How Do Pore Coordination Number & Dihedral Angle Influence Pore Shrinkage Vs Growth In Sintering Furnaces?

A pore’s fate during sintering—whether it shrinks away or grows into a permanent defect—is determined by a geometric tug-of-war between its coordination number and the dihedral angle at solid-solid-pore junctions. Simply put, the pore coordination number (N, the number of grains surrounding a pore) dictates the sign of its surface curvature, while the dihedral angle (ψ) sets the critical threshold that separates stable shrinkage from runaway growth. In a typical ceramic fired at 120° dihedral angle, pores with fewer than 6 neighboring grains (in 2D cross-section) shrink spontaneously, whereas those with more than 6 grow, resisting final densification. High-temperature sintering furnaces influence this balance by controlling thermal profiles and atmosphere, which in turn modify grain growth and interfacial energies to prevent pores from exceeding the critical coordination number.

The interplay between pore coordination number and dihedral angle creates a thermodynamic boundary: every dihedral angle defines a critical coordination number (Nc). Pores with N < Nc are convex toward their center and shrink; pores with N > Nc are concave and grow. Precise furnace control manipulates this boundary by stabilizing grain structures and interfacial energies, keeping pores on the shrinking side.

The Dual Forces That Decide Pore Stability

What the Dihedral Angle Tells You About Interfacial Energy

The dihedral angle ψ is not just a geometric curiosity—it’s a direct window into the ratio of grain boundary energy to solid-vapor surface energy. The force balance at the junction of a grain boundary and a pore surface is given by:

γ_gb = 2γ_sv cos(ψ/2)

A large ψ means the grain boundary energy is low relative to the solid-vapor energy. This configuration favors pores with sharply angled surfaces that are convex toward the pore center. When surfaces are convex inward, the Laplace pressure compresses the pore, driving shrinkage.

A small ψ indicates a high-energy grain boundary that forces the pore to adopt a more lenticular, concave shape. Concave pore surfaces resist elimination because the capillary pressure works against densification, or the pore can even grow by consuming vacancies from surrounding boundaries.

How the Coordination Number Sets the Curvature

The pore coordination number N counts how many grains meet at the pore. It’s a structural fingerprint of the powder compact and evolves during sintering.

When N is low, the pore sits at the junction of few grains. Their surfaces can bow inward, creating the convex curvature that promotes shrinkage. The free-energy reduction from eliminating pore surface outweighs any increase in grain boundary area, so the system naturally moves toward full density.

When N is high, the pore is hemmed in by many grains. Their boundaries create a near-polyhedral void with concave surfaces. Expanding the pore—or at least preventing its closure—becomes thermodynamically favorable, because it would require building costly new grain boundary area to shrink the pore. The result is a thermodynamic barrier to densification, where large, high-N pores grow while small ones vanish.

The Critical Coordination Number: Where the Switch Happens

The critical coordination number Nc is the pivot point. For any given dihedral angle, there is a threshold N above which the pore surface curvature flips from convex to concave.

In a 2D cross-section, the relationship is straightforward: ψ = (180°·Nc – 360°) / Nc, or equivalently Nc = 360°/(180° – ψ).
For the classic 120° dihedral angle, Nc = 6. Pores with N < 6 shrink; N > 6 grow.
In real 3D polycrystalline microstructures, the same logic holds, but Nc scales higher. At ψ = 120°, 3D pores require Nc = 12—a far more demanding threshold that explains why large pores in powder metallurgy and ceramic processing are so difficult to eliminate.

Thus, the coordination number and dihedral angle are interdependent. A furnace that can increase ψ (e.g., through atmosphere control that lowers γ_gb) will raise Nc, making even relatively large pores unstable and prone to shrinkage.

How High-Temperature Furnaces Manipulate This Balance

Thermal Profiles Tame Coordination Numbers

Controlled heating schedules prevent abnormal grain growth, which can quickly push pores beyond Nc. When grains enlarge normally, they reduce the effective number of grains bounding a pore. A pore that started with N > Nc can see its coordination drop below the critical threshold as surrounding grains grow together, flipping it from growth mode to shrinkage mode.

But there’s a catch: if grain growth is too rapid, pores detach from boundaries before they can shrink, becoming trapped inside grains where they are virtually impossible to remove. Furnace thermal cycles must therefore balance grain coarsening with pore mobility.

Atmosphere Engineering Shifts the Dihedral Angle

Atmospheres in high-temperature furnaces—whether vacuum, hydrogen, or inert gas—directly alter the solid-vapor surface energy γ_sv and the grain boundary energy γ_gb.

Reducing atmospheres or oxygen partial pressure control can lower γ_gb relative to γ_sv, increasing ψ. A higher ψ yields a larger Nc and makes pores easier to shrink.
Carefully selected dopants or sintering aids that segregate to grain boundaries can also raise ψ, shifting the critical coordination number upward and enabling densification even for initially unfavorable pore configurations.

The furnace operator’s ability to hold a precise atmosphere and temperature window is what turns this thermodynamic theory into a practical densification strategy.

The 3D Reality Check

Most sintering models simplify pores to 2D shapes, but real ceramic compacts host 3D pore networks. At ψ = 120°, the 3D critical coordination number is 12—not 6. This means a pore surrounded by fewer than 12 grains can still shrink, but the barrier is higher.

Powder processing techniques that produce high green density and uniform pore distribution (cold isostatic pressing, slip casting) minimize the population of large, high-N pores before the part ever enters the furnace. By starting with a compact where most pores are already well below Nc, even a modest increase in dihedral angle during sintering can yield near-full density.

Understanding the Trade-offs and Pitfalls

Excessive Grain Growth Traps Pores

While controlled grain growth can rescue some large pores by lowering their N, abnormal grain growth does the opposite. A few huge grains can swallow smaller ones, leaving pores stranded in grain interiors. Once detached, a pore’s dihedral angle no longer matters—it can only shrink by bulk diffusion, which is orders of magnitude slower, halting densification.

Dihedral Angles Near 180° Make Pores Immobile

An extremely high ψ (approaching 180°) sounds ideal because it maximizes Nc. However, at 180°, the pore has no preference for grain boundaries; it becomes spherical and can easily detach. The peak pore migration velocity plummets, and pores are left behind by moving boundaries. There is an optimum ψ (often around 120°–150°) that balances high Nc with sufficient pore mobility.

Agglomerates Create Local High-N Regions

Even in a well-processed compact, isolated agglomerates can produce pockets of large, high-N pores that cannot be eliminated by any realistic furnace schedule. No amount of atmosphere or temperature tuning can overcome a packing defect that starts with N > Nc in 3D. The only solution is upstream processing to break up agglomerates and achieve a homogeneous particle arrangement.

Making the Right Choice for Your Sintering Goal

The influence of coordination number and dihedral angle is not an abstract phase diagram—it’s a daily decision-making framework for furnace operators and ceramic engineers. Here’s how to apply it depending on your priorities:

If your primary focus is achieving maximum final density: Start with a high green density and uniform pore structure to keep N low, then use a controlled atmosphere that raises the dihedral angle (lowers γ_gb) to increase Nc. This combination ensures the largest possible fraction of pores are thermodynamically unstable and shrink.
If your primary focus is controlling grain size while eliminating pores: Employ a stepped thermal profile that first promotes normal grain growth to reduce the N of large pores below the 3D Nc threshold, then holds at a temperature where densification outpaces grain coarsening.
If your primary focus is preventing pore detachment and trapped defects: Avoid atmosphere conditions that push ψ too close to 180°. Maintain a dihedral angle in the 120°–150° range, and pair it with a heating rate that keeps pores mobile enough to stay attached to migrating boundaries.
If your primary focus is troubleshooting incomplete densification: Inspect your microstructure for high-N pores (often at triple points of large grains). If you see them, your thermal cycle may be allowing abnormal grain growth before N falls below Nc, or your green compact may require improved dispersion and pressing.

The path to a fully dense ceramic is not a single setting—it’s a choreography of temperature, time, and interfacial chemistry, all designed to keep every pore firmly on the shrinking side of its critical coordination number.

Summary Table:

Factor / Metric	Shrinkage Condition (Densification)	Growth Condition (Defect Formation)	Furnace Optimization Strategy
Pore Curvature	Convex inward (High internal pressure drives closure)	Concave inward (Resists closure; consumes vacancies)	Control thermal cycles to balance grain coarsening and pore mobility
2D Critical Threshold ($N_c$ @ $\psi=120^\circ$)	$N < 6$	$N > 6$	Prevent abnormal grain growth to keep coordination low
3D Critical Threshold ($N_c$ @ $\psi=120^\circ$)	$N < 12$	$N > 12$	Pair uniform green density with controlled heating rates
Dihedral Angle ($\psi$) Effect	Large $\psi$ (Low grain boundary energy ratio)	Small $\psi$ (High grain boundary energy ratio)	Engineer atmosphere (vacuum, $H_2$, inert) to lower $\gamma_{gb}$ and raise $\psi$

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