temperature, Tgm. The entropy drop for this first order liquid/glass transition is ~2/3 of the entropy of fusion of the crystallized eutectic alloy. Below Tgm, the configurational entropy of the frozen glass continues to fall rapidly approaching that of the crystallized eutectic solid in the low T limit. The so-called Kauzmann paradox, with negative liquid entropy (versus the crystalline state), is averted and the liquid configurational entropy appears to comply with the third law of thermodynamics, Despite their ultra-fragile character, the liquids at x = 14 and 16 are bulk glass formers yielding fully glassy rods up to 2-and 3mm diameter on water quenching in thin wall silica tubes. To the authors' knowledge, the low Cu content alloys are the first definitive examples of glasses that exhibit first order melting.
A liquid near its glass transition is a metastable state of matter that ultimately crystallizes given adequate time to explore its available configurational phase space.
Crystallization is triggered by a relatively improbable fluctuation whereby the liquid crosses a crystal nucleation barrier. If the time required for this improbable fluctuation sufficiently exceeds the time required for the liquid to explore its available non-crystalline configurations, one may define a specific metastable configurational entropy for the liquid as sC = kB ln (WC) where WC enumerates the non-crystalline configurations per atom available. Stable liquid configurations or inherent states are defined as atomic arrangements that minimize the specific liquid configurational potential energy per atom, f. Interpreting WC(f) as a density of inherent states (per atom) vs. f leads to sC(f) = kB lnWC(f) as the definition of liquid configurational entropy. Since the early work of Goldstein [1], this picture has been the basis for liquid theories based on the potential energy landscape (PEL) approach [2][3][4][5][6]. PEL theory maintains that thermodynamic state functions (e.g. entropy, enthalpy, free energy, etc.) of a liquid are approximately separable into additive configurational and vibrational contributions that are weakly coupled via anharmonic effects. [2,3] Defining and/or measuring sC(f) or sC(T) requires that the liquid explore all noncrystalline inherent states of given f over the relevant experimental time scale. In short, the liquid should behave ergodically within the non-crystalline configurational phase space.
Crystallization of a metastable undercooled liquid always limits the available experimental configurational equilibration time. The transformation to a fully crystallized sample involves both crystal nucleation and growth. The former is statistical, and the latter is often kinetically sluggish [7] so that progression of the liquid-crystal transformation is often spread over time scales spanning orders of magnitude [8]. To further complicate measurements of sC(T), liquids near Tg relax toward equilibrium following "stretched" exponential behavior [9] where DsC(t) ~ exp[-(t/t) b ] with a stretching exponent b < 1.
Typically, b ~1/2 or less for fragile undercooled liquids. Achieving metastable configurational equilibrium may therefore require times exceeding the Maxwell relaxation time tM by orders of magnitude. Configurational relaxation involves a spectrum of activated processes that include a slow a-relaxation and a fast b-relaxation. These appear to merge into a single process above a dynamic merging or cross-over temperature, TC [10,11].
In this report, we present an experimental study of the configurational thermodynamic state functions for a series high fragility Pt80-xCuxP20 bulk metallic glass forming liquids [12]. The bulk metallic glasses form along a binary eutectic line that terminates at a ternary eutectic composition. The eutectic solid is a mixture of 3 crystalline phases each having a distinct composition that differs from that of the liquid. We use this multi-phase crystallized eutectic solid as a well-defined thermodynamic reference state.
The crystalline eutectic solid exhibits a sharp melting transition to a homogeneous singlephase liquid at the eutectic composition and temperature, cE and TE with a well-defined enthalpy and entropy of fusion.
For the metallic glasses investigated, crystal nucleation from the undercooled liquid is a transient process whereby the observed nucleation rate rises very steeply following extended configurational relaxation of the liquid over a relatively long incubation time tLX(T). Nucleation is followed by rapid coupled eutectic growth where the liquid to crystal transformation is completed in a relatively shorter time scale DtLX << tLX(T). We show that tLX(T) exceeds the configurational a-relaxation time (Maxwell Relaxation time), ta, of the liquid by orders of magnitude; typically, tLX(T)/ta ~ 10 3 -10 6 . This ratio quantifies the extent of configurational equilibration of the liquid prior to the detectable onset of crystallization, and thereby the extent to which the liquid achieves metastable configurational equilibrium. The temporally sharp release of enthalpy associated with the short DtLX permits an accurate calorimetric determination of the total heat of crystallization at ambient pressure P0 and fixed T, hLX(T,P0,cE). We argue that hLX(T,P0,cE) is an accurate measure of the liquid configurational enthalpy, i.e. that hC(T,P0,cE)» hLX(T,P0,cE). This follows since the difference in vibrational enthalpy and entropy between the liquid and crystallized samples is observed to be quite small and essentially neglectable in bulk metallic glasses as reported recently for Cu-Zr and Al-Cu-Zr bulk glasses [13]. To simplify notation, we henceforth drop ambient pressure P0 and liquid composition cE as independent thermodynamic variables, since both are fixed in each of our experiments, and report hLX(T) data.
It is common in the glass literature to measure the configurational contribution to the liquid heat capacity. For metallic glasses, this is commonly assumed to be well approximated by difference in liquid and crystal heat capacity cLX(T) [14][15][16]. This heat capacity difference is often assumed to follow a "1/T 2 " temperature dependence as originally proposed by Kubaschewski et. al. [17]. In PEL theory, this T-dependence follows directly from assuming a Gaussian distribution of inherent states, WC(f) [2][3][4]. By integrating dhLX/T = cLX(T)dT/T, one obtains a specific configurational enthalpy that follows a "1/T" temperature dependence with an integration constant ℎ "# (∞) representing hLX(T) in the limit T®¥ . We describe our data using a more general empirical form for the liquid configurational enthalpy :
where qh is an alloy-dependent characteristic "isenthalpic" temperature and further introduce the exponent n, which takes the value n=1 for the special case of a Gaussian distribution of W(f). We refer to n as a thermodynamic fragility index.
As shown in SM (see SM section B), eqn.(1) is equivalent to a microcanonical configurational entropy sC(f) = kB ln[WC(f)] of the form: 𝑠 ( (𝜙) ∝ 𝑠 ( (𝜙 7 ) -𝐶(𝜙 7 -𝜙) 9:; 9
where C is a normalization constant and f0 is the limiting value of configurational potential energy in the high T limit. For n =1, the expression reduces to a Gaussian distribution for WC(f) [2][3][4][5][6]. From the fits to hC(T) » hLX(T) using eqn.(1), we compute the specific liquid configurational entropy, sC(T), by integration. For the Pt-Cu-P alloys, we find sC(T)® 0, at a "Kauzmann" temperature [18] , TK, that lies near or even slightly above the laboratory glass transition. While the configurational entropy at TK vanishes, the configurational enthalpy (relative to the crystallized eutectic solid) remains finite. This residual enthalpy, hR= hC(TK), is found to be 30-38 % of hC(¥) over the alloy series. For x = 14 and 16, the bulk glasses display a distinct latent heat and a discontinuous first order transition at an apparent melting temperature Tgm. In addition to first order melting, the x = 14 and 16
glasses display: (1) ultra-high Angell fragility parameter [19] m > 90, and non-Newtonian viscosity behavior at very low strain rates (~10 -6 s -1 ), (2) a systematic reduction in crystal nucleation time tLX(T) as strain rate increases. Despite their ultra-high fragility, the alloys at x = 14 and 16 form bulk glasses ( > 2mm diameter glassy rods) if quenched under relatively quiescent conditions.
Table I lists the alloy compositions studied and summarizes important properties of the Pt80-xCuxP20 bulk glass forming alloys. The crystalline alloys exhibit sharp near-eutectic melting. Instrumental broadening of the melting transition depends on the heating rate used in the DSC melting scan. This rate dependence was quantified by observing the melting transition of a pure metal (e.g. Sn, Al) as seen in Fig.S7 of the SM. Table I gives the liquidus and solidus temperatures at the lowest heating rate used for a given alloy. Based on the melting data, the composition x = 23 appears to be a ternary eutectic composition with a melting transition of width £ 4K at a heating rate of 0.2 K/m, essentially indistinguishable Cu content [12] is included for reference in Fig. S2. The bulk glasses at all compositions display an extended metastable liquid region with TX -Tg > 50 K in DSC scans at heating rates of 5-20 K/min. Table I lists the calorimetric onset of the glass transition at 10 K/min, the measured total heat of fusion (averaged over 5 or more measurements) at the indicated heating rate (ranging between 0.2 K/min and 1 K/min), the measured melting onset temperature (solidus) TS, and melting completion temperature (liquidus) TL, both at the lowest heating rate used. Finally, the table gives the critical casting thickness, dC, for bulk glass formation determined by water quenching from the high temperature melt in a thin walled (1mm) silica tube [21,22]. The dC values provide one measure of alloy glass forming ability than can be related to tLX(Tnose), the minimum crystallization time at the "nose" of the TTT-diagram [21,22] as discussed in ref. 21.
Fig. 1a shows isothermal DSC scans at different temperatures above Tg for the most extensively studied ternary eutectic Pt57Cu23P20 alloy. The scans were obtained using disks (~ 60-80 mg) cut from a 2 mm glassy rod prepared by water quenching. To bring the ascast glass samples to a well-defined initial state, all samples were first equilibrated by preannealing in the DSC at the calorimetric onset Tg (~ 230 ˚C ) for 15 hours. At Tg, the time for the onset of crystallization is observed to be tLX(T) > 10 6 s ( ~ 1 week). Following the pre-anneal, each sample was slowly heated at 0.5 K/min from Tg to a designated holding temperature T and held there until crystallization occurred. Crystallization is indicated in Fig. 1a by a single sharp exothermic peak in the isothermal scans. The data are plotted on a logarithmic time scale in Fig. 1a. When the sharp exothermic peak is normalized by its height (in W/g) and the curves at different temperatures are shifted to align along the logarithmic time axis at the peak time, tpeak, a single peak shape versus log(t/tpeak) is seen.
The plot illustrates key features of the liquid/crystal transformation: (1) there is a long incubation period tLX(T) ~10 4 -10 6 s preceding the detectable onset of exothermic heat release, (2) crystallization emerges abruptly with an exothermic signal rising sharply to a peak and then decaying rapidly, (3) crystallization is completed over a relatively short time "nose time" of ~ 1s as seen in Fig. 1c. Here, "nose time " refers to the minimum time to crystal nucleation in the C-shaped TTT-diagram as discussed in ref. [22]. This extrapolated time roughly agrees with that inferred from the critical casting thickness of the glass, dC =15mm (see Table I) of ~1s [22]. To assess the degree of liquid configurational relaxation prior to crystallization, we carried out viscosity measurements. The viscosity data are provided for reference in Fig. S3. We used the viscosity data to estimate a Maxwell configurational relaxation time for the undercooled liquid, ta = h(T)/G, where G » 32 GPa, is the shear modulus of the glass (measured ultrasonically, see SM). Fig. 1c compares log(ta) with log(tpeak) and shows that the ratio tpeak/ta ranges from 10 3 -10 6 , increasing with T above Tg. This ratio quantifies the extent of liquid relaxation towards metastable configurational equilibrium prior to crystallization. It establishes that the measured hLX(T) is representative of a configurationally relaxed undercooled liquid that in metastable equilibrium as illustrated in Fig. 1c.
To ensure the liquid/crystal transformation is completed (no glass remains) following the isothermal segment, the samples were subsequently heated (at 5 K/min) through the melting transition. For Pt57Cu23P20, a relatively small heat release (~ 4 J/g) is consistently observed during the heating segment in the range of 320 -340 ˚C. This is believed to result from relaxation and/or coarsening of the initially crystallized structure.
No other calorimetric events are observed through the eutectic melting transition. The heat of fusion was measured during each reheating segment and found to have an average value of 68.5 ± 1 J/g. High resolution SEM and chemical mapping were used to determine that the as-cast glass is chemically homogeneous on the length scale of nanometers and lacks any observable microstructure as shown Fig. S5 of the SM. X-ray diffraction was used to establish that the as-quenched, pre-annealed (at Tg for 15h), and isothermally held sample prior to the exothermic event, were all fully amorphous. A series of diffraction scans on a single 3mm diameter disk sample were taken at various steps in the thermal history as shown in Fig. 1d for a sample isothermally crystallized at 240 ˚C. The x-ray sample was processed in the DSC to ensure that its thermal history was identical to that of other calorimetric samples. The DSC segment scans were interrupted at each respective step and the sample was cooled to room temperature for x-ray. The diffraction scans indicate a fully amorphous structure for the initial as-cast sample (scan #1), the pre-annealed state for 15h at 230 ˚C (scan #2), and the isothermally equilibrated sample (after heating to 240 ˚C and holding for 6h), but before onset of the crystallization event (scan #3). Following the crystallization event (26 h at 240 ˚C) the x-ray scan shows a fully crystallized alloy (scan #4). On subsequent heating to 350 ˚C and holding there for 2h, x-ray scan #5 is essentially identical with scan #4. This demonstrates that the crystalline phases produced during the exothermic event at 240 ˚C (scan #4) are present and unchanged on further heating to higher temperature and following the small heat release at 320 -340 ˚C. No new diffraction peaks are associated with the small exothermic heat release at 320 -340 ˚C. The sharp exothermic heat release in the isothermal segment at 240 ˚C therefore corresponds to a transformation from a homogeneous and equilibrated undercooled liquid to a fully crystallized sample.
The measured heats of crystallization during all isothermal segments (including the small excess contribution at ~320 -340 ˚C observed upon subsequent heating) are compiled for reference in Table S1 of the SM. For temperatures above ~270 ˚C, it was not possible to obtain accurate isothermal measurements of hLX(T) since tLX(T) becomes too short to permit stabilization of the DSC control loop prior to the onset of crystallization.
Data above 270 ˚C were acquired by pre-annealing at 230 ˚C (15 hours as in the isothermal scans), and then continuous heating at varied rates ranging from 0.1 K/min to 10 K/min.
These constant heating scans also exhibit a single very sharp crystallization event as shown in Fig. 1e. With increasing heating rate, crystallization occurs at progressively higher temperatures and the small exothermic heat release between 320 -340 ˚C disappears and apparently merges with the single sharp exothermic event for heating rates above 1 K/min.
No other detectible heat release is observed through the eutectic melting transition. To quantify the width of the crystallization event, consider the difference between the onset and the completion temperatures. At lower heating rates (below 5 K/min), the peak width is of order 2K. The entire heat of crystallization is released during this event. The crystallization peak temperature thus accurately reflects the temperature at which the heat hLX(T) is released. At the highest heating rates, the exothermic peak becomes broadened by instrumental effects (see Fig. S7 and discussion). Corrections for the rate dependent peak shift (see discussion below) are indicated in Table S1. The constant heating rate data were combined with the isothermal data to give an hLX(T)-curve over a broader range of T (from 230 ˚C up to ~ 290 ˚C) as displayed in Fig. 2a. a maximum liquid undercooling down to ~365 ˚C (~180 ˚C below TE) was attained as seen in Fig.S4 of the SM. The undercooling nearly reaches the nucleation nose temperature (~350 ˚C). The undercooling scans also display a single sharp exothermic crystallization peak (provided that liquid undercooling > 50 K is attained before crystallization). These undercooling data provide a measure of hLX(T) from just above the nucleation nose to temperatures within 50 K of TE as shown in Fig2a. Following each crystallization event, continued cooling to 200 ˚C resulted in no further detectable calorimetric events. The sample was then reheated through melting and back to 900 ˚C to complete a cycle. The heat of fusion during the reheating was measured and was repeatable with an average value of 68.5 ± 1 J/g.
Finally, Fig. 2a shows two additional measurements done using the Rapid Discharge Heating (RDH) method previously described [23,24]. Here, a glassy sample rod of 4 mm diameter was heated rapidly and uniformly using ohmic dissipation at a rate of ~ 10 5 K/s to a target temperature and held there until crystallization occurred. The rod temperature at its center was measured using a high-speed infrared pyrometer (5 µs response time).
Following rapid heating to 350 ˚C, the sample configurationally relaxes, and the temperature drops and stabilizes at T ~ 312 ˚C as shown in Fig. S8 of the SM. . Following an incubation time of tLX(T » 312 ˚C) of ~ 0.6 s, the sample abruptly crystallizes, as seen in Fig. S8, accompanied by a sharp recalescence event and temperature rises sharply to ~ 515 ˚C , below but near the alloy eutectic temperature. This suggests the sample was constitutionally undercooled at 312 ˚C [25]. A high speed (up to 1300 frames/s) infrared camera (FLIR SC4000) was employed to simultaneously image the eutectic crystallization front. Selected movie frames at ~ 1/8 second intervals are shown in Fig. S8. A coupled eutectic growth front is seen to propagate from a single nucleation site over the entire sample at an average speed of ~1.5 cm/sec. This eutectic growth speed is very high when viewed in terms of models of crystal growth such as those described by Orava and Greer [7]. The result suggests that the short range chemical order of the crystalline phases must already be present in the liquid thereby mitigating the need for chemical partitioning kand diffusion along the crystallization front. This demonstrates why the liquid-crystal transformation in the Pt-Cu-P alloys (at and below 312 ˚C) is completed in very short time scales (as in Fig. 1e). Crystallization is not growth limited, but rather controlled by the sharp onset of nucleation. The temperature rise of ~200 K on recalescence was used to estimate a heat release of hLX(312 ˚C) of ~52 J/g as shown in Fig, 2a. The reader is referred to the SM for details.
The complete set of hLX(T) data obtained using the four different methods above are shown in Fig. 2a. The collective data were fitted using eqn.
The analytic expressions for sC(T) and gC(T) for the Cu23 alloy are shown in Figs. 2b and 2c. The entropy of fusion (see Table I) was evaluated at TE (taken as the average of the experimental TS and TL). From the computed sC(T), one gets an analytic expression for the Kauzmann temperature TK by setting sC(TK) = 0. Using a dimensionless DsF (in units of [hC(¥)/qh]), TK is given by:
1 9:; From the h(T) data as provided in Fig. S3, the rheological glass transition temperature defined by h(Tg) = 10 12 Pa-s, is Tg = 501.5 K, slightly below TK. The experimentally determined TK is, within uncertainty, essentially indistinguishable from the rheologically defined laboratory Tg. From the viscosity data, an Angell fragility parameter [19] for the alloy was determined to be m = 72 ± 3; the alloy is rheologically very fragile. At ambient pressure, one may ignore the "Pv" term in the configurational Gibbs free energy, i.e. gC(T)» hC(T)-TsC(T). To calculate gC(T) in Fig. 2c, we assumed that at TK, the fully ordered glass is configurationally frozen. Below TK, we have taken sC(T)=0 and gC(T) = hC(TK). The apparent entropy from our fits would become negative (sub-ensemble) below TK, suggesting the frozen system effectively runs out of configurational states. A similar picture was introduced used by Derrida [26] to describe the freezing of spin glasses. Notice that gC(T) remains linear vs. T down to deep undercooling temperatures thereby following the Turnbull-Spaepen approximation [27] to surprisingly high accuracy. This indicates that configurational freezing sets in only at very deep liquid undercooling. Following the same methods described above for the Cu23 alloy, we determined the hLX(T) curves for the other metallic glasses in the Pt80-xCuxP20 alloy series. Both the isothermal and constant heating data for all alloys of the series have been compiled for reference in Table S1 of the SM. Fig. 3a compares the hC(T) = hLX(T) plots for 3 representative alloy compositions and illustrates the variation of the hC(T)-curves as Cu content varies. Table II summarizes the fitting parameters hC(¥), qh, and n obtained using eqn,(1) for these three cases x = 20, 23 and 27. For the two cases x = 14 and 16 with the lowest Cu content, a separate discussion and analysis as given below. Table II includes the Kauzmann temperatures, TK, obtained from eqn.(4). While the computed configurational entropy of the liquids vanishes at TK, the configurational enthalpy does not. The ordered glasses at their respective TK possess a finite residual enthalpy hR = hC(TK) vs. the crystallized eutectic reference state. This residual enthalpy was evaluated using eqns. (3) and (1). Table II gives the ratio hR/hLX(¥) for x = 20, 23, and 27. This varies from 0.28 -0.36. Note that hR represents the heat of crystallization of a fully ordered or ideal glass at TK. The configurational enthalpy or equivalently potential energy of the ordered glass state lies hR above that of the crystalline state. Fig. 3b shows the overall variation of 1/n versus Cu content x. The solid curve in Fig. 3b is a simple linear fit vs. the composition x. The plot suggests that n diverges at x ~ 17. For completeness, x-ray diffraction data for x = 16 before and following crystallization and on subsequent heating are included in Fig. S6 of the SM (similar to the x-ray scans for the x = 23 case in Fig. 1d). As seen in Table II, TK increases as x decreases from TK = 472 ± 10 K at x = 27 to an apparent value of TK = 517.4 K at x = 18. For x £ 18, determining an accurate TK requires a further discussion to be presented below.
The diverging n near x ~ 17, indicated by Fig. 3b, suggests the glass transition becomes a first order freezing transition. On heating at a fixed rate, the calorimeter response to increasing T will always lag behind the measured calorimeter temperature, i.e. glass melting is heat flow limited and the calorimeter has a finite response time (~ tens of seconds). This systematic lag will broaden the calorimeter response to a first order transition. The lag effect may be assessed and quantified by observing the melting transition of a pure metal. To assess the systematic peak shift due to a finite heating rate, we measured the melting transition of pure Sn (Tm = 501 K) at heating rates varying from 0.5 to 20 K/min. The melting transition of Sn (Tm = 229 ˚C) was chosen for its proximity to the present temperature range of interest. Fig. S7 in the SM shows the melting peak shift and broadening corrections as determined from the Sn melting data. This peak shift correction was used to provide an instrumental heating rate correction to the peak temperature (a correction ranging from less than ~ 1 K up to ~ 10 K as heating rate increases from below 1 K/min up to 20 K/min) for the glass-liquid hLX(T) data. The peak shift corrections versus heating rate, as determined from the melting of Sn, are listed in Table SI. The corrected data for Cu14 and Cu16 are plotted in Fig. 4a. For Cu14, the corrected hLX(T) data display a clear vertical step at a glass melting temperature of Tgm = 533 K. For Cu16 one observes a smaller vertical step at Tgm = 548 K. Both curves imply a latent heat and first order glass-liquid melting transition. For the Cu14 sample, we obtain a latent heat of ~27 J/g at 533 K with Tgm lying ~30 K above the nominal laboratory onset Tg. For Cu16, the latent heat, ~20 J/g, is somewhat smaller but with somewhat higher Tgm = 547 K By contrast with Figs. 2a and 3a, the hC(T)-curves for Cu14 and Cu16 presented in Fig. 4a change curvature becoming concave downward along with showing a clear discontinuous enthalpy jump. The jump occurs at a temperature Tgm that lies distinctly above the nominal laboratory Tg . From below Tgm, one approaches this freezing transition from the solid-like or "glass" side of the enthalpy discontinuity. It follows that eqn.( 1) is no longer appropriate. Recognizing that the residual enthalpy of the glass, hR, is an additive constant to the enthalpy of the liquid-glass freezing transition, it is natural to describe the glass below Tgm using a modified version of eqn.(1) describing the approach to the discontinuity from below the step. To analyze this data, we introduce a power law form:
to describe the approach to melting of the glass. Notice that qh now represents the temperature where hC(T) for the low temperature glass phase crosses hC(¥), or equivalently
TK * is effectively an upper bound for the glass melting temperature. Note that TK * for the Cu14 alloy is slightly lower than for Cu16 implying a more restrictive upper bound.
Note that the data for x = 14 (16) in the glass region below Tgm = 533 K (546 K)
were generally obtained from isothermal crystallization scans. The glass enthalpy approaches a residual value of hR ~26.0 J/g (24.5 J/g) for the Cu14 (Cu16) compositions.
The temperature dependence of the glass enthalpy below Tgm suggests that the equilibrium glass is in a configurationally excited state when equilibrated at finite T < Tgm. While slow kinetics might limit relaxation and equilibration of the glass below Tgm, this seems unlikely as the data were obtained under isothermal conditions whereby the glass was relaxed for time scales far exceeding the configurational Maxwell relaxation time as discussed earlier.
Assuming the data represent a glass in configurational equilibrium, one may use the enthalpy fits from eqn.( 5) to compute the T-dependent configurational entropy of the glass below Tgm. Combining the fits with the measured latent heat of glass melting at Tgm and the measured entropy of fusion of the crystalline eutectic one may now generate a piece-wise continuous configurational entropy vs. T plot for the glass/liquid system as displayed in Fig. 4b. The figure shows the configurational entropy curves (entropy in units of the gas constant R) versus T for both the x = 14 and 16 glass-liquid systems. Within the accuracy of the eqn,(5) fitting and the assumption of configurational equilibrium, one finds the configurational entropy of the Cu14 glass at Tgm = 533 K to be 0.0308 J/g-K or 0.533 R.
The fit from eqn.( 5) then provides an extrapolation of the glass entropy from Tgm to T = 0 K. The predicted entropy at T = 0 K vanishes within an estimated uncertainty of ~ ± 0.03 R. The glass apparently obeys the third law of thermodynamics within the stated uncertainties. This provides an immediate resolution to Kauzmann's apparent paradox. The equilibrium glass configurational entropy never becomes negative but rather rapidly approaches that of the crystallized eutectic solid as T falls below Tgm. The configurational entropy ultimately vanishes only at 0K. In this respect, the glass behaves much like a crystalline solid but with a relatively smaller heat of formation for configurational excitations (or defects in the case of crystals).
It should be noted that we have ignored possible vibrational contributions to the entropy throughout our analysis. In fact, our data for the heat of crystallization implicitly include any contribution arising from differences in anharmonicity between the glass and crystalline eutectic since we measure the total heat release during crystallization. In this regard, the form of eqn, (5) and the high values of the exponent n obtained strongly suggest that anharmonic contributions are relatively small. This follows since anharmonicity, to leading order in T, would display a "T 2 " contribution to the heat of crystallization. This contrasts sharply with the large values of n indicated from fitting the data. It is concluded that configurational degrees of freedom must dominate the temperature dependence observed for the heat of crystallization below Tgm.
Essentially the Cu14 and Cu16 glasses melt in an apparent manner similar to crystal melting, but very differently than generally envisioned for a traditional glass transition.
The ordered glass might be usefully viewed as a configurationally ordered crystalline solid albeit with an infinite unit cell. It should not be surprising that this configurationally ordered glass has a unique configurational ground state enthalpy, hR, that differs from that of the eutectic crystalline solid state. Allotropic or polymorphic crystalline phases also have characteristic ground state energies that differ by phase.
The 1)/n ]. Within measurement capability, we observe a latent heat of melting at a well-defined Tgm for the x = 14 and 16 samples. This signals unambiguous first order melting. The glasses apparently melt in a similar manner to a crystal, but very differently than expected
for a traditional glass transition. The ordered glasses melt to a fully disordered liquid in its high temperature limit. Finally, the equilibrium glass configurational entropy remains positive and finite below Tgm and approaches zero in the limit of low T. The Kauzmann paradox is averted and the configurational entropy contribution appears to obey the third law.
Based on MD simulations, Berthier et.al. [28,29] and others [30] have recently reported that ultra-stable (configurationally relaxed) atomic glasses with a polydisperse atomic size distribution exhibit first order melting behavior on rapid heating through the glass transition. Such ultra-stable glasses are prepared experimentally (and in simulations) using layer by layer atomic deposition onto substrates held near Tg, thereby allowing a degree of configurational relaxation not achievable on cooling a monolithic liquid [28,29].
In the MD simulations, such glasses melt by propagation of a first order melting front in the same manner as crystals melt. The authors attribute this first-order melting behavior to the highly ordered, low configurational enthalpy state associated with the ultra-stable glasses. In the ultra-fragile Pt-Cu-P alloys, we have shown that extensive configurational relaxation of the liquid state is achieved near Tgm prior to the onset of crystallization. We thereby achieve a low enthalpy "ideal-glass" state of zero configurational entropy lying near the lowest achievable glass enthalpy hC(T)= hR. We suggest that the first order transition at Tgm is analogous to that seen in the ultra-stable glasses in MD simulations.
If we consider alloy composition x to a tunable external parameter, then our results lead to a picture wherein the configurational entropy of the undercooled liquids sC(T,x) develops a folding behavior versus T at a critical value of xc ~16-17. The sC(T,x) isobars become folded when x < xc giving rise to an entropy of glass fusion. At xc, one expects a critical point as 1/n ® 0 accompanied by diverging heat capacity. Below xc, the glass transition is first order with a progressively increasing latent heat as x decreases. This picture is reminiscent of Van der Waal's description of the liquid/gas transition where the P-v isotherms become folded below the liquid-gas critical temperature. In the present case,
x correlates directly with liquid fragility. Essentially, liquid fragility plays the role of an externally tunable control parameter. The first order glass transition is associated with the ultra-high fragility limit. The rheological Angell fragility parameter m is also expected to diverge, and liquid viscosity should show a discontinuous jump at Tgm. It is noteworthy that such behavior has been reported in bulk metallic glass forming liquids (e.g. Vitreloy 1)
well above the laboratory Tg [38,39]. For the Van der Waal's picture of the liquid-gas transition, the liquid bulk modulus vanishes at a critical point. The authors suggest the present first order glass transition be interpreted as transition from a fluid lacking long range elasticity to a solid with a finite quasi-static macroscopic shear modulus. A study of ultrasonic elastic properties at the first order glass-liquid transition in the low Cu alloys is currently underway.
To conclude, we comment on why the present Pt-Cu-P glasses display such unusual first order freezing/melting behavior. Equivalently, one may question the microscopic origins of liquid fragility. If the glass transition is connected with the emergence of longrange elastic rigidity, one may speculate that the present liquids lack inherent local rigidity.
Local rigidity of atomic clusters is related to atomic potentials or force fields. A lack of local rigidity of atomic clusters can be traced to a soft interatomic potential that provides little resistance to bond length changes and therefore to configurational rearrangements of the atomic cluster. This leads naturally to a narrow distribution of inherent states and thereby to a low freezing temperature. The onset of long range elasticity is accompanied by long range elastic interactions, of the type discussed by Eshelby [31,32], that broaden the configurational excitation spectrum on freezing. If elastic percolation in three dimensions is inherently a first order transition, this will lead to a sharp glass transition.
We speculate that the noble metals Pt and Cu and particularly P are characterized by soft interatomic interactions that lead to a narrow PEL at high T which then broadens substantially with the onset of long range elastic rigidity during freezing at low T. The relatively low freezing temperature (Tg) of the present Pt-Cu-P glasses by comparison with other similar transition metal-metalloid glasses provides a clue supporting this conjecture.
The calorimetry work reported here was done using calibrated Netzsch 404C F3 DSC with separate calibrations for each constant heating rate reported. A sapphire standard is used for calibration. X-ray diffraction scans were acquired using a Bruker D2 Phaser diffractometer. Viscosity measurements were carried out using a Perkin-Elmer Thermo Mechanical Analyzer in the beam-bending configuration, The Rapid Discharge Heating experiments were performed using 15 kJ capacitive discharge heating system as described in ref. [23]. The system permits rapid and very uniform heating of rods of diameter d =3-5 mm length L = 3 cm at heating rates ranging up to ~ 10 5 K/s. This is due to the fact that Ohmic heat dissipation is uniform throughout the sample. The reader is referred to ref. 23 for details. For rapid heating experiments, temperatures were measured using a non-contact high speed Impac series 5 infrared pyrometer with 5µ time resolution. A FLIR SC-4000 infrared camera (having 256,000 pixels, frame rates up 1300 frames/s) was employed in conjunction with the rapid discharge heating system to provide the infrared images of couple-eutectic crystallization of rapidly heated glassy rods. Ultrasonic elastic constant measurements were done using the pulse-echo method with 25 MHz quartz transducers to determine shear and longitudinal sound velocities on 3mm diameter glassy rods. Density measurements were done using the Archimedes method.
To measure the heat of crystallization for a configurationally equilibrated liquid, the crystallization time tLX must exceed the time required for liquid relaxation. Timedependent cP studies [33][34][35] demonstrate that this may require up to 10 3 a-relaxation steps of duration ta = Maxwell relation time = h(T)/G to ensure the liquid reaches metastable equilibrium. In the present work, this requires tLX > 10 3 -10 4 ta. Following the onset of crystal nucleation, the liquid-crystal transition occurs in a single step by rapid coupled eutectic growth. Coupled-eutectic growth occurs for alloys in sufficiently close proximity to the eutectic composition [7,8,36,37]. At higher temperatures (e.g. Fig. 2a, RDF data @ 312 ˚C) we have used high speed infrared imaging to directly measure the speed of eutectic crystallization front to be ~1.5 cm/s (see Fig. S8 and discussion). This high growth speed explains the small scale of DtLX seen in calorimetry. A single nucleation event leads complete crystallization of a sample with "cm" dimensions within a time of ~1 sec. as seen in Fig. S8. It is important to appreciate that this is not the general mode of crystallization in metallic glasses. To accurately measure hLX , the typical calorimetric power signal generated by the liquid to crystal transformation DPLX(t) ~ hLX/DtLX must lie well above the sensitivity level of the instrument. The sensitivity/stability of the Netzsch 404 was determined to be roughly ±10-15 µW. With our typical sample size of ~ 60 -80 mg and total heat release of hLX ~ 50 J/gram, this requires DtLX < 5 x 10 3 s to ensure an accurate measurement (< 5% error) of hLX. At the lowest temperatures (longest DtLX) calorimeter noise and drift limit accuracy as seen for example, see Fig. 1a for the lowest isothermal crystallization temperature of 239 ˚C. Finally, for isothermal measurements, the incubation time tLX(T) must be long enough to ensure stabilization of the DSC temperature and PID loop following heating to a given constant T. In our experiments, the Netzsch 404c F3 response time is typically 30 -100 s. This explains why isothermal measurements were done below ~ 270 ˚C while constant heating rate scans were used to measure hLX for higher temperatures.
TABLE I Summary of basic properties of Pt80-xCuxP20 bulk metallic glass forming alloys in the present study. Table includes onset of laboratory Tg at 10 K/m, melting data for solidus TS and liquidus TL temperatures at the indicated heating rates, heat of fusion, and the critical casting thickness (maximum rod diameter) for glass formation by water quenching in thin wall silica tubes. Alloy (at.%) Tg(K) (onset) DSC TS (K) Solidus [rate K/m] TL (K) Liquidus [rate K/m] Heat of fusion [J/g] Critical glass Thickness dC [mm] Pt66Cu14P20 504 828.9 [1 K/m] 879.4 [1 K/m] 67.7 1.5-2* Pt64Cu16P20 504 827.8 [1 K/m] 877.5 [1 K/m] 68.7 3 Pt62Cu18P20 505 816.9 [0.5 K/m] 864.1 [0.5 K/m] 71.8 4 Pt60Cu20P20 505 825.6 [1 K/m] 833.7 [1 K/m] 67.8 6-7 Pt57Cu23P20 eutectic composition 505 823.8 [0.2 K/m] 827.9 [0.2 K/m] 68.5 15 Pt53Cu27P20 506 819.5 [0.1 K/m] 840.1 [0.1 K/m] 67.5 27 *glass formed when quenched without mechanical agitation (see text) in a thin wall silica capillary tube. Table II. Compilation of Pt100-xCuxP20 metallic glass fitting parameters obtained from eqn.(1) for x ³18 (eqn.(5) for x =14,16), along with Angell Fragility parameter m from viscosity data (see SM), the calculated Kauzmann temperature TK from eqn.(4) or TK * from eqn.(6) as described in the text, and the normalized residual configurational enthalpy, hR/hC(¥), for the fully ordered glass. Alloy hC(¥) (J/g) q h (K) n m Fragility (Angell) TK (*) (K) 𝒉 𝑹 𝒉 𝑪 (∞) Pt66Cu14P20 67.9 561.1 * -19.2 * >90? TK * =562.1 0.383 Pt64Cu16P20 68.5 560.5 * -28.3 * >90? TK * =560.7 0.360 Pt62Cu18P20 72.4 508.3 25.7 >90? 517.4 0.360 Pt60Cu20P20 68.3 492.2 13.3 >82 # 511.4 0.341 Pt57Cu23P20 70.2 484.8 8.36 73 505.5 0.334 Pt53Cu27P20 69.9 433.6 4.30 --472 ±10** 0.266 # Viscosity is highly non-Newtonian and strain rate sensitive. Crystallization is induced by flow. It was not possible to determine an accurate value for Angell parameter m. Value given is a lower bound. *Values obtained using eqn.(5) and eqn.(6). See text for discussion of fitting procedures used for samples with Cu content x =14, 16. ** Value has large error due to large uncertainty in eqn.(1) fitting parameters Fig. 1 (a) Isothermal DSC scans vs. log(t-sec.) at various fixed temperatures illustrating the sharp exothermic crystallization event for eutectic Pt57Cu23P20. All isothermal scans were done following a pre-anneal of the as-cast sample for 15 hours at Tg =230 ˚C; (b) Normalized signal versus ln(t/tpeak); (c) Plot of log(tpeak) versus Tg/T compared with the logarithm of the Maxwell relaxation time, log(tM) = log(h(T)/G) as described in the text. Vertical dashed lines indicate the configurational equilibration region; (d) X-ray diffraction scans of Pt57Cu23P20 sample beginning in the as-cast state and at various steps employed for isothermal crystallization at 240 ˚C. The sample remains fully glassy prior to the exothermic heat release at 240 ˚C (scans #1-3) while it is fully crystallized following this event (scan #4). On further heating to 350 ˚C for 2h (scan #5), the diffraction pattern remains essentially identical to that of scan (#4), (e) Constant heating rate scans for Pt57Cu23P20 from Tg = 230 ˚C through melting over a wide range of heating rates. All constant heating rate samples were pre-annealed for 15 h at 230 ˚C prior to the scan as in the isothermal scans in (a). (b) -4 -2 0 2 4 6 8 0.8 0.85 0.9 0.95 1 1.05 1.1 log(t) (s) Tg/T Metastable Liquid log(t peak ) log(t M ) (c) Figures 1 (d) and 1 (e) (e) (d) Figures 2(a-c) -30 -20 -10 0 10 20 30 400 500 600 700 800 900 1000 Free Energy (J/g) T (K) T K Pt 57 Cu 23 P 20 -0.02 0 0.02 0.04 0.06 0.08 0.1 400 500 600 700 800 900 1000 Entropy (J/g-K) T (K) Pt 57 Cu 23 P 20 T K T E (c) (b) (a) 0 10 20 30 40 50 60 70 80 400 500 600 700 800 900 1000 Configurational Enthalpy (J/g) T (K) Isotherms Constant HR Undercooling T E RDH High T limit eqn.(1) x = 23 h R T K (a) Figure 3(a) and 3(b) -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 30 35 1/n Cu content (%) First Order (b) 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 n = 1 Gaussian n = 4.3 8.4 13.3 x = 20 x = 23 x = 27 hC(T) hC(¥) q/T (a) Figure 4a 0 10 20 30 40 50 60 70 80 450 470 490 510 530 550 570 590 Enthalpy (J/g) T (K) Heat of Fusion Isothermal Constant HR T gm T gm x = 14 x = 16 h R (a) (b) Figure 4b 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 400 450 500 550 600 650 700 750 800 850 900 Configurational Entropy (units of R) Temperature (K) Glass (finite T) Entropy of fusion (Glass) T gm T E Disordered Liquid (high T limit) Entropy of fusion (Eutectic Solid) Ideal Glass x = 14 x = 16 temperature using a 25 MHz transducer to determine the longitudinal and shear sound velocities. The sample mass density at ambient T was measured using the Archimedes method. For x =23, one obtains r = 15.22 g/cc. This yields a shear modulus of G = 30.7 GPa for the cast glassy rod. Following annealing at 230 C for 15 hours, the ambient T shear modulus increases to G = 32.5 GPa due to relaxation of the as-cast glass during the annealing. The latter value was used to estimate the Maxwell relaxation times shown in Fig.1c of the main text. As such, we designate this as the eutectic composition. The onset temperature of the glass transition increases very slightly with increasing Cu content.
Glass forming ability versus Cu content for the series of Pt80-xCuxP20 alloys studied in this report. The glass forming ability is defined by the maximum diameter, dC, of a fully glassy rod that can be produced by melting the alloy at ~900 ˚C is a sealed silica tube of wall thickness 1mm and then quenching into water at room temperature. The reader is referred to reference 1 for a detailed summary of the glass forming ability of the ternary alloys along with quaternary alloys formed by adding 1-2 at.% of Ag or B to the ternary alloys. Such additions are observed to increase the glass forming ability to as high as dC ~ 6 cm,
0 5 10 15 20 25 30 10 12 14 16 18 20 22 24 26 28 30 Critical Rod Diameter (mm)
Copper Content (at. %) Fig. S3. Viscosity data obtained from beam bending using a Perkin-Elmer TMA for Cu23 (blue) and Cu20 (red) samples. For the Cu20 sample, the viscosity was observed to be non-Newtonian at the smallest loading force available in the TMA. Data below 10 10 Pa-s could not be obtained due to apparent strain-rate induced crystallization. The fragility, m, for the Cu20 sample should be considered as a lower bound. Liquid undercooling measurements of the heat of crystallization were performed on each of the alloy compositions [4]. For both x=20 and x=23, it was possible to achieve deep undercooling of the liquid below the eutectic temperature TE following cyclic overheating to 900 ˚C and cooling down to 200 ˚C For samples with Cu content x < 20, only shallow undercooling (< 80 K) were achieved. The experiments were performed in the same Netzsch 404c F3 DSC using silica crucibles.
Fig. S6 shows a sequence of x-ray diffraction scans taken during and following preannealing at 230 ˚C of the Pt64Cu16P20 followed by isothermal crystallization at 245 ˚C. The sample was subsequently heated in steps to 350 ˚C, 390 ˚C, and 480 ˚C and then cooled following each step to room temperature to obtain the diffraction scans. On heating to 350 ˚C, the scan indicates that an order-disorder transition occurs for the Pt-Cu-rich fcc phase to an ordered Pt7Cu. Fig. S6. Sequence of x-ray diffraction curves for Pt64Cu16P20 glass in its initial glassy condition and following crystallization and exothermic heat release during an isothermal crystallization at 245 ˚C (scans #1 and #2). Upon subsequent heating to 350 ˚C and 390 ˚C, the intensity of several diffraction peaks changes (e.g. peak at 2q ~ 44 deg. disappears) as observed from scans #2 through scan #3. This is attributed to chemical order/disorder transition for the Pt7Cu-phase [5]. From the data, one also identifies the monoclinic Pt5P2 -type phase containing Cu in solution on the Pt-sites. [6]. Heating to 480 ˚C results in sharpening of diffraction peaks (scan #5) suggesting that significant grain growth of the crystalline phases occurs.
Fig. S7 shows the constant heating rate scans of the melting transition of pure Sn that were used to assess the melting peak broadening and peak shift corrections associated with instrumental broadening in the Netzsch 404 calorimeter. The data were used to correct crystallization peak temperatures for constant heating rate broadening. S1. This correction was specifically used for the Cu14 and Cu16 crystallization peak temperature in Fig. 4 of the main text.
A capacitive discharge heating system described in ref. [7] was used to uniformly and rapidly heat glassy rods of diameter 4 mm and length 2.5 cm of the Pt57Cu23P20 and Pt60Cu20P20 alloys. The rods are clamped at the ends using a Cu electrode to deliver a current pulse over ~ 3 ms that heats the rod with a fixed energy input [7]. An IMPAC high speed pyrometer with response time of 5 µs was used to measure temperature over a ~0.75 mm spot at the rod center. The rod temperature at the centerline versus time is shown in Fig. S8(a). The rod initially heats to ~350 ˚C in about 3 ms, then configurational relaxation results in heat absorption and temperature decay to about 312 ˚C at about 0.2 s elapsed time. Heat absorption, in contrast to heat release, arises since the initial as-cast glass is in
220 230 240 250 260 0.1 1 10 100 Temperature (C) Heating Rate (K/m) Pure Sn