Freely Jointed Chain Calculator

Mode * Force → Extension Extension → Force Statistics only

Segments N *

Number of rigid links.

Segment length b (nm) *

Often treated as Kuhn length.

Temperature T (K) *

Used through kBT.

Force F (pN)

Typical single-molecule forces: 0-100 pN.

Extension x (nm)

Must satisfy |x| < N b.

Calculate Reset

How to use this calculator

1. Select a mode: force-to-extension, extension-to-force, or statistics only.
2. Enter N, b (nm), and T (K).
3. Provide either force F (pN) or extension x (nm), depending on mode.
4. Click Calculate. Results appear above the form.
5. Use Download CSV or Download PDF to export the result tables.

Example data table

Article

1. Freely jointed chain in polymer physics

The freely jointed chain (FJC) is a baseline model for flexible polymers. It represents a molecule as N rigid links of length b connected by frictionless joints. The model captures entropic elasticity: stretching reduces the number of available conformations, creating a restoring force even without energetic springs.

2. Inputs and physically meaningful ranges

Your key inputs are N, b, and temperature T. The contour length is Lc = N b and sets the maximum extension. b is often used as an effective Kuhn length. Choose N and b so Lc matches your polymer.

3. Dimensionless force and the Langevin function

The calculator forms the dimensionless force y = F b/(kB T). The mean fractional extension is L(y) = coth(y) - 1/y, and the mean extension is <x> = Lc L(y). At small y, L(y) ≈ y/3, giving a near-linear spring. At larger y, L(y) approaches 1, reflecting finite extensibility.

4. Useful reference numbers for interpretation

At 298 K, kB T ≈ 4.11 pN·nm. With b = 1.0 nm and F = 1 pN, y ≈ 0.24 and <x>/Lc ≈ 0.08. At F = 10 pN, y ≈ 2.4 and the chain nears high extension, so extra force yields smaller extension gains.

5. Small-force elasticity and effective stiffness

In the weak-stretch regime, the FJC reduces to a Hookean response with k0 = 3 kB T/(N b^2). This is helpful when comparing to bulk moduli or when checking consistency across datasets. Because k0 scales as 1/N, longer chains are softer, while increasing b stiffens the response in this regime.

6. Thermodynamic output and constant-force free energy

The calculator also reports the constant-force free energy G(F) = -N kB T ln(sinh(y)/y). It is useful for work estimates, temperature scaling, and fitting force-clamp data where extension and free-energy trends are analyzed together.

7. Inverse mode for fitting experimental curves

Many experiments measure extension and need the force. Inverse mode estimates y ≈ L-1(x/Lc) using a stable rational approximation, then returns F = y kB T/b. This is suitable for quick parameter sweeps and initial fits. Near |x/Lc| -> 1, the inverse becomes sensitive, so keep the extension fraction below unity.

8. Assumptions, limitations, and reporting

The FJC ignores bending stiffness, self-avoidance, and bond stretching. Semiflexible chains may fit better with worm-like chain models. At very high force, energetic stretching and instrument compliance matter. Export results to archive inputs, units, and derived values.

FAQs

1) What does N represent in this model?

N is the number of rigid segments. Increasing N increases contour length Lc = N b and reduces small-force stiffness k0. Choose N so Lc matches the physical chain length you are modeling.

2) Why is b sometimes called the Kuhn length?

For coarse-grained descriptions, b can represent an effective segment length that reproduces the chain's large-scale statistics. In that sense it plays the role of a Kuhn length, even if the microscopic bonds are smaller.

3) What units should I use for force and length?

Use pN for force, nm for lengths, and K for temperature. The calculator converts kB T into pN·nm internally, keeping y dimensionless and outputs consistent for common single-molecule datasets.

4) Why must |x| be less than Lc?

Lc is the maximum end-to-end length if every segment aligns. The FJC cannot exceed this limit, so the inverse calculation requires |x/Lc| < 1. Values near 1 become numerically sensitive.

5) When is the linear spring approximation reliable?

When y = F b/(kB T) is small, typically below ~0.2, the Langevin function is nearly linear. In that regime, x ≈ F/k0 and the calculator also reports a small-force check for comparison.

6) Does this model include excluded volume effects?

No. The basic FJC assumes ideal statistics with freely crossing segments. In good solvents or crowded environments, self-avoidance can change scaling and extension. Use the FJC as a baseline or fit with corrections if needed.

7) How should I cite or report results from this tool?

Report the chosen mode, inputs (N, b, T, and F or x), and units. Include Lc and the fractional extension x/Lc. Export CSV for records and PDF for sharing with collaborators or lab notes.