electrical and thermal conductivity

Chapter 7Electrical and Thermal Conductivity

Abstract After a Sect. 1.1 devoted to electrical conductivity and a section thatdeals with magnetic and dielectric losses (1.2), this chapter explores the theory ofthermal conduction in solids. The examined categories of solids are: metals Sect.1.3.2, Dielectrics Sects. 1.3.3 and 1.3.4 and Nanocomposites Sect. 1.3.5. In Sect.1.3.6 the problem of thermal and electrical contact between materials is consideredbecause contact resistance occurring at conductor joints in magnets or other highpower applications can lead to undesirable electrical losses. At low temperature,thermal contact is also critical in the mounting of temperature sensors, where badcontacts can lead to erroneous results, in particular when superconductivity phe-nomena are involved.

7.1 Electrical Conductivity

7.1.1 Relation Between Thermal and Electrical Conductivity

It is well known that when applying a gradient of temperature to a metal, a heatflow is observed. Moreover, an electric field applied to a metal produces both acurrent flow and a heat flow. This evidence indicates that thermal conductivity andelectrical conductivity are strongly related, and thus they can be treated in parallel(see Sect. 7.3.3.2).

In general, the electrical and thermal conductivities of pure metals are higherthan those of alloys, see Table 7.1; this is due to the presence of defects which actas scattering centers for electrons and phonons. On the other hand, insulatingmaterials and most composites have extremely low thermal conductivities. Somespecial crystalline insulators, such as quartz, diamond and sapphire, have highthermal conductivities. They are useful for electrical insulating connections thatrequire good thermal contact. Several books deal with these items in detail, see,e.g., Refs. [1, 2].

G. Ventura and M. Perfetti, Thermal Properties of Solids at Roomand Cryogenic Temperatures, International Cryogenics Monograph Series,DOI: 10.1007/978-94-017-8969-1_7, � Springer Science+Business Media Dordrecht 2014

131

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

Tab

le7.

1E

lect

rica

lre

sist

ivit

yof

som

em

etal

san

dal

loys

(lX

cm).

All

data

are

from

Ref

.[1

46]

10K

20K

50K

77K

100

K15

0K

200

K25

0K

295

K

Met

als

Ag

(RR

R18

00)

110

-4

310

-3

1.03

10-

12.

710

-1

4.2

10-

17.

210

-1

1.03

1.39

1.60

Al

(RR

R35

00)

–7

10-

44.

710

-2

2.2

10-

14.

410

-1

1.01

1.59

2.28

2.68

Au

(RR

R34

00)

610

-4

1.2

10-

22.

010

-1

4.2

10-

16.

210

-1

1.03

1.44

1.92

2.20

Cu

(RR

R34

00)

–1.

010

-3

4.9

10-

21.

910

-1

3.4

10-

17.

010

-1

1.05

1.38

1.69

Cu

(OF

HC

)(R

RR

100)

1.5

10-

21.

710

-2

8.4

10-

22.

110

-1

3.4

10-

17.

010

-1

1.07

1.41

1.70

Cu

(OF

HC

)(6

0co

lddr

awn)

3.0

10-

23.

210

-2

1.0

10-

12.

310

-1

3.7

10-

17.

210

-1

1.09

1.43

1.73

Fe

(RR

R10

0)1.

510

-3

710

-3

1.35

10-

15.

710

-1

1.24

3.14

5.3

7.55

9.8

In(R

RR

5000

)1.

810

-2

1.6

10-

19.

210

-1

1.67

2.33

3.80

5.40

7.13

8.83

Nb

(RR

R21

3)–

6.2

10-

28.

910

-1

2.37

3.82

6.82

9.55

12.1

214

.33

Ni

(RR

R31

0)–

910

-3

1.5

10-

15.

010

-1

1.00

2.25

3.72

5.40

7.04

Pb

(RR

R14

000)

–0.

532.

854.

78–

––

––

Pb

(RR

R10

0000

)–

––

–6.

359.

9513

.64

17.4

320

.95

Pt

(RR

R60

0)2.

910

-3

3.6

10-

27.

210

-1

1.78

2.74

24.

786.

768.

7010

.42

Ta

(RR

R77

)3.

210

-3

5.1

10-

29.

510

-1

2.34

3.55

6.13

8.6

11.0

13.1

Ti

(RR

R20

)–

2.0

10-

21.

44.

457.

916

.725

.734

.843

.1W

(RR

R10

0)2

10-

44.

110

-3

1.50

10-

15.

610

-1

1.03

2.11

3.20

4.33

5.36

All

oys

Al

1100

-08

10-

28

10-

21.

610

-1

3.2

10-

15.

110

-1

1.07

1.72

2.37

2.96

Al

5083

-03.

033.

033.

133.

333.

554.

154.

795.

395.

92A

l60

61-T

61.

381.

391.

481.

671.

882.

463.

093.

684.

19B

eryl

co(C

u 97.7

Be 0

.02

Co 0

.003)

6.92

6.92

7.04

7.25

7.46

7.96

8.48

8.98

9.43

Car

trid

gebr

ass

(70

%C

u30

%Z

n)4.

224.

224.

394.

664.

905.

425.

936.

426.

87H

aste

lloy

C12

312

312

312

4–

–12

6–

127

Inco

nel

625

124

124

125

125

––

127

–12

8In

cone

l71

810

810

810

810

9–

–11

413

415

6

(con

tinu

ed)

132 7 Electrical and Thermal Conductivity

Tab

le7.

1(c

onti

nued

)

10K

20K

50K

77K

100

K15

0K

200

K25

0K

295

K

Inva

r(F

e 0.6

4N

i 0.3

6)

50.3

50.5

52.1

54.5

57.0

63.3

70.0

76.5

82.3

Mon

elC

uNi3

036

.436

.536

.636

.736

.937

.437

.938

.338

.5P

hosp

hor

bron

zeA

8.58

8.58

8.69

8.89

9.07

9.48

9.89

10.3

10.7

Sta

inle

ssS

teel

(304

L)

49.5

49.4

50.0

51.5

53.3

58.4

63.8

68.4

73.3

Sta

inle

ssS

teel

(310

)68

.668

.870

.472

.574

.478

.482

.385

.788

.8S

tain

less

Ste

el(3

16)

53.9

53.9

54.9

56.8

58.8

63.8

68.9

73.3

77.1

V0.9

Ti 0

.06

Al 0

.04

–14

714

815

015

215

716

216

616

9

7.1 Electrical Conductivity 133

As we mentioned in other parts of this book, the knowledge of the thermalproperties of matter is mandatory to carry out cryogenic experiments. In thissection and in Chap. 9, useful values of electrical and thermal conductivities formany technical materials are given over the cryogenic range of temperatures.

7.1.2 Electrical Resistivity of Metals

Electric charge is transported through metals by electrons, belonging to the con-duction band, which are free to move within the crystal lattice of the solid.Electron motion is generally described by a collective wave model in which theelectron clouds move through the material as waves [3]. It is useful, however, toexplain the temperature dependence of the electrical resistivity and heat conduc-tion due to electrons by using the elementary kinetic theory of transport in metals[3, 4].

The free electron model describes the electrons like a ‘‘Fermi gas’’ (a gas madeof free fermions). Let us assume that the thermal energy (kBT) is sufficiently lowcompared to the Fermi energy (EF, the energy of the electrons which occupy thehighest-occupied state at T = 0 K). This is, at cryogenic temperatures, a very goodapproximation since EF is about 104 K in most metals [5].

Only electrons with energy very close to EF can contribute to the conductionbecause they are the only ones that receive a thermal energy sufficient to jump inthe conduction band. Their average velocity (\v[) is

\v [ ffi vF ¼�hkF

með7:1Þ

where kF is the radius of the Fermi sphere and me is the mass of the electron [5].This velocity is commonly written as

vF ¼4:20n

106 m

sð7:2Þ

where n is a dimensionless parameter ranging between 2 and 6 for most metals [5].Considering that n is of the order of unit, vF is about 1 % of the velocity of light

and is thus several orders of magnitude higher compared to the velocity of theparticles which form a classical gas.

We can further define the relation between the mean scattering time (the timebetween two collisions) and the mean free path (\k[) as

\s[ ¼\k[vF

: ð7:3Þ


http://dx.doi.org/10.1007/978-94-017-8969-1_9

The mean scattering time is related to the electrical resistivity q, and thus to theelectrical conductivity r by the equation

1q¼ r ¼ nc � e2

me\s[ ¼ nc � e2

mevF\k[ ð7:4Þ

where nc is the number of electrical carriers (only electrons for metals) and e is thecharge of electron. In metals, n is approximately constant; in fact, nc can beobtained knowing the valence of the metal, and so the only temperature-dependentterm that can influence the electrical resistivity is \k[.

The mean free path between collisions is dominated by two different scatteringmechanisms:

1. At very low temperatures, only a few phonons are present. Thus,\k[is mainlylimited by scattering processes due to chemical or physical crystal-latticeimperfections (impurities, vacancies, interstitials) and, therefore, is indepen-dent on temperature. Thus, near liquid helium temperature, q approaches aconstant value which is referred to as ‘‘residual resistivity’’ q0.

2. Near room temperature, the electrical resistivity of most pure metals decreasesmonotonically with temperature following an approximately linear relation-ship. This trend is the result of electron-phonon scattering and is the dominanttemperature-dependent contribution to the resistivity.

It can be useful to introduce Matthiessen’s rule. This empirical rule assumesthat if more than one scattering source is present (e.g., electron-phonon andelectron-impurities), the total q is simply the sum of the resistivities one wouldhave if each scattering process was present alone [6].

Taking into account the electron-impurities and electron-phonon scatteringcontributions, we obtain

qtot ¼ q0 þ qðTÞ: ð7:5Þ

As an example of the behavior of electrical resistivity, Fig. 7.1 shows a plot ofq (T) for various purities of copper defined in terms of the residual resistivity ratio(RRR = q (273 K)/q (4.2 K), see, e.g., Refs. [1, 7]). The more pure and defectfree the metal, the higher its RRR value. It should also be noted that the tem-perature at which a near constant resistivity is obtained decreases with increasingpurity. This is obviously due to the fact that, for small amount of impurities, thedefect contribution to scattering became dominant only at very low temperatures.At very low temperature in high purity samples,\k[may become very large, evenapproaching the sample size, such that scattering off the surface of the sample cancause a ‘‘size effect’’ dependence of the q [8]. Note that at high temperatures, qcurves of all grades of purity collapse in one curve (dashed), representing theelectron-phonon scattering dependence.

In Fig. 7.2., the electrical resistivity of some metals of comparable purity isreported. A linear behavior is observed from about 50 K to room temperature. It is


worth noting that for metallic elements, a concentration of impurities of about1 ppm can have a significant effect on electron transport, as can the amount ofcold-worked generated imperfections [9].

The universal form for the q of pure metals makes them very useful as tem-perature sensors, such as platinum resistance thermometers which are used forprecise measurements in the intermediate temperature regime (30–100 K) wheretheir sensitivity, dR/dT, is roughly constant [10].

The electrical resistivity is often (but not always (see, e.g., [11] and the exampleof Sect. 8.5.1) one of the easiest properties to measure and, as a result, q (T) isknown and tabulated for many elements and alloys of interest [12–17].

Electrical resistivities of metals, technical alloys and common solders arereported in Table 7.1.

Fig. 7.2 Electrical resistivityof some metals of comparablepurity. Plot from data ofTable 7.1

Fig. 7.1 Electrical resistivityversus temperature for copperof differing purities [12]


http://dx.doi.org/10.1007/978-94-017-8969-1_8

7.1.3 Electrical Conductivity of Semiconductors

Pure semiconductors are a class of materials which have an energy gap (Eg)between the valence and conduction band of the order of 1 eV at roomtemperature.

In (7.4), we found that for metals, the only nonconstant parameter controlling qwas\k[. However, in the case of semiconductors, nc (the number of carriers) alsovaries drastically with temperature. At T = 0 K, semiconductors are perfectinsulators: in fact, the conduction band is entirely empty, while the valence band isfull. The number nc of carriers in the conduction band increases with increasingthermal energy because the fraction of electrons which can ‘‘jump’’ to the con-duction band is higher at higher temperatures; in particular, it is possible to find anapproximate dependence:

nc � eEG=kBT : ð7:6Þ

Pure (or ‘‘intrinsic’’) semiconductors can be distinguished from insulatorssimply by the value of the energy gap between conduction and valence band, e.g.,diamond (EG & 6.3 104 K at 300 K) is also an insulator at high temperatures. It isworth noting that the energy gap is temperature-dependent. In fact, it can vary byabout 10 % from 0 to 300 K. This behavior is due to the thermal expansion whichmodifies the periodic potential experienced by electrons and to the temperature-dependent phonon distribution.

The resistivity of pure semiconductors covers a large range of values (from10-4 to 107 Xm) depending on the chemical nature of the material. These valuesare orders of magnitude higher than that of most metals (q * 10-8 Xm).

The conductivity of semiconductors can be increased by ‘‘doping’’ them withimpurities which introduce energy levels inside the gap, and hence charge carrierswhose number increases with temperature. Even small concentrations of impuritiescan change the conductivity of a semiconductor by several orders of magnitude atroom temperature, as reported in Fig. 7.7.

The typical dopants are elements from the 13th group (called ‘‘acceptors,’’ e.g.,B, Al, Ga) and from the 15th group (called ‘‘donors,’’ e.g., P, As, Sb) of theperiodic table. This choice is obviously related to the external electronic config-uration of those elements, which have one electron less (the acceptors) or more(the donors) compared to the semiconductors belonging to the 14th group (e.g., Si,Ge). Generally (but not always!), the doping element differs from the semicon-ductor in terms of total electronic configuration by only one electron: Ge (atomicnumber 32) is often doped with Ga (atomic number 31) or with As (atomic number33) because a similar atomic radius favors the creation of a uniform structurewhich minimizes the increased scattering due to the insertion of impurities that canact as scattering centers.


Note that in Fig. 7.3, all curves tend to collapse into one at high temperatures.The temperature at which curves become one increases with increasing donorconcentration. One might think that this phenomenon is due to the increase ofimpurity concentration, but this is not true. In fact, we have to remark that therange of doping reported in Fig. 7.3 is really limited. If we consider the densityand the atomic weight of pure Ge (gGe = 5.323 g/cm3 and p.a. = 72.64 gmol-1,respectively) and calculate the number of Ge atoms in a volume of 1 cm3, we get

nGe ¼gGe � V � NA

p:a:¼ 4:3 � 1022atoms: ð7:7Þ

Comparing this number with the doping range (1014–1016 atoms/cm3), wededuce that in the most doped sample, the ratio between Ge atoms and Sb atoms isgreater than 5 105! The main reason for this behavior is instead due to the differentnumber of carriers (see (7.4)) in the different samples; this ‘‘extrinsic’’ effect ispresent even if the impurity concentration only changes by a factor 102.

Due to the strong temperature dependence of their resistivity, semiconductorsare most commonly encountered in cryogenic applications as temperature sensorswith high negative temperature coefficients of q. For example, a high sensitivity atT \ 1 K can be achieved using doped Ge as a sensor [18]. Electrical character-istics of these sensors (frequently called thermistors) often show a strong depen-dence on magnetic field (magnetoresistance) which can be either a useful orharmful characteristic, depending on the type of measurement that one wants toperform [7].

Fig. 7.3 Resistivity of Gedoped with Sb versus 1/T.The number refers to thedonor concentration (cm-3)[136]


7.2 Magnetic and Dielectric Losses

To understand the problem of losses, we start reminding the reader of the for-mulation of two of Maxwell’s equations, namely,

r� E!¼ � o B

!

otð7:8Þ

r � H!¼ J

!þ oD!

ot; ð7:9Þ

with E = electrical field, B = magnetic induction, H = magnetic field,J = current density, D = electric displacement field.

It is easy to see that a change of B or D can produce a change in the electricalproperties of the material that are related to the thermal conductivity, thus causinga generation of heat. For example, the power dissipated in a cylinder of radiusr length L is

Pe ¼pr4L dB=dtð Þ2

8q: ð7:10Þ

The factor 1/q in (7.10) leads to the choice of low-conductivity materials for themixing chamber of dilution refrigerators if high magnetic fields and vibrations arepresent [7].

7.2.1 Losses in Dielectric Materials

Losses in dielectric materials are seldom considered in cryogenics. In steadyoperating conditions, ‘‘dc’’ losses are extremely small at cryogenic temperatures,but ‘‘ac’’ losses cannot be neglected because of the low values of c at low tem-peratures. As with a capacitor, losses are described either in terms of a complexform of the dielectric constant e(x) = e0 + e00, whose real part (e0) is responsiblefor heating, or by tand = e0/e. The simple lumped constant model which sche-matizes the phenomenon by a pure capacitance Ce paralleled with a pure resistanceR is physically unsatisfactory. Fortunately, when the capacitor impedance ismeasured, a good bridge (e.g., Andeen–Hagerling [19]) supplies the correct valueof both e0 and e00.

In the case of polymers, losses are due to dynamic mechanical relaxationcaused by heat transfer between the intermolecular mode (strain-sensitive mode)and the intramolecular mode (strain-insensitive mode) [20]. Since heat is trans-ferred into the intramolecular modes with a characteristic relaxation time, that is, a

7.2 Magnetic and Dielectric Losses 139

function of temperature, the physical properties of polymer materials heavilydepend on the frequency of the excitation.

If a wide range of temperatures for polymer materials is considered, the exis-tence of various transitions is very important. The simplest is a phase transitionwhich occurs in the crystalline region and is a first order transition. The physicalproperties of polymeric materials change significantly before and after this phasetransition point. A polymeric substance has many subtransitions originated inmolecular motions beside first order transitions. Since the molecular structure ofthe polymer is very complex, many degrees of freedom exist.

The most important transition is the amorphous region in the glass transitionthat occurs at a characteristic temperature called Tg. It is well recognized thatpolymers possess considerable molecular mobility, even below their glass transi-tion temperature. Hereafter, molecular mechanisms for relaxations below Tg areclassified by:

(1) internal rotation of an end group in the side chain, as shown in Fig. 7.4 forpoly(ethy1 methacrylate);

(2) proton tunneling, as shown in Fig. 7.5 for polyethylene;(3) motion of a methyl group, see Refs. [21–23];(4) molecular motion at defect regions, see Refs. [24–28];(5) effects due to impurities or additives [29] as shown in Fig. 7.6.

A comparison between the contribution of dielectric and thermal conductionlosses can be found for Upilex R in Refs. [7, 30].

Fig. 7.4 Dielectric loss ofpoly(ethy1 methacrylate) atthree frequencies plottedagainst temperature [28]


7.3 Thermal Conductivity

7.3.1 Introduction

The heat flow through a material is the energy transport phenomenon due to athermal gradient. The thermal current density is defined as heat flow qQ/qt per area(A), and can be expressed (in the x direction) as

jxðx; y; z; tÞ ¼1A

oQ

ot¼ �jðTÞ oTðx; y; z; tÞ

oxð7:11Þ

Fig. 7.6 Dielectric losstangent of high-densitypolyethylene with 0.2 %Ionox B; high-densitypolyethylene with 0.2 %BHT D; polystyrene with0.2 % Ionox C together withthe loss for oxidized high-density polyethylene A. Fullsymbols refer tomeasurements at 4.2 K, whileempty ones refer to 1.56 K[29]

Fig. 7.5 Temperaturedependence of the dielectricloss tangent for high-densitypolyethylene at 1 kHz [27]

7.3 Thermal Conductivity 141

where j is the thermal conductivity and the minus sign accounts for the fact thatheat Q moves from warmer to colder zones. This formula is valid for otherdirections and both in the stationary and nonstationary case. It is important to notethat the thermal gradient is generally time- and direction-dependent, and (7.11) isvalid, point by point, since carriers do not follow a simple linear path, but candiffuse in all directions.

In a steady situation, the time dependence disappears; when heat conduction ismainly in x direction, (7.11) becomes

oQ

ot¼ �jðTÞA dT

dx¼ 0: ð7:12Þ

If T is time-dependent, differentiation of (7.12) gives

oT

ot¼ w

o2T

ox2ð7:13Þ

where w, called thermal diffusivity, is related to thermal conductivity by

w ¼ jg � c ð7:14Þ

where g is the mass density and c is the specific heat of material.Thermal carriers determining thermal conductivity are lattice vibrations (pho-

nons) and electric-charge carriers (electrons or holes). To estimate the temperaturedependence of thermal conductivity, a very simplified model which considersthermal carriers as particles of a gas diffusing through a material is often used.

For phonons, the thermal conductivity of an isotropic material can be expressedas [5]

jðTÞ ¼ 13

XZxD

0

cixðx; TÞviðx; TÞ\k [ ðx; TÞdx ð7:15Þ

where the sum runs over the modes ‘‘i’’ up to xD, v is the velocity of phonons,\k[is the mean free path and cix (differential specific heat) is the contribution ofphonons of frequency between x and x + dx. We can simplify this expression inthe ‘‘dominant phonon’’ approximation [3–5], obtaining

j ¼ 13

c �\v [ �\k[ ð7:16Þ

where c is the specific heat per unit volume, \v[ is the average velocity ofparticles and\k[ is the mean free path of a carrier inside a material. The velocityof phonons is the velocity of sound in the material with typical values of about(3–5) 105 cm/s.


This simple formalism can also be used in the case of electrons; however, forelectrons, the velocity can be assumed to be the Fermi velocity because onlyelectrons near the Fermi energy can contribute to thermal transport (they are theonly ones which can give rise to a transition to higher energy levels, as explainedin Sect. 7.1.1 for electrical conductivity). The typical value of Fermi velocity is107–108 cm/s.

Note that in (7.16), thermal carriers do not move in a ballistic path: hence,thermal conductivity is determined by scattering processes between carriers andpoint-defects, dislocations, other thermal carriers, boundaries, and crystallitesboundaries. Each scattering event gives a thermal resistance contribution Ri. Weobtain the total thermal conductivity by Matthienssen’s rule [6]

1j¼X

i

Ri: ð7:17Þ

Figure 7.7 shows the trend of j for various materials for T [ 2 K. The highestj is registered for metals and ‘‘special’’ insulators (diamond, sapphire, quartz),while the lowest can be found in polymers like nylon and polystyrol.

7.3.1.1 Thermal Conductivity of Metals

In analogy with the process of electrical conductivity, the behavior of j can beunderstood in terms of a kinetic theory model for gases of electrons and phonons[31]. In the frame of such a simple model, j is in the form (7.16).

Remembering that for free electrons, the expressions of C and vF,

Fig. 7.7 Thermalconductivity of variousmaterials at2 K \ T \ 300 K [7]


C ¼ p2k2Bne

2EFT ð7:18Þ

vF ¼ffiffiffiffiffiffiffiffi2EF

me

r; ð7:19Þ

are the electronic contribution to the thermal conductivity, and je can be easilycalculated by inserting (7.18) and (7.19) into (7.16) as

je ¼p2k2

Bne\s[3me

T: ð7:20Þ

At high temperatures (T [ hD), the main free path (and thus also \s[, asdescribed in (7.4)) is proportional to T-1, due to the increase in the latticevibrations, and j approaches a constant value. At low temperatures, \s[ isapproximately constant since impurity scattering dominates; hence, the thermalconductivity should be proportional to T.

As in the case of q, j depends on the chemical nature of the metal and on thegrade of purity of the sample. In Fig. 7.8a, we report the values of j for somemetals with comparable purity, while in Fig. 7.8b, j of copper specimens withdifferent purities. The asymptotic value near room temperature in Fig. 7.8b gives anear constant j = 4 W/cm K. With decreasing temperature, the thermal conduc-tivity passes through a maximum that is typical of almost all metals, whichdepends on the purity of the sample, followed by a linear region at the lowesttemperatures, a behavior also visible for all metals in Fig. 7.8a.

As mentioned, the electronic thermal and electrical conductivities in puremetals have similar scattering processes, thus a correspondence clearly shouldexist between these two properties.

Fig. 7.8 Thermal conductivity versus temperature: a Of some metals with comparable purity (Al99.994 %, Ir = 99.995 %, Sn = Zn = 99.997 %, Cu = Ag = 99.999 %) [137]. b Of Cu atdiffering purities [12]


The empirical formula which relates these two quantities is known as theWiedemann–Franz Law [32]. For the free-electron model, the ratio betweenelectron thermal and electrical conductivities is given by

je

r¼ p2k2

B

3e2T ¼ L0T ð7:21Þ

where e is the electron charge (e & 1.6 10-19 C).The quantity L0 & 2.45 10-8 V2/K2 is the free electron Lorenz number which

is almost independent of material properties and temperature.Experiments have shown that the value of L0 is not exactly the same for all

materials. Kittel [33] gives some values of L0 ranging from L = 2.23 9 10-8 W XK-2 for copper at 273 K to L = 7.2 9 10-8 W X K-2 for tungsten at 373 K. TheWiedemann–Franz law is generally valid near room temperature and for lowtemperatures (T \\ hD), but may not hold at intermediate temperatures [5]. Incertain materials (such as Ag or Al), however, the value of L0 also may decreasewith increasing temperature. In the purest samples of Ag and at very low tem-peratures, L0 can drop by as much as an order of magnitude [34]. The overallbehavior of the Lorenz ratios with sample purity are plotted in Fig. 7.9 [35].

7.3.2 Lattice Thermal Conductivity

Also, the lattice contribution to the thermal conductivity of metals, semiconductorsand insulators may be explained in terms of kinetic theory, although the thermalcarriers in this case are phonons. It is still possible to apply (7.16). Note thatsometimes, \k[ becomes as large as the specimen size (i.e., the ‘‘size effect’’).

Fig. 7.9 Electronic Lorentzratio for pure metals anddefect-free metals [35]


Most insulators and intrinsic semiconductors have thermal conductivities sev-eral orders of magnitude lower than common pure metals. At temperatures above*20 K, generally the thermal conductivity decreases monotonically with tem-perature. At low temperatures, below *10 K, where the scattering becomesapproximately independent of temperature, the thermal conductivity decreasesmore rapidly, approaching zero as T7. For more details, see, e.g., Ref. [36].

7.3.3 Thermal Conductivity of Dielectrics

Materials presenting very low electric conductivity (e0 B 10-2 at room tempera-ture, frequency = 1 kHz, see Sect. 7.2.1) and low thermal conductivity(B1 Wm-1 K-1) are called dielectric materials; in this case, no electronic con-tribution to thermal conductivity exists, so only phonon scattering phenomena areto be considered. We shall divide materials in two different subclasses: purecrystals and amorphous materials. This choice is due to the fact that differentpeculiar scattering events occur in the two cases.

7.3.3.1 Pure Crystals

The most important scattering mechanisms which have been observed are:

(a) Phonon-phonon scattering (umklapp processes).(b) Phonon-boundaries of specimen (or crystallites).(c) Phonon-point defects scattering.(d) Phonon-dislocations scattering.

Detailed and formal treatment of phonon scattering processes may be found,e.g., in Ref. [3].

At high temperatures (T [ hD), the main contribution is due to the phonon-phonon scattering. In fact, in this range of temperatures, the number of phonons islarge enough to give rise to umklapp processes (u-processes, [3]) since theprobability of this kind of events is proportional to the number of phonons. Whentwo phonons interact, a wave vector which falls outside the first zone of bound-aries is obtained: the total effect is a reduction of heat flow or, in other words, athermal resistance [4]. This phenomenon gives a resistance contribution, forT C hD,

Ru /T

\m [ ahDð7:22Þ

where \m[ and a are the mean atomic mass and spacing, respectively.Instead, for T � hD,


Ru / e�hD=T : ð7:23Þ

We obtain the boundary contribution from (7.15)

RB / T�3: ð7:24Þ

For materials with point defects, such as vacant lattice sites, interstitial atoms,impurity atoms, or isotopes of the specimen, starting from (7.14), we obtain alinear dependence on temperature. Callaway [37] gives a more rigorous treatment,obtaining

RD / T3=2: ð7:25Þ

Dislocations, i.e., imperfections in the crystal lattice with one dimensionextension, give a contribution with a temperature dependence

RD / T�2: ð7:26Þ

Summing all these contributions, the effect on thermal conductivity leads to thegraphs of Fig. 7.10.

In Fig. 7.11, we report thermal conductivity data of some pure crystals.A special material is graphite for the thermal behavior due to its peculiar

structure. In fact, graphite behaves as a very good conductor at high temperature,but below 1 K is a very good insulator [38].

This fact is very interesting from a cryogenic point of view because graphitecan be employed as a thermal switch. In Fig. 7.7, we report the comparison amongdata of some types of graphite (Figs. 7.12, 7.13).

Fig. 7.10 Qualitative representation of the conductivity of a pure crystalline material (a) and of acrystal with isotopes or impurities (b)


Fig. 7.13 Comparisonbetween the thermalconductivity of anamorphous, semi-crystallineand crystalline material

Fig. 7.11 Thermalconductivities for some purecrystals: LiF, KCl, TiO2

[138]; KBr, KBr0.53I0.47

[139]; Li3N [140]; Al2O3

[141]; La2CuO4,La1.9Sr0.1CuO4 [142]

Fig. 7.12 Comparison of theconductivity of graphites [38]


7.3.3.2 Amorphous and Semi-crystalline Materials

Amorphous materials have a peculiar temperature dependence of thermal con-ductivity. This behavior is shown in Fig. 7.17.

We observe a T 2 dependence at very low temperature, a plateau between about5–15 K and a weak positive slope above 20 K. A semi-crystalline sample shows ahigher conductivity compared to amorphous above 15 K, and lower j at lowtemperature. We shall now concisely analyze contributions to j of the scatteringprocesses. For further information, see, e.g., Ref. [39].

(a) Amorphous materials

T B 1 K: Tunneling Processes

The measurements of thermal conductivity carried out by Zaitlin and Andersonin 1975 demonstrated for the first time that below 1 K, the acoustic phonons arethe main responsibility of the heat transfer [40]. Instead, the excitations whichproduce the excess (the almost linear contribution) of specific heat cannot carrythermal energy because they are to be considered as localized excitations. Themeasurements of Zaitlin and Anderson confirmed the tunneling model proposedindependently by Anderson et al. [41] and Phillips [42] in 1972 with the aim ofexplaining the measured thermal and acoustic properties of amorphous materials.

According to this ‘‘two-level state’’ (TLS) theory, because of the structuraldisorder, groups of atoms have more than one possible position, each differingfrom the other for a very small energy, of the order of E B 10-4 eV (see alsoSect. 1.7). The quantum tunneling transition between the two levels can only takeplace with absorption or emission of phonons in order to conserve energy. Fromthis theory, the mean free path is a function of frequency and temperature

\k[ / x�1 coth�hx

2kBT: ð7:27Þ

The two-level systems present separation energy arranged in a wide range; thus,phonons involved in this kind of scattering can present a wide range of frequency.In the ‘‘dominant phonon’’ approximation, an estimation of the mean free path inthe function of temperature gives

\k[ ðxÞ / x�1 ð7:28Þ

\k[ ðTÞ / T�1: ð7:29Þ

As we discussed in Sect. 1.7, the specific heat contribution of TLS is propor-tional to T. Recalling that the mean free path is proportional to j-1, we obtain

j / T2: ð7:30Þ


http://dx.doi.org/10.1007/978-94-017-8969-1_1

http://dx.doi.org/10.1007/978-94-017-8969-1_1

4 K \ T \ 15 K: Plateau Region

This plateau is typical of amorphous materials in this indicative temperaturerange. There is not a really satisfactory explanation of this behavior. A formalsolution is given by choosing an opportune\k[(x) in order to obtain a constant j.The specific heat can be calculated as

cx /o

oT�hxDðxÞ\ f ðx; TÞ[ð Þ: ð7:31Þ

The value is determined by the product D(x)k(x) which can be expressed by anexponential law xd. In the acoustic approximation (T \\ hD), we obtain

j / T1þd: ð7:32Þ

In this zone, polymer materials act as a low-pass filter for phonon, and domi-nant-phonon approximation is not applicable. For further information, see Refs. [3,43–47].

T [ 30 K

For T [ 30 K, the mean free path does not depend on temperature or frequency:in fact, \k[ is the order of a few atomic spaces. In this case, the term ‘‘phonon’’has no significance. In this range, we experimentally observe a weak dependenceof j on temperature. For polymers,

j / Td ð7:33Þ

with d = 0.3–0.5, depending on the chemical composition of the material.Figure 7.14 shows the thermal conductivity of some amorphous solids.

(b) Semi-crystalline polymeric materials

The thermal conductivity of semi-crystalline materials show quite a differentdependence on temperature compared with amorphous materials because it isstrictly dependent on the quantity and size of crystalline inclusions (crystallites)[48]. Normally, compared to pure amorphous samples, semi-crystalline materialsshow a lower conductivity below 30 K, and a higher conductivity above. Aparameter of paramount importance is the crystallinity (fc) [49] of the sample,defined as the weight percentage of crystal phase over the total weight of thesample because a low degree of order in the polymer (disordered chains) candrastically decrease \k[, thus lowering j, according to (7.16).

T \ 20 K: Interface Scattering

Semi-crystalline materials do not show a plateau region because they have anadditional resistive term due to discontinuity between amorphous and crystallinezones [50]. They show a temperature dependence Tc with 0.5 \ c\ 3. Interfacescattering resistivity increases with fc, leading to a decrease of j with temperature


and to an increase of j(T) slope. For temperatures below 1 K, thermal conductivitydecreases and converges to a T2 dependence for temperatures low enough(T \ 0.1 K). At such temperatures, in fact, phonons have a wavelength longenough to see the crystalline zones as point defects; hence, scattering processesdue to TLS remain dominant. Above *30 K, thermal conductivity increases withfc and it sometimes can show a peak around 100 K [49].

In Fig. 7.15, we report the thermal conductivity of (BaF2)1-x(LaF3)x as a typicalexample. The pure BaF2 (x = 0) and the pure LaF3 (x = 1) have a thermal con-ductivity typical of perfect crystals. By increasing doping, the thermal conductivityis lower, slowly approaching a minimum value (jmin, the straight line in Fig. 7.15)for x = 0.33 at high temperatures. It is worth noting that for this value of doping,the conductivity assumes a trend similar to that of a-SiO2 (dashed line in Fig. 7.15);however, a complete amorphous-like behavior is never observed [51].

In Table 7.2, we report a summary of thermal behavior for amorphous poly-meric and semi-crystalline materials.

Fig. 7.15 Thermalconductivity of (BaF2)1-

x(LaF3)x [51]

Fig. 7.14 Thermalconductivity of someamorphous solids [143]


7.3.4 Thermal Conductivity of Nanocomposites

When reduced to nanoscale, a lot of materials radically change their physical andchemical properties. This is principally due to the fact that the smaller is thedimension of the nanomaterial, the bigger is the ratio between surface and volumeatoms, thus providing high and sometimes unexpected reactivity. To emphasizethat atoms at the surface possess high reactivity (due to their unsaturated coor-dination sphere) compared to bulk atoms, and also behaviors very difficult torationalize, we can cite the famous sentence by Wolfgang Pauli, that is, ‘‘God -made the bulk; surfaces were invented by the devil.’’ The nanoparticles andnanomaterials have unique mechanical, electronic, magnetic, thermal, optical, andchemical properties, thus providing a wide spectrum of new possibilities ofengineered nanostructures and nanocomposites for communications, biotechnol-ogy and medicine, photonics and electronics. For all of these reasons, nanotech-nology is a research field of growing importance [52].

The first remarkable talk about nanotechnology was given by Richard Feynmanin 1959 [53]. However, even if the terms nanomaterial and nanocomposite wereintroduced in the 20th century, such materials have actually been used for cen-turies and have always existed in nature [54]. One of the first and most famousexamples is the Lycurgus cup, made in the 4th century AD. The opaque green cupturns to a glowing translucent red if illuminated. Chemical analysis of thisextraordinary artwork indicates that the glass contains approximately 330 ppm ofsilver and 40 ppm of gold with an average particle size of approximately 70 nm.However, it is not the presence of these particular elements that is responsible forthe effect, but rather the way the initial glass composite was produced [55–57].Another example where enhanced properties were not obtained from under-standing but from empirical experiments is the Damascus steel. The swords thatwere made of these alloys were very flexible, sharp and stiff. Many centuries later,it was discovered that ancient Muslim smiths, in the 17th century, were inadver-tently using carbon nanotubes within the metallic matrix of the blade [58]. Otherprimitive nanocomposites were created in the 1860s. Experiments with vulcanized

Table 7.2 Summary of thermal behavior for amorphous polymeric and semi-crystallinematerials

Temperature range Temperature dependence Scattering processes

Amorphous polymersT B 1 K j / T2 TLS scattering4 B T B 15 K j ¼ kos Rayleigh scatteringT [ 30 K j / Tn n = 0.5–0.3 Structural defects scattering

Semi-crystalline polymersT B 1 K j / T2 TLS scattering1 B T B 2 K j / Tn n = 1–2 TLS scattering and crystalline zones2 B T B 20 K j / Tn n = 0.5–3 Crystalline zones scatteringT [ 30 K j / Tn Depends on degree of crystallinity


rubber and carbon black led to significant enhancements of the mechanicalproperties of rubber tires [59].

A nanocomposite (NC) may be defined as a composite system consisting of apolymer matrix and homogeneously dispersed filler particles which have at leastone dimension below 100 nm. Polymers are the most common materials that areused for NCs fabrication. During the past decades, polymer NCs have attractedconsiderable interest both in science and industry [60].

The first nanoclay composite, which was produced to reinforce the macroscopicproperties of an elastomer, was described in a patent from the National LeadCompany in 1950 [61]. The discovery of carbon nanotubes (CNT) by Iijima in1991 [62] and Buckminsterfullerene (C60) by R. F. Curl, Sir H. W. Kroto and R.E. Smalley in 1995 (Nobel Prices in Chemistry in 1996) were the first stepstowards a production of single- and multi-walled carbon nanotubes and newnanoscale materials and devices based on CNT [63]. For the great versatility ofchemical and physical behaviors that characterize the nanomaterials, the fields ofapplication range from agriculture and food production to space science andmedicine [64].

There is no satisfactory explanation for the origin of the change of the prop-erties of polymer NCs. It is generally accepted that the large surface-to-volumeratio of the nanoscale inclusions plays a significant role [65, 66]. Smaller particlesdisplay a much larger surface area for interaction with the polymer for the samemicroscopic volume fraction than larger particles, so it is generally better tominimize the dimension, even providing the desired properties [55]. It is currentlythought [66] that many of the characteristics of NCs are determined by theinteractions that occur at nanoparticle-matrix interfaces. The creation of ahomogeneous distribution of nanoparticles is not an easy task because particleshave a strong tendency to agglomerate: in fact, almost all the nanomaterials arekinetically and thermodynamically unstable objects. To prevent aggregation, it iscommon to cover the surface of these objects with single molecules or polymers,thus limiting the interaction because of hindrance and/or electrostatic repulsion[59, 67]. The formation of chemical bonds between the inorganic and organiccomponents is of great importance for a homogeneous dispersion of the filler inhost polymers [68–70]. A coupling agent is a chemical substance that is applied tothe surface of a material that has to be modified to make it compatible with anothermaterial of a different nature. The molecular structure enables the coupling agentto work as an intermediary in bonding organic and inorganic materials [71]. Avariety of coupling agents, such as silanes, zirconates, titanates and zircoalumi-nates have been introduced to the market since then in order to improve theinterface between the polymer and the filler [72].

One of the critical aspects of nanotechnology research is how to modify thesurface of different nanoparticles to make them compatible with polymer matricesand more useful for different applications [73]. The most important changes inproperties of NC are not caused by the order of magnitude in size reduction, but bythe phenomena such as size confinement, predominance of interfacial phenomenaand quantum mechanisms [74–76].


Fillers may be classified as inorganic or organic substances, and are furthersubdivided according to their chemical family [77]. Stable dispersion of filler inthe final composite is necessary to eliminate filler agglomerates that would act asweak points that might induce electrical or mechanical failure.

Nanoparticle composite properties at room temperature have received enor-mous attention in the last decade, but studies at low temperature are very rare.Room temperature thermal conductivities of insulating polymer materials areusually 1–3 orders of magnitude lower than those of ceramics and metals. Due tothe chain-like structure of polymers, the heat capacity consists of the contributionof both lattice vibrations and other type of vibrations, characteristic of the con-sidered material, which originate from internal motions of the repeating unit. Thelattice (skeleton) vibrations are acoustic vibrations which give the main contri-bution to the thermal conductivity at low temperatures. The characteristic vibra-tions of the side groups of the polymer chains are instead optical vibrations whichbecome visible at temperatures above 100 K [78]. As we have seen inSect. 7.3.3.2, generally, the thermal conductivity of amorphous polymers increaseswith increasing temperature if the temperature is in the glassy region and decreasesslowly or remains constant in the rubbery region.

Numerous applications in the field of electrical engineering require high ther-mal conductivity, such as insulating materials for power equipment, electronicpackaging and encapsulations, computer chips, satellite devices and other areaswhere good heat dissipation is needed. For polymers reinforced with differenttypes of fillers, this is even more important. Improved thermal conductivity inpolymers may be achieved either by molecular orientation or by the addition ofhighly heat-conductive fillers [79, 80].

Temperature, pressure, density of the polymer, orientation of chain segments,crystal structure, crystallinity and many other factors may affect the thermalconductivity of polymers [7, 81].

Figure 7.16 shows the schematic representation of the higher-order structure ofa resin to achieve macroscopic isotropy and high thermal conductivity. The pro-posed resin has three characteristic features:

(a) microscopic anisotropic crystal-like structures obtained via local alignment,e.g., via oriented mesogens (see, e.g., [82]);

(b) macroscopic isotropy of the epoxy due to disorder of the domains of thecrystal-like structures;

(c) the oriented mesogens are connected with the amorphous structure viacovalent bonds.

The thermal conductivity values of the new developed resin were up to fivetimes higher than those of conventional epoxy resins because the mesogens formhighly ordered crystal-like structures which suppress phonon scattering.

To improve the thermal conductivity of the polymer composites, Ekstrand andco-authors [83] proposed three approaches that can also be realized in parallel:


(a) decreasing the number of thermally resistant junctions;(b) forming conducting networks by suitable packing;(c) minimize filler-matrix interfacial defects.

To enhance the thermal conductivity of polymeric structures the main scientificapproach is to fill them with particles of materials with high thermal conductivitysuch as a-Al2O3, b-SiO2, SiC, diamond, SiN and BN [84–86]. In particular, boronnitride [87–89] led to the best candidate to effectively improve the thermal con-ductivity of epoxy-based composites. A study published by Han et al. [89] provedthat the size of dispersed particles is not crucial until a high doping ratio isreached, thus allowing for an easier composites preparation. Industrial companiesspecializing in the production of polymer-based insulating materials use a fill-grade up to 60 wt% of SiO2 or Al2O7. The thermal conductivity of these materialsis not significantly higher than that of pure polymers, but the very low pricejustifies their production [84, 85].

7.3.5 Composite Materials

Different theoretical and empirical approaches are available to predict and fit thethermal conductivity of two-phase systems. Here, we present a simple overview ofthe principal theories about the thermal conductivity of composite materials.

Fig. 7.16 Schematicrepresentation of amacroscopically isotropicepoxy resin [144]


The simplest three are the rule of mixture (parallel model, arithmetic mean):

jC ¼ f � jF þ ð1� f Þjm; ð7:34Þ

the inverse rule of mixture (series model, harmonic mean):

1jC¼ f

jFþ ð1� f Þ

jm; ð7:35Þ

and the geometric mean

jC ¼ ðjFÞf � ðjmÞð1�f Þ: ð7:36Þ

In all formulas, jc, jF and jm are the thermal conductivities of composite, fillermaterial and polymer matrix, respectively, and f is the volume fraction of the filler[90, 91].

The upper or lower boundaries of the thermal conductivity are given when fillerparticles are arranged either parallel to or in series with the heat flow. As soon asthe particles have a random distribution and are not aligned in the direction of theheat flow in the polymer, the parallel and series model do not give a good pre-diction of the thermal conductivity of the composites. The parallel model typicallyoverestimates the thermal conductivity of a composite and thus shows the upperlimit, while the series model tends to predict the lower limit of the thermal con-ductivity of a two-component system [92, 93].

Maxwell obtained a formula for the electrical conductivity of randomly dis-tributed and noninteracting homogeneous spheres in a homogeneous medium [94].Eucken adapted the electrical conductivity equation to thermal conductivity [95].Frieke extended Maxwell’s model and derived an equation for ellipsoidal particlesin a continuous phase [96]. Using different assumptions for permeability and fieldstrength than Maxwell, Bruggeman derived the theoretical model for a dilutesuspension of noninteracting spheres dispersed in a homogeneous medium [97].

However, most of the experimental results show that Maxwell–Eucken andBruggeman models as well as the Frieke model do not predict the thermal con-ductivity of a composite correctly [86, 98–100].

Tsao developed a model relating the thermal conductivity of a composite to twoexperimentally determined parameters which describe the spatial distribution ofthe two phases [101]. Cheng and Vachon extended Tsao’s model by assuming thediscrete phase in the continuous matrix [102].

Sundstrom and Lee reported that the Cheng–Vachon model shows a reasonableagreement with experimental data obtained from polystyrene or polyethylenesystems filled with glass, calcium oxide (CaO), aluminum oxide (Al2O3) andmagnesium oxide (MgO) [103]. Contrary to Sundstrom and Lee, Hill and Supancicshowed that the results predicted by the Cheng–Vachon model have much lowerthermal conductivity values compared to experimental results [99].


Aforementioned models are based on the amount of filler loading and do nottake into account the geometry of the particles and the size of filler particles. Thework of Hamilton and Crosser [104, 105] is based on Maxwell’s and Frieke’stheoretical models. They take into consideration the sphericity of particles (thesphericity is defined as the surface area of a sphere with the same volume as theparticle divided by the surface area of the particle). Hatta and Taya [106] proposeda model which can be applied to systems filled with particles having a high aspectratio or unidirectional fillers.

Meredith and Tobias suggested a model for high-loaded systems [107]. Lewisand Nielsen [108–110] adopted the Halpin–Tsai [111] mechanical model to obtaina model for the thermal conductivity. Agari and Uno [112–114] proposed a modelwhich is based on the generalization of both parallel and series models for filledcomposites. Generally, the Agari and Uno semi-empirical model fits experimentaldata well. However, it does not predict the thermal conductivity, but is basically afit function. To extend the overview of the thermal conductivity modeling, manydifferent models can be mentioned, for example, Russell [115], Topper [116],Jefferson–Witzell–Sibitt [117], Springer–Tsai [118], Budiansky [119], Baschirowand Selenew [120], McCullough [90], and McGee [121], and many others,including mathematical numerical methods [122, 123].

Summarizing, we can conclude that the thermal properties become morecomplicated with the addition of fillers to polymers, and thus no single theory ortechnique accurately predicts the thermal conductivity for all types of composites,but all of them can be successfully applied in particular cases.

Table 7.3 shows the list of models which can be used to describe the thermalconductivity of composite systems and when the particular model can be applied.

As mentioned, the number of references to measurements of thermal conduc-tivity at room temperature is huge. We only wish to cite two low temperatureexamples: Ref. [124] reports on the measurements of thermal properties ofnanosystems at very low temperatures by the 3x method. Authors discuss the

Table 7.3 List of thermal conductivity models and when they can be successfully applied

Model Terms of use

Series, parallel Particles are aligned either parallel or perpendicular to heat flowMaxwell–Eucken, Frieke,

BruggemanIdeal system, noninteracting spherical/spheroidal particles in

homogeneous mediumGeometric mean, Cheng–

VachonDiscrete phase in continuous matrix, only taking into account

filler loadingHamilton–Crosser Sphericity is taken into considerationHatta–Taya For the systems filled with high-aspect ratio particlesMeredith–Tobias For high-loaded compositesLewis–Nielsen Size, geometry and manner of particle packing is taken into

accountAgari–Uno Fitting function with adjustable constantsRussell, Topper For porous composites, containing voids of gas


intrinsic limitations of these methods when the thermal properties of nano-objectsare studied at temperatures below 1.2 K. In Ref. [125], SiO2/epoxy NCs roomtemperature tensile properties are reported. The effects of silica nanoparticlecontent is studied on the cryogenic thermal properties of the NCs.

Reference [126] reports the measurement of the thermal conductivity of a NCmaterial made of a Nylon-6 matrix in which metallic copper nanoparticle (5 % inweight) are uniformly dispersed. Nevertheless, data measured can differ substan-tially from the one obtained for pure polymers, also showing interesting features,in particular, a sharp dip at 1.4 K as shown in Fig. 7.17.

This is a unique features in thermal conductivity interpreted as a resonantscattering of phonons by copper nanoparticles. The temperature at which phononfrequency equals nanoparticle resonant frequency is

T ¼ �h

2pkBL

ffiffiffiffiffiffiffiffiffiffiffiffiffiE þ G

2q

s

ð7:37Þ

where L is the peak of size distribution, E is longitudinal elastic modulus (Youngmodulus), G is the shear modulus for tangential strain and q is the (Cu) density.Hence, relying on (a), the temperature of the negative notch in conductivity can bemodulated by changing the parameters of the chemical synthesis. The practicalapplications of notch in the conductivity of composite materials have not yet beenexplored and are beyond the goals of this book.

7.3.5.1 Contact Resistance

Thermal and electrical contact between materials (also two pieces of the samematerial) is an important subject in cryogenics (particularly at very low temper-atures), and yet it is still only qualitatively understood. Contract resistance

Fig. 7.17 Low-temperaturethermal conductivity ofNylon-6/Cu obtained byMartelli et al. [126] comparedto the one obtained by Scottet al. for pure Nylon-6 [145]


occurring at conductor joints in magnets or in other instruments that require highpower can lead to undesirable electrical losses. At low temperature, thermalcontact is also critical in the mounting of temperature sensors, where bad contactscan lead to erroneous results, in particular, when superconductivity phenomena areinvolved.

When two materials are joined together for the purpose of transporting heat orelectrical current, a localized resistance appears at the boundary. The magnitude ofthis resistance depends on a number of factors, including the properties of the bulkmaterials, the preparation of the interface between the two materials, whether thereare bonding or interface agents present, and external factors such as the appliedpressure.

The electrical contact resistance is of greatest interest in the production of jointsbetween high purity metals such as copper, where its value can contribute or evendominate the overall resistance of an electrical circuit. Generally, the contactresistance in pure metals has a temperature dependence that scales with theproperties of the bulk material. For electrical contacts between pure metals withoutbonding materials like solder, the value of the electrical contact resistancedecreases with applied pressure normal to the joint interface. This tendency is dueto an increase with pressure in the effective contact area between the two bulksamples. In fact, the two surfaces have microscale roughness due to how thesurfaces were prepared: as the pressure is increased normal to the surfaces, theasperities tend to mechanically yield and deform, increasing the effective area ofcontact. As the bulk material has high conductivity, the contact resistance ismostly due to the constriction of current or heat flow that occurs at the smallcontact points [127]. By increasing the contact pressure, the amount of constrictionfor current flow decreases, thus reducing the contact resistance. At very lowtemperatures, the aforementioned phenomenon should be investigated morethoroughly since mechanical stress in the contact zone may change the bulkproperties of materials (see, e.g., Ref. [128]).

A summary of the measured electrical contact resistivity for various unbondedsamples as a function of applied pressure can be found in Ref. [129].

Values of contact resistance can be obtained by, RB = qB/A, where A is theapparent contact area. Note that at a particular contact pressure, there is still a widevariation in the contact resistivity, a result that is probably due to variations insample preparation, treatment and oxidation. The contact resistance generallydecreases with applied pressure as

qB �np

ð7:38Þ

where the pressure is expressed in Pa and the resistivity in Xm2. The parameter n isexperimentally determined and is often about 3 104 [Kg m s-2 X] for metals [1],but can vary significantly for insulators [129].


For thermal contact resistance, there are two cases:

(a) the thermal contact resistance between metals, which is expected to correlatewith the electrical contact resistance as much as with bulk metals. Thiscorrelation is approximately correct for contacts between identical metals.This means that Wiedemann–Franz law must be used with great caution. Ifthe contact is between dissimilar metals or if there are solders or otherinterface metals involved, the thermal contact resistance can no longer bescaled with qB. This latter point is particularly significant at low temperatureswhere many alloys are superconducting (see Fig. 7.18);

(b) for thermal contact resistance between nonconducting materials, the funda-mental limit, even for ideal contacts, is the mismatch in the phonon transportacross the interface [7, 130]. Since the phonon spectra for the two types ofmaterials are not the same, there is an impedance mismatch that leads to aresistance occurring within roughly one phonon wavelength at the interface.

This effect is known as Kapitza conductance which initially referred to the heattransfer between liquid helium and metals (see, e.g., Ref. [7]). The theory ofKapitza conductance predicts [131] a Kapitza conductivity (jK)

jK � const � T3: ð7:39Þ

For most solids, const is on the order of 1 kW/m2 K4 [129]. Equation (7.39)puts an upper limit on the magnitude of the thermal contact conductance forinsulating contacts; real contacts between nonideal surfaces are more complex andtheir understanding is still qualitative.

For joints between real materials, the interface is irregular with random pointsof contact. In this case, the thermal contact conductance mostly depends on theconstriction resistance at the asperities similarly to the electrical contact resistance

Fig. 7.18 Thermal contactconductance as a function oftemperature for a variety ofcontact preparations andconditions. The contact areais assumed to be of 1 cm2.The apex s indicates a solder.Dashed lines are estimates ofthe conductance intemperature regions where nodata were available. Datawere taken from [132]


in metals. Thus, particularly for deformable materials, the thermal contact con-ductance increases with interface pressure.

It is important to pay attention because for metals at very low temperature, thethermal contact resistance may be higher than that of a low-resistivity thermalconnection [128].

Experimentally, the correlation between contact conductivity and pressure canbe written as

jK � w � pn ð7:40Þ

where n & 1 and w is an empirical coefficient [129].Thermal contact conductance varies over a wide range, depending on whether

the contact is insulating or conducting. Figure 7.17 shows data for low temperaturethermal contacts [132]. Some general features can be observed:

(a) the thermal contact conductance values at low temperatures can range oversix orders of magnitude, depending on materials and surface preparation;

(b) solder-bonded contacts,with solder of similar agents that fill the asperitiesgenerally have higher thermal conductance than bare contacts. However, thebonding agents can also contribute to the interface resistance, particularly ifthe bond region is thick. In the low temperature region (T \ 5 K), most of thedata agree with a power law, q * T-n, but there are two distinct charac-teristic behaviors:

(1) pure metal-metal contacts have a temperature dependence that correlateswith that of the bulk metal, hence, at low temperature q * T-1, with thecoefficient of proportionality being mainly determined by sample purityand contact pressure, but varying between 10-1 and 10-3 W/cm2K2;

(2) if the contact is bonded with solder or indium, the conductance can bemuch higher, but at low temperatures, such contacts may becomesuperconducting.

Finally, if the interface is between two nonconducting materials, the thermalconductance is generally lower, following the correlation scaling with the bulkthermal conductivity, q * T-n, where n * 7.

In addition to Fig. 7.18, Ref. [133] reports the experimental values of thethermal boundary resistance occurring at interfaces between two solids at su-bambient temperatures. Data are in the 4–300 K range and report the thermalresistance between different metals (Cu, stainless steel), interlayered by variouscryogenic bonding agents (Apiezon-N, Cryocon grease, In and InGa), ormechanically connected (dry) contacts.

In Ref. [134], the thermal contact conductance of several demountable copperjoints below 1 K is reported. Joints were made by bolting together either two goldflat surfaces or by a clamp around a rod. A linear dependence on temperature wasseen. Most of the measured conductance values fall into a narrow range:0.1–0.2 WK-1 at 1 K. Results in the literature for similar joints consist of


predictions based on electrical resistance measurements using the Wiedemann–Franz law. However, there is little evidence of the validity of this law in the case ofjoints. Nevertheless, the results are in agreement with the literature predictions,suggesting that such predictions are a reasonable approximation.

In Ref. [135], thermal conductance measurements of different types of boltedjoint at sub-Kelvin temperatures are presented. Joints containing sapphire surfacesprovided good thermal isolation in the 100 mK and 4 K temperature range. Thebest joint contained sapphire discs separated by diamond powder and had a con-ductance of 0.26 lWK-1. A mechanical support structure constructed from similarjoints, but using alumina powder, had a measured heat leak of 2.57 lW between80 mK and 1.1 K and was capable of supporting a mass of over 10 kg. Jointsbetween metal surfaces provided good thermal conduction; a bolted joint betweencopper and a beryllium-copper alloy (C17510 TF00) had a measured conductanceof 46 mW K-1 at 100 mK, increasing linearly with temperature. The paper alsoreports measurements made on a copper-copper compression joint using differ-ential thermal contraction to provide the clamping force: the performance is aboutan order of magnitude worse than for the bolted joint.

References

1. Van Sciver, S.W.: Helium Cryogenics. Springer, New York (2012)2. Tritt, T.M.: Thermal Conductivity: Theory, Properties, and Applications. Springer, New

York (2004)3. Ziman, J. (ed.): Electrons and Phonons. Clarendon Press, Oxford (1972)4. Rosenberg, H.M. (ed.): The Solid State. Clarendon Press, Oxford (1984)5. Ashcroft, N.W., Mermin, N.D.: Solid State Physics Holt. Rinehart and Winston, New York

(1976)6. Matthiessen, A., Vogt, C.: On the influence of temperature on the electric conducting-power

of alloys. Philos. Trans. R. Soc. Lond. 154, 167–200 (1864)7. Ventura, G., Risegari, L.: The Art of Cryogenics: Low-Temperature Experimental

Techniques. Elsevier, Amsterdam (2007)8. Olson, J.: Thermal conductivity of some common cryostat materials between 0.05 and 2 K.

Cryogenics 33(7), 729–731 (1993)9. DeGarmo, E.P., Black, J.T., Kohser, R.A., Klamecki, B.E.: Materials and Process in

Manufacturing. Macmillan Publishing Company, New York (1984)10. Moiseeva, N.: Methods of constructing an individual calibration characteristic for working

platinum resistance thermometers. Meas. Tech. 44(5), 502–507 (2001)11. Woodcraft, A.L.: Zirconium copper—a new material for use at low temperatures? In: AIP

Conference Proceedings 2006, p. 1691 (2006)12. Powell, R., Fickett, F.: Cryogenic properties of copper vol. 1. In: Proceedings of INCRA

REP (1979)13. Clark, A., Childs, G., Wallace, G.: Electrical resistivity of some engineering alloys at low

temperatures. Cryogenics 10(4), 295–305 (1970)14. Ledbetter, H., Reed, R., Clark, A.: Materials at Low Temperatures, vol. 1. American Society

for Metals, Metals Park, OH (1983)15. Meaden, G.T.: Electrical Resistance of Metals, vol. 2. Plenum press, New York (1965)16. Hall, L.: Survey of Electrical Resistivity Measurements on 16 Pure Metals in the

Temperature Range 0 to 273 K (1968). http://www.getcited.org/pub/101292840


http://www.getcited.org/pub/101292840

17. Matula, R.A.: Electrical resistivity of copper, gold, palladium, and silver. J. Phys. Chem.Ref. Data 8, 1147 (1979)

18. Haller, E.: Advanced far-infrared detectors. Infrared Phys. Technol. 35(2), 127–146 (1994)19. Andeen, C.G., Hagerling, C.W.: High Precision Capacitance Bridge. In. Andeen-Hagerling

Inc., Ohio (1988)20. Hayakawa, R., Tanabe, Y., Wada, Y.: A thermodynamic theory of mechanical relaxation

due to energy transfer between strain-sensitive and strain-insensitive modes in polymers.J. Macromol. Sci. Part B Phys. 8(3–4), 445–461 (1973)

21. Powles, J.: Nuclear magnetic resonance absorption in polymethyl methacrylate andpolymethyl a-chloroacrylate. J. Polym. Sci. 22(100), 79–93 (1956)

22. Odajima, A., Woodward, A., Sauer, J.: Proton magnetic resonance of some a-methyl group-containing polymers and their monomers. J. Polym. Sci. 55(161), 181–196 (1961)

23. Tanabe, Y., Hirose, J., Okano, K., Wada, Y.: Methyl group relaxations in the glassy phase ofpolymers. Polymer J 1, 107–115 (1970)

24. Armeniades, C., Baer, E.: Structural origin of the cryogenic relaxations in poly (ethyleneterephthalate). J. Polym. Sci. Part A-2: Polym. Phys. 9(8), 1345–1369 (1971)

25. Hiltner, A., Baer, E.: A dislocation mechanism for cryogenic relaxations in crystallinepolymers. Polym. J. 3(3), 378–388 (1972)

26. Arisawa, H., Yano, O., Wada, Y.: Dielectric loss of poly (vinylidene fluoride) at lowtemperatures and effect of poling on the low temperature loss. Ferroelectrics 32(1), 39–41(1981)

27. Yano, O., Wada, Y.: Dynamic mechanical and dielectric relaxations of polystyrene belowthe glass temperature. J. Polym. Sci. Part A-2: Polym. Phys. 9(4), 669–686 (1971)

28. Shimizu, K., Yano, O., Wada, Y.: Dielectric relaxations in polymers with pendent phenyl orpyridine groups at temperatures from 4 �K to 80 �K. J. Polym. Sci. Polym. Phys. Ed. 13(12),2357–2368 (1975)

29. Yano, O., Yamaoka, H.: Cryogenic properties of polymers. Prog. Polym. Sci. 20(4),585–613 (1995)

30. Barucci, M., Gottardi, E., Peroni, I., Ventura, G.: Low temperature thermal conductivity ofKapton and Upilex. Cryogenics 40(2), 145–147 (2000)

31. Berman, R. (ed.) Thermal Conduction in Solids. Oxford University Press, Oxford (1976)32. Franz, R., Wiedemann, G.: Ueber die Wärme-Leitungsfähigkeit der Metalle. Ann. Phys.

165(8), 497–531 (1853)33. Kittel, C. (ed.): Introduction to Solid State Physics, 8th edn. Wiley, New York (2005)34. Gloos, K., Mitschka, C., Pobell, F., Smeibidl, P.: Thermal conductivity of normal and

superconducting metals. Cryogenics 30(1), 14–18 (1990)35. Hust, J., Sparks, L.: Lorenz Ratios of Technically Important Metals and Alloys, vol. 634.

US Government printing office, Washington (1973)36. Pobell, F.: Matter and Methods at Low Temperatures. Springer, Berlin (2007)37. Callaway, J., Wang, C.: Energy bands in ferromagnetic iron. Phys. Rev. B 16(5), 2095

(1977)38. Woodcraft, A.L., Barucci, M., Hastings, P.R., Lolli, L., Martelli, V., Risegari, L., Ventura,

G.: Thermal conductivity measurements of pitch-bonded graphites at millikelvintemperatures: finding a replacement for AGOT graphite. Cryogenics 49(5), 159–164 (2009)

39. White, G., Meeson, P.: Experimental Techniques in Low-Temperature Physics. ClarendonPress, Oxford (2002)

40. Zaitlin, M.P., Anderson, A.: Phonon thermal transport in noncrystalline materials. Phys.Rev. B 12(10), 4475 (1975)

41. Anderson, P.W., Halperin, B., Varma, C.M.: Anomalous low-temperature thermalproperties of glasses and spin glasses. Phil. Mag. 25(1), 1–9 (1972)

42. Phillips, W.: Tunneling states in amorphous solids. J. Low Temp. Phys. 7(3–4), 351–360(1972)

43. Lubchenko, V., Wolynes, P.G.: Intrinsic quantum excitations of low temperature glasses.Phys. Rev. Lett. 87(19), 195901 (2001)

References 163

44. Lubchenko, V., Wolynes, P.G.: The origin of the boson peak and thermal conductivityplateau in low-temperature glasses. Proc. Natl. Acad. Sci. 100(4), 1515–1518 (2003)

45. Talon, C., Zou, Q., Ramos, M., Villar, R., Vieira, S.: Low-temperature specific heat andthermal conductivity of glycerol. Phys. Rev. B 65(1), 012203 (2001)

46. Parshin, D.: Interactions of soft atomic potentials and universality of low-temperatureproperties of glasses. Phys. Rev. B 49(14), 9400 (1994)

47. Buchenau, U., Galperin, Y.M., Gurevich, V., Parshin, D., Ramos, M., Schober, H.:Interaction of soft modes and sound waves in glasses. Phys. Rev. B 46(5), 2798 (1992)

48. Reese, W.: Thermal properties of polymers at low temperatures. J. Macromol. Sci. Chem.3(7), 1257–1295 (1969)

49. Choy, C., Greig, D.: The low temperature thermal conductivity of isotropic and orientedpolymers. J. Phys. C: Solid State Phys. 10(2), 169 (1977)

50. Pobell, F. (ed.) Matter and Methods at Low Temperature, 2nd edn. Springer, Berlin (1991)51. Cahill, D.G., Watson, S.K., Pohl, R.O.: Lower limit to the thermal conductivity of

disordered crystals. Phys. Rev. B 46(10), 6131 (1992)52. Roy, R., Komarneni, S., Parker, J., Thomas, G.: Nanophase and Nanocomposite Materials.

Mater. Res. Soci. 241 (1984)53. Feynman, R.P.: There’s plenty of room at the bottom. Eng. Sci. 23(5), 22–36 (1960)54. Advani, S.G.: Processing and Properties of Nanocomposites. World Scientific, Singapore

(2007)55. Green, C., Vaughan, A.: Nanodielectrics-How Much Do We Really Understand?[Feature

Article]. IEEE Electr. Insul. Mag. 24(4), 6–16 (2008)56. Tait, H.: 5 Thousand Years of Glass. University of Pennsylvania Press, Philadelphia (2004)57. Vaughan, A.: Raman nanotechnology-the Lycurgus Cup-letter to the editor. IEEE Electr.

Insul. Mag. 24(6), 4 (2008)58. Andritsch, T.: Epoxy based nanocomposites for high voltage DC applications. Synthesis,

dielectric properties and space charge dynamics. PhD thesis, Delft University ofTechnology (2010)

59. Gupta, R.K., Kennel, E., Kim, K.-J.: Polymer Nanocomposites Handbook. CRC Press(2010)

60. Mark, J., Wen, J.: Inorganic-organic composites containing mixed-oxide phases. In:Macromolecular Symposia 1995, pp. 89–96. Wiley Online Library (1995)

61. Carter, L.W., Hendricks, J.G., Bolley, D.S.: U.S. Patent 2.531.396. USA Patent62. Iijima, S.: Helical microtubules of graphitic carbon. Nature 354(6348), 56–58 (1991)63. Tans, S.J., Devoret, M.H., Dai, H.J., Thess, A., Smalley, R.E., Geerligs, L.J., Dekker, C.:

Individual single-wall carbon nanotubes as quantum wires. Nature 386(6624), 474–477(1997)

64. Twardowski, T.E., Twardowski, T.A.: Introduction to Nanocomposite Materials: Properties,Processing, Characterization. DEStech publications, Inc U.S.A (2007)

65. Liao, J., Ren, Y., Xiao, T., Mai, Y., Yu, Z.: Polymer Nanocomposites. WoodheadPublishing Limited, Cambridge (2006)

66. Pissis, P., Kotsilkova, R.: Thermoset Nanocomposites for Engineering Applications. RapraTechnology, UK (2007)

67. Kango, S., Kalia, S., Celli, A., Njuguna, J., Habibi, Y., Kumar, R.: Surface modification ofinorganic nanoparticles for development of organic-inorganic nanocomposites–a review.Prog. Polym, Sci (2013)

68. Kurimoto, M., Watanabe, H., Kato, K., Hanai, M., Hoshina, Y., Takei, M., Okubo, H.:Dielectric properties of epoxy/alumina nanocomposite influenced by particle dispersibility.In: Electrical Insulation and Dielectric Phenomena, 2008. CEIDP 2008. IEEE AnnualReport Conference on 2008, pp. 706–709 (2008)

69. Nelson, J.K.: Dielectric Polymer Nanocomposites. Springer, New York (2010). http://link.springer.com/book/10.1007/978-1-4419-1591-7


http://link.springer.com/book/10.1007/978-1-4419-1591-7


70. Singha, S., Thomas, M.J.: Polymer composite/nanocomposite processing and its effect onthe electrical properties. In: Electrical Insulation and Dielectric Phenomena, 2006 IEEEConference on 2006, pp. 557–560 (2006)

71. Yung, K., Wang, J., Yue, T.: Thermal management for boron nitride filled metal coreprinted circuit board. J. Compos. Mater. 42(24), 2615–2627 (2008)

72. Levering, A.W.: Interphases in Zirconium Silicate Filled High Density Polyethylene andPolypropylene (1995)

73. Schulz, M.J., Kelkar, A.D., Sundaresan, M.J.: Nanoengineering of Structural, Functionaland Smart Materials. CRC Press, Boca Raton (2005)

74. Capek, I.: Nanocomposite Structures and Dispersions, vol. 27. Access Online via Elsevier(2006)

75. Roy, M., Nelson, J., MacCrone, R., Schadler, L., Reed, C., Keefe, R.: Polymer nanocompositedielectrics-the role of the interface. IEEE Trans. Dielectr. Electr. Insul. 12(4), 629–643(2005). http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1511089&tag=1

76. Smith, R., Liang, C., Landry, M., Nelson, J., Schadler, L.: The mechanisms leading to theuseful electrical properties of polymer nanodielectrics. IEEE Trans. Dielectr. Electr. Insul.15(1), 187–196 (2008)

77. Xanthos, M.: Polymers and Polymer Composites. Functional Fillers for Plastics, pp. 1–16.Wiley, New York (2005)

78. Godovsky, Y.K.: Thermodynamic Behavior of Solid Polymers in Plastic Deformation andCold Drawing. Springer, Berlin (1992)

79. Tekce, H.S., Kumlutas, D., Tavman, I.H.: Effect of particle shape on thermal conductivity ofcopper reinforced polymer composites. J. Reinf. Plast. Compos. 26(1), 113–121 (2007)

80. Hansen, D., Bernier, G.: Thermal conductivity of polyethylene: the effects of crystal size,density and orientation on the thermal conductivity. Polym. Eng. Sci. 12(3), 204–208 (1972)

81. Mark, J.E.: Physical Properties of Polymers Handbook. Springer, Berlin (2007). http://link.springer.com/book/10.1007/978-0-387-69002-5

82. Kesava Reddy, M., Subramanyam Reddy, K., Yoga, K., Prakash, M., Narasimhaswamy, T.,Mandal, A., Lobo, N.P., Ramanathan, K., Rao, D.S., Krishna Prasad, S.: Structuralcharacterization and molecular order of rodlike mesogens with three-and four-ring core byXRD and 13C NMR spectroscopy. J. Phys. Chem. B 117(18), 5718–5729 (2013)

83. Ekstrand, L., Kristiansen, H., Liu, J.: Characterization of thermally conductive epoxy nanocomposites. In: Electronics Technology: Meeting the Challenges of Electronics TechnologyProgress, 2005. 28th International Spring Seminar on 2005, pp. 35–39 (2005)

84. Kim, W., Bae, J.W., Choi, I.D., Kim, Y.S.: Thermally conductive EMC (Epoxy MoldingCompound) for microelectronic encapsulation. Polym. Eng. Sci. 39(4), 756–766 (1999)

85. Hsieh, C.Y., Chung, S.L.: High thermal conductivity epoxy molding compound filled with acombustion synthesized AlN powder. J. Appl. Polym. Sci. 102(5), 4734–4740 (2006)

86. Wong, C., Bollampally, R.S.: Comparative study of thermally conductive fillers for use inliquid encapsulants for electronic packaging. IEEE Trans. Adv. Packag. 22(1), 54–59 (1999)

87. Okamoto, T., Sawa, F., Tomimura, T., Tanimoto, N., Hishida, M., Nakamura, S.: Propertiesof high-thermal conductive composite with two kinds of fillers. In: Properties andApplications of Dielectric Materials, 2007. Proceedings of the 7th International Conferenceon 2003, pp. 1142–1145 (2007)

88. Xu, Y., Chung, D.: Increasing the thermal conductivity of boron nitride and aluminumnitride particle epoxy-matrix composites by particle surface treatments. Compos. Interfaces7(4), 243–256 (2000)

89. Han, Z., Wood, J., Herman, H., Zhang, C., Stevens, G.: Thermal properties of compositesfilled with different fillers. In: Electrical Insulation, 2008. ISEI 2008. Conference Record ofthe 2008 IEEE International Symposium on 2008, pp. 497–501 (2008)

90. McCullough, R.L.: Generalized combining rules for predicting transport properties ofcomposite materials. Compos. Sci. Technol. 22(1), 3–21 (1985)

91. Progelhof, R., Throne, J., Ruetsch, R.: Methods for predicting the thermal conductivity ofcomposite systems: a review. Polym. Eng. Sci. 16(9), 615–625 (1976)

References 165

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1511089&tag=1



92. Tavman, I.: Effective thermal conductivity of isotropic polymer composites. Int. Commun.Heat Mass Transf. 25(5), 723–732 (1998)

93. Agarwal, S., Khan, M.M.K., Gupta, R.K.: Thermal conductivity of polymer nanocompositesmade with carbon nanofibers. Polym. Eng. Sci. 48(12), 2474–2481 (2008)

94. Maxwell, J.C.: A Treatise on Electricity and Magnetism, vol. 1. Clarendon Press, Oxford(1881)

95. Pal, R.: On the Lewis-Nielsen model for thermal/electrical conductivity of composites.Compos. A Appl. Sci. Manuf. 39(5), 718–726 (2008)

96. Fricke, H.: A mathematical treatment of the electric conductivity and capacity of dispersesystems I. The electric conductivity of a suspension of homogeneous spheroids. Phys. Rev.24(5), 575 (1924)

97. Bruggeman, V.D.: Berechnung verschiedener physikalischer Konstanten von heterogenenSubstanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropenSubstanzen. Ann. Phys. 416(7), 636–664 (1935)

98. Lee, E.S., Lee, S.M., Shanefield, D.J., Cannon, W.R.: Enhanced thermal conductivity ofpolymer matrix composite via high solids loading of aluminum nitride in epoxy resin.J. Am. Ceram. Soc. 91(4), 1169–1174 (2008)

99. Hill, R.F., Supancic, P.H.: Thermal conductivity of platelet-filled polymer composites.J. Am. Ceram. Soc. 85(4), 851–857 (2002)

100. Stevens, G., Herman, H., Han, J., Wood, J., Mitchell, A., Thomas, J.: The role of nano andmicro fillers in high thermal conductivity electrical insulation systems. In: 11th InsuconConference, Birmingham, UK 2009, pp. 286–291

101. Tsao, G.T.-N.: Thermal conductivity of two-phase materials. Ind. Eng. Chem. 53(5),395–397 (1961)

102. Cheng, S., Vachon, R.: The prediction of the thermal conductivity of two and three phasesolid heterogeneous mixtures. Int. J. Heat Mass Transf. 12(3), 249–264 (1969)

103. Sundstrom, D.W., Lee, Y.D.: Thermal conductivity of polymers filled with particulatesolids. J. Appl. Polym. Sci. 16(12), 3159–3167 (1972)

104. Hamilton, R.: Thermal conductivity of two phase materials. Dissertation, University ofOklahoma (1960)

105. Hamilton, R., Crosser, O.: Thermal conductivity of heterogeneous two-component systems.Ind. Eng. Chem. Fundam. 1(3), 187–191 (1962)

106. Hatta, H., Taya, M.: Effective thermal conductivity of a misoriented short fiber composite.J. Appl. Phys. 58(7), 2478–2486 (1985)

107. Meredith, R.E., Tobias, C.W.: Conduction in heterogeneous systems. Advances inelectrochemistry and electrochemical engineering 2(II), 15–47 (1962)

108. Nielsen, L.E.: Mechanical properties of particulate-filled systems. J. Compos. Mater. 1(1),100–119 (1967)

109. Lewis, T., Nielsen, L.: Dynamic mechanical properties of particulate-filled composites.J. Appl. Polym. Sci. 14(6), 1449–1471 (1970)

110. Landel, R.F.: Mechanical Properties of Polymers and Composites, vol. 90. CRC Press,(1994)

111. Halpin, J.: Stiffness and expansion estimates for oriented short fiber composites. J. Compos.Mater. 3(4), 732–734 (1969)

112. Agari, Y., Uno, T.: Thermal conductivity of polymer filled with carbon materials: effect ofconductive particle chains on thermal conductivity. J. Appl. Polym. Sci. 30(5), 2225–2235(1985)

113. Agari, Y., Uno, T.: Estimation on thermal conductivities of filled polymers. J. Appl. Polym.Sci. 32(7), 5705–5712 (1986)

114. Agari, Y., Ueda, A., Nagai, S.: Thermal conductivity of a polymer composite. J. Appl.Polym. Sci. 49(9), 1625–1634 (1993)

115. Russell, H.: Principles of heat flow in porous insulators*. J. Am. Ceram. Soc. 18(1–12), 1–5(1935)


116. Topper, L.: Industrial design data—analysis of porous thermal insulating materials. Ind.Eng. Chem. 47(7), 1377–1379 (1955)

117. Jefferson, T., Witzell, O., Sibbitt, W.: Thermal conductivity of graphite—silicone oil andgraphite-water suspensions. Ind. Eng. Chem. 50(10), 1589–1592 (1958)

118. Springer, G.S., Tsai, S.W.: Thermal conductivities of unidirectional materials. J. Compos.Mater. 1(2), 166–173 (1967)

119. Budiansky, B.: Thermal and thermoelastic properties of isotropic composites. J. Compos.Mater. 4(3), 286–295 (1970)

120. Baschirow, A., Selenew, J.: Thermal conductivity of composites. Plaste Kaut 23, 656 (1976)121. McGee, S., McGullough, R.: Combining rules for predicting the thermoelastic properties of

particulate filled polymers, polymers, polyblends, and foams. Polym. Compos. 2(4),149–161 (1981)

122. Privalko, V., Novikov, V.: Model treatments of the heat conductivity of heterogeneouspolymers. In: Thermal and Electrical Conductivity of Polymer Materials, pp. 31–77.Springer (1995)

123. Dul’Nev, G., Novikov, V.: Conductivity determination for a filled heterogeneous system.J. Eng. Phys. Thermophys. 37(4), 1184–1187 (1979)

124. Heron, J.-S., Souche, G.M., Ong, F.R., Gandit, P., Fournier, T., Bourgeois, O.: Temperaturemodulation measurements of the thermal properties of nanosystems at low temperatures.J. Low Temp. Phys. 154(5–6), 150–160 (2009)

125. Huang, C., Fu, S., Zhang, Y., Lauke, B., Li, L., Ye, L.: Cryogenic properties of SiO2/epoxynanocomposites. Cryogenics 45(6), 450–454 (2005)

126. Martelli, V., Toccafondi, N., Ventura, G.: Low-temperature thermal conductivity of Nylon-6/Cu nanoparticles. Physica B 405(20), 4247–4249 (2010)

127. Batchelor, G., O’Brien, R.: Thermal or electrical conduction through a granular material.Proc. R. Soc. Lond. A Math. Phys. Sci. 355(1682), 313–333 (1977)

128. Risegari, L., Barucci, M., Bucci, C., Fafone, V., Gorla, P., Giuliani, A., Olivieri, E., Pasca,E., Pirro, S., Quintieri, L.: Use of good copper for the optimization of the cooling downprocedure of large masses. Cryogenics 44(3), 167–170 (2004)

129. Van Sciver, S.W., Nellis, M.N., Pfotenhauer, J.: Thermal and electrical contactConductance between metals at low temperatures. In: Proceedings Space CryogenicsWorkshop 1984, Berlin (DE) (1984)

130. Peterson, R., Anderson, A.: The Kapitza thermal boundary resistance. J. Low Temp. Phys.11(5–6), 639–665 (1973)

131. Little, W.: The transport of heat between dissimilar solids at low temperatures. Can. J. Phys.37(3), 334–349 (1959)

132. Radebaugh, R.: Thermal conductance of indium solder joints at low temperatures. Rev. Sci.Instrum. 48, 93 (1977)

133. Gmelin, E., Asen-Palmer, M., Reuther, M., Villar, R.: Thermal boundary resistance ofmechanical contacts between solids at sub-ambient temperatures. J. Phys. D Appl. Phys.32(6), R19 (1999)

134. Didschuns, I., Woodcraft, A., Bintley, D., Hargrave, P.: Thermal conductancemeasurements of bolted copper to copper joints at sub-Kelvin temperatures. Cryogenics44(5), 293–299 (2004)

135. Bintley, D., Woodcraft, A.L., Gannaway, F.C.: Millikelvin thermal conductancemeasurements of compact rigid thermal isolation joints using sapphire–sapphire contacts,and of copper and beryllium–copper demountable thermal contacts. Cryogenics 47(5),333–342 (2007)

136. Fritzsche, H.: Resistivity and hall coefficient of antimony-doped germanium at lowtemperatures. J. Phys. Chem. Solids 6(1), 69–80 (1958)

137. Rosenberg, H.M.: The thermal conductivity of metals at low temperatures. Philos. Trans. ofthe R. Soc. A Math. Phys. Sci. 247(933), 441–497 (1955)

138. Berman, R., Foster, E., Ziman, J.: The thermal conductivity of dielectric crystals: the effectof isotopes. Proc. R. Soc. Lond. A 237(1210), 344–354 (1956)

References 167

139. Nathan, B., Lou, L., Tait, R.: Low temperature thermal properties of mixed crystal KBr KI.Solid State Commun. 19(7), 615–617 (1976)

140. Guckel, H.: Silicon microsensors: construction, design and performance. Microelectron.Eng. 15(1), 387–398 (1991)

141. Locatelli, M., Arnaud, D., Routin, M.: Thermal conductivity of some insulating materialsmaterials below 1 K. Cryogenics 16(6), 374–375 (1976)

142. Morelli, D., Doll, G., Heremans, J., Peacor, S., Uher, C., Dresselhaus, M., Cassanho, A.,Gabbe, D., Jenssen, H.: Thermal conductivity of single crystal lanthanum cuprates at verylow temperature. Solid State Commun. 77(10), 773–776 (1991)

143. Stephens, R.: Low-temperature specific heat and thermal conductivity of noncrystallinedielectric solids. Phys. Rev. B 8(6), 2896 (1973)

144. Fukushima, K., Takahashi, H., Takezawa, Y., Hattori, M., Itoh, M., Yonekura, M.: Highthermal conductive epoxy resins with controlled high-order structure [electrical insulationapplications]. In: Electrical Insulation and Dielectric Phenomena, 2004. CEIDP’04. 2004Annual Report Conference on 2004, pp. 340–347 (2004)

145. Scott, T.A., de Bruin, J., Giles, M.M., Terry, C.: Low-temperature thermal properties ofnylon and polyethylene. J. Appl. Phys. 44(3), 1212–1216 (1973)

146. Ekin, J. (ed.) Experimental Techniques for Low Temperature Measurements. OxfordUniversity Press, Oxford (2006)


本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/

http://www.xuebalib.com/

http://www.xuebalib.com/cloud/


http://www.xuebalib.com/vip.html

http://www.xuebalib.com/db.php

http://www.xuebalib.com/zixun/2014-08-15/44.html


electrical and thermal conductivity

Documents