表面吸附与离子迁移：科学说明与审核

Surface Adsorption & Migration: scientific basis and review

2026-10-27 · Version 1.0 · Synthetic analytic potential

1. 模型、单位与吸附位点

1. Model, units and adsorption sites

u = 2πx
v = 2π[y − c sin(2πx)]
V(x,y) = A[1 − cos(u)]/2 + B[1 − cos(v)]/2
r = a(x,y)
A = 0.35 eV (adjustable 0.15–0.80 eV)
B = 1.20 eV; c = 0 or 0.18; a = 3.00 Å (adjustable 2.5–5 Å)

x、y 是无量纲周期分数坐标，能量单位 eV；真实长度映射为 a(x,y)。整数坐标是等能极小值，显示于示例方形格子的几何空位中心。相邻 x 方向位点为 A=(0,0)、B=(1,0)。鞍点 (½,0) 的能量严格为 A，横向稳定方向与反应坐标不必平行于显示坐标轴。端点能量相同，正反势垒相同。示例周期不参与势能公式，调整 a 只改变路径物理长度及以 eV/Å 表示的力，不改变无量纲路径和 eV 势垒。

x and y are dimensionless fractional coordinates; energies are in eV and physical lateral positions are a(x,y). Integer coordinates are equal-energy minima, shown at geometric hollow centres of the illustrative square grid. Adjacent x-direction sites are A=(0,0) and B=(1,0). The saddle (½,0) has energy exactly A; its stable and reaction directions need not align with the displayed axes. Equal endpoint energies give equal forward and reverse barriers. The illustrative period does not enter V: changing a changes physical path length and force in eV/Å, but not the fractional path or barrier in eV.

本模型只包含横向两个自由度。吸附粒子高度固定于表面原子中心平面上方 2.00 Å；没有垂直束缚势、吸附能或脱附能计算。粒子始终位于外表面之上，不穿透基底，不演示层间迁移。三层方形原子阵列和仪器台是几何显示，不参与势能计算；球半径与光照不代表电子密度或真实化学键。

Only two lateral degrees of freedom are included. Adsorbate height is fixed at 2.00 Å above the top substrate-atom centre plane. No vertical binding potential, adsorption energy or desorption energy is calculated. The particle remains above the outer surface without penetration or interlayer migration. The three square atom layers and mounting stage are geometric displays that do not enter V. Sphere radii and lighting do not represent electron density or chemical bonds.

2. 实际执行的数值弦求解

2. Numerical string actually executed

g = ∇V; τ = normalized central-difference path tangent
qᵢ ← qᵢ − η[gᵢ − (gᵢ · τᵢ)τᵢ]
η = 0.0008 (fractional-coordinate numerical step)
65 images; endpoints fixed; redistribute by arc length after each step
Stop: max |g⊥| < 10⁻⁴ eV/fractional coordinate, or 5000 iterations
Reported residual: max |g⊥| / a in eV/Å
Reported barrier: maxᵢ V(qᵢ) − V(q₀)

求解器以 y=c sin(2πx) 为初始猜测；逐步消除离散路径法向的势能梯度，再按弧长重分布。能量剖面、俯视图、三维路径与导出共用求解图像和同一解析势。65 个图像包含对称鞍点；残差与最大迭代数均在导出中记录。该离散弦算法演示零温最低能路径的思想，不声称对任意势面保证全局最低路径，也不求自由能面。它与 NEB 的切向弹簧实现不同，页面不将它标为 NEB 结果。

The initial guess is y=c sin(2πx). The solver descends the discrete perpendicular potential gradient and redistributes the string by arc length. The energy profile, map, 3D path and export share these images and the same analytic V. The 65 images include the symmetry saddle; residual and iteration count are exported. This discrete string demonstrates a zero-temperature minimum-energy-path construction, without claiming a globally optimal path for arbitrary landscapes or a free-energy surface. It differs from NEB's tangential-spring formulation and is not labelled an NEB result.

3. 示例速率与动画的区别

3. Illustrative rate and playback

k(T) = ν₀ exp[−ΔE‡/(kB T)]
ν₀ = 10¹³ s⁻¹ (assumed)
kB = 8.617333262145 × 10⁻⁵ eV K⁻¹
T = 200–700 K

此处将解析势垒代入固定尝试频率的 Arrhenius 形式，仅用于比较温度与势垒对示例速率的指数影响；不是材料的定量跃迁速率。忽略活化熵、再穿越、量子隧穿、多粒子作用、溶剂、电场和电荷转移。播放是沿已求路径的平滑位置插值，速度由观看需要设定，不是分子动力学或按上述速率等比例显示的热运动。

The analytic barrier is inserted into a fixed-prefactor Arrhenius expression only to compare exponential sensitivity to temperature and barrier. This is not a quantitative material hopping rate. Activation entropy, recrossing, quantum tunnelling, many-particle interactions, solvent, electric fields and charge transfer are omitted. Playback smoothly interpolates an already calculated path at a viewing speed; it is neither molecular dynamics nor thermal motion displayed on the calculated rate timescale.

4. 数值审核与适用范围

4. Numerical review and scope

模型测试独立核对：整数平移周期性；极小值与已知鞍点能量；解析梯度与中央有限差分；直线通道的闭式剖面；弯曲路径的固定端点、非零弯曲与离散垂直残差；路径插值的端点；零势垒极限；温度倍增的速率比；非法温度与能量输入。6 项 Node 测试通过。独立 AI 交叉审核复核了全部 6 项测试，并额外检查 1000 组中央差分（梯度最大误差 7.34×10⁻⁹ eV/分数坐标）。A=0.15/0.35/0.80 eV、c=0/0.18 时，数值路径势垒均等于 A，残差不超过 3.32×10⁻⁵ eV/Å，最多 439 次迭代收敛。还通过鞍点 Hessian 与跨越 x=0.5 的下界独立核对势垒。未发现剩余实质科学阻塞；上线交互验证另行记录。独立 AI 审核不等同于表面科学专家认证。

Model tests independently check integer-translation periodicity; minima and the known saddle energy; analytic gradients against central differences; the straight-channel closed-form profile; curved-path fixed endpoints, nonzero curvature and discrete perpendicular residual; interpolation endpoints; the zero-barrier limit; rate ratios on doubling temperature; and invalid temperature/energy inputs. Six Node tests pass. Independent AI cross-review reran all six tests and checked 1,000 central differences (maximum gradient error 7.34×10⁻⁹ eV per fractional coordinate). For A=0.15/0.35/0.80 eV and c=0/0.18, path barriers equal A, residuals remain ≤3.32×10⁻⁵ eV/Å, and convergence takes ≤439 iterations. The saddle Hessian and the lower bound at x=0.5 independently confirm the barrier. No material scientific blocker remains; live interaction verification is recorded separately. Independent AI review is not certification by a surface-science expert.

用途：理解表面位点、势能剖面、路径优化与热激活的关系。不可据此报告特定物质的吸附能、迁移势垒、扩散系数、离子电导、实验机理或真实材料最优路径。

Use this to understand adsorption sites, potential profiles, path optimization and thermal activation. Do not report material adsorption energies, migration barriers, diffusion coefficients, ionic conductivity, experimental mechanisms or real-material optimal pathways from this example.

model.mjs SHA-256: 8f797e04d4b528aad6be814a9f2ff8d402cc7899cd417a5b5b9a73c558337f3d

5. 一手依据

5. Primary sources

1. E, W.; Ren, W.; Vanden-Eijnden, E. String method for the study of rare events. Phys. Rev. B 66, 052301 (2002).
2. Henkelman group: Nudged Elastic Band, Transition State Tools for VASP.
3. IUPAC Gold Book: Arrhenius equation, A00446.

来源仅支持算法与速率表达的理论背景，不是本页面示例势能参数的材料来源。解析势公式和示例参数是本教学模型自行明确选择的定义。

These sources support the theoretical background of pathway methods and the rate expression; they are not material-data sources for this potential. The formula and illustrative parameter values are explicitly chosen definitions of this teaching model.

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