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Diffusion mechanisms at the Pb solid–liquid interface: Atomic level point of view
Chemical Physics Letters 634 (2015) 108–112
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Diffusion mechanisms at the Pb solid–liquid interface: Atomic level point of view Xuegui Sun a,b , Shifang Xiao b , Huiqiu Deng b , Wangyu Hu a,b,∗ a b
College of Materials Science and Engineering, Hunan University, Changsha 410082, China Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China
a r t i c l e
i n f o
Article history: Received 25 September 2014 In final form 15 May 2015 Available online 9 June 2015
a b s t r a c t The mobility of atoms at a Pb solid–melt interface has been studied using molecular dynamics simulation. Three cooperative diffusive mechanisms were directly observed: immobilization, vacancy diffusion and desorption-mediated jumps. We hold that the in-plane structural ordering and desorption-mediated diffusion mechanism result in the formation and movement of vacancies at the solid–melt interfaces. A one-dimensional random walk model was proposed to describe the atomic movement at the solid–melt interfaces. In addition, we suggested that in-plane structural ordering plays an important role in the diffusion process. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The mobility of molecules/atoms at the homogeneous solid–liquid (S–L) interfaces is a controlling factor in a number of technologically relevant fields, such as self-assembled monolayer growth and nucleation [1–6]. However, the mechanism of the liquid molecules/atoms moving on solid surfaces remains mysterious and a matter of debate. Now there are lots of experiments and theories which have contributed to fundamental studies on the mobility of the molecules at heterogeneous S–L interfaces [7–17]. Because of the significant challenges inherent in measuring dynamic properties for homogeneous S–L interfaces, experimental data are much limited in this case. For convenience narration, S–L interface represents heterogeneous and homogeneous S–L interface, while solid–melt interface only represents homogeneous S–L interface in this Letter. In the simplest picture of molecules moving at the S–L interfaces, the migration of molecules occurs by desorption-mediated jumps. That is, the molecules first desorb from the surface, and then undergo Fickian diffusion in the bulk liquid, and re-adsorb surface lastly [12]. This diffusion process has been verified in the experiments, such as the diffusion of polymers at heterogeneous S–L interfaces [13,14]. Recently, it has been shown that the movement of the solute molecules on a solid surface involves two diffusion mechanisms,
∗ Corresponding author at: College of Materials Science and Engineering, Hunan University, Changsha 410082, China. E-mail address: [email protected] (W. Hu). http://dx.doi.org/10.1016/j.cplett.2015.05.062 0009-2614/© 2015 Elsevier B.V. All rights reserved.
the immobilization (oscillates around the equilibrium site) and the desorption-mediated jumps [14,18]. With the continuous time random walk model and two-dimensional periodic potential theory model, Schwartz and co-workers [18,19] reasonably elucidated the experimental phenomena and the diffusion mechanisms at the S–L interfaces, and the results show that two different kinds of diffusion modes result in the deviation of the self-part of van Hove function from Guassian [18]. To date, the structure of solid–melt interfaces on the atomic scale has been systematically studied [20–22]. In contrast, little is known about the mass transport properties in the solid–melt interfacial region, especially how the microscopic liquid structure near solid–melt interface influences surface transports. For example, Geysermans et al. [23] suggest that the diffusion in the interfacial layers proceeds via vacancies, which accommodate the density misfit between solid and liquid. In this Letter, the molecular dynamics (MD) method is utilized to investigate the movement model of Pb atoms at the Pb solid–melt interface. Based on the analysis from MD simulations, we observed three different modes of diffusion: immobilization, the desorption-mediated jumps and vacancy diffusion. Subsequently, a one-dimensional random walk model was proposed to describe the atomic movement at the solid–melt interfaces. 2. Model and simulation detail In this study, the simplified model of the Pb solid–melt interface is constructed by the fluid Pb filling the space between two fixed walls. Here, the fixed solid wall provides a semi-periodic potential field for liquid Pb. To this end, a pseudo FCC lattice composed
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ensemble (NVT) with a Nosé–Hoover thermostat [26]. Equilibration for 10 ns preceded the data collection stage at which the atomic coordinates are output every 1 ps. It is important to note that if the atom has passed through the periodic boundary, the value is output for what the coordinate would be if it had not been wrapped back into the periodic box [24]. 3. Results and discussion As illustrated in Figure 1, the liquid Pb atoms adjacent to the interface show both layering and in-plane structural ordering. The liquid atoms in the first layer (shown in Figure 1(b)) close to the interface, forming a thin crystal-like film, are localized in equilibrium sites corresponding to the substrate, which agrees well with the previous experiments [27] and MD simulations [28]. The ˚ that is equal to bulk crystal-like film has a lattice spacing (5.06 A) solid Pb. A crystal-like film was also observed at the solid–melt interface [20] and heterogeneous Pb/Cu S–L interface [28]. Mark et al. suggest that the diffusion coefficients for the crystal-like films adjacent to the Pb/Cu S–L interface are very close to zero [28]. Close to S–L interfaces, however, the diffusion coefficient is not sufficient to describe the dynamic property of particles. Schwartz et al. tracked the motion of molecules at S–L interfaces and they found that some molecular trajectories appeared immobilized while others were highly mobile [14,18]. We will discuss the dynamic properties at the Pb solid–melt interface in more detail below. Figure 2 displays the distribution of atomic displacements calculated by the self-part of the van Hove correlation function Gs (x, t) [29], where t equal 0.9 ns. The Gs (x, t) can be expressed as follow: 1 Gs (x, t) = N
Figure 1. (Color online). (a) Schematic boxes, (b) projection of a snapshot of simulated Pb system taken after 10 ns on y-axis, (c) snapshot illustrating layer where ordering occur at S–L interface in x–y plane (dotted box in (b)). (The cyan circles represent equilibrium sites, black rectangle represents vacancy, red circles represent liquid atoms.)
of fixed Pb atoms is constructed as the solid wall. A huge number of works concluded that the liquid is ordered at the S–L or solid–melt interface. Figure 1 showed representative snapshots at start of equilibrated simulation after 10 ns. This ordering in liquids can be seen in this simulation as shown in Figure 1, and proved the validity of our model. The simulation is performed using the LAMMPS [24] code. The embedded atom method (EAM) potential fitted by Lim et al. has been applied to describe the interaction between Pb atoms [25]. The system under consideration is a liquid slab confined by two parallel solid walls with the lateral sizes of 80a × 80a and the thickness of a single wall is 3a, where a is the lattice constant equaling to ˚ The solid surfaces terminating the crystalline slab are cho5.06 A. sen to be the (0 0 1) surface and the z-axis is in the direction normal to the interface. The liquid slab contains 2 048 000 atoms which filled the space between two solid walls with the separation of 80a. Periodic boundary conditions are applied in the x and y directions, parallel to the interfaces. Figure 1(a) shows the schematic boxes for the solid–melt interface. The MD time step is set as 1.0 fs. The simulation is performed at the temperature of 800 K in the canonical
N i
ı(x + xi (t) − xi (t + t))
,
(1)
where · indicates ensemble averaging. The function Gs (x, t) describes the probability that an atom moves the distance of x along the x axis during the time of t. Here, the x axis parallel to [1 0 0] orientation of crystal Pb. As indicated in Ref. [29], the random displacements of particles follow the Gaussian distribution for simple liquid or gas systems. In our simulation, the Gs (x, t) is non-Gaussian distribution and shows multiple peaks. As shown in Figure 2, there is a narrow central Gaussian peak within the range of |x| ≤ 1.3 A˚ (about 0.25a). The narrow central peak associates with atomic displacements during the thermal vibration of the
Figure 2. (Color online). Distributions of atomic surface displacements. Cyan line is simulated data using the MD and black line is simulated data using the random walk model described in the text.
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crystal-like atoms, which accords with the macroscopic experiment results at S–L interfaces [18]. Varying from the situation at silica/water interfaces [18], there are multiple peaks in the ˚ Kim et al. [30] suggest that Gs (x, t) often range of |x| ≥ 1.3 A. shows multiple peaks due to mobile particles undergoing hopping motions. Here, the positions of the second peaks in Gs (x, t) are at |x| = 2.5 A˚ which exactly equal to 0.5a. The nearest neighbor (NN) √ distance in FCC is a/ 2, while along [1 0 0] orientation, the distance is 0.5a. The observation proves that some atoms jumping to their nearest neighbor sites. As shown in Figure 1(b), a vacancy can be found in the crystallike layer. We infer that crystal-like layer contains vacancies due to the desorption-mediated diffusion, where liquid atoms detach from the crystal-like layer leaving vacancies on it. Consequently, it is possible for atoms to jump to NN sites by the vacancy diffusion mechanism. The other peaks in Gs (x, t) are not caused by the atoms jumping on surface. And we will discuss the reason for this in a later section. The Gs (x, t) can roughly be described by an exponential curve for a long distance (|x| ≥ 4.0). The mechanism of the long distance motion is the desorption-mediated diffusion [18], where a liquid atom desorbs from the surface, and diffuses through the adjacent liquid phase, and then readsorbs on the surface. Figure 3(a) depicts a representative trajectory of an atom which oscillates around its equilibrium position. As illustrated in Figure 3(b) and inset, an atom jumps between NN sites and does not leave the crystal-like layer (z coordinate value does not change evidently with time). Hence, it moves in the quasi-two-dimensional plane. Figure 3(b) directly proves that the vacancy diffusion at the solid–melt interface does exist. The in-plane structural ordering and desorption-mediated diffusion mechanism result in the formation and movement of vacancies in the crystal-like layer, which has never been observed in the previous experiments at S–L interfaces. We assume there are two reasons for the lack of vacancy diffusion observation in experiments at S–L interfaces. The one is the vacancy diffusion could not be distinguished because of the limitation of experimental resolution [18]. The other one is the tagged solute molecules, used in experiments [10,11,13,18], are too big to occupy the vacancies in the crystal-like layer. It is reasonable for us to believe that the movement of solvent or small solute molecules/atoms via vacancy diffusion will be experimentally confirmed at crystal/liquid interfaces in the future. Figure 3(c) illustrates the desorption-mediated diffusion process. The trajectory switching between periods of immobilization and mobility corresponds to the experimental results [18]. By analyzing the atomic trajectories and Gs (x, t), we can propose that there are three kinds of diffusional modes at the S–L interface, there are: immobilization, vacancy diffusion and desorption-mediated diffusion. Additional support for the evidence of the three mechanisms comes from the quantitative analysis of the cumulative squareddisplacement distribution C(r2 , t) [31]. It represents the probability of finding the atoms outside a circle with a radius of r from their original positions after the time of t. The C(r2 , t) can be expressed as follow: C(r2 , t) = n(r2 , t)/N, where n(r2 , t) represents the number of atoms outside a circle with a radius of r from their original potions after the time of t, and the N represents the total number of traced atoms. At the time t0 (t0 greater than 10 ns), we record the coordinate of atoms in the crystal-like layer, and output these atoms’ coordinate at time t0 + t. The radius squared can be expressed as follow: ri2 = xi2 + yi2 (i represent ith atom). Here xi equal to xi (t0 + t) − xi (t0 ), yi equal to yi (t0 + t) − yi (t0 ). According to the definition of cumulative squared-displacement distribution, The C(r2 , t) can be computed. For a regular 2D random walk, the formula of this distribution can be expressed by C(r2 , t) = exp(−r2 /4Dt), where D is the diffusion coefficient. At the S–L interfaces [13,14], however,
Figure 3. (Color online). A representative trajectory of an atom undergoes (a) immobilization, (b) vacancy diffusion and (c) desorption-mediated diffusion, respectively. The inset in (a)–(c) shows the z-coordinate (Z(t)). The green circles in (b) represent equilibrium sites.
the distribution has been described by a dual-exponential function, which elucidates two independent diffusive modes. Figure 4 shows the C(r2 , t) curve of the liquid atoms on the surface, where t is 0.9 ns. In this simulation, the C(r2 , t) cannot be fitted as a single exponential or a dual-exponential function, which indicates that there are more than two kinds of diffusion modes existed at the solid–melt interface. The C(r2 , t) function can be taken as a linear superposition of three parts: C = pA CA + pB CB + pC CC , where the subscripts (A, B, C) correspond to immobilization, vacancy diffusion and desorption-mediated diffusion modes, respectively
X. Sun et al. / Chemical Physics Letters 634 (2015) 108–112
Figure 4. (Color online). Cumulative squared-displacement distribution for liquid atoms on solid surface (black line). Cumulative squared-displacement distribution for subdiffusion mode (immobilization), mediadiffsuion mode (vacancy diffusion) and superdiffusion mode (desorption-mediated diffusion) from Fig. 3.
and p represents the fraction of different modes. CA contains only the information of atomic trajectories of atoms vibrating around equilibrium sites, CB means atoms jumping to the NN site by vacancy diffusion and CC describes atoms undergoing a long jump via desorption-mediated diffusion. The fraction of each type in the diffusion process can be derived from the statistical results of trajectories. We can categorize different diffusion mechanisms according to features of trajectories. If an atom leaves the crystallike layer in the time interval 0.9 ns, then a desorption-mediated jumping event occurred. If an atom always does not leave the ˚ then an crystal-like layer and |x| is less than or equal to 1.5 A, immobilization event occurred. If an atom always does not leave ˚ then a vacancy the crystal-like layer and |x| is greater than 1.5 A, diffusion event occurred. The statistical values of pA , pB and pC equal to 50%, 9% and 41% respectively. The vacancy diffusion mechanism cannot be neglected at solid–melt interfaces, although it is not the primary one. Based on the analysis of Gs (x, t), C(r2 , t) and atomic trajectories, we propose three kinds of distinct diffusion modes at the solid–melt interface. In order to test this hypothesis, we have established a one-dimensional random walk model to describe the atomic motion. Each particle in our model presents three possible states with different probabilities (list in Table 1). The onedimensional random walk model contains N = 108 particles and the time interval spans 0.9 ns. If an atom is in the state of immobilization, a random displacement is assigned for it. The displacement is randomly generated according to the Gaussian distributions [32] √ √ with the expression of fim (x) = 1/( × 2 × ) × exp(−x2 /2 2 ), ˚ where equals to 0.4 A. If an atom is in the state of vacancy diffusion, then displacement is randomly selected from the distribution: fvac (x) = 0.5 × ı(x ± a/2), which represent atoms jumping to one of the NN sites. Then the atom oscillates around the new equilibrium site. If an atom stays in desorption-mediated diffusion state, the displacement is selected randomly from the distribution of
Table 1 Probability and distribution function of different diffusion modes.
Probability Distribution function
Immobilization
Vacancy diffusion
Desorption-mediated diffusion
0.5 √ √ 1/( × 2 × ) × exp(−x2 /2 2 )
0.09 0.5 × ı(x ± a/2)
0.41 0.5 × × exp(−|x|)
111
fdesorp (x) = 0.5 × × exp(−|x|), where is obtained from the MD displacement distributions at long tail and its value is 0.1124. When a particle diffuses to the bulk liquid, we are unsure that it will return to the surface or not. So, a fitting parameter, preadsorb , is proposed. Here, preadsorb represents the probability of the desorbed atoms that readsorb to the surface. This parameter affects the height of the Gs multi-peak and does not affect the location of peaks. The MD data agree best with the random walk model data when this parameter equals 0.3. In this case, we assume that the desorbed atoms readsorb to the surface with a probability of 0.3 during the time interval of 0.9 ns. Then the resorbing atom oscillates around a new equilibrium site as well. After resetting the atomic positions, the atomic displacements are measured. As is shown in Figure 2, this random walk model can excellently reproduce the MD data. It can be easily confirmed that three different motion patterns are available in the process of surface transport at solid–melt interface. The existence of multiple peaks in Gs (x, t) is also observed in the random walk model. The phenomenon originates from the fact that atom desorbs from the crystal-like layer, then resorbs and oscillates around a new equilibrium site in the crystal-like layer with probability of 30% during time interval of 0.9 ns. If not taking this situation into consideration, the multiple peaks will disappear. These peaks in Gs (x, t) (|x| ≥ 4.0) locate around the integral multiples of the half lattice constant. This phenomenon is caused by the following process: a desorbed atom readsorbs to the crystal-like layer and then oscillates around a new equilibrium site in this layer. 4. Conclusion In summary, we have performed MD simulations to investigate the diffusion dynamics of liquid atoms near Pb solid–melt interface. Both the van Hove displacement distribution and the single-atomic trajectories suggest that the liquid atoms at solid–melt interface experience three kinds of transport motions. Two of these cases (immobilization and desorption-mediated diffusion) have been borne out by previous experiments [18]. Interestingly, we have reasonably confirmed the existence of vacancy diffusion at the solid–melt interface, which is less-commonly reported in literature. Furthermore, we have proposed a simple random walk model to describe the atomic displacement at solid–melt interface. It is slightly different from the current theory model (continuous time random walk) [18]. In our model, we have introduced the effect of vacancies diffusion and taken into account the condition which atoms readsorb on surface. The Gs (x, t) predicted from the random walk model agrees well with the MD simulation results. This newly proposed model will play an important role in helping us understanding the mass transport properties near the solid–melt interface. Acknowledgements This research is supported by Chinese National Fusion Project for ITER with Grant No. 2013GB114001 and National Natural Science Foundation of China with Grant Nos. NSFC 51271075 and NSFCNSAF11076012. References [1] [2] [3] [4] [5] [6]
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[7] O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, R. Voituriez, Phys. Rev. Lett. 105 (2010) 150606. [8] M. Ricci, P. Spijker, K. Voïtchovsky, Nat. Commun. 5 (2014) 4400. [9] M. Schunack, T.R. Linderoth, F. Rosei, E. Lægsgaard, I. Stensgaard, F. Besenbacher, Phys. Rev. Lett. 88 (2002) 156102. [10] F. Zaera, Chem. Rev. 112 (2012) 2920. [11] N. Schwierz, D. Horinek, S. Liese, T. Pirzer, B.N. Balzer, T. Hugel, R.R. Netz, JACS 134 (2012) 19628. [12] O.V. Bychuk, B. O’Shaughnessy, Phys. Rev. Lett. 74 (1995) 1795. [13] A. Honciuc, D.K. Schwartz, JACS 131 (2009) 5973. [14] R. Walder, N. Nelson, D.K. Schwartz, Phys. Rev. Lett. 107 (2011) 156102. [15] R. Zangi, S.A. Rice, Phys. Rev. Lett. 92 (2004) 035502. [16] A. Honciuc, A.W. Harant, D.K. Schwartz, Langmuir 24 (2008) 6562. [17] A. Honciuc, A.L. Howard, D.K. Schwartz, Nat. Commun. 2 (2011) 515. [18] M.J. Skaug, J. Mabry, D.K. Schwartz, Phys. Rev. Lett. 110 (2013) 256101. [19] M.J. Skaug, A.M. Lacasta, L. Ramirez-Poscina, J.M. Sancho, K. Lindenberg, D.K. Schwartz, Soft Matter 10 (2014) 753.
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Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Diffusion mechanisms at the Pb solid–liquid interface: Atomic level point of view Xuegui Sun a,b , Shifang Xiao b , Huiqiu Deng b , Wangyu Hu a,b,∗ a b
College of Materials Science and Engineering, Hunan University, Changsha 410082, China Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China
a r t i c l e
i n f o
Article history: Received 25 September 2014 In final form 15 May 2015 Available online 9 June 2015
a b s t r a c t The mobility of atoms at a Pb solid–melt interface has been studied using molecular dynamics simulation. Three cooperative diffusive mechanisms were directly observed: immobilization, vacancy diffusion and desorption-mediated jumps. We hold that the in-plane structural ordering and desorption-mediated diffusion mechanism result in the formation and movement of vacancies at the solid–melt interfaces. A one-dimensional random walk model was proposed to describe the atomic movement at the solid–melt interfaces. In addition, we suggested that in-plane structural ordering plays an important role in the diffusion process. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The mobility of molecules/atoms at the homogeneous solid–liquid (S–L) interfaces is a controlling factor in a number of technologically relevant fields, such as self-assembled monolayer growth and nucleation [1–6]. However, the mechanism of the liquid molecules/atoms moving on solid surfaces remains mysterious and a matter of debate. Now there are lots of experiments and theories which have contributed to fundamental studies on the mobility of the molecules at heterogeneous S–L interfaces [7–17]. Because of the significant challenges inherent in measuring dynamic properties for homogeneous S–L interfaces, experimental data are much limited in this case. For convenience narration, S–L interface represents heterogeneous and homogeneous S–L interface, while solid–melt interface only represents homogeneous S–L interface in this Letter. In the simplest picture of molecules moving at the S–L interfaces, the migration of molecules occurs by desorption-mediated jumps. That is, the molecules first desorb from the surface, and then undergo Fickian diffusion in the bulk liquid, and re-adsorb surface lastly [12]. This diffusion process has been verified in the experiments, such as the diffusion of polymers at heterogeneous S–L interfaces [13,14]. Recently, it has been shown that the movement of the solute molecules on a solid surface involves two diffusion mechanisms,
∗ Corresponding author at: College of Materials Science and Engineering, Hunan University, Changsha 410082, China. E-mail address: [email protected] (W. Hu). http://dx.doi.org/10.1016/j.cplett.2015.05.062 0009-2614/© 2015 Elsevier B.V. All rights reserved.
the immobilization (oscillates around the equilibrium site) and the desorption-mediated jumps [14,18]. With the continuous time random walk model and two-dimensional periodic potential theory model, Schwartz and co-workers [18,19] reasonably elucidated the experimental phenomena and the diffusion mechanisms at the S–L interfaces, and the results show that two different kinds of diffusion modes result in the deviation of the self-part of van Hove function from Guassian [18]. To date, the structure of solid–melt interfaces on the atomic scale has been systematically studied [20–22]. In contrast, little is known about the mass transport properties in the solid–melt interfacial region, especially how the microscopic liquid structure near solid–melt interface influences surface transports. For example, Geysermans et al. [23] suggest that the diffusion in the interfacial layers proceeds via vacancies, which accommodate the density misfit between solid and liquid. In this Letter, the molecular dynamics (MD) method is utilized to investigate the movement model of Pb atoms at the Pb solid–melt interface. Based on the analysis from MD simulations, we observed three different modes of diffusion: immobilization, the desorption-mediated jumps and vacancy diffusion. Subsequently, a one-dimensional random walk model was proposed to describe the atomic movement at the solid–melt interfaces. 2. Model and simulation detail In this study, the simplified model of the Pb solid–melt interface is constructed by the fluid Pb filling the space between two fixed walls. Here, the fixed solid wall provides a semi-periodic potential field for liquid Pb. To this end, a pseudo FCC lattice composed
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ensemble (NVT) with a Nosé–Hoover thermostat [26]. Equilibration for 10 ns preceded the data collection stage at which the atomic coordinates are output every 1 ps. It is important to note that if the atom has passed through the periodic boundary, the value is output for what the coordinate would be if it had not been wrapped back into the periodic box [24]. 3. Results and discussion As illustrated in Figure 1, the liquid Pb atoms adjacent to the interface show both layering and in-plane structural ordering. The liquid atoms in the first layer (shown in Figure 1(b)) close to the interface, forming a thin crystal-like film, are localized in equilibrium sites corresponding to the substrate, which agrees well with the previous experiments [27] and MD simulations [28]. The ˚ that is equal to bulk crystal-like film has a lattice spacing (5.06 A) solid Pb. A crystal-like film was also observed at the solid–melt interface [20] and heterogeneous Pb/Cu S–L interface [28]. Mark et al. suggest that the diffusion coefficients for the crystal-like films adjacent to the Pb/Cu S–L interface are very close to zero [28]. Close to S–L interfaces, however, the diffusion coefficient is not sufficient to describe the dynamic property of particles. Schwartz et al. tracked the motion of molecules at S–L interfaces and they found that some molecular trajectories appeared immobilized while others were highly mobile [14,18]. We will discuss the dynamic properties at the Pb solid–melt interface in more detail below. Figure 2 displays the distribution of atomic displacements calculated by the self-part of the van Hove correlation function Gs (x, t) [29], where t equal 0.9 ns. The Gs (x, t) can be expressed as follow: 1 Gs (x, t) = N
Figure 1. (Color online). (a) Schematic boxes, (b) projection of a snapshot of simulated Pb system taken after 10 ns on y-axis, (c) snapshot illustrating layer where ordering occur at S–L interface in x–y plane (dotted box in (b)). (The cyan circles represent equilibrium sites, black rectangle represents vacancy, red circles represent liquid atoms.)
of fixed Pb atoms is constructed as the solid wall. A huge number of works concluded that the liquid is ordered at the S–L or solid–melt interface. Figure 1 showed representative snapshots at start of equilibrated simulation after 10 ns. This ordering in liquids can be seen in this simulation as shown in Figure 1, and proved the validity of our model. The simulation is performed using the LAMMPS [24] code. The embedded atom method (EAM) potential fitted by Lim et al. has been applied to describe the interaction between Pb atoms [25]. The system under consideration is a liquid slab confined by two parallel solid walls with the lateral sizes of 80a × 80a and the thickness of a single wall is 3a, where a is the lattice constant equaling to ˚ The solid surfaces terminating the crystalline slab are cho5.06 A. sen to be the (0 0 1) surface and the z-axis is in the direction normal to the interface. The liquid slab contains 2 048 000 atoms which filled the space between two solid walls with the separation of 80a. Periodic boundary conditions are applied in the x and y directions, parallel to the interfaces. Figure 1(a) shows the schematic boxes for the solid–melt interface. The MD time step is set as 1.0 fs. The simulation is performed at the temperature of 800 K in the canonical
N i
ı(x + xi (t) − xi (t + t))
,
(1)
where · indicates ensemble averaging. The function Gs (x, t) describes the probability that an atom moves the distance of x along the x axis during the time of t. Here, the x axis parallel to [1 0 0] orientation of crystal Pb. As indicated in Ref. [29], the random displacements of particles follow the Gaussian distribution for simple liquid or gas systems. In our simulation, the Gs (x, t) is non-Gaussian distribution and shows multiple peaks. As shown in Figure 2, there is a narrow central Gaussian peak within the range of |x| ≤ 1.3 A˚ (about 0.25a). The narrow central peak associates with atomic displacements during the thermal vibration of the
Figure 2. (Color online). Distributions of atomic surface displacements. Cyan line is simulated data using the MD and black line is simulated data using the random walk model described in the text.
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crystal-like atoms, which accords with the macroscopic experiment results at S–L interfaces [18]. Varying from the situation at silica/water interfaces [18], there are multiple peaks in the ˚ Kim et al. [30] suggest that Gs (x, t) often range of |x| ≥ 1.3 A. shows multiple peaks due to mobile particles undergoing hopping motions. Here, the positions of the second peaks in Gs (x, t) are at |x| = 2.5 A˚ which exactly equal to 0.5a. The nearest neighbor (NN) √ distance in FCC is a/ 2, while along [1 0 0] orientation, the distance is 0.5a. The observation proves that some atoms jumping to their nearest neighbor sites. As shown in Figure 1(b), a vacancy can be found in the crystallike layer. We infer that crystal-like layer contains vacancies due to the desorption-mediated diffusion, where liquid atoms detach from the crystal-like layer leaving vacancies on it. Consequently, it is possible for atoms to jump to NN sites by the vacancy diffusion mechanism. The other peaks in Gs (x, t) are not caused by the atoms jumping on surface. And we will discuss the reason for this in a later section. The Gs (x, t) can roughly be described by an exponential curve for a long distance (|x| ≥ 4.0). The mechanism of the long distance motion is the desorption-mediated diffusion [18], where a liquid atom desorbs from the surface, and diffuses through the adjacent liquid phase, and then readsorbs on the surface. Figure 3(a) depicts a representative trajectory of an atom which oscillates around its equilibrium position. As illustrated in Figure 3(b) and inset, an atom jumps between NN sites and does not leave the crystal-like layer (z coordinate value does not change evidently with time). Hence, it moves in the quasi-two-dimensional plane. Figure 3(b) directly proves that the vacancy diffusion at the solid–melt interface does exist. The in-plane structural ordering and desorption-mediated diffusion mechanism result in the formation and movement of vacancies in the crystal-like layer, which has never been observed in the previous experiments at S–L interfaces. We assume there are two reasons for the lack of vacancy diffusion observation in experiments at S–L interfaces. The one is the vacancy diffusion could not be distinguished because of the limitation of experimental resolution [18]. The other one is the tagged solute molecules, used in experiments [10,11,13,18], are too big to occupy the vacancies in the crystal-like layer. It is reasonable for us to believe that the movement of solvent or small solute molecules/atoms via vacancy diffusion will be experimentally confirmed at crystal/liquid interfaces in the future. Figure 3(c) illustrates the desorption-mediated diffusion process. The trajectory switching between periods of immobilization and mobility corresponds to the experimental results [18]. By analyzing the atomic trajectories and Gs (x, t), we can propose that there are three kinds of diffusional modes at the S–L interface, there are: immobilization, vacancy diffusion and desorption-mediated diffusion. Additional support for the evidence of the three mechanisms comes from the quantitative analysis of the cumulative squareddisplacement distribution C(r2 , t) [31]. It represents the probability of finding the atoms outside a circle with a radius of r from their original positions after the time of t. The C(r2 , t) can be expressed as follow: C(r2 , t) = n(r2 , t)/N, where n(r2 , t) represents the number of atoms outside a circle with a radius of r from their original potions after the time of t, and the N represents the total number of traced atoms. At the time t0 (t0 greater than 10 ns), we record the coordinate of atoms in the crystal-like layer, and output these atoms’ coordinate at time t0 + t. The radius squared can be expressed as follow: ri2 = xi2 + yi2 (i represent ith atom). Here xi equal to xi (t0 + t) − xi (t0 ), yi equal to yi (t0 + t) − yi (t0 ). According to the definition of cumulative squared-displacement distribution, The C(r2 , t) can be computed. For a regular 2D random walk, the formula of this distribution can be expressed by C(r2 , t) = exp(−r2 /4Dt), where D is the diffusion coefficient. At the S–L interfaces [13,14], however,
Figure 3. (Color online). A representative trajectory of an atom undergoes (a) immobilization, (b) vacancy diffusion and (c) desorption-mediated diffusion, respectively. The inset in (a)–(c) shows the z-coordinate (Z(t)). The green circles in (b) represent equilibrium sites.
the distribution has been described by a dual-exponential function, which elucidates two independent diffusive modes. Figure 4 shows the C(r2 , t) curve of the liquid atoms on the surface, where t is 0.9 ns. In this simulation, the C(r2 , t) cannot be fitted as a single exponential or a dual-exponential function, which indicates that there are more than two kinds of diffusion modes existed at the solid–melt interface. The C(r2 , t) function can be taken as a linear superposition of three parts: C = pA CA + pB CB + pC CC , where the subscripts (A, B, C) correspond to immobilization, vacancy diffusion and desorption-mediated diffusion modes, respectively
X. Sun et al. / Chemical Physics Letters 634 (2015) 108–112
Figure 4. (Color online). Cumulative squared-displacement distribution for liquid atoms on solid surface (black line). Cumulative squared-displacement distribution for subdiffusion mode (immobilization), mediadiffsuion mode (vacancy diffusion) and superdiffusion mode (desorption-mediated diffusion) from Fig. 3.
and p represents the fraction of different modes. CA contains only the information of atomic trajectories of atoms vibrating around equilibrium sites, CB means atoms jumping to the NN site by vacancy diffusion and CC describes atoms undergoing a long jump via desorption-mediated diffusion. The fraction of each type in the diffusion process can be derived from the statistical results of trajectories. We can categorize different diffusion mechanisms according to features of trajectories. If an atom leaves the crystallike layer in the time interval 0.9 ns, then a desorption-mediated jumping event occurred. If an atom always does not leave the ˚ then an crystal-like layer and |x| is less than or equal to 1.5 A, immobilization event occurred. If an atom always does not leave ˚ then a vacancy the crystal-like layer and |x| is greater than 1.5 A, diffusion event occurred. The statistical values of pA , pB and pC equal to 50%, 9% and 41% respectively. The vacancy diffusion mechanism cannot be neglected at solid–melt interfaces, although it is not the primary one. Based on the analysis of Gs (x, t), C(r2 , t) and atomic trajectories, we propose three kinds of distinct diffusion modes at the solid–melt interface. In order to test this hypothesis, we have established a one-dimensional random walk model to describe the atomic motion. Each particle in our model presents three possible states with different probabilities (list in Table 1). The onedimensional random walk model contains N = 108 particles and the time interval spans 0.9 ns. If an atom is in the state of immobilization, a random displacement is assigned for it. The displacement is randomly generated according to the Gaussian distributions [32] √ √ with the expression of fim (x) = 1/( × 2 × ) × exp(−x2 /2 2 ), ˚ where equals to 0.4 A. If an atom is in the state of vacancy diffusion, then displacement is randomly selected from the distribution: fvac (x) = 0.5 × ı(x ± a/2), which represent atoms jumping to one of the NN sites. Then the atom oscillates around the new equilibrium site. If an atom stays in desorption-mediated diffusion state, the displacement is selected randomly from the distribution of
Table 1 Probability and distribution function of different diffusion modes.
Probability Distribution function
Immobilization
Vacancy diffusion
Desorption-mediated diffusion
0.5 √ √ 1/( × 2 × ) × exp(−x2 /2 2 )
0.09 0.5 × ı(x ± a/2)
0.41 0.5 × × exp(−|x|)
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fdesorp (x) = 0.5 × × exp(−|x|), where is obtained from the MD displacement distributions at long tail and its value is 0.1124. When a particle diffuses to the bulk liquid, we are unsure that it will return to the surface or not. So, a fitting parameter, preadsorb , is proposed. Here, preadsorb represents the probability of the desorbed atoms that readsorb to the surface. This parameter affects the height of the Gs multi-peak and does not affect the location of peaks. The MD data agree best with the random walk model data when this parameter equals 0.3. In this case, we assume that the desorbed atoms readsorb to the surface with a probability of 0.3 during the time interval of 0.9 ns. Then the resorbing atom oscillates around a new equilibrium site as well. After resetting the atomic positions, the atomic displacements are measured. As is shown in Figure 2, this random walk model can excellently reproduce the MD data. It can be easily confirmed that three different motion patterns are available in the process of surface transport at solid–melt interface. The existence of multiple peaks in Gs (x, t) is also observed in the random walk model. The phenomenon originates from the fact that atom desorbs from the crystal-like layer, then resorbs and oscillates around a new equilibrium site in the crystal-like layer with probability of 30% during time interval of 0.9 ns. If not taking this situation into consideration, the multiple peaks will disappear. These peaks in Gs (x, t) (|x| ≥ 4.0) locate around the integral multiples of the half lattice constant. This phenomenon is caused by the following process: a desorbed atom readsorbs to the crystal-like layer and then oscillates around a new equilibrium site in this layer. 4. Conclusion In summary, we have performed MD simulations to investigate the diffusion dynamics of liquid atoms near Pb solid–melt interface. Both the van Hove displacement distribution and the single-atomic trajectories suggest that the liquid atoms at solid–melt interface experience three kinds of transport motions. Two of these cases (immobilization and desorption-mediated diffusion) have been borne out by previous experiments [18]. Interestingly, we have reasonably confirmed the existence of vacancy diffusion at the solid–melt interface, which is less-commonly reported in literature. Furthermore, we have proposed a simple random walk model to describe the atomic displacement at solid–melt interface. It is slightly different from the current theory model (continuous time random walk) [18]. In our model, we have introduced the effect of vacancies diffusion and taken into account the condition which atoms readsorb on surface. The Gs (x, t) predicted from the random walk model agrees well with the MD simulation results. This newly proposed model will play an important role in helping us understanding the mass transport properties near the solid–melt interface. Acknowledgements This research is supported by Chinese National Fusion Project for ITER with Grant No. 2013GB114001 and National Natural Science Foundation of China with Grant Nos. NSFC 51271075 and NSFCNSAF11076012. References [1] [2] [3] [4] [5] [6]
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