Characteristic free volume change of bulk metallic glasses

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I. INTRODUCTION
Free volume is a widely used concept related to the structure and properties of bulk metallic glasses (BMGs) such as glass-forming ability (GFA), 1,2 mechanical properties, 3 inelastic deformation, 4 and the Invar effect. 5The absolute value of the free volume V f ðTÞ contained in BMGs at a given temperature is, however, difficult to measure because it is impossible to obtain a reference amorphous state without any free volume in an experiment. 6Therefore, rather than the absolute free volume value V f ðTÞ, the free volume change DV f ðTÞ relative to the crystalline or relaxed amorphous state is usually measured on the basis of the variation in some other physical parameters with a change in temperature.However, a number of difficulties remain in the measurement of DV f ðTÞ.For example, Beukel et al. [7][8][9][10][11] showed that the specific heat capacity DC p is proportional to the temperature derivative of free volume, although DC p by itself cannot be employed to compare the DV f ðTÞ of different compositions for the undetermined composition-dependent proportional coefficient.3][14][15] Room temperature density q (Refs.8, 15-17) is a convenient, but not real-time, parameter.It can be employed to determine DV f after heating but not during the heating process, particularly in the glass transition process.The change in the maximum wave-vector ðQ max ðT room ÞÞ=Q max ðTÞ ½ 3 , which is measured by real-time high-energy synchrotron radiation x-ray diffraction (XRD), is equal to the volume change ðVðTÞÞ=VðT room Þ ½ . The DV f ðTÞ between the as-cast and relaxed states of BMGs can be quantified in situ as the ðQ max ðT room ÞÞ=Q max ðTÞ ½ 3 difference between the heating and reheating processes at glass transition temperature T g , according to the work carried out by Yavari et al. [18][19][20][21] However, this method does not clearly articulate how the free volume changes in the glass transition process because both the as-cast and reference relaxed amorphous states undergo glass transition.The development of a DV f ðTÞ measurement that is real-time and allows convenient comparisons among different compositions thus remains a challenge.
Regardless of the complexity of the original definition of free volume, as elaborated upon by Cohen and Turnbull, [22][23][24][25] the free volume of BMGs is actually a part of their volume and changes with temperature.7][28] It is thus possible to measure the free volume change using the thermal dilatation (DIL) test, and a DIL test-based method for measuring DV f ðTÞ was recently developed. 1The characteristic free volume change, i.e., the free volume released in structural relaxation DV f-sr , is identified quantitatively from the DV f ðTÞ curve.For a series of Fe-(Er)-Cr-Mo-C-B BMGs, DV f-sr is sensitive to and closely correlated with GFA. 1,2This paper presents a more detailed description of this measurement method.The sample diameter and heating rate effects on the DV f-sr of Fe-based BMGs are investigated and analyzed.A systematic investigation of the correlation between the DV f-sr and GFA of 12 typical BMGs with 6 base metals, and that between the peak temperature of thermal expansion coefficient T a-p and the ending temperature of the glass transition process T g-end , is also conducted.Finally, the validity of DV f-sr for measurement purposes is confirmed via comparison between DV f-sr and the free volume change measured with the specific heat capacity, room temperature density, and positron annihilation lifetime methods.

III. RESULTS AND DISCUSSION
A. Definitions of DV f-sr and DV f-gt Figure 1(a) is a sketch of the typical absolute free volume V f ðTÞ and the equilibrium free volume line, which is valid only in a narrow region around T g (Ref.7).The initial free volume V f0 does not change until the onset temperature of structural relaxation T sr-on is reached.At T sr-on , the free volume begins to be released due to atomic mobility. 39bove T sr-on , V f ðTÞ decreases continuously until it intersects the equilibrium line at the point of (T sr-end , V f-re ).T sr-end is both the ending temperature of structural relaxation and the onset temperature of the glass transition.V f-re is the residual free volume contained in the amorphous solid after structural relaxation and before glass transition.To return to the equilibrium state, V f ðTÞ must increase above T sr-end , finally coinciding with the equilibrium line at the point of (T g-end , V f-scl ).T g-end is both the ending temperature of the glass transition and the onset temperature of the super-cooled liquid region.At T g-end , the glassy solid is completely transformed into a super-cooled liquid with a free volume of V f-scl .The glass transition from point (T sr-end , V f-re ) to point (T g-end , V f-scl ) is a free volume generation process, which occurs only with continuous heating. 39As shown in Fig. 1, DV f-sr and DV f-gt are defined as quantifying the characteristic free volume changes in the structural relaxation and glass transition processes, respectively.The temperature intervals of these two processes are DT sr and DT gt .They are formulated as follows: (1) The reference state of Fig. 1(a) is the amorphous state with absolute zero free volume at any temperature, i.e., V f ðTÞ ¼ 0. However, it is impossible to obtain an amorphous state without any free volume in the temperature region around T g even after a very long period of annealing below T g (Refs.6 and 40).Hence, in practice, the alternative crystalline state is adopted in both the specific heat capacity and positron lifetime methods, 7,8,14 although the packing density of this state differs from that of the amorphous state.6][17][18][19][20][21] The free volume generated in the glass transition process of the relaxed amorphous state, however, is greater than that of the as-cast amorphous state.
The other alternative reference state is the amorphous state with a constant free volume V f0 at any temperature, i.e., the straight dashed line V f ðTÞ ¼ V f0 shown in Fig. 1(a).The free volume change with temperature relative to this reference state is DV f ðTÞ ¼ V f ðTÞ À V f0 , as shown in Fig. 1(b).Although the absolute values of V f0 , V f-re , and V f-scl remain unknown, DV f-sr , DV f-gt , DV sr , and DT gt can be determined from the DV f ðTÞ curve in Fig. 1(b).The key point to measuring the two characteristic free volume changes is thus to find a reference amorphous state with a constant DV f0 at any temperature.Certainly, this kind of reference amorphous state cannot be obtained by experimentation, but Sec.III B articulates a way to hypothesize it.
][26][27][28]41 Fig. 2(b) shows the similarity of the DSC traces and the thermal expansion coefficient a L ðTÞ trace above the onset crystallization temperature T x-on .In the crystallization process, the atomic rearrangement from disorder to order results in the simultaneous energy release shown in the DSC trace and the volume shrinkage shown in the a L ðTÞ trace.Fig. 2(c) shows a peak in the a L ðTÞ trace around the glass transition point.On the one hand, a L ðTÞ obviously increases due to the free volume generation in the glass transition process.On the other hand, the amorphous solid is gradually transformed into the supercooled liquid in this process.The increase in liquid content results in a decrease in viscosity.Under the compressive load of 0.3 N exerted by the DIL instrument, a L ðTÞ drops dramatically, and the softening begins at the peak temperature of T a-p .T a-p depends on how fast the viscosity decreases with temperature in the glass transition process and how great the compressive stress is.In other words, T a-p is determined by the composition and diameter of the sample.For the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG rod with a diameter of 3.0 mm, T a-p (867:461:6 K) is slightly lower than T g-end (868:162:3 K) in the DSC trace, as shown in Fig. 2(c).The difference between T a-p and T g-end will be discussed in Secs.III C 3 and III D 1.
In addition, with the free volume release shown in Fig. 2(c), a L ðTÞ decreases steadily above T sr-on .The small free volume change in the low-temperature region below T sr-on that is due to the variation in the chemical short range order 7,26 is not considered in this research.This research focuses only on the obvious free volume change in the hightemperature region above T sr-on , which is related primarily to the variation in the topological short range order. 7T sr-on thus corresponds to the onset temperature of the main structural relaxation.At T sr-on , the free volume contained in the sample is assumed to be the initial free volume V f0 .Above T sr-on , if the free volume does not change with temperature, then V f0 remains the same, a L ðTÞ has a constant value a sr-on , as shown in the hypothesized a L ðTÞ trace in Fig. 2(c), and the sample expands linearly at constant slope a sr-on as shown in the hypothesized DIL trace in Fig. 2(a).The hypothesized DIL trace is thus the thermal expansion of the reference amorphous state with a constant V f0 , which is exactly the reference amorphous state depicted in Fig. 1.Consequently, the free volume change DV f relative to this hypothesized reference amorphous state as a function of temperature, i.e., from T sr-on to T a-p , can be calculated by integrating the difference between the experimental a L ðTÞ trace and hypothesized a L ðTÞ trace or, more simply, by subtracting the hypothesized DIL trace DL 0 ðTÞ=L 0 from the experimental trace DLðTÞ=L 0 , as follows: where the hypothesized DIL trace DL 0 ðTÞ=L 0 is The premise of Eq. ( 3) is where the volume thermal expansion coefficient a V ðTÞ is a V ðTÞ ¼ 1=VðTÞ Â dVðTÞ=dT.This linear relation premise, however, is invalid above T a-p because the sample begins to soften at T a-p . 28Therefore, the free volume generation above T a-p could not be observed in the DIL test.Although the DV f ðTÞ measured with the DIL method describes only the free volume change from T sr-on to T a-p , the change trend in Fig. 2(d) is in good agreement with that of the sketch in Fig. 1(b).As shown in Fig. 2(d), the lowest temperature T sr-end , which is also the intersection temperature between the experimental a L ðTÞ and hypothesized a L ðTÞ traces, corresponds to the ending temperature of structural relaxation and the beginning temperature of the glass transition.Both the temperature interval of the structural relaxation process DT sr and the first characteristic free volume change DV f-sr can be quantified as in Fig. 2(d).The quantification of DV f-sr and DT sr in Fig. 2(d) is the same as that in Fig. 1(b).The second characteristic free volume change DV f-gt , in contrast, cannot be accurately quantified by the DV f ðTÞ curve in Fig. 2(d), and its investigation is thus omitted from this research.The detailed reasons for this omission are provided in Secs.III C 3 and III D 1.
The composition-dependent proportional coefficient is required to measure the free volume change using the specific heat capacity method, 8 although no additional parameters are needed to determine the DV f ðTÞ with the DIL method, as previously noted.DV f ðTÞ is thus convenient for comparing the measurement results of different BMGs.I, and Fig. 4 present the measurement results for different D and R h .In each group of tests, one parameter was changed and the other kept the same.As shown in Table I, a sr-on undergoes a slight increase and T sr-on changes little with a change in D and R h .For the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG, the hypothesized DIL trace, which is determined by a sr-on and T sr-on based on Eq. ( 4) and deemed to be the thermal expansion of the hypothesized reference amorphous state, shows little variation across tests.Hence, the DV f ðTÞ in different conditions is dependent primarily on the experimental thermal expansion above T sr-on .Five tests were carried out for the U 3.0 mm sample measured at 5 K/min and two tests for the other conditions.The repeatability errors of DV f-sr were 0:03-0:04 Â 10 À3 for most of the tests, but were 0:05-0:06 Â 10 À3 for the test on (U 5.0 mm, 5 K/min) and 0:08-0:1 Â 10 À3 for that on (U 3.0 mm, 20 K/min).Different from the 20 $ 30 mg sample used in the DSC test, the mass of the Fe-based BMG rod samples used in the DIL tests ranged from 0:2 $ 3:6 g, depending on the D. Furthermore, different from the non-contact dilatometer developed by Ye et al., which is able to support a high R h of 80 K/min, 12 the thermomechanical dilatometer used in the current study (DIL 402 C, Netzsch) becomes unstable at any R h larger than 30 K/min.A large sample mass and higher heating rate would affect the uniformity of the sample temperature and cause a large degree of error.However, when considering the 0:5 Â 10 À3 order of the characteristic free volume changes, the error is acceptable.
1. Sample diameter effect Figure 3(a1) shows the divergence between the experimental and hypothesized DIL traces to increase with a decrease in D. It can be seen from Fig. 3(a3) that the smaller samples have a smaller a L ðTÞ value above T sr-on .The structural difference between the samples with a different D lies primarily in their different initial V f0 .The initial V f0 contained in the amorphous solid is closely related to the cooling process of liquid metal.A faster cooling rate allows a greater free volume of liquid to be frozen into the amorphous solid. 39,42cording to the approximate inverse square relation between the cooling rate and sample dimensions proposed by Johnson et al., 43 the cooling rate employed in fabricating the samples with a D of 1.0, 2.0, 3.0, and 5.0 mm is estimated to be 4:0 Â 10 3 K/s, 1:0 Â 10 3 K/s, 4:4 Â 10 2 K/s, and 1:6 Â 10 2 K/s, respectively.The absolute value of V f0 cannot be determined on the basis of the DV f ðTÞ curves, despite the smaller samples prepared with a faster cooling rate having a larger  initial V f0 (Ref.42).In addition, the absolute value of the residual V f-re remains unknown, although Fig. 4(a1) shows the smaller diameter sample to have a higher T sr-end and thus a larger residual V f-re .Furthermore, Fig. 4(a3) shows the smaller sample also to have a larger DV f-sr .After structural relaxation, a part of the initial V f0 is released as DV f-sr , and the residual component is V f-re .Based on Eq. ( 1), the foregoing process can be further quantified as If the ratios k of the four samples differ greatly, then it is impossible for V f0 , DV f-sr , and V f-re to all increase at the same time with a decrease in D, which means either that V f0 and DV f-sr increase while V f-re decreases or that V f0 and V f-re increase while DV f-sr decreases.Hence, it is reasonable to deduce that the four samples with different D have similar k.Only under this condition can V f0 , DV f-sr and V f-re increase at the same time with a decrease in D. This ratio k represents the degree of structural relaxation, which is dependent primarily on how long such relaxation is sustained.As shown in Fig. 4(a2), the time interval of structural relaxation Dt sr changes little with D, which confirms the similarity of k in the different samples tested with the same R h .In addition, with the same R h , the smaller sample with a larger V f0 requires a wider temperature interval DT sr to release a larger DV f-sr , as shown in Fig. 4(a2).

Heating rate effect
Figure 3(b1) shows the experimental DIL traces measured with a faster R h to be closer to the hypothesized DIL traces.It can be seen from Fig. 3(b2) that a L ðTÞ decreases less during structural relaxation and increases more during the glass transition in the test with a faster R h .The samples with the same D had the same initial V f0 and compressive stress applied to them.In the test with a faster R h , T sr-end shifted to a higher temperature, as shown in Fig. 4(b1), and the residual V f-re consequently grew larger.The ratio k in Eq. ( 5) thus decreases with a faster R h .The temperature interval DT sr increases slightly, whereas the time interval Dt sr undergoes an obvious decrease with a faster R h , as shown in Fig. 4(b2), which also indicates a less insufficient degree of structural relaxation and a smaller ratio k with a faster R h .Whether based on Eq. ( 1) or ( 5), a smaller DV f-sr will be obtained in a test with a faster R h , as shown in Fig. 4(b3).5][46] Among the samples with a D of 3 mm tested with different R h , T a-p has good agreement with T g-end with an acceptable degree of experimental error.Fig. 4(a4) shows that the smaller sample prepared at a faster cooling rate has a slightly higher T g-end .This weak positive dependence of T g-end on the fabrication cooling rate is another natural characteristic of glass. 44The T a-p of the samples with a D of 1.0 and 2.0 mm, however, is obviously lower than the T g-end .The compressive stress (0:3=ðpðD=2Þ 2 Þ) applied to the samples with a D of 1.0, 2.0, 3.0, and 5.0 mm was 382, 95, 42, and 15 KPa, respectively.Compared to the strength of the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG with 3500 MPa, 36 the compressive stress exerted by the instrument was too small to affect the thermal expansion of the solid sample.Therefore, the measurement results for DV f ðTÞ below T sr-end and DV f-sr are reliable.Above T sr-end , a glassy solid is gradually transformed into a super-cooled liquid via the glass transition process.In this transition process, the sample structure consisted of a glassy solid and a super-cooled liquid, a type of structure that is a non-Newtonian fluid whose viscosity depends on the compressive stress. 28Before the glass transition had finished, i.e., when some of the solids had not yet been transformed into liquids, the viscosities of the samples with diameters of 1.0 and 2.0 mm to which a relatively large compressive stress had been applied were sufficiently low to facilitate the softening occurring at T a-p .For the small samples, the T a-p was thus obviously lower than the T g-end , which constitutes the major difference between the observed glass transition behavior in the DIL and DSC tests.Furthermore, in addition to D, BMG composition also affects the T a-p , which will be discussed in Sec.III D 1. Hence, it does not make sense to use the DIL method to measure incomplete free volume generation.
In general, both D and R h were found to have clear effects on DV f ðTÞ.The DIL method is unable to measure the entire free volume generation in the glass transition process accurately because of the sample softening effect.DV f-sr , in contrast, describes the free volume change in an amorphous solid, and is thus independent of compressive stress.For the same D and R h , DV f-sr , as measured using the DIL method, can be reliably employed to make comparisons among different BMGs.Such comparison allows the correlation between the structure and properties of different BMGs, including GFA, to be investigated quantitatively.
D. Study of GFA by (T a-p -T g-end ) and DV f-sr GFA is one of the most important issues in BMG research.A number of physical parameters have been found to be correlated with GFA, such as the fragility of liquid metal, [47][48][49] the viscosity at the melting point, 50 the volumetric change from glass transition to melting, 51 and the density change upon crystallization. 2 The free volume concept is often involved in these findings.To date, however, there has been no systematic investigation of the correlation between free volume and GFA, due to the lack of a free volume measurement method that provides comparable results for different compositions.Although the DIL method is unable to determine the absolute value of the initial free volume V f0 contained in as-cast samples, a comparison of DV f-sr , which reflects the amorphous solid structural features of different BMGs, is useful for investigating GFA.As studied in Sec.III C, the DV f-sr of the sample with a smaller D measured at a slower R h had a larger value, which allows more effective comparisons among different BMGs.The smallest critical diameter (D c ) among the 12 typical Pd-, Mg-, Cu-, Zr-, Ti-, and Fe-BMGs was 2.5 mm for the Ti 42:5 Zr 2:5 Hf 5 Cu 42:5 Ni 7:5 BMG (Ref.35).Therefore, samples with a D of 2.0 mm were used in the DIL tests.An R h of 5 K/min was adopted to save time.The tests were performed two to three times, and the results are shown in Fig. 5 and Table II.The DV f-sr errors are 0:03-0:06 Â 10 À3 for most compositions, except for 0:10-0:11 Â 10 À3 for the two Mg-based BMGs.It is difficult to control the volatilization amount of Mg when preparing Mg-based BMGs using the injection method.Hence the large errors for these BMGs may have been caused by the unsatisfactory repeatability in sample preparation.

083523-6
Hu, Zeng, and Fu J. Appl.Phys.111, 083523 (2012)  separately in Fig. 6.It is clear that T a-p is lower than T g-end for the 2.0 mm-diameter samples.Hence, the DIL method is unable to determine the entire free volume generation in the glass transition process.One interesting finding is that the ðT g-end À T a-p Þ of the composition with a larger GFA (D c is given in Table II) is smaller than that with the same base metal but a smaller GFA.The implication is that, for BMGs of the same base metal, the viscosity of composition with a larger GFA decreases more slowly in the glass transition process, and T a-p is thus closer to T g-end under the same compressive stress.

Sensitive correlation between DV f-sr and GFA
As shown in Figs.5(a3b3)-5(k3l3), all of the samples have a clear T sr-on above which the a L ðTÞ decreases continuously.DV f ðTÞ curves obtained using Eq. ( 3) are shown in Figs.5(a4b4)-5(k4l4).It is clear that the compositions with a larger GFA display a smaller free volume release during structural relaxation than those with the same base metal but a smaller GFA.Table II lists several of the parameters related to GFA: the reduced glass transition temperature T rg , which Turnbull 52 proposed as the first quantitative criterion for glass formation, a series of GFA indicators, c, c m , and c c , developed from T rg , as proposed by Liu et al., [53][54][55][56] and the DV f-sr quantified from the DV f ðTÞ curves in Figs.5(a4b4)-5(k4l4).To allow quantitative comparison of the sensitivity of these parameters in terms of their ability to reflect a variation in GFA, their relative changes in each pair of BMGs with the same base metal are also listed.Table II shows that DV f-sr is much more sensitive than the other criteria.The addition of a large amount of Cu to Pd-(Cu)-Ni-P (Ref.30) leads to a remarkable decrease in T l , and consequently increases T rg , c, c m , and c c by about 10%, although DV f-sr changes even more dramatically, by about À40%.In addition, in the cases of a minor addition of Y to Cu-Zr-Al-(Y), 33 a similar composition but different rare earth elements for Mg-Cu-(Gd, Dy) 31,32 and a small variation in Cr and Mo for Fe-Er-(Cr, Mo)-C-B, 37 there is a significant increase in GFA, but the increase in T rg and its derivatives is not very significant.However, there is still a quite obvious decrease in DV f-sr .Furthermore, for ðZr 58 Nb 3 Cu 16 Ni 13 Al 10 Þ 98:5 Y 1:5 (Ref.34) and Ti 42:5 Zr 2:5 Hf 5 Cu 42:5 Ni 7:5 (Ref.35), which has the greatest GFA among the Ti-based BMGs without the addition of toxic beryllium and precious palladium, the addition of minor amounts of Y and Si changes T g , T x , and T l only slightly.T rg and its derivatives thus undergo a subtle change.GFA enhancement in these two compositions is deemed by their developers to be attributable to the change in their local atomic structure, 34,35 which is well supported by the obvious decreases in DV f-sr .
As discussed in Sec.III C, under the same sample diameter and heating rate, the DV f-sr released during structural relaxation is determined primarily by the initial V f0 that is frozen from the free volume contained in the liquid metal.How much of the initial V f0 is contained in the amorphous solid depends on the amount of free volume in the liquid metal and how fast that free volume is annihilated with a change in temperature in the fast cooling process.These two factors are related to the viscosity 50 and fragility 48 of glassforming liquid, respectively.Glass-forming liquid with a , DV f-sr and D c of the 12 BMGs.The diameter of the samples was 2 mm.The heating rates in the DSC and DIL tests were 20 K/min and 5 K/min, respectively.The relative change is calculated as ðX LargerGFA À X SmallerGFA Þ=X SmallerGFA , where X denotes T rg , c, c m , c c , and DV f-sr .higher viscosity and less fragility is more stable.1][52][53][54][55][56] For this kind of liquid, the smaller initial V f0 is frozen into the amorphous solid after fast cooling.During the constant heating in the DIL test, a smaller DV f-sr is thus released in the structural relaxation process.From the perspective of glass formation, DV f-sr reflects the BMG structural feature inherited from the liquid metal, and T rg and its derivatives are related primarily to the thermodynamic features.Compared to thermodynamic features, structural feature are more sensitive to the enhancement of GFA in BMGs with the same base metal. 51 Close correlation between DV f-sr and GFA To investigate the effect of DV f-sr on GFA for the BMGs with different base metals, the correlation between DV f-sr and D 2 c was also investigated.As shown in Fig. 7, there is a good linear relationship between DV f-sr and Log D 2 c .In addition, DV f-sr and Log D c also have a linear relationship with the same accuracy (R ¼ 0:936).DV f-sr thus exhibits a close correlation with the GFA of these 12 BMGs.Furthermore, the regularity in Fig. 7 is similar to that of our previous results, and can be compared with the regularity obtained by Park et al., 51 as discussed in our previous work. 1

E. Comparison with other methods
Although our definition of DV f-sr differs from that of the free volume change in other research, [7][8][9][10][11]13,16,17 the results of this research are still comparable to the free volume changes measured with the DSC, room temperature density, and positron annihilation lifetime methods. Accordin to free volume theory, 7 the reduced free volume x can be expressed as x ¼ v f =cv Ã , where v f is the free volume per atom volume and cv Ã is a constant of order 0.1.The definite value of cv Ã for most BMGs is lacking in the available literature.Hence, for convenience of discussion, cv Ã is assumed to be 0.1 in the following comparison.
Based on the specific heat capacity data, Beukel and Sietsma 7 reported the Dx released in structural relaxation for as-quenched Pd 40 Ni 40 P 20 to be 0:96 Â 10 À2 (see Fig. 3 in Ref. 7), corresponding to a DV f of 0:96 Â 10 À3 , which is obviously larger than our result of 0:237 Â 10 À3 .This inconsistency can be attributed to the ribbon sample used in Ref. 7. Using room temperature density data, Haruyama et al. 16,17  BMGs, T sr-end was 15-25 K below T g-on and the DV f-sr was 1:365 Â 10 À3 and 0:773Â 10 À3 , respectively.The data of these four Zr-based BMGs, which have a smaller GFA than the Pd-based BMG and Vitreloy 1 b, are close to or larger than 0:8 Â 10 À3 and can be compared with each other, in spite of their different compositions, sample fabrication conditions and measurement heating processes.
No comparisons of the Mg-, Cu-, Ti-and Fe-based BMGs were conducted because there are few available reports on the free volume measurements of these BMGs, although the aforementioned comparisons of the Pd-and Zrbased BMGs demonstrate the validity of the first characteristic free volume change DV f-sr .Unfortunately, this research was unable to measure the free volume generation during the entire glass transition process, i.e., the second characteristic free volume change DV f-gt presented in Fig. 1(b), because of sample softening above T a-p in the DIL test.The thermal expansion in the glass transition process and subsequent super-cooled liquid region could be measured without the sample softening effect using the synchrotron radiation XRD method [18][19][20][21] or a non-contact DIL instrument. 4120][21]41 Hence, if the hypothesized thermal expansion rather than the relaxed sample thermal expansion is taken as the reference, then it is hoped that DV f-gt and the subsequent free volume generation with a constant rate in the super-cooled liquid region shown in Fig. 1(b) can be quantified accurately using a non-contact DIL instrument.

IV. CONCLUSIONS
The free volume change DV f ðTÞ relative to a hypothesized amorphous reference state was measured using the DIL method.A characteristic free volume change, i.e., the free volume released in structural relaxation DV f-sr , was obtained on the basis of the DV f ðTÞ curve.The sample diameter and heating rate were found to have clear effects on DV f-sr .For the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG, DV f-sr increased with decreases in the sample diameter and heating rate.DV f-sr measured by the DIL method under the same sample diameter and heating rate conditions allows convenient comparison among different BMGs.The comparison showed the GFA enhancement for Pd-, Mg-, Cu-, Zr-, Ti-, Fe-based BMGs to be sensitive to decreases in DV f-sr and ðT g-end À T a-p Þ.In addition, a good linear relationship between the DV f-sr and Log D 2 c or Log D c of the 12 BMGs was found.It can therefore be concluded that DV f-sr is sensitive to and closely correlated with GFA.Furthermore, the DV f-sr measurement results for the Pdand Zr-based BMGs are in good agreement with the free volume change measured with the DSC, room temperature density and positron annihilation lifetime methods.Finally, another characteristic free volume change, i.e., the free volume generated in the glass transition process DV f-gt , could not be quantified accurately in this research because of the sample softening effect.However, the DV f-gt shows promise for measurements with other non-contact DIL instruments using the hypothesized amorphous reference state proposed herein.
It is very difficult to measure the absolute free volume of BMGs.Hence, it is feasible to adopt the characteristic free volume change to explore the relationship between the structure and properties of BMGs.In this exploration, DV f-sr , which is measured using the DIL method and reflects the structural features of amorphous solids, can play an important role given its comparability and convenience.

Fe 41 Co 7
Cr 15 Mo 14 C 15 B 6 Y 2 (Ref.36) BMG rods with diameters (D) of 1.0, 2.0, 3.0, and 5.0 mm and Pd-(Refs.29 and 30), Mg-(Refs.31 and 32), Cu-(Ref.33), Zr-(Ref.34), Ti-(Ref.35), and Fe-(Ref.37) based BMGs rods with a D of 2.0 mm were prepared using copper-mold casting (RAPID QUENCH MACHINE SYSTEM VF-RQT50, MAKABE).The original length of the as-cast rods was 50 mm.The amorphous states of the as-cast samples were confirmed by XRD (D8 ADVANCE, BRUKER) and differential scanning calorimetry (DSC; SETSYS Evolution 1750, SETARAM).Rod samples with a length L 0 of 25:060:1 mm were cut from the middle of the as-cast rods, with their two ends carefully polished to ensure parallelity.DIL tests were performed in a thermal dilatometer (DIL 402 C, NETZSCH) with a resolution of 1.25 nm.The equipment was calibrated with an alumina standard sample at heating rates of 2.5, 5, 10, and 20 K/min, and flushed with a high-purity argon flow at 100 ml/min for 10 min before the test and at 50 ml/min during it.The applied load was 0.3 N. Accuracy was ensured during the measurement process by the very low-thermal expansion of the Invar measurement system at a constant temperature (20.0 6 0.1 C) maintained by the circulating coolant liquid. 38Data on linear thermal expansion DLðTÞ=L 0 and the linear thermal expansion coefficient a L ðTÞ (a L ðTÞ ¼ 1=LðTÞ Â dLðTÞ=dT) were recorded in the tests.

FIG. 1 .
FIG. 1. Sketch of (a) equilibrium free volume line and absolute free volume V f ðTÞ relative to the reference state V f ðTÞ ¼ 0 and (b) free volume change DV f ðTÞ relative to the reference state V f ðTÞ ¼ V f0 .

FIG. 2 .
FIG. 2. (a) Experimental and hypothesized DIL traces.DSC and a L ðTÞ traces with (b) large-and (c) small-scale vertical coordinates.(d) The corresponding DV f ðTÞ curve.

Figure 3 ,
Figure 3, TableI, and Fig.4present the measurement results for different D and R h .In each group of tests, one parameter was changed and the other kept the same.As shown in TableI, a sr-on undergoes a slight increase and T sr-on changes little with a change in D and R h .For the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG, the hypothesized DIL trace, which is determined by a sr-on and T sr-on based on Eq. (4) and deemed to be the thermal expansion of the hypothesized reference amorphous state, shows little variation across tests.Hence, the DV f ðTÞ in different conditions is dependent primarily on the experimental thermal expansion above T sr-on .Five tests were carried out for the U 3.0 mm sample measured at 5 K/min and two tests for the other conditions.The repeatability errors of DV f-sr were 0:03-0:04 Â 10 À3 for most of the tests, but were 0:05-0:06 Â 10 À3 for the test on (U 5.0 mm, 5 K/min) and 0:08-0:1 Â 10 À3 for that on (U 3.0 mm, 20 K/min).Different from the 20 $ 30 mg sample used in the DSC test, the mass of the Fe-based BMG rod samples used in the DIL tests ranged from 0:2 $ 3:6 g, depending on the D. Furthermore, different from the non-contact dilatometer developed by Ye et al., which is able to support a high R h of 80 K/min,12 the thermomechanical dilatometer used in the current study (DIL 402 C, Netzsch) becomes unstable at any R h larger than 30 K/min.A large sample mass and higher heating rate would affect the uniformity of the sample temperature and cause a large degree of error.However, when considering the 0:5 Â 10 À3 order of the characteristic free volume changes, the error is acceptable.

Figures 4 (
Figures 4(a4) and 4(b4) show the T a-p determined by the DIL tests and T g-end determined by the DSC tests under different conditions of D and R h .The increase in T g-end with a faster R h shown in Fig.4(b4) is a natural characteristic of glass.[44][45][46]Among the samples with a D of 3 mm tested with different R h , T a-p has good agreement with T g-end with an acceptable degree of experimental error.Fig.4(a4) shows that the smaller sample prepared at a faster cooling rate has a slightly higher T g-end .This weak positive dependence of T g-end on the fabrication cooling rate is another natural characteristic of glass.44The T a-p of the samples with a D of 1.0 and 2.0 mm, however, is obviously lower than the T g-end .The compressive stress (0:3=ðpðD=2Þ 2 Þ) applied to the samples with a D of 1.0, 2.0, 3.0, and 5.0 mm was 382, 95, 42, and 15 KPa, respectively.Compared to the strength of the Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG with 3500 MPa,36 the compressive stress exerted by the instrument was too small to affect the thermal expansion of the solid sample.Therefore, the measurement results for DV f ðTÞ below T sr-end and DV f-sr are reliable.Above T sr-end , a glassy solid is gradually transformed into a super-cooled liquid via the glass transition process.In this transition process, the sample structure consisted of a glassy solid and a super-cooled liquid, a type of structure that is a non-Newtonian fluid whose viscosity depends on the compressive stress.28Before the glass transition had finished, i.e., when some of the solids had not yet been transformed into liquids, the viscosities of the samples with diameters of 1.0 and 2.0 mm to which a relatively large compressive stress had been applied were sufficiently low to facilitate the softening occurring at T a-p .For the small samples, the T a-p was thus obviously lower than the T g-end , which constitutes the major difference between the observed glass transition behavior in the DIL and DSC tests.Furthermore, in addition to D, BMG composition also affects the T a-p , which will be discussed in Sec.III D 1. Hence, it does not make sense to use the DIL method to measure incomplete free volume generation.In general, both D and R h were found to have clear effects on DV f ðTÞ.The DIL method is unable to measure the entire free volume generation in the glass transition process accurately because of the sample softening effect.DV f-sr , in contrast, describes the free volume change in an amorphous solid, and is thus independent of compressive stress.For the

FIG. 7 .
FIG. 7. Correlation between the GFA (D 2 c ) and DV f-sr of the twelve BMGs.The 95% confidence bands are also shown.
To render this comparison more accurate, it is necessary to investigate how factors other than BMG composition affect DV f ðTÞ.The most important experimental parameters in the DIL test are the sample diameter D and heating rate R h .Their effects are discussed in Sec.III C.
C. Sample diameter and heating rate effects on DV f (T)

TABLE I .
T sr-on and a sr-on of Fe 41 Co 7 Cr 15 Mo 14 C 15 B 6 Y 2 BMG rods under different conditions of sample diameter (D) and heating rate (R h ).

TABLE II .
Summary of the thermal analyses by DSC (T rg