3
Rheology Chapter and Ultrasonic Properties of Pt57.5Ni5.3Cu14.7P22.5 Liquid
Key words: Bulk metallic glass, Non-Newtonian flow, Liquid fragility, Ultrasonic measurement, Compression test, Amorphous
3.1 Abstract
The equilibrium and non-equilibrium viscosity and isoconfigurational shear modulus of Pt57.5Ni5.3Cu14.7P22.5 supercooled liquid are evaluated using continuous-strain-rate compression experiments and ultrasonic measurements. By means of a thermodynamically consistent cooperative shear model, variations in viscosity with both temperature and strain rate are uniquely correlated to variations in isoconfigurational shear modulus, which leads to an accurate prediction of the liquid fragility and to a good description of the liquid strain rate sensitivity.
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3.2 Introduction
The Pt-Ni-Cu-P bulk-glass forming system [1] is known to form one of the toughest bulk metallic glasses to date [2-4], characterized by a fracture toughness value of ~ 80 MPa-m1/2. The inherent toughness of this system is shown to be a consequence of its tendency to undergo extensive shear-band networking prior to fracture [2]. This tendency has been primarily attributed to its high Poisson's ratio (~ 0.42), which designates that the material favors accommodation of stress by shear. Interestingly, Poisson's ratios of metallic glasses were recently shown to be directly correlated to the rheology of their undercooled liquid state, and specifically to the liquid fragility [5]. In a recent rheological study [6] it was demonstrated that Pt-based liquid is indeed one of the most fragile metallic-glass-forming liquids, which to some extent explains its inherently tough nature. In the present study we employ continuous-strain-rate compression experiments and acoustic measurements in conjunction with a recently developed cooperative shear model [7-9] to assess the rheology and ultrasonic properties of Pt57.5Ni5.3Cu14.7P22.5 liquid under equilibrium and non-equilibrium conditions.
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3.3 Experimental
The rheology of the supercooled liquid was assessed using the continuous-strain-rate compression setup as described in Ref. [10]. The alloy ingot was prepared by first prealloying Pt (99.9mass%), Ni (99.9mass%), and Cu (99.99mass%) by induction melting, and then alloying P (99.999mass%) by step-wise furnace heating. The specimens were prepared by first fluxing the alloy with B2O3, and subsequently casting it into 4 mm diameter rods, whose amorphous nature was verified by thermal analysis. The rods were cut and polished to form 4 mm tall cylindrical specimens. The typical 2-to-1 geometric ratio was not adopted here, as the 1-to-1 ratio was found more appropriate in geometrically constraining the specimen against excessive barreling (possibly related to a high Poisson's ratio). The compression experiments were performed for at least 5 Maxwellian relaxation times after the transient stress strain response to ensure that a steady-state flow stress was attained. The strain-rate dependent viscosity was measured at temperatures ranging from 473 K to 523 K, and is presented in Fig. 1. The strain-rate dependence exhibits the typical trend observed in other glass forming systems: in the low strain rate limit, viscosity is stabilized at the Newtonian limit characterized by a strainrate sensitivity exponent of 1.0; In the high strain rate limit, viscosity is stabilized at a non-Newtonian limit characterized by a strain-rate sensitivity exponent of ~ 0.1. The isoconfigurational shear modulus at the high-frequency "solid-like" limit was evaluated using ultrasonic measurements along with density measurements [11]. Ultrasonic shear wave speeds were measured using the pulse-echo overlap set-up described in Ref. [8]. Densities were measured by the Archimedes method, as given in the American Society for Testing and Materials standard C693-93. Measurements were
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Figure 3.1. Viscosity of Pt57.5Ni5.3Cu14.7P22.5 at the indicated temperatures and strain rates assessed from continuousstrain-rate compression experiments. Lines are fits to the data using the kinetic balance formulation given in Eq. (3.4).
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performed ex situ on the amorphous specimens at room temperature, after being quenched rapidly from the processing temperature. The isoconfigurational shear moduli at the processing temperatures were estimated by extrapolating the room temperature measurements using a linear Debye-Grneisen constant to account for the thermal expansion effect on the shear modulus of the frozen glass. A measured linear DebyeGrneisen coefficient of ~ 13 MPa/K for Pt57.5Ni5.3Cu14.7P22.5 was utilized [12]. We measured the temperature-dependent equilibrium isoconfigurational shear modulus by performing ultrasonic measurements on relaxed undeformed specimens annealed at temperatures between 472 K and 503 K. We also measured the strain-rate dependent non-equilibrium isoconfigurational shear modulus by performing ultrasonic measurements on specimens deformed at 473 K and strain-rates between 110-5 and 3.410-4 s-1. The thermal annealing process, as well as the steady-state deformation process, was performed for several Maxwellian relaxation times to ensure that a steady configurational state had been attained, while the succeeding quenching process was performed as rapidly as possible in order to freeze that configurational state. The results for the ultrasonically measured shear modulus, corrected for the Debye-Grneisen effect, are presented in Fig. 3.2 for the equilibrium liquid annealed at the indicated temperatures, and in Fig. 3.3 for the non-equilibrium liquid deformed at the indicated rates. For a more complete description of the experimental methods used in this chapter, please refer to the experimental section in Chapter 2.3.
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3.4 Discussion
In several recent studies [7-9], a thermodynamic link between the isoconfigurational shear modulus G and viscosity has been proposed, as follows:
G T ln ( ) = Gg Tg ln g
(
)
q
(3.1)
where Tg is the glass transition temperature, g 1012 Pa-s is the equilibrium viscosity at Tg , Gg is the equilibrium shear modulus at Tg , and is the Born limit of viscosity. The exponent q is defined in previous studies as q = n ( n + p ) , where n and p are the reduced "elastic" and "cooperative volume" fragility indices, respectively; this quantifies the contributions of isoconfigurational shear modulus and cooperative volume shear to the softening of the shear flow barrier. In a similar analysis for the Zr-based bulk-glass forming liquid [8] it was determined that the best correlation between G and is obtained when n p (i.e., q 1 2 ). In the present analysis we take q = 1 2 to hold for the Pt57.5Ni5.3Cu14.7P22.5 liquid as well. For this liquid we also take Tg = 489 K
(interpolated value for which g 1012 Pa-s ), Gg = 30.5 GPa (interpolated value at Tg ), and 4.55 10-5 Pa-s (taken as the Planck's limit of viscosity). We can therefore use Eq. (3.1) to correlate the liquid viscosity evaluated from mechanical tests to the liquid shear modulus evaluated acoustically. In Fig. 3.2 we superimpose the equilibrium shear moduli predicted from Newtonian viscosities, while in Fig. 3.3 we superimpose the
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Figure 3.2. Acoustically measured shear modulus corrected for the DebyeGrneisen effect of the relaxed equilibrium liquid annealed at the indicated temperatures ( ). Predicted equilibrium shear modulus from measured equilibrium viscosity data using Eq. (3.2) ( ). The line is a fit to the data using the temperature dependence relationship given in Eq. (3.2).
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Figure 3.3. Acoustically measured shear modulus corrected for the DebyeGrneisen effect of the non-equilibrium liquid deformed at 473 K at the indicated strain rates ( ). Predicted non-equilibrium shear modulus from measured non-equilibrium viscosity data using Eq. (3.2) ( ). The line is a fit to the data using the kinetic balance formulation given in Eq. (3.4).
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non-equilibrium shear moduli predicted from non-Newtonian viscosities. As evidenced from these plots, the liquid viscosity can be very well correlated to shear modulus via Eq. (3.1). A proposed thermodynamic relation for the temperature dependence of the equilibrium isoconfigurational shear modulus, Ge (T ) , can be utilized to fit the experimental data of Fig. 3.2 and determine the reduced elastic fragility index n for this liquid [9]. This relation is given by:
Ge (T ) = Gg exp n 1 - T Tg .
(
)
(3.2)
A fit to the equilibrium data of Fig. 3.2 yields n = 1.29 . Such a high value for the reduced elastic fragility places Pt57.5Ni5.3Cu14.7P22.5 among the most fragile liquids investigated using this treatment [8, 9]. Specifically, the Angell fragility m can be estimated by relating m to n using m = (1 + n q ) log g [8], which gives m = 59 , a
(
)
value consistent with the fragilities reported previously for similar Pt-based liquids [6]. In the context of this analysis, non-Newtonian flow has been treated as a steady nonequilibrium flow state arising from a balance between the rate of mechanical work and the rate of barrier crossing [9]. The barrier energy density W can be related to and G as follows:
G W = kT ln ( ) = Wg Gg
1q
(3.3)
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where Wg = kTg ln g is the equilibrium barrier energy density at Tg . The kinetic
balance that determines the non-Newtonian flow state is given by [8]: -
(
)
( n q )Wg
Tg c p
& 2 =
(W - We ) (W
Gg
Wg
)
q
(3.4)
where & is strain rate, c p is the specific heat capacity change at Tg , We (T ) = Wg exp ( n q ) 1 - T Tg is the equilibrium barrier energy density, and is a model parameter. This parameter arises from treating the irreversible barrier crossing events as uni-molecular Maxwellian relaxation processes, and in essence quantifies the deviation from that assumption. For Pt57.5Ni5.3Cu14.7P22.5, the measured
(
)
c p = 2.56 MJ/m3 can be employed [13]. The quantity c p was estimated as the change
in specific heat evaluated at the calorimetric glass transition temperature as measured using a Differential Scanning Calorimetry (DSC) trace. Fig. 3.4 presents the DSC trace used for this estimate. As shown in Figs. 3.1 and 3.3, Eq (3.4) is capable of capturing the non-equilibrium viscosity and shear m...