Kutz M. Handbook of materials selection

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638 CERAMICS TESTING

materials expand when heated and contract when cooled. This characteristic is

valuable to the engineer or scientist involved in ceramic design or research. The

coefﬁcient of thermal expansion (COTE),

␣, is strongly related to the strength

of the atomic bonds. Energy must be put into the material for the atoms to move

from their equilibrium positions. Typically, ceramics, which have strong ionic

or covalent bonds, have lower thermal expansion coefﬁcients than metals. The

average COTE is simply the change in the length of the material per unit length

per unit temperature:

⌬L

␣

⫽

L ⌬T

The volume COTE,

␣

, is the change in volume of the material for a given

temperature. Consequently, for an isotropic material the volume COT is given

as:

␣

⫽ 3

␣

Thermal expansion is a tensor property that is different along individual crys-

tallographic axes.

This varying expansion can cause residual stresses in some

test specimens that can actually cause cracks between the crystal faces in some

rare situations. This thermal cycling cracking will manifest itself in the form of

a hysteresis curve from expansion data during thermal cycling tests.

Typically, the COTE is lower for materials having a high melting point. The

linear COTE is a more exact deﬁnition and continuously changes with regard to

temperature. The COTE for materials is usually listed in handbooks as a com-

plicated temperature-dependent function or is shown as a constant that is valid

only for a speciﬁed temperature range. Taking into account the temperature

dependence of the linear COTE, experimental measuring gives the more exact

equation

⭸L(T)1

␣

(T ) ⫽

冉冊

⭸TL(T)

⫽

where L(T

) is the experimentally measured length of the specimen at a given

temperature T

. The difference in the linear COTE and the average COTE is

shown graphically in Fig. 10.

Understanding the thermal expansion of a given ceramic is especially impor-

tant in design applications such as composites that combine structurally distinct

constituents (ﬁber, matrix, etc.) of different materials (ceramic, metal, etc.) with

different COTE. For example, a ceramic with a low COTE combined with a

metal possessing a high COTE can result in critical stresses for a given temper-

ature change.

Dilatometry

The most popular method for measuring thermal expansion is dilatometry.

The

test setup is simple, consisting of a cylindrical test specimen placed between a

3 THERMAL TESTING 639

Fig. 10 Difference in linear COTE and average COTE. (From Ref. 15 Engineered Materials

Handbook, Vol. 4: Ceramics and Glasses, ASM International,

Materials Park, OH 44073-0002, Fig. 1, page 612.)

ﬁxed base and a movable push rod. The ceramic test specimen is heated at a

ﬁxed rate, which displaces the push rod a given distance that is recorded by a

sensing device. Software is readily available to record the data and determine

the temperature-dependent COTE and the average COTE record.

Dilatometers can be of the single or double push rod variety. The single push

rod method involves ﬁrst calibrating the test apparatus in terms of the change

in length versus temperature-sensing device with a test specimen possessing a

known COTE. Next, the ceramic test specimen is tested compared to the pre-

viously measured standard to determine its COTE. Calibrating the test apparatus

with a standard reference specimen ﬁrst in this manner can correct for problems

such as dilatometer material expansion, thermal gradients, and nonlinearity in

the heating rate.

The double push rod design tests the ceramic test specimen

and the standard reference material side by side. The thermal expansion data is

then calibrated by comparing it with the standard reference material with a

known COTE.

Other Techniques

X-ray diffraction can give insight into the structure of ceramics. The tensor

components of thermal expansion can be determined with X-ray diffraction

through the measuring of changes of the interplanar spacing with relation to the

temperature. One distinct advantage with this method is that only a very small

640 CERAMICS TESTING

sample of the ceramic is required. Another advantage is that measuring the tensor

components of thermal expansion of a specimen is possible with X-ray diffrac-

tion. This can be important when trying to estimate residual stresses that will

occur during thermal cycling.

Interferometry is a technique that involves placing the test specimen between

two reﬂecting surfaces and measuring the displacement through the movement

of the surfaces. The basis for this being that parallel reﬂective surfaces a short

distance apart will show interference fringes when illuminated by monochro-

matic light. As the reﬂective surfaces move apart from the specimen expansion,

the interference fringes move past a reference point on one of the reﬂecting

planes. The expansion of the sample in terms of length can be expressed as

⌬L

␭

⫽⫹

L 2LL

where N is the number of interference fringes that pass the reference point, L is

the specimen length, ⌬L is the change in the specimen length, ␭ is the wave-

length of the light source, and A is the correction for the light source based on

the atmosphere in the measuring environment. Restrictions on the test specimen

geometry make this technique less popular than dilatometry methods. One ad-

vantage to this technique, however, is that ceramics with volatile components

can be tested by interferometry inside a closed, heated test chamber where the

volatile component’s vapor pressure can be controlled.

3.2 Thermal Conductivity

Thermal conductivity is a measure of the rate of heat transfer in a given material

by conduction. The heat ﬂowrate through a material is proportional to the heated

area of the material and the temperature gradient across the specimen. This

proportionality yields the coefﬁcient of thermal conductivity,

␬

, which is shown

in the following equation

dQ dT

⫽⫺

␬

dt dx

The negative sign on the right side indicates that heat ﬂows from a higher to a

lower temperature. The rate of heat ﬂow is dQ/dt, A is the cross-sectional area

of the material, and dT/dx is the temperature gradient.

Two important mechanisms involved in transferring thermal energy are lattice

vibrations (or phonons) and transfer of free electrons. Valence electrons travel

from hot to cold areas with the energy gained, and then transfer their energy to

other atoms. Many electrons in ceramics cannot be excited into the conduction

band except at quite high temperatures because the energy gap is typically too

large.

Consequently, heat transfer in ceramics is primarily a result of lattice

vibrations. This results in ceramics typically having a much lower thermal con-

ductivity than metals. However, not all ceramics have low thermal conductivity.

Advanced ceramics such as AlN and SiC have good thermal conductivity and

low electrical conductivity. Consequently, they are good for electronic appli-

3 THERMAL TESTING 641

Fig. 11 Guarded hot plate test setup. (From Ref. 15 Engineered Materials Handbook, Vol. 4:

Ceramics and Glasses, ASM International, Materials Park, OH 44073-0002, Fig. 2, page 613.)

cations where dissipating heat is a requirement. Thermal conductivity is an

important aspect of materials for microelectronic substrates and electronic

packaging materials. The development of higher density circuits makes heat

dissipation an increasingly difﬁcult problem to overcome.

Guarded Hot Plate

The guarded hot plate technique is a comparative method that measures thermal

conductivity through the application of thermocouples to a cylindrical ceramic

test specimen sandwiched between two cylindrical sections of a reference ma-

terial with the same diameter and with a known thermal conductivity. The test

setup is shown in Fig. 11. Thermally conductive paste is applied to the mating

surfaces to ensure adequate heat transfer between materials. Using a reference

material with a thermal conductivity similar to that of the ceramic test specimen

will yield the best results. Stack heaters at the top and bottom of the cylinder

stack introduce heat into the materials. The thermocouples are positioned near

the mating surfaces on the reference and test cylinders. The thermocouples de-

termine the temperature gradient and then the steady-state heat ﬂux is established

between the stack heaters. Using the thermal conductivity of the reference ma-

terial and the measured temperature gradients in the two test specimen cylinders

sandwiching the reference cylinder, the heat ﬂux through the entire stack can be

calculated using the thermal expansion equation previously given. These two

independent calculations of heat ﬂux through the stack are then averaged, and

the average is used as the heat ﬂux value for the ceramic test specimen. In

addition, the difference in the two calculated values give the heat loss value for

the stack. The coefﬁcient of thermal conductivity,

␬

, can then be determined

642 CERAMICS TESTING

Fig. 12 Schematic of laser ﬂash method test setup. (From Ref. 15 Engineered Materials

Handbook, Vol. 4: Ceramics and Glasses, ASM International,

Materials Park, OH 44073-0002, Fig. 3, page 614.)

because the average heat ﬂux through the stack, the area of the test specimen

cylinder, and the measured temperature gradient along the ceramic test speci-

men’s length are all already known. To eliminate convective heat losses, mea-

surements can be made in a vacuum or reduced atmosphere environment.

Laser Flash Method

Another method for determining thermal conductivity is the laser ﬂash method,

which involves quickly heating a thin ceramic specimen on one side via a quick

‘‘thermal pulse’’ from a laser and then using the measured temperature values

over time of the back face of the specimen to calculate thermal diffusivity and

conductivity. Thermal diffusivity,

␣, is a different means of expressing a mate-

rial’s heat conduction properties. Thermal diffusivity takes into account that heat

can diffuse through a material subject to different boundary conditions, causing

both spatial and temporal variations of temperature.

Thermal diffusivity is a

temperature-dependent material tensor property that is deﬁned by the equation

⫽

␣

(T)ⵜ T

The relationship between thermal diffusivity and thermal conductivity is given

by the equation

k (T)

␣

(T) ⫽

␳

(T)c (T)

where

␳

(T) is the temperature-dependent density and c

(T) is the speciﬁc heat

capacity of the test specimen.

The ceramic test specimen is in the shape of a thin disk and is at a constant

temperature T, when the laser ﬂash exerts a thermal pulse on the front face.

Figure 12 shows a schematic of the test setup.

The assumptions here are that the thermal energy is deposited uniformly over

the front face of the test specimen, that the heat travels along the thickness of

the specimen only to the back face, and that the pulse heats the sample only.

Then, knowing the temperature of the back face at times after the pulse is

introduced into the test specimen, the thermal diffusivity of the test specimen

3 THERMAL TESTING 643

can be calculated using various algorithms. Different available algorithms use

various temperatures of the back face to calculate thermal diffusivity.

An advantage of the laser ﬂash method is that, in addition to having a simple

test setup, it can be used over a large range of thermal conductivities and ceramic

specimen temperatures. Also, it is quite popular for measuring thermal conduc-

tivities at high temperatures as maintaining a steady-state condition is difﬁcult

at higher temperatures.

Other thermal pulse methods using different sources such as electron beams

or quartz ﬂash lamps are also possible utilizing very similar test procedures.

3.3 Heat Capacity

Heat capacity is a property that refers to the amount of energy that must be

added or subtracted from a material to raise or lower its temperature. The amount

of energy required to raise the temperature of a material by a degree varies from

material to material based on its properties. Speciﬁc heat at a constant pressure,

, is the most common expression of a material’s heat capacity and is deﬁned

as the amount of heat needed to raise the temperature of one gram of a substance

by one kelvin at a constant pressure. The speciﬁc heat is derived from the

enthalpy, H, and this relation is shown in the following equation:

⭸H

c ⫽

冉冊

⭸T

where T is the temperature and the subscript p indicates constant pressure. From

this equation, we can see that the speciﬁc heat is the slope of the enthalpy versus

temperature change. The speciﬁc heat value can be considered a constant for a

short range of temperature, but the actual measured speciﬁc heat term is ex-

pressed as a polynomial to express the nonlinearity of the speciﬁc heat with the

temperature. This more deﬁned speciﬁc heat capacity and its form are given in

the equation

⫺

1/2

C (T) ⫽ a ⫹ bT ⫹ cT ⫺ dT ⫹ eT

However, the form of equation above does not provide a speciﬁc heat for all

temperature ranges of many materials. Various equations of this form must be

determined for various temperature ranges. There are various methods for de-

termining the heat capacity of ceramics.

Calorimetry

Calorimetry can be used to determine the heat capacity of a ceramic material.

A speciﬁc amount of a material test specimen is heated to an initial temperature

with an external furnace and then deposited into a calorimeter of a lower tem-

perature. The calorimeter measures the heat energy that the test specimen gives

off while cooling to the equilibrium temperature that is between the specimen

and calorimeter temperature. Measurements of the enthalpy, H, at various tem-

peratures gives a plot of H(T) versus T. Consequently, the speciﬁc heat, c

, can

be calculated from this data.

644 CERAMICS TESTING

An advantage of calorimetry is that it can calculate speciﬁc heats over a wide

temperature range. A drawback, however, is that the method is insensitive to

transitions with minor changes in enthalpy.

Differential Scanning Calorimetry

Differential scanning calorimetry is the most popular method for measuring the

speciﬁc heat of a ceramic. This type of calorimeter measures the heat ﬂowrate

to a ceramic test specimen while it is heated at a given constant rate. A computer

monitors the temperature of the test specimen and makes adjustments to keep

the temperature of the ceramic test specimen increasing at a constant rate. Con-

sequently, the speciﬁc heat, c

, of a ceramic test specimen of known mass m

can be calculated from the following equation

dH(T) dT

⫽ mc (T)

dt dt

This method is commonly used and there are various commercial differential

scanning calorimeters available. One of the major drawbacks to this technique

is that commercial setups can only be used up to a temperature of about 800

⬚C.

In addition, testing errors resulting from thermal resistance between the test

specimen and heat ﬂow sensing device and other sources requires the use of a

comparative process when testing. A different material with a known heat ca-

pacity is tested under the same conditions and any difference is noted and in-

corporated into the error evaluation of the ceramic test specimen being

investigated.

4 NONDESTRUCTIVE EVALUATION TESTING

Many nondestructive evaluation (NDE) testing techniques are commonly used

for evaluating metals and other materials. Applying these techniques, however,

to ceramics does not always provide adequate results because of the unique

nature of ceramics that impede many of the typical testing techniques. The in-

creasing use of advanced ceramic materials in critical applications along with

properties that make them sensitive to quite small defects increases the need to

deﬁne NDE testing methods directly applicable to ceramics.

Advanced ceramics are typically brittle in nature. Defects as small as 10

␮

can be critically detrimental and must be avoided. They can be prevented with

careful process control of the fabrication of the ceramic materials from ﬁne

powders or with careful inspection of the ﬁnished ceramic parts. Both strategies

can be applied with NDE testing as a major component.

4.1 Ultrasonography

Ultrasonic testing is a common NDE testing technique that can detect and de-

scribe ﬂaws and material conditions that other methods are unable to do. Ultra-

sonic waves are propagated through the material and the waves are disrupted at

the discontinuities in the material such as defects, voids, or cracks. The waves

are scattered or partially reﬂected at these discontinuities and from this action,

a measure of the discontinuity characteristics such as location, size, and shape

4 NONDESTRUCTIVE EVALUATION TESTING 645

are revealed. The ultrasonic method does not always reveal everything about the

discontinuity to the degree desired but is still an invaluable tool.

The distinguishing characteristic of ceramics that make typical ultrasonic test-

ing techniques less successful are the high ultrasonic velocities and the smaller

critical defect size that must be detected. To detect and characterize small de-

fects, the equipment must be modiﬁed so that the ultrasonic wave frequency is

increased. An ultrasonic wave with a wavelength of similar size to the defect is

required to detect the defect. Consequently, with a ceramic material where a

critical defect can be a very small value such as 10

␮

m, a wavelength of ap-

proximately the same size is needed.

This indicates a very high frequency, ƒ,

is required based on the frequency equation

ƒ ⫽

␭

where

␭

is the wavelength and v is the longitudinal wave velocity.

An important property to be discerned in a ceramic material is its bulk po-

rosity. Ultrasonic testing can reveal the material’s bulk porosity even though the

individual pore size is much smaller than the investigating wavelength. The

porosity can be characterized by measuring the ultrasonic velocity and attenu-

ation. Measuring the ultrasonic velocity typically entails using a transducer to a

face of a ceramic specimen with parallel faces and a known thickness. The

round-trip travel time of the acoustic pulse is measured with the transducer in a

pulse-echo mode. The ultrasonic velocity

v is given by the equation

v ⫽

where d is the thickness of the test specimen and t is the time for round-trip

travel. It has been found that there is a linear relationship between ultrasonic

velocity and porosity. Once the ultrasonic velocity has been measured, the var-

iations in porosity in a specimen can be shown with the aid of ultrasonic C-

scans.

Ultrasonic attenuation is the process by which the ultrasonic beam, when

propagating through the test specimen, loses energy. Grain boundaries, pores,

voids, and other internal defects cause scattering of the waves, resulting in a

lower energy beam. This attenuation in the ultrasonic beam is expressed by the

equation

⫽ I exp(⫺

␣

where I is the beam intensity, x is the distance into the test specimen (x ⫽ 0at

surface), and

␣

is the ultrasonic attenuation coefﬁcient of the material. Once the

attenuation of a material is characterized, various imaging equipment can map

the amplitude image of a specimen, which visually describes the porosity of a

test specimen.

646 CERAMICS TESTING

A standard that applies directly to ultrasonic testing is ASTM E494, ‘‘Stan-

dard Practice for Measuring Ultrasonic Velocity in Materials.’’

4.2 Radiography

Radiography uses radiation to characterize a material’s structure by examining

the interaction between the electromagnetic wave and the material itself. The

detection of voids, cracks, pores, and other defects is the primary goal of this

technique. The detection of these defects is a result of the attenuation and scatter

of the radiation as it passes through the material. Various radiation sources can

be used in the application of radiography but the most versatile has been found

to be X-ray sources. There are two common methods for employing X-ray

sources to discover discontinuities and defects in a material’s structure. One

method is to create a two-dimensional image of the test specimen or component

where the variation of the image intensity indicates the degree of attenuation.

Another approach is to use many attenuation image ‘‘slices’’ and an algorithm

to develop a three-dimensional image.

X-ray microradiography techniques employ a divergent X-ray beam to pro-

duce a two-dimensional image of the test specimen or component being inves-

tigated. High contrast and clarity are required when trying to detect the

inherently small defects in most ceramics. One common method of applying X-

ray microradiography is using a contact method where the test specimen or

component being studied contacts an imaging medium, which results in a high-

resolution image. Another method uses X-ray sources that are typically less than

100

␮

m, which allows direct magniﬁcation of the component or test specimen

image.

The ability of these methods to detect defects is dependent on the

degree of contrast and resolution that is obtained in the image. One disadvantage

to this two-dimensional technique is that cracks and other linear defects orien-

tated transverse to the beam direction will be much more difﬁcult to detect.

X-ray computed tomography utilizes many two-dimensional ‘‘slice’’ images

of the X-ray attenuation to form a three-dimensional representation of the object

being investigated. This is especially useful when trying discover defects and

discontinuities in a complex three-dimensional component. It is used primarily

as a research tool and a few of the main drawbacks to the technique are its high

cost and complexity.

5 ELECTRICAL TESTING

Ceramic materials have important functions in various electrical and electronic

applications. Ceramics, with their unique properties, provide capacitive, insula-

tive, conductive, resistive, and other functions in electronic circuitry. An example

of the important role ceramics have in electronics is the use of advanced ceram-

ics such as alumina oxide (Al

) as the substrate material in electronic pack-

aging. Increases in circuit density have consequently resulted in more stringent

requirements being placed on substrate materials. A substrate is desired that has

thermal expansion characteristics that closely match with silicon to prevent crit-

ical thermal stresses. A low dielectric constant is also needed to improve signal

processing. In addition, thermal conductivity is a high priority to dissipate heat

from the high-density circuit. Based on stringent requirements such as these, the

development of new advanced ceramics and subsequent testing to quantify the

6 SUMMARY 647

development and selection of ceramic materials for important electrical/elec-

tronic applications is of increasing importance. Directly adapting current elec-

trical testing techniques to ceramics serves to improve the process of ceramic

integration into electronics and other electrical applications.

5.1 Electrical Resistance at Elevated Temperatures

An important aspect of ceramic use in electronic applications is the variation of

electrical resistance as the temperature rises. With the trend toward higher and

higher density circuits, characterizing substrate and other chip component prop-

erties at elevated temperatures is quite important.

Measuring electrical resistance at high temperature involves applying a volt-

age across a test specimen contained in a heating furnace. A ceramic test spec-

imen is mounted between the electrodes in the furnace that has heated the test

specimen to the desired temperature. A voltage of 500 V DC is applied across

the test specimen for one minute and the volume resistance is measured. The

process is repeated for various temperatures until maximum test temperature is

reached. The volume resistivity,

␳

, is then calculated from the following equa-

tion:

␳

⫽ R

where R

is the measure volume resistance, A is the area of the electrode, and

h is the average thickness of the ceramic specimen in the area covered by the

electrode. Further speciﬁcs are given in ASTM D1829—90, ‘‘Electrical Resis-

tance of Ceramic Materials at Elevated Temperatures.’’

5.2 Flexural Strength of Electronic-Grade Ceramics

Another important test for characterizing electronic ceramic components is the

ﬂexural strength test. Flexural strength testing is already standardized, but ap-

plication to electronic-grade ceramics requires slight modiﬁcation. The testing

sample sizes are much smaller and commonly referred to as ‘‘microbars.’’

Changes in the size of the bending test ﬁxtures, load application, and material

preparation are also required. The speciﬁcs are outlined in ASTM F417—78,

‘‘Standard Test Method for Flexural Strength of Electronic-Grade Ceramics.’’

The ﬂexural strength is still calculated using the standard equation:

3PL

S ⫽

2bd

where S is the ﬂexural strength, P is the force at fracture, L is the distance

between supports, and b and d are the width and thickness, respectively, of the

specimen.

6 SUMMARY

Testing of ceramics has been standardized or formalized for mechanical, thermal,

NAE, and electrical aspects of these brittle materials. Although testing for many

properties and performance related to these aspects is accepted, methodologies