Acta Materialia 53 (2005) 4111–4120

www.actamat-journals.com

Molecular dynamics simulation of Y-doped R37 grain boundaryin alumina

Jun Chen a,b,*, Lizhi Ouyang a, W.Y. Ching a

a Department of Physics, University of Missouri-Kansas City, Kansas City, MO 64110, USAb Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received 1 March 2005; received in revised form 26 April 2005; accepted 7 May 2005

Available online 22 June 2005

Abstract

A molecular dynamics technique with a multi-atom potential was used to simulate the behavior of Y-doping in R37 grain bound-

ary (GB) in a-Al2O3. Results on melting process, elastic constants, and GB diffusivity in pure a-Al2O3, R37 GB, and Y-doped R37GB have been obtained and compared with available experimental data. Our simulation indicates that the pure GB pre-melts at a

temperature of 1417 K before the bulk melts at 2105 K. When doped with Y, a melting nucleate around Y is formed and the pre-

melting temperature is increased to 1536 K. The diffusivities of Al and O in the GB region have the same order of magnitude. In

comparison, Y-doping decreases diffusivity and increases diffusion activation energy of both Al and O at the GB. Our results are

consistent with the notion of ‘‘site-blocking’’ effect to explain the decrease in GB diffusivity with Y-doping. At low temperature

(<1536 K), ‘‘site-blocking’’ is due to the larger size of Y ions, but at a high temperature (>1536 K), ‘‘site-blocking’’ is related to

possible formation of nucleates surrounding the Y ions.

� 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundary; Alumina; Y-doping; Diffusivity; MD simulation

1. Introduction

a-Alumina is a much utilized high-temperature struc-

tural ceramic in many industrial sectors. It has been

known for some time that when alumina is doped with

Y, the creep rate of the material is reduced by almost

two orders of magnitude, the so-called ‘‘Y-effect’’

[1–8]. Furthermore, alumina grain growth and inter-

granular fracture behavior are modified by Y-doping.A variety of experimental methods based primarily on

electron microscopy and spectroscopic methods have

been used to investigate the structural environment of

the dopants in the grain boundary (GB) region. These

include Auger electron spectroscopy (AES) [4], X-ray

photoelectron spectroscopy (XPS) [5], high resolution

1359-6454/$30.00 � 2005 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2005.05.012

* Corresponding author.

E-mail address: chenju@umkc.edu (J. Chen).

scanning transmission electron microscopy (HR-STEM)[6,7], high resolution secondary ion mass spectroscopy

(HR-SIMS) [8,9], electron energy-loss spectroscopy

(EELS) [10], and extended X-ray absorption fine struc-

ture (EXAFS) spectroscopy [11]. These experimental

studies have established that Y ions segregate to GBs

in alumina. On the basis of these studies, several

strengthening mechanisms have been postulated to ex-

plain the improved creep resistance in Y-doped alumina[1]. They are: (i) reduced diffusivity along the GBs due to

the interaction of segregants with diffusing species (an-

ions, cations, or both); (ii) solute and/or precipitate drag

on the moving grain front; (iii) interference of segregants

with GB dislocation motion; (iv) the influence of segre-

gants on the generation or annihilation of lattice dislo-

cations at the GBs; (v) enhanced resistance to grain

sliding due to the existence of a monolayer or a sub-monolayer of a secondary phase; (vi) a combination of

ll rights reserved.

4112 J. Chen et al. / Acta Materialia 53 (2005) 4111–4120

any of the above mechanisms. Not all the mechanisms

listed above are mutually exclusive and most of them

are related in some subtle ways. Several studies [12,13]

have suggested that the reduced diffusivities of ionic spe-

cies with segregants (mechanism (i)) is the most viable

mechanism responsible for the improved creep resis-tance in Y-doped alumina. A simple atomic model, the

so-called ‘‘site-blocking’’ effect was proposed [14] based

on the different in ionic radii of Y3+ ion (0.89 A) and

Al3+ ion (0.50 A). It is envisioned that the larger sized

Y dopants effectively block the fast diffusive paths at

the GB. Cho et al. [15] had carried out a modeling study

on the segregation behavior of a large number of GB

models in alumina. Using a Voronoi type local volumeconstruction for Al sites and interstitial sites in the GB

region, they analyzed the statistical distribution of the

free volume in relation to doping with rare earth (RE)

ions of different sizes. The result supports the ‘‘site

blocking’’ mechanism to explain the enhanced creep

resistance due to RE doping. More recently, ab initio

calculations on doped GB models in alumina [16–18]

and the crystalline phases of the Y–Al–O system[19,20] have shown that the Y–O bond is stronger than

the Al–O bond due to increased covalency. These results

suggested that the enhanced creep resistance of doped

alumina may be attributed to the stronger bonding of

the segregated dopants at the GB.

The sintering, creeping, and grain growth of alumina

depend to a large extent on the diffusivity of Al and O.

Experimentally, there are two difficulties in directly mea-suring the diffusion coefficients of Al and O [21–25].

Firstly there is only one suitable isotope 26Al, which

has an inconveniently long half-life. Secondly, the pres-

ence of impurities dominates the transport properties,

making the diffusion coefficients of Al and O difficult

to determine. Le Gall and Prot [22–24] reported a series

of experimental measurements on the diffusivity of O in

crystal alumina, their sub-boundary and grain bound-ary. Heuer and Lagerlor [25] reviewed the self-diffusion

coefficients of Al and O in a-Al2O3. Harding et al. [26]

performed theoretical simulations and obtained the dif-

fusion coefficients and activation energies of Al and O in

the R7 GB in alumina using the Monte Carlo method

[26]. However, there is a lack of experimental and theo-

retical data on the diffusion coefficients of Al and O in

the Y-doped GB in alumina in general. Recently, Yos-hida et al. [27,28] started systematic studies of the GB

diffusivity and their dependence on various divalent

and tetravalent cation doping. Even more encouraging,

careful sliding experiments were carried out on bicrys-

tals containing [0001] symmetric tilt R7 GB using

HRTEM [29]. Such work opens the way for quantitative

study in RE doped GBs under carefully controlled

conditions.A detailed knowledge of the melting behavior, elastic

properties, and diffusion coefficients of Y-doped GBs

will certainly aid understanding of the ‘‘Y-effect’’. Re-

cently, it was reported that Y concentration in the R37GB is about four Y atoms/nm2 [7]. In this paper, we re-

port the results of molecular dynamic (MD) simulations

of the melting process, temperature-dependent elastic

constants, and GB diffusivity in a Y-doped ð0 1 �1 8Þ=½0 4 �4 1�=180� ðR37Þ GB model. The MD method is a

suitable tool to study the melting process, and for esti-

mating the diffusion coefficients and elastic constants

based on statistical fluctuations [30,31]. In our simula-

tion, the melting process is carried out from 300 to

2400 K using the constant number, constant pressure

and constant temperature (NPT) ensemble. After the

stable structure at a given high temperature (>1000 K)is obtained, the elastic constants and the diffusion coef-

ficients of the ions near the GB are calculated using the

constant number, constant volume and constant temper-

ature (NVT) ensemble. The paper is organized as fol-

lows. In the following section, Section 2, we briefly

describe the computational method used and the con-

struction of the initial simulation models. In Section 3,

our results are presented and analyzed in detail. A briefsummary is given in Section 4.

2. Computational method and initial model construction

2.1. Potential

In molecular dynamics, the key issue is the accuracyof the interatomic potential. Most researchers use pair

potentials whose parameters are fixed by simulating

the property of crystalline a-Al2O3 [32,33]. However,

pair potentials can not properly represent the angular ef-

fect of bonding. In the present work, we used a modified

Born–Mayer–Huggins (BMH) pair potential plus a

three-body term. This type of multi-atom potential has

been extensively used by Blonski and Garofalini[34,35] to study surface and bulk properties of

a-Al2O3. The BMH potential is given by

V ð2Þij ¼

qiqjrij

erfcðrij=bijÞ þ Aij exp �cijBij

� �; ð1Þ

where rij is the separation distance between ions i and j,

qi and qj are the ionic charges, Aij, Bij are potential

parameters, and erfc is the complementary error func-

tion. The three-body interactions are described by the

following term:

V ð3Þjik ¼ kjik exp

cijrij�r0ij

þ cikrik�r0ik

� �Xjik

� �if rij < Rij and rik < Rik;

V ð3Þjik ¼ 0 if rij P Rij and rik P Rik

ð2Þ

with kjik ¼ c1=2ij c1=2jk where the angular factor Xjik is given

by

Table 1

The potential parameters used for the Y–Al–O system

Two-body potential Three-body term

Atoms Atoms

i j Aij (eV) Bij (A�1) bij (A

�1) i j k kij (eV) kij (A) Rij (A) h0jik (�)

O–O 453.12 3.44 2.34 Al/Y–O–Al/Y 6.242 2.0 2.6 109.5

Al–O 1556.25 3.44 2.30 O–Al–O 149.324 2.8 3.0 109.5

Y–Oa 1345.11 3.44 2.30 O–Y–Oa 168.250 2.8 3.0 109.5

Al–Al 312.11 3.44 2.35

Y–Ya 215.14 3.44 2.35

Y–Ala 256.55 3.44 2.30

Ion charge qi: O (�2); Al (+3); Y (+3).a This paper.

J. Chen et al. / Acta Materialia 53 (2005) 4111–4120 4113

Xjik ¼ ½ðcos hjik þ cos h0jikÞ sin hjik cos hjik�2; ð3Þ

where hjik is the angle formed by the ions j, i, and k with

the ion i placed at the vertex. The parameters we usedfor Al and O were taken from [34] and are listed in

Table 1. These parameters were obtained by fitting the

coherence energy and the elastic constants of the crystal,

and have been successfully used to calculate the proper-

ties of bulk and interfacial a-Al2O3 [34,35].

Similar potential parameters involving Y are not

available in the literature. We obtained the necessary po-

tential parameters for interactions between Y–O andY–Al by fitting the lattice constant and elastic moduli

of crystalline Y3Al5O12 (YAG) using the ‘‘GULP’’ pro-

gram [36]. The lattice constant and elastic constants for

the YAG crystal obtained from the MD simulation

using these fitted potential parameters are listed in

Table 2. The deviation of the lattice constant from the

experimental value is less than 1%. The atomic coordi-

nates of YAG from the MD simulation are: Al1:(0.00, 0.00, 0.00), Al2: (0.375, 0.00, 0.25), O: (0.0298,

0.045, 0.653) and Y: (0.125, 0.00, 0.25) which are consis-

tent with the experimentally determined values [37]. We

therefore expect the potential parameters involving the

Y ion to be reasonably accurate.

2.2. Initial model and simulation process

The initial model for the R37 GB was constructed

using pair potential and the MIDAS code [38]. This

Table 2

Calculated and measured equilibrium lattice constants and elastic

constants of YAG

a (A) C11 (GPa) C12 (GPa) C44 (GPa) Young

modulus

(GPa)

Theor. 12.11 376.13 129.61 137.62 300.54

Exp. [35] 12.00 339.00 114.00 116.00 300.00

Error (%) 0.8 12.7 16.5 18.6 0.1

model was then refined to make it periodic by matching

the bulk crystalline planes away from the GB line. The

model contains a total of 760 atoms (304 Al and 456

O ions) and two oppositely oriented GBs. For compar-ison, a supercell of pure a-Al2O3 of the same size (760

atoms) was also constructed and is shown in Fig. 1(a).

The x-axis of this perfect crystal supercell is along the

surface ð0 1 �1 8Þ direction of a-Al2O3. The R37 GB

model is shown in Fig. 1(b). We divide the model into

four regions: two bulk regions and two GB regions.

Each GB region contains three layers of atoms as shown

in Fig. 1(b) and (c). Both the GB model and the perfectcrystalline supercell model were relaxed at T = 300 K

and P = 1 MPa using the NPT ensemble. The relaxed

supercell parameters are listed in Table 3. The GB model

has a cross-sectional area of 55.283 A2. Each GB region

contains 18 O atoms and 12 Al atoms. The bonding con-

figurations of the 12 Al ions in the GB region is such

that 2 have 5-fold, 8 have 6-fold and 2 have 8-fold

Al–O bonds. The longest Al–O bond is 1.992 A andthe shortest Al–O bond is 1.786 A.

Fig. 1. The initial atomic structure models. Each model contains 760

atoms: (a) pure a-Al2O3 crystal; (b) R37 GB; (c) Y + R37 GB. Note

that there are two oppositely directed GBs in (b) and (c) due to the

periodicity of the model.

Table 3

The MD calculated equilibrium lattice constant and elastic constants

of supercell models of pure a-Al2O3, R37 GB and Y + R37 GB at

T = 300 K and P = 1 MPa

a-Al2O3 Error (%) R37 GB Y + R37 GB

Theor. Exp. [39]. Theor. Theor.

a (A) 118.045 121.925 122.175

b (A) 4.714 4.711 4.735

c (A) 11.724 11.735 11.235

C11 (GPa) 549.194 496.00 10.7 498.237 512.981

C12 (GPa) 189.362 163.00 13.2 187.991 181.906

C44 (GPa) 175.157 147.00 13.9 154.218 161.720

4114 J. Chen et al. / Acta Materialia 53 (2005) 4111–4120

To study the effect of Y-doping on ionic diffusivity inthe GB region, we introduced Y to the R37 GB model at

two different concentrations: 1.8 and 3.6 Y atoms/nm2.

This level of concentration is consistent with the exper-

imentally observed dopant concentration of 4 Y atoms/

nm2 [7]. In the case of low dopant concentration of 1.8 Y

atoms/nm2, we replaced two Al atoms having 5-fold

Al–O bonding by two Y ions (labeled as Y1, Y3 in

Fig. 1(c)). For the higher dopant concentration of 3.6Y atoms/nm2, we replaced four Al atoms that were 5-

fold bonded by four Y ions (labeled as Y1, Y2, Y3,

Y4). Again, the structures were relaxed using the NPT

ensemble with T = 300 K and P = 1 MPa. The results

are listed in Table 3. At the higher Y concentration,

the lattice constant of the R37 GB supercell is increased

from a = 121.925 A in the undoped case to

a = 122.175 A.In the present MD simulations, we used the program

�MOLDY� [39]. The above three models were extended

to 1 · 2 · 2 supercells each containing 3040 atoms. The

simulations were carried out for temperatures ranging

from 300 to 2400 K. Two simulation ensembles, NVT

and NPT (P = 1 MPa), were chosen for two different

processes. The NPT ensemble was used to simulate the

melting process and to obtain stable structures at differ-ent temperatures. We used a time step of Dt = 0.002 ps

with 50,000 MD steps so that the total simulation time

was 0.1 ns for each temperature. The heating was done

in a step-by-step procedure. At a given temperature T

and after 50,000 MD steps, the temperature was in-

creased by 100 K and the next 50,000 MD steps were

carried out.

After the stable structures for the three modelsa-Al2O3, R37 GB, and Y + R37 GB under different

temperatures were obtained, we calculated the temper-

ature-dependent elastic constants by using the NVT

ensemble. For the NVT ensemble simulation, the MD

parameters were chosen to be Dt = 0.002 ps and

N = 25,000 with a total simulation time of 0.05 ns.

Due to the fact that the NVT ensemble converges much

faster than the NPT ensemble, a simulation time of0.05 ns was quite sufficient for our purpose. The elastic

constants Cpqrs are obtained from the statistical fluctu-

ation formula [30]

V 0h�10iph

�10jqh

�10lrh

�10msCpqrs

¼ � 4

kBTdðMijMlmÞ þ 2NkBT ðG�1

mi G�1jl þ G�1

li G�1jm Þ

þXa;b

gðrabÞsabisabjsablsabm

* +. ð4Þ

In Eq. (4), the first term is the fluctuation term (pres-

sure), the second term is the kinetic term (thermal),and the third term is the Born term (static). V0 is the vol-

ume containing the N particles, h0 is a 3 · 3 cell matrix,

kB is the Boltzmann constant, T is the temperature,

G�1is the inverse of the metric G = h 0h, sab is the vector

joining atom a and b of length rab, and g(rab) and Mij are

related to momentum and potential, respectively. De-

tailed descriptions concerning g(rab) and Mij can be

found in [30].The inclusion of a three-body term in the potential

implies that the Born term in Eq. (4) now consists of

two parts, the two body part and the three body part

such that

V 0h�10iph

�10jqh

�10lrh

�10msCpqrs

¼� 4

kBTdðMijMlmÞ þ 2NkBT ðG�1

mi G�1jl þG�1

li G�1jm Þ

þ hðtwo-body-BornÞijlmi þ hðthree-body-BornÞijlmi:ð5Þ

In accordance with classical diffusion theory, the diffu-

sion coefficient D is obtained from the mean-square dis-

placement (MSD) data as a function of time, or the

Einstein relationXi

hjriðtiÞ � riðt0Þj2i ¼ 6DNt; ð6Þ

where ri(ti) is the position of atom i at a given time ti,

and ri(t0) is the initial position. D is obtained from the

slope of an MSD curve vs. time. The activation energy,

Q, for GB diffusion is obtained by running MD simula-

tions at different temperatures T and fitting the results tothe Arrhenius relation

D ¼ D0 expð�Q=kBT Þ; ð7Þwhere D0 is a pre-exponential factor, Q is the activation

energy. From the Arrhenius plot of D vs. 1/T, the slope

yields the activation energy Q and the intersection withthe vertical axis gives D0.

The total radial distribution function (RDF), or

equivalently the pair correlation function (PCF), is ob-

tained from the sum of partial RDF. The partial RDF

gab(r) is calculated as [39]

gabðrÞ ¼1

q2

Xi

Xj 6¼i

dðr þ ria � rjbÞ* + !

;

where q is the density of system, a and b are ion species, iand j are different ions.

16

18

20

22

400 800 1200 1600 2000 2400

16

18

20

22

16

18

20

22

(a)

(c)

Temperature(K)

Tbulk

Tbulk

TGB

TGB

Tbulk

(b)

Vol

ume(

Å3 x1

03 )

Fig. 2. Volume vs. temperature in the MD simulations at a pressure of

1 MPa. The volume is for the enlarged system containing 3040 atoms:

(a) Pure a-Al2O3; (b) undoped R37 GB; (c) Y-doped R37 GB.

J. Chen et al. / Acta Materialia 53 (2005) 4111–4120 4115

3. Results and discussion

3.1. Melting temperature

The data for volume change from the MD simula-

tions for the three models a-Al2O3, R37 GB andY + R37 GB (3.6 Y atoms/nm2) at P = 1 MPa and tem-

perature T from 300 to 2400 K are shown in Fig. 2. For

crystalline a-Al2O3, the volume increases linearly with

temperature until T = 2234 K. At T = 2234 K, the

supercell volume shows a sudden upward jump, indicat-

ing the first-order melting of the crystal. Compared with

the experimental equilibrium melting temperature of

2313.16 K [40], our result is lower by 3.1% which mayindicate that the potential we used is a little too soft.

Fig. 2(b) shows the volume vs. temperature curve for

the pure R37 GB. This curve shows two jumps which we

interpret as evidence for two melting points. One is at

TGB = 1417 K for the GB and the other at

Tbulk = 2105 K for the bulk region. At TGB = 1417 K,

the volume jump is relatively small because the GB re-

gion is relatively small. For the bulk melting atTbulk = 2105 K, the volume change is considerably lar-

ger. The GB pre-melting temperature of 1417 K can be

compared with the experimental sintering temperature

of about 1400 K for a-Al2O3 [41]. The bulk melting tem-

perature of 2105 K in the R37 GB model is lower than

the bulk melting temperature of 2234 K in the pure

a-Al2O3 model. This can be interpreted as meaning that

in ceramic sintering, the effect of the GB is to lower thebulk melting temperature [42].

In Fig. 2(c), the volume vs. temperature curve for the

Y-doped R37 GB is shown. Similar to Fig. 2(b), the GB

pre-melting is at TGB = 1536 K and the bulk melting

temperature is at Tbulk = 2094 K. These numbers are

to be compared with the experimental sintering temper-

ature of 1550–1650 K for Y-doped alumina [41]. Com-

pared with the undoped R37 GB of Fig. 2(b), it isclear that Y-doping increases the pre-melting tempera-

ture at the grain boundary, but has little influence on

the bulk melting temperature. Recently, we have carried

out an ab initio calculation of the electronic structure

and bonding of a Y-doped R3 GB model in a-Al2O3

[16]. Previously, the electronic structure and bonding

in crystalline phases of the Y–Al–O system have been

investigated [19,20]. It was concluded that the Y–Obond is stronger than the Al–O bond because of in-

creased covalency. So, we believe that the increased

TGB in the Y-doped GB model is somehow related to

the stronger interaction between Y and O, although

the present MD simulation is a classical simulation.

For analyzing the melting process of undoped and

Y-doped R37 GB in a-Al2O3, snapshots of atomic con-

figurations at different temperatures are shown in Fig. 3.Fig. 3(a) and (b) displays the snapshots of undoped R37GB at T = 1417 K and Y-doped GB at T = 1536 K after

0.1 ns of simulation time, respectively. Compared with

the initial models of Fig. 1(b) and (c), the GB region be-

comes wider and the atomic structures are clearly much

more disordered. Furthermore, it is clear from Fig. 3(b)

that there exists a melting nucleate around the Y ions.

The formation of this melting nucleate is consistent with

the notion of a stronger Y–O bond [16–20]. Due to thestronger interaction between Y and O, Y will form a

cluster with surrounding O ions during the GB melting

process. In contrast, there is no such clustering of ions

in the GB region in the undoped R37 GB model of

Fig. 3(a). Of course, one can also argue that the GB re-

gion itself is a big melting nucleate during the melting

process as has been pointed out in [42].

To better illustrate the GB melting process, we havecalculated the RDF for the atoms in the GB region be-

fore and after the melting for the pure and undoped R37GB. The results are shown in Fig. 4. It is clear from

Fig. 4 that the first peak is the Al–O pair in the undoped

GB. After Y-doping, it also includes the Y–O pair at a

Fig. 3. Snapshot for atoms after 0.1 ls of MD simulation time in the

R37 GB: (a) undoped R37 GB model at T = 1417 K; (b) Y + R37 GB

model at T = 1536 K (with dopant concentration of 3.6 Y atoms/nm2).

4116 J. Chen et al. / Acta Materialia 53 (2005) 4111–4120

slightly larger separation. The Al–Al and Al–O peaks

are also indicated. Because of the small number of

atoms in the GB region, these peaks are rather spiky.

However, we can still see that the Al–Al and the O–O

peaks become broader or disappear at the temperature

2 4 6 8 100

2

4

6

8

10

120

2

4

6

8

10

12

2 4 6 8 10

Y- O(Al-O)

O-O

(c)

(a)

O-O

Al-O Al-O

Al-Al

RD

F (

arbi

trar

y un

it)

Y- O(Al-O)

Al-Al

Radius (Å)Radius (Å)

(d)

O-OAl-Al

O-OAl-Al

(b)

Fig. 4. Calculated RDF for atoms in the GB region: (a) undoped GB

at T = 300 K; (b) undoped GB at T = 1417 K; (c) Y-doped GB at

T = 300 K; (d) Y-doped GB at T = 1536 K.

above the melting temperature. Thus, the RDF dia-

grams fully support the description of the GB melting

process described above.

3.2. Elastic constants of three models

We next report the results of elastic constant cal-

culation for the three models. Shear moduli C0 = (C11 �C12)/2 andC44 are used for monitoring themelting process.

It should be pointed out that the elastic constantsC11,C12,

C44 discussed here refer to the orthorhombic supercell of

Fig. 1 and should not be confused with theCij of crystalline

a-Al2O3 in the corundum structure [43]. Fig. 4(a) shows the

temperature dependence of the elastic constants of thesupercell model of pure a-Al2O3. A linear decrease in C11

as temperature increases from 300 to 2234 K is observed.

Near the melting point of T = 2234 K, the shear moduli

C0 = (C11 � C12)/2 and C44 are still positive but they de-

crease very rapidly. This signals that the crystal starts to

soften and eventually melts. Our results are consistent with

the Born instability rule that melting is preceded by a con-

tinuous softening of the lattice with increasing temperatureand volume. The crystal no longer has adequate rigidity to

resist melting when its elastic shear modulus is sufficiently

small [44].

Fig. 5(b) and (c) shows the temperature dependence

of the elastic constants for the undoped R37 GB model

and the Y-doped R37 GB (3.6 Y atoms/nm2), respec-

tively. Similar to the volume variations in Fig. 2(b),

there are two transition points for elastic constants inboth models. They occur at TGB = 1417 K and

Tbulk = 2105 K for the undoped R37 GB, and

TGB = 1536 K and Tbulk = 2094 K for Y + R37 GB. At

the pre-melting temperature of 1417 K for the undoped

GB and 1536 K for the Y-doped GB, the shear moduli

C 0 and C44 still have fairly large values. They start to de-

crease rapidly after the pre-melting temperature. At the

temperature that corresponds to bulk melting, the shearmoduli C 0 and C44 decrease steeply for both the un-

doped and the Y-doped GB, indicating that the bulk re-

gion starts to melt.

Comparing the elastic constants of the three models,

the pure crystalline a-Al2O3 always has the largest elas-

tic constants at all temperatures, and the undoped R37GB has the smallest elastic constants, with Y-doped

R37 GB somewhere in between. The numerical valuesof the elastic constants of the three models at

T = 300 K are listed in Table 3. These results indicate

that Y-doping increases the elastic constants of the

GB model. Hence, the present MD results support

the conclusion of our ab initio strain–stress calculation

of the Y-doped R3 GB in a-Al2O3 [16]. In [16], the the-

oretical strain–stress data show that the Y-doped GB

has a higher stress than the undoped GB under thesame strain which was attributed to the stronger Y–O

bond.

100

200

300

400

500

400 800 1200 1600 2000 2400

100

200

300

400

500

100

200

300

400

500

C11

C22

C44

C '

(b)

C11

C22

C44

C '

Temperature(K)

(c)

Ela

stic

Mod

uli (

GP

a)

C11

C22

C44

C '

Tbu lk

Tbu lk

Tbulk

TG B

TG B

(a)

Fig. 5. Elastic constants as a function of temperature from MD

simulation: (a) pure a-Al2O3 model; (b) undoped R37 GB model;

(c) Y-doped R37 GB model.

5000 10000 15000 20000 25000 300000

20

40

60

80

100

120

5000 10000 15000 20000 25000 300000

20

40

60

80

(a)O-1812k

Al-1010kO-1010kAl-1505kO-1505kAl-1598kO-1598k

Al-1812k

The

mea

n-sq

uare

d di

spla

cem

ent (

Å2 )

(b) Y-1794k

O-1794kAl-1794k

Y-1630kY-1536k

O-1630k

O-1536kAl-1536k

Al-1630k

No. of MD time steps

Fig. 6. Mean-square displacements vs. simulation time: (a) undoped

R37 GB model at T = 1010, 1598 and 1812 K; (b) Y-doped R37 GB at

T = 1536, 1630, and 1794 K.

0

20

40

60

80

100

120

140

4 5 6 7 8 9 10

0

10

20

30

40

50

(a)

Al in undoped GB O in undoped Gb O in undoped GB-experiment [24]

D (

cm2 /s

x10

-10 )

(b)

1/T (104/k)

Al in Y-doped Σ37 GB(3.6 Y atom/nm2)O in Y-doped Σ37 GB(3.6 Y atom/nm2)Y in Y-doped Σ37 GB(3.6 Y atom/nm2)

Al in Y-doped Σ37 GB(1.8 Y atom/nm2)O in Y-doped Σ37 GB(1.8 Y atom/nm2)Y in Y-doped Σ37 GB(1.8 Y atom/nm2)O in Y-doped GB -experiment [24]

Fig. 7. Grain boundary diffusion coefficients D of O, Al, and Y as a

function of temperature (>1000 K): (a) undoped R37 GB; (b) Y-doped

R37 GB at two different dopant concentrations. The experimental

values are taken from [24].

J. Chen et al. / Acta Materialia 53 (2005) 4111–4120 4117

3.3. Diffusion coefficients and activation energy

We have also calculated the diffusion coefficients of

the ions in the GB for temperatures ranging from 1000to 2200 K. The MSD of atoms in the GB region as a

function of MD simulation steps at temperatures 1010,

1505, 1598, 1812 K for the undoped GB, and at 1536,

1603, 1794 K for Y-doped R37 GB are displayed in

Fig. 6(a) and (b), respectively. The slopes of the curves

are proportional to the diffusion coefficients D through

the Einstein relation of Eq. (6) (see Fig. 6).

The calculated diffusion coefficients D as a functionof 1/T for Al, Y and O at the GB region are shown in

Fig. 7(a) and (b) for the undoped and Y-doped R37GB, respectively. For the Y-doped GB, data for two dif-

ferent concentrations (1.8 and 3.6 Y atoms/nm2) are

shown. Also shown in Fig. 7 are the experimental values

for O diffusion in GB [24]. It should be pointed out that

these experimental values are for the averaged behavior

involving all types of GBs, not for a particular grainboundary in Al2O3. In general, diffusion coefficients of

Al and O have the same order of magnitude. However,

Y ions have a higher value of D at the GB region than

Al and O. From the temperature dependence of D

shown in Fig. 7(a), we can identify two distinctively dif-

ferent regions. In the undoped R37 GB, the low temper-

ature region is from 1010 to 1417 K and the high

-63.0 -62.5 -62.0 -61.5 -61.0-6.0-5.5-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.5

-63.0 -62.5 -62.0 -61.5 -61.0-13.0

-12.5

-12.0

-11.5

-11.0

-10.5

-10.0

-9.5

-9.0(a)

o

A

B

y-ax

is (

Å)

x-axis (Å)

(b)

o

A

B

z-ax

is (

Å)

x-axis (Å)

Fig. 8. Trace of diffusion of one Y ion in the GB at T = 1700 K and

pressure of 1 MPa (dopant concentration 3.6 Y atoms/nm2). Point A is

the initial position, point B is the end position: (a) trace in the XY

plane; (b) trace in the XZ plane.

4118 J. Chen et al. / Acta Materialia 53 (2005) 4111–4120

temperature region is from 1417 to 2105 K. In the

Y-doped R37 GB, the corresponding two temperature

regions are from 1002 to 1536 K, and from 1536 to

2094 K. Experimental data show that O diffusion coeffi-

cient decreases after the GB is doped with Y [24]. Our

results in Fig. 7(a) and (b) show that both Al and O dif-fusion coefficients decrease after Y-doping. Our results

in Fig. 7(b) also show that the diffusion coefficients are

larger at the concentration of 1.8 Y atoms/nm2 than at

the 3.6 Y atoms/nm2. This implies that a higher dopant

concentration in the grain boundary region results in the

decrease of ionic diffusivity.

Activation energy can be estimated from tempera-

ture-dependent curve of D by fitting to the Arrheniusequation. In Table 4, we list the activation energies Q

and pre-diffusion parameters D0 for O, Al, and Y in

the low and high temperature regions. We found that

for undoped R37 GB, QO > QAl and for Y-doped R37GB QO > QAl > QY. The distinct difference in diffusion

activation energy in the low and high temperature re-

gions suggests that there should be at least two diffusion

mechanisms for the two different temperature ranges.There is very limited work in the literature on the diffu-

sion activation energy in Al2O3 GBs. In the high temper-

ature region, comparable theoretical and experimental

results exist. Using a Monte Carlo method, Harding

et al. [26] estimated the activation energy for Al and O

in R7 GB in Al2O3 to be 1.2 and 1.3 eV, respectively.

They proposed that the diffusion mechanism at the GB

is a vacancy-induced type. These values are in goodagreement with our calculated values of 1.43 and

1.50 eV for Al and O in R37 GB in the high temperature

region. The experimentally measured activation energy

of 1.8 eV for Al in poly-crystal samples of a-Al2O3 [21]

is also in line with our calculated values. In the low tem-

perature region, there is no comparable data available in

the literature. It is likely that the diffusion mechanism is

the interstitial-induced type, which requires a lower acti-vation energy than that of the vacancy-induced type.

Table 4 also shows that the activation energies of Al

and O in the GB region are increased after being doped

with Y. This is consistent with the experimental facts

that Y-doping decreases ion diffusion in the GBs [1–8].

Table 4

Calculated activation energies Q and pre-diffusion parameters D0 for O, Al

O

D0 (cm2/s) Q (eV)

R37 GB (low Ta) 8.315 · 10�10 0.139

R37 GB (high Ta) 1.695 · 10�5 1.503

Y + R37 GB (1.8 Y/nm2, low T) 1.547 · 10�11 0.185

Y + R37 GB (1.8 Y/nm2, high T) 1.315 · 10�6 1.786

Y + R37 GB (3.6 Y/nm2, low T) 1.612 · 10�11 0.197

Y + R37 GB (3.6 Y/nm2, high T) 3.138 · 10�6 1.811

a Low T means low temperature region; high T means high temperature r

Chao et al. [14] have proposed a model based on a

‘‘site-blocking’’ effect to explain this phenomenon.

Chao�s mechanism is based on the difference in ionic ra-dii of Y3+ (0.89 A) and Al3+ (0.50 A). The Y dopant will

effectively block the fast diffusive paths at GBs due to its

size, which in turn will cause the decreased diffusion of

other ions. This model is mainly from the point of view

of geometry and does not take the temperature effect

into consideration. Our MD simulation results appear

to partially support this mechanism. Our previous work

shows that Y forms a stronger bond with O than Al, andmay also interact with Al [16]. Thus, it will require a

higher energy for the migration of Al and O after

Y-doping. We believe that at low temperature

(T < 1536 K), the ‘‘site-blocking’’ effect is due to the rel-

atively larger Y ion. However, at high temperature

(T > 1536 K), the occurrence of a melting nucleate in-

side the GB region after Y-doped GB as shown in Fig.

3(b) is the main reason. Here, the ‘‘site-blocking’’ effectis related to the melting nucleate around the dopant Y

and is caused by the specific Y–O bonding that depends

on the local geometry.

In Fig. 8, we trace the movements of a Y ion

at T = 1536 K and P = 1 MPa. It can be seen that the

motion is primarily confined to the lateral plane parallel

to the GB with little vertical motion off the GB plane.

and Y at the R37 GB

Al Y

D0 (cm2/s) Q (eV) D0 (cm

2/s) Q (eV)

5.284 · 10�9 0.153

3.419 · 10�5 1.431

3.481 · 10�10 0.189 3.657 · 10�10 0.124

4.873 · 10�5 1.844 1.284 · 10�5 1.112

6.445 · 10�11 0.162 8.241 · 10�11 0.101

3.419 · 10�6 1.721 4.592 · 10�7 1.035

egion.

J. Chen et al. / Acta Materialia 53 (2005) 4111–4120 4119

The distance between the initial position A and the final

position B for Y is about 4.5 A at 50 ps simulation time.

From the results of the sub-diffusivity Dx (x-component)

of the Y ion, all Dx have very small values at all temper-

atures. This indicates that it is difficult for Y to diffuse

into the bulk region. They mostly segregate along theGB.

4. Conclusion

In this paper, we report the results of MD simulation

of Y-doping in R37 GB in alumina at high temperature.

The melting process, the temperature dependence of theelastic constants, and the GB diffusion coefficients of

pure a-Al2O3, R37 GB, and Y-doped R37 GB models

were calculated and compared. Our results have pro-

vided further insights into the nature of Y-doping in

the GBs of alumina. We have shown that the ceramic

melting process of alumina can be divided into two

parts: a GB pre-melting part and a bulk melting part.

Y-doping at the GB increases the GB pre-melting tem-perature and they form a melting nucleate in the GB re-

gion around the Y ion. Y-doping also slightly increases

the elastic constants of model over the undoped GB

model. These effects can be explained by the stronger

Y–O bonds compared to the Al–O bonds.

Our results also show that there should be two differ-

ent diffusion mechanisms for both Al and O in the GB.

We suggest that in the low temperature region (lowerthan the GB pre-melting temperature) the diffusion is

interstitial and in the high temperature region (higher

than the GB pre-melting temperature), it is mainly va-

cancy-induced. Y-doping reduces the diffusion coeffi-

cients and improves the activation energies of both Al

and O in the GB. Our results partially support the

‘‘site-blocking’’ model as an explanation for the decrease

in the diffusion coefficient. However, at higher tempera-tures, the decrease is due to the creation of a melting

nucleate around Y dopant. Finally, we must point out

that the present simulation is a classical one with no

quantum effect included. Ideally, accurate free energy

calculation based on ab intitio approach would be desir-

able. However, such calculations on large systems as in

the present study are still computationally prohibitive

but will be attempted in the near future.

Acknowledgements

Work is supported by the US Department of Energy

under the Grant No. DE-FG02-84DR45170. This

research used the resources of NERSC supported by

the Office of Science of DOE under the Contract No.DE-AC03-76SF00098.

References

[1] Gulgun MA, Putlayev Valery, Ruhle Manfred. J Am Ceram Soc

1999;82(7):1849.

[2] Li Y, Wang CM, Chan HM, Rickman JM, Harmer MP, Chabala

JM, et al. J Am Ceram Soc 1999;82:1497.

[3] Thompson AM, Soni KK, Chan HM, Harmer MP, et al. J Am

Ceram Soc 1997;80(2):373.

[4] Cook RF, Schrott AG. J Am Ceram Soc 1988;71:50.

[5] Taylor RI, Cood JP, Brook RJ. J Am Ceram Soc 1974;57:539.

[6] Atsunaga K, Nishimura H, Muto H, Yamamoto T, Ikuhara Y.

Appl Phys Lett 2003;82:1179;

Nishimura H, Matsunaga K, Sato T, Yamamoto T, Ikuhara Y. J

Am Ceram Soc 2003;86:574.

[7] Gemming T, Nufer S, Kurtz W, Ruhle M. J Am Ceram Soc

2003;86(4):590.

[8] Gall M Le, Huntz AM, Lesage B, Monty C, Bernadini J. J Mater

Sci 1995;30:201.

[9] Thompson AM, Soni KK, Chan HM, Harmer MP, Williams DB,

Chabala JM, et al. J Am Ceram Soc 1997;80:373.

[10] Bruley J, Cho J, Chan HM, Harmer MP, Rickman JM. J Am

Ceram Soc 1999;82(10):2865.

[11] Wang CM, Cargill GS, Chan HM, Harmer MP. Acta Mater

2000;48:2579.

[12] Yoshida H, Ikuhara Y, Sakuma T. Philos Mag A 1999;79(5):249.

[13] Cho J, Wang C-M, Chan HM, Rickman JM, Harmer MP. J

Mater Sci 2002;37(1):59.

[14] Cho J, Chan MP, Harmer MP, Rickman JM, Thompson AM. J

Am Ceram Soc 1997;80(4):1013;

Cho J, Chan HM, Harmer MP, Rickman JM, Thompson AM. J

Am Ceram Soc 1998;81:3001.

[15] Cho J, Rickman JM, Chan HM, Harmer MP. J Am Ceram Soc

2000;83(2):344–52.

[16] Jun Chen, Rulis Paul, Xu Y-N, Ouyang Lizhi, Ching WY. Acta

Mater 2005;53(2):403.

[17] Fabris Stefano, Nufer S, Marinopoulos AG, Elsasser C. Phys Rev

B 2002;66:155415.

[18] Fabris Stefano, Elsasser C. Acta Mater 2003;71:51.

[19] Xu Y-N, Ching WY, Xu Y-N. Phys Rev B 1999;59:10530.

[20] Ching WY, Xu Y-N. Phys Rev B 1999;59:12815.

[21] Stubican VS, Osenbach JW. Solid State Ionics 1984;12:375.

[22] Prot D, Monty C. Philos Mag A 1996;73:899.

[23] Gall M Le, Huntz AM, Monty C. Philos Mag A 1996;73:919.

[24] Prot D, Gall MLe, Lesage B, Huntz AM, Monty C. Philos Mag A

1996;73:935.

[25] Heuer AH, Lagerlof KPD. Philos Mag Lett 1999;79(8):619.

[26] Harding JH, Atkinson KJW, Grimes RW. J Am Ceram Soc

2003;86(4):554.

[27] Yoshida H, Ikuhara Y, Sakuma T. Acta Mater 2002;50:2955.

[28] Yoshida H, Hashimoto S, Yamamoto T. Acta Mater 2005;53:433.

[29] Matsunaga K, Nishimura H, Hanyu S, Muto H, Yamamoto T,

Ikuhara Y. Appl Surf Sci 2005;241:75.

[30] Cagin T, Ray JR. Phys Rev B 1988;38:7940.

[31] Ray John R, Moody Michael C. Phys Rev B 1985;32(2):733.

[32] Matsui M. Miner Mag A 1994;58:571.

[33] Gutierrez G, Johansson B. Phys Rev B 2002;65:104202.

[34] Blonski S, Garofalini SH. J Phys Chem 1996;100:2201.

[35] Blonski S, Garofalini SH. Catal Lett 1994;25:325.

[36] Gale Julian. ‘‘GULP’’, a general utility lattice program. Available

from: http://www.ch.ic.ac.uk/gale/Research/gulp.html.

[37] Stoddart PR, Ngoepe PE, Mjwara PM, Comins JD, Saunders

GA. J Appl Phys 1993;73:7298.

[38] W.Y. Ching [unpublished].

[39] Refson K. ‘‘MOLDY’’, release 2.13, a general-purpose molecular

dynamics code; 1998. Available from: http://www.earth.ox.uk/

~keith/moldy.html.

4120 J. Chen et al. / Acta Materialia 53 (2005) 4111–4120

[40] See for example: Hart LD, editor, Alumina chemicals, science and

technology handbook. Associate Editor Lense E, 1990. Wester-

ville (OH): The American Ceramics Society, Inc.

[41] Row IM, Cannon RM, Gulgun MA, et al. J Am Ceram Soc

2003;86(4):658.

[42] Lutsko JF, Wolf D, Phillpot SR, Yip S. Phys Rev B 1989;40(5):

2841.

[43] Gieske JH, Barsch GR. Phys Status Solidi 1968;29:121.

[44] Jin ZH, Gumbsch P, Lu K, Ma E. Phys Rev Lett 2001;87(5):

055703.