# Gravity Induced Diffusion: Sedimentation in Condensed Matter

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## Abstract

In the paper the mathematical model describing the gravity induced sedimentation process in multicomponent system is presented. The model is based on the generalized interdiffusion method (bi-velocity method): (1) the volume continuity equation, (2) equation of motion, (3) Cauchy stress tensor and (4) Nernst Planck flux formulae. The method is applied to simulate the sedimentation processes of the selenium isotopes and the ternary InSbGe system.

### Keywords

gravity interdiffusion pressure sedimentation thermodynamics volume continuity## Introduction

The sedimentation process, i.e. gravity induced diffusion, is generally known as the transport phenomena of macroscopic solutes induced by gravitational or centrifugal field.[1] The sedimentation of isotope atoms in liquids is important from the point of view of separation of isotopes.[2] Enriched isotopes are crucial in the fields of atomic energy ^{235}U, ^{6}Li, etc.[3] medical treatment ^{50}Cr, ^{168}Yb, etc.[4] and in the information technology field for quantum computing ^{29}Si, etc.[5] In the 1970s, Barr and Smith and Anthony[6,7] investigated the sedimentation of Au atoms in elemental metals (K, In and Pb) with low melting temperatures under maximum acceleration fields of 1-2 × 10^{5} g. In 1996, Mashimo et al. developed an ultracentrifuge apparatus that can generate an acceleration field up to 10^{6} g for long times at high temperatures.[8] The second-generation ultracentrifuge was newly developed, in 2001.[9] Using this apparatus he studied the sedimentation the substitutional solute atoms in different elements[10, 11, 12] e.g. ultracentrifuge experiments at high temperature on elemental selenium to examine the sedimentation of isotope atoms in liquid matter.[2]

Almost all sedimentation processes are described by the Lamm equation proposed in 1929.[13] This classical equation was formulated for axially symmetric macroscopic particles on the basis of macroscopic mechanics and thermodynamics. The fundamental idea of Lamm is that the driving force between the centrifugal field acting on the macro solute is given by the difference between the centrifugal force and the buoyant force caused by the surrounding liquid solvent.[1] Several approaches to the Lamm equation were made. Brenner and Condiff extended method for particles of arbitrary shape and considered the rotation effect.[14] However in these approaches the self-interaction effects among macroscopic solutes caused by the centrifugal field are not considered, therefore, the driving force is not expressed as a function of solute concentration, i.e. the density of the mixture solvent does not change with concentration.[1]

In 1988 Mashimo proposed an equation for describing the diffusion of atoms induced by the centrifugal field in a two component system.[1] His theory is based on linear irreversible thermodynamics (Onsager relations) and the Nernst-Einstein relation considering the effect of concentration change.[1] The Mashimo’s theory considers the chemical activity of the simulated system by introducing the Q parameter (the ratios of the diffusion coefficient matrix). The disadvantage of the Mashimo theory is the assumption of the unknown diffusivity under the combined stress and gravity forces.

In the previous paper the bi-velocity method, the rigorous mathematical derivation of the mass, momentum and energy conservations laws and the results of the method for binary Bi-Sb diffusion couple were presented.[15] In this work the model is applied to simulate the multicomponent sedimentation process when the ideality sweeping statement is applied (i.e. chemical activity is equal to the concentration). Using the bi-velocity method for the first time the application of the formalism in modeling the ternary interdiffusion of selenium isotopes is shown.

## Mashimo et al.[16,17] Sedimentation Model

*J*

_{i},

*c*

_{i}and

*M*

_{i}denote the overall flux (diffusion and convection), the concentration and atomic mass of

*i*th component, respectively. \( M_{j}^{*} \) is the effective mass of the surrounding solvent mixture per molar volume.

*R*,

*T*and

*g*denotes the gas constant, temperature and gravity force, respectively. \( D_{il}^{1} \) denotes the intrinsic diffusion coefficient, \( D_{ij}^{2} \) is the unknown (experimentally determined) sedimentation coefficient under the gravity force.

*j*th component.

*L*

_{ij}, and chemical potential,

*μ*

_{j}

^{ch}:[16]

The presented model was used by Mashimo to simulate the sedimentation of binary systems, mainly the Bi-Sb, Se-Te and In-Pb. In the next section the mathematical model describing the gravity induced sedimentation process in multicomponent system is presented.

## The Conservation of Momentum and Volume Continuity (Bi-velocity Method)

*i*th component,

*υ*

_{i}, is a sum of diffusion, Darken (convection) and deformation velocities, respectively:

*B*

_{i}and

*F*

_{i}denote the mobility and the forces acting on the

*i*th component, respectively;

_{i}is the diffusion potential;

*μ*

_{i}

^{ch}and

*μ*

_{i}

^{m}denote the chemical and mechanical potentials.

*F*

_{i}

^{centr}is the centrifugal force acting on the

*i*th component (\( F_{i}^{\text{centr}} \) =

*M*

_{i}

*g*, where

*M*

_{i}is the molar mass of the

*i*th component and

*g*is gravitation force).

*i*th component and

*p*

^{g}is a pressure, Eq 9.

*F*

^{ext}is induced by the buoyant force, thus:

*i*-component momentum change in a subregion \( \left| {\Upomega_{i} \left( t \right)} \right| \):

*r*is a radius and

*ω*the angular speed.

### Volume Continuity Eq 15

*t*, \( \left| {\Upomega \left( t \right)} \right| = \int_{\Upomega \left( t \right)} {dx} \), is affected by the distribution of every mixture component and stress field. In this paper the limiting situation, when the volume changes due to the external stress field only (gravity), is analyzed, thus:

### Cauchy Stress Tensor

### Initial Conditions

The initial density distribution of the *i*th component: \( \mathop \rho \limits^{0}_{i} \).

### Boundary Conditions

The diffusion flux of the *i*th component at the mixture boundary, \( \rho_{i} \upsilon_{i}^{d} \left( t \right) \cdot n = 0\quad {\text{on}}\quad \partial \left| \Upomega \right| \), where *n* represents the unit vector normal to the boundary i.e., in the closed system analyzed here, the mass flow through the mixture boundaries does not occur.

### Numerical Solution

The solution of the presented model was obtained in several steps: (1) discretization of the problem (mathematical reformulation), (2) numerical solution using Finite Difference Method (FDM), (3) solving the resulting system of ordinary differential equations and computer implementation of method. The differential equations solver based on the Runge-Kutta-Fehlberg method with adaptive stepsize control. Fehlberg discovered a fifth-order method with six function evaluations where another combination of this six functions gives a fourth-order method. The difference between these two estimates can then be used as an estimate of the truncation error to adjust the stepsize.[23]

## Results

The data used to simulate the sedimentation process in Se-Te binary systems

Component | Atomic mass ( | Molar volume (Ω | Initial composition, at.% |
---|---|---|---|

SeTe system, | |||

Se | 78.96 | 17.99 | 70 |

Te | 127.6 | 20.45 | 30 |

^{82}Se/

^{76}Se is compared with the experimental results.[25] The simulations were performed in three-components system. The two components were the isotopes and the third was the reference ideal component. The data used in calculations is shown in Table 2.

The data used to simulate the sedimentation process for composition ratio of ^{82}Se/^{76}Se

Component | Atomic mass ( | Molar volume (Ω | Initial composition, at.% |
---|---|---|---|

| |||

| 82 | 16.45 | 9.2 |

| 76 | 16.45 | 9.0 |

Se | 78.96 | 16.45 | 81.8 |

^{82}Se/

^{76}Se compared with the (a) experimental results and (b) Mashimo model[2] are shown in Fig. 3.

The data used to simulate the sedimentation process of InGeSb ternary system

Component | Atomic mass ( | Molar volume (Ω | Initial composition, at.% |
---|---|---|---|

| |||

In | 114.818 | 15.71 | 30 |

Sb | 121.769 | 18.18 | 30 |

Ge | 72.64 | 13.65 | 40 |

## Conclusions

In this paper a mathematical model describing the gravity induced sedimentation process in multicomponent system is presented. The model is based on the generalized interdiffusion method (bi-velocity method): (1) the volume continuity equation, (2) equation of motion, (3) Cauchy stress tensor and (4) Nernst Planck flux formulae. In the present approach the plastic deformation is neglected. The method was applied to simulate the sedimentation processes of the selenium isotopes and the ternary InSbGe system.

*p*

^{g}and (2) pressure generated by the differences in composition (chemical forces),

*p*

^{d}. The

*p*

^{d}can be calculated from overall dilatation of the system:[27]

*E*denote the Young modulus and

*v*Poisson number. In such a case the diffusion velocity can be rewritten as:

## Notes

### Acknowledgments

This work has been supported by the Ministry of Higher Education and Science in Poland, project N N507 505 138.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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