Electrostatic Depletion in Complex Systems

• Electrostatic depletion is the exclusion of mobile charges or particles from specific regions due to unfavorable electrostatic forces and geometrical constraints.
• Quantitative models reveal that depletion effects, such as varying depletion widths and scaling laws, are critical in semiconductor interfaces and curved charged surfaces.
• Experimental and theoretical studies across colloids, polar solvents, and coacervates demonstrate that controlled depletion directly influences device performance and material phase behavior.

Searching arXiv for recent and relevant papers on electrostatic depletion and closely related usages across interfaces, semiconductors, colloids, and coacervates. Electrostatic depletion denotes, across several arXiv literatures, the reduction or complete removal of mobile particles, carriers, ions, solvent molecules, or guest macromolecules from a spatial region because electrostatic fields, image interactions, charge correlations, or electrostatically mediated geometry make occupancy unfavorable. The depleted entity and the governing mechanism are strongly context-dependent. In curved charged interfaces it can mean a particle-free waist on a pinched sphere; in semiconductors it usually means a space-charge region depleted of mobile carriers; in electrolytes and polar liquids it can refer to ion or solvent exclusion near dielectric boundaries; and in dense polymeric condensates it can denote an effective attraction generated by electrostatic correlations in the host medium (Chen et al., 2018, Sallese, 2015, Liu et al., 2015, Wu et al., 2 Aug 2025).

1. Terminological scope and unifying features

The term is not used uniformly. In semiconductor electrostatics, depletion is the formation of a non-neutral region near a surface or junction where mobile carriers are depleted and fixed ionized dopant charges remain. In the pinched-sphere problem, depletion is a particle-free zone produced by the interplay between unscreened Coulomb repulsion and negative Gaussian curvature. In electrolyte and solvent theories, depletion often means image-force-driven exclusion of ions or dipoles from an interface, or Donnan-mediated partitioning of polyelectrolytes between a slit and a reservoir. In complex coacervates, electrostatic depletion refers to an effective attraction between guest macromolecules caused by disruption of favorable electrostatic correlations in the host network (Sallese, 2015, Chen et al., 2018, Buyukdagli et al., 2012, Landman et al., 2021, Sin et al., 2022, Wu et al., 2 Aug 2025).

Context	Depleted quantity	Immediate driver
Pinched charged surface	surface particles	Coulomb repulsion plus concave geometry
Semiconductor or field-effect structure	mobile carriers	band bending, gates, and space charge
Electrolyte or polar solvent	ions and/or solvent dipoles	image self-energy, Donnan partitioning, orientational ordering
Colloidal or coacervate medium	polymers, colloids, or guest macromolecules	electrostatic correlations or long-range repulsion balanced by short-range attraction

A recurrent misconception is that all depletion phenomena are variants of classical Asakura–Oosawa excluded-volume attraction or of Debye screening. The cited work does not support that equivalence. The depletion zone on a pinched sphere has no neutralizing background and disappears in the conductor limit $N\to\infty$ (Chen et al., 2018). Semiconductor depletion is governed by Poisson’s equation with ionized dopants and carrier statistics (Sallese, 2015). In complex coacervates, the attraction does not require a size mismatch between solute and host polymers (Wu et al., 2 Aug 2025). This suggests a shared phenomenological theme—electrostatically driven exclusion under geometrical or thermodynamic constraints—rather than a single microscopic law.

2. Geometry-induced depletion on curved charged interfaces

On a pinched sphere, the surface is parameterized by

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$

with waist radius $r(\pi/2)=(1-\kappa)R$. For $0<\kappa<1$ the shape is peanut-like, and the Gaussian curvature at the waist is

$K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$

The waist becomes negatively curved for $1/7<\kappa<1$, while the total curvature integral remains $4\pi$ by Gauss–Bonnet–Chern. Charged particles are constrained to this surface and interact through the three-dimensional Coulomb kernel $V_{ij}\sim 1/r_{ij}$, with $r_{ij}$ the Euclidean distance in the embedding space rather than the geodesic distance on the surface (Chen et al., 2018).

As $\kappa$ increases at fixed $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$0, the ground-state crystallography changes in a sequence. At mild pinching, elongated scars align with the long axis; at intermediate pinching, scars cleave into pleats whose seven-fold ends anchor at the negatively curved waist and whose five-fold ends lie in positively curved regions; around $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$1, seven-fold disclinations proliferate and crystalline order near the waist is disrupted; for $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$2, a sharply defined depletion zone appears at the waist, completely void of particles. The excess topological charge $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$3 in the waist region is positive for $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$4 and surges between $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$5 and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$6, coincident with pleat formation and crystalline disruption (Chen et al., 2018).

The local mechanism is electrostatic rather than purely topological. When the two sides of the waist contain equal particle numbers, the surface electrostatic potential has a local maximum at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$7, so a positively charged test particle cannot be stabilized there and is expelled along the tangent direction away from the waist. When the side populations differ slightly, the potential increases monotonically with $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$8, and the test charge migrates to the side with fewer particles. The depletion width $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$9, normalized as $r(\pi/2)=(1-\kappa)R$0, follows a power law $r(\pi/2)=(1-\kappa)R$1 for $r(\pi/2)=(1-\kappa)R$2, with $r(\pi/2)=(1-\kappa)R$3, $r(\pi/2)=(1-\kappa)R$4, and $r(\pi/2)=(1-\kappa)R$5 increasing approximately linearly with $r(\pi/2)=(1-\kappa)R$6. Because $r(\pi/2)=(1-\kappa)R$7, the depletion zone shrinks with $r(\pi/2)=(1-\kappa)R$8 more slowly than the mean particle spacing $r(\pi/2)=(1-\kappa)R$9, which supports its identification as a finite-size effect. The same qualitative phenomenon also appears for biconcave shapes with $0<\kappa<1$0, including depletion zones at $0<\kappa<1$1 and $0<\kappa<1$2 (Chen et al., 2018).

The continuum limit removes the effect. Under conductor boundary conditions, a mobile charged surface becomes equipotential as $0<\kappa<1$3, and a genuine void region would impose both the potential and its normal derivative, which is incompatible with Laplace’s equation. The same study shows that the electrostatic energy decreases monotonically as $0<\kappa<1$4 increases under both fixed-volume and fixed-area protocols, implying an increase of the effective capacitance $0<\kappa<1$5. This identifies curvature-controlled depletion as a consequence of finite-$0<\kappa<1$6 electrostatics on concave geometry rather than as a continuum screening phenomenon (Chen et al., 2018).

3. Semiconductor depletion and electrostatic free energy

In semiconductors, electrostatic depletion is classically the formation of a space-charge region in which mobile carriers are removed, leaving fixed ionized dopants. In p-type depletion, holes are removed and ionized acceptors provide negative fixed charge; in n-type depletion, electrons are removed and ionized donors provide positive fixed charge. The full depletion approximation assumes a constant ionized dopant density $0<\kappa<1$7 or $0<\kappa<1$8, negligible mobile carriers in the depleted region, and a sharp transition to neutrality at depletion width $0<\kappa<1$9. This approximation is treated as a singular case in which standard electrostatic energy definitions remain valid (Sallese, 2015).

The central revision proposed for semiconductor nanostructures is that the customary electrostatic energy,

$K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$0

or equivalently

$K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$1

does not generally yield the correct force when carrier statistics generate realistic accumulation, inversion, or depletion profiles. In a coupled metal–vacuum–semiconductor capacitor problem, the force obtained by differentiating the conventional field energy does not match the Coulomb force. To restore consistency, an additional free-energy term $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$2 is introduced, so that the total electric free energy becomes

$K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$3

The extra term vanishes for insulators with potential-independent space charge and for the full depletion approximation, but not for general semiconductor statistics (Sallese, 2015).

For a non-degenerate p-type semiconductor, the paper expresses the surface field $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$4 and surface potential $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$5 through semiconductor statistics, and shows that the extra free energy can dominate the dipole energy. The practical consequence is that force predictions in MEMS and NEMS can be wrong by large factors when standard formulas are used. At $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$6 nm, the reported ratio $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$7 deviates by nearly an order of magnitude for $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$8, and by a factor of $K_{\textrm{G,min}}=\frac{1-7\kappa}{(1-\kappa)^3}.$9 even at $1/7<\kappa<1$0. The discrepancy is most pronounced when the electrode gap is comparable to the extension of the non-neutral region, and $1/7<\kappa<1$1 becomes especially important in accumulation and strong inversion (Sallese, 2015).

This energetic reformulation matters because semiconductor depletion is often treated as a purely geometric capacitor problem. The cited analysis instead makes the carrier-statistical origin explicit: once the charge density depends nonlinearly on the local potential, the electrostatic free energy cannot always be reduced to field and polarization terms alone. In that sense, depletion in semiconductors is not merely an emptying of bands but part of a broader self-consistent thermodynamic problem (Sallese, 2015).

4. Field-effect depletion in low-dimensional and gated devices

A later self-consistent formulation, the Pure Electrostatic Self Consistent Approximation (PESCA), treats depletion in low-dimensional semiconducting devices as the local reduction of mobile carrier density until a region becomes partially or fully empty. The key small parameter is

$1/7<\kappa<1$2

the ratio of geometrical to quantum capacitance. For a uniform 2DEG under a planar gate,

$1/7<\kappa<1$3

Typical estimates in the paper are $1/7<\kappa<1$4 for a GaAs 2DEG at $1/7<\kappa<1$5 nm and $1/7<\kappa<1$6 for a Si 2DEG at $1/7<\kappa<1$7 nm. PESCA imposes a two-branch constitutive rule inside the conducting semiconductor region: depleted cells satisfy $1/7<\kappa<1$8, while metallic cells satisfy a pinned potential condition. This provides a convergent electrostatic algorithm for screening, partial depletion, pinch-off phase diagrams, and, after extension to Landau levels, edge reconstruction in the quantum Hall regime (Lacerda-Santos et al., 21 Feb 2025).

Several device classes realize depletion operationally rather than only as an equilibrium space-charge region. In depletion-mode quantum dots in intrinsic silicon, fixed negative charge in a $1/7<\kappa<1$9 nm $4\pi$0/$4\pi$1 nm $4\pi$2 stack induces a $4\pi$3DHG at the $4\pi$4 interface, with $4\pi$5 corresponding to $4\pi$6–$4\pi$7 and an effective built-in drive $4\pi$8–$4\pi$9 V. Positive gate biases then locally deplete the pre-existing hole gas to form tunnel barriers and a dot. Transport spectroscopy gives charging energies of $V_{ij}\sim 1/r_{ij}$0–$V_{ij}\sim 1/r_{ij}$1 meV in the few-hole regime and $V_{ij}\sim 1/r_{ij}$2–$V_{ij}\sim 1/r_{ij}$3 meV in the many-hole regime. Deep-UV exposure at $V_{ij}\sim 1/r_{ij}$4 nm in cumulative doses up to $V_{ij}\sim 1/r_{ij}$5 s neutralizes the fixed charge and restores the two-terminal resistance to the intrinsic-Si value at $V_{ij}\sim 1/r_{ij}$6 K (Amitonov et al., 2017).

In junctionless oxide transistors on paper, electrostatic depletion spans the full ITO channel thickness when the depletion width

$V_{ij}\sim 1/r_{ij}$7

satisfies $V_{ij}\sim 1/r_{ij}$8. A chitosan-based proton conductor forms electric double layers with a maximum specific capacitance of $V_{ij}\sim 1/r_{ij}$9 at $r_{ij}$0 Hz, enabling low-voltage depletion, threshold modulation from $r_{ij}$1 V to $r_{ij}$2 V as $r_{ij}$3 is swept from $r_{ij}$4 V to $r_{ij}$5 V, and OR logic based on overlap or removal of depletion regions. In depletion-mode buried-channel InGaAs/InAs QWFETs, the gate raises the conduction-band profile and subband energies in a strained InAs quantum well, reducing occupancy as $r_{ij}$6 becomes more negative. The self-consistent Poisson–Schrödinger treatment shows that inversion capacitance and ballistic current increase with In content, while higher-$r_{ij}$7 dielectrics and thinner oxides sharpen the depletion-to-inversion crossover (Dou et al., 2012, Ahmed et al., 2012).

Electrostatic depletion also appears in strongly inhomogeneous systems. In doped topological insulators, the surface-state charge repels bulk carriers and produces an intrinsic depletion zone near the surface; for typical bulk carrier densities of order $r_{ij}$8, the band bending is about $r_{ij}$9 meV for electron doping and $\kappa$0 meV for hole doping in the absence of surface dopants, and the depletion width can exceed $\kappa$1 quintuple layers. Applying a gate potential can generate a depletion zone with vanishing carrier density, and the density profile in the transition zone between the depleted region and the bulk is reported to be independent of the applied potential. In back-gated LAO/STO, a non-linear dielectric environment focuses electric-field lines at the edges of the conducting channel, so depletion is stronger at the edges than at the center. For $\kappa$2, $\kappa$3, and $\kappa$4 V, the center density can be up to $\kappa$5 the edge density; for $\kappa$6, the simulated center-density change scales as $\kappa$7, and scanning SQUID microscopy directly images the resulting current narrowing (Galanakis et al., 2012, Persky et al., 2020).

5. Electrolytes, polar solvents, and Donnan-depleted fluids

At low-permittivity boundaries in electrolytes, electrostatic depletion often originates in image-charge and self-energy effects. A self-consistent field theory augments Poisson–Boltzmann with an ion self-energy obtained from a generalized Debye–Hückel Green’s function. In this framework, dielectric mismatch near an interface raises the potential of mean force of mobile ions, depleting them from the interfacial region over a range that can extend to the Debye length. When two like-charged interfaces approach to a distance comparable to $\kappa$8, the depleted ion density in the gap lowers the osmotic pressure and generates an entropic attraction that can overcome mean-field electrostatic repulsion. For a $\kappa$9 M $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$00 electrolyte in water, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$01 nm; in the examples with $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$02, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$03, and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$04 nm, attraction appears when $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$05 is comparable to $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$06, and the attraction–repulsion phase boundary follows approximately $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$07 up to $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$08 M (Liu et al., 2015).

A related but distinct interfacial exclusion appears for polar solvents themselves. At carbon–water interfaces, image dipole interactions deplete water molecules from a thin interfacial layer and lower the local dielectric screening. The resulting dielectric “dead layer” acts in series with the diffuse layer, so the low-voltage differential capacitance becomes

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$09

where $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$10 is the solvent-depletion thickness and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$11 the ionic-depletion thickness. The reported values are $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$12–$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$13 Å and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$14–$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$15 Å, with the solvent depletion supplying the dominant capacitance reduction. The extended dipolar Poisson–Boltzmann theory matches measured capacitances of carbon-based materials in water without fitting parameters, whereas classical PB overestimates them by about an order of magnitude (Buyukdagli et al., 2012).

In mixtures containing nonadsorbing polyelectrolytes, depletion is modified by Donnan partitioning. For two neutral plates separated by a slit of width $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$16 in the Donnan limit $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$17, the potential inside the slit is uniform and the Donnan potential at close contact is

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$18

If polymer is excluded, the slit develops an attractive osmotic pressure imbalance. If the slit widens to $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$19, polymer can enter at the cost of the Casassa confinement free energy, and the competition between Donnan-driven entry and chain confinement generates a repulsive regime at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$20, followed by an attractive regime. In the counterion-dominated limit, the position of the maximal repulsion is

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$21

The zero-field theory matches self-consistent field calculations with full Poisson–Boltzmann electrostatics over a broad parameter range (Landman et al., 2021).

In binary polar solvent mixtures, the electric double layer depletes the two solvents asymmetrically. The mean-field theory of ions plus two polar solvents imposes the incompressibility constraint

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$22

and couples solvent orientational ordering to the local field through Langevin polarization. The principal control parameter is the ratio $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$23, the solvent dipole moment divided by molecular volume. The species with smaller $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$24 depletes more strongly near the charged surface, while the species with larger $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$25 can become locally enriched. This lowers the local permittivity through dielectric saturation and modifies the differential capacitance. The model predicts that increasing the bulk fraction of a low-$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$26 solvent lowers the maximum differential capacitance, whereas increasing the added-solvent dipole moment raises it (Sin et al., 2022).

6. Colloidal, nanocrystal, liquid-crystalline, and coacervate manifestations

In charge-stabilized nanocrystal dispersions, electrostatics can determine whether depletion-type attractions generate open gels or dense aggregation. Ligand-stripped tin-doped indium oxide nanocrystals in acetonitrile have a measured zeta potential of approximately $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$27 mV, radius $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$28 nm, and hydrodynamic diameter $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$29 nm in DMF. Adding PEG of $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$30 g/mol with $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$31 nm produces a reentrant phase diagram with two gelation windows: a low-$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$32 window at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$33 mM attributed to bridging and a high-$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$34 window at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$35 mM attributed to depletion after surface saturation. The unified model uses $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$36, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$37, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$38, and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$39, and predicts bridging gels for $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$40–$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$41 and depletion gels for $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$42. SAXS, HR-TEM, and XRD show that the nanocrystals remain discrete in the gel, while the infrared LSPR red-shifts by about $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$43 because of interparticle coupling in the network (Cabezas et al., 2018).

Anisotropic nanoparticles show the same competition in a different geometry. Upconversion nanorods dispersed in water are charge-stabilized after removal of oleic acid, with baseline ionic strength $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$44 mM and Debye length $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$45 nm; adding NaCl up to $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$46 M reduces $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$47 to about $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$48 nm and screens electrostatic repulsion. Nonadsorbing dextran then supplies depletion attractions. For UCNR1, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$49 nm and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$50 nm; for UCNR2, $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$51 and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$52. The phase diagrams contain isotropic, nematic, isotropic–nematic coexistence, and isotropic–smectic-membrane coexistence regions. At high dextran concentration, I–N coexistence occurs at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$53 and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$54, and the resulting aligned films have $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$55–$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$56 and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$57–$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$58. Here depletion is not itself electrostatic, but electrostatic repulsion determines the conditions under which depletion stabilizes ordered, fluid-like mesostructures rather than uncontrolled aggregation (Xie et al., 2017).

A more direct dense-fluid use of the term appears in complex coacervates. There, electrostatic depletion is defined as a strong effective attraction between guest macromolecules caused by electrostatic correlations in the host polycation–polyanion network. The coarse-grained simulations use a Gaussian core model with smeared electrostatics at $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$59, host chain lengths $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$60 or $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$61, and a measured electrostatic correlation length $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$62. The interfacial-energy picture gives

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$63

and for $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$64 neutral guest chains of length $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$65,

$r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$66

The guest–guest PMF in coacervates becomes attractive for sufficiently long or weakly charged guests: neutral $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$67 chains have a well depth of about $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$68, neutral $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$69 chains about $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$70, and low-charge-density polyelectrolytes with $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$71 about $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$72. Fully charged IDP-like polyampholytes are attractive for low sequence blockiness $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$73, with a PMF minimum around $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$74, and become repulsive by $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$75. Unlike Asakura–Oosawa depletion, this mechanism requires no size mismatch between guest and host polymers (Wu et al., 2 Aug 2025).

7. Limitations, boundary conditions, and open directions

The literature also specifies when depletion does not survive or when current models should not be overextended. On pinched spheres, the depletion zone is explicitly a finite-size effect that vanishes as $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$76 under conductor boundary conditions; the published calculations use unscreened $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$77 interactions and $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$78 energy minimization, so screening, dielectric heterogeneity, thermal fluctuations, and membrane elasticity remain outside the model (Chen et al., 2018). In semiconductor energetics, the extra free-energy term was derived for linear dielectric response, quasi-static charging, and non-degenerate p-type statistics, while PESCA is controlled by the small parameter $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$79 and is not intended for regimes dominated by strong quantum confinement, tunneling, or large quantum-capacitance corrections (Sallese, 2015, Lacerda-Santos et al., 21 Feb 2025).

For electrolytes and polar liquids, several approximations are similarly explicit. The self-consistent field theory for like-charge attraction neglects nonlocal dielectric response and ion-specific hydration and is reported to be accurate from weak to moderate coupling, up to $r(\theta)=R\left[1+\kappa(3\cos^2\theta-1)\right],$80 (Liu et al., 2015). The binary-solvent EDL theory neglects short-range interactions such as preferential solvation (Sin et al., 2022). The dipolar Poisson–Boltzmann theory for carbon interfaces treats ions as point charges and solvent molecules as rigid dipoles, omitting steric layering, roughness, nanopore curvature, and higher multipoles (Buyukdagli et al., 2012). The coacervate simulations employ a coarse-grained model with fixed dielectric constant, no explicit water structure, and no systematic variation of salt or temperature, so the reported scaling arguments for screening effects remain extrapolations rather than directly simulated trends (Wu et al., 2 Aug 2025).

These limitations are not incidental. They show that electrostatic depletion is highly sensitive to boundary conditions, compressibility, dielectric contrast, and the level at which charge correlations are resolved. The term therefore identifies a recurrent exclusion phenomenon, but its operational content is set by the specific field theory, self-consistent electrostatics, or correlation model appropriate to the material system under study.