Acta Materialia 52 (2004) 1271–1281

www.actamat-journals.com

Aluminization of high purity iron by powder liquid coating

Koji Murakami a,*, Norihide Nishida a, Kozo Osamura b, Yo Tomota c

a Department of Materials Engineering, Industrial Technology Center of Okayama Prefecture, 5301 Haga, Okayama 701-1296, Japanb Department of Materials Science and Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

c Department of Materials Science, Faculty of Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan

Received 26 November 2002; received in revised form 9 November 2003; accepted 13 November 2003

Abstract

A new powder liquid coating method is proposed for the aluminization of Fe. Mixed powder slurries of Al +Ti or Al +Al2O3 are

pasted onto Fe specimens, and the specimens are then dried and heated in a vacuum. Unlike hot dipping or powder pack ce-

mentation, this technique can be used to aluminize specimens selectively without the need for special equipment or halides. The

amount of Al adhering to the substrate is determined by the Al–Ti reaction or coalescence of molten Al in Al2O3 powder during

heat treatment. The Al concentration profile of the modified layer can be controlled by adjusting the powder mixing ratio or heat

treatment conditions. The properties of the modified layer are analyzed using a new formulation, where the diffusion equation is

treated numerically with consideration of the concentration dependence of the interdiffusion coefficient. The calculated profiles are

stable and in good agreement with the experimental data.

� 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Powder processing; Aluminizing; Iron; Titanium; Kinetics

1. Introduction

Aluminum is highly resistant to oxidation, sulfidation

[1], and degradation in chloride-containing aqueous so-

lutions [2]. However, as steels containing more than 20

at.%Al exhibit poor elongation and impact properties [3],

Al is usually applied to steel by diffusion as a surface

treatment to provide corrosion resistance. In addition tothis corrosion protection effect, Al is one of the most

useful elements to improvewear and fatigue resistance for

steel nitriding. This effect is due to the high hardness of

aluminum nitride and high internal stress derived from

the formation of nitrides.

A process combining aluminization with ion nitriding

has recently been reported [4–6] and samples modified by

this process have been shown to have very high hardness

* Corresponding author. Tel.: +81-862-869-600; fax: +81-862-869-

630.

E-mail addresses: kouji_murakami@pref.okayama.jp (K. Mura-

kami), norihide_nishida@pref.okayama.jp (N. Nishida), osamura@

hightc.mtl.kyoto-u.ac.jp (K. Osamura), tomota@mx.ibaraki.ac.jp (Y.

Tomota).

1359-6454/$30.00 � 2003 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2003.11.011

of about HV1500. An appropriate concentration of Al is

crucial in this process, as an excessive Al concentration

will hinder nitriding, causing themodified layer to become

fragile and spall easily. In order to avoid over-aluminiz-

ing, Tsuji et al. [4] employed sputtering to apply a 2–3 lm-

thick Al thin film on steel substrates. However, it is

difficult to use this method effectively onmechanical parts

with complex shapes, and is also very expensive.To solve these problems and take full advantage of

the potential benefits of Al, the authors propose a new,

low-cost aluminization method for Fe surfaces. The

method involves heat treating specimens in a vacuum

using Al +Ti or Al +Al2O3 mixed powders. In this re-

port, the detailed mechanism of the proposed alumini-

zation process is investigated through phenomenological

and kinetic analyses.

2. Experimental

Disks of high-purity Fe (99.9%) were used for sub-

strates. The surfaces of the substrates were polished with

abrasive papers to #600 grade and washed twice with

ll rights reserved.

1272 K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281

acetone in an ultrasonic cleaning bath each for 0.3 ks.

Various slurries were prepared and pasted onto the

disks. The slurries consisted of atomized Al powder (<3

lm in diameter), crushed Ti (<38 lm) and ethylene

glycol (5:0� 103 mm2 to give mixed powders of1.0� 10�2 kg). Slurries of Al +Al2O3 (crushed powder,

10–20 lm) were also prepared in order to examine the

influence of the Al–Ti reaction on the formation of the

aluminized layer. Slurries were stirred with an impeller

at 50 revolutions/second for 0.6 ks. After pasting the

slurry onto the disks, the specimens were heated in an

oven at 473 K for 3.6 ks to remove ethylene glycol.

Aluminization was performed by heating the powder-coated disk in a quartz cylinder using an infrared heat-

ing source under a pressure of about 1.3� 10�3 Pa.

The optimal slurry composition for providing a uni-

form aluminized layer was determined by measuring the

amount of powder on the disk that remained adhered to

the surface after aluminization. The mass of each sub-

strate was measured before and after aluminization, and

the amount of Al adhered to the surface was calculatedfrom the results. This measurement was performed un-

der various conditions of heating rate, holding temper-

ature, and holding time.

The microstructure of the modified layer was ob-

served by optical microscope and electron probe micro

analysis (EPMA), and the Al concentration was mea-

sured quantitatively by EPMA with ZAF matrix cor-

rection. The constituent phases formed in thealuminizing process were identified by X-ray diffraction

(XRD).

3. Experimental results

3.1. Powder composition

The optimal mixing ratio of Al-to-Ti was determined

based on the properties of the aluminized layer. Fig. 1(a)

is a photograph of a high-purity Fe disk aluminized with

a pure Al slurry applied to a thickness of 0.45 mg/mm2

(heating rate 0.33 K/s, held at 1273 K for 3.6 ks). The

Fig. 1. Photographs of various specimens aluminized under conditions of 0.45

and (c) Al:Ti¼ 6:4.

treatment conditions are denoted in the form of

Al:Ti¼ 10:0–0.45 mg/mm2–0.33 K/s–1273 K–3.6 ks

throughout this report. In this case, molten Al con-

densed and adhered non-uniformly on the steel surface.

This kind of modified layer is not appropriate for thepresent purposes. Figs. 1(b) and (c) show the specimens

aluminized using powder compositions of Al:Ti¼ 2:8

and 4:6 (by mass) under the same conditions as the

sample in Fig. 1(a). After aluminization, a thin porous

disk could be easily removed from the surface, and the

exposed surface was visibly modified. The porous disks

formed on specimens aluminized with slurry composi-

tions of Al:Ti¼ 6:4 and 8:2 were deformed around thefringe, corresponding to a region of non-uniform

aluminization as shown in Fig. 1(c).

The X-ray diffraction patterns of the aluminized

substrates revealed the formation of a-phase material

for the specimen aluminized at Al:Ti¼ 2:8. The surfaces

of the specimens treated at Al:Ti¼ 4:6, 6:4 and 8:2

contained Fe3Al(D03, ordered phase in a), FeAl(B2),

and FeAl2 +Fe2Al5, respectively. The porous diskscontained some Al–Ti intermetallic compounds includ-

ing thermodynamically unstable phases. The porous

disks for Al:Ti¼ 2:8, 4:6, 6:4 and 8:2 were identified as

Ti3Al +Ti2Al +TiAl, Ti3Al +TiAl, TiAl3 +TiAl2, and

TiAl3+Ti2Al5+TiAl2, respectively.

As shown in Fig. 2, where the porous thin disk was

removed, the specimen aluminized at Al:Ti¼ 2:8 exhib-

ited polishing scratches and islands, which are thoughtto be the traces of molten Al adhered to the disk during

heat treatment. In contrast, the specimens aluminized at

Al:Ti¼ 4:6 and 6:4 had a uniform porous surface with-

out scratches. Area analysis of the modified surface by

EPMA revealed that the Ti content of the surface of the

aluminized substrate increased with increasing mixing

ratio of Al in the slurry. The porous disk removed after

aluminizing at Al:Ti¼ 4:6 and 6:4 contained Fe in someareas of the surface that was in contact with the sub-

strate. Fe was not detected at Al:Ti¼ 2:8.

A cross-sectional optical micrograph of the alumi-

nized specimen after etching with Nital failed to reveal a

clear interface between the aluminized layer and the base

mg/mm2–0.33 K/s–1273 K–3.6 ks. (a) Al-only powder, (b) Al:Ti¼ 4:6,

Fig. 2. Secondary electron micrograph of the surface of specimens aluminized under conditions of 0.45 mg/mm2–0.33 K/s–1273 K–3.6 ks. (a)

Al:Ti¼ 2:8 and (b) Al:Ti¼ 4:6.

K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281 1273

Fe. The corresponding area analysis by EPMA is shown

in Fig. 3. While the specimen aluminized at Al:Ti¼ 2:8

exhibited a non-uniform, wavy aluminized layer, the

specimen treated with a slurry of Al:Ti¼ 4:6 and 6:4

(Fig. 3) had a uniform aluminized surface, except the

fringes of the sample treated with a Al:Ti¼ 6:4 slurry.Al concentration profiles for the specimens alumi-

nized using slurries of Al:Ti¼ 4:6 and 6:4 are shown in

Fig. 4. As Ti was detected only at the surface of the

aluminized substrate, only Al and Fe were considered in

the quantitative analysis. Referring to the phase dia-

gram in Fig. 5, the c region corresponds to 0–0.88 at.%

Al, and the a region corresponds to 1.41–53.0 at.% Al at

1273 K. The a=c interface occurs 104 lm from the

Fig. 3. EPMA area analysis of specimens aluminized under conditions of Al:T

and (c) Fe map.

surface for the Al:Ti¼ 4:6 specimen, and at 150 lm for

Al:Ti¼ 6:4.

To further investigated the aluminization mechanism,

Al2O3 powder was used instead of Ti. After pasting

slurries of Al:Al2O3 ¼ 2:8, 4:6, 6:4 and 8:2 on substrates

to a thickness of 0.45 mg/mm2, the specimens weresubjected to heat treatment (0.33 K/s, 1273 K, 3.6 ks).

The topmost layer could be removed as a thin disk for

the specimens aluminized with slurries of Al:Al2O3 ¼ 2:8

and 4:6 (Fig. 6(a)), whereas specimens treated with

Al:Al2O3 ¼ 6:4 (Fig. 6(b)) and 8:2 slurries did not pro-

duce removable disks.

Cross-sectional optical micrographs of these speci-

mens are shown in Fig. 7. Condensed Al particles of 20–

i¼ 6:4–0.45 mg/mm2–0.33 K/s–1273 K–3.6 ks. (a) Al map, (b) Ti map

Fig. 4. Al concentration profile for specimens aluminized with slurries

of Al:Ti¼ 4:6 and 6:4 under conditions of 0.45 mg/mm2–0.33 K/s–1273

K–3.6 ks.

Fig. 5. Fe–Al phase diagram [10].

1274 K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281

30 lm in diameter can be seen trapped in the Al2O3

powder for specimens treated with slurries of

Al:Al2O3 ¼ 2:8 and 4:6 (Fig. 7(a)). In the specimens

treated with slurries of Al:Al2O3 ¼ 6:4 (Fig. 7(b)) and

Fig. 6. Photographs of specimens aluminized under conditions of 0.45

8:2, the molten Al was not effectively trapped by Al2O3

particles, and the residual powders mainly consisted of

Al2O3 particles. In this case, almost all of the pasted Al

appears to have adhered onto the substrates during heat

treatment.The properties of the Ti and Al2O3 powders were

then investigated by varying the amount of powder

pasted onto the substrate. Fig. 8 shows the mass gain

change of the substrate as a function of the amount of

Al powder pasted onto the substrate (Al:Ti¼ 4:6 and

Al:Al2O3 ¼ 4:6, 0.25–0.85 mg/mm2). It is clear that the

amount of Al adhered to the substrate exhibits greater

variance in the case of Al +Al2O3 compared to Al +Ti.The mass gain increased with increasing amount of Al

powders pasted onto the substrate for Al +Ti slurries,

while such a relationship was unclear for Al +Al2O3.

For Al +Ti, at maximum of 25% Al powder adhered to

the substrate as the amount of Al was increased to 0.35

mg/mm2. The corresponding Al concentration and

thickness of the aluminized layer tAl, as shown in Fig. 3,

ranged from 22.5 to 35 at.% and 104 to 120 lm,respectively.

3.2. Heating conditions

Specimens prepared with a slurry of Al:Ti¼ 4:6 ap-

plied to a thickness of 0.45 mg/mm2 were heated under

various conditions to investigate the effect of heat

treatment. Fig. 9 shows the results of area analysis forsamples heated at a heating rate of 1.3 K/s to 1073, 1173

or 1273 K. Molten Al was observed to adhere to the

substrate (Fig. 9(a)), and the residual reacted with Ti

particles (Fig. 9(b)). Subsequently, the particle surface

transformed into Al–Ti intermetallic compounds

(Fig. 9(c)), and the thin sintered porous disk became

detached from the substrate.

Fig. 10 shows Al concentration profiles for specimensheated at 1.3 K/s–1273 K and held at that temperature

for 0.3, 0.9, 1.8, 3.6 or 7.2 ks. These data will be used

later to investigate the kinetics of the aluminization

process. When the samples were aluminized at 0.33 K/s

up to 973 or 1073 K and held for 3.6 ks, Fe2Al5, FeAl2,

mg/mm2–0.33 K/s–1273 K–3.6 ks. (a) Al:Al2O3 ¼ 4:6 and (b) 6:4.

Fig. 7. Cross-sectional optical micrographs of specimens shown in Fig. 6 (unetched). (a) Al:Al2O3 ¼ 4:6 and (b) 6:4.

Fig. 8. Substrate mass gain as a function of the amount of applied Al.

K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281 1275

FeAl, and a were identified by EPMA analyses. In

contrast, only FeAl and a were observed for the sample

aluminized at 0.33 K/s–1173 K and held for 3.6 ks. The

amount of Al adhered to the substrate did not vary

significantly among these samples. The 0.33 K/s–973 K–3.6 ks sample exhibited a wavy interface between the

aluminized layer and the base Fe, with morphology al-

most identical to that shown in Fig. 9(b). This wavy

interface became flat as the aluminization temperature

was raised to 1173 K or higher.

4. Discussion

4.1. Phenomenological features of the aluminization

process

Based on the experimental results, the Al +Ti alum-

inization process is thought to occur as shown in Fig. 11.

The process proceeds as follows:

(a) After the slurry is pasted onto the steel disk and al-lowed to dry, Al particles fill open spaces between Ti

particles.

(b) When the Al powder is melted, the Al is thermally

activated, breaking the thin Al2O3 film coating the

particles and causing the surface of the Fe substrate

and Ti particles to become coated with molten Al.

(c) As the Fe concentration in the molten Al increases,

the layer transforms into a composite of Fe–Al inter-

metallic compounds and solidifies according to the

phase diagram (Fig. 5). In the early stage of interdif-fusion, a wavy interface is formed between the inter-

metallic compounds and the base Fe. At the same

time, the molten Al surrounding the Ti powder

transforms into intermetallic Ti–Al.

(d) A reaction between Al and Ti proceeds, and the ac-

tual Al supply to the substrate is thought to cease

when no molten Al exists in the applied layer. In this

stage, contacts between the aluminized substrateand the residual powder may still exist, as observed

by EPMA area analysis of the removed porous disk.

The wavy interface becomes straight as the temper-

ature is increased to more than 1173 K, and effective

diffusion of Al atoms into the Fe substrate begins.

During cooling, the residual powder is thought to

leave the aluminized substrate, forming a porous

disk, due to differences in thermal expansioncoefficients.

The largest standard Gibbs free energy changes in the

various reactions of Al with Ti at 973 K is given by DG�

ðhTii þ3fAlg ¼ hTiAl3iÞ ¼ �107 kJ/mol–TiAl3 [7–9],

while that for Al with Fe is DG�ðhFei þ3fAlg ¼hFeAl3iÞ ¼ �22:8 kJ/mol–FeAl3 [6]. Ti particles promote

the formation of a uniformly dispersed sheet of molten Al

on the substrate, and rapid solidification of excess Al byAl–Ti reaction is considered to determine the amount of

Al that adheres to the substrate.

For the Al +Al2O3 slurries, Al2O3 particles also dis-

perse molten Al on the substrate. However, the amount

of Al that adheres to the substrate is determined solely

by the coalescence of molten Al between Al2O3 particles

due to surface tension (Fig. 12). Excess molten Al par-

ticles are trapped effectively only when the open spacesbetween Al2O3 particles are sufficiently narrow to pre-

vent Al from flowing to the substrate due to thermal

Fig. 9. EPMA area analysis of specimens aluminized with Al:Ti¼ 4:6 slurry under conditions of 0.45 mg/mm2–1.3 K/s. (a) 1073 K–0 s, (b) 1173 K–

0 s, and (c) 1273 K–0 s.

1276 K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281

agitation. Therefore, the controllability of theAl +Al2O3 system is poor compared to the Al +Ti

system.

4.2. Numerical analysis of kinetics of aluminization

process

4.2.1. Formulation and calculation procedure

To quantitatively predict the Al concentration profileafter heat treatment, the diffusion equation is solved

numerically. As shown in Fig. 13, the interdiffusion

coefficient of Al and Fe, ~D, varies from 10�1 to 100 lm2/

s at 1273 K [11]. This means that taking the dependence

of ~D on Al concentration, c, into account is indispens-able for precise prediction of the Al concentration pro-

file. Therefore, the diffusion equation to be used is as

follows:

ocðt; xÞot

¼ o

ox~Docðt; xÞox

� �; 06 x < 1; 06 t < 1:

ð1Þ

The value of c (in mol/mm3) is calculated using Eq. (2)

with the atomic percentage of Al (xAl), and molar vol-

umes of Al (vAl ¼ 1:00� 104 mm3/mol) and Fe

(vFe ¼ 7:11� 103 mm3/mol) for simplicity

Fig. 11. Schematic of the aluminization p

Fig. 12. Schematic of the aluminization pro

Fig. 10. Al concentration profiles for specimens aluminized under

various conditions.

K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281 1277

c ¼ xAl

xAlvAl þ ð100� xAlÞvFe: ð2Þ

The model for this calculation is depicted in Fig. 14. No

further supply of Al in the semi-infinite system is con-

sidered in the calculation. Therefore, the boundary

conditions are as follows:

Jðt; x ¼ 0Þ ¼ � ~Docðt; xÞox

� �����x¼0

¼ 0; ð3Þ

limx!1

cðt; xÞ ¼ 0: ð4Þ

For the initial conditions, the Al concentration profile

observed for the Al:Ti¼ 4:6–0.45 mg/mm2–1.3 K/s–1273

K–0.3 ks sample (Fig. 10) was used because the entire

aluminized layer of this specimen was a and the a/cinterface was linear. As Eq. (1) with boundary condi-

tions (3) and (4) cannot be solved analytically, conver-

rocess using mixed Al+Ti powder.

cess using mixed Al +Al2O3 powder.

Fig. 13. Interdiffusion coefficient for the Fe–Al system at 1273 K [11].

Fig. 14. Model used for calculation.

1278 K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281

sion into difference equations is required for numerical

treatment.

In the numerical analysis of the diffusion equation, it

is well known that an appropriately small time slice must

be found to obtain stable solutions for a given spatial

resolution and ~D. However, the number of iterations

required to complete the calculation for the detailed Alconcentration profile tends to be large, resulting in a

large accumulated round-off error. To avoid this prob-

lem, the Crank–Nicholson scheme, which exhibits good

stability for high spatial resolution and rough time

increment, is adopted and modified for the above

equations.

By Taylor expansion, the LHS of Eq. (1) can be

converted to the following difference equation with errorof OððDtÞ2Þ:cðt þ Dt; xÞ � cðt; xÞ

Dt¼ oc

otþ 1

2

o2cot2

Dt þ � � � ;

ocot

¼ cðt þ Dt; xÞ � cðt; xÞDt

� 1

2

o2cot2

Dt þOððDtÞ2Þ: ð5Þ

In the same way, the RHS of Eq. (1) can be converted by

the forward difference method to give

o

ox~Docox

� �

¼ 1

ðDxÞ2~DðxÞcðt; x� DxÞ � ~DðxÞ þ ~Dðx

��þ DxÞ�cðt; xÞ

þ ~Dðxþ DxÞcðt; xþ DxÞg � 1

2

o

oxo~Dox

ocox

� �Dx

þOððDxÞ2Þ: ð6Þ

From Eqs. (5) and (6), the local round-off error is on the

order of OðDt þ DxÞ. Here, a backward scheme is in-

troduced to eliminate the term of OðDtÞ in Eq. (5). By

replacing t with t þ Dt in the forward difference repre-

sentation, Eq. (6), the backward difference scheme and

its Taylor expansion series are given by

o

ox~Docox

� �

¼ 1

ðDxÞ2~DðxÞcðt�

þ Dt; x� DxÞ

� ~DðxÞ�

þ ~Dðxþ DxÞ�cðt þ Dt; xÞþ ~Dðxþ DxÞcðt þ Dt; xþ DxÞg

� o2cot2

Dt � 1

2

o

oxo~Dox

ocox

� �Dx

þO ðDtÞ2�

þ DtDxþ ðDxÞ2�: ð7Þ

Averaging Eqs. (6) and (7) leads to

o

ox~Docox

� �

¼ 1

2

1

ðDxÞ2~DðxÞcðt;x��

�DxÞ� ~DðxÞ�

þ ~DðxþDxÞ�cðt;xÞ

þ ~DðxþDxÞcðt;xþDxÞgþf~DðxÞcðtþDt;x�DxÞ

� ~DðxÞ�

þ ~DðxþDxÞ�cðtþDt;xÞ

þ ~DðxþDxÞcðtþDt;xþDxÞg�1

2

o2cot2

Dt�1

2

o

oxo~Dox

ocox

� �DxþO ðDtÞ2

�þDtDxþðDxÞ2

�:

ð8Þ

Using Eqs. (5) and (8), the difference expression of thediffusion equation (1) is obtained as follows, where

OððDtÞ2Þ is eliminated:

K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281 1279

cðtþDt;xÞ�cðt;xÞ

¼1

2

Dt

ðDxÞ2~DðxÞcðt;x��

�DxÞ� ~DðxÞ�

þ ~DðxþDxÞ�cðt;xÞ

þ ~DðxþDxÞcðt;xþDxÞgþf~DðxÞcðtþDt;x�DxÞ� ~DðxÞ�

þ ~DðxþDxÞ�cðtþDt;xÞþ ~DðxþDxÞcðtþDt;xþDxÞg�1

2

o

oxo~Dox

ocox

� �DtDxþO ðDtÞ3

�þðDtÞ2DxþDtðDxÞ2

�:

ð9Þ

For the numerical calculation, Eq. (9) should be ex-

pressed with subscripts i ð06 i6MÞ and j ð06 j6NÞ,corresponding to t and x, respectively,

ciþ1;j� ci;j

¼ a2

nDjci;j�1�ðDjþDjþ1Þci;jþDjþ1ci;jþ1

� þ Djciþ1;j�1

��ðDjþDjþ1Þciþ1;jþDjþ1ciþ1;jþ1

o; ð10Þ

a � Dt

ðDxÞ2: ð11Þ

It should be noted that the local round-off error for Eq.(10) is OðDtDxÞ.

For the boundary conditions, a modification of the

diffusion equation Eq. (1) at the surface is adopted in-

stead of converting Eq. (3) into a difference equation:

ocot

����x¼0

¼ o~Dox

ocox

�þ ~D

o2cox2

�����x¼0

¼ ~Do2cox2

� �����x¼0

: ð12Þ

Here, the boundary condition (3) was used in the

modification. The average of the forward and backward

difference equations for Eq. (12) is given by

ciþ1;0�ci;0¼aD0

2

hðci;0�2ci;1þci;2Þ

þðciþ1;0�2ciþ1;1þciþ1;2Þi: ð13Þ

From Eqs. (10) and (13), the computation is reduced to

a simple set of simultaneous equations to be solved for

ðciþ1;0; ciþ1;1; . . . ; ciþ1;N Þ from ðci;0; ci;1; . . . ; ci;N Þ.Movement of the a=c interface, caused by c ! a

transformation, is treated as follows. By setting DX as

the distance of a=c interface movement in Dt, the ma-

terials balance around the interface (between j ¼ k and

k þ 1, shown in Fig. 14) is expressed as

ðJi;k � Ji;kþ1ÞDt ¼ ðci;k � ci;kþ1ÞDX : ð14ÞTo improve the accuracy of the calculation, Ji;k and

Ji;kþ1 are given by Eqs. (15) and (16), respectively. These

equations are obtained using four points of ci;j adjacentto the interface [12,13], where the coefficients are de-

termined in the same way as for Savitzky–Golay

smoothing, that is, a cubic curve is fitted to the four

points, and the first derivative of the curve at the in-

terface is calculated with respect to x ð¼ jDxÞ.

Ji;k ¼ �Dk�2ci;k�3 þ 9ci;k�2 � 18ci;k�1 þ 11ci;k

6Dx; ð15Þ

Ji;kþ1 ¼ �Dkþ1

�11ci;kþ1 þ 18ci;kþ2 � 9ci;kþ3 þ 2ci;kþ1

6Dx:

ð16Þ

Substituting Eqs. (15) and (16) into Eq. (14) gives the

velocity of a=c interface movement:

DX ¼ Dt6Dx

1

ci;k�ci;kþ1

Dk 2ci;k�3ð½ �9ci;k�2þ18ci;k�1�11ci;kÞ

þDkþ1ð�11ci;kþ1þ18ci;kþ2�9ci;kþ3þ2ci;kþ4Þ�: ð17Þ

The full computation procedure is summarized as

follows:

1. Set the time increment and spatial resolution (Dt andDx).

2. Load the initial condition fc0;jg and initial a=c inter-

face X0.

3. Set fDjg from Fig. 13.

4. Calculate fciþ1;jg from fci;jg (solve Eqs. (10) and

(13)).

5. Calculate Ji;k, Ji;kþ1, and DX with new fci;jg (Eqs.

(15)–(17)! new fci;jg).6. Search for a new k.7. Set ci;k ¼ ca;min and ci;kþ1 ¼ cc;max.

8. Repeat from setp (3).

4.2.2. Calculation results

In order to obtain stability at the a=c interface for

large Dt, the profile (Fig. 10, Al:Ti¼ 4:6–0.45 mg/mm2–

1.3 K/s–1273 K–0.3 ks) was first smoothed by calcula-

tion with parameters of Dt ¼ 0:1 s, M ¼ 600, Dx ¼0:5 lm, N ¼ 400, ck ¼ ca;min (corresponding to 1.41

at.%), ckþ1 ¼ cc;max (0.88 at.%) and ~Dc ¼ 2:9�10�4 lm2/s [14]. For the smoothed initial condition, Dtwas changed from 0.1 to 10.0 s, and M was changedcorrespondingly such that MDt ¼ 3:24 ks (Dx ¼ 0:5 lm,

N ¼ 400). As shown in Fig. 15, the thickness of the re-

sultant aluminized layer (tAl) was 107–119 lm (Dt ¼ 0:1to 5.0 s). Numerical solutions were unstable for

DtP 6:0 s.

Fig. 16(a) shows the calculated Al concentration

profile for ~D in Fig. 13 after 0.54, 1.44, 3.24 and 6.84 ks

from the initial state, where Dt ¼ 1:0 s, Dx ¼ 0:5 lm,and N ¼ 400. The calculated Al concentration profile

for the average value of ~D in Fig. 13 given by

hDi ¼Z ca;max

ca;min

dcDðcÞ !,Z ca;max

ca;min

dc ð18Þ

is shown in Fig. 16(b).

Fig. 15. Change in thickness of the aluminized layer tAl with time in-

crement Dt (MDt ¼ 3:24 ks, Dx ¼ 0:5 lm, N ¼ 400).Fig. 17. Measured and calculated thickness of aluminized layer tAl

(Dt ¼ 1:0 s, Dx ¼ 0:5 lm, N ¼ 400).

1280 K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281

The position of the a=c interface (arrows in Fig. 16)

calculated using DðcÞ is in good agreement with the

observed data, while the result using hDi is quite dif-

ferent, as shown in Fig. 17. Here, tAl is expressed as a

linear function of the square root of heat treatment time.

This simple relationship suggests the existence of a

constant effective diffusion coefficient for the movement

of the a=c interface even when there is a strong depen-dence of ~D on Al concentration.

In Fig. 16(a), the calculated Al concentration profile

for 0.9 and 1.8 ks exhibits a characteristic step at around

30 at.%, which does not appear in the profile calculated

using hDi. The origin of this step was investigated by

changing the Al concentration at x ¼ 20 lm as a func-

tion of heating time, as plotted in Fig. 18. The Al con-

centration calculated using DðcÞ exhibits a steeperdescent at first than that for hDi, indicating that the step

is attributed to the local maximum of DðcÞ in Fig. 13,

Fig. 16. Calculated Al concentration profiles calculated using (a) DðcÞin Fig. 13, and (b) hDi ¼ 0:279 lm2/s (Dt ¼ 1:0 s, Dx ¼ 0:5 lm,

N ¼ 400).

that is, fast Al diffusion at around 30 at.% to give the

characteristic profile in Fig. 10 (1273 K–0.9 or 1.8 ks).

The calculated Al concentration profiles, however,

exhibited discrepancies from the measured profile, par-ticularly in terms of a near the a=c interface. This may

be because the reported data for ~D differs from the ac-

tual values. Incorrect conversion between at.% and mol/

mm3 (Eq. (2)) can also be suspected, but the results

obtained through the use of the observed density of a[1,3] and c-Fe gave only slight discrepancies (at most 0.5

at.% Al concentration and 2 lm thickness).

Despite these discrepancies, the present calculation isuseful for estimating the thickness of the modified layers

and the Al concentration near the surface. For example,

these results show that the Al concentration at the sur-

face should be no higher than about 5 at.% in order to

obtain a good nitrided layer without cracks resulting

from excessive residual stress. Since prolonged heat

Fig. 18. Change in Al concentration at x ¼ 20 lm as a function of time

(Dt ¼ 1:0 s, Dx ¼ 0:5 lm, N ¼ 400).

K. Murakami et al. / Acta Materialia 52 (2004) 1271–1281 1281

treatment results in a low-Al aluminized layer and

possibly grain coarsening, the amount of Al that adheres

to the steel surface must be reduced to a certain extent.

A rule for estimating an appropriate amount of Al is

given below.First, a rectangular initial condition of 50 at.% Al

and 40 lm layer thickness is set. The Al concentration of

50 at.% was chosen to avoid the treatment of other in-

termetallic compounds, and the thickness was deter-

mined such that the total amount of Al in the observed

aluminized layer (1273 K–0.3 ks) would be the same as

that in the rectangular profile. From the result that the

two initial conditions produced almost the same profiles,rectangular initial conditions are thought to be appli-

cable to roughly predict the properties of the aluminized

layer after heat treatment. Next, two rectangular initial

conditions (50 at.%–10 lm and 50 at.%–5 lm) are cal-

culated under the 1273 K–3.6 ks condition. The first

initial condition gives a layer of 6.5 at.% Al (at the

surface) and 55 lm (tAl) in thickness, and the second

gives 8.6 at.% Al and 63 lm. These values correspondsto Al films of 3.2 and 4.4 lm thick deposited by physical

vapor deposition or sputtering, and are in agreement

with Tsuji�s results [4,5].In the aluminization of iron and steel to provide anti-

corrosion properties, the higher the Al content in the

modified layer, the more fragile and susceptible to

cracking the layers become. The proposed calculation

can be applied to determine the optimum coating con-ditions for austenitic stainless steel or Fe–Cr alloy

[15,16], provided that the method is refined to treat all

phases (Fe–Al, Ni–Al, etc.) and the correct values of

interdiffusion coefficients are used for each phase.

5. Conclusions

High-purity Fe substrates were aluminized by heat-

treating specimens coated with a slurry composed of

Al +Ti or Al +Al2O3 powders and ethylene glycol. The

Al concentration and thickness of the aluminized layer

can be controlled by changing the powder mixing ratio

and the heat-treatment conditions. During heat treat-

ment, molten Al adheres to the steel surface to form

intermetallic Al–Fe. In the Al +Ti slurry, residual mol-ten Al immediately reacts with Ti particles to form in-

termetallic Al–Ti compounds, and this solidifying

reaction effectively prevents further supply of Al to the

substrate. In the Al +Al2O3 system, excess Al is trapped

between Al2O3 particles at a certain mixing ratio of

Al to Al2O3. The amount of Al contributing to the

aluminization process was found to depend on the mass

of the slurry applied to the specimen in both techniques,

although the Al+Ti method provided better control

over aluminization than Al +Al2O3.

Al concentration profiles were calculated by solving adiffusion equation taking into account the dependence

of the interdiffusion coefficient on Al concentration and

a=c interface movement. A modified Crank–Nicholson

scheme was formulated for the diffusion equation to

obtain stability and precision instead of converting the

equation into a simple forward difference equation. The

solution was in good agreement with the experimental

results in terms of thickness of the aluminized layer, andthe step in the observed Al concentration profile, dem-

onstrating the validity of taking into account the

dependency of interdiffusion coefficient on Al concen-

tration. The numerically calculated Al concentration

profile results differed from the measured results, par-

ticularly in terms of a near the a=c interface, attributed

to a discrepancy in the data used in the calculation. This

simple calculation, however, is considered applicable forpredicting the amount of Al that should be supplied to

the substrate and the appropriate heat treatment con-

ditions to obtain aluminized layers suitable for nitriding

or as an anti-corrosion coating.

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