IDGHLY ANISOTROPIC CRYSTALS l. Introduction

We usually associate the most extreme cases of anisotropy with easily cleaved crystals such as mica and graphite, or with the fibrous chain-like crystals of asbestos and some polymers. Their anisotropy arises from having weak Van der Waals forces between planes (or chains) and stronger covalent bonding within planes (or chains).

Generally this anisotropy shows up as high compressibility in one direction and low compressibility in the other. For example, in layered crystals of axial symmetry such as graphite XII

»

^X.l; while for chain-like crystals such as Te XII «X.l. Indeed, for Te and Se at room temperature XII has a small negative value; under hydrostatic pressure the spiral chains decrease in radius and expand slightly in length. Another measure is the ratio of the elastic compliances along and perpendicular to the symmetry axis,

$33/$11, which is about 28 for graphite and 0.6 for Te.

The vibrations that depend primarily on the weak force-constants have much lower frequencies than those that are governed by the strong force constants, and at low temperatures they are excited preferentially and dominate the heat capacity and thermal expansion, with low values of

e

^{C .} Above the low temperature region

e

^{C (T)}

increases with T until the high frequency vibrations become excited at much higher temperatures. Likewise changes in the Griineisen 'Y occur, because the uniaxial strain dependences of low and high frequency modes are very different and may differ also

Non-Metals 189

with orientation.

5.8.2. Layer Structures

A simple layer structure has strong elastic anisotropy like that of graphite, with only small cross compliance between the

II

^and -L directions. The low frequency modes are polarized roughly perpendicular to the layers and the high frequency modes roughly perpendicular to the axis. The high frequencies are therefore weak-ened by stretching the crystal perpendicular to the axis but scarcely affected by stretching along the axis, so that 'Y.l is large and positive and 'YII is small. In contrast, the lower frequencies are weakened by stretching along the axis (which reduces the restoring forces for motion in this direction) and are strengthened by stretching perpendicular to the axis (which increases the restoring force perpen-dicular to the layers because of the tension between neighbors), so that for these modes 'Y.l is negative and 'YII large and positive. At low temperatures when only low frequency modes are excited there is a relatively large expansion in the soft direction along the axis, and a small contraction within the layers. As T increases the excitation of the high frequency modes will have little effect on the expansion along the axis, but their large 'Y.l will drive the expansion within the layers positive, though it remains small because of the high stiffness in the layers. A theoretical example is provided by one form of the rhombohedral model discussed in Section 2.6.3.

Graphite. In graphite, each carbon atom is bonded to three other carbons forming a network of planar hexagonal rings. The distance between carbon atoms in a plane is 0.142 nm (1.42 A), and planes are about 0.33 om (3.3 A) apart. It is a good electronic conductor within the planes, giving an electronic fT contribution to the heat capacity at low temperatures. The weak bonding between planes results in a compressibility at room temperature of XII = 27 x 10-¹² Pa -1, compared with X.l = 0.5 x 10-12 . This layer structure led to an early theoretical prediction that the vibrational heat capacity at low temperatures might vary as T2 rather than T3, and indeed the first definitive measurements on a natural crystal from 13 to 300 K [DeS53] showed Cp ex: T2 from 13 to 50 K. But at low temperatures the weak interlayer forces become important, and later measurements [Van63] extending down to 0.4 K on a similar natural crystal gave Cp = 13.8T+27.7T³,...J·g-acl .

K-¹ below 1.2 K, confirming that in the low temperature or long wave limit the Debye T3 law applies with

eo ::::::

413 K. These and other measurements of Cp on various graphites (Canadian and Madagascar natural graphites, pile graphite, pyrolitic graphite, etc.) showed that the values obtained below 20 K were highly dependent on crystallite size, stacking faults, etc., with the natural crystals giving the lowest values of heat capacity (Fig. 5.24). Data in [Tou70b] suggests that at higher temperatures, above say 50 K, differences are small. At 300 K, Cp

=

8.58 J.g_ac¹.K-¹ [DeS58].

190 Cbapter 5

3.0

2.5

"j 2.0

"j ~

Qj _I 1.5

~

bO

U 1.0

0.5

0 0 2 3 4 5

T(K)

Fig. 5.24. Cp(T) for representative samples of graphite below 5 K [OeS53, OeS58]. -(top curve):

graphitized lampblack, crystallites", 100m - -(middle): pile graphite, crystallites", 25 om - - - -(bottom): natural crystals ~ 100,... m

Note that the low temperature values of

e

C ('"

eo)

are not a good guide to the heat capacity of graphite (or boron nitride) at higher temperatures; the high frequency modes do not become excited and increase Cp until much higher tem-peratures than predicted by the Debye model. For example, measured values of Cp at about 100 K and 300 K respectively correspond to Debye temperatures of 1050 K and 1500 K.

The thermal expansion of a highly oriented pyrolytic graphite sample from 30 to 270 K [Bai70] is shown in Fig. 5.25 and Table 5.S. A sample of hexagonal BN (of graphite structure) showed similar and even greater anisotropy, see [BarSO, p. 676].

There are very many high temperature expansion measurements on graphites from many sources (not single crystal) because of its importance as a high temperature refractory, but relatively few at low temperatures [Tou77].

Bailey and Yates [Bai70] calculate approximate values for the principal Griineisen parameters with accuracy limited by lack of low temperature elastic moduli. They find 'Yl.. ~ -5 at lowest temperatures rising to ~ -1 at 270 K and becoming positive at higher temperatures. In the soft direction, 'Y~ ~ 3 and 'YfO ~ 1.

Values of a and !l.1/1293 = (1293 -

iT

)/h93 are given in the American Institute of Physics Handbook [Kir72] for graphite. Values of a (in units of 1O-6K-1) may

Non-Metals

be compared with those for 'glassy' carbon GC-20 (crystallites::; 10 nm):

T 293 K 200K lOOK anisotropy of both the elasticity and the Griineisen functions combine to give large positive expansion perpendicular to the planes and small negative expansion in the planes. But this is not necessarily true of crystals with more complex layers. Thus arsenic has Griineisen functions which are almost isotropic, and the anisotropy of its thermal expansion stems largely from its elasticity (see Section 6.3.1). And InBi, with compound layers consisting of Bie.slnBie.s sandwiches, has reversed anisotropy in thermal expansion, with positive expansion in ab directions and con-traction perpendicular to the planes (see Section 8.4.6).

192 ChapterS Table S.S. Data for some layer and chain-like crystals. Sources are

reviews [BarSO, Tou77]

Material 80 ^{Bo a293} 1- ^a293 II ^{'Y 1-,293} 'YiI,293 'Y1-,O 'Yil,o (K) (GPa) (1O-61K) (l0-6/K)

Graphite 413 36 -1.3 27 -1 -5 3

BN 485 -2.8 38

Se 160 12.5 69 -13 1.5 -1.6 0.85 1.0

±1O approx

Te 152 22 29.5 -2.3 1.8 -0.7 1.0 1.1

POM 271 11 75 2.4 1.35 -1.2 0.7 0.7

5.8.3. Chain Structures

Selenium. Se crystallizes in the trigonal system in the form of spiral chains of atoms arranged in a hexagonal array, so that each chain has six neighbors: the space group is Dl (D3121) or D~ (D3231). There are also amorphous and monoclinic forms as for sulphur.

Low temperature values of heat capacity for trigonal Se show some variation depending on the sample preparation. Values for Cp/T3 below 10 K lie within about 10% of 0.5 mJ·mol-¹·K-4 , corresponding to eo ~ 160 K [Mei7S, Las69]. Their values of eo for monoclinic Se are about 12S K. The thermal expansion data from 10 to 300 K [Gr07S] show the expected large anisotropy, with a.l

=

69.8 x 10-⁶ K-¹ and all = -13.4 x 10-⁶ K-¹ at 300 K. Other data are given in Table 5.8.

Tellurium. Te crystallizes in the same trigonal structure of spiral chains as See There are more consistent thermal data for Te than for See The heat capacity, measured from 1.5 to 20 K [Lea73] and above 14 K [Sla39b], gives a

e

^C (T) curve similar in shape to those of other crystals represented in Figs. 5.4 and 5.7: eo = 152 K, e~in = 131 Kat T = 10 K (eo/IS) and e~ ~ ISO K. This contrasts with the shape of the

e

^C (T) curves for graphite and BN mentioned above.

The thermal expansion of single crystals determined down to 2 K (Fig. 5.26, refs.

in [BarSO, p. 691]) shows anisotropy but less marked than for selenium: at 293 K, a.l

=

29.6 x 10-⁶ K-¹ and all

=

-2.3 x 10-⁶ K-1• Principal 'Y values are included in Table 5.S. A theoretical model indicates that the anisotropy may be reduced by radial contraction of the spiral chains with increasing temperature [Gib73].

5.9. POLYMERS 5.9.1. Introduction

Polymers are widely used in cryogenic applications, particularly when fiber re-inforced, because of their strength/weight ratio and nonmagnetic nature. They are generally in the amorphous (glassy) or semicrystalline condition. If the crystallites

Non-Metals ₁₉₃

20r---_r----~--_r----~--_r--~~~_r----~

10 2

- ^-

^I _~

-

_~

oQ

0

0 ~--~~---~---,~---~O

~I~,'

')r.. ^~ --

^_I

/ " ' 2 10 2050100

all

~" ~(K)

~~X_~

-10 L--..J....----l--....I..----I:--r.--6 40

o

^{10 20T} (K) 30

Fig. 5.26. a(T) for Te with y(T) in inset. From [BarSO, p. 692].

194 ChapterS

are oriented by the growth process or by drawing, the properties are then anisotropic.

In the fully crystallized state, which is not usually achieved with bulk samples, the polymers consist of chains (linear, planar zigzag, helical, etc.); the crystals can have axial, orthorhombic or lower symmetry. Whatever the structure, the covalent in-trachain linkage (i.e., within chains) is much stronger than the interchain linkage (i.e., between chains), so that we may expect the lattice dynamics of crystalline poly-mers to resemble that of tellurium or selenium, with the added complication of side groups of atoms. In some polymers such as epoxy resins and vulcanized rubbers the interchain linkage is strengthened by cross bonding. At low temperatures, the thermal properties are largely determined by low frequency modes governed by in-terchain forces and weak intrachain torsional forces; at higher temperatures the high frequency intrachain modes become important. However, polymers normally melt before the highest frequencies are excited. *

For both isotropic and oriented samples, the heat capacity and thermal expan-sion depend on the degree of crystallinity, i.e., the fraction of the material that is crystalline rather than amorphous, and also on whether the samples are isotropic or have crystallites and/or chains oriented preferentially (texture). We shall discuss first the experimental behavior of the few examples available of single or quasi-single crystal specimens, before examining the properties of the partially crystalline and amorphous materials.

Data on heat capacity have been reviewed and tabulated in a series of papers by Gaur et al. [Gau81] in the Journal of Physical and Chemical Reference Data. A Cryogenic Monograph by Hartwig [HarJ4] entitled Polymer Properties at Room and Cryogenic Temperatures includes a review of heat capacity and thermal expansion, as well as dielectric and elastic behavior and thermal conductivity. Hartwig tabulates values for linear thermal expansion and heat capacity of many common polymers.

His values for Cp (per gram) correlate closely with Cp values (per mole) given by Gaur et al. [Gau81]. Hartwig does not include data for the powder or fiber-filled polymer composites. Some of the latter are included in the chapters by Clark and others in Materials at Low Temperatures [Cla83]. Other good sources of thermal data on polymers and composites are the proceedings of the International Cryogenic Materials Conferences published by Plenum Press as Advances in Cryogenic Engi-neering Materials and specialist ICMC conferences on Nonmetallic Materials and Composites at Low Temperatures [Cla79, Har82, Har88b, Oka95a}.

Table 5.9 gives selected values for cp and tll/l293 = (1293 -l4)/1293 for the following polymers in amorphous (a), crystalline (c) or semicrystalline (sc) state [Gau81, Har94]:

• POM (polyoxymethylene ... (CH20)n)

• PE (polyethylene ... (CH2)n)

• PTFE (polytetraftuorethylene ... (CF2)n)

'Vulcanized rubbers continue to be solids above the glass transition region and exhibit interesting thenna) expansion [Bar98]. but this does not usually occur at cryogenic temperatures.

In document at Low Temperatures (pagina 194-200)