Scripta Materialia 49 (2003) 379–386

www.actamat-journals.com

A theoretical study of Gasarite eutectic growth

Liu Yuan *, Li Yanxiang

Department of Mechanical Engineering, Tsinghua University, Room 204, Welding Hall, Beijing 100084, PR China

Received 15 October 2002; received in revised form 22 May 2003; accepted 28 May 2003

Abstract

A theoretical model has been developed to get the relation between the porosity, the inter-pore spacing and the

processing parameters in the metal–gas eutectic (Gasarite eutectic) solidification. The theoretical relation between the

inter-pore spacing and the solidification rate also can be described as a simple expression (v � L2 ¼ A) as that presentedin the classical Jackson–Hunt eutectic growth model.

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

Keywords: Materialia; Metal–gas eutectic; Directional solidification; Porous material; Medeling

1. Introduction

Porous materials with a regular distribution of

pores can be produced by using a novel method

named Gasar developed by the Dnepropetrovsk

Metallurgical Institute in Ukraine [1]. Most metal–

hydrogen binary systems also have a eutectic de-

composition as the classic eutectic systems. Thesemetals can be melted, saturated with hydrogen,

and then directionally solidified into a porous

structure, as shown in Fig. 1 [2]. During solidifi-

cation, solid metal and hydrogen simultaneously

form by a gas-eutectic reaction, resulting in a po-

rous structure with elongated gaseous pores filled

by hydrogen. In Japan, castings with elongated

pores were named Lotus-type porous metals, be-cause they look like the lotus roots.

* Corresponding author. Tel.: +86-10-6277-3640; fax: +86-

10-6277-3637.

E-mail address: yuanliu@mail.tsinghua.edu.cn (Y. Liu).

1359-6462/03/$ - see front matter � 2003 Acta Materialia Inc. Publis

doi:10.1016/S1359-6462(03)00330-0

Gasar structures offer a number of attractive

properties [3], such as significantly improved

thermal conductivity over conventionally pro-

cessed porous materials. In addition, these mate-

rials combine good mechanical properties with

reduced weight. The ultimate tensile strength and

the yield strength of Gasar materials with the cy-

lindrical pores orientated parallel to the tensiledirection decrease linearly with increasing poro-

sity. In other words, the gas pores do not cause

stress concentration during the stretching process

[4]. Due to these attractive properties, the Gasar

materials have many potential applications.

These initial studies about the Gasar process are

focused on the nucleation mechanism of gaseous

hydrogen bubbles in Gasar solidification [5,6], andthe experimental studies on the processing condi-

tions to produce Gasar structures [7–12]. How-

ever, no systematic investigations have been made

to determine the relation between the porosity, the

inter-pore spacing and the processing parameters

such as the gas pressure during melting and

hed by Elsevier Science Ltd. All rights reserved.

Fig. 1. Gasar process and the obtained Gasar structure [2].

380 Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386

solidification processes, the solidification rate andthe melt temperature. In this paper, this problem is

examined through a solution of the solute distri-

bution in front of the solidifying front. The solu-

tion procedure is similar to that adopted in

Jackson–Hunt model [13]. These theoretical ana-

lyses are helpful for understanding the Gasarite

eutectic growth.

2. Model analysis

2.1. Solute field

To simplify the model analysis, three hypothe-

ses are supposed:

1. The pore size and inter-pore spacing are homo-

geneous.

Fig. 2. A schematic diagram of the Gasarite growth

2. No fluid convection occurs in the direction par-

allel to the solidifying front.

3. There is a uniform temperature in the gas pores

that equals the melting point of the metal.

Fig. 2 is a schematic diagram of the Gasarite

growth and the corresponding cylindrical coordi-

nate system used for solution of the solute field. rgis the radius of the gas pore and rs is one-half of

the inter-pore spacing. The porosity can be ex-

pressed as:

e ¼ r2g=r2s ð1Þ

When the escaping rate of H2 from the melt is

negligible, under the steady-state solidification

conditions the porosity, e, is only dependent upon

the gas pressure and the melt temperature.

The solute distribution in the melt can be de-

scribed with the following steady-state diffusion

equation:

o2Cor2

þ 1

roCor

þ o2Coz2

þ vDL

oCoz

¼ 0 ð2Þ

in which v is the solidification rate, DL is the dif-

fusion coefficient of the gas atom in the melt. The

corresponding boundary conditions are expressed

as:

oCoz

����z¼0

¼ vDL

PRTm

ðat the gas phase front; 06r6rgÞ

ð3Þ

oCoz

����z¼0

¼� vDL

ðC�ls�CsÞ

ðat the solid phase front; rg < r6rsÞ ð4Þ

and the corresponding coordinate selection.

Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386 381

where P is the gas pressure in the pore. R is the gas

constant and Tm is the melting point of the metal.

C�ls is the average content of H2 in the melt at the

solidifying front. Cs is the average content of H2

in the solid metal, which depends upon the gas

pressure, and the average temperature of the so-

lidified metal.

After a strict mathematical manipulation, the

general solute distribution can be written as:

C ¼ C1 þX1n¼0

Bn e�xnzJ0ðknr=rsÞ ð5Þ

C�ls ¼

C1 � ð1� eÞCs � PRTm

� e þ vDL

PRTm

� Cs

� �P1n¼1

4eJ21ðffiffie

pknÞ

ð1�eÞk2nxnJ20 ðknÞ

e � vDL

P1n¼1

4eJ21ðffiffie

pknÞ

ð1�eÞk2nxnJ20 ðknÞ

ð9Þ

xn ¼ ðv=2DLÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðv=2DLÞ2 þ ðkn=rsÞ2

qð6Þ

where C1 is the solute concentration at z ! 1 and

equal to Cm––the saturated solute concentration in

the melt (see Section 3.1). J0ðxÞ is the zero-order

Bessel function and kn is the root of J1ðxÞ ¼ 0

(J1ðxÞ is the first-order Bessel function). Bn can be

derived from the following boundary conditions:

B0 ¼ ðC�ls �CsÞð1� eÞ� P

RTm� e

h iðn¼ 0Þ

Bn ¼� 2ffiffie

pJ1ð

ffiffie

pknÞ

knxnJ20 ðknÞ

� �vDL

ðC�ls �CsÞþ P

RTm

h iðnP1Þ

8<:

ð7Þ

The classical Jackson–Hunt model simplified the

solute field by assuming v=2DL � 0 in Eq. (6), andC

�ls � CE (the eutectic composition) in Eq. (4). In

this paper, these assumptions are relaxed.

2.2. Average content of hydrogen at solid phase

growth front

The average content of H2 at the growing solid

front can be obtained through integrating thesolute field C at z ¼ 0.

C�ls ¼

R rsrg2prCjz¼0 dr

pðr2s � r2gÞ

¼R rsrg2pr � C1 þ B0 þ

P1n¼1 BnJ0ðknr=rsÞ

� �dr

pðr2s � r2gÞ

¼ C1 þ B0 þvDL

ðC�ls

�� CsÞ þ

PRTm

�

X1n¼1

4eJ 21 ð

ffiffie

pknÞ

ð1� eÞk2nxnJ 2

0 ðknÞð8Þ

Substituting B0 into Eq. (8) above gives:

It can be seen from Eq. (9) that the solidification

rate, the gas pressure and the melting temperature

all have effects on C�ls.

3. Results and discussion

The Cu–H2 system is used as an example to testand evaluate the model established above. Fig. 3 is

the phase diagram of Cu–H2 eutectic system.

Table 1 lists the used parameters related to Cu–H2

system.

3.1. Porosity

If all of the formed bubbles are trapped as poresduring the steady-state solidification process, a

simple equation based on the ideal gas law can be

used to calculate the porosity in the final casting

[10]:

e ¼ ðCm � CsÞRTm

ðCm � CsÞRTm þ 1000000 � P �MH2

qs

� � ð10Þ

where Cm and Cs are the hydrogen concentra-tions (in ppm) in the liquid and the solid at the

0.000 0.001 0.002 0.003 0.004 0.0051340

1345

1350

1355

1360

1.01x105 Pa

L

S

S+L

Tem

pera

ture

(K

)

XH

1.01x107 Pa

Fig. 3. Phase diagram of copper-hydrogen eutectic system [6].

382 Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386

solidification front. qs is the density of the solid

metal and MH2is the molecular weight of H2. P is

the gas pressure (in Pa) in the pore. For Cu–H2

system, Cm and Cs can be expressed as [8]:

Cm ¼ nðT ÞffiffiffiffiffiffiffiPH2

p¼ 0:72159 expð�5234=T Þ

ffiffiffiffiffiffiffiPH2

pð11Þ

0.0 2.0x105 4.0x105 6.0x105 8.0x1050.0

0.1

0.2

0.3

0.4

0.5

0.6 Pg=P

H2, P

Ar=0 Pa, T=1523K

Pg=P

H2+P

Ar=8.08x105Pa, T=1523KPo

rosi

ty (

e)

PH2 /Pa

Fig. 4. Influence of the partial pressure of hydrogen on the

porosity of Gasar Cu.

Cs ¼ n0ðT ÞffiffiffiP

p

¼ 0:43399 expð�5888=T ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPg þ Pc þ Ph

pð12Þ

where PH2is the partial pressure of H2. Pg is the

total gas pressure in the furnace atmosphere. In

general, the height of the casting is not higher than

0.2 m in the Gasar solidification. If the average

height is selected as 0.125 m, then the average

hydrostatic pressure Ph is about 104 Pa for Cu–H2

system (Ph ¼ qlgh ¼ 8000 10 0:125 � 104 Pa).

The pore diameter ranges from tens of microns toseveral millimeters in the Gasar structures [17]. If

the average radius is selected as 30 microns, the

average capillary pressure Pc is 8.5 · 104 Pa for

Cu–H2 system (Pc ¼ 2rl=rg ¼ 2 1:28=30 10�6 �8:5 104 Pa).

Table 1

The parameters of Cu–H2 system used for calculation

ml (K/molm3) DHðliquid CuÞ (cm2/s) rls (1356

(J/m2)

0.01026 [6] 1:46 10�2 expð�18900=RT Þ [14] 0.178 [15

It can be seen from Fig. 4 that, if the pure hy-

drogen is used singly, the porosity first increases

up to a maximum, then starts to drop with in-

creasing hydrogen pressure. In most cases, theGasar process uses the mixture of hydrogen and

argon because that the hydrogen bubbles are not

easy to form and float away before the solidifying

front under this kind of condition [10]. In other

words, the use of the mixed gases favors the cou-

pled growth between the solid and the gaseous

phase, therefore favors the achievement of Gasar

structures. In this situation, the porosity increasescontinually with rising hydrogen partial pressure

at a constant argon partial pressure. This depen-

dence of porosity on the partial pressures of argon

and hydrogen arises because the amount of hy-

drogen dissolved in the melt is proportional to the

square root of hydrogen pressure, whereas the

density of hydrogen in the pore is proportional to

the total applied pressure.As for the influence of the melt temperature on

the porosity, it is apparent that the porosity in-

creases continually with the rise of the melt tem-

perature because that a higher melt temperature

K) DHm (J/g) qs (g/cm3) ql (1356 K)

(g/cm3)

] 211.85 [16] 8.9 8.0

Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386 383

will lead to a higher solubility of hydrogen in the

molten metal (see Eq. (11)).

3.2. Solute distribution

According to Eqs. (5)–(10), the solute distribu-

tion in front of the solidifying front and the av-

erage hydrogen content at the solidifying front canbe calculated. The two-dimensional solute distri-

butions at different solidification rates are shown

in Fig. 5(a), (b). It can be found that there exists a

slight hydrogen concentration gradient in the

radial direction, but the strongest concentration

gradient develops in the z axial direction when the

solidification rate is as small as 50 lm/s. With in-

creasing solidification rate, a sharp concentrationgradient also occurs in the radial direction. It is

obvious that the diffusion of the hydrogen atom in

the radial direction will become a restrictive factor

for the growth of the hydrogen pore. In this situ-

ation, rs and rg will be automatically adjusted to

maintain the cooperative growth between the solid

and the gaseous phase. This point is the major

reason why the inter-pore spacing decreases withrising solidification rate discussed in Section 3.3.

3.3. Inter-pore spacing

By using the minimum undercooling criterionas that in the Jackson–Hunt model, the required

undercooling for the growth of the solid phase and

the inter-pore spacing can be calculated. However,

for the metal–gas eutectic solidification, the

Fig. 5. Solute distribution in front of the solidification front (T ¼

growth of the gas pore does not need any und-

ercooling, so that only the undercooling for the

growth of the solid phase is calculated here.

Fig. 6 shows the interfacial energies balancebetween the solid, the liquid and the gas phases.

Generally speaking, the surface energy of the solid

phase ðrsÞ is much higher than that of the liquid

(rl) and the interfacial energy of liquid/solid (rls).

With rs P rls þ rl, the average curvature radius

(rsl) of the solid/liquid interface should be written

as rsl ¼ ðrs � rgÞ ¼ ð1�ffiffie

pÞrs.

The required undercooling for the growth of thesolid can be described as:

DT ¼ mlðCE � C�lsÞ þ

rlsTEDHmð1�

ffiffie

pÞrsqs

ð13Þ

where ml is the slope of liquidus in metal–gas eu-

tectic phase diagram. DHm and qs are the fusion

enthalpy and the density of the solid metal, re-

spectively. CE and TE are the eutectic compositionand the eutectic temperature of the metal–gas eu-

tectic system, respectively.

It can be seen from Eq. (9) that C�ls is not a

simple function of rs, so that it is not possible to

obtain a simple relationship––L ¼ A � v�n (L ¼ 2rs,L is the inter-pore spacing) as that in the Jackson–

Hunt model through directly applying oDT=ors ¼0 (minimum undercooling criterion) to Eq. (13).But the calculated results in Fig. 7 show that, al-

though the form of Eq. (9) is complex, actually, C�ls

has a linear relation with rs at a constant solidi-

fication rate. In this case:

1400 K, rs ¼ 200 lm, PH2¼ PAr ¼ 5:05 105 Pa, e ¼ 0:25).

rs /µ m

v=500µm/s

v=100µm/s

v=200µm/s

* lsC

0 20 40 6031.0

31.5

32.0

32.5

33.0

33.5

34.0

/ppm

Fig. 7. Relationship between C�ls and rs (z ¼ 0, PH2

¼ PAr ¼5:05 105 Pa, T ¼ 1400 K).

1000 200 300 400 5000.000

0.015

0.030

0.045

0.060

0.075

T=1400K,

PH2=4.04x105Pa

PAr

=2.02x105Pa

T=1500K,

PH2=P

Ar

T=1400K,

PH2=P

Ar=5.05x10

5Pa

=5.0 Pa5x105x105

Fig. 8. Relationship between the parameter, F , in Eq. (14) and

the solidification rate.

Fig. 6. A schematic diagram of the interfacial energies balance.

384 Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386

C�ls ¼ E þ F � rs ð14Þ

in which E and F are constants dependent upon Pand T . As shown in Fig. 8, the parameter F not

only has a linear relation with v, but also the in-

terceptions are zero, so Eq. (14) can be simplified

as:

C�ls ¼ E þ H � v � rs ð15Þ

in which H is a constant dependent on the gas

pressure (P ) and the melt temperature (T ).Substituting Eq. (15) into Eq. (13) and applying

the minimum undercooling criterion gives:

v � L2 ¼ A ð16Þ

where A is a constant dependent on the selected

system, the gas pressure (P ) and the melt temper-

ature (T ). It is apparent that this equation is sim-ilar to the relation between the lamellar spacing

and the solidification rate (L ¼ A � v�n) presented

in the classical Jackson–Hunt eutectic growth

model.

Fig. 9(a)–(c) show the effects of the solodifica-

tion rate, the gas pressure and the melt tempera-

ture on the inter-pore and the pore size. It is

obvious that the inter-pore spacing and the poresize decrease with increasing solidification rate.

Besides the solidification rate, the gas pressure and

the melt temperature also influence the inter-pore

spacing and the pore size. However, their influ-

ences are small, compared with that of the solidi-

fication rate. The dropping trend of the inter-pore

spacing with increasing hydrogen partial pressure

is arrested at high partial pressures. However, therising trend of the pore size continues with in-

creasing hydrogen partial pressure at a constant

total pressure. The inter-pore spacing and the pore

size all increase with rising melt temperature.

Comparatively, the solidification rate is the

major factor influencing the inter-pore spacing and

the pore size in the Gasarite eutectic solidification.

Only limited quantitative measurements [18] at asolidification rate of �500 lm/s show the average

pore size in Gasarite copper is about 30 lm. This

value is close to the predicted values of �40 lmunder the same condition. In addition, much more

qualitative experimental results [19] also show that

the inter-pore spacing decreases with increasing

solidification rate. However, the lack of more

quantitative experimental results prevents theevaluation of the accuracy of Eq. (16).

Fig. 9. Effects of the solidification rate, the gas pressure and the melt temperature on the inter-pore spacing (L) and the pore size (2rg):(a) effect of the solidification rate, (b) effect of the gas pressure and (c) effect of the melt temperature.

Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386 385

Unlike the classical solid–solid eutectics, the

metal–gas eutectics present a special characteristic

in which the inter-pore spacing and the pore size

are determined not only by the diffusion self-

adjustment, but also are functions of the gas

pressure and the melt temperature. In addition, the

nucleation conditions, the temperature gradient

GT and the different thermal conductivities of thesolids and the gaseous phases also should have an

important influence on the solute distribution and

the inter-pore spacing [6,9]. The latter two factors

may lead to the solute transferring along the so-

lidification front, therefore enhance the inter-pore

spacing. The nucleation will adjust the inter-pore

spacing to a smaller level. Further detailed inves-

tigations on the influence of these factors remainfor future work.

4. Conclusions

1. The gas pressure and the melt temperature in

Gasar solidification determine the porosity of

the produced Gasar structures. If the pure hy-

drogen is used singly, the porosity first increases

up to a maximum, then starts to drop with in-

creasing hydrogen pressure. If the mixture ofhydrogen and inert gas are used, the porosity

increases continually with rising hydrogen par-

tial pressure. The porosity increases continually

with the melt temperature.

2. A theoretical model has been developed to de-

scribe the Gasarite eutectic growth. The calcu-

lated results show that the relation between

the inter-pore spacing and the solidificationrate can be described as a simple expression

386 Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386

(v � L2 ¼ A) as that presented in the Jackson–

Hunt model.

3. Unlike the classical solid–solid eutectics, the

metal–gas eutectics present a special character-istic in which the inter-pore spacing and the

pore size are determined not only by the diffu-

sion self-adjustment, but also are functions of

the gas pressure and the melt temperature.

The inter-pore spacing decreases and the pore

size increases with increasing hydrogen partial

pressure. The inter-pore spacing and the pore

size all increase with rising melt temperature.

Acknowledgements

The authors are very grateful to the Postdoc-

toral Foundation and the Doctoral Foundation of

China for the financial support.

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