Cole–Cole Plot Parameters Calculator

The Cole–Cole relaxation model for complex permittivity is: ε*(ω) = ε∞ + (εs − ε∞) / (1 + (iωτ)^1−α).

Using the standard separation into real and imaginary parts with x = ln(ωτ), the hyperbolic form is:

• ε′ = ε∞ + 0.5(εs−ε∞)·[1 − sinh((1−α)x)/(cosh((1−α)x)+sin(απ/2))]
• ε″ = 0.5(εs−ε∞)·[cos(απ/2)/(cosh((1−α)x)+sin(απ/2))]

The loss peak occurs at ωτ = 1. The peak magnitude becomes ε″max = 0.5(εs−ε∞)·cos(απ/2)/(1+sin(απ/2)), which this calculator uses to estimate α.

1. Select Estimate parameters if you know εs, ε∞, f_p, and ε″_max.
2. Enter values carefully and keep frequency in Hz.
3. Press Calculate to get α, τ, and the Cole–Cole point.
4. Use Download CSV for spreadsheets or Download PDF for reports.
5. Select Forward calculation to compute ε′ and ε″ at a chosen frequency.

1) What a Cole–Cole plot shows

A Cole–Cole plot maps dielectric loss ε″ against storage ε′ over frequency. Each point represents a complex permittivity value at a specific angular frequency ω = 2πf. The curve shape reveals relaxation processes, interfacial polarization, and material heterogeneity. The arc position is bounded by ε∞ and εs on the ε′ axis.

2) Key parameters used by this calculator

The model uses εs, ε∞, α, and τ. The relaxation strength Δε = εs − ε∞ sets the arc height and contrast. The broadening factor α (0 to <1) spreads the relaxation times, producing a depressed arc; many real materials fall between 0.05 and 0.60. The time constant τ sets the frequency scale and typically decreases as temperature rises.

3) Typical measurement data and units

Dielectric spectroscopy reports ε′ and ε″ versus frequency in hertz. Permittivity terms are dimensionless, while τ is in seconds. For many polymers and electrolytes, τ ranges from microseconds to seconds. Use consistent electrode geometry and stable contact pressure, because stray capacitance and leakage can bias ε∞ and εs.

4) Using the loss peak to estimate τ

For the Cole–Cole form used here, the maximum loss occurs at ωτ = 1. If you identify the peak frequency fp from ε″(f), the calculator estimates τ = 1/(2πfp). For example, fp = 1 kHz gives τ ≈ 1.59×10⁻⁴ s. Smooth noisy data and avoid electrode-dominated regions near very low frequencies.

5) Using the peak height to estimate α

The peak value ε″max depends on α and Δε. The calculator forms r = 2ε″max/Δε, which must satisfy 0 < r ≤ 1. Debye relaxation gives r = 1 and α ≈ 0. Smaller r indicates broader relaxation and larger α. If your peak is flattened, check for DC conductivity contributions.

6) Forward calculation for plotted points

If α and τ are known from fitting, forward mode computes ε′ and ε″ at any frequency. It evaluates x = ln(ωτ) and applies the hyperbolic expressions. You can generate points to recreate the Cole–Cole arc, then plot them in a spreadsheet to compare against measured points across decades of frequency.

7) Data quality checks and troubleshooting

Ensure εs is greater than ε∞; otherwise Δε becomes negative and the arc is unphysical. If r falls outside the valid range, recheck ε″max, baseline subtraction, and units. Multiple relaxations can distort peaks and may require multi-process fitting. In temperature sweeps, fp should shift smoothly with each step.

8) Where these parameters are applied

Cole–Cole parameters support polymer curing studies, electrolyte aging, ceramics characterization, and moisture monitoring. α tracks dispersion and structural disorder, while τ shifts with environment and processing. Tracking τ over time can flag aging, while Δε can reflect water uptake or porosity changes. Exported CSV and PDF outputs support lab notebooks, QA reports, and comparisons across batches.

1) What does α physically represent?

α represents the spread of relaxation times. α = 0 corresponds to a single Debye time constant. Larger α depresses the Cole–Cole arc and indicates broader dynamics from heterogeneity, interfaces, or distribution of local environments.

2) Why must εs be greater than ε∞?

εs is the low-frequency limit and usually exceeds the high-frequency limit ε∞ for a relaxation. If εs ≤ ε∞, the relaxation strength Δε is nonpositive and the standard Cole–Cole arc interpretation becomes invalid.

3) How do I find ε″max from my dataset?

Plot ε″ versus frequency and locate the maximum value. Use a smoothed curve or a local polynomial fit if noise is high. Record both the peak height and the corresponding peak frequency for best estimates.

4) Does the loss peak always occur at ωτ = 1?

For the Cole–Cole form implemented here, the maximum ε″ occurs at ωτ = 1. If your system has multiple relaxations or electrode polarization, the observed peak can shift, and a single-process model may not fit perfectly.

5) Can this calculator model multiple arcs?

This version models one Cole–Cole relaxation. If your data shows two arcs or shoulders, fit each process separately or use a multi-term model. Use forward mode to compare candidate parameters against each frequency region.

6) What frequency range is recommended?

Choose a range that captures the low-frequency plateau near εs and the high-frequency plateau near ε∞. A span of at least two decades around the loss peak usually improves stability of α and τ estimation.

7) Why do my fitted and measured ε″max differ?

Differences can arise from baseline drift, conductivity contributions, or a non-Cole–Cole response. Ensure ε∞ is taken from the high-frequency limit, and verify the selected peak belongs to the same relaxation process.