Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering

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cause the movement of the atom. Table 5-1 also shows values of D

, the pre-exponent

term, and the constant c

from Equation 5-1 where the rate process is di¤usion. We will

see later that D

is the di¤usion coe‰cient when 1=T ¼ 0.

5-5 Rate of Diffusion (Fick’s First Law)

The rate at which atoms, ions, particles or other species di¤use in a material can be

measured by the ﬂux (J ). Here we are mainly concerned with di¤usion of ions or atoms.

The ﬂux is deﬁned as the number of atoms passing through a plane of unit area per unit

time (Figure 5-7). Fick’s ﬁrst law explains the net ﬂux of atoms:

TABLE 5-1 9 Diffusion data for selected materials

Diffusion Couple (cal/mol) (cm

/s)

Interstitial diffusion:

C in FCC iron 32,900 0.23

C in BCC iron 20,900 0.011

N in FCC iron 34,600 0.0034

N in BCC iron 18,300 0.0047

H in FCC iron 10,300 0.0063

H in BCC iron 3,600 0.0012

Self-diffusion (vacancy diffusion):

Pb in FCC Pb 25,900 1.27

Al in FCC Al 32,200 0.10

Cu in FCC Cu 49,300 0.36

Fe in FCC Fe 66,700 0.65

Zn in HCP Zn 21,800 0.1

Mg in HCP Mg 32,200 1.0

Fe in BCC Fe 58,900 4.1

W in BCC W 143,300 1.88

Si in Si (covalent) 110,000 1800.0

C in C (covalent) 163,000 5.0

Heterogeneous diffusion (vacancy diffusion):

Ni in Cu 57,900 2.3

Cu in Ni 61,500 0.65

Zn in Cu 43,900 0.78

Ni in FCC iron 64,000 4.1

Au in Ag 45,500 0.26

Ag in Au 40,200 0.072

Al in Cu 39,500 0.045

Al in Al

114,000 28.0

OinAl

152,000 1900.0

Mg in MgO 79,000 0.249

O in MgO 82,100 0.000043

Data from several sources, including Adda, Y. and Philibert, J.,

La Diffusion dans les Solides, Vol. 2, 1966.

C H A P T E R 5 Atom and Ion Movements i n Materials130

J ¼D

ð5-3Þ

where J is the ﬂux, D is the di¤usivity or di¤usion coe‰cient



, and dc=dx is the

concentration gradient

atoms

 cm



. Depending upon the situation, concentration may

be expressed as atom percent (at%), weight percen t (wt%), mole percent (mol%), atom

fraction, or mole fraction. The units of concentration gradient and ﬂux will also change

respectively.

Several factors a¤ect the ﬂux of atoms during di¤usion. If we are dealing with dif-

fusion of ions, electrons, holes, etc., the units of J, D, and

will reﬂect the appropriate

species that are being considered. The negative sign in Equation 5-3 tells us that the ﬂux

of di¤using species is from higher to lower concentrations, making the

term negative,

and hence J will be positive. Thermal energy associated with atoms, ions etc. causes the

random movement of atoms. At a microscopic scale the thermodynamic driving force

for di¤usion is concentration gradient. A net or an observable ﬂux is created depending

upon temperature and concentration gradient.

Concentration Gradient The concentration gradient shows how the composition of the

material varies with distance: Dc is the di¤erence in concentration over the distance Dx

(Figure 5-8). The concentration gradient may be created when two materials of di¤er-

ent composition are placed in contact, when a gas or liquid is in contact with a solid

material, when nonequilibrium structures are produced in a materia l due to processing,

and from a host of other sources.

Figure 5-7

The flux during diffusion is defined as the number of atoms

passing through a plane of unit area per unit time.

Figure 5-8 Illustration of the concentration gradient.

5-5 Rate of Diffusion (Fick’s First Law) 131

The ﬂux at a particular temperature is constant only if the concentration gradient is

also constant—that is, the compositions on each side of the plane in Figure 5-7 remain

unchanged. In many practical situations, however, these compositions vary as atoms

are redistributed, and thus the ﬂux also changes. Often, we ﬁnd that the ﬂux is initially

high and then gradually decreases as the concentration gradient is reduced by di¤usion.

The example that follows illustrates calculations of ﬂux and concentration gradients for

di¤usion of dopants in semiconductors, but only for the case of constant concentration

gradient. Later in this chapter, we will consider non-steady state di¤usion with the aid

of Fick’s second law.

EXAMPLE 5-3 Semiconductor Doping

One way to manufacture transistors, which amplify electrical signals, is to dif-

fuse impurity atoms into a semiconductor material such as silicon (Si). Suppose

a silicon wafer 0.1 cm thick, which originally contains one phosphorus atom

for every 10 million Si atoms, is treated so that there are 400 phosphorous (P)

atoms for every 10 million Si atoms at the surface (Figure 5-9). Calculate the

concentration gradient (a) in atomic percent/cm and (b) in

atoms

 cm

. The lattice

parameter of silicon is 5.4307 A



SOLUTION

First, let’s calculate the initial and surface compositions in atomic percent:

1 P atom

atoms

 100 ¼ 0: 00001 at% P

400 P atoms

atoms

 100 ¼ 0: 004 at% P

0:00001  0:004 at% P

0:1cm

¼0:0399

at% P

To ﬁnd the gradient in terms of

atoms

 cm

, we must ﬁnd the volume of the unit cell:

cell

¼ð5:4307  10

8

cmÞ

¼ 1:6  10

22

cell

The volume occupied by 10

Si atoms, which are arranged in a diamond cubic

(DC) structure with 8 atoms/cell, is:

Figure 5-9

Silicon wafer showing variation

in concentration of P atoms

(for Example 5-3).

C H A P T E R 5 Atom and Ion Movements i n Materials132

V ¼

atoms

cell

1:6  10

22

cell

hi

¼ 2  10

16

The compositions in atoms/cm

are:

1 P atom

2  10

16

¼ 0:005  10

atoms



400 P atoms

2  10

16

¼ 2  10

atoms



0:005  10

 2  10

atoms



0:1cm

¼1:995  10

atoms

 cm

5-6 Factors Affecting Diffusion

Temperature and the Diffusion Coefficient The kinetics of the process of di¤usion are

strongly dependent on temperature. The di¤usion coe‰cient D is related to temperature

by an Arrhenius-type equation,

D ¼ D

exp

Q



ð5-4Þ

where Q is the activation energy (in units of cal/mol) for di¤usion of species under

consideration (e.g., Al in Si), R is the gas constant 1:987

cal

mol  K



, and T is the

absolute temperature (K). D

is the pre-exponential term, similar to c

in Equation 5-1.

It is a constant for a given di¤usion system and is equal to the value of the di¤usion

coe‰cient at 1=T ¼ 0orT ¼ y. Typical values for D

are given in Table 5-1, while the

temperature dependence of D is shown in Figure 5-10 for some metals and ceram ics.

Covalently bonded materials, such as carbon and silicon (Table 5-1), have unusually

high activation energies, consistent with the high strength of their atomic bonds.

In ionic materials, such as some of the oxide ceramics, a di¤using ion only enters a

site having the same charge. In order to reach that site, the ion must physically squeeze

past adjoining ions, pass by a region of opposite charge, and move a relatively long

distance (Figure 5-11). Consequently, the activation energies are high and the rates of

di¤usion are lower for ionic materials than those for metals.

When the temperature of a material increases, the di¤usion coe‰cient D increases

(according to Equation 5-4) and, therefore, the ﬂux of atoms increases as well. At higher

temperatures, the thermal energy supplied to the di¤using atoms permits the atoms to

overcome the activation energy barrier and more easily move to new sites in the atomic

arrangements. At low temperatures—often below about 0.4 times the absolute melting

temperature of the material—di¤usion is very slow and may not be signiﬁcant. For this

reason, the heat treatments of metals and the processing of ceramics are done at high

5-6 Factors Affecting Diffusion 133

temperatures, where atoms move rapidly to complete reactions or to reach equilibrium

conditions. Because less thermal energy is required to overcome the smaller activation

energy barrier, a small activation energy Q increases the di¤usion coe‰cient and ﬂux.

The following example illustrates how Fick’s ﬁrst law and concepts related to temper-

ature dependence of D can be applied to design an iron membrane.

Figure 5-10 The diffusion coefficient D as a function of reciprocal temperature for some

metals and ceramics. In this Arrhenius plot, D represents the rate of the diffusion process.

A steep slope denotes a high activation energy.

Figure 5-11

Diffusion in ionic compounds. Anions can only enter other

anion sites. Smaller cations tend to diffuse faster.

C H A P T E R 5 Atom and Ion Movements i n Materials134

EXAMPLE 5-4 Diffusion in Yttrium in Chromia Film Formed by Oxidation of

Ni-Cr Alloy

The di¤usion of yttrium ions in chromium oxide or chromia (Cr

) has been

studied by Lesage and co-workers. This was done by forming a thin ﬁlm of

yttrium oxide on a nickel chromium alloy. The base alloy substrate formed a

chromium oxide layer on the surface after oxidation and the yttrium ions from

yttrium oxide layer subsequently di¤used into the chromium oxide ﬁlm formed

by alloy oxidation. Their experiments show that the di¤usion coe‰cient of

yttrium ions while di¤using through the bulk chromia exhibited the following

temperature dependence.

Temperature

(˚ C)

Diffusion Coefficient (D)

(cm

/s)

800 2:0  10

18

850 1:8  10

18

900 3:2  10

18

950 7:9  10

18

1000 2:4  10

17

(Source: J. Li, M.K. Loudjani, B. Lesage, A.M.

Huntz, Philosophical Magazine A, 1997, 76[4],

pp. 857–69).

(a) What is the activation energy for this bulk di¤usion of yttrium ions in

chromia?

(b) What is value of the pre-exponential term D

between lnðDÞ and 1=T ?

SOLUTION

(a) We assume that the Arrhenius relationship is applicable.

D ¼ D

exp

Q



Note that the reported D value at 850



C is actually a bit lower than that for

800



C. This is unexpected, and the value may be within the experimental

measurement error and hence not much di¤erent. There may have been

some experimental artifact or some other unknown factor that caused this

value to be lower. We will use these values, since the di¤erence between the

values of di¤usion coe‰cients at 800 and 850



C is not extremely large.

Taking logarithms of both sides:

ln D ¼ ln D



5-6 Factors Affecting Diffusion 135

Therefore, the di¤usion data given are rewritten as

Temp (K) 1/T (K

) D cm

/s ln D

1073 0.000932 2E–18 40.9159

1123 0.00089 1.7E–18 40.7534

1173 0.000853 3.2E–18 40.2834

1223 0.000818 7.9E–18 39.3797

1273 0.000786 2.4E–17 38.2685

We can then either plot the data or use a spreadsheet such as

Excel

to calculate the slope. In this case, we have used the SLOPE

(known_y’s,known_x’s) function in Excel

to calculate the slope of the

expected straight line. The known y-axis values are those under the ln D

column, and the known x-values are under the 1=T column. This calcula-

tion gives us a slope value of 17415.7. Note the negative sign for the slope

value.

This value of slope is equal to ðQ=RÞ. If we use gas constant value of

R ¼ 8:314 Joules/K-mol, then the value of activation energy Q will be

Q ¼ ðÞ  ð17415:7Þð8:314Þ Joules=mol

¼ 144794:5 Joules=mol or 144:8kJ=mol

If we use the value of R ¼ 1:987 cal/K-mol, then the value of activation

energy Q will be

Q ¼ð1Þð144794:5Þð1:987Þ cal=mol

¼ 34605:07 cal=mol or 34:6 kcal=mol

(b) We also calculate the slope of the straight line ﬁtted to the lnðDÞ versus

1=T data. This can be done using graph paper or as is done here, using a

spreadsheet. Here we used the INTERCEPT(known_y’s,known_x’s) func-

tion in Excel

using the ln D values as known y-axis data and the 1=T

values as the x-axis data. This gives us a value of the intercept as 25.01.

Thus, the value of D

will be exp(intercept value) or expð25:01Þ¼

1:36  10

11

, and the units will be cm

/s.

D ¼



1:36  10

11



exp 

34:6

kcal

mol

D ¼



1:36  10

11



exp 

144:8

mol

C H A P T E R 5 Atom and Ion Movements i n Materials136

EXAMPLE 5-5 Design of an Iron Membrane

An impermeable cylinder 3 cm in diameter and 10 cm long contains a gas that

includes 0:5  10

N atoms per cm

and 0:5  10

H atoms per cm

on one

side of an iron membrane (Figure 5-12). Gas is conti nuously introduced to the

pipe to assure a constant concentration of nitrogen and hydrogen. The gas on

the other side of the membrane includes a constant 1  10

N atoms per cm

and 1  10

H atoms per cm

. The entire system is to operate at 700



C, where

the iron has the BCC structure. Design an iron membrane that will allow no

more than 1% of the nitrogen to be lost through the membrane each hour,

while allowing 90% of the hydrogen to pass through the membrane per hour.

SOLUTION

The total number of nitrogen atoms in the container is:

ð0:5  10

N=cm

Þðp=4Þð3cmÞ

ð10 cmÞ¼35:343  10

N atoms

The maximum number of atoms to be lost is 1% of this total, or:

N atom loss per h ¼ð0:01Þð35:34  10

Þ¼35:343  10

N atoms=h

N atom loss per s ¼ð35:343  10

N atoms=hÞ=ð3600 s=hÞ

¼ 0:0098  10

N atoms=s

The ﬂux is then:

J ¼

ð0:0098  10

ðN atoms=sÞ



ð3cmÞ

¼ 0:00139  10

atoms

 s

The di¤usion coe‰cient of nitrogen in BCC iron at 700



C ¼ 973 K is:

Figure 5-12 Design of an iron membrane (for Example 5-5).

5-6 Factors Affecting Diffusion 137

From Equation 5-4 and Table 5-1:

D ¼ D

exp

Q



¼ 0:0047

exp

18;300

cal

mol

1:987

cal

K  mol

ð973ÞK

¼ 0:0047 expð9:4654Þ¼ð0:0047Þð7:749  10

5

Þ¼3:64  10

7

From Equation 5-3:

J ¼D



¼ 0:00139  10

N atoms

s

Dx ¼DDc=J ¼

ð3:64  10

7

=sÞ 1  10

 50  10

N atoms

hi

0:00139  10

N atoms

 s

Dx ¼ 0:0128 cm ¼ minimum thickness of the membrane

In a similar manner, the maximum thickness of the membrane that will permit

90% of the hydrogen to pass can be calculated:

H atom loss per h ¼ð0:90Þð35:343  10

Þ¼31:80  10

H atom loss per s ¼ 0:0088  10

J ¼ 0:125  10

H atoms

 s

From Equation 5-4:

¼ 0:004

exp

18;300

cal

mol

1:987

cal

K  mol

ð973ÞK

¼ 1:86  10

4

Since

Dx ¼D Dc=J

Dx ¼

1:86  10

4



49  10

H atoms



0:125  10

H atoms

 s

¼ 0:0729 cm ¼ maximum thickness

An iron membrane with a thickness between 0.0128 and 0.0729 cm will be

satisfactory.

Types of Diffusion In volume di¤usion, the atoms move through the crystal from one

regular or interstitial site to another. Because of the surrounding atoms, the activation

energy is large and the rate of di¤usion is relatively slow.

However, atoms can also di¤use along boundaries, interfaces, and surfaces in the

material. Atoms di¤use easily by grain boundary di¤usion because the atom packing is

poor at the grain boundaries. Because atoms can more easily squeeze their way through

C H A P T E R 5 Atom and Ion Movements i n Materials138

the disordered grain boundary, the activation energy is low (Table 5-2). Surface dif-

fusion is easier still because there is even less constraint on the di¤using atoms at the

surface.

Time Di¤usion requires time; the units for ﬂux are

atoms

 s

! If a large number of atoms

must di¤use to produce a uniform structure, long times may be required, even at high

temperatures. Times for heat treatments may be reduced by using higher temperatures

or by making the di¤usion distances (related to Dx) as small as possible.

We ﬁnd that some rather remarkable structures and properties are obtained if we

prevent di¤usion. Steels quenched rapidly from high temperatures to prevent di¤usion

form nonequilibrium structures and provide the basis for sophisticated heat treatments.

Similarly, in forming metallic glasses (Chapter 4) we have to quench liquid metals at a

very high cooling rate (@10

C/second). This is to avoid di¤usion of atoms by taking

away their thermal energy and encouraging them to assemble into nonequilibrium

amorphous and chemically homogeneous arrangements. Melts of silica te glasses, on the

other hand, are viscous and di¤usion of ions through these is slow. As a result, we do

not have to cool these melts very rapidly.

TABLE 5-2 9 The effect of the type of diffusion for thorium in tungsten and for self-diffusion in

silver*

Diffusion Coefficient (D)

Diffusion Type Thorium in Tungsten Silver in Silver

/s Q cal/mole D

/s Q cal/mole

Surface 0.47 66,400 0.068 8,900

Grain boundary 0.74 90,000 0.24 22,750

Volume 1.00 120,000 0.99 45,700

* Given by parameters for Equation 5-4.

EXAMPLE 5-6 Tungsten Thorium Diffusion Couple

Consider a di¤usion couple setup between pure tungsten and a tungsten alloy

containing 1 at.% thorium. After several minutes of exposure at 2000



C, a

transition zone of 0.01 cm thickness is established. What is the ﬂux of thorium

atoms at this time if di¤usion is due to (a) volume di¤usion, (b) grain boundary

di¤usion, and (c) surface di¤usion? (See Table 5-2.)

SOLUTION

The lattice parameter of BCC tungsten is 3.165 A



. Thus, the number of tung-

sten atoms per cm

is:

W atoms

2 atoms=cell

ð3:165  10

8

=cell

¼ 6:3  10

In the tungsten-1 at.% thorium alloy, the number of thorium atoms per cm

is:

¼ð0:01Þð6:3  10

Þ¼6:3  10

Th atoms=cm

5-6 Factors Affecting Diffusion 139

Askeland D.R., Fulay P.P. Essentials of Materials Science &amp; Engineering

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Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering