Hamaker Constant¶

A material-and-medium-dependent coefficient sets the strength of nonretarded van der Waals dispersion interaction between idealized macroscopic bodies.

Version: v1 · 2026-10-31 · History
Domain-specific #: 14506
Domain group: Natural Sciences
Origin domain: Chemistry & Materials Science
Subdomains: Dispersion Forces, Interfacial Physics → Chemistry & Materials Science
Aliases: Hamaker coefficient

Core Idea¶

The Hamaker constant A is an energy-scale coefficient used to express the strength and, under a signed convention, the direction of dispersion interaction between idealized macroscopic bodies. In Hamaker's pair-additive model, a molecular −C₁₂/r⁶ potential is integrated over bodies of number densities ρ₁ and ρ₂, giving A₁₂ = π²ρ₁ρ₂C₁₂. Geometry and separation then supply a factor: for two flat half-spaces in the nonretarded idealization, interaction energy per area is −A/(12πD²); a sphere of radius R near a plane has approximate energy −AR/(6D) when D ≪ R. The coefficient is not a universal constant of one substance detached from its partner and environment.^[1]

The simple separation of “material coefficient” and “geometric factor” is useful but conditional. Hamaker's original 1937 paper itself considers two spheres immersed in liquid. A medium changes the effective interaction, and Lifshitz's later electromagnetic treatment uses dielectric response rather than a simple sum of independent pair potentials. At larger separations retardation, and at real contacts coatings, roughness and other forces, can make a single distance-independent A or smooth-surface geometry an inadequate fit.^[1]^[2]^[3]

How would you explain it like I'm…

Super-Close Sticky Number

When two things get super, super close, there's a tiny pull between them, coming from all their little bits. The Hamaker constant is a number that tells how strong that pull is for those two things, and sometimes whether it pulls or pushes. It depends on both things and on what's in between them, like air or water.

Strength of the Tiny Pull

All objects have a very weak attraction to each other when they are extremely close, coming from their molecules. Scientists use the Hamaker constant, a number, to describe how strong that pull is between two objects. To find the total pull, they combine the Hamaker constant with the objects' shapes and the distance between them — for example, two flat surfaces or a ball near a flat surface. The number depends on both materials and on whatever is between them, like water. It isn't a fixed number for just one material, and at bigger distances or on rough surfaces the simple formulas stop working well.

Dispersion Interaction Coefficient

The Hamaker constant A is an energy-scale coefficient describing the strength (and, with a sign convention, the direction) of the dispersion, or van der Waals, interaction between macroscopic bodies. In Hamaker's pair-additive model, the attraction between molecules, which falls off as 1/r⁶, is added up over all the molecules in two bodies, giving A₁₂ = π²ρ₁ρ₂C₁₂. Geometry and distance then contribute a separate factor: for two flat surfaces, energy per area is −A/(12πD²); for a sphere of radius R near a plane, about −AR/(6D) when D is much smaller than R. A belongs to a pair of materials and their environment, not to a single substance. A liquid between the bodies changes it, and Lifshitz's later theory uses the materials' dielectric response instead of summing pairs. At large distances or for rough, coated surfaces, a single constant may not fit well.

The Hamaker constant A is an energy-scale coefficient expressing the strength — and, under a signed convention, the direction — of dispersion interactions between idealized macroscopic bodies. In Hamaker's pair-additive model, a molecular pair potential −C₁₂/r⁶ is integrated over two bodies with number densities ρ₁ and ρ₂, yielding A₁₂ = π²ρ₁ρ₂C₁₂. A geometry- and separation-dependent factor then completes the interaction: in the nonretarded idealization, two flat half-spaces have interaction energy per unit area −A/(12πD²), and a sphere of radius R near a plane has energy approximately −AR/(6D) for D ≪ R. A is not a universal constant of one substance independent of its partner and environment. The factorization into material coefficient and geometric factor is useful but conditional. Hamaker's original 1937 paper already considered two spheres immersed in a liquid, and an intervening medium alters the effective interaction. Lifshitz's later electromagnetic theory computes the interaction from dielectric response rather than summing independent pair potentials. At larger separations retardation, and at real contacts coatings, roughness, and other forces, can make a single distance-independent A or smooth-surface geometry inadequate.

Structural Signature¶

Sig role-phrases: interacting bodies; dispersion-interaction law; material/medium coefficient; geometry and gap; inferred force/energy; validity boundary.

Bodies: specify the two surfaces/particles, their composition and the intervening medium. Silica–water–silica is a different interaction system from fluorocarbon-coated surfaces in dry air.^[3]^[4]
Microscopic or continuum model: pair-additive London attraction provides Hamaker's simplest derivation; Lifshitz models macroscopic dielectric fluctuation response.^[1]^[2]
Coefficient A: in the simple pair model, π²ρ₁ρ₂C₁₂ collects density and pair strength and has energy units. In measured practice an “effective A” is a fitted parameter under chosen force geometry and surface condition.^[1]^[3]
Geometric factor: gap D, radii and surface shape determine how the coefficient appears in energy or force. Do not confuse the sphere–plane energy −AR/(6D) with its derivative force, approximately −AR/(6D²) in this limit.^[1]
Range: nonretarded, smooth, homogeneous, close-body conditions license the simple forms; medium, retardation and roughness can alter the relationship.^[2]^[3]
Inference: a force-distance profile can be used to estimate A only after other contributions, notably double-layer or capillary forces, are accounted for.^[3]^[4]

Condensed: specified body–medium–body system + dispersion model + geometry/gap + valid force regime → Hamaker coefficient in an interaction law.

What It Is Not¶

Not a fundamental constant like c: numerical A depends on the two bodies and intervening medium, and fitted values can depend on surface preparation.^[3]
Not the force itself: A has energy units; geometry and D turn it into interaction energy or force.
Not a guarantee of attraction for every three-material combination: a dielectric-response treatment can change sign in some media; one must evaluate the actual combination rather than insert a positive air value.
Not proof that all measured attraction is dispersion: capillary bridges and other surface forces can contribute; Ohnishi and colleagues distinguish vapor-induced capillary effects from van der Waals attraction in prepared fluorocarbon layers.^[4]
Not exact for rough, coated or very distant bodies: real force profiles can depart from ideal smooth-surface and nonretarded assumptions.^[3]^[2]
Not simply DLVO theory: A supplies the dispersion component; electrostatic double-layer repulsion and other terms are separate.

Scope of Application¶

The original Hamaker calculation treats adhesive attraction of spheres by integrating London pair interactions, including a section on liquid immersion. The half-space and sphere–plane limiting equations express the same coefficient under ideal nonretarded geometry, useful for contact and colloid estimates. A stated “Hamaker constant of silica” is incomplete unless the other surface and medium are identified.^[1]

Valmacco and coauthors' original 2016 atomic-force study measured forces between pairs of silica particles across aqueous KCl. Strong electrolyte screened double-layer forces so the longer-range measured profiles were dominated by dispersion under their conditions. After 1200°C treatment, the authors fitted A ≈ 2.4×10⁻²¹ J; untreated particles had a much weaker fitted value (7±5)×10⁻²³ J. Their explanation was nanoscale surface roughness, not a change of silica's chemical name. This is a test of the smooth-geometry assumption as much as a measurement of material response.^[3]

Ohnishi, Yaminsky and Christenson's original 2000 surface-force study prepared fluorocarbon monolayers by different routes. In dry air, they found van der Waals attraction for some but not all preparations and an estimated A of about 7×10⁻²⁰ J in the reported attractive setting. In saturated water vapor, some preparations showed capillary bridging instead of a clean dispersion-force profile. A named surface chemistry plus gap alone did not specify one uncomplicated force mechanism.^[4]

Clarity¶

Keep three levels separate: (i) the molecular C₁₂ and number densities in a pair-additive nonretarded derivation, (ii) a theoretical A from dielectric spectra for a specified body–medium–body configuration, and (iii) an effective A fitted from a real force curve under a chosen geometric/surface model. They need not coincide exactly. The 2016 silica experiment demonstrates roughness sensitivity; its “effective Hamaker constant” is not a tabulated bulk number that all silica contacts must reproduce.^[1]^[2]^[3]

State whether a formula is for energy, energy per area or force, and give its geometry and small-gap condition. For the plate formula, doubling D reduces the magnitude of energy per area fourfold. For the sphere–plane approximate energy, doubling D halves its magnitude, while the corresponding force magnitude falls fourfold. These are deductions from the stated ideal formulas, not new empirical results.^[1]

Manages Complexity¶

A summarizes many microscopic interactions into one coefficient that can be carried between ideal shapes through analytic factors. It makes surface-force estimates tractable without summing every atom pair at every evaluation. The compression loses material dispersion spectra, medium response, roughness and other competing force components. Lifshitz's treatment addresses collective electromagnetic response more generally, while the silica and fluorocarbon measurements show why an experimentally fitted coefficient must be accompanied by preparation, gap and medium information.^[2]^[3]^[4]

Abstract Reasoning¶

Given a proposed A value, ask first: which two bodies, across which medium, in which distance regime? Identify whether A is pair-additive, Lifshitz-calculated or fit from a force profile. Write the geometry-specific energy equation, then differentiate if a force is needed. Check whether roughness or a capillary/electrostatic contribution could explain the measured profile. A successful fit to one setup does not authorize a universal property claim for the named material.^[1]^[3]^[4]

Diagnostic: Does the same body–medium–body and surface geometry underwrite both the quoted A and the force law in which it is being used?

Knowledge Transfer¶

The portable reasoning is a factorization: a coefficient packages interaction strength while geometry determines how it manifests. Similar-looking coefficients appear elsewhere in physics, but their portability depends on separability assumptions. Here the accent is London/Lifshitz dispersion, material dielectric response, colloidal media and contact geometry. The 2016 silica case demonstrates a caveat: even when bulk composition is held fixed, roughness can change the effective parameter obtained from force data. The 2000 fluorocarbon case adds a mechanism boundary: capillary attraction cannot be silently relabeled van der Waals force.^[3]^[4]

Examples¶

Silica particles across water: effective A and roughness¶

Valmacco and colleagues used AFM force profiles for pairs of silica particles in concentrated aqueous KCl. Heating to 1200°C produced relatively smooth surfaces and a fitted A about 2.4×10⁻²¹ J; untreated particles yielded (7±5)×10⁻²³ J. Their paper attributes the difference to nanoscale roughness after excluding other proposed changes under their experiments. Both pairs were “silica in water,” yet the smooth-sphere model's effective attraction differed greatly.^[3]

Mapped back: bodies = silica spheres; medium = aqueous electrolyte; signal = measured force versus separation; coefficient = fitted effective Hamaker A; geometry = sphere–sphere; boundary = nanoscale roughness changes observed profile and the inference.

Fluorocarbon monolayers in air and vapor: mechanism competition¶

Ohnishi, Yaminsky and Christenson prepared fluorocarbon surfaces by Langmuir–Blodgett deposition and two adsorption routes. Their dry-air force profiles showed van der Waals attraction except in one preparation; in saturated water vapor, some preparations instead showed capillary bridging. The reported approximately 7×10⁻²⁰ J A applies to the attractive dry-air profile, not to every surface preparation or the vapor-bridge force.^[4]

Mapped back: bodies = prepared fluorocarbon-coated surfaces; medium = dry air versus saturated vapor; coefficient = inferred for the dispersion-attractive profile; geometry = surface-force measurement; boundary = preparation and capillary condensation change the operative mechanism.

Force from an unrelated cause near miss¶

An electrostatically charged pair of surfaces can attract, but a measured attractive force by itself does not identify a Hamaker coefficient. The competing force must be separated or modeled. This is a constructed contrast, not a third reported experiment.^[3]

Mapped back: bodies and gap remain, but dispersion attribution and its A inference are not established.

Structural Tensions¶

Simple factorization versus material/geometry fidelity. A single A times a smooth-shape law makes force estimates easy to compare, but assumes enough homogeneity, smoothness and distance-independence that real contacts may violate. Lifshitz calculations represent dielectric response, while separate surface models can account for roughness, at the cost of spectra, geometry and calibration inputs. The silica study's fitted effective A values show the practical consequence of idealization. Diagnostic: Is the simple model's error smaller than the decision-relevant force difference?^[2]^[3]

Suppress competing forces versus preserve the use setting. Concentrated salt in the silica experiment screens electrical double layers so dispersion can be isolated. But an unscreened suspension could have a different total interaction and aggregation behavior; the isolated A cannot predict it alone. Similarly, changing water vapor changes capillary contribution in the fluorocarbon study. Diagnostic: Is the goal to isolate dispersion, or to predict net force under actual medium conditions?^[3]^[4]

Structural–Framed Character¶

The coefficient is structural within a model: it mediates between material interaction strength and geometry-specific force or energy. Its practical value is framed by a declared pair, medium, separation and surface condition. “Strong adhesion,” “stable colloid” and “good fit” are evaluative judgments tied to an application, not definitions of A. Human analysts choose a force model, infer gap zero, screen charge or fit a profile; surface-science laboratories and standard reporting conventions make measurements comparable. The name traces to Hamaker's 1937 pair-summation treatment, but its vocabulary traveled into Lifshitz dielectric calculations and effective experimental coefficients, so historical formula and modern fit must not be conflated. Recognizing the shared dispersion-strength role is justified; importing a dry-air or smooth-silica A into a different liquid or rough contact is not. Its character: a conditional surface-interaction coefficient that simplifies macroscopic force laws while exposing the model, medium and surface assumptions it packages.^[1]^[2]^[3]^[4]

Structural Core vs. Domain Accent¶

The portable skeleton is separation of a relation's strength parameter from geometric dependence. The domain-bound mechanism is nonretarded London pair summation or dielectric dispersion theory, with specified body–medium–body optical properties and close-surface geometry. The Hamaker A has defined energy units and surface-force formulas; generic coefficients in economics, networks or mechanics do not inherit its dielectric or dispersion meaning.

The coefficient can be inferred from force measurements or calculated theoretically. It can also help analyze colloid stability when other relevant forces are included.

Neighborhood in Abstraction Space¶

Hamaker Constant sits in a sparse region of the domain-specific corpus (68^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Statistical Mechanics & Particle Phenomena (19 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

Not to Be Confused With¶

London C₁₂: a microscopic pair-potential coefficient, not the integrated macroscopic A. Lifshitz-calculated coefficient: can serve an analogous strength role using dielectric spectra, but is not identical to the naive pair sum under all separations. DLVO net interaction: combines dispersion with electrostatic terms. Surface energy or work of adhesion: related quantities requiring contact and geometry assumptions, not interchangeable units with A. Capillary bridging: a different attraction mechanism that can dominate humid-surface measurements.^[1]^[2]^[4]

References¶

[1] Hamaker, H. C. (1937), “The London–van der Waals Attraction between Spherical Particles”, Physica 4: 1058–1072, doi:10.1016/S0031-8914(37)80203-7. Publisher indexed original and intermittently accessible original PDF. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[2] Lifshitz, E. M. (1956), “The Theory of Molecular Attractive Forces between Solids”, Soviet Physics JETP 2: 73–83. Original dielectric-response treatment. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[3] Valmacco, Valentina, et al. (2016), “Dispersion forces acting between silica particles across water, influence of nanoscale roughness”, Nanoscale Horizons 1: 325–330, doi:10.1039/C6NH00070C. Original publication and force measurements. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q

[4] Ohnishi, Satomi, Vassili Yaminsky, and H. K. Christenson (2000), “Measurements of the Force between Fluorocarbon Monolayer Surfaces in Air and Water”, Langmuir 16: 8360–8367, doi:10.1021/la000262e. Original abstract in author institution repository; full paper not read in this pass. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k