ABSTRACT
This chapter constructs the grand potential phase field functional for solidification with multiple order parameters in the spirit of the approach developed by Ofori-Opoku et al. [13]. To proceed, we first clarify some notion and introduce some variables. Let N denote the number of distinct ordered phases or orientations in the system. Define an order parameter vector, ϕ ( r → ) = ( ϕ 1 ( r → ) , ϕ 2 ( r → ) , ⋯ , ϕ N ( r → ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_1.tif"/> , the components of which vary from 0 < ϕ i < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_2.tif"/> ( i = 1 , 2 , 3 , ⋯ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_3.tif"/> ) and represent the order of one of N solid phases (or orientations) at any location in space, where ϕ i ( r → ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_4.tif"/> represents liquid region and ϕ i ( r → ) > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_5.tif"/> and ordered regions covered by grain/phase i. Where order parameters interact (e.g. grains merging), they will always be constrained to satisfy ϕ 1 + ϕ 2 + ⋯ + ϕ N ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_6.tif"/> . For an n-component mixture, we define μ ( r → ) = ( μ 1 ( r → ) , μ 2 ( r → ) , ⋯ , μ n − 1 ( r → ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_7.tif"/> and c ( r → ) = ( c 1 ( r → ) , c 2 ( r → ) , ⋯ , c n − 1 ( r → ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_8.tif"/> , which are vector fields representing, respectively, n − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_9.tif"/> independent chemical potentials and impurity concentration fields at any location in space. In what follows, we also define a set of interpolation functions denoted by g α ( ϕ ) ≡ g ( ϕ α ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_10.tif"/> (where α indexes some component of ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_11.tif"/> ), whose specific form used here is chosen such as to satisfy g ( ϕ α ) = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_12.tif"/> when ϕ α = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_13.tif"/> (i.e., the component α of ϕ ( r → ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_14.tif"/> equals one) and g ( ϕ α ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_15.tif"/> when ϕ α = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_16.tif"/> . Note that in this description where ϕ α = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_17.tif"/> , all ϕ β ≠ α ≡ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math4_18.tif"/> .
