Pengunci Kecepatan Partikel pada Grafena dengan Gelombang Permukaan Bergerak

Abstrak

Difusi cepat yang diinduksi oleh fluktuasi termal dan getaran telah terdeteksi pada skala nano. Dalam artikel ini, gerakan partikel pada lapisan grafen dengan gelombang permukaan bergerak dipelajari dengan simulasi dinamika molekuler dan model teoritis. Dibuktikan bahwa partikel akan terus bergerak pada kecepatan gelombang dengan syarat-syarat prasyarat tertentu, yaitu efek pengunci kecepatan. Dengan mengekspresikan potensi van der Waals (vdW) antara partikel dan permukaan bergelombang sebagai fungsi dari kelengkungan, mekanisme dijelaskan berdasarkan puddle potensial dalam koordinat frame gelombang relatif. Dua syarat prasyarat diajukan: posisi awal partikel harus terletak dalam puddle potensial, dan energi kinetik awal tidak dapat mendorong partikel untuk melompat keluar dari puddle potensial. Analisis parameter menunjukkan bahwa wilayah pengunci kecepatan akan dipengaruhi oleh panjang gelombang, amplitudo, dan pasangan potensi antara partikel dan gelombang. Dengan panjang gelombang yang lebih kecil, amplitudo yang lebih besar, dan potensi vdW yang lebih kuat, wilayah pengunci kecepatan menjadi lebih luas. Penelitian ini mengungkapkan jenis gerakan koheren baru untuk partikel pada materi berlapis berdasarkan teori puddle potensial, yang dapat menjadi penjelasan bagi fenomena difusi cepat pada skala nano.

Pendahuluan

Sekitar akhir, serangkaian fenomena transportasi dan difusi cepat yang diinduksi oleh gelombang permukaan/fonon terdeteksi pada skala mikro/nano. Pada awalnya, fenomena termoforik sepanjang nanotubus karbon [1,2,3,4,5] atau pita grafen [6,7,8,9,10] telah diteliti secara ekstensif. Fluktuasi termal dikonfirmasi memungkinkan aliran air yang kontinu melalui nanotubus karbon (CNT) dengan mengenakan gradien termal axial di seluruh permukaannya [11,12,13]. Simulasi dinamika molekuler nonekuilibrium dilakukan untuk menjelajahi kelayakan penggunaan gradien termal pada substrat grafen besar untuk mengontrol gerakan flake grafen nano kecil [6]. Selain itu, transportasi titik air yang diinduksi termal pada permukaan grafen dan nitrida bor heksagonal (h-BN) dipelajari dengan simulasi dinamika molekuler [8, 9]. Fenomena ini disarankan berkaitan dengan mode fonon tertentu [14,15,16,17,18,19]. Misalnya, Schoen et al. mengaitkan gerakan termoforik di dalam nanotubus karbon dengan mode bernafas tabung [1, 20]. Panizon et al. [21] menunjukkan bahwa gelombang/fonon berjalan lentur pada grafen dapat mentransfer momentumnya ke adsorbat dan menyebabkan transportasi. Mirip dengan fenomena termoforik, Angelos et al. menunjukkan bahwa riak merambat yang diinduksi suhu pada grafen dapat menyebabkan difusi cepat titik air nano yang 2–3 orde magnitud lebih cepat dari difusi sendiri molekul air dalam air cair [22, 23].

Selain fluktuasi termal, studi mengonfirmasi bahwa getaran juga dapat mentransportasi partikel dan titik air di dalam dan di luar nanotubus karbon (CNT) [24,25,26,27]. Misalnya, titik air nano ditransportasi sepanjang nanotubus dengan kecepatan mendekati 30 nm/ns ketika gelombang akustik silang terpolarisasi linier melewatkan momentum linier ke titik air nano [24, 28]. Guo et al. menunjukkan bahwa molekul air di dalam ayunan kantilever didorong oleh gaya sentrifugal dan dapat mengalami aliran kontinu dari ujung tetap ke ujung bebas CNT dengan simulasi dinamika molekuler [26, 29]. Transportasi unidireksional nano baru dari molekul air melalui nanotubus karbon tunggal dinding (SWCNT) dirancang dengan menggunakan muatan getaran dan SWKT komposit dengan energi permukaan asimetris [30]. Zhou et al. [31] menyelidiki inversi arus dalam pompa air berukuran nano berdasarkan SWKT yang dipenggerakkan oleh getaran mekanis dan mengonfirmasi bahwa arus air sangat tergantung pada frekuensi getaran mekanis. Chang dan Guo [32] menemukan gelombang dominosi dalam nanotubus karbon yang dapat menembakkan molekul di dalamnya dengan kecepatan besar hingga 1 km/s. Proses dominosi yang dapat dibalik juga terbukti dalam nanotubus karbon tunggal dinding [33].

Karena berbagai fenomena difusi dan transportasi cepat yang diinduksi oleh fluktuasi termal dan getaran terdeteksi pada skala nano, dikonfirmasi bahwa gerakan naik dan turun pada permukaan dapat meningkatkan difusi dan transportasi. Koneksi antara gelombang dan gerakan partikel masih tidak jelas dan tidak dapat disatukan. Penjelasan utama adalah bahwa momentum permukaan dapat ditransfer ke partikel atau titik air di luar permukaan [22, 24]. Namun, hubungan antara amplitudo, frekuensi dan interaksi antara partikel dan permukaan tidak dapat diungkap dari penjelasan ini. Selain itu, Angelos et al. menunjukkan bahwa preferensi yang jelas untuk satu tanda kelengkungan grafen diperlukan untuk difusi cepat adsorbat pada permukaan grafen [22], yang menunjukkan bahwa potensi interaksi yang diinduksi oleh morfologi permukaan bergelombang berkaitan erat dengan difusi cepat. Oleh karena itu, menjelajahi interaksi antara permukaan bergelombang dan partikel di luar adalah sangat penting untuk memahami mekanisme transportasi dan difusi cepat pada skala nano.

Dalam artikel ini, dengan mempelajari partikel di luar permukaan grafen bergelombang berdasarkan interaksi vdW yang digambarkan oleh pasangan potensi Lennard–Jones (L–J), hubungan koheren antara gerakan bergelombang dan kecepatan partikel ditunjukkan dengan simulasi MD. Kecepatan keseluruhan partikel yang dilemparkan pada permukaan bergelombang dikonfirmasi tetap sama dengan gelombang berjalan dengan syarat-syarat prasyarat tertentu, yaitu efek pengunci kecepatan. Kemudian, teori puddle potensial dibangun berdasarkan potensi interaksi antara partikel dan permukaan gelombang yang diekspresikan sebagai fungsi dari kelengkungan [34,35,36]. Dengan teori ini, dua syarat prasyarat untuk efek pengunci kecepatan diajukan, dan trajetori serta kecepatan yang diprediksi oleh teori puddle potensial setuju dengan hasil simulasi MD. Selain itu, efek panjang gelombang dan amplitudo serta parameter interaksi vdW dianalisis, yang menunjukkan kesepakatan yang baik dengan regulasi yang terdeteksi untuk fenomena surfing titik air pada permukaan grafen [22]. Mekanisme efek pengunci kecepatan yang diinduksi gelombang mengungkapkan hubungan koheren baru antara kecepatan partikel dan permukaan bergelombang.

Metode

Simulasi MD diimplementasikan dalam paket perangkat lunak Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). Permukaan bergelombang diasumsikan adalah lapisan grafen, yang memiliki densitas nomor atom $\rho = 3.85 \times 10^{19} \,{\text{m}}^{ - 2}$. Lembar grafen awalnya datar dengan z = 0 Å dan panjang 6344 Å sepanjang arah x, sehingga ukuran sel unit adalah 6000 atom. Di sepanjang y-axis kondisi batas periodik digunakan dengan panjang periode 12,2 Å. Di sini, partikel bola dipertimbangkan dengan massa $m = 0.83 \times 10^{ - 25} \,{\text{kg}}$, untuk menyederhanakan model dan fokus pada efek geometris permukaan bergelombang. Pada awalnya, partikel ditempatkan pada z = 7 Å dan x = 200 Å. Partikel memiliki kecepatan awal − 50 m/s di arah z-direction dan sekitar 2000 m/s di arah x-direction. Dengan menetapkan waktu awal untuk kecepatan awal di arah z-direction, posisi awal dapat dikontrol untuk partikel jatuh pada permukaan bergelombang.

Potensi urutan ikatan empiris reaktif (REBO) digunakan untuk memodelkan atom grafen [37]. Sementara itu, potensi Lennard–Jones dipilih untuk memodelkan interaksi antara partikel $P$ dan setiap atom karbon dalam grafen sebagai,

$$u\left( R \right) = \varepsilon \left( {{\sigma \mathord{\left/ {\vphantom {\sigma R}} \right. \kern-\nulldelimiterspace} R}} \right)^{12} - \varepsilon \left( {{\sigma \mathord{\left/ {\vphantom {\sigma R}} \right. \kern-\nulldelimiterspace} R}} \right)^{6}$$ (1)

where $\varepsilon = 5.92 \times 10^{ - 21} \,{\text{J}}$ and $\sigma = 4 \times 10^{ - 10} \,{\text{m}}$. Tinggi keseimbangan antara partikel $P$ dan permukaan bergelombang diambil sebagai $h = 4.2 \times 10^{ - 10} \,{\text{m}}$, ditentukan oleh kondisi gaya normal sebagai nol dan hasil simulasi, yang dijelaskan secara detail dalam Berkas Tambahan 1: 1.

Fungsi gelombang berjalan mengambil bentuk sinus sebagai,

$$y = A\sin \left( {\frac{2\pi }{\lambda }x - \omega t} \right)$$ (2)

where the amplitude is taken as $A = 1 \times 10^{ - 9} \,{\text{m}}$ and the wavelength is $\lambda = 21.75 \times 10^{ - 9} \,{\text{m}}$ unless otherwise noted. The angular frequency is taken as $\omega = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {10^{ - 12} }}} \right. \kern-\nulldelimiterspace} {10^{ - 12} }}$ corresponding to a period of 10 ps; thus, the wave speed is $v_{{{\text{wave}}}} = 2175\,{\text{m}}/{\text{s}} = \lambda \omega /2\pi$. To trigger the travelling wave, the left 10 Å of graphene (i.e. y $\in$ [− 10, 0] Å) is wiggled in z-direction with the amplitude and frequency mentioned above. Moreover, carbon atoms with x > 6010 Å is clamped to keep the graphene sheet stable. In particular, if a flat graphene sheet is to be simulated, unclamped graphene atoms will also be tethered to their initial positions along z-axis with a weak spring constant of 0.0938 eV//Å2 (besides A is set to 0).

An initial temperature of 5 K is assigned to unfixed carbon atoms. This temperature is set to eliminate the thermally activated ripples caused by a harmonic coupling between the bending and stretching modes of graphene and focus on the effect of the travelling wave caused by mechanical excitation [22]. The structure then evolves in micro-canonical ensemble (NVE) with a time step of 1 fs. We monitored this evolution and found temperature nearly unchanged during the whole simulation.

Hasil dan Pembahasan

The trajectory of particles on wavy graphene surface as well as flat graphene surface is illustrated in Fig. 1. The time interval is taken as the period of wavy surface. It is found that the relative position of particle does not change in reference to the wave crests or troughs, which means the particle is locked on the wavy surface with its speed equal to the wave speed. As a comparison, the particle’s overall motion on the flat surface is apparently slower than that on a wavy surface with the same initial position. The speed of particle diminishes quickly on the flat surface due to friction, whereas the friction seems not working for particles on wavy graphene surface. More MD simulation cases with different simulation temperatures and parameters of wave function are shown in Additional file 1: 1. The simulations of atom Xe and molecule C₆₀ moving on wavy and flat surface are done to confirm the generalizability of this phenomenon and shown in Additional file 1: 2.

Pengunci Kecepatan Partikel pada Grafena dengan Gelombang Permukaan Bergerak

To understand the mechanism of speed-locking effect at nanoscales, a model is built by considering the interaction between wavy surface S and an external particle P, which is shown in Fig. 2a, b. Assuming that the wavelength and amplitude of wavy surface are $\lambda$ and A, respectively, the nearest height between P and S is h, the number density of S is ${\rho }_{s}$. In MD simulation, the interaction between particle P and wavy surface is taken as vdW interaction, which is depicted by L–J potential,

$$U_{{{\text{L}} {-} {\text{J}}}} = \varepsilon \left[ {\left( {\frac{\sigma }{r}} \right)^{12} - \left( {\frac{\sigma }{r}} \right)^{6} } \right]$$

Then, the interaction between P and S is proved to be written as a function of mean curvature and Gauss curvature based on L–J pair potential [34,35,36],

$$\begin{aligned} U_{6 - 12} & = \frac{{4\pi \rho_{s} \varepsilon \sigma^{12} }}{{5h^{10} }}\left[ {1 - hH + h^{2} H^{2} + \frac{{9h^{2} }}{16}\left( {H^{2} - K} \right)} \right] \\ & \quad - \,\frac{{2\pi \rho_{s} \varepsilon \sigma^{6} }}{{h^{4} }}\left[ {1 - hH + h^{2} H^{2} + \frac{{3h^{2} }}{4}\left( {H^{2} - K} \right)} \right] \\ \end{aligned}$$ (3)

Here, point $P_{1}$ is the nearest point on surface S to particle P, and H and $K$ are the mean curvature and the Gauss curvature at point $P_{1}$ (Fig. 2a) [20], respectively. Through this curvature-based potential [Eq. (3)] has been used in explaining many abnormal phenomena at micro/nano scales [38, 39], the reliability of Eq. (3) in this case is validated by compared with the surface potential in MD simulation for parameters given above and displayed in Fig. 2c.

Before analysing the influence of wavy surface on particle P, friction should be investigated and taken account. The friction between particles and the wave surface can be very complicated at nanoscales [39,40,41,42,43]. A primitive estimation of the friction is made by simulating the motion of a particle on a flat graphene layer by MD as detailed in Additional file 1: 3. For convenience, here a flat surface instead of wavy one is taken. This approximation is estimated in Additional file 1: 3 combined with further potential puddle mechanism. With parameters given above, the friction is estimated as $f = - 5.2 \times 10^{ - 13} \,{\text{N}}$.

Then, the relative potential between surface S and particle P is investigated by considering friction. Firstly, a relative wave-frame coordinate $PXY$ is built as shown in red colour in Fig. 2b, which moves at the wave speed thus keeps stationary to the travelling wave. So, the travelling wave is “frozen” in $PXY$. Since the particle keeps moving rightward in reference to the graphene, the friction acting on it will constantly be leftward along the surface. As a result, the relative wave-frame potential will be the curvature-based potential minus the work done by friction,

$$P = U_{n} + f * x$$ (4)

Substituting the curvature-based potential U_n and friction into Eq. (4), the relative wave-frame potential can be evaluated and is drawn in Fig. 2d.

Since the wave-frame coordinate PXY moves along with the travelling wave, the initial location of particle P in the potential determines the particle’s trajectory. Assuming that the initial velocity of particle P is $v_{0}$ and the wave speed is $v_{{{\text{wave}}}}$, two prerequisite conditions can be proposed based on Fig. 2d: the initial position of particle $P$ locates in potential puddle of the red zone $\lambda_{1}$; the initial wave-frame kinetic energy satisfies $\frac{1}{2}m\left( {v_{0} - v_{{{\text{wave}}}} } \right)^{2} \le \Delta U$. Then, the particle will not be able to hop out the puddle but instead be trapped and wobble inside the puddle. In perspective of an absolute coordinate, particle $P$ will oscillate in the potential puddle but keep moving with propagating wave by the speed locked around the wave speed, hence the speed-locking effect. Otherwise, if the initial location of particle $P$ falls inside the blue zone $\lambda_{2}$ or the relative initial kinetic energy $\frac{1}{2}m\left( {v_{0} - v_{{{\text{wave}}}} } \right)^{2} > \Delta U$, the particle $P$ will not stay inside a single puddle but hops leftward to lower puddles along the wave-frame potential surface. In the perspective of an absolute coordinate, the particle will lag behind the propagating wave until another equilibrium of forces is met. One possibility of such equilibrium is that the particle stops moving on graphene thus the friction vanishes. Interestingly, in lit [21]. Panizon et al. reveal that when there is a velocity difference, the travelling wave will be scattered by the particle and offer a propulsion force, suggesting the final speed of particle will be larger than zero.

To formulate and better illustrate our theory, the movement equation of particle P is further established by Newton's laws of motion. The driving forces exerted on particle P include two parts, normal force $F_{{\text{n}}}$ and tangential force $F_{{\text{t}}}$, namely (Fig. 2b),

$$F_{{\text{n}}} = \frac{{\partial U_{6 - 12} }}{\partial h}; \, F_{{\text{t}}} = \frac{{\partial U_{6 - 12} }}{\partial H}\nabla H + \frac{{\partial U_{6 - 12} }}{\partial K}\nabla K$$ (5)

For L–J potential, both attractive and repulsive interactions exist between atoms, the external particle $P$ will stay at a height h where the normal force $F_{{\text{n}}}$ is zero, the determination calculation of height h is put in Additional file 1: 2. Then, the equation of motion of particle $P$ in $x$ direction is,

$$m\ddot{x} = F_{x} - f$$ (6)

Here, $F_{x}$ is the component of tangential force $F_{{\text{t}}}$ in $x$ direction (Fig. 2b). Calculating Eq. (6) gives the particle trajectory. For the sinusoidal wave surface, Gaussian curvature is zero and the mean curvature equals to the curvature of curve in $Ozx$ surface, i.e. $K = 0$ and $H = \kappa$ [52], substituting Eq. (5) into (6), the moving trajectory of particle P can be solved numerically.

The locking and unlocking examples are shown in Fig. 3. For the initial location (Fig. 3a) corresponding to the locking region $\lambda_{1}$ in Fig. 2d, the trajectories from theoretical and MD simulation results are compared in Fig. 3b. It shows that the particle stops moving on the flat graphene surface in a very short time due to friction, while the particle keeps moving rightward on the wave surface. And the theoretical trajectory approximates to MD simulation results. This tendency is further confirmed in Fig. 3c for particle’s velocity shown in ten times of simulation time. Since the particle falls in the locking zone and the initial speed equals to the wave speed, it will oscillate in the potential puddle and its overall speed will equal to the wave speed, which is in accordance to our speculation. For particle with its initial location (Fig. 3d) falling in unlocking region $\lambda_{2}$ in Fig. 2d, trajectory of particle on wave surface tends to a constant (Fig. 3e) and is confirmed further by the velocity distribution. It is interesting that the travelling wave can enhance particle’s motion even when it falls on speed unlocking region compared to the motion of particle on flat graphene surface. Figure 3f illustrates that the velocity will decrease to zero for time longer than simulation time. More examples are illustrated in Additional file 1: 3.

According to the potential puddle mechanism, the speed-locking effect of particle is dominated by the potential wave surface. The effect of parameters can be discussed based on potential puddle theory. Obviously, these include wavelength $\lambda$, amplitude A, frequency $\omega$ and the L–J potential parameters. It is noted that the friction is assumed to remain the same regarding to different parameters in following analysis. The potential distributions for different wavelength A, amplitude $\lambda$ and the L–J potential parameter $\varepsilon$ are illustrated in Fig. 4, respectively. Figure 4a reveals that the potential puddle depth decreases with an increase of $\lambda$, and there will be no speed-locking range when wavelength exceeds a critical value. Also, since a lower frequency $\omega$ relates to a larger $\lambda$, the speed-locking range decreases with an increase of $\omega$. Figure 4b illustrates that the potential puddle depth increases with an increase of A, and the speed-locking effect vanishes when the amplitude is too small. It is noted that the ratio A/$\lambda$ should not be too large to prevent from damages. Usually, both $\lambda$ and A increase when the scale of wave or particles increases. To study the scale effect, we keep the ratio $\lambda$/A fixed and examine the influence of varying $\lambda$ or A. Figure 4c shows the potential puddle depth decrease swiftly with an increasing $\lambda$ or A. This indicates that the curvature-based driving force decrease quickly with an increasing scale, so the speed-locking effect for particles will vanish on surface with large-scale wave. For L–J potential parameter $\varepsilon$, it is confirmed that the speed-locking region will be broader when the pair interaction potential is strong and the speed-locking effect will disappear when the pair interaction potential is weak (Fig. 4d).

It is noted that the stiffness and L–J potential parameters are different for other 2D nanomaterials, which lead to different frequency and wave speed [44]. According to parameter analysis, the potential puddle will appear by choosing proper wavelength and amplitude for wavy surface. As the potential puddle is the precondition for particles moving with wavy surface, this speed-locking effect will also establish for many 2D nanomaterial layers under the short-range interaction.

Although the movement of one particle is discussed in this paper, it is still in the frame of thermo environment. The potential puddle is the essential condition for the coupling movement between particle and surface. For multiple particles, if they all locate in potential puddle region and satisfy the prerequisite conditions, they will be trapped and moves with the wavy surface. According to the parameters effect, the movement of particles can be controllable by adjusting the wavelength and amplitude. As the speed-locking region will be larger for surface wave with smaller wavelength, larger amplitude and higher frequency, the fast diffusion on the wavy surface will also be enhanced. The parametric analysis is also in accordance to the fast diffusion regulation detected in many other literatures. For example, Angelos et al. pointed out that the diffusion coefficient increases with the ripple amplitude of graphene surface [22]. They confirmed that the amplitude of the ripples increases revealing an increased preference of the droplets for the valleys, which can be explained by Fig. 4b. When the amplitude increases enough, the speed-locking region would probably cover the whole wavelength and enhance the diffusion. In addition, they pointed out that the potential for the valley is always smaller than the potential for crest [22] (Fig. 4), which is responding to lower potential for the crest region shown in Fig. 4. Cao et al. studied the flow of fluid inside nanochannel in the presence of travelling surface waves and confirmed that the velocity increases with the increasing of amplitude and frequency [45], which is also in accordance to the parametric analysis.

MD simulation can only reflect the property in a very short time, more potential application of this speed-locking effect can be deduced from the potential puddle mechanism. For example, by adjusting the amplitude and frequency, it is possible to realize almost locking or unlocking region, which can drive particles to move or stop. It is noted that the wavy movement of surface in vertical direction can be transformed into the movement of particle in transverse direction, which is similar to a kind of ratchet movement and may be used in nano-electro-mechanical system. In addition, as the interaction between particle and surface will affect the movement, the trajectory enhanced by wavy surface will be different for particles with different pair potentials, which can lead to phrase separation.

Kesimpulan

In conclusion, we demonstrate a distinctive relation between particle and graphene layer with travelling surface wave, i.e. speed-locking phenomenon. By MD simulation, it confirmed that the speed of particle can be kept around wave speed with certain conditions. A theoretical model is built to elucidate the mechanism, where the puddle of potential surface dominates the locking effect. The locking conditions are proposed based on this model, i.e. the initial position of particle locates in the potential puddle and initial kinetic energy cannot drive particle to hop out of potential puddle. The particle trajectory predicted by theoretical predictions agrees well with MD simulations results. The effect of wavelength and amplitude as well as L–J potential parameter is discussed. The work also provides a new perspective to understand the fast diffusion and transport on wavy surface and potential applications for phrase separations.

Ketersediaan data dan bahan

All data generated or analysed during this study are included in this published article [and its Additional files].

Singkatan

MD:: Dinamika molekuler
vdW:: Van der Waals
CNT:: Nanotubus karbon
h-BN:: Nitrida bor heksagonal
SWCNT:: Nanotubus karbon tunggal dinding
L–J:: Lennard–Jones
LAMMPS:: Large-scale Atomic/Molecular Massively Parallel Simulator
REBO:: Reactive empirical bond order
NVE:: Ensemble mikro-kanonis

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