# Surface roughness

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Surface roughness often shortened to roughness, is a component of surface texture. It is quantified by the deviations in the direction of the normal vector of a real surface from its ideal form. If these deviations are large, the surface is rough; if they are small, the surface is smooth. In surface metrology, roughness is typically considered to be the high-frequency, short-wavelength component of a measured surface. However, in practice it is often necessary to know both the amplitude and frequency to ensure that a surface is fit for a purpose.

Surface finish, also known as surface texture or surface topography, is the nature of a surface as defined by the three characteristics of lay, surface roughness, and waviness. It comprises the small, local deviations of a surface from the perfectly flat ideal. In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

Surface metrology is the measurement of small-scale features on surfaces, and is a branch of metrology. Surface primary form, surface fractality and surface roughness are the parameters most commonly associated with the field. It is important to many disciplines and is mostly known for the machining of precision parts and assemblies which contain mating surfaces or which must operate with high internal pressures.

## Contents

Roughness plays an important role in determining how a real object will interact with its environment. In tribology, rough surfaces usually wear more quickly and have higher friction coefficients than smooth surfaces. Roughness is often a good predictor of the performance of a mechanical component, since irregularities on the surface may form nucleation sites for cracks or corrosion. On the other hand, roughness may promote adhesion. Generally speaking, rather than scale specific descriptors, cross-scale descriptors such as surface fractality provide more meaningful predictions of mechanical interactions at surfaces including contact stiffness  and static friction. 

Tribology is the science and engineering of interacting surfaces in relative motion. It includes the study and application of the principles of friction, lubrication and wear. Tribology is highly interdisciplinary. It draws on many academic fields, including physics, chemistry, materials science, mathematics, biology and engineering. People who work in the field of tribology are referred to as tribologists. Wear is the damaging, gradual removal or deformation of material at solid surfaces. Causes of wear can be mechanical or chemical. The study of wear and related processes is referred to as tribology. Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

Although a high roughness value is often undesirable, it can be difficult and expensive to control in manufacturing. For example, it is difficult and expensive to control surface roughness of fused deposition modelling (FDM) manufactured parts.  Decreasing the roughness of a surface usually increases its manufacturing cost. This often results in a trade-off between the manufacturing cost of a component and its performance in application. Manufacturing is the production of products for use or sale using labour and machines, tools, chemical and biological processing, or formulation. The term may refer to a range of human activity, from handicraft to high tech, but is most commonly applied to industrial design, in which raw materials are transformed into finished goods on a large scale. Such finished goods may be sold to other manufacturers for the production of other, more complex products, such as aircraft, household appliances, furniture, sports equipment or automobiles, or sold to wholesalers, who in turn sell them to retailers, who then sell them to end users and consumers.

Roughness can be measured by manual comparison against a "surface roughness comparator" (a sample of known surface roughness), but more generally a surface profile measurement is made with a profilometer. These can be of the contact variety (typically a diamond stylus) or optical (e.g.: a white light interferometer or laser scanning confocal microscope). A profilometer is a measuring instrument used to measure a surface's profile, in order to quantify its roughness. Critical dimensions as step, curvature, flatness are computed from the surface topography.

However, controlled roughness can often be desirable. For example, a gloss surface can be too shiny to the eye and too slippery to the finger (a touchpad is a good example) so a controlled roughness is required. This is a case where both amplitude and frequency are very important.

## Parameters

A roughness value can either be calculated on a profile (line) or on a surface (area). The profile roughness parameter (Ra, Rq,...) are more common. The area roughness parameters (Sa, Sq,...) give more significant values.

### Profile roughness parameters 

The profile roughness parameters are included in BS EN ISO 4287:2000 British standard, identical with the ISO 4287:1997 standard.  The standard is based on the ″M″ (mean line) system.

There are many different roughness parameters in use, but $Ra$ is by far the most common, though this is often for historical reasons and not for particular merit, as the early roughness meters could only measure $Ra$ . Other common parameters include $Rz$ , $Rq$ , and $Rsk$ . Some parameters are used only in certain industries or within certain countries. For example, the $Rk$ family of parameters is used mainly for cylinder bore linings, and the Motif parameters are used primarily in the French automotive industry.  The MOTIF method provides a graphical evaluation of a surface profile without filtering waviness from roughness. A motif consists of the portion of a profile between two peaks and the final combinations of these motifs eliminate ″insignificant″ peaks and retains ″significant″ ones. Please note that $Ra$ is a dimensional unit that can be micrometer or microinch. The micrometre or micrometer, also commonly known by the previous name micron, is an SI derived unit of length equalling 1×10−6 metre ; that is, one millionth of a metre.

Since these parameters reduce all of the information in a profile to a single number, great care must be taken in applying and interpreting them. Small changes in how the raw profile data is filtered, how the mean line is calculated, and the physics of the measurement can greatly affect the calculated parameter. With modern digital equipment, the scan can be evaluated to make sure there are no obvious glitches that skew the values.

Because it may not be obvious to many users what each of the measurements really mean, a simulation tool allows a user to adjust key parameters, visualizing how surfaces which are obviously different to the human eye are differentiated by the measurements. For example, $Ra$ fails to distinguish between two surfaces where one is composed of peaks on an otherwise smooth surface and the other is composed of troughs of the same amplitude. Such tools can be found in app format. 

By convention every 2D roughness parameter is a capital R followed by additional characters in the subscript. The subscript identifies the formula that was used, and the R means that the formula was applied to a 2D roughness profile. Different capital letters imply that the formula was applied to a different profile. For example, Ra is the arithmetic average of the roughness profile, Pa is the arithmetic average of the unfiltered raw profile, and Sa is the arithmetic average of the 3D roughness.

Each of the formulas listed in the tables assume that the roughness profile has been filtered from the raw profile data and the mean line has been calculated. The roughness profile contains $n$ ordered, equally spaced points along the trace, and $y_{i}$ is the vertical distance from the mean line to the $i^{\text{th}}$ data point. Height is assumed to be positive in the up direction, away from the bulk material.

#### Amplitude parameters

Amplitude parameters characterize the surface based on the vertical deviations of the roughness profile from the mean line. Many of them are closely related to the parameters found in statistics for characterizing population samples. For example, $Ra$ is the arithmetic average value of filtered roughness profile determined from deviations about the center line within the evaluation length and $Rt$ is the range of the collected roughness data points.

The arithmetic average roughness, $Ra$ , is the most widely used one-dimensional roughness parameter.

ParameterDescriptionFormula
Ra,  Raa, Ryni Arithmetical mean deviation of the assessed profile $Ra={\frac {1}{n}}\sum _{i=1}^{n}\left|y_{i}\right|$ Rq, Rms  Root mean squared $Rq={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}y_{i}^{2}}}$ RvMaximum valley depth$Rv=\min _{i}y_{i}$ RpMaximum peak height$Rp=\max _{i}y_{i}$ Rt, RyMaximum Height of the Profile$Rt=Rp-Rv$ Rsk Skewness $Rsk={\frac {1}{nRq^{3}}}\sum _{i=1}^{n}y_{i}^{3}$ Rku Kurtosis $Rku={\frac {1}{nRq^{4}}}\sum _{i=1}^{n}y_{i}^{4}$ RzDIN, RtmAverage distance between the highest peak and lowest valley in each sampling length, ASME Y14.36M - 1996 Surface Texture Symbols$RzDIN={\frac {1}{s}}\sum _{i=1}^{s}Rt_{i}$ , where $s$ is the number of sampling lengths, and $Rt_{i}$ is $R{\text{t}}$ for the $i^{\text{th}}$ sampling length.
RzJISJapanese Industrial Standard for $Rz$ , based on the five highest peaks and lowest valleys over the entire sampling length.$R{\text{zJIS}}={\frac {1}{5}}\sum _{i=1}^{5}Rp_{i}-Rv_{i}$ , where $Rp_{i}$ and $Rv_{i}$ are the $i^{\text{th}}$ highest peak, and lowest valley respectively.

Here is a common conversion table with also roughness grade numbers:

Roughness, NRoughness values, RaRMSCenter line avg., CLARoughness, Rt
ISO grade numbersmicrometers (µm)microinches (µin.)(µin.)(µm)
N1250200022002000200
N1125100011001000100
N1012.550055050050
N96.325027525025
N83.2125137.512513
N71.66369.3638
N60.83235.2324
N50.41617.6162
N40.288.881.2
N30.144.440.8
N20.0522.220.5
N10.02511.110.3

#### Slope, spacing, and counting parameters

Slope parameters describe characteristics of the slope of the roughness profile. Spacing and counting parameters describe how often the profile crosses certain thresholds. These parameters are often used to describe repetitive roughness profiles, such as those produced by turning on a lathe.

ParameterDescriptionFormula
$Rdq,R\Delta q$ the RMS of the profile within the sampling length$Rdq={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\Delta _{i}^{2}}}$ $Rda,R\Delta a$ the average absolute slope of the profile within the sampling length$Rda={\frac {1}{N}}\sum _{i=1}^{N}|\Delta _{i}|$ $\Delta _{i}\!$ where delta i is calculated according to ASME B46.1 and is a 5th order Savitzky–Golay smoothing filter $\Delta _{i}={\frac {1}{60dx}}(y_{i+3}-9y_{i+2}+45y_{i+1}-45y_{i-1}+9y_{i-2}-y_{i-3})$ Other "frequency" parameters are Sm, $\lambda$ a and $\lambda$ q. Sm is the mean spacing between peaks. Just as with real mountains it is important to define a "peak". For Sm the surface must have dipped below the mean surface before rising again to a new peak. The average wavelength $\lambda$ a and the root mean square wavelength $\lambda$ q are derived from $\Delta$ a. When trying to understand a surface that depends on both amplitude and frequency it is not obvious which pair of metrics optimally describes the balance, so a statistical analysis of pairs of measurements can be performed (e.g.: Rz and $\lambda$ a or Ra and Sm) to find the strongest correlation.

Common conversions:

#### Bearing ratio curve parameters

These parameters are based on the bearing ratio curve (also known as the Abbott-Firestone curve.) This includes the Rk family of parameters. Sketches depicting surfaces with negative and positive skew. The roughness trace is on the left, the amplitude distribution curve is in the middle, and the bearing area curve (Abbott-Firestone curve) is on the right.

#### Fractal theory

The mathematician Benoît Mandelbrot has pointed out the connection between surface roughness and fractal dimension.  The description provided by a fractal at the microroughness level may allow the control of the material properties and the type of the occurring chip formation. But fractals cannot provide a full-scale representation of a typical machined surface affected by tool feed marks, it ignores the geometry of the cutting edge. (J. Paulo Davim, 2010, op.cit.). Fractal descriptors of surfaces have an important role to play in correlating physical surface properties with surface structure. Across multiple fields, connecting physical, electrical and mechanical behavior with conventional surface descriptors of roughness or slope has been challenging. By employing measures of surface fractality together with measures of roughness or surface shape, certain interfacial phenomena including contact mechanics, friction and electrical contact resistance,can be better interpreted with respect to surface structure. 

### Areal roughness parameters

Areal roughness parameters are defined in the ISO 25178 series. The resulting values are Sa, Sq, Sz,... Many optical measurement instruments are able to measure the surface roughness over an area. Area measurements are also possible with contact measurement systems. Multiple, closely spaced 2D scans are taken of the target area. These are then digitally stitched together using relevant software, resulting in a 3D image and accompanying areal roughness parameters.

## Soil-surface roughness

Soil-surface roughness (SSR) refers to the vertical variations present in the micro- and macro-relief of a soil surface, as well as their stochastic distribution. There are four distinct classes of SSR, each one of them representing a characteristic vertical length scale; the first class includes microrelief variations from individual soil grains to aggregates on the order of 0.053–2.0 mm; the second class consists of variations due to soil clods ranging between 2 and 100 mm; the third class of soil surface roughness is systematic elevation differences due to tillage, referred to as oriented roughness (OR), ranging between 100 and 300 mm; the fourth class includes planar curvature, or macro-scale topographic features  .

The two first classes account for the so-called microroughness, which has been shown to be largely influenced on an event and seasonal timescale by rainfall and tillage, respectively. Microroughness is most commonly quantified by means of the Random Roughness, which is essentially the standard deviation of bed surface elevation data around the mean elevation, after correction for slope using the best-fit plane and removal of tillage effects in the individual height readings.  Rainfall impact can lead to either a decay or increase in microroughnesss, depending upon initial microroughness conditions and soil properties.  On rough soil surfaces, the action of rainsplash detachment tends to smoothen the edges of soil surface roughness, leading to an overall decrease in RR. However, a recent study which examined the response of smooth soil surfaces on rainfall showed that RR can considerably increase for low initial microroughness length scales in the order of 0 - 5 mm. It was also shown that the increase or decrease is consistent among various SSR indices  .

## Practical effects

Surface structure plays a key role in governing contact mechanics,  that is to say the mechanical behavior exhibited at an interface between two solid objects as they approach each other and transition from conditions of non-contact to full contact. In particular, normal contact stiffness is governed predominantly by asperity structures (roughness, surface slope and fractality) and material properties.

In terms of engineering surfaces, roughness is considered to be detrimental to part performance. As a consequence, most manufacturing prints establish an upper limit on roughness, but not a lower limit. An exception is in cylinder bores where oil is retained in the surface profile and a minimum roughness is required. 

Surface structure is often closely related to the friction and wear properties of a surface.  A surface with a higher fractal dimension, large $Ra$ value, or a positive $Rsk$ , will usually have somewhat higher friction and wear quickly. The peaks in the roughness profile are not always the points of contact. The form and waviness (i.e. both amplitude and frequency) must also be considered.

## Related Research Articles In physics, a standing wave, also known as a stationary wave, is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes. In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. In mathematics, Lyapunov fractals are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, thus represent a universality. In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions spliced together back-to-back, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution. A leadscrew, also known as a power screw or translation screw, is a screw used as a linkage in a machine, to translate turning motion into linear motion. Because of the large area of sliding contact between their male and female members, screw threads have larger frictional energy losses compared to other linkages. They are not typically used to carry high power, but more for intermittent use in low power actuator and positioner mechanisms. Common applications are linear actuators, machine slides, vises, presses, and jacks. X-ray reflectivity is a surface-sensitive analytical technique used in chemistry, physics, and materials science to characterize surfaces, thin films and multilayers. It is related to the complementary techniques of neutron reflectometry and ellipsometry. An optical flat is an optical-grade piece of glass lapped and polished to be extremely flat on one or both sides, usually within a few tens of nanometres. They are used with a monochromatic light to determine the flatness of other surfaces, whether optical, metallic, ceramic, or otherwise, by interference. When an optical flat is placed on another surface and illuminated, the light waves reflect off both the bottom surface of the flat and the surface it is resting on. This causes a phenomenon similar to thin-film interference. The reflected waves interfere, creating a pattern of interference fringes visible as light and dark bands. The spacing between the fringes is smaller where the gap is changing more rapidly, indicating a departure from flatness in one of the two surfaces. This is comparable to the contour lines one would find on a map. A flat surface is indicated by a pattern of straight, parallel fringes with equal spacing, while other patterns indicate uneven surfaces. Two adjacent fringes indicate a difference in elevation of one-half wavelength of the light used, so by counting the fringes, differences in elevation of the surface can be measured to better than one micrometre.

Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindrical roller for a bearing. In geometric dimensioning and tolerancing, control of a cylinder can also include its fidelity to the longitudinal axis, yielding cylindricity. The analogue of roundness in three dimensions is sphericity. Contact mechanics is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces and frictional stresses acting tangentially between the surfaces. This page focuses mainly on the normal direction, i.e. on frictionless contact mechanics. Frictional contact mechanics is discussed separately. Normal stresses are caused by applied forces and by the adhesion present on surfaces in close contact even if they are clean and dry.

ISO 25178: Geometric Product Specifications (GPS) – Surface texture: areal is an International Organisation for Standardisation collection of international standards relating to the analysis of 3D areal surface texture.

Smoothing splines are function estimates, , obtained from a set of noisy observations of the target , in order to balance a measure of goodness of fit of to with a derivative based measure of the smoothness of . They provide a means for smoothing noisy data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where is a vector quantity. In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

Waviness is the measurement of the more widely spaced component of surface texture. It is a broader view of roughness because it is more strictly defined as "the irregularities whose spacing is greater than the roughness sampling length". It can occur from machine or work deflections, chatter, residual stress, vibrations, or heat treatment. Waviness should also be distinguished from flatness, both by its shorter spacing and its characteristic of being typically periodic in nature.

In laser science, laser beam quality defines aspects of the beam illumination pattern and the merits of a particular laser beam's propagation and transformation properties. By observing and recording the beam pattern, for example, one can infer the spatial mode properties of the beam and whether or not the beam is being clipped by an obstruction; By focusing the laser beam with a lens and measuring the minimum spot size, the number of times diffraction limit or focusing quality can be computed. In probability theory, an exponentially modified Gaussian (EMG) distribution describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component. As described here, white light interferometry is a non-contact optical method for surface height measurement on 3-D structures with surface profiles varying between tens of nanometers and a few centimeters. It is often used as an alternative name for coherence scanning interferometry in the context of areal surface topography instrumentation that relies on spectrally-broadband, visible-wavelength light.

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