- Email: [email protected]
Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming alloys
Author’s Accepted Manuscript Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming alloys Tuo Wang, Lina Hu, Yanhui Liu, Xidong Hui
www.elsevier.com/locate/msea
PII: DOI: Reference:
S0921-5093(18)31685-X https://doi.org/10.1016/j.msea.2018.12.004 MSA37276
To appear in: Materials Science & Engineering A Received date: 22 October 2018 Revised date: 28 November 2018 Accepted date: 1 December 2018 Cite this article as: Tuo Wang, Lina Hu, Yanhui Liu and Xidong Hui, Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming a l l o y s , Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2018.12.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming alloys Tuo Wanga, Lina Hub, Yanhui Liuc*, Xidong Huia*
a
State Key Laboratory for Advanced Metals and Materials, University of Science and
Technology Beijing, Beijing 100083, China b
Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials
(Ministry of Education), Shandong University, Jinan 250061, China c
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
[email protected] [email protected]
*
Corresponding author.
ABSTRACT Nowadays, plasticizing has become a critical issue for the engineering application of bulk metallic glasses (BMGs). What is the origin of plastic flow and how to evaluate the plasticity of BMGs with a unified standard are still open questions. Here we report a universal fragility criterion to scale the plasticity of BMGs from the equilibrium liquids and supercooled liquids. Based on our experimentally measured and previously reported viscosities at superheated and supercooled region for 51 glasses, the plasticity () of BMGs has been found to positively correlate to the supercooling liquid fragility (m) with the equation of = 92exp[-73/(m-15)], where is the plastic strain and m the fragility near glass transition temperature, but negatively correlate to the fragile-to-strong (F-S) transition tendency characterized by F-S transition factor and temperature. The intrinsic mechanism of the plasticity can be well interpreted in the frame of structural relaxation theory. This work provides a new insight into the 1
microscopic essence of the plasticity of glassy solids and a universal approach to scale the fluidity of metallic glasses. Keywords： Bulk metallic glasses; Plasticity; Fragility; Fragile-to-strong transition
1. Introduction Bulk metallic glasses (BMGs) possess a number of remarkable properties, such as high strength and hardness, excellent corrosion resistance and soft magnetic properties etc [1]. Unfortunately, they typically lack of macroscopic plasticity at room temperature. In order to develop plastic BMGs by composition design, it is essential to establish the correlations between plasticity and physical parameters, so that the mechanism that governs deformation can be comprehensively understood. Up to now, a number of correlations have been proposed between BMG plasticity and physical and structural quantities, such as Poisson’s ratio [2, 3], nanoscale structural heterogeneity [4], phase separation [3, 5] and density of flowing units [6-8] etc. Interestingly, for non-metallic glasses, Novikov and Sokolov [9] suggested that fragility is linearly correlated with the ratio of bulk modulus (B) to shear modulus (G), B/G, or equivalently Poisson’s ratio ν. Although such correlation is nonlinear for BMGs [10], it implies that BMG plasticity may be associated with liquid fragility. BMG deformation is known to proceed by shear band formation and propagation as consequence of shear transformation zones (STZ), i.e. the localized collective atomic rearrangement under shear stress [11, 12], Recently, a number of studies suggested that the formation of shear bands in BMGs can be considered as stress-driven glass-to-liquid transition [13]. For example, the yield shear stress is propositional to glass transition 2
temperature Tg [14]. The viscosity within shear band is found to be on the same order as that of supercooled liquid [15, 16]. All these mean that the material within shear bands behaves as supercooled liquid, so the way to solve the puzzles in the plastic flow of BMGs may be steered to the viscous flow that accompanies supercooling of liquid and glass formation. Phenomenologically, when the molten liquid is cooled down to the glass-transition temperature, the viscosity of the liquid is sharply increased to 12 powers by 10 poise [9], and this process has been characterized by the Angell’s theory [17] of liquid fragility, m, which is defined as m
d log d (T g / T )
(1) T Tg
where is shear viscosity, Tg is the glass transition temperature and T is the absolute temperature. The values of m are obviously different between different glassy systems, e.g., the m’s of Pt-based BMGs [18] may reached 68, Zr-based BMGs range from 30 to 56 [19-23] and fused silica [22] is only 15. The equilibrium viscosity of a glass-forming liquid can be described by the Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equation [24, 25], which is described by: log log
B C exp( ) T T
(2)
where η∞ is viscosity at the high temperature limit, B and C are constants. In 1999, Ito et al. [26] has found that the change of the viscosity around molten temperature in water is likely the fragile liquid OPT, whereas the behavior around Tg is similar to strong liquid SiO2. This phenomenon is so-called fragile-to-strong (F-S) transition. The F-S transition was also found in other metallic glass forming liquids (MGFLs), which might be a general dynamic phenomenon in GFLs [24-27]. In order to describe this physical phenomenon, a generalized MYEGA expression can be written as:
3
log log
1 C C T [(W1 exp( 1 ) W2 exp( 2 )] T T
(3)
where C1, C2, W1 and W2 are constants. This equation contains the fragility of the molten and the supercooled liquid. Investigating the behavior of supercooled liquid which can be quantified by fragility m and the progress of F-S transition can shed light on the deformation and plasticity of BMGs. In this work, we demonstrate that plasticity of BMGs is closely correlated with the property of glass forming liquid which can be characterized by fragility m. We found that the plasticity of BMGs is positively correlated with the fragility of supercooling liquid, but negatively correlated to the temperature of F-S transition temperature of supercooled liquid. This finding has implication for understanding the origin of plasticity and deformation mechanism of BMGs from the perspective of liquid. 2. Materials and methods Master alloy ingots were prepared by alloying pure Zr, Fe, Al, Cu and Nb metals with a purity of over 99.5 mass% in an arc-melter in a Ti-gettered argon atmosphere. To ensure compositional homogeneity, each ingot was remelted at least four times. Subsequently, cylindrical samples with 2 mm in diameter were obtained by suction casting in a copper mold. The amorphous nature of the alloy rods were confirmed by X-ray diffractometer (XRD) using a Bruker D8 ADVANCE (Cu Kα radiation). The compression specimens with 2 mm in diameter and 4 mm in length were cut from the rods, and the ends were polished carefully to ensure parallelism. The Tg of the bulk alloy rods were checked by Netzsch STA 449C differential scanning calorimeter (DSC). The uniaxial compression test was performed with a CMT4305 universal testing machine at room temperature with a strain rate of 2*10-4s-1. For the brittle BMGs, the test was stopped when the samples were fractured. And for the BMGs with microscopic 4
compressive plasticity, the test was stopped at the stress obviously dropped. The plastic strain was defined as the difference value of the intersection point at strain-axis between the elastic line and its parallel line at the maximum stress. The viscosities of MGFLs were measured in a high vacuum atmosphere with an oscillating viscometer. The viscosities were measured for four times at a temperature, and the average value was regarded as the equilibrium viscosity for a certain MGML. 3. Results and disscussion 3.1 Discrete distribution of the plastic strains among different BMGs system In the first place, let us begin with the compositional dependence of plastic strains for the different 9 BMGs. The amorphous nature of the alloy rods were confirmed by XRD as shown in Fig. 1. It is seen that all the curves have only one halo peak, which is the feature of glassy structure of alloys.
Fig. 1 XRD patterns of the glassy rod specimens with the diameter of 2mm.
The compressive stress-strain curves of the interested Zr- and CuZr- based alloys are shown in Fig. 2a. From these curves, the substantial plastic strains can be 5
determined and listed in Table 1. It is seen that the plastic strains for these 9 BMGs are obviously affected by the composition, which vary from 0.9% to 14.3% for (Zr66.5Cu22.5Fe5Al10)100-xNbx (x=0, 2, 4, 6) and from 0.2% to 10.2 % for (Zr66Cu17.5Fe6Al10.5)100-xNbx (x=0, 1, 1.5). The plastic strain of these BMGs seems to mainly be determined by the addition of Nb content in similar systems. However, it is seen that the plastic strains against Nb content are not linear for the 9 BMGs, as shown in Fig. 2b. As for the Zr47.75Cu47.75Al4.5 and Zr47.25Cu47.25Al5.5 BMGs, however, their plastic strains are 4.5% and 7.4%, respectively, which indicates that the plastic strain is affected by the Al content when Zr and Cu have same concentration. However, the plastic strains against Cu, Zr, Al contents are not closely correlative for the 9 BMGs (Fig. 2c-e). In order to find the correlation of plasticity to the composition and other parameters, we collected the data of plastic strains of more than 40 BMGs and other type of glasses from literatures [2, 5, 8, 10, 18-23, 28-46], and listed in Table 1.
6
Fig. 2 (a) Room-temperature compressive engineering stress-strain curves for 9 as-cast BMGs; (b)-(e) the plastic strains against Nb (b), Zr (c), Cu (d), and Al (e) content.
It is found that most of the BMGs with plastic strains above 10% belong to Zr-, CuZr-, Pd- and Pt- based alloy systems, while Fe-, Mg-, Hf-, Ca-based BMGs exhibit small plastic strains with some of them being completely brittle. Apparently, the seresults demonstrate that the compositional dependence of BMG plastic strain is complicated. There appears no single expression that can describe such strong dependence.
Table 1 Plastic strain (ε), fragility (m), the activation energy of relaxation (Eα), the activation energy of β relaxation (Eβ) and the ratio of Eα/Eβ for MGs reported in the literature and this work. Systems
Zr-based
Composition
ε (%)
m
Tg (K)
Eα
Eβ
(kJ/mol) (kJ/mol)
Eα/Eβ
Ref.
Zr62.5Cu22.5Fe5Al10
0.9
30
654
376
141
2.7
This work
Zr61.25Cu22.05Fe4.9Al9.8Nb2
1.9
34
651
424
141
3
This work
Zr60Cu21.6Fe4.8Al9.6Nb4
5.2
39
656
490
142
3.5
This work
Zr58.75Cu21.15Fe4.7Al9.4Nb6 14.3 46
660
581
143
4.1
This work
Zr66Cu17.5Fe6Al10.5
0.2
31
638
379
138
2.7
This work
Zr65.3Cu17.3Fe6Al10.4Nb1
4.1
36
649
447
140
3.2
This work
Zr65Cu17.3Fe5.9Al10.3Nb1.5 10.2 43
655
539
142
3.8
This work
Zr47.75Cu47.75Al4.5
4.5
39
686
512
148
3.5
This work
Zr47.25Cu47.25Al5.5
7.4
41
690
542
149
3.6
This work
Zr41.25Ti13.75Cu12.5Ni10Be22.5 1.5
39
648
484
140
3.5
19, 28
Zr46.75Ti8.25Cu7.5Ni10Be27.5
2
34
633
412
137
3
29, 30
Zr57Nb5Cu15.4Ni12.6Al10
3.3
35
670
449
145
3.1
20, 28
Zr65Cu15Al10Ni10
2
30
637
366
138
2.7
21, 31
Zr57Ti5Cu20Ni8Al10
1.1
40
672
515
145
3.6
22, 32
Zr50Cu35Al7Pb5Nb3
5.1
37
710
503
153
3.3
23
Zr55Cu30Al7Pb5Nb3
9.8
41
693
544
150
3.6
23
Zr60Cu25Al7Pb5Nb3
10.1 43
674
555
146
3.8
23
Zr62Cu23Al7Pb5Nb3
18.3 56
670
718
145
5
23
Zr50Cu48Al2
15
47
676
608
146
4.2
33
Zr50Cu44.5Al5.5
2.6
39
678
506
147
3.4
8
Zr50Cu44Al5.5Mo0.5
5.7
41
682
535
147
3.6
8
7
Zr50Cu43Al5.5Mo1.5
10.9 47
685
616
148
4.2
8
Zr50Cu41.5Al5.5Mo3
20.4 50
688
658
149
4.4
8
Zr64Cu26Al10
6.8
44
662
558
143
3.9
22, 34
Cu47.5Zr47.5Al4
7.5
44
683
575
148
3.9
35
Cu43Zr43Al7Ag7
8
45
700
603
151
4
18
Cu43Zr43Al7Be7
3.5
42
696
560
150
3.7
18
Cu46.25Zr45.25Al7.5Sn1
4.1
40
718
550
155
3.5
22
Cu50Zr50
4
39
664
496
144
3.4
22
Cu46Zr42Al7Y5
5
49
713
669
154
4.3
5, 36
3
40
431
330
93
3.5
37, 38
Mg65Cu25Gd10
2
39
423
316
91
3.5
10
Mg58.5Cu30.5Y11
0.4
30
421
242
91
2.7
22, 37
Mg57Cu31Y6.6Nd5.4
1.2
29
427
237
92
2.6
22, 37
10.4 52
673
670
145
4.6
19, 22
Pd40Ni40P20
1.5
46
593
522
128
4.1
39, 40
Pt57.5Cu14.7Ni5.3P22.5
20
68
508
661
110
6
2, 18
Fe63Mo14C15B6Er2
0
28
778
417
168
2.5
22, 41
Fe65Mo14C15B6
0.8
33
790
499
171
2.9
22, 41
La66Al14Cu10Ni10
0
32
459
281
99
2.8
22, 42
La55Al25Ni20
0.1
35
491
329
106
3.1
21
Hf50Ni25Al25
1.3
33
858
542
185
2.9
22, 34
Hf55Ni25Al20
2.5
36
828
571
179
3.2
22, 34
Gd-based
Gd55Al25Co20
0
25
597
286
129
2.2
35, 43
Ce-based
Ce70Al20Ni10Cu10
0
29
358
199
78
2.6
42, 44
Ca-based
Ca65Mg15Zn20
0
21
373
150
81
1.9
10, 45
B2O3
0
32
554
339
120
2.8
46
Fused silica
0
15
-
-
-
-
22
Calcium phosphate
0
24
-
-
-
-
22
Window
0
17
-
-
-
-
22
ZBLAN
0
28
-
-
-
-
22
Cu(Zr)-based
Mg65Cu7.5Ni7.5Zn5Ag5Y5Gd 5
Mg-based
Pd-based Pt-based Fe-based La-based Hf-based
Others
Pd77.5Si16.5Cu6
3.2 Fragility extended from high temperature liquid For many MGFLs, the temperature dependence of η for temperature intervals lying between the liquidus temperature TL and Tg can be described by the Angell’s theory [17]. According to Angell, the fragility scales with activation energy of glass transition, 8
Eg, by [27] m
Eg
(4)
RTg ln 10
To obtain the m values for our 9 BMGs, we carried out DSC measurements at various heating rate (Fig. 3a). The Eg for the BMGs can be gotten by Kissinger equation[3], ln(
Tg
)
Eg RTg
C
(5)
where β is the heating rate, Tg is the glass transition temperature at a certain heating rate, R is the ideal gas constant, C is a constant. So, -Eg/R is the slope of the line on the plot of the heating rate versus 1/Tg. The Kissinger curves of the BMGs is shown in Fig. 3b.
Fig. 3 (a) DSC curves measured at different heating rate for Zr61.25Cu22.05Fe4.9Al9.8Nb2 BMG as a representative of 9 BMGs; (b) activation energies fitted by Kissinger equation for this work.
In addition to the 9 BMGs, we also collected the m values of 42 BMGs and other type of glasses reported in literatures, as listed in Table 1. It is seen that the m values are in the range of 30≤m≤46, indicating that they are intermediate glasses [17]. Fig. 4a shows plastic strain vs. fragility for all the glasses. As can be seen, the 9
glasses can be roughly classified into two categories according to the plastic strain, i.e. brittle and plastic ones. Here, we define the brittle glasses as those showing plastic strain less than 0.2%. It can be seen that Zr-, CuZr-, Pt-, Pd-based BMGs are plastic, whereas fused silica, calcium phosphate, ZBLAN, B2O3, Ca-, Mg-, La-, Ce-, Gd-based BMGs are brittle. According to Angell [17], glass forming liquids can be classified as strong, intermediate, and fragile ones based on the fragility parameter. The BMGs shown in Table 1 are therefore in the category of intermediate glasses. Fig. 4a presents the plot of plastic strain, , as a function of fragility. Fitting to the data points yields = 92exp[-73/(m-15)] with a correlation coefficient of 0.74. From this equation, one can be derive the theoretical minimum value of m=15. This limiting value represents the strongest liquid, which have been verified by the DSC, DMA, or viscosity measurements on different GFLs. It is also indicated that the equation obtained from Fig. 4a is basically in agreement with the recognized theory of fragility and a fragile glass
Fig. 4 (a) Plastic strain, ε, versus the fragility, m, near the supercooling region reported in the literature and measured this work; (b) the fracture toughness, KC, versus m in the literature.
forming liquid is favorable for the achievement of higher plasticity. The argument is further supported by the correlation between fracture toughness, KC, and fragility, as 10
shown in Fig. 4b in which various BMGs and oxides glasses are included [1, 2, 10, 18-20, 22, 47-52]. It appears that fracture toughness is linearly dependent on fragility, e.g. KC = 2m-25 with a correlation coefficient of 0.85. 3.3 Fragile-to-strong transition during the formation of plastic BMGs Since it is technically difficult to directly measure the realistic viscosities, η, near the glass transition temperature, for the glass formers with low glass forming ability (GFA), in this work, the viscosities of the interested bulk metallic glass liquids (BMGLs) above liquidus temperature, TL, were measured by oscillating viscometer [24, 53].The link between plasticity and fragility indicates that BMG deformation is closely associated with the property and behavior of glass forming liquid.
Fig. 5 (a) Experimentally measured viscosities, , and fitting curves as functions of the temperature; (b) the curves of the viscosities as the functions of the glass transition temperature, Tg, scaled by temperature, T, fitted by Eq. 2 for Zr58.75Cu21.15Fe4.7Al9.4Nb6 MGFL; (c) The curve fitted by Eq. 3 for the Zr58.75Cu21.15Fe4.7Al9.4Nb6 MGFL which 11
includes two terms of contributions to the overall viscous behavior of metallic glass-forming liquids: a fragile term dominant at high temperatures and a strong term dominant at low temperatures.
Fig. 5a shows the viscosities of different glass forming liquids as a function of temperature T. Two kinds of fitting curves are extrapolated to Tg, where the viscosity values
are
1012
Pois
as
indicated
in
Fig.
5b.
According
to
the
Mauro-Yue-Ellison-Gupta-Allan MYEGA equation (See Eq. 2 in the introduction) [24, 25]. Here, the slope of the red solid line at Tg is the fragility parameter m’ which can be obtained from the high-temperature liquid (The temperature is above liquidus temperature TL) viscosity. Based on this theory, m’ presents the fragility of the equilibrium state (the temperature above TL), i.e. m quantifies the fragility of the liquids above TL. The m values and fitting parameters B, C calculated by Eq. 2 based on the visicosities data as shown in Fig. 5a are listed in Table 2. According to Eq. 1, 2 and the value of logη∞, we can obtain the red dashed curve in Fig. 5b. Those two terms can be written as MYEGA equation (See Eq. 3 in the introduction) [24, 25]. Taking Zr58.75Cu21.15Fe4.7Al9.4Nb6 as an example, we carried out fitting to the viscosity data. As shown in Fig. 5c, the dashed-dotted curve is for the fragile term W1 exp(
C1 in Eq. 3, and the dashed curve for strong term C ) W2 exp( 2 ) . The F-S T T
transition temperature can be identified when W1 exp(
C1 C ) W2 exp( 2 ) and are listed T T
in Table 2 for different BMGs. Table 2 The parameters of B, C and the fragility of high temperature m’ and parameters of C1, C2, W1, W2, the temperatures of the fragile-to-strong liquid, Tf-s, for the samples.
Samples
B
C
m’ 12
C1
C2
W1
W2
Tf-s (K)
Zr62.5Cu22.5Fe5Al10
10.54 4436 111
17004
726
3670
0.00034
1005
Zr61.25Cu22.05Fe4.9Al9.8Nb2
6.30
4747 118
18228
651
15361
0.00031
992
Zr60Cu21.6Fe4.8Al9.6Nb4
3.67
5144 126
19024
1148
45732
0.00061
986
Zr58.75Cu21.15Fe4.7Al9.4Nb6
1.91
5607 134
19140
1478
72727
0.00100
976
Zr66Cu17.5Fe6Al10.5
8.45
4449 113
17226
766
6113
0.00038
991
Zr65.3Cu17.3Fe6Al10.4Nb1
4.85
4896 121
19859
1038
107396 0.00052
983
Zr65Cu17.3Fe5.9Al10.3Nb1.5
1.76
5612 135
21615
1376
700763 0.00076
981
Zr47.75Cu47.75Al4.5
2.61
5654 134
20580
1166
212828 0.00058
985
Zr47.25Cu47.25Al5.5
3.08
5576 131
20700
1311
213043 0.00058
983
Tf-s can be seen as the most competitive temperature for the fragile and strong terms, and a parameter f (= m/m) was induced to represent this competition. It can be seen that f is positively related to Tf-s from the red line in Fig. 6a. Then, is there any relationship between Tf-s and plasticity? Fig. 6b depicts the dependence of plastic strains on Tf-s for the Zr-based BMGs. As can be seen, the plasticity of the BMGs is inversely related to Tf-s. According to Fig. 6a, f should be also inversely related to plasticity, as shown in Fig. 6b. How to understand it? Here, we use the model of α and β relaxations
Fig. 6 (a) f as function of r (black) and Tf-s (red) for the present 9 MGFLs; (b)The plastic strains as function of f (black) and Tf-s (red) for the present 9 MGFLs.
to understand this relationship. The ratio between the activation energies for the α and slow β relaxations, r (i. e. r = Eα/Eβ) can be defined as a relaxation competition parameter. The closer the Eβ and Eα values are, the more comparable the structural units 13
involved in the slow β relaxation and those in α relaxation are. It is known that Eβ = (26 ±2)RTg [54] and Eα = 2.303mRTg [35]. Therefore, the ratio r is solely related to fragility by r = Eα/Eβ = 0.089m, i.e. a larger fragility index m indicates an enlarged difference between Eα and Eβ. Kinetically, the F-S transition is accompanied by different relaxation processes in supercooled liquids upon cooling, which is governed by these two characteristic structures i. e. the primary α and the slow β relaxations [27, 35, 55]. As the black line in Fig. 6a shows that r is inversely related to f, i.e. the more intense competition (i.e. f is more larger) between fragile and strong term in MGFLs, the more comparable (i.e. r is more lower) between the α and β structural units. Based on the above perception, the schematic of α and β structural units with brittle and ductile BMGs is as shown in Fig. 7. The blue and red circles present α and β structural units, respectively. When the units are more comparable (Fig. 7a), the structure of the BMG is more homogeneous, resulting in a poor plasticity. On the contrary, if the units are more distinguishable (Fig. 7b), the structure of the BMG is more heterogeneous, resulting in a good plasticity. Based the above discussion, we propose two main relationships, i.e. plasticity is positively correlated with fragility values, but inversely related with Tf-s. It can be understood that m is proportional to Eα, the large activation energy, Eα, will lead to a longer relaxation time, resulting in a long time to achieve internal equilibrium for the system, and the temperature of fragile-strong liquid transition Tf-s being delayed, i. e. more lower Tf-s. Therefore, more β relaxation and more distinguishable (Fig. 7b) structure will be preserved which may enhance the plasticity. From the viewpoint of F-S transition, the MGFLs in low temperature inherit more fragile feature when the MGFLs in high temperature are more fragile, and the tendency of the transition degree is more gentle, resulting in a larger m and a good plasticity. 14
Fig. 7 Schematic diagrams of α and β structural units in (a) brittle and (b) ductile BMGs. The blue and red circles represent the α and β structural units, respectively.
Conclusion In summary, The correlation of fragility to the plasticity for 51 glass materials has been inspected. Macroscopic plasticity has been evidenced in many BMG systems such as Zr-, CuZr-, Pt-, and Pd-based BMGs. No universal compositional dependence of the plasticity of BMGs has been reached across the whole BMG alloy systems. Based on the fragilities and plasticities of 51 glass materials measured in this work and previously reported in the literature, universal criterion for plasticity can be established with the equation of = 92exp[-73/(m-15)], which indicates that there is a positive relationship between plasticity and fragility. There is fragile-to-strong (F-S) transition during the cooling process from superheated liquid to supercooling liquid. The plasticity of BMGs is negatively correlated to the F-S transition factor and temperature. The intrinsic mechanism of the plasticity of BMGs can be well interpreted by the competition of relaxation with relaxation mode. This deduction can be supported the correlation of plasticity to the ratio of activation energy of relaxation to relaxation.
Acknowledgments The authors are grateful for the financial support of National Key Basic Research 15
Program (No. 2016YFB0300502), and National Natural Science Foundation of China (Nos. 51571016 and 51531001).
References [1] W.H. Wang, C. Dong, C.H. Shek, Bulk metallic glasses, Mater. Sci. Eng., R 44 (2004) 45-89. [2] J. Schroers, W.L. Johnson, Ductile Bulk Metallic Glass, Phys. Rev. Lett. 93 (2004) 255506. [3] T. Wang, Y. Wu, J. Si, Y. Liu, X. Hui. Plasticizing and work hardening in phase separated
Cu-Zr-Al-Nb
bulk
metallic
glasses
by
deformation
induced
nanocrystallization, Mater. Des. 142 (2018) 74-82. [4] Z.W. Zhu, L. Gu, G.Q. Xie, W. Zhang, A. Inoue, H.F. Zhang, Z.Q. Hu, Relation between icosahedral short-range ordering and plastic deformation in Zr-Nb-Cu-Ni-Al bulk metallic glasses, Acta Mater. 59 (2011) 2814-2822. [5] E.S. Park, D.H. Kim, Phase separation and enhancement of plasticity in Cu-Zr-Al-Y bulk metallic glasses, Acta Mater. 54 (2006) 2597-2604. [6] Z. Lu, W. Jiao, W.H. Wang, H.Y. Bai, Flow unit perspective on room temperature homogeneous plastic deformation in metallic glasses, Phys. Rev. Lett. 113 (2014) 045501. [7] Q. Wang, J.J. Liu, Y.F. Ye, T.T. Liu, S. Wang, C.T. Liu, J. Lu, Y. Yang, Universal secondary relaxation and unusual brittle-to-ductile transition in metallic glasses, Mater. Today 20 (2017) 293-300. [8] T. Wang, L. Wang, Q. Wang, Y. Liu, X. Hui, Pronounced Plasticity Caused by Phase Separation and β-relaxation Synergistically in Zr-Cu-Al-Mo Bulk Metallic Glasses, Sci. Rep. 7 (2017) 1238. [9] V.N. Novikov, A.P. Sokolov, Poisson's ratio and the fragility of glass-forming liquids, Nature 431 (2004) 961-963. [10] E.S. Park, J.H. Na, D.H. Kim, Correlation between fragility and glass-forming ability/plasticity in metallic glass-forming alloys, Appl. Phys. Lett. 91 (2007) 031907. 16
[11] A.S. Argon, Plastic deformation in metallic glasses, Acta Metall. 27 (1979) 47-58. [12] P. Schall, Structural rearrangements that govern flow in colloidal glasses, Science 318 (2007) 1895-1899. [13] P. Guan, M. Chen, T. Egami, Stress-temperature scaling for steady-state flow in metallic glasses, Phys. Rev. Lett. 104 (2010) 205701. [14] Y.H. Liu, C.T. Liu, W.H. Wang, A. Inoue, T. Sakurai, M.W. Chen, Thermodynamic origins of shear band formation and the universal scaling law of metallic glass strength, Phys. Rev. Lett. 103 (2009) 065504. [15] S.X. Song, T.G. Nieh. Flow serration and shear-band viscosity during inhomogeneous deformation of a Zr-based bulk metallic glass, Intermetallics 17 (2009) 762-767. [16] J. Bokeloh, S.V. Divinski, G. Reglitz, G. Wilde, Tracer measurements of atomic diffusion inside shear bands of a bulk metallic glass, Phys. Rev. Lett. 107 (2011) 235503. [17] C.A. Angell, Formation of glasses from liquids and biopolymers, Science 267 (1995) 1924-1935. [18] J.H. Na, E.S. Park, Y.C. Kim, E. Fleury, W.T. Kim, D.H. Kim, Poisson’s ratio and fragility of bulk metallic glasses, J. Mater. Res. 23 (2008) 523-528. [19] D.N. Perera, Compilation of the fragility parameters for several glass-forming metallic alloys, J. Phys.: Condens. Matter 11 (1999) 3807-3812. [20] Z. Evenson, S. Raedersdorf, I. Gallino, R. Busch, Equilibrium viscosity of Zr-Cu-Ni-Al-Nb bulk metallic glasses, Scr. Mater. 63 (2010) 573. [21] Y. Kawamura, T. Nakamura, H. Kato, H. Mano, A. Inoue, Newtonian and non-Newtonian viscosity of supercooled liquid in metallic glasses, Mater. Sci. Eng., A 304 (2001) 674-678. [22] J.D. Plummer, I. Todd, Implications of elastic constants, fragility, and bonding on permanent deformation in metallic glass, Appl. Phys. Lett. 98 (2011) 021907. [23] S. Zhu, G. Xie, F. Qin, X. Wang, A. Inoue, Ni- and Be-free Zr-based bulk metallic glasses with high glass-forming ability and unusual plasticity, J. Mech. Behav. Biomed. Mater. 13 (2012) 166-173. [24] C. Zhang, L. Hu, Y. Yue, J.C. Mauro, Fragile-to-strong transition in metallic glass-forming liquids, J. Chem. Phys. 133 (2010) 014508. 17
[25] J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. 106 (2009) 19780. [26] K. Ito, C.T. Moynihan, C.A Angell, Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water, Nature 398 (1999) 492-495. [27] J.J.Z. Li, W.K. Rhim, C.P. Kim, K. Samwer, W.L. Johnson, Evidence for a liquid-liquid phase transition in metallic fluids observed by electrostatic levitation, Acta Mater. 59 (2011) 2166-2171. [28] B.A. Sun, H.B. Yu, W. Jiao, H.Y. Bai, D.Q. Zhao, W.H. Wang, Plasticity of ductile metallic glasses: a self-organized critical state, Phys. Rev. Lett. 105 (2010) 035501. [29] J.L. Zhang, H.B. Yu, J.X. Lu, H.Y. Bai, C.H. Shek, Enhancing plasticity of Zr46.75Ti8.25Cu7.5Ni10Be27.5 bulk metallic glass by precompression, Appl. Phys. Lett. 95 (2009) 071906. [30] Z.F. Zhao, Z. Zhang, P. Wen, M.X. Pan, D.Q. Zhao, W.H. Wang, W.L. Wang, A highly glass-forming alloy with low glass transition temperature, Appl. Phys. Lett. 82 (2003) 4699-4701. [31] S.H. Xie, X.R. Zeng, H.X. Qian, Correlations between the relaxed excess free volume and the plasticity in Zr-based bulk metallic glasses, J. Alloys Compd. 480 (2014) 37-40. [32] L.Q. Xing, Y. Li, K.T. Ramesh, J. Li, T.C. Hufnagel, Enhanced plastic strain in Zr-based bulk amorphous alloys, Phys. Rev. B 64 (2001) 607. [33] L. Xia, K.C. Chan, S.K. Kwok, P. Yu, Enhanced plasticity of a Zr50Cu48Al2 bulk metallic glass, J. Non-Cryst. Solids 357 (2011) 1469-1472. [34] L. Zhang, Y.Q. Cheng, A.J. Cao, J. Xu, E. Ma, Bulk metallic glasses with large plasticity: Composition design from the structural perspective, Acta Mater. 57 (2009) 1154-1164. [35] Q. Sun, C. Zhou, Y. Yue, L. Hu, A direct link between the Fragile-to-Strong transition and relaxation in supercooled liquids, J. Phys. Chem. Lett. 5 (2014) 1170-1174. [36] G.J. Fan, J.J.Z. Li, W.K. Rhim, D.C. Qiao, H. Choo, P.K. Liaw, W.L. Johnson, Thermophysical properties of a Cu46Zr42Al7Y5 bulk metallic glass, Appl. Phys. Lett. 88 18
(2006) 221909. [37] Q. Zheng, H. Ma, E. Ma, J. Xu, Mg-Cu-(Y,Nd) pseudo-ternary bulk metallic glasses: The effects of Nd on glass-forming ability and plasticity, Scr. Mater. 55 (2006) 541-544. [38] E.S. Park, J.Y. Lee, D.H. Kim, A. Gebert, L. Schultz, Correlation between plasticity and fragility in Mg-based bulk metallic glasses with modulated heterogeneity, J. Appl. Phys. 104 (2008) 023520. [39] Y. Kawamura, A. Inoue, Newtonian viscosity of supercooled liquid in a Pd40Ni40P20 metallic glass, Appl. Phys. Lett. 77 (2000) 1114. [40] N. Chen, D.V. Louzguine-Luzgin, G.Q. Xie, T. Wada, A. Inoue, Influence of minor Si addition on the glass-forming ability and mechanical properties of Pd40Ni40P20 alloy, Acta Mater. 57 (2009) 2775-2780. [41] X.J. Gu, A.G. Mcdermott, S. Joseph Poon, G.J. Shiflet, Critical Poisson’s ratio for plasticity in Fe-Mo-C-B-Ln bulk amorphous steel, Appl. Phys. Lett. 88 (2006) 211905. [42] S. Li, R.J. Wang, M.X. Pan, D.Q. Zhao, W.H. Wang, Bulk metallic glasses based on heavy rare earth dysprosium, Scr. Mater. 53 (2005) 1489-1492. [43] H. Shen, H. Wang, J. Liu, D. Xing, F. Qin, F. Cao, D. Chen, Y. Liu, J. Sun, Enhanced magnetocaloric and mechanical properties of melt-extracted Gd55Al25Co20 micro-fibers, J. Alloys Compd. 603 (2014) 167. [44] D.W. Johnson, K. Samwer, A universal criterion for plastic yielding of metallic glasses with a (T/Tg)2/3 temperature dependence, Phys. Rev. Lett. 95 (2005) 195501. [45] G. Wang, P.K. Liaw, O.N. Senkov, D.B. Miracle, M.L. Morrison, mechanical and fatigue behavior of Ca65Mg15Zn20 bulk-metallic glass, Adv. Eng. Mater. 11 (2010) 27-34. [46] Q. Qin, G.B. Mckenna, Correlation between dynamic fragility and glass transition temperature for different classes of glass forming liquids, J. Non-Cryst. Solid 352(2006) 2977. [47] J. Xu, U. Ramamurty, E. Ma, The fracture toughness of bulk metallic glasses, JOM 62 (2010) 10-18. [48] Z.F. Zhao, Z. Zhang, P. Wen, M.X. Pan, D.Q. Zhao, W.H. Wang, A highly 19
glass-forming alloy with low glass transition temperature, Appl. Phys. Lett. 82 (2003) 4699-4701. [49] S.V. Madge, D.V. Louzguine-Luzgin, J.J. Lewandowski, A.L. Greer,Toughness, extrinsic effects and Poisson’s ratio of bulk metallic glasses, Acta Mater. 60 (2012) 4800-4809. [50] J.J. Lewandowski, W.H. Wang, A.L. Greer, Intrinsic plasticity or brittleness of metallic glasses, Philos. Mag. Lett. 85 (2005) 77-87. [51] G. Wang, P.K. Liaw, O.N. Senkov, D.B. Miracle, The duality of fracture behavior in a Ca-based bulk-metallic Glass, Met. Mater. Trans. A 42 (2011) 1499-1503. [52] N.A. Shamimi, X.J. Gu, S.J. Poon, G.J. Shiflet, J.J. Lewandowski, Chemistry (intrinsic) and inclusion (extrinsic) effects on the toughness and Weibull modulus of Fe-based bulk metallic glasses, Philos. Mag. Lett. 88 (2008) 853-861. [53] R. Roscoe, Viscosity determination by the oscillating vessel method I: theoretical considerations, Proc. Phys. Soc. 72 (1958) 576-584. [54] H.B. Yu, W.H. Wang, H.Y. Bai, Y. Wu, M.W. Chen, Relating activation of shear transformation zones to β relaxations in metallic glasses, Phys. Rev. B 81 (2010) 220201. [55] J. Hedströ m, J. Swenson, R. Bergman, H. Jansson, S. Kittaka, Does Confined Water Exhibit a Fragile-to-Strong Transition? Eur. Phys. J.: Spec. Top. 141 (2007) 53-56.
20
www.elsevier.com/locate/msea
PII: DOI: Reference:
S0921-5093(18)31685-X https://doi.org/10.1016/j.msea.2018.12.004 MSA37276
To appear in: Materials Science & Engineering A Received date: 22 October 2018 Revised date: 28 November 2018 Accepted date: 1 December 2018 Cite this article as: Tuo Wang, Lina Hu, Yanhui Liu and Xidong Hui, Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming a l l o y s , Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2018.12.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Intrinsic correlation of the plasticity with liquid behavior of bulk metallic glass forming alloys Tuo Wanga, Lina Hub, Yanhui Liuc*, Xidong Huia*
a
State Key Laboratory for Advanced Metals and Materials, University of Science and
Technology Beijing, Beijing 100083, China b
Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials
(Ministry of Education), Shandong University, Jinan 250061, China c
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
[email protected] [email protected]
*
Corresponding author.
ABSTRACT Nowadays, plasticizing has become a critical issue for the engineering application of bulk metallic glasses (BMGs). What is the origin of plastic flow and how to evaluate the plasticity of BMGs with a unified standard are still open questions. Here we report a universal fragility criterion to scale the plasticity of BMGs from the equilibrium liquids and supercooled liquids. Based on our experimentally measured and previously reported viscosities at superheated and supercooled region for 51 glasses, the plasticity () of BMGs has been found to positively correlate to the supercooling liquid fragility (m) with the equation of = 92exp[-73/(m-15)], where is the plastic strain and m the fragility near glass transition temperature, but negatively correlate to the fragile-to-strong (F-S) transition tendency characterized by F-S transition factor and temperature. The intrinsic mechanism of the plasticity can be well interpreted in the frame of structural relaxation theory. This work provides a new insight into the 1
microscopic essence of the plasticity of glassy solids and a universal approach to scale the fluidity of metallic glasses. Keywords： Bulk metallic glasses; Plasticity; Fragility; Fragile-to-strong transition
1. Introduction Bulk metallic glasses (BMGs) possess a number of remarkable properties, such as high strength and hardness, excellent corrosion resistance and soft magnetic properties etc [1]. Unfortunately, they typically lack of macroscopic plasticity at room temperature. In order to develop plastic BMGs by composition design, it is essential to establish the correlations between plasticity and physical parameters, so that the mechanism that governs deformation can be comprehensively understood. Up to now, a number of correlations have been proposed between BMG plasticity and physical and structural quantities, such as Poisson’s ratio [2, 3], nanoscale structural heterogeneity [4], phase separation [3, 5] and density of flowing units [6-8] etc. Interestingly, for non-metallic glasses, Novikov and Sokolov [9] suggested that fragility is linearly correlated with the ratio of bulk modulus (B) to shear modulus (G), B/G, or equivalently Poisson’s ratio ν. Although such correlation is nonlinear for BMGs [10], it implies that BMG plasticity may be associated with liquid fragility. BMG deformation is known to proceed by shear band formation and propagation as consequence of shear transformation zones (STZ), i.e. the localized collective atomic rearrangement under shear stress [11, 12], Recently, a number of studies suggested that the formation of shear bands in BMGs can be considered as stress-driven glass-to-liquid transition [13]. For example, the yield shear stress is propositional to glass transition 2
temperature Tg [14]. The viscosity within shear band is found to be on the same order as that of supercooled liquid [15, 16]. All these mean that the material within shear bands behaves as supercooled liquid, so the way to solve the puzzles in the plastic flow of BMGs may be steered to the viscous flow that accompanies supercooling of liquid and glass formation. Phenomenologically, when the molten liquid is cooled down to the glass-transition temperature, the viscosity of the liquid is sharply increased to 12 powers by 10 poise [9], and this process has been characterized by the Angell’s theory [17] of liquid fragility, m, which is defined as m
d log d (T g / T )
(1) T Tg
where is shear viscosity, Tg is the glass transition temperature and T is the absolute temperature. The values of m are obviously different between different glassy systems, e.g., the m’s of Pt-based BMGs [18] may reached 68, Zr-based BMGs range from 30 to 56 [19-23] and fused silica [22] is only 15. The equilibrium viscosity of a glass-forming liquid can be described by the Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equation [24, 25], which is described by: log log
B C exp( ) T T
(2)
where η∞ is viscosity at the high temperature limit, B and C are constants. In 1999, Ito et al. [26] has found that the change of the viscosity around molten temperature in water is likely the fragile liquid OPT, whereas the behavior around Tg is similar to strong liquid SiO2. This phenomenon is so-called fragile-to-strong (F-S) transition. The F-S transition was also found in other metallic glass forming liquids (MGFLs), which might be a general dynamic phenomenon in GFLs [24-27]. In order to describe this physical phenomenon, a generalized MYEGA expression can be written as:
3
log log
1 C C T [(W1 exp( 1 ) W2 exp( 2 )] T T
(3)
where C1, C2, W1 and W2 are constants. This equation contains the fragility of the molten and the supercooled liquid. Investigating the behavior of supercooled liquid which can be quantified by fragility m and the progress of F-S transition can shed light on the deformation and plasticity of BMGs. In this work, we demonstrate that plasticity of BMGs is closely correlated with the property of glass forming liquid which can be characterized by fragility m. We found that the plasticity of BMGs is positively correlated with the fragility of supercooling liquid, but negatively correlated to the temperature of F-S transition temperature of supercooled liquid. This finding has implication for understanding the origin of plasticity and deformation mechanism of BMGs from the perspective of liquid. 2. Materials and methods Master alloy ingots were prepared by alloying pure Zr, Fe, Al, Cu and Nb metals with a purity of over 99.5 mass% in an arc-melter in a Ti-gettered argon atmosphere. To ensure compositional homogeneity, each ingot was remelted at least four times. Subsequently, cylindrical samples with 2 mm in diameter were obtained by suction casting in a copper mold. The amorphous nature of the alloy rods were confirmed by X-ray diffractometer (XRD) using a Bruker D8 ADVANCE (Cu Kα radiation). The compression specimens with 2 mm in diameter and 4 mm in length were cut from the rods, and the ends were polished carefully to ensure parallelism. The Tg of the bulk alloy rods were checked by Netzsch STA 449C differential scanning calorimeter (DSC). The uniaxial compression test was performed with a CMT4305 universal testing machine at room temperature with a strain rate of 2*10-4s-1. For the brittle BMGs, the test was stopped when the samples were fractured. And for the BMGs with microscopic 4
compressive plasticity, the test was stopped at the stress obviously dropped. The plastic strain was defined as the difference value of the intersection point at strain-axis between the elastic line and its parallel line at the maximum stress. The viscosities of MGFLs were measured in a high vacuum atmosphere with an oscillating viscometer. The viscosities were measured for four times at a temperature, and the average value was regarded as the equilibrium viscosity for a certain MGML. 3. Results and disscussion 3.1 Discrete distribution of the plastic strains among different BMGs system In the first place, let us begin with the compositional dependence of plastic strains for the different 9 BMGs. The amorphous nature of the alloy rods were confirmed by XRD as shown in Fig. 1. It is seen that all the curves have only one halo peak, which is the feature of glassy structure of alloys.
Fig. 1 XRD patterns of the glassy rod specimens with the diameter of 2mm.
The compressive stress-strain curves of the interested Zr- and CuZr- based alloys are shown in Fig. 2a. From these curves, the substantial plastic strains can be 5
determined and listed in Table 1. It is seen that the plastic strains for these 9 BMGs are obviously affected by the composition, which vary from 0.9% to 14.3% for (Zr66.5Cu22.5Fe5Al10)100-xNbx (x=0, 2, 4, 6) and from 0.2% to 10.2 % for (Zr66Cu17.5Fe6Al10.5)100-xNbx (x=0, 1, 1.5). The plastic strain of these BMGs seems to mainly be determined by the addition of Nb content in similar systems. However, it is seen that the plastic strains against Nb content are not linear for the 9 BMGs, as shown in Fig. 2b. As for the Zr47.75Cu47.75Al4.5 and Zr47.25Cu47.25Al5.5 BMGs, however, their plastic strains are 4.5% and 7.4%, respectively, which indicates that the plastic strain is affected by the Al content when Zr and Cu have same concentration. However, the plastic strains against Cu, Zr, Al contents are not closely correlative for the 9 BMGs (Fig. 2c-e). In order to find the correlation of plasticity to the composition and other parameters, we collected the data of plastic strains of more than 40 BMGs and other type of glasses from literatures [2, 5, 8, 10, 18-23, 28-46], and listed in Table 1.
6
Fig. 2 (a) Room-temperature compressive engineering stress-strain curves for 9 as-cast BMGs; (b)-(e) the plastic strains against Nb (b), Zr (c), Cu (d), and Al (e) content.
It is found that most of the BMGs with plastic strains above 10% belong to Zr-, CuZr-, Pd- and Pt- based alloy systems, while Fe-, Mg-, Hf-, Ca-based BMGs exhibit small plastic strains with some of them being completely brittle. Apparently, the seresults demonstrate that the compositional dependence of BMG plastic strain is complicated. There appears no single expression that can describe such strong dependence.
Table 1 Plastic strain (ε), fragility (m), the activation energy of relaxation (Eα), the activation energy of β relaxation (Eβ) and the ratio of Eα/Eβ for MGs reported in the literature and this work. Systems
Zr-based
Composition
ε (%)
m
Tg (K)
Eα
Eβ
(kJ/mol) (kJ/mol)
Eα/Eβ
Ref.
Zr62.5Cu22.5Fe5Al10
0.9
30
654
376
141
2.7
This work
Zr61.25Cu22.05Fe4.9Al9.8Nb2
1.9
34
651
424
141
3
This work
Zr60Cu21.6Fe4.8Al9.6Nb4
5.2
39
656
490
142
3.5
This work
Zr58.75Cu21.15Fe4.7Al9.4Nb6 14.3 46
660
581
143
4.1
This work
Zr66Cu17.5Fe6Al10.5
0.2
31
638
379
138
2.7
This work
Zr65.3Cu17.3Fe6Al10.4Nb1
4.1
36
649
447
140
3.2
This work
Zr65Cu17.3Fe5.9Al10.3Nb1.5 10.2 43
655
539
142
3.8
This work
Zr47.75Cu47.75Al4.5
4.5
39
686
512
148
3.5
This work
Zr47.25Cu47.25Al5.5
7.4
41
690
542
149
3.6
This work
Zr41.25Ti13.75Cu12.5Ni10Be22.5 1.5
39
648
484
140
3.5
19, 28
Zr46.75Ti8.25Cu7.5Ni10Be27.5
2
34
633
412
137
3
29, 30
Zr57Nb5Cu15.4Ni12.6Al10
3.3
35
670
449
145
3.1
20, 28
Zr65Cu15Al10Ni10
2
30
637
366
138
2.7
21, 31
Zr57Ti5Cu20Ni8Al10
1.1
40
672
515
145
3.6
22, 32
Zr50Cu35Al7Pb5Nb3
5.1
37
710
503
153
3.3
23
Zr55Cu30Al7Pb5Nb3
9.8
41
693
544
150
3.6
23
Zr60Cu25Al7Pb5Nb3
10.1 43
674
555
146
3.8
23
Zr62Cu23Al7Pb5Nb3
18.3 56
670
718
145
5
23
Zr50Cu48Al2
15
47
676
608
146
4.2
33
Zr50Cu44.5Al5.5
2.6
39
678
506
147
3.4
8
Zr50Cu44Al5.5Mo0.5
5.7
41
682
535
147
3.6
8
7
Zr50Cu43Al5.5Mo1.5
10.9 47
685
616
148
4.2
8
Zr50Cu41.5Al5.5Mo3
20.4 50
688
658
149
4.4
8
Zr64Cu26Al10
6.8
44
662
558
143
3.9
22, 34
Cu47.5Zr47.5Al4
7.5
44
683
575
148
3.9
35
Cu43Zr43Al7Ag7
8
45
700
603
151
4
18
Cu43Zr43Al7Be7
3.5
42
696
560
150
3.7
18
Cu46.25Zr45.25Al7.5Sn1
4.1
40
718
550
155
3.5
22
Cu50Zr50
4
39
664
496
144
3.4
22
Cu46Zr42Al7Y5
5
49
713
669
154
4.3
5, 36
3
40
431
330
93
3.5
37, 38
Mg65Cu25Gd10
2
39
423
316
91
3.5
10
Mg58.5Cu30.5Y11
0.4
30
421
242
91
2.7
22, 37
Mg57Cu31Y6.6Nd5.4
1.2
29
427
237
92
2.6
22, 37
10.4 52
673
670
145
4.6
19, 22
Pd40Ni40P20
1.5
46
593
522
128
4.1
39, 40
Pt57.5Cu14.7Ni5.3P22.5
20
68
508
661
110
6
2, 18
Fe63Mo14C15B6Er2
0
28
778
417
168
2.5
22, 41
Fe65Mo14C15B6
0.8
33
790
499
171
2.9
22, 41
La66Al14Cu10Ni10
0
32
459
281
99
2.8
22, 42
La55Al25Ni20
0.1
35
491
329
106
3.1
21
Hf50Ni25Al25
1.3
33
858
542
185
2.9
22, 34
Hf55Ni25Al20
2.5
36
828
571
179
3.2
22, 34
Gd-based
Gd55Al25Co20
0
25
597
286
129
2.2
35, 43
Ce-based
Ce70Al20Ni10Cu10
0
29
358
199
78
2.6
42, 44
Ca-based
Ca65Mg15Zn20
0
21
373
150
81
1.9
10, 45
B2O3
0
32
554
339
120
2.8
46
Fused silica
0
15
-
-
-
-
22
Calcium phosphate
0
24
-
-
-
-
22
Window
0
17
-
-
-
-
22
ZBLAN
0
28
-
-
-
-
22
Cu(Zr)-based
Mg65Cu7.5Ni7.5Zn5Ag5Y5Gd 5
Mg-based
Pd-based Pt-based Fe-based La-based Hf-based
Others
Pd77.5Si16.5Cu6
3.2 Fragility extended from high temperature liquid For many MGFLs, the temperature dependence of η for temperature intervals lying between the liquidus temperature TL and Tg can be described by the Angell’s theory [17]. According to Angell, the fragility scales with activation energy of glass transition, 8
Eg, by [27] m
Eg
(4)
RTg ln 10
To obtain the m values for our 9 BMGs, we carried out DSC measurements at various heating rate (Fig. 3a). The Eg for the BMGs can be gotten by Kissinger equation[3], ln(
Tg
)
Eg RTg
C
(5)
where β is the heating rate, Tg is the glass transition temperature at a certain heating rate, R is the ideal gas constant, C is a constant. So, -Eg/R is the slope of the line on the plot of the heating rate versus 1/Tg. The Kissinger curves of the BMGs is shown in Fig. 3b.
Fig. 3 (a) DSC curves measured at different heating rate for Zr61.25Cu22.05Fe4.9Al9.8Nb2 BMG as a representative of 9 BMGs; (b) activation energies fitted by Kissinger equation for this work.
In addition to the 9 BMGs, we also collected the m values of 42 BMGs and other type of glasses reported in literatures, as listed in Table 1. It is seen that the m values are in the range of 30≤m≤46, indicating that they are intermediate glasses [17]. Fig. 4a shows plastic strain vs. fragility for all the glasses. As can be seen, the 9
glasses can be roughly classified into two categories according to the plastic strain, i.e. brittle and plastic ones. Here, we define the brittle glasses as those showing plastic strain less than 0.2%. It can be seen that Zr-, CuZr-, Pt-, Pd-based BMGs are plastic, whereas fused silica, calcium phosphate, ZBLAN, B2O3, Ca-, Mg-, La-, Ce-, Gd-based BMGs are brittle. According to Angell [17], glass forming liquids can be classified as strong, intermediate, and fragile ones based on the fragility parameter. The BMGs shown in Table 1 are therefore in the category of intermediate glasses. Fig. 4a presents the plot of plastic strain, , as a function of fragility. Fitting to the data points yields = 92exp[-73/(m-15)] with a correlation coefficient of 0.74. From this equation, one can be derive the theoretical minimum value of m=15. This limiting value represents the strongest liquid, which have been verified by the DSC, DMA, or viscosity measurements on different GFLs. It is also indicated that the equation obtained from Fig. 4a is basically in agreement with the recognized theory of fragility and a fragile glass
Fig. 4 (a) Plastic strain, ε, versus the fragility, m, near the supercooling region reported in the literature and measured this work; (b) the fracture toughness, KC, versus m in the literature.
forming liquid is favorable for the achievement of higher plasticity. The argument is further supported by the correlation between fracture toughness, KC, and fragility, as 10
shown in Fig. 4b in which various BMGs and oxides glasses are included [1, 2, 10, 18-20, 22, 47-52]. It appears that fracture toughness is linearly dependent on fragility, e.g. KC = 2m-25 with a correlation coefficient of 0.85. 3.3 Fragile-to-strong transition during the formation of plastic BMGs Since it is technically difficult to directly measure the realistic viscosities, η, near the glass transition temperature, for the glass formers with low glass forming ability (GFA), in this work, the viscosities of the interested bulk metallic glass liquids (BMGLs) above liquidus temperature, TL, were measured by oscillating viscometer [24, 53].The link between plasticity and fragility indicates that BMG deformation is closely associated with the property and behavior of glass forming liquid.
Fig. 5 (a) Experimentally measured viscosities, , and fitting curves as functions of the temperature; (b) the curves of the viscosities as the functions of the glass transition temperature, Tg, scaled by temperature, T, fitted by Eq. 2 for Zr58.75Cu21.15Fe4.7Al9.4Nb6 MGFL; (c) The curve fitted by Eq. 3 for the Zr58.75Cu21.15Fe4.7Al9.4Nb6 MGFL which 11
includes two terms of contributions to the overall viscous behavior of metallic glass-forming liquids: a fragile term dominant at high temperatures and a strong term dominant at low temperatures.
Fig. 5a shows the viscosities of different glass forming liquids as a function of temperature T. Two kinds of fitting curves are extrapolated to Tg, where the viscosity values
are
1012
Pois
as
indicated
in
Fig.
5b.
According
to
the
Mauro-Yue-Ellison-Gupta-Allan MYEGA equation (See Eq. 2 in the introduction) [24, 25]. Here, the slope of the red solid line at Tg is the fragility parameter m’ which can be obtained from the high-temperature liquid (The temperature is above liquidus temperature TL) viscosity. Based on this theory, m’ presents the fragility of the equilibrium state (the temperature above TL), i.e. m quantifies the fragility of the liquids above TL. The m values and fitting parameters B, C calculated by Eq. 2 based on the visicosities data as shown in Fig. 5a are listed in Table 2. According to Eq. 1, 2 and the value of logη∞, we can obtain the red dashed curve in Fig. 5b. Those two terms can be written as MYEGA equation (See Eq. 3 in the introduction) [24, 25]. Taking Zr58.75Cu21.15Fe4.7Al9.4Nb6 as an example, we carried out fitting to the viscosity data. As shown in Fig. 5c, the dashed-dotted curve is for the fragile term W1 exp(
C1 in Eq. 3, and the dashed curve for strong term C ) W2 exp( 2 ) . The F-S T T
transition temperature can be identified when W1 exp(
C1 C ) W2 exp( 2 ) and are listed T T
in Table 2 for different BMGs. Table 2 The parameters of B, C and the fragility of high temperature m’ and parameters of C1, C2, W1, W2, the temperatures of the fragile-to-strong liquid, Tf-s, for the samples.
Samples
B
C
m’ 12
C1
C2
W1
W2
Tf-s (K)
Zr62.5Cu22.5Fe5Al10
10.54 4436 111
17004
726
3670
0.00034
1005
Zr61.25Cu22.05Fe4.9Al9.8Nb2
6.30
4747 118
18228
651
15361
0.00031
992
Zr60Cu21.6Fe4.8Al9.6Nb4
3.67
5144 126
19024
1148
45732
0.00061
986
Zr58.75Cu21.15Fe4.7Al9.4Nb6
1.91
5607 134
19140
1478
72727
0.00100
976
Zr66Cu17.5Fe6Al10.5
8.45
4449 113
17226
766
6113
0.00038
991
Zr65.3Cu17.3Fe6Al10.4Nb1
4.85
4896 121
19859
1038
107396 0.00052
983
Zr65Cu17.3Fe5.9Al10.3Nb1.5
1.76
5612 135
21615
1376
700763 0.00076
981
Zr47.75Cu47.75Al4.5
2.61
5654 134
20580
1166
212828 0.00058
985
Zr47.25Cu47.25Al5.5
3.08
5576 131
20700
1311
213043 0.00058
983
Tf-s can be seen as the most competitive temperature for the fragile and strong terms, and a parameter f (= m/m) was induced to represent this competition. It can be seen that f is positively related to Tf-s from the red line in Fig. 6a. Then, is there any relationship between Tf-s and plasticity? Fig. 6b depicts the dependence of plastic strains on Tf-s for the Zr-based BMGs. As can be seen, the plasticity of the BMGs is inversely related to Tf-s. According to Fig. 6a, f should be also inversely related to plasticity, as shown in Fig. 6b. How to understand it? Here, we use the model of α and β relaxations
Fig. 6 (a) f as function of r (black) and Tf-s (red) for the present 9 MGFLs; (b)The plastic strains as function of f (black) and Tf-s (red) for the present 9 MGFLs.
to understand this relationship. The ratio between the activation energies for the α and slow β relaxations, r (i. e. r = Eα/Eβ) can be defined as a relaxation competition parameter. The closer the Eβ and Eα values are, the more comparable the structural units 13
involved in the slow β relaxation and those in α relaxation are. It is known that Eβ = (26 ±2)RTg [54] and Eα = 2.303mRTg [35]. Therefore, the ratio r is solely related to fragility by r = Eα/Eβ = 0.089m, i.e. a larger fragility index m indicates an enlarged difference between Eα and Eβ. Kinetically, the F-S transition is accompanied by different relaxation processes in supercooled liquids upon cooling, which is governed by these two characteristic structures i. e. the primary α and the slow β relaxations [27, 35, 55]. As the black line in Fig. 6a shows that r is inversely related to f, i.e. the more intense competition (i.e. f is more larger) between fragile and strong term in MGFLs, the more comparable (i.e. r is more lower) between the α and β structural units. Based on the above perception, the schematic of α and β structural units with brittle and ductile BMGs is as shown in Fig. 7. The blue and red circles present α and β structural units, respectively. When the units are more comparable (Fig. 7a), the structure of the BMG is more homogeneous, resulting in a poor plasticity. On the contrary, if the units are more distinguishable (Fig. 7b), the structure of the BMG is more heterogeneous, resulting in a good plasticity. Based the above discussion, we propose two main relationships, i.e. plasticity is positively correlated with fragility values, but inversely related with Tf-s. It can be understood that m is proportional to Eα, the large activation energy, Eα, will lead to a longer relaxation time, resulting in a long time to achieve internal equilibrium for the system, and the temperature of fragile-strong liquid transition Tf-s being delayed, i. e. more lower Tf-s. Therefore, more β relaxation and more distinguishable (Fig. 7b) structure will be preserved which may enhance the plasticity. From the viewpoint of F-S transition, the MGFLs in low temperature inherit more fragile feature when the MGFLs in high temperature are more fragile, and the tendency of the transition degree is more gentle, resulting in a larger m and a good plasticity. 14
Fig. 7 Schematic diagrams of α and β structural units in (a) brittle and (b) ductile BMGs. The blue and red circles represent the α and β structural units, respectively.
Conclusion In summary, The correlation of fragility to the plasticity for 51 glass materials has been inspected. Macroscopic plasticity has been evidenced in many BMG systems such as Zr-, CuZr-, Pt-, and Pd-based BMGs. No universal compositional dependence of the plasticity of BMGs has been reached across the whole BMG alloy systems. Based on the fragilities and plasticities of 51 glass materials measured in this work and previously reported in the literature, universal criterion for plasticity can be established with the equation of = 92exp[-73/(m-15)], which indicates that there is a positive relationship between plasticity and fragility. There is fragile-to-strong (F-S) transition during the cooling process from superheated liquid to supercooling liquid. The plasticity of BMGs is negatively correlated to the F-S transition factor and temperature. The intrinsic mechanism of the plasticity of BMGs can be well interpreted by the competition of relaxation with relaxation mode. This deduction can be supported the correlation of plasticity to the ratio of activation energy of relaxation to relaxation.
Acknowledgments The authors are grateful for the financial support of National Key Basic Research 15
Program (No. 2016YFB0300502), and National Natural Science Foundation of China (Nos. 51571016 and 51531001).
References [1] W.H. Wang, C. Dong, C.H. Shek, Bulk metallic glasses, Mater. Sci. Eng., R 44 (2004) 45-89. [2] J. Schroers, W.L. Johnson, Ductile Bulk Metallic Glass, Phys. Rev. Lett. 93 (2004) 255506. [3] T. Wang, Y. Wu, J. Si, Y. Liu, X. Hui. Plasticizing and work hardening in phase separated
Cu-Zr-Al-Nb
bulk
metallic
glasses
by
deformation
induced
nanocrystallization, Mater. Des. 142 (2018) 74-82. [4] Z.W. Zhu, L. Gu, G.Q. Xie, W. Zhang, A. Inoue, H.F. Zhang, Z.Q. Hu, Relation between icosahedral short-range ordering and plastic deformation in Zr-Nb-Cu-Ni-Al bulk metallic glasses, Acta Mater. 59 (2011) 2814-2822. [5] E.S. Park, D.H. Kim, Phase separation and enhancement of plasticity in Cu-Zr-Al-Y bulk metallic glasses, Acta Mater. 54 (2006) 2597-2604. [6] Z. Lu, W. Jiao, W.H. Wang, H.Y. Bai, Flow unit perspective on room temperature homogeneous plastic deformation in metallic glasses, Phys. Rev. Lett. 113 (2014) 045501. [7] Q. Wang, J.J. Liu, Y.F. Ye, T.T. Liu, S. Wang, C.T. Liu, J. Lu, Y. Yang, Universal secondary relaxation and unusual brittle-to-ductile transition in metallic glasses, Mater. Today 20 (2017) 293-300. [8] T. Wang, L. Wang, Q. Wang, Y. Liu, X. Hui, Pronounced Plasticity Caused by Phase Separation and β-relaxation Synergistically in Zr-Cu-Al-Mo Bulk Metallic Glasses, Sci. Rep. 7 (2017) 1238. [9] V.N. Novikov, A.P. Sokolov, Poisson's ratio and the fragility of glass-forming liquids, Nature 431 (2004) 961-963. [10] E.S. Park, J.H. Na, D.H. Kim, Correlation between fragility and glass-forming ability/plasticity in metallic glass-forming alloys, Appl. Phys. Lett. 91 (2007) 031907. 16
[11] A.S. Argon, Plastic deformation in metallic glasses, Acta Metall. 27 (1979) 47-58. [12] P. Schall, Structural rearrangements that govern flow in colloidal glasses, Science 318 (2007) 1895-1899. [13] P. Guan, M. Chen, T. Egami, Stress-temperature scaling for steady-state flow in metallic glasses, Phys. Rev. Lett. 104 (2010) 205701. [14] Y.H. Liu, C.T. Liu, W.H. Wang, A. Inoue, T. Sakurai, M.W. Chen, Thermodynamic origins of shear band formation and the universal scaling law of metallic glass strength, Phys. Rev. Lett. 103 (2009) 065504. [15] S.X. Song, T.G. Nieh. Flow serration and shear-band viscosity during inhomogeneous deformation of a Zr-based bulk metallic glass, Intermetallics 17 (2009) 762-767. [16] J. Bokeloh, S.V. Divinski, G. Reglitz, G. Wilde, Tracer measurements of atomic diffusion inside shear bands of a bulk metallic glass, Phys. Rev. Lett. 107 (2011) 235503. [17] C.A. Angell, Formation of glasses from liquids and biopolymers, Science 267 (1995) 1924-1935. [18] J.H. Na, E.S. Park, Y.C. Kim, E. Fleury, W.T. Kim, D.H. Kim, Poisson’s ratio and fragility of bulk metallic glasses, J. Mater. Res. 23 (2008) 523-528. [19] D.N. Perera, Compilation of the fragility parameters for several glass-forming metallic alloys, J. Phys.: Condens. Matter 11 (1999) 3807-3812. [20] Z. Evenson, S. Raedersdorf, I. Gallino, R. Busch, Equilibrium viscosity of Zr-Cu-Ni-Al-Nb bulk metallic glasses, Scr. Mater. 63 (2010) 573. [21] Y. Kawamura, T. Nakamura, H. Kato, H. Mano, A. Inoue, Newtonian and non-Newtonian viscosity of supercooled liquid in metallic glasses, Mater. Sci. Eng., A 304 (2001) 674-678. [22] J.D. Plummer, I. Todd, Implications of elastic constants, fragility, and bonding on permanent deformation in metallic glass, Appl. Phys. Lett. 98 (2011) 021907. [23] S. Zhu, G. Xie, F. Qin, X. Wang, A. Inoue, Ni- and Be-free Zr-based bulk metallic glasses with high glass-forming ability and unusual plasticity, J. Mech. Behav. Biomed. Mater. 13 (2012) 166-173. [24] C. Zhang, L. Hu, Y. Yue, J.C. Mauro, Fragile-to-strong transition in metallic glass-forming liquids, J. Chem. Phys. 133 (2010) 014508. 17
[25] J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. 106 (2009) 19780. [26] K. Ito, C.T. Moynihan, C.A Angell, Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water, Nature 398 (1999) 492-495. [27] J.J.Z. Li, W.K. Rhim, C.P. Kim, K. Samwer, W.L. Johnson, Evidence for a liquid-liquid phase transition in metallic fluids observed by electrostatic levitation, Acta Mater. 59 (2011) 2166-2171. [28] B.A. Sun, H.B. Yu, W. Jiao, H.Y. Bai, D.Q. Zhao, W.H. Wang, Plasticity of ductile metallic glasses: a self-organized critical state, Phys. Rev. Lett. 105 (2010) 035501. [29] J.L. Zhang, H.B. Yu, J.X. Lu, H.Y. Bai, C.H. Shek, Enhancing plasticity of Zr46.75Ti8.25Cu7.5Ni10Be27.5 bulk metallic glass by precompression, Appl. Phys. Lett. 95 (2009) 071906. [30] Z.F. Zhao, Z. Zhang, P. Wen, M.X. Pan, D.Q. Zhao, W.H. Wang, W.L. Wang, A highly glass-forming alloy with low glass transition temperature, Appl. Phys. Lett. 82 (2003) 4699-4701. [31] S.H. Xie, X.R. Zeng, H.X. Qian, Correlations between the relaxed excess free volume and the plasticity in Zr-based bulk metallic glasses, J. Alloys Compd. 480 (2014) 37-40. [32] L.Q. Xing, Y. Li, K.T. Ramesh, J. Li, T.C. Hufnagel, Enhanced plastic strain in Zr-based bulk amorphous alloys, Phys. Rev. B 64 (2001) 607. [33] L. Xia, K.C. Chan, S.K. Kwok, P. Yu, Enhanced plasticity of a Zr50Cu48Al2 bulk metallic glass, J. Non-Cryst. Solids 357 (2011) 1469-1472. [34] L. Zhang, Y.Q. Cheng, A.J. Cao, J. Xu, E. Ma, Bulk metallic glasses with large plasticity: Composition design from the structural perspective, Acta Mater. 57 (2009) 1154-1164. [35] Q. Sun, C. Zhou, Y. Yue, L. Hu, A direct link between the Fragile-to-Strong transition and relaxation in supercooled liquids, J. Phys. Chem. Lett. 5 (2014) 1170-1174. [36] G.J. Fan, J.J.Z. Li, W.K. Rhim, D.C. Qiao, H. Choo, P.K. Liaw, W.L. Johnson, Thermophysical properties of a Cu46Zr42Al7Y5 bulk metallic glass, Appl. Phys. Lett. 88 18
(2006) 221909. [37] Q. Zheng, H. Ma, E. Ma, J. Xu, Mg-Cu-(Y,Nd) pseudo-ternary bulk metallic glasses: The effects of Nd on glass-forming ability and plasticity, Scr. Mater. 55 (2006) 541-544. [38] E.S. Park, J.Y. Lee, D.H. Kim, A. Gebert, L. Schultz, Correlation between plasticity and fragility in Mg-based bulk metallic glasses with modulated heterogeneity, J. Appl. Phys. 104 (2008) 023520. [39] Y. Kawamura, A. Inoue, Newtonian viscosity of supercooled liquid in a Pd40Ni40P20 metallic glass, Appl. Phys. Lett. 77 (2000) 1114. [40] N. Chen, D.V. Louzguine-Luzgin, G.Q. Xie, T. Wada, A. Inoue, Influence of minor Si addition on the glass-forming ability and mechanical properties of Pd40Ni40P20 alloy, Acta Mater. 57 (2009) 2775-2780. [41] X.J. Gu, A.G. Mcdermott, S. Joseph Poon, G.J. Shiflet, Critical Poisson’s ratio for plasticity in Fe-Mo-C-B-Ln bulk amorphous steel, Appl. Phys. Lett. 88 (2006) 211905. [42] S. Li, R.J. Wang, M.X. Pan, D.Q. Zhao, W.H. Wang, Bulk metallic glasses based on heavy rare earth dysprosium, Scr. Mater. 53 (2005) 1489-1492. [43] H. Shen, H. Wang, J. Liu, D. Xing, F. Qin, F. Cao, D. Chen, Y. Liu, J. Sun, Enhanced magnetocaloric and mechanical properties of melt-extracted Gd55Al25Co20 micro-fibers, J. Alloys Compd. 603 (2014) 167. [44] D.W. Johnson, K. Samwer, A universal criterion for plastic yielding of metallic glasses with a (T/Tg)2/3 temperature dependence, Phys. Rev. Lett. 95 (2005) 195501. [45] G. Wang, P.K. Liaw, O.N. Senkov, D.B. Miracle, M.L. Morrison, mechanical and fatigue behavior of Ca65Mg15Zn20 bulk-metallic glass, Adv. Eng. Mater. 11 (2010) 27-34. [46] Q. Qin, G.B. Mckenna, Correlation between dynamic fragility and glass transition temperature for different classes of glass forming liquids, J. Non-Cryst. Solid 352(2006) 2977. [47] J. Xu, U. Ramamurty, E. Ma, The fracture toughness of bulk metallic glasses, JOM 62 (2010) 10-18. [48] Z.F. Zhao, Z. Zhang, P. Wen, M.X. Pan, D.Q. Zhao, W.H. Wang, A highly 19
glass-forming alloy with low glass transition temperature, Appl. Phys. Lett. 82 (2003) 4699-4701. [49] S.V. Madge, D.V. Louzguine-Luzgin, J.J. Lewandowski, A.L. Greer,Toughness, extrinsic effects and Poisson’s ratio of bulk metallic glasses, Acta Mater. 60 (2012) 4800-4809. [50] J.J. Lewandowski, W.H. Wang, A.L. Greer, Intrinsic plasticity or brittleness of metallic glasses, Philos. Mag. Lett. 85 (2005) 77-87. [51] G. Wang, P.K. Liaw, O.N. Senkov, D.B. Miracle, The duality of fracture behavior in a Ca-based bulk-metallic Glass, Met. Mater. Trans. A 42 (2011) 1499-1503. [52] N.A. Shamimi, X.J. Gu, S.J. Poon, G.J. Shiflet, J.J. Lewandowski, Chemistry (intrinsic) and inclusion (extrinsic) effects on the toughness and Weibull modulus of Fe-based bulk metallic glasses, Philos. Mag. Lett. 88 (2008) 853-861. [53] R. Roscoe, Viscosity determination by the oscillating vessel method I: theoretical considerations, Proc. Phys. Soc. 72 (1958) 576-584. [54] H.B. Yu, W.H. Wang, H.Y. Bai, Y. Wu, M.W. Chen, Relating activation of shear transformation zones to β relaxations in metallic glasses, Phys. Rev. B 81 (2010) 220201. [55] J. Hedströ m, J. Swenson, R. Bergman, H. Jansson, S. Kittaka, Does Confined Water Exhibit a Fragile-to-Strong Transition? Eur. Phys. J.: Spec. Top. 141 (2007) 53-56.
20