### Abstract

The thermodynamics of nonuniform solutions usually employs a gradient energy term in the expression for the local free energy. This arises from retaining the first nonvanishing terms of a MacLaurin expansion of the free energy in the composition and its derivatives. The nature of this expansion is discussed, and the formulation is extended to include derivatives higher than the second. General, a priori arguments suggest that a reasonable next approximation is f*=f(c) +κ_{1}∇^{2}C+κ_{2}(∇c) ^{2}+κ_{3}∇^{4}c+κ_{4}(∇ ^{2}c)^{2}+κ_{5}∇c·∇ ^{3}c+κ_{6}(∇c)^{2}∇ ^{2}c+κ_{7}(∇c)^{4}, for an isotropic solution. The results for a cubic system are also presented. The significance of the coefficients are discussed using a simple mean field pair interaction analysis. This interpretation provides physical insight into the nature of the terms and suggests that in the absence of specific knowledge about a particular situation, the derivative terms may be ordered in the following sequence of decreasing importance: κ_{1}∇^{2}c, κ_{3}∇^{4}c, [k_{2}(∇c)^{2}, κ_{5}∇c·∇^{3}c], [κ _{4}(∇^{2}c)^{2}, κ^{6}(∇c) ^{2}∇^{2}c, κ_{7}(∇c)^{4}]. The effect on the profile and interfacial tension of a flat diffuse interface of the K3V*c and the -4(Vc)*terms are determined. The pair interaction analysis indicates that the κ_{3}∇^{4}c term sharpens the interface profile and decreases the interface tension. When K? is positive, the κ_{7}(∇c)^{4} term increases the interfacial tension and makes the interface more diffuse.

Original language | English (US) |
---|---|

Pages (from-to) | 4036-4042 |

Number of pages | 7 |

Journal | The Journal of Chemical Physics |

Volume | 56 |

Issue number | 8 |

State | Published - 1972 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*56*(8), 4036-4042.

**Higher derivatives in the thermodynamics of nonuniform solutions. I. Basic interface theory.** / Hopper, R. W.; Uhlmann, Donald R.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 56, no. 8, pp. 4036-4042.

}

TY - JOUR

T1 - Higher derivatives in the thermodynamics of nonuniform solutions. I. Basic interface theory

AU - Hopper, R. W.

AU - Uhlmann, Donald R

PY - 1972

Y1 - 1972

N2 - The thermodynamics of nonuniform solutions usually employs a gradient energy term in the expression for the local free energy. This arises from retaining the first nonvanishing terms of a MacLaurin expansion of the free energy in the composition and its derivatives. The nature of this expansion is discussed, and the formulation is extended to include derivatives higher than the second. General, a priori arguments suggest that a reasonable next approximation is f*=f(c) +κ1∇2C+κ2(∇c) 2+κ3∇4c+κ4(∇ 2c)2+κ5∇c·∇ 3c+κ6(∇c)2∇ 2c+κ7(∇c)4, for an isotropic solution. The results for a cubic system are also presented. The significance of the coefficients are discussed using a simple mean field pair interaction analysis. This interpretation provides physical insight into the nature of the terms and suggests that in the absence of specific knowledge about a particular situation, the derivative terms may be ordered in the following sequence of decreasing importance: κ1∇2c, κ3∇4c, [k2(∇c)2, κ5∇c·∇3c], [κ 4(∇2c)2, κ6(∇c) 2∇2c, κ7(∇c)4]. The effect on the profile and interfacial tension of a flat diffuse interface of the K3V*c and the -4(Vc)*terms are determined. The pair interaction analysis indicates that the κ3∇4c term sharpens the interface profile and decreases the interface tension. When K? is positive, the κ7(∇c)4 term increases the interfacial tension and makes the interface more diffuse.

AB - The thermodynamics of nonuniform solutions usually employs a gradient energy term in the expression for the local free energy. This arises from retaining the first nonvanishing terms of a MacLaurin expansion of the free energy in the composition and its derivatives. The nature of this expansion is discussed, and the formulation is extended to include derivatives higher than the second. General, a priori arguments suggest that a reasonable next approximation is f*=f(c) +κ1∇2C+κ2(∇c) 2+κ3∇4c+κ4(∇ 2c)2+κ5∇c·∇ 3c+κ6(∇c)2∇ 2c+κ7(∇c)4, for an isotropic solution. The results for a cubic system are also presented. The significance of the coefficients are discussed using a simple mean field pair interaction analysis. This interpretation provides physical insight into the nature of the terms and suggests that in the absence of specific knowledge about a particular situation, the derivative terms may be ordered in the following sequence of decreasing importance: κ1∇2c, κ3∇4c, [k2(∇c)2, κ5∇c·∇3c], [κ 4(∇2c)2, κ6(∇c) 2∇2c, κ7(∇c)4]. The effect on the profile and interfacial tension of a flat diffuse interface of the K3V*c and the -4(Vc)*terms are determined. The pair interaction analysis indicates that the κ3∇4c term sharpens the interface profile and decreases the interface tension. When K? is positive, the κ7(∇c)4 term increases the interfacial tension and makes the interface more diffuse.

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M3 - Article

VL - 56

SP - 4036

EP - 4042

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 8

ER -