Polymer Persistence Length Calculator

Calculator

Calculation mode

Pick the data you actually have.

Temperature

Used in k_BT for thermal bending.

Contour length (optional)

Enables L/l_p and predicted R_rms.

Bending stiffness, κ

WLC: l_p = κ / (k_BT).

Kuhn length, b

Relation: b = 2l_p (WLC ↔ FJC).

End-to-end R_rms

Provide contour length too, then invert WLC numerically.

Add electrostatic persistence length correction (optional)

Useful for charged polymers in electrolyte solutions.

Bjerrum length, l_B

Typical water near room temperature ≈ 0.7 nm.

Debye length, λ_D

Shorter λ_D means stronger screening.

Charge spacing, A

Distance between effective charges along the chain.

Formula used

Worm-like chain (bending stiffness): l_p = κ / (k_BT)
Kuhn relation: b = 2l_p
WLC end-to-end mean-square distance:
⟨R²⟩ = 2l_pL \left[1 - \frac{l_p}{L}\left(1-e^{-L/l_p}\right)\right]

In “statistics” mode, the calculator numerically inverts this relation using your R_rms and L.
Optional electrostatics (OSF-style): l_e = l_Bλ_D² / (4A²), so l_p = l_p,0 + l_e

Units are handled so your final l_p is reported in nm and m.

How to use this calculator

Select a calculation mode based on what you measured.
Enter temperature in Kelvin to define k_BT.
Provide the mode-specific input (κ, b, or R_rms with L).
Optionally enable electrostatics and enter l_B, λ_D, and A.
Press Submit to display results above the form.
Use Download CSV or Download PDF for reporting.

Example data table

Scenario	Mode	Inputs	Expected l_p (nm)
Moderately stiff chain	Bending stiffness	κ = 500 pN·nm², T = 298 K	≈ 121.4
Equivalent Kuhn representation	Kuhn length	b = 200 nm	100
Measured end-to-end size	Statistics	L = 1000 nm, R_rms = 600 nm	Estimated by inversion

Use “Load Example Values” to populate a working set of inputs.

Professional article

What persistence length represents

The persistence length l_p is the distance over which a polymer’s local tangent direction stays correlated. In the worm-like chain picture, tangent–tangent correlations decay approximately as exp(−s/l_p). Larger l_p means a stiffer chain and smoother bending over longer segments.

Connecting stiffness to thermal energy

When you know bending stiffness κ, this calculator uses l_p = κ/(k_BT). The ratio κ/k_BT is a competition between elastic resistance to curvature and thermal agitation. At 298 K, k_BT ≈ 4.11 pN·nm, so κ in pN·nm² maps directly into nanometer-scale stiffness.

Temperature sensitivity and comparisons

Because l_p scales as 1/T for fixed κ, raising temperature reduces the apparent persistence length. For example, if κ is constant, moving from 298 K to 310 K lowers l_p by about 3.9%. This matters when comparing experiments taken at different temperatures or when using solvent conditions that shift κ.

Kuhn length as a coarse-grained step

The Kuhn length b is an effective step size for a freely jointed chain that reproduces large-scale statistics. For a worm-like chain, b = 2l_p. Reporting b alongside l_p helps connect mechanical stiffness to polymer-physics scaling laws, such as estimates of coil size and entropic elasticity.

Using end-to-end statistics to infer l_p

If you measured a contour length L and an end-to-end root-mean-square distance R_rms, the calculator inverts the worm-like chain ⟨R²⟩ relation numerically. This approach is useful for microscopy or scattering-derived sizes, and it naturally accounts for intermediate regimes between fully flexible (L/l_p ≫ 1) and nearly rigid (L/l_p ≪ 1).

Charged polymers and ionic screening

For polyelectrolytes, electrostatic repulsion can stiffen the chain. The optional OSF-style correction uses l_e = l_Bλ_D²/(4A²), so weaker screening (larger λ_D) increases l_e quadratically. As a rough guide in water, λ_D is about 1 nm near 0.1 M salt and about 3 nm near 0.01 M.

Reference values to sanity-check inputs

Typical persistence lengths span many orders of magnitude: flexible synthetic polymers can be below 1 nm, double-stranded DNA is often near 50 nm in moderate salt, actin filaments are on the order of 10 µm, and microtubules can approach millimeter scales. Use these ranges to catch unit mistakes or unrealistic parameters.

Reporting results and uncertainty

When documenting l_p, include the method (κ-based, Kuhn-based, or statistical inversion), temperature, and any electrostatic parameters used. If you provide L, the L/l_p ratio helps interpret regime and model validity. Exporting CSV/PDF supports traceable reporting and consistent comparison across samples.

FAQs

1) Can l_p be larger than the contour length?

Yes. If l_p ≫ L, the chain behaves nearly like a rigid rod over its full contour. In that regime, R_rms approaches L and bending fluctuations are small.

2) Why does temperature change the result in κ mode?

In κ mode, l_p = κ/(k_BT). Higher temperature increases thermal energy, making the same κ appear less stiff, so the computed persistence length decreases.

3) What units should I use for κ?

You can enter κ in pN·nm² or J·m. The calculator converts internally and reports l_p in nm and m, so you can keep your lab’s preferred mechanical units.

4) When should I enable the electrostatics correction?

Enable it for charged polymers when ionic strength varies across samples. The correction estimates additional stiffness from screened electrostatic repulsion and is most informative when λ_D and charge spacing A are known or well-estimated.

5) Why do I need contour length in statistics mode?

The inversion uses both L and R_rms to solve the worm-like chain relation. Without L, the same R_rms could correspond to many different stiffness and length combinations.

6) What does L/l_p tell me?

It indicates regime: large values mean flexible, coil-like behavior, while small values mean rod-like behavior. This ratio helps assess whether WLC assumptions and your measurement scale are consistent.

7) How can I validate the output quickly?

Check that R_rms never exceeds L, compare l_p to known ranges for similar polymers, and verify that unit conversions (nm, µm, m) match your experimental inputs.