Mark G. Vangel

*Statistical Engineering Division, ITL*

Anthony J. Bur

*Polymers Division, MSEL*

The objective of this project is to develop
methods for indirectly measuring the temperature
and pressure of a polymer during fabrication. A
dye which fluoresces when
excited by a laser, is added to the polymer.
Empirically, it has been
shown previously that the ratio of the intensities of a
fluorescence spectrum at two wavelengths can,
for a practically useful range of temperatures,
materials, and dyes be nearly linearly related to the
temperature within the polymer. Thus, linear
calibration curves for temperature can often
be determined. By measuring the intensity at two
wavelengths, a user can then use these curves to
estimate temperature - at atmospheric pressure.

However, polymers are typically under pressure
during fabrication, so we have begun this year to
attempt to measure temperature and pressure
simultaneously from fluorescence spectra.
Experiments were designed to obtain spectra for a
rectangular grid of temperatures and pressures,
with 10 levels of pressure and 5 to 9
levels of temperature. Data were obtained for
several combinations of polymers and dyes.

Perhaps the most promising result obtained thus far is
based on a multivariate additive ANOVA. For each (of typically
hundreds) of values of wavelength ,
the responses
are a two-way table in temperature and pressure,
to which a simple additive ANOVA model can be fit.
This model fits extremely well for the polymer/dye
combinations for which data are available. The
temperature effect is more complicated, but the
pressure effect appears to be very nearly equal
to the product of a function of wavelength with
a function of pressure. Using an obvious notation,
we have

The functions in the above expression are easily
estimated pointwise by least squares and smoothed
afterwards, if necessary. Conveniently, *u*_{2}(*P*) is
nearly linear over the pressure range in the experiments.
Once the above functions have been estimated, one
can develop a model for estimating temperature, independent
of pressure, using an orthogonal trigonometric basis of
modest dimension (say, 10 to 30). Each basis function
is constructed to also be orthogonal to
.
By multiplying each basis function by each spectrum and
integrating over ,
a matrix results,
with as many rows as spectra and as many columns as
basis functions. Multiple regression, with this matrix
as a design matrix and *T* as response, is then used
to determine a linear combination of basis functions which
predicts temperature well, for all pressures. Excellent
fits to temperature have been obtained in this way. To
better assess predictive capability, cross-validation
will be used in future work.

Figure 7: The figure illustrates (for an example with
,
,
and
)
estimates of
the 10 (unsmoothed) pressure effects from both a
multivariate additive ANOVA with independent
errors (points) and from a least-squares fit of
the above model, which assumes a multiplicative
pressure effect. The bold curve, proportional to
the estimate of
,
is approximately
rotated about the horizontal axis as pressure
is increased.