The brightener is being used up

Every other clock on this site belongs to the observer. A gain has a time constant; an afterimage is an adaptation; the slowest one is chemical and it is a pigment recovering on a hundred-second time constant.

This is the first clock that belongs to the sample, and it does not recover at all.

What the curve is walking down is visible at one moment rather than over a life, and the pair a proofing job has to hold together says it most sharply.

A more heavily brightened stock starts higher on the same curve and has further to fall, which is the version a specifier is most likely to have chosen.

The claim #

A brightener is consumed by exactly the light that makes it work, so a sheet’s whiteness is a function of its exposure history and not of its formulation. The consequence for a specification is that a whiteness figure needs a date on it, and none of them has one.

• The mechanism is photodegradation. An excited fluorophore usually returns its photon and occasionally does something else — reacting with oxygen, principally — and the something else is irreversible.
• The whiteness falls from 122.5 to 87.6 over the dose range modelled here, which is most of the way back to the unbrightened base at 82.
• The fall is convex from the beginning, not sigmoid. The absorption saturates in the loading, so the first molecules lost are the ones doing the least work — and the sheet still loses whiteness fastest at the start.
• A just-noticeable change arrives at the half-dose, which is the point at which half the brightener is gone and the sheet has moved ΔE00 2.1.
• And the clock is stopped by the same thing that stops the glow. A sheet kept under LED light or behind a filter loses nothing, because it is receiving no excitation — so it keeps a whiteness it cannot display.

Why the molecule breaks #

The photophysics is short and it is the reason the effect exists rather than being a manufacturing defect.

A fluorophore absorbs a photon into an excited singlet state. From there the overwhelmingly likely outcome is emission — that is the quantum yield, which for a good brightener is above 0.8. The remaining probability is split between non-radiative relaxation, which is harmless, and intersystem crossing to a triplet state, which is not.

A triplet state is long-lived by molecular standards — microseconds to milliseconds rather than nanoseconds — which gives it time to meet something. What it usually meets is molecular oxygen, and the reaction produces singlet oxygen, which then attacks the double bonds the fluorophore’s conjugated system is made of. The molecule that comes out the other side does not fluoresce.

So the destruction rate is proportional to the excitation rate, at least to first order, which makes the dose rather than the elapsed time the natural variable. A sheet in a drawer ages chemically for other reasons and does not lose its brightener; a sheet in a window loses it at a rate set by the window.

The same chemistry is why the additive is not simply used at higher concentration. Doubling the loading roughly doubles the number of molecules available to break, and the emission it buys is less than double because the absorption saturates — so the sheet with twice the brightener starts higher and loses proportionally faster.

The shape of the curve, and why it is not the obvious one #

A quantity decaying exponentially produces an exponential curve in whatever it drives. This one does not, and the reason is worth following because it runs opposite to the intuition.

The loading falls exponentially with dose: half of it is gone at the half-dose, a quarter at twice that. But the whiteness is not proportional to the loading. The absorption saturates through a Beer–Lambert exponential, so the first units of brightener absorb most of what is available and later units absorb the remainder of a diminishing supply.

The obvious prediction from that is a plateau — the sheet should lose very little at first, because it starts with more brightener than it needs. The measurement says otherwise. The whiteness falls by 2.5 points over the first fifth of the half-dose and by 6 points over the second, then accelerates: 122.5 to 121.9 to 121.0 to 118.5 to 112.3, and on to 103.4, 92.4 and 87.6.

The curve is convex the whole way, which means the fractional rate of loss is highest at the end and the absolute rate is highest in the middle. There is no plateau because the loading here is on the shoulder of the saturation rather than deep in it — a commercial sheet is not loaded to the point of diminishing returns, because the additive costs money and the tint bound stops it anyway.

So the practical statement is unhelpful in the useful direction. A brightened sheet does not hold its whiteness for a while and then fall off; it starts losing immediately, and the loss becomes noticeable at about the point where half the additive is gone.

The published sequence shows the plateau the essay says is absent #

The section on the curve’s shape sets up a prediction and reports that the measurement refuses it. The eight numbers it prints do not refuse it.

Taking the losses between consecutive readings of 122.5, 121.9, 121.0, 118.5, 112.3, 103.4, 92.4 and 87.6:

The largest step is eighteen times the first, and the first three readings together give up 1.5 points while the fifth and sixth give up 19.9 between them. That is a plateau at the start followed by an acceleration — which is precisely the shape the saturating absorption predicts and the essay reports as ruled out.

Three sentences depend on the opposite reading and none of them survives the column. The fall is steepest at the start appears in the dek and in the hero caption; the sheet still loses whiteness fastest at the start appears in the argument; and the curve is convex the whole way appears immediately after a sentence saying the absolute rate is highest in the middle, which is a description of a curve with an inflection rather than a convex one. The losses grow for five steps and then fall, so the inflection is real and sits between the sixth and seventh readings.

The likely origin of the error is visible in the same paragraph. The whiteness falls by 2.5 points over the first fifth of the half-dose and by 6 points over the second — 2.5 and 6.2 are steps three and four, not one and two. Two intervals were read off the middle of the sequence and described as its beginning, and everything about steepness follows from that.

What the corrected shape says instead #

The repair strengthens the essay’s own mechanism rather than weakening it, which is why it is worth making rather than eliding.

The saturating absorption predicts that the first molecules lost are doing the least work, so the sheet should hold its whiteness at first. It does: 1.5 points over the first two intervals, against a total fall of 34.9. The acceleration that follows is the saturation being exhausted — once the loading drops below the shoulder, every further molecule lost costs the full amount. And the final step slows again because the sheet is approaching its base and has almost nothing left to lose: 87.6 against a base of 82, having given up 86 per cent of everything that was there.

So the curve is a sigmoid after all, and the practical statement inverts. A brightened sheet does hold its whiteness for a while and then falls off, which is more useful to a buyer than the essay’s version and is a good deal less comfortable for a supplier: the sheet passes inspection, passes it again some months later, and then loses eleven points in the interval after that.

It also changes what the just-noticeable threshold means. Arriving at ΔE00 2.1 by the half-dose is consistent with the corrected shape — the sheet is by then past the shoulder and into the steep part — and the earlier reading, which would have had it drifting steadily from day one, is not.

Which leaves one claim needing a different defence #

The essay’s closing generalisation says the property is consumed in proportion to how much of it is being displayed, and that is untouched: the destruction rate is proportional to the excitation rate whatever the whiteness curve looks like.

What the corrected shape does undermine is the reassurance a supplier might take from a stable early measurement. A sheet re-measured after a modest exposure and found unchanged has not demonstrated that it is durable; it has demonstrated that it is still on the plateau, and the plateau is where a saturating system spends its first fifth of a lifetime whatever its half-life is. A flat reading early is exactly what this mechanism predicts and is no evidence at all about the rest of the curve, which is a sharper warning than the version where the fall is steepest at the start and any measurement would have caught it.

What a dated measurement would have to say #

Every whiteness figure on every data sheet is a measurement made once, on a sample, at the mill. The number is correct at the moment it is taken and there is no mechanism by which it stays correct.

A specification that took this seriously would have to carry three things it does not.

The date of measurement, which is the cheapest and is occasionally present as a batch code.

The exposure since. This is the one that cannot be supplied, because it depends on where the sheet has been — a pallet in a dark warehouse and a display sample in a shop window have received doses differing by orders of magnitude.

And the measurement condition, without which the number is a joint property of the sheet and a lamp in any case. An ultraviolet-excluded measurement is nearly immune to this whole essay, because it reports the reflected component and the reflected component barely changes as the brightener degrades.

That last observation is the practical one. The M₂ figure is stable in time and does not describe the sheet a person sees; the M₁ figure describes what a person sees and has a half-life. There is no single number that is both, and the industry’s habit of quoting one number is a choice between the two that nobody makes explicitly.

Where it is visible without an instrument #

Three cases where the effect is large enough to see, and each isolates a different variable.

A poster or a book jacket in a shop window, half of which is behind a display fitting. The exposed half yellows relative to the shaded half over months, and the boundary is sharp because the shading is. What is being seen is not the ink fading — that happens too and more slowly — but the substrate’s brightener going.

A stack of paper stored on edge. The top sheet and the edges of the stack receive light and the interior does not, so fanning a ream that has sat in an office for a year shows a gradient of a few whiteness points from the outside in.

And a shirt washed many times against one washed few. Detergents replenish the brightener at every wash, so a laundered garment is in a steady state between deposition and destruction rather than on a decay curve. That is why the effect is invisible in laundry and visible in paper: one of them has a source term.

The last case is the useful contrast. The decay is only a decay when nothing is putting the additive back, and the two industries that use brighteners most heavily differ in exactly that respect.

What the decay costs is best read against the sheet a proof is actually made on, since that is the comparison a print buyer sees.

The two clocks that run together #

There is a second decay in the same sheet running the other way, and the pair is what makes an aged sheet look the way it does.

Cellulose yellows. Residual lignin oxidises, hemicelluloses degrade, and the products absorb in the blue — which is the same band the brightener emits in and the same direction the base was already leaning. That process is driven by light, heat and acidity rather than by ultraviolet alone, so it runs in a dark warehouse as well as in a window.

The brightener decays only where there is excitation. So the two clocks are driven by different things, and a sheet’s history decides which dominates: a document in a drawer yellows and keeps its brightener, and a poster in a window loses its brightener and yellows.

The two are indistinguishable in a single measurement and separable in two. Under an ultraviolet-excluded condition the brightener contributes nothing, so what is measured is the base and its yellowing alone; under an included one both terms appear. Two measurements a year apart under both conditions decompose the change exactly.

That is a small piece of practical metrology and it is the useful consequence of everything in this essay: the measurement condition that is nearly useless for describing how a sheet looks is the one that isolates how the sheet is ageing, and the two questions want different instruments.

What was computed, and how #

The loading decays as a half-life in accumulated dose, with the sheet rebuilt at each dose and re-measured under the ultraviolet-included condition. The dose axis is in arbitrary units whose half-life is stated; converting it to hours of sunlight would need a photodegradation quantum yield and a spectral overlap integral, and neither is a number this collection has.

What is not arbitrary is the shape, and the shape follows from two things the model does contain: an exponential decay in the loading and a saturating absorption. Both are stated, and the convexity of the whiteness curve is a consequence rather than an input.

Whiteness is the CIE formula under the measurement condition’s own white, so the curve is directly comparable with every other whiteness figure here. The colour difference from the starting sheet is CIEDE2000, and the dose at which it first exceeds one is reported rather than interpolated.

The assertion is a floor on the total loss, requiring the sheet to give up more than ten points of whiteness over the modelled range. It is written that way rather than on the shape because a version of the model with a badly chosen half-life would still produce the right shape and the wrong amount of decay, and the amount is what makes the phenomenon worth an essay.

Where the model stops #

The half-life is chosen, not measured. Real photodegradation rates for stilbene brighteners depend on the substrate, the coating, the oxygen availability and the spectral distribution of the exposure, and vary by more than an order of magnitude. The number here sets the horizontal scale of every figure and nothing else.

The reflected component is held fixed. Real paper yellows as well, through separate reactions in the cellulose and any residual lignin, and that yellowing runs the same direction as the brightener loss. The model separates the two so that the fluorescent term can be seen on its own; a real sheet’s decline is both.

And the destruction is treated as first-order in the excitation. At high intensities the triplet population saturates and the rate becomes sub-linear, which is exactly the regime an accelerated weathering test runs in — so an accelerated test overstates the lifetime by a mechanism this model does not have.

Who found it, and when #

Photodegradation of fluorescent whitening agents has been studied since the additives were commercialised, and the singlet-oxygen mechanism was established through the 1970s. It is the reason weathering standards for brightened materials exist and the reason accelerated tests use xenon arcs — which supply the ultraviolet.

The consequence for measurement is documented in the paper industry and rarely propagates to a data sheet. What is standard practice is to measure a sample soon after manufacture; what is not standard is to say when, or to state the reference condition in a way that would let a later measurement be compared.

The parallel with the observer’s own chemistry is worth noting rather than pressing. A photopigment bleaches and recovers because a regeneration cycle puts it back; a fluorophore has no such cycle in a sheet of paper, and does in a wash cycle. The presence or absence of a source term is what decides whether a photochemical process is a clock or a decline, and it is a property of the system rather than of the molecule.

The generalisation #

The pattern is a property measured once and quoted as though it were constant, in a system where the measurement’s own mechanism consumes it.

The awkward part is not that the sheet changes; everything changes. It is that the mechanism producing the number is the same mechanism destroying it. A brightener that is glowing is being used up at a rate proportional to how much it is glowing, so the property is being consumed in proportion to how much of it is being displayed.

That shape recurs: a catalyst that is poisoned by its own product, a battery whose capacity test discharges it, a reference sample that fades under the lamp used to measure it. The diagnostic is to ask whether the act of exhibiting the property draws on a finite stock, and if it does, whether anything replenishes it. A specification for such a property is not a number; it is a number and a rate, and quoting the first alone is a statement that the second is zero.

Where the ladder goes next #

If the clock is driven by the excitation, then everything that changes the excitation changes the clock — so the window a sheet is behind sets both how bright it looks and how long it stays that way, in opposite directions.

The other direction is the observer rather than the sheet. A sheet losing whiteness slowly is a stimulus changing under an adaptation state that changes faster, so the loss is nearly invisible in the moment and obvious in a comparison — and what a brightened white looks like at all is a question the colorimetry in this essay does not answer.