# Difference between revisions of "Thermodynamics of Solid and Fluid Surfaces"

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==Plane Fluid Interfaces - Area Work Term== | ==Plane Fluid Interfaces - Area Work Term== | ||

− | For surfaces between fluids, Gibbs defined surface tension <math>\sigma</math> in a work term. The quantity <math>\sigma</math> is taken as a force per unit length of surface perimeter. When a portion of the perimeter moves an infinitesimal distance in the | + | For surfaces between fluids, Gibbs defined surface tension <math>\sigma</math> in a work term. The quantity <math>\sigma</math> is taken as a force per unit length of surface perimeter. When a portion of the perimeter moves an infinitesimal distance in the plane of the surface, the area change <math>dA</math> is the product of perimeter length and distance moved. Thus <math>\sigma dA</math> is a forces-times-distance work term, and it appears in the combined first and second laws of thermodynamics as follows: |

<center><math>dU = TdS - PdV + \displaystyle \sum_{i} \mu_i dN_i + \sigma dA\ [1]</math> </center> | <center><math>dU = TdS - PdV + \displaystyle \sum_{i} \mu_i dN_i + \sigma dA\ [1]</math> </center> | ||

− | Strictly speaking, <math>\sigma</math> is defined as the change in internal energy when the area is reversibly | + | Strictly speaking, <math>\sigma</math> is defined as the change in internal energy when the area is reversibly increased at constant entropy and volume in a closed system. |

− | Gibbs | + | Gibbs went on to show that, in equilibrium, temperature and chemical potentials are constant everywhere within the system. For the special case of plane boundaries in equilibrium, pressure is also constant. For a system containing a plane surface, Equation 1 can be integrated to give: |

<center><math>U = TS - PV + \displaystyle \sum_{i} \mu_i dN_i + \sigma A\ [2] </math></center> | <center><math>U = TS - PV + \displaystyle \sum_{i} \mu_i dN_i + \sigma A\ [2] </math></center> | ||

− | This can be rearranged to give <math>\sigma</math>, the excess free energy due to the presence of the surface | + | This can be rearranged to give <math>\sigma</math>, the excess free energy due to the presence of the surface: |

<center><math> \sigma = (1/A) [U - TS + PV - \displaystyle \sum_{i} \mu_i dN_i]\ [3]</math></center> | <center><math> \sigma = (1/A) [U - TS + PV - \displaystyle \sum_{i} \mu_i dN_i]\ [3]</math></center> | ||

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Here <math>U - TS + PV</math> is the Gibbs free energy of the system with the interface, and <math>\displaystyle \sum_{i} \mu_i dN_i</math> is the Gibbs free energy without the interface. | Here <math>U - TS + PV</math> is the Gibbs free energy of the system with the interface, and <math>\displaystyle \sum_{i} \mu_i dN_i</math> is the Gibbs free energy without the interface. | ||

− | Equations [2] and [3] pertain to the entire system. Because most of this system is occupied by homogeneous phases, it is convenient to subtract out their contribution. | + | Equations [2] and [3] pertain to the entire system. Because most of this system is occupied by homogeneous phases (which we are not so interested in), it is convenient to subtract out their contribution. For homogeneous phases <math>\alpha</math> and <math>\beta</math>, we have: |

<center><math> 0 = [U^{\alpha} - TS^{\alpha} + PV^{\alpha} - \displaystyle \sum_{i} \mu_i dN_i^{\alpha}]\ [4]</math></center> | <center><math> 0 = [U^{\alpha} - TS^{\alpha} + PV^{\alpha} - \displaystyle \sum_{i} \mu_i dN_i^{\alpha}]\ [4]</math></center> | ||

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<center><math> 0 = [U^{\beta} - TS^{\beta} + PV^{\beta} - \displaystyle \sum_{i} \mu_i dN_i^{\beta}]\ [5]</math></center> | <center><math> 0 = [U^{\beta} - TS^{\beta} + PV^{\beta} - \displaystyle \sum_{i} \mu_i dN_i^{\beta}]\ [5]</math></center> | ||

− | It is apparent that the homogeneous phases contribute nothing to the RHS of Equation 3, and Equation 3 continues to hold if we remove from our consideration as much of the homogeneous phases as we wish. What is left then is a layer which is | + | It is apparent that the homogeneous phases contribute nothing to the RHS of Equation 3, and Equation 3 continues to hold if we remove from our consideration as much of the homogeneous phases as we wish. What is left then is a layer which is thick enough to extend into both homogeneous phases. The location of the bounds of this layer parallel to the surface do not matter, except that they must reach far enough into each phase such that there is no longer any influence of the surface. Equations 4 and 5 hold for any incremental addition to the thickness of this layer. If we define the layer content per unit surface area of extensive quantities and denote them by the symbols [U] [S] [V] and [<math>N_i</math>], then <math>\sigma</math>, which was defined as a force per unit length in a work term, is identically equal to the Gibbs free energy of forming this layer containing the surface: |

+ | |||

+ | <center><math> \sigma = [U] - T[S] + P[V] - \displaystyle \sum_{i} \mu_i [dN_i]\ [6]</math></center> |

## Revision as of 15:55, 16 April 2012

Entry by Emily Redston, AP 226, Spring 2012

Work in Progress

## Reference

*Thermodynamics of Solid and Fluid Surfaces* by J. W. Cahn. Segregation to Interfaces, ASM Seminar Series (1978) pp. 3-23.

## Keywords

thermodynamics, interfaces, excess free energy

## Introduction

In this paper, Cahn provides a slightly different take on Gibbs' original development of the thermodynamics of fluid and solid surfaces. His goal was to develop a mathematical system that was physically equivalent to Gibbs' formulation while being more easily applicable. The key change he made is that he concentrated on the system itself rather than defining excess quantities by comparing the actual system to a hypothetical system not containing the surface. This avoids problems with the identification of the location of the dividing surface.

## Plane Fluid Interfaces - Area Work Term

For surfaces between fluids, Gibbs defined surface tension <math>\sigma</math> in a work term. The quantity <math>\sigma</math> is taken as a force per unit length of surface perimeter. When a portion of the perimeter moves an infinitesimal distance in the plane of the surface, the area change <math>dA</math> is the product of perimeter length and distance moved. Thus <math>\sigma dA</math> is a forces-times-distance work term, and it appears in the combined first and second laws of thermodynamics as follows:

Strictly speaking, <math>\sigma</math> is defined as the change in internal energy when the area is reversibly increased at constant entropy and volume in a closed system.

Gibbs went on to show that, in equilibrium, temperature and chemical potentials are constant everywhere within the system. For the special case of plane boundaries in equilibrium, pressure is also constant. For a system containing a plane surface, Equation 1 can be integrated to give:

This can be rearranged to give <math>\sigma</math>, the excess free energy due to the presence of the surface:

Here <math>U - TS + PV</math> is the Gibbs free energy of the system with the interface, and <math>\displaystyle \sum_{i} \mu_i dN_i</math> is the Gibbs free energy without the interface.

Equations [2] and [3] pertain to the entire system. Because most of this system is occupied by homogeneous phases (which we are not so interested in), it is convenient to subtract out their contribution. For homogeneous phases <math>\alpha</math> and <math>\beta</math>, we have:

It is apparent that the homogeneous phases contribute nothing to the RHS of Equation 3, and Equation 3 continues to hold if we remove from our consideration as much of the homogeneous phases as we wish. What is left then is a layer which is thick enough to extend into both homogeneous phases. The location of the bounds of this layer parallel to the surface do not matter, except that they must reach far enough into each phase such that there is no longer any influence of the surface. Equations 4 and 5 hold for any incremental addition to the thickness of this layer. If we define the layer content per unit surface area of extensive quantities and denote them by the symbols [U] [S] [V] and [<math>N_i</math>], then <math>\sigma</math>, which was defined as a force per unit length in a work term, is identically equal to the Gibbs free energy of forming this layer containing the surface: