DMT Adhesive Contact Simulator — Derjaguin-Muller-Toporov Theory
For hard elastic spheres, the DMT model adds the long-range attraction outside the contact area to a Hertzian profile. The pull-off force F_po = -2 pi R gamma is 4/3 times that of JKR. Use the Tabor parameter to verify the model validity range.
Parameters
Sphere radius R
mm
Reduced modulus E*
GPa
Surface energy γ
mJ/m²
Normal load F (negative = tension)
μN
Defaults reflect a glass ball (E* about 100 GPa). The Tabor equilibrium spacing z0 = 0.2 nm is assumed. Set F < 0 to observe adhesive holding under tensile load.
Results
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DMT contact radius a_D
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Pull-off force F_po^DMT = -2πRγ
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Compare: F_po^JKR = -(3/2)πRγ
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Tabor parameter μ (validity)
Schematic + Contact Radius vs Load Curves
Top = sphere-on-flat schematic (DMT: attraction halo outside contact / JKR: stress concentration at edge) / Bottom = DMT (green) vs JKR (red) vs Hertz (blue) curves, yellow dot = current state
Theory & Key Formulas
The DMT theory describes a hard elastic sphere on a flat by superposing the long-range attractive force outside the contact area onto a Hertzian contact profile.
DMT applies to stiff spheres of small radius. The ratio F_po^DMT / F_po^JKR = 4/3 ≈ 1.33. The Maugis-Dugdale model bridges the intermediate regime (not handled by this tool).
What is the DMT Adhesive Contact Simulator
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I've played with a JKR adhesive contact simulator. What is different about DMT? Aren't they both about adhesive contact?
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Roughly: if JKR is for "soft and large" spheres, DMT is for "hard and small" spheres. Both deal with surface energy gamma, but they place the adhesion in different regions. In DMT the sphere barely deforms, so we integrate the long-range attraction (van der Waals etc.) over the sphere outside the contact area. In JKR the adhesion is concentrated inside the contact, especially at the edge where stress concentrates. Set E* to 100 GPa (glass-like) in the simulator above and look at the green halo in the schematic — that is the picture of the attraction outside the contact.
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I see. So the pull-off forces differ by a factor of 4/3 — -628 micronewtons (DMT) vs -471 (JKR). The same gamma yet that different?
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Yes — the derivations are completely different, so this must be the case. DMT assumes a Hertzian shape and integrates the attraction outside the contact; the result is $F_\text{po}^\text{DMT}=-2\pi R\gamma$. JKR minimizes elastic energy plus surface energy and gets $F_\text{po}^\text{JKR}=-(3/2)\pi R\gamma$. The ratio is 2/(3/2)=4/3≈1.33. When a real measurement on the same gamma and R disagrees with one of them, you need to first decide which model is appropriate. That is what the Tabor parameter is for.
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The "Tabor parameter mu = 5.0" — what does that mean? It says "validity".
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$\mu = (R\gamma^2/(E^{*2}z_0^3))^{1/3}$, where z0 is the equilibrium spacing of about 0.2 nm. Small mu means "barely deforms — DMT applies", large mu means "deforms readily — JKR applies". The thresholds are: DMT for mu less than 0.1, JKR for mu greater than 5, with Maugis-Dugdale in between. At the default settings mu is 5, right on the boundary. Try lowering R to 0.01 mm (10 micrometres) and mu drops, putting you into the pure DMT regime — much like a typical AFM tip.
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In the lower plot, green (DMT) and red (JKR) differ a lot at low load, but both converge to the blue Hertz curve as F grows.
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Good catch. When F is much larger than pi R gamma, the adhesion term becomes negligible and all three models agree with Hertz. In practice — AFM, MEMS, powder handling, adhesive tape design — at low load with small contacts, the adhesion term dominates and choosing DMT or JKR has a big effect on the result. Reading the Tabor diagnostic alongside this curve gives you an immediate feel for which model to use.
Frequently Asked Questions
DMT (Derjaguin-Muller-Toporov) is for hard spheres of small radius and adds the long-range attraction outside the contact area to a Hertzian profile. JKR (Johnson-Kendall-Roberts) is for soft, large spheres where adhesion is concentrated inside the contact. The discriminator is the Tabor parameter mu = (R·γ²/(E*²·z₀³))^(1/3). Use DMT for μ<0.1, JKR for μ>5, and the Maugis-Dugdale interpolation in between. This tool targets hard materials (E*>10 GPa) under DMT, and the Tabor value is shown so you can verify the validity range. For soft contacts use the JKR simulator (jkr-adhesive-contact.html).
DMT integrates the long-range attractive forces (van der Waals) over the sphere outside the contact area while keeping the Hertzian contact shape; this gives F_po^DMT = -2πRγ. JKR concentrates adhesion inside the contact and derives the detachment condition from the singular stress at the contact edge; the result is F_po^JKR = -(3/2)πRγ. The ratio is 2 / (3/2) = 4/3 ≈ 1.33, so DMT predicts a pull-off force about 25 percent larger than JKR. The difference arises because the two models assume completely different contact shapes for hard versus soft spheres.
Yes. AFM probes (Si₃N₄ or Si, E*=tens to hundreds of GPa) are hard, and the radius R=10–100 nm is very small, which keeps Tabor μ small and places the experiment in the DMT regime. Setting R as small as 0.01 mm (10 μm) and E* around 100 GPa in this tool reproduces a typical AFM situation. The standard workflow of measuring F_po from the force curve and back-solving the surface energy from F_po = -2πRγ can be simulated directly. For soft samples such as polymers, use the JKR simulator (jkr-adhesive-contact.html) instead.
The defaults (R=1 mm, E*=100 GPa, γ=100 mJ/m²) give μ of about 5 to illustrate the boundary between DMT and JKR. Strict DMT validity requires μ<0.1 (for example R=10 nm with E*=200 GPa). The presets reflect glass / ceramic balls of typical millimetre size and stiffness, which sit close to the validity boundary. Move R down to 0.01 mm (10 μm), or raise E* toward 500 GPa, to enter the pure DMT regime. Accurate work in the intermediate region requires the Maugis-Dugdale model, which is outside the scope of this tool.
Real-World Applications
AFM and nanoindentation surface-energy measurement: Force curves taken with an AFM yield the pull-off force F_po between tip and sample, which is back-solved with F_po = -2πRγ (DMT) or F_po = -(3/2)πRγ (JKR) to extract γ. For hard samples (Si, SiO₂, metals, ceramics) DMT is the standard choice. This tool lets you enter a tip radius and modulus to simulate the expected pull-off force.
MEMS / NEMS stiction design: In micro- and nano-electromechanical systems (micromirrors, accelerometers, RF switches), "stiction" of moving parts to the substrate is a serious failure mode. Si-to-Si contact (E* about 170 GPa) is well described by DMT. Use this tool to estimate the pull-off force from contact-tip radius and surface energy, and compare it with the restoring spring force to decide whether the device will release. The effect of surface treatments such as self-assembled monolayers (which lower γ) can also be evaluated.
Powder handling and agglomeration control: Hard, small particles such as ceramic powders, pharmaceutical actives, or toner are well described by DMT for cohesion. With particle radii of tens of micrometres and E* of tens of GPa, μ is small and the system sits in the DMT regime. Manufacturing processes use these estimates for dispersibility, flowability and dust-explosion risk. Capillary forces from humidity must be added separately.
Ceramic and glass ball mechanical contact: In bearing-ball analyses for steel balls (E* about 200 GPa) or sapphire / ruby balls (E* above 300 GPa), DMT is the baseline model when estimating the adhesion contribution. Under normal loads adhesion is negligible, but in ultra-low-load precision measurements (probe microscopes, atomic-spacing metrology) the adhesion term can dominate.
Common Misconceptions and Cautions
The most common misconception is to think that either DMT or JKR is "the correct one". In reality the two models occupy opposite ends of a spectrum. For Tabor μ<0.1 (hard, small spheres) DMT is rigorous, and for μ>5 (soft, large spheres) JKR is rigorous. In the intermediate region neither is rigorous, and the Maugis-Dugdale interpolation or a numerical contact analysis is required. With this tool's defaults (γ=100 mJ/m², R=1 mm, E*=100 GPa) μ is about 5 — right at the boundary — so the true pull-off force should lie between the DMT and JKR values. Always inspect the Tabor diagnostic and decide which regime your problem belongs to.
The next most common error is to assume that "hard means adhesion can be ignored". At high loads the adhesion term in F + 2πRγ is dominated by F and a_D matches the Hertz radius. But in the low-load regime where F is much smaller than 2πRγ (with the defaults, 2πRγ = 628 μN, so for F below about 200 μN) adhesion dominates, and ignoring it badly under-estimates the contact radius. In AFM, MEMS, precision metrology and space environments (low gravity), even hard materials must include the adhesion term.
Finally, note that F_po as computed here is for a "clean and dry" idealised surface. In real environments, water (capillary force from humidity) can amplify the pull-off force by a factor of 10 to 100. Surface roughness reduces the effective contact area below the geometric radius R, lowering the apparent pull-off force. Use this tool as the "entry point for Tabor diagnosis and model selection" — final design values should come from measurements, or from numerical analysis that includes roughness, humidity and contamination.