# N0751 — a differential swelling sensor: exact response and limits

**Proved:** the linear two-path model results below. **Not proved:** that a real swelling material obeys this model, chemical specificity, universal detection of arbitrarily fast changes, or novelty.

## Original proposal

A mechanical differentiator could detect rapid chemical changes through the mismatch of two swelling pathways.

A concrete interpretation is a mechanically read difference of two displacement paths responding to the same concentration c(t), with equal calibrated equilibrium gain g>0 and different positive time constants τ_f<τ_s:

    τ_f dx_f/dt+x_f=g c(t),
    τ_s dx_s/dt+x_s=g c(t),
    y=x_f−x_s.

The material, input delivery, mechanical readout, and operating range must be experimentally shown compatible with this approximation. The two paths need equal equilibrium gain, not merely equal dimensions.

## Theorem 1: rejection of a constant concentration

For a constant concentration c₀ at equilibrium, both paths equal gc₀, so y=0. For a step c(t)=A from an initially equilibrated zero baseline,

    y(t)=gA[exp(−t/τ_s)−exp(−t/τ_f)], t≥0.

This follows by solving each first-order equation with an integrating factor and subtracting. For A>0, y is positive at every finite t>0, and tends to zero both initially and at long times. Thus a concentration change produces a transient while a constant equilibrated input is rejected. ∎

The peak occurs at

    t*=τ_f τ_s ln(τ_s/τ_f)/(τ_s−τ_f).

Differentiating y and solving dy/dt=0 gives this expression; the derivative changes from positive to negative there, so it is the unique peak.

For g=A=1, τ_f=1, τ_s=4, t*=(4/3)ln4 and peak y=(3/4)4^(−1/3)≈0.47247. These are arbitrary-unit model values, not material measurements.

## Theorem 2: the exact frequency band

For a steady sinusoid of angular frequency ω, substitution of a complex exponential yields

    H(iω)=g iω(τ_s−τ_f)/[(1+iωτ_f)(1+iωτ_s)].

Therefore

    |H|=g |ω|(τ_s−τ_f)/sqrt[(1+ω²τ_f²)(1+ω²τ_s²)].

The response vanishes at ω=0 and tends to zero as |ω|→∞. This device is a band-pass change sensor, not an unlimited-bandwidth differentiator.

To find the maximum, set z=ω². Apart from a positive constant, squared gain is

    F(z)=z/[(1+τ_f²z)(1+τ_s²z)].

Its derivative has the sign of 1−τ_f²τ_s²z², so the unique positive maximum occurs at

    ω*=1/sqrt(τ_f τ_s),
    |H(iω*)|=g(τ_s−τ_f)/(τ_s+τ_f). ∎

## Theorem 3: a controlled derivative approximation

Assume the stable causal responses are driven by a bounded history with sufficiently smooth c. Integrating the convolution by parts gives

    x_τ(t)=g c(t)−g∫₀∞ exp(−s/τ)c′(t−s)ds.

Thus

    y(t)=g∫₀∞ [exp(−s/τ_s)−exp(−s/τ_f)]c′(t−s)ds.

If |c″|≤K over the relevant history, subtract g(τ_s−τ_f)c′(t). The mean-value bound |c′(t−s)−c′(t)|≤Ks gives

    |y−g(τ_s−τ_f)c′(t)|
       ≤gK∫₀∞ s[exp(−s/τ_s)−exp(−s/τ_f)]ds
       =gK(τ_s²−τ_f²).

The kernel is nonnegative because τ_s>τ_f. This proves a bounded approximation to a scaled derivative for inputs whose derivative changes sufficiently slowly. It is not an approximation for arbitrary sharp inputs; use the exact step or frequency response there. ∎

## Counterexamples and physical limits

- If the time constants are equal, the matched paths produce y=0 for every input: no change signal exists.
- If gains differ, a constant concentration leaves y=(g_f−g_s)c₀. Calibration mismatch mimics a change or drift.
- Arbitrarily high-frequency changes are attenuated, not amplified.
- Temperature sensitivity, hysteresis, diffusion modes, input-delivery lag, changing geometry, and mechanical inertia can invalidate the two-exponential model.
- A synthetic transient does not demonstrate chemical specificity or detection at a stated noise level. Both require independent physical calibration and blinded tests.

The exact output is therefore predictable only within the specified model. Existing swelling kinetics and linear filtering principles are the foundation; no new physical law or verified original device is claimed.