Kozak John J, Garza-López Roberto A, Abad Enrique
DePaul University, 243 South Wabash, Chicago, Illinois 60604-2301, USA and Beckman Institute, Caltech, Pasadena, California 91125, USA.
Department of Chemistry and Seaver Chemistry Laboratory, Pomona College, Claremont, California 60604-2301, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032147. doi: 10.1103/PhysRevE.89.032147. Epub 2014 Mar 31.
We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.
我们设计了一种二维的、类分形晶格，并通过数值和解析方法研究了在这种晶格上的随机游走与规则的平面欧几里得晶格之间的差异。我们研究了扩散控制过程从外部位点流向中心对称反应中心以及相反地从中心对称源流向边界位点的效率。在这两种情况下，我们发现，为类分形晶格上的平均游走长度推导的解析表达式对系统尺寸具有代数依赖性，而对于规则欧几里得晶格，这种依赖性可能是超越性的。将这些表达式与使用经典扩散理论在连续极限下推导的表达式进行了比较。我们的分析和数值结果量化了一类典型的空间不均匀性在多大程度上会损害固体支撑物上吸附原子扩散以及金属有机材料中表面辅助自组装的效率。
