Molecular Transport in Sub-Nano-Scale Systems
1
S. Yu. Krylov
Institute of Physical Chemistry, Russian Academy of Sciences, 119991 Moscow, Russia
Abstract. Over the last several years, there has been an increasing interest in the problem of molecular transport in atomic-size
channels. This is caused, on one hand, by the manufacturing of new materials with fine channel structure (carbon nanotubes)
and, on the other hand, by the non-triviality of the physical mechanisms involved. In this paper we try to give insight into
the problem, concentrating on the following three aspects. First, the deep physical analogy between the problem of molecular
transport in ”sub-nano-channels” and the problem of surface diffusion of adsorbed particles. A great deal of knowledge
can be transferred from the latter field (relatively well investigated during last several decades) to the former. Second, the
applicability of ideas and methods of physical kinetics, and RGD in particular, to analyze many aspects of both particle-insubstrate and particle-on-substrate diffusion. This seems to be underestimated in the literature. Finally, the appearance of new
transport phenomena specific for sub-nano-channels, like quantum effects of the confinement (e.g., quantum sieving) and
phonon-molecule drag.
INTRODUCTION
Considering a gas in a channel and decreasing size of the system (say, at constant pressure and temperature), one
passes the hydrodynamic regime, the intermediate and the Knudsen gas regimes (the traditional fields of interest of
RGD), and then the situation in which surface diffusion at the channel walls coexists with or prevails over the bulk gas
flow. Finally one ”enters” channels whose width is comparable with the size of the molecule, to be called "sub-nanochannels". Here one deals with an extreme case of surface diffusion.
The problem of molecular transport in nano-channels was extensively investigated by chemists who developed
thermodynamical and/or mechanistic models in application to zeolites and related microporous materials [1]. Surprisingly, physicists did not ”enter” sub-nano-channels until very recent times, although some important observations
(like a ”universal” increase of diffusivity with occupancy in atomic-size pores of zeolites [1]) remained unexplained
for many years, thus suggesting the existence of unusual physical mechanisms. Perhaps for the first time, the physical
nontriviality of the problem was pointed out in the middle of 90s [2-4]. A few years later, an increasing stream of
publications on the behavior of molecules in sub-nano-channels has appeared in the literature (see, e.g., [5-12]). The
stream was apparently activated by the manufacturing of carbon nanotubes, materials that have well defined microscopic structure, outstanding mechanical and electronic characteristics, and hold considerable promise as sieves for
molecule and isotope separation [7], and as molecular containers with applications to hydrogen fuel cells [13] and
rechargeable Li batteries [12].
What are the mechanisms that determine mobility of molecules in an atomic-size channel?
Essentially one deals with an ensemble of particles that are in permanent interaction with the substrate. Generally
this interaction can be considered as being split into two parts. The first is a static potential V that confines the
particle in the transverse direction but allows (quasi-one-dimensional) axial motion along the channel. The second
part is the interaction with phonons and, generally speaking, other thermal excitations of the substrate. Relaxation of
molecular momentum and energy to the phonon bath takes place on the time scale of the order of ;12 s [2].
Consequently, even if the channel is assumed to be absolutely smooth (i.e. the static potential V is independent of
the coordinate x along the channel), the molecules will exhibit diffusive motion with mean free path of the order of
several tenth of nanometer. The diffusion coefficient D (sometimes called collective diffusion coefficient 2 ) is usually
( )
10
1
2
Invited paper
One should distinguish [14] between the transport diffusion and the tracer (or self-) diffusion that characterize mobility of a marked particle.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
735
=
defined by the equation G ;Drn; with G the particle flux along the channel and n the density (per unit length) 3 .
For the simple case of a smooth channel, the diffusion coefficient in the low density limit is easily calculated [2] to be
D0(SC ) vT2 = ; with the characteristic rate of molecule–phonon ”collisions” and vT
kB T=m 1=2 the thermal
molecular velocity. The situation in this case can be thought of as diffusion in a molecule–phonon mixture.
In reality the potential V is a periodic function of the coordinate x; in view of atomic structure of the solid and/or due
to specific character of the interaction bonds. Corresponding potential minima represent ”adsorption sites”. Usually
the height of the barriers between the sites, Vb Vmax ; Vmin ; is larger than temperature kB T : Consequently, the
diffusion is an activated process that is seen as being realized by consecutive jumps between the sites. This is the
origin of the lattice gas model that is often used in this field [1, 14]. Still the phonon subsystem tacitly plays the key
role: it is the source of energy and momentum needed for molecules to overcome the potential barrier to escape from
the adsorption well, and the sink of those leading to trapping to the neighboring (or, possibly, to a distant) well. The
simplest way [14] is to take into account only single hops to the neighboring adsorption sites, and to calculate the jump
rates using the transition state theory (TST). This leads to a density-independent (in contrast to experiments) diffusion
coefficient
D(TST ) l2 ;Vb =kB T ;
(1)
=
2
= (2
=
)
(
= exp(
( ))
)
(10
)
10 )
13 12 s;1 the frequency of
with l the distance between the sites (the period of the potential V x ; and the molecule vibration in the adsorption well. The TST is of course a rough model because it is based on equilibrium
treatment of the system and ignores the role of the particle’s coupling to the substrate excitations in the dynamics of
jumps. More sophisticated—kinetic—theory is needed to correctly account for the effects of the particle–substratebath coupling, and for the effects of interparticle interactions (see below).
The above discussion tacitly assumed the axial and radial motions of the molecule in the channel to be decoupled,
which seems a reasonable approximation. Alternatively, one may consider diffusion due to ”deterministic Hamiltonian
chaos” [15] caused by the coupling between these types of motion in realistic potential V x;y;z : However, this cannot
be the acting mechanism of diffusion [16] since it will always be dominated by the molecule–phonon coupling.
One finds a deep physical analogy between the transport in a sub-nano-channel and the ”ordinary” surface diffusion
of adsorbed species [14, 17] (in its one-dimensional variant observed on certain channeled faces of crystals). In
the latter case adparticles are confined to the surface by attractive forces acting on a sub-nanometer scale, so that
the analogy is direct. A good part of knowledge can hopefully be transferred from the latter field (relatively well
investigated due to the use of advanced experimental techniques like STM and FIM) to the former. The list of nontrivial
problems that one meets there includes, in particular, the occurrence of ”long” jumps between non-neighboring
adsorption sites (see, e.g., [18]); the problem of anomalous preexponential factors of the diffusion coefficient (that can
deviate by many orders of magnitude [19] from the ”normal” value given by the TST result (1)); and the diffusion of
clusters [20, 21]. A more complicated question is the density dependence of the diffusion coefficient [1, 14]: depending
on the situation, both a ”universal ” behavior and a variety of trends have been observed.
The main fundamental aspect of our problem (for both in-substrate and on-substrate diffusion) is the evolution of
an ensemble of particles residing in a one-dimensional (1D) periodic potential V x and relaxing to a bath formed by
the substrate. Kinetic treatments of this problem have so far been less popular than the TST model, and nearly all, with
very few exceptions, have been restricted to the Fokker-Plank (FP) model of particle–substrate-bath coupling [17]. On
the one hand, FP approach is rather complicated mathematically, so that it generally requires numerical calculations,
and its use is usually restricted to the low density limit only. On the other hand, the approach is as yet not general; in
particular, it implies diffusive motion in phase space, that is not necessarily the case.
In the next Section we generalize the results of a series of our recent works [5, 6, 22-25] in which a new kinetic
approach to the theory of 1D surface diffusion has been developed. We will show how, under some simplifying
assumptions, one can reach a relatively simple, analytical description that goes far beyond the abilities of the TST
model. Importantly, the theory allows, for the first time, kinetic treatment of diffusion at finite occupancy, resulting in
a new sight on mechanisms determining the density dependence of diffusivity. Remarkably, the theory is very much
based on the ideas and methods of physical kinetics of gases (and RGD in particular), although their applicability in
this field might seem unexpected.
In the subsequent Sections we consider a set of specific questions concerned with the molecular transport in subnano-channels. One of the questions is how far the phonon subsystem can be considered as being in equilibrium. We
(
)
()
r
Instead of n one sometimes considers gradients of chemical potential or pressure; the corresponding diffusion coefficient differs from D by an
obvious (density dependent) factor called thermodynamic factor [1, 14].
3
736
will show that the phonon–molecule drag [26] modifies the transport in a temperature-nonuniform system essentially
[2] and leads to a thermo-molecular pressure difference effect (or thermal transpiration) which is strongly different
from that known for wider channels.
Other questions of interest are how the molecule enters the channel from the ”outside world”, and how far the axial
and radial motion can be considered decoupled. We will show that for light particles like He and H 2 there are quantum
effects of the channel confinement [3] that lead, in particular, to the possibility to realize truly-one-dimensional gas,
and to a ”quantum sieving” effect. Note that the idea of quantum sieving has enjoyed much interest lately [7, 8, 11] in
view of its potential applicability to light molecule and isotope separation.
ANALYTICAL KINETIC THEORY OF ONE-DIMENSIONAL TRANSPORT
Kinetic equation and physical picture
(
)
Consider the evolution of the quasi-classical, one-particle distribution function (probability density) f v;x;t of
molecules residing in a periodic potential field V x formed by the channel and interacting with thermal excitations
of the substrate. Neglecting memory effects, the kinetic equation can generally be written in the form
()
@f + v @f ; 1 dV @f = I
(2)
@t @x m dx @v M;Ph ff g + IM;M ff g
with v and m the velocity and mass of the particle, IM;Ph ff g a dissipative term taking into account the interaction
with the phonon bath, and I M;M ff g accounting for the effect of interparticle interactions. We start with the case
of low density, IM;M ff g = 0: Assuming for simplicity the particle–substrate-bath coupling to be independent of x;
IM;Ph ff g can generally be written as
Z
Z
IM;Ph ff g = f (E 0 ;x)W (E 0 ! E )dE 0 ; f (E;x) W (E ! E 0 )dE 0
=
2+ ( )
(
(3)
)
V x the total energy of the particle, and W E 0 ! E the probability of transition per unit time
with E mv 2 =
0
between levels E and E accompanied by creation or annihilation of phonons (or other thermal excitations) in the
solid. The superscripts denote direction of motion lost in the E -representation. In view of the detailed balance
principle, the transition probability obeys the reciprocity relation
exp(;E=kB T )W (E ! E 0) = exp(;E 0=kB T )W (E 0 ! E )
(4)
If the characteristic value () of jE 0 ; E j is sufficiently small, the general expression (3) can be reduced to the
diffusive Fokker-Plank form used by many authors [17]. As a matter of tradition which dates back to Kramers [27]
(rather than a matter of microscopical foundation), the FP model, together with its analog the generalized Langevin
approach, is sometimes believed to be one of the most sophisticated ways to analyze activated surface phenomena
[27]. However, the question of the role of finite
in diffusion kinetics remains open. In fact, for particle–phonon
interactions one expects
to be of the order of the Debye temperature T D of the substrate,
kB TD : This
quantity is not much smaller, but often even larger than the temperature of the experiment, k B T: An alternative
way of treating IM;Ph ff gis to use a BGK-like model [28] which tacitly implies a very large value of : Both in FP
and BGK variants, the kinetic equation (2) remains complicated and generally requires numerical solution.
Instead of exactly solving essentially approximate equations, one may try to develop an approximate treatment of
the problem on the basis of ”exact” equations (2), (3). Our aim is to achieve a relatively simple analytical treatment
of surface diffusion that would keep—in contrast to TST—more or less correct dynamics of adparticle–substrate-bath
interaction.
As long as kB T Vb ; adparticles spend most of the time in bound states (E < Vb in the potential wells, so that
one observes a lattice gas. Jumps between the adsorption sites result from excitation of adparticles to unbound states
(E > Vb where they have enough energy to overcome the potential barrier. An excited (unbound) particle can fall
back into the well of its origin (i.e. no jumps occur), or it can fly over several lattice distances until it will be trapped
(a long jump). Furthermore, once being trapped, the particle can fall down the well, but it can also be excited again
(re-trapping). Basically, this is the physics behind any kinetic approach, including the FP model.
( )
)
)
737
Remarkably, this picture, though complicated in its mathematical description [17, 27] when
is small, can be
substantially simplified in case of finite ; namely when kB T < Vb : Expecting kB TD (in view of the
multiphonon character of the interaction, can be even larger than k B TD ), one may believe this condition to cover
nearly the whole temperature interval of interest in experiments. If k B T < ; transitions downward in energy are much
more probable than upward transitions, as follows from (4). Consequently, re-trapping is not very important and can be
neglected. Furthermore, one can neglect unbound-unbound transitions E;E 0 > Vb in a narrow zone of width about
kB T above Vb (a ”conduction zone” responsible for intersite jumps). Consequently, the effective friction experienced
by an unbound particle will be determined by the unbound-bound transitions only. With these simplifications, the
equation (2) with (3) turns out to be analytically solvable.
(
)
Intersite jumps and diffusion at low density
Using the above simplifications, one can easily obtain the jump length distribution from an integral form of the
kinetic equation (2) with ( 3) for evolution of an unbound particle [22, 23]:
pk = (1 ; )k;1 ; = 1 ; exp(;tfl = )
(5)
Here pk is the probability of a jump of length k in units of the lattice spacing l, and can be interpreted as the
probability of being trapped in a well. t fl is the mean time of the (unbound) particle flight between two saddle
points,
while is determined by the summed frequency of downward transitions from a certain energy state, ;1 =
R
0 0
E <E W (E ! E )dE ; which is nearly independent of E since Vb :
The jump length distribution (5) is completely determined by a dimensionless parameter t fl = which characterizes
the effective friction experienced by an unbound particle. The time of flight t fl is explicitly expressed [6, 23] in terms
of V (x); and it can be easily calculated for any given potential. This quantity is typically of the order of 10 ;12 s,
and it only weakly depends on temperature (weaker than T ;1=2 ). As far as the particle-surface relaxation time is
0
concerned, it is not a simple problem to calculate this quantity under actual conditions of multiphonon coupling to
the substrate [29]. However, information about analogous quantities of the same physical nature is available from
extensive studies of various molecule-surface interaction phenomena (see [29] and references therein). Accordingly,
one expects to be an increasing function of temperature, with typical values around ;13 ; ;12 s:
With these characteristic values of the parameters involved, eq. (5) predicts the occurrence of extended jumps to vary
from system to system between situations where it is negligible and where it is of the order of several tens percent.
The result (5) readily suggests simple explanations [22, 23] for the occurrence of double and triple jumps recently
observed (see, e.g., [18]) in direct experiments for quasi-one-dimensional surface diffusion.
Calculation of the diffusion coefficient is a more complicated problem and it needs an additional assumption. Let
us now assume that the population of bound states (E < V b is as in equilibrium. This statement seems obvious for
”deep” states near the bottom of the wells, where the particles spent a time sufficiently large for equilibration. Since
re-trapping is neglected (see above), the assumption seems reasonable also for all bound states 4 . With this and the
above assumptions, and using the reciprocity relation (4 ), the general expression (3) can be easily transformed—for
unbound particles—to a simple form I M;Ph jEVb ; f ; f0 ;1 ; with f0 the equilibrium distribution, and with
defined above. Note that this is not a BGK-like model for the full collision integral, but a direct consequence of the
master equation (3) for unbound particles. Now the kinetic equation (2) can be solved analytically [6, 23] with respect
to the deviation of the distribution function of the unbound particles from the equilibrium one, f f 0
rn :
Then the diffusion coefficient in the low density limit is obtained as
10
10
)
= (
)
= (1+
)
;1
D = D0 exp(; kVbT ) ; D0 = 2 tfl l2 (6)
B
In addition to the trivial dependence on l 2 ; and exp(Vb =kB T ) (as predicted by TST, see (1)) there appears in a natural
way an effective friction parameter t fl =; the same as in ( 5). The physics behind the appearance of this parameter in
(6 ) is clear. If the particle–bath coupling (as described by t fl = ) is strong, an excited (unbound) particle prefers to fall
4
Strictly speaking, this assumption is restrictive, and it bounds the consideration to the case of not too strong effective friction, presumably
tfl = < 10 [23].
738
back into the well of its origin, so that the jump rate and hence D are small. In the opposite case of weak coupling, D
is large as a result of contribution of long jumps, see (5), although the jump rate in this case is small again [23].
One old puzzle in the field of surface diffusion is concerned with a strong discrepancy between the theoretical
expectation and the experimental results for the preexponential factor (D 0 ) of the diffusion coefficient. Experimental
data for D0 obtained for various systems [19] are scattered in an interval that spans several orders of magnitude
around a ”normal” value D 0 l2 ;4 ; ;3 cm2 =s resulting from all TST-based theories. The result (6) gives a
possible key to solve the problem of ”anomalous” prefactors [25] taking into account, in particular, the dependence of
the relaxation time on the spectrum of surface phonons (and hence on the morphology of the surface) and, possibly,
on other types of substrate excitations like electron-hole pairs.
10
10
Density dependence of one-dimensional molecular transport
The principal achievement of this approach is the possibility to develop—for the first time—a kinetic theory of
surface diffusion at finite occupancy. Averaging the distribution function over one potential unit of length l; the kinetic
equation (2) can be reduced to a form characteristic for gases. This gives the opportunity to use a collision integral
IM;M ff g of the type introduced by Enskog to take into account the finite size () of the molecules [5]. Assuming the
adparticles to interact as hard particles, the final expression for the collective diffusion coefficient can be written as 5
Dc (n) = DY (n)Z (n)
with D given by (6). The two factors given by
(7)
;1
Y (n) = (1 ; n);2 ; Z (n) = 1+2 t 1 ;nln
(8)
fl
determine the dependence of the diffusivity on density n: The factor can vary between 0 and 1 depending on =l (for
truly hard particles); = 1=2 is expected to be a good approximation.
The origin of the factor Y in (7), (8) can be treated in two ways. In the course of the kinetic derivation, Y originates
from the nondissipative part of the collision integral I M;M ff g and reflects an effect of the finite size of adparticles on
diffusivity in the presence of a density gradient. However, Y can also be derived [23] directly from the equation of state
of the system (without any additional assumptions at all) as a thermodynamic factor [14, 17] that relates the particle
fluxes under the action of density and pressure gradients. The factor Z in (7), (8) has an essentially kinetic nature,
because it originates form the dissipative part of the collision integral IM;M ff g in (2). Physically, it describes an
additional contribution to particle-substrate friction concerned with interaction (collisions) between the adparticles 6 .
The developed theory allows us [23] to describe collective diffusion also in traditional (but, in fact, more complicated)
terms of jumps. In this way, the factor Z is seen as an effect of the reduction of the effective jump rates due to adparticle
interactions, while Y reflects an anisotropy of the jump rates in the presence of the density gradient.
While Y gives rise to an increase of the diffusivity with increasing density, Z has the opposite effect. Depending on
the value of the effective friction parameter t fl = , expressions (7),(8) predict either a monotonous increase of D n or
nonmonotonous dependence with a minimum [23]. For values of the parameters characteristic for physisorbed particles
l; tfl one has D = ; ; with nl the occupancy. This readily gives microscopic explanation [6]
to the ”universal” increase of diffusivity with occupancy (close to = ; ) systematically observed in zeolites [1].
Further development of kinetic theory is needed in order to incorporate the finite-distance interaction between
adparticles into the kinetic scheme. This should bring up an additional density dependence, the dependence of the
effective activation barrier of diffusion on occupancy (the physics that is usually introduced by lattice gas models in
an ad hoc way [14]).
()
(
)
( ) 1 (1 )
=
1 (1 )
5
We refer to the papers [5, 6] for details that are too cumbersome to be given here. In [23] it is shown how the result (7),(8) can be reproduced from
simple physical arguments.
6 A result very close to that given by Y (n) in (8) was recently obtained [30] using a density functional formalism in an oversimplified model that
tacitly assumes Z (n) = 1:
739
Mobility of clusters
An interesting question, which we will touch here only briefly, is the diffusion of atomic clusters. One of the most
exciting experimental observations in the field of surface diffusion is concerned with the ”gliding” motion of a twodimensional cluster as a whole, reported for Ir 7 and Ir19 clusters on an Ir(111) surface [21]. The process is characterized
by at least two puzzling features [21], both in strong contradiction with intuitive expectations. First, the pre-exponential
factor of the cluster diffusion coefficient was found to be unusually large (about orders of magnitude larger than
that for one-particle diffusivity). Second, larger clusters are inclined to exhibit longer jumps along the surface. Since
gliding seems to be the only possible mechanism of 1D cluster diffusion (in contrast to a variety of mechanisms known
in 2D case [20], like diffusion of atoms along the cluster edge), the observed behavior may be characteristic also for
sub-nano-channels. The developed kinetic approach allows us to explain these puzzling observations attributing them
to the manifestation of two interesting effects [24].
Considering atomic cluster as a particle with internal degrees of freedom, one can use the above results (5) and
(6), with the parameters involved still to be specified. Interaction of the cluster as a whole with the substrate should be
averaged over fast intracluster vibrations. This leads to the appearance of a temperature-dependent term in the diffusion
barrier Vb : This term is linear in T; and hence it manifests itself not in the activation energy, but in the prefactor of (6),
leading to its unusually large value. This effect is entropic in nature, and it can be considered as a consequence of the
”dynamical misfit” between the cluster and substrate lattices.
Considering the relaxation time for a cluster one has to take into account a phonon discrimination effect [24].
Motion of the cluster as a whole is significantly coupled only to long-wave phonon modes, with the wave length larger
than the size of the cluster. As a result, should be an increasing function of the size. In this case, as follows from (5),
larger clusters should be inclined to make longer jumps indeed.
4
SPECIFIC EFFECTS IN SUB-NANO CHANNELS
Above we have seen how one can treat a set of key problems of quasi-1D molecular transport. One general feature is
still to be mentioned. In an atomic-size channel molecules cannot pass each other: if particle 1 is to the left of particle
2, this order will remain forever, in contrast to 3D and 2D systems. This fact does not manifest itself in the transport
(collective) diffusion considered above, but it strongly complicates description of the tracer (self-) diffusion at finite
occupancy [31], and it can lead to interesting effects in mixtures, like ”negative” osmosis [9].
Now we will briefly touch upon a number of interesting problems concerned with the behavior of molecules in
atomic-size channels under specific conditions.
Phonon-molecule drag
Usually, describing molecule–surface interactions, one can consider the phonon subsystem of the substrate as being
in equilibrium. Special consideration, however, is needed when there is a temperature gradient along the surface. In
view of the momentum exchange between the molecules and the phonons, the thermally induced phonon flux leads
to the appearance of an additional contribution to the flow of the particles. This phonon-molecule drag effect [2]
is a kind of interphase kinetic phenomenon [26]: thermodynamic force in one medium rT in the solid) causes a
nonequilibrium response in the adjusted medium (in our case this is the molecule flow in the nano-channel). This
corresponds to the appearance of an additional term (of the order of ph rT=T; with ph the phonon mean free
path in the solid) in the molecule-phonon ”collision integral” (3). Assuming for simplicity the channel to be smooth
V x const), and considering the case of low density, one can calculate the molecule flux in the channel [2] to be
(
( ( )=
G = ;(1=2)nvT2 rn=n ; (1=2)nvT2 rT=T ; cphnvT ph rT=T ;
(9)
with the dimensionless parameter cph 1; and with defined above. The characteristic value of the phonon mean free
path ph in crystals at T 300 K is about 10 nm, and it increases with decreasing T: In the worse case of a solid
with big amount of defects ph is of the order of several tenth of nanometer, as is the molecule mean free path in the
740
channel, vT (see above). One observes that the phonon-molecule drag effect—the third term in (9 )—gives a very
significant contribution to transport in our case 7 , although in macroscopic channels it was negligible.
It is reasonable to consider the thermo-molecular pressure difference (TMPD) as the experimentally most easily
accessible quantity. Considering the pressure difference established between two vessels, connected by our channel
(or by a system of such channels) and being kept at different temperatures, and using (9), one obtains
p = V ; 2cph ph T
p
k T
v T
B
(1 2)
(10)
T
This is to be compared with the classical result p=p =
T=T valid in a Knudsen gas. The first term in the
r.h.s. of (10) is the effect of the ”ordinary” thermal creep. In our case it is strongly enhanced due to a low potential
energy ;V of the molecule adsorbed in the channel with respect to a free molecule, since this leads to an increase in
in-channel density with respect to that in the vessels. (This is in itself an important feature of transport in nano-pores
with respect to that in wider channels.) The phonon–molecule drag effect—the second term in (10)—has the opposite
sign, and its role increases with decreasing T (due to a strong increase of ph : At a certain temperature [2] the TMPD
can change its sign. Observation of this would be an undoubted manifestation of the phonon–molecule drag effect.
( )
)
Truly 1D gas and quantum sieving
Surface diffusion of atoms and molecules in most cases can be considered as a classical process, as discussed above;
exception is given by quantum tunneling diffusion of light particles like He at low temperatures. In sub-nano-channels,
however, quantum effects of a new type can arise, already at room temperatures, as a result of the quantization of radial
motion [3].
As a simple but nontrivial approximation, one can consider the potential for radial motion in a cylindrical channel
to be a circular square well with depth V: Importantly, the width of the well is not the diameter d of the channel, as
one might think, but d ; ; with the hard core of the molecule. For energy levels in such a well one has [3]
(
)
Ei = 2i2h2 m;1 (d ; );2 ; i = 0; 1; 2;::: ;
(11)
with i simply related to zeros of Bessel functions, with, e.g., 0 = 2:4; 1 = 3:8: Since (d ; ) is a small quantity,
both the zero point energy E 0 and the level splitting (E1 ; E0 ) turn out to be relatively large for light particles, thus
leading to two interesting effects as shown below.
What happens if the level splitting E1 ; E0 kB T ? The first and upper energy levels are not populated. One
meets the situation that all particles in the channel are for their radial motion in the ground state. The system will
behave as a truly one-dimensional gas: two degrees of freedom are frozen out and only one—the axial motion—
remains. Apparently, this is the only case of truly reduced dimensionality 8 in a molecular system. For hydrogen
: : nm] this situation can be reached already at T close to room temperatures, since for, say,
: nm one obtains from (11) E 1 ; E0 =kB
K.
characteristic value of d ; The other curious effect is concerned with the value of the zero point energy. What happens if E 0 > V ? Zero
point radial motion overcompensates the attraction to the walls, and the binding energy V ; E 0 of the molecule in
the channel becomes negative. This means that the molecule meets an energy barrier at the entrance to the channel,
that can be overcome only with additional activation. As follows from (11), for helium V=k B K, : nm)
this situation is realized for channels with diameter d slightly smaller than 0.4 nm. The medium acts as a ”quantum
sieve”, and the behavior in this extreme case is paradoxical: Smaller molecule (He) cannot enter the pores large enough
from the geometrical point of view, while larger particles, say, Ne : nm, V K) are still welcome for
adsorption.
For isotopes (having equal values of both V and but different m quantum sieving is realized in a form more easily
accessible in experiment. Isotopes 1 and 2 have different binding energies in the channel only due to the dependence
of E0 on m; eq. (11). For the ratio of densities in the channel n (1) =n(2) with respect to that in the gas phase
(
[
(0 24 0 31)
(
)
)=01
(
)
= 428
(
(
( 0 32
)
( =
)
)
200
03
1000
This conclusion is not changed by the presence of a periodic dependence of V on x in a real channel, since this leads (as the main effect) to the
appearance of the same Boltzman factor exp( Vb =kB T ) in all the three terms of (9).
8 One should not confuse this ideal situation with widely investigated (see, e.g., [10]) quasi-one-dimensional gas/liquid where the (classical)
transverse motion in the channel, although it may be decoupled from the problem of interest, is still there.
7
;
741
h i
h
i
(2)
(1)
(2) ;1 ;1 = exp 2 2 h2 (d ; );2 k ;1 T ;1 m;2 m :
(V = n(1)
0
V =nV ) one obtains =V = exp ; E0 ; E0 kB T
B
Due to its exponential dependence on m; the separation coefficient can be very high [3]. Note that the quantum
separation effect is opposite to that in the usual Knudsen gas flow: heavier particles penetrates into the pores better
than lighter ones.
The idea of quantum sieving has enjoyed much interest lately. The challenge of its applicability to light molecule and
isotope separation is concerned with advanced computations taking into account realistic potential field in sub-nano
pores [7, 8] and rotational degrees of freedom of the molecules [11].
ACKNOWLEDGMENTS
This work has been done during the stay of the author at Huygen Laboratory, Leiden University, The Netherlands.
Special thanks are due to professor Jo Hermans for initiating this work and helpful discussions. This paper concludes a
series of works started with professor Jan Beenakker, who passed away in 1998, and whose scientific spirit permanently
stimulates the author.
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