Finding numbers for sparkle and shine

Three-dimensional measurements characterise appearance of effect coatings

Reinhard Feld*

Coatings designers, formulators and applicators have struggled for years to find a way to reliably measure the colour and appearance of special effect coatings. Traditional in-plane spectrophotometric measurements are inadequate to characterise modern automotive colours and the like.

A method has now been developed to precisely measure the colour and appearance qualities that have confounded optical instruments for more than 60 years. Using advanced instruments and software to interpret the data, companies now have the tools required to reliably institute quality control for products ranging from automotive body panels through to large and small appliances coated with special effect paints.
Attempts have been made to relate measurements from hand-held spectrophotometers that collect colorimetric data from several in-plane angles to events occurring in production processes. However, these instruments by their nature could not collect the data points needed to yield reliable results.
Others have tried to skirt around the problem by creating a coarseness or "sparkle" index that relies on photographic images taken in-plane, but this method is easily confused and is inadequate for rigorous analysis.
Existing laboratory-based instruments can accurately characterise special effect paints, but they are bulky, take hours to process a single sample and operate only in a sheltered environment. To merge the best of both worlds, X-Rite, the world’s largest designer and manufacturer of colour measurement instruments and software, applied established principles of quantum electrodynamics, optics and statistics to simplify the way that laboratory spectrophotometers perform their work.

Advanced mathematical principles were used to create a three-dimensional model for the behaviour of any special effect paint. This can be used as a distinguishing ‘fingerprint’ for designers, paint manufactures and their end-users. Production personnel can now immediately identify and troubleshoot defects that are not detected by other methods.
The mathematical model can be applied to virtually any product that uses pigmented and flaked ingredients for its colour and appearance. This includes automotive paints, metallic printing inks, nacreous pigments in plastics, textured and patterned fabrics, prints on glossy paper and even cosmetics.
The new method detects and quantifies what is essentially a three-dimensional spectral curve by adding more sensors and illuminators to a spectrophotometer to gather information out of the plane of illumination relative to the test surface. Using this technique, it was found possible to unravel in two days a problem involving matching parts coated with effect paints that troubled a major automotive company for more than two months.
The term xDNA has been coined to describe the company’s three-axis technology that is based on an advanced spectrophotometer and software package to interpret data. Two illuminators and 11 sensors are used that measure 31 bands of the visible spectrum, from blue representing the shortest waves in the 400 nanometre range to red representing the longest at the 700 nanometre range.The illumination sources are gas-filled tungsten lamps colour corrected to approximately 4000 °K that flash intense white light at 15 ° and 45 ° angles to the normal of the test surface. Light emanating from the test surface is collected at ten angles deviating from the spectral angles at -15 °, 15 °, 25 °, 45 °, 75 °, 110 ° in the plane of illumination and 25 °az90, 25 °az-90, 60 °az125.3, and 60 °az-125.3, where the az notation refers to the azimuthal rotation from the aspectral reference (see Figure 1).

The foundation of the new method is the BiDirectional Reflectance Distribution Function (BRDF), a function first proposed at the College of Optical Sciences at the University of Arizona in 1977. This is used today in applications as wide-ranging as analysing climate change on Earth, the fabrication of semiconductors and computer chips, through to the computer-generated graphics seen in movies.
The BRDF can be used to learn much about the material nature of an object by directing light of known characteristics onto the test surface, then measuring and analysing the returning light. According to the BRDF concept, light returned from the object must by definition carry encoded in it the transformation that it underwent inside the object. Because of the law of conservation of energy, the energy of the illuminating light must equal the total energy of light reflected, refracted, absorbed and scattered.One aspect of BRDF rests on the fact that all materials are dispersive, meaning that the response of any coating or material will change as a function of the wavelength of light that illuminates it. For instance, the tendency of any material to bend light (its refractive index) is different for blue light and for red light. This change in bending power exists independently of the apparent colour of the material.All materials also have unique dielectric constants, which can be thought of as a way to measure their dispersive properties. Reliable information about the composition of an object can be obtained by using the equation that the bending power and absorption of light by a material (its "complex refractive index") is proportional to the square root of the dielectric constant.
Another aspect of BRDF rests on the fact that blue light is scattered differently from red light in the same object. Smaller particles scatter light of different wavelengths differently from larger particles. However, all materials scatter light to some degree. Again, this tendency to scatter light is independent of the apparent colour of the material.

The use of the BRDF to accurately and reliably measure special effect coatings is not new. The function is the basis on which large laboratory-bound instruments define and characterise such coatings. But to dramatically reduce both the size of the instrument and the time required for measurement, the mathematical dimensionality had to be reduced.Laboratory-bound instruments take many measurements over a long time frame and generate an unwieldy amount of data for practical use on the factory floor. The mathematical dimensionality of data from an instrument that generates 19 spectral curves with 31 data points is considerable. Simply reporting perceptual colorimetric values would not help customers.
To reduce the dimensionality of the problem but preserve the integrity of the results, the instrument designers employed the theories of Professor Richard Feynman, a Nobel Prize-winning physicist. Feynman proposed that many problems in optics and light can be solved by always treating light as a particle and describing it as a vector with magnitude and direction [1].

By applying statistical sampling to data derived from the BRDF theory and using Feynman’s analysis of light, a hand-held instrument was developed that could determine in two seconds a reading that yielded highly reliable and repeatable characterisations of special effect coatings.Feynman’s theory also states there is a statistical probability for each direction that a single photon at a particular energy level can take as it strikes a test surface. As an observer continues to add photons by increasing the intensity of light, eventually the observer will begin to get data counts from sensors that are strategically situated around the object.These data counts essentially represent the statistical probability of how photons at specific energy levels (wavelengths) will reflect, absorb and scatter as they pass into, through and out of the material back to the sensors.The photons can travel in only a specific number of vector directions to reach the strategic locations of the instrument’s pickups, while the magnitude of those vectors is represented by the number of photons collected at each pickup. Taken as a whole, the vectors represent the response function for the coating being analysed.
By examining a set of xDNA measurements (Table 1), it can be seen how the data is arranged to develop xDNA coordinates. In the rows on the left hand side of the table are 31 different wavelengths of light in ten nanometre increments. Across a particular row representing a wavelength, one can see the intensity of light at ten particular angles to the normal. These intensity and angle measurements can be converted to vectors.

Though converting the spectral curves into vectors helps matters, it still does not reduce the mathematical dimensionality of the problem for easy analysis by end users. The table still comprises a total of 589 reflectance values {31 wavelengths X [(10 sensors x 2 illuminators) 1 reading point]} which itself can be unwieldy for anyone trying to describe a particular coating or analyse a process problem.To reduce the mathematical dimensionality of the problem, a vector addition method is used that maintains the integrity of the two key pieces of information: coating response by wavelength, and degree of change of coating response when the wavelength is changed.
Rather like building a single, long, very crooked stick by gluing 19 toothpicks end-to-end, the computer software places the end of one vector on the tip of another vector until the combined 19 vectors point to a single data point in three-dimensional space for each wavelength. In Table 1, the last three columns on the far right hand side represent the x, y and z coordinates of the final data point at that particular wavelength.
If the computer software were to draw a line from the origin of the first vector to the tip of the last vector after they are added together, that final vector represents the direction with highest probability for a single photon to travel after it interacts with the coating or material. Sample plots of the x, y co-ordinates across the spectrum for two different coatings are shown in Figure 2.

The theory in physics and material science which confirms that this vector addition will provide meaningful results is termed Effective Medium Theory. In carrying out the vector summation, a coating that contains a number of ingredients has been treated as if it were a bulk material. This theory states that it is permissible to treat a complex combination of ingredients as a simple (but new and unknown) material so long as its constituents are uniform in concentration.
The above theory also forms the basis for detecting a change in formula, or its structural characteristics. If any one ingredient or concentration of ingredient is changed substantially, the coating will behave as a different new material.
If it is assumed that manufacturers largely understand the characteristics of the coatings and the process they use to apply those coatings, then it is reasonable to transform xDNA curves mathematically in a linear fashion over limited distances. As part of its analysis, the computer software draws lines between the endpoints of the 31 final vectors creating an xDNA curve in three-dimensional space that is entirely unique to that coating.

The xDNA system addresses two aspects of the coating: the perceptual domain and the production process and formulation domain. In the perceptual domain, companies trying to control appearance need to set bounds and limits of perceptual differences. This is commonly done through the use of colorimetry, where limits are set that relate to human perception through the use of tristimulus spaces such as CIELAB colour space and colour difference formulas such as CIE DE 2000. But xDNA can also perform this function admirably.
In essence, the xDNA curve is a spectral curve in three dimensions because it still has all its original 31 wavelength points obtained from testing the coating. What is notable about this domain is that all the same colour calculations that would be applied conventionally can be utilised through the xDNA curve.
For instance, spectral curves may be constructed for each angle of an earlier generation five-angle instrument, followed by calculation of CIELAB values and setting of DE tolerances. However, using the five angles to hold tolerances for colour and appearance would be very difficult.
The xDNA curve can be used to compute CIELAB values and set a single DE tolerance that represents a probability distribution of perception for all inspection angles. The colour difference metric based on DE and computed from a three-dimensional xDNA curve has been named DF.
Because xDNA is a robust method for tracking perceptual differences, it eliminates the need for a coarseness index or sparkle metric that is essentially the outgrowth of an image processing routine with no clear perceptual equivalent.

With regard to the process and formulation domains, the xDNA curves drawn from the endpoints of the vector summations represent the statistical probability distribution of the scatter, and scatter will only change if the structure of the coating or its ingredients changes. Dimensional tolerances can therefore be applied directly on the xDNA curve in this three-dimensional space to allow process and formulation to be controlled.
As the structure of a coating changes, so does its scatter. As the ingredients in a coating change, the response across the wavelengths changes. So the end result is that changes to process relate to mathematical transformations (translations and rotations) of an xDNA curve, while changes in the composition will cause changes in the overall shape of the xDNA curve across the spectrum.
However, it is not possible to completely separate the effects of ingredients and process on the final appearance of a coating. This is understandable in that the appearance of a coating derives from the distribution of the ingredients and the tolerances applied to them, such as size tolerances of metallic flakes.
For instance, a manufacturer may permit a tolerance of plus or minus 10 µm for a batch of metallic flakes of nominal 20 µm diameter. The question then becomes, how close the centre of the standard normal distribution curve of flakes can approach 25 µm diameter before the manufacturer is essentially receiving a batch of 25 µm diameter flakes.Thus, while there are some grey areas in which a process can be altered so dramatically that it may be termed a change in formulation, xDNA has already proven itself to be an effective tool to determine in real-world manufacturing situations whether problems are process or formula related. In addition, xDNA offers a reliable and repeatable standard for all companies in a supply chain to use for special effect paints and may provide guidance on how far a process can be adjusted to create the desired colour and appearance.

[1] R P. Feynman, QED: the Strange Theory of Light and Matter, Princeton Science Library, 1971