Phase-Field Modeling of Step Dynamics on Growing Crystal Surface: Direct Integration of Growth Units to Step Front

We propose a new formulation for numerically simulating step dynamics on growing crystal surfaces in the framework of a phase-field technique. The step advancement rate is proportional to a supersaturation at the crystal surface when the growth units in the ambient phase are integrated to the step front directly (direct integration hypothesis). We conduct numerical simulations of some standard step dynamics problems: the advancement of a straight step, the growth or dissolution of a two-dimensional island, and the vertical growth of the crystal surface due to single or multiple screw dislocations. During evaluations, our phase-field model accurately calculated the rate of advancement of a straight step for various supersaturations. The calculated time variation of the radius of the two-dimensional island showed good agreement with the exact solution. The vertical growth rate due to screw dislocations qualitatively agreed with the predictions of the classical theory of Burton, Cabrera, and Frank. Our simple formulation requires only a single parabolic partial differential equation to be solved numerically. Thus, our phase-field model provides a simple numerical tool for a quantitative step-by-step trajectory calculation, when the advancing velocity of each step follows the direct integration hypothesis

Echo amplitude representation procedure with four omni-directional microphones.

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Relationship between <i>I</i>[<i>M</i>] and <i>E</i>[<i>M</i>] under various degradation levels of ILD resolution.

(A) Relationship between the degree of injection and the detection error for the 36 ear motions without the degradation of the ILD resolution. (B) Example of change in the sinusoidal signal for each degradation level. (C) Relationship between the degree of injection and the detection error for each ear motion under the degraded ILD resolutions. Note that these evaluations were conducted for ear motions with relatively small detection errors (E[M]<20°) in the no degradation condition (A). The length of the vertical black line corresponds to the increase in the detection error when the ILD discretization level changes from 0 dB to 3 dB.</p

Colormaps of <i>U</i>_<i>M</i>(<i>θ</i>, <i>φ</i>) and the degrees of injection for various ear motion patterns.

The pitch angle functions are fixed to according to actual bat behavior. Blue and orange lines indicate the angle functions of the left and right ears, respectively. The left and top array panels display the roll angle functions and the yaw angle functions , respectively.</p

Colormaps of degree of injection <i>I</i>[<i>M</i>] of all combinations of <i>ψ</i>_<i>e</i>-<i>φ</i>_<i>e</i>-<i>θ</i>_<i>e</i> angle functions.

The fixation of the pitch angle functions to is removed, so that the degrees of injection were evaluated for 63 = 216 motion patterns.</p

Examples of direction detection performance with appropriate ear motions.

The formation of Fig 5 is same as Fig 4. Blue color map and less-visible error lines mean the good performance of direction detection.</p

Examples of the direction detection performance without ear motion using supervised machine learning.

The ear motion condition was chosen as [: 0, : 0, : ]. Blue ‘x’ markers indicate test data (θ, φ) and red ‘+’ markers indicate output data (θguess, φguess). Black lines denote the error lines connecting points (θ, φ) and (θguess, φguess). Each detection error line tends to stretch vertically, indicating that the elevation angle is difficult to detect while the azimuth angle can be accurately detected. (TIF)</p

Explanations of the evaluation function U_F and degree of injection I[F].

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Expression of spatial orientation of the directional ear.

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Five types of dimension pairs of the convex hull and each ear’s orbit.

The blue lines indicate the left ear’s orbit and the orange lines indicate the right ear’s orbit . When both orbits coincide, only the orange line is displayed. The convex hull of the union of both ears’ orbits is displayed in each case.</p