Surface Energetics of ZnS Nanoparticles

• The paper reveals that surface energies derived from both DFT and calorimetric measurements dictate phase stability and morphology control in ZnS nanoparticles.
• Methodologies include pseudo-hydrogen passivation and tetrahedral-cluster approaches to quantify dangling bond effects and evaluate facet-dependent energies.
• Results demonstrate that ligand capping can render effective surface energies negative, suppressing Ostwald ripening and enabling tailored nanocrystal engineering.

Zinc sulfide (ZnS) nanoparticles are prototypical II–VI semiconductor nanomaterials with a rich surface physics arising from their polymorphism, strong ionic bonding, and sensitivity to surface termination, adsorbates, and ligand environment. The surface energetics of ZnS nanoparticles controls their phase stability, morphology, growth mechanisms, and interactions with surrounding media, underpinning both fundamental nanoscale phenomena and applications in optoelectronics, geochemistry, and planetary science.

1. Definitions and Thermodynamic Framework

The surface energy, γ, of a ZnS nanoparticle facet is formally defined as the excess free energy per unit area required to create that surface from the bulk. For a slab geometry, as used in first-principles calculations, the absolute surface energy is evaluated by:

$$\gamma = \frac{E_\mathrm{slab} - n_\mathrm{bulk} E_\mathrm{bulk} - n_H \mu_H}{A}$$

where $E_\mathrm{slab}$ is the total energy of the passivated or partially-passivated slab, $n_\mathrm{bulk}$ and $E_\mathrm{bulk}$ are the number and energy per formula unit in bulk ZnS, $n_H$ is the number of pseudo-hydrogen atoms used to passivate dangling bonds, $\mu_H$ denotes the pseudo-chemical potential of these passivators, and $A$ is the surface area of the relevant facet (Zhang et al., 2015).

For nanoparticles, surface energy must be considered in conjunction with the bulk Gibbs free energy and the significant surface-to-volume ratio. The total free energy per mole for a polymorph $i$ (e.g., sphalerite or wurtzite) becomes:

$$G_\mathrm{total,i}(D, T) = G_\mathrm{bulk,i}(T) + \gamma_i(T) \cdot \frac{A(D)}{N_f}$$

where $A(D)$ is the particle surface area and $N_f$ is the number of formula units (Subramani et al., 4 Jan 2026). This framework underpins size-dependent phase stability and morphological evolution in ZnS nanocrystals.

2. First-Principles Calculations and the Role of Dangling Bonds

For ideal, unreconstructed surfaces of ZnS, density functional theory (DFT) combined with pseudo-hydrogen passivation accurately quantifies absolute surface energies. Two primary methodologies for determining the pseudo-hydrogen chemical potential $\mu_H$ are employed:

• Pseudo-molecule method: Constructs a tetrahedrally-coordinated molecule (e.g., Zn[H^*]_4) and derives $\mu_H$ from its total energy, capturing intrinsic bond strengths but neglecting the specific local environment (Zhang et al., 2015).
• Tetrahedral-cluster method: Models a finite tetrahedral ZnS cluster with all surface dangling bonds passivated, fitting energies for site-specific $\mu_H$ (corner, edge, face) by varying cluster size. This approach closely reproduces real surface coordination and reduces errors to ≲1 meV/Ų.

For zinc-blende ZnS, calculated absolute surface energies (meV/Ų):

Facet	γ (meV/Ų)	Termination
(111)	88.6	Zn-terminated
($\bar 1\bar1\bar1$)	87.5	S-terminated
(110)	23.9	nonpolar

These values confirm that polar (111) facets are much higher in energy than the nonpolar (110) face, due to a greater density of unsaturated dangling bonds and electrostatic polarity (Zhang et al., 2015).

The surface energy is quantitatively decomposed as:

$$\gamma = N_\mathrm{dangling} \cdot \frac{\epsilon_\mathrm{dang}}{A}$$

where $N_\mathrm{dangling}$ is the number of dangling bonds per surface cell, and $\epsilon_\mathrm{dang}$ is the effective energy per dangling bond, directly linked to the passivation energy $\mu_H$.

3. Experimental Surface Energetics: Polymorphs and Size-Dependent Stability

Direct calorimetric measurements of ZnS nanocrystals, both for sphalerite (cubic) and wurtzite (hexagonal) polymorphs, reveal distinct and size-dependent surface energetics (Subramani et al., 4 Jan 2026). Using solution calorimetry and differential water-adsorption methods, unsolvated (bare) surface energies at 298 K are determined:

Polymorph	γ (J/m²)	Surface Condition
Sphalerite	1.76 ± 0.32	Unsolvated (bare)
Wurtzite	1.50 ± 0.50	Unsolvated (bare)
Sphalerite	1.25 ± 0.21	Chemisorbed H₂O
Wurtzite	0.99 ± 0.32	Chemisorbed H₂O

The lower surface energy of wurtzite compared to sphalerite drives a thermodynamic reversal of phase stability at the nanoscale. For diameters $D \lesssim 10$ nm, the surface term dominates, stabilizing wurtzite even though sphalerite is bulk stable below 1020 °C.

The phase boundary for polymorphic stability is given by:

$$D_c = \frac{6(\gamma_\mathrm{wurtzite} - \gamma_\mathrm{sphalerite})}{\Delta G_\mathrm{bulk}}$$

With $\Delta G_\mathrm{bulk} \approx 3.6$ kJ/mol and $\Delta\gamma \approx -0.26$ J/m² at 298 K, $D_c \approx 9.8$ nm (Subramani et al., 4 Jan 2026).

4. Surface Modification by Ligands: Negative Surface Energies

Colloidal ZnS quantum dots capped with carboxylate ligands (e.g., oleate) exhibit fundamentally distinct thermodynamics. Dissolution calorimetry demonstrates that the effective surface energy of ligand-capped ZnS QDs is negative at room temperature (Calvin et al., 2022):

$$\gamma_\mathrm{surf}(\text{ZnS–oleate}) = -1.51 \pm 0.10\ \mathrm{J/m^2}$$

This counterintuitive result arises when the enthalpic stabilization from strong Zn–O(carboxylate) binding exceeds the energy required to break ZnS lattice bonds at the surface. The binding enthalpy of dense oleate shells is quantified at approximately –1.8 J/m², overwhelming the "bare" positive surface energy. The measured $\gamma_\mathrm{surf}$ is constant across the 1.1–1.3 nm QD radius range, indicating negligible size-dependence and curvature correction within experimental uncertainty.

For core–shell systems such as InP/ZnS QDs, the interfacial energy is even more favorable:

$$\gamma_\mathrm{interface}(\text{InP/ZnS}) = -10.07 \pm 0.64\ \mathrm{J/m^2}$$

despite a substantial lattice mismatch, signifying extremely robust chemical driving forces at core–shell interfaces (Calvin et al., 2022).

5. Implications for Nanoparticle Morphology and Growth

The facet dependence of $\gamma$ directly informs equilibrium and kinetic morphologies via Wulff construction. For "bare" or solvated ZnS nanoparticles, the {110} facets predominate due to their much lower surface energy relative to {111} (Zhang et al., 2015), producing rod-like or plate-like shapes at equilibrium. For ligand-capped QDs, facet energies are further modulated by the ligand–surface binding enthalpy, leading to selective facet stabilization and dramatic modifications of equilibrium shape.

Negative or near-zero $\gamma$ profoundly affects growth kinetics, suppressing Ostwald ripening and enhancing colloidal stability. Nucleation theory predictions, such as classical LaMer model barriers and critical radii, require reevaluation in the context of negative $\gamma$ values (Calvin et al., 2022).

Surface energetics also mediates phase selection: any process (thermal annealing, ligand exchange, solvent removal) that alters $\gamma$ can drive phase transitions between wurtzite and sphalerite below the bulk transition temperature, as observed in hydrothermal and planetary environments (Subramani et al., 4 Jan 2026).

6. Environmental and Technological Repercussions

In geological systems, ZnS nanoparticles typically nucleate as wurtzite for $D < 10$ nm, transition to sphalerite on coarsening (thermal or pressure-induced), and may remain kinetically trapped in either phase depending on surface passivation and environmental conditions. In exoplanetary atmospheres, the lower $\gamma$ of wurtzite lowers nucleation barriers, influencing cloud microphysics and planetary albedo (Subramani et al., 4 Jan 2026).

In technological contexts, the ability to tune surface energetics through ligand chemistry provides routes to engineer colloidal stability, narrow size distributions, and robust heterointerfaces, critical for applications in photonics, catalysis, and bioimaging (Calvin et al., 2022). However, surface energetics are temperature-sensitive; ligand detachment at elevated temperatures (>200 °C) can revert $\gamma$ toward positive values and restore classical coarsening pathways.

7. Comprehensive Synthesis and Outlook

Surface energetics of ZnS nanoparticles is determined by an interplay of crystal termination, dangling-bond density, passivation scheme, and environmental binding. DFT-based pseudo-hydrogen methods accurately quantify absolute surface energies for ideal facets, revealing a pronounced facet anisotropy in zinc-blende nanoparticles (Zhang et al., 2015). Direct calorimetric experiments on nanocrystals and powders confirm that wurtzite surfaces are energetically favored at the nanoscale, with surface energies in the range 1.0–1.8 J/m² depending on solvation and adsorbed species (Subramani et al., 4 Jan 2026). In colloidal QDs capped by strongly binding ligands, the effective $\gamma$ can be rendered negative, challenging classical nucleation and growth paradigms (Calvin et al., 2022).

A unified thermodynamic model incorporating both bulk and surface contributions precisely rationalizes size–temperature phase stability, transition boundaries, and observed morphologies under diverse laboratory and natural conditions. Ligand and solvent effects must be explicitly incorporated in predictive models of nanocrystal stability, growth, and phase selection. Surface energetics remains the controlling factor for ZnS nanoparticles across disciplines, from quantum dot engineering to planetary mineralogy.