Appendix 1: Quick Reference to Most Frequently Used Important FVM Related Equations

Transcription

1 Appendix 1: Quick Reference to Most Frequently Used Important FVM Related Equations Chapter 3 Equation of Uhlmann for the time needed for a small volume fraction ξ to crystallize: 8 9 t ¼ 9:3ηðTÞ < a o ξ exp 1:024 = T 3 r ΔT3 r kt : f 3 N v 1 exp ΔH mδt r 3 ; 0:25 ; (3:1) where ΔT r = (T m T), T r = T/T m,a o is the average atomic diameter, N v is the number of atoms per unit volume, f is the fraction of sites at the melt/crystalline interface where atoms are preferentially added or removed, and ΔH m is the molar enthalpy of fusion. Critical cooling rate for vitrification is _T cr ¼ ðt m T n Þ=t n ; (3:2) where T n and t n are the temperature and the time at the nose of the timetemperature-transformation diagram. Arrhenian type of viscosity temperature dependence: η ¼ η o exp Q a : (3:3) Vogel-Fulcher-Tammann type of viscosity temperature dependence: η ¼ η VFT B o exp : (3:4) T T o where η VFT o and B are empirical constants and T o is known as ideal glass transition temperature. Newtonian relation of homogeneous viscous flow: # Springer-Verlag Berlin Heidelberg 2016 K. Russew, L. Stojanova, Glassy Metals, DOI /

2 248 Appendix 1: Quick Reference to Most Frequently Used Important FVM... η ¼ τ _e ; (3:5) where _e is the strain rate of the specimen under applied shear stress τ. Chapter 4 Einstein equation for flow of mixtures: η eff ¼ ηð1 þ 2:5ςÞ: (4:1) where ζ is the volume fraction of the suspended particles, η o is the viscosity of the viscous medium, and η eff is the viscosity of a mixture consisting of a small volume fraction of spherical particles suspended in the viscous medium. Chapter 5 Probability for appearance of a void of volume υ is Pðυ 0 Þ ¼ γ exp γυ0 ; (5:1) υ f υ f where γ is geometric overlap factor, which value is between 0.5 and 1. The full probability P(υ*) for appearance of a void of a volume greater than υ* is ð / Pðυ Þ ¼ P υ ð Þdυ ¼ exp γυ =υ f. This probability is called concentration of v structural defects c f : c f ¼ exp γυ ¼ exp 1 ; (5:2) υ f x where x ¼ υ f γυ is the so-called reduced free volume. Quasi-equilibrium atomic free volume: υ f,eqðtþ ¼ x eq ðtþγυ ¼ T T o B where B and T o are two model parameters. The rate of tangential deformation (viscous flow): γυ ; (5:3)

3 Appendix 1: Quick Reference to Most Frequently Used Important FVM _e ¼ c f Ω υ oe o k r ; (5:4) where c f is the concentration of structural flow, υ o is the volume of the flow defect, and k r is a frequency factor of atomic rearrangements in the vicinity of flow defects. Temperature dependence of k r : k r ¼ ν r exp Q f kt sinh τe oυ o kt ; (5:5a) where ν r is vibrational (attempt) frequency and Q f is the activation energy for atomic jump over the potential barrier. At << 1, Eq. 5.5a reduces to The viscosity η: The product (e o υ o ): k r ¼ ν r τe o υ o kt τe o υ o kt exp Q f : (5:5b) Kt η ¼ exp Q f ktω 1 kt ðe o υ o Þ 2 : (5:6) ν r c f General form of the viscosity temperature dependence: The hybrid equation for quasi-equilibrium viscosity is e o υ o ¼ Aexp Q s : (5:7) kt η ¼ η o Texp Q n 1 : (5:8) η eq ¼ η o Texp Q n B exp : (5:9) T T o Time dependence of the concentration c f of flow defects: c f dc f dt ¼k rc 2 f ; (5:10) where k r ¼ ν r exp Q r and Q r is the activation energy of annihilation. Equation 5.10 supposes that the process of defect annihilation is a bimolecular reaction:

4 250 Appendix 1: Quick Reference to Most Frequently Used Important FVM... dc f dt ¼k rc 2 f þ k p P; (5:11) where Р is a constant and k p is the rate constant of defect production process. Objectively, it should be equal to k r. When the system comes to equilibrium: dc f dt ¼k r c 2 f c 2 f,eq : (5:12) Supposing that the production of defects occurs at already existing defects: dc f dt ¼k rc f c f c f, eq : (5:13) Supposing that the sites for annihilation and production of defects depend upon the number of defects in excess: τexp Qn eðþ¼ t η o T dc f dt ¼k r c f c f,eq cf c f,eq : (5:14) c f, eqt þ 1 ln 1 c f, o c f,eq k r c f,o 1 exp k r c f, eqt ln c f, eq k r c f,o : (5:15) Under continuous heating conditions with a constant heating rate q, the differential describing the concentration change of structural flow defects as a function of the isothermal annealing time: where PT, ð qþ ¼ ν r q exp Q r Its solution is cf,high 1 T,q dc f dt þ PT ð Þc f ¼ c 2 f QT B TT 0 ð Þ; (5:16) and QT, ð qþ ¼ ν r q exp Q r ð T ð θ ð T B ð Þ¼ c1 f,0 B Qðθ,qÞexp@ Pðθ 00,qÞdθ 00 Adθ 00 C B Aexp Pðθ 0,qÞdθ 0 A: T 0 T 0 T 0 (5:17) The parameter Т о is considered as the starting temperature of heating. The combination of Eq. 5.8 with Eq represents the FVM description of glassy alloys viscosity temperature dependence under continuous heating conditions with a constant heating rate q. At temperatures, considerably lower than the glass transition temperature Т g,

5 Appendix 1: Quick Reference to Most Frequently Used Important FVM dc f dt ¼k rc f c f c f,eq ffikr c 2 f : (5:18) Its solution for nonisothermal annealing conditions with a constant heating rate q is f,low ðt, qþ ¼ c1 f, 0 þ ν ð T r Q exp r q 0 dt 0 ; (5:19) T B c 1 where Т B is the starting temperature of heating. At temperatures, considerably lower than the glass transition temperature Т g, 2 ηðt, qþ ¼ η 0 Texp Q η f, 0 þ ν ð T r Q exp r q 0 dt 0 7 5: (5:20) T B c 1 Chapter 6 Classical definition of Angell s melt fragility number: 2 3 m A ¼ 4 dð logη Þ : (6:1) 5T¼Tg d T g T FVM definition of Angell s melt fragility number: " BT g m A ¼ 0:434 2 þ Q # η 1 : (6:2) T g T o g Moynihan s interpretation of the melt fragility number: ΔT g T g ffi 2 m M ; (6:3) where ΔT g ¼ T g T on ; Т g is the glass transition temperature, and Т on is the onset temperature of crystallization of the glass forming material. FVM interpretation of Moynihan s melt fragility number:

6 252 Appendix 1: Quick Reference to Most Frequently Used Important FVM... " m 0 M ¼ 0:434 BT on T g T o ð Ton T o Þ þ Q n 2 1 # μ 1 : (6:4) R T g T on Chapter 7 The maximal value of bending stress at the surface of the ribbon: σ o ¼ E d ; (7:1) 2r o where Е denotes the Young modulus and d is the ribbon thickness. The initial overall (elastic only) deformation: Under isothermal conditions: e о ¼ σ о E ¼ d 2r o (7:2) e 0 ¼ σðþ t E þ e f ðþþe t a ðþ t (7:3a) where σ(t)/e is the elastic deformation, e f (t) is the plastic deformation, and e a (t) is the anelastic deformation. For nonisothermal experimental conditions: e 0 ¼ σ ð T F, qþ þ e f ðt F, qþþe a ðt F, qþ (7:3b) E where e(t F,q)/E is the elastic deformation, e f (T F,q) is the plastic deformation, e a (T F,q) is the anelastic deformation, T F is the final temperature of heating, and q is the heating rate. The experimentally determined ratios under isothermal and nonisothermal conditions are e f ðt 1 Þþe a ðt 1 Þ e o ¼ r o r 1 (7:4a) and e f ðt F1, q 1 Þþe a ðt F1, q 1 Þ e o ¼ r o r 1, respectively (7:4b) The dependence of bend stress relaxation σ/σ o is

7 Appendix 1: Quick Reference to Most Frequently Used Important FVM σðt 1 Þ ¼ 1 r o, and σ o r 1 σðt F1, q 1 Þ σ o ¼ 1 r o r 1 : (7:5a) (7:5b) The irreversible bend strain caused by the viscous flow: e f t e o ðþ E ¼ 3η o exp Q η ð t о 1 r o dt 0 r 1 ðt 0 Þ t 0 : (7:6a) For nonisothermal constant heating rate conditions: e f e f ðt F, qþ ¼ 1 q ðt F, qþ ¼ E q e o ðt F T B ðt F T B σðt 0, qþ 3ηðT 0, qþ dt0, and 1 r o r 1 1 3ηðT, qþ dt; (7:6b) where Т F denotes the maximal temperature of annealing and T B denotes the starting temperature of the nonisothermal heat treatment. By isothermal or nonisothermal annealing, respectively, the deformations due to the viscous flow only are e f ðt 1 Þ ¼ r o and (7:7a) e o r 2 e f ðt F1, q 1 Þ e o ¼ r o r 2 : (7:7b) The anelastic part of bend deformation is e a ðt 1 Þ 1 ¼ r o 1 e o r 1 r 2 e a ðt F1, q 1 Þ 1 ¼ r o e o The time derivative of Eqs. 7.3a and 7.3b is r 1 1 r 2 and (7:8a) : (7:8b)

8 254 Appendix 1: Quick Reference to Most Frequently Used Important FVM... dσðþ t dt ¼E _e f ¼E σðþ t 3ηðÞ t or (7:9a) dσðt F, qþ ¼E _e f ¼E σ ð T F, qþ dt 3ηðT F, qþ : (7:9b) The integration of Eq. 7.9a for isothermal experimental conditions results in ln σðþ t E ¼ σ o 3k r η o Texp Q ln 1 þ k r c f,ot ; (7:10a) η where k r = ν r exp(q r /) denotes the rate constant of relaxation. For nonisothermal conditions of the bend stress relaxation experiment under constant heating rate q, the solution of Eq. 7.9b is ln σ T ð F, qþ ¼ E σ o 3qη o ðt F T B 2 3 c 1 f,o þ ν ð T 6 r Q exp r q 0 dt T B 1 Texp Q 1 η dt (7:10b) where T B and Т F are the starting temperature of heating and the maximal temperature. The ratio is σ 2 ðþ t ¼ 1 r o and (7:11a) σ o r 2 σ 2 ðt F, qþ σ o ¼ 1 r o r 2 ; (7:11b) where r 2 denotes the sample radius of curvature after the second stress-free annealing of the amorphous sample. The empirical equation of Kohlrausch-Williams-Watts (KWW) describing the anelastic deformation is " e a ðþ t ¼ exp t # b ; (7:12) e o τ r where τ r ¼ τ o exp Q r : (7:13)

9 Appendix 1: Quick Reference to Most Frequently Used Important FVM Under nonisothermal experimental conditions, equation of Kohlrausch- Williams-Watts is " # e a ðt, qþ ðt T B Þ=q b ¼ exp ; (7:14) τ o expðq r =Þ e o where Т В is the starting temperature of annealing and q is the heating rate. As a function of the time t and the heating rate q used 2 0 e a ðt, qþ ¼ exp4 h e o τ o exp Q r RqtþT ð B Þ 1 3 b ia 5: (7:15) with fraction exponent 0.1 < b < 1. Chapter 9 Low temperature linear temperature dependence of sample length: L 0 ðtþ ¼ L 0 ðt B Þ 1 þ α 0 l ðt T B Þ ; (9:1) where Т B is the starting temperature of heating, L o (T B ) is the initial sample length at Т B, α l o is the coefficient of linear thermal expansion in the low-temperature range Т B T o, and T o is the ideal glass temperature. Deviation from the real temperature dependence of the sample length: ΔL f ðt, qþ ¼ L f ðt, qþl 0 ðtþ: (9:2) Taking into account that the x, Refs. 2 5, correlates to the real mean free volume υ f according to the expression x = υ f /γυ*, change of the reduced free volume along with increasing temperature: xt, ð qþx 0 ¼ L3 f ðt, qþl 3 0 ðtþ γυ ; (9:3) N where N is the number of atoms in a cube of the glassy alloy studied with edge length L 0 (T B ) at temperature T = T B. N ¼ L3 0ð T BÞ N A ; (9:4) V mol

10 256 Appendix 1: Quick Reference to Most Frequently Used Important FVM... When expanding Eq. 9.3 into a power series, one obtains Or V mol xt, ð qþx 0 ffi 3L 2 0 ðtþδl f ðt, qþ γυ N A L 3 0ð T BÞ : (9:5) 3L 2 0 ðtþδl V mol f ðt, qþ γυ N A L 3 0ð T BÞ ¼ 1 1 lnc f,0 lnc f ðt, qþ ; (9:6) where c f (T,q) is defined by Eq The temperature dependence for the experimentally observed sample length by heating the sample with constant heating rate q is (a) T < T o : (b) T > T o : L f ðt, qþ ¼ L 0 ðtþ ¼ L 0 T B 1 þ α 0 l ðt T B Þ ; (9:7) L f ðt, qþ ¼ L 0 ðt B Þ 1 þ α 0 γυ N A L 0 ðt B Þ l ðt T B Þ þ 3V mol 1 þ α 0 2 x 2 l ðt T B Þ ; (9:8) x þ 0 ð lnc f, θ 1 0 ð 1 T Pðθ 0, qþdθ 0 ð T 6 lnb Qðθ, qþe T 0 dθc c f, A þ Pðθ 0, qþdθ T 0 T 0 where the functions P(Т,q) and Q(T,q) were defined in Chap. 5. The anomaly of thermal expansion γυ N A L 0 ðt B Þ ΔL f ðt, qþ ¼ 3V mol 1 þ α 0 2 x 2 l ðt T B Þ x þ 0 ð lnc f, θ 1 : 0 ð 1 T Pðθ 0, qþdθ 0 ð θ 6 lnb Qðθ, qþe T 0 dθc c f, A þ Pðθ 0, qþdθ T 0 T 0 (9:9) The anomalous contribution ΔC p to the specific heat C p is presented as

11 Appendix 1: Quick Reference to Most Frequently Used Important FVM ΔC p ðt, qþ ¼ β dx dt ; (9:10) where β is a material-specific coefficient of proportionality. In the temperature range considerably lower than the glass transition temperature T g, ΔC p, lowðt, q Þ ¼ β c f, lowðt, qþqt, ð qþ 2 : (9:11) lnc f,lowðt, qþ In the temperature range in the vicinity of the glass transition temperature T g, ΔC p, highðt, q Þ ¼ β c f,highðt, qþqt, ð qþpt, ð qþ 2 : (9:12) lnc f,highðt, qþ The functions P(Т,q) and Q(T,q) were defined in Chap. 5. Chapter 10 The change of structural defects concentration along with increasing the time of annealing at constant temperature T is dc f ðt, tþ ¼k r c f c f c f, e ; (10:5) dt where k r ¼ ν r exp Q r is the rate constant of relaxation, c f,e (T) is the equilibrium defect concentration, and R is the universal gas constant. Its solution is c f ðt, tþ 1 ¼ c 1 f, e þ c1 f,o c1 f,e exp k r c f, et ; cf ðt ¼ 0sÞ ¼ c f,0: (10:6) c f,o is here the defect concentration at the start of the isothermal heat treatment. The temperature dependence of c f,e (T) is c f,eðtþ ¼ exp B ; (10:7) T T o where B and T o are constants. The density change of a cube of initial edge length L o, caused by nonisothermal or isothermal structural relaxation, is presented as

12 258 Appendix 1: Quick Reference to Most Frequently Used Important FVM... W ΔρðT, qþ ¼ ρðt, qþρ o ¼ 3 W L o þ ΔL f ðt, qþ L 3 o ¼ 3W ΔL f ðt, qþ=l o L 3 o 1 þ 3ΔL ; (10:8) f ðt, qþ=l o ΔρðT, qþ ¼ 3 ΔL f ðt, qþ=l o ; (10:9) ρ o 1 þ 3 ΔL f ðt, qþ=l o under nonisothermal relaxation conditions, and ΔρðÞ t ¼ 3 ΔL f ðt, tþ=l o ; (10:10) ρ o 1 þ 3 ΔL f ðt, tþ=l o under isothermal relaxation conditions. Chapter 11 Crystal growth rates G: G ¼ fkt ½ 3πa o η 1 exp ð ΔG m=þ; (11:1) where η denotes the coefficient of viscous flow responsible for atomic motion required for growth, a o is the mean atomic diameter, ΔG m is the molar free energy change, and f is the fraction of sites at the interface where crystal growth occurs. General form of the equation of Kolmogorov, Avrami, Johnson, and Mehl: ζ ¼ 1 expðv e Þ; (11:8) where ξ ¼ V cr =V o is the volume fraction of crystallized regions, V cr is the overall volume of crystallized regions, V o is the total volume of the sample, and V e is the so-called extended volume of growing crystalline particles per unit volume of the matrix. The extended volume of growing crystalline particles is V e ¼ 4 3 πnt ðþg3 ðtþt 3 ; (11:9) where t is the time of crystallization and G(T) is the crystal growth rate at a temperature T.

13 Appendix 1: Quick Reference to Most Frequently Used Important FVM Arrhenian type equation of the crystal growth rate: GT ð Þ ¼ G o exp Q eff ; (11:10) where G o is a pre-exponential factor and Q eff is the effective activation energy of crystallization. By nucleation site saturation: ζ ¼ 1 exp KðTÞt 3 ; (11:11) where K(T) = NG 3 (T). Its temperature dependence can be presented as lnðkt ð ÞÞ ¼ ln NG 3 Q eff o 3 : (11:12) Chapter 12 The energy density per unit volume: τ γ γ o ¼ ðt g T o ρc p dt; (12:1) where τ γ is the maximal stress of share deformation yielding deformation γ o 1 of the basic share unit, ρ is the density of the material, C p is the molar specific heat, T g is the glass transition temperature, and Т о is the ambient temperature. The creep strength equals The material density: σ γ ¼ 2τ γ : (12:2) ρ ¼ ½1 3αðT T o Þ ; (12:3) where ρ o is the density at ambient temperature and α is the coefficient of thermal expansion. ρ o

14 260 Appendix 1: Quick Reference to Most Frequently Used Important FVM... Equation of B. Yang for the fracture strength: σ f ¼ 55 ΔT g ; (12:4) V m where σ f is the fracture strength, ΔT g is the temperature difference T g T o, and V m is the molar volume of the glassy alloy.

15 Appendix 2: Concluding Remarks In this book, we have tried to make use of the free volume theory of Cohen and Turnbull to give an insight into the relaxation phenomena in glassy metals around the glass transition temperature. It was taken into account that the glassy metals are a particular kind of condensed matter, i.e., matter in the amorphous state. The free volume model (FVM) describes the atomic mobility in liquids, undercooled melts, and amorphous solids via the free volume available for a single atom within the material structure. By rapid quenching from the melt into the structure of glassforming metallic melts, non-equilibrium excess free volume is frozen in. The structural relaxation of a non-equilibrium amorphous structure with concentration of structural defects higher than the quasi-equilibrium concentration of structural defects is identified as structure, striving to reach the quasi-equilibrium for a given temperature structural state, via annihilation or production of free volume. These free volume changes along with the temperature/time changes are, of course, not instantaneous, but possess a definite mechanism and kinetics, about which the free volume theory of Turnbull and Cohen does not propose any indications and which should be specified experimentally. The authors have endeavored to collect most of the available experimental evidence from the bibliography and from own experimental studies upon the viscous flow behavior, thermal expansion and specific heat peculiarities, bend stress relaxation behavior, density changes, etc. of amorphous metallic alloys, showing that the relaxation of glassy metals around the glass transition temperature proceeds via annihilation and/or production of structural defects as carriers of free volume. The kinetics of this process, both under isothermal and non-isothermal experimental conditions, seems to be best described as a kinetics of a bimolecular reaction. The authors believe that their attempt to combine the FVM basic concepts with the proper kinetic equations for description of free volume annihilation and/or production in the amorphous metallic alloys was successful and convincing. It describes well the glassy metal property changes caused by relaxation processes during continuous heating and/or cooling. This approach provides a useful tool for understanding the relaxation phenomena in glassy metals. The authors recommend to the studious reader to pay a special attention to the developed in this work method for the study of the bend stress relaxation phenomena in ribbonlike glassy metals, especially to the experimental possibility to # Springer-Verlag Berlin Heidelberg 2016 K. Russew, L. Stojanova, Glassy Metals, DOI /

16 262 Appendix 2: Concluding Remarks separate the fully irreversible bend stress phenomena caused by the viscous flow from the fully reversible anelastic bend stress deformation. The last one can be described very well by the empirical stretch exponent equation of Kohlrausch- Williams-Watts. In this way the possibility arises to open a new window for future studies upon the interrelation between the irreversible relaxation phenomena and the fully reversible relaxation processes in glassy metallic alloys. The nature and mechanism of the reversible anelastic deformation in glassy metals are still not well understood and studied. Last but not least, the authors of this book would like to point out that they share the belief and conviction of the famous Austrian physicist and philosopher Ludwig Boltzmann, namely, that the task of theoretical modeling consists in constructing a picture of the external world that exists purely internally and must be a guiding star for the researcher: It follows that it cannot be our task find out an absolutely correct theory but rather a picture, which is as simple as possible while representing the phenomena as well as possible The FVM of Cohen and Turnbull seems to be such a picture.