Materials Sciences and Applications, 2011, 2, 314-318
doi:10.4236/msa.2011.25041 Published Online May 2011 (
Copyright © 2011 SciRes. MSA
Vibrations of a One-Dimensional Host-Guest
Amelia Carolina Sparavigna
Dipartimento di Fisica, Politecnico di Torino Duca degli Abruzzi 24, Torino, Italy.
Received January 14th, 2011; revised March 17th, 2011; accepted March 24th, 2011.
A simple model shows how it is possible to create a gap in the vibrational spectrum of a one-dimensional lattice. The
proposed model is a host-guest chain having, instead of point-like masses connected by spring, massive cages hosting
particles inside. We imagine the cage as a rigid box containing a mass linked by a spring to the box inner wall. The
presence of guests creates an energy gap in the dispersion of vibrational frequencies. The gap is about the internal
resonance of the mass hidden in the cage. The model is proposed to help understanding the macroscopic behaviour of
some phononic materials and the properties of materials with microscopic rattling modes.
Keywords: Metamaterials, Phonons, Band Gap, Thermal Properties
1. Introduction
In spite of their simplicity, the study of models with
masses or rods connected by springs can be quite helpful
in understanding the properties of metamaterials as of
other nano-engineered structures. The term “metamate-
rial”, credited to Rodger Walser, indicates a material,
which gains its properties from its structure, rather than
from the properties of components [1,2]. In fact, the term
“metamaterial” is commonly used for composites, which
are distinguishing themselves for unusual properties.
There are several examples regarding electromagnetism
and elastic properties. We have, for instance, the left-
handed materials possessing negative refractive index,
able to affect in an uncommon way the passage of elec-
tromagnetic waves near them [3-6]. In the case of elastic
materials, a property considered as unusual is a negative
Poisson’s ratio. Materials with negative Poisson coeffi-
cient are named auxetics [7-12]. Among them, natural
auxetics occur in biological systems too.
Metamaterials usually share similar behaviours with
photonic and phononic crystals [13,14]. For instance,
some metamaterials have been prepared, which are able
to act as total wave reflector within certain sonic fre-
quency ranges. These sonic materials, which are then
behaving as phononic crystals, are mainly fabricated in-
cluding in a hosting component some localized resonant
structures [15].
As in the case of electromagnetic metamaterials, we
can prepare some composites displaying an effective “ne-
gative” elastic constant, analogous to the negative refrac-
tive index [16,17]. In [16], the author is discussing the
case of metamaterials, which are guest-host systems,
having units possessing hidden resonant masses inside.
Figures in Reference [16] are quite stimulating to study
and discuss the vibrational properties of such structures.
Among the many models composed of rigid cages with
moving particles inside, let us use the simplest one we
can imagine, that is a one-dimensional chain composed
of rigid host-guest units.
2. A Host-Guest Model
Figure 1 shows the model. It is a simple spring model
describing an interacting system of host cages of mass
interconnected by springs having constant
guest atoms of mass m attached to the cage inner walls
by means of springs with constant
. The one-dimen-
sional model we consider has then rigid units and spring
connections, with distance L between cages. The unit
cell of the lattice has a position given by the lattice indi-
ces 1, , 1, 2,iiii
. If the cage is imagined as a
closed box, a mass can be hidden in it. Its presence is
revealed by the frequencies of the system.
Let us define o
which are the
natural angular frequencies of cage and hidden masses.
In the following we will use the dimensionless ratios:
Vibrations of a One-Dimensional Host-Guest System
Copyright © 2011 SciRes. MSA
Figure 1. The unit of the chain is composed by a cage with a
hidden mass inside.
, mmM
, o
 and o
 .
Let us investigate the harmonic vibrations of the chain
supposed to be infinite with displacements of masses in
longitudinal direction. ,bi
is the displacement from
equilibrium of one of two masses, that are the cage and
the hidden mass: b can have two possible determina-
and m for the reticular position i of equi-
In the case of small displacements, equations are:
MiMiMiMimi Mi
mimiM i
MxKxKxKxK xx
mxK xx
that is, using .,,
, we have:
iiii i
ii i
 
 
 
  (2)
If we are looking for Bloch waves with wavevector q,
it is possible to write for each lattice site:
 
exp; exp
itiqxBit iqx
 (3)
and then the dispersion relations for frequency ω can be
easily obtained from the dynamical Equations (2); in the
following way. We defined L the distance between cages;
then 1ii
Lx x
, is the distance between the sites of the
chain. Inserting (3) in (1) we have:
2e eee0
ee 0
 
and then:
2e e0
iqL iqL
 
 (5)
These equations have a non-trivial solution when:
In Equation (6),
21cosCK qL . This equation
gives the dispersion relation
of the angular
frequency as a function of a given wavenumber q.
Let us consider for plotting, the reduced frequencies
, o
. Dispersion relations of the chain
as a function of the wavenumber q are shown in Figure 2.
Let us remember that the dispersion relation is showing
the angular frequency
(in our case the reduced an-
gular frequency
), as a function of the wavenumber
q. Note the existence of a phononic gap between the two
branches. This gap is about the natural frequency of the
mass inside the cage. The figure is obtained assuming
14, 14mk
. The horizontal line represents
the reduced natural frequency of the hidden oscillator. At
the edge of the Brillouin Zone, we have a frequency of
the system almost corresponding to that of the natural
. This is an important result for the engi-
neering of phononic materials, because it means that the
propagation of waves with a frequency equal to that of
the internal resonance is not allowed. Using specific val-
ues of , mk
, it is possible tailoring the phononic prop-
erties of the metamaterial.
Figure 3 shows the dispersion of vibrational frequent-
cies in several conditions. In the upper part of the figure,
Figure 2. Dispersion of vibrational frequencies for model in
Figure 1 ' is the reduced natural frequency of the hidden
oscillator. Note that at the edge of the Brillouin Zone (en-
circled) we have a frequency almost corresponding to that
of the natural oscillation of the hidden mass. This means
that the propagation of waves with a frequency equal to
that of the internal resonance is not allowed.
Vibrations of a One-Dimensional Host-Guest System
Copyright © 2011 SciRes. MSA
Figure 3. In the upper part of the figure, it is shown the behaviour of dispersion for three fixed values of k'. In each panel, m'
is changing from 0.25 to 2.5. It is possible to see that the gap increases and that the acoustic branch has a long wavelength
limit possessing a sound speed decreasing with the increase of the hidden mass. On the right, for k' = 1.5, the gap corre-
sponding to m' = 2.5 is explicitly shown as a grey band, overlying the curves. In the lower part of the image, it is the value of
m' to be fixed and each panel shows the dispersions as the constant of the spring k' is varying from 0.1 to 1. Note the different
behaviour of the branches as the elastic constant changes. On the right, the panel is showing the gap for m' = 1, k' = 1.
we see the behaviour of dispersion for three fixed values
of k. In each panel, m is changing. It is possible to
observe that the gap increases and that the acoustic
branch has a long wavelength limit possessing a sound
speed decreasing with the increase of the hidden mass.
The right panel of the upper part of the figure is showing
the gap for a specific choice of m and k
. In the
lower part of the image, it is the value of m to be fixed
Vibrations of a One-Dimensional Host-Guest System
Copyright © 2011 SciRes. MSA
and each panel shows the dispersions as the constant of
the spring k is varying. Again, the right panel is
showing the gap. As we can see from the figure, adjust-
ing the ratios of masses or of spring constants or both it
is possible to tailor the gap. We have then a band of fre-
quencies where the propagation of the waves in the ma-
terial is not allowed. It means that the material is not
transmitting these frequencies: the waves are reflected by
the composite material.
3. Discussion
The model shows, from a macroscopic point of view, that
a structure with a cage hosting a mass displays a gap in
the allowed frequencies. It is therefore illustrating how
some phononic crystals could be created by means of
host-guest systems. Moreover, we have seen that an in-
crease of the hidden mass reduces the speed of the acous-
tic long wavelengths, an interesting result for engineering
materials with very low thermal conductivity and for the
development of more efficient thermoelectric devices.
In fact, a low thermal conductivity is required for the
thermoelectric conversion in solid-state heat engines. In
these devices, the electron gas serves as the working
fluid, converting the heat flow in electric power [18]. For
thermoelectric applications, materials must have a high
figure-of-merit, which is a goodness factor including the
Seebeck coefficient and the electrical and thermal con-
ductivities. A decrease of thermal conductivity means an
increase of the figure-of-merit. In the case of crystalline
materials, it is enough to disturb the phonon paths by
disorder or lattice defects [19-21] to have a low conduc-
tivity. Unfortunately, defects decreased the charge trans-
port too.
Therefore, the figure-of-merit can only be moderately
improved by reducing the lattice thermal conductivity: to
have a significantly larger goodness parameter it is nec-
essary to improve the electrical properties [22]. The aim
of recent researches is to employ the Phonon Glasses -
Electronic Crystals, PGECs, where the lattice is disor-
dered and then phonons are strongly scattered, but the
electrons remain free to move. To create such structures,
a possibility is the use of materials containing weakly
bound atoms, “rattling” within an atomic cage. These
materials have a low thermal conductivity, as that dis-
played by glasses, but have an electric conductivity as
high as in crystals [23]. Typical of these materials are the
filled skutterudites [24] and the clathrates [25], which are
host-guest systems at the atomic, microscopic scale.
In host-guest lattices, the guest entities are supposed to
have oscillations, so-called rattler modes, which scatter
the acoustic phonons and reduce the thermal conductivity.
In a resonant scattering model [26], it was hypothesized
an “avoided crossing” between acoustic phonons and
localized guest modes, that coming from a mixing of
guest and host modes with an energy exchange as a con-
sequence. Avoided crossing was found in hydrates [27,28]
and recently in a PGEC material. In Reference [29], the
phonon dispersion relations of Ba8Ga16Ge30 are showing
unambiguously the theoretically predicted avoided cross-
ing of the rattler modes and the acoustic-phonon bran-
ches. Ba8Ga16Ge30 is a clathrate type-I structure with a
host cage framework of Ga and Ge atoms holding Ba
guest atoms inside the cages. The phenomenon referred
as the “avoided crossing”, is the same as that we show in
Figures 2 and 3, that is, the presence of a gap separating
the two branches of frequency dispersion. This gap is
created about the natural frequency of the guest.
As a conclusion, we can tell that the use of host-guest
systems, from macroscopic to microscopic scales, can be
an interesting method for engineering new materials. The
host-guest model we proposed is showing that the struc-
ture is able to create some phononic band-gaps. It is also
changing the sound speed in the material. The proposed
model can hopefully stimulate new engineering methods
for metamaterials with improved vibrational properties or
reduced thermal conductivity. Applied to microscopic
materials, it helps understanding the microscopic proper-
ties of those materials with rattling modes.
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