Phasefield Modelling

1. In condensed systems, why is H approximately equal to U?

2. What are the assumptions when deriving $\Delta G_{mix}$ for ideal solution? Is thermal entropy factored?

3. How is chemical potential different from partial molar free energy? ¹

4. How will the G-X curve change when $\Delta H_{mix}$ is non-zero? Plot G-X curve for $\Omega$ < 0 and $\Omega$ > 0 cases and compare. What is the final expression for G?

5. Why does the Free Energy curve make a W like pattern and not an M like pattern? How can this be explained physically? Why will the $\Delta S_{mix}$ term dominate at low temperatures leading to a W type curve?

6. What are the models used to calculate $\Delta H_{mix}$ in non-ideal solutions? What determines the sign of $\omega$ in the model?

7. What is the assumptions regarding $\Delta H_{mix}$ when deriving $\Delta G_{mix}$ for regular solution? Is it valid for large values of $\Delta H_{mix}$?

8. How does congruent melting appear in a phase diagram? What determines the shape of such a phase diagram i.e. in which cases does the inversion happen in such phase diagrams?

9. If the higher temperature portion of the phase diagram opens downward, what type of behaviour will be observed at lower temperatures? Will there be phase separation or ordering?

10. What do Cahn-Hillard and Allen-Cahn equations explain?

11. How is $\Omega$ related to crystal structure? Derive the relation ²

12. Why is Stirling’s approximation good enough for large numbers? Plot and show using MATLAB

13. Show that molar property of a mixture is given by assuming it to be a mechanical mixture

14. Show that $\mu_{a} = G - X_{b} \frac{dG}{dX_{b}}$. How can this be used to geometrically prove that the intercepts on the axes when a tangent is drawn to the free energy curve give the chemical potential?

15. How do you prove the same result for a generic thermodynamic property? (that the tangent at a point on the curve gives the partial molar quantity on the intercept)

16. How does the diffusion happen in a double well G-X curve? How are diffusion and concentration gradient related?

17. What is Fick’s Law? What does the negative sign indicate?

18. How do you show that $\frac{\partial c}{\partial t}=-\nabla.\vec{J}$ using mass conservation? How does it lead to Fick’s “second” law?

19. Will $\frac{\partial c}{\partial t}=-D\frac{\partial^{2} c}{\partial^{2} t}$ be valid for the double well case? Why?

20. Can Diffusivity be negative? How is the first law modified to accommodate the idea of uphill diffusion?

21. What were the shortcomings in the model with negative D? Does it explain the wavelength of the modulation to be around 100 angstroms?

22. How is mobility (M) defined? What is the need to introduce mobility? How is it related to the concavity of the GX curves?

23. What is the physical significance of the spinodal line? Which two regions does it separate? Which portion is metastable and which one is unstable?

24. How do you determine the unstable or metastable portion from a GX curve?

25. How does this idea lead to the fact that there is no nucleation step in spinodal decomposition?

26. How do we calculate the change in free energy for a generic GX curve? How does it vary when it is an open system? How can this be used as a general idea for any property which varies with composition? (Hilliard’s derivation)

27. How is the stability determined by tangent construction? At which compositions (or regions of GX curve) will the system be unstable and phase separate? How is this related to positive deviation condition?

28. How is the form of the solution to the diffusion equation derived?

29. How is a Matano interface defined? ³

30. What is the flux w.r.t the Matano interface? How can this be derived?⁴

31. How are diffusivity and mobility connected? How does it vary with the curvature of the GX curve?

32. What happens when we substitute the relation between Diffusivity and Mobility into the general solution of the sinusoidal diffusion equation solution?

33. How does this result imply that the sinusoidal fluctuations in composition die down with time when second derivative of free energy is negative? How can this information be used to deduce that nucleation does not occur in spinodal decomposition?

34. How does $R(\beta)$ vary with $\beta$ and with $\lambda$?

35. What is the contradiction in the models prior to Cahn-Hilliard equation? Why does the earlier model conclude that ordering will happen at atomic level if we assume spinodal conditions?

36. Why cannot the classical diffusion equation with negative diffusivity not explain phase separation? How does the prediction of wavelength from the model compare against the observed wavelengths of around 100 angstroms?

37. What is the missing information in the classical model that was incorporated in the Cahn-Hilliard model?

38. Why is it a good idea to use non-dimensionalisation and deal with non-dimensional quantities?

39. Why does entropy dominate at the end points of the GX curve? Why does the entropy dominate at the pure ends? How does this relate to the characteristic W shape?

40. How do we get the phase diagram from the GX plot using MATLAB? How do we plot the spinodal points in the phase diagram using MATLAB?

41. How is the diffusion equation non-dimensionalised using characteristic length and characteristic time?

42. How does a constitutive law refer to a stimuli and a response? How does this show itself in the Fourier Law of Heat Conduction and Classical Diffusion equation?

43. What is the conservation law equivalent for the classical diffusion equation and Fourier law respectively?

44. How do you analytically solve the classical diffusion equation? How are the co-efficients $A_{n}$ determined in the derivation?

45. How does the solution of the diffusion equation depend on the boundary conditions chosen?

46. How is the error function solution derived? How is the analytical solution different from the error function solution?

47. What is the diffusion distance? How is it useful for back of the envelope calculations for diffusion time?

48. How is the classical diffusion equation formulated for numerical solution? What is $\alpha$ used in this calculation?

49. How is the implicit formulation different from the explicit one?

50. How are the equations written in a matrix form to be solved numerically? How is matrix inversion used?

51. Why is the system size taken as $2^{n}$ in spectral techniques?

52. Why is a Fourier transform used in spectral analysis? What is the transformation in this case?

53. How is Fourier transform used to analytically solve the diffusion equation?⁵

54. Where does Fast Fourier Transform come into the picture?

55. How is the classical diffusion equation solved using FFT?

56. What is the problem of stability vs. accuracy? When is explicit method preferred considering this aspect?

57. Is composition field a vector or scalar? Is Diffusivity a vector or scalar? ⁶

58. How are scalars, vectors and tensors defined in terms of the indices used to describe them?

59. How is the Einstein summation convention used to write the classical diffusion equation?

60. How do you check if the free indices match on both sides of the equation? How is the dummy index managed while writing the equation?

61. How do you get rid of the dependence on the frame of reference when writing the equation? How is the transformation matrix written in such a case? How is a vector defined using co-ordinate transformation?

62. How is a property tensor different from the transformation matrix? Can the transformation matrix exist in the absence of more than one co-ordinate system?

63. How do you define a second rank tensor in terms of multiplication needed by transformation tensors? How are vectors defined in this context?

64. What is a symmetric tensor? What is a skew symmetric tensor? Can this idea be applied to vectors? Why?

65. What is the Neumann’s principle? What are its implications?

66. How is a group defined? What essential properties does it satisfy?

67. What is the order of a finite group? What is a cyclic group?

68. What is point group symmetry? What is the difference between rotation and inversion operations?

69. How do you show that for a cubic system, the second rank tensor (Diffusivity) will be isotropic?

70. How is the condition for maxima and minima derived from Taylor series expansion? What are the assumptions here?

71. How does this lead to the fact that the curvature is positive for opening up and curvature is negative for opening down curves?

72. How do you find out extremum of a functional?

73. How you find the shortest distance between two points on a rubber ball? How is the functional written in this case? (Geodesic problem)

74. What is the brachistochrone problem? How is the functional written for this case?

75. What is the condition for finding extremum in functionals?

76. How is the Euler-Lagrange equation derived? How do you determine the first variation for a given functional?

77. How is the delta operator defined? What is variational derivative computed?

78. How does the boundary condition come up as a consequence of defining the variational derivative?

79. Elaborate : “There is a cost associated with making the interface diffused instead of keeping it discontinuous”. How does the composition gradient come into the picture and give intuition about the CH equation?

80. How does including the gradient include the shortcomings of the classical diffusion model?

81. How is the free energy functional written in terms of the concentration gradients and higher order derivatives?

82. Why do all odd rank tensors become zero when inversion symmetry is assumed? How do you approach this argument considering that we are looking for a free energy minima?

83. Cubic tensors with symmetry $\kappa_{i,j}=\kappa_{j,i}$ will be isotropic. How?

84. How is chemical potential written in terms of variational derivative? Why is the $1/N$ needed?

85. How can the diffusion equation be written using the chemical potential?

86. What is the Cahn-Hilliard equation? How is the $1/N$ got rid of from the derivation?

87. Where does the the $\nabla^{2}(\frac{\partial f}{\partial C})$ term originate in the diffusion equation?

88. Which is the modification term to the diffusion equation? What effects are incorporated in the modification term? ⁷

89. What determines the thickness of the inter-diffusion region? ⁸

90. How is the non-classical form different from the classical form of diffusion equation?

91. How does the modification term ensure that the growth of smaller wavelengths is inhibited?

92. In solidification problems why can a free energy functional not be used? Why is an entropy functional used?

93. How does the sign of the $R(\beta)$ term decide the critical growth which decides which wavelength leads to growth? How does a plot of $R(\beta)$ vs $\beta$ look like?

94. How is the excess free energy (interfacial energy) calculated using FFTW and inverse FFTW?

95. How do you derive the analytical solution to Cahn-Hilliard equation in 1D?

96. How do you verify that the solution changes with variation of A and $\kappa$ as would be expected with the change in barrier or concentration gradients?

97. How do you analytically show that the interfacial energy is given as $\sigma = \frac{\gamma}{3}$ where $\gamma = \sqrt{A \times \kappa}$? How is the width of the interface calculated?

98. How does width of the interface depend on A and $\kappa$? ⁹

99. How does the Cahn-Hilliard solution by FDM compare against Spectral techniques in terms of accuracy for a given computational capability? Why?

100. Why is B2 called ordered BCC? How is the order over and above the order of regular BCC?

101. What is the difference between spinodal decomposition and ordering in terms of nature of field variables?

102. What is the Allen-Cahn equation? Why is it known as reaction-diffusion equation?

103. Why is grain growth modelled using Allen-Cahn equation? Why does this case not consider mass conservation?

104. How is concentration a conserved order parameter?

105. What are the steps involved in putting together a phase-field model?¹⁰

106. What effect does Gibbs-Thomson effect describe?

107. The equilibrium concentration of $\beta$ phase will be different in the case of a curved interface vs. a flat interface. How do we quantify this difference in concentration?

108. Which will be greater? Equilibrium concentration of curved interface or flat interface?

109. How can this quantification be shown using a GX diagram? How is it related to the second derivative of free energy?

110. Why can the Gibbs-Thomson effect be simulated using the same code as that of Cahn-Hilliard equation?

111. How can we show that the relevant radius where Gibbs-Thomson effect becomes important is in the order of nanometers? Or in other words, why will nano-sized particles have a strong effect?

112. How can this be used to explain the fact that gold nano-particles will melt at a temperature that is different from bulk gold?

113. How can grain growth be simulated using Allen-Cahn equation? How can the data from this be used for validation based on the trends of the analytical solution i.e. $R^{2}-R_{o}^{2} \propto t$ ?

114. What is the need to take an additional order parameter when studying grain growth using the Zener-Frank model?¹¹

115. What are the constraints in the formulation of the order parameter in the Fan-Chen model?

116. What would be the genesis of the argument for varying $\kappa$ in the Fan-Chen model? ¹²

117. What is the need to have a grain boundary function in the Fan-Chen model?

118. “Fourier Transform implicitly assumes periodic boundary conditions” How is this idea used to code Fan-Chen model without explicitly considering periodic boundary conditions?

119. How can you show the different grains with different colours such that the microstructure looks like the EBSD maps?

120. What are the different ways of formulating the grain boundary grooving problem using periodic boundary conditions?

Footnotes