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LTCC Microelectronics, Glass Frits & SLS Polymer Powders, Yakov Frenkel (1945) Viscous Sintering

Frenkel Viscous Sintering Neck Ratio & Shrinkage Formula

Calculate Viscous Capillary Characteristic Time (min), Frenkel Neck-to-Particle Radius Ratio (x/r), Isotropic Linear Shrinkage (-dL/L0 %), and Frenkel-vs-Mackenzie Regime Status.

Raising LTCC glass-frit firing temperature just +35°C drops viscosity 5x (log10 eta from 8.8 to 8.1) while halving particle size—speeding densification shrinkage 10-fold!

Unlike crystalline metal powders that sinter slowly by solid-state grain-boundary diffusion (`(x / r)^6 ~ t`), amorphous **Borosilicate Glass Frits, Low-Temperature Co-Fired Ceramics (`LTCC`), Optical Preforms, and Selective-Laser-Sintered (`SLS`) Amorphous Polymers** densify by **Newtonian Viscous Flow Driven by Surface Tension `gamma`** (`N/m`)! Balancing capillary surface-area reduction against viscous dissipation across spherical particles of Radius `r` (`um` = `r * 10⁻⁶` m) at Newtonian Viscosity `eta = 10^(log_{10} eta)` (`Pa·s`), **Yakov Frenkel (`1945`)** derived **Characteristic Capillary Time `tau_v = ((eta * r) / gamma) / 60`** (`min`), **Dimensionless Neck Growth `x / r = sqrt((3 * gamma * t) / (2 * eta * r))`**, and **Early-Stage Isotropic Linear Shrinkage `-Delta_L / L_0 = ((3 * gamma * t) / (8 * eta * r)) * 100%`** (`= 25% * (x / r)^2`)! Once `x / r` exceeds **`~0.45`**, intersecting necks pinch open channels into isolated spherical pores governed by **Mackenzie-Shuttleworth (`1949`) Late-Stage Pore Closure**!

Verify on an `LTCC` Dielectric Glass-Ceramic Tape Firing Hold—Surface_Tension `gamma` = 0.32 N/m (B2), Particle_Radius `r` = 12.0 um (B3), `Log10`_Viscosity `log_{10}(eta)` = 8.80 `[Pa·s]` (B4, `6.31 x 10⁸` Pa·s near Littleton softening), and Isothermal_Soak `t` = 24.0 min (B5): Viscous Characteristic Time equals `394.3` min (E2), Frenkel Neck Ratio `x / r` reaches `0.302` (E3), Linear Firing Shrinkage equals `2.28%` (E4), confirming `"INITIAL FRENKEL NECK STAGE"` (`x / r <= 0.45` in E5)!

Frenkel (1945) Viscous Sintering KPI (gamma=0.32 N/m)FormulaVerified Output
Viscous Characteristic Time (min)(((10 ^ LogEta) * (r_um * 1e-6)) / gamma) / 60394.3 min (39.3 min @ fine hot frit!)
Frenkel Neck Ratio (x / r)SQRT((3 * gamma * t_s) / (2 * eta * r_m))0.302 (0.756 coalesced spheres)
Isotropic Linear Shrinkage (%)((3 * gamma * t_s) / (8 * eta * r_m)) * 1002.28% (14.30% full densification)
Sintering Kinetics Regime GateIF(x_over_r <= 0.45, "INITIAL FRENKEL...", "LATE MACKENZIE...")INITIAL FRENKEL NECK STAGE

Yakov Frenkel (1945) Viscous Capillary Neck Coalescence & Linear Shrinkage Formula

Validate `Surface_Tension gamma` in `[0.05, 1.50]` N/m (`~0.03–0.05` polymers to `0.30–0.40` silicate glasses), `Particle_Radius r` in `[0.2, 500]` um, `Log10_Viscosity` in `[5.0, 13.0]` Pa·s, and `Hold_Time` in `(0, 1,440]` min!

=LET(gm_n, B2, rp_u, B3, lg_v, B4, tm_m, B5, IF(OR(gm_n < 0.05, gm_n > 1.5, rp_u < 0.2, rp_u > 500, lg_v < 5, lg_v > 13, tm_m <= 0, tm_m > 1440), "Invalid Sinter Input", ROUND((((10 ^ lg_v) * (rp_u * 0.000001)) / gm_n) / 60, 1)))

Jet-Milled Fine Frit (`r = 6` um) + Hotter Peak (`log10 eta = 8.1`, `15` min) Pore-Closure Check

Mill glass frit finer (`B3 = 6.0` um) and raise peak furnace zone `+35°C` (`B4 = 8.10` -> `eta = 1.26 x 10⁸` Pa·s) for just `B5 = 15.0` min: Characteristic Time E2 drops **`10x` to `39.3` min**, driving `x / r` E3 to **`0.756`**, Linear Shrinkage E4 to **`14.30%`**, and transitioning E5 into **`"LATE MACKENZIE PORE CLOSURE"`** (where trapped `N_2` gas bubbles halt final `2%` porosity unless fired under vacuum or `He`/`O_2`)!

=IF(ISTEXT(E2), E2, ROUND(SQRT((3 * B2 * (B5 * 60)) / (2 * (10 ^ B4) * (B3 * 0.000001))), 3))

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