Competitive formation of ternary metallic glasses

Competitive formation of ternary metallic glasses

Acta Materialia 54 (2006) 1927–1934 www.actamat-journals.com Competitive formation of ternary metallic glasses D. Ma *, Y.A. Chang Department of Mate...

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Acta Materialia 54 (2006) 1927–1934 www.actamat-journals.com

Competitive formation of ternary metallic glasses D. Ma *, Y.A. Chang Department of Materials Science and Engineering, University of Wisconsin-Madison, 1509 University Avenue, Madison, WI 53706, USA Received 4 November 2005; received in revised form 9 December 2005; accepted 13 December 2005 Available online 20 February 2006

Abstract In an effort to elucidate the formation of metallic glasses and glass-matrix composites in ternary eutectic systems, we have applied the competitive growth principle with the premise that glass formation competes with the growth of all the possible crystalline structures. It is found that each crystalline structure defines a characteristic glass-forming velocity beyond which its growth will be completely suppressed by glass formation. A glass-forming composition diagram, as thus derived, indicates that the formation of a full glass is limited by the complete suppression of growth of the primary competing structure that has the greatest characteristic glass-forming velocity. Moreover, the formation of two types of glass-matrix composites, i.e., single-phase dendrites plus a glass, and single-phase dendrites plus a two-phase eutectic plus a glass, has been predicted as a result of the incomplete suppression of growth of the primary and/or secondary competing structures. A new criterion gauging the globally optimum glass-forming ability in a ternary eutectic system is also derived. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dendrite; Eutectic; Metallic glass; Composite; Competitive growth

1. Introduction It has long been recognized that avoiding crystallization is a prerequisite for glass formation from a deeply undercooled liquid. As a crystallization process has two steps, i.e., nucleation and subsequent crystal growth, earlier theories of glass formation were established on the basis of either bypassing nucleation [1,2] or suppressing crystal growth [3,4]. A long-standing criterion for gauging the glass-forming ability (GFA) is the reduced glass transition temperature Trg (=Tg/Tl, where Tg is the glass transition temperature and Tl is the liquidus temperature), proposed by Turnbull [1] in 1969 who considered the crystal nucleation effect on the glass formation. Although this criterion has been the best ‘‘rule of thumb’’ for GFA so far [5], Trg cannot rationalize the experimental findings over the years that in many glass-forming systems the best glass formers are essentially off-eutectic alloys rather than eutectics as inferred from this criterion itself, and, moreover, it is *

Corresponding author. E-mail address: [email protected] (D. Ma).

unable to explain the existence of crystalline phases of various forms in a glassy matrix, i.e., the formation of glassmatrix in situ composites [6]. However, on the other hand, the idea based on suppressing crystal growth has been providing an insight during the past two decades into the formation of a variety of microstructures accompanying glass formation [3,4,7–11]. Boettinger was among the first who recognized the importance of crystal growth effects on glass formation and proposed that suppressed crystal growth may promote glass formation [3,4]. Using this concept, he intuitively proposed a phase/microstructure selection diagram showing qualitatively the optimum growth velocities and alloy compositions viable for the formation of a monolithic glass or glass-matrix composites in a binary regular eutectic system. His directional-quenching experiments demonstrated that a bulk metallic glass (BMG) was formed on a preexisting crystalline substrate at a controlled growth velocity of 2.5 mm/s for the glass-forming alloy Pd77Cu6Si17 [3]. Later, BMG formation by suppressing crystal growth was also reported by Inoue et al. [12]. In an arc-melted alloy (e.g., Zr65Cu15Ni10Al10) they observed that a bulk glassy phase

1359-6454/$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.12.015

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D. Ma, Y.A. Chang / Acta Materialia 54 (2006) 1927–1934

occupied nearly the whole button, while a layer of columnar crystalline grains occurred unexpectedly at the bottom of the button where cooling was faster due to immediate contact with a water-cooled copper hearth. Recently, in a series of works, Li and co-workers [7–11] have developed the phase/microstructure selection diagrams for binary systems with regular and irregular eutectics, using the competitive growth principle with the consideration that the formation of an amorphous (or glassy) phase competes with the growth of various forms of crystalline phases. On the basis of their theoretical analyses, they have proposed a new GFA criterion and formulated a microstructure-based strategy to pinpoint the best bulk glass former, especially effective for systems where the best glass formers are not at eutectic compositions. This has been successfully demonstrated in the Zr–Cu system [10,11]. In this paper, we attempt to extend the concept of competitive growth to a glass-forming ternary eutectic system, in order to elucidate the competitive formation of ternary glasses and glass-matrix composites. As most multi-component bulk glass-forming systems such as (Zr,Ti)–(Cu,Ni)–Al, (Zr,Ti)–(Cu,Ni)–Be, La–(Cu,Ni)–Al, and Pd–(Cu,Ni)–P might be, to some extent, considered as pseudo-ternaries, our analysis appears to be also useful for shedding light on their tendencies to form BMGs. 2. Competitive growth and full glass formation During the past few decades the competitive growth theory has been established to understand the phase/microstructure selection in solidification of eutectic alloys [13,14], peritectic alloys [15], and even organic materials [16]. The existence of eutectic-coupled zones, for instance, is well known as a result of the competition between eutectic and dendritic growth, and has been validated experimentally in many eutectic systems such as Al–Cu [17], Fe–C [18] and Pb–Sn–Gd [19], and Cu–Ag–Al [20]. Being phenomenological, the competitive growth theory requires comparison of the interface temperatures ðT xi Þ of crystalline phases with various growth forms and predicts that the phase/structure with the highest interface temperature ðT xi Þ is kinetically the most stable one, which will be selected and experimentally observed in the solidified microstructure [3,4,13–16]. In solidification, the interface temperature ðT xi Þ of a crystalline phase/structure (x) is determined by its thermodynamic equilibrium temperature (e.g., melting temperature or liquidus temperature) less the undercooling required for its growth or tip advancing. The interface undercooling, in many cases, is determined by the interplay between the solute diffusion that tends to minimize the scale of the morphology and the capillarity effects that tend to maximize the scale [21]. Since, depending on the solidification conditions, a crystalline phase can grow either individually in a variety of morphologies (e.g., dendritic and cellular), or cooperatively in hybrid structures consisting of multiple phases (e.g., eutectic), one should consider all the possibilities of solidification structures

when using this approach. For example, in a ternary eutectic system consisting of three crystalline phases (e.g., a, b, and c), crystal growth structures normally exhibit seven forms, i.e., single-phase cellular/dendritic a, b, and c, two-phase cellular/dendritic eutectic a + b (designated as Eu1 hereafter), a + c (Eu2), and b + c (Eu3), and threephase planar eutectic a + b + c (Eu) [22]. As for a glassforming ternary eutectic system, we can treat the glass as another competing phase, which may grow (or form) with a planar interface [3]. Thus, applying the competitive growth principle, we can expect complete suppression of crystal growth when the temperature of full glass formation Tg is higher than T xi of any crystal growth structure in a ternary eutectic system [10]. In mathematical formulation, this is T g P T xi

ð1Þ

where x represents any competing crystalline phase, e.g. a, b, c, a + b (Eu1), a + c (Eu2), b + c (Eu3), a + b + c (Eu). According to Mccartney et al. [20,22] and Himemiya et al. [23], in ternary alloy solidification under a positive temperature gradient (G), the growing interface temperatures of single-phase cells/dendrites (e.g., a), two-phase cellular/dendritic eutectics (e.g., a + b (Eu1)), and the three-phase planar eutectic (e.g., a + b + c (Eu) in a lamelabc lar form), being denoted as T ai ; T ab i ; T i , respectively, can be written as GDL  K a V 1=2 V GDL ab  K ab V 1=2 T ab i ¼ T e þ M c C 1C  V T abc ¼ T e  K e V 1=2 i

T ai ¼ T al 

ð2Þ ð3Þ ð4Þ

where, under smaller G and higher V,      1=2 pffiffiffi k aB  1 k aC  1 K a ¼ 2 2 CmaB C 1B þ CmaC C 1C DL DL 

K ab

  1=2 pffiffiffi  1=2 kC  1 ¼ 2 2 CM C þ 2M aL QL C 1C DL

ð5Þ ð6Þ

and Ke ¼

pffiffiffiffiffiffiffiffiffiffiffiffi C=DL uðfx ; mxj Þ

ð7Þ

In the above equations, T al is the liquidus temperature at which the liquid is in equilibrium with a (similarly T ab l ab and T ac l ), T e is the melting temperature of the pure binary bc eutectic a + b (similarly T ac e and T e ), Te is the ternary invariant eutectic temperature, V is the growth velocity (or the isotherm advancing speed) that scales with the cooling rate, DL is the diffusion coefficient in the liquid, MC is the slope of the eutectic valley (¼ dT =C lC < 0, similarly MB and MA), C1C is the bulk alloy composition with respect to the third component C (similarly C1B and C1A), C is the Gibbs–Thomson coefficient, maB and maC are the liquidus slopes in the a phase region with respect to components

D. Ma, Y.A. Chang / Acta Materialia 54 (2006) 1927–1934

B and C, respectively, kaB and kaC are the coefficients of solute partition between a and the liquid with respect to components B and C, respectively, kC is the weighted partition coefficient between a + b and the liquid with respect to the third component C, M is the weighted liquidus slope (=MaMb/(Ma + Mb)) where Ma and Mb are the effective liquidus slope defined by a and b, respectively, aL = 2C(1 + n)(1/nMb + 1/Ma), QL = P(1 + n)2C0B/nDL where n is the ratio of the volume of b to that of a, P and C0B are materials constants. Ka is the growth constant for cellular/dendritic a, as given in Eq. (5) and similarly Kb and Kc; Kab is the growth constant for cellular/dendritic eutectic a + b (Eu1), as expressed in Eq. (6) and similarly Kac and Kbc. Ke is the growth constant for planar ternary eutectic, which is weakly dependent on alloy compositions, while in Eq. (7), fx is the volume fraction of the x phase (a, b, c), mxj is the liquidus slope with respect to the j component (A, B, or C) in the x phase, uðfx ; mxj Þ is a function of fx and mxj . The reader is referred to Himemiya and Umeda [22] for more details on Ke. Similarly, one can obtain the growth temperatures for cellular/dendritic b and c (i.e., T bi and T ci ) from Eq. (2) by replacing Ka with Kb and Kc, respectively, and those for cellular/dendritic two-phase eutectic a + c (Eu2) and b + c (Eu3) (i.e., T ac and T bc i i ) from Eq. (3) by replacing Kab with Kac and Kbc,

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respectively. Figs. 1(a)–(c) show schematically, as a function of growth velocity, the interface temperatures for the single-phase cellular/dendritic growth (as described in Eq. (2)), the two-phase cellular/dendritic eutectic growth (Eq. (3)), and the ternary planar eutectic growth (Eq. (4)), respectively. Omitting the negligible G effects at high V in Eqs. (2)–(4) and using Eq. (1) by taking the equity (i.e., T g ¼ T xi ), one can define seven characteristic velocities, i.e., ac bc e V ac ; V bc ; V cc ; V ab c ; V c ; V c , and V c for any glass-forming ternary alloy composition (C1A, C1B, C1C), beyond which glass formation will suppress the growth of a, b, c, a + b (Eu1), a + c (Eu2), b + c (Eu3), and a + b + c (Eu), respectively. These characteristic velocities can be expressed as 2

V ac ¼ ðT al  T g Þ =K 2a

ð8aÞ

V bc ¼ ðT bl  T g Þ2 =K 2b

ð8bÞ

2

V cc ¼ ðT cl  T g Þ =K 2c

ð8cÞ

2 ab 2 V ab c ¼ ðT l  T g Þ =K ab

ð8dÞ

2

ð8eÞ

2

ð8fÞ

ac 2 V ac c ¼ ðT l  T g Þ =K ac bc 2 V bc c ¼ ðT l  T g Þ =K bc 2

T

V ec ¼ ðT e  T g Þ =K 2e Tl

a

V T

Tl

b

where T ab l bc T bc l ¼ Te 

ð8gÞ T ac l

¼ T ab ¼ T ac and e  M C C 1C , e  M B C 1B , M A C 1A . Note that the K terms (e.g., Ka, Kb, Kc, Kab, Kac, and Kbc) and the Tl terms (e.g., ac bc T al ; T bl ; T cl ; T ab l ; T l , and T l ) are all dependent on comab ac position, thus the Vc terms (e.g., V ac ; V bc ; V ac c ; Vc ; Vc , bc and V c ) vary as composition changes. In contrast, as Ke, Te, and Tg are only weakly composition dependent [24], V ec remains constant for all the compositions. Thus, to make a full glass from a ternary alloy, the externally imposed growth velocity or the isotherm advancing velocity (V) due to the heat extraction must exceed the maximum value (Vc) of these seven characteristic velocities in order to suppress the growth of all forms of crystalline structures, i.e. ac bc e V P V c ¼ maximumfV ac ; V bc ; V cc ; V ab c ; V c ; V c ; V cg

V T

Te

c

V Fig. 1. Schematic variation of the growth temperature with velocity at a constant gradient for: (a) a single-phase dendritic structure; (b) a twophase cellular/dendritic eutectic structure; (c) a planar ternary eutectic structure.

ð9Þ

In Eq. (9), Vc defines the critical glass-forming velocity for the given ternary alloy (C1A, C1B, C1C). Although, in principle, one can use Eqs. (8a)–(8g) and (9) to determine the critical glass-forming velocities for any ternary eutectic systems, we will focus our present analysis on the simplest case, i.e., the regular ternary eutectic system, where the relative values of K and Tl are readily estimated. Here, ‘‘the regular ternary eutectic’’ refers to the case where a, b, and c solidify in a similar behavior (i.e., similar dependency of Ka, Kb, and Kc on composition) and so do a + b (Eu1), a + c (Eu2), and b + c (Eu3) (i.e., similar dependency of Kab, Kac, and Kbc on composition) while the three-phase eutectic (Eu) growth is in a lamellar structure. In a following paper, we will treat the irregular ternary eutectic

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systems where one of the three phases solidifies in a substantially different manner. Eq. (9) indicates that, although all the seven crystalline structures are potentially competing with the glass formation, the critical glass-forming velocity for the alloy in question is determined solely by the structure that has the largest characteristic velocity. Here, we define this crystalline structure as the primary competing structure that limits the formation of a full glass. As each crystalline structure could be dominating in a range of compositions, we expect seven composition regions corresponding to ac bc e V ac ; V bc ; V cc ; V ab c ; V c ; V c , and V c , respectively, within a ternary eutectic tie triangle bounded by the three solid phases, i.e., a, b, and c, as shown schematically in Fig. 2. In the following, we discuss each situation in detail. When V c ¼ V ec , the three-phase eutectic is the primary structure competing with glass formation. This is expected to happen for alloys near the ternary eutectic composition ac bc since T al ffi T bl ffi T cl ffi T ab and Ke < l ffi Tl ffi Tl ffi Te Kab  Kac  Kbc < Ka  Kb  Kc. When V c ¼ V ab c , the two-phase eutectic a + b limits the full glass formation. This may happen for alloys near the a monovariant eutectic (a + b) valley since T ab l > Tl  b c ac bc Tl > Te > Tl  Tl  Tl and Kab  Ke < Ka  Kb < Kac  Kbc  Kc. Similarly, we can expect V c ¼ V ac c for the compositions near the monovariant eutectic (a + c) valley and V c ¼ V bc c for the compositions near the b + c valley. When V c ¼ V ac , the single-phase dendritic a is the primary competing structure. This would happen for alloys near the a phase corner in the ternary phase diagram since ac b c bc T al > T ab and Ka  Ke < l  Tl > Te > Tl  Tl  Tl Kab  Kac < Kb  Kbc  Kc. Similarly, we can obtain

α

Vcα

e2

Vcα β

e1

Vce Vcα γ

Vcβγ

Vcγ

Vcβ

γ

β e3

Fig. 2. Schematic representations of critical glass-forming velocities for ternary alloys within the ternary eutectic tie triangle bounded by the three solid phases.

V c ¼ V bc for the compositions near the b phase corner and V c ¼ V cc for the compositions near the c phase corner. ac As indicated in Eqs. (8a)–(8g), V ac ; V bc ; V cc ; V ab c ; Vc , bc and V c are strongly dependent on composition although V ec remains constant. Thus, one can anticipant that the topology of the three-dimensional Vc-composition plot might, more or less, resemble the liquidus surface of the ternary phase diagram. 3. Competitive growth and partial glass formation In Eq. (9), we define the structure having the greatest ac bc value of the Vc terms (V ac ; V bc ; V cc ; V ab c ; V c ; V c , and e V c ) as the primary competing structure. Similarly, we can assign the structure with the second greatest value of the Vc terms as the secondary competing structure, and then tertiary, and so on. As discussed in Section 2, a full glass forms if the isotherm advancing velocity V is greater than the greatest value of the Vc terms, indicative of suppression of all the crystalline structures. However, if V is less than the greatest value but greater than the second greatest value of the Vc terms, all the crystalline structures would be overgrown by the glass formation, except the primary competing structure. This condition leads to the formation of a partial glass, or a glass-matrix composite where the primary competing structure coexists with the glass. Here, the primary competing structure could be either a singlephase dendritic structure or a two-phase cellular/dendritic eutectic structure. The reason is that, in both cases, dendritic solidification must be accompanied by solute rejection regardless of growth velocity [3,4] and the interdendritic liquid will have composition closer to the ternary eutectic composition, which can be readily vitrified into a glass. It is also possible that V is less than the second greatest value but greater than the third greatest value of the Vc terms; thus one can obtain, owing to microsegregation again, another type of glass-matrix composite consisting of the glass and the primary and secondary competing crystalline structures. In this situation, the primary competing structure must be a single-phase dendrite and the secondary structure is a two-phase cellular/dendritic eutectic. The reason is likely that solute rejection in the single-phase dendritic solidification will facilitate dendritic growth of a two-phase eutectic, and then both types of dendritic solidification will promote solute buildup in the interdendritic regions; thus the interdendritic liquid will have a composition approaching the ternary eutectic composition, and eventually will be frozen into a glass at the given growth velocity. In addition, knowing that glass-matrix composite formation results from microsegregation during primary and/or secondary dendritic solidification [3,4], one can accordingly conclude that coexistence of a ternary eutectic and a glass is impossible in solidification microstructures of a ternary alloy because of no solute segregation during ternary eutectic growth. Thus, if V < V ec , no glass is able to form. As V ec remains constant for all the

D. Ma, Y.A. Chang / Acta Materialia 54 (2006) 1927–1934

ternary compositions, it defines the globally lowest growth velocity for glass formation in a ternary eutectic system. 4. Microstructure selection In order to elucidate the formation of glasses and glassmatrix composites in a more straightforward manner, we have drawn schematically the interface temperatures of the competing structures altogether as a function of growth velocity so as to apply the competitive growth principle. Microstructure selection as a function of alloy composition and growth velocity can thus be predicted. A representative diagram expected for a given growth velocity V* (slightly greater than V ec as indicated in Fig. 3(b)) is shown in Fig. 3(a), which depicts four types of composition regions within the eutectic tie triangle. Here, we denote these regions as A, B, C, and D, and will discuss the formation of glasses and glass-based composites in these regions as follows. Region A. Near the ternary eutectic composition a full glass is expected where the primary competing structure, i.e., the ternary planar eutectic, is overgrown by glass formation at V  > V ec . The advancing interface for glass formation appears to be planar, as observed by Boettinger [3] and is shown schematically in Fig. 4(a). Fig. 5(a) illustrates schematically the interface temperatures of the seven competing crystalline structures, i.e., a, b, c, a + b (Eu1), a + c (Eu2), b + c (Eu3), and a + b + c (Eu), as a function of growth velocity for a representative ternary alloy in region A (i.e., alloy 1 in Fig. 3(a)). Also plotted in Fig. 5(a) is the temperature for glass formation (Tg), which appears to be a horizontal line (labeled by Tg) as Tg is only weakly dependent on alloy compositions and growth/cool-

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ing conditions [24]. The solid lines represent the highest interface temperatures, indicative of the phase/structure being observed at the corresponding velocities, while, in contrast, the dashed lines denote the interface temperatures of the structures being overgrown. It is clear from Fig. 5(a) that, if omitting the Tg line temporarily, at all V the ternary eutectic growth temperature ðT ei Þ is always higher than the interface temperatures of all the single- and two-phase solidification structures (shown as dashed lines), indicating that the ternary eutectic is the only crystalline structure being observed. This situation is consistent with the condition that Ke < Kab  Kac  Kbc < Ka  Kb  Kc, and has been validated by the observation of eutectic coupled zone in many eutectic systems [19,20]. However, in an easy glassforming system, the Tg line will intercept with all these curves at their high V end that define seven characteristic velocities as given in Eqs. (8a)–(8g) and shown in Fig. 5(a). Here, V ec , as the largest among these characteristic velocities, is the critical glass-forming velocity for this ternary alloy. Thus, according to the competitive growth principle, one can expect that a ternary eutectic (Fig. 4(e)) is the dominant structure at V < V ec where the ternary eutectic temperature is the highest, while a monolithic glass forms at V > V ec where the glass transition temperature is the highest. Region B. Near the two-phase eutectic valley (e.g., B1, B2, and B3) the two-phase cellular/dendritic growth prevails due to its relatively higher liquidus temperature and smaller undercooling required for growth (e.g., smaller Kab in region B1, Kac in region B2, and Kbc in region B3). But a glass can form at the interdendritic regions due to microsegregation caused by dendritic solidification as addressed in Section 3 provided that V > V ec . Thus, at a

Fig. 3. (a) Schematic representations of the composition boundaries of the various structural regions (dashed and solid lines) for a given velocity V  ð> V ec Þ and temperature gradient. The numbers and letters refer to the regions described in the text. (b) Schematic of microstructure variation with velocity for alloy compositions along the a–a 0 line as shown in (a).

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Fig. 4. Schematics of growth interface morphologies in a ternary eutectic system: (a) glass; (b) glass plus cellular/dendritic two-phase eutectic; (c) glass plus single-phase dendrite; (d) glass, cellular/dendritic two-phase eutectic, and single-phase dendrite; (e) planar ternary eutectic; (f) ternary eutectic plus cellular/dendritic two-phase eutectic; (g) ternary eutectic plus single-phase dendrite; (h) ternary eutectic, cellular/dendritic two-phase eutectic, and singlephase dendrite.

Alloy3 Alloy 1

T

Tl

T

Te

Te

Eu

Eu

Eu1 Eu1 Eu2

Tg

Tg

Eu3

Vc

a

T

Eu3

Vc Vc Vce Vc Vc Vc

V

Alloy 2

T

V

Vce Vc

c

Alloy 4

Tl

Tl

Te

Te Eu

Eu Eu1

Eu1

Tg

b

Vce Vc

V

Tg

d

Vce Vc Vc

V

Fig. 5. Schematic representations of the growth temperatures of various crystalline structures competing with glass formation for: (a) alloy 1; (b) alloy 2; (c) alloy 3; (d) alloy 4.

given velocity V* (slightly greater than V ec ), the glass-matrix composite is expected to be present in this composition region and the solidification interface is shown schematically in Fig. 4(b). Fig. 5(b) shows the interface temperatures of the primary and secondary structures (e.g., Eu1 and Eu) competing with glass formation as a function of growth velocity for a representative alloy in region B1 (i.e., alloy 2 in Fig. 3(a)). Based on the competitive growth

principle, it can be expected that a full glass be formed at V > V ab c , a composite consisting of glass and Eu1 at e V ec < V < V ab c , and Eu1 + Eu at V < V c . Thus, as long as e ab * V is greater than V c but less than V c , compositions in region B1 yield a composite of Eu1 and the glass. Similar arguments apply to regions B2 and B3. Region C. Near the tie lines between the liquid and the solid phases (e.g., C1, C2, and C3), the single-phase den-

D. Ma, Y.A. Chang / Acta Materialia 54 (2006) 1927–1934

dritic growth can be followed by the formation of a glass, as shown schematically in Fig. 4(c). A typical growth temperature–velocity diagram for alloy 3 in region C1 is shown in Fig. 4(c), indicating that Eu and a are the primary and secondary structures competing with the glass formation, respectively. It is clear that a full glass forms at V > V ac while a composite of the glass plus a forms when V ec < V < V ac . Region D. For alloys away from the eutectic valleys and the tie lines (e.g., D1, D2, D3, D4, D5, and D6), single-phase dendrites can be followed firstly by two-phase cells/dendrites and eventually by a glass. The interface morphology is shown schematically in Fig. 4(d). A typical growth temperature–velocity diagram for a representative alloy 4 in region D1 is shown in Fig. 4(d), indicating that Eu, Eu1, and a are the primary, secondary, and tertiary structures, respectively, competing with the glass formation. It is expected that a full glass forms when V > V ac , a composite a of the glass plus a (in Fig. 3(c)) at V ab c < V < V c while another composite consisting of the glass, Eu1, and a (in Fig. 3(d)) at V ec < V < V ab c . Fig. 3(b) shows the solidification structures obtained as a function of growth velocity and alloy composition, where the alloy compositions are assumed to pass through the a– eutectic tie line and the b + c eutectic valley (e3), i.e., the ratio of CB to CC is kept constant. Fig. 3(b) is essentially an isopleth of Fig. 3(a) along the a–a 0 composition line. Six types of solidification microstructures can be expected such that: when V > V ec , the microstructure evolves, with decreasing CA, from the a + glass composite, to a full glass and finally the Eu3 + glass composite, while for V < V ec from the a + Eu composite (Fig. 4(g)), to the entirely ternary eutectic Eu (Fig. 4(e)) and finally the Eu2 + Eu composite (Fig. 4(f)). Fig. 3(b) is fairly similar to Fig. 1(b) in Ref. [10] as well as Fig. 9(c) in Ref. [4], both of which represent the glass-forming diagrams of binary eutectic systems. Compositions of alloy 1–3 are also indicated in this diagram and their temperature–velocity diagrams are shown schematically in Fig. 5(a)–(c), respectively. 5. Glass-forming ability (GFA) Inspecting Eqs. (8a)–(8g), one can use a single equation to represent the critical glass-forming velocity (VC) for a ternary eutectic system, i.e.  x 2 Tl  Tg VC ¼ / ð1  T rg Þ2 ð10Þ Kx where T xl is the liquidus temperature in terms of the thermodynamic equilibrium between the liquid and the primary competing structure/phase x, Kx is the growth constant for x, and Trg (= T g =T xl ) is the reduced glass transition temperature. Eq. (10) indicates that for a given ternary alloy, its critical glass-forming velocity is determined solely by the suppression of the primary competing crystalline structure. It is also noted that the critical glass-forming velocity is essentially associated with the reduced glass

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transition temperature, i.e., Trg. Noted that Trg also determines the nucleation rate [1], and we can expect that both nucleation and crystal growth upon cooling a glassforming liquid be controlled by Trg. As Trg exhibits maximum at the ternary eutectic composition, Eq. (10) demonstrates that the ternary eutectic alloy has the greatest GFA. In another words, the critical glass-forming velocity, i.e., V ec , is the minimum value attainable for a ternary eutectic system. This is consistent with the hints obtained from Figs. 3(b) and Figs. 5(a)–(d). Thus, the globally minimum glass-forming velocity ðV min c Þ can be written as 2

¼ V ec ¼ ðT e  T g Þ =K 2e V min c

ð11Þ

As given in Eq. (11), the value of V min is determined by c the difference between Te and Tg (i.e., Te  Tg) as well as the growth constant for the ternary eutectic (i.e., Ke in Eq. (7)). Accordingly, an easier glass-forming system (requiring smaller V min c ) must have a smaller Te  Tg, and, concurrently, a larger Ke. This can be rationalized from both thermodynamic and kinetic points of view. The thermodynamic stability of the glass relative to the eutectic is enhanced when Tg is approaching Te, while a larger Ke is indicative of the greater kinetic difficulty of eutectic growth, which in turn promotes easier formation of a glass. Here, Ke, as given in Eq. (7), is determined by the interplay/compromise between the solute diffusion that tends to minimize the scale of the morphology (e.g., eutectic interphase spacing), and the capillarity effect that tends to maximize the scale [3,4,21]. In terms of the Ke value, the easiest glass-forming system (requiring a large Ke) is always associated with deep eutectics (large mxj ), high viscosities in the liquid (e.g., smaller diffusion coefficient DL), and large solid/liquid interfacial energies (e.g., large C). Using the competitive growth principle, Ma et al. [10] derived a formulation for V min for binary eutectic systems c exactly in the same form as Eq. (11). Thus, Eq. (11) can be used to explain the many experimental observations [9–11,25,26] that a ternary eutectic system always has a better GFA than do its constituent binary eutectics. For instance, the GFA of Zr–Cu–Al is much better than that of its constituent binaries, e.g., Zr–Cu [9–11]. The reason is twofold. Thermodynamically, the eutectic temperature (Te) of a ternary system is always lower than that of its constituent binary eutectics, owing to the requirement of phase equilibrium [27]. For example, it was reported that Pd–Cu– Si, Pd–Ni–P, and Pt–Ni–P have better GFAs than their constituent binaries since their ternary eutectic temperatures are lowered by 50–300 K from those of their constituent binaries [26]. On the other hand, kinetically, the Ke value of the ternary eutectic is always greater than those of its constituent binary eutectics. This is because ternary eutectic growth is accompanied by simultaneous separation of three crystalline phases from the eutectic liquid, which is geometrically more difficult than that of accommodating two phases required by binary eutectic growth. In addition, atomic interdiffusion in a ternary eutectic liquid is more

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complex than that in a binary eutectic, due to multiple chemical interactions involved. In general, Eq. (11) suggests that bulkier glasses tend to form toward higher-order deeper eutectics.

tion is due to the suppression of the primary competing crystalline structure, while glass-matrix composites result from the coexistence of primary and/or secondary competing structures with the glass. A new criterion that elucidates the GFA in a ternary eutectic system is also derived.

6. Comparisons with available experiments Acknowledgments Recently, Li and co-workers [3] reported their experimental observations of bulk glass and glass-matrix composite formation associated with a ternary eutectic, i.e., L ! s5 + ZrCu + Cu10Zr7, in the Zr–Cu–Al system. Using a water-cooled copper mold, they cast alloy rods of 5 mm in diameter, which may generate an average cooling rate of 200 K/s or an effective solidification velocity of the order of magnitude of several mm/s [28]. They observed a full glass-forming region inclusive of the ternary eutectic composition (Zr45Cu49Al6), most likely to be region A as shown in Fig. 3(a). They also reported the existence of three glass-matrix composite-forming regions (i.e., s5 + glass, ZrCu + glass, Cu10Zr7 + glass), consistent with the predicted regions C1, C2, and C3, respectively, as also indicated in Fig. 3(a). It is notable that the interface temperature–growth velocity diagrams in Fig. 5(a)–(d) can be used to explain the microstructure evolution in Bridgman solidified alloys [7–9]. An example is taken for the well-known bulk glassforming alloy Pd42(Ni10Cu30)P18. Hu et al. [8] reported that with decreasing the velocity from 0.13 to 0.04 mm/s, the solidification microstructure of Pd42(Ni10Cu30)P18 evolves as a full glass, and a composite of glass + primary dendrite, then another composite of glass + primary dendrite + twophase eutectic colony, and, eventually, primary dendrite + interdendritic eutectic. From Fig. 5(d), we can predict the same sequence of microstructure formation for an alloy in region D. Thus, the Pd42(Ni10Cu30)P18 alloy is likely located in region D which is away from the eutectic tie lines and eutectic valleys, and its ability to form glass and glass-based composites is limited primarily by the growth of a primary dendritic phase and a two-phase eutectic. 7. Conclusions The competitive growth principle has been applied to study the formation of glasses and glass-matrix composites in ternary eutectic systems. It is found that full glass forma-

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