An update of the equation for predicting the dissolution rate of glass fibers from their chemical compositions

Sir,

In 2000 we published an article which describes a method to predict from a glass fiber’s chemical composition its dissolution rate in-vitro in a physiological solution that mimics the near-neutral environment in the lung (Eastes et al., 2000). The prediction is in the form of Equation (1)(1) , for which the coefficients, P_i, are determined by applying standard linear regression techniques to a set of measured dissolution rates for fibers of known chemical composition. (1) where:

k_dis is the dissolution rate constant in ng/cm²/h

P_i is the coefficient for oxide i

W_i is the mass percent of oxide i in the glass

Σ_iW_i is for all oxides for which a P_i has been determined

The data used to derive the coefficients published in the 2000 article are a set of 62 compositions with dissolution rates measured in our laboratory and published in the open literature.

In 2000 we had additional unpublished dissolution rate data, and in the years since we have generated considerably more using the mass loss and solution analysis techniques (Potter & Mattson, 1991; Mattson, 1994) and the optical technique (Potter, 2000). This letter presents an updated set of coefficients based on all the dissolution rate data we have generated in our laboratory in our near-neutral mimic solution. Use of the larger data set improves the accuracy of the coefficients in the original article and, perhaps more importantly, allows us to determine coefficients for oxides we could not include previously.

Our database contains 1547 individual determinations of fiber dissolution rate in a pH 7.4 mimic solution (solution composition in Mattson, 1994). Prior to deriving coefficients from these data, we reviewed the database as follows:

1. Express all rates as k_dis in units of ng/cm²/h: Expressing the dissolution rate in this way for incongruently-dissolving fibers measured by the optical or the SEM method, requires that we know or approximate the weight fraction of the material removed from the leached layer (X_l). We have assumed throughout that all silica, alumina, and titania in the unaltered fiber remain in the leached layer and that all other components leach completely. This certainly underestimates X_l, but it provides a reasonable approximation for k_dis. This approximation was necessary for 68 of the 1547 k_dis values. Incongruent dissolution is more typical of rapidly dissolving fibers; only 13 of the 68 have k_dis values less than 10,000.

2. Eliminate data with identifiable experimental errors: We critically reviewed each measurement to see if there is some reason, regardless of the k_dis value, that it should be excluded from further analysis such as, for example, a flow rate that is too low in the mass loss or solution analysis methods.

3. Eliminate questionable data: We reviewed all measurements for each fiber or set of fibers with essentially the same composition to identify any outlying measurements. We eliminated such outlying measurements even if no reason was found to exclude them.

4. Reduce the database to one measurement per fiber sample: In some cases we had made many measurements on one fiber sample. Including all these individually in the regression process would weight this composition too heavily even though its k_dis value is presumably more accurate due to the many measurements. To assure reasonably equal weighting for all fiber compositions, we reduced the final database to one measurement per sample by averaging all measurements on a given sample. We did not combine measurements on different samples of essentially the same composition into one.

This review resulted in a database of 391 compositions with an average measured k_dis for each. The compositional space covered is extremely broad since the samples include a great variety of fiber types – glass, mineral, and slag wools; refractory ceramic fibers; and a wide range of experimental glass compositions. There are three problems with applying liner regression techniques uncritically to the entire database: (1) For such a large compositional range the assumption of linear dependence of dissolution rate on composition, on which the regression process is based, is invalid, and the fit is poor. (2) Some oxide components vary in the database over too small a range to allow accurate definition of their effect on k_dis. (3) Even though some oxide components vary over a large range, there may be too few data to control the fit and define a reasonable dependence.

It is therefore necessary to determine for what fiber components there are enough data to define a dependence and over what range this dependence is sufficiently linear to give the most useful predictive ability. The process by which we accomplished this is a somewhat subjective repetition of the regression analysis on a continually edited database: (1) looking for outlying points where the fit is poor and eliminating the particular compositions from the database, (2) identifying oxides where the compositional dependence is poorly defined due to limited oxide variation and eliminating compositions which contain significant concentrations of those particular oxides, (3) looking for overall trends in the quality of fit, which may indicate basic non-linearity in the compositional region and the need for a contracted overall compositional range. This reduced the database from 391 compositions to a final set of 237 compositions shown graphically in Figure 1. This final data set is included in the Supplementary material to this letter. The coefficients fit from these data by linear regression are in Table 1. Figure 2 compares the value of k_dis predicted by the new coefficients with the measured value.

Figure 1. Compositional ranges of the data from which the new coefficients in Table 1 were derived. Each circle shows the weight percent of one measured oxide in one fiber composition in the final database.

Figure 2. The dissolution rate constant, k_dis, calculated by Equation (1)(1) using the new coefficients compared to the measured value. The line indicates an ideal fit.

Table 1. Coefficients for the calculation of the fiber dissolution rate constant, k_dis, in near-neutral solution.

	Eastes et al. 2000			New
Oxide	Coefficient	Standard error	Range	Coefficient	Standard error	Range
SiO2	−0.0120	0.0028	50–70	−0.0231	0.0015	45–70
Al2O3	−0.2141	0.0110	0–7.5	−0.2517	0.0078	0–10
CaO	0.1081	0.0112	0–15	0.1312	0.0065	0–20
MgO	0.1376	0.0126	0–15	0.1181	0.0120	0–8
Na2O	0.0939	0.0087	0–20	0.1356	0.0042	5–23
K2O	–	–	–	0.1526	0.0137	0–15
B2O3	0.1467	0.0091	0–15	0.1711	0.0030	0–24
BaO	0.0692	0.0310	0–3	0.0894	0.0077	0–24
P2O5	–	–	–	0.2030	0.0314	0–4
ZnO	–	–	–	0.0995	0.0149	0–9
ZrO2	–	–	–	−0.1365	0.0447	0–4
F2	0.1187	0.0613	0–1.5	–	–	–

The earlier set of coefficients are included for comparison. The ranges are defined as the compositional ranges of the data used in regression analyses.

Much of the reduction in the database to the final 237 compositions is elimination of the mineral wools due to their low soda, high alumina, and/or high iron oxide content. Inclusion of these compositions yielded a poor fit due to non-linearity of the dissolution rate dependence. Thus, while the coefficients developed here are applicable to a wide compositional range, which includes most areas of interest for glass wool manufacture, they are not necessarily applicable to other synthetic vitreous fiber types such as mineral wools, refractory ceramic fibers, and some continuous filament compositions.

Disclosure statement

No potential conflict of interest was reported by the authors.