• Title/Summary/Keyword: Smoluchowski

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Study of Diffusion Controlled Reactions in Liquids: A Perturbation Series Solution and a Numerical Solution of the Smoluchowski Equations

  • Mino Yang;Sangyoub Lee;Kim Yung Sik;Kook Joe Shin
    • Bulletin of the Korean Chemical Society
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    • v.10 no.6
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    • pp.529-535
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    • 1989
  • A general perturbation series solution of the Smoluchowski equation is applied to investigate the rate of recombination and the remaining probability of a pair of particles in liquids. The radiative boundary condition is employed and the convergence of the perturbation series is analyzed in terms of a convergene factor in time domain. The upper bound to the error introduced by the n-th order perturbation scheme is also evaluated. The long time behaviors of the rate of recombination and the remaining probability are found to be expressed in closed forms if the perturbation series is convergent. A new and efficient method of purely numerical integration of the Smoluchowski equation is proposed and its results are compared with those obtained by the perturbation method. For the two cases where the interaction between the particles is given by (i) the Coulomb potential and (ii) the shielded Coulomb potential, the agreement between the two results is found to be excellent.

Application of Reaction Path Smoluchowski Equation Formalism to the Photoisomerization of Trans-Stilbene

  • Kim, Dong-Sup;Lee, Sang-Youb
    • Bulletin of the Korean Chemical Society
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    • v.12 no.6
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    • pp.692-698
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    • 1991
  • The reaction path Smoluchowski equation approach developed in a recent work to calculate the rate constant for a diffusive multidimensional barrier crossing process is extended to incorporate the configuration-dependent diffusion matrix. The resulting formalism is then applied to the investigation of stilbene photoisomerization dynamics. Adapting a model two-dimensional potential and a model diffusion matrix proposed by Agmon and Kosloff [J. Phys. Chem.,91 (1987) 1988], we derive an eigenvalue equlation for the relaxation rate constant of the stilbene photoisomerization. This eigenvalue equation is solved numerically by using the finite element method. The advantages and limitations of the present method are discussed.

Study of Diffusion-controlled Processes. Solution of the Smoluchowski Equation with a Step Potential

  • Kim, Dae-Young;Shin, Seok-Min;Shin, Kook-Joe
    • Bulletin of the Korean Chemical Society
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    • v.7 no.4
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    • pp.271-275
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    • 1986
  • The Smoluchowski equation with a step potential is solved in one-dimensional case and three-dimensional case with spherical symmetry. Exact analytic expressions for the solution and the remaining probability are obtained in one-dimensional case for the reflecting boundary condition and the long time behavior of the remaining probability is compared with the earlier work. In three-dimensional case, only the long time behavior is evaluated. More general case with the radiation boundary condition is also investigated and the results are shown to approach correct limits of the reflecting boundary condition.

분자동역학 모의실험을 이용한 Lennard-Johns 입자계의 확산계수 계산

  • Jeong, Gu-Hwan
    • Proceeding of EDISON Challenge
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    • 2015.03a
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    • pp.171-176
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    • 2015
  • 일반적으로 어떠한 계의 크기가 커지면 확산계수 (Diffusion coefficient, D) 는 증가하는 것으로 알려져 있다. 본 연구에서는 원자의 개수와 계의 크기를 증가시키면서 정방계와 직방계에서의 확산계수를 계산하였다. 확산계수를 계산하는 방법으로 Einstein-Smoluchowski 관계식을 사용하였다. 정방계에서 x, y, z축의 확산계수를 계산해본 결과, 계의 크기와 원자의 개수가 증가할 때 각 축의 확산계수도 같이 증가하는 것을 확인할 수 있었다. 그리고 직방계에서 x, y축의 셀 길이를 고정시키고 z축의 셀 길이를 늘여가며 확산계수를 계산해본 결과, x, y축의 확산계수는 정방계와 비슷하게 증가하는 경향을 보였으나 z축의 확산계수는 변화가 거의 없음을 확인할 수 있었다.

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Experimental Study on Electrokinetic Streaming Potential in Micropore Channels of Hollw-Fiber Based on General Helmholtz-Smoluchowski's Principle (일반적 Helmholtz-Smoluchowski 원리에 따른 중공사 미세기공 채널에서의 계면동전기 흐름전위에 관한 실험연구)

  • 전명석;조홍일
    • Membrane Journal
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    • v.12 no.1
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    • pp.41-50
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    • 2002
  • The streaming potential generated by the electrokinetic flow within electric double layer of charged microchannel is applied to determine the zeta potential of hollow-fiber membrane pore by using the general Helmholtz-Smoluchowski equation. The streaming potential is know to provide a useful real-time information on the surface property and the interaction between pore and particles in actual situations and physicochemical conditions. The influence of physicochemical parameters upon the filtration with hollow-fibers has been examined with an in-situ and simultaneously monitoring the streaming potential as well as permeate flux. In particular, the present study examined an experimental method to identify the effect of cake layer which can vary according to the axial position of a hollow-fiber and the progress of membrane fouling by measuring the position-dependent streaming potential. As the latex concentration increases, the permeate flux decreased but the streaming potential increased. The growth of cake layer has been mire developed with increasing latex concentration, however, the effect of surface charges of latexes deposited on the membrane surface leads to increase the streaming potential. With increasing ionic concentration of KCI, both the permeate flux and the streaming potential decrease. The increase of ionic concentration provides a compact cake layer due to the shrinkage of Debye length and the decreased streaming potential results from the weakened ionic flows owing to a thin diffusive double layer.

Point Particle Approximation for Single and Two Species Diffusion-Influenced Reactions

  • Kim, Hyojoon
    • Bulletin of the Korean Chemical Society
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    • v.34 no.5
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    • pp.1454-1456
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    • 2013
  • The fluctuations in concentrations of reactants dominate the long-time dynamics of the single (A + A ${\rightarrow}$ 0) and two species (A + B ${\rightarrow}$ 0) diffusion-influenced annihilation reactions. Although hierarchical Smoluchowski approaches can provide a systematic and flexible framework to deal with the fluctuation effects, their results are too complicated to be analytically solved. For the efficient numerical calculation of the complicated fluctuation effect terms, we show that the presented point particle approximation is not only practical but also quite accurate for most conditions in diffusion-influenced reaction systems.

Study of Diffusion-Controlled Processes. Potential Shape Dependence in One-dimension

  • Shin, Seok-Min;Shin, Kook-Joe
    • Bulletin of the Korean Chemical Society
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    • v.8 no.2
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    • pp.83-88
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    • 1987
  • The Smoluchowski equations with a linear and a parabolic potentials in one-dimensional case are solved for the reflecting boundary condition. Analytic expressions for the long-time behaviors of the remaining probabilities are obtained. These results, together with the previous result for a step potential, show the dependence of the desorption process on the form of potential. The effect of the radiation boundary condition is also investigated for three types of potentials.

FRACTAL DIMENSION OF SIMULATED SEDIMENTS (모의류사의 쪽거리 차원)

  • 김형수;윤용남
    • Water for future
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    • v.27 no.3
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    • pp.115-121
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    • 1994
  • Cohesie sediment movement in estuarine systems is strongly affected by the phenomena of aggregation and flocculation. Aggregation is the process where primary particles are clustered together in tightly-packed formations; flocculation is the process where aggregates and single particles are bonded together to form large particle groups of very low specific density. The size, shape and strength of the flocculants control the rate of deposition and the processes of pollutant exchange between suspended sediments and ambient water. In estuarine waters, suspended sediments above the lutocline form the mobile suspension zone while below the lutocline they form the stationary suspension zone. Suspended particles in the mobile zone are generally in a dispersed state and the controlling forces are the Brownian motion and the turbulent flow fluctuations. In the stationary suspension zone, the driving force is the gravity. This paper discusses the settling and particle flocculation characteristics under quiescient flow conditions. Particles are entering the study domain randomly. Particles in the mobile suspension zone are simulated by using the Smoluchowski's model. Flocs created in the mobil suspension zone are moving into the stationary suspension zone where viscosity and drag effects are important. Utilizing the concepts of the maximum Feret's diameter and the Minkowski's sausage logic, the fractal dimension of the flocs within the stationary suspension is estimated and then compared with results obtained by other studies.

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Memory Equations for Kinetics of Diffusion-Influenced Reactions

  • Yang, Mino
    • Bulletin of the Korean Chemical Society
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    • v.27 no.10
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    • pp.1659-1663
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    • 2006
  • A many-body master equation is constructed by incorporating stochastic terms responsible for chemical reactions into the many-body Smoluchowski equation. Two forms of Langevin-type of memory equations describing the time evolution of dynamical variables under the influence of time-independent perturbation with an arbitrary intensity are derived. One form is convenient in obtaining the dynamics approaching the steady-state attained by the perturbation and the other in describing the fluctuation dynamics at the steady-state and consequently in obtaining the linear response of the system at the steady-state to time-dependent perturbation. In both cases, the kinetics of statistical averages of variables is found to be obtained by analyzing the dynamics of time-correlation functions of the variables.

Experimental study of electro-osmotic hydraulic conductivity changes with voltage gradients (전압경사에 따른 전기삼투 투수계수의 변화에 대한 실험적 연구)

  • Yoon, Dong-Wook;Kim, Hyun-Ki;Cho, Nam-Jun
    • Proceedings of the Korean Geotechical Society Conference
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    • 2010.09a
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    • pp.1002-1007
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    • 2010
  • This study evaluates the electro-osmotic hydraulic conductivity for the clay specimen by applying the series of the voltage gradient simultaneously with different stress conditions. The test results shows that the shrinkage of voids corresponds to the linear decrease in the electro-osmotic seepage velocity, and the changes of electro-chemical characteristics in the specimen induces the gradual decrease of the electro-osmotic seepage velocity with the constant voltage gradient.

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