Adv. Manuf. (2013) 1:293 304 DOI 10.1007/s40436-013-0040-3 Deterination of accurate theoretical values for therodynaic properties in bulk etallic glasses Pei-You Li Gang Wang Ding Ding Jun Shen Received: 15 July 2013 / Accepted: 23 Septeber 2013 / Published online: 31 October 2013 Ó Shanghai University and Springer-Verlag Berlin Heidelberg 2013 Abstract Deviation values of specific heat difference DC p ; the Gibbs free energy difference DG; enthalpy difference DH; and entropy difference DS between the supercooled liquid and corresponding crystalline phase produced by the linear, hyperbolic, and Dubey s expressions of DC p and the corresponding experiental values are deterined for sixteen bulk etallic glasses (BMGs) fro the glass transition teperature g to the elting teperature : he calculated values produced by the hyperbolic expression for DC p ost closely approxiate experiental values, indicating that the hyperbolic DC p expression can be considered universally applicable, copared to linear and Dubey s expressions for DC p ; which are accurate only within a liited range of conditions. For instance, Dubey s DC p expression provides a good approxiation of actual experiental values within certain conditions (i.e., n ¼ DCp g=dc p \2; where DCg p and DCp represent the specific heat difference at teperatures g and ; respectively). Keywords Bulk etallic glass (BMG) Specific heat Linear expression Hyperbolic expression P.-Y. Li J. Shen School of Materials Science and Engineering, Harbin Institute of echnology, Harbin 150001, People s Republic of China e-ail: junshen@hit.edu.cn G. Wang (&) D. Ding Laboratory for Microstructures, Shanghai University, Shanghai 200444, People s Republic of China e-ail: g.wang@shu.edu.cn 1 Introduction Due to the presence of a large supercooled liquid region, bulk etallic glasses (BMGs) usually exhibit high theral stability against crystallization. As a result, a large range of experiental tie and teperatures for nucleation and crystalline growth processes exist in etallic glass foring elts. Characterization of the three therodynaic paraeters including Gibbs free energy difference DG, entropy difference DS; and enthalpy difference DH; are iportant in evaluation of nucleation and crystal growth processes in BMGs between the supercooled liquid and corresponding crystalline phases [1, 2]. Nucleation rates have been shown to have an exponential dependence on DG [3], acting as a driving force of nucleation. When DG is sall, the critical nucleation work is iproved, and nucleation rates are reduced [4]. As a result, the glass foring ability (GFA) of these aterials is iproved. he values of DG; DS; and DH are routinely calculated by easuring changes in the specific heat difference, DC p ; between the supercooled liquid and corresponding crystalline phases across a range of teperatures. he etastable nature of supercooled liquids, however, akes accurate experiental values for DC p difficult to deterine [5]. hus, ost DC p values for the supercooled liquid regions of various BMGs are only rough approxiations generated by fitting liited experiental data to the elting teperature, ; in the vicinity of the glass transition teperature g : Because the accurate specific heat data in the supercooled region are notably absent, the functional dependences of DG; DS; and DH on teperature are generally estiated theoretically [5]. Several odels for calculating DG; DS; and DH values have been previously proposed based on different expressions for DC p [6 14].
294 P.-Y. Li et al. In these expressions, hopson et al. [12] and Hoffan et al. [13] assued that DC p was constant with teperature. Whereas Mondal et al. [10] and Patel et al. [11] suggested that DC p value depended linearly or hyperbolically on teperature, respectively. Each of these expressions, however, is deduced strictly fro experiental data [6, 10 14] without theoretical support. Recently, Dubey et al. [7, 8] proposed a theoretical expression for DC p based on the hole theory of the liquid state, thus calculating ore accurate values for DG, DH, and DS fro experiental results collected aong the teperature range fro g to in the Zr 57 Cu 15.4 Ni 12.6 Al 10 Nb 5 BMG [8]. Furtherore, a hyperbolic expression for DC p was deduced that provided an optial atheatical odel for elucidating GFA based on the theoretical expression for DC p proposed by Dubey et al. [7, 8]. According to the hyperbolic expression for DC p based on the hole theory of the liquid state [15], the current study further deduced a linear expression for DC p : he values of DG; DH; DS and for BMGs in the teperature range fro g to were calculated based on the hyperbolic, linear, and Dubey s expression for DC p : Sixteen BMG aterials [16 26] were selected as odels for using in experiental evaluation of the accuracy of these 3 expressions for DC p (Dubey s, hyperbolic, linear). he deviations observed in therodynaic paraeters between experiental results and these three odels [7, 8, 15] were coparatively evaluated. 2 Expressions for the therodynaic paraeters DC p, DG, DS, and DH Since DG is vital to the study of GFA in BMGs, expressions for DC p used in the calculation of DG values are iportant. he authors [15] previously proposed a hyperbolic expression for DC p based on the hole theory of the liquid state [7, 8], shown as follows: DC p ¼ DCp r h 1 þ ð2 r h Þ! ¼ DCp 1 r h 1 þ ð2 r h Þ 1 D ; ð1þ where DCp is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at ; r h is a coefficient related to the hole foration energy in the hyperbolic express; D is the degree of supercooling (D ¼ ; where is the teperature). In BMG systes, the values are saller than when is decreased fro to g ; suggesting that the value of D= is less than one. In this case, it is reasonable to approxiate that the hyperbolic ter in Eq. (1) can be expanded using a aylor series, as follows: ¼ 1 1 D ¼ 1 þ D þ D 2 þþ D n ; ð2þ thus producing the expression by neglecting a portion of the higher-order ters (n [ 1), DC p ¼ DCp 3 r l ð2 r l Þ ; ð3þ where r l is a coefficient related to the hole foration energy in the linear expression. Since the portion of the higher-order ters (n [ 1) in aylor s series, i.e., Eq. (2), is neglected, which can odify the coefficient of ð2 r h Þ in Eq. (1), r l is used to replace r h : he linear expression of DC p shown in Eq.(3) is siilar to the linear for of DC p ¼ A þ B; where A (A ¼ gdcp DCp g g ¼ DCp ð 3 r lþ) and B (B ¼ DCg p DCp g ¼ DC p ð r l 2Þ ) are the coefficients for linear expression, proposed by Patel et al. [11]. Evaluation of the paraeter r l also results in a ethod siilar to that proposed by Dubey et al. [7, 8]. Since experiental values of DCp g are usually easured in the vicinity of g; the DCp value can be eployed in conjunction with Eq. (3) to yield r l ¼ 2 þ 1 n ; ð4þ 1 rg where rg ¼ g = is the reduced glass transition teperature; and n ¼ DCp g=dc p ; where DCg p is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at g. DG; DH and DS are the differential values between the supercooled liquid and the corresponding crystalline phase, which can be expressed as Z DH ¼ DH DC p d; ð5þ and Z DC p DS ¼ DS d; DG ¼ DH DS; ð6þ ð7þ where DH is the enthalpy of fusion, and DS ¼ DH = is the entropy of fusion. Substituting Eq. (3) into Eqs. (5) (7), the novel expressions for DH, DS and DG can be obtained as
herodynaic properties in bulk etallic glasses 295 able 1 herodynaic paraeters for evaluation of DG, DH, and DS in the 16 BMGs Alloys A/(J ol -1 K -2 ) B/(J ol -1 K -3 ) C/(J ol -1 K -2 ) g /K /K rg DH /(kj ol -1 ) DC p /(J ol-1 K -1 ) DC g p /(J ol-1 K -1 ) La 62 Al 14 Cu 24 [16] 0.03071 4.16 9 10 5-1.49 9 10-5 401 673 0.60 6.835 14.840 12.510 La 55 Al 25 Ni 20 [17] 0.02190 1.24 9 10 6-1.01 9 10-5 491 712 0.69 7.477 12.940 13.460 Cu 47 i 34 Zr 11 Ni 8 [18] 0.01650 2.83 9 10 6-6.82 9 10-6 673 1114 0.60 11.300 12.187 14.257 Zr 46 Cu 46 Al 8 [19] 0.12850 4.61 9 10 6-5.59 9 10-6 715 979 0.73 8.035 13.490 16.410 La 55 Al 25 Cu 10 Ni 5 Co 5 [17] 0.02520 1.69 9 10 6-1.18 9 10-5 466 661 0.70 6.095 12.610 15.580 La 62 Al 14 (Cu 5/6 Ag 1/6 ) 24 [16] 0.03340 7.54 9 10 5-3.04 9 10-5 404 656 0.62 6.118 10.600 13.160 Mg 65 Cu 25 Y 10 [20] 0.01750 1.8 9 10 6-1.02 9 10-5 410 730 0.56 8.650 10.730 15.700 Zr 57 Cu 15.4 Ni 12.6 Al 10 Nb 5 [18] 0.01630 6.32 9 10 6-8.37 9 10-6 682 1091 0.63 9.400 13.130 20.810 Zr 46 (Cu 4.5/5.5 Ag 1/5.5 ) 46 0.01620 8.17 9 10 6-2.08 9 10-5 620 937 0.66 8.200 11.060 17.870 Al 8 [19] Pt 57.3 Cu 14.6 Ni 5.3 P 22.8 [21] 0.01010 5.77 9 10 6-1.20 9 10-5 488 776 0.63 11.400 10.190 26.280 Zr 52.5 Cu 17.9 Ni 14.6 Al 10 0.00260 6.43 9 10 6-16.8 9 10-6 675 1072 0.63 8.200 7.520 19.830 i 5 [18] i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 0.02140 1.02 9 10 7-0.97 9 10-5 678 1093 0.62 10.949 8.400 27.920 [22] i 37.65 Cu 43.25 Ni 9.6 Zr 9.5 [22] 0.00870 1.17 9 10 7-9.25 9 10-6 673 1097 0.61 11.040 8.150 27.510 Zr 41.2 i 13.8 Ni 10 Cu 12.5 Be 22.5 0.01620 8.17 9 10 6-2.08 9 10-5 620 937 0.66 8.200 6.220 23.300 [23, 24] Zr 58.5 Cu 15.6 Ni 12.8 Al 10.3 Nb 2.8 0.01280 5.70 9 10 6-1.37 9 10-5 660 1083 0.61 8.700 2.655 15.550 [25] Pd 43 Ni 10 Cu 27 P 20 [26] 0.03320 4.89 9 10 6-5.00 9 10-5 576 790 0.73 7.200 2.860 17.280 DH ¼ DH DCp ð3 r l Þð Þ 1 r l 1 ð 2 2 2 Þ ; DS ¼ DS DCp r l 2 D þð3 r l Þ ln DG ¼ DS D DCp ðr l 2Þ D2 2 þð3 r l Þ D ln ð8þ ; ð9þ : ð10þ Coparative studies were conducted on the expressions for DG, DS, anddh produced by the hyperbolic expression and Dubey s expression for DC p using the fraework of the hole theory of the liquid state as a basis. his technique allowed for further characterization of the therodynaic behaviors of BMGs. Based on the hyperbolic expression for DC p ; the expressions for DG, DS,andDH,respectively,are[15] DG ¼ DS D DCp ð3 r h ÞD þ ðð3 r h Þ ðr h 1ÞDÞ ln ð11þ ; DH ¼ DH DC p ðr h 1Þð Þþð2 r h Þ ln ; ð12þ DS ¼ DS DC p and ð2 r h Þ þðr h 1Þ ln ; ð13þ r h ¼ 1 þ 1 rgn 1 rg : ð14þ Based on the hole theory of the liquid state, Dubey et al. [7, 8] provided an expression for the DC p as DC p ¼ DCp 2 D exp r D ; ð15þ where r D is a coefficient related to the hole foration energy in the Dubey s expression. Substituting Eq. (15) into Eqs. (5) (7), the siplified Dubey s expressions for DG; DH and DS are [7, 8] DG ¼ DS D DCp D 2 D 1 r D 2 3 DH ¼ DH DCp D 1 r DD 2 and DS ¼ DS DC p D 1 þ D 2 ; ð16þ ; ð17þ 1 r D 2 D! 1 þ 2D 3 1 þ D : 2 ð18þ
296 P.-Y. Li et al. Fig. 1 Specific heat difference DC p a, b, and the deviation values of D DC p c, d as functions of teperature derived for 2 of 16 representative BMGs, Zr 46 Cu 46 Al 8 and La 55 Al 25 Cu 10 Ni 5 Co 5 alloys he value r D is given by [7, 8]! r D ¼ rg 1 ln 1 rg rg 2 n : ð19þ he deviation percentage D, between the theoretical calculated value and the experiental value can be expressed as D ¼ VðodÞ VðexpÞ VðexpÞ ; ð20þ where VðodÞ and VðexpÞ represent the calculated values and experiental values of DC p ; DG; DH and DS respectively. As shown in Eqs. (4), (14), and (19), ost of the BMG aterials in this study exhibit rg values that can be considered to be a constant equal to 0.65 (discussed in detail in later sections). hus the deviation values, D, for DC p ; DG; DH; and DS occurring between teperatures fro g to correlate with the r l ; r h and r D values as well as the n value. 3 Deviation between theoretical odel-based calculated and experiental values of DC p in BMGs he therodynaic behavior of BMGs is studied using expressions for therodynaic paraeters DC p ; DG; DS; DH; based on experiental results fro 16 different BMGs. he values of DG, DS, and DH are calculated using experientally easured DC p values, which can be expressed as [16 26] DC p ¼ A þ B 2 þ C 2 ; ð21þ where A, B, and C are constant. hese constants and paraeters can be found in Refs. [16 26] and are suarized in able 1. he calculated DC p value, experiental DC p value, and deviation value between calculated and experiental values fro g to can be generated by Eqs. (1), (3), (15), (20), and (21). It is prolix that all figures of DC p ; DG; DS; DH are described for the 16 alloys. In consideration of the different n values and alloy
herodynaic properties in bulk etallic glasses 297 Fig. 2 Specific heat difference DC p a, b, and the deviation values of D DC p c, d as functions of teperature derived for 2 of 16 representative BMGs, i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 and Pd 43 Ni 10 Cu 27 P 20 alloys copositions in the present study, the 4 BMGs Zr 46 Cu 46 Al 8 [19], La 55 Al 25 Cu 10 Ni 5 Co 5 [17], i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 [22], and Pd 43 Ni 10 Cu 27 P 20 [26] were representatively plotted in Figs. 1 and 2, respectively (2 aterials per figure). hese iages are representative of results fro all 16 BMGs (see able 1). Figures 1 and 2 show experientally fitted DC p values and calculated values deduced fro the hyperbolic, Dubey s, and linear expressions. Initially, the deviations in DC p for these 4 BMGs increased, followed by an iediate reduction with further teperature increased fro g to : he deviations in DC p achieved a axiu deviation value, D ax ; at an uncertain teperature in the range of g and : Notably, this value was achieved approxiately at the idpoint between g and in each saple. For Zr 46 Cu 46 Al 8 and La 55 Al 25 Cu 10 Ni 5 Co 5, the D ax values of DC p ; D ax DC p ; fro hyperbolic, linear, and Dubey s expressions were each saller than 4 %, indicating that calculated values of DC p closely approxiated experiental values, as copared with the D ax DC p values for i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 and Pd 43 Ni 10 Cu 27 P 20 BMGs (see Fig. 2). A coparison of the calculated and experiental values of DC p in BMGs is provided by the D ax DC p values for 16 BMGs (see able 2). he relationship between D ax DC p values and n values for the 16 alloys is suarized in Fig. 3. In addition, D ax DC p values derived fro the linear expression deonstrated a axiu value of 16 % of the initial value, and the values derived fro Dubey s expression increased with increasing n value for n [ 2. Notably, the axiu value of Dubey s expression approached 38 % of the initial value for a n value of 6. For the hyperbolic expression, D ax DC p values were generally less than 11 % of the initial values. Cuulatively, these findings indicated that the hyperbolic expression for DC p fitted well with the experiental values copared with both linear and Dubey s expressions. When n \ 2, the D ax DC p derived fro all 3 expressions was saller
298 P.-Y. Li et al. able 2 he axiu deviation values Dax DCp of DCp; deviation values D DG; D DH; and D DS; of DG; DH; and DS at g; and n values for 16 alloys (Dax DCp; D DG; D DH; and D DS denote the deviation values of DCp; DG; DH; and DS; respectively, between the calculated and experiental values) Alloys Dax DCp Hyper.-D ( = g ) Line-D ( = g ) Dubey-D ( = g ) n Hyper. Line Dubey D DG D DH D DS D DG D DH D DS D DG D DH D DS La62Al14Cu24 2.7 0.6 9.3 0.7 2.0 3.8 0.1 0.2 0.5 4.4 32.8 74.7 0.84 La 55 Al 25 Ni 20 1.6 1.9 5.1 0.3 0.5 0.9 0.3 0.8 1.1 0.4 4.2 6.7 1.04 Cu 47 i 34 Zr 11 Ni 8 1.3 3.2 7.5 0.3 0.8 1.4 0.8 2.1 3.9 1.0 10.8 23.8 1.17 Zr 46 Cu 46 Al 8 1.1 2.7 3.1 0.3 0.7 2.0 0.6 1.7 2.3 0.2 1.3 2.3 1.22 La 55 Al 25 Cu 10 Ni 5 Co 5 0.6 2.4 3.1 0.1 0.3 0.4 0.5 1.2 1.8 0.1 2.0 3.3 1.24 La 62 Al 14 (Cu 5/6 Ag 1/6 ) 24 2.5 0.7 3.3 0.7 1.5 2.5 0.0 0.2 0.3 1.4 9.1 18.4 1.24 Mg 65 Cu 25 Y 10 4.2 10.3 10.8 0.9 2.3 4.5 2.4 5.9 11.3 0.5 4.1 11.2 1.46 Zr 57 Cu 15.4 Ni 12.6 Al 10 Nb 5 1.3 6.7 4.2 0.4 1.9 7.4 2.8 10.7 41.0 0.3 4.1 21.1 1.58 Zr 46 (Cu 4.5/5.5 Ag 1/5.5 ) 46 Al 8 0.7 5.5 2.0 0.2 0.8 2.5 2.2 8.7 25.0 0.1 2.2 8.0 1.62 Pt 57.3 Cu 14.6 Ni 5.3 P 22.8 2.4 13.8 3.5 0.4 1.2 2.0 2.7 6.6 10.7 0.6 1.5 2.4 2.58 Zr 52.5 Cu 17.9 Ni 14.6 Al 10 i 5 5.4 9.0 11.0 1.8 5.5 14.0 2.0 7.0 18.0 3.8 12.5 32 2.64 i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 5.3 9.1 15.4 1.7 5.2 14.6 3.1 10.9 31.9 5.5 19.5 57.1 3.28 i 37.65 Cu 43.25 Ni 9.6 Zr 9.5 1.1 15.9 11.1 0.2 0.9 2.7 5.1 16.8 45.7 3.8 13.2 36.7 3.38 Zr 41.2 i 13.8 Ni 10 Cu 12.5 Be 22.5 4.1 9.6 18.8 1.0 2.5 4.4 2.4 7.2 13.0 5.4 18.1 33.4 3.74 Zr 58.5 Cu 15.6 Ni 12.8 Al 10.3 Nb 2.8 10.5 10.4 34.0 1.7 3.9 6.4 1.9 4.9 8.4 7.1 20.4 36.0 5.85 Pd43Ni10Cu27P20 10.6 2.4 37.5 1.1 2.5 3.2 0.3 0.7 0.9 5.4 15.6 20.7 6.04
herodynaic properties in bulk etallic glasses 299 Fig. 3 Relationships between the axial deviation values D ax DC p (the axiu deviation of DC p between the calculated and experiental values), and the n values for the 16 alloys in able 2 than 11 % of the initial value, suggesting that all 3 theoretical odels were applicable. 4 Deviations, calculated values, and experiental values of DG, DS, and DH in BMGs he deviation values D of paraeters DG; DS; DH; D DG; D DS; and D DH; respectively, between calculated and experiental values for 4 BMGs were deterined. As shown in Figs. 4 7, the values of D DG; D DH; and D DS exhibited axiu deviations at g in the teperature range fro g to : A notable exception to this trend was the deviation value of DG fro Dubey s expression for DC p for Zr 46 Cu 46 Al 8 (see Fig. 4d) and Mg 65 Cu 25 Y 10 (not shown). hus, deviation values for DG; DS; and DH at g could reasonably denote the degree of fit between DG; DS; and DH values in both calculated expressions and experiental results. D DG; D DH; and D DS values at g for all 16 etallic glasses are listed in able 2. he D DG; D DH; and D DS values at g and n values for 16 BMGs are listed in able 2. he axiu D DG values were achieved in the hyperbolic expression, whereas Dubey s and linear expression values for DC p were saller than 8 % of initial values, indicating the accuracy of calculated values using these 3 expressions for DG relative to experiental values. he axiu D DH and D DS values achieved by the hyperbolic expression for DC p in 16 BMGs were less than 10 % and 15 % of initial values (see able 2), respectively. he axiu D DH and D DS values derived fro linear and Dubey s expressions for DC p (see able 2) presented bigger values of 16.8 % and 45.7 %, respectively, copared with those derived fro the hyperbolic expression. hus, calculations for DG; DH; and DS values also suggested that the hyperbolic expression for DC p was the ost accurate predictor of experiental values. Results and analysis of D DH and D DS derived fro Dubey s expression for DC p produced siilar findings to those of D DC p derived fro Dubey s expression. he ajority of D DH and D DS values derived fro Dubey s expression for n \ 2 were very close to the D DH and D DS values derived fro the linear and Fig. 4 Paraeters DG a, DH b, and DS c as well as deviation values D DG d, D DH e, and D DS f as functions of teperature derived for the Zr 46 Cu 46 Al 8 alloy using reported experiental results and three different expressions
300 P.-Y. Li et al. Fig. 5 Paraeters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of teperature derived for the La 55 Al 25 Cu 10 Ni 5 Co 5 alloy using reported experiental results and 3 different expressions Fig. 6 Paraeters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of teperature derived for the i 36.89 Cu 43.87 Ni 9.36 Zr 9.88 alloy using reported experiental results and 3 different expressions hyperbolic expressions, suggesting that the difference between D DH and D DS values in each of the three expressions was very sall for n \ 2 (see able 2). he ajority of D DH and D DS values derived fro Dubey s expression for n [ 2 were uch larger than the D DH and D DS values derived fro the linear and hyperbolic expressions, suggesting that ost DH; and DS values derived fro Dubey s expression did not accurately predict experiental values for n [ 2. hus, the hyperbolic expression for DC p represents are relatively universal expression, copared to the linear expression and Dubey s expression for DC p which are only
herodynaic properties in bulk etallic glasses 301 Fig. 7 Paraeters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of teperature derived for the Pd 43 Ni 10 Cu 27 P 20 alloy using reported experiental results and 3 different expressions Fig. 8 Relationship between rg and n values accurate under certain conditions. Notably, experiental values ore closely fit values produced by the linear expression for DC p than values produced by Dubey s expression for DC p : Dubey s expression for DC p was, however, a good approxiation of experiental values for n \ 2, though not for n [ 2. 5 Discussion D ax between the calculated and experiental values for DC p ; DG; DH; and DS fro g to was found to be associated with the n paraeter according to Fig. 3 and able 2. Equations (4), (14), and (19) showed that in addition to the effects of the n paraeter, D values were also influenced by the rg values, which ranged fro 0.56 to 0.73 and could be expressed as a function of the n value (see Fig. 8). hus, equidistant low, ediu, and high rg values of 0.56, 0.65, and 0.73 were selected to further characterize the effect of rg on the value of D (or D ax ). Figure 9a shows the relationship between the r h ; r D ; and r l values and the n values for rg values of 0.56, 0.65, and 0.73. For n \ 2, the change in the 3 r h ; r D ; and r l values with rg was negligible, suggesting that DC p ; DG; DH; and DS values calculated using different odels were virtually identical (see Fig. 3 and able 2). For n [ 2, however, the values of r h ; r D ; and r l at the 3 different rg values revealed a decreasing trend. As n values increased fro 0.5 to 7, r D values exhibited only sall decreases, while larger decreases were exhibited by r h values. Moderate decreases in r l values were observed in between those of r D and r h copared with r h ; r D ; and r l values at rg = 0.65. Notably, this change can be neglected to siplify analysis. hus the ajority of BMGs rg can be considered constant ( rg ¼ 0:65). Figure 9b deonstrates the relationship between r h ; r D ; and r l values and the n paraeter at rg = 0.65. When rg = 0.65, the changes in r h ; r D ; and r l values are revealed to be very large, leading to variation in the deviation values for DC p ; DG; DH; and DS between calculated and experiental values. As shown in Fig. 3 and able 2, the D ax DC p values can be used to reveal the fit of DG; DH; and DS between the calculated and experiental values (see Fig. 3 and able 2). Due to this observation, only D ax DC p values are discussed.
302 P.-Y. Li et al. Fig. 9 Relationship between r h ; r D ; and r l ; and n values for different rg a and rg ¼ 0:65 b Graphs of C p as a function of teperature when n values equal to 2 and 3 (n ¼ 2; n ¼ 3) are shown in Fig. 10a, in which Cp l and Cs p represent the heat capacity of the supercooled liquid and crystal, respectively. Cp l and Cs p values were derived by fitting the experiental data. Figure 10b shows the DC p value evolution and teperature increases for n values of 2 and 3, where the n paraeter is the change rate of DCp s and DC p : In order to characterize the difference in D ax values produced by Dubey s expression and the hyperbolic expressions for DC p fro g to ; corresponding teperatures for D ax DC p values can be inferred. eperatures corresponding to the D ax DC p values for Zr 46 Cu 46 Al 8,La 55 Al 25 Cu 10 Ni 5 Co 5,i 36.89 Cu 43.87 Ni 9.36 Zr 9.88, and Pd 43 Ni 10 Cu 27 P 20 BMGs are generally between g and (see Figs. 1 and 2). hus ax ¼ 0:5 g þ ; where ax is the hypothetical axiu teperature of D ax DC p : All 16 studied BMGs also exhibit trends siilar to those of the 4 representative aterials shown (data not shown). DC p values calculated using the linear Fig. 10 Graphs of C p a and DC p b for n = 2 and n = 3, respectively (he inset shows the fit of the linear expression for DC p at different n values in the teperature range fro g to ) expression closely approxiate experiental values for DC p (see Fig. 3). hus, for teperature of 0:5 g þ ; a calculated value closely fits to the experiental DC p ; DC 0:5ð gþ Þ p ðexp :Þ; can be expressed as (see Fig. 10b) DC 0:5ð gþ Þ p ðexp :Þ ¼ DCg p þ DC p 2 ¼ 1 þ n DCp 2 : ð22þ Substitution of ax into Eqs. (1) and (15), the hyperbolic expression and Dubey s expression for DC p ; DC 0:5ð gþ Þ p deduced as ðhyper:þ and DC 0:5ð gþ Þ p DC 0:5ð gþ Þ p ðhyper:þ ¼ r h 1 þ 2 r h 0:5ð1 þ rg Þ DC 0:5ð gþ Þ p ðdubeyþ can be DCp ; ð23þ DC p 1 rg ðdubeyþ ¼ 2 exp r D : 0:5ð1 þ rg Þ 1 þ rg ð24þ
herodynaic properties in bulk etallic glasses 303 6 Conclusions Fig. 11 Relationship between D ax fro the hyperbolic expression and Dubey s expression for DC p and the n paraeter for rg = 0.65 using Eqs. (25) and (26) hus, expressions of D ax ðhyper:þ and D ax ðdubeyþ generated by the hyperbolic expression and Dubey s expression for DC p between the calculated and experiental values, respectively, can be written as D ax ðhyper:þ ¼ r h 1 þ 2 r h 0:5ð1þ rg Þ 1 0:5ð1 þ nþ ; ð25þ 1 1 exp r rg ð0:5ð1þ rg ÞÞ 2 D 1þ rg D ax ðdubeyþ ¼ 1 ; ð26þ 0:5ð1 þ nþ where rg is treated as a constant with a value of 0.65. Figure 11 shows the relationship between D ax fro the hyperbolic expression and Dubey s expression for DC p as well as the n paraeter based on Eqs. (25) and (26).When n \ 2, the ajority of D ax ðdubeyþ and D ax ðhyper:þ values are less than 10 %, indicating that these values closely approxiate the D ax DC p values in Fig. 3. When 2 \ n \ 7, the D ax ðdubeyþ values draatically increase fro 10 % to 43.7 %, and the D ax ðhyper:þ values gradually increase fro 8 % to 16 %. hese findings suggest that the accuracy of values calculated using the hyperbolic expression for DC p are higher than those calculated using Dubey s expression for DC p : hus, calculated D ax values (see Fig. 11) fro the hyperbolic expression and Dubey s expression vary according to trends very siilar to those of experiental D ax values (see Fig. 3). Based on the error scale deterined by these findings, the expressions of D ax ðhyper:þ and D ax ðdubeyþ can be used to indicate changes in D ax DC p according to the n paraeter values shown in Fig. 3 and able 2 for all 16 exained BMGs in the current study. A linear expression for DC p derived fro the hyperbolic expression for DC p was deduced and used to obtain a novel expression for DG; DH; and DS: According to the experientally deterined therodynaic paraeters of the 16 exained BMGs in the current study, ore accurate calculations of DC p ; DG; DH; and DS were obtained using the linear, hyperbolic, and Dubey s expression for DC p : hese results suggest that the hyperbolic expression for DC p can be applied as a universal expression for DC p ; while linear and Dubey s expressions for DC p are condition-dependent. Notably, Dubey s expression for DC p also closely approxiated experiental values when n \ 2, though values were shown to deviate fro experiental values for n [ 2. Acknowledgents he work described in this paper was supported by the grant fro the National Natural Science Foundation of China (Grant No. 51025415). References 1. Zallen R (1973) he physics of aorphous solids. Wiley, New York 2. Machlin E (2007) An introduction to aspects of therodynaics kinetics relevant to aterials science. Elsevier, Science or echnology Books, Asterda 3. Stillinger FH (1988) Supercooled liquids, glass transitions and the Kauzann paradox. J Che Phys 88:7818 7825 4. urnbull D (1950) Foration of crystal nuclei in liquid etals. J Appl Phys 21:1022 1028 5. Paul A (1982) Cheistry of glasses. Chapan and Hall, London 6. Singh HB, Holz A (1983) Stability liit of supercooled liquids. Solid State Coun 45:985 988 7. Dubey KS (2010) herodynaic and viscous behaviour of glass foring elts and glass foring ability. AIP Conf Proc 1249:211 232 8. Singh PK, Dubey KS (2012) herodynaic behaviour of bulk etallic glasses. herochi Acta 530:120 127 9. Jones D, Chadwick G (1971) An expression for the free energy of fusion in the hoogeneous nucleation of solid fro pure elts. Philos Mag 24:995 998 10. Mondal K, Chatterjee UK, Murty BS (2003) Gibb s free energy for the crystallization of glass foring liquids. Appl Phys Lett 83:671 673 11. Patel A, Pratap A (2010) Study of therodynaic properties of Pt 57.3 Cu 14.6 Ni 5.3 P 22.8 bulk etallic glass. AIP Conf Proc 1249:161 165 12. hopson CV, Spaepen F (1979) On the approxiation of the free energy change on crystallization. Acta Metall 27:1855 1859 13. Hoffan JD (1958) herodynaic driving force in nucleation and growth processes. J Che Phys 29:1192 1193 14. Ji X, Pan Y (2007) Gibbs free energy difference in etallic glass foring liquids. J Non-Cryst Solids 353:2443 2446 15. Li PY, Wang G, Ding D et al (2013) Characterizing therodynaic properties of i-cu-ni-zr bulk etallic glasses by hyperbolic expression. J Alloys Copd 550:221 225
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