PECULIARITIES OF HEAT TRANSFER IN LOW – DIMENSIONAL SYSTEMS
Oleg Gendelman*,
Senior Lecturer,
Faculty of Mechanical Engineering,
Technion – Israel Institute of Technology,
Haifa, Israel
Horev Fellow – supported by Taub and Shalom foundations
*
The problem of heat conduction in dielectrics was considered as solved since
1920s. Explanation proposed by R. Peierls had involved fast mixing and phonon –
phonon scattering due to weak nonlinearity of the interatomic interactions. Recent
numerical simulations has demonstrated that both conjectures are wrong, at least for one
– dimensional and some two-dimensional models – nonlinearity does not imply efficient
mixing and equipartition of energy and the equipartition, if present, does not necessarily
imply normal heat conductivity. Moreover, even if these models demonstrate normal heat
conductivity, the behavior of heat conduction coefficient is strongly model – dependent
and very different from results of many known physical experiments.
Some nontrivial consequences for real physical systems may be already
discussed. Anomalies of heat conduction consistent with those discovered in lowdimensional models are suggested for explanation of anomalous transport properties of
carbon nanotubes and polymer crystals.
INTRODUCTION
Customary approach to heat transport in solids is based on the concept of thermal
diffusion. The diffusion coefficient, referred to as heat conductivity, is a
phenomenological parameter related to microscopic structure of the material under
discussion. The above conjecture is not straightforward, as conservative microscopic
dynamics of a specimen should be reduced and averaged down to diffusive macroscopic
bulk process. In dielectric crystals, being the scope of present paper, the heat transfer is
due to propagating vibrations of the crystal lattice. Famous work of R. Peierls [1] in the
first time has related the finite heat conductivity of solids with intrinsic nonlinearities of
phonon – phonon interaction in the crystals of dielectrics. The following implicit
propositions were adopted in order to explain the finite heat conductivity of solids:
a)
Weak nonlinearity of phonon – phonon interaction
inevitably leads to effective mixing of the phonon
modes and ensures formation of local thermal
equilibrium;
b)
The above local thermal equilibrium is accompanied
by diffusive thermal transport with well – defined
finite phenomenological constant of heat conductivity.
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These two long – believed statements became open for verification with the help
of simplified low – dimensional models as a result of development of numerical
simulation techniques. To day t is clear that, generally speaking, both of them are wrong.
The problem started from the work of Fermi, Pasta, and Ulam (FPU) [2], where
the absence of fast mixing in a chain with weakly nonlinear potential of nearest –
neighbor interaction was detected for the first time. It was afterwards understood that
non-integrability of the system is a necessary condition for normal heat conductivity. As
it was demonstrated recently for the FPU lattice [3, 4], disordered harmonic chain [5, 6,
7], diatomic 1D gas of colliding particles [8, 9, 10, 11], and the diatomic Toda lattice
[12], the non-integrability is not sufficient in order to get normal heat conductivity. It
leads to a linear distribution of temperature along the chain for small gradient, but the
value of heat flux is proportional to 1/Nk, where N is the number of particles in the chain
and number exponent 0 < k < 1. Thus, the coefficient of heat conductivity diverges in the
thermodynamic limit N→∞. Analytical estimations [4] have demonstrated that any chain
possessing an acoustic phonon branch should have infinite heat conductivity in the limit
of low temperatures.
From the other side, there are some artificial systems with on-site potential (dinga-ling and related models) having normal heat conductivity [13, 14]. The heat
conductivity of the Frenkel-Kontorova chain was first considered in Ref. [15]. Finite heat
conductivity for certain parameters was obtained for Frenkel-Kontorova chain [16], for
the chain with sinh-Gordon on-site potential [17], and for the chain with φ4 on-site
potential [18, 19]. These models are not invariant with respect to translation and the
momentum is not conserved. It was supposed that the on-site potential is extremely
significant for normal heat conduction [18] and that the anharmonicity of the on-site
potential is sufficient to ensure the validity of Fourier’s law [20]. A recent detailed
review of the problem is presented in Ref. [21].
Probably the most interesting question related to the heat conductivity of
1Dmodels (which actually stimulated the first investigation of Fermi, Pasta and Ulam [1])
is whether small perturbation of the integrable model will lead to convergent heat
conduction coefficient. It is supposed that for the one-dimensional chains with conserved
momentum the answer will be negative [22]. Still, normal heat conduction is observed in
some special systems with conserved momentum [23, 24, 25], but it may be clearly
demonstrated only well apart from the integrable limit. It means that mere nonintegrability is insufficient to ensure normal heat conduction if additional integral is
present.
In the systems with on-site potential the situation is not so clear. Although it was
supposed that non-integrable system without additional integrals of motion would have
convergent heat conductivity [22], no rigorous proof was presented. From the other side,
recent attempt of numerical simulation of the heat transfer in Frenkel-Kontorova model
[26] demonstrated that because of computational difficulties no unambiguous conclusion
can be drawn whether the heat conduction remains convergent for all finite values of
perturbation of the integrable limit system (linear chain or continuous sine-Gordon
system).
It seems that the computational difficulties of investigation of the heat conduction
in the vicinity of the integrable limit are not just issue of weak computers or ineffeective
procedures. In the systems with conserved momentum the divergent heat conduction is
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fixed by power-like decrease of the heat flux autocorrelation function with power less
than unity. Still, for the systems with on-site potential exponential decrease is more
typical [26]. For any fixed value of the exponent the heat conduction converges; if the
exponent tends to zero with the value of the perturbation of the integrable case, then for
any finite value of the perturbation the characteristic correlation time and length will be
finite but may become very large. Consequently, they will exceed any available
computation time or size of the system and still no conclusion on the convergence of the
heat conduction will be possible.
In this respect it is reasonable to mention the findings of papers [24, 25, 26]
concerning the transitions from infinite to finite heat conduction in the chain of rotators
and Frenkel - Kontorova system. These findings were criticized in paper [27] by
providing the computation results for larger systems. We agree with the authors that the
results of papers [24, 25, 26] do not prove the reality of the above transition; actually, in
paper [26] it is stated explicitly. From the other side, it is clear that such more extended
numerical simulations cannot prove that the transitions do not exist at all. Probably, such
conclusion may be driven if, with the help of some not yet developed method, the
simulation length or effective time will be extended by many orders of magnitude.
Present paper is aimed to illustrate the above – mentioned developments and the
state – of – art in the field of heat conduction in low – dimensional systems. Few groups
of models will be considered and major findings will be outlined. Relation of the results
for low – dimensional systems to the heat conductivity of real 3D objects as well as
possible applications will be also discussed.
CONSERVED - MOMENTUM 1D MODELS
The first group of models (including FPU model mentioned above) investigated
from the viewpoint of the heat conductivity was the group of models with conserved
momentum. More exactly, the investigators have considered various models with the
Hamiltonian
p2
1
   i   V ( xi  xi 1 )
(1)
2 i mi
i
where pi are momenta of the particles, mi – their masses, xi – their displacements and v –
potential of nearest - neighbor interactions. Such model obviously conserves the total
momentum of the system P=  pi .
i
For this group of models two distinctly different scenarios of the heat conduction
were revealed. The first one [21] is related to majority of the models investigated and is
characterized by linear temperature distribution along the chain subject to thermal
gradient; local thermal equilibrium also can be achieved. Still, the heat conduction
coefficient diverges in the thermodynamical limit of very large system. Two examples of
this sort are presented at Fig. 1-2 (see [21] for more information).
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Fig. 1 Thermal conductivity of the FPU model with quartic nonlinearity versus lattice
length N for T+ =011; T_ =0,09. The inset shows the effective growth rate of the scaling
exponent of the heat conduction versus N. Circles and diamonds correspond to free and
fixed boundary conditions respectively.
Fig. 2 Thermal conductivity of the FPU model with quartic nonlinearity, T+= 150; T- =
15, fixed boundary condition.
The result presented above (thermalization with divergent heat conduction
coefficient) is typical for big variety of models. Actually, only one interesting exception
is known, and it is a system with periodic nearest neighbor potential. The simplest form
of this potential is expressed as
V ( x)  1  cos x
(2)
Chain with potential (2) was the first one – dimensional lattice with conserved
momentum which demonstrated convergent heat conductivity. Fig. 3 illustrates this
finding.
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Fig. 3 . Dependences of the natural logarithm of the thermal conductivity coefficient on
the natural logarithm of the chain fragmennt with length N1. Circles 1 correspond to the
numerically calculated values at temperature T = 0.2 (T+ = 0.21 and T-= 0.19). Straight
line 2 approximating this dependence has the slope d = 0.26. Squares 3 indicate the
numerically calculated values at temperature T = 0.3 (T+ = 0.33 and T- = 0.27). Straight
line 4 approximating this dependence has zero slope (k = 31.8).
The reason for unusual behavior of the heat conduction coefficient is that
potential (2) allows formation of high – frequency discrete rotational modes
(rotobreathers) which prevent free propagation of the heat flux. Their effect on the heat
transport is illustrated at Fig. 4.
Fig.4 Energy relaxation of the thermalized chain at temperature T = 3. Dependences of
the local energy En on the number n of the chain unit and on the time t. At the initial
instant, the system has the temperature T; then, the dynamics of the chain with the
absorbing ends is considered (N = 300, and T+= T-= 0).
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Thus, it is possible to conclude that in general one – dimensional chain with
conserved momentum cannot reproduce the properties of real 3D systems with diffusive
heat transport, unless some very special cases are considered. This circumstance
eventually led to consideration of more complicated models with on-site potential.
1D MODELS WITHOUT MOMENT CONSERVATION
Typical Hamiltonian for this sort of models is as follows:
pi2
1
 
  V ( xi  xi 1 )  U ( xi )
(3)
2 i mi
i
Models with Hamiltonian (3) do not conserve global momentum and generally
have no integrals of motion besides total energy. These discrete models with intrinsic
nonlinearity were believed to have convergent heat conduction. Still, it turned out to be
not easy to demonstrate that by means of numerical simulation. Exampel of these
difficulties is presented at Fig. 5 for a model with harmonic V(x) and sinusoidal U(x)
(Frenkel – Kontorova model).
Fig.5.Dependence of the heat conductivity coefficient on the reduced temperature
for the chain with periodic on-site potential and different discreteness ratio.
It is unclear whether the heat conduction diverges or remains finite but b=very
large. In order to elucidate this problem other model has been proposed, based on impact
nonlinearity.
Let us consider the one-dimensional system of hard particles with equal masses
subject to periodic on-site potential. Interaction of absolutely hard particles is described
by the following potential V (r) = ∞ if r < d and V (r) = 0 if r > d, where d is the diameter
of the particle. This potential corresponds to pure elastic impact with unit recovery
coefficient.
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It is well-known that the elastic collision of two equal particles with collinear
velocity vectors leads to exchange of their velocities. The one-dimensional chain of
equivalent hard particles without external potential is a paradigm of the integrable
nonlinear chain, since all interactions are reduced to exchange of velocities. Presence of
external potential does not change this fact, since the collision takes zero time and thus
the effect of the external force on the energy and momentum conservation is absent. If the
diameter of the particles is zero, then the energies of the quasiparticles associated with
“moving pulses” are preserved in the collision. Therefore effectively the interaction
between the quasiparticles disappears and the chain of equal particles with zero size
subject to any on-site potential turns out to be completely integrable system. Thus
contrary to many previous statements it is possible to construct an example of strongly
nonlinear one-dimensional chain without momentum conservation, which will have
clearly divergent heat conductivity (even linear temperature profile will not be formed).
The situation differs if the size of the particles is not zero, as the energies of the
particles are not preserved in the collisions. In order to consider the effect of such
interaction we proposed simplified semi-phenomenological analytical model [28] based
on standard singular perturbation technique. Main result is presented at Fig.6.
Fig.6 Dependence of the coefficient of the exponential decrease of the autocorrelation
function (a) and the coefficient of the heat conduction ·(b) on the particle diameter d of
1D gas at T=1. Curves 1 and 2 correspond to piecewise linear on-site potential, curve 3
represents theoretical predictions.
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The above results mean that there exists a new class of universality of 1D chain
models with respect of their heat conductivity. The limit case of zero-size particles is
integrable, but the slightest excitation of this integrable case by introducing the nonzero
size leads to drastic change of the behavior – it becomes diffusive and the heat
conduction coefficient converges. It should be stressed that this class of universality,
unlikely the systems with conserved momentum, cannot be revealed by sole numerical
simulation. The reason is that the correlation length (as well as the heat conduction
coefficient) diverges as the system approaches the integrable limit; therefore any finite
capacity of the numerical installation will be exceeded. That is why the analytical
approach is also necessary.

MODELS IN HIGHER DIMENSIONS
Much less is known on the behavior of simple dynamical models in two and three
dimensions. Generally adopted point of view is that for two – dimensional case the heat
conduction of momentum – conserving model is logarithmically divergent. This result is
illustrated at Fig. 7 for the case of two –dimensional symmetric FPU lattice and lattice
with Lennard – Jones interactions.
Fig. 7 Heat conductivity versus the system size Nx for the 2d FPU (a) and
Lennard–Jones (b) models. In panel (a) T+ =20 and T- =10; in panel (b) T+ =1 and T=0:5. In both cases, statistical errors have the size of the symbols.
Still, the above results were obtained for square lattice with pertinent dynamical
instability. More refined investigations are required in order to establish the heat
conduction peculiarities in two dimensions.
The situation in three dimensions may be characterized as almost complete
ignorance, besides some preliminary indications of finite heat conduction in extremely
simplified model [29].
DISCUSSION: POSSIBLE APPLICATIONS
In spite of very preliminary and incomplete understanding of the heat conduction
in low – dimensional models it is already possible to figure out few real physical objects
which are supposed to exhibit anomalies related to low effective dimensionality.
The first example of this sort is a polymer crystal [30]. Thermal conductivity is
dependent on the length of polymer chain, as presented at Fig. 8.
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Fig. 8 Thermal conductivity as a function of pressure, T=273 K. (1) C9H20 (2)C11H24 (3)
C13H28 (4) C15H32 (5)C17H36 (6)C19H40
The discrepancies in heat conduction for various chain length are attributed to one
– dimensional aspects of the chain dynamics.
The other very interesting example with suggested two – dimensional dynamics is
carbon nanotubes. These objects are schematically presented at Fig.9
Fig. 9 a) C60 b) C70 c) C80 d) C260
Heat conduction in these systems is now a subject of numerous extensive studies.
To mention just the most recent result, in paper [31] it was demonstrated that the heat
conduction coefficient in single – walled nanotubes diverges as approximately L0.3 where
l is a length of the tube.
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Thus, it is possible to conclude that the low – dimensional anomalies which were
previously related only to curious problem of grounding Fourier law are completely
pertinent for describing peculiarities of heat transport in real physical systems.
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