Once the primaries are accurately measured and stored in the database, the data must be evaluated for consistency using specialized numeric and graphic calculations.

The Mathematics of Computer Color Matching

For a computer formulation program to work effectively, there must be a mathematical relationship between the concentrations of each dye used to produce a sample and the sample’s color. It is not the purpose of this paper to discuss in complete detail the mathematics of color matching, but a brief description will follow. More detailed information may be obtained from a large assortment of textbooks and articles.

The mathematical basis for all color matching software is the Kubelka-Munk series of equations. These equations state that for opaque samples such as textile materials, the ratio of total light absorbed and scattered by a mixture of dyes is equal to the sum of the ratios of light absorbed and scattered by the dyes measured separately. Where absorption is defined as "K" and scattering is defined as "S", Kubelka-Munk states that _:

K/S is not a readily measurable quantity, but it can be calculated from the reflectance of a sample -- "R" -- by the Kubelka-Munk equation that states _:

As an example, if a sample has a reflectance of 20% at a wavelength of 500nm, then the K/S can be calculated as:

If the K/S of a target color is measured at several wavelengths, the concentrations of each dye can be calculated by trial and error from primary dyeings to achieve the closest match. A computer color matching program is capable of performing hundreds of iterations in a short period of time to produce the initial dye concentrations.

Graphical Analysis of Primaries Using K/S

K/S calculations are invaluable in evaluating the build characteristics of dyestuffs because there is a direct relationship between K/S and dye concentration. In Figure 1, the K/S values of several samples are plotted versus the sample concentrations.

The uniformity of this curve will determine the software’s ability to accurately generate new formulas, especially in the unknown areas between the known concentrations. Two curves are displayed, one based on K/S for each dye at the wavelength of maximum light absorbence and one based on K/S integrated over the range of 400-700nm. The shapes of these curves may differ due to hue changes as dye concentration increases.

The quality of the dye primaries is indicated by the smoothness of the K/S curves. Most dyes will exhibit linear behavior in the lower concentrations and will begin to flatten as the dye saturation point is approached. The point at which the K/S curve stops increasing corresponds to the maximum achievable depth of shade for a dye, approximately 5% for the dye pictured, and the concentration used should not exceed this amount in shade matching. It is often the case that, due to physical properties such as light fastness and wash fastness, the practical use range of a dye is well below its saturation concentration.

A graph of log K/S versus concentration (Figure 2) is useful for evaluating the linearity of lighter concentrations.

Any samples that appear to deviate from the linearity of the majority of data points should be remeasured for confirmation and redyed if proven to be in error. It is acceptable to delete as many as two points if eight or more concentrations have been dyed as long as they are in the linear portion of the graph and are not one of the two strongest concentrations. If three or more points appear questionable, it is preferable to redye the entire set of primaries for best results from the formulation software. Small variations as indicated in Figure 3 can be corrected using mathematical smoothing techniques to produce the modified curve in Figure 4, but only if the changes are less than 10%.

At first glance, the K/S curve for the black dye in Figure 5 appears to be acceptable. If the formulation program was asked to generate a concentration of this dye corresponding to a sample K/S value of 17, it would predict approximately 2.4%. Notice, however, the large gap in the concentration levels between 1% and 4%. If an additional point was dyed at 2.5% and added to the database, the computer’s predicted concentration would change to approximately 1.6%.

As a rule, it is best not to leave gaps of more than 1.0% between primary dyeings. This will be helpful in preventing the highly variable concentrations predicted in the previous example.

Graphical Analysis of Primaries Using Reflectance

Graphs of percent light reflectance (%R) versus wavelength for a set of primaries are useful in detecting the presence of contamination, especially in the lighter samples. Figure 7 is a graph of %R versus wavelength for a set of violet primaries:

The reflectance curves for the set of primary dyeings should all have the same basic shape with no lines that cross. There will be occasions, however, when some line crossing occurs at higher reflectance values, especially above 600nm for yellow to red dyes. As long as the crossing is not significant, there is no need to remove or redye the sample. If the reflectance curve exhibits significant crossing as in Figure 8 for the 0.15% level of the red dye, then the sample has become contaminated during the dyeing process or by some other means. This point should definitely be redyed or, if the remaining points have a linear K/S versus concentration build, simply deleted.

For most dyes, the percent reflectances for the darker concentrations are so low relative to that of the light concentrations that it is difficult to determine from the %R graph if there are any problems with the darker dyeings. To make evaluation easier, a plot of K/S versus wavelength is used. In Figure 9, K/S versus wavelength is plotted for the violet dye seen previously in Figure 7.

Reexamining Figure 7 does indicate that for the darker primaries, the reflectance curves seem to merge between 525nm and 600nm. It can be seen in the K/S versus wavelength graph that the reflectance curves have been reversed, with the darker concentrations at the top of the graph while the lighter concentrations are at the bottom. It can be concluded from the graph that the darker primaries are also consistent.

A point to remember when evaluating reflectance versus wavelength graphs is that some dyes do exhibit a change in hue as the concentration increases. For these dyes, the reflectance curves of all concentrations will not be identical in shape, but this does not mean that a particular sample is necessarily contaminated. For these dyes, absence of crossing is the critical characteristic.

Confirmation of Primary Accuracy

K/S and reflectance graphs are essential tools in determining the quality of the primary data. After any questionable points have been deleted or replaced, the accuracy of the database should be confirmed by generating match predictions for a series of dyed "knowns".

A "known" is simply a piece of material that has been dyed using the same techniques and substrate that were used to dye the primaries. When asked to predict a formula for a known sample, the computer formulation software should produce a formula reasonably close to the actual concentrations that were used to produce the sample. Small variations are common and are typically due to dye interaction that cannot be accounted for when the primaries are dyed separately.

If the computer prediction is significantly different from the actual formula, then there may be a problem with one or more sets of primary data. A secondary technique for confirmation of primary quality is to dye a known that contains only one dye. This will eliminate dye interaction as a possible source of the error. If the software cannot accurately predict the concentration of the single dye, then a problem does exist with the set of primary data and the set should be redyed. This test is also useful when changing to a new lot of dyestuff to determine the degree of error that will be introduced when the new lot is used in the lab or in production.

In most cases, variations between predicted and actual formulas are consistent within shade families and among particular dye combinations. To take advantage of this fact, some computer formulation systems have the ability to "learn" about dye interaction and to compensate for it by modifying the initial formula prediction. This means that the first lab dyeing based on a modified prediction will be much closer to the target color than a formula based only on primary data. This process works well if the materials being dyed are consistent for an extended period of time. Application of this technology can significantly reduce the number of lab dyeings required to match new shades.



Primaries Provided by Dye Manufacturers

Many companies choose to use primaries provided by dye manufacturers. This data is usually very consistent and offers a way for a lab to immediately begin shade matching or to evaluate a new dye or family of dyes. The dye manufacturer’s primaries are usually available on a computer diskette in a form suitable for a large number of color matching programs.

Although these primaries are usually very consistent when evaluated using the techniques discussed earlier, this does not guarantee that the formulas predicted will be accurate. This is caused by the use of different materials, different chemicals, different equipment and dyeing procedures, and even by different water supply. This does not mean that the supplier primaries cannot be useful, it simply means that the initial computer predictions will not be as accurate as ones based on primaries dyed in the lab’s own environment. One solution to this problem, especially if the primaries are not required immediately, is to provide the supplier with fabric, chemicals, dyeing procedures, and water to use to produce the dyeings. The only remaining variable will be the equipment.

A computer color matching system is not a "black box" that magically produces the exact formula for a sample on demand. The accurate preparation of dye primaries and the careful evaluation of the stored data are essential if a color matching system is to be able to predict formulas with any consistency. As this is accomplished, significant reductions in the number of dyeings required to match new shades are made possible.

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