Supramolecular Polymer Length Calculator

Example data table

Formula used

For isodesmic supramolecular assembly, each association step uses the same equilibrium constant K (M⁻¹). The distribution of chain sizes is geometric with ratio y = K·[M]_free.

• a = K · C_total
• y = ((2a + 1) − √(4a + 1)) / (2a)
• [M]_free = y / K
• DP_n = 1 / (1 − y)
• DP_w = (1 + y) / (1 − y)
• L_n = DP_n · l₀ (contour length)

If you enter ΔG° instead of K, the calculator uses K = exp(−ΔG°/(R·T)). If you enable temperature adjustment, it uses the van’t Hoff relation with ΔH°.

How to use this calculator

1. Enter total monomer concentration and select its unit.
2. Choose how you want to provide thermodynamics: K or ΔG°.
3. Set the temperature; optionally adjust K from a reference temperature.
4. Enter the monomer repeat length to convert DP into nanometers.
5. Optional: add uncertainties and a sample count for confidence bands.
6. Press Submit; results appear above, ready for export.

Isodesmic Framework

This calculator uses an isodesmic supramolecular polymerization model, where each monomer addition has the same association constant K. Define a = K*Ctotal. The form solution gives y = K*[M]free = ((2a+1)-sqrt(4a+1))/(2a). From y, DPn = 1/(1-y) and DPw = (1+y)/(1-y), so PDI = DPw/DPn = 1+y. The polymerized fraction is 1 - [M]free/Ctotal. Use this when growth is stepwise without a distinct nucleation barrier.

Concentration Scaling

At fixed K, raising Ctotal increases a and pushes y toward 1, which amplifies DPn sharply. In the low-a limit, y≈a and DPn stays close to 1, indicating mostly monomer and short oligomers. In the high-a regime, small concentration changes can shift DPn by orders of magnitude. The estimated chain concentration is Ctotal/DPn, so longer chains imply fewer total chains at constant mass balance. This helps interpret dilution series and compare formulation windows.

Thermodynamic Inputs

You can enter K directly or compute it from free energy with K = exp(-DG/(R*T)). The tool assumes a 1 M reference state, so DG should match that convention. For temperature shifts, the optional van’t Hoff adjustment uses ln K(T) = ln Kref - (DH/R)*(1/T - 1/Tref). Negative DH typically increases K at lower temperature, raising DPn and predicted length. Treat these corrections as approximations when heat capacity changes are significant.

Length Conversion

Degree of polymerization becomes a physical contour length by multiplying by the repeat length l0. The calculator converts angstrom to nanometer and reports Ln = DPn*l0 and Lw = DPw*l0. These are contour lengths for an extended backbone, not end-to-end distances, hydrodynamic radius, or persistence-length controlled dimensions. Use Ln and Lw to rationalize trends in viscosity, scattering intensity, and length-dependent binding, then refine with structural models for your motif.

Uncertainty and Reporting

Measured inputs often include uncertainty from concentration preparation, fitting K, and estimating l0. When enabled, the uncertainty module draws random values from normal distributions and recomputes DPn, Ln, and polymerized fraction across many trials. Use 1000 to 5000 samples for stable intervals. Export options create a CSV for analysis and a PDF snapshot for lab records.

FAQs

1) What does the calculator mean by polymer length?

It reports contour length, calculated as degree of polymerization times repeat length. This is an upper-bound geometric length for an extended chain, useful for comparing conditions and setting expectations for length-sensitive measurements.

2) Why can I input either K or ΔG°?

Some experiments fit equilibrium constants, while others report free energies. The calculator converts ΔG° to K using temperature and the gas constant, so both routes produce the same equilibrium y and polymer length metrics.

3) What is the difference between DP_n and DP_w?

DP_n weights all chains equally and reflects the typical chain count. DP_w weights longer chains more strongly and is always larger or equal. Their ratio is the dispersity, indicating how broad the size distribution is.

4) How should I choose the repeat length l₀?

Use the distance added per monomer along the backbone, from molecular models, crystal structures, or literature. If your monomer stacks with spacing near 3.5 Å, enter 3.5 Å and refine if the assembly geometry differs.

5) When should I use the van’t Hoff option?

Use it when you have K measured at a reference temperature and an estimated enthalpy change. It provides an approximate K at your chosen temperature, helping you compare datasets collected under different thermal conditions.

6) Why did the uncertainty run produce too few valid samples?

If uncertainties are large, random draws can create nonphysical negative concentration, K, or repeat length, which are discarded. Reduce the uncertainty inputs, increase concentration, or raise the sample count to obtain stable confidence intervals.