為瞭解材料中多變的物理現象，對於「結構-性質」關係的瞭解是不可或缺的。在材料科學中一項重大挑戰即為透過在微觀尺度下控制材料表現，使其具有全新性質與功能。相較於晶體材料的高度對稱性，無序材料內額外的熱力學與結構自由度為其提供了許多優勢。無序系統內固有的隨機性使得其行為無法如同晶體一般簡單的透過其組成粒子於空間中排列的對稱性描述。然而，然而「無序」一詞不代表其結構缺乏特徵。由於存在不可忽視的粒子間交互作用，在無序系統中每個粒子的排列或多或少的與其周遭環境關聯。而此類微妙的關聯性只能透過統計學變數描述，其總結了足夠大量粒子行為的的集體行為。在此觀點下討論任意單獨粒子的性質是無意義的。

散射是常見於研究無序系統塊材特性的技術。藉由分析中子束與材料互動後發生的動能與方向變化，散射頻譜可供研究者一窺其中組成成分的排列形式。相反的，給定材料於微觀尺度下之互動，可透過電腦模擬取樣得出其於巨觀下之結構、行為。傳統上分析散射頻譜的方法為藉由遞迴地更新已知解析形式的數學函數中參數以比對測量所得數據，並以最終取得最佳化之參數組合描述材料特性。然而發展具解析形式的散射函數以精確地捕捉散射頻譜中特徵並連結材料性質通常是一項極為艱鉅的挑戰，我們因而尋求資料探勘技術協助，比對自散射實驗與電腦模擬中得到的粒子分布關聯函數，透過機率性的描述連結散射函數與其關聯之材料物理特性。

於本論文中我們提出一種泛用的策略，藉由基於高斯過程回歸算法的機器學習模型協助定量的由散射頻譜推測控制軟物質材料構象之參數。其中由分子動力學模擬產生的數據將作為基準訓練集用於訓練回歸模型。另外，為解決散射數據中粒子間與粒子內關聯難以分離的棘手問題，訓練集中的粒子間關聯函數將透過變分自動編碼器算法生成低維度代表，用於協助形塑人造散射函數以擬合與材料構象直接相關的粒子間關聯函數。

我們選擇兩個具代表性的無序系統：1. 軟物質，包含帶電膠體懸浮液與二元長鏈共聚物，以及2. 一類具成分無序性的晶體材料：高熵合金，作為研究對象，以測試本研究中提出用於結構鑑定之非參數性算法的可行性與泛用性。最後，我們成功展示了這些新發展的算法相較傳統方法具有更高的精準度以及更優良的運算效率。

One of the grand challenges and opportunities in materials science is to harness the new phenomena, properties, and function that emerge by de novo design of materials at the nanoscale. In this pursuit, materials characterized by inherent structural disorder offer many advantages due to their additional thermodynamic degrees of freedom in comparison to the crystalline materials.

The signature feature shared by numerous disordered systems is their inherent disorderliness. As a result, their structure and behavior cannot be simply characterized by the periodicity in spatial arrangement of particles as crystalline materials. Nevertheless, the term disorder does not necessarily mean that the structure in such systems are entirely random as in gas. Due to the presence of interaction at the atomic and molecular level, the constituent particles in disordered systems are highly correlated in position and momentum. Such coherence can only be described in terms of statistical variables that portray the behavior of a collection of sufficiently large amount of particles. Attribute of any single particle is not relevant in this descriptive framework.

Prominent among the tools for addressing the structure-property relationships of disordered systems are computer simulation and elastic scattering. In both techniques, the quantity of interest is the two-point static correlation function in reciprocal Q space, which manifests the spatial arrangement of particles in real space.

In this mathematical setting, the routine approach for obtaining the microstructural insights, from either computational trajectories or experimental spectra, is to first identify the relevant structural parameters, establish the theoretical model of two-point correlator accordingly, and use it as a basis of regression analysis in order to obtain the optimized parameters for structural description.

However, more than often the structure of disordered systems cannot be satisfactorily modelled analytically due to the complexity in structural collectivity. As a result, the disordered structure cannot be reconstructed in an unbiased manner. To circumvent this inherent deficiency of the existing deterministic methods, in this thesis we establish a non-parametric strategy, based on the principle of machine learning, to address the structural inversion problem in a probabilistic manner. Specifically, we deliberately select two representative soft condensed systems, colloidal suspension and copolymer solutions, and one hard matter system, the high entropy alloy, which is treated as heated solids with frozen atomic configurations in our study, to demonstrate the feasibility and usefulness of this new non-parametric method for investigating the structure of disordered systems.

In this thesis we have successfully demonstrated the superior performance of this method in numerical accuracy, computational efficiency, and general applicability over the existing parametric approaches. We are therefore optimistic to abut the prospect of our method and hope it will render a new window of opportunity for facilitating the structural inversion problems commonly encountered in the various topics of materials study.

摘要 i

Abstract ii

1 Introduction 1
1.1 Characterization of Disordered Matter 2
1.2 Small Angle Scattering 4
1.2.1 Effective Inter Particle Interaction 9
1.2.2 Scattering Functions for Polymer Chain 14
1.3 Atomic Level Materials Deformations 20
1.3.1 Anisotropy and Heterogeneity 20
1.3.2 Local Configuration Units 22
1.4 Machine Learning Based Regression and Classification Algorithms 27
1.4.1 Singular Value Decomposition, Principal Component Analysis and Linear Discriminant Analysis 29
1.4.2 Gaussian Process Regression 33
1.4.3 Variational Auto Encoder 37

2 Inversion Problem of Effective Inter-Particle Interaction 41
2.1 Background 41
2.2 Preparation of the Training Sets 43
2.3 Data Distributions and the Kernel Function 46
2.4 Validation of the Gaussian Process (GP) Regressor 52
2.5 Extraction of the Inter-particle Correlations from Elastic Scattering Spectrum 56

3 Inversion Problem of Block Copolymer Conformation 65
3.1 Background 65
3.2 Preparation of the Training Sets 69
3.3 Validation of the Gaussian Process (GP) Regressor 75

4 Structural Essence of Plasticity in High-Entropy Alloys 79
4.1 Background 79
4.2 Preparation of the Atomic Trajectories 82
4.3 Independent Variables Characterizing the Local Configuration of HEA Identified by PCA 83
4.3.1 Influence of Size and Composition of LCU on the Plasticity of strained HEA 84
v4.3.2 Influence of orientation of LCU on the plasticity of strained HEA. 89
4.3.3 Influence of LCU Shapes in the Quiescent State on the Plasticity of strained HEA 95
4.4 Classification of Configurational Features Using Linear Discriminant Analysis 97

5 Conclusions 103

References 105

[1] P. G. de Gennes, “Soft matter,” Rev. Mod. Phys., vol. 64, pp. 645–648, Jul 1992.
[2] F. Otto, A. Dlouhý, C. Somsen, H. Bei, G. Eggeler, and E. P. George, “The influences
of temperature and microstructure on the tensile properties of a cocrfemnni high-entropy alloy,” Acta Mater., vol. 61, no. 15, pp. 5743–5755, 2013.
[3] Z. Zhang, M. Mao, J. Wang, B. Gludovatz, Z. Zhang, S. X. Mao, E. P. George, Q. Yu,
and R. O. Ritchie, “Nanoscale origins of the damage tolerance of the high-entropy alloy crmnfeconi,” Nat. Commun., vol. 6, no. 1, pp. 1–6, 2015.
[4] C. Varvenne, G. P. M. Leyson, M. Ghazisaeidi, and W. A. Curtin, “Solute strengthening in random alloys,” Acta Mater., vol. 124, pp. 660–683, 2017.
[5] S. I. Rao, C. Woodward, T. A. Parthasarathy, and O. Senkov, “Atomistic simulations of dislocation behavior in a model fcc multicomponent concentrated solid solution alloy,” Acta Mater., vol. 134, pp. 188–194, 2017.
[6] Q. Ding, Y. Zhang, X. Chen, X. Fu, D. Chen, S. Chen, L. Gu, F. Wei, H. Bei, and Y. Gao,
“Tuning element distribution, structure and properties by composition in high-entropy alloys,” Nature, vol. 574, no. 7777, pp. 223–227, 2019.
[7] L. Li, Q. Fang, J. Li, B. Liu, Y. Liu, and P. K. Liaw, “Lattice-distortion dependent yield strength in high entropy alloys,” Mater. Sci. Eng., A, p. 139323, 2020.
[8] S.-H. Chen, “Small angle neutron scattering studies of the structure and interaction in micellar and microemulsion systems,” Annual Review of Physical Chemistry, vol. 37, no. 1, pp. 351–399, 1986.
[9] R. Henderson, “A uniqueness theorem for fluid pair correlation functions,” Physics Letters A, vol. 49, no. 3, pp. 197–198, 1974.
[10] H. Wang, F. H. Stillinger, and S. Torquato, “Sensitivity of pair statistics on pair potentials in many-body systems,” The Journal of Chemical Physics, vol. 153, no. 12, p. 124106, 2020.
[11] K. S. Schmitz, Macroions in solution and colloidal suspension. Wiley-VCH Verlag GmbH, 1993.
[12] T. Zemb and P. Charpin, “Micellar structure from comparison of x-ray and neutron small-angle scattering,” Journal de Physique, vol. 46, no. 2, pp. 249–256, 1985.
[13] A. Tardieu, A. Le Verge, M. Malfois, F. Bonnete, S. Finet, M. Ries-Kautt, and L. Belloni,“Proteins in solution: from x-ray scattering intensities to interaction potentials,” Journal of Crystal Growth, vol. 196, no. 2-4, pp. 193–203, 1999.
[14] J.-P. Hansen and I. R. McDonald, Theory of simple liquids: with applications to soft matter. Academic press, 2013.
[15] J. Lebowitz and J. Percus, “Mean spherical model for lattice gases with extended hard cores and continuum fluids,” Physical Review, vol. 144, no. 1, p. 251, 1966.
[16] J. A. Barker and D. Henderson, “What is” liquid”? understanding the states of matter,” Reviews of Modern Physics, vol. 48, no. 4, p. 587, 1976.
[17] E. Waisman, “The radial distribution function for a fluid of hard spheres at high densities: mean spherical integral equation approach,” Molecular Physics, vol. 25, no. 1, pp. 45–48, 1973.
[18] W. R. Smith, D. Henderson, and Y. Tago, “Mean spherical approximation and optimized cluster theory for the square-well fluid,” The Journal of Chemical Physics, vol. 67, no. 11, pp. 5308–5316, 1977.
[19] M. Heinen, P. Holmqvist, A. J. Banchio, and G. Nägele, “Pair structure of the hard-sphere yukawa fluid: An improved analytic method versus simulations, rogers-young scheme, and experiment,” The Journal of chemical physics, vol. 134, no. 4, p. 044532, 2011.
[20] J.-P. Hansen and J. B. Hayter, “A rescaled msa structure factor for dilute charged colloidal dispersions,” Molecular Physics, vol. 46, no. 3, pp. 651–656, 1982.
[21] J. Van Leeuwen, J. Groeneveld, and J. De Boer, “New method for the calculation of the pair correlation function. i,” Physica, vol. 25, no. 7-12, pp. 792–808, 1959.
[22] J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Physical Review, vol. 110, no. 1, p. 1, 1958.
[23] P.-J. Derian, L. Belloni, and M. Drifford, “Contribution of small ions to the scattered intensity in the hypernetted chain approximation: Application to micellar solutions,” The Journal of chemical physics, vol. 86, no. 10, pp. 5708–5715, 1987.
[24] M. Wertheim, “Exact solution of the percus-yevick integral equation for hard spheres,” Physical Review Letters, vol. 10, no. 8, p. 321, 1963.
[25] J. Méndez-Alcaraz, B. D’Aguanno, and R. Klein, “Structure of binary colloidal mixtures of charged and uncharged spherical particles,” Langmuir, vol. 8, no. 12, pp. 2913–2920, 1992.
[26] F. J. Rogers and D. A. Young, “New, thermodynamically consistent, integral equation for simple fluids,” Physical Review A, vol. 30, no. 2, p. 999, 1984.
[27] M. Huš, M. Zalar, and T. Urbic, “Correctness of certain integral equation theories for core-softened fluids,” The Journal of Chemical Physics, vol. 138, no. 22, p. 224508, 2013.
[28] L. Belloni, “Inability of the hypernetted chain integral equation to exhibit a spinodal line,” J. Chem. Phys., vol. 98, p. 8080, 1983.
[29] B. Beresford-Smith, D. Y. C. Chan, and D. J. Mitchell, “The electrostatic interaction in colloidal systems with low added electrolyte,” J. Colloid Interface Sci., vol. 105, p. 216, 1985.
[30] L. Belloni, “Attraction of electrostatic origin between colloids,” Chem. Phys., vol. 99, p. 43, 1985.
[31] L. Belloni, Neutron, X-Ray and Light Scattering: Introduction to an Investigative Tool for Colloidal and Polymetric Systems. edited by Th. Zemb and P. Lindner, North-Holland, Amsterdam, 1991.
[32] G. Fritz, A. Bergmann, and O. Glatter, “Evaluation of small-angle scattering data of charged particles using the generalized indirect fourier transformation technique,” J. Chem. Phys., vol. 113, p. 9733, 2000.
[33] P.-G. De Gennes and P.-G. Gennes, Scaling concepts in polymer physics. Cornell university press, 1979.
[34] A. Guinier, G. Fournet, and K. L. Yudowitch, Small-angle scattering of X-rays. Wiley New York, 1955.
[35] G. Strang, Linear algebra and learning from data, vol. 4. Wellesley-Cambridge Press Cambridge, 2019.
[36] L. D. Landau and E. M. Lifshitz, Statistical Physics. Reading, MA: Addison-Wesley,
1958.
[37] J. A. Schellman, “Flexibility of dna,” Biopolymers: Original Research on Biomolecules, vol. 13, no. 1, pp. 217–226, 1974.
[38] W. Kuhn and H. Kuhn, “Die frage nach der aufrollung von fadenmolekeln in strömenden lösungen,” Helvetica Chimica Acta, vol. 26, no. 5, pp. 1394–1465, 1943.
[39] A. Y. Grosberg, A. R. Khokhlov, H. E. Stanley, A. J. Mallinckrodt, and S. McKay, “Statistical physics of macromolecules,” Computers in Physics, vol. 9, no. 2, pp. 171–172, 1995.
[40] H.-P. Hsu, W. Paul, and K. Binder, “Estimation of persistence lengths of semiflexible polymers: Insight from simulations,” Polymer Science Series C, vol. 55, no. 1, pp. 39–59, 2013.
[41] P. J. Flory, Principles of polymer chemistry. Cornell university press, 1953.
[42] J. S. Pedersen and P. Schurtenberger, “Scattering functions of semiflexible polymers with and without excluded volume effects,” Macromolecules, vol. 29, no. 23, pp. 7602–7612, 1996.
[43] K. Binder, Monte Carlo and molecular dynamics simulations in polymer science. Oxford University Press, 1995.
[44] J. M. Ziman, Models of disorder: the theoretical physics of homogeneously disordered systems. Cambridge: Cambridge University Express, 1979.
[45] Z. Wang and W.-H. Wang, “Flow units as dynamic defects in metallic glassy materials,” Natl. Sci. Rev., vol. 6, no. 2, pp. 304–323, 2019.
[46] J. Yeh, S. Chen, S. Lin, J. Gan, T. Chin, T. Shun, C. Tsau, and S. Chang, “Nanostructured high entropy alloys with multiple principal elements: novel alloy design concepts and outcomes,” Adv. Eng. Mater., vol. 6, no. 5, pp. 299–303, 2004.
[47] B. Cantor, I. Chang, P. Knight, and A. Vincent, “Microstructural development in
equiatomic multicomponent alloys,” Mater. Sci. Eng., A, vol. 375, pp. 213–218, 2004.
[48] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, “Icosahedral bond orientational order in supercooled liquids,” Phys. Rev. Lett., vol. 47, no. 18, p. 1297, 1981.
[49] T. Ninomiya, “Topological disorder in condensed matter,” in Topological Disorder in Condensed Matter (F. Yonezawa and T. Ninomiya, eds.), Berlin: Springer, 1983.
[50] E. A. Lazar, J. Lu, and C. H. Rycroft, “Voronoi cell analysis: The shapes of particle
systems,” American Journal of Physics, vol. 90, no. 6, pp. 469–480, 2022.
[51] H. Peng, M. Li, and W. Wang, “Structural signature of plastic deformation in metallic glasses,” Phys. Rev. Lett., vol. 106, no. 13, p. 135503, 2011.
[52] J. Ding, S. Patinet, M. L. Falk, Y. Cheng, and E. Ma, “Soft spots and their structural
signature in a metallic glass,” Proc. Natl. Acad. Sci. U.S.A., vol. 111, no. 39, pp. 14052–
14056, 2014.
[53] X. Yang, R. Liu, M. Yang, W.-H. Wang, and K. Chen, “Structures of local rearrangements in soft colloidal glasses,” Phys. Rev. Lett., vol. 116, no. 23, p. 238003, 2016.
[54] K. Šolc and W. H. Stockmayer, “Shape of a random flight chain,” J. Chem. Phys.,
vol. 54, no. 1, p. 2756, 1971.
[55] D. N. Theodorou and U. W. Suter, “Shape of unperturbed linear polymers: polypropylene,” Macromolecules, vol. 18, no. 6, pp. 1206–1214, 1985.
[56] K. P. Murphy, Machine learning: a probabilistic perspective. Cambridge: MIT press, 2012.
[57] R. B. Cattell, “The scree test for the number of factors,” Multivariate behavioral research, vol. 1, no. 2, pp. 245–276, 1966.
[58] M. Zhu and A. Ghodsi, “Automatic dimensionality selection from the scree plot via the use of profile likelihood,” Computational Statistics & Data Analysis, vol. 51, no. 2, pp. 918–930, 2006.
[59] W. Schommers, “Pair potentials in disordered many-particle systems: A study for liquid gallium,” Physical Review A, vol. 28, no. 6, p. 3599, 1983.
[60] A. P. Lyubartsev and A. Laaksonen, “Calculation of effective interaction potentials from radial distribution functions: A reverse monte carlo approach,” Physical Review E, vol. 52, no. 4, p. 3730, 1995.
[61] W. G. Noid, J.-W. Chu, G. S. Ayton, V. Krishna, S. Izvekov, G. A. Voth, A. Das, and
H. C. Andersen, “The multiscale coarse-graining method. i. a rigorous bridge between
atomistic and coarse-grained models,” The Journal of chemical physics, vol. 128, no. 24, p. 244114, 2008.
[62] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. The MIT Press, Cambridge, 2006.
[63] C. M. Bishop, Pattern Recognition and Machine Learning. Springer, New York, 2006.
[64] G.-R. Huang, C.-H. Tung, D. Chang, C. N. Lam, C. Do, Y. Shinohara, S.-Y. Chang,
Y. Wang, K. Hong, and W.-R. Chen, “Determining population densities in bimodal micellar solutions using contrast-variation small angle neutron scattering,” The Journal of Chemical Physics, vol. 153, no. 18, p. 184902, 2020.
[65] C.-H. Tung, G.-R. Huang, S.-Y. Chang, Y. Han, W.-R. Chen, and C. Do, “Revealing the
influence of salts on the hydration structure of ionic sds micelles by contrast-variation small-angle neutron scattering,” The Journal of Physical Chemistry Letters, vol. 11, no. 17, pp. 7334–7341, 2020.
[66] C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization,” ACM Transactions on mathematical software (TOMS), vol. 23, no. 4, pp. 550–560, 1997.
[67] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980, 2014.
[68] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” arXiv preprint arXiv:1312.6114, 2013.
[69] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, 1998.
[70] R. G. Horn and J. N. Israelachvili, “Direct measurement of structural forces between two surfaces in a nonpolar liquid,” The Journal of Chemical Physics, vol. 75, no. 3, pp. 1400–1411, 1981.
[71] W. Poon, “Colloids as big atoms,” Science, vol. 304, no. 5672, pp. 830–831, 2004.
[72] V. N. Manoharan, “Colloidal matter: Packing, geometry, and entropy,” Science, vol. 349, no. 6251, p. 1253751, 2015.
[73] M. Girard, S. Wang, J. S. Du, A. Das, Z. Huang, V. P. Dravid, B. Lee, C. A. Mirkin, and M. Olvera de la Cruz, “Particle analogs of electrons in colloidal crystals,” Science, vol. 364, no. 6446, pp. 1174–1178, 2019.
[74] W. B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions. Cambridge University Press, Cambridge, 1992.
[75] K. S. Schmitz, Macroions in Solution and Colloidal Suspension. Wiley-VCH, New York, 1992.
[76] L. Belloni, “Electrostatic interactions in colloidal solutions: Comparison between primitive and one-component models,” The Journal of Chemical Physics, vol. 85, no. 1, pp. 519–526, 1986.
[77] S.-H. Chen and E. Y. Sheu, Micellar Solutions and Microemulsions: Structure, Dynamics, and Statistical Thermodynamics. edited by S.-H. Chen and R. Rajagopalan, Springer-Verlag, New York, 1990.
[78] L. Belloni, “Colloidal interactions,” Journal of Physics: Condensed Matter, vol. 12, no. 46, p. R549, 2000.
[79] R. Klein, Neutron, X-rays and Light. Scattering Methods Applied to Soft Condensed Matter. edited by Th. Zemb and P. Lindner, North Holland, Amsterdam, 2002.
[80] K. S. Schmitz, Introduction to Dynamic Light Scattering by Macromolecules. Academic Press, San Diego, 1990.
[81] C. Caccamo, “Integral equation theory description of phase equilibria in classical fluids,” Physics reports, vol. 274, no. 1-2, pp. 1–105, 1996.
[82] B. W. van de Waal, “On the origin of second-peak splitting in the static structure factor of metallic glasses,” Journal of non-crystalline solids, vol. 189, no. 1-2, pp. 118–128, 1995.
[83] S. Nosé, “A unified formulation of the constant temperature molecular dynamics methods,” The Journal of chemical physics, vol. 81, no. 1, pp. 511–519, 1984.
[84] W. G. Hoover, “Canonical dynamics: Equilibrium phase-space distributions,” Physical review A, vol. 31, no. 3, p. 1695, 1985.
[85] L. Verlet, “Computer” experiments” on classical fluids. i. thermodynamical properties of lennard-jones molecules,” Physical review, vol. 159, no. 1, p. 98, 1967.
[86] J.-M. Y. Carrillo and A. V. Dobrynin, “Polyelectrolytes in salt solutions: Molecular dynamics simulations,” Macromolecules, vol. 44, no. 14, pp. 5798–5816, 2011.
[87] S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” Journal of computational physics, vol. 117, no. 1, pp. 1–19, 1995.
[88] G. Nägele, “On the dynamics and structure of charge-stabilized suspensions,” Physics Reports, vol. 272, no. 5-6, pp. 215–372, 1996.
[89] F. Pedregosa and et al., “Scikit-learn: Machine learning in Python,” J. Mach. Learn. Res., vol. 12, pp. 2825–2830, 2011.
[90] O. Glatter, Neutron, X-Ray and Light Scattering: Introduction to an Investigative Tool for Colloidal and Polymetric Systems. edited by Th. Zemb and P. Lindner, North Holland, Amsterdam, 1991.
[91] O. Glatter, Neutron, X-rays and Light. Scattering Methods Applied to Soft Condensed Matter. edited by Th. Zemb and P. Lindner, North Holland, Amsterdam, 2002.
[92] J. A. Anta and S. Lago, “Self-consistent effective interactions in charged colloidal suspensions,” The Journal of chemical physics, vol. 116, no. 23, pp. 10514–10522, 2002.
[93] F. Chollet et al., “Keras,” 2015.
[94] P. Alexandridis, “Amphiphilic copolymers and their applications,” Current Opinion in Colloid & Interface Science, vol. 1, no. 4, pp. 490–501, 1996.
[95] T. P. Lodge, “Block copolymers: past successes and future challenges,” Macromolecular chemistry and physics, vol. 204, no. 2, pp. 265–273, 2003.
[96] K. Binder, “Phase transitions in polymer blends and block copolymer melts: Some recent developments,” Adv. Polym. Sci., vol. 112, pp. 181–299, 1994.
[97] K. E. Doncom, L. D. Blackman, D. B. Wright, M. I. Gibson, and R. K. O＇Reilly, “Dispersity effects in polymer self-assemblies: a matter of hierarchical control,” Chemical Society Reviews, vol. 46, no. 14, pp. 4119–4134, 2017.
[98] T. Zemb and P. Lindner, Neutron, X-rays and light. Scattering methods applied to soft condensed matter. North Holland, 2002.
[99] H.-P. Hsu, W. Paul, and K. Binder, “Scattering function of semiflexible polymer chains under good solvent conditions,” The Journal of Chemical Physics, vol. 137, no. 17, p. 174902, 2012.
[100] G. Porod, “X-ray and light scattering by chain molecules in solution,” Journal of Polymer Science, vol. 10, no. 2, pp. 157–166, 1953.
[101] P. Sharp and V. A. Bloomfield, “Light scattering from wormlike chains with excluded volume effects,” Biopolymers: Original Research on Biomolecules, vol. 6, no. 8, pp. 1201–1211, 1968.
[102] M. Bawendi and K. F. Freed, “A wiener integral model for stiff polymer chains,” The Journal of chemical physics, vol. 83, no. 5, pp. 2491–2496, 1985.
[103] M. Rawiso, R. Duplessix, and C. Picot, “Scattering function of polystyrene,” Macromolecules, vol. 20, no. 3, pp. 630–648, 1987.
[104] A. Kholodenko, “Persistence length and related conformational properties of semiflexible polymers from dirac propagator,” The Journal of chemical physics, vol. 96, no. 1, pp. 700–713, 1992.
[105] S. Stepanow, “Statistical mechanics of semiflexible polymers,” The European Physical Journal B-Condensed Matter and Complex Systems, vol. 39, no. 4, pp. 499–512, 2004.
[106] W.-R. Chen, P. D. Butler, and L. J. Magid, “Incorporating intermicellar interactions in the fitting of sans data from cationic wormlike micelles,” Langmuir, vol. 22, no. 15, pp. 6539–6548, 2006.
[107] O. Kratky and G. Porod, “Röntgenuntersuchung gelöster fadenmoleküle,” Recueil des Travaux Chimiques des Pays-Bas, vol. 68, no. 12, pp. 1106–1122, 1949.
[108] C. Svaneborg and J. S. Pedersen, “A formalism for scattering of complex composite structures. i. applications to branched structures of asymmetric sub-units,” The Journal of Chemical Physics, vol. 136, no. 10, p. 104105, 2012.
[109] A. De Myttenaere, B. Golden, B. Le Grand, and F. Rossi, “Mean absolute percentage error for regression models,” Neurocomputing, vol. 192, pp. 38–48, 2016.
[110] W. Marshall and S. Lovesey, Theory of Thermal Neutron Scattering: The Use of Neutrons for the Investigation of Condensed Matter. Clarendon Press, Oxford, 1971.
[111] E. P. George, W. Curtin, and C. C. Tasan, “High entropy alloys: A focused review of mechanical properties and deformation mechanisms,” Acta Mater., vol. 188, pp. 435–474, 2020.
[112] E. P. George, D. Raabe, and R. O. Ritchie, “High-entropy alloys,” Nat. Rev. Mater., vol. 4, no. 8, pp. 515–534, 2019.
[113] C. Varvenne, A. Luque, and W. A. Curtin, “Theory of strengthening in fcc high entropy alloys,” Acta Mater., vol. 118, pp. 164–176, 2016.
[114] J. Li, Q. Fang, B. Liu, and Y. Liu, “Transformation induced softening and plasticity in high entropy alloys,” Acta Mater., vol. 147, pp. 35–41, 2018.
[115] L. Zhang, Y. Xiang, J. Han, and D. J. Srolovitz, “The effect of randomness on the strength of high-entropy alloys,” Acta Mater., vol. 166, pp. 424–434, 2019.
[116] F. G. Coury, M. Kaufman, and A. J. Clarke, “Solid-solution strengthening in refractory high entropy alloys,” Acta Mater., vol. 175, pp. 66–81, 2019.
[117] C.-H. Tung, G.-R. Huang, Z. Bai, Y. Fan, W.-R. Chen, and S.-Y. Chang, “Structural origin of plasticity in strained high-entropy alloy,” arXiv, vol. 2005.07088, 2020.
[118] R. Mari, F. Krzakala, and J. Kurchan, “Jamming versus glass transitions,” Physical review letters, vol. 103, no. 2, p. 025701, 2009.
[119] G. Barkema and N. Mousseau, “Event-based relaxation of continuous disordered systems,” Physical review letters, vol. 77, no. 21, p. 4358, 1996.
[120] D. Rodney and C. Schuh, “Distribution of thermally activated plastic events in a flowing glass,” Physical review letters, vol. 102, no. 23, p. 235503, 2009.
[121] C. Liu, P. Guan, and Y. Fan, “Correlating defects density in metallic glasses with the distribution of inherent structures in potential energy landscape,” Acta Materialia, vol. 161, pp. 295–301, 2018.
[122] A. Cottrell, “The nabarro equation for thermally activated plastic glide,” Philosophical Magazine, vol. 86, no. 25-26, pp. 3811–3817, 2006.
[123] W.-M. Choi, Y. H. Jo, S. S. Sohn, S. Lee, and B.-J. Lee, “Understanding the physical metallurgy of the cocrfemnni high-entropy alloy: an atomistic simulation study,” Npj Comput. Mater., vol. 4, no. 1, pp. 1–9, 2018.
[124] B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, and R. O. Ritchie, “A fracture-resistant high-entropy alloy for cryogenic applications,” Science, vol. 345, no. 6201, pp. 1153–1158, 2014.
[125] M. L. Falk and J. S. Langer, “Dynamics of viscoplastic deformation in amorphous solids,” Phys. Rev. E, vol. 57, no. 6, p. 7192, 1998.
[126] R. Balluffi, “Structure and properties of point defects in grain boundaries in metals,” Le Journal de Physique Colloques, vol. 43, no. C6, pp. C6–71, 1982.
[127] J. P. Snyder and P. M. Voxland, An album of map projections. US Government Printing Office, 1989.
[128] G. Marsaglia, “Choosing a point from the surface of a sphere,” Ann. Math. Stat., vol. 43, no. 2, pp. 645–646, 1972.
[129] M. J. Buerger, Elementary Crystallography: An Introduction to the Fundamental Geo-metric Features of Crystals. Cambridge: The MIT Press, 1978.
[130] T. Ogawa, “Problems in a digital description of a configuration of atoms and some other geometrical topics in physics,” in Topological Disorder in Condensed Matter (F. Yonezawa and T. Ninomiya, eds.), Berlin: Springer, 1983.
[131] G.-R. Huang, Y. Wang, C. Do, L. Porcar, Y. Shinohara, T. Egami, and W.-R. Chen,
“Determining gyration tensor of orienting macromolecules through their scattering signature,” J. Phys. Chem. Lett., vol. 10, no. 14, pp. 3978–3984, 2019.