UN核燃料烧结致密化过程的相场模拟

图1 烧结模拟中空位扩散示意图

Fig.1 Schematic of vacancy diffusion simulated in sintering simulation (Arrows show the paths of vacancy diffusion along free surfaces and grain boundaries)

M 为迁移率张量函数，表达形式为^[23]：

M = \frac{v_{m} D^{s}}{R T}

(8)

D^{s} = D^{s} ρ^{2} (1 - ρ^{2}) T^{s}

(9)

Τ^{s} = I - n_{s} \otimes n_{s}

(10)

n_{s} = \frac{\nabla ρ}{|\nabla ρ|}

(11)

式中，ν_m为摩尔体积；R为气体常数；T为温度； D^s为表面扩散系数张量；D^s为表面扩散率，取决于材料的性质； T^s为表面投影张量，保证表面扩散在晶粒与气孔接触的表面切线方向； I 为单位张量；符号⊗为张量积； n_s为垂直于曲面的单位法向量。

η_i 的演化动力学方程由Allen-Cahn方程^[21]引入平流通量项后表示为：

\begin{array}{l} \frac{\partial η_{i}}{\partial t} = - L \frac{δ F}{δ η_{i}} - \nabla \cdot [η_{i} v_{a d v i} (r)] = - L [\frac{\partial f}{\partial η_{i}} - κ_{η} \nabla^{2} η_{i}] - \\ \nabla \cdot [η_{i} v_{a d v i} (r)] (i = 1, 2, \dots, p) \end{array}

(12)

式中， v_adv_i ( r )为 r 位置处晶粒i的平流速度；L为表征晶界迁移率的系数，在各向同性假设下为常数，与材料的具体性质关系如下^[24]：

L = \frac{γ_{g b} M_{b}}{κ_{η}}

(13)

M_{b} = \frac{v_{m} D^{g b}}{10 R T δ}

(14)

式中，M_b为晶界迁移率；D^gb为晶界扩散率，取决于材料的性质。

根据式(7)修正的Cahn-Hilliard方程与式(12)修正的Allen-Cahn方程，可以求得相场变量随时间的演化过程。通过对相场变量的变化求解，可以计算出演化任意时刻的微观组织与空间分布。本工作采用的求解相场方程的数值方法为有限差分法。对于相互关联的非线性偏微分方程组，时间上采用显示Euler算法；空间上采用五点差分法求解Laplace项^[25]。

为了直观地表征烧结过程，本工作中可视化变量φ为^[20]：

φ = \sum_{i} η_{i}^{2}

(15)

1.4 平流速度场 v_adv(r)

烧结过程中粒子的刚体运动由平移和转动2部分组成， v_adv( r )表示为^[22]：

v_{a d v} (r) = \sum_{i} v_{a d v i} (r) = \sum_{i} [v_{t i} (r) + v_{r i} (r)]

(16)

式中， v_t_i ( r )为在 r 位置处晶粒i的平移速度，表达式为^[22]：

v_{t i} (r) = m_{t} \frac{η_{i} (r)}{V_{i}} \int_{V} κ_{F} \sum_{i \neq j} (ρ - ρ_{0}) \cdot f_{V} (η_{i}, η_{j}) (\nabla η_{i} - \nabla η_{j}) d^{3} r

(17)

V_{i} = \int_{V} η_{i} (r) d^{3} r

(18)

式中，m_t为平移迁移率，m²/s；V_i 为晶粒i的体积；积分项为作用在晶粒i质心上的合力；κ_F为质量密度相对于在晶界平衡值变化的刚度系数，本工作取κ_F = 100；常数ρ₀为晶界质量密度的平衡值，本工作取ρ₀ = 0.98；r为空间位置；函数f_V 用来识别2颗粒晶界之间的区域，表达式为：

f_{V} (η_{i}, η_{j}) = \{\begin{array}{l} 1 (η_{i} η_{j} \geq c^{*}) \\ 0^{} (o t h e r w i s e) \end{array}

(19)

式中，c^*为晶界阈值。式(17)右侧的梯度运算在相邻粒子之间产生大小相等、方向相反的局域力。

v_r_i ( r )为在 r 位置处晶粒i的旋转速度，表达式为^[22]：

v_{r i} (r) = \frac{m_{r}}{V_{i}} T_{i} \times (r - r_{c i}) η_{i} (r)

(20)

式中，m_r为转动迁移率，m²/s； T_i 为作用在晶粒i的扭矩，表达式为：

T_{i} = \int_{V} (r - r_{c i}) \times κ_{F} \sum_{i \neq j} (ρ - ρ_{0}) f (η_{i}, η_{j}) (\nabla η_{i} - \nabla η_{j}) d^{3} r

(21)

式中， r_c_i 为晶粒i质心的位置，表达式为：

r_{c i} = \frac{1}{V_{i}} \int_{V} r η_{i} (r) d^{3} r

(22)

2 无量纲化处理

无量纲化后的动力学方程与量纲控制方程具有相同的形式，可以在不改变驱动力的情况下缩放模型参数，将模拟的数量级统一，减小计算误差。本工作相场模型的变量与参数采用无量纲形式进行描述。

根据式(7)，保守场演化方程含有量纲的量为t、 M 、F、∇、 v_adv( r )，选取基准值参考空间长度l^*、参考时间t^* 和参考能量密度ε^*，分别得到无量纲Hamilton算符 $\tilde{\nabla} = l^{*} \nabla$ 、无量纲时间τ = t / t^* 和无量纲自由能 $\tilde{F}$ = F / ε^*。因此，式(7)变形得：

\begin{array}{l} \frac{\partial ρ}{\partial τ} = \tilde{\nabla} \cdot (\frac{M ε^{*} t^{*}}{l^{* 2}} \tilde{\nabla} \frac{δ \tilde{F}}{δ ρ} - \frac{t^{*}}{l^{*}} ρ v_{a d v} (r)) = \tilde{\nabla} \cdot \frac{M ε^{*} t^{*}}{l^{* 2}} \cdot \\ \tilde{\nabla} [\frac{1}{ε^{*}} \frac{\partial f}{\partial ρ} - \frac{κ_{ρ}}{ε^{*} l^{* 2}} {\tilde{\nabla}}^{2} ρ] - \tilde{\nabla} \cdot [\frac{t^{*}}{l^{*}} ρ v_{a d v} (r)] \end{array}

(23)

由此可得无量纲迁移率张量 $\tilde{M}$ = Mε^*t^* / l^*2；无量纲平移迁移率 ${\tilde{m}}_{t} = m_{t} t^{*} / l^{*}$ ；无量纲转动迁移率 ${\tilde{m}}_{r} = m_{r} t^{*} / l^{*}$ 。

根据式(12)，非保守场演化方程含有量纲的量为t、L、F、∇和 v_adv_i ( r )。因此，式(12)变形得：

\begin{array}{l} \frac{\partial η_{i}}{\partial τ} = - L ε^{*} t^{*} \frac{δ \tilde{F}}{δ η_{i}} - \tilde{\nabla} \cdot [\frac{t^{*}}{l^{*}} η_{i} v_{a d v i} (r)] = \\ - L ε^{*} t^{*} [\frac{1}{ε^{*}} \frac{\partial f}{\partial η_{i}} - \frac{κ_{η}}{ε^{*} l^{* 2}} \nabla^{2} η_{i}] - \\ \tilde{\nabla} \cdot [\frac{t^{*}}{l^{*}} η_{i} v_{a d v i} (r)] (i = 1, 2, \dots, p) \end{array}

(24)

由此可得无量纲化迁移系数 $\tilde{L}$ = Lε^*t^*。

本工作取t^* = 1 / (Lε^*)，ε^* = B；则 $\tilde{L}$ = 1，B = 1；l^*的取值与δ有关，对应无量纲化之后一个单位长度占有的界面宽度，本工作中，l^* = δ/m (其中，m代表扩散界面在模拟中占据的格点数)。

根据所选取的参考物理量，无量纲迁移率张量 $\tilde{M}$ 与无量纲自由能函数 $\tilde{F}$ 中A、B、κ_ρ 、κ_η 等参数对应的无量纲形式分别为：

\{\begin{array}{l} \tilde{M} = \frac{15 D_{s} m^{2}}{2 D_{g b}} \\ {\tilde{m}}_{t} = \frac{m m_{t}}{δ}, {\tilde{m}}_{r} = \frac{m m_{r}}{δ} \\ \tilde{A} = \frac{A}{ε^{*}} = \frac{12 γ_{s} - 7 γ_{g b}}{γ_{g b}} \\ \tilde{B} = \frac{B}{ε^{*}} = 1 \\ {\tilde{κ}}_{ρ} = \frac{κ_{ρ}}{ε^{*} l^{* 2}} = \frac{3}{4} (\frac{2 γ_{s} - γ_{g b}}{γ_{g b}}) m^{2} \\ {\tilde{κ}}_{η} = \frac{κ_{η}}{ε^{*} l^{* 2}} = \frac{3}{4} m^{2} \end{array}

(25)

本工作利用相场模型对UN陶瓷粉末在1823 K温度下的烧结过程进行了数值模拟，模拟中用到的UN物理参数如表1^[26,27]所示。

表1 UN的物理参数^[26,27]

Table 1 Physical parameters of UN at 1823 K^[26,27]

Parameter	Value	Unit	Ref.
D^s	7.5 × 10^-12	m²·s^-1	[26]
D^gb	0.01D^s	m²·s^-1
γ_s	1.6	J·m^-2	[27]
γ_gb	0.8	J·m^-2	[27]
δ	6	nm	[27]

Note:D^s—surface diffusivity, D^gb—grain-boundary diffusivity, γ_s—surface energy, γ_gb—grain-boundary energy, δ—diffuse interface width

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将UN的物理参数代入上述方程中，求得各无量纲参数的具体数值如表2所示。

表2 模拟中的无量纲参数

Table 2 Non-dimensional parameters used in simulation

Parameter	Value	Parameter	Value
$\tilde{A}$	17	${\tilde{m}}_{t}$	30-100
$\tilde{B}$	1	${\tilde{m}}_{r}$	1
${\tilde{κ}}_{η}$	6.75	$\tilde{L}$	1
${\tilde{κ}}_{ρ}$	20.25	Δx = Δy	1
$\tilde{M}$	6750	Δt	2 × 10^-5

Note: $\tilde{A}$ , $\tilde{B}$ , ${\tilde{κ}}_{η}$ , ${\tilde{κ}}_{ρ}$ —non-dimensional parameters of free energy function; $\tilde{M}$ —non-dimensional surface mobility; ${\tilde{m}}_{t}$ —non-dimensional translation mobility; ${\tilde{m}}_{r}$ —non-dimensional rotation mobility; $\tilde{L}$ —non-dimensional Allen-Cahn mobility; Δx, Δy—space scale; Δt—time scale

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3 结果与讨论

3.1 2个颗粒烧结

本工作采用相场模型，模拟了双晶粒结构中UN陶瓷粉末烧结过程，将该相场模型的模拟结果与未加平流通量相场模型的模拟结果进行对比分析。所选模拟区域差分网格尺寸为150 × 150，选取周期性边界条件。分析平流通量对UN陶瓷粉末烧结组织演变过程、烧结颈形成、平衡二面角和致密化的影响。

3.1.1 烧结颈的形成

图2为未引入平流通量和引入平流通量2个等尺寸的圆形颗粒在烧结过程中的组织演变。其中颗粒的半径R = 30，与平流通量模型模拟相关的参数m_t = 30，c^* = 0.05，κ_F = 100，ρ₀ = 0.98。从图2a~c与d~f对比可以看出，未引入平流通量的模拟烧结颈形成较慢，颗粒位置无明显移动；含有平流通量的模拟烧结颈的形成明显更快，颗粒位置发生移动，且稳态时烧结颈的长度更长。这与文献[19]给出的致密化特征十分吻合。烧结致密化取决于颗粒-孔隙相互作用，表面扩散主导的未加平流通量模型对烧结过程孔隙收缩没有贡献，所以烧结颈形成较慢；引入平流通量模型含有颗粒刚体运动过程，空位可以被晶界吸收湮灭，所以不仅促进烧结颈增长，且随着烧结颈生长，颗粒彼此接近。

图2

图2 平流通量对2个晶粒烧结形貌演化的影响

Fig.2 Simulated effects of advection flux on morphologies of two grains with the boundary between them by the phase-field methods of without (a-c) and with (d-f) advection flux (a, d) 2 × 10⁴ step (b, e) 50 × 10⁴ step (c, f) 100 × 10⁴ step

图3为未引入平流通量与引入平流通量2种模拟数据的烧结颈对数增长曲线。可以看出，未引入平流通量模拟的烧结颈增长曲线近似为一条直线，斜率为0.21左右，满足烧结颈曲线增长规律^[11]。引入平流通量模拟的烧结颈曲线分为3个时期：烧结前期，烧结颈的增长较快，斜率为0.68左右，这是因为颗粒的致密化移动使得烧结颈更快增长，从A点到B点平流通量起主要作用导致晶界宽度变窄；烧结中期，从B点到C点平流通量作用减弱，晶界宽度恢复，导致烧结颈长度下降；烧结后期，烧结颈不再增长且相比于未引入平流通量模型提前达到稳态，斜率为0.06左右。引入平流通量模型对烧结颈的形成在烧结前期作用较强，后期作用不明显。

图3

图3 平流通量对2个颗粒烧结颈增长曲线的影响

Fig.3 Logarithmic growth curves of particle sintered neck methods for without and with advection flux (l—neck length, t'—time step)

3.1.2 平衡二面角

界面性质在烧结过程中的气孔收缩、致密化和粗化过程中起关键作用。相邻的2个粒子在相界处由于表面能和晶界能形成一个平衡二面角(Ψ)，在各向同性条件下，晶界处平衡二面角方程为^[28]：

Ψ = 2 a r c c o s (\frac{γ_{g b}}{2 γ_{s}})

(26)

图4分别为未引入平流通量与引入平流通量2种烧结模型的平衡二面角，演化的总时间为2 × 10⁶ step。本工作UN烧结晶界能与界面能的比值为γ_gb / γ_s = 0.5，对应计算值为Ψ = 150°。从图4可以看出，未引入平流通量模拟平衡二面角与理论计算值吻合较好，引入平流通量模拟最终平衡二面角与理论计算值有较小误差。证明了平流通量的引入会加快烧结颈的形成，但最终平衡二面角仍然不变，只由材料的特性所决定。这与只考虑表面扩散^[29]的结论一致，平衡二面角与晶界能和界面能的比值保持一致。

图4

图4 平流通量对平衡二面角的影响

Fig.4 Simulated equilibrium dihedral angles by two sintering models

(a) without advection flux (b) with advection flux

3.1.3 晶界处气孔的收缩

图5为模拟双晶粒结构中晶界上含有气孔的致密化过程，将引入平流通量相场模型的模拟结果与未引入平流通量相场模型的模拟结果对比分析。从演化图像中可以看出，未引入平流通量模型晶界上的气孔稳定存在，引入平流通量模型晶界上的气孔逐渐收缩最终致密化。这与模型预期完全符合，表面扩散对致密化气孔收缩没有贡献。引入平流通量模型时2个颗粒晶界上气孔收缩随时间演化的曲线如图6所示。从曲线可以看出，平流通量作用分为2个阶段，演化前期由表面扩散主导，平流通量致密化气孔收缩作用较小，气孔收缩较慢；演化后期平流通量致密化主导，致密化气孔收缩较快。

图5

图5 平流通量对2个颗粒烧结过程中晶界气孔演化的影响

Fig.5 Simulated evolutions of a pore in the grain boundary between two particles by the phase-field methods of without (a-c) and with (d-f) advection flux (a, d) 10 × 10⁴ step (b, e) 20 × 10⁴ step (c, f) 30 × 10⁴ step

图6

图6 平流通量作用下2个颗粒烧结过程中晶界的气孔收缩率

Fig.6 Simulated pore shrinkage curve of grain boundary between two particles by the advection flux model

3.2 致密化烧结

3.2.1 三叉晶界处的气孔收缩

图7为引入平流通量后，3个颗粒烧结的相场模拟结果。从图7b~d可以看出，颗粒发生了平移，逐渐完成致密化收缩。为了更加清楚直观地理解平流通量的作用，图8给出了与图7对应的平流通量的空间分布和时间演变，其中蓝绿色的部分表示平流通量的作用，越接近蓝色表示平流通量的作用越强。从图8b~d可以看出，平流通量作用于晶界与自由表面上，气孔收缩时平流通量逐渐增强，致密化完成后平流通量迅速减弱。

图7

图7 引入平流通量后3个颗粒烧结时的形貌演化过程

Fig.7 Simulated evolutions of three particles by the phase-field methods of advection flux

(a) 0 step (b) 2 × 10⁴ step

图8

图8 3个颗粒烧结时形貌演化过程所对应的平流通量

Fig.8 Simulated evolutions of advection flux on morphologies of three particles (The blue-green part shows the effect of the advective flux, and the closer to the blue, the stronger the effect of the advective flux)

(a) 0 step (b) 2 × 10⁴ step

引入平流通量模型3个颗粒气孔收缩随时间演化的曲线如图9所示。结合图8平流通量的空间分布与时间演变，气孔收缩过程存在2个斜率的变化，可以将3个颗粒平流通量气孔收缩分为3个阶段：第1阶段，颗粒间形成颈缩；第2阶段，相邻颗粒通过平移运动导致粒子中心相互靠近；第3阶段，气孔完成收缩。在烧结过程中，涉及到2种机制：表面扩散机制和致密化平流通量机制。烧结初期，物质通过表面扩散主导传输，形成烧结颈；烧结中期烧结颈形成后，平流通量机制开始逐渐取代表面扩散主导，空位沿着晶界迁移，逐渐完成致密化过程；烧结后期最终致密化完成，达到稳态，平流通量迅速减弱。3个阶段斜率逐渐减小，表明随着烧结进行，烧结动力学逐渐减弱，平流通量随着烧结进行作用降低。

图9

图9 平流通量作用下3个颗粒烧结时晶界上封闭气孔的收缩率

Fig.9 Simulated pore shrinkage curve of grain boundary among three particles by the advection flux model

3.2.2 平流通量平移分量对气孔收缩的影响

为进一步探究颗粒平移和旋转对致密化气孔收缩的影响，图10比较了不同平移迁移率下3个颗粒的气孔收缩。从图10可以明显看出，颗粒平移对致密化气孔收缩具有显著促进作用，随着m_t的增大，致密化的速率增加，最终致密化的程度增加；当m_t超过80时，致密化程度相对不变，最终完成致密化；当m_t超过100时，致密化速率相对不变。与多相场(MPF)模型^[30]的结果对比发现，本工作模拟颗粒平移迁移率对致密化影响更加明显。

图10

图10 不同平移迁移率下的气孔收缩率

Fig.10 Pore shrinkage curves for different translation mobilities (m_t)

3.3 稳定三叉晶界的形成

图11为直径为60个格点的4个圆形颗粒形貌演变的相场模拟。可以看出，在演化初期，颗粒逐渐靠拢，气孔收缩完成致密化，形成角度为90°的四叉晶界；之后随着演化的进行，晶界发生变化，颗粒发生偏移，形成稳定的120°三叉晶界，这与大量的UN烧结实验^[3~6,8,9]结果相吻合。这是因为演化中期晶界为不稳定状态(相邻晶界间的夹角处于90°)，当相邻晶界演化至夹角为120°时，三叉晶界形状保持稳定。结合图7中3个颗粒烧结相场模拟结果，可以得出致密化气孔收缩完成趋向于形成稳定的120°三叉晶界。

图11

图11 引入平流通量后4个颗粒烧结时的形貌演化过程

Fig.11 Simulated evolutions of four particles by the phase-field methods of advection flux

(a) 1 × 10⁴ step (b) 25 × 10⁴ step

3.4 多晶烧结过程

图12为颗粒尺寸单峰分布的多晶烧结相场模拟，模拟所采用的组织为30个不同尺寸的颗粒(图12a)。为与实验条件一致，选择颗粒的平均半径为150 nm，整个模拟区域的网格尺寸为256 × 256，采用耦合平流通量的致密化模型进行相场模拟。从图12可以看出，气孔对晶界移动起阻碍作用^[31]，气孔越大阻碍作用越强；相邻颗粒之间先形成烧结颈；颗粒之间形成气孔，气孔逐渐收缩，小气孔处晶界曲率较大、驱动力较强，优先完成致密化；致密化完成的部分逐渐形成稳定的120°三叉晶界。将本工作的演化模拟结果与真空热压烧结2 h的实验观察结果^[4]进行对比可得，UN烧结形貌演化形成三叉晶界、气孔收缩致密化的行为与实验结果趋势一致。

图12

图12 多晶烧结形貌演化的相场模拟

Fig.12 Simulated evolutions of multi-grain by the phase-field methods with unimodal particle size distribution

(a) 1 × 10⁴ step (b) 40 × 10⁴ step

(e) 160 × 10⁴ step (f) 200 × 10⁴ step

4 结论

(1) 引入了描述粒子刚体运动的平流通量作用项，建立了固相烧结致密化过程的相场模型，动态再现了UN晶粒烧结过程中内部气孔完成收缩的全过程。研究发现：引入平流通量模型对烧结颈的形成，在烧结前期作用较强，后期作用不明显；平流通量刚体运动的加入会加快烧结颈的形成，但最终平衡二面角的取值仍然不变，只由晶界能和界面能的比值所决定。

(2) 烧结过程中气孔的收缩分为3个阶段：初始阶段由于表面扩散占主导地位，颗粒间形成颈缩；中间阶段，平流通量刚体运动引起的致密化逐渐占据主导地位，相邻晶粒通过平移运动导致粒子中心相互靠近；最终阶段，气孔完成收缩，即致密化完成阶段。随着烧结进行，烧结动力学逐渐减弱，平流通量刚体运动随着烧结进行作用降低。随着平移迁移率(m_t)的增大，致密化速率逐渐增加，最终致密化程度增加；m_t超过100时致密化速率相对不变。

(3) 烧结相场模拟结果验证了相交120°的三叉晶界是能量上稳定的晶界，4个颗粒组成的晶界最终也演变成了稳定的三叉晶界；多晶UN颗粒烧结经历了烧结颈形成生长、气孔收缩致密化、稳定的三叉晶界形成等一系列微观过程；多晶烧结形成稳定的三叉晶界和气孔收缩致密化的现象与实验结果一致。

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