Representative layer theory

For the special case where the incident radiation is normal (perpendicular) to a surface and the absorption is negligible, the intensity of the reflected and transmitted beams can be calculated from the refractive indices η₁ and η₂ of the two media, where r is the fraction of the incident light reflected, and t is the fraction of the transmitted light:

${\displaystyle r={\frac {(\eta _{2}-\eta _{1})^{2}}{(\eta _{2}+\eta _{1})^{2}}}}$ , ${\displaystyle t={\frac {4\eta _{1}\eta _{2}}{(\eta _{2}+\eta _{1})^{2}}}}$ , with the fraction absorbed taken as zero ( ${\displaystyle a}$ = 0 ).

For a beam of light traveling in air with an approximate index of refraction of 1.0, and encountering the surface of a material having an index of refraction of 1.5:

${\displaystyle r={\frac {(\eta _{2}-\eta _{1})^{2}}{(\eta _{2}+\eta _{1})^{2}}}={\frac {0.5^{2}}{2.5^{2}}}=0.04}$ , ${\displaystyle t={\frac {4\eta _{1}\eta _{2}}{(\eta _{2}+\eta _{1})^{2}}}={\frac {(4)(1)(1.5)}{2.5^{2}}}=0.96}$

There is a simplified special case for the spectroscopic parameters of a sheet. This sheet consists of three plane parallel layers (1:front surface, 2:interior, 3:rear surface) in which the surfaces both have the same remission fraction when illuminated from either direction, regardless of the relative refractive indices of the two media on either side of the surface. For the case of zero absorption in the interior, the total remission and transmission from the layer can be determined from the infinite series, where ${\displaystyle r_{0}}$ is the remission from the surface:

${\displaystyle A=0,\qquad }$ ${\displaystyle R=r_{0}+(1-r_{0})^{2}\sum _{n=1}^{\infty }r_{0}^{(2n-1)},\qquad }$ ${\displaystyle T=(1-r_{0})^{2}\sum _{n=0}^{\infty }r_{0}^{2n}.}$

These formulas can be modified to account for absorption.[15][18] Alternatively, the spectroscopic parameters of a sheet (or slab) can be built up from the spectroscopic parameters of the individual pieces that compose the layer: surface, interior, surface. This can be done using an approach developed by Kubelka for treatment of inhomogeneous layers. Using the example from the previous section: { A₁ = 0, R₁ = 0.04, T₁ = 0.96 } {A₃ = 0, R₃ = 0.04, T₃ = 0.96 }.

We will assume the interior of the sheet is composed of a material that has Napierian absorption coefficient k of 0.5 cm⁻¹, and the sheet is 1 mm thick (d = 1 mm). For this case, on a single trip through the interior, according to the Bouguer-Lambert law, ${\textstyle T=\exp(-kd)}$, which according to our assumptions yields ${\textstyle T=\exp(-0.5\ {\text{cm}}^{-1}\cdot 0.1\ {\text{cm}})=0.95}$ and ${\textstyle A=0.05}$. Thus { A₂ = 0.05, R₂ = 0, T₂ = 0.95 }.

Then one of Benford's equations[19] can be applied. If A_x, R_x and T_x are known for layer x and A_y R_y and T_y are known for layer y, the ART fractions for a sample composed of layer x and layer y are:

${\displaystyle T_{x+y}={\frac {T_{x}T_{y}}{1-R_{(-x)}R_{y}}},\qquad }$ ${\displaystyle R_{x+y}=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}},\qquad }$ ${\displaystyle A_{x+y}=1-T_{x+y}-R_{x+y}}$

(The symbol ${\displaystyle R_{(-x)}}$ means the reflectance of layer ${\displaystyle x}$ when the direction of illumination is antiparallel to that of the incident beam. The difference in direction is important when dealing with inhomogeneous layers. This consideration was added by Paul Kubelka in 1954.[20] He also pointed out that transmission was independent of the direction of illumination, but absorption and remission were not.)

Step 1: We take layer 1 as x, and layer 2 as y. By our assumptions in this case, { ${\displaystyle R_{1}=R_{(-1)}=0.04,R_{2}=0,T_{1}=0.96,T_{2}=0.95}$ }.${\displaystyle T_{x+y}={\frac {T_{x}T_{y}}{1-R_{(-x)}R_{y}}}={\frac {(0.96)(0.95)}{1-(0.04)0}}=0.912\quad }$ ${\displaystyle R_{x+y}=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}}=0.04+{\frac {(0.96^{2})0}{1-(0.04)0}}=0.04\quad }$ ${\displaystyle R_{y+x}=R_{y}+{\frac {T_{y}^{2}R_{x}}{1-R_{(-y)}R_{x}}}=0+{\frac {(0.95^{2})(0.04)}{1-0(0.04)}}=0.0361}$

Step 2: We take the result from step 1 as the value for new x [ x is old x+y; (-x) is old y+x ], and the value for layer 3 as new y.

${\displaystyle T_{x+y}={\frac {T_{x}T_{y}}{1-R_{(-x)}R_{y}}}={\frac {(0.912)(0.96)}{1-(0.0361)(0.04)}}=0.877=T_{123}}$ ${\displaystyle R_{x+y}=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}}=0.04+{\frac {(0.912^{2})(0.04)}{1-(0.0361)(0.04)}}=.0733=R_{123}}$

${\displaystyle A_{123}=1-R_{123}-T_{123}=1-0.877-.073=0.05}$

Dahm has shown that for this special case, the total amount of light absorbed by the interior of the sheet (considering surface remission) is the same as that absorbed in a single trip (independent of surface remission).[21] This is borne out by the calculations.

The decadic absorbance (${\displaystyle {\mathsf {\mathrm {A} b}}_{10}}$) of the sheet is given by: ${\displaystyle {\mathsf {\mathrm {A} b}}_{10}=-log(1-A_{123})=0.0222}$

The Stokes Formulas can be used to calculate the ART fractions for any number of layers. Alternatively, they can be calculated by successive application of Benford's equation for "one more layer".

If A₁, R₁, and T₁ are known for the representative layer of a sample, and A_n, R_n and T_n are known for a layer composed of n representative layers, the ART fractions for a layer with thickness of n + 1 are:

${\displaystyle T_{n+1}={\frac {T_{n}T_{1}}{1-R_{n}R_{1}}},\qquad }$ ${\displaystyle R_{n+1}=R_{n}+{\frac {T_{n}^{2}R_{1}}{1-R_{n}R_{1}}},\qquad }$ ${\displaystyle A_{n+1}=1-T_{n+1}-R_{n+1}}$

In the above example, { ${\displaystyle A_{1}=0.05,R_{1}=0.0733,T_{1}=0.877}$ }. The Table shows the results of repeated application of the above formulas.

Within a homogeneous media such as a solution, there is no scatter. For this case, the function is linear with both the concentration of the absorbing species and the path-length. Additionally, the contributions of individual absorbing species are additive. For samples which scatter light, absorbance is defined as "the negative logarithm of one minus absorptance (absorption fraction: ${\displaystyle \alpha }$) as measured on a uniform sample".[22] For decadic absorbance,[23] this may be symbolized as: ${\displaystyle \mathrm {A} _{10}=-log_{10}(1-\alpha )}$ .   Even though this absorbance function is useful with scattering samples, the function does not have the same desirable characteristics as it does for non-scattering samples. There is, however, a property called absorbing power which may be estimated for these samples. The absorbing power of a single unit thickness of material making up a scattering sample is the same as the absorbance of the same thickness of the materiel in the absence of scatter.[24]

Suppose that we have a sample consisting of 14 sheets described above, each one of which has an absorbance of 0.0222. If we are able to estimate the absorbing power (the absorbance of a sample of the same thickness, but having no scatter) from the sample without knowing how many sheets are in the sample (as would be the general case), it would have the desirable property of being proportional to the thickness. In this case, we know that the absorbing power (scatter corrected absorbance) should be: {14 x the absorbance of a single sheet} ${\displaystyle =(14)(0.0222)=0.312}$ . This is the value we should have for the sample if the absorbance is to follow the law of Bouguer (often referred to as Beer's law).

In the Table below, we see that the sample has the A,R,T values for the case of 14 sheets in the Table above. Because of the presence of scatter, the measured absorbance of the sample would be: ${\displaystyle {\mathsf {\mathrm {A} b}}_{10}=-log(1-A_{S})=-log(0.466)=0.2728}$. Then we calculate this for the half sample thickness using another of Benford's equations. If A_d, R_d and T_d are known for a layer with thickness d, the ART fractions for a layer with thickness of d/2 are:

${\displaystyle R_{d/2}={\frac {R_{d}}{1+T_{d}}},\qquad }$ ${\displaystyle T_{d/2}={\sqrt {T_{d}(1-R_{d/2}^{2})}},\qquad }$ ${\displaystyle A_{d/2}=1-T_{d/2}-R_{d/2},}$

In the line for half sample [S/2], we see the values which are the same as those for 7 layers in the Table above, as we expect. Note that ${\displaystyle -log(1-A_{S/2})=-log(1-0.292)=0.150}$. We desire to have the absorbance be linear with sample thickness, but we find when we multiply this value by 2, we get ${\displaystyle (2)(0.150)=0.300}$ , which is a significant departure from the previous estimate for the absorbing power.

The next iteration of the formula produces the estimate for A,R,T for a quarter sample: ${\displaystyle -log(1-0.162)\times 4=0.307}$. Note that this time the calculation corresponds to three and a half layers, a thickness of sample that cannot exist physically.

Continuing for the sequentially higher powers of two, we see a monotonically increasing estimate. Eventually the numbers will start jumping with round off error, but one can stop when getting a constant value to a specified number of significant figures. In this case, we become constant to 4 significant figures at 0.3105, which is our estimate for the absorbing power of the sample. This corresponds to our target value of 0.312 determined above.

To be representative, the layer must meet the following criteria:[25]

• The volume fraction of each type of particle is the same in the representative layer as in the sample as a whole.

• The surface area fraction of each type of particle is the same in the representative layer as in the sample as a whole.

• The void fraction of the representative layer is the same as in the sample.

• The representative layer is nowhere more than one particle thick. Note this means the “thickness” of the representative layer is not uniform. This criterion is imposed so that we can assume that a given photon of light has only one interaction with the layer. It might be transmitted, remitted, or absorbed as a result of this interaction, but it is assumed not to interact with a second particle within the same layer.

In the above discussion, when we talk about a “type” of particle, we must clearly distinguish between particles of different composition. In addition, however, we must distinguish between particles of different sizes. Recall that scattering is envisioned as a surface phenomenon and absorption is envisioned as occurring at the molecular level throughout the particle. Consequently, our expectation is that the contribution of a “type” of particle to absorption will be proportional to the volume fraction of that particle in the sample, and the contribution of a “type” of particle to scattering will be proportional to the surface area fraction of that particle in the sample. This is why our “representative layer” criteria above incorporate both volume fraction and surface area fraction. Since small particles have larger surface area-to-volume ratios than large particles, it is necessary to distinguish between them.

Under these criteria, we can propose a model for the fractions of incident light that are absorbed (${\displaystyle A_{1}}$), remitted (${\displaystyle R_{1}}$), and transmitted (${\displaystyle T_{1}}$) by one representative layer.[26]

${\displaystyle A_{1}=\sum S_{j}(1-exp(-k_{j}d_{j}))}$ , ${\displaystyle R_{1}=\sum S_{j}b_{j}d_{j}}$ , ${\displaystyle T_{1}=1-R_{1}-T_{1}}$

in which:

• ${\displaystyle S_{j}}$ is the fraction of cross-sectional surface area that is occupied by particles of type ${\displaystyle j}$.

• ${\displaystyle K_{j}}$ is the effective absorption coefficient for particles of type ${\displaystyle j}$.

• ${\displaystyle b_{j}}$ is the remission coefficient for particles of type ${\displaystyle j}$.

• ${\displaystyle d_{j}}$ is the thickness of a particle of type ${\displaystyle j}$ in the direction of the incident beam.

• The summation is carried out over all of the distinct “types” of particle.

In effect, ${\displaystyle S_{j}}$represents the fraction of light that will interact with a particle of type ${\displaystyle j}$, and ${\displaystyle K_{j}}$and ${\displaystyle b_{j}}$ quantify the likelihood of that interaction resulting in absorption and remission, respectively.

Surface area fractions and volume fractions for each type of particle can be defined as follows:

${\displaystyle v_{i}={\frac {\frac {w_{i}}{\rho _{i}}}{\sum {\frac {w_{j}}{\rho _{j}}}}}}$ , ${\displaystyle s_{i}={\frac {\frac {w_{i}}{\rho _{i}d_{i}}}{\sum {\frac {w_{j}}{\rho _{j}d_{j}}}}}}$ , ${\displaystyle V_{i}=(1-v_{0})v_{i}}$ , ${\displaystyle S_{i}=(1-v_{0})s_{i}}$

in which:

• ${\displaystyle w_{i}}$ is the mass fraction of particles of type i in the sample.

• ${\displaystyle v_{i}}$ is the fraction of occupied volume composed of particles of type i.

• ${\displaystyle s_{i}}$ is the fraction of particle surface area that is composed of particles of type i.

• ${\displaystyle V_{i}}$ is the fraction of total volume composed of particles of type i.

• ${\displaystyle S_{i}}$ is the fraction of cross-sectional surface area that is composed of particles of type i.

• ${\displaystyle \rho _{i}}$ is the density of particles of type i.

• ${\displaystyle v_{0}}$ is the void fraction of the sample.

This is a logical way of relating the spectroscopic behavior of a “representative layer” to the properties of the individual particles that make up the layer. The values of the absorption and remission coefficients represent a challenge in this modeling approach. Absorption is calculated from the fraction of light striking each type of particle and a “Beer’s law”-type calculation of the absorption by each type of particle, so the values of ${\displaystyle K_{i}}$used should ideally model the ability of the particle to absorb light, independent of other processes (scattering, remission) that also occur. We referred to this as the absorbing power in the section above.