Deborah, Chirp, and Resilience

In rheology, the Deborah number (De) carries a timeless message:

everything flows — if you wait long enough.

Named after the prophetess Deborah, it stands out as a rare feminine presence in the language of rheology, and a poetic one. The number reminds us that what appears solid and unchanging may, over long enough timescales, soften, yield, and flow.

It’s a perfect metaphor for International Women’s Day, a celebration built on stories of resilience, progress, and the women who powerfully shape our world.

To understand how materials behave across different timescales, the Deborah number gives us a simple but powerful recipe:

Deborah number = (relaxation time) / (observation time)

The observation time is the easy part; it’s simply how long an event takes in real life: a quick tap, a slow spread, a minute of settling.

But the relaxation time, the time it takes a material to „forget“ a deformation, is something we must measure in the rheology lab.

And that’s where frequency sweeps come in.

In an oscillatory frequency sweep, we gently oscillate the material back and forth at different speeds within the linear viscoelastic region; where the material’s structure stays intact and undamaged. At high frequencies the deformation is too fast for the material to rearrange, so it behaves more like a solid. At low frequencies we give the material plenty of time to respond, revealing its fluid-like character. Where the elastic and viscous curves cross, the famous G′–G″ crossover, we find the material’s characteristic relaxation time.

For quick responding materials, this works beautifully. But for slower, more structured materials, think thick creams or gels, reaching the frequencies needed to capture long relaxation times become painfully slow. To reach long relaxation times, the instrument must probe very low frequencies, sometimes so low that:

• one single data point can take minutes and minutes
• a full sweep can take hours

So, to measure the Deborah number properly, especially for stubborn, slow-flowing materials, we needed a faster way to explore long timescales.

That’s where frequency chirps change everything.

A conventional frequency sweep measures one frequency point at a time. A chirp replaces the point-by-point sweep with a continuous glide through frequencies, like sliding your finger across a piano instead of pressing each key individually. In a single, smooth, rapid oscillation, the rheometer covers the entire frequency range. This allows us to access extremely low frequencies, capturing long-timescale behavior that conventional sweeps would need hours to reveal.

With relaxation time in hand, we can finally compute the Deborah number in ways that connect directly to everyday experience. Now, let’s apply this to two familiar materials, shampoo and a thick body lotion, and see how their personalities change with the timescale of interaction.

The Deborah number simply compares them:

• De << 1 → the material behaves mostly like a liquid
• De ≈ 1 → the material is viscoelastic, showing both solid-like and liquid-like responses
• De >> 1 → the material behaves mostly like a solid

Let’s start with something we all meet before our first cup of coffee: shampoo.

We ran a frequency sweep on a common commercial shampoo (Figure 1).

The crossover was at:

ω_c = 14.69 rad/s

That means the material’s characteristic relaxation time is:

But Deborah number is only meaningful when we compare λ with a real-life observation time, something people actually experience outside the rheology lab.

When you tap your finger quickly into a mound of shampoo in your palm, this quick motion happens in roughly 20–30 milliseconds.

Let’s take T = 0.025 s.

With De ≈ 2.7, shampoo is behaving in a mostly solidlike, elastic manner at the moment of impact:

• It briefly resists your finger
• You feel a soft, elastic „bounce“ at the surface

When you tilt the bottle and let shampoo pour into your hand, the flow typically takes around 3–4 seconds.

Let’s use T = 3.5 s as a representative value.

With De ≈ 0.02, the shampoo is now clearly in the liquid‑dominated regime:

• It flows freely under gravity
• Viscous behavior controls the motion

This is why shampoo pours smoothly and consistently, forming a ribbon or stream rather than a blob that resists movement.

Next up is a thick body lotion, a material that behaves much more „stubbornly“ than shampoo. A standard frequency sweep (Figure 2) was pushed down to 0.628 rad/s (0.1 Hz). Still no crossover. G′ stayed above G″ the entire time.

Going below 0.1 Hz with a conventional sweep?

That’s when the rheometer starts acting like it’s on a coffee break… a very long one.

So, we switched to frequency chirp.

In the chirp data (light blue), suddenly…

A crossover appears at 0.015 rad/s.

This value corresponds to a relaxation time of:

Observation 1: First touch (T ≈ 0.5 s)

This is an extremely elastic, solid-like response.

This is why lotion initially feels thick, structured, and resistant.

Observation 2: Slow Application / Gentle Spreading (T ≈ 45 s)

A slow, gentle massage-type spreading motion across a larger area typically takes around 45 seconds.

We are now near the viscoelastic transition:

• It still maintains structure
• Yet it is spreadable and can be worked across the skin
• It deforms under your hands but does not fully flow on its own

This mixed personality, structured yet cooperative, is exactly what gives lotions their luxurious, creamy feel during slower application.

And before we wrap up, let me introduce one more sample, though for its own good, we won’t mention names. Let’s just say the frequency chirp went all the way down to 0.0001 Hz and still…

No crossover. Not even close. Figure 3 says it all.

Think of someone who absolutely refuses to relax at a spa. You dim the lights, put on calming music, bring out chamomile tea, and they’re still sitting there like, „Nope. I’m holding my structure.“

Shampoo, body lotion, and the unknown stiff guy all taught us something through their Deborah numbers:

• Some respond quickly.
• Some take their time.
• Some hold their structure for a very long time.

And all are correct, valid, and meaningful, depending on the timescale.

This is exactly the message Deborah number gives us about life:

Strength is not defined by how fast you respond, but by how you endure over time.

On this International Women’s Day, we celebrate exactly that:

The resilience, patience, courage, and strength embodied by the women who inspire us every day.