Manufacturers that design, formulate and apply coatings have struggled for years to find a rigorous and easy-to-use method that reliably measures the color and appearance of special-effect paints. Companies tried to relate measurements taken with handheld spectrophotometers that collect colorimetric data from several in-plane angles to events that occurred in production processes, but these instruments, by their nature, could not collect the essential data points to yield reliable results. Other companies have tried to skirt the problem by creating a coarseness index or “sparkle” metric that relies on photographic images taken in-plane, but this method is easily confused and insufficient for rigorous analysis.

Existing laboratory-based instruments can accurately characterize special-effect paints, but the devices are bulky, take hours to process a single sample, and operate only in a sheltered environment. To meld the best of both worlds, X-Rite Inc., applied established principles of quantum electro dynamics, optics, and statistics to essentially simplify the way that laboratory spectrophotometers perform their work.

#### A New Approach

Engineers employed some fundamental scientific principles to create a three-dimensional mathematical model for any special-effect paint that can be used as a distinguishing fingerprint for designers, paint manufacturers, and their end-users. Production personnel now can immediately identify and troubleshoot defects that are not detected using other methods. The mathematical model can be applied to virtually any product that uses pigmented and flaked ingredients for its color and appearance: automotive paint, 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 3 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, X-Rite was able to unravel in two days a problem involving matching parts coated with effect paints that troubled a major automaker for more than two months. X-Rite coined the term xDNA — similar to the way every person has a unique DNA structure — to describe its three- axis technology that includes an advanced spectrophotometer and software package that interprets data.

X-Rite’s new spectrophotometer uses two illuminators and 11 sensors that measure 31 bands of the visible spectrum, from blue representing the shortest waves in the 400 nanometer range to red representing the longest waves at the 700 nanometer range. The illumination sources are gas-filled tungsten lamps color corrected to approximately 4000°K that flash intense white light at 15° and 45° angles to nominal of the test surface. Light emanating from the test surface is collected at 10 angles from spectral angles at -15°, 15°, 25°, 45°, 75°, 110°, 25°az90, 25°az-90, 60°az125.3, and 60°az-125.3. The “az” notation refers to the azimuthal rotation from aspectral reference.

### BiDirectional Reflectance Distribution Function

The foundation of the new method rests on the BiDirectional Reflectance Distribution Function (BRDF), a function first proposed at the College of Optical Sciences at the University of Arizona in 1977 that is used today in applications as wide-ranging as analyzing climate change on Earth, the fabrication of semiconductors and computer chips, and the computer-generated graphics seen in movies. Observers can use the BRDF to learn much about the material nature of an object by directing light of known characteristics onto the test surface, then measuring and analyzing the returning light. Under the BRDF, light coming back from the object must, by definition, have 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 energy of light reflected, refracted, absorbed and scattered.

One leg 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 wavelength of light that illuminates it. For instance, any material’s tendency to bend light (its refractive index) is different for blue light than it is for red light. This change in bending power exists independently of the apparent color of the material. All materials also have unique dielectric constants, which can be thought of as a way to measure their tendency to be dispersive. We obtain reliable information about the composition of an object by using the equation that the bending power and absorption of light by a material (complex refractive index) is proportional to the square root of the dielectric constant.

Another leg of BRDF rests on the way in which light scatters as it strikes an object. Blue light scatters differently than red light in the same object. Smaller particles of material scatter light of different wavelengths differently than larger particles. Regardless, all materials scatter light to some degree. Again, this tendency to scatter light is independent of the apparent color of the material.

The use of BRDF to accurately and reliably measure special-effect coatings is not new. The function is the cornerstone of how large laboratory-bound instruments define and characterize such coatings. But to dramatically reduce both the size of the instrument and time required for measurements, engineers and designers decided to reduce the mathematical dimensionality of the problem.

Laboratory-bound instruments essentially take many measurements over a long time frame and generate an unwieldy amount of data for practical use on the factory floor. But by applying statistical sampling to data derived from the BRDF theory and analysis of light developed by Prof. Richard Feynman, the engineers were able to develop a handheld instrument that could determine a reading in two seconds that yielded highly reliable and repeatable characterizations of special-effect coatings.

#### Feynman’s Theory

The mathematical dimensionality of an instrument that generates 19 spectral curves with 31 data points is considerable. To reduce dimensionality of the problem, the instrument designers employed the theories of Feynman, a Nobel Prize winning physicist known for his work in quantum mechanics and particle physics. Feynman proposed that many problems concerning optics and light can be solved by always treating light as a particle and describing it as a vector with magnitude and direction. Each detector would then represent the statistical probability of a photon reaching it.

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.

In short, X-Rite designers have created only a specific number of vectors by direction that the photons can travel though the strategic location of the instrument’s pickups, and the magnitude of those vectors is simply represented by the number of photons collected at each pickup. Taken as a whole, the vectors represent the response function for the coating being analyzed.

So for a given material or coating, the new handheld instrument now yields two extremely valuable pieces of information: 19 vectors at any given wavelength which represents the coating response at that wavelength (its structural characteristics), and the changes across the 19 vectors as the wavelengths are changed (its perceptual characteristics). Software engineers use a vector addition method that maintains the integrity of these two key pieces of information.

There is a theory in physics and materials science called Effective Medium Theory that allowed X-Rite to perform this vector addition and obtain meaningful results. In doing this vector summation, engineers have treated a coating that contains a number of ingredients as if it were a bulk material. Effective Medium Theory states that it is permissible to treat a complex combination of ingredients as a new and unknown material so long as its constituents are uniform in concentration. 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.

While there are some gray areas where 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 for root cause analysis in real world manufacturing situations as to whether problems are process and 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 serve as guidance on how far manufacturing engineers can adjust a process to evoke desired color and appearance of products.

*This article was written by Jon Nisper, Director of Research and Development, X-Rite (Grand Rapids, MI). For more information, contact Mr. Nisper at This email address is being protected from spambots. You need JavaScript enabled to view it., or visit http://info.hotims.com/22920-201.*