Sufficient conditions for thermal rectification in graded materials

0.1 MB·English

by  Emmanuel Pereira

Sufficient conditions for thermal rectification in graded materials
∗
Emmanuel Pereira
Departamento de F´ısica–ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil
(Dated: January 25, 2011)
Weaddressafundamentalproblemfortheadvanceofphononics: thesearchofafeasiblethermal
diode. Weestablish sufficient conditionsfor theexistenceof thermalrectification in general graded
materials. By starting from simple assumptions satisfied by the usual anharmonic models that
describe heat conduction in solids, we derive an expression for the rectification. The analytical
1 formula shows how to increase the rectification, and the conditions to avoid its decay with the
1 system size, a problem present in the recurrent model of diodes given by the sequential coupling
0 of two or three different parts. Moreover, for these graded systems, we show that the regimes
2 of non-decaying rectification and of normal conductivity do not overlap. Our results indicate the
graded systems as optimal materials for a thermal diode, thebasic component of several devices of
n
phononics.
a
J
PACSnumbers: 05.70.Ln;05.40.-a;44.10.+i
4
2

INTRODUCTION thefirstmicroscopicsolid-statethermalrectifierbyusing
]
i a graded material: nanotubes externally and inhomoge-
c
s neously mass-loaded with heavy molecules. It is worth
The study of the macroscopic laws of thermodynamic
- to recall that graded materials, i.e., inhomogeneous sys-
l transportfromthe underlyingmicroscopicmodelsisstill
r tems whose compositionand/or structure change gradu-
t a challenge in statistical physics. In particular, the in-
m ally in space, are abundant in nature, can also be man-
vestigation and control of the energy transport, which
. ufactured, and have attracted great interest in many ar-
t mainly involves conduction of heat or electricity, is a
a eas [13]: there are many works devoted to the study of
fundamental problem of huge theoretical and practical
m the electric, optical, mechanical and other properties of
interest. The invention of transistor and other devices
- gradedmaterials,but there arefew studies in relationto
d usedtocontroltheelectricchargeflowhasledtothewell
their heat conduction investigation.
n known developmentof modern electronics. Its much less
o developed counterpart - the study and control of heat
Inthepresentwork,weaddressthisfundamentalprob-
c
current - has, recently, presented interesting progress,
[ lem of phononics: the built of an appropriate thermal
promising to establish, in addition to electronics, a new
diode, namely, a simple system that may be constructed
1 physical branch in energy and information processing -
v in practice, and with a rectification that does not decay
the phononics [1, 2]: researchers have proposed nanode-
0 with the system size. We start from simple conditions
vices such as thermal diodes or rectifiers [3–5] (already
9 for the local thermal conductivity, conditions that are
5 builtinpractice[6]),thermaltransistors[7],thermallogic
quite general and that are satisfied by anharmonic crys-
4 gates [8] and memories [9]. The most fundamental com-
talmodelsusedtodescribeheatconductioninsolids,and
1. ponent of these instruments is the thermal diode, a de- then we show that they are sufficient to lead to rectifi-
0 vice in which heatflows preferably in one direction. In a
cationin gradedmodels. Moreover,we deriveanexpres-
1 short analysis, we may say that this promising advance
sion for such rectification that allows us to see how to
1
of phononics is directly dependent on the development
: make it larger, and how to avoid its decay with the sys-
v of its basic component: a thermal diode with suitable
tem size. In short, we show that properly manipulated
i properties.
X gradedmaterialshavesuitablepropertiesofrectification,
r There are analytical attempts to investigate the phe- and so, they shall play a central role in the building of
a
nomenon of thermal rectification such as the works on thermalnano-devices. Thesimplicityoftheinitialcondi-
spin-boson junctions [10], billiard systems [11], etc., but tionsandofthe argumentsto establishthe resultsshows
most of the results are by means of computer simula- the ubiquity of thermal rectification in graded systems.
tions, see e.g. the work of B. Li and collaborators [12]. Moreover,the existenceofsimple ingredientsforthe rec-
Themostcommonandrecurrentdesignofdiodesisgiven tification, as described here, deserves attention: as well
bythesequentialcouplingoftwoormorechainswithdif- known, in the literature, the mechanism behind rectifi-
ferent anharmonic potentials [3–5]. Although frequently cation in graded models is far from being clear. E.g., we
studied, this procedure is criticized [5] due to the dif- recall the comment of G. Casati [2] on the explanation
ficulty to construct such diode in practice, and due to of Chang. et al [6]: “the authors speculate that solitons
the significative decay of the rectification with the sys- might be involved in the rectification process, but this
tem size. Recently, a different procedure was considered is still to be confirmed”. Here, we do not have to make
by Chang et al. [6], who built, in an experimental work, any speculation about the vibrational spectra or other
2
intricate property. depends on temperature (which does not follow in the
harmonic case), and changes as we run the chain.
From the fact that the heat current comes into the
EXISTENCE OF THERMAL RECTIFICATION
systembythe firstsite,passestroughthe chainandgoes
out by the last site, we have
Let us introduce our assumptions and derive our re-
sults. F1,2 =F2,3 =...=FN−1,N ≡F. (2)
WeconsiderachainwithN sites,wherethefirstsiteis These equations together with eq.(1) give us
connected to a thermal bath at temperature T1, and the
last site is connected to a bath at temperature TN. It is F(C1T1α+C2T2α) = T1−T2
possible to extend our analysis also for a d-dimensional F(C2T2α+C3T3α) = T2−T3
lattice with two thermal baths at the boundaries: the
... = ...
chain structure is represented by the axis (direction) of
the heat flow. We assume that it is possible to build F(CN−1TNα−1+CNTNα) = TN−1−TN.
a temperature gradient in the system. Such condition Summing up the equations, we find
always happens if the Fourier’s law holds, but we do not
demandthislawhere(anyway,wewillstudycaseswhere F =K(T1−TN),
Fourier’s law holds). Precisely, we assume that the heat N −1
flow from site j to j+1 is given by where
Fj,j+1 =−Kj(∇T)j = CjTjα+C1j+1Tjα+1 (Tj −Tj+1)(1,) K=+{2CC1NT−1α1T+Nα2−C12+T2αC+NT.N.α. −1·(N −1), (3)
(cid:9)
i.e., with, say, the local thermal conductivity given by that is the Fourier’s law for the case of the thermal con-
the average of a function of the local temperatures and ductivityKremainingfiniteasN →∞. Fromeq.(1)and
other parameters of the system. For the homogeneous eq.(2), it follows that
model, such expression reads
T1−T2 T2−T3
= =...
FjH,j+1 = C(Tjα+1Tjα+1)(Tj −Tj+1)= C′1T¯jα (Tj −Tj+1), C1T1α+C2T2α = C2T2αTN+−C13−T3αTN . (4)
where T¯α = Tα+Tα /2, C′ = 2C, which is exactly CN−1TNα−1+CNTNα
j j j+1
theformulade(cid:0)scribedbys(cid:1)everalresults(onhomogeneous Thus,giventhetemperaturesofthebathsT1 andTN,by
models) from the literature: e.g., in ref.[14], we have using the equations above we determine the inner tem-
α = 2, C′T2 = 1/K = λ2T2/ω9µ3, where λ is the coeffi- peratures T2, T3, ..., TN−1. For ease of computation,
cient of the quartic anharmonic potential, ω is the coef- let us consider the system submitted to a small gradient
ficientof the interparticle quadraticinteraction,andµ is of temperature: T1 = T +a1ǫ, TN = T +aNǫ, ǫ small.
the harmonic pinning coefficient. Still for this φ4 model, Hence, Tk = T + akǫ + O(ǫ2). We will carry out the
indifferentconditionsandmethods,BricmontandKupi- computations only up to O(ǫ). And so, up to O(ǫ), we
ainen, [15] and Spohn et al. [16] found Kj = Tj−2. And, have Tkα =Tα+αTα−1ǫak (that comes from the Taylor
inreferencetoworkswithdetailedcomputersimulations, series), and
Aoki and Kusnezov [17] obtain for this one-dimensional
φ4 model, K∝T−1,35; similarly,N. Liand B.Li[18] ob- Tk−Tk+1 = (ak−ak+1)ǫ ,
tainK∝T−1,5, withslightchangesinthe exponentthat CkTkα+Ck+1Tkα+1 (Ck+Ck+1)Tα
depend on the values of the pinning and anharmonic- as said, up to O(ǫ). From this equation and eq.(4), we
ity. It is also worthto recallthat, by using an analytical obtain
simplified scheme (derived from a rigorous and more in-
a1−a2 a2−a3 aN−1−aN
tricate approach [19]), we obtain a similar formula for = =...= . (5)
C1+C2 C2+C3 CN−1+CN
the localthermalconductivityof the gradedanharmonic
self-consistentchain[20],i.e.,oftheanharmonic,inhomo- We may rewrite these equations as
geneousmodelgivenbyachainofoscillatorswithquartic a1−a2 a1−a2
on-site potential, quadratic nearest-neighbor interparti- =
C1+C2 C1+C2
cle interaction, particles with different masses and inner
a1−a2 a2−a3
stochastic reservoirsconnected to each site. =
C1+C2 C2+C3
Letus nowprovethe existence ofthermalrectification ... = ...
foragradedanharmonicsystemwithatemperaturegra- a1−a2 aN−1−aN
= .
dient in the bulk, and whose local thermal conductivity C1+C2 CN−1+CN
3
Summing them up, we obtain sites (i.e., for k replacing 3 in the relation above), and
then we show that it follows for k+1. In fact, by using
a1−a2
· (C1+2C2+...+2CN−1+CN)=a1−aN the definitions we see that
C1+C2
⇒a2 =a1+ (aNC˜(−N)a1)(C1+C2), C˜(k+1) == CC˜1(k+)+2CC2k++..C.k++12,Ck+Ck+1
where C˜(N)≡(C1+2C2+...+2CN−1+CN). Similarly, Q˜(k+1) = C˜(2)C2+...+C˜(k)Ck =Q˜(k)+C˜(k)Ck.
writing(ak−1−ak)/(Ck−1+Ck)insteadof(a1−a2)/(C1+ Then, a direct computation shows that
C2) in the LHS of the list of equations above, we obtain
ak =a1+ (aNC˜(−N)a1)C˜(k), (6) H(Ce˜n(Kce,+fo1r))t2he−d4iQff˜e(rken+ce1)b−etw2Ceke+n1tC˜h(ekt+he1r)m=alCc12on−dCukc2+ti1v.-
ities of the system with N sites, we obtain
fork =2,...,N−1. Andso,forthethermalconductivity
(3) it follows that 1 − 1 = αTα−1ǫ(a1−aN) C2−C2 , (11)
K K′ (N −1)C˜(N) 1 N
K = (N −1)· TαC˜(N)+αTα−1ǫ(a1C1+2a2C2+... (cid:2) (cid:3)
n −1 where, we recall, αTα−1ǫ(a1 − aN) in the numerator
+2aN−1CN−1+aNCN)} . (7) above is Tα − Tα up to O(ǫ). Thus, the existence of
1 N
thermal rectification for anisotropic, e.g. graded, mate-
Toinvestigatetheexistenceorabsenceofrectification,
rials is transparent.
we need to analyze the heat flow for the system with
inverted thermal baths, that is, we compute the new
thermalconductivityforthesamesystem,butwithtem-
RECTIFICATION PROPERTIES
′ ′ ′
peratures T , where T1 = TN and TN = T1. Following
the previous manipulations, we see that, in the system
Now, let us examine the rectification in details and
with inverted baths, the new temperature for the site k
′ ′ searchforconditionsleadingtosuitableproperties. First,
is T =T +a ǫ, where, for k =2,3,...,N −1,
k k we write the expression for the rectification factor f
r
a′ =a − (aN −a1)C˜(k). (8) |K−K′| |Tα−Tα||C2−C2|
k N C˜(N) fr ≡ K′ ≈ 1Tα N (C1˜(N))N2 .
′ ′
Obviously: a1 =aN,andaN =a1. Hence,theexpression Hence,fixedthe temperaturesatthe boundaries,the be-
for the “inverted” thermal conductivity becomes
havior of f with N is given by |C2−C2|/(C˜(N))2. We
r 1 N
K′ = (N −1)· TαC˜(N)+αTα−1ǫ(a′1C1+2a′2C2+... recall that
n
+2a′N−1CN−1+a′NCN −1. (9) C˜(N)=C1+2C2+...+2CN−1+CN ≈2Z NCxdx.
(cid:1)(cid:9) 1
And, with simple manipulations, we get
And, for a small gradient of temperature in the system,
1 − 1 = αTα−1ǫ(a1−aN) K=(N −1)/{TαC˜(N)+O(ǫ)}.
K K′ (N −1)C˜(N)
× C˜(N)2−4Q˜(N)−2C C˜(N) ,(10) Thus, to get a normal conductivity (Fourier’s law) we
n N o must have C˜(N) ∼ N, i.e., C ∼ constant. That is, for
N
whereQ˜(N)≡C˜(2)C2+...+C˜(N−1)CN−1. Asasimple tdhieensetsg,rifadtheedcsoynsdteumctsi,vaittyleisanstoramtasmltahlelntetmhepereracttiufirceagtiroan-
test for the expression above, note that it vanishes (as
factor decays to zero as N → ∞. To avoid the decay of
expected) in the case of a homogeneous system (C1 =
therectificationfactor,forexample,tomakeitfiniteand
C2 =...=CN). nonzero as N → ∞, we need to take C ∼ cexp(γN).
To continue the analysis, we take a chain with three N
And so, C˜(N)∼c(exp(γN)−1)/γ. For γ >0, C˜(N) has
sites (say, the smallest possible system). A direct com-
exponential growth and K(N) → 0 as N → ∞. For
putation gives us
γ < 0, C˜(N) → constant and K ∼ N, i.e., we have
(C˜(3))2−4Q˜(3)−2C3C˜(3)=C12−C32. an abnormal conductivity. That is, the regimes of non-
decaying rectification and of normalconductivity do not
Now we prove, by induction, that such relation is valid overlap. Thepossibilityofanon-decayingrectificationis
for any number of sites: we assume that it is valid for k a very important property: as recalled before, the decay
4
ofrectificationisaproblemforthe usualdiodesgivenby FINAL REMARKS
the sequential coupling of different parts.
Moreover,stillfromthepreviousexpression(takeT1 > We have some remarks. First, we stress that we have
TN and CN > C1), we see that the thermal conductivity presented here sufficient, not necessary, conditions for
is smaller when the heat flows from the sites with larger manifesting thermal rectification in anisotropic systems.
C to the sites with smaller C. Inref.[22],bycomputersimulations,theauthorsdescribe
Inshort,wehaveshownthatinalatticesystemwhere rectificationinagradedmassFermi-Pasta-Ulamchain,a
it is possible to build a temperature gradient, i.e., with model with an invariant translational potential and ab-
the heat flow from site j to j +1 given by eq.(1), with normal conductivity (even for the case of homogenous
graded structure (i.e. graded C ) and with local thermal masses). We alsorecallthat,for the (verydifferent)case
j
conductivity dependent on temperature (see eq.(1)), we of a system of two-terminal junctions, sufficient condi-
will always have thermal rectification. To be precise, we tions for rectification have been described in a recent
need to recall that in our proof, for ease of computa- work by Wu and Segal [23].
tion, we have assumed a system with small temperature A further investigation of great interest is the behav-
gradient (however, we believe that it is not a necessary iorofthe gradedsystemassubmittedtoalargegradient
condition - more comments ahead). It is interesting to of temperature: we believe that it shall lead to a signi-
note that such conditions - temperature gradient in the ficative rectification. In ref.[24], for some specific models
bulk,localconductivitydependentontemperatureanda given by chaotic billiard systems, the authors claim that
gradedstructure - appear in the quantumharmonic self- there is a significative rectification “provided the tem-
consistentchainofoscillators[21],asystemthatpresents peratures (of the two sides of the system) are strongly
rectification in opposition to its classical version (with a different...”
conductivity that does not depend on temperature).
Toconclude,weemphasizethatduetotheirsimplicity,
To give a concrete example, we turn to the chain theassumptionsandargumentsdescribedherefollowfor
with homogeneous anharmonic potential, homogeneous many of the usual systems modeling heat conduction in
interparticle interactions, etc, but with graded masses. solids: it shows the ubiquity of rectification in graded
For the model with inner self-consistent reservoirs,weak systems. Moreover,theexistenceofsimpleconditionsfor
nearest neighbor interactions, quartic anharmonicity, in theexistenceofanefficientrectification,andthefactthat
an approximate calculation [20], we have graded systems may be constructed in practice (and are
even abundant in nature) indicate that they are optimal
C materialtobeusedintheconstructionofathermaldiode
Fj,j+1 = (mj+1Tj1/2+mjTj1+/21)(Tj −Tj+1), (caenrtdaianllsyoctohnertmribaultteratnositshteorasd,veatcn)c,eaonfdpshoo,nthoneiircsu.seshall
Work supported by CNPq (Brazil).
where C involves the coefficients for the anharmonic-
ity, interparticle interaction, etc. The denominator of
the expression above may be written as [(Tj1/2/ρj,j+1)+
(Tj1+/21/ρj+1,j)], where ρj,j+1 = mj/mj+1mj, ρj+1,j = ∗ Electronic address: emmanuel@fisica.ufmg.br
mj+1/mj+1mj. To follow, we define [1] L. Wang, and B. Li, Physics World 21, 27 (2008).
[2] G. Casati, Nature Nanotech.2, 23 (2007).
ρ¯j ≡ ρj,j−1+ρj,j+1 = 1 · (mj+1+mj−1), [3] M88.,T0e9r4r3a0n2eo(,2M00.2P).eyrard,andG.Casati,Phys.Rev.Lett.
2 2 (mj−1mj+1) [4] B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93,
184301 (2004).
i.e., ρ¯ is proportional to the inverse of a reduced mass. [5] B. Hu, L. Yang, and Y. Zhang, Phys. Rev. Lett. 97,
j
Hence,consideringtheentiresystemj =1,...,N,weap- 124302 (2006).
[6] C.W. Chang et al., Science 314, 1121 (2006).
proximatelyhaveFj,j+1givenbyeq.(1)withCj =1/ρ¯jC. [7] B. Li et al.,Appl.Phys. Lett. 88, 143501 (2006).
And the analysis follows as previously described: now
[8] L.Wang,andB.Li,Phys.Rev.Lett.99,177208 (2007).
with the bigger flow in the direction from the larger to [9] L.Wang,andB.Li,Phys.Rev.Lett.101,267203(2008).
the smaller masses. It is worth to recall that such prop- [10] D. Segal, and A. Nitzan, Phys. Rev. Lett. 94 034301
erty, a bigger heat flow from the larger to smaller densi- (2005).
ties, as described here, has been already experimentally [11] J.P. Eckmann, and C. Mejia-Monasterio, Phys. Rev.
described [6]. Lett. 97, 094301 (2006).
[12] N. Yang et al., Appl. Phys. Lett. 95 033107 (2009); N.
Similar properties appear in a system with homoge- Yang, G. Zhang, B. Li, Appl. Phys. Lett. 93 243111
neousparticlemasses,butgradedanharmonicon-sitepo- (2008);M.Huetal.,Appl.Phys.Lett.92211908(2008).
tentials or graded interpaticle interactions. [13] J. P. Huang,K. W. Yu,Phys.Rep.431, 87 (2006).
5
[14] R. Lefevere, A. Schenkel, J. Stat. Mech.: Theory Exp. Phys. Rev.Lett. 96, 100601 (2006).
02, L02001 (2006). [20] E. Pereira, Phys.Rev.E 82, 040101 (R) (2010).
[15] J. Bricmont, A.Kupiainen, Phys. Rev. Lett. 98, 214301 [21] E. Pereira, Phys.Lett. A 374, 1933 (2010).
(2007); Commun. Math. Phys. 274, 555 (2007). [22] N.Yang,N.Li,L.Wang,B.Li,Phys.Rev.B76020301
[16] K. Aoki, J. Lukkarinen, H. Spohn, J. Stat. Phys. 124, (R) (2007).
1105 (2006). [23] L.-A.Wu,D.Segal,Phys.Rev.Lett.102095503(2009).
[17] K.Aoki, D. Kusnezov,PhysLett. A 265, 250 (2000). [24] G. Casati, C. Mej´ıa-Monasterio, T. Prosen, Phys. Rev.
[18] N.Li, B. Li, Phys. Rev.E 76, 011108 (2007). Lett. 98 (2007) 104302.
[19] E. Pereira, R. Falcao, Phys. Rev. E 70, 046105 (2004);