УДК 538.931



PhD in Physics and Mathematics, associate professor Yarmolenko M.V.

Kyiv National University of Technologies and Design, Ukraine, Cherkasy


In the article it was proved theoretically and experimentally that the interface curvature can either accelerate or slow down the diffusion phase layer growth in cylindrical and spherical samples and the Kirkendall shift when compared with a planar sample depending on the average phase concentration only. It is shown that internal stress, arising due to dilatation during phase growth, can either accelerate or slow down the growth in addition to the above-mentioned effect, depending on the difference in mobilities of different atoms within each phase and independently on the sign of dilatation.

Key words: Kirkendall shift, Matano plane, reactive diffusion, vacancies, interfaces, intermetallic compounds, kinetics.


Кандидат фізико-математичних наук, доцент Ярмоленко М.В. Твердофазні реакції у процесі дифузії: кінетика фазоутворення / Київський національний університет технологій та дизайну, Україна, Черкаси.

У статті доведено теоретично та підтверджено експериментально, що кривизна міжфазної границі може як пришвидшувати, так і уповільнювати дифузійне утворення шарів фаз у циліндричних та сферичних зразках та зміщення Кіркендалла в залежності лише від середньої концентрації однієї з речовин. Додатково впливати на кінетику можуть також внутрішні механічні напруги, які виникають у процесі фазоутворення.

Ключові слова: зміщення Кіркендалла, площина Матано, реакційна дифузія, вакансії, міжфазні границі, інтерметаліди, кінетика.


Кандидат физико-математичних наук, доцент Ярмолэнко М.В. Твердофазные реакции в процессе диффузии: кинетика фазообразования / Киевский национальный университет технологий и дизайна, Украина, Черкассы.

В статье доказано теоретически и подтверждено экспериментально, что кривизна межфазной границы может как ускорять, так и замедлять диффузионное образование слоев фаз в цилиндрических и сферических образцах и смещение Киркендалла в зависимости только от средней концентрации одного из веществ. Дополнительно влиять на кинетику могут тоже внутренние механические напряжения, которые возникают в процессе фазообразования.

Ключевые слова: смещение Киркендалла, плоскость Матано, реакционная диффузия, вакансии, межфазные границы, интерметаллиды, кинетика.


Introduction. The shift of the crystal lattice during mutual diffusion in solids was first discovered by Kirkendall [1] and theoretically described by Darken [2]. There are a lot of experimental data of the Kirkendall effect for different binary systems. But still not all peculiarities of the process have been found. Treatments of diffusion in metallic solid solutions require an accurate, convenient analysis that takes account of the actual variations of atomic size with concentration. There are reference systems of two types: conventional reference systems (Fick and molecular), and the lattice (Kirkendall) reference system [3]. The Kirkendall effect is now considered as excellent evidence for the validity of a vacancy mechanism of diffusion in metals [4]. For this reason, was solved a random-walk vacancy problem with appropriate boundary conditions. The results of computer modeling and the analytical solution of the Kirkendall effect agree with the real experimental data [5,6]. Describing the growth of intermediate phase layers during chemical diffusion in cylindrical and spherical samples offers some difficulty, since the change in interface area S(R) should be taken into account. In addition, there is a considerable concentration dependence of the interdiffusion coefficient D(C) and an exact knowledge of D(C) is needed for each phase of a binary system. Moreover, if a phase grows with volume change, internal stress arises, influencing growth kinetics of the phases. Therefore, the problem can not be solved in a general form, no matter how modern the computer systems are.

Purpose. Solid state reactions (SSRs) are governed by two magic powers – thermodynamics and kinetics. Common understanding is that kinetics determines only the rate of fulfilment (implementation) of thermodynamic laws. Actually, the only concept, which had been taken from the nucleation theory, is the existence of critical nuclei [8]. The prevalent erroneous concept of an “invariant” for diffusion demonstrates the need for a clearer understanding of the role of reference planes in treatments of diffusion. The diffusion velocity, v, of a given component is used as the basis for defining the diffusion flux. Absolute values of v can be determined from data on the Kirkendall shift, but only relative values (v − Ω) can be employed if only a conventional reference system is used in the analysis [3].

Methods. For describing the growth kinetics of the phases and the Kirkendall shift kinetics, an approximation of constant diffusion flux along the diffusion direction within the width of each phase is used (so-called constant flux method) which is theoretically grounded in [7]. This technique necessitates no allowance for the concentration dependence of D(C). The relative change of the diffusion flux within the width of each phase is approximately equal to dC«1 , where dC is the range of phase homogeneity, while the interdiffusion coefficient may vary by more than a factor of 10 over the region of homoheneity dC.

Originality. Computer simulation can not describe fairly well interface curvature influence on intermediate phase layers kinetics during chemical diffusion and the Kirkendall shift kinetics [5, 6, 8]. So we have to use the mathematical equations.

Full text Stattya_Yarmolenko.doc


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