Elsevier

Materials Chemistry and Physics

Volume 153, 1 March 2015, Pages 145-154
Materials Chemistry and Physics

Localization of electromagnetic field on the “Brouwer-island” and liquid metal embrittlement

https://doi.org/10.1016/j.matchemphys.2014.12.046Get rights and content

Highlights

  • A new theoretical model of liquid metal embrittlement has been developed.

  • Light localization has a strong influence on liquid metal embrittlement.

  • Light is localized in folds at interfaces between phases and components.

Abstract

Liquid metal embrittlement (LME) manifests itself as a sudden destruction of a metal sample if it is covered by a thin liquid film of eutectic mixture of specially selected metals. The proposed theoretical model of this phenomenon is based on an assumption related to the possibility of electromagnetic field localization in folds of interface between the phases or components of eutectic mixture filling cracks in solid metal surface (the typical example is In–Ga eutectic on Al-surface). Based on simultaneous presence of three different components in each space point of eutectic mixture (homogeneous In + Ga melt, solid In, and solid Ga), the system of interface folds could be simulated by the Brouwer surface – well known in topology. This surface separates three different components presented at each of its point. Such fractal surfaces posses by a finite volume. The volume occupied by the surface is defined as a difference between the eutectic mixture volume and the sum of volumes of its components. We investigate localization of external electromagnetic radiation in this system of folds. Due to very large magnitude of effective dielectric permeability of the considered system, at relative small volume change and fractal dimension of interface close to the value 3, the wave length of incident radiation inside the system is considerably decreased and multiscale folds are filled with localized photons. A probability of this process and the life time of the localized photons are calculated. The localized photons play crucial role in destruction of primary cracks in the metal surface. They are capable “to switch of” the Coulomb attraction of charge fluctuations on opposite “banks” of the crack filled with the eutectic. As a result, the crack could break down.

Introduction

Solidity decrease of a macroscopic sample covered by a thin liquid film of specially selected substance is called the Rehbinder Effect (RE) [1], [2]. Sometimes, a destruction and total dispersing of the sample is observed. As a rule, some insignificant mechanical stress is desirable. RE is widely used in mining industry, in machining of materials and etc. Usually, the reagents corresponding to one or another material are selected empirically.

The most brightly RE is observed if solid metals are coated with some other liquid metals. In this case, RE is commonly called a Liquid Metal Embrittlement (LME). In turn, the above mentioned destruction process manifests itself in full if the covering liquid is eutectic mixture of metals. The classic example is coating of Al-surface by In + Ga eutectic. There is no clear physical explanation of this phenomenon available so far [3]. Some experimental results describing various aspects of the problem are presented in [4], [5].

The eutectic point is the peculiar point on a phase equilibrium curve of two components arbitrary mixed in liquid state and are not mixed in solid one (see Fig. 1), where the phase diagram of In–Ga mixture is schematically shown). At this point, liquid phase (In + Ga melt) and two solid phases are simultaneously coexisted. The crystallization temperature is in order of 19 °C.

At concentration corresponding to the eutectic point, we have three different components in each space point of the system. The interface of these components has a very unusual shape. It separates them at each point throughout the bulk.

Such surfaces are well known in the topology. Their two-dimensional variants are known as set of the Julia and “island” of the Brouwer [6]. In both cases, this is a closed non-self-intersecting line separating three and more numbers of “countries” in each point.

To illustrate this phenomenon, let us imagine an island with two lakes in the ocean. One of them is filled with warm water and the second is cold. At the first day, channels sourcing from the warm lake and not connected with both cold lake and ocean are dug. The channels are produced in a way ensuring that no point of land on the island is located further that 1 m from the channel. The next day, a similar system of channels sourcing from the cold lake is dug ensuring that it is not intercepting with the first channels. At the third day, the channels from the ocean are produced ensuring all above conditions (not intercepting with the first two channel systems and no point of land on the island is located further than 1 m from the channels). Then, over the next three days, all channels are extended to ensure that they do not intercept and all points of land are located closer than 0.5 m from the water. Next step is to extend channels to ensure that no point of land is further than 0.25 m, and so on. As the result, after repeating these steps indefinite number of times, the land between channels would turn into a line separating these three water types at each point.

Generalization of this scheme for three-dimensional case is trivial. Being at the eutectic point, one could replace the warm water, for example, by solid In, the cold water by the solid Ga, and the salt ocean water by homogeneous liquid melt In + Ga.

The lines similar to coastline of the Brouwer island possess one more remarkable property. These lines have a finite area. Analogically, the interface of three-dimensional Brouwer-construction has a finite volume.

The surfaces with a finite volume are interesting per se, however the light localization in these systems is more interesting. Light localization is a possibility of a photon cycling in a system of low-absorption heterogeneities [7]. Localization is related to closed loops on trajectories of photons. If the photon moves along a closed loop, the phase shift of its wave function is equal to zero [8]. The probability amplitudes corresponding to both possible ways of the loop passing (clockwise and counterclockwise) interfere constructively. Every loop means indispensable return of the photon to the starting point. Thanks to increase of the probability of the loop generation caused by the interference, the light flux scattered to the back hemisphere also increases. This, in turn, stimulates the production of new loops on the photon trajectory, and so on. The strong localization appears as the result of this self-sustaining process.

Fractal system of folds of the considered interface possesses by space self-similarity or scale invariance. The folds with wide range of sizes are present in the system, including both folds comparable with characteristic size of the system and vanishingly small ones. This is a reason for renormalization or decrease of incident radiation wave-length λ as the latter penetrates inside the system. The photon wavelength λint becomes much smaller than λ, and the photon frequency ω remains unchanged because the effective photon velocity v simultaneously decreases in accordance with the relation v = ωλint/(2π).

Renormalization of λ occurs as follows; let the photon with the wavelength λ to be captured by some relatively large fold (primary resonance cavity). This capture results in an increase in the effective dielectric permeability ε of the system because ε increases near any electromagnetic resonance [9]. The increase of ε¯ initiates, in turn, a decrease in the photon wavelength becauseλint=λ/ε.

The photon acquires the ability to “scan” more and more minor details of the structure. The photon with a renormalized wavelength λint finds another smaller resonance cavity. New capture again stimulates an increase in ε¯ and a new decrease in λint, and so on. As the result, all interface folds could become filled with renormalized virtual photons, including those with λint → 0. Effective velocity of such photons is zero. It is the localized light.

In addition, it could be shown that the localized radiation is the reason for generation of new folds at the interface. These new folds serve as reservoirs for new localized photons etc. It is self-sustained process that forms such complex interface. Moreover, the physical reason for appearance of the volume defect could be explained by light localization positions. This volume is filled by renormalized localization photons “smeared” by a thin layer over the interface between phases or components. Usually, the volume of eutectic mixture is large than sum of separate volumes of its components, due to so called volume defect ΔV appearance. It is quite possible, that there is a certain finite limit of the product λint → 0 and the infinite area of interface between phase S → ∞, and just this limit is equal to volume defect ΔV.

Section snippets

Effective dielectric permeability of interface folds

We assume that the system of consideration is defined by simultaneous presence at each space point of three components with dielectric permeabilities ε1, ε2, ε3, and that such system could be characterized by effective dielectric permeability ε. The potential of a photon interaction with any element of the system with dielectric permeability, for example ε1, has the following form [10]:παβa(r,r)=ε1ε4πω2c2δαβηa(r)δ(rr),where r and r are the initial and final photon coordinates; a is the

Light localization

Total cross section, accounted for the localization in a system of particles, is obtained by averaging following expression over coordinates c of particles [11], [12], [13]:σab=caccb*dnf,where summation is carried out over all system particles c, and nf is the unit polarization vector along the direction of scattered quantum. Indexes a and b mean that initial and final particles on the photon trajectory could be different. If a = b, we deal with classic cross section which is being a square

Scattering cross section

The differential scattering cross section is obtained by addition of wave functions of incident and scattered photons to the block K:dσdnf=2πc2ωiV2πc2ωfVeiαefβefσeiν×Kαβσν(r,r,r¯',r¯)exp[i(kir+kfrkfr¯'+kir¯)]drdrdr¯'dr¯.

Let us sum the perturbation series for K. In accordance with Fig. 8, the kernel L can be presented in the form:Lαβσν(r,r,r¯',r¯)=(ΔVV)2Nc(ω4π)4|ε|2gNc2(rr¯)αβQ(r,r)σ¯ν¯Q(r¯',r¯).

The equation for K has the formKαβσν(r,r,r¯',r¯)=Lαβσν(r,r,r¯',r¯)+Lαγνμ(r,r1,r¯',r¯

Localization and liquid metal embrittlement

Let us consider the simplest model of crack in a metal – a plain slot dividing two semi-infinite metals. Let us fill this crack with In + Ga-eutectic. Our goal is to understand how light localized in folds can initiate disappearance of Van der Waals attraction between crack banks.

For a description of the electromagnetic properties of metal we adopt the jellium model, i.e. conduction electron scattering by the ion cores is neglected. The ions are spread out into a uniform distribution of

Conclusion

Liquid eutectic mixture is a structure with unusual topology. Interface structure between its phases and components is isomorphic to that of three-dimensional Brouwer's construction separating three different components at each point. Such surfaces possess by a finite volumes. So, the volume of eutectic mixture is greater than sum of volume of its components. The difference of these volumes is occupied by the interface.

Eutectic mixture formation is complex dynamical process in which the main

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This work was supported by RSCF (No 14-03-00507).

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