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Self-Propulsion of Capped Carbon Nanotubes: A Molecular Dynamics Study Francesco Delogu* Dipartimento di Ingegneria Chimica e Materiali, UniVersita` degli Studi di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy ReceiVed: February 17, 2010; ReVised Manuscript ReceiVed: April 26, 2010
This work focuses on the molecular dynamics of assemblies formed by a carbon nanotube and a small metallic cluster. One nanotube end is capped, and the cluster is located there. When the assemblies are embedded in a liquid phase, this enters the nanotube through its open end. A rapid effusion of solvent from the nanotube interior is induced by submitting the metallic clusters to specific processes consisting of a fast thermal excitation, a coupling with a magnetic field, and a thermally induced phase transition. The solvent effusion generates a thrust on the whole assembly that finally results in a definite linear displacement. A reversible cycle of operation was obtained in all of the cases, which give rise to a novel kind of motor-like behavior on the nanoscale. I. Introduction Harnessing nanoscale phenomena represents one of the grand challenges facing basic sciences.1 In fact, after a century of constant improvements in the capability of observing and rationalizing the interactions between atoms and molecules, scientists are now expected to begin directing such interactions and governing the physical and chemical behavior.1 Within this framework, achieving the control of locomotion of nanometer-sized objects through a fluid environment is perceived as a fundamental advance.2 Two different strategies are currently explored. The first one is essentially inspired to the functionalities of biochemical systems, which often exceed by a great margin those of artificial nanomachines and could be usefully exploited in nanotechnology.1,2 Directing the capabilities of living systems to specific purposes and synthesizing artificial functionalities similar to biological ones are both under scrutiny.3-7 In contrast, the second strategy relies on the design and fabrication of nanodevices based on ideas and methodologies independent of biological examples.1,2 A few of the most important results obtained along this line include the chemically powered linear motion of bimetallic nanorods,8-10 the construction of nanoengines exploiting photochemistry and photocatalysis,11-13 and the fabrication of artificial nanomachines based on metal and semiconductor nanowires as well as on carbon nanotubes (CNs).14-16 Precisely CNs are attracting considerable interest as relatively simple and versatile building blocks for nanomotor architectures,1,17 where their unique electronic and mechanical properties can find valuable application. For example, it has been shown that the motion of individual atoms, molecules, and clusters located in the interior of CNs can be activated by temperature gradients.15,16,18 Similarly, short outer CNs can be moved along the track provided by coaxial inner CNs.15 Linear displacements can also couple with angular ones, the relative rotation of double wall CNs being the result of either mechanical forces operating on the CNs or the combination of helicity and phononic excitation.14,15,19-22 In all of the above-mentioned cases, the use of a CN as a track is a necessary condition for the operation of the nanomotor. However, in various situations such a feature can represent a * To whom correspondence should be addressed. E-mail: delogu@ dicm.unica.it.
limitation, being instead highly desirable to obtain directional motion in fluid environments independent of any track. This would readily allow a number of applications including the transport of chemical systems, the agitation of liquid layers close to surfaces, and the capture specific chemicals in a liquid phase.2 Taking advantage of molecular dynamics (MD) methods, this work shows that relatively simple systems including a CN with a capped end can in principle undergo such kind of track-free directional motion. The fundamental idea, already discussed in a preliminary work,23 is that a CN with a single end-capped could be propelled by the effusion of a fluid initially contained in its interior. The present work aims at showing that effusion can be successfully induced by exploiting the properties of a metallic cluster located in the CN interior, provided that it is accessible to the liquid phase in which the CN is embedded. Here, three different cases are investigated. In the first one, the cluster is submitted to a fast thermal excitation mimicking the conditions attainable with laser irradiation. In the second case, the cluster position inside the CN is governed by means of an external magnetic field. In the third example, the cluster undergoes a thermally activated structural transition resulting in a significant length change. All of the above-mentioned processes induce a relatively fast effusion of the solvent phase initially inside the CN, thus producing the directional motion of the assemblies. In addition, the systems exhibit a considerable degree of reversibility, which allow them to operate as nanoscale motors. Although systems and processes investigated in this work are closer to a possible experimental realization than the very first ones,23 the degree of idealization is still quite high. As a consequence, any attempt of moving from the numerical to the experimental ambit is probably destined to go frustrated. Yet, it is not the intention of this work of directing experimental research. Rather, it could hopefully represent source of inspiration for experimentalists who decided to tackle the challenging questions posed by nanoscale locomotion. The present calculations should be viewed according to such perspective. Their details are given below. II. Numerical Simulations The study of the possible motor-like behavior of the CNcluster assemblies required the preparation of suitable initial
10.1021/jp101456s 2010 American Chemical Society Published on Web 05/12/2010
Self-Propulsion of Capped Carbon Nanotubes atomic configurations. Despite the differences between the investigated systems, their initial configurations were constructed according to a general procedure briefly summarized in the following. First, a rigid single-wall CN with selected helicity, radius, and length was generated. A fullerene-derived cap of suitable size was also generated and placed at one end, leaving the other completely or partially open. Then, a metallic cluster of selected size was placed in the CN interior as closest as possible to the capped end. Afterward, the rigid CN-cluster assembly was embedded in a relaxed liquid phase at the desired temperature and pressure. The system was enclosed by periodic boundary conditions (PBCs) along the three Cartesian directions and equilibrated for about 0.5 ns in the NPT ensemble with number N of atoms, pressure P, and temperature T constant.24 Equations of motion were solved with a fifth-order predictorcorrector algorithm24 and a time step of 1 fs. Coherent with the aim of the work, simulations rely upon the use of relatively simple potential schemes. These are not expected to provide a realistic representation of the forces that operates in experimental systems, but rather an approximate description of general qualitative features. Accordingly, the pure interactions of solvent molecules with each other as well as the cross-interactions between the different species were described by 12-6 Lennard-Jones potentials.24 The potential parameters were either taken from literature or evaluated according to the Lorentz-Berthelot rules.25 These were also applied to the case of cross interactions involving metallic species. To such aim, a Lennard-Jones description of interactions between metallic atoms was used exclusively to allow the application of the above-mentioned combination rules.25 However, the dynamics of metallic cluster was described by resorting to a semiempirical potential based on the second-moment approximation of the tight-binding band energy.26,27 During the relaxation stage in NPT, the solvent atoms or molecules entered the internal volume of the rigid CN as a consequence of the concentration gradient and occupied all the available space, attaining a density similar to the one exhibited by the bulk liquid. The solvent species inside the CN were identified by a simple check of Cartesian coordinates, which were compared with the Cartesian coordinates of the C atoms located at the open end. Once the system exhibited fully relaxed values of potential and kinetic energies as well as of pressure and volume, the rigid CN configuration was replaced by a flexible one. This was done by allowing the C atoms to interact with each other by a Tersoff-Brenner potential, the parameters of which were taken from the literature.28,29 The system was then relaxed for 0.1 ns. The resulting configuration was finally employed to initialize all the simulations concerning the behavior of the CN-cluster assemblies embedded in the liquid phase. Once the assemblies attained their initial configuration, the clusters were submitted to a selected action. Additional details of calculations regarding the different cases are given in the following. For details on interaction potentials, see the Supporting Information. II.1. Fast Thermal Excitation. A CN with (18,0) helicity, radius of about 0.7 nm, and length of about 5 nm was employed together with an Al cluster of 19 atoms in an icosahedral arrangement. The CN-cluster assembly was embedded in an Ar liquid phase at the temperature of 94 K and the pressure of 0.1 MPa. The whole system occupied a cubic volume of about 9 × 7 × 7 nm3. At the end of equilibration, the number of Ar atoms inside the rigid CN amounted to 32, and it did not change after the transformation of the rigid CN-cluster assembly into a
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Figure 1. Initial (top) and relaxed (bottom) atomic configurations of the CN-Al cluster (a), CN-shuttle (b), and CN-Ag nanorod (c) assemblies. Metal and solvent atoms or molecules are shown in dark and light gray, respectively. The assembly orientation is indicated by the Cartesian axis in (a). The arrow and the dashed lines in (b) indicate the hole size in the open end of the outer CN.
flexible one. A relaxed configuration of the flexible CN-cluster assembly is shown in Figure 1a. The cluster was submitted to a rapid thermal excitation aimed at reproducing the one obtainable by ultrashort pulsed laser irradiation.30-32 This is a powerful experimental method that provides a unique tool to prepare and study condensed matter in extreme states.30-32 Optical energy is initially deposited in the electronic degrees of freedom, with no detectable consequence for the lattice.30-32 The subsequent excitation of the phonon spectrum typically occurs within a few femtoseconds.30-32 Thus, the heating rates attainable by irradiation with lasers of suitable intensity is on the order of 100 K ps-1.30-34 The process is extremely complex and cannot be reproduced by MD methods except for the temperature rise following irradiation. This process was simulated by increasing the kinetic energy of Al atoms at a nominal heating rate of 100 K ps-1, which allowed the Al atoms to reach a temperature of 194 K in 1 ps. Heating was interrupted when temperature attained such value. The response of the system to the rapid temperature rise was studied by monitoring the energies of Ar atoms inside and outside the capped CN interior as well as of C atoms belonging to the CN itself. The root-mean square (rms) displacements of Al and Ar atoms were also evaluated.24 II.2. Oscillating Magnetic Field. A rigid-capsule-like single wall CN with (36,0) helicity, radius of about 1.4 nm, and length of about 5 nm was generated. One of its ends was completely capped, whereas a roughly circular hole, centered along the capsule main axis and with radius of about 0.4 nm, was left in the other. A smaller rigid CN with (28,0) helicity, radius of about 1.1 nm, length of about 2 nm, and both ends open was placed inside the CN capsule. The distance between inner and
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outer CN walls is around 0.34 nm, thus comparable with the one of graphite interlayers.35 No chemical species can access the space between the concentric cylindrical walls. A metallic cluster of 65 Co atoms was placed inside the inner CN, completely filling its interior. As a consequence, no chemical species can diffuse from one side to the other. Inner CN and metallic cluster form a shuttle that can move along the main axis of the outer CN capsule. The rigid capsule-shuttle assembly was embedded in a liquid Ar phase occupying a volume equal to about 9 × 7 × 7 nm3 at the temperature of 94 K and the pressure of 0.1 MPa. The shuttle was initially placed as closest as possible to the capped end, which leaved the Ar atoms free to enter the available space inside the outer CN. After 0.5 ns, about 283 Ar atoms entered the CN assembly by thermal diffusion. No significant modification of the atomic configuration is observed when the inner shuttle was left free to move inside the outer capsule. Analogously, no modification occurred when rigid CNs were replaced by flexible ones. The relaxed flexible CN-shuttle assembly is shown in Figure 1b. The cluster is thought here to possess a magnetic dipole moment m directed along the assembly main axis in the sense of positive x coordinates. The dipole moment allows the cluster to interact with a superimposed external oscillating magnetic field. The fundamental idea is that the shuttle could move back and forth along the capsule main axis in response to the force generated by the oscillating field. Being the attrition between inner and outer CNs negligible, the shuttle motion would find no hindrance. In principle, the magnetic field also induces a torque on the magnetic dipole that can determine its reorientation along the field direction. However, it is assumed that this process takes place on a time scale longer than the one required for the force to operate.36-38 To induce the shuttle motion, an intense magnetic field B with modulus B equal to about 3.2 × 104 T and pointing in the x Cartesian direction in the sense of positive x coordinates was used. The metallic cluster was given a magnetic dipole moment m of modulus m equal to about 1.2 × 10-21 J T-1, which roughly corresponds to 130 µB. This value is reasonable on the basis of previous experimental findings regarding Co clusters.39 Of course, the approximation is quite rough and can be motivated only in the light of the highly qualitative character of the present work. The net force F ) ∇(m · B) operating on the cluster40 has modulus F equal to about 3.0 × 10-19 N, large enough to induce the shuttle axial motion. II.3. Thermally Induced Structural Transition. A rigid CN with (36,0) helicity, radius of about 1.4 nm, and length of about 7 nm was constructed. One end of the CN was capped, whereas the other was left open. An Ag nanorod with square cross section of side length s of about 2 nm and total length l of about 5 nm was created starting from a face-centered-cubic (fcc) bulk lattice. The nanorod includes roughly 1200 atoms. Its main axis coincides with the 〈100〉 crystallographic direction, so that the free surfaces have (100), (010), and (001) structure. The CN-cluster assembly was embedded in a liquid CCl4 phase, prepared starting from a simple cubic lattice of CCl4 molecules. These were regarded as spheres interacting with each other via a 12-6 Lennard-Jones potential.41 The system includes about 50 000 molecules in a volume of about 20 × 20 × 20 nm3. Following previous work,42 cross-interactions between Ag and CCl4 were described by a 12-6 Lennard-Jones potential by representing Ag as Xe, which has a similar atomic mass. In fact, the energy exchanges between Ag atoms and CCl4 molecules are substantially governed by mass, the fine details
Delogu of the potential curve determining effects negligible in the first approximation.41,42 The nanorod structure was studied by comparing the mutual distances of groups of neighboring atoms. By simple crystallographic arguments,43 this allows to point out the departure from the fcc crystalline arrangement and to determine the attained crystalline structure. Actually, fcc Ag nanorods with side length shorter than 2 nm can undergo a transition to a bodycentered tetragonal structure (bct) due to surface stresses.44 In fact, these generate a compressive stress along the nanorod main axis large enough to destabilize the initial fcc phase. As the system is left free to evolve, the fcc nanorod transforms into a bct one characterized by elementary cell parameters equal to 0.347 and 0.286 nm. Being dependent on mechanical stresses related to surface contributions, the bct-fcc transition can be governed by applying suitable strain fields.44 However, the transition behavior can be also affected by temperature, as will be shown in the following. Therefore, depending on temperature, the nanorod can exhibit a bct or a fcc structure, with total length difference of about 25%. The transition allows the nanorod to behave as a piston, increasing or decreasing the total space available to solvent molecules inside the CN. Once embedded in CCl4, the assembly was relaxed at 300 K and 0.1 MPa for 0.5 ns. During this stage, about 110 CCl4 molecules entered the assembly. After additional 50 ps, the constraints on the CN-nanorod assembly were removed and the rigid CN was transformed into a flexible one. An atomic configuration of the CN-nanorod assembly is shown in Figure 1c. The lattice structure of the Ag nanorod was periodically changed by submitting the system to cycles of temperature increase and decrease at a nominal rate of 1 K ps-1. The temperature change was prolonged also after either bct-fcc or fcc-bct transitions have started. In fact, this permits the transformation to reach completion in shorter times, thus saving computational power. Heating or cooling were interrupted only after the transformation attained completion. Additional simulations were also carried out on systems analogous to the ones described above, but with no metallic cluster inside the CNs. This allowed ascertaining that the presence of metallic clusters is a necessary conditions to move CNs in the fluid. In fact, in the absence of metallic clusters the different CNs did not undergo any displacement when submitted to the same above-mentioned conditions. III. Results All of the processes undergone by the investigated CN-cluster assemblies produce a relatively fast effusion of the solvent phase outside the outer CN. In turn, such effusion generates a thrust, which allows the whole assembly to cover a certain distance in the axial direction. For clarity, the results concerning the three different cases will be separately discussed. III.1. Response to Fast Thermal Excitation. The numerical simulations aimed at investigating the behavior of the CN-cluster assembly started with the rise of the average temperature TAl of Al atoms up to 194 K in 1 ps, a value significantly lower than the cluster melting point. This was identified by the small plateau exhibited by TAl when plotted as a function of time t as well as by the displacement of atoms from their initial positions and was found to amount to about 320 K. The observed melting point is markedly lower than the ones reported in the literature for similar unsupported clusters,45-47 but differences can be explained by the interactions of the Al cluster with the supporting CN, which can provide alternative paths to the
Self-Propulsion of Capped Carbon Nanotubes
Figure 2. Temperatures TAr of the Ar atoms inside the CN, TCN of C atoms, TAr,out of the Ar atoms outside the CN, and TAl of the Al cluster as a function of time t. The vertical dotted line marks the instant at which the Al cluster is thermally excited to 194 K, whereas the horizontal dotted line indicates the boiling point TAr,bp of Ar atoms. The arrow indicates instead the small plateau, indicating the occurrence of the liquid-gas transition for the Ar atoms inside the CN. Data regard exclusively the stage in which the different temperatures increase.
nucleation of melting.48 At the same time, a temperature value of 194 K is considerably higher than the boiling point of liquid Ar, TAr,bp, which amounts approximately to 138 K.49 Then, the heat exchange between Al cluster and surrounding Ar atoms inside the CN can induce a local evaporation process of the solvent phase. The system response to the rise of cluster temperature was monitored by evaluating the CN temperature TCN and the temperatures TAr and TAr,out respectively of the Ar atoms inside and outside the CN. For simplicity, TAr,out was estimated by considering only the Ar atoms distant less than 1.5 nm from the CN. The TAr, TAr,out, and TCN values are shown in Figure 2 as a function of time t. The TAr,out and TCN curves only describe the stage of temperature rise, so that the last plotted values represent the maximum temperatures reached by the given sets of Ar or C atoms. Data indicate that TAr undergoes a roughly linear increase up to about 160 K, becoming higher than TAr,bp after about 0.45 ps. The occurrence of the liquid-gas transition is clearly pointed out by the small plateau in the TAr curve at the TAr,bp value, by the definite increase of rms displacements of the Ar atoms inside the CN, and by the contemporary linear increase of the pressure inside the CN from 0.1 to 5 MPa in about 50 ps. In fact, the time scale governing the liquid-gas transition of the Ar atoms inside the CN is so short that the system is unable to equilibrate the pressures inside and outside the CN, where Ar atoms are in gaseous and liquid phase, respectively. The Ar atoms outside the CN are not in direct thermal contact with the Al cluster. Therefore, they can only acquire energy through the expansion of gaseous Ar outside the CN and the vibration of C atoms. In both cases, the kinetic energy transfer is relatively fast. However, the liquid Ar phase is considerably efficient in removing the excess local heat via a rapid energy redistribution among all the different species. Dissipation is fast and scarcely influenced by the Nose` thermostat, taking place on a time scale shorter than the one governing the coupling between system and thermostat. As a consequence, the maximum values reached by the average temperatures TCN of C atoms and TAr,out of the Ar atoms outside the CN roughly amount to 128 and 105 K, respectively, which makes the Ar atoms outside the CN unable to undergo the liquid-gas transition. The transient imbalance between the pressures inside and outside the CN determines the relatively fast effusion of the gaseous Ar atoms through the CN open end, with consequent generation of a thrust along the axis of the CN-Al cluster assembly in the direction of its cap, i.e., of negative x Cartesian coordinates. The atomic configuration indicates that most of gaseous Ar atoms have already effused after about 80 ps from
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Figure 3. (a) Number nAr of Ar atoms inside the CN as a function of time t. The vertical dotted line marks the instant in which the Ar atoms inside the CN have undergone the liquid-gas transition. (b) The x coordinate of the CM of the CN-Al cluster assembly, xCM, scaled to the initial CM position, xCM,0, as a function of time t.
the completion of the liquid-gas transition (see Figure S1, Supporting Information). After 100 ps, only two Ar atoms are still inside the CN, as shown in Figure 3a by the plot of the number nAr of Ar atoms inside the CN as a function of time t. The effects of the Ar atom effusion on the CN-Al cluster assembly were monitored by following the x Cartesian coordinate of the center of mass (CM) of the assembly, xCM, as a function of time t. The CM displacement was quantified by scaling xCM to the initial CM x coordinate, xCM,0. As shown in Figure 3b, the quantity xCM - xCM,0 undergoes a definite decrease. Thus, the assembly displacement rate is maximum at the beginning of the effusion process, amounting to about 4.3 nm ns-1, and then diminishes progressively due to both frictional forces in the liquid and effusion deceleration. As a whole, the CN-Al cluster assembly covers a distance of about 0.5 nm in roughly 0.4 ns. Then, the average displacement rate is around 1.2 nm ns-1. The excess kinetic energy possessed by the effused Ar atoms is rapidly removed by the ones surrounding the CN-Al cluster assembly. In fact, the TAr data regarding the effused Ar atoms shown in Figure 2 indicates that TAr drops below the Ar boiling point TAr,bp in about 0.11 ns, which corresponds to a cooling rate of about 200 K ns-1. The temperature TAl of the Al cluster, also shown in Figure 2 for comparison, points out that the fast initial thermal excitation of the Al cluster is followed by a progressive TAl decrease occurring at rates on the order of 150 K ns-1. Once the effusion of Ar atoms has occurred, only two Ar atoms remain inside the CN. However, after only 20 ps the Ar flow reverses its direction and Ar atoms refill the CN-Al cluster assembly under the effect of the concentration gradient. It can be seen from Figure 3a that the nAr increase is quite rapid, the initial value of 32 ( 2 atoms being recovered after about 0.2 ns. The recovery of the initial temperatures requires instead longer times. In particular, TAl and TAr decrease to 94 K after about 1.0 and 0.7 ns, respectively, from the instant in which they took their maximum values. The configuration attained by the system at the end of the reequilibration stage after the effusion of gaseous Ar atoms is substantially indistinguishable from the initial one. It follows that the behavior of the CN-Al cluster assembly can be described in terms of cycles of operation with a reversible sequence of steps. To specifically address the issue of reversibility, three additional consecutive cycles of operation were performed starting from the relaxed system obtained after the very first cycle. To save
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computational time, the Al cluster was thermally excited when the temperature TAr attained values smaller than TAr,bp and the number nAr of Ar atoms inside the CN was equal to 32. As these two requirements were fulfilled, the Al cluster temperature TAl was raised up to 194 K at 100 K ps-1. The result of the aforementioned processes is the net displacement of the CN-Al cluster assembly in the direction of negative x Cartesian coordinates. Four consecutive drops are clearly distinguishable in correspondence of each cycle performed (see Figure S2, Supporting Information). On the whole, the CN-Al cluster assembly has moved about 2.0 nm along the x Cartesian axis in about 2.5 ns, which corresponds to an average displacement rate roughly equal to 0.8 nm ns-1. III.2. Response to an Oscillating Magnetic Field. In the initial configuration, the shuttle formed by the short CN with the Co cluster inside is located close to the capped end of the outer CN and the remaining volume available in the CN interior is occupied by 283 Ar atoms. When the magnetic field B is superimposed to the system, the shuttle moves along the main axis of the outer CN in the direction of positive x coordinates, i.e., of the CN open end. The force operating on the shuttle is sufficiently intense to overcome any possible resistance to compression of the Ar atoms inside the CN. As the space available to Ar atoms progressive decreases, the Ar atoms are pushed outside the outer CN through the hole in its open end. This is large enough to permit the passage of Ar atoms, but not of the shuttle. Thus, it can operate like an almost attrition-free piston defining the volume accessible to Ar in the outer CN. Consequent to the force applied, the shuttle moves at an average rate of about 13 nm ns-1, which allows the effusion process to attain completion within about 0.2 ns. It must be also noticed that the hole is large enough to permit the expulsion of the Ar atoms as a consequence of the shuttle displacement with no detectable increase of pressure inside the assembly. This is an important aspect. In fact, should a pressure develop in the interior of the outer CN, the resulting force would operate against the shuttle displacement limiting the effusion efficiency. As effusion terminates, the magnetic field B is reversed. Correspondingly, the shuttle moves in the direction of negative x coordinates and returns to its starting position close to the capped end of the outer CN. At the same time, Ar atoms enter the available space and refill the CN-shuttle assembly, which attains a final configuration virtually undistinguishable from the initial one. The system has operated a complete cycle. It is here worth noticing that the precision with which the magnetic field B is reversed is of outstanding importance. Indeed, the collision of the shuttle with either the capped end or the partially open one of the outer CN could in principle determine its rupture. Therefore, avoiding such energetic collisions is a necessary condition to ensure the integrity of the CN-shuttle assembly. According to the observations above, the sequence of consecutive stages representing the response of the CN-shuttle assembly to an oscillating magnetic field is completely reversible. This latter issue was demonstrated by performing a simulation spanning three consecutive cycles. The data regarding the position xCM,s - xCM,s,0 of the shuttle center of mass (CM) relative to the initial one, xCM,s,0, and the number nAr of Ar atoms located in the interior of the assembly are shown in Figure 4a as a function of time t. Depending on the direction of the shuttle displacement inside the outer CN, nAr exhibits a periodic change in time roughly between 280 and 20. As the Ar atoms effuse under the effect of the shuttle displacement, a thrust acting on the CN-shuttle assembly is generated along the x Cartesian
Delogu
Figure 4. (a) Position xCM,s - xCM,s,0 of the shuttle CM relative to the initial one, xCM,s,0 (full line), and the number nAr of Ar atoms located in the interior of the assembly (dotted line) as a function of time t. (b) Relative displacement of the assembly CM with respect to its initial position along the x Cartesian direction, xCM - xCM,0, as a function of time t.
Figure 5. Number nbct of Ag atoms with bct arrangement as a function of time t.
direction. Correspondingly, the assembly moves in the direction of negative x Cartesian coordinates. The relative displacement of the assembly CM with respect to its initial position along the x Cartesian direction, xCM - xCM,0, is shown in Figure 4b as a function of time t. Three consecutive displacements are observed. The displacements exhibit approximately the same length and take place in correspondence of the Ar atom effusions. The total displacement undergone by the assembly amounts to about 1 nm. Being this distance covered in about 1 ns, the average displacement rate roughly amounts to 1 nm ns-1. This value is quite smaller than the maximum rate of displacement, which is approximately equal to 4 nm ns-1. III.3. Response to a Structural Transition. The Ag nanorod included in the initial configuration of the CN-nanorod assembly has bct structure. The elementary cell is described by the two parameters c and a, indicating respectively the lattice spacing along the 〈100〉 crystallographic direction and perpendicular to it. The a and c values are equal to about 0.347 and 0.286 nm. The comparison with the lattice spacing of fcc Ag, equal to about 0.408 nm,27 indicates that the bct nanorod is about 25% shorter than the fcc one. Starting from the initial value of 300 K, the system temperature was gradually raised at a nominal rate of 1 K ps-1. The temperature rise induces a corresponding increase of the thermal motion of all the chemical species with no indication of structural modification up to about 370 K. At such temperature, a phase transformation starts involving first the atoms located at the (100) free surface perpendicular to the 〈100〉 crystallographic direction. The transformation rapidly spreads over the neighboring atomic layers, proceeding along the nanorod axis in the direction of the capped CN end. The number nbct of Ag atoms with bct arrangement is shown in Figure 5 as a function of time t. It can be seen that nbct decreases almost linearly up to
Self-Propulsion of Capped Carbon Nanotubes zero, which corresponds to the complete transformation. This is attained in about 12 ps. It is here worth noting that the temperature of 370 K is higher than the CCl4 boiling point, equal to about 350 K.50 Nevertheless, the application of PBCs to the simulation cell determines as usual a significant increase of the transition temperature due to the absence of free surfaces that can act as preferential nucleation sites for the phase transformation.51 In the present case, the effect of PBCs is such that the CCl4 keeps a liquidlike character up to a temperature of about 395 K. Although in principle the CN-nanorod assembly could represent a heterogeneous nucleation site for the molten phase, simulations do not point out any detectable effect on the CCl4 behavior. As the Ag nanorod undergoes the bct-fcc transition, its total length changes roughly from 3.6 to 5 nm. Correspondingly, the total volume available to CCl4 molecules inside the CN-nanorod assembly decreases from about 20 to 12 nm3. As a consequence, the number of CCl4 molecules inside the assembly changes from 112 to 61 in about 12 ps. Correspondingly, the system temperature attains the value of 382 K. Such value was then kept for 10 ps to allow the full CN-nanorod assembly relaxation. Once the bct-fcc nanorod transition has completed, and the nanorod length has reached its maximum value of about 5 nm, the system temperature was decreased at a cooling rate of 1 K ps-1. The nanorod keeps the fcc structure until a temperature of 369 K is attained. At such temperature, the reverse fcc-bct transformation takes place, substantially with the same mechanism of the bct-fcc one. In fact, the phase transformation starts at the free (100) surface and then progressively involves the neighboring atomic planes. The bct structure is recovered within a time interval of about 14 ps, when the system temperature is equal to 355 K. As the Ag nanorod takes the bct structure, the space available to CCl4 molecules inside the CN-nanorod assembly increases again. Correspondingly, the number nCCl4 of CCl4 molecules inside the assembly reattains a value of 113. Therefore, the final atomic configuration is virtually indistinguishable from the initial one. It must be noted that the time intervals associated with phase transformations, either from bct to fcc or from fcc to bct, are quite short. In fact, they range between 10 and 15 ps. In this regard, at least two points are worth noting. First, such short times of phase transformation are observed in the simulation of nanorods of different chemical nature and employing different force schemes.52-54 In all of the cases, a phase transformation front nucleating at free surfaces perpendicular to the main nanorod axis is observed. Second, the short times of transformation are not related to an exceptionally high displacement rate of the phase transformation front. This is equal to about 300 nm ns-1, which is on the same order of magnitude of the displacement rate of transition front in the case of heterogeneous melting under superheating conditions for a variety of systems.51,55-60 Rather, the short times can be associated with the reduced dimensionality of the simulated system. Rich of free surfaces, it easily undergoes nucleation processes determining a rapid increase of structural disorder and favoring local atomic rearrangements. Related to such reduced dimensionality, it should also be noted that the influence of small superheating degrees is considerably marked with respect to the case of bulk materials. Therefore, the rapid phase transformations undergone by the nanorod can be regarded as a manifestation of reduced dimensionality effects. According to the above-mentioned observations, the cycle of operation of the CN-nanorod assembly is completely reversible. Reversibility has been demonstrated by simulating three consecutive cycles. The data regarding the system tem-
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Figure 6. (a) System temperature T (full line) and the number nCCl4 of CCl4 molecules inside the assembly (dotted line) as a function of time t. (b) Relative displacement of the assembly CM with respect to its initial position along the x Cartesian direction, xCM - xCM,0, as a function of time t.
perature T and the number nCCl4 of CCl4 molecules inside the assembly are shown in Figure 6a as a function of time t. It can be seen that a strong coupling between T and nCCl4 exists, with the latter quantity following a roughly periodic variation as the temperature T is cyclically changed approximately between 355 and 370 K. At each cycle, the rapid effusion of CCl4 molecules generates a thrust along the axis of the CN-nanorod assembly, pushing it in the direction of negative x Cartesian coordinates. The relative displacement of the assembly CM with respect to its initial position along the x Cartesian direction, xCM - xCM,0, is shown in Figure 6b as a function of time t. Data indicate the occurrence of three successive displacements, approximately of the same length, starting in coincidence with the effusions of CCl4 molecules. The initial rate of displacement is always equal to about 50 nm ns-1, but its average value is significantly smaller, being around 4 nm ns-1. IV. Discussion The numerical findings heretofore described point out that all the different CN-cluster assemblies can actually exhibit a motor-like behavior when embedded in a liquid phase. Three fundamental features underlie such peculiar behavior. First, the availability of space inside the outer CN that can be occupied by solvent molecules by diffusion. Second, the capability of inducing a relatively fast effusion of the solvent either in liquid or in gaseous form. Third, the reversibility of the whole cycle of operation. Regarding this latter point, it should be noted that precisely the reversibility allows these motors to work with no fuel supply. In fact, the driving force for the assembly motion along a given direction is only due to local processes not involving chemical modifications in the assembly or in the solvent. At the same time, it is useful remembering that a fully reversible cycle of operation is not possible in principle.61 As demonstrated by the thermal engine idealization represented by the Carnot cycle, any given cyclical sequence of operations determines an increase of overall entropy.61 However, such entropy excess is absorbed by ideal thermostats, the temperature of which keeps constant whatever the amount of heat exchanged.61 In the case of simulations carried out, the role of thermostat is played by the solvent surrounding the nanometersized assembly. Far from stating that no thermal effect takes place in the solvent phase, it is nevertheless possible to say that such effects are negligible on the time scales spanned by calculations. As a consequence, no detectable modification of the general system dynamics can occur, which leads to an apparently complete reversibility of motor-like behavior.
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Taking into due account all the previously cited observations, it must be inferred that every process and assembly satisfying the three above-mentioned conditions could in principle produce a motor-like behavior. Here, only three examples have been investigated by numerical simulation. In the first case, a CN-Al cluster assembly was embedded in liquid Ar. The effusion of Ar atoms from the interior of the outer CN was obtained by submitting the Al cluster to a rapid thermal excitation, which in turn induced the liquid-gas transition of the Ar atoms inside the CN. The effusion process produced a thrust able to displace the assembly in the direction opposite to its open end. In the second case, the cycle of operation of the CN-cluster assembly is virtually the same, but the effusion of Ar atoms is driven by the motion of a shuttle governed by a magnetic field. It is simply the compression of the Ar atoms inside the outer CN to determine their fast effusion. In the third case, it is the fast elongation of an Ag nanorod due to a transition between two different crystalline phases to push the solvent molecules outside the assembly. The thrust generated by the effusion and the distance covered by the assemblies are a function of a variety of parameters including the effusion rate, the nature of the solvent, the intensity of its interactions with the outer CN, and the size of the whole assembly. No systematic exploration of their effects has been carried out. However, a few additional simulations have confirmed the most general expectations. In particular, the thrust increases with the effusion rate, whereas the total distance covered by a given assembly during its displacement as well the displacement rate decrease as the intensity of interactions with the solvent increase and as the assembly size increases. In other words, the assembly displacement becomes increasingly difficult as the assembly size and the solvent viscosity increase. All of the examples investigated here, and other ones that could be easily proposed, have a highly speculative character. It can be hardly imagined at present that experimentalists could be able to readily fabricate CN-based assemblies equal to the ones investigated or of comparable complexity. Nevertheless, the realization of similar assemblies, or of others based on the same ideas, could not be so far in time. It is already possible to manipulate CNs as well as porous frameworks to insert nanometer-sized systems in their interior.62 In a variety of cases, porosity of known size provides a template to control the size of inclusions.62 Therefore, the synthesis of a CN with a nanoparticle inside is in principle feasible. Directing the properties of the nanoparticle to obtain a rapid effusion of a solvent phase can be more complex, and the degree of complexity will depend on the particular process to be exploited. However, it seems as a whole that the systems and dynamics underlying the ideas reported in this work are not exceedingly far from experimental realization. In view of the full achievement of the necessary experimental capabilities, the present work demonstrates the validity of a general strategy for obtaining directional motion in a liquid environment. Regarding the materials and methods considered in this study, it is worth noting that they have been chosen exclusively in the light of the feasibility of computations. It follows that the results discussed must be regarded as quite rough and at best qualitative. It has been chosen to privilege illustration purposes rather than a detailed comparison between the different cases. Coherent with this, the CN-based assemblies studied in the present work differ in both chemical nature and size, apart from the processes to which metallic clusters are submitted. In addition, solvent phases are also different. The idea underlying such choices is that of reporting a few examples of nanometer-sized assemblies that can exhibit motor-like behavior under suitable conditions. The
Delogu ones showed in this work are only preliminary results, and a detailed comparison of efficiencies of different assemblies and a characterization of effects of possible importance related to temperature, chemical nature of solvent phase, size of CNs, and metallic clusters are left to future investigation. Regarding this, it is also worth noting that the calculations performed required considerable computational power. At present, it was not possible to carry out a systematic comparison of systems of similar size in the light of the different processes undergone by the metallic clusters. Therefore, only a few additional simulations were performed to have a rough idea, where strictly necessary, of the way a given factor can affect the motor-like behavior. Even though the results of such additional simulations are not discussed here in detail, it is nevertheless possible to state that such results support the conclusions drawn from the discussed cases. In particular, the numerical findings suggest that a similar motor-like behavior can be expected for larger systems with analogous architecture. Before concluding this work, at least another point deserves attention, which substantially concerns the chemical reactivity of CN-based assemblies. This point has not been explicitly dealt with in the previous sections in view of the speculative and illustrative character of the study. However, it is of outstanding importance in connection with the possible experimental realization of nanometer-sized systems similar to the investigated ones. To suitably motivate the importance of chemical reactivity, it must be first noted that both CNs and metallic clusters can undergo relatively extreme conditions, for example, very high heating rates, localized mechanical stresses, and intense external fields. All of these factors are in principle able to promote an enhancement of chemical reactivity, with a consequent degradation of the CN-cluster assemblies, at least in terms of functionalities. The assembly formed by the open CN containing the Al cluster can be used as an example of the above-mentioned situation. Al and C are relatively inert with regard to each other except for relatively high temperatures, at which they tend to form carbides.63 The reaction is relatively slow, the dissolution of C in molten Al and the nucleation of the carbide species requiring long times.64 On the basis of this information, a reaction between the Al cluster and the CN cannot be excluded. However, it must be noted that the melting of Al is not a condition necessary to solvent phase effusion. In addition, the reactivity of isolated perfect CNs is quite different from the one of graphite, which can be highly defective.64 Therefore, it can be reasonably inferred that Al and C atoms do not react under the simulated conditions, also consequent to the short times of local excitation compared with usual reaction times of bulk systems.64 Analogous considerations can be done for the mechanical stresses to which the CN is submitted as a result of the effusion process. It follows that a suitable choice of solvent phase, metallic species, and processing conditions can actually permit avoiding undesired chemical transformations. V. Conclusions The obtained numerical evidence indicate that nanometersized assemblies consisting of a CN containing a metallic cluster can undergo directional motion when embedded in a liquid phase and submitted to specific processes. Aimed at pointing out the potential of the general strategy employed to produce the assembly displacement, three different systems were considered. In all of the cases, the CN has one of its ends completely or partially open to allow the solvent phase to occupy the available space. At the same time, the cluster is somewhat able to induce a fast effusion of the solvent phase under suitable conditions.
Self-Propulsion of Capped Carbon Nanotubes The results show that the effusion of the solvent phase in liquid or gaseous form from the interior of the CN-cluster assembly always determines the progressive displacement of the assembly along a given direction. This occurs in the absence of any track, and the efficiency of the whole process depends on the characteristic features of effusion, assembly, and solvent phase. The effusion process was triggered by the fast thermal excitation of the metallic cluster and the consequent local solvent evaporation, by the displacement of the cluster inside the outer CN, and by the modification of the cluster length by thermally activated structural transitions. Effusion and subsequent refilling of the available space inside the outer CN form a reversible cycle of operation. Therefore, a motor-like behavior can be obtained by suitably controlling the fundamental physical parameters governing the process to which the clusters are submitted. Despite their rough character, the numerical findings here discussed show that the general strategy underlying the examples studied is in principle feasible. Hopefully, it could be source of inspiration for future experimental research. Acknowledgment. Financial support has been given by the University of Cagliari. A. Ermini, ExtraInformatica s.r.l., is gratefully acknowledged for his kind assistance and technical support. Supporting Information Available: Detailed description of interatomic potentials, tables of potential parameter values, Figure S1 showing the atomic configuration of the CN-Al cluster assembly after 80 ps from the beginning of effusion, and Figure S2 showing four consecutive linear displacements of the above-mentioned assembly. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Directing Matter and Energy: FiVe Challenges to Science and the Imagination; Report from the Basic Energy Sciences Advisory Committee; Hemminger, J., Ed.; U.S. Department of Energy: Washington, DC, 2007. (2) Ozin, G. A.; Manners, I.; Fournier-Bidoz, S.; Arsenault, A. C. AdV. Mater. 2005, 17, 3011. (3) Balzani, V.; Credi, A.; Venturi, M. Molecular DeVices and Machines: A Journey into the Nanoworld; Wiley-VCH: Weinheim, Germany, 2003. (4) Kinbara, K.; Aida, T. Chem. ReV. 2005, 105, 1377. (5) Astumian, R. D. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 1843. (6) Browne, W. R.; Feringa, B. L. Nature Nanotechnol. 2006, 1, 25. (7) Kay, E. R.; Leigh, D. A.; Zerbetto, F. Angew. Chem., Int. Ed. 2007, 46, 72. (8) Paxton, W. F.; Kistler, K. C.; Olmeda, C. C.; Sen, A.; St. Angelo, S. K.; Cao, Y.; Mallouk, T. E.; Lammert, P. E.; Crespi, V. H. J. Am. Chem. Soc. 2004, 126, 13424. (9) Fournier-Bidoz, S.; Arsenault, A. C.; Manners, I.; Ozin, G. A. Chem. Commun. 2005, 441. (10) Ru¨ckner, G.; Kapral, R. Phys. ReV. Lett. 2007, 98, 150603. (11) Balzani, V.; Clemente-Leo`n, M.; Credi, A.; Ferrer, B.; Venturi, M.; Flood, A. H.; Stoddart, J. F. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 1178. (12) Vicario, J.; Walko, M.; Meetsma, A.; Feringa, B. J. Am. Chem. Soc. 2006, 128, 5127. (13) Torras, J.; Rodrı`guez-Ropero, F.; Bertran, O.; Alema`n, C. J. Phys. Chem. C 2009, 113, 3574. (14) Fennimore, A. M.; Yuzvinsky, T. D.; Han, W.-Q.; Fuhrer, M. S.; Cumings, J.; Zettl, A. Nature 2003, 424, 408. (15) Barreiro, A.; Rurali, R.; Herna`ndez, E. R.; Moser, J.; Pichler, T.; Forro`, L.; Bachtold, A. Science 2008, 775, 320. (16) Somada, H.; Hirahara, K.; Akita, S.; Nakayama, Y. Nano Lett. 2009, 9, 62. (17) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; World Scientific: Singapore, 1998; Chapter 4. (18) Shiomi, J.; Maruyama, S. Nanotechnology 2009, 20, 055708. (19) Tuzun, R. E.; Noid, D. W.; Sumpter, B. G. Nanotechnology 1995, 6, 52.
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