Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN DUC HIEN STUDY ON ELASTIC AND NONLINEAR DEFORMATION OF METALS, BINARY AND TERNARY INTERSTITIAL ALLOYS Speciality: Mathematical and Theoretical Physics Code: 9.44.01.03 SUMMARY OF PhD THESIS HANOI – 2023 This work is completed at Hanoi National University of Education Scientific supervisor 1: Assoc Prof PhD Nguyen Quang Hoc Scientific supervisor 2: Assoc Prof PhD Hoang Van Tich Reviewer 1: Prof PhD Bach Thanh Cong – University of Science, Vietnam National University, Hanoi Reviewer 2: Prof PhD Vu Van Hung – University of Education, Vietnam National University, Hanoi Reviewer 3: Assoc Prof PhD Nguyen Hong Quang – Institute of Physics - Vietnam Academy of Science and Technology The thesis will be defended at the University-level thesis evaluation Council at Hanoi National University of Education at the hour date month year The thesis is available at - Hanoi National Library - Library of Hanoi National University of Education INTRODUCTION Reasons for choosing the topic Metals and alloys are materials with a long history of development and are widely used in industries and practical daily life There are two types of alloys as substitutional alloys and interstitial alloys Steel is a typical form of exportation with special importance in construction, transportation, machine manufacturing An interstitial alloy Fe is FeC It is called carbon steel and accounts for a large proportion in the steel industry Fe and its interstitial alloys such as FeSi, FeC, FeH make up most of the Earth's core and celestial bodies Their thermodynamic, elastic and melting properties provide us with information about the composition, structure, evolution of the Earth, celestial bodies Information about metals such as Au, Cu, Fe, Cr and some of their alloys are especially important for designing and manufacturing equipments to serve the needs of human life The study of mechanical properties including the deformation process of materials such as metals and alloys, including interstitial alloys, has attracted the attention of many researchers, both theoretically and experimentally These research results play a very important role in science, technology and human life Statistical moment method (SMM) has been successfully applied to study the deformation of metals and binary substitutional alloys However, the study of elastic and nonlinear properties of cubic ternary and binary interstitial alloys by SMM is still an open problem For the above reasons, we choose the topic “Study on elastic and nonlinear deformation of metals, binary and ternary interstitial alloys” Purpose, object and scope of research The purpose of the thesis is to develop SMM to study elasticnonlinear deformation, effect of strain on diffusion for ternary and binary interstitial alloy taking into account the influence of temperature, pressure, concentrations of substitutional and interstitial atoms The object of the study is the elastic-nonlinear deformation, the effect of the strain on the diffusion of cubic ternary and binary interstitial alloy Research scope: Studied temperature range from zero to the melting temperature of the main metal; the studied substitutional atomic concentration range from zero to 10%; the studied interstitial atomic concentration range is from zero to 5% and the studied pressure range is from zero to 100 GPa Research methods The main research method is the statistical moment methold (SMM) In addition, in numerical calculations we use the Maple sofware and approximate methods Scientific and practical significance The results from the thesis provide the information about the nonlinear-elastic deformation and the influence of the deformation on the diffusion for metals and alloys The thesis contributes to the development of SMM in studying properties of materials in general and in studying the deformation and the influence of the deformation on the diffusion for interstitial alloy materials in particular Some numerical results of the thesis can be references for forecasting and orienting experimental studies New contributions of thesis Build analytic expressions for nonlinear and elastic deformation quanties of cubic ternary and binary interstitial alloy using SMM The thesis contributes to supplementing and perfecting the deformation theory and cubic interstitial alloy Apply the obtained theory to calculate numerically for some metals and alloys The obtained numerical results are compared with the experimental data and the results calculated by other theoretical methods Some numerical results can be predictive and guide future experiments Thesis outline In addition to the Introduction, the Conclusion, the References and the Appendix the thesis content is presented in chapters The content of the thesis has been reported at national and international specialized scientific conferences and published in articles in domestic and international scientific journals CHAPTER 1: OVERVIEW ON DEFORMATION OF METALS AND ALLOYS 1.1 Interstitial alloys 1.2 Deformation theory 1.2.1 Elastic deformation Under the action of an external force, a solid is deformed, that is, it changes shape and size When deformed, the points or atoms of the solid move Elastic deformation is the deformation of a solid object under the action of an external force that, after unloading, the solid body returns to its original shape and size 1.2.2 Nonlinear deformation Nonlinear strain (inelastic strain, residual strain) is the deformation of a solid body under the action of an external force, but after unloading, the deformation is not lost and the solid body does not return to shape, size and shape original size That happens when the external force (load) must be large enough Nonlinear strain does not change the deformation volume 1.2.3 Elastic wave in solids The longitudinal and transverse wave velocities have the form Vd = 2C44 + C12 , Vn = C44 (1.28) 1.2.4 Influence of deformation on diffusion The dependence of the diffusion coefficient D on the biaxial tensile stress has the form 2 r m m V + V − V/ / , Dx ( ) = Dx ( = 0) exp (1.31) kBT where Dx ( ) is the diffusion coefficient in the x direction of the system subjected to biaxial stress , V r is the volume of recovery and V/m/ is the volume component displaced in the direction parallel to the direction of diffusion 1.3 Some major methods of research There are many different theoretical methods in studying nonlinear and elastic deformation of metals and alloys such as the harmonic theory, the pseudoharmonic theory, the molecular dynamics method, the finite element method, the calculations from first principles (ab initio), the tight-binding Hamiltonian method, the density functional theory, the machine learning method, the lattice Green function method, the calculation of phase diagram, the modified embedded atomic method, etc Although there are many theoretical methods used to study the properties of objects during elastic and nonlinear deformation process, these methods are still limited in one aspect or the other Most of the methods not take into account the anharmonicity effect in the lattice vibrations, not to mention the effect of pressure on the deformation processes of the object, not to mention the dependence of the deformation quantities on the concentrations of substitutional and interstitial atoms, the results of deformation studies are mostly for metals and limited for alloys, not taking into account the effect of stress or strain on the diffusion in the alloy 1.4 Statistical moment method Kˆ n +1 a = Kˆ n a Qˆ n +1 a + Kˆ n an +1 a − B2m i 2m )! m =0 ( 2m Kˆ n (2m) an +1 (1.57 ) a Conclusion of Chapter In chapter 1, we present an overview of strain theory and methods of studying elastic - nonlinear deformation of metals and alloys, which refers to the theory of elastic deformation, nonlinear strain, and elastic deformation linear, elastic wave velocity and effect of strain on diffusion in the material; Basic content and advantages and disadvantages of popular deformation research methods Introduce a traceability formula that relates higher moments to lower moments and use it to determine the free energy of the system CHAPTER 2: STUDY ON NOLINEAR AND ELASTIC DEFORMATION OF INTERSTITIAL ALLOY AC 2.1 Alloy model and Helmholtz free energy Fig 2.1 Model of interstitial alloy AC with BCC structure (a) and FCC structure(b) AC = c X X − TScAC , (2.1) X 2 Y X = U X + X + X YX2 − 1X 1 + X + k X + 2 k X4 4 X YX 3 YX YX 1 + − 1X + 2 1X X 1 + (1 + YX ) , ( ( ) ) X = 3 x X + ln − e−2 xX , YX x coth x, X = A, C, A1, A2 , (2.2) cA = − 7cC , cA1 = 2cC , cA2 = 4cC for BCC lattice, cA = − 15cC , cA1 = 6cC , cA2 = 8cC for FCC lattice 2.2 Cohesive energy, alloy parameters and mean nearest neighbor distance between two atoms For BCC lattice, u0 A = 4 AA ( r1A ) + 3 AA ( r2 A ) , r2 A = r1 A , (2.4) 2 d AA ( r1A ) d AA ( r1A ) d AA ( r2 A ) kA = + + + 3r1A dr1A dr12A dr22A + d AA ( r2 A ) , r2 A dr2 A (2.5) d AA ( r1A ) d AA ( r1A ) d AA ( r1 A ) d AA ( r1 A ) + + + 2 54 9r1A dr1A dr1A dr1A 9r1A dr1A 9r1A 1A = + 2A = d AA ( r2 A ) d AA ( r2 A ) d AA ( r2 A ) + - , 24 dr2 A dr24A 4r22A dr22A 4r2 A (2.6) d AA ( r1A ) d AA ( r1A ) d AA ( r1A ) d AA ( r2 A ) + + ( 2.7 ) dr1A 2r2 A dr14A 3r12A dr12A 3r13A dr23A u0C = AC ( r1C ) + 2 AC ( r2C ) , r2C = 2r1C , d AC ( r1C ) d AC ( r2C ) d AC ( r2C ) + + , r1C dr1C r2C dr2C dr2C (2.8) kC = C = (1C + 2C ) , 1C = + (2.10) d AC (r1C ) d AC (r1C ) d AC (r2C ) - + + dr1C 48 8r12C dr12C 8r1C dr24C d 3 AC (r2C ) d 2 AC (r2C ) d AC (r2C ) + , 2 8r2C dr2C dr2C 16r2C dr2C 16r23C 2C = + (2.9) (2.11) d 3 AC (r1C ) d 2 AC (r1C ) d AC (r2C ) - + + 4r1C dr1C dr1C 2r1C dr1C 2r1C d 3 AC (r2C ) d 2 AC (r2C ) d AC (r2C ) − + , 2 4r2C dr2C dr2C 4r2C dr2C 4r2C ( ) u0 A1 = u0 A + 3 A1C r1 A1 , k A1 = k A + ( ) d 2 A1C r1A1 dr12A1 + ( (2.13) ( ) d A1C r1A1 , r1 A1 dr1 A1 ) A1 = 1A1 + A1 , 1A1 = 1A + (2.12) (2.14) (2.15) d A1C d A1C − + 24 dr14A 6r1 A1 dr13A 1 + d A1C d A1C − , 2 4r1A1 dr1A1 4r1A1 dr1A1 (2.16) d A1C (r1A1 ) , 4r1 A1 dr13A1 γ 2A1 = A + (2.17) ( ) u0 A2 = u0 A + 6 A2 C r1 A2 , k A2 = k A + ( )+ d 2 A2 C r1 A2 dr12A2 ( (2.18) ( ) d A2 C r1 A2 , r1 A2 dr1 A2 ) A2 = A2 + A2 , 1A2 = 1A + (2.19) (2.20) ( ) d A2 C r1A2 + 24 dr14A 2 15 d A2 C (r1A2 ) 15 d A2 C (r1A2 ) + − , dr1 A2 dr12A2 4r1A2 4r1A2 A2 = A + d A2 C (r1A2 ) d A2 C (r1A2 ) + + 4r1A2 dr14A2 dr13A2 + d A2 C (r1A2 ) d A2 C (r1A2 ) - , dr1 A2 dr12A2 8r12A2 8r1A2 For BCC lattice, u0 A = 6 AA ( r1A ) + 3 AA ( r2 A ) , r2 A = kA = (2.21) d 2 AA ( r1A ) dr12A + 2r1A , (2.22) (2.23) d AA ( r1 A ) d AA ( r2 A ) d AA ( r2 A ) + + , r1A dr1A r2 A dr2 A dr2 A (2.24) 1A = d AA ( r1A ) d AA ( r1A ) d AA ( r1A ) d AA ( r1A ) + - + + 24 4r1A dr1A dr1A dr1A 8r1A dr1A 8r1A + 2A = d AA ( r2 A ) d AA ( r2 A ) d AA ( r2 A ) + - , (2.25) 24 dr2 A dr24A 4r22A dr22A 4r2 A d AA ( r1 A ) d AA ( r1 A ) d AA ( r1 A ) d AA ( r1 A ) + + − + 2 4r1A dr1A dr1A dr1A 8r1A dr1A 8r1A + d AA ( r2 A ) d AA ( r2 A ) d AA ( r2 A ) + (2.26) 2 2r2 A dr2 A dr2 A 4r2 A dr2 A 4r2 A u0C = 3 AC ( r1C ) + 4 AC ( r2C ) , r2C = kC = d 2 AC ( r1C ) dr12C + 3r1C , 2 d AC ( r1C ) d AC ( r2C ) d AC ( r2C ) + + , r1C dr1C 3r2C dr2C dr2C (2.28) (2.29) C = (1C + 2C ) , 1C = + (2.27) d AC (r1C ) d 2 AC (r1C ) d AC (r1C ) d 4 AC (r2C ) + - + + 24 dr1C 54 dr14C 4r12C dr12C 4r1C dr24C d 3 AC (r2C ) d 2 AC (r2C ) d AC (r2C ) + , 9r2C dr2C dr23C 9r22C dr22C 9r2C 2C = (2.30) d 3 AC (r1C ) d 2 AC (r1C ) d AC (r1C ) - + + 2r1C dr1C dr1C 4r1C dr1C 4r1C + d 4 AC (r2C ) d 2 AC (r2C ) d AC (r2C ) + - , 2 dr2C dr2C 3r2C dr2C 3r2C ( ) u0 A1 = u0 A + A1C r1 A1 , k A1 = k A + (2.32) ( ) d A1C r1A1 , r1A1 drr1 A (2.33) ( ) A1 = 1A1 + A1 , 1A1 = 1A + A1 = A + (2.34) d A1C (r1A1 ) d A1C (r1A1 ) - , dr1A1 8r12A1 dr12A1 8r1A1 (2.35) d A1C (r1A1 ) d A1C (r1A1 ) d A1C (r1A1 ) − + , 2 4r1A1 dr1A1 dr1 A1 2r1 A1 dr1 A1 2r1 A1 ( ) uOA2 = uOA + 4 A2 C r1 A2 , k A2 (2.31) ( ) (2.36) (2.37) ( ) d A2 C r1A2 d A2 C r1A2 = kA + + , 3r1A2 dr1 A2 dr12A2 (2.38) r1FX ( P, T ) = r1X ( P, T ) + r1X ( P, 0)( + ) F AC = c − TScACF , F X X (2.62) (2.63) X AC = AC f AC ( ) = c X + X X 1 F − v v AC AC 2 XF F 2v AC AC , 1+ (2.64) F 2 r01 + F X v AC ) f AC ( F ) = f AC max = CAC AC max F (2.70) (2.72) (2.73) f ACmax , C AC F (2.74) AC max f AC max = 1+ F C AC F (1 + F ) (2.75) ACmax = 1AC max = + T 2r01 X , T F F F 2r01 + FX X r r1X T 1X f AC ( ) = CAC AC , ( XF F r1X F = AC − AC lAC max = AC C AC = , ( F ) AC 1+ F f AC ( 0,2 ) , (2.77) , (2.78) e AC , + e (2.80) AC 0,2 0,2 ACe = AC (2.76) e AC (2.81) + e 2.5 Numerical results and discussion for metals and alloys Using the Mie-Lennard-Jones n-m, the Finnis – Sinclair and the Morse potentials The numerical results for Fe, FeSi, FeH, FeC, Au, Cu, and CuSi are summarized in Table 2.2 to 2.10, Table 2.13 to 2.20, illustrated from Figure 2.2 to 2.19 EYAC e = AC 10 7000 6800 6600 Vd (ms-1) 6400 6200 6000 5800 SMM Experiments of Shibazaki at al (2016) Experiments of Antonangeli and Ohtani (2015) Experiments of Decremps at al (2014) Experiments of Liu at al (2014) 5600 5400 5200 5000 10 P (GPa) Figure 2.3 Vd(P) for Fe at T = 300 K calculated by SMM and from experiments of Antonangeli (2015)[11], Decremps (2014)[25], Liu (2014)[80] Shibazaki (2016)[111] According to Table 2.3 for FeSi at P = and the same temperature when increasing Si concentration, the elastic modulus, elastic constant and elastic wave velocity all decrease sharply For example, for FeSi at P = and T = 1000K with increasing Si concentration from to 5%, E decreases 65.48% For FeSi at P = and the same Si concentration, with increasing temperature, the elastic modulus, elastic constant and elastic wave velocity all decrease For example, for FeSi at P = and cSi = 5% with increasing temperature from 100 to 1000K, E decreases 61.88% According to Table 2.4 for FeSi at T = 300K and the same Si concentration with increasing pressure, the elastic modulus, elastic constant and elastic wave velocity all increase For example, for FeSi at T = 300K and cSi = 5% with increasing pressure from to 10GPa, E increases 35.85% Table 2.7 EY(cC) (GPa) for FeC at P = and T = 300 K calculated by SMM and from experiments [115] E c cC (%) 1,4 2,3 (1010Pa/%) SMM 208.2 198.5 194.6 188.8 185.9 -0.6309 Experiments[115] 208.2 204.8 201.5 197.9 193.8 -0.6180 δ (%) 3.1 3.4 4.6 4.1 2.1 11 Bảng 2.8 EY(cH) (GPa) for FeH at P = and T = K calculated by SMM and ab initio of Psiachos at al (2011)[104] cH (%) PPTKMM 222.8 216.9 211.1 205.4 199.8 194.2 ab initio[104] 229.2 225.9 222.6 219.4 216.1 212.8 (%) 2.79 3.98 5.17 6.38 5.01 8.74 180 PPTKMM Experiments of Santra at al (2014) Experiments of Ledbetter and Naimon (1974) 160 EY (GPa) 140 120 100 80 cSi(%) Figure 2.11 EY (cSi ) for CuSi at T = 300K and P = calculated by SMM and from experiments of Ledbetter and Naimon (1974)[75], Santra at al (2014)[109] 18 1200 16 cSi = cSi = 2% cSi = 5% 14 800 1(MPa) f(GPa) 12 1000 10 600 400 Fe, SMM Fe, XEXPT of Smith et al.(2020) Fe, DEXPT of Smith et al.(2020) FeSi2%, SMM FeSi5%, SMM 200 (a) 0 0 7 (%) (%) Figure 2.13 (a) f ( , cSi ) and (b) 1 ( , cSi ) for FeSi at T = 300K, P = calculated by SMM and from experiments of Smith at al (2020)[113] Conclusion of Chapter 12 In chapter 2, we build the theory of elastic and nonlinear strain of interstitial alloy AC with cubic structure in which general analytic expressions of free energy, displacement of particles from lattice are drawn, mean nearest neighbor distance between two atoms, elastic deformation quantities such as isothermal and adiabatic elastic moduli, Young's modulus, mass compression modulus, shear modulus, elastic constants, longitudinal wave velocity, transverse wave velocity, nonlinear strain quantities such as strain energy density, maximum real stress, elastic strain limit to determine stress - strain curve Analytical results of the resulting strain-specific quantities depend on the temperature, pressure and the interstitial atom concentration The obtained theoretical results are numerically calculated for metals Fe, Au, Cu and interstitial alloy FeSi, FeH, FeC, AuSi, CuSi The law of deformation depends on temperature and pressure of interstitial alloy is consistent with the law of deformation depending on temperature and pressure of the main metal in interstitial alloy Young E modules, mass compression modulus K, shear modulus G, elastic constants C11, C12, C44 and elastic wave propagation velocities Vd, Vn of metal and interstitial alloy decrease with increasing temperature and increase with increasing temperature pressure increase The elastic strain quantities of interstitial alloy all decrease with the increase of interstitial atom concentration The maximum real stress and elastic strain limit of metal and interstitial alloy decrease with the increase of temperature and increase with the increase of pressure The maximum real stress and elastic strain limit of interstitial alloy decrease with the increase of interstitial atom concentration Many numerical results obtained by SMM, especially elastic and nonlinear deformation of metals, have been compared with numerical results by other theoretical methods and experimental data and have some very consistent results good Many numerical results obtained by SMM on elastic and nonlinear strain of interstitial alloy are new, predictive, experimentally oriented 13 CHAPTER 3: STUDY ON NOLINEAR AND ELASTIC DEFORMATION OF INTERSTITIAL ALLOY ABC 3.1 Alloy model Fig 3.1 Model of substitutional and interstitial alloy ABC with BCC structure (a) and FCC structure(b) 3.2 Helmholtz free energy ABC = AC + cB ( B − A ) + TScAC − TScABC , 3.3 Mean nearest neighbor distance a ABC = c AC a AC BTAC B + cB aB TB , BT = cAC BTAC + cB BTB BT BT (3.1) (3.2) 3.4 Elastic deformation EYABC = EYAC + cB ( EYA − EYB ) , (3.9) Determining Young's modulus, we can also determine other elastic modules, elastic constants and elastic wave propagation speed according to formulas (3.12) to (3.19) 3.5 Nonlinear deformation (3.19) r1FX ( P, 0) = r1X ( P, 0)(1 + ) , r1FX ( P, T ) = r1X ( P, T ) + r1X ( P, )( + ) , ( ) F F ABC = AC + cB BF − AF + TScACF − TScABCF , ABC = ABC ABC , 1+ 14 (3.20) (3.21) (3.22) f ABC ( ) = X + 2 F 2v ABC c X X 2 F F X2 r1 X 1 F − v ABC v ABC F 2r01 X T ( ) −cB A F − v v ABC ABC + 2 F 2v ABC 2 r01F X + F v ABC F + FX r 1X F 2 r01 + F A v ABC AF F r1A + T 2r01 A , T ) f ABC ( F ) = f ABC max = CABC ABC max F (3.26) (3.27) (3.28) f ABCmax , C ABC F (3.29) ABC max f ABC max = 1+ F C ABC F (1 + F ) (3.30) ABCmax = 1ABC max = + 2r01 X − T 2 F F F A2 2r01F A + FA r r1A T 1A f ABC ( ) = CABC ABC , ( XF F r1X F = ABC − ABC AC 1+ F f ABC ( 0,2 ) , (3.32) , (3.33) e ABC , + e (3.35) ABC 0,2 0,2 ABCe = ABC (3.31) ( F ) lABC max = ABC C ABC = , e ABC (3.36) + e 3.6 Numerical results and discussion for alloys The numerical results for FeCrSi, AuCuSi are summarized in Table 3.1 to Table 3.18 and illustrated from Figure 3.3 to Figure 3.10 EYABC e = ABC 15 340 240 320 220 300 280 200 K (GPa) 240 220 200 180 160 140 180 SMM Ab initio of Zhang (2010) Ab initio of Olsson (2003) Experiments of Heintze (2009) Experiments of Speich (1972) 160 140 (a) 120 SMM Ab initio of Zhang (2010) Ab initio of Olsson (2003) Experiments of Heintze (2009) Experiments of Specich (1972) 120 (b) 100 100 80 10 cCr (%) cCr (%) Figure 3.2 (a) E ( cCr ) and (b) K ( cCr ) for FeCr at T = 298K, P = calculated by SMM, ab initio of Olsson [100], ab initio of Zhang [131] and from experiments of Heintze [40] and Specich [115] 35 cSi = cSi = 1% cSi = 3% cSi = 5% 30 EY (1010 Pa) EY (GPa) 260 25 20 15 10 10 20 30 40 50 60 70 P (GPa) Figure 3.3 EY(P, cSi) of AuCuSi at cCu = 10% and T = 300K 16 10 18 750 16 cSi=0 cSi=2% cSi=5% 14 700 f(GPa) f(GPa) 12 10 650 600 cSi=0 cSi=2% cSi=5% 550 (a) 2 (b) 500 (%) Figure 3.4 (a) f ( , cSi ) and (b) (%) ( , cSi ) for FeCrSi at cCr =10%, T = 300K and P = 25 800 P=0 P = 2,55 GPa P = 4,88 GPa P = 9,47 GPa P = 18,78 GPa 20 P=0 P = 2,55 GPa P = 4,88 GPa P = 9,47 GPa P = 18,78 GPa 700 600 500 f(GPa) 1(MPa) 15 10 400 300 200 100 (a) (b) 0 10 12 10 12 (%) (%) Figure 3.9 (a) f ( , P) and (b) ( , P ) for AuCuSi at cCu =10%, cSi = 1% T = 300K Conclusion of Chapter In chapter 3, we build the theory of elastic-nonlinear deformation of substitutional and interstitial ternary alloy ABC with the cubic structure, which derives the general analytic expressions of free energy, magnitude displacement of the particle from the lattice, average nearest neighbor distance between two atoms, elastic deformation quantities such as isothermal and adiabatic elastic moduli, Young's modulus, mass compression modulus, shear modulus, 17 constants elastic, longitudinal wave velocity, transverse wave velocity, nonlinear strain quantities such as strain energy density, maximum real stress, elastic strain limit to determine the stress - variation curve form The analytical results of the strain-specific quantities obtained depend on the temperature, pressure, substitution atom concentration and interstitial atom concentration The obtained theoretical results are numerically obtained for the alloy FeCrSi and AuCuSi The law of temperature and pressure dependent deformation of the trigonal alloy is consistent with the law of temperature and pressure dependent deformation of the main metal in alloy The law of interstitial atom concentration dependent deformation of trigonal alloy is consistent with the interstitial atom concentration dependent deformation law of the corresponding binary alloy The substitution atomic concentration dependent strain law of trigonal alloy is consistent with the substitution atomic concentration dependent strain law of the corresponding binary alloy Elastic deformation quantities such as Young's modulus E, mass compression modulus K, shear modulus G, elastic constants C11, C12, C44 and elastic wave propagation velocities Vd, Vn of trigonal alloy decrease with increases in temperature, increases with increasing pressure, decreases with increasing concentration of interstitial atoms, and increases slowly with increasing concentration of substituted atoms The maximum real stress and elastic strain limit of trigonal alloy decrease with the increase of temperature, increase with the increase of pressure, decrease with increasing concentration of interstitial atoms, and decrease slowly with the increase of concentration alternative etymology Some numerical results obtained by SMM for triplicates are compared with numerical results by SMM for corresponding binary options, including comparison with numerical results by legal methods other theories and experimental data and have some very good results Many numerical results obtained by SMM on elastic and nonlinear strain of trigonal alloy are new and predictive, experimentally oriented 18 CHAPTER 4: STUDY ON INFLUENCE OF STRAIN ON DIFFUSION FOR METALS AND INTERSTITIAL ALLOYS AC AND ABC 4.1 Influence of strain on diffusion of interstitial alloy AC (4.35) Q(0, T ) D( ) = D0 ( ) exp − , kBoT ( AC V r + V m AC AC AC D0 ( ) = D0 (0, T ) exp kBoT ) , (4.36) Q(0, T ) D( P) = D0 ( P) exp − , kBoT PV a D0 ( P) = D0 (0, T ) exp − AC k T Bo (4.41) (4.42) 4.2 Influence of strain on diffusion of interstitial alloy ABC We also have all the results as shown in Section 4.1 However, with interstitial alloy ABC we have the following expressions c A gvf ( A) + c A gvf ( A1 ) + c A gvf ( A2 ) c g f ( B) exp − B v nv = exp − , kBoT kBoT RABC = N 1 − n n + n ( B v v X (4.43) − 1) c X X + nv n1c X X(1) − X ( ) − 1 − nv n1 + nv ( BA − 1) cB A + nv n1cB A(1) − T ScABC * + ScABC , (4.44) with BCC structure V r (X ) = N ( ) (4.46) ( ) (4.47) R3 a ABC − a3ABC with FCC structure V r (X ) = N R3 a ABC − a3ABC 4.3 Numerical results and discussion for metals and alloys The numerical results are summarized in Table 4.1 to Table 4.10 and illustrated from Figure 4.1 to Figure 4.5 19 2000 Au, SMM Au, Experiments of Markin (1957) Au, experiments of Herzig (1978) AuSi0,01, SMM 1750 D (10-11 m2s-1) 1500 1250 1000 750 500 250 950 1000 1050 1100 1150 1200 1250 1300 1350 T (K) Figure 4.3 D(T)(D(cm2/s)) for Au, AuSi0,01 at P = calculated by SMM and from experiments of Makin [83] and Herzig [41] 0.30 0.050 Au AuSi0,01 AuSi0,03 0.045 0.040 0.28 D0 (10-2 cm2/s) 0.035 D0 (cm2/s) Cu CuSi0,01 CuSi0,03 0.29 0.030 0.025 0.020 0.015 0.27 0.26 0.25 0.24 0.010 (a) 0.23 0.005 0.000 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.22 1.0 (b) 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 (%) (%) Figure 4.4 D0(, cSi) for (a) AuSi and (b) CuSi at T = 900K and P = calculated by SMM Conclusion of Chapter In chapter 4, we consider the effect of stress or strain on diffusion in binary and ternary alloys We construct the theory of self diffusion (vacuum diffusion) of interstitial alloy AC and substitutional and interstitial ternary alloy ABC with the cubic structure which derives the analytic expressions of the diffusive quantities properties such as activation energy, pre-exponential coefficient, diffusion coefficient depending on temperature, pressure and stress or strain The obtained theoretical results are numerically calculated for Au, Cu metals and interstitial ternary alloy AuSi, CuSi The law of diffusion depending on temperature, pressure, stress or strain of the 20 binary HK is consistent with the law of diffusion depending on temperature, pressure, stress or strain of the parent metal in alloy Calculation results of activation energy Q of Au and Cu using SMM are in good agreement with experimental data, CALPHAD and ab initio calculations The error when calculating the pre-exponential factor D0 by SMM, the CALPHAD and ab initio calculations compared with the experiment is larger than the error when calculating the activation energy Q by SMM, the CALPHAD, ab initio calculations compared with the experiment The Si intercalated atom decreases the diffusion coefficients D, D0 and increases Q Both D, D0 and Q increase with temperature The value of the diffusion coefficient is only significant in the high temperature region In the low temperature region, the diffusion coefficient is very small The interstitial atom strongly influences the diffusion in AuSi but has little effect on the diffusion in CuSi The diffusion coefficient decreases with increasing pressure and strain or stress Some results of numerical calculation of diffusion quantities of metals by SMM are in good agreement with experiments and other calculations by CALPHAD, ab initio, Results of numerical calculation of diffusive quantities of interstitial binary alloy are generally new and directional, experimentally predictive GENERAL CONCLUSION The thesis uses the SMM to study the properties of elastic – nonlinear deformation and the influence of strain on diffusion for metals, binary interstitial alloys and ternary substitutional and interstitial alloys with BCC and FCC structures at zero pressure and under pressure The main results of the thesis include: Build analytic expressions for the elastic moduli E, K, G, elastic constants C11, C12, C44 and the longitudinal wave velocities Vd and Vn for metals and ternary and binary interstitial alloys with BCC and FCC structures Build analytic expressions of strain energy density, dependence of real stress on strain, maximum real stress, elastic strain limit for metals, and ternary and binary interstitial alloys with BCC and FCC structures Build the theory of diffusion by vacancy mechanism in metals and alloys alternating two and three components with cubic structure 21 in which analytic expressions of quantities characteristic for diffusion can be derived such as activation energy, pre-exponential coefficient, diffusion coefficient and activation volume depend on temperature and pressure An analytic expression is built that allows to determine the dependence of the deformation of the diffusion coefficient and the preexponential coefficient The analytical results obtained on the characteristic quantities for elastic and nonlinear deformation were applied numerically, discussed and derived the laws of variation according to temperature, pressure and component concentration for the three metals types of Fe, Au, Cu, ternary interstitial alloys FeSi, FeC, FeH, CuSi and binary interstitial alloys FeCrSi and AuCuSi Apply numerical calculation, discuss and derive the laws of variation with temperature and pressure to quantities characteristic of diffusion for metals Au, Cu and two ternary interstitial alloys CuSi, AuSi We draw three rules as follows: First, the elastic strain quantities, elastic wave propagation velocities and nonlinear strain quantities of the two- and threecomponent HKXK with cubic structure all decrease as the interstitial atom concentration increases Second, the elastic deformation quantities, the elastic wave propagation velocities and the nonlinear deformation quantities of metals, two- and three-component HKXK with cubic structure all decrease with increasing temperature, all increase when the pressure is increased The third is that the elastic strain quantities, the elastic wave propagation velocities and the nonlinear strain quantities of the threecomponent HKXK with cubic structure all change slowly with the substitution atom concentration The expressions obtained by SMM in the thesis have an analytic form that is convenient for numerical calculation, data evaluation using Maple, Origin, Most of the results calculated by SMM in the thesis are has an error of less than 10% compared with experiment and is consistent with calculations by other methods such as ab initio, MD, CALPHAD, and is consistent with experimental data Most numerical results by SMM for ternary and binary interstitial alloys at different temperatures, pressures and component concentrations are new, predictive, and oriented to future experiments 22 FURTHER RESEARCH DIRECTIONS Applying the theoretical results of the thesis to calculate the numbers for the quantities characteristic of elastic and nonlinear deformation for ternary and binary interstitial alloys different from the numerically calculated alloys in the thesis Building the theory of elastic and nonlinear strain for ternary interstitial alloys hexagonal structures with hexagonal closed-pack structures Developing the theory of elastic and nonlinear strain for ternary and binary interstitial alloys when considering point defects in alloy crystal lattice LIST OF PUBLISHED PAPERS RELATED TO THE THESIS Hoc N.Q and Hien N.D (2018), Study on elastic deformation of substitution alloy AB with interstitrial atom C and BCC structure under pressure, 42nd Vietnam National Conference on Theoretical Physics (NCTP-42), IOP Conf Series: Journal of Physics: Conf Series, 1034(2018)012005, IOP Publishing DOI: 10.1088/1742-6596/1034/1/012005 Hoc N.Q, Tinh B.D and Hien N.D (2019), Elastic moduli and elastic constants of alloy AuCuSi with FCC structure under pressure, High Temperature Materials and Processes, 38, 264-272 DOI: 10.1515/htmp-2018-0027 Hoc N.Q, Hoa N.T and Hien N.D (2019), Build the theory of nonlinear deformation for BCC and FCC substitutional alloys AB with interstitial atom C under pressure, HNUE Journal of Science, Natural Sciences, 64(6), 45-56 DOI: 10.18173/2354-1059.2019-0030 Nguyễn Quang Học, Nguyễn Đức Hiền, Nguyễn Thị Hòa, Lê Lan Hương Nguyễn Ngọc Lê (2022), Sự khuếch tán kim loại Au, Cu hợp kim xen kẽ AuSi, CuSi, HNUE Journal of Science, Natural Sciences, 67(1), 27-37 DOI: 10.18173/2354-1059.2022-0004 Nguyễn Quang Học, Nguyễn Đức Hiền, Nguyễn Thị Hòa, Phạm Phương Uyên Trịnh Hồng Ngọc (2022), Biến dạng đàn hồi vận tốc sóng đàn hồi hợp kim tam nguyên FeCrSi AuCuSi, HNUE Journal of Science, Natural Sciences, 67(1), 38-53 DOI: 10.18173/2354-1059.2022-0005 23 Hoc N.Q and Hien N.D (2022), Nonlinear deformation of Au, AuSi and AuCuSi with FCC structure under pressure, HNUE Journal of Science, Natural Sciences, 67(2), 40-54 DOI: 10.18173/23541059.2022-0021 Hoc N Q, Hien N D and Vi T K (2022), Elastic deformation and elastic wave velocity of iron and its binary interstitial alloys, Temperature, pressure and interstitial atom concentration dependences, Physica B: Condensed Matter, 644, 414134 DOI: https://doi.org/10.1016/j.physb.2022.414134 24