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- Table of Contents
Journal of Nanomaterials
Volume 2013 (2013), Article ID 682832, 43 pages
Correlation Effects in Kinetics of One-Dimensional Atomic Systems
National Research Nuclear University, MEPhI, Kashirskoye Sh. 31, Moscow 115409, Russia
Received 10 January 2013; Accepted 7 March 2013
Academic Editor: Yun Zhao
Copyright © 2013 V. D. Borman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper is devoted to the analysis of the correlation effects and manifestations of general properties of 1D systems (such as spatial heterogeneity that is associated with strong density fluctuations, the lack of phase transitions, the presence of frozen disorder, confinement, and blocked movement of nuclear particle by its neighbours) in nonequilibrium phenomena by considering the four examples. The anomalous transport in zeolite channels is considered. The mechanism of the transport may appear in carbon nanotubes and MOF structures, relaxation, mechanical properties, and stability of nonequilibrium states of free chains of metal atoms, non-Einstein atomic mobility in 1D atomic systems. Also we discuss atomic transport and separation of two-component mixture of atoms in a 1D system—a zeolite membrane with subnanometer channels. We discuss the atomic transport and separation of two-component mixture of atoms in a 1D system—zeolite membrane with subnanometer channels. These phenomena are described by the response function method for nonequilibrium systems of arbitrary density that allows us to calculate the dynamic response function and the spectrum of relaxation of density fluctuations 1D atomic system.
In recent years, one-dimensional objects became available and actively investigated due to their unique properties. They are free chains of metals just one atom thick (Au, Ag, Pt, In, Mn, Li, and Ni) [1–13], chains on specially prepared surfaces of crystals with high crystalline indices [14–19], the molecules (atoms) in subnanometer channels of complex oxides, zeolites [20–22], carbon nanotubes [23–27], and in some metal organic frameworks (MOF).
According to modern concepts, the common feature of one-dimensional objects as statistical systems is spatially inhomogeneous state and density fluctuations. These fluctuations are caused by the crucial role of many-particle correlations in 1D systems. This feature is connected with the absence of phase transitions in 1D systems of particles of one kind with short-range potential of the interparticle interaction (see van Hove Theorem [28, 29]). In this case, for certain values of the density the susceptibility of 1D system has a maximum but stays finite in contrast to 3D systems, where susceptibility becomes infinite in vapour-liquid and liquid-solid phase transitions .
The absence of phase transition means that stable nuclei of a new phase are not formed, and the one-dimensional clusters nuclei that are formed in a system have a finite lifetime and are the density fluctuations. 1D systems are characterized by frozen disorder because the atomic particles cannot be exchanged with their neighbours and block the movement of each particle neighbors. For kinetic phenomena in atomic 1D confinement system, the role of confinement is not only to preserve the geometry of the system, but also the system relaxation by phonons. Relaxation of the system in confinement due to strong interparticle correlations depends on the ratio of the energy of particle interaction with confinement and interparticle interaction .
This paper is devoted to the analysis of correlation effects and manifestations of these general properties of nonequilibrium phenomena in the four examples. The anomalous transport in zeolite channels is considered. The mechanism of the transport may appear in carbon nanotubes and MOF structures, relaxation, mechanical properties, and stability of nonequilibrium states of free chains of metal atoms, non-Einstein atomic mobility in 1D atomic systems. Also we discuss atomic transport and separation of two-component mixture of atoms in a 1D system—a zeolite membrane with subnanometer channels.
These phenomena are described below using the response function method for nonequilibrium systems of arbitrary density . This method allows us to calculate the dynamic response function and the spectrum of relaxation of density fluctuations of 1D atomic system. The spectrum contains diffusion and hydrodynamic modes. In the latter case, the relaxation of the system and the nuclear transport occurs through the distribution of density waves and not by the diffusion of particles. This relaxation mechanism becomes the determining by increasing the density that allows one to explain the abnormally high rate of atomic transport in subnanometer channels. The change of relaxation mechanism by increase of the density is associated with the observed non-Einstein character of mobility of atomic particles (the law for the mean square displacement is violated ( is the diffusion coefficient, and is the time of observation) ). The found spectrum of relaxation allows to calculate the probability of decay of 1D free clusters of various lengths  and to explain the dependence of the stability of clusters of metal atoms on the chain atoms properties, the mechanical state of a cluster under the influence of an external force, and the number of particles in a stable cluster. These values were measured simultaneously in experiments .
The response function method was used to describe the practically important transport of two-component mixture in membranes with subnanometer channels.
It was shown that a phase transition in an inhomogeneous state may occur in 1D system of particles of two kinds with the increase of filling factor. Such a phase transition occurs due to an effective particle attraction and is characterised by the formation of clusters of a finite size . The amplitude of the fluctuations in the density corresponding to the formation of such a state is the order parameter. An equation for the order parameter for a mixture of particles in the channels is derived. In contrast to the well-known phenomenological Landau-Ginzburg equation , the coefficients of the equation are calculated for a known interparticle interaction potential depending on the density and temperature. The analysis of this equation allows us to describe the observed increase in the separation factor with increasing flux through 1D channels at their closing by a stable clusters with a certain concentration of the components and determine the dependence of the separation on the concentration, temperature, and density.
2. Materials and Methods
2.1. Molecular Transport in Subnanometer Channels
Molecular transport in nanometer and subnanometer channels in porous bodies is currently drawing a great deal of interest from the standpoint of fundamental science [35–39] and because of the many applications of membrane and nanotechnologies in various fields ranging from nuclear power to ecology [40, 41]. Polycrystalline ceramic membranes consisting of complex oxides (zeolites), possessing subnanometer channels ranging in diameter from 0.3 to 1.4 nm, have been synthesized recently . On account of the high selectivity of molecular transport in zeolite membranes, as compared with the well-known polymeric membranes [41–43], new technologies for separation, reprocessing, and utilization of materials are being intensively developed on the basis of zeolite membranes [41–43].
When the channel diameters in membranes decrease to the nanometer scale, molecular transport is determined by the Knudsen flow in the central zone of the channel, free of interaction of molecules with walls, and particle diffusion in the surface force field . In subnanometer channels, the interaction potentials between molecules and the opposite walls overlap, and molecular transport occurs under conditions of a constant interaction of molecules with a solid. Consequently, the diffusion coefficient in the limit of small filling numbers of a channel is determined by the relaxation of particles on phonons and surface defects . For molecules, with the exception of light particles (H, He), the channel walls are impenetrable, and consequently molecular transport is possible only along the channel axes. In this sense, it is different from diffusion in solids and may be assumed to be one-dimensional. In contrast to surface diffusion in channels with diameter nm, a fundamentally new property of molecular transport should appear in subnanometer channels. As the filling factor of a channel or the diameter () of molecules increases, so that , each molecule can block the motion of two molecules. Consequently, it can be expected that as the diameter of the molecules or the external gas pressure increases, the molecular flux in membranes with subnanometer channels should decrease. However, experiments have shown that the diffusion flux of a number of molecules (, , and others) in a ZSM-5 zeolite membrane with channel diameters 0.54–0.57 nm does not decrease but rather increases as the external gas pressure increases . It has also been established that the diffusion coefficient for these gases increases by more than an order of magnitude as the filling factor of the channels with molecules increases. It has been found that for a number of gases, the temperature dependence of the flux possesses a maximum and a minimum.
In this section, the transport of a single-component molecular gas (system of atomic particles) in subnanometer-diameter channels is investigated theoretically.
2.1.1. Sorption Isotherm in a 1D Channel
Let us consider a surface in contact with an ideal single-component gas with temperature and pressure . Let us assume that particles located on the surface do not interact with one another. We will also assume that the energy of the gas molecules on the surface is . Let there be adsorption centers on the surface. Then the average number of particles adsorbed on the surface as a function of the gas pressure and temperature has the form (Langmuir isotherm) where is the mass of a gas molecule, is Planck’s constant, and is the reciprocal of the temperature. The method used to derive the Langmuir isotherms (1) admits wide extensions. Specifically, this method can be used to solve the problem of the filling factor of a cylindrical channel with diameter taking into account the interparticle interaction in the channel. Let the channel diameter be comparable to the maximum diameter of a gas molecule. Let us consider the equilibrium of the gas with the surface on which of the channels described previously emerge. Let be the binding energy of a particle at the entrance to the channel. If the energy is negative, then it is energetically favorable for the gas molecule to enter the channel. Let be the total number of particles in a channel of length , the total number of particles in the channel and on the surface, and the number of settling locations in the channel. Then the partition function for the grand canonical ensemble, taking into account the interaction of the gas particles in the channel, is Here is the binding energy of the particles in the channel and is the partition function, corresponding to taking account of the interaction of the gas particles in the channel. Since depends on the number of particles in the channel, it is impossible to calculate the partition function (2) in the grand canonical ensemble in the general case. However, in the problem of filling of a channel, the states for which the number of particles in the channel should make the main contribution to the partition function (2). Then the quantity can be replaced by the partition function of interacting particles in the channel, calculated with the average number of particles in the channel: The approximation (3) corresponds to the thermodynamic limit () for gas (system of atomic particles) molecules located in the channel. Using the relation (3), the partition function (2) can be easily calculated. Using the condition for equilibrium between the gas and the surface, we obtain for the average number of particles in the channel Here is the free energy of interaction per atomic particle in the channel. Instead of the number of settling locations in the channel, it is convenient to introduce the average distance between settling locations in the channel () and to replace by the filling factor of particles in the channel. Then, we obtain from (4) an equation determining the pressure and temperature dependences of the filling factor of the channel: The relations (4) show that the problem of obtaining the equation for the adsorption , determining the filling factor at various pressures and temperatures of the gas above the surface, reduces to calculating the quantity , determining the “correction” to the pressure as a result of the interaction of the particles in the channel. Thus, to obtain the adsorption isotherms in the system under study, it is necessary to calculate the total free energy of the particles in a subnanometer channel taking into account their interaction with the channel wall and with one another. For this, we will consider particles in a channel whose size is comparable to the average particle diameter. In the general case, the total potential energy of such a system can be written as Here is the potential energy of the pair interaction of the particles located at points with coordinates and ; is the interaction energy between a particle located at the point and the channel walls. We will transform the expression for the pair interaction potential of the particles using the fact that the channel diameter is comparable to the particle diameter. Then the interaction potential depends only on the particle coordinates along the channel . We write the expression for in the form This relation holds when the channel diameter is comparable to the particle diameter, and the pair interaction potential of the particles is not a long-range potential. It is convenient to represent the interaction energy between a particle and the channel walls in the form Here is a dimensionless two-dimensional radius vector in a plane perpendicular to the channel. It is convenient to write the partition function of such a system in the form Here The integration over the particle coordinates in a plane perpendicular to the channel can be performed exactly: Here is the effective potential in which a particle moves in the channel as a result of the interaction of the particle with the walls: Let us assume that the energy of a gas molecule interacting with the channel surface is constant and equal to . Then The energy physically corresponds to the binding energy of a particle in the surface potential . Thus, under the assumptions made previously, the gas in the channel can be assumed to be one-dimensional (the channel diameter is comparable to the maximum diameter of an atomic particle). It is well known [44, 45] that the partition function of a one-dimensional gas with an arbitrary interaction potential can be calculated exactly under certain assumptions, which are formulated in the following. Indeed, let us consider an equilibrium system of particles in a channel which possesses only one degree of freedom per particle that are located in the segment of the axis. The total partition function of the system has the form where Here is the pair interaction potential of the particles in the channel. To calculate , the cofactor responsible for the partition function of an ideal gas must be eliminated from the partition function (14). In our problem, the cofactor corresponding to a gas of particles in a channel without a pair interaction potential between the particles is already included in the expression (2). Thus, since the partition function of a one-dimensional gas without an interaction has the form we obtain from (2) Thus, the problem has been reduced to calculating the configuration integral In studying the configuration space of a system of particles in a one-dimensional channel, it should be kept in mind that the quantities are not independent. They are related as where is the total length of the channel. Physically, this relation corresponds to the impossibility of particles penetrating one another and the condition of “blocking” of particles with diameter in a channel whose diameter . An explicit expression for the configuration integral can be obtained if the explicit form of the pair interaction potential of the particles is known. We will consider a simple but nontrivial case: a system of hard spheres with diameter . The expression for the interatomic interaction energy in this case is We obtain for in the hard-sphere approximation Thus, in the hard-sphere approximation, the adsorption isotherm for a gas in a subnanometer channel has the form The relation (22) makes it possible to construct the dependence of the filling factor on the gas pressure for various molecules. It should be kept in mind that the sign of the energy determines the possibility or impossibility of a particle entering a channel: for , it is energetically favorable for a gas particle to enter the channel for any external pressure greater than . For , the particles must overcome a potential barrier to enter the channel. The configuration integral can be calculated exactly for an arbitrary interaction between the particles in the channels . This makes it possible to obtain an equation of state of a 1D gas for an arbitrary interaction. Thus, for an interparticle interaction potential of the form where is the effective radius of attraction, the equation of state of a one-dimensional gas is In the limits and , this equation becomes the equation of state of a system of hard spheres. The isotherms calculated numerically starting from (24) are presented in Figure 1. It is also evident in this figure that at high temperatures the gas behaves almost as an ideal gas (). At low temperatures the isotherms seem to consist of two parts. For high density , which is typical for a condensed phase, whereas at pressure less than a characteristic value a gas-like phase for which appears.
It can be shown that at a transition from one regime to another, the free energy of the system has no singularities. For this reason, there is no exact analogy with phase transitions. This assertion agrees completely with the Landau-van Hove Theorem , according to which any one-dimensional model of a gas with a finite interaction radius does not undergo phase transitions. On the other hand, it should be expected that for sufficiently low temperatures, as the filling factor increases, the gas in the channel tends to form clusters whose size increases with the filling factor. The number and size of the clusters grow with the total number of particles in the system in a manner so as to ensure the existence of such clusters in the thermodynamic limit, where , , and . This agrees completely with the formation of nuclei of a new phase in three-dimensional systems with first-order phase transitions [34, 46]. On the other hand, in our one-dimensional system, in contrast to three-dimensional systems, a continuous behavior of thermodynamic quantities determined as derivatives of the free energy of the system with respect to the temperature, pressure, and filling factor should be expected in the entire range of variation of the thermodynamic parameters. For example, the specific heat calculated using the relation (24) will possess a maximum at the “critical” point. This fact has been observed experimentally , which attests to the applicability of the one-dimensional model employed previously.
2.1.2. Transport in a Dense 1D System
The gas flux through a membrane is the main experimentally measurable parameter (Section 2.1.3). Particle transport is usually described using a relation between the outgoing particle flux and the parameters of the problem, such as, for example, the difference of the gas pressures on different sides of the membrane. This description is based on Fick’s relation between the gas flux density , the gradient of the gas concentration, and the diffusion coefficient , using the conservation law for the number of particles: In the relations (25) and (26), is the flux, is the diffusion coefficient, and is the gradient of the particle concentration. The relation (25) is actually a definition of the diffusion coefficient . Thus, the problem of determining the particle flux through a channel reduces to calculating the diffusion coefficient. A more general method of describing transport is to calculate the characteristic relaxation time of density fluctuations arising or specially created in the system under study . Formally, the problem of determining this time reduces to calculating the characteristic frequency or spectrum for the system under study. Thus, when the particle density is low and there is no interparticle interaction, an explicit expression for the spectrum can be easily obtained from the relation (25). Indeed, let be the average particle concentration in the system. From (25), we obtain an equation determining the dynamics of the density fluctuations : Switching to a Fourier representation of the density fluctuations we obtain an expression determining the reciprocal of the relaxation time of the density fluctuations in the case at hand: It is obvious that a spectrum of this type is characteristic for systems where relaxation to equilibrium occurs by diffusion. Let us consider a system of interacting particles with arbitrary density. Assume that we have been able to calculate the characteristic relaxation time of density fluctuations in the system. If the spectrum of relaxation times in such a system has the form then it is natural to consider the quantity to be the diffusion coefficient in the system, provided that in the limit of low densities the quantity becomes the diffusion coefficient for a low-density system of particles: This definition of the diffusion coefficient means that the system possesses a diffusion characteristic mode. Proceeding from the definition (30), to calculate the diffusion coefficient , it is necessary to know the characteristic modes of a weakly nonequilibrium system. If these characteristic modes have the form (30), then will be the diffusion coefficient of the system. It is obvious that the diffusion coefficient in the general case is nonlocal and is a functional of the density of the diffusing particles. The expression for the flux of diffusing particles will have a form generalizing equation (25):
The equation, obtained using this relation, for the particle number density in the system is in general a nonlinear integrodifferential equation. When studying the diffusion of a single-component gas in subnanometer channels, it should be kept in mind that the pair interaction potential between the particles is of short range, and no phase transitions occur in the system of particles in the channel (see the preceding section). In this case, it can be assumed that the diffusion coefficient is local: The corresponding expression for the flux of diffusing particles and the equation for their density become For low densities, the relation (34) becomes (25), which is the first term in the expansion of the flux in odd powers of the density gradient. Thus, for systems where the relaxation spectrum of the fluctuations is of the form (30), the diffusion coefficient can be determined starting from the explicit form of this spectrum. The expression obtained for the diffusion coefficient in this case is a generalization of the expressions obtained for the diffusion coefficient in various models, specifically, in the Maxwell-Stefan model .
It is convenient to calculate the relaxation spectrum using the response functions [31, 48, 49]. Let us formulate the basis of this formalism. Consider the system of the atomic particles, which can interact with each other as well as the environment such as liquid, solid, or surface. Let us show, following , that many-particles distribution functions of such a system are the density functionals. Assume that each particle at the coordinate interacts with time-dependent external field . In order to describe such a system, one can use Liouville equation for many-particle distribution function , where are the coordinates and th momentum of the particles. Solving the Liouville equation with the fixed initial distribution function and for the different external potentials , one can obtain one-to-one conformance between and . On the other hand, introducing the density of the particles where is the phase space element, and is the density operator, one can obtain one-to-one conformance between the external field and the density . From this it directly follows that many-particles distribution functions are density functionals: Following , let us write the action for the Liouville equation in the form where is the Lagrangian of the system, which is equal to the difference between the kinetic and potential energy of the particles. It follows from (36) and (37) that the action is the density functional: Functional has an extrema for the equilibrium state of the system [48, 49]. Due to the linearity of the (38) by the potential energy, system of the particles interacting with the external field corresponds to the functional , which can be expressed as Functional (39) has an extrema for the stationary or quasi-stationary state of the system [48, 49] and represents the time average of the free energy functional : The functional was for the first time introduced in such a form in the papers [31, 48]. Equation (40) determines the density of the atomic particles . Let us express the free energy using the response functions defined as follows. Consider the system in the external field ( - external field amplitude). The response function of such a system couples the density fluctuation with the external field , which generate this fluctuation [31, 48] at : It is obvious that in the case of the linear response and , the response function is independent on . Let us rewrite the definition (42) in the operator form Such a form (43) will be used further in the paper. In the presence of the external field, the functional at is determined by the relation (39): Here is the functional at , determined by the formula (38). In order to obtain linear response function, let us use the relation (39). One can rewrite the solution of the (39) at with the sum . In this case, one can obtain, in the first order and with respect to (39), or At , the quantity is independent on . Comparison of the relations (43) and (46) gives the following form of the response function : It follows from the functional definition that it can be expressed as Here corresponds to the functional of the particles without their interaction with each other, and corresponds to the functional (38), which represents particle-particle interactions. Two times variation of the (49), with respect to (48), gives the following: where is the response function of the noninteracting particles It is shown in the paper [31, 48] that the function is connected to the effective lagged interaction between the particles. Multiplying the relation (45) on the in the left and on the in the right, one can obtain the equation on the response function. In the case of the homogeneous quasi-stationary state of the atomic system, the functions , depend only on the differences , so (50) can be rewritten as where , are the Fourier transform of the response functions , , and , respectively; for example, Equation (53) allows us (when the function is known) to determine the response function of the system of interacting particles and relaxation spectrum of the system [31, 48].
In order to determine the relaxation spectrum, let us note, that the expression follows directly from (43). Due to the response function definition (as the response of the system on the arbitrary small external field), one can write . This equation has the nontrivial solutions only if . So, we can write that the relaxation spectrum of the system is determined by the relation [31, 48] We will use the response function formalism to calculate the response function and the diffusion coefficient of a dense system of atomic particles in a subnanometer channel. We will calculate first the response function and the relaxation spectrum of such a gas when there is no interaction between the particles. We write the diffusion equation for the filling factor of the channel with molecules in the presence of a weak perturbing external field in the form Here is the diffusion coefficient for noninteracting particles. Let be the equilibrium filling factor of the channel. We will seek the solution of (56) in the form Then, we obtain from (56), assuming the external field to be weak, an equation for : In deriving (58), only first-order infinitesimals in and were retained in all terms. We will seek the solution of (58) in the form of a Fourier integral Substituting the expressions (59) into (58) and solving the resulting linear equation for , we obtain Here is the response function of a gas of noninteracting diffusing particles. Indeed, as , the function reduces to the well-known response function of an ideal equilibrium gas with density at temperature [38, 47]: The relaxation spectrum of such a gas is determined from the relation (55) and has the form In accordance with the definition (26), the quantity is the diffusion coefficient. The relations obtained make it possible to calculate the relaxation spectrum by solving (54). Thus, we find for the response function of a one-dimensional system Using the relations (61) and (64), we obtain an equation determining the relaxation spectrum of the system under study: Equation (65) can be solved in a general form in the quasistatic case . Indeed, in this case, we find from (65) It is convenient to rewrite the relation (67), introducing the pair distribution function : Then we obtain from relations (53) and (61) Substituting the expression (69) into (67) gives a relation determining the relaxation spectrum of a dense gas in a one-dimensional channel: We will now calculate the diffusion coefficient of a gas in a one-dimensional channel taking into account the interaction between hard-sphere particles. In this case, the correlation function has been calculated exactly  with an arbitrary filling factor of the channel: Using the relations (70) and (71), we find It follows from (72) that the general equation describing particle transport in a dense 1D system, in accordance with (34), has the form This equation can be obtained from (72) if the characteristic local equilibration times in a dense system of particles in a channel are short compared with the characteristic propagation times of disturbances along such a system. For the cases considered in the following, this assumption obviously holds. We will consider solutions of (73) in the case where complete equilibrium has been established in the channel and the filling factor , with the function being determined by the relation (23). Transport in a channel in this case will be determined by the dynamics of the motion of disturbances of the filling factor. The solution of (73) in this case should be sought in the form . It follows from (73) that the relaxation spectrum of disturbances in this case has the form (72), where . Thus, taking account of the interaction of hard-sphere particles in a channel does not change the character of the relaxation of its weakly nonequilibrium state: the relaxation spectrum (72) remains a diffusion spectrum with diffusion coefficient . However, it is important to underscore that transport in the channel in this case is a collective effect and proceeds via transport of disturbances of the equilibrium density . The diffusion coefficient in this case is the diffusion coefficient of disturbances of the equilibrium density. It is convenient to rewrite the relation (72) for the diffusion coefficient in a different form. Using the Arrhenius character of the diffusion coefficient for noninteracting particles ( is proportional to the product of the squared lattice constant of the wall material of the channel and the relaxation frequency of particles in a channel on defects and lattice phonons and is calculated in , and (13) is the activation energy of diffusing of noninteracting particles in the channel), we will rewrite (72) in the form This way of writing the diffusion coefficient for a gas of interacting particles in a channel makes it possible to give a physical interpretation for the change in the diffusion coefficient accompanying a change in the filling factor . The relation (75) shows that the interaction of particles in a channel decreases the activation energy of the motion of particles, even in the hard-sphere model. For an interaction between hard-sphere particles such that there is no direct attraction between the particles, the diffusion activation energy decreases with increasing filling factor as a result of the effective interaction (see the following). Physically, this corresponds to a change because of the presence of another particle in a neighboring potential well, in the parameters of the potential in which a gas particle moves. Since the gas particles are assumed to be indistinguishable, diffusion with the filling factor (when the effective diffusion activation energy becomes comparable to the temperature of the system) can be interpreted as a transfer of “excitation” of the density along a chain of close gas particles. It is obvious that the motion of such an “excitation” will occur with substantial velocities. This leads to a large increase in the diffusion coefficient when . For , the possibility of such a diffusion mechanism is played out in . We note that for all gases investigated, the barrier is different from 0. This is indicated by the presence of a maximum in the temperature dependence of the flux in a zeolite membrane . It can be assumed that the increase in the diffusion coefficient in a channel observed with filling factors is related to the formation of clusters in the channel whose sizes increase with the filling factor. The transport of gas in the channel containing such clusters is determined by the motion of “excitation” in a finite-size cluster. The formation of clusters in a gas consisting of particles with a hard-sphere pair interaction potential is related to the well-known  appearance of an effective attraction between such particles. This effect is manifested in the appearance of a maximum in the correlation function , describing the probability of finding the “first” particle at a distance from the “second” particle.
The relaxation spectra of the system for an arbitrary wave vector can be found from (70) and (71). It is found that the spectrum contains a real part, which corresponds to transport of an “excitation” in the system. The imaginary part of the spectrum obtained has a minimum at a definite value of the wave vector , which depends on the filling factor . It is natural to interpret the value as the characteristic size of a cluster with a given filling factor and the value as its characteristic lifetime. As the filling factor increases, the value of the effective attraction between the particles increases ; this could lead to the appearance of clusters consisting of two or more particles. However, it is found that as a result of the one-dimensionality of the channel, the lifetime of the clusters that are formed is finite, which corresponds to the absence, as noted previously, of a phase transition in such systems. The dependences of the lifetime and size of the clusters formed on the filling factor are presented in Figure 2.
It is evident in the figure that increasing the filling factor increases the lifetime and the size of the clusters formed. In finite-size channels, for a definite value of the filling factor , a cluster equal in size to the length of the channel is formed. It is obvious that the diffusion coefficient in such a channel increases without bound as , even though the lifetime of such a cluster is finite. If the range of relaxation frequencies is known, then an expression that defines the relaxation of the Fourier components of the particle density in the channel in the case of arbitrary density is given by
From (76), we can get that the equation for the amplitude of the fluctuations can be written as
Equation (77) describes the relaxation of component of the density fluctuation at arbitrary wave vector. In particular, at , this equation allows to describe the relaxation of density fluctuations and propagation of disturbances on the cluster of finite size in the case of the formation of the cluster. At , the equation describes the relaxation of density fluctuations on large spatial scales. This value is related to the macroscopic flux. Let us write the equation of continuity to calculate the flux :
After applying the Fourier transform, (78) takes the form
Using the relation (70) we find that and therefore the spectrum contains both real and imaginary parts, corresponding to the “hydrodynamic” (real part of ) and diffusive (imaginary part of ) transport of particles. Therefore, (80) for the flux contains terms relating to hydrodynamic transport of particles in a cluster and diffusive transport of particles in the space between clusters. We note that the response function method can be used to take into account the interaction of particles in a channel for a potential that is different from a hard-spheres interaction, similarly to the way this was done previously.
Analysis of the experimental data showed that for high filling factors , the dependence of the diffusion coefficient , calculated from the relation (72), leads to a discrepancy between theory and experiment (Section 2.1.3). This discrepancy could be due, in our opinion, to the above-described influence of a finite channel length, the asphericity of the gas molecules, and the possible breakdown of one dimensionality of the problem. These effects lead to the following dependence of the diffusion coefficient on the filling factor: Here is a coefficient that takes into account the finite size of the channel, the possible deviation of the channel from one-dimensionality, and the asphericity of the molecules. The relations (72), (81), and (52) make it possible to describe transport in a subnanometer channel for different values of parameters such as the gas pressure and the pressure difference outside the membrane, the temperature of the membrane, and the type of gas molecules. As an example, we present the dependence, calculated according to (72), (81), and (23), of the diffusion coefficient in a subnanometer channel on the relative temperature for various relative pressures (Figure 3).
It is evident in the figure that the character of the temperature dependence of the relative diffusion coefficient in subnanometer channels is substantially different at different pressures . For example, if the pressure is high (), the dependence of on has a pronounced maximum at temperatures (curve 1). As pressure decreases, the magnitude of this maximum decreases and the maximum itself shifts into the region of lower temperatures (curves 2, 3). A further decrease of pressure to values results in the appearance of a minimum in the dependence of on in the temperature range . The maximum occurs at . Since for , the flux depends on and the average distance between particles in the channel is much greater than the diameter of the particles, the increase in the diffusion coefficient for is due to the temperature dependence of the diffusion coefficient on individual molecules in the channel. At high temperatures, all curves saturate at a value corresponding to . In the next section, the theoretical laws obtained previously will be compared with the experimental data.
2.1.3. Analysis of the Experimental Data: Comparison of Theory with Experiment
Quite extensive experimental investigations of the sorption and transport properties of a series of organic molecules (, n-, i-, and others) and the inert gases Ar, Ne, and Kr in ZSM-5 zeolite mem- branes (MFI, silicate-1) have now been performed. The results are presented in [42, 50, 51]. According to , ZSM-5 zeolite membranes have a complicated chemical and crystalline structure. The chemical structure of zeolite membranes is given by the formula For small values of , zeolite membranes of this type are called silicate-1 membranes. The crystalline structure of ZSM-5 membranes has been studied quite well. It consists of a 3D structure, consisting of sinusoidal channels with a circular cross section (nm) parallel to the axis , which intersect straight channels with an elliptical cross section nm2 parallel to the axis [50, 54]. Cavities ~0.9 nm in size form at the intersection of the channels. The sorption capacity of ZSM-5 zeolite is determined by the number of sorbed (entering the channels) molecules per cell of a crystal. According to [50, 55], the cell parameters are nm, nm, and nm. A single cell is a structure consisting of four segments of 0.46 nm linear channels, four segments of 0.66 nm sinusoidal channels, and four 0.54 nm intersections. Depending on the structure of a molecule, sorption of one or two molecules per intersection is possible. In , the sorption capacity of ZSM-5 was measured for a number of molecules. It was concluded that the molecules , , and n- are sorbed with one molecule per intersection, while nitrogen, n-hexane, and p-xylene are sorbed with two molecules per intersection. Thus, for gases of the type , n-, and , channels in the crystal structure of ZSM-8 can be treated as one dimensional. For nitrogen-type molecules, two molecules can be arranged at an intersection, and therefore molecules can change places at an intersection (absence of blocking). Nonetheless, since there are significantly more nitrogen molecules in a channel (24 molecules per unit cell ), the one-dimensional model can also be used for this gas. Analysis of the experimental data presented [43, 55–58] shows that all gases investigated can be conditionally divided into two groups. The first group of gases, containing, specifically, Ar, Kr, Ne, and , is characterized by a linear pressure dependence of the filling factor ; that is, Henry’s law holds for them . For these gases, the pressure dependence of the flux is nearly linear, and the diffusion coefficient is essentially independent of the filling factor . The temperature dependence of the flux is characterized by the presence of a minimum at temperatures K (e.g., for Ar and Kr), while a maximum is not observed in the diffusion coefficient [57, 58]. Here neon, whose diffusion coefficient increases with temperature , is an exception. The second group of gases, containing, specifically, i-, , , and , is characterized by a dependence of the filling factor on the pressure in the form of a curve that saturates . For these gases, the pressure dependence of the flux is likewise characterized by a curve that emerges at a value that is pressure independent . For this group of gases, the diffusion coefficient depends strongly on the filling factor . The temperature dependence of the flux is characterized by the existence of a maximum and minimum at high (K) temperature . The model, constructed in this paper, of the sorption and transport properties makes it possible to describe these dependences of transport in one-dimensional channels on the basis of general assumptions. Figure 4 shows the dependence of the diffusion coefficient on the filling factor (solid line), calculated using (81) and the experimental data obtained in  for various gases. We note that the dependence (81) presented in Figure 4 takes into account the asphericity of the molecules, equal to . For inert gases and methane (first group of gases), the diffusion coefficient measured in the range of filling factors depends weakly on . A large increase in the diffusion coefficient at is observed for gases of the type and n- (second group of gases). This behavior of the diffusion coefficient can be satisfactorily described on the basis of the model proposed, which takes account of the pair interaction of particles with a very simple form of the interaction potential for hard spheres. It is obvious that this coefficient is different for different molecules.
The physical mechanism leading to an increase of the diffusion coefficient consists in a decrease of the activation barriers for diffusion, which is due to an interaction of the particles for high filling factors. Indeed, it follows from the relation (75) for the effective diffusion activation energy that in this case, when the filling factor is low, , gas diffusion in the channel can be studied in the single-particle approximation neglecting the pair interaction of the gas particles with one another. As the filling factor increases, the diffusion activation energy decreases as a result of the effective attraction of the gas particles to one another. This can be interpreted as the formation of widely separated clusters. Diffusion in each cluster can be treated as a transfer of “excitation” along a chain of close gas particles. A further increase of the filling factor increases the cluster lifetime and decreases the average intercluster distance and, in consequence, increases the diffusion coefficient. For , the diffusion process can be regarded as a motion of this “excitation” along the entire channel, when the arrival of a particle at the channel entrance results in the end-most particle leaving the channel.
Figure 5(a) shows the temperature dependences of the gas flux through a membrane for ethane and argon, calculated using the formulas (5) and (81) and a relation following from (81): . These dependences were compared with the experimental values obtained in . It is evident that the theoretical and experimental results agree satisfactorily with one another. The temperature dependence can also be understood on the basis of the model developed in the present work. Indeed, it follows from the relation (5) that at low temperatures (), the filling factor .
The effective diffusion activation energy is at zero temperature (56). The diffusion coefficient of the gas particles in the channel (81) approaches zero as a result of the “freezing out” of the thermal motion of the particles in the potential field of the channel surface. Increasing the temperature decreases without appreciably changing the filling factor. The value of reaches a minimum at a certain temperature . It is obvious that the diffusion coefficient reaches its maximum value precisely at this temperature. It follows from (54) and (75) that a further increase of temperature decreases the filling factor , and this is accompanied by an increase in the effective diffusion activation energy right up to the value which is attained for at some temperature . The temperature corresponds to a minimum in the function . A further increase of the diffusion coefficient with increasing temperature corresponds to the variation of the coefficient according to the Arrhenius law . Figure 5(b) shows the temperature dependence of the flux for argon, calculated using (81) and the data from . We note that the theory developed in the present paper predicts that the presence of a minimum at K should be accompanied by the appearance of a maximum at K. In this connection, for further elaboration of the theory, it is of interest to make an experimental search for a maximum in the temperature dependence of the flux for argon near K. Thus, two types of behavior of particles in a channel, which have different diffusion mechanisms, can be distinguished in the pressure and temperature ranges investigated. For the first group, this is the diffusion of single particles in a channel, and for the second group it is diffusion as a result of a collective interaction of particles in a completely filled channel. Such behavior of single-component gases is a direct consequence of the real one-dimensionality of subnanometer channels, where molecules cannot change places with one another. The proposed model is based on two basic assumptions: (1) the pair interaction between gas particles plays the decisive role in the description of the state and transport phenomena in a one-component gas in subnanometer channels, and (2) as a result of the fact that the channel diameter is comparable to the diameter of the gas molecules, such channels can be assumed to be one dimensional. The one-dimensionality of the system studied is actually due to the fact that the gas molecules do not have classical transverse degrees of freedom. This difference from the conventional systems plays a fundamental role in the analysis of all phenomena described above.
We note that (73) derived previously possesses, besides the solution described in the text, strongly nonuniform nonstationary soliton-like solutions. It should be expected that an experimental consequence of the existence of such solutions will be the appearance of substantial fluctuations of the gas flux through a membrane. The proposed model satisfactorily describes the equilibrium properties of a gas in a channel (specifically, the filling factor) and the weakly nonequilibrium properties of the gas, such as the diffusion coefficient and flux. It should be noted that a satisfactory description of the equilibrium properties, for example, the adsorption isotherm, has been obtained previously on the basis of a number of phenomenological models . A qualitative analysis of the maximum (but not minimum) in the temperature dependence of the relative diffusion coefficient is presented in . A model describing the dependence of the diffusion coefficient on the filling factor on the basis of a one-particle approximation by introducing into the diffusion rate jumps of individual particles a phenomenological correction factor has been proposed in . However, even though each of the phenomena indicated previously can be described on the basis of individual models, the approximations used in so doing either are not physically substantiated or completely contradict one another.
2.2. Particle Mobility and Diffusion in 1D System
Although, in many cases, the parameters of 1D systems can be calculated exactly, their complete theoretical description has not yet been accomplished. 1D systems belong to the class of strongly fluctuating systems ; thus, in contrast to systems of higher dimensionalities, they cannot unambiguously be described only by mean values such as mean density and compressibility . The transport properties of 1D systems can also be rather specific (see [43, 57, 60] and Section 2.1). Recent ab initio calculations  have shown that water flux through hydrophobic nanotubes is more than one order of magnitude higher than the estimated ordinary diffusion flux. The permeation experiments on zeolite membranes  have shown that for gases, whose particle diameter is larger than half the zeolite channel diameter, the increase of the diffusion coefficient with density is more than one order of magnitude larger than would be expected for one-particle diffusion. On the other hand, there are also experimental results, for example, obtained using nuclear magnetic resonance (NMR) and quasi-elastic neutron scattering (QENS) techniques, indicating that the particle mobility in 1D systems can decrease and the mean square displacement (MSD) can become proportional not to time, as it is in the case of a normal (Einstein) diffusion, but to the square root of time, (so-called single-file diffusion mode (SFD)) [43, 62–67]. Moreover, the experiments [64, 66] show that both the Einstein and SFD diffusion modes can exist for the same particles but at different densities and observation times. Furthermore, for certain investigated gases, a density increase leads to a transition from the Einstein diffusion to the SFD mode . Thus some experiments suggest that the particle mobility in 1D systems must decrease with increasing density, whereas others show that diffusion accelerates as density becomes larger. An explanation for this contradiction is absent at present.
Diffusion in 1D systems has been studied in numerous theoretical works [36, 57, 67–73]. The dependence has been obtained for a 1D system within the screened-gas model, which partly takes into account correlations between the particles [68–73]. It has been suggested [70–73] that the appearance of the two diffusion modes, characterized by the different time dependences of MSD, can be explained by a transition from the one-particle diffusion mode, occurring at short times, to SFD appearing at longer times when the particle mobility is limited by its neighbors. In this case at the dependence of MSD is always . Although this approach can describe the experimentally observed dependences of MSD, corresponding to SFD, it fails to explain the increase of the diffusion coefficient and transport acceleration observed in dense 1D systems . The physical reason for this failing is that the proposed model  considered the system of individual tagged particles, whereas in experiments one does not follow the diffusion of a particular particle but rather deals with the system of many of identical particles. We propose a description for the particle diffusion  and mobility [43, 62–65] in 1D systems of identical particles with blocking effect (BE) when the motion of a particle is blocked by other particles or clusters. According to our model, different mobility and diffusion transport mechanisms can be realized in a 1D system depending on its density. At low densities, an ordinary diffusion of single particles takes place. As density increases, a formation of 1D clusters of a finite size and lifetime takes place. In this case, there are two competing mechanisms prevalent in the system: a barrier-free propagation of density perturbations (“collective diffusion” (CD))  and a blocking effect. When BE dominates, the SFD mode is realized, whereas when the CD mechanism prevails, we predict an essential transport acceleration, as compared to the one-particle diffusion mode. In this case, MSD is Einstein-like with the diffusion coefficient increasing with density. The proposed model consequently takes into account all the time-space correlations occurring in a 1D system composed of identical particles with an arbitrary interaction potential. This model is able to describe the experimentally observed MSD dependencies at different densities as well as the transport acceleration with increasing density of a 1D system. To describe the transport properties of a 1D system, we apply the response function method (see Section 2.1 and [31, 32, 48]). This approach allows one to consistently take into account many-body correlations and obtain the diffusion coefficient and MSD for a 1D system of arbitrary density. Within response function method, we calculate the response function , which determines the Fourier image of the density fluctuation occurring in the 1D system in the external field of frequency and wave number . The relaxation spectrum of a density fluctuation () can be calculated from the relation . Note that in response function method, can be interpreted as a distribution function if , where is the particle diameter and is the one-particle diffusion coefficient . A density fluctuation in a 1D system is determined by the relation (87), and the MSD can be expressed as It is evident from (87) and (83) that MSD is determined by the relaxation spectrum . The evaluation of is reduced to the solution of Dyson equation for the response function (55) and for the 1D system takes the form (70), where is the Fourier image of the pair correlation function of interacting particles. The function can be obtained for a 1D system with an arbitrary interaction potential : Here is the temperature. The last expression in (84) helps us to obtain the dependence of pressure on the filling factor (). It follows from (70) and (84) that at , the relaxation spectrum describes the one-particle diffusion . For , the Fourier image of the pair correlation function approaches zero for any arbitrary short-range potential . This corresponds to the absence of correlations between the particles at the length scale , where is the average particle path between two collisions. In this case, the relaxation spectrum also describes the one-particle diffusion mode . The analysis of the relaxation spectrum performed in Section 2.1 for the hard-core potential has shown that due to collective effects and density fluctuations, the 1D system can transform into a nonhomogeneous state as density increases. This state is characterized by the formation of 1D clusters with a finite lifetime (Section 2.3) (Section 2.1). In this case at high density (), long distances (), and (see (70), (84)) the spectrum becomes diffusion-like with the diffusion coefficient increasing with density (72) that corresponds to the CD mode existing in a nonhomogeneous, clustered 1D system . In order to describe the relaxation of density fluctuations at times (that corresponds to the long-time limit ) for an arbitrary filling factor , one should do series expansions of (70) and (84) with respect to the parameter and keep the expansion terms following the first one. It is convenient to rewrite this expansion with respect to parameter . Having done this, the relaxation spectrum for can be written as where the expansion coefficients are , , and . The values with apostrophes represent the corresponding derivatives of with respect to wave number , calculated at . Equations (83) and (85) can be used to evaluate the time dependence of at any length scale, starting from a diffusion path of just few particle jumps up to those comparable to the total length of the channel. Note that the real part of the spectrum , which at corresponds to the sound mode, describes the collective effect of density perturbation propagation (), and its imaginary part determines the diffusion mode for the relaxation of a density fluctuation taking into account the length-time correlations between the particles (). Equations (87), (83), and (85) allow us to study the mobility and distribution function of the particles in a 1D system for different length/time scales, filling factors and interaction potentials . Figure 6 presents the dependences of the distribution function on the dimensionless coordinate , calculated for different dimensionless times () using the square-well interaction potential with the well depth () and width .
It is important to mention that the obtained results do not depend on the specific form of the pair interaction potential ; we have obtained similar results using rectangular potentials, the Lennard-Jones and Morse potentials. We also note that the shape of the distribution function is independent on density as both dimensionless coordinate and time are determined by the filling factor. The distribution functions at different time scales clearly differ from each other. The distribution is a Gauss-like function at short times (, ) that corresponds to the one-particle diffusion (Figure 6). At , , the distribution function varies faster than the Gauss distribution (Figure 6) that corresponds to limited mobility of a particle due to its blocking by other particles. At longer times (, ), the distribution function becomes Gauss-like again (Figure 6) as the transport in a dense 1D system is due to the CD mechanism . In this case, the relaxation spectrum is diffusion-like with a density-dependent diffusion coefficient .
MSDs calculated at different time and length scales (Figure 7) using the distribution functions (Figure 6) noticeably differ from each other confirming the existence of several diffusion modes in the system. At short length-time scales (, ), one should observe the one-particle diffusion mode with (Figure 7(a)). When is comparable to and time , the correlations between the particles become essential that lead to the appearance of blocking effect. In this case, MSD has a non-Einstein character (Figure 7(b)). At longer times , , the particle mobility is mostly determined by the second term in (85). This leads to the reappearance of the Einstein-type 110 dependence of MSD, , where is an effective diffusion coefficient. Notice that the characteristic times at which different diffusion modes can be observed depend on the density of the system. Indeed, the average distance between the particles and, consequently, the time decrease with increasing density.
Further we compare our results to those of the QENS, NMR [43, 62–65], and permeation  experiments as well as to the results of the molecular dynamics (MD) calculations  summarized in Table 1. Notice that such a system can be considered as one dimensional with the presence of BE as the diameters of the experimentally investigated molecules are less than the doubled diameter of the membrane channel. Our estimates for zeolite ZSM-48 obtained using the data from  show that for the low filling factor () and times ns, the normal diffusion mode, , should occur while at higher density () the dependence should be observed, which is in a perfect agreement with the experimental data (Table 1). In Table 1, experimental and MD results are presented; and are the particle and membrane channel diameters, respectively, is a temperature, and is a degree of filling of the channel. The fifth column represents the data on the motion mode of a particle in a channel (“SFD" means , , “normal" corresponds to ). The observation time of corresponding modes is represented in the sixth column.
The experimental results obtained for ethane molecules in zeolite and for methane molecules in zeolite ZSM-48 (Table 1) are successfully described by our model. Indeed, using (85), one can write MSD for the SFD mode at as Equation (86) with is identical to the relation derived in  with the accuracy of the numeric coefficient . Note that describes the correlations effects in a 1D system of interacting particles. Using (86) and the parameters for , we obtain μm for times ~100 ms that is in good agreement with the values reported in [62, 64]. At short times s, the main contribution comes from the first term in (85), and, in this case, the normal diffusion mode is realized. Our results obtained using (83) and (85) at and the experimental data for ethane molecules in zeolite are compared in Figure 8.
The dependence of the diffusion coefficient on the filling factor was investigated for several gases (, , and ) in silicalite-1 membrane at times s . The diffusion coefficients for these gases were found to increase more than 20 times as the filling factor varied from 0 to 0.9. This observation can be explained by the collective effect of density perturbation propagation through a 1D clustered system. The calculated dependence of the relative diffusion coefficient on the filling factor in comparison with experimental results is presented in Figure 8 (insert). The figure shows that the diffusion coefficient is only weakly dependent on the filling factor for (Figure 8 insert) but it rapidly increases when the filling factor becomes larger. This behavior of can be easily understood as at low filling factors, the particles inside the membrane channel do not interact with each other, no clusters are formed, and particle motion is purely diffusive. The increase of the diffusion coefficient for is due to the cluster formation inside the channel and activation of the CD mode (Section 2.1).
2.3. Formation of High- and Low-Density Clusters in a 1D System
In recent experiments , the one-dimensional (1D) Au chains formed on the NiAl surface have been investigated at low temperature (K). This surface exhibits 1D grooves made of the rows of Ni atoms, which suppress the transverse movements of Au adatoms allowing them to move only along the groove. The measured Au-Au distances in such chains were about Å that is very close to those in bulk gold . The properties of 1D Au chains were also investigated in experiments [8, 12, 75–77] where they were either pulled out of a gold surface by an STM tip or created in a process of thin film etching. Such chains were stable during the observation time and consisted of several gold atoms. The Au-Au distances in some of them appeared to be strikingly large, up to 3.5–4.0 Å [75–77], whereas in others much smaller Au-Au separations (2.5–2.9 Å) were observed [8, 12, 78–80]. These results suggest that a 1D gold chain can exist in two different states characterized by different interatomic distances.
The ab initio calculations performed at zero temperature have shown that a 1D gold wire should break when the Au-Au separations exceed ~2.9 Å [81–84]. Being in agreement with the experiments reporting the Au-Au distances of 2.5–2.9 Å, this result, however, does not explain the existence of large Au-Au separations. It has lately been demonstrated that the presence of hydrogen or other light atoms in the wire structure can stabilize gold chains and significantly alter their properties [3, 4, 82–88]. Assuming a somewhat ordered distribution of impurities within the wire, one can easily explain the experimentally observed large Au-Au separations [6, 82, 83, 87]. Although this finding offers an intriguing possibility to modify the wire properties by an intentional incorporation of impurities (Im), a spontaneous formation of locally ordered Au-Im-Au structures seems unlikely and remains to be proven experimentally. Another explanation has been proposed by Sun et al.  who suggested that the large Au-Au distances might result from chain melting. It is known, however, that phase transitions do not occur in 1D systems ; thus, the proposed model should be taken with precaution.
Monoatomic chains can disintegrate due to atomic movements in both longitudinal and transverse directions caused by thermal fluctuations. The probability of disintegration is determined by the characteristic frequencies of atomic vibrations in the chain, and the energy barrier atoms need to be overcome in order to leave the chain, which is comparable with the energy per atom in the chain. A change of interatomic distances due to transversal or longitudinal motion of one of the atoms leads to a transition to a nonequilibrium state, and the energy per atom increases. As the energy barriers in the longitudinal and transversal directions are comparable , the disintegration probabilities for these two mechanisms appear to be comparable as well. Therefore, further on we consider only one mechanism occurring due to longitudinal fluctuations.
The probability of disintegration of a chain due to longitudinal fluctuations is determined by the frequency spectrum of relaxation of density fluctuations (relaxation spectrum) of the quasi-1D system. To study the relaxation spectrum of a quasi-1D system, we apply the previously stated response functions for density functional method, which allows one to describe the equilibrium and dynamical properties of a system if its interaction potential is known. Within this method, one can find the response function that determines the relation between the density fluctuation and the external field as well as all the space-time correlations in the system. Additionally, one can find the free energy of the quasi-1D system taking into account density fluctuations ( and Section 2.1). A density fluctuation appearing in the quasi-1D system can be determined as Here is the modulus of the wave number, and is the amplitude of the density fluctuation, which in the analysis of cluster relaxation can be arbitrary and adequately considered to be a constant for in the interval ; . Clusters forming in the system are characterized by the parameter , which is defined as a Fourier integral (87) in the interval , where is the cluster length, and is the atomic diameter. The cluster lifetime () is determined by the maximal magnitude of the imaginary part of the spectrum () in the interval .
The relaxation spectrum of the 1D system is determined by (70). If the relaxation of such a system is determined by diffusion with the coefficient , then . is a dimensionless density. is the average number of particles per length unit. is a Fourier image of the pair correlation function for interacting particles, which is related to the Fourier component of the pair distribution () as , where is the Dirac delta-function. Note that the divergence must be removed from before any analysis. Otherwise reverse Fourier transformation and the transition from wave numbers to coordinates are impossible due to the integral divergence. For a quasi-1D system, the pair distribution can be calculated exactly for an arbitrary short-range interaction potential : Here and are 1D pressure and temperature, respectively. Equation (88) allows one to evaluate the pair distribution and consequently the relaxation spectrum of the 1D system. We have performed an extensive analysis, which has shown that the properties of quasi-1D clusters are determined by the presence of the repulsive and attractive terms in the potential but not by the specific form of . In particular, we have obtained similar results using rectangular potentials, and the Lennard-Jones as well as Morse potentials. We notice that this finding correlates well with the known results of the liquid state theory [45, 46, 91]. For the sake of simplicity, we present here the calculations of 1D pressure ( is dimensionless density) and pair distribution done for the square-well potential with the well width and depth : The performed calculations give the dependence: Equations (88) and (90) allow one to analyze the possible states of the quasi-1D system of the interacting particles of an arbitrary density. They also let us to estimate the relaxation time of a density fluctuation (cluster) as well as the interatomic distances in the clusters. We choose to perform this analysis using the parameters of the pair potential (89), eV, Å, and Å, corresponding to the gold atoms . For these parameters, we calculate the compressibility (Figure 9), pair distribution (Figure 10), and dimensionless spectrum (Figure 11).
At low densities (), the compressibility increases with increasing density and it approaches its first maximum at (Figure 9). The compressibility has a second maximum at . The derivative of the compressibility equals 0 at extrema. Therefore, the presence of extrema can indicate changes in the system state. As we show in the following, the presence of maxima does not depend on the specific form of the pair interaction potential , but it is determined by the ratio ( is a characteristic depth of an arbitrary potential). The second maximum of the compressibility disappears if . When , the second maximum always exists if the condition is satisfied regardless of the specific form of the potential . Also we have to note that the ratio provides the existence of the second maximum whereas the position of this maximum does not depend on the magnitude of this ratio. This allows us to conclude that in the limiting case , the variation of the temperature of a system does not lead to any qualitative changes in a state of a system.
To demonstrate how the physical state of the system changes at the compressibility maxima, let us analyze the pair distribution that determines the probability to find atoms at a certain distance from each other. It has been shown in Section 2.1 that the function when . The distances between the maxima of are equal to 1, 2, 3, and so forth, atomic diameters. Hence, the separation between these maxima corresponds to the interatomic distance in the clusters .
In the vicinity of the first compressibility maximum (), the distance between the peaks of and, correspondingly, the interatomic distances in the clusters are ~3.6 Å (Figure 10(a)). At densities close to the second maximum, interatomic distance in the cluster ≤3 Å (Figure 10(b)). As it was mentioned above in the limiting case , the variation of the temperature of a system does not lead to any changes in the state of this system. Therefore, at and 12 K, the magnitude of the gold cluster interatomic distance will be the same.
As it has been noticed previously the qualitative shape and the distance between the peaks of the pair distribution function are independent of the concrete form of the interaction potential. The pair distributions calculated for with square-well potential (solid line) and Morse potential for the Au atoms (dashed line) are compared in Figure 11.
Figure 11 demonstrates that the distances between the peaks are the same for both potentials, but the peaks for the Morse potential are higher compared to the ones obtained with the square-well potential. This difference is due to the different characteristic length scales of the Morse and square-well potentials. The characteristic length scale of the Morse potential is larger than that of the square-well potential that leads to a larger atomic attraction at long distances. Note that in the case of the Morse potential in (89) is a potential well depth and is the characteristic length scale of the potential.
Further, we analyze the stability of the 1D clusters. The change of the free energy of the 1D system () due to the cluster disintegration determines the probability of such a disintegration . can be divided into the following contributions: that is the change of the free energy due to the pair interactions in the cluster; that is the change of the free energy coming from changes in the cluster electronic states; and the fluctuation part that is an average fluctuation of the free energy of the 1D system caused by cluster disintegration calculated for the stationary state of the 1D system. Therefore, the probability of cluster disintegration can be written as To estimate the cluster lifetime , we analyze the contributions to . Since there are no phase transitions in 1D systems, is always positive and its magnitude is determined by the spectrum  (Figure 12):
It follows from (92) that , which is related to the probability of cluster disintegration due to the loss of one or several atoms, is determined by the maximal magnitude of in the interval (; ). In the vicinity of the first maximum of the compressibility at (Figure 9), . Here is the wave number corresponding to the maximum of , (Figure 12(a)), and is the number of particles in the cluster. It follows from (91) that in this case the magnitude of is large for all the clusters with the length exceeding several atomic radii. In the vicinity of the second maximum of the compressibility at (Figure 9), becomes larger as the cluster length increases (Figure 12(b)). In this case for all clusters with the length exceeding , for which , ) and the probability is large. Thus at all densities, only clusters of a certain maximal length can exhibit a noticeable lifetime, and the formation of long clusters appears to be unfavorable.
is the sum of the changes of the free energy of the pair interaction between atoms in the cluster and the free energy of the cluster electronic states ). In the limiting case , , is negative  and it is determined by the overlap of the atomic orbitals. This overlap is determined by the penetrability coefficient of the potential barrier separating the electronic potential wells of the neighboring atoms (Section 2.4). In the quasi-classical approximation, depends exponentially on the width and height of the potential barrier (Section 2.4). It has been shown  that due to collective effects, the penetrability coefficient depends not only on the interatomic distances, but also on the difference between the maximum and minimum values () of the pair distribution (Figure 10). The interatomic distances in the clusters formed at the densities about are larger than those for the clusters existing at (Figure 10), but . Using the approach of , we find that in this case the magnitudes () for the clusters of both kinds appear to be similar. Thus, for our analysis of cluster stability, we can use the value eV obtained from the measurements of the cluster conductivity in the scanning tunneling spectroscopy (STS) experiments for the Au chains on the NiAl surface . The energy per atom can be written as . Calculations show that is negative and eV and eV for the clusters at and , respectively.
Let us now compare the lifetime and interatomic distances calculated for the 1D clusters to the known properties of the 1D chains fabricated in experiments [74–80]. The results of the STS experiments  as well as the ab initio study of the electronic structure of gold wires on NiAl  suggest that the main role of the substrate is to be a natural template insuring the one-dimensionality of the Au chains. This allows one to suggest that the observed Au-Au distances (~2.9 Å) in such chains to a large extent might be determined by the one-dimensionality of the system. Taking into account the minimal possible distance between Au atoms (~2.5 Å ), we can estimate the density in these chains: . Thus, the fabrication of Au chains on NiAl can be understood as an artificial formation of 1D clusters with the density fulfilling the condition (). As we have shown previously, the Au-Au separations in such 1D clusters should range from 3 Å () to 2.7 Å () or 2.5 Å (). The lifetime of such clusters of the length can be estimated as the characteristic time () during which an atom stays in the potential well of the depth (see (91)). Here s. The calculations of the spectrum at show that, for example, for the cluster of the length Å, containing ~5 Au atoms . In this case using the estimates obtained previously ( eV), one can estimate the lifetime of such chains to be s (K). Notice that in the absence of collective effects and attractive interaction between atoms in a quasi-1D system the lifetime of the chain of Å would be very short, s (here s m2/s is the diffusion coefficient for Au atoms at the surface of NiAl  at K).
The formation of suspended Au chains by pulling them out of the Au surface or by etching an Au film with an electron beam [2, 8, 12, 75–79] corresponds to conditions where considerable tensile stress is applied to the chain. The presence of an external force corresponds to the appearance of the one-dimensional pressure , where is a free energy of the 1D chain. One can see from (90) that 1D chains of finite length are affected by two types of forces: forces corresponding to the effective repulsion of the atoms from each other as a result of their movement (the first term in (90)) and forces corresponding to their mutual attraction that is determined by interaction potential of atoms with each other. The attractive force depends on depth of potential well and radius of the potential and equals in the van der Waals approximation. Thus a cluster is in a compressed state. The value of compression is . The presence of such a compression can lead to loss of stability of the cluster in the transverse direction and formation of a zigzag configuration of a chain. Therefore, to form a stable linear chain, it is necessary to attach an external force to the end of a chain to compensate the compression . The presence of an external force corresponds to the appearance of one-dimensional pressure , where is the free energy of a one-dimensional chain. Let us estimate the lifetime of these chains.
The decrease of the tensile force corresponds to the decrease of the density (90). The observation of Au chains [75–77] containing three or more atoms and exhibiting large Au-Au separations ( Å) can be considered as a formation of a 1D cluster at the pressure for which satisfies the condition (). The lifetime of such chains can be estimated as , where , . For eV, eV and (Figure 12(a)), it turns out that s (K). The experimentally obtained chains [1, 8, 12, 78–80] with the Au-Au separations of 2.5–2.9 Å are formed at the pressure for which fulfills the condition . According to our estimations, the Au-Au separations in such clusters should be Å (Figures 10(b) and 12(b)) and their lifetime is s (K, Å, ) (Figure 11).
Within the framework of these assumptions, we can also estimate minimal lifetime of chains of other metals, such as Ag, Ni, and Pt. Due to the lack of experimental data on the characteristic values of for these materials let us assume , where is the binding energy per atom for dimer . The calculated values of minimal lifetime of one-dimensional chains are presented in Table 2. As one can see, the material of a one-dimensional chain greatly affects its stability.
Let us estimate the minimal length of stable chains and the force that is necessary for their formation. Note that if we consider the chain as a rod of length with a radius equal to the atomic radius , then such a rod loses its stability when it is compressed with a force greater than , where is the Youngs modulus. So, if the compression pressure of the chain is , then the chain is unstable. Therefore, the minimum compressive force that ensures stability of the chain is . Chains of metal atoms have a conductivity comparable to the metal one , so the range of the interatomic potential is comparable to the length of the chain , where and varies from metal to metal. Relations and allow to estimate the minimum number of atoms in a cluster , where is an interatomic distance in the cluster. The value is the force that compresses a stable cluster without external influences on it. In this case, is a force required to break the cluster with the minimum number of atoms in it. Table 2 shows the results of the estimations of the minimum number of particles in the chain and the corresponding force required to break the chains for chains of different material. It was assumed that for the chains of Ag. For the rest of metal chains, this coefficient was estimated by using the relative conductivities , , Ni, Pt.
2.4. Transport of a Two-Component Mixture in One-Dimensional Channels
The dependence of transport selectivity on the pressure in a two-component system of atomic particles is nonmonotonic: the selectivity attains its maximal value and then decreases instead of increasing. Thus, the mechanism of diffusion enhancement in 1D channel, which was proposed in Section 2.1, is inapplicable for two-component mixtures.
The transport of a two-component mixture in subnanometer channels was considered earlier [51, 97] on the basis of the generalized phenomenological Stephan-Maxwell equation. The authors of these publications used the dependence of the chemical potential on the fill factor, taking into account the finite size of particles, but disregarding their interaction. The dependences obtained in [51, 97] also indicate a monotonous increase in the diffusion coefficients, fluxes, and selectivity upon an increase in and, hence, fail to describe the experimental data for two-component mixtures.
2.4.1. Ground State of a System in 1D Channels
Let us consider the surface of a porous body in contact with a two-component gas mixture at temperature and pressure . Suppose that adsorption centers are located on the surface. We assume that the particles on the outer surface do not interact with one another. We also assume that the energy of a gas molecule on the surface is equal to , , depending on the species of the molecule. We also suppose that cylindrical channels () of diameter and length emerge at the surface. We assume that the diameter of a cylinder is comparable to the maximal diameter of the gas molecule. Let us assume that is the binding energy of the th particle at the mouth of a channel, is the number of particles of the th species above the membrane, is the total number of particles of the th species in the channel, is the total number of th particles in the channel and on the surface, and is the number of “seats” in the channel. Then the partition function of the grand canonical ensemble taking into account the interaction of system of atomic particles in the channel has the form Here, the second sum corresponds to summation over configurations