**6. Traditional surface texture parameters and functions (parameters R and S)**

The roughness can be characterized by several parameters and functions (such as height parameters, wavelength parameters, spacing and hybrid parameters (Gadelmawla et al., 2002). The following parameters and functions related to the height and spacing (also called parameters R and S) will be discussed as well as their calculation.

#### **6.1 Height parameters (R)**

The most significant parameters in the case of roughness are the Height Parameters.

#### **6.1.1 Roughness average**

Among Height Parameters, the roughness average (Ra) is the most widely used because it is a simple parameter to obtain when compared to others. The roughness average is described as follows (Park, 2011).

$$\mathcal{R}\_{\mathbf{a}} = \frac{1}{\mathcal{L}} \Big| \prescript{\mathcal{L}}{Z(\mathbf{x})} | Z(\mathbf{x}) | \,\mathrm{d}\mathbf{x} \tag{2}$$

Where Z(x) is the function that describes the surface profile analyzed in terms of height (Z) and position (x) of the sample over the evaluation length "L" (Figure 12).

Thus, the Ra is the arithmetic mean of the absolute values of the height of the surface profile Z(x). Many times the roughness average is called the Arithmetic Average (AA), Center Line Average (CLA) or Arithmetical Mean Deviation of the Profile. The average roughness has advantages and disadvantages. The advantages include: ease of obtaining the same average roughness of less sophisticated instruments, for example, a profilometer can provide (Ra); possibility of repetition of the parameter, since it appears very stable statistically, recommended as a parameter for the characterization of random surfaces, it is usually used to describe machined surfaces (B.C. MacDonald & Co, 2011).

can be used as second attachment. The skid is one foot blunt that has a large radius of curvature, and it's placed either beyond or behind the stylus. The transducers sense the difference in level between the stylus and the skid. The skid acts as a mechanical filter to attenuate the long spatial wavelength of the surface. As a result of slippage, the wavelength information is long lost. If the long wavelengths are functionally relevant, then the use of a slide should be avoided. In addition, the use of a skid surfaces or surfaces with periodic discrete peaks can result in distortion of the measured profile (Vorburguer & Raja, 1990).

The numerical evaluation of roughness is always preceded by removal of waviness from the measured profile. This is achieved in a surface texture measuring instrument by using an analog or digital filter. In order to exclude waviness, a limiting wavelength has to be specified. This limiting wavelength is referred to as cutoff. The cutoff is given in mm or Inch and the following values are available in many instruments, 0.08, 0.25, 0.8, 2.5, and 8 nm. The cutoff selected must be short enough to exclude irrelevant long wavelength and at the same time long enough to ensure that enough texture has been included in the assessment to give meaningful results. Usually five cutoff settings are used for assessment, and overall

**6. Traditional surface texture parameters and functions (parameters R and S)**  The roughness can be characterized by several parameters and functions (such as height parameters, wavelength parameters, spacing and hybrid parameters (Gadelmawla et al., 2002). The following parameters and functions related to the height and spacing (also called

The most significant parameters in the case of roughness are the Height Parameters.

Among Height Parameters, the roughness average (Ra) is the most widely used because it is a simple parameter to obtain when compared to others. The roughness average is described

L

0

Where Z(x) is the function that describes the surface profile analyzed in terms of height (Z)

Thus, the Ra is the arithmetic mean of the absolute values of the height of the surface profile Z(x). Many times the roughness average is called the Arithmetic Average (AA), Center Line Average (CLA) or Arithmetical Mean Deviation of the Profile. The average roughness has advantages and disadvantages. The advantages include: ease of obtaining the same average roughness of less sophisticated instruments, for example, a profilometer can provide (Ra); possibility of repetition of the parameter, since it appears very stable statistically, recommended as a parameter for the characterization of random surfaces, it is usually used

a

and position (x) of the sample over the evaluation length "L" (Figure 12).

to describe machined surfaces (B.C. MacDonald & Co, 2011).

<sup>1</sup> R |Z x |dx <sup>L</sup> (2)

traverse length is seven cutoffs (Vorburguer & Raja, 1990).

parameters R and S) will be discussed as well as their calculation.

**6.1 Height parameters (R)** 

**6.1.1 Roughness average** 

as follows (Park, 2011).

Fig. 12. Profile of a surface (Z). It represents the average roughness Ra and Rq is the RMS roughness based on the mean line (B.C. MacDonald & Co, 2011).

The average roughness, as already said, is just the mean absolute profile, making no distinction between peaks and valleys. Thus it becomes a disadvantage to characterize an average surface roughness if these data are relevant.

The average roughness can be the same for surfaces with roughness profile totally different because it depends only on the average profile of heights. Surfaces that have different undulations are not distinguished (Figure 13). We may have an even surface and some other with peaks (or valleys) with small contributions presenting the same value of average roughness. For this reason, more sophisticated parameters can be used to fully characterize a surface when more significant information is necessary, for example, distinguish between peaks and valleys.

Fig. 13. Different profiles of surfaces, with the same roughness average (adapted from Predev, 2011).

#### **6.1.2 Root mean square roughness (Rq)**

The root mean square (RMS) is a statistical measure used in different fields. We cite, as an example, the use of the RMS amplitude applied to harmonic oscillators, such as on an alternating electric current. The root mean square of roughness (Rq) is a function that takes the square of the measures. The RMS roughness of a surface is similar to the roughness average, with the only difference being the mean squared absolute values of surface roughness profile. The function Rq is defined as (Gadelmawla et al., 2002):

$$\mathcal{R}\_{\mathbf{q}} = \sqrt{\frac{1}{L} \Big| \int\_{0}^{L} |Z^{2}(\mathbf{x})| \, \mathrm{d}x} \tag{3}$$

Measurement of the Nanoscale Roughness by

profile (x) on an evaluation length (L), given by (Park, 2011):

Atomic Force Microscopy: Basic Principles and Applications 159

These parameters are useful when trying to find some very sharp peak, which could affect any application of the sample, a scratch or an unusual crack on the material. Often we use the average of these parameters, RTm, Rpm and Rvm and comprising the averages along the

> L pm pi i x

> L vm vi i x

(7)

(8)

(9)

<sup>1</sup> R R L

<sup>1</sup> R R L

Tm Ti pm vm

These average parameters have the same advantage that the extreme parameters described

The Rz (ISO) is the arithmetic mean of the five highest peaks added to the five deepest

L

i x <sup>1</sup> R RR R L

in (4), (5) and (6), but lose accuracy when searching for singularities.

valleys over the evaluation length measured. It is a parameter similar to RTm.

Fig. 15. Rz average of the sum of the five highest peaks in the five deepest valleys of

The Third Highest Peak to Third Lowest Valley Height is a parameter that overlooks the two highest peaks and the two deepest valleys, taking as measure the vertical variation of the third highest peak from the third deepest valley. The advantage of using this parameter is that it disregards any discrepancies that do not interfere in a meaningful way in the characterization of the surface profile. This ignores the extreme values thus reducing the instability of peaks and valleys. It is a commonly used parameter for the characterization of

**6.1.4 Ten point average roughness - Rz (ISO)** 

sample's profile (Zygo Corporation, 2011).

**6.1.5 Third highest peak to third lowest valley height (R3Zi)** 

porous surfaces and sealing surfaces (B.C. Macdonald & Co, 2011).

The software does not need to be sophisticated in order to obtain Rq. For this reason much of the surface analysis equipment (profilometer and SPMs) provides Rq. In SPM, the Rq depends on the swept area of the sample, the scan size.

The Rq is more sensitive to peaks and valleys than the average roughness due to the squaring of the amplitude in its calculation.

#### **6.1.3 Maximum height of the profile (RT), maximum profile valley depth (Rv) and maximum profile peak height (Rp)**

The Maximum Profile Peak Height (Rp) is the measure of the highest peak around the surface profile from the baseline. Likewise the Maximum Profile Valley Depth (Rv) is the measure of the deepest valley across the surface profile analyzed from the baseline (Park, 2011). We can write:

$$\mathbf{R\_{p}} = \left| \max \mathbf{Z(x)} \right| \text{ for } \ 0 \le x \le L \tag{4}$$

$$\mathbf{R}\_{\mathbf{v}} = \left| \min \mathbf{Z}(\mathbf{x}) \right| \text{ for } \ 0 \le \mathbf{x} \le L \tag{5}$$

Thus the Maximum Height of the Profile (RT) can be defined as the vertical distance between the deepest valley and highest peak.

$$\mathbf{R}\_{\rm T} = \mathbf{R}\_{\rm p} + \mathbf{R}\_{\rm v} \tag{6}$$

Fig. 14. Illustration of Maximum Height of the Profile (RT), Maximum Profile Valley Depth (Rv) and Maximum Profile Peak Height (Rp) for the surface profile (adapted from Predev, 2011).

The root mean square (RMS) is a statistical measure used in different fields. We cite, as an example, the use of the RMS amplitude applied to harmonic oscillators, such as on an alternating electric current. The root mean square of roughness (Rq) is a function that takes the square of the measures. The RMS roughness of a surface is similar to the roughness average, with the only difference being the mean squared absolute values of surface

> L 2

<sup>1</sup> R |Z x |dx <sup>L</sup> (3)

R max <sup>p</sup> Z(x) for 0 *x L* (4)

R minZ(x) <sup>v</sup> for 0 *x L* (5)

RRR Tpv (6)

0

The software does not need to be sophisticated in order to obtain Rq. For this reason much of the surface analysis equipment (profilometer and SPMs) provides Rq. In SPM, the Rq

The Rq is more sensitive to peaks and valleys than the average roughness due to the

The Maximum Profile Peak Height (Rp) is the measure of the highest peak around the surface profile from the baseline. Likewise the Maximum Profile Valley Depth (Rv) is the measure of the deepest valley across the surface profile analyzed from the baseline (Park,

Thus the Maximum Height of the Profile (RT) can be defined as the vertical distance between

Fig. 14. Illustration of Maximum Height of the Profile (RT), Maximum Profile Valley Depth (Rv) and Maximum Profile Peak Height (Rp) for the surface profile (adapted from Predev, 2011).

**6.1.3 Maximum height of the profile (RT), maximum profile valley depth (Rv) and** 

roughness profile. The function Rq is defined as (Gadelmawla et al., 2002):

q

**6.1.2 Root mean square roughness (Rq)** 

depends on the swept area of the sample, the scan size.

squaring of the amplitude in its calculation.

**maximum profile peak height (Rp)** 

the deepest valley and highest peak.

2011). We can write:

These parameters are useful when trying to find some very sharp peak, which could affect any application of the sample, a scratch or an unusual crack on the material. Often we use the average of these parameters, RTm, Rpm and Rvm and comprising the averages along the profile (x) on an evaluation length (L), given by (Park, 2011):

$$\mathbf{R\_{pm}} = \frac{1}{\mathbf{L}} \sum\_{i=\times}^{\rm L} \mathbf{R\_{pi}} \tag{7}$$

$$\mathbf{R}\_{\rm vm} = \frac{1}{\mathbf{L}} \sum\_{i=\mathbf{x}}^{\mathbf{L}} \mathbf{R}\_{\rm vi} \tag{8}$$

$$\mathbf{R}\_{\rm Tm} = \frac{1}{\mathbf{L}} \sum\_{i=\infty}^{\rm L} \mathbf{R}\_{\rm Ti} = \mathbf{R}\_{\rm pm} + \mathbf{R}\_{\rm vm} \tag{9}$$

These average parameters have the same advantage that the extreme parameters described in (4), (5) and (6), but lose accuracy when searching for singularities.

#### **6.1.4 Ten point average roughness - Rz (ISO)**

The Rz (ISO) is the arithmetic mean of the five highest peaks added to the five deepest valleys over the evaluation length measured. It is a parameter similar to RTm.

Fig. 15. Rz average of the sum of the five highest peaks in the five deepest valleys of sample's profile (Zygo Corporation, 2011).

#### **6.1.5 Third highest peak to third lowest valley height (R3Zi)**

The Third Highest Peak to Third Lowest Valley Height is a parameter that overlooks the two highest peaks and the two deepest valleys, taking as measure the vertical variation of the third highest peak from the third deepest valley. The advantage of using this parameter is that it disregards any discrepancies that do not interfere in a meaningful way in the characterization of the surface profile. This ignores the extreme values thus reducing the instability of peaks and valleys. It is a commonly used parameter for the characterization of porous surfaces and sealing surfaces (B.C. Macdonald & Co, 2011).

Measurement of the Nanoscale Roughness by

**6.2.3 High spot count** 

(Adapted from Park, 2011)

**6.2.4 Mean spacing** 

**6.2.5 Average wavelength** 

Wavelength is given by:

Atomic Force Microscopy: Basic Principles and Applications 161

The High Spot Count (HSC) is a parameter similar to the peak count. The main difference between these two parameters is in the defined peak. To be considered a peak in determining the Peak Count, the peak must be followed by a valley that crosses the entire band width (upper and lower threshold). For the High Spot Count, a threshold is set above the average roughness and only the peaks that exceed this one threshold are considered.

Fig. 18. Illustration of the High Spot Counting and the threshold determining a peak.

surfaces is of fundamental importance (B.C. Macdonald & Co, 2011).

The mean spacing (Sm) is generally described in μm or mm.

The use of parameters such as Peak Count, Peak Density and High Spot Count has its main application in sheet metal production where the quality control of coatings and paint

The Mean Spacing (Sm) is the average spacing between peaks in the length of evaluation. In this case the peak is defined as the highest point, along the profile, between a line crossing over the midline and returning below the midline. The spacing between peaks is the horizontal distance between the points where two peaks cross above the midline (Gadelmawla et al., 2002). Thus the mean spacing (Sm) is defined as the average of spacing individual (Si):

> L m i i 1 <sup>1</sup> S S L

The Average Wavelength (λa) is a parameter that relates the spacing between local peaks and valleys weighted by their individual frequencies and amplitudes. Thus, the Average

> <sup>a</sup> <sup>a</sup> a

<sup>R</sup> <sup>λ</sup>

(10)

<sup>Δ</sup> (11)

#### **6.2 Roughness spacing parameters**

The Roughness Spacing Parameters are the parameters that relate the roughness to the profile of curling and repetition over a surface.

Fig. 16. The two highest peaks (RP1 and RP2) and the two deepest valleys (RV1 and RV2) are disregarded. The R3zi is counted from the third highest peak (RP3) and the third deepest valley (RV3) (adapted from Predev, 2011).

#### **6.2.1 Peak count**

The Peak Count (Pc) is a parameter that provides the count of peaks analyzed along the length L of a surface profile. In this case, the computed peak is the "peak" crossing above an upper threshold and then below the lower threshold. Therefore only values of extreme peaks are significant and establishes a bandwidth in which only peaks and valleys beyond this range will be computed (Park, 2011). The Peak Count is expressed in peaks/inch or peaks/cm (Zygo Corporation, 2011).

Fig. 17. To determine the peak count only the peaks and valleys that exceed the bandwidth and return are considered (Zygo Corporation, 2011).

#### **6.2.2 Peak density**

Peak Density (PD) represents the density of peaks, i.e., the number of peaks per unit area (Zygo Corporation, 2011).

#### **6.2.3 High spot count**

160 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

The Roughness Spacing Parameters are the parameters that relate the roughness to the

Fig. 16. The two highest peaks (RP1 and RP2) and the two deepest valleys (RV1 and RV2) are disregarded. The R3zi is counted from the third highest peak (RP3) and the third deepest

The Peak Count (Pc) is a parameter that provides the count of peaks analyzed along the length L of a surface profile. In this case, the computed peak is the "peak" crossing above an upper threshold and then below the lower threshold. Therefore only values of extreme peaks are significant and establishes a bandwidth in which only peaks and valleys beyond this range will be computed (Park, 2011). The Peak Count is expressed in peaks/inch or

Fig. 17. To determine the peak count only the peaks and valleys that exceed the bandwidth

Peak Density (PD) represents the density of peaks, i.e., the number of peaks per unit area

**6.2 Roughness spacing parameters** 

profile of curling and repetition over a surface.

valley (RV3) (adapted from Predev, 2011).

peaks/cm (Zygo Corporation, 2011).

and return are considered (Zygo Corporation, 2011).

**6.2.1 Peak count** 

**6.2.2 Peak density** 

(Zygo Corporation, 2011).

The High Spot Count (HSC) is a parameter similar to the peak count. The main difference between these two parameters is in the defined peak. To be considered a peak in determining the Peak Count, the peak must be followed by a valley that crosses the entire band width (upper and lower threshold). For the High Spot Count, a threshold is set above the average roughness and only the peaks that exceed this one threshold are considered.

Fig. 18. Illustration of the High Spot Counting and the threshold determining a peak. (Adapted from Park, 2011)

The use of parameters such as Peak Count, Peak Density and High Spot Count has its main application in sheet metal production where the quality control of coatings and paint surfaces is of fundamental importance (B.C. Macdonald & Co, 2011).

#### **6.2.4 Mean spacing**

The Mean Spacing (Sm) is the average spacing between peaks in the length of evaluation. In this case the peak is defined as the highest point, along the profile, between a line crossing over the midline and returning below the midline. The spacing between peaks is the horizontal distance between the points where two peaks cross above the midline (Gadelmawla et al., 2002). Thus the mean spacing (Sm) is defined as the average of spacing individual (Si):

$$\mathbf{S\_m} = \frac{1}{L} \sum\_{i=1}^{L} \mathbf{S\_i} \tag{10}$$

The mean spacing (Sm) is generally described in μm or mm.

#### **6.2.5 Average wavelength**

The Average Wavelength (λa) is a parameter that relates the spacing between local peaks and valleys weighted by their individual frequencies and amplitudes. Thus, the Average Wavelength is given by:

$$
\lambda\_{\mathbf{a}} = \frac{\mathbf{R\_{a}}}{\Delta\_{\mathbf{a}}} \tag{11}
$$

Where Ra is the roughness average and Δa the mean slope of profile (Park, 2011).

Fig. 19. Three individual spaces in a surface profile. The mean spacing is the average of the three individual spaces on the evaluation length (Zygo Corporation, 2011).

#### **6.2.6 RMS average wavelength**

Similar to the Rq and Ra, RMS Average Wavelength (λq) takes as reference the root mean square of the spacing between peaks and valleys weighted by their individual frequencies and amplitudes (Park, 2011). RMS Average Wavelength can be calculated by:

$$
\lambda\_{\mathbf{q}} = 2\text{m}\frac{\mathbf{R}\_{\mathbf{q}}}{\Delta\_{\mathbf{q}}}\tag{12}
$$

Measurement of the Nanoscale Roughness by

of the tooth enamel (Takikawa et al., 2006).

at lower energies (Yoriyaz et al., 2009).

**7.4 Polymeric membranes** 

**7.2 Dental enamel** 

necessarily be considered.

(Heintze et al., 2006).

**7.3 X-ray anode** 

Atomic Force Microscopy: Basic Principles and Applications 163

The surface roughness acts as a gateway to allocate the ions in the network structure of the material. In collating materials, height and spacing parameters related to roughness are

The bleaching process can cause variation in the surface roughness of human tooth enamel.

The surface roughness is responsible for diffuse scattering of light incident on tooth enamel. Thus the surface roughness is an important variable in the bleaching process and must

The search for materials that do not significantly increase the roughness of the tooth surface is a challenge in dentistry. Bleaching agents based on nanoparticles of hidroxiapatatia have been shown to be effective by reducing the surface roughness and increasing the brightness

The brightness and surface roughness are closely associated by an inverse correlation. As shown in the figure 20, the gloss increases with decreasing roughness of tooth enamel

The anode is the positive electrode in an X-ray tube. It receives the impact of electrons accelerated by the potential difference due to the high voltage applied. The anode is generally made of materials that have high thermal dissipation, such as copper, molybdenum or rhenium. Depending on the x-ray application (i.e. energy of x-ray) a metal coating such as tungsten (W) or molybdenum (Mo) is placed over these thermally

A change in the spectral distribution of X-rays in a cathode ray tube with increasing roughness of the anode has been observed. The increased surface roughness implies an increase of characteristic peaks and a decrease corresponds to the lower energies of the bremsstrahlung spectrum, and an increase in the average energy of beams of X-rays (Nagel, 1988; Stears et al., 1986). The increased surface roughness implies an increase of characteristic peaks and a decreasing in the part corresponding to the lower energies of the bremsstrahlung spectrum. Unlike the filtration process for tungsten (W) where a dip occurs

In the area of environmental protection, a very significant technology is the process of separation by polymeric membranes. Polymeric membranes, such as Polysulfone / Blend Membrane PLURONIC F127, are used to separate the undesired solute in solution. Thus, the active area of the polymer membrane to carry out the process is the surface. The properties related to the surface are important for performing the separation process. Properties such as the pores size distribution, long-range electrostatic interactions and surface roughness are

factors that determine the efficiency of polymer membrane for this application.

essential to achieve the efficiency of the material (Cruz et al., 2002).

dissipative metals at the impact area of the accelerated electrons.

These changes are responsible for color changes, glare reduction, opacity.

Where Rq is the RMS roughness and Δq the RMS slope of profile (Park, 2011).

#### **7. Applications (materials)**

The roughness is a very significant parameter for various applications. The characterization of materials through its roughness allows one to obtain information on the efficiency of materials in various application areas.

#### **7.1 Electrochemical intercalation**

Some materials, usually transition metals oxides, are classified as intercalation materials. These materials are able to receive short radius ions (H+, Li+...) in its network structure via electrochemical techniques (such as cyclic voltammetry and electrochemical impedance spectroscopy). This process is known as electrochemical intercalation.

The electrochemical intercalation is a reversible process, making these materials very interesting in applications where the control over the ability of insertion/extraction of an ion in a structure is essential.

Intercalation materials are commonly used in micro-batteries, smart windows, smart mirrors, displays, gas sensors and other applications. These materials have potential applications due to the possibility of control of their electronic and optical properties.

The electro-physical-chemical properties of intercalation materials are strongly dependent on surface roughness.

The surface roughness acts as a gateway to allocate the ions in the network structure of the material. In collating materials, height and spacing parameters related to roughness are essential to achieve the efficiency of the material (Cruz et al., 2002).
