheres of the nearest neighbor of the adjacent turns.

To represent the dependence of the total force F(ri) on the other forces of the whole segmented particles except i-th one, here we introduce a new representation F(ri;{r}j) for the total force

(2.26)

where {r}j denotes the set of positions of whole spheres except the i-th one and the whole grid points of the cylindrical head.

To perform the MD simulation, the total force F(ri;{r}j) is also extended to be the time dependent one as F(ri(t);{r(t)}j), then the following Newtonian equation is used for the determination of the shrinkage of the nanohelix for the i-th spheres:

(2.27)

where the first term on the right side of Eq. (2.27) is the Stokes viscous force (which is termed as the fifth force in this MD simulation), R is the radius of a sphere of the segmented nanohelix defined by R = L1/(2Nsg), η is the viscosity of the fluid that surrounds the sphere, and msg is the mass of the sphere defined using weight W1 of the one turn nanohelix shown in Eq. (S3.8) (should not be confused with magnetic moment, msg, in Eq. (2.1)) as

(2.28)

and where the position ri(t) is related to the velocity vi(t) defined by

(2.29)

To solve numerically the Newtonian equations of Eqs. (2.27) and (2.29), time t is discretized in time step (increment) Δt and Eq. (2.28) is rewritten as

(2.30)

To obtain the position vector ri(t+Δt), the semi-implicit Euler method is used as

(2.31)

By taking the initial position of ri(0)=r¯i=(x¯i,y¯i,z¯i) of the free nanohelix, Eqs. (2.30) and (2.31) are repeated iteratively to obtain the stable configuration until time-independent positions ri(t) for the whole NTsg segmented spheres are realized.

As the boundary condition in this model calculation, we fix the 0-th segmented sphere of nanohelix which coincides with the right most part of the cylindrical head in the initial position as

(2.32)

and the motions in x- and y-variables are assumed to be neglected because the axis of the FePd nanorobot is set to be in the same direction as the applied magnetic field H along the z-axis.

To solve the time-sequential evolution of Eqs. (2.30) and (2.31) with adequate accuracy, the time step Δt is set for whole time steps; Δt = 10−11 [s]

The reason for the above choice of Δt is that the MD simulation results become stable if Δt is equal to 1.0 × 10−11 [s] or less. The viscosity η of water defined by η = 10−3 [Pa·s] is used for evaluating the shrinkage of the nanohelix.46 We will use the viscosity of blood, η = 3.5 × 10−3 [Pa·s] (Ref. 4) later on for the MD model simulation.

The MD model simulation is performed for evaluating the shrinkage of the nanohelix of the FePd nanorobot accounting for the four forces defined in Eq. (2.26) and the fifth force of viscous drag in Eq. 2.27); the results of the time evolution of the shrinkage of the nanorobot are shown in Fig. 6 where the upper figures are the configurations (shrinkage and relaxation) of the nanorobot over time-course, and the lower figure shows the total shrinkage (Stot) of the nanohelix of the nanorobot as a function of time Stot = zNTsg(t)z0(t). Because the gap between the two adjacent turns of the nanohelix is very small, g = 3nm (taken after the real dimension of the TEM photo, see Fig. 3), the magnetic interaction between the spheres is very strong. It follows from Fig. 6 that the shrinkage of the nanohelix begins from its right end which is free from the mechanical constraint. At the other end connected to the cylindrical head, the shrinkage begins gradually as shown in Fig. 6. Finally, the nanohelix shrinks completely after the saturation time of around 80 ns with the total shrinkage of 47 nm.

FIG. 6.

Shrinkage and relaxation of the FePd nanorobot made of the cylindrical head and nanohelix. The upper figure has two parts, left showing the sequential configurations during the applied H-field, while the right showing the fully shrunk configuration at time (tE = 1.9) to the fully relaxed (free nanohelix) configuration (tE = 5.0) after the magnetic field is switched off where tE = 0 corresponds to tS = 0.19, and thus tE = t − 0.19. Lower figure: total shrinkage (Stot) shown as the solid line upon the applied constant magnetic field and relaxation shown as the dashed line vs time (t) curve after the magnetic field is off.

FIG. 6.

Shrinkage and relaxation of the FePd nanorobot made of the cylindrical head and nanohelix. The upper figure has two parts, left showing the sequential configurations during the applied H-field, while the right showing the fully shrunk configuration at time (tE = 1.9) to the fully relaxed (free nanohelix) configuration (tE = 5.0) after the magnetic field is switched off where tE = 0 corresponds to tS = 0.19, and thus tE = t − 0.19. Lower figure: total shrinkage (Stot) shown as the solid line upon the applied constant magnetic field and relaxation shown as the dashed line vs time (t) curve after the magnetic field is off.

Close modal

Next, we performed the MD simulation for the case of only FePd nanohelix (without the FePd cylindrical head) subjected to the constant magnetic field under the boundary conditions that the left end of the nanohelix is fixed while the right end of the nanohelix is free. The results are shown in Fig. 7 which indicates that the nanohelix starts shrinking from the free end (right side) arriving at finally full shrinkage over the entire length. The case of the nanohelix only with both ends of the nanohelix being free under the applied constant magnetic field is also simulated, and the results of which are given in Fig. 8. It follows from Fig. 8 that both ends of the nanohelix start shrinking first, ending up with full shrinkage over the entire nanohelix length. The results of the shrinkage–time curves of Figs. 6–8 indicate that the final shrinkages of these three cases remain the same, i.e., 47 nm, and the actuation speed of Fig. 8 corresponding to the free-free nanohelix only is the fastest, while the speed of the actuation for the case of the nanohelix only with one-end fixed (Fig. 7) is the second fastest, and the case of the nanorobot made of the cylindrical head and nanohelix (Fig. 6) is the slowest.

FIG. 7.

The total shrinkage (Stot) of the free nanohelix with left end fixed subjected to the applied magnetic field, showing the time-sequential shrinkage (solid curve) and relaxation (dashed curve) motions, first the free (right) end nanohelix part, ending up with fully shrunk stage over the entire nanohelix length.

FIG. 7.

The total shrinkage (Stot) of the free nanohelix with left end fixed subjected to the applied magnetic field, showing the time-sequential shrinkage (solid curve) and relaxation (dashed curve) motions, first the free (right) end nanohelix part, ending up with fully shrunk stage over the entire nanohelix length.

Close modal
FIG. 8.

Total shrinkage (Stot) of the free nanohelix with both ends free subjected to magnetic field, in the upper part of the figure, the left showing time-sequential configurations for shrinkage of nanohelix with time spanning from tS = 0 to tS = 0.618 while the right showing those from tE = 0.044 to tE = 1.1 for relaxation (expansion) of the nanohelix after H field was off. tE = 0 corresponds to tS = 0.618, and thus, tE = t – 0.618.

FIG. 8.

Total shrinkage (Stot) of the free nanohelix with both ends free subjected to magnetic field, in the upper part of the figure, the left showing time-sequential configurations for shrinkage of nanohelix with time spanning from tS = 0 to tS = 0.618 while the right showing those from tE = 0.044 to tE = 1.1 for relaxation (expansion) of the nanohelix after H field was off. tE = 0 corresponds to tS = 0.618, and thus, tE = t – 0.618.

Close modal

Here, we state three experimental works which are required in support of this molecular dynamics modeling: (i) processing of FePd nanohelices and nanorobots, (ii) processing of FePd nanoparticles, and (iii) measurement of the M-H curve of as-annealed FePd nanoparticles.

The processing route that we are using currently for Fe70Pd30 nanohelices is based on 6 sequential steps: (i) porous alumina anodizing (PAA) template,42 (ii) confined mesoporous SiO2 template within the PAA to generate nanohelical holes,43 (iii) attachment of the electrode made of Au, conducting polymer, and indium tin oxide (ITO) substrate, (iv) removal of the aluminum and barrier alumina layer at the bottom, (v) electro-deposition of Fe70Pd30 into the nanohelical holes in the PAA-SiO2 template,44 and (vi) removal of the PAA-SiO2 template.

As to step (v), we followed the procedure of Haehnel et al.44 who made deposition of Fe70Pd30 straight nanorods and did some modifications in our deposition conditions.

As to the processing route for the combined Fe70Pd30 cylindrical head and Fe70Pd30 nanohelix, we modified the second half part of processing route, namely, the removal of the upper SiO2 template is made to create a hollow cylinder which is deposited with a solid cylinder of Fe70Pd30 by the subsequent electrodeposition while the lower part of the SiO2 template still contains nanohelical holes. The final step is to remove the template materials (PAA and SiO2) by chemical etching. After the final step, we will subject the FePd nanohelices or combination of the cylindrical head and nanohelix to annealing to convert as-processed Fe70Pd30 to the correct PSMA grade where the annealing condition is 770 °C for 5 min in the vacuum condition, followed by rapid quenching to induce the correct PSMA properties of Fe70Pd30.

We processed Fe70Pd30 nanoparticles (NPs) first, followed by annealing of them. Fe70Pd30 NPs are synthesized by reduction of palladium acetylacetonate (II) and succeeding thermal decomposition and reaction of Fe(CO)5 following the polyol method.45 It is noted that the original processing route based on Ref. 45 resulted in Fe43Pd57.

As-synthesized Fe70Pd30 NPs are subjected to annealing at 770 °C for 1 min in the environment of Ar + 5%H2 gas where sample Fe70Pd30 NPs are seated inside a quartz tube which is in turn inside a furnace that can be movable in the horizontal direction to induce the rapid cooling rate. The total weight of the as-annealed FePd nanoparticles is 0.00749 g. The average diameter of the annealed Fe70Pd30 NPs that is measured by small angle x-ray scattering (SAXS) (Rigaku XRD system) is 45 nm.

The magnetization (M) of as-annealed Fe70Pd30 NPs is measured by VSM (Veralab model, Quantum Design) as a function of the applied magnetic field (H) at various temperatures (280, 300, 340, and 380 K). The M-H curve at room temperature (300 K) is used for Fig. 4 (shown as a dashed line) from which Langevin function (shown as a solid line in Fig. 4) is fitted (see supplementary material S1).

We performed a parametric study on the effects of the gap (g) on the total shrinkage of the FePd nanorobot made of only free nanohelix without any constraint under the applied field of 2.24 T. The results of the MD simulation indicate that the total shrinkage of the nanorobot with nanohelix only (Stot) decreases with an increase in the gap (g): Stot = 47, 20, and 10 nm for g = 3, 6, and 9 nm, respectively.

The results of the above MD simulation are based on the assumption that the magnetization of the FePd nanohelix is equal to the saturation magnetization (Ms) of FePd nanoparticles, 120 emu/g, which is realized under the applied H field of 2.24 T or above. If the applied field is less than 2.24 T, we need to use the M values (not Ms) depending on the applied H field. To this end, we fitted the measured M-H curve of Fig. 4 by the Langevin function (see supplementary material S1). By using the Langevin function of the M-H curve of Fig. 4, we examined the effects of the applied H field on the total shrinkage of the FePd nanohelix (free-end condition at both ends), the results of which are shown in Fig. 9 where four different applied H fields are used, H = 2.24, 0.448, 0.224, and 0.112 T. Under the applied field of 0.448 T, the total shrinkage of the FePd nanohelix is close to 47 nm, complete closure of the initial gaps under H = 2.24 T with g = 3 nm. However, if the applied field is reduced to 0.224 T and 0.112 T, the total shrinkage of the FePd nanohelix becomes only 30 nm and 20 nm, respectively, about 67% and 22% of that under applied field of 2.24 T, respectively.

FIG. 9.

The total shrinkage (Stot) of FePd nanorobots made of the nanohelix only as a function of time under various applied H-fields; H = 2.24 (thick solid curve), 0.448 (thin solid curve), 0.224 (dashed curve), and 0.112 T (dashed-dotted curve).

FIG. 9.

The total shrinkage (Stot) of FePd nanorobots made of the nanohelix only as a function of time under various applied H-fields; H = 2.24 (thick solid curve), 0.448 (thin solid curve), 0.224 (dashed curve), and 0.112 T (dashed-dotted curve).

Close modal

We examined the effects of viscosity η on the motions of two different FePd nanorobots: (i) made of the cylindrical head (fixed) and nanohelix tail (free) and (ii) nanohelix only with both ends free to find that the times for final displacement (47 nm) of (i) and (ii) surrounded with water (η = 0.001 Pa·s)46 are nearly 3.5 and 3.3 times faster than that with blood (η = 0.0035 Pa·s)4, respectively. This suggests that the proposed FePd nanorobots even in more viscous fluid (like in blood) provide us with reasonably fast actuation speed.

In view of the current MD model based on Eq. (2.27), the effects of induced mass need to be considered.47–49 However, the real mass of a segmented sphere in the MD model is msg which is equal to 4πR3ρFePd/3 while the induced mass of the segmented sphere is 2πR3ρwater/3 where R is the radius of the segmented sphere, ρFePd is the density of FePd (equal to 8 g/cm3), and ρwater is the density of water (1 g/cm3). Thus, the net mass in Eq. (2.27) is the sum of πR3ρFePd/3 and 2πR3ρwater/3. Therefore, the ratio of the induced mass to the real mass is 2{ρFePd/ρwater}, which is estimated as 1/16, and thus, we can ignore the effect of induced mass in the current MD model.

In this MD model simulation, we used the case of applied H-field coincident with the axis of the FePd nanohelix, namely, along the z-axis, see Fig. 2. If the applied H-field is not along the z-axis, but inclined with the axis of the FePd nanohelix by ϕ, we consider such case here. The results are shown in Fig. 10 where the total shrinkage of the FePd nanorobot made of the cylindrical head and nanohelix tail is plotted in the negative value while the elongation of the nanohelix is plotted along the positive axis of the vertical axis in Fig. 10 where the case of ϕ = 0 corresponds to the results of Fig. 6. The MD simulation results of Fig. 10 are obtained based on the assumption that the nanohelix under the applied magnetic field will deform axially without the bending of the nanohelix. Let us consider the magnetic interaction energy U of the simplified nanohelix subjected to applied H-field whose direction is ϕ with respect to the axis of the nanohelix is given by

(5.1)

Eq. (5.1) is the simplified case of the nanohelix represented by only two identical magnetic moments with its magnetic moment of M that are separated by distance of r. The case of ϕ = 0 and π/2 of Fig. 10 gives the minimum and maximum energy, corresponding to the stable and unstable configurations, respectively. If we let the magnetic energy of Eq. (5.1) be zero, we can solve for the corresponding angle of ϕcrit given by

FIG. 10.

The total displacement of the FePd nanohelix (Stot) in the FePd nanorobot made of the head and tail subjected to applied H-field whose axis makes angle ϕ (in degree) with the axis of the FePd nanohelix. The negative value of Stot corresponds to the shrinkage of the FePd nanohelix while the positive to the elongation of the nanohelix. The results of Stot with ϕ = 0 correspond to those of Fig. 6.

FIG. 10.

The total displacement of the FePd nanohelix (Stot) in the FePd nanorobot made of the head and tail subjected to applied H-field whose axis makes angle ϕ (in degree) with the axis of the FePd nanohelix. The negative value of Stot corresponds to the shrinkage of the FePd nanohelix while the positive to the elongation of the nanohelix. The results of Stot with ϕ = 0 correspond to those of Fig. 6.

Close modal
(5.2)

Eq. (5.2) gives us ϕcrit = 54.7° which is very close to ϕ = 55° in Fig. 10 for which there is no shrinkage nor elongation of the FePd nanohelix. This case of ϕ = 55° corresponds to the case where the total magnetic attractive and repulsive forces are in balance, and thus, the net force becomes zero.

In this MD simulation, we used the Young's modulus of martensite, E = 30 GPa, see Fig. 1(b) with the corresponding shear modulus, G = 10.71 GPa. Thus, the spring constant k is the smallest corresponding to the 100% martensite phase of FePd. At room temperature, FePd is most likely 100% austenite phase, as the martensite start temperature (Ms) of bulk FePd as shown in Fig. 1(b) is 4 °C.36 We ran the MD simulation for the case of the 100% austenite phase of FePd for which E = 90 GPa and G = 32.13 GPa corresponding to the highest value of spring constant k. The results of total shrinkage (Stot) of the FePd nanohelix are shown as a function of time in Fig. 11 where two cases of G = 10.71 GPa (100% martensite) and 32.13 GPa (100% austenite) are shown as solid and dashed curves, respectively. It follows from Fig. 11 that the maximum shrinkage of the FePd nanohelix is reduced from 47 nm to 16 nm if the FePd is in the 100% austenite phase. If we consider the stress-induced martensite (SIM) phase transformation, the maximum shrinkage of the FePd nanohelix even at room temperature may be increased from 16 nm to 47 nm. For under the compression of the FePd nanohelix, the maximum shear stress in the periphery (τmax) of FePd nanohelix wire is increased according to the following formula (neglecting the higher order term):

(5.3)

where d (=10 nm) and D are the diameter of the FePd nanohelix wire and the mean diameter of the nanohelix (D= (D0+ DI)/2 = 50 nm), Nturn is the number of turns (Nturn = 16), umax is the maximum shrinkage of the FePd nanohelix (umax = 16 nm for the 100% austenite phase,) and G is taken as 32.13 GPa (for 100% austenite), and then, τmax is calculated as 40.93 MPa. If we assume SIM taking place in the compressed FePd nanohelix which is shown schematically in the three-dimensional phase transformation diagram, Fig. 1(a), as the vertical dashed line with arrow marker, the phase of the FePd nanohelix in its periphery would become the martensite due to this SIM. The volume fraction of the martensite phase in the compressed FePd nanohelix in the periphery region, however, cannot be determined unless the boundary line between austenite and martensite phases in Fig. 1(a) (shown as the inclined line in the σ – T plane) is known for nano-sized FePd. Despite some unknown properties of the nano-sized FePd, we can conclude that the shear modulus (or Young's modulus) of the FePd nanohelix under compressive stress would be between the 100% austenite and 100% martensite due to SIM, and therefore, the realistic maximum shrinkage of the FePd nanohelix is considered to lie between 47 nm (100% martensite phase) and 16 nm (100% austenite phase), see Fig. 11.

FIG. 11.

The total shrinkage (Stot) of the FePd nanohelix robot based on 100% martensite (G = 10.71 GPa, solid curve) and 100% austenite phase of FePd (G = 32.13 GPa).

FIG. 11.

The total shrinkage (Stot) of the FePd nanohelix robot based on 100% martensite (G = 10.71 GPa, solid curve) and 100% austenite phase of FePd (G = 32.13 GPa).

Close modal

The properties of the macro-sized FePd spring are considered to be different from those of the nano-sized FePd nanohelix. Due to the lack of the mechanical properties of nano-sized FePd (except for the magnetic properties of the nanohelix), in this paper, however, we used the mechanical properties of the macro-sized FePd as the input data for the MD simulations of nano-sized FePd.

Convergence of the MD simulation is examined for the case of the FePd nanohelix with both ends free in terms of the three key parameters: number of segments of the one-turn of the FePd nanohelix (Nsg), strength of LJ potential (fLJ), and time increment (Δt). The effects of Nsg on the total shrinkage vs time relation of the FePd nanohelix are shown in Fig. 12 where the case of Nsg = 16, 21, 25, 31, 35, and 41 is examined. Fig. 12 reveals that the effects of Nsg are considered to be modest with most stable at Nsg = 25, and thus, we used this value as the standard value of Nsg. Next, we studied the effects of the strength of LJ potential (fLJ) on the Stot-time relation, the results of which are shown in Fig. 13 where the cases of fLJ = 10–15–10−22 [J] are examined. Fig. 13 indicates that for smaller values of fLJ (10−20–10−22 [J]), the overpenetration (Stot > 47 nm) takes place, namely, one turn of FePd nanohelix segments is observed to penetrate to the adjacent turn, which is unrealistic. On the other hand, for the larger values of fLJ (=10−15 and 10−16 [J]), the sudden overflow with extremely large values of Stot is observed while the solutions of Stot become stable for fLJ = 10−17 [J], and thus, we employed this as the standard value. Finally, we studied the effects of time increment, Δt, on the Stot-time relation. If Δt is taken be larger than Δt = 10−11 s, for example, Δt = 10−10 s, the solutions become unstable namely, exhibiting a very large number. However, the solutions become stable for the case of Δt < 10−11 s, although the computation time becomes much longer. Therefore, we employed Δt = 10−11 s as the standard value.

FIG. 12.

Total shrinkage as a function of time of the FePd nanohelix with both ends free under various values of Nsg (16, 21, 25, 31, 35, and 41) while using the standard values of fLJ = 1 × 10−17 [J] and Δt = 10−11 [s], indicating that the case of Nsg = 25 gives rise to the most convergent solution (shown in red color line).

FIG. 12.

Total shrinkage as a function of time of the FePd nanohelix with both ends free under various values of Nsg (16, 21, 25, 31, 35, and 41) while using the standard values of fLJ = 1 × 10−17 [J] and Δt = 10−11 [s], indicating that the case of Nsg = 25 gives rise to the most convergent solution (shown in red color line).

Close modal
FIG. 13.

Total shrinkage as a function of time of the FePd nanohelix with both ends free under various values of fLJ (10−15–10−22 [J]) while using the standard values of Nsg = 25 and Δt = 10−11 [s], indicating that the case of fLJ = 10−17 [J] gives rise to the most stable solution (shown in red color line).

FIG. 13.

Total shrinkage as a function of time of the FePd nanohelix with both ends free under various values of fLJ (10−15–10−22 [J]) while using the standard values of Nsg = 25 and Δt = 10−11 [s], indicating that the case of fLJ = 10−17 [J] gives rise to the most stable solution (shown in red color line).

Close modal

The molecular dynamics (MD) model is proposed to simulate the shrinkage and relaxation (elongation) motions of the FePd nanohelix for two designs of FePd nanorobots, (i) FePd cylindrical head and nanohelix tail and (ii) FePd nanohelix only, where both components of the cylindrical head and nanohelix are discretized into a number of FePd segmented spheres, each having its own magnetic moment. The geometry and dimensions of the FePd nanorobots are taken after the as-processed FePd nanorobots shown in Fig. 3. The MD model accounts for five different forces: (i) magnetic interaction forces between segmented FePd spheres of the FePd cylinder head and FePd nanohelix, (ii) mechanical spring force of harmonic oscillator potential, (iii) magnetic interaction forces among the segmented spheres within FePd spring, (iv) repulsive forces between the two adjacent segmented spheres across the two neighboring turns of the spring that can be derived by using Lenard-Jones potential, and (v) drag resistance force due to viscous fluid that surrounds the FePd nanorobot. We performed a parametric study to examine the effects of the gap (g) of nanohelix and applied magnetic field (H) to find that the use of the gap of the as-processed FePd nanohelix, g = 3, gives rise to complete closure of the all gaps, and thus, the total shrinkage (Stot) of the nanohelix is 47 nm. If g is increased to 6 and 9 nm, the values of Stot are decreased to 20 and 10 nm, respectively. The predicted Stot of the nanorobot made of only FePd nanohelix under various applied constant magnetic fields by the MD model suggests that if the applied H-field remains reasonably high, at least above 0.4 T, Stot is close to 47 nm, while those under the H-field of 0.224 and 0.112 T give rise to smaller Stot. We considered two boundary conditions of FePd nanorobots: one end fixed and both ends being free. The first boundary condition is important if we use the FePd nanorobots to be bonded to target cells. In conclusion, the results of this MD simulation give us a new direction in that the proposed FePd nanorobots may be useful in applying the dynamic oscillation force on the target cells under time-varying magnetic field in making the target cells mechanical stress induced cell death.

See supplementary material for the following: S1: The experimental M-H curve fitted by Langevin type curve, S2: Discretization of FePd cylindrical head by spherical segmented elements, S3: Segmented spheres in FePd nanohelix.

The authors (Taya and Xu) are thankful to Nabtesco Corporation and NSF NRI grant (1637535) for their supports. Taya is thankful to the Japan Society for the Promotion of Science for his sabbatical leave at Shinshu University, Japan, where a part of this paper was written. Matsuse and Taya are thankful to Professor M. Kimura of Shinshu University for providing access to his laboratory and the computer system. The authors are thankful to Mr. Sawyer Morgan of University of Washington, Chemical Engineering for his efforts in proof reading the manuscript.

1.
S. P.
Fletcher
,
F.
Dumur
,
M. M.
Pollard
, and
B. L.
Feringa
,
Science
310
,
80
(
2005
).
2.
P.
Ketterer
,
F. M.
Willner
, and
H.
Dietz
,
Sci. Adv.
2
,
e1501209
(
2016
).
3.
M.
Iwaki
,
S. F.
Wickham
,
K.
Ikezaki
,
T.
Yanagida
, and
W. M.
Shih
,
Nat. Commun.
7
,
13715
(
2016
).
4.
P.
Pouponneau
,
J. C.
Leroux
, and
S.
Martel
,
Biomaterials
30
(
31
),
6327
(
2009
).
5.
D.
Schamei
,
A. G.
Mark
,
J. G.
Gibbs
,
C.
Miksch
,
K. I.
Morzov
,
A. M.
Leshansky
, and
P.
Fischer
,
ACS Nano
8
(
9
),
8794
(
2014
).
6.
K.
Ishiyama
,
M.
Sendoh
,
A.
Yamazaki
, and
K. I.
Arai
,
Sens. Actuators, A
91
,
141
(
2001
).
7.
R.
Dreyfus
,
J.
Baudry
,
M. I.
Roper
,
M.
Fermiger
,
H. A.
Stone
, and
J.
Bibette
,
Nat. Lett.
437
(
6
),
862
(
2005
).
8.
S.
Sudo
,
S.
Segawa
, and
T.
Honda
,
J. Intell. Mater. Syst. Struct.
17
,
729
(
2006
).
9.
G.
Kósa
,
M.
Shoham
, and
M.
Zaaroor
,
IEEE Trans. Rob.
23
(
1
),
137
(
2007
).
10.
P.
Tierno
,
R.
Golestanian
,
I.
Pagonabarraga
, and
F.
Sagués
,
J. Phys. Chem. B
112
,
16525
(
2008
).
11.
Z.
Nagy
,
O.
Ergeneman
,
J. J.
Abbott
,
M.
Hutter
,
A. M.
Hirt
, and
B. J.
Nelson
,
Proceedings of IEEE International Conference on Robotics and Automation, Pasadena, CA, 19–23 May 2008
, p.
874
.
12.
S.
Kim
,
F.
Qiu
,
S.
Kim
,
A.
Ghanbari
,
C.
Moon
,
L.
Zhang
,
B. J.
Nelson
, and
H.
Choi
,
Adv. Mater.
25
(
41
),
5863
(
2013
).
14.
J. J.
Abbott
,
K. F.
Peyer
,
M. C.
Lagomarsino
,
L.
Zhang
,
L.
Dong
,
I. K.
Kaliakatsos
, and
B. J.
Nelson
,
Int. J. Rob. Res.
28
(
11–12
),
1434
(
2009
).
15.
A.
Servant
,
F.
Qiu
,
M.
Mazza
, and
K.
Kostarelos
,
Adv. Mater.
27
(
19
),
2981
(
2015
).
16.
P.
Pouponneau
,
J. C.
Leroux
,
G.
Soulez
,
L.
Gaboury
, and
S.
Martel
,
Biomaterials
32
(
13
),
3481
(
2011
).
17.
P.
Pouponneau
,
G.
Bringout
, and
S.
Martel
,
Ann. Biomed. Eng.
42
(
5
),
929
(
2014
).
18.
H. W.
Huang
,
H. S.
Sakar
,
A. J.
Petruska
,
S.
Pane
, and
B. J.
Nelson
,
Nat. Commun.
7
,
12263
(
2016
).
19.
K.
Kobayashi
and
K.
Ikuta
,
Appl. Phys. Lett.
92
,
262505
(
2008
).
20.
M.
Yasui
,
I.
Ikeuchi
, and
K.
Ikuta
, in
IEEE Proceedings of MEMS Paris
, January 29–February 2
2012
.
21.
G.
Cheng
,
J.
Tse
,
R. K.
Jain
, and
L. L.
Munn
,
PLoS One
4
(
2
),
e4632
(
2009
).
22.
J. M.
Tse
,
G.
Cheng
,
J. A.
Tyrrell
,
S. A.
Wilcox-Adelman
,
Y.
Boucher
,
R. K.
Jain
, and
L. L.
Munn
,
Proc. Natl. Acad. Sci. U.S.A.
109
(
3
),
911
(
2012
).
23.
G.
Helmlinger
,
P. A.
Netti
,
H. C.
Lichtenbeld
,
R. J.
Melder
, and
R. K.
Jain
,
Nat. Biotechnol.
15
(
8
),
778
(
1997
).
24.
T.
Stylianopoulos
,
J. D.
Martin
,
V. P.
Chauhan
,
S. R.
Jain
,
B.
Diop-Frimpong
,
N.
Bardeesy
,
B. L.
Smith
,
C. R.
Ferrone
,
F. J.
Hornicek
,
Y.
Boucher
,
L. L.
Munn
, and
R. K.
Jain
,
Proc. Natl. Acad. Sci. U.S.A.
109
(
38
),
15101
(
2012
).
25.
M.
Ogiue-Ikeda
,
Y.
Sato
, and
S.
Ueno
,
IEEE Trans. Magn.
40
(
4
),
3018
(
2004
).
26.
S.
Yamaguchi
,
M.
Ogiue-Ikeda
,
M.
Sekino
, and
S.
Ueno
,
IEEE Trans. Magn.
41
(
10
),
4182
(
2005
).
27.
S.
Yamaguchi
,
Y.
Sato
,
M.
Sekino
, and
S.
Ueno
,
IEEE Trans. Magn.
42
(
10
),
3581
(
2006
).
28.
M.
Domenech
,
I.
Marrero-Berrios
,
M.
Torres-Lugo
, and
C.
Rinaldi
,
ACS Nano
7
(
6
),
5091
(
2013
).
29.
E.
Zhang
,
M. F.
Kircher
,
M.
Koch
,
L.
Eliasson
,
S. N.
Goldberg
, and
E.
Renström
,
ACS Nano
8
(
4
),
3192
(
2014
).
30.
D. H.
Kim
,
E. A.
Rozhkova
,
I. V.
Ulasov
,
S. D.
Bader
,
T.
Rajh
,
M. S.
Lesniak
, and
V.
Novosad
,
Nat. Mater.
9
(
2
),
165
(
2010
).
31.
S.
Leulmi
,
X.
Chauchet
,
M.
Morcrette
,
G.
Ortiz
,
H.
Joisten
,
P.
Sabon
,
T.
Livache
,
Y.
Hou
,
M.
Carrière
,
S.
Lequien
, and
B.
Dieny
,
Nanoscale
7
(
38
),
15904
(
2015
).
32.
N.
Wang
,
J. D.
Tytell
, and
D. E.
Ingber
,
Nat. Rev. Mol. Cell Biol.
10
(
1
),
75
(
2009
).
33.
M. D.
Tomasini
,
C.
Rinaldi
, and
M. S.
Tomassone
,
Exp. Biol. Med.
235
(
2
),
181
(
2010
).
34.
T.
Wada
and
M.
Taya
,
Proc. SPIE
4699
,
294
302
(
2002
).
35.
M.
Taya
,
E.
Van Volkenburgh
,
M.
Mizunami
, and
S.
Nomura
,
Biosinspired Actuators and Sensors
(
Cambridge University Press
,
2016
), Chap. 5.
36.
Y.
Toi
,
J. B.
Lee
, and
M.
Taya
,
JSME J. Comput. Sci. Technol.
2
(
1
),
11
(
2008
).
37.
H.
Kato
,
T.
Wada
,
T.
Tagawa
,
L.
Liang
, and
M.
Taya
, in
Proceedings of the 50th Anniversary of Japan Society of Materials Science, Osaka, Japan, May 21–26, 2001
, Vol.
A
, p.
296
.
38.
T.
Yamamoto
,
M.
Taya
,
Y.
Sutou
,
Y.
Liang
,
T.
Wada
, and
L.
Sorensen
,
Acta Mater.
52
,
5083
(
2004
).
39.
U.
Allenstein
,
Y.
Ma
,
A.
Arabi-Hashemi
,
M.
Zink
, and
S. G.
Mayr
,
Acta Biomater.
9
,
5845
(
2013
).
40.
S.
Chikazumi
,
Physics of Magnetism
(
Krieger Publishing Company
,
Malabar, Floria
,
1964
), translated by S. H. Charap, Chap. 1.
41.
K.
Kremer
and
G. S.
Grest
,
J. Chem. Phys.
92
,
5057
(
1990
).
42.
H.
Masuda
and
K.
Fukuda
,
Science
268
,
1466
(
1995
).
43.
Y.
Wu
,
G.
Cheng
,
K.
Katsov
,
S. W.
Sides
,
J.
Wang
,
J.
Tang
,
G. H.
Fredrickson
,
M.
Moskovits
, and
G. D.
Stucky
,
Nat. Mater.
3
,
816
(
2004
).
44.
V.
Haehnel
,
C.
Mickel
,
S.
Fähler
,
L.
Schultz
, and
H.
Schörb
,
J. Phys. Chem. C
114
,
19278
(
2010
).
45.
N. S.
Gajbhiye
,
S.
Sharma
, and
R. S.
Ningthoujam
,
J. Appl. Phys.
104
,
123906
(
2008
).
46.
Y.
Moriyama
,
H.
Hiyama
, and
H.
Asai
,
Biophysical J.
74
,
487
(
1998
).
47.
W. Y.
Shih
,
X.
Li
,
H.
Gu
,
W. H.
Shih
, and
I. A.
Aksay
,
J. Appl. Phys.
89
(
2
),
1497
(
2001
).
48.
P. I.
Oden
,
G. Y.
Chen
,
R. A.
Steele
,
R. J.
Warmack
, and
T.
Thundat
,
Appl. Phys. Lett.
68
(
26
),
3814
(
1996
).
49.
L. D.
Landau
and
F. M.
Lifshitz
,
Fluid Mechanics
(
Pergamon Press
,
London
,
1959
), p.
95
.

Supplementary Material