1 · Specify which pressure and which velocity

1 · 指定是哪一种压力与速度

An isolated spherical bubble has no preferred axis and hence no directed microjet. It can generate radial flow and a strong shock. To discuss jet speed, introduce a boundary, a neighbour, a pressure gradient or a shaped interface and solve that asymmetry. “Supersonic” must specify the sound speed of the gas, the carrier, or the shocked material.

孤立球形气泡没有优选轴，因此不会产生有方向的微射流，但可产生径向流动和强冲击波。讨论射流速度时，需要引入边界、邻泡、压力梯度或特定界面形状，并求解这种非对称性。“超声速”还必须说明参照的是气体、载液，还是受冲击材料的声速。

2 · Verified high-end examples, with the measurement type attached

2 · 经核查的高值实例，同时注明测量类型

Reuter & Ohl (2021), APL, DOI 10.1063/5.0045705 — PDF pp. 2–3; the >850 m/s value is a lower bound.

Brujan et al. (2001), JFM, DOI 10.1017/S0022112000003347 — PDF p. 1 abstract.

Fan et al., author manuscript; PRL 132, 104004 (2024), DOI 10.1103/PhysRevLett.132.104004 — author manuscript PDF pp. 4–7, especially p. 5; final publication verified by DOI.

Liang et al. (2022), JFM, DOI 10.1017/jfm.2022.202 — PDF pp. 30–34, Figures 7–8.

The last entry follows the primary Nature Communications article, specifically its Results, Figure 3 and Discussion. The indexed full article text was available; its PDF could not be acquired. The experiment used 1064 nm, 6 ns, 24 mJ pumping, far above the inherited sub-µJ pulse. These checked examples do not establish a worldwide record or a PFH limit. A 2026 electrically generated tandem-bubble study was excluded from this laser-specific table.

最后一项依据 Nature Communications 原始论文的结果、图 3 及讨论。可获取已索引的完整论文文本，但未能下载 PDF。实验采用 1064 nm、6 ns、24 mJ 激发，远高于沿用的亚微焦脉冲。这些经核查实例不能确定世界纪录或 PFH 极限。2026 年一篇电驱动串联气泡研究未列入这个激光专属表。

3 · Solve the simplest bubble and locate its false infinity

3 · 求解最简单的气泡，并定位其非物理无穷大

ρ(RR̈ + 3Ṙ²/2) = −Δp
Δp = p_∞ − p_v

Ṙ = −√{[2Δp/(3ρ)][(R_max/R)³ − 1]}
t_R = 0.914681R_max√(ρ/Δp)

For the inherited 8 µm example, ideal collapse takes 0.73482 µs. At half radius the wall speed is 21.51 m/s; at one tenth it is 256.99 m/s. The formula diverges at zero radius because no compressible or internal-pressure mechanism arrests it. The divergence is a model-failure signal, not a physically achievable infinite jet speed or load.

对于沿用的 8 µm 例子，理想塌缩时间为 0.73482 µs。在半径减半时，泡壁速度为 21.51 m/s；减至十分之一时，为 256.99 m/s。由于模型中没有可压缩性或内部压力机制终止塌缩，零半径处速度发散。这个发散是模型失效信号，并非可实现的无限射流速度或载荷。

M_w = |Ṙ|/c
R/R_max = {1 + 3ρ(M_wc)²/(2Δp)}^−1/3

At a declared wall-Mach cutoff of 0.1, the ideal radius fraction is 0.14424. This is an escalation convention, not a universal physical discontinuity. An incompressible model should not be extended to obtain collapse-shock peaks.

若声明泡壁马赫数截止值为 0.1，则理想半径比例为 0.14424。这是模型升级约定，不是普适的物理突变点。不能将不可压缩模型继续延伸，用于获取塌缩冲击波峰值。

4 · Add gas, acoustics and the liquid equation of state

4 · 加入气体、声学及液体状态方程

p_w = p_B − 2σ/R − 4μṘ/R
(1 − Ṙ/c)RR̈ + (3/2)(1 − Ṙ/(3c))Ṙ²
= (1 + Ṙ/c)(p_w − p_∞)/ρ + [R/(ρc)](dp_w/dt)

A compressible equation still needs a bubble-content law. To verify sensitivity, the supplied executable uses a deliberately pedagogical, fixed-mass gas with κ = 1.4 plus constant water vapor, not a PFH closure. It stops at the first rebound or at wall Mach 0.1. The 1 kPa-at-maximum-radius case reaches the Mach cutoff before rebound; its stopped pressure is not a predicted peak. The other cases remain in the declared low-Mach screen.

可压缩方程仍需要泡内物质规律。为核验敏感性，所附程序明确采用教学性的固定质量气体、κ = 1.4 和恒定水蒸气，而非 PFH 闭合关系。计算在首次回弹或泡壁马赫数达到 0.1 时停止。最大半径处气压为 1 kPa 的算例在回弹前达到马赫数截止值，因此停止时的压力并非预测峰值。其他算例位于声明的低马赫数筛查区间内。

H = ∫_p∞^pw dp/ρ(p)
(1 − Ṙ/C)RR̈ + (3/2)(1 − Ṙ/(3C))Ṙ²
= (1 + Ṙ/C)H + (1 − Ṙ/C)(R/C)(dH/dt)

At strong collapse, use an experimentally appropriate liquid EOS, a nonideal bubble-content EOS, heat/mass transfer and a shock propagation calculation. A Tait liquid law is useful within its fitted range but is not guaranteed at tens of GPa. Gas concentration, condensation and the hard-core/nonideal state can change the pressure drastically; increasing a gas pressure does not necessarily increase collapse intensity.

强塌缩时需要实验适用的液体状态方程、非理想泡内状态方程、热质传递及冲击波传播计算。Tait 液体规律在拟合范围内有用，但不保证适用于数十 GPa。气体浓度、凝结以及硬核／非理想状态可能大幅改变压力；提高气压并不一定增强塌缩。

Liang et al. (2022), JFM, DOI 10.1017/jfm.2022.202 — Sections 3–4; extended model and limitations; gas/condensation dependence.

5 · Replace an unspecified gas with actual PFC inventory

5 · 用真实 PFC 存量替代未指定气体

m_d = ρ_PFC(4πa³/3)
E_phase,d = m_d[c_p,PFC(T_b − T₀) + L_v]
V_v,d = (m_d/M)R_uT_b/p

The inherited illustrative core is PFH, not an identified project core: a = 250 nm, density 1670 kg/m³, molar mass 0.3380418 kg/mol, T_b = 332.2 K, c_p = 710 J/(kg·K), and L_v = 93200 J/kg. The molar mass and phase data trace to the NIST Chemistry WebBook; the chosen heat capacity/density and temperature extrapolations remain estimates as in the original project contract.

沿用的示例液核为 PFH，但本项目真实液核尚未确定：a = 250 nm，密度 1670 kg/m³，摩尔质量 0.3380418 kg/mol，T_b = 332.2 K，c_p = 710 J/(kg·K)，L_v = 93200 J/kg。摩尔质量及相变数据可追溯至 NIST Chemistry WebBook；所选热容、密度及温度外推仍与原始项目参数一样属于估计。

One such droplet contains 1.093 × 10⁻¹⁶ kg and requires about 13.217 pJ of sensible-plus-latent energy from 293.15 K. Its ideal vapor radius at 1 atm and the declared temperature is 1.2814 µm. An 8 µm effective vapor volume therefore requires about 243.325 continuous droplet equivalents and 3.2161 nJ. The project’s effective 8 µm bubble is not the vapor volume of one 250 nm droplet. Coalescence into a single bubble is an additional dynamical assumption.

这样一颗液滴含有 1.093 × 10⁻¹⁶ kg 物质，从 293.15 K 起所需显热及潜热约为 13.217 pJ。在 1 atm 和所给温度下，其理想蒸气半径为 1.2814 µm。因此，8 µm 的有效蒸气体积需要约 243.325 个连续液滴当量及 3.2161 nJ。项目中的有效 8 µm 气泡，并不是单个 250 nm 液滴的蒸气体积。是否汇合为一个气泡还是额外的动力学假设。

p_{liquid core} = p_∞ + 2σ_eff/a + Δp_shell
p_BV_B = R_uT_BΣ n_s

Using the water-like 0.0728 N/m merely as a teaching value gives a 582.4 kPa Laplace term at 250 nm. The actual droplet shell and interfaces must supply σ_eff and the shell stress. The 1 atm vapor-volume calculation omits the bubble’s own curvature pressure, gel work and mass transfer; it is not an equilibrium radius prediction. Vaporization enthalpy already includes constant-pressure expansion work, so adding that same pV work again double counts it.

仅将水样的 0.0728 N/m 用作教学值时，250 nm 处的 Laplace 项为 582.4 kPa。真实液滴壳层及界面须提供 σ_eff 和壳层应力。1 atm 蒸气体积计算忽略了气泡自身曲率压力、凝胶功及质传递，因此不是平衡半径预测。汽化焓已包含定压膨胀功，再加入同一项 pV 会重复计算。

dm_PFC,v/dt = 4πR²j″_PFC
dU_B/dt = −p_B(dV_B/dt) + Q̇ + Σ h_s(dm_s/dt)

A real PFC forward model needs the identified core/shell/absorber, the stochastic activation time, vapor and noncondensable masses, temperature, interfacial heat/mass rates and a suitable EOS. Setting the entire PFC vapor pressure constant throughout violent collapse, or treating all PFC vapor as permanent gas, supplies two different extreme closures. Neither is automatically correct.

真实 PFC 前向模型需要确定液核、壳层、吸收体、随机激活时刻、蒸气及不凝气体质量、温度、界面热质传递速率与适当状态方程。在猛烈塌缩中令所有 PFC 蒸气压恒定，或把所有 PFC 蒸气当作永久气体，是两种不同的极端闭合关系，不能自动认定其中任何一种正确。

6 · Finite energy gives conditional bounds, not a universal maximum

6 · 有限能量给出有条件的界限，而非普适最大值

Δp(4πR_max³/3) ≤ η_bE_abs
U_j ≤ √(2E_j/m_j)

E_sh ≈ [4πr²/(ρc)]∫p′(r,t)² dt
p_rect = √[E_shρc/(4πr²τ_sh)]

Without a specified pulse width or shape, an energy integral bounds neither a unique peak nor an arbitrarily narrow spike. For illustration only, allocate 1% of the inherited absorbed energy to a 50 ns rectangular acoustic pulse at 1 mm: the pressure scale is 83.27 kPa. This is not the pressure of the actual PFC event or a general peak bound. Near a nonlinear shock, use its EOS and energy flux instead of the weak acoustic formula.

没有指定脉冲宽度或形状时，能量积分既不能确定唯一峰值，也不能限制任意窄的尖峰。仅作示例，将沿用吸收能量的 1% 分配给在 1 mm 处、宽度为 50 ns 的矩形声脉冲，压力尺度为 83.27 kPa。这并非真实 PFC 事件的压力，也不是一般峰值上界。非线性冲击波附近应采用相应状态方程和能流，而非弱声学公式。

q_j = ρU_j²/2
p_wh ≈ ρcU_j
p_contact ≈ [Z_lZ_s/(Z_l + Z_s)]U_relative

At U_j = 1400 m/s in the inherited carrier, the arithmetic gives q_j = 978.24 MPa and linear rigid-stop water hammer 2071.07 MPa. But the ambient-water Mach number is 0.945, so the linear water-hammer estimate has no reliable quantitative status here. A measured jet velocity is not a measured target pressure; compressible impact, target impedance, footprint and pulse duration decide the load.

若在沿用载液中取 U_j = 1400 m/s，算术上得 q_j = 978.24 MPa，线性刚性停止水锤为 2071.07 MPa。但相对常温水的马赫数为 0.945，所以此处线性水锤估算不具备可靠的定量地位。实测射流速度不等于实测目标压力；可压缩冲击、目标阻抗、作用区域及脉冲宽度共同决定载荷。

7 · Three to five bubbles: solve the coupling, then check its limits

7 · 三至五个气泡：求解耦合，再检查边界

R_iR̈_i + 3Ṙ_i²/2
+ Σ_j≠i[R_j²R̈_j + 2R_jṘ_j²]/d_ij
= [p_B,i − p_∞ − 2σ/R_i − 4μṘ_i/R_i]/ρ

This leading interaction model follows from the liquid potential of separated spherical sources. For instantaneous coupling it assumes travel times are short relative to source variation. It excludes translation, shape modes and contact. For identical synchronized empty cavities in a geometry with equal neighbour sums, the conserved liquid kinetic energy includes cross terms. The corresponding integrated speed is below the isolated value.

该主导相互作用模型来自分离球形源的液体势。瞬时耦合要求传播时间短于源变化时间，并排除平移、形状模态及接触。若同步的相同空腔处于邻泡求和相等的几何中，守恒的液体动能包含交叉项，相应积分速度低于孤立气泡值。

S = Σ_j≠i1/d_ij
Ṙ² = [2Δp/(3ρ)](R_max³ − R³)/[R³(1 + SR)]
C(χ) = √(3/2)∫₀¹x^3/2√[(1 + χx)/(1 − x³)] dx
χ = SR_max

These are verified ideal-cavity calculations with equal 8 µm maxima, not three to five individually vaporized nanodroplets. They show that even a synchronous symmetric cluster changes the collapse coefficient. Asymmetric or delayed pairs can instead focus a fast jet, as the Fan experiment demonstrates. The instantaneous-coupling approximation worsens near the fast endpoint; its physical error is not bounded by the numerical convergence test.

这些是最大半径均为 8 µm、经核验的理想空腔计算，并非三至五个分别汽化的纳米液滴。它们说明即使同步且对称，气泡群也会改变塌缩系数。非对称或延迟产生的气泡对则可能聚焦出高速射流，Fan 的实验展示了这一点。瞬时耦合近似在高速末期变差；数值收敛检验不能限定其物理误差。

8 · Pressure from several bubbles needs travel delays and a shared budget

8 · 多泡压力需要传播延迟与共同能量预算

V_i = 4πR_i³/3
V̈_i = 4π(2R_iṘ_i² + R_i²R̈_i)
p′(x,t) ≈ [ρ/(4π)]Σ V̈_i(t − t_i − d_i/c)/d_i

Equal pulses arriving in phase at a point can add coherently; arrival-time spread or phase differences can reduce the peak. Prescribing the same isolated bubble history for every nearby source can violate energy conservation because neighbours also change source work and radiation. At fixed total laser energy, increasing the count is not permission to multiply a single-bubble pressure by that count.

相同脉冲同相到达某一点时可以相干叠加；到达时间离散或相位差异会降低峰值。给每个邻近气泡规定相同的孤立泡历程可能违反能量守恒，因为邻泡也改变源功及辐射。总激光能量固定时，增加数量并不意味着可以用数量直接乘单泡压力。

This table uses the coupled ideal trajectories, an observer at (0,0,1 mm), travel delays in water-like liquid, and only the arrival interval for which all source histories remain available below the Mach cutoff. It is not the full-event collapse maximum. The tetrahedron and cube have different arrival paths and source dynamics; their entries illustrate why count alone cannot order pressure. No result is extrapolated to the target at 8.5 µm from the bubble centre.

本表采用耦合的理想历程，观测点为 (0,0,1 mm)，传播延迟按水样液体计算，并只使用所有源历程均有数据且未超过马赫数截止值的到达时间区间。它并非完整事件的塌缩峰值。四面体与立方阵列具有不同到达路径及源动力学，说明不能仅凭数量给压力排序。任何结果都未外推至距气泡中心 8.5 µm 的实际目标位置。

9 · Uniform droplets, uniform bubble arrays and a uniform mixture are different

9 · 均匀液滴、均匀气泡阵列及均匀混合介质不同

n_b(x,t) = n_d(x)P_act(x,t)
α(x,t) = n_b(x,t)(4πR³/3)
n_b = 1/ℓ³   (one bubble per cubic cell)

For 8 µm bubbles at 40 µm cubic pitch, α = 0.03351, or 3.351%. A 27-cell volume has an equivalent spherical cloud radius 74.44 µm. The classical interaction screen below is about 2.90, so a low void fraction does not ensure weak collective effects. This parameter comes from monodisperse linear cloud theory; it is a screen, not a laser-collapse pressure formula.

若 8 µm 气泡按 40 µm 的立方节距排列，则 α = 0.03351，即 3.351%。27 个单元的体积对应等效球形云半径 74.44 µm。下述经典相互作用筛查量约为 2.90，因此较低空隙率不保证群体效应较弱。该量来自单分散线性气泡云理论，仅用于筛查，不是激光塌缩压力公式。

B = α(R_c/R)²
F(r)/F₀ = exp(−2r²/w₀²)

At the inherited w₀ = 25 µm, a grid node 40 µm off axis receives only 0.005976 of central fluence; a (40,40) µm corner receives 0.00003571. Uniform PFC concentration under this beam therefore does not supply uniform bubble size or timing. A uniform-array calculation requires a different illumination or an explicitly imposed source distribution. The illustrative inventory concentration from the original focal volume would yield about 21.85% ideal vapor volume if all equivalents vaporized without constraints—already a dense, interacting state.

沿用 w₀ = 25 µm 时，离轴 40 µm 的节点仅获得中心能量面密度的 0.005976；(40,40) µm 角点仅为 0.00003571。因此，该光束下均匀的 PFC 浓度不能给出均匀的气泡尺寸或时序。均匀阵列计算需要改变照明方式，或明确规定源分布。原焦域体积的示例存量浓度若所有当量均无约束汽化，理想蒸气体积将占约 21.85%，已属于密集的相互作用状态。

Maeda & Colonius (2019), JFM, DOI 10.1017/jfm.2018.968 — cloud interaction parameter, Section 3; nonlinear cloud anisotropy.

10 · Solve a finite uniform cloud in the acoustic limit

10 · 在声学极限下求解有限均匀气泡云

p̂(r,ω) = −[ρω²N_bV̂/(4πr)]e^ikrF(kR_c)
F(z) = 3[sin z − z cos z]/z³
k = ω/c

This follows by integrating the retarded Green function over a sphere; angular integration and the radial integral produce the factor F. In the compact limit F tends to one. Finite cloud size introduces cancellation instead of unlimited coherent multiplication. It is a solved radiation problem for prescribed identical sources, not a solved source-generation problem. Strong interaction requires self-consistent bubble susceptibility and pressure.

该式由对球体积分延迟 Green 函数得到；角向及径向积分产生因子 F。紧致极限下 F 趋于一，有限云尺寸引入相消，而不是无限的相干倍增。它求解的是规定相同源的辐射问题，不是源生成问题。强相互作用需要气泡响应率与压力自洽求解。

∇²p′ − (1/c_l²)∂²p′/∂t² = −ρ ∂²α′/∂t²
α̂′ = n_bχ_V(ω)p̂′ + α̂_drive
k_eff² = ω²/c_l² − ρn_bω²χ_V(ω)

χ_V(ω) = −4πR_e/{ρ[ω₀² − ω² − 2iβω]}
ω₀² = [3κp_g,e − 2σ/R_e]/(ρR_e²)

Once the susceptibility, activation source, boundaries and geometry are specified, this is a closed linear pressure problem that can be solved in frequency or time. It contains resonance, attenuation and wave interference. It applies to the acoustic perturbation regime after bubbles exist; it does not describe their nonlinear nucleation or violent collapse. A nanodroplet-rich liquid before activation must not be assigned the gas-mixture compressibility after activation.

给定响应率、激活源、边界及几何后，这是可在频域或时域求解的闭合线性压力问题，包含共振、衰减及波干涉。它适用于气泡已经存在后的声学扰动状态，不描述非线性成核或猛烈塌缩。激活前富含纳米液滴的液体，不能被赋予激活后含气混合介质的可压缩性。

1/K_eff = (1 − α)/K_l + α/(κp_g,e)
c_eff = √(K_eff/ρ_mix)

For the 3.351% void example, κ = 1.4 and equilibrium gas pressure including curvature at 8 µm give about 71.2 m/s under this deliberately simplified limit. Its large difference from 1482 m/s shows why pure-water travel time cannot automatically be used through a bubbly layer. Rapid phase change, frequency-dependent bubble response and a gel scaffold require a different susceptibility.

对于 3.351% 空隙率示例，取 κ = 1.4，且 8 µm 平衡气压包含曲率项，则在这个刻意简化的极限下约为 71.2 m/s。它与 1482 m/s 的巨大差异说明，穿过含泡层时不能自动采用纯水传播时间。快速相变、随频率变化的气泡响应及凝胶骨架要求不同的响应率。

Worked spatial solution with pressure feedback

包含压力反馈的空间解

Consider a finite spherical cloud with a uniform small harmonic drive and uniform bubble susceptibility. Use the time convention exp(−iωt), the same carrier density inside and outside, regularity at the centre, and an outgoing wave outside. Pressure and normal velocity are continuous at Rc. This solves the spatial pressure problem once the PFC susceptibility and drive are supplied; it does not estimate those formulation-specific functions.

考虑具有均匀小幅谐波驱动及均匀气泡响应率的有限球形气泡云。采用时间约定 exp(−iωt)，内外载液密度相同，中心正则、外部波向外传播，并在 Rc 处保持压力及法向速度连续。给定 PFC 响应率及驱动后，此解完成空间压力问题，但不会估计这些依赖配方的函数。

k_i² = ω²/c_l² − ρn_bω²χ_V
k_l = ω/c_l
p₀ = ρω²α̂_drive/k_i²
j₀(z) = sin z/z
q = ik_l − 1/R_c

A = qp₀/[k_ij′₀(k_iR_c) − qj₀(k_iR_c)]
p̂_in(r) = p₀ + Aj₀(k_ir)
p_b = p₀k_ij′₀(k_iR_c)/[k_ij′₀(k_iR_c) − qj₀(k_iR_c)]
p̂_out(r) = p_b(R_c/r)e^{ik_l(r−R_c)}

The executable example uses hypothetical equilibrium 8 µm fixed-gas bubbles, αdrive = 10⁻⁷ and β = 0.1ω₀. It checks the pressure/velocity match to 2.2 × 10⁻¹³ relative error and recovers the independently integrated no-feedback Green-function field to 2.4 × 10⁻¹⁵. Maximum fractional radius perturbation stays below 7.4 × 10⁻⁶, within its small-amplitude assumption. The frequency sweep is a conditional response, not a laser-collapse peak or a calibration of PFC.

可执行示例采用假想的平衡 8 µm 固定气体气泡、αdrive = 10⁻⁷ 及 β = 0.1ω₀。压力／速度匹配的相对误差为 2.2 × 10⁻¹³；在无反馈情况下，独立积分的 Green 函数场重现误差为 2.4 × 10⁻¹⁵。最大半径相对扰动低于 7.4 × 10⁻⁶，符合小幅假设。频率扫描为有条件的响应，并非激光塌缩峰值或 PFC 标定。

Finite-cloud solution and independent verification · Frequency-response data.

有限气泡云解及独立核验 · 频率响应数据。

11 · The rigorous nonlinear PFC pressure problem

11 · 严格的非线性 PFC 压力问题

For strong collapse, the final tier resolves compressible liquid and bubble/gas domains, their moving interfaces, phase exchange and the actual solid stack. In each fluid bulk domain the equations below are conservation laws; interfacial mass, momentum and heat jump conditions join them. Species and EOS identify water, PFC vapor and permanent gas. A subgrid Eulerian–Lagrangian model is appropriate only while its bubbles remain separated and its stress/thermal closures have been verified.

对于强塌缩，最终模型层级需解析可压缩液体及气泡／气体区域、运动界面、相态交换和实际固体层。在每个流体区域，下述方程为守恒规律，并通过界面的质量、动量及热量跳跃条件连接。物种和状态方程区分水、PFC 蒸气及永久气体。只有气泡保持分离且应力／热闭合关系已核验时，才适合采用亚网格 Euler–Lagrange 模型。

∂ρ/∂t + ∇·(ρu) = 0
∂(ρu)/∂t + ∇·(ρu⊗u) = −∇p + ∇·τ
∂(ρe_tot)/∂t + ∇·[(ρe_tot + p)u]
= ∇·(τu) − ∇·q + Q‴

u_fluid·n = ẇ_solid·n
t_fluid = t_solid
σ_fluid = −pI + τ
p_F = −n·σ_fluidn − p_∞
P_max,film = max_{x∈A, t∈[t0,t1]} p_F(x,t)

A target-specific maximum additionally needs a fixed admissible pulse-energy range and a failure rule: usable pressure may be limited by membrane rupture, gel fracture, droplet depletion, film damage or receiver rejection before the largest fluid pressure is reached. The current project does not identify the core/shell/absorber, activation law, bubble history, gel dynamic constitutive law, or stack/interface properties sufficiently to assign a unique numerical PFC-array peak. The theory above specifies a solvable conditional model and rejects the unsupported substitution of a water-bubble record for that number.

目标体系的最大值还需要固定允许的脉冲能量范围及失效规则：达到最高液体压力之前，膜破裂、凝胶断裂、液滴耗尽、薄膜损伤或接收失败就可能限制可用压力。当前项目尚未充分确定液核、壳层、吸收体、激活规律、气泡历程、凝胶动态本构及层间／界面性质，因而不能给出唯一的 PFC 阵列峰值数值。上述理论给出了可求解的有条件模型，也排除了用水气泡纪录替代该数值的不受支持做法。

Technical appendix · Derivation and verification ledger

技术附录 · 推导与核验记录

T = 2πρΣR_i³Ṙ_i²
+ 4πρΣ_i<jR_i²R_j²Ṙ_iṘ_j/d_ij
Π = (4πΔp/3)ΣR_i³

Apply the Lagrange equation to T − Π. Differentiation of each cross term cancels the product containing the local velocity and leaves the neighbour acceleration/velocity terms in Section 7. For an equal-radius symmetric mode, energy conservation immediately gives the integrated speed and the collapse integral. Independently, direct ODE integration conserves this energy; the same numerical convergence does not validate omitted compressibility, phase change or shape dynamics.

对 T − Π 应用 Lagrange 方程。每个交叉项求导后，含本地速度的乘积相消，剩下第七节中的邻泡加速度／速度项。对于等半径对称模态，能量守恒直接给出积分速度及塌缩积分。独立的直接 ODE 积分也守恒此能量；相同的数值收敛并不验证被忽略的可压缩性、相变或形状动力学。

0.914681 = [√(3/2)/3] B(5/6, 1/2)

The quadrature and beta coefficient agree to about 1.3 × 10⁻¹¹ relative error. The largest ideal-cluster energy residual is 2.2 × 10⁻¹¹; tightening solver tolerances/step sizes changes cutoff time by less than 2 × 10⁻¹⁰ relative and the sampled pre-cutoff pressure maximum by about 0.0131% at worst. The files contain the complete values and stopping rules. These are numerical verification statements. Physical validation requires synchronized target measurements and held-out cases.

数值积分与 Beta 系数的相对误差约为 1.3 × 10⁻¹¹。理想气泡群的最大能量残差为 2.2 × 10⁻¹¹；收紧求解容差及步长后，截止时刻的相对变化小于 2 × 10⁻¹⁰，采样的截止前压力最大值最坏变化约 0.0131%。文件包含完整数值及停止规则。这些结论属于数值核验，物理验证仍需同步的目标测量及独立验证算例。

Calculation source · Inherited and derived values · 1/3/4/5/27-bubble summary · Verification results · Source provenance.

计算源代码 · 沿用及推导数值 · 1／3／4／5／27 泡结果 · 核验结果 · 来源记录。