Thermodynamics of diffusion (extracurricular material - not tested)

Size: px

Start display at page:

Download "Thermodynamics of diffusion (extracurricular material - not tested)"

Natalie Butler
5 years ago
Views:

1 Thermodynamics of diffusion (etracurricular material - not tested) riving force for diffusion iffusion in ideal and real solutions Thermodynamic factor iffusion against the concentration gradient Spinodal decomposition vs. nucleation and growth SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

2 riving force for diffusion In general, it is common for atoms to diffuse from regions of high concentration towards the regions of low concentration. Thus, the phenomenological Fick s laws describe the diffusion in terms of the relationships between the diffusion flu and concentration gradient. E.g., consider ideal solution: toms here jump randomly both right and left ut there are not many atoms here to jump to the left s a result there is a net flu of atoms from left to right. The thermodynamic properties of solid solutions, however, play an important role in diffusion and, under certain conditions, may even induce the diffusion against the concentration gradient (<)! SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

3 riving force for diffusion (I) The empirical Fick s first law assumes proportionality between the diffusion flu and the concentration gradient. ut thermodynamics tells us that any spontaneous process should go in the direction of minimization of the free energy. s we can see from the eamples below, atoms can diffuse from regions of high concentration towards the regions of low concentration down the concentration gradient (left) as well as from the regions of low concentration towards the regions of high concentration up the concentration gradient? (right) -rich -rich -rich -rich SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

4 riving force for diffusion (II) SE 35, Phase iagrams and Kinetics, Leonid Zhigilei iffusion occur so that the free energy is minimized and is therefore driven by the gradient of free energy. The chemical potential of atoms of type can be defined as the free energy per mole of atoms. toms migrate so as to remove differences in chemical potential. iffusion ceases at equilibrium, when and Therefore, the free energy gradient can be epressed through the chemical potential gradient: where is the atomic mobility of atoms. In both cases the and atoms are diffusing from the regions where chemical potential is high to the regions where chemical potential is lower. Thedrivingforcefor diffusion is gradient of chemical potential.

5 riving force for diffusion (III) hemical potential gradient is the driving force for diffusion: where is mobility of atoms if, then - gradient of chemical potential is in the same direction as the concentration gradient. if, then - diffusion occurs against the concentration gradient! For eample, we can identify regions with negative / in a system with miscibility gap: We will discuss the behavior of homogeneous solution cooled within the miscibility gap later, after deriving equations for diffusion flu in ideal and regular solutions. SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

6 SE 35, Phase iagrams and Kinetics, Leonid Zhigilei Lets consider diffusion driven by the chemical potential gradient for ideal and regular solutions. riving force for diffusion (IV) For an ideal solution: The factor in brackets is termed the thermodynamic factor F. It defines how inter-atomic interaction affects the diffusion of the atoms in the presence of concentration gradient. ln For a regular solution: Ω ln Ω Ω F F

7 riving force for diffusion (V) s shown below, the thermodynamic factor is the same for both species and at a given composition and is related to the curvature of the free energy curve. reg ln- Ω ln ln ln Ω Ω - ln- -ln - ln - ln - - ln Ω - Ω Ω F F Ω SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

8 SE 35, Phase iagrams and Kinetics, Leonid Zhigilei The presence of a strain energy gradient, an electric field, or a temperature gradient can also affect the diffusion and, in particular, can induce diffusion of atoms against the concentration gradient. For eample, in the presence of a strain energy gradient the equation for the chemical potential will include an elastic strain energy term E(). For a regular solution we have riving force for diffusion (VI) E F E ln Ω E F E F F

9 iffusion against the concentration gradient: Spinodal ecomposition When the free energy curvature is negative, the thermodynamic factor F is negative, and the diffusion is directed against the concentration gradient: F F T hemical spinodal iscibility gap T SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

10 Spinodal ecomposition Homogeneous solution cooled into the miscibility gap will decompose into and so that the total free energy of the system decreases. The mechanism of decomposition into and is different within the chemical spinodal region and outside (in the nucleation regions). Let s consider small fluctuations around the average composition : - spatial coordinate Free energy decreases as a result of an arbitrary infinitesimal fluctuation in composition the system is unstable Free energy increases as a result of an infinitesimal fluctuation in composition the system is stable with respect to small fluctuations (fluctuations are small) - - SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

11 Spinodal ecomposition lthough the system within the miscibility gap but outside the spinodal region is stable (metastable) with respect to small fluctuations, it is unstable to the separation into and determined by the common tangent construction. There is large difference in composition between and and large composition fluctuations are required in order to decrease the free energy. process of formation of a large composition fluctuation is called nucleation. The phase separation is occurring in this case by nucleation and growth (will be discussed later). Spinodal decomposition Nucleation and growth atoms atoms atoms atoms spatial coordinate SE 35, Phase iagrams and Kinetics, Leonid Zhigilei spatial coordinate

12 Region of spinodal decomposition on a phase diagram with a miscibility gap T T Nucleation and growth Spinodal decomposition T T SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

13 omputer simulation of spinodal decomposition in a binary alloy SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

omputer simulation of laser overheating & eplosive boiling Short pulse laser irradiation leads to strong superheating and rapid decomposition of a surface region of the

14 omputer simulation of laser overheating & eplosive boiling Short pulse laser irradiation leads to strong superheating and rapid decomposition of a surface region of the target into a miture of gas phase atoms and liquid droplets SE 35, Phase iagrams and Kinetics, Leonid Zhigilei

Spinodal decomposition

Part V. Solid-solid transformations II Module 1 : Spinodal decomposition Part V. Solid-solid transformations II In this part, we discuss two transformations namely spinodal decomposition and ordering.