3. Let’s Learn About the Kubelka–Munk Function
I am actually writing this part after finishing the entire post, but unless you are studying DRS or the Kubelka–Munk theory professionally, you probably do not need to study it in this much detail, haha.
Still, the more you know, the easier it becomes to understand and use the instrument!
This time, while organizing UV-Vis DRS data, I decided to take a closer look at an equation that frequently appeared in the analysis.
It is the Kubelka–Munk function.
When reading papers related to semiconductors or photocatalysts, the following equation appears very often. Those who write research papers may also have seen terms such as K/M or K–M when using DRS data!!!!!!!!!!!!!!
F(R) = (1 − R)² / 2R
What does this equation mean, and why is it used to analyze diffuse reflectance data?
In this post, we will go through everything step by step, from why the Kubelka–Munk function is needed to how it is used to estimate the band gap!!!!
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Figure 1. Absorption and scattering of light inside a powder sample
What Is the Kubelka–Munk Theory?
The Kubelka–Munk theory was developed to describe the optical properties of materials in which light is scattered in many directions, such as powders, pigments, paper, and coating layers.
In solutions or transparent samples, light can travel relatively straight through the sample.
However, when light is directed at a powder sample, some of it is absorbed, while the rest collides with the particles and is scattered in different directions.
The Kubelka–Munk theory simplifies this complicated movement of light and explains the absorption and scattering behavior of a material.
In simple terms, it can be described as:
A method for using reflected light measured from a powder sample to investigate the light absorption properties of the material
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Kubelka–Munk in One Sentence
The Kubelka–Munk function converts the diffuse reflectance measured from a powder or opaque sample into a value related to its absorption properties.
To put it more simply:
It is a calculation method that converts reflectance data into a form that can be compared in a way similar to absorbance.
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Key Terms
Diffuse reflectance, absorption coefficient, scattering coefficient, Kubelka–Munk function, and band gap
Why Do We Not Use Ordinary Absorbance?
Light can pass through liquids and transparent thin films, so their transmittance can be measured.
Absorbance is then calculated by comparing the intensity of light before and after it passes through the sample.
However, light cannot easily pass directly through powders or thick, opaque samples.
This is because the light collides with particles, scatters in different directions, is partially absorbed by the sample, and is partially reflected back out of the sample.
Therefore, it is difficult to determine the optical properties of these materials using a conventional transmission-mode UV-Vis measurement.
In such cases, UV-Vis diffuse reflectance spectroscopy, or UV-Vis DRS, is used.
DRS measures light that is reflected from the sample in many different directions.
The Kubelka–Munk function is then used to convert the measured reflectance into a value related to light absorption.
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Figure 2. Comparison between transmission measurement of a transparent sample and diffuse reflectance measurement of a powder sample
The Kubelka–Munk Function
For an optically thick sample, the Kubelka–Munk function can be expressed as follows:
F(R∞) = (1 − R∞)² / 2R∞ = K/S
Each term has the following meaning:
R∞: Diffuse reflectance measured from a sufficiently thick sample
K: Absorption coefficient
S: Scattering coefficient
F(R∞): Kubelka–Munk function
The most important point here is that the Kubelka–Munk function does not represent the absorption coefficient K itself.
More precisely, F(R) represents:
The ratio of the absorption coefficient K to the scattering coefficient S, or K/S
Therefore, the Kubelka–Munk value should not be considered exactly the same as ordinary absorbance!!!!
This is because it contains the effects of both absorption and scattering.
What Happens to F(R) When Reflectance Decreases?
Let us calculate it using some simple numbers.
Suppose the reflectance at a certain wavelength is 50%.
When substituting the value into the Kubelka–Munk equation, we must use 0.50, not 50.
F(R) = (1 − 0.50)² / (2 × 0.50)
The result is:
F(R) = 0.25
Now, suppose the reflectance is 20%.
F(R) = (1 − 0.20)² / (2 × 0.20)
The result is:
F(R) = 1.60
As shown above, the Kubelka–Munk value generally increases as reflectance decreases.
When less light is reflected out of the sample, it may indicate that more light has been absorbed by the sample.
However, reflectance is affected not only by absorption but also by scattering caused by particle size and surface conditions.
Therefore, a decrease in reflectance should not automatically be interpreted as an increase in absorption alone.
Important Points When Performing the Calculation
Instrument results often display reflectance as a percentage.
For example, if the reflectance is 70%, you should not substitute 70 directly into the equation.
Instead, it must be converted as follows:
70% → 0.70
Similarly:
50% → 0.50
20% → 0.20
5% → 0.05
Using the percentage value directly in the equation will produce a completely different result, so this must always be checked!!!!
How to Estimate the Band Gap Using the Kubelka–Munk Function
The main reason the Kubelka–Munk function frequently appears in semiconductor and photocatalyst research is that it is used to estimate the optical band gap.
The overall analysis process is as follows.
1. Measure Reflectance Using UV-Vis DRS
First, measure the diffuse reflectance R of the sample as a function of wavelength.
If the instrument displays reflectance as a percentage, divide the values by 100 to convert them into values between 0 and 1.
2. Convert Reflectance into the Kubelka–Munk Function
Substitute the reflectance measured at each wavelength into the following equation:
F(R) = (1 − R)² / 2R
This process converts the reflectance data into a form that is easier to compare with the absorption properties of the sample.
3. Convert Wavelength into Photon Energy
The relationship between wavelength and photon energy is expressed as follows:
hν (eV) = 1240 / λ (nm)
For example, if the absorption edge is located near 400 nm:
1240 ÷ 400 = 3.10 eV
4. Create a Tauc Plot
In DRS analysis, F(R) is assumed to be proportional to the absorption coefficient and is therefore applied to the Tauc relationship.
For an allowed direct transition, the following graph is generally used:
[F(R)hν]² versus hν
For an allowed indirect transition, the following graph is used:
[F(R)hν]¹ᐟ² versus hν
A linear region connected to the absorption edge is selected and extrapolated to the x-axis.
The point where the extrapolated line intersects the x-axis is interpreted as the optical band gap, Eg.
Figure 3. Process of converting diffuse reflectance into the Kubelka–Munk function and constructing a Tauc plot
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Direct and Indirect Band Gaps
When creating a Tauc plot, it is first necessary to determine whether the material undergoes a direct or indirect transition.
In a direct-transition semiconductor, an electron can move from the valence band to the conduction band without requiring a large change in momentum.
Therefore, the following plot is generally used:
Allowed direct transition: [F(R)hν]²
In contrast, an indirect-transition semiconductor requires a momentum change involving a phonon when the electron moves between the valence and conduction bands.
In this case, the following plot is used:
Allowed indirect transition: [F(R)hν]¹ᐟ²
There is one potentially confusing point here.
Different papers may use different symbols and definitions for the exponent in the Tauc equation.
Therefore, rather than checking only the numerical value of the exponent written in the equation, it is important to confirm what quantity is actually plotted on the y-axis.
How Should the Linear Region Be Selected?
Creating a Tauc plot does not mean that the band gap is automatically calculated with perfect accuracy.
The value of Eg may change depending on which region of the graph is selected as the linear region.
If a region at excessively low energy is selected, the result may include the effects of defect states or the Urbach tail.
In contrast, if a region at excessively high energy is selected, other optical transitions or strongly absorbing regions may be included.
Therefore, the following points should be considered when selecting the linear region:
Whether the region is connected to the actual absorption edge
Whether it shows reasonable linearity
Whether it agrees with the direct or indirect transition characteristics of the material
Whether it differs excessively from band-gap values reported in previous studies
If the most linear-looking region is selected without careful consideration, the resulting band gap may differ from the actual value.
Conditions for Applying the Kubelka–Munk Function
The Kubelka–Munk function cannot be applied to every sample in exactly the same way.
Several conditions should be considered to obtain relatively reliable results.
1. The Sample Must Be Sufficiently Thick
If the sample is too thin, light may pass through it and reach the holder or substrate behind it.
In that case, the measured reflectance will include contributions from materials other than the sample itself.
Therefore, the sample must be optically thick enough that adding more material does not significantly change the measured reflectance.
2. The Sample Must Be Uniform
If the composition or density varies significantly within the sample, absorption and scattering may differ depending on the measurement position.
This is also why the surface of a powder sample should be made as even as possible when it is placed in the sample holder.
3. The Scattering Coefficient Should Not Change Significantly
To use F(R) in a manner similar to the absorption coefficient, it is necessary to assume that the scattering coefficient S does not change greatly over the analyzed wavelength range.
If S changes significantly with wavelength, it becomes difficult to determine whether changes in F(R) are caused by absorption or scattering.
4. The Effect of Specular Reflection Should Be Minimized
If strong mirror-like reflection occurs at the sample surface, it becomes difficult to accurately observe the diffuse reflectance originating from within the sample.
This is particularly important for smooth thin films and glossy samples.
Why Is Particle Size Important?
Even when the same material is measured, the DRS result may change depending on the particle size and surface condition.
This is because changing the particle size also changes the way light is scattered inside the sample.
Reflectance can also be affected by how strongly the sample is packed, the amount of empty space between particles, and how flat the surface is.
Because the Kubelka–Munk function represents K/S, a change in the scattering coefficient S can alter F(R), even when the actual absorption remains the same.
When comparing multiple samples, it is preferable to keep the following conditions as similar as possible:
Similar particle sizes
The same amount of sample
The same sample holder
A similar degree of compression
The same measurement range and reference material
Special care is required when directly comparing F(R) values from samples with completely different forms, such as powders and thin films.
Does a Large Kubelka–Munk Value Mean a Better Photocatalyst?
A large Kubelka–Munk value means that the K/S ratio is large at a particular wavelength.
It may indicate that the material strongly absorbs light, but this alone does not prove that the photocatalytic performance is good.
For a photocatalytic reaction to proceed efficiently:
The material must absorb light and generate electrons and holes.
The generated charge carriers must avoid recombination.
The electrons and holes must move to the surface.
The desired oxidation and reduction reactions must occur at the surface.
In other words, absorbing a large amount of light is only the first step in a photocatalytic reaction.
To properly evaluate the actual performance, other results such as PL, photocurrent, EIS, reaction rates, and product analysis should also be examined.
Advantages of the Kubelka–Munk Function
One major advantage of the Kubelka–Munk function is that it allows powders and opaque samples, which are difficult to analyze using transmission measurements, to be studied.
It also converts DRS reflectance into a form related to absorption properties, making it easier to compare absorption edges and changes in band gaps among different samples.
The equation is not particularly complicated, so the data can be processed relatively easily using programs such as Excel or Origin.
Limitations of the Kubelka–Munk Function
The Kubelka–Munk function represents K/S rather than the actual absorption coefficient itself.
Therefore, if the assumption of a constant scattering coefficient is not appropriate, the results cannot be interpreted solely as changes in absorption.
The following factors may also affect the results:
Sample thickness
Particle size
Surface roughness
Degree of sample compression
Specular reflection
Nonuniformity within the sample
For composite materials or materials containing many defects, the linear region of the Tauc plot may not be clearly defined.
Therefore, the calculated band gap should be understood as an optical estimate based on specific measurement conditions and assumptions, rather than as a perfectly accurate absolute value.
Kubelka–Munk in One Sentence
The Kubelka–Munk function converts the diffuse reflectance measured from a powder or opaque sample into K/S, the ratio of the absorption coefficient to the scattering coefficient.
In semiconductor and photocatalyst analysis, this value is used to construct a Tauc plot and estimate the optical band gap.
However, when interpreting the result, the sample thickness, scattering characteristics, particle size, direct or indirect transition type, and selection of the linear region must all be considered.
Summary
The Kubelka–Munk theory explains the absorption and scattering of light in materials that strongly scatter light.
The Kubelka–Munk function is expressed as:
F(R) = (1 − R)² / 2R = K/S
This function is used to convert diffuse reflectance obtained from UV-Vis DRS into a value that can be more easily compared with the absorption properties of a material.
In semiconductor and photocatalyst research, the converted value is used to create a Tauc plot and estimate the band gap.
However, F(R) contains the effects of both absorption and scattering.
Therefore, it is important to keep the sample conditions and measurement parameters as consistent as possible and to avoid interpreting the calculated value as an absolute result.
When I first saw this equation, it looked like a complicated optical theory.
However, it becomes much easier to understand when we think of it as:
A process for converting diffuse reflectance into a form that is easier to analyze
Let us keep learning one concept at a time!!!!!!
