A central positive charge in a plasma is surrounded by a cloud of mobile
electrons that screen its electric field. The potential
decays as φ(r) = (q/4πε₀r)·e−r/λD rather than
the bare Coulomb law. Adjust temperature T and density n to see how the
Debye length λD changes.
Adjust temperature and density to see how the Debye length changes ·
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Plasma Parameters
Live Stats
Debye length λD—
φ at r = λD—
φCoulomb at λD—
Screening ratio—
Electrons—
Display
Show potential field
Show Debye circle
Legend
🔴 Central positive ion
🔵 Negative electrons (mobile)
🟡 Positive background ions (fixed)
Heatmap: warm = high φ, cool = screened
What is Debye Screening?
In a plasma — an ionised gas of electrons and ions — free electrons redistribute
themselves around any excess charge to shield it from the rest of the plasma.
The resulting potential is the Yukawa / screened-Coulomb potential:
φ(r) = (q / 4πε₀r) · exp(−r / λD)
The Debye length λD sets the scale of screening:
λD = √(ε₀ kB T / n e²)
Beyond ~3 λD the potential is reduced to less than 5 % of the bare Coulomb value.
This is why plasmas are quasi-neutral on scales larger than λD:
any local charge imbalance is neutralised within a Debye sphere.
Key Dependencies
Higher temperature → electrons have more kinetic energy → screening cloud
spreads out → larger λD. Higher density → more electrons available to screen → tighter cloud →
smaller λD.
Real-world examples
Solar wind plasma: λD ≈ 10 m. Fusion tokamak plasma: λD ≈ 70 µm.
Ionosphere: λD ≈ 1 cm. Understanding Debye screening is essential for
modelling spacecraft charging, ion thrusters, and magnetic confinement fusion.
About Debye Screening in Electrolytes
In an ionic solution or plasma, mobile charges redistribute around any fixed charge to neutralise it, reducing the Coulomb potential from a bare 1/r form to a screened Yukawa form φ(r) = (q/4πε₀εr)·exp(−r/λD). The Debye length λD = √(ε₀εkBT / 2NAe²I) determines how far the influence of the charge extends — shorter in concentrated solutions (high ionic strength I) and longer in dilute or high-temperature systems. Debye-Hückel theory is the foundation of electrolyte thermodynamics and underlies corrections to the activity coefficients of ions used in electrochemistry and biochemistry.
Adjust the salt concentration and temperature to watch the screening cloud contract or expand around the central test charge, and compare electrolyte screening with its plasma counterpart where free electrons play the role of the mobile ions.
Frequently Asked Questions
What is the Debye length and what does it represent?
The Debye length λD is the characteristic distance over which electrostatic effects are screened in an ionic medium. Beyond roughly 3λD, the potential of a point charge falls to less than 5% of its unscreened value. In a 100 mM NaCl solution (physiological saline), λD ≈ 0.97 nm; in pure water at 25 °C it is about 960 nm; in a typical laboratory plasma at 10⁴ K and 10¹⁶ m⁻³ it is around 70 µm.
How does ionic strength affect Debye screening?
Ionic strength I = ½ Σ cizi² (in mol L⁻¹) accounts for both the concentration and charge number of all ionic species. Because λD ∝ 1/√I, doubling the ionic strength reduces the Debye length by a factor of √2. This is why proteins unfold or DNA strands lose their repulsion in high-salt buffers: the electrostatic barriers between charged groups are screened away when I is large.
What is the Debye-Hückel approximation and when does it break down?
Debye-Hückel theory linearises the Poisson-Boltzmann equation, assuming the electrostatic energy qφ is much smaller than kBT. This is valid only when eφ/kBT ≪ 1, roughly corresponding to dilute solutions (I < 0.01 M) and monovalent ions. For concentrated electrolytes, multiply-charged ions (Ca²⁺, Al³⁺), or near the surface of highly charged colloids, higher-order corrections (extended Debye-Hückel, Pitzer equations) or full Poisson-Boltzmann solutions are required.
How does Debye screening differ in a plasma versus an electrolyte?
In both cases the physics is formally identical: mobile charges screen a test charge on a length scale λD. In a plasma all mobile particles are electrons and ions at (often) very high temperature and low density, giving λD in the micrometre range. In an electrolyte the carriers are dissolved ions at ~300 K; their much higher concentration gives nanometre-scale Debye lengths. The Debye criterion for a valid plasma — the number of particles within a Debye sphere (4πnλD³/3 ≫ 1) — ensures statistical smoothness of the screening cloud.
What role does Debye screening play in biology?
Almost every biological process involving charged macromolecules is controlled by Debye screening. Protein-protein binding affinity, DNA double-helix stability, ion-channel gating, and the stability of lipid membranes all depend on the balance between electrostatic attraction/repulsion and screening. The physiological ionic strength (~150 mM, λD ≈ 0.8 nm) is thought to be evolutionarily tuned to permit specific molecular recognition while preventing non-specific aggregation.
What is the DLVO theory of colloidal stability?
DLVO theory (Derjaguin, Landau, Verwey, Overbeek) describes whether colloidal particles aggregate or remain dispersed. It balances the screened electrostatic repulsion between like-charged particles (Debye-Hückel) against the attractive van der Waals force. Adding salt compresses the Debye layer, reducing the repulsion and eventually allowing van der Waals attraction to dominate — causing the colloid to flocculate. This is why adding salt to clay-bearing water causes the clay to settle.
How is Debye screening relevant to semiconductor transistors?
In a metal-oxide-semiconductor (MOS) device, the Debye length in the semiconductor determines how deeply gate-field effects penetrate. For silicon doped to 10¹⁷ cm⁻³, λD ≈ 4 nm — setting the minimum thickness of the depletion layer in a MOSFET. As transistors shrink below 10 nm, quantum confinement effects supersede classical Debye screening, requiring entirely different models.
What is charge inversion and does Debye theory predict it?
Charge inversion occurs when a charged surface (or macroion) attracts counterions so strongly that it acquires an apparent charge of opposite sign — measured, for example, by the reversed direction of electrophoretic motion. Standard Debye-Hückel theory cannot predict charge inversion because it ignores ion-ion correlations. Strongly correlated Coulomb systems (e.g., multivalent ions, room-temperature ionic liquids) require Poisson-Boltzmann with correlation terms or Monte Carlo simulation.
What happens to Debye screening at very high temperatures?
As temperature rises, kBT grows and the Debye length λD ∝ √T increases: ions are harder to localise by electrostatic energy, so the screening cloud spreads. In a plasma heated to millions of kelvin (as in a tokamak fusion reactor), λD can reach hundreds of micrometres, and the plasma behaves almost like an ideal gas — pairwise interactions become negligible compared to thermal kinetic energy.
How is Debye screening measured experimentally?
Atomic force microscopy (AFM) can measure the exponentially decaying force between a charged silica tip and a charged flat surface as a function of separation, directly fitting λD. Surface force apparatus (SFA) measurements between mica sheets in electrolyte solution were some of the earliest direct validations of Debye-Hückel theory. More recently, scanning ion-conductance microscopy measures the Debye layer around single charged nanoparticles in solution.
What is the Debye-Waller factor and is it related?
Despite the shared name, the Debye-Waller factor (used in X-ray and neutron diffraction to quantify how thermal vibrations reduce Bragg peak intensity) is conceptually different from Debye screening. Both were introduced by Peter Debye, who made major contributions to statistical mechanics, electrolyte theory, X-ray scattering, and polymer physics throughout the early twentieth century — hence the recurrence of his name across physics and chemistry.