Studies on undercooled metallic melts

Progress ,,I Mutcrrals Scrente Vol. 42. pp 301-309. 1997 s(‘ 1997 Eisewer Smnce Ltd All rights reserved Prrnted KI Great Britam 0079-6425197 $32.00

PII:80079~6425(97)00020-O

STUDIES ON UNDERCOOLED METALLIC MELTS P. Ramachandrarao National Metallurgical Laboratory,

Jamshedpur,

831 007, India

CONTENTS I. INTRODUCTION

LIQUIDS 2. THERMODYNAMIC BEHAVIOUR OF UNDERCOOLED 3. TEMPERATURE DEPENDENCE OF VISCOSITY BASED ON THE FREE VOLUME THEORY 4 RELATIONSHIPS BETWEEN THERMODYNAMIC AND VISCOUS BEHAVIOUR 4.1. Enthalpy and Viscous Behaviour 4.2. Configurational Entropy and Viscosity 4.3. The Ideal Glass Transition Temperature 4.4. The Second Kauzmann Temperarure 4.5. Amorphous Aluminium Alloys 5 CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

301 301 303 304 304 306 307 308 308 309 309 309

1. INTRODUCTION The author had the opportunity and pleasure of working with Professor Robert Cahn at the University of Sussex for a considerable length of time. During this association several areas of investigation were pursued, notable amongst them being application of the hole theoretical model of liquids for modelling their viscous behaviour(‘,2) and the use of diamond as a splat cooling substrate. (3) The enhancement of thermal conductivity through the use of diamond for the substrate enabled us to produce the glassy state in an aluminium alloy, preceding the current alloys by nearly two decades. In this paper, we review the subsequent efforts of the author and his co-workers in the areas of thermodynamic and viscous behaviour of undercooled liquids and the present status of amorphous aluminium alloys. 2. THERMODYNAMIC

BEHAVIOUR

OF UNDERCOOLED

LIQUIDS

An undercooled liquid exists in a state of metastable equilibrium. Consequently, any attempt at determining its thermodynamic properties through experiment is beset with problems. The liquid would tend to crystallise at the slightest disturbance. As a result, one can determine the thermodynamic properties of such liquids experimentally only at temperatures close to the glass transition temperature where the incubation times for crystallisation may be sufficiently large. The thermodynamic properties over the rest of the undercooled temperature domain can only be derived through appropriate models. 301

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Progress in Materials Science

One of the well-known models of the liquid state is the hole model of Hirai and Eyring.(4) In this model the liquid is assumed to possess a lattice interspersed with a large number of holes or vacant sites which are in a dynamic equilibrium. The number of such holes present at a given temperature can be obtained through standard procedures by a minimisation of the free energy of the system. A liquid possesses an additional heat capacity over that of the solid which has no holes and this extra heat capacity, AC,, can be shown to follow the relationship AC, = nR(t,lRT)‘g

(1)

where g = exp{ - [(eh +

PWRTI - A}

(2)

and A = 1 - l/n

(3a)

with n=

e,le,

(3b)

In the above, R is the gas constant, E,,stands for the energy expended in forming a hole, p is the pressure, and 8, and (3, are the hole and atomic volumes, respectively. In general,

n will be greater than unity owing to the relaxation accompanying hole formation. Ramachandrarao(5) has shown that the value of n can be obtained by considering the hole to be of the same size as that of an ion in its highest ionisation state. If there is more than one species in an alloy, the hole size can be taken to be that of the smallest ion. Knowledge of the hole formation energy and hole volume enables us to obtain AC, as a function of temperature. This in turn can be used to estimate the free energy difference, AG, enthalpy difference, AH, and entropy difference, AS, between an undercooled liquid and its equilibrium crystalline form through a standard thermodynamic formulation. Dubey and Ramachandrarao@) have shown that: AG = A&AT{ 1 - (AC;/AS,)(AT/2T)[l

- (yS/3)(AT/T)]}

(4)

AH = AH, - m,,(g, - g)

(5)

and AS = AS, - (AC;/y2S2)((1 + 78) - [(l + yST,,)/T] exp[(-ydAT)/T]}

(6)

An alternative method of arriving at the thermodynamic properties of an undercooled liquid is to proceed on the assumption that the heat capacity of an undercooled liquid is derivable as a continuous extrapolation of its value at the melting temperature. Under such a premise, Lele et a/.(‘) used the Taylor series expansion technique to show that AG = AS,AT - [AC,m(AT’)/(T, + T)] + 1/2[8AC,,/8T],JAT3)/(T,

+ T)

(7)

where ln(T,/T) = (2AT)/(T, + T) was used. Differentiation

of the above expression with respect to T yields

AS = AS, - 2AC;(AT)/(T,

+ T) + [aACp/aT],(AT2)/(T,

+ T)

(8)

Studies on Undercooled

303

Metallic Melts

Table 1. Free energy difference between the liquid and equilibrium solid phases of Au,,,,.& 186(melting temperature, r, = 636 K; glass transition temperature, & = 290 K) AG (J mol-‘) Temperature (K) 600 500 400 300 290

Eq. (4)

From AC,

549 1971 3183 4067 4131

550 1973 3194 4097 4163

Excellent agreement has been found between values of AG obtained from eq. (4) and those determined from measured values of AC, near the glass transition and melting temperatures. One example is shown in Table 1. Other substances spanning organic, inorganic and metallic materials yield similar excellent agreement. In all these calculations yS of eq. (4) is taken as 0.5. It may also be remarked that AG values obtained from eq. (7) are identical to those from eq. (4). 3. TEMPERATURE

DEPENDENCE OF VISCOSITY FREE VOLUME THEORY

BASED ON THE

A physical situation where all the constituent atoms or molecules of a liquid are so tightly packed that the total volume is composed of only the volume occupied by the constituents and that of the interstices is left out would immediately suggest that fluidity would be at its lowest. In other words, we tend to associate the availability of free space or free volume between atoms to be an essential requirement for the flow of liquids. It is, therefore, not surprising that Doolittle@) correlated the viscosity, q, of a liquid to the free volume through the equation: ? = A exp(Blf,)

(9)

j-r= (VT- IQ/v,

(10)

where

VTand K, represent the volume of the liquid at temperature T and a reference temperature, TR. A and B in the above equation are constants for a given material. When we consider polymers, the chains have the tendency to fold and entangle and obstruct flow. Such a situation was visualised by Adam and Gibbs.c9) According to their model, the viscosity is a function of the configurational entropy, AS,, and its temperature dependence is given by q = A exp(C/TAS,) where A and C are constants. The value of AS, can be estimated from AS, =

[C&liquid) - C&glass)]6 In T s

(11)

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Progress in Materials Science

where C, stands for the heat capacity. It is generally observed that C, of a glass is very close to that of the crystalline solid and has a similar temperature dependence. Another well-known method of expressing the viscous behaviour of liquids is due to Williams, Landel and Ferry. (lo)Their equation, known as the WLF equation, relates the relaxation time, z, at any temperature T to that at a reference temperature, r, through the equations

--~o&(WGN

= MT - KWU - In + Gl

(12)

C, and C, are constants which vary in a correlated fashion when x is changed to r such that C,’ = c, - (T - 7)

(13a)

C,’ = (C,C*)/[C, - (Z - T?l

(13b)

and

At temperatures close to the glass transition temperature, Tg, Vogel”‘) and Fulcher”*) have shown that the temperature dependence of viscosity is fairly well represented by an equation of the form In q = In q. + C”/(T - Z-J

(14)

In practice, when dealing with undercooled liquids over a wide temperature range between T, and Z&one observes that qo, z and C will vary with the temperature interval over which the data are fitted with an equation of the above form. In the next section an attempt is made to relate the thermodynamic properties of an undercooled liquid (as modelled in Section 2) with its viscous behaviour as described here. Attempts made by the author and his co-workers in this direction are summarised and compared with experimental data. The approaches developed are shown to yield new insights into the correlation between the thermodynamic and viscous behaviour of liquids below their normal freezing temperature. 4. RELATIONSHIPS

BETWEEN THERMODYNAMIC VISCOUS BEHAVIOUR

AND

4.1. Enthalpy and viscous Behaviour Following the Doolittle expression, eq. (9), and assuming that all the free volume arises due to the creation of holes, it is possible to express q as ? = A exp[B(l - g)/@ -

go)1= 4 expP& - goI1

(19

where A, and B, are constants. It is also possible to eliminate g and go in terms of the thermodynamic parameters. (13)Such an exercise leads to two significant equations: In q = In A, + n.z,,B,/(AH - AH,)

(16)

In 0 = In A3 + {B,[l - (AC~/AS,)]e,,/RT} + B3(AC~/AS,,,)/(T,/T)*

(17)

In eq. (16), AH, is the enthalpy difference between the undercooled liquid and the corresponding equilibrium solid at a reference temperature To. eq. (17) reveals the dominant role of the ratio between AC; and AS,,, in determining the viscous behaviour,

Studies on Undercooled

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Metallic Melts

(AH-AH,)-' (MOLE CAL-' x 10') 35

45

55

65

75

10.0-

F-' 6.00-TERPHENYL

F

2.0-

O-

-2.0L 0.5

I

I

I

1.0

1.5

2.0

2.5

(AH-AH&-' (MOLE CAL-' x lo31 Fig. I. Variation of viscosity as a function of the reciprocal of residual enthalpy (AH - A&).

while eq. (16) establishes a novel relationship between AH and q. It is also shown that in the event of AC, being temperature-independent, the Vogel-Fulcher equation [eq. (14)] can be recovered with C” being (n@,/AC,,). In general, AC, would be strongly dependent on temperature, and Dubey and Ramachandrarao (I31had earlier demonstrated the validity of eq. (16) in such cases. The results are shown in Fig. 1. In the hole theoretical model, the enthalpy AH at any temperature T, given by eq. (5), can also be rewritten as AH, = AH,,, - nE,,[e-Eh’Rr]F

(18)

Ramachandraraoo4) employed the above equation to estimate the heat of crystallisation, AH,, of a glass at its crystallisation temperature, T,. The necessary values of the parameters (E,, and n) of the hole model were evaluated from viscosity data of the alloy melts studied. The following equations based on the Doolittle equation [eq. (9)] were employed: In r] = In q,, + l/n exp(&/RT)

(19)

and dln(q)/d(l/T)

= ln(&/nR) + (&/R)l/T

(20)

306

Progress in Materials Science Table 2. Viscosity and crystallisation data for metallic glasses

Alloy Au&el, .Ch Pd,, Ge,Si,, 5

n 3.524 2.714

(kJ I%-‘) 11.30 22.27

In tlo”

(2)

(2)

-6.6686 - 1.9264

300 675

625 1010

(kJAz-‘) 10.63 7.26

(k:::;-‘) 6.19 + 0.58 3.87

AH?’ (kJ mol-‘) 6.53 4.14

“Viscosity is in kg m-’ s-‘.

Excellent agreement was obtained (Table 2) between the behaviour of heat of crystallisation obtained from viscosity data and differential scanning calorimetory, confirming the close relationship between thermodynamic and viscous behaviour of a liquid.

4.2. Conjigurational

Entropy and viscosity

Adam and Gibbsf9) have derived a relationship between 11 and the configurational entropy, AS, [eq. (ll)]. On the assumption that AS, equals AS given by eq. (14), one can derive an expression for the temperature dependence of q. Kauzmann@) has argued that AS goes to zero at T = T,, the ideal glass transition temperature. By using this additional information and some algebraic manipulations, it is possible to show that In rl = In A + [B(l + T,/T)I/((T - T,)[l + A0

- To)

+ A*(T-

T,)‘+

A3(T-

K)3 + . . .I}. . .

(21)

where B = C/(2ACpm) and A,, AZ, A,, etc. are constants that depend on AC: and derivatives of AC, with respect to Tat T = T,. (I’)In the case that AC, varies linearly with temperature, eq. (21) reduces to In q = In A + (C/2AC;)[l/(T

- T,)](l + T,/T)/(l - [cc(T,,,- &)/TJ}

(22)

a = T,[~AC,/dT], If AC, is independent

of temperature, In q = In A + C’( 1 + T,/T)/2( T - TJ

(23)

with C’ = C/AC; and AC, = AC;. At temperatures close to the glass transition temperature, Z/T is approximately 0.74 and eq. (23) reduces to the well-known Vogel-Fulcher equation [eq. (14)]. However, we note that the constant c” in eq. (14) should be temperature-dependent at least to the extent of c” = (1 + T,IT)C’/2

(24)

It is therefore concluded that the Vogel-Fulcher equation is in fact an approximate form arising from the more general expression of the form stated in eq. (21). The complexity of the viscous behaviour and consequently the curvature in the In q versus l/T plot will thus be a reflection of the temperature dependence of AC,. This aspect was established earlier by us in the case of several glass-forming liquids.

Studies on Undercooled

Metallic Melts

307

4.3. The Ideal Glass Transition Temperature Professor Cahn has always been fascinated by undercooling and the Kauzmann ideal glass transition temperature. (16)It is pertinent to estimate this important parameter, defined as the temperature at which the entropy of a liquid equals that of its solid, since it has played a very significant role in dealing with both the viscous behaviour and the thermodynamics of undercooled liquids. The value of To can be derived from both eqs (6),(8) by equating AS to zero. From eq. (6) we get AS,,, = AC;[(I - 6*)/26* + S( 1 - 6)“- ’/Ni(N

- 1 + 6)/6N]($)N-*

(25)

with 6 = z/T,. A numerical analysis of eq. (25) has shown that the summation term contributes little to the total value of AS,,,. It is also not possible to obtain an analytical expression for AS,,, from eq. (25). Ignoring the insignificant terms of eq. (25) yields AS, = AC;[ 1 - 6*/2?12(yS/6)(2+ 6)( 1 - 6)?/b3]

(26)

We observe that 6 or the ratio Z&/T,is governed by the ratio A&/AC;, a conclusion which was intuitively reached by Kauzmann(‘s’ in his classic paper of 1948. Conducting a similar analysis with eq. (8) leads us to the conclusion To= T,{(2AC; + AS,) + [(2AC,m+ AS,,,)’ - 8T,A&@AC,/6T)]“*}/2(6AC,/6T)

(27)

where the derivative is taken at T,. We had earlier obtained To from such procedures(‘4’ and found excellent agreement between Tocalculated from thermodynamic data at the melting temperature only and by extrapolation of available data. Table 3 summarises the results. The values of & obtained from thermodynamic data are also compared with those obtained from viscosity data in cases where both types of data are available and can be relied upon. The results are shown in Table 4. The close correspondence between the two values is a reflection on the relationship between the thermodynamic and viscous behaviour of liquids.

Table

3. The ideal glass transition Melting

Substance

W, Ca(N0&4HI0 Cd(NO&.4HZ0 ZnCI: A& HS04.3HJO LiCH,COO Mg(CH,C00)2 Glyceral Ethanol 2-Methylpentane 0-Terphenyl Mgo,,,Ga, 186 A%814Si0 I86 Au, 77Ge0 &iO ry4

temperature,

temperature, r, (K) 123 315.5 333.4 591 585 236.72 560 335 293 158.5 119.55 328 696 636 625

T, obtained materials

from eq. (23) and ACpm/AS,,, of the glass-forming

AC.“IAS,

6 = T/T,

Calculated value of % (K)

Extrapolated value of To (K)

0.6373 2.0196 1.7730

0.4695 0.6434 0.6081 0.4488 0.4423 0.5497 0.6705 0.5943 0.4598 0.3891 0.5091 0.6121 0.3800 0.3145 0.3168

339 203 202 265 259 130 376 199 135 62 61 200 265 200 198

411 202 198 260 270 135 381 209 132 58 59 248

1.3687 1.2466 1.2510 2.1292 1.8810

1.2667 0.7927 0.9819 1.4180 0.3845 0.5010 0.3956

200 241.3

308

Progress in Materials Science Table 4. The ideal glass transition temperature, T, for polymers; rr’ is the value based on thermodynamic data and TrLF is the value derived from the conventional WLF equation Material Polyethylene Hevea rubber Polystyrene Poly(propylene oxide)

T (K)

T, (K)

T$ (K)

TrLF (K)

231 200 354 198

414 421 513 348

143.9 132.8 299.3 171.0

157.8 146.4 306.4 174.0

4.4. The Second Kauzmann Temperature In recent years, one other Kauzmann temperature has been identified for the solid-liquid transition. Fecht and Johnson (17)have pointed out that, at a temperature greater than the melting point, the crystal may have an entropy which is smaller than that of the liquid. This is the conceivable physically and passes another entropy catastrophe. Cahn”‘) considered the implications of this for superheating. Lele et al.“‘) have demonstrated that the new Kauzmann temperature is also a solution to eq. (8). In general, the higher Kauzmann temperature occurs at about 1.9 to 2.1 times the absolute melting temperature while the ideal glass transition temperature occurs at about 0.4-0.5 times T,. Lele et al.“” were of the opinion that reaching the upper Kauzmann temperature may be impossible at ordinary premium owing to the intervention of instabilities such as vaporisation, the elastic constants reaching zero values and changes in electronic structure. 4.5. Amorphous Aluminium Alloys In 1971, Professor Cahn suggested that we exploit the high thermal conductivity of diamond (40 times that of copper) at liquid nitrogen temperatures for rapid solidification of metals. He also arranged to buy the suitably cut diamond from Amsterdam. The author and Laridjani chose to prove the efficacy of the diamond substrate by selecting AI-Cu and Al-Ge alloys. Eutectic compositions of the former were known to yield lamellar eutectic structure even at very high rates of cooling and the lamellar spacing could be used to estimate the cooling rate. Al-Ge alloys, on the other hand, yielded nanocrystalline single-phase solid solutions in earlier studies and posed a challenge in terms of their complex behaviour. Accordingly, experiments were conducted on these two alloys by embedding the diamond in a block of copper cooled to liquid nitrogen temperature and splatting the liquid alloys on to the diamond surface. Electron microscopic studies on the resultant foils gave the following results.(20) 1. There was a notable extension of the solid solubility of copper in aluminium. Even the eutectic alloy solidified into a single-phase solid solution. 2. The Al-Ge eutectic alloy yielded a glassy phase in the thinnest regions. The formation of the amorphous phase was undoubtedly the first, non-controversial detection of an amorphous phase in aluminium alloys. Interest in aluminium-based amorphous alloys was revived recently by the discovery of easy glass-forming compositions in Al-rare-earth systems. A number of alloy compositions in the Al-Ln-Ni and AI-Ln-Ni-M systems, with Ln being lanthanum, ytterbium or cerium and M being manganese, iron or cobalt, have been quenched into the glassy state.(22,23)Suitable heat treatments yield a dispersion of nanocrystalline particles of aluminium in an amorphous

Studies on Undercooled

Metallic Melts

309

matrix. Their strength can reach values of over 1300 MPa with better ductility than the corresponding crystalline compositions. The earliest reports of complete amorphous-phase formation in ternary aluminium alloys was in Al-Fe-B and Al-Co-B systems.‘24’They could not, however, be exploited further owing to their brittleness. Two alloys involving rare earths have also been fabricated into components, opening up a new field of activity with exciting possibilities. 5. CONCLUSIONS The author’s association with Professor Cahn has been extremely fruitful and enjoyable. It has introduced him to exciting activity and has kept his group active in areas of thermodynamic modelling of undercooled liquids with applications in the estimation of viscosity, glass-forming ability and evaluation of ideal glass transition temperatures. Early studies by the author in association with Professor Cahn demonstrated the possibility of generating amorphous phases in aluminium alloys. Recent efforts of other investigators have led to the development of commercially exploitable amorphous aluminium alloys. ACKNOWLEDGEMENTS The author is grateful to Professor T. R. Anantharaman for introducing him to rapid solidification and his colleagues and co-workers Professor S. Lele, Dr K. S. Dubey, Dr G. V. S. Sastry, Dr K. Chattopadhyay and Dr S. N. Ojha for many stimulating discussions and contributions in this area. He is also grateful to Professor S. Ranganathan for constant encouragement and stimulus.

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