### Abstract

Diffusion and reaction in heterogeneous media plays an important role in a variety of processes arising in the physical and biological sciences. The determination of the relaxation times T_{n} (n = 1,2,...) and the mean survival time τ is considered for diffusion and reaction among partially absorbing traps with dimensionless surface rate constant κ̄. The limits κ̄= ∞ and κ̄ = 0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. Rigorous lower bounds on the principal (or largest) relaxation time T_{1} and mean survival time τ for arbitrary κ̄ are derived in terms of the pore size distribution P(δ). Here P(δ)dδ is the probability that a randomly chosen point in the pore region lies at a distance δ and δ + dδ from the nearest point on the pore-trap interface. The aforementioned moments and hence the bounds on T_{1} and τ are evaluated for distributions of interpenetrable spherical traps. The length scales 〈δ〉 and 〈δ^{2}〉^{1/2}, under certain conditions, can yield useful information about the times T_{1} and τ, underscoring the importance of experimentally measuring or theoretically determining the pore size distribution P(δ). Moreover, rigorous relations between the relaxation times T_{n} and the mean survival time are proved. One states that τ is a certain weighted sum over the T_{n}, while another bounds τ from above and below in terms of the principal relaxation time T_{1}. Consequences of these relationships are examined for diffusion interior and exterior to distributions of spheres. Finally, we note the connection between the times T_{1} and τ and the fluid permeability for flow through porous media, in light of a previously proved theorem, and nuclear magnetic resonance (NMR) relaxation in fluid-saturated porous media.

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

Pages (from-to) | 6477-6489 |

Number of pages | 13 |

Journal | The Journal of chemical physics |

Volume | 95 |

Issue number | 9 |

State | Published - 1991 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of chemical physics*,

*95*(9), 6477-6489.

**Diffusion and reaction in heterogeneous media : Pore size distribution, relaxation times, and mean survival time.** / Torquato, S.; Avellaneda, Marco.

Research output: Contribution to journal › Article

*The Journal of chemical physics*, vol. 95, no. 9, pp. 6477-6489.

}

TY - JOUR

T1 - Diffusion and reaction in heterogeneous media

T2 - Pore size distribution, relaxation times, and mean survival time

AU - Torquato, S.

AU - Avellaneda, Marco

PY - 1991

Y1 - 1991

N2 - Diffusion and reaction in heterogeneous media plays an important role in a variety of processes arising in the physical and biological sciences. The determination of the relaxation times Tn (n = 1,2,...) and the mean survival time τ is considered for diffusion and reaction among partially absorbing traps with dimensionless surface rate constant κ̄. The limits κ̄= ∞ and κ̄ = 0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. Rigorous lower bounds on the principal (or largest) relaxation time T1 and mean survival time τ for arbitrary κ̄ are derived in terms of the pore size distribution P(δ). Here P(δ)dδ is the probability that a randomly chosen point in the pore region lies at a distance δ and δ + dδ from the nearest point on the pore-trap interface. The aforementioned moments and hence the bounds on T1 and τ are evaluated for distributions of interpenetrable spherical traps. The length scales 〈δ〉 and 〈δ2〉1/2, under certain conditions, can yield useful information about the times T1 and τ, underscoring the importance of experimentally measuring or theoretically determining the pore size distribution P(δ). Moreover, rigorous relations between the relaxation times Tn and the mean survival time are proved. One states that τ is a certain weighted sum over the Tn, while another bounds τ from above and below in terms of the principal relaxation time T1. Consequences of these relationships are examined for diffusion interior and exterior to distributions of spheres. Finally, we note the connection between the times T1 and τ and the fluid permeability for flow through porous media, in light of a previously proved theorem, and nuclear magnetic resonance (NMR) relaxation in fluid-saturated porous media.

AB - Diffusion and reaction in heterogeneous media plays an important role in a variety of processes arising in the physical and biological sciences. The determination of the relaxation times Tn (n = 1,2,...) and the mean survival time τ is considered for diffusion and reaction among partially absorbing traps with dimensionless surface rate constant κ̄. The limits κ̄= ∞ and κ̄ = 0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. Rigorous lower bounds on the principal (or largest) relaxation time T1 and mean survival time τ for arbitrary κ̄ are derived in terms of the pore size distribution P(δ). Here P(δ)dδ is the probability that a randomly chosen point in the pore region lies at a distance δ and δ + dδ from the nearest point on the pore-trap interface. The aforementioned moments and hence the bounds on T1 and τ are evaluated for distributions of interpenetrable spherical traps. The length scales 〈δ〉 and 〈δ2〉1/2, under certain conditions, can yield useful information about the times T1 and τ, underscoring the importance of experimentally measuring or theoretically determining the pore size distribution P(δ). Moreover, rigorous relations between the relaxation times Tn and the mean survival time are proved. One states that τ is a certain weighted sum over the Tn, while another bounds τ from above and below in terms of the principal relaxation time T1. Consequences of these relationships are examined for diffusion interior and exterior to distributions of spheres. Finally, we note the connection between the times T1 and τ and the fluid permeability for flow through porous media, in light of a previously proved theorem, and nuclear magnetic resonance (NMR) relaxation in fluid-saturated porous media.

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M3 - Article

VL - 95

SP - 6477

EP - 6489

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 9

ER -