Biermann Ch. Handbook of Pulping and Papermaking

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494 24. OPTICAL PROPERTIES OF PAPER

two objects have the same tristimulus values

(described in the following section) under one set

of lighting conditions, but different tristimulus

values under a second set of lighting conditions.]

It is impossible to know how another person

perceives color, but there are some known phe-

nomena such as yellowing of

the

lens with age and

certain types of color blindness which affect the

way a person perceives color. Furthermore, one's

experiences alter one's perception of color.

Fig. 24-7 shows that dark brown paper

(unbleached kraft paper) has a low absorbance in

the blue range and a fairly high absorbance in the

yellow range. (Remember, however, that TAPPI

brightness is measured in the blue region at 457

nm.) This is typical of brown materials. Gray

objects, however, would have a fairly uniform

absorbance across the spectrum. Dark gray has a

uniformly high absorbance (low reflectance or

transmittance), and light gray has a low absor-

bance (high reflectance or transmittance). Yellow-

ish papers, such as newsprint, fall between gray

and brown in their spectral curve.

Fig. 24-8 shows the reflectance curves (ob-

tained with a reverse spectrophotometer) of vari-

ous dyed papers (top). [The absorbance curve

(see page 342 for a mathematical development) is

included for interest because the eye has a loga-

rithmic response that is more like the absorbance

curve.] One observes that red construction paper

reflects a high percentage of the red light (80%),

but very little of the blue light (about 10%). The

opposite is true of blue paper. Because the shapes

of these curves are much different, they are said

to have different hues, or shades. Red, orange,

yellow, green, yellowish green, blue, indigo,

violet, andreddishorangeare examples of different

hues.

Purple (and related colors such as mauve)

is a hue that does not exist in the spectrum and

corresponds to blue and red added together.

Saturation refers to the "spectral purity" of a

color. A pure blue light mixed with some white

light still appears to be blue (a somewhat different

blue),

but has lost some of its saturation.

Monochromatic light has perfect saturation. Hue

and saturation define the

chromaticity

of an object.

One also observes that the shape of the blue

bond is the same as that of the blue report cover,

but the blue report cover has a much deeper blue

appearance. These two samples differ in their

strength or brightness values. Brightness is the

least critical of the three definitions of color since

it is easily changed by the level of incident light-

ing. (In other words, brightness is always chang-

ing anyway, and the eye—brain system compen-

sates for various levels of lighting automatically.)

Colorimeters, Because the spectral curves in

the visible region are relatively smooth, it is not

necessary to scan the entire spectrum. A series of

filters can be used across the visible spectrum.

For example, the Elrepho (photoelectric reflect-

ance photometer made by Carl Zeiss Co.) has a

series of seven filters that have dominant wave-

lengths of 420, 460, 490, 530, 570, 620, and 680

nm; it is an example of an abridged spectropho-

tometer. Special filters can also be used to mea-

sure the tristimulus values described below.

24.4 TRISTIMULUS SYSTEMS

Introduction

Since there are three types of color sensors in

the human eye, it should be possible to define

color by three coefficients corresponding to the

primary colors. The exact wavelengths of these

primary colors must be agreed upon and are the

basis of the coefficients. Any color of any bright-

ness could then be plotted on a three—dimensional

coordinate system (sometimes called a color

space).

If one ignores brightness (since humans

tend to perceive color independent of the level of

lighting), then a color plane can be defined that

intersects through the space; such a diagram is

called a chromaticity chart and takes into account

hue and saturation. The CIE chromaticity diagram

was developed in 1931.

[A certain color stimulus (what a spectro-

photometer sees from a surface) absolutely defines

a certain color sensation (in a standard observer),

but a certain color sensation can be caused by an

infinite number of possible color stimuli.] It

stands to reason that much more information is

contained in a spectral curve of (potentially) an

infinite number of points than in three numbers of

a tristimulus system.

Munsell

The Munsell system (1929) is a

nonquantitative three—dimensional system for

TRISTIMULUS SYSTEMS 495

plotting colors. It was specifically derived such other words, the distance required such that the

that there is uniform spacing of perception. In observer can recognize two separate shades is

100%-

80%H

0%-

,,««"-«::"""-""-""’"’’’""""’"’

400

^

450

500

,

550

600

650

700 750 800

Wavelength, nm

1.2-

Red Construction

Paper

Fig.

400

24-8.

-T

750

450 500 550 600 650 700 750 800

Wavelength, nm

Spectral responses of dyed papers as reflectance curves (top) and absorbance curves

496 24. OPTICAL PROPERTIES OF PAPER

about the same throughout the system. The

Munsell value corresponds to brightness from

black (0) to perfect diffuse reflectance (10). The

Munsell hue and chroma are the other axes.

CIE system

The first quantitative tristimulus system was

developed by the CIE (Commission Internationale

de I'Eclairage), which defines the internationally

accepted standard observer (since perception of

color is observer—dependent), used in the other

systems. (Older literature calls this the ICI sys-

tem.) The 1931 observer is based on the average

2° vision of 17 observers, while this was expand-

ed to 76 observers in 1964. The sample is illumi-

nated at a 45° angle with observations at 0°. (A

45° illumination with 45° observation is an obser-

vation of incident light, which has a higher pro-

portion of the color source characteristics.)

Since the color of an object is dependent

upon the source of the illumination, several stan-

dard illuminants were also specified (Section 24-

5).

Illuminant A corresponds to a tungsten light

operating at a temperature of 2854 K. Illuminant

B corresponds to direct sunlight, and Illuminant C,

perhaps the most common reference illuminant,

corresponds to north sky (in the northern

hemisphere) average diffuse daylight (as on an

overcast day); obviously these illuminants are

simulated in the laboratory with a specified ar-

rangement. Illuminant Dgs is also daylight but

specifies the UV and visible region for use in

fluorescence; it is also widely used as the refer-

ence for color television

monitors.

Illuminant E is

a theoretical source of pure white light.

CIE tristimulus values are obtained for a

given set of lighting conditions (for expected

further illumination of the object, such as paper

expected to be viewed under diffuse sunlight)

since, as has been said, two objects that appear the

same color under one set of lighting conditions

may appear different under a second set of lighting

conditions. The light source used when the CIE

values are determined must also be specified since

the light shining on the object partly determines

the spectral reflectance of the object. Usually the

light source is Illuminant E, perfectly white light,

(unless fluorescence is being considered in which

case Illuminant D is specified), since a spectro-

photometer operates as if the light source is pure

white light, because the light reflected off a sam-

ple is always compared to the light incident on the

sample for each wavelength under consideration.

For example, an object with Illuminant E

may have several different sets of tristimulus

values corresponding to a set for use in tungsten

light illumination, direct sunlight illumination, or

diffuse sunlight illumination. If one has a

reflectance curve then the tristimulus values can be

obtained with a computer that automatically factors

in the effect of any light source used to generate

the reflectance curve.

The 1931 CIE RGB system was defined with

primaries of red, R, at 700.0 nm; green, G, at

546.1 nm; and blue, B, at 435.8 nm. While the

exact selection of wavelengths is arbitrary, they

were selected from regions of the primary colors,

according to Grassman's (published in 1853) laws

of color mixing (Brunner et al., 1990). When the

relative amounts of each of these colors were

added together to match monochromatic light of

each wavelength across the spectrum (based on the

agreement of a group of standard observers), the

result in Fig. 24-9 was obtained, which is a plot of

the tristimulus values. However, this necessarily

led to negative values of the red primary in order

to match some colors (the mathematical equivalent

of having to add red primary to the sample to

obtain perceived agreement).

In order to avoid the use of negative values,

primaries (color matching functions) that had no

physical meaning were selected in the CIE XYZ

system so that positive values of each could be

combined to make any color of the spectrum (the

first criterion). (It was more important that certain

criteria be met rather than have color matching

functions that could be physically reproduced.)

The coefficients of the three color matching

functions at any particular wavelength are desig-

nated jc, y, and ?, respectively. The green

primary function, y , was chosen to match the

standard luminosity function (sometimes called

visual efficiency in older literature) of

the

standard

observer (the second criterion). The blue primary

function, ?, was chosen to be zero over as much

of the visible spectrum as possible (the third

criterion). The tristimulus values of each of the

three XI^ primaries are given in Fig. 24-10 at 10

TRISTIMULUS SYSTEMS 497

380 420

620

660

700

580

Wavelength, nm

Fig. 24-9. Tristimulus values (color matching functions) of the CIE RGB system.

740

nm intervals. The tabular values will be used

presently and are included in Table 24-2. [The

standard specifies the values from 360 nm to 830

nm at 1 nm increments to five decimal places, but

such precision and range is seldom required;

these values are available in Wyszecki and Stiles

(1967).] Separate integration (for each tristimulus

function over the visible spectrum) of the energy

of the light source

(E^)

multiplied by the spectral

reflectance from the object expressed from 0 to 1

{R^

multiplied by tristimulus values (x ,y , oil)

gives X, y, and Z as follow:

420

540

580

Wavelength, nm

Fig. 24-10. Tristimulus values (color matching functions) for perfectly white light of the CIE XYZ

system .

498

24.

OPTICAL PROPERTIES OF PAPER

C{E.ZR. dk

where

C=100/({E^y d\)

(By these definitions, Y will always be 100

for perfectly white light.) The integration is

approximated by a summation, such as by taking

values at every 10 nm from 380 nm to 720 nm (dX

« AX = 10 nm); high precision, of course,

requires a smaller increment. High precision is

obtained by taking smaller increments where the

functions have high values in the selected

ordinate

integration

method.

Tables that give intensities for

each wavelength (or every 10 nm) for a particular

standard illuminant simplify the calculations.

Many computerized spectrophotometers have the

software to makes these calculations automatically.

The fourth criterion of the three color func-

tions is that white light corresponds to X = 7 =

The three chromaticity coordinates are the

ratio of each of the tristimulus values divided by

the sum of all three tristimulus values. This gives

the ratios of the three tristimulus values with loss

of the overall magnitude information. The three

equations are:

X = XI{X + y + Z)

y = YI{X ^ Y -¥Z)

z = ZI{X -^Y+Z)

Any two of these specify the third since x +

y + z = 1. The use of x and y alone in a plot

(since the value of z is implicit mx + y) allows

all colors to be plotted in a two—dimensional

system where the overall brightness is ignored.

(The selection of x and y is arbitrary; a plot of

any two of the variables would have worked.)

Therefore, the chromaticity

diagram

is a plot

and y values as shown in Fig. 24-11 [with the

data obtained from Wyszecki and Stiles (1967)].

The data are plotted every 10 nm from 380 nm to

720 nm; plotting every 1 nm would have made

the curve smoother. Let us examine how the

points on a chromaticity chart are determined and

what they represent.

The simplest example

monochromatic light.

For example, let us determine the point for blue

light at 450 nm. Since overall brightness is not

considered, consider all light to be in units from 0

to 1 transmittance. Monochromatic light at 450

nm corresponds to 0 at all wavelengths and 1 at

450 nm. From Table 24-2, X, 7, and Z are

calculated as 0.3362, 0.0380, and 1.7721, respec-

tively. The solutions of x and y are then

0.3362/2.1463 and 0.0380/2.1463 or 0.1566 and

0.0177, respectively. Fig. 24-11 shows the

location of this point at those coordinates. Thus,

points on the outer "circle" represents monochro-

matic light.

White light is another useful example, but

takes a little more work. One takes the light

output for each wavelength and multiplies by the

factors in Table 24-2. It is important that the

spacing of the wavelengths be equal so that one

wavelength does not have more weight than

another in the calculations (unless one wants to

divide each increment by the wavelength interval).

But using a relative light output at each wave-

length of 1, a simple sum is taken as shown in

Table 24-2. Since all of the sums are equal, it

follows that

= y = z = 0.3333.

The lower line connects monochromatic blue

and red lights in various proportions to obtain

various shades of

purple.

Obviously this does not

occur when light is separated with a prism, but

can nevertheless occur to make colors.

If monochromatic green light of wavelength

520 nm is mixed with various parts of white light

the line is obtained that goes from 520 nm to

white (achromatic) light. This line corresponds to

various shades (various excitation purities) of the

same hue (dominant wavelength). A color, there-

fore,

can be described by its dominant wavelength

and excitation purity or its x and y values.

The reduced CIE Y, x, y system uses x and

y and retains 7 for a measurement of brightness.

A problem of the system demonstrated in

Fig. 24-11 is that the sensitivity to changes in

color by the human eye varies greatly in different

regions of the plot. A perception of color change

in the green area corresponds to a much larger

distance than in the red area. This should not be

surprising since the CIE XYZ space is not a

uniform color space as is the Munsell system.

[See McAdam, J. Opt. Soc. Am. 32:247(1942).]

Also,

there is no area that corresponds to black.

TRISTIMULUS SYSTEMS 499

0.3 0.4 0.5 X 0.6

Fig.

24-11.

The 1931 chromaticity chart.

a, b systems

Several systems use L, a, and b variables.

These systems are designed to have a relatively

uniform color space, which makes them useful for

determining perceptual color differences. The

variables have similar physical meaning in all of

these systems, but somewhat different numerical

values. It is an example of an opponent color

scale. Adams coordinates is one system that uses

this notation. Two others are discussed presently.

As shown in Fig. 24-12, L has values of 0

(corresponding to black) to 100 (perfect

whiteness); positive values of a correspond to

redness, while negative values correspond to

greenness; and positive values of b correspond to

yellowness, while negative values correspond to

blueness. When a and b are both zero, the sample

is achromatic (black, gray, or white).

The CIE L*fl*Z?* (1976) system (pronounced

as "sealab") defines the variable as follows for

Illuminant C (Z, 7, and Z are each > 0.01):

L* = 116 Y"^ - 16

a* = 500 [(1.02X)^^^-Y'^]

b* = 200[Y^^-(OM1Z)^^]

For use in perfectly white light the coefficients

1.02 and 0.847 are changed to 1.

L = 10 or 100

(white)

L =0

(black)

Fig. 24-12. The L, a, b color systems.

500 24. OPTICAL PROPERTIES OF PAPER

Table 24-2. Tristimulus values (color matching

functions) for perfectly white light of the CIE

XYZ system.

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

770

Totals

0.0014

0.0042

0.0143

0.0435

0.1344

0.2839

0.3483

0.3362

0.2908

0.1954

0.0956

0.0322

0.0049

0.0093

0.0633

0.1655

0.2904

0.4334

0.5945

0.7621

0.9163

1.0263

1.0622

1.0026

0.8544

0.6424

0.4479

0.2835

0.1649

0.0874

0.0468

0.0227

0.0114

0.0058

0.0029

0.0014

0.0007

0.0003

0.0002

0.0001

10.68

0.0000

0.0001

0.0004

0.0012

0.0040

0.0116

0.0230

0.0380

0.0600

0.0910

0.1390

0.2080

0.3230

0.5030

0.7100

0.8620

0.9540

0.9950

0.9520

0.8700

0.7570

0.6310

0.5030

0.3810

0.2650

0.1750

0.1070

0.0610

0.0320

0.0170

0.0082

0.0041

0.0021

0.0010

0.0005

0.0003

0.0001

0.0000

10.69

0.0065

0.0201

0.0679

0.2074

0.6456

1.3856

1.7471

1.7721

1.6692

1.2876

0.8130

0.4652

0.2720

0.1582

0.0782

0.0422

0.0203

0.0087

0.0039

0.0021

0.0017

0.0011

0.0008

0.0003

0.0002

0.0000

10.68

The Hunter (1958) L, a, b values are related

to the CIE XYZ tristimulus values as follow:

L = 10 Y^'

a = 17.5 (1.02 X- y)/7«-^

h = 7.0(y-0.847 2:)/7«-^

Other color systems

There are numerous color systems that have

been developed for various purposes. TIS 0804-

04 gives formulae for interconversion of many

color systems.

Determination of

tristimulus

values

Tristimulus values can be obtained by the

comparison method where the color is matched by

mixing three primary standards until visual obser-

vation gives a match, spectrophotometric analysis,

or use of trichromatic methods employing analysis

with three separate filters.

Video color systems

Video color—based systems (such as broad-

cast television) are tristimulus systems. Video

based systems, however, have additional con-

straints (Brunner et al., 1990). For example,

these systems have to use efficient encoding of

information because their transmissions have

limited bandwidth during broadcast. Also, they

must be compatible with black and white systems.

This point is mentioned because the use of

video based color is important in computer vision

and other techniques, which will have more

importance to the industry in the future. For

example, broadcast color television can reproduce

only about 50% of the area of the CIE chromatici-

ty chart (Fig. 24-11).

24.5 BLACKBODY RADIATION AND

OTHER LIGHT SOURCES

Since the source of light is very important in

the appearance of colored objects, blackbody

radiation will be considered. A blackbody is an

object that absorbs all electromagnetic radiation

that falls on it. The rate of radiation emitted from

a blackbody is a function of temperature only and

is independent of

the

type of

material.

The energy

the

emitted radiation at a given wavelength R(v)

was derived by Planck in 1900 and is given as a

function of frequency (v), Planck's constant (/z),

the speed of light in a vacuum (c), temperature

(7),

and Boltzmann's constant

Qc):

BLACKBODY RADIATION AND OTHER LIGHT SOURCES 501

2.5

••-»

V,-

1.5-

0.5H

200

600

1000 1400 1800 2200 2600

Wavelength, nm (decreasing energy-> )

Fig. 24-13. Blackbody radiation at several temperatures.

3400

(4-4

TAPPI T

525

Brightness

(457 nm filter)

380

460

TOO"

740

500 540 580 620

Wavelength, nm (decreasing energy-> )

Fig. 24-14. Blackbody radiation in the visible region at several temperatures and the normalized

transmittance of the blue filter used in paper brightness determinations.

R(v)

lirhv^

(24-3)

Fig. 24-13 shows the energy distribution as a

function of wavelength over a broad range for

several temperatures. Fig. 24-14 considers only

the visible region of the electromagnetic spectrum.

The number of photons per wavelength drops off

faster at lower wavelengths since each photon has

more energy so fewer photons "carry" the energy.

One observes that with increasing temperature

more energy is given off and the frequency of

maximal energy increases. As one heats a piece

of metal it begins to glow faint red, then brighter

yellow, then white hot, then blue—white. A tung-

502 24. OPTICAL PROPERTIES OF PAPER

sten incandescent light typically operates at 2800

K; one can see that most of the energy is given off

as heat in the infrared region (i.e., it is an ineffi-

cient producer of light) and most of the visible

light is toward the red end of the spectrum. Light

sources are often described by their color

tempera-

ture, i.e., the spectral distribution of a blackbody

radiator of a specified temperature that most

closely matches the light source. CIE lUuminant

B is approximated by 4870 K and lUuminant C by

6740 K.

The use of halogen gases allows the tempera-

ture at which a tungsten light operates to increase.

This gives a whiter distribution of light and in-

creases the efficiency of the light. Halogen lights

operate in the temperature range of 2800—3400 K.

Light can be produced by the excitation of

molecules or atoms (often by an electric charge,

but also by the processes of fluorescence and

combustion) in addition to blackbody radiation.

This mechanism gives different energy distribu-

tions.

For example, arc lights tend to give a high

percentage of their energy at a few selected wave-

lengths. Sodium lights give out a lot of light in

the yellow region of the spectrum at 589 nm.

They may be approximated by a color temper-

ature, but this is often not satisfactory.

Light is modified by filters to achieve certain

purposes. The blue filter centered at 457 nm for

paper brightness is well known. Heat filters are

also used to remove the IR energy from blackbody

radiation sources. UV filters are used for a

similar

purpose,

especially with high—temperature

blackbody radiation, where large amounts of UV

light (which can cause eye and skin damage) are

produced. The UV c band is defined as 375—405

nm, UV b is defined as 345—375 nm, and the UV

a band is defined as 300—345 nm (presented in

order of increasing energy).

variations in the composition of paper will affect

the optical properties. A generalized approach to

the optical properties of paper allows one to

predict the properties of paper to avoid a labor-

intensive trial—and—error work approach. The

concepts of brightness and color of paper have

already been considered in this chapter.

A few notes on common conventions used in

this chapter. Reflectance values that are deter-

mined relative to MgO should be reported as a

percentage. Absolute reflectances can be calculat-

ed if

the

reflectance of the MgO standard is known

or the photometer is calibrated as such. Absolute

reflectance is reported from 0 to 1. Paper with

85.0%

reflectance relative to a certain MgO

standard that is known to have an absolute reflect-

ance of 0.980 corresponds to an absolute reflect-

ance of 0.833 (calculated from 0.850 x 0.980).

Most equations use absolute reflectance values.

The relationships expressed in this section

strictly apply to only a single wavelength (or a

relatively narrow band of wavelengths) of light

(and preferably measurements taken with the same

light source and geometry). Difficulty may arise

in nonwhite (or nongray) papers since TAPPI

opacity is determined with the CIE Y function as

a weighing factor (centered near 567 nm) while

brightness is determined at 457 nm. This is a

common occurrence that is usually ignored, but

(usually unwittingly) implies that k and s are equal

under the two conditions!

There are many other reflectance models in

the literature with various underlying assumptions.

The dichromatic reflection model is one, but it

does not work well with paper at present. The

Kubelka—Munk theory is relatively simple in

terms of the number of constants involved, works

very well for many papers, and is well document-

ed for use by the pulp and paper industry.

24.6 KUBELKA-MUNK THEORY

Introduction

The optical properties of paper are extremely

important for many of its uses, especially when

paper is used for printing. The important optical

properties include reflectance (color and bright-

ness,

especially diffuse reflectance) and opacity.

It is important to know how processing and

Opacity

Opacity is the property of a material that

indicates the ability to hide what is behind it.

Opacity is usually evaluated as a contrast ratio, C,

such as TAPPI opacity, which is the reflectance of

a single sheet with a perfectly black backing (i?o)

divided by the reflectance with a white (or nearly

white) backing (such as 89% reflectance, /^Q.SQ)-

[See Judd (1934) below to see how the value of

KUBELKA-MUNK THEORY 503

89%

reflectance accidentally came to be for

TAPPI opacity.] The reflectance of

(effectively

infinite) thick pad of paper is R„ and is known as

reflectivity. The reflectance of

single sheet with

a perfectly white backing is given as R^ and the

corresponding opacity is called the ideal opacity.

TAPPI opacity = Qsp = Or =

Ro/Ro,s9

Printing opacity = C«, = 0 =

RQ/RO.

Judd (1937, 1938) describes (based on

Kubelka—Munk theory) how any opacity value

may be converted to any other opacity value by

formula (see Table 24-4 for TAPPI opacity from

any other opacity) or graphical aid. If a 0.8900

backing is not available, TAPPI opacity could be

determined by using another backing (of known

reflectance) near 0.89 and applying a correction.

R is the reflectance of a layer (e.g., of paper

or coating on paper) on top of a background with

reflectance ofRg. For example,

corresponds to

a black background where

= 0. The use of a

general form of R allows the theory to have much

more general application, such as the effect of a

coating layer on paper.

Uses of

Kubelka—Munk Theory

Kubelka—Munk theory is used in paper-

making in several ways. If one knows the optical

properties of each pulp, filler, and dye used in

papermaking, then the optical properties of

a paper

made with any combination of

the

materials can be

predicted. If one knows the contrast ratio and

reflectivity of a paper, the changes in these prop-

erties with a change in basis weight can be pre-

dicted. If any two variables of

RQ,

Ro^g,

and

R^,

are known, the third can be determined. Thus,

one can calculate R^ from a single sheet of paper

by measuring

RQ^^

and

RQ.

(However, if one uses

jRo.89

2^^

Roo, and if these are close in value, then

the error in determining

will be large.) Simi-

larly, for nearly opaque papers it is difficult to

determine other parameters with high precision.

Historical

context

Judd (1937) documents much of the early

work on the relationships between incident,

reflected, and emergent light from

thin,

homoge-

neous layer of absorbing and scattering materials

such as that of G.G. Stokes in 1860; Channon,

Renwick, and Storr in 1918; and Silberstein in

1927. Judd points out that many of these methods

have ignored light that escapes from the sides of

these layers because of the added complexity.

The Kubelka—Munk (1931) theory of light

allows one to predict quantitatively the behavior of

light in coatings such as paint for which it was

developed; this work has become the basis of

much

of this field. Steele (1935) reproduced many

of the derivations of equations from the Kubelka

and Munk derivation (for an English language

audience), gave a discussion of their work, and

provided additional work for routine application of

this theory to paper. Kubelka (1948) points out

that other derivations are available that indepen-

dently lead to many of

the

same equations present-

ed in this section.

The variables

k and s

Two parameters are the basis of the theory:

S, the scattering coefficient, and K, the absorption

coefficient, each for an infinitely thin layer of

material

(dX),

These variables, then, are intensive

properties and do not depend on the amount of

material (such as thickness of the sheet).

[Coffee has a high value of K and a low

value of 5. Conversely, cream has a low value of

K and a high value of S. Therefore, when one

puts creamer in coffee it turns lighter in color; S

the

coffee is increased dramatically, ^increases

only negligibly (or decreases with dilution), and

S/K

increases dramatically.]

This theory was developed to define proper-

ties based on the thickness of

material (X), but in

the case of porous materials like paper, workers

quickly

started using

basis weight. Later, Van den

Akker (1949) proved that basis weight (W) can be

used with

many

advantages; he recommended that

the lowercase variables s and k be used if the

thickness of material is reported as a basis weight

(the practice used

in this

chapter). These substitu-

tions are mathematically equivalent to the original

theory, and all derived relationships remain the

same. For example, SX, scattering power, is

dimensionless and equal in magnitude to sW;

therefore, s/k = S/K, etc.

While these variables are often reported for

a particular pulp, processing variables such as