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Material Jetting
Material jetting (MJ)
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Material jetting also uses photopolymers, but instead of a vat, MJ jets droplets of material layer by layer from a nozzle onto the build surface in a similar fashion to 2D ink jet printers.
Initially, the liquid is heated to roughly 30-60°C to reach an optimum viscosity for jetting the liquid. The print head travels over the build surface, depositing droplets of liquid. The material is then cured by a UV light to solidify and harden the photopolymer. This process is repeated layer by layer. Deposition of droplets in this way results in less waste than either SLA or DLP.MJ requires support structures for any overhangs and usually doesnt require post curing, unlike SLA, due to smaller layer heights.
Benefits and Limitations
Potentially the largest attraction to MJ is that multiple nozzles can be set up which then jet different plastics (of different colourings of plastic) to either give different properties or colour to different sections of a print. Different colourings can also be mixed to give specific hues. In addition, its common for support structures to be made from a secondary dissolvable material that can be removed by either pressurised water or an ultrasonic bath. Dissolvable support structures like this can leave no mark once removed, maintaining high quality surface finish.
Other benefits include lower waste than SLA or DLP due to using jetted droplets rather than a vat (as mentioned earlier). Also, print quality is very high, meaning MJ has very smooth surface finishes and dimensionally accurate prints.
MJ shares some of the same disadvantages as SLA and DLP, such as the properties of the plastic making it unsuitable for many applications due to the brittle nature of printed objects. Polymers used are also photosensitive and break down over time and, finally, printing is expensive.
Powder bed fusion
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Powder bed fusion (PBF) refers to a number of methods where, common for each process, material powder is heated in a chamber and fused a layer at a time. Once a layer is formed, the build platform is lowered by an amount equal to one layer and new powder is spread over previous layers either by a blade or roller. For the case of polymer AM, there are two common methods of powder bed fusion - Selective Laser Sintering (SLS) and Multi Jet Fusion (MJF).
Selective Laser Sintering (SLS)
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SLS is analogous to SLA, using a moveable laser to selectively sinter polymer powder in layers. Initially, a thin layer of powder covers a platform inside a chamber. The chamber is held just under the melting point of the polymer so that when a laser is applied, the powder begins to melt, sintering and fusing together. Multi Jet Fusion (MJF)
MJF, instead of using a laser, uses nozzles to drop a “binding agent” onto the surface of the powder bed. Just like MJ, this is done in a similar way to how 2D printers jet ink. Additional agents can be added to help define boundaries or give specific colour to individual voxels (a 3D pixel). However, currently, the choice of colour is limited. The binding agents will define whether a voxel is part of the structure or will remain as powder. The agents have high absorption of IR radiation, so after the agents are jetted, an IR light passes over the powder bed to locally heat the powder in locationscontaining the binding agent, causing the powder to melt and fuse. Post processing is minimal, unlike other AM methods which require the removal of support structures, and mainly consists of cleaning excess powder.
Benefits and Limitations
The main benefit of PBF is that there is no need for support structures, as the surrounding powder supports the forming object, enabling more design options to manufacturers. This links to a second benefit, which is that as no material is wasted on support structures and, as powder is reusable, little waste is produced.
MJF specifically can more easily and quickly produce a larger number of objects at once by utilising the entire print volume. While it cant match injection moulding for high volumes of objects, at low volume production, MJF is cost equivalent.
PBF processes have a number of downsides. Both methods result in rough surface finishes, with roughness depending on powder particle size (but almost no visible layer lines as an upside). This is because powder particles at the edge of a voxel that are being heated by a laser or IR radiation have a reasonable chance to partially sinter, binding to the surface of the desired shape.
The choices of material available to SLS and MJF are mostly limited to various nylons, in turn limiting properties. Printed objects also tend to be fairly weak, reducing the potential uses of any printed object.
All the PBF methods are also very energy intensive. This is due to having to keep the powder in a heated chamber (so the polymer melts readily when more heat is applied) which must be reheated for each print. This can lead to further problems such as potentially affecting unused powder in the chamber, rendering it unusable. Also, as the prints experience heating and cooling, it is possible for prints to warp.
Finally, similarly to the vat photopolymerisation methods, hollow shapes cannot be formed as powder would be unable to drain away in an enclosed shape.
Material Extrusion
Material extrusion AM methods operate under the principal of using a nozzle to extrude and eject hot plastic onto a build surface. There are two slightly varying methods of material extrusion which are Fused Deposition Modelling (FDM) and Arburg Plastic Freeforming (APF) (translated from the German name - Arburg Kunststoff Freiformen).
Fused Deposition Modelling (FDM)
FDM is what most people imagine as 3D printing due to being the most common type of additive manufacturing, mainly due to its low cost making it accessible to industry and hobbyists. It also has a relatively large selection of materials available to use, as numerous thermoplastics can be printed using FDM. The most common plastics used are ABS and PLA.
FDM uses a spool of plastic that feeds a thread of plastic (known as a filament) through a nozzle. The nozzle is on a print head which has the operation of mechanically forcing the filament through the print head in a cold section, then heating and melting the plastic in a hot section before extruding the plastic through the nozzle. Now melted and extruded, the plastic is printed directly onto the surface of a build in a continuous stream. The print nozzle moves in the x-y plane to control where the plastic is placed. Once the layer is completed, the platform the object is on can move downwards in the z direction by a small amount for the next layer to be built.As the deposited plastic is hot, it is able to fuse to layers below, softening and binding to the surface of the previous layer. This leads to anisotropic properties as this fuse is weaker than the extruded thread of material, meaning printed objects are generally weaker in the z-direction. For more on anisotropy in , see the properties of FDM section
Arburg Plastic Freeforming (APF)
APF works similarly to FDM with a few notable differences. The first of which is that the plastic supplied is in pellets. These melted pellets are then forced along to a nozzle using a rotating screw, similar to injection moulding. Once in the nozzle, the nozzle periodically opens and closes, letting out individual droplets of melted plastic onto the build surface.. The platform being printed on can move in the xyz directions to control where the droplets are placed. Droplet size can be controlled by changing the nozzle diameter. APF and FDM usually print onto a hot surface (the exact temperature may depend on the plastic being used) to reduce warping (plastic can warp due to shrinkage on rapid cooling), and give better adhesion between the print and print bed.
Both process need support structures, so post processing mostly involves the removal of these structures. Some more advanced printers are able to print multiple plastics at once so support structures can be printed in a material, such as PVA, that can be easily removed (usually using water). Further post processing can involve smoothing the surface of the object as especially FDM can give poor surface finish, however surface finish can vary quite a lot between printer models.
Benefits and Limitations
As mentioned earlier, material extrusion (or specifically FDM) printing is cheap and widely accessible. The common plastics used are also cheap meaning FDM is ideal for amateurs.
Also mentioned earlier, material extrusion has many plastics available to it, including polymers infused with other materials. This enables more choice over properties, with each polymer having its own pros and cons when used to print.
On the other hand, material extrusion can suffer from lower resolution and worse quality prints than photopolymerisation or PBF processes. Surface finishes are bumpy with usually easily visible layer lines. This can lead to more time post processing to achieve a smooth surface.
Binding between layers can be poor, especially for some polymers. To explain what this means for the print, read the section.
Print failure can also be quite common. This is a problem for all AM methods, but FDM is susceptible to quite a few potential faults in printing e.g. fast heating then cooling of plastic can lead to thermal contraction as the object is printed, resulting in the print warping. *For more on problems that can occur during printing, you can read this article:* *.*
Properties of FDM Prints
Additive manufacturing methods currently struggle with recreating the mechanical properties of parts made by traditional production methods, such as injection moulding. This section explores the material properties of FDM prints and what may be causing these properties.
Bonding
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Fundamental to understanding the properties of FDM prints is understanding how the printed lines of plastic (an individual line also being known as a raster) bond together. Bonding is caused when hot plastic is ejected from the nozzle and is applied to the print surface, partially melting neighbouring plastic and entangling polymer chains. The strength of the bond can be modelled as a temperature-dependant diffusion model, where greater temperature for a longer time results in greater diffusion and, subsequently, greater entanglement. The processes of entangling like this can be referred to as the plastic adhering to itself.
This method of bonding has a of couple problems. Firstly, the extent of entanglement of chains will be far less **between** rasters than **within** rasters. Less entanglement means less force is required to break apart the chains so the bonding is weaker. The second problem is that gaps, known as voids, can form and reduce the overall cross-sectional area that contributes to the strength of a build. Voids develop due to the nature of the geometry of the rasters and layers which, when stacked in a print, leave holes from inefficient packing. In addition, printing errors can mean rasters or layers dont fully adhere, also creating voids between lines. The colder a neighbouring line of plastic is to the one being printed, the more likely the bonding between lines will be weaker, with a greater chance of void formation.
Diagram showing section of printed FDM lines with voids from lack of full adhesion and line geometry
The result is that strength parallel to a set of printed lines is high, while perpendicular is weak (so is anisotropic), with values depending on polymer strength and the extent of adhesion between rasters and layers.
### Infill Pattern
To combat anisotropy in the x-y plane, FDM printers print specific structures that have the same strength in both x and y by increasing the strength of one direction while decreasing it in the other. A common structure would be a crosshatch pattern with roughly half the material pointing in the x direction, and half in the y, meaning each direction has equal amounts of the strength from rasters parallel to that direction, and the weak bonding from rasters perpendicular.
Diagram showing rasters printed in alternating directions parallel to x and y. No lines are drawn vertically in z direction
Below is an SEM picture of an FDM printed sample, showing voids that have formed between between rasters. Also seen through the void is the layer below, which is pointing perpendicular to the viewed top layer.
SEM picture of an FDM print. A void can be seen in the top layer. Through the void, rasters of the layer below can be seen pointing perpendicular to the top layer.
It is worth noting is that most printers arent made to produce objects that are 100% filled with plastic. Instead, a printer will print the outside of a layer as a solid line, and then fill in the gaps in the centre with a predetermined “infill pattern” so a specific percentage of the area inside is filled. This is beneficial as it saves plastic so is more economical, but the structure will lose some of its strength. A reduction in fill percentage will mean a reduction in cross sectional area of plastic providing strength under tension. Therefore, as a general rule, lower fill percentage means lower strength strength (assuming other properties of the print are the same). The magnitude of this decrease will depend on the fill pattern used. The print will also be lighter, which may be important depending on the application of the object.
An example set of fill patterns, known as grid patterns, are shown below at different fill percentages, followed by an example vertical cross section showing where the infill would go.
Diagram showing the shape of infill patterns in the x-y plane
Diagram showing vertical cross section of a sample
The infill pattern can be described by its “raster angle”: the angle between the path of the nozzle and the x-axis of the printing platform during FDM. . For the 30% and 50% infills shown above, the raster angle alternates between ±45° for a layer, then alternates between 45° and 135° for the next. The difference between the patterns is how closely spaced the lines are printed. This isnt the only way to produce a pattern like this, however. The two patterns could be printed by keeping raster angle at 45° for a layer, then at -45° for the next, but this may affect properties. The 90% fill differs by having raster angle alternating between 0° and 90° layer by layer.
The exact effect fill pattern has on strength can be quite complex, with many different factors coming into play. For instance, different patterns can result in different adhesion strengths. As mentioned previously, higher temperature of neighbouring filament will increase the extent of adhesion by improving the ability of plastic to entangle. Therefore, a fill pattern that prints lines that can adhere soon after being printed will be hotter when bonding and, therefore, form stronger bonds. An example of such a pattern would be a Hilbert Curve, which has very short times before printed lines are able to bond to each other.
Diagram of Hilbert curve pattern
Print angle
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Of course, the pattern will also affect how much material is aligned at particular angles with respect to a loading direction, which can in turn affect strength. It has been observed that rasters at 0° to loading direction are strongest, and then weaken with increasing angle up to 90°. This can be explained by considering how shear and normal stresses change with angle and the resulting force acting parallel and perpendicular to the rasters. At 0°, rasters are loaded parallel to their printed direction where they are strong (failure requires the rasters to break which requires high stress to overcome the high levels of entanglement). As angle increases, a greater proportion of stress is applied perpendicular to the raster, where failure can occur by rasters breaking from each other (often known as delamination). As tensile strength between rasters is significantly lower, failure will occur at lower stresses. At 90°, the force will only act perpendicular to the rasters so strength is at a minimum.
Analysis like this can become complex when considering patterns with multiple directions printed, such as the crosshatch and Hilbert curve, which have rasters at 0° and 90°. Rotating print pattern relative to the loading direction will now increase raster angle for some rasters but decrease it for the rest. On rotating 0° or 900, the pattern will have the same strength (raster angle for both cases will be 0°/90° so strength is now symmetrical about rotation by 45° where raster angle is -45°/45°). However, the angle at which the rasters are strongest can vary. One argument is that at 45° (raster angle -45°/45°), all the rasters are under normal and shear stresses so all rasters are at risk of delamination so strength is low, while at 0° (raster angle 0°/90°) half the rasters very easily delaminate as they face perpendicular to the loading direction, but the other half are orientated parallel so produce a strong sample. However, it could be argued for 0° that the perpendicular rasters can fail before the ones parallel, thereby increasing the stress on the parallel rasters from the reduction of cross-sectional area, leading the parallel rasters to fail as well, meaning a print with raster angles at 0° and 90° could be the weakest orientation. Sometimes, it can even be seen that strength doesnt vary much with angle. Ultimately, which direction is strongest would depend on many variables, including adhesion strength, type of infill pattern and infill percentage.
While these patterns help prevent anisotropy in the x-y plane, they do not combat anisotropy in the z direction as, in all cases for FDM, no lines are printed vertically so strength is entirely dependent on adhesion between layers, making the z direction significantly weaker than the other two.
Diagram showing two samples fracturing, one printed so rasters are parallel to loading direction (printed flat and loaded in the x/y direction), and one printed so rasters are facing perpendicular to loading direction (printed vertically then loaded in the z direction).
The may be useful to read as it goes into further detail about similar concepts described here.
Choice of material
Choice of material can also have large effects on properties, and may even cause different challenges on printing. PLA is the most commonly used plastic for FDM printers mainly due to ease of printing. As the glass transition (*T*g) and melting (*T*m) temperature are relatively low (~60°C and ~160°C respectfully), nozzle temperature can be held quite low on printing (~215°C) meaning risk of warping is reduced. PLA is also biodegradable so, being made from plant starch, is an environmentally friendly choice of material. However, PLA is quite brittle, and its low *T*g means it is unsuitable for high temperature applications.
The second most common plastic used is ABS. ABS has a *T*g of ~105°C, making it more suitable for higher temperature applications. However, this comes at a cost. ABS must be printed at a higher temperature than PLA so cools more rapidly when printing and, therefore, ABS is more likely to warp or crack during a print. Overall, ABS is more ductile, durable, and can be used at higher temperatures than PLA, which can make it a desirable choice of plastic in some cases. However, despite having similar tensile strengths to PLA when made using other methods, ABS doesnt adhere to itself as well as PLA, therefore meaning the tensile strength of ABS prints are usually lower than equivalent PLA prints.
Simulation
Below is a simulation that shows a stress strain curve for an FDM printed plastic object under tension where you can choose certain variables about the print. The values shouldnt be taken as exact as many variables are not being considered here which may affect properties. Variables such as print speed, print temperature, and even the exact batch of plastic can all affect the final properties of a print. The simulation also shows prints at 45° being weaker, but this may not be the case for all printers, plastics or infill patterns. *(When rotating print direction, the infill pattern is still printed at the same raster angle relative to the print bed, meaning it changes the angle relative to the print body so effectively gives a new raster angle when loading the sample e.g. a print with original raster angles ±45° rotated by 45° now has **effective** raster angles of 0°/90°).*
For comparison, general purpose PLA is quoted at having a tensile strength between 47-70MPa while injection moulded ABS has tensile strength 42-46MPa *(CES EduPack)*.
Additive manufacturing other materials
Additive manufacturing methods extends past polymers, also being available to metals and other materials using processes that are fairly analogous to those used for polymers. A list of these processes with brief descriptions is given below.
Metal
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* Fused Deposition Modelling, **FDM**
+ This is directly analogous to FDM for polymers, except using molten metal instead.* Selective Laser Melting, **SLM**
+ Type of powder bed fusion. Uses a laser to melt and bind metal in a powder bed. Analogous to SLS.* Electron Beam Melting, **EBM**
+ Type of powder bed fusion. Like SLS but uses an electron beam to melt the powder instead, and therefore, this process must be done in a vacuum.* Laser Engineering Net Shape, **LENS**
+ Type of direct energy deposition, DED. This can be seen as a mix between FDM and soldering. Metal wire is supplied to a build where is it then heated and melted by a laser, locally depositing metal onto the build surface* Electron Beam Additive Manufacturing, **EBAM**
+ Type of DED. Similar to LENS but an electron beam is used to heat the metal instead and therefore, this process must be done in a vacuum.* Binder Jetting, **BJ**
+ Binding agent added to metal powder, sticking the powder together. The bound powder is then later sintered once out of the powder bed. This is fairly similar to MJF without the use of IR radiation to immediately sinter the object.* Nano Particle Jetting, **NPJ**
+ Metal particles dissolved in a solvent liquid are applied to a print surface by nozzles. The object is heated immediately, evaporating the solvent leaving the metal particles. The object is then sintered later. The printing process is similar to MJ, but requires extra sintering and doesnt use UV.
Other Materials
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* Fused Deposition Modelling, **FDM**
+ Same process as for metals and polymers. Material filament is melted and extruded through a nozzle.* Paste Extrusion Modelling, **PEM**
+ Very similar to FDM but used for materials that are a paste at room temperature, such as cement paste. The printing works in the same way except there is no heating element. Also, instead if filament, simply a paste supply is extruded through a nozzle onto the print surface.* Binder Jetting, **BJ**
+ Similar to metal BJ and polymer MJF. Binding agent droplets are applied layer by layer to a material powder bed, causing powder to stick. Generally used for sand or gypsum.* Drop on demand, **DOD**
+ DOD is a type of material jetting (MJ) and is similar to the MJ process for polymers. Hot material is jetted dropwise through nozzles, layer by layer onto the print surface. Material then solidifies on cooling. An example material used for this is wax.* Laminated Object Manufacturing, **LOM**
+ Nozzles apply an adhesive to the top of a build surface. A new layer of material is then *laminated* onto the previous layer, bound by the adhesive. The layer is then cut to give the correct cross section by a knife, laser or wire. The process is repeated to produce an object.
The table below shows a summary of processes that are similar to those used for polymers.
Polymer | Metal | Ceramic/Other || MJF | BJ | BJ |
| SLS | SLM / EBM | |
| FDM | FDM | FDM / PEM |
| APF | | |
| MJ | NPJ | DOD |
| SLA | | |
| DLP | | |
| | | LOM |
| | LENS / EBAM | |
And the following table organises each method into broader AM methods.
Powder Bed Fusion | Direct Energy Deposition | Material Extrusion | Binder Jetting | Material Jetting | Photo-polymerisation | Sheet Lamination || MJF | LENS | All FDM types | Metal BJ | Polymer MJ | SLA | LOM |
| SLS | EBAM | Sand or gypsum BJ | NPJ | DLP | |
| SLM | | APF | DOD | | | |
| EBM | | | | | | |
Looking to the future, the number of materials available to additive manufacturing will continue to increase. One such material is organic matter (printing organic matter is sometimes known as bio printing), with ambitions of printing tissue to test drugs, and entire organs potentially for transplants.
Summary
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Prints have separate resolution in the x-y plane (determined by minimum movement of a printing nozzle/laser) and z direction (determined by layer height). This is then separate from minimum feature size which is determined by diameter of the nozzle/laser point.
To help distinguish the various methods of polymer additive manufacturing, the benefits, limitations and usage of each are summarised in the table below.| | | | |
| - | - | - | - |
| **Type of AM** | **Benefits** | **Downsides** | **Best for** |
| **Material Extrusion (geared towards FDM)** | * Cheap
* Many materials available
| * Bad quality print
* Significant anisotropic effects
* Good chance of warping from heat
| * Amateur use
* Rapid prototyping
|
| **Photopolymerisation** | * High quality prints
* Isotropic
| * Very limited materials available
* Lots of post-processing
* No hollow sections
* Photosensistive
| * Functional prototyping
* Moulds
|
| **Material Jetting** | * Multiple materials possible at once
* Good quality prints
* Very smooth surface finish
| * Very expensive
* Limited materials
* Photosensitive
| * Rapid prototyping
* Coloured, custom models
* Moulds
|
| **Powder Bed Fusion** | * No support structures
* Mass production easier with MJF
* Relatively good quality prints
* Limited post-processing
| * Limited materials available
* Rough surface finish
* No hollow sections
* Chance of warping from heat
| * Functional prototyping
* Custom end-use models
|It is important to note that all additive manufacturing methods struggle to control properties with the precision and variety that traditional methods have. Limited material pool size and printing effects such as the anisotropy in FDM prints prevent additive manufacturing being seen as a better alternative to existing methods. Combined with the fact that additive manufacturing is generally quite slow and unable to produce products in bulk, AM is mostly being used in pre-production for prototyping, to produce parts that are used to make moulds which then allow fast production, and in printing one-off custom objects such as hearing aids (which require specific shapes to fit in the users ear).
Looking specifically at FDM, properties are highly anisotropic and can vary greatly depending on variables such as infill percentage, fill pattern used and print direction. How you create a print should be considered when making a product with its purpose in mind so these variables can be chosen appropriately.
Questions
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### Select the correct order of the AM manufacturing process.### Quick questions*You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!*1. Which of the following processes uses material extrusion to print? | | | |
| - | - | - |
| |
a |
Arburg Plastic Freeforming, APF |
| |
b |
Selective Laser Sintering, SLS |
| |
c |
Material Jetting, MJ |
| |
d |
Digital Light Processing, DLP |### Deeper questions*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*2. How does Stereolithography bond layers together? | | | |
| - | - | - |
| |
a |
Powder heats up, melting and fusing with the layer below |
| |
b |
Newly deposited material is hot, partially melting the surface of the previous layer and entangles polymer chains, fusing |
| |
c |
The previous layer is left in a green state so that the surface is left partially reacted. When the new layer is cured, the previous layer also reacts and binds |
| |
d |
New layers are bound by adhesive which is applied between each layer as its printed |
3. Which of the following prints of a tensile test sample would you expect to have the highest ultimate tensile strength? | | | |
| - | - | - |
| |
a |
PLA, 90% fill, loaded in z direction |
| |
b |
PLA, hollow, printed at 45° to the x direction and loaded in that direction. |
| |
c |
ABS, 30% fill, loaded in x direction |
| |
d |
PLA, 30% fill, printed on its side and loaded in the x direction. |
4. You wish to print a hollow section for a prototype. Which method of printing should you use? | | | |
| - | - | - |
| |
a |
Stereolithography, SLA |
| |
b |
Electron Beam Melting, EBM |
| |
c |
Multi Jet Fusion, MJF |
| |
d |
Fused Deposition Modelling, FDM |
5. Which factor usually limits resolution in FDM? | | | |
| - | - | - |
| |
a |
Layer height |
| |
b |
Minimum movement of nozzle in x-y plane |
| |
c |
Nozzle diameter |
| |
d |
Print temperature |
6. What is a key limitation that applies to all additive manufacturing processes? | | | |
| - | - | - |
| |
a |
Expensive |
| |
b |
Lack of control over material properties |
| |
c |
Printing has visible layer lines |
| |
d |
Warping on printing |
Going further
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**Books**
*Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital Manufacturing*, I.Gibson**Other**
is a good site which goes into a little more detail about using 3D printing.
I recommend watching videos of each process to fully grasp manufacturing, and also viewing images of objects made by each process to see how final print quality differs.
###
|
Aims
On completion of this TLP you should:
* Understand the basic principles of atomic force microscopy (AFM), including the different modes it can be used in.
* Understand how AFM can be used in materials science.
* Be aware of some of the problems that can be encountered, and how to overcome them.
Before you start
You should have a basic understanding of the behaviour of to understand how the piezo-scanner works in AFM.
Introduction
Atomic force microscopy (AFM) is part of the family of techniques known as scanning probe microscopy, and has proved itself extremely valuable and versatile as an investigative tool. The AFM invented by Gert Binnig and others in the mid 1980s differed in many ways from todays instruments, but its basic principles remain the same. Binnig had already received the Nobel Prize in Physics for his creation of the scanning tunnelling microscope (STM), and the first AFMs in fact relied on an integrated STM tip. But the AFM had a major advantage over STM; it could be used for insulating as well as conducting samples.
Over the years, AFM has already had a significant impact in many disciplines, from surface science to biological and medical research. Because of its ability to image samples on an atomic scale, it has been vital to the advance of nanotechnology.
This AFM image shows the surface of a thin film of GaN. The surface morphology is dominated by terraces and steps. The step heights are approximately 0.25 nm, corresponding to one layer of gallium and nitrogen atoms. This illustrates the ability of AFM to measure very small height changes on surfaces.
The figure above is a topographic AFM image of a collagen fibril. The fibril is the striped structure running diagonally across the middle of the image. The periodicity of the narrow stripes or bands seen in the image is 64 nm. AFM can be used to image biological samples such as collagen without requiring a conductive coating to be added. It is even possible to take images of live cells in a fluid environment
In simple terms, the atomic force microscope works by scanning a sharp probe over the surface of a sample in a raster pattern. By monitoring the movement of the probe, a 3-D image of the surface can be constructed. Below is a schematic diagram of an AFM.
Tip Surface Interaction
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When the tip is brought close to the sample, a number of forces may operate. Typically the forces contributing most to the movement of an AFM cantilever are the *coulombic* and *van der Waals* interactions.
* **Coulombic interaction:** This strong, short range repulsive force arises from electrostatic repulsion by the electron clouds of the tip and sample. This repulsion increases as the separation decreases.
* **Van der Waals interactions:** These are longer range attractive forces, which may be felt at separations of up to 10 nm or more. They arise due to temporary fluctuating dipoles.
The combination of these interactions results in a force-distance curve similar to that below:
Plot of force against distance
As the tip is brought towards the sample, van der Waals forces cause attraction. As the tip gets closer to the sample this attraction increases. However at small separations the repulsive coulombic forces become dominant. The repulsive force causes the cantilever to bend as the tip is brought closer to the surface.
There are other interactions besides coulombic and van der Waals forces which can have an effect. When AFM is performed in ambient air, the sample and tip may be coated with a thin layer of fluid (mainly water). When the tip comes close to the surface, *capillary forces* can arise between the tip and surface. These effects are summarised in the animation below.
It is also possible to detect other forces using the AFM, such as magnetic forces to map the magnetic domains of a sample.
Modes of Operation
AFM has three differing modes of operation. These are contact mode, tapping mode and non-contact mode.
Contact mode
In contact mode the tip contacts the surface through the adsorbed fluid layer on the sample surface. The detector monitors the changing cantilever deflection and the force is calculated using Hookes law:
F = − *k* x (*F* = force, *k* = spring constant, *x* = cantilever deflection)
The feedback circuit adjusts the probe height to try and maintain a constant force and deflection on the cantilever. This is known as the *deflection setpoint*.
Tapping mode
In tapping mode the cantilever oscillates at or slightly *below* its resonant frequency. The amplitude of oscillation typically ranges from 20 nm to 100 nm. The tip lightly “taps” on the sample surface during scanning, contacting the surface at the bottom of its swing.
Because the forces on the tip change as the tip-surface separation changes, the resonant frequency of the cantilever is dependent on this separation.
\[\omega = \omega\_0 \sqrt{ 1 - \frac{1}{k} \frac{\mathrm{d}F}{\mathrm{d}z} }\]
The oscillation is also damped when the tip is closer to the surface. Hence changes in the oscillation amplitude can be used to measure the distance between the tip and the surface. The feedback circuit adjusts the probe height to try and maintain a constant amplitude of oscillation i.e. the *amplitude setpoint*.
Non-contact mode
In non-contact mode the cantilever oscillates near the surface of the sample, but does not contact it. The oscillation is at slightly *above* the resonant frequency. Van der Waals and other long-range forces decrease the resonant frequency just above the surface. This decrease in resonant frequency causes the amplitude of oscillation to decrease.
In ambient conditions the adsorbed fluid layer is often significantly thicker than the region where van der Waals forces are significant. So the probe is either out of range of the van der Waals forces it attempts to measure, or becomes trapped in the fluid layer. Therefore non-contact mode AFM works best under ultra-high vacuum conditions.
Comparison of modes
-
| | | |
| - | - | - |
| | **Advantage** | **Disadvantage** |
| Contact Mode | * High scan speeds
* Rough samples with extreme changes in vertical topography can sometimes be scanned more easily
| * Lateral (shear) forces may distort features in the image
* In ambient conditions may get strong capillary forces due to adsorbed fluid layer
* Combination of lateral and strong normal forces reduce resolution and mean that the tip may damage the sample, or vice versa
|
| Tapping Mode | * Lateral forces almost eliminated
* Higher lateral resolution on most samples
* Lower forces so less damage to soft samples or tips
| * Slower scan speed than in contact mode
|
| Non-contact Mode | * Both normal and lateral forces are minimised, so good for measurement of very soft samples
* Can get atomic resolution in a UHV environment
| * In ambient conditions the adsorbed fluid layer may be too thick for effective measurements
* Slower scan speed than tapping and contact modes to avoid contacting the adsorbed fluid layer
|
The Scanner
=
The scanner moves the probe over the sample (or the sample under the probe) and must be able to control the position extremely accurately. In most AFMs are used to achieve this. These change dimensions with an applied voltage. The diagram below shows a typical scanner arrangement, with a hollow tube of piezoelectric material and the controlling electrodes attached to the surface.
Diagram of a typical piezo scanner (cut into two parts). Separate pairs of electrodes control movement in the x, y and z directions
Tip and Cantilever
The cantilever is a long beam with a tip located at its apex. In most AFMs the motion of the tip is detected by reflecting a laser off the back surface of the cantilever.
Tip
-
The tip is generally pyramidal or tetrahedral in shape, and usually made from silicon or silicon nitride. Silicon can be doped and made conductive, allowing a tip-sample bias to be applied for making electrical measurements. Silicon nitride tips are not conducting.
The geometry of the tip greatly affects the lateral resolution of the AFM, since the tip-sample interaction area depends on the tip radius. The radius of the apex of a new tapping mode tip is around 5–15 nm, but this increases quickly with wear. In general the sharper the tip, the higher the resolution of the AFM image.
Cantilever
For contact mode AFM the cantilever needs to deflect easily without damaging the sample surface or tip. Therefore it should have a low spring constant, this is achieved by making it *thin* (0.3–2 μm). It also needs a high resonant frequency to avoid vibrational instability, so is typically *short* (100–200 μm). V-shaped cantilevers are often used for contact mode as these can provide low resistance to vertical deflection, whilst resisting lateral torsion.
Optical microscopy image of a triangular cantilever
For tapping mode AFM a high spring constant is required to reduce noise and instabilities. Rectangular cantilevers are often used for tapping mode.
Optical microscopy image of a rectangular cantilever
Detection of cantilever deflection
There are a number of ways to detect the deflection of the cantilever in an AFM. The most common method is using a laser beam. A *diode laser* is focused onto the back reflective surface of the cantilever, and reflects onto a photodetector. This is position sensitive, and usually has four sectors. The vertical deflection of the cantilever is determined by the difference in light intensity measured by the upper and lower sectors. It is also possible to measure the lateral deflection of the cantilever by the difference between the left and right sectors of the photodetector; this technique is known as (LFM).
How the deflection of the cantilever is measured
Feedback
When the tip contacts the surface directly the tip and/or surface may be damaged. If the tip is blunted or damaged, then the imaging capability of the AFM is reduced. Soft surfaces (e.g. on biological samples) can also be easily damaged.
In almost all operating modes, a feedback circuit is connected to the deflection sensor and attempts to keep the tip–sample interaction constant by controlling the tip–sample distance. This protects both the tip and the sample. Either the cantilever deflection (in static mode) or oscillation amplitude (in dynamic mode) is monitored by the feedback circuit, which attempts to keep this at a setpoint value by adjusting the z height of the probe. The height of the probe is what is recorded to produce a topographic image.
In practice however feedback is never perfect, and there is always some delay between measuring a change from the setpoint and restoring it by adjusting the scanning height. In tapping mode for example this can be measured by the difference between the instantaneous amplitude of oscillation and the amplitude setpoint. This is known as the amplitude error signal, and highlights changes in surface height.
| | |
| - | - |
| Topography map | Amplitude error |
| Graph of the topography through a slice of the acquired image | Graph of the amplitude error through a slice of the acquired image |
| Example images showing the relationship between topography and amplitude error signal. The two line plots demonstrate a slice through the acquired image. |
The feedback system is affected by three main parameters:
* *Setpoint* – this is the value of the deflection or amplitude that the feedback circuit attempts to maintain. This is usually set such that the force on the cantilever is small, but the probe remains engaged with the surface.
* *Feedback gains* – the higher these are set, the faster the feedback system will react. However if the gains are too high then the feedback circuit can become unstable and oscillate, causing high frequency noise in the image.
* *Scan rate* – scanning the probe over the surface more slowly gives the feedback circuit more time to react and results in better tracking, but this increases the time needed to acquire an image.
Scanner Related Artefacts
=
There are a number of problems and artefacts that can arise during atomic force microscopy. This page and the following pages will discuss some of them, and how they can be overcome.
Hysteresis
The piezoelectrics response to an applied voltage is not linear. This gives rise to *hysteresis*. Since the scanner makes more movement per volt at the beginning of a scan line than at the end, this can cause artefacts in the images, especially at large scan sizes. This is overcome by using a non-linear voltage waveform calculated during a calibration procedure.
Example of a voltage waveform calibrated to overcome hysteresis
Scanner creep
-
If the applied voltage suddenly changes e.g. to move the scanning position, then the piezo-scanners response is not all at once. It moves the majority of the distance quickly, then the last part of the movement is slower. If this is done during scanning, then the slow movement will cause distortion. This is known as *creep*.
When a change in x-offset is applied, features are distorted in the x-direction
When a change in y-offset is applied, features are distorted in the y-direction
The scan size is changed abruptly, and features are distorted
Bow and tilt
Because of the construction of the piezo-scanner, the tip does not move in a perfectly flat plane. Instead its movement is in a parabolic arc, as shown in the image below. This causes the artefact known as *scanner bow*. Also the scanner and sample planes may not be perfectly parallel, this is known as *tilt*. Both of these artefacts can be removed by using post-processing software.
Diagram of scanner bow
Tip Related Artefacts
=
For densely packed features the tip size can cause errors in determining the heights and the sizes of the “islands” or the overall appearance of the surface. Sidewall angles of the tip can also lead to inaccurate lateral resolution measurements for high aspect ratio features.
The tip may pick up loose debris from the sample surface. This may be reduced by cleaning the sample with compressed air or N2 before use. Or the tip can be damaged during scanning, which degrades the images. This may be blunting of the tip, as shown in the SEM image below:
Below is an example of an image taken with a severely damaged tip. The shape due to tip damage appears several times over the image, effectively the sample is imaging the tip rather than the other way round.
| | |
| - | - |
| Sample imaged with sharp tip | The same sample imaged using a severely damaged tip |
| Sample imaged with sharp tip | The same sample imaged using a severely damaged tip |
One easy way to check for tip artefacts is to rotate the sample (**not** just the scanning direction) by 90 degrees. This is demonstrated in the following animation:
Other Artefacts
=
Feedback related
The feedback is supposed to keep the tip-sample interaction at a fixed setpoint by adjusting the z height of the probe, as discussed earlier. However if the scan speed across the sample is fast, then the feedback may not be able to react quickly enough and tracking is poor. This can be seen by comparing the trace and retrace (forward and backward direction) for a single line in the scan. The following image shows the height and amplitude trace (white) and retrace (yellow) when tracking is good. The height trace and retrace are almost identical, and the amplitude retrace is a mirror image of the trace because it is in the opposite direction.
When tracking is poor, the trace and retrace of height no longer overlap. Blurred images result. This can happen because the gains are set too low, or the scan speed is too high.
The images below are examples of poor tracking
| | |
| - | - |
| | |
| Topography | Amplitude error |
With sharp slopes, poor tracking may result in overshoot giving rise to “comet tails” in the image. The following images show indium aluminium nitride with small balls of indium on the surface. On the left the gains are set high enough for the scan rate, and tracking is good. On the right the gains are too low for the scan rate, and the tracking is poor. This results in overshooting off the edges of the indium dots, appearing in the image as comet tails. This can also be seen as the trace and retrace not overlapping.
| | |
| - | - |
| | |
| Good tracking | Poor tracking, resulting in “comet tails |
However if the gains are set too high, then the feedback circuit can begin to oscillate. This causes high frequency noise.
Amplitude error image for a scan with the gains set too high
The precise values used for feedback gains will vary between instruments. A good rule of thumb is to increase the gain until excess noise begins to appear, and then reduce it slightly to get good tracking with low noise.
Vibrations
AFMs are very sensitive to external mechanical vibrations, which generally show up as horizontal bands in the image.
Evidence of external vibrations in an amplitude error image
These vibrations may be transmitted through the floor, for example from footsteps or the use of a lift. These can be minimised by the use of a vibrational isolation table, and locating the AFM on a ground floor or below.
Acoustic noise such as people talking can also cause image artefacts, as can drafts of air. An acoustic hood can be used to minimise the effects of both of these.
Acoustic hood open
Acoustic hood closed
Summary
=
Atomic force microscopy may be used to image the micro- and nano-scale morphology of a wide range of samples, including both conductive and insulating materials, and both soft and hard materials. Successful imaging requires optimisation of the feedback circuit which controls the cantilever height, and an understanding of the artefacts which may arise due to the nature of the instrument and any noise sources in its immediate environment. Despite these issues, atomic force microscopy is a powerful tool in the emerging discipline of nanotechnology.
Questions
=*Note: This animation requires Adobe Flash Player 8 and later, which can be .*
### Quick questions*You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!*1. Which operating mode allows for the fastest scanning speeds? | | | |
| - | - | - |
| |
a |
Contact mode |
| |
b |
Tapping mode |
| |
c |
Non-contact mode |### Deeper questions*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*2. If high frequency noise is seen in an image, what should be done? | | | |
| - | - | - |
| |
a |
Increase the feedback gains |
| |
b |
Decrease the feedback gains |
| |
c |
Change the tip |
| |
d |
Recalibrate the AFM |
Going further
=
### Books
* Meyer, Hug and Bennewitz, *Scanning Probe Microscopy: The Lab on a Tip*, Springer, 2003
Websites
* , SPM Principles (NT-MDT) including animations
* (Pacific Nanotechnology)]
* (nanoHUB.org)] – a presentation/podcast introducing AFM
|
Aims
On completion of this TLP you should:
* Understand the concept of anisotropy, and appreciate that the *response*
(e.g. displacement) need not be parallel to the *stimulus* (e.g. force)
* Understand the nature of anisotropic behaviour in a range of properties, including electrical and thermal conductivity,
diffusion, dielectric permittivity and refractive index, and be aware of a range of everyday examples
* Be familiar with the use of representation surfaces
Before you start
There are no specific prerequisites for this TLP, but you will find it useful
to have a basic knowledge of crystal structures, as this will enable a better
understanding of the structural origins of anisotropy. Take a look at the and the .
Introduction
Some physical properties, such as the density or heat capacity of a material, have values independent of direction; they
are *scalar* properties. However, in contrast, you will see that many properties vary with direction within a material.
For example, thermal conductivity relates heat flow to temperature gradient, both of which need to be specified by direction
as well as magnitude - they are *vector* quantities. Therefore thermal conductivity must be defined in relation to
a direction in a crystal, and the magnitude of the thermal conductivity may be different in different directions.
A perfect crystal has long-range order in the arrangement of its atoms. A solid
with no long-range order, such as a glass, is said to be *amorphous*. Macroscopically,
every direction in an amorphous structure is equivalent to every other, due to
the randomness of the long-range atomic arrangement. If a physical property relating
two vectors were measured, it would not vary with orientation within the glass;
i.e. an amorphous solid is *isotropic*. In contrast, crystalline materials
are generally *anisotropic*, so the magnitude of many physical properties
depends on direction in the crystal. For example, in an isotropic material, the
heat flow will be in the same direction as the temperature gradient and the thermal
conductivity is independent of direction. However, as will be demonstrated in
this TLP, in an anisotropic material heat flow is no longer necessarily parallel
to the temperature gradient, and as a result the thermal conductivity may be different
in different directions.
The occurrence of anisotropy depends on the symmetry of the crystal structure. Cubic crystals are isotropic for many properties,
including thermal and electrical conductivity, but crystals with lower symmetry (such as tetragonal or monoclinic) are anisotropic
for those properties.
Many (but not all) physical properties can be described by mathematical quantities called *tensors*. A non-directional
property, such as density or heat capacity, can be specified by a single number. This is a *scalar*, or *zero
rank tensor*. Vector quantities, for which both magnitude and direction are required, such as temperature gradient,
are *first rank tensors*. Properties relating two vectors, such as thermal conductivity, are *second rank tensors*.
Third and higher rank tensor properties also exist, but will not be considered here, since the mathematical descriptions
are more difficult.
Mechanical analogy of anisotropic response
A *stimulus* (such as a force or an electrical field) does not necessarily
induce a response (such as a displacement or a current) parallel to it.
This can be demonstrated with a simple mechanical model, consisting of a mass supported by two springs.
Mechanical model: a mass supported by two springs
A force, *F*, applied at an angle *θ*, to the central mass acts as the stimulus.
The response is the displacement, *r*, of the mass, at an angle *φ*.
Diagram showing applied force and resulting displacement
For second rank tensor properties in anisotropic materials, parallel responses
occur along orthogonal directions known as the *principal directions*.
The following photographs show the response of the model under the application of various forces. (Click on an image to
view a larger version.)
| | |
| - | - |
|
Model with no force applied |
Model with horizontal force producing horizontal displacement (parallel response)
(*θ* = *φ* = 90º) |
|
Model with vertical force producing vertical displacement (parallel response) (*θ* = *φ* = 0º) |
Model with 45º displacement from non-45º
force (non-parallel anisotropic response) (*θ* = approx 35º,
*φ* = 45º) |
Note that the displacement of the mass is only parallel to the force when the
force acts parallel or perpendicular to the springs. These are the directions
of the *principal axes*.
### Symmetry
As a rule, the symmetry present in crystalline materials (such as mirror planes
and rotational axes) determines or restricts the orientation of the principal
axes. In this model, there exist two orthogonal mirror planes perpendicular to
the plane of the model, one parallel and the other perpendicular to the springs,
and a third mirror plane exists in the plane of the model. The principal axes
lie along the intersections of these mirror planes. Real crystals typically show
more complicated symmetry, but the orientation of the principal axes is still
determined by the main symmetry elements.
Anisotropic properties may be analysed by resolving onto these principal axes.
The symmetry elements of any physical property of a crystal must include the
symmetry elements of the of the crystal (Neumanns Principle). Thus crystals that, for
example, display spontaneous polarization (see later section on anisotropic dielectric
permittivity) can belong to only a few symmetry classes. It is worth noting that
the absence of a centre of symmetry does not necessarily imply anisotropic second
rank tensor properties, nor does the presence of a centre of symmetry rule out
anisotropy in such properties.
Anisotropic thermal conductivity
When a temperature gradient is present in a material, heat will always flow from the hotter to the colder region to achieve thermal equilibrium. As mentioned in the introduction, thermal conductivity is the property that relates heat flow to the temperature gradient. In an isotropic material:
\[J = k{{dT} \over {dr}}\]
where J = heat flow, k = thermal conductivity, and dT/dr = temperature gradient.
### Anisotropic thermal conductivity in quartz**.**
In quartz, perpendicular to the c-axis, the thermal conductivity is 6.5 Wm-1K-1. However, the thermal conductivity parallel to c is 11.3 Wm-1K-1.
The anisotropic thermal conductivity of quartz can easily be seen using a simple
demonstration. Two sections cut from a quartz crystal, one perpendicular to the
c-axis, and one parallel to it, are in turn mounted as shown in the diagram below.
Pieces of plastic containing a heat sensitive liquid crystal are then glued to
the top surfaces and the sections are heated from a point at their centre, using
a soldering iron. As the quartz heats up, the heat sensitive film changes colour,
which allows us to see how quickly the heat is conducted away from the centre.
The colours indicate contours of constant temperature.
Diagram of experimental apparatus
When heating the section cut perpendicularly to the c-axis, the observed shape is a circle, showing that the thermal conductivity is the same in all directions in this plane. However, when using the section cut parallel to the c-axis, the shape seen is an ellipse, which shows that the thermal conductivity in this plane is direction-dependent.
Your browser does not support the video tag.Video of a section of quartz cut perpendicular to the c axis being heated from a point at its centre
Your browser does not support the video tag.Video of a section of quartz cut parallel to the c axis being heated from a point at its centreThe heat flow does not have to be parallel to the thermal gradient. A result
of this can be seen by considering one-dimensional conduction in a long rod and
a thin plate, both made of the same anisotropic material, arranged so that the
normal to the plate and the length of the rod are oriented in an arbitrary general
direction.
### Thin Plate
 Here the geometry of the set-up constrains the temperature gradient to be perpendicular
to the plate. Due to the anisotropic nature of the material, the heat flux, **J**,
will be in the direction shown, say. However, the thermal *conductivity*
perpendicular to the plate is defined as the component of the heat flux parallel
to the temperature gradient, j||, divided by the magnitude
of that gradient. Thus:
\[{k\_{||}} = {{{j\_{||}}} \over {gradT}}\]
### Rod
 Now the heat must flow along the rod, and the temperature gradient will be in a different
direction, as shown. Here the thermal *resistivity* is defined as the component
of the temperature gradient parallel to the rod, *gradT*||,
divided by the magnitude of the heat flux. Thus:
\[{\rho \_{||}} = {{grad{T\_{||}}} \over J}\]
where ρ is the resistivity.
It is important to realise that in anisotropic materials
\[{\rho \_{||}} \ne {\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 {{k\_{||}}}}}\right.\kern 0.01em}
\!\lower0.7ex\hbox{${{k\_{||}}}$}}\]
except along the principal axes. Only in isotropic materials is the resistivity always the reciprocal of the conductivity, and vice versa.
*Note*: By using a large, thin plate and a long rod, the effects of
the alterations in the directions of heat flow and temperature gradient close
to the edges (of the plate) or ends (of the rod) - "edge effects" and
"end effects" - affect only a very small proportion of the sample and
can be ignored.
Derivation of the anisotropy ellipsoid
The variation of anisotropic properties such as conductivity can conveniently be illustrated by a "representation
surface". In many cases this is an ellipsoid.
Suppose a three-dimensional temperature gradient, *gradT*, lies along
a direction specified by direction cosines l, m and n>,
where, for example, l is the cosine of the angle between the x-axis
and the temperature gradient vector. Then the components of the temperature gradient
parallel to the principal axes will be:
\[grad{T\_x} = (gradT)l grad{T\_y} = (gradT)m grad{T\_z} = (gradT)n\]
The components of the heat flux are:
\[{j\_x} = {k\_1}(gradT)l {j\_y} = {k\_2}(gradT)m {j\_z} = {k\_3}(gradT)n\]
where k1, k2 and k3
are the values of thermal conductivity along the principal axes, x, y and z, and are called the principal values.
Hence, resolving back along the direction of the temperature gradient, the
heat flux is:
\[{j\_{||}} = {j\_x}l + {j\_y}m + {j\_z}n = ({k\_1}{l^2} + {k\_2}{m^2} + {k\_3}{n^2})gradT\]
Thus the value of the thermal conductivity, klmn, defined
by
\[{k\_{lmn}} = {{{j\_{||}}} \over {gradT}}\]
is related to the principal values and the directional cosines by:
\[k = {k\_1}.{l^2} + {k\_2}.{m^2} + {k\_3}.{n^2}\]
\[l = {\raise0.7ex\hbox{$x$} \!\mathord{\left/
{\vphantom {x r}}\right.\kern 0.01em}
\!\lower0.7ex\hbox{$r$}}\;\;\;
m = {\raise0.7ex\hbox{$y$} \!\mathord{\left/
{\vphantom {y r}}\right.\kern 0.01em}
\!\lower0.7ex\hbox{$r$}}\;\;\;
n = {\raise0.7ex\hbox{$z$} \!\mathord{\left/
{\vphantom {z r}}\right.\kern 0.01em}
\!\lower0.7ex\hbox{$r$}}\]Substituting in our equation for k gives:
\[k = {{{k\_1}{x^2}} \over {{r^2}}} + {{{k\_2}{y^2}} \over {{r^2}}} + {{{k\_3}{z^2}} \over {{r^2}}} = {1 \over {{r^2}}}({k\_1}{x^2} + {k\_2}{y^2} + {k\_3}{z^2})\]
Setting
\[r = {\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 {\sqrt k }}}\right.\kern 0.01em}
\!\lower0.7ex\hbox{${\sqrt k }$}}\]
then
k>1*x*2 + k2*y*2 + k3*z*2
= 1
If all the principal values are positive (as they must be for thermal conductivity),
then this equation describes the surface of an ellipsoid. The general equation
of an ellipsoid (with semi-axes a,b,c) is:
\[{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} + {{{z^2}} \over {{c^2}}} = 1\]
Thus for this *representation ellipsoid*, the semi-axes are:
\[{1 \over {\sqrt {{k\_1}} }},{1 \over {\sqrt {{k\_2}} }},{1 \over {\sqrt {{k\_3}} }}\]
The radius of this ellipsoid in a general direction is equal to the value of \({1 \over {\sqrt {{k}} }}\)in
that direction. Thus the value of k in a particular direction - the ratio of the component of the heat flow in
that direction to the magnitude of the temperature gradient in that direction - can easily be calculated from the radius
in that direction.
An equivalent representation surface exists for electrical conductivity, and
a similar representation surface exists for refractive index - the optical indicatrix.
Both of these are discussed later in this TLP.
### Using the representation surface for thermal conductivity
As shown above, the representation surface for thermal conductivity is an ellipsoid
with semi-axes \({1 \over {\sqrt {{k\_1}} }}\),
\({1 \over {\sqrt {{k\_2}} }}\)
and \({1 \over {\sqrt {{k\_3}} }}\).
The distance between the centre of the ellipsoid and a point, P, on its surface,
is equal to \({1 \over {\sqrt {{k}} }}\)
at this point.
As well as determing the conductivity, the representation surface can be used
to relate the directions of heat flow and the temperature gradient. If the temperature
gradient is applied radially from the centre of the ellipsoid, then the direction
of resulting heat flow is perpendicular to the tangential plane constructed from
the point at which the thermal gradient meets the surface of the ellipsoid.
In an isotropic material, the representation surface will be a sphere, and the heat flow will be in the same direction
as the temperature gradient. However, in an anisotropic material, heat flow is no longer necessarily parallel to the temperature
gradient.
We will now revisit the **anisotropic thermal conductivity of quartz**.
Because of the crystal symmetry of quartz, *k*1 = k2  k3,
and so the representation surface for the thermal conductivity of quartz is a
uniaxial ellipsoid of revolution. Depending on the relative values of k1
and k3, this is a shape either like a rugby ball (k3 < k1)
or a Smartie (k3 > k1).
Consider sections of the ellipsoid:
1. *Perpendicular to the c-axis:*
Since k1 = k2,
this section is a circle, and the direction of heat flow is parallel to the temperature
gradient.
2. *Perpendicular to the b-axis (i.e. parallel to the c-axis).*
Since k1 k3,
this section is an ellipse, and the direction of heat flow is no longer parallel
to the temperature gradient, except in the directions of the principal axes (which
here correspond to the semi-axes of the ellipse).
The direction of the heat flux is always parallel to the normal to the tangential
plane drawn at the point at which the direction of the temperature gradient intersects
the representation ellipsoid. This is called the radius-normal property.
.
Anisotropic electrical conductivity
=
The current density, ***J***, is related to the electric field, ***E***, by
***J***= *σ**E***
In an analogous way to thermal conductivity, the current density does not have to be parallel to the electric field. Three
different examples of anisotropic electrical conductivity are described here. These show how the anisotropy is related to
the crystal structure.
In metals, conductivity occurs by transport of delocalised electrons through
the crystalline lattice, under the influence of an applied electric field. The
conductivity is limited by the scattering of the electrons by imperfections in
the periodicity of the structure (vibrations, impurities, etc). Because of the
high symmetry in cubic metals, the overall drift velocity is parallel to the electric
field, i.e. there is an isotropic response. However in hexagonally "close
packed" metals, the nature of the symmetry in the crystalline array allows
the conductivity to be anisotropic. For example in cadmium, it varies from 1.3 x 107Sm-1
along the six-fold axis to 1.5 x 107 Sm-1
perpendicular to that axis.
Graphite consists of layered planes of carbon atoms with a structure as shown below. The layers are stacked above one another
in a staggered fashion, the spacing between layers being about 2.3 times the distance between the adjacent carbon atoms
in a layer.
The Hexagonal Structure of Graphite Planes
Here the hexagonal carbon rings provide the delocalised electrons, allowing
easy conduction within the planes. Conduction is very much less perpendicular
to the planes (around three orders of magnitude smaller) - this is highly anisotropic
behaviour. The structure also creates anisotropy in other properties of graphite,
such as thermal conductivity and thermal expansion.
This planar anisotropy is also seen in high temperature superconductors like
BiSrCaCuO. The copper oxide "ab" planes provide superconducting pathways
for electrons, but such pathways are not available perpendicular to the planes.
Anisotropic diffusion
=
The rate of diffusion of a specific atomic species is measured in terms of
the coefficient of diffusion, *D*, which relates the flux of atoms (number
crossing unit area in unit time) to the concentration gradient. In an isotropic
material:
J = -D(*gradc*)
where J = flux (number per unit
area per unit time) of an atomic species across a plane normal to the concentration
gradient, *gradc*. The negative sign indicates that the flux is from high
to low concentrations. For an isotropic material, such as an amorphous solid,
the diffusion coefficient is independent of temperature.
Flux and concentration gradient are both vectors, so the coefficient of diffusion
is a second rank tensor in an anisotropic material. The diffusion is anisotropic
and can be described by three principal values of D, in the same way
as thermal and electrical conductivity. (*Note*: we are not concerned here
with the dependence of diffusion on temperature).
### Example: Diffusion of Ni in Olivine, (Mg,Fe)2SiO4
Olivine is the name for a series of minerals between two end members, fayalite
(Fe2SiO4) and forsterite (Mg2SiO4).
The two minerals form a solid solution where the iron and magnesium atoms can
be substituted for each other without significantly changing the crystal structure.
Olivine has an orthorhombic structure. The lattice parameters depend on the precise
composition, but a typical set of values is: a = 0.49 nm,
b = 1.04 nm, c = 0.61 nm.
The principal values of D for diffusion of nickel atoms in olivine also depend on the precise composition of the olivine. One set of values for an unspecified composition at 1423 K (1150ºC) is:
Dx = 4.40 x 10-18 m2s-1 Dy = 3.35 x 10-18 m2s-1 Dz = 124.0 x 10-18 m2s-1
where the values correspond to the diffusion coefficients along the *x*, *y* and *z* crystallographic axes respectively. The exact values are unimportant for this discussion, but it is important to appreciate that diffusion occurs much faster parallel to the *z*-axis than in directions in the plane perpendicular to it. This happens because of the way in which the atoms are arranged in the crystal structure, a plan view of which is shown below (projected down the *x*-axis).
Note that x = 0, 25, 50 and 75 represent the *x*-coordinates of the atoms (as a percentage of the unit cell dimension *a*).Crystal structure of olivine (click on image to view a larger version)
As you can see, there are chains of M2+ sites (where M2+ represents a metal ion, in this case either Mg or Fe) parallel to the *z*-axis. Diffusion occurs by Ni2+ substituting for M2+ along these chains, making diffusion in this direction much faster than in any other.
Your browser does not support the video tag.Rotatable model of the olivine structure### Fast ion conduction
When the structure of a solid material contains a large number of vacant sites
(as a consequence of its composition), then it is likely to show high ionic mobility
for some species of ion (sometimes an anion, sometimes a cation), even at modest
temperatures. A high ionic mobility means that charge can be transferred very
easily, and conductivities can approach those of aqueous electrolyte solutions
or molten salts.
For a material to be a fast ion conductor, there should be:
* a high concentration of charge carriers
* a high concentration of vacant sites in the structure
* a low activation energy for ionic migration
There are a number of different types of fast ion conductors. The material
may be either a cationic or an anionic conductor, depending on the charge of the
mobile ion. The material may also have a fully ionic structure, or the mobile
ions may be in a covalent host structure. The dimensionality of the mobility can
also vary: the ions may move through channels (1D), within layers of a structure
(2D), or throughout the whole structure (3D).
### Examples of fast ion conductors
#### 1D fast ion conductors: Tungsten Bronzes, MxWO3
In these materials, WO3 tetrahedra or octahedra form a covalent network. Usually either M+ or M2+ (for example Na+ or Cu2+) are the mobile ions, and these move along channels in the structure. As a result, the conductivity is very high in this direction. An example of a tungsten bronze is shown below, projected along the *z*-axis, which is parallel to the channel direction.
Structure of tungsten bronze
#### 2D fast ion conductor: Sodium beta-alumina, Na β-Al2O3
Sodium beta-alumina consists of blocks of γ-alumina
(which has a spinel structure, the details of which need not be considered here)
connected by a layer of bridging oxygen and sodium ions. Not all the Na+
sites are occupied, and conduction occurs by the movement of the ions within this
layer.
Structure of sodium beta-alumina
Anisotropic dielectric permittivity
=
When an electric field, ***E***, is applied to a dielectric solid, positive and negative charges are displaced in opposite directions within the solid, creating *polarisation*, ***P***. This is defined as the net dipole moment per unit volume. (An electrical dipole is created by a small separation of equal and opposite charges.)
In an isotropic material, these vectors are related by:
***P*** = (*e* - 1)*e*o***E***
*e*o is the permittivity of free space, and *e* is the relative dielectric permittivity (a scalar constant in this case). As with the other examples, in anisotropic materials this scalar has to be replaced by a tensor. Often the occurrence of highly anisotropic dielectric permittivity is associated with *ferroelecticity* (spontaneous polarisation reversible by an electric field) and *pyroelecticity* (temperature dependent generation of polarisation).
### Example: barium titanate
The high temperature form of BaTiO3 has the cubic perovskite structure with a primitive cubic lattice. At 150°C, a = 0.401 nm. In the temperature range 0ºC to 120ºC, BaTiO3 is tetragonal. At 100°C it has *a*= *b* = 0.400 nm and *c* = 0.404 nm.
| | |
| - | - |
| Your browser does not support the video tag.Video of cubic perovskite structure | Your browser does not support the video tag.Rotatable model of cubic perovskite structure |
| Your browser does not support the video tag.Video of tetragonal perovskite structure | Your browser does not support the video tag.Rotatable model of tetragonal perovskite structure |The tetragonal-cubic phase transition is highlighted in the following video.
It shows a thin section of barium titanate viewed between crossed-polars, which
is heated through the transition temperature, and then allowed to cool naturally.
Initially, the sample is below the transition temperature, and since the domains
of the anisotropic tetragonal phase exhibit birefringence, it is brightly coloured
when viewed between crossed-polars. When the sample reaches the transition temperature,
the isotropic cubic phase forms, which appears black. The heat source is then
removed, so the sample cools down and again undergoes a phase transition to return
to the anisotropic tetragonal phase.Your browser does not support the video tag.
Video of barium titanate phase transitionIn the tetragonal form, the Ti ion is displaced by a small distance, from the
centre of the surrounding octahedron of nearest neighbour oxygen ions, along the
*z*-direction. A spontaneous polarisation along the *z*-axis is
generated, but by symmetry there is no polarisation in the *x-y* plane.
Note that the polarsation can be orientated forwards or backwards along the tetragonal
axis. This polarisation is easily changed by applying an electric field parallel
to the *z*-axis, but a field applied to the *x-y* plane has little
effect on the polarisation. Consequently the dielectric permittivity is anisotropic.
The refractive index, *n*, is given by the square root of the relative dielectric permittivity, i.e. .
The resulting optical effects are considered in the next section.
Optical anisotropy and the optical indicatrix
=
In transparent materials with anisotropic dielectric permittivity, important
optical effects can be observed. Recall that a light wave may be considered in
terms of oscillating transverse electric and magnetic fields. Here we concentrate
on the effects of the electric field. When discussing optical properties it is
important to remember that this field is in a direction lying in the wavefront.
It is *not* necessarily perpendicular to the direction of propagation.
The interaction between the electric field and the material is governed by the
dielectric permittivity discussed in the previous section. A large value of the
permittivity gives rise to a large refractive index, and consequently the wave
travels relatively slowly. (The refractive index n is related to the
velocity of light in the medium, v, and the velocity in a vacuum, c,
by n = c/v.)
In an anisotropic material the refractive indices can again be illustrated by a representation surface - the *optical indicatrix*. For each (orthogonal) principal direction in the anisotropic material, there is an associated principal refractive index. The variation of the refractive index with the plane of the wavefront can be represented by an ellipsoid. The semi-axes of this optical indicatrix are directly proportional to the principal refractive indices.
Optically isotropic materials (e.g. cubic crystals) have one refractive index, with a spherical indicatrix.
Crystals with one 3, 4 or 6 fold axis of symmetry have a principal axis of
the ellipsoid along this symmetry axis. These *uniaxial* crystals have
an indicatrix which is an ellipsoid with a circular cross-section perpendicular
to the major symmetry axis – an ellipsoid of revolution. They have two principal
refractive indices and one *optic axis* (parallel to the symmetry axis
and so perpendicular to the circular section).
In general, the electric field of a light wave experiences two *permitted
vibration directions*, known as the fast and slow directions, both in the
plane of the wavefront, and determined by the shape of the indicatrix. Consider
a section passing through the origin of the indicatrix for a uniaxial crystal,
and orientated parallel to the wavefronts, as shown by the dotted line in the
left-hand figure below. The two permitted vibration directions are given by the
major and minor axes of this section. The corresponding refractive indices are
the lengths of these axes. The section will be elliptical unless the light is
travelling along the optic axis so that the plane of the wavefront coincides with
the circular section of the ellipsoid.
The observation of two refractive indices for a general orientation of the wavefront is known as *birefringence*. Related effects such as stress-induced birefringence and photoelasticity are discussed in the .
| | |
| - | - |
| Diagram | Diagram |
The *ordinary* vibration direction lies in the circular section of the
indicatrix (i.e. perpendicular to the optic axis) with refractive index no.
Light travelling along the optic axis experiences just this refractive index -
the *ordinary refractive index*.
The *extraordinary* vibration direction lies in the plane of the wavefront
and perpendicular to the ordinary vibration direction, and has refractive index
n'e. The value of n'e is determined from
the ordinary refractive index and the principal extraordinary refractive index
ne, as follows.
Consider a cross-section of the indicatrix (as shown in the diagram on the right above), containing the optic axis and
the extraordinary vibration direction. The equation for this ellipse will be:
\[{{{x^2}} \over {n\_o^2}} + {{{z^2}} \over {n\_e^2}} = 1\]
For the point P, x = ne'cosθ,
and y = ne'sinθ.
Therefore
\[{{{{(n{'\_e}\cos \theta )}^2}} \over {n\_o^2}} + {{{{(n{'\_e}\sin \theta )}^2}} \over {n\_e^2}} = 1\]
For a general extraordinary wave, the direction in which the light energy travels,
the ray direction, is no longer perpendicular to the wavefront. For an explanation
see, for example, the link to Optical Birefringence in .
As a side note, the relative magnitudes of no and ne
determine whether a material is defined as optically positive or negative, the
optical sign.
### Example: calcite rhomb
The birefringence (defined as \(|{n\_0} - {n\_e}|\))
in calcite is so large that two images can easily be observed when viewing an
object through a suitable crystal with the naked eye. One of which is due to
the ordinary wave (with electric field vibrating parallel to the ordinary vibration
direction) and the other is due to the extraordinary wave (with electric field
vibrating parallel to the extraordinary vibration direction).The DoITPoMS logo viewed through a calcite rhomb (click on image to view larger version)
In calcite, the planar carbonate groups all lie in planes normal to the three-fold
axis - the optic axis. The groups are well separated in the direction of the axis.
This makes the crystal less polarisable parallel to the axis, so the refractive
index for vibrations parallel to the triad axis is smaller than for vibrations
perpendicular to it (making the crystal optically negative: ne
< no).
Your browser does not support the video tag.Video of calcite structure
Liquid crystals
=
When most solids melt, they form an isotropic fluid, whose optical, electrical
and magnetic properties do not depend on direction. However, when some materials
melt, over a limited temperature range they form a fluid that exhibits anisotropic
properties. These materials generally consist of organic molecules that have an
elongated shape, with a rigid central region and flexible ends. The molecules
in a *liquid crystal* do not necessarily exhibit any positional order,
but they do possess a degree of orientational order.
The anisotropic behaviour of liquid crystals is caused by the elongated shape of the molecules. The physical properties
of the molecules are different when measured parallel or perpendicular to their length, and residual alignment of the rods
in the fluid leads to anisotropic bulk properties. This residual alignment occurs as a result of preferential packing arrangements,
and also electrostatic interactions between molecules that are most favourable (lowest in energy) in aligned configurations.
There are three types of liquid crystal: nematic, smectic and cholesteric. In the liquid crystalline phase, the vector
about which the molecules are preferentially oriented, **n**, is known as the "director".
The long axes of the molecules will tend to align in this direction.
Three types of liquid crystal
In addition to the long range orientational order of nematic liquid crystals, smectic liquid crystals also have one dimensional
long range positional order, the molecules being arranged into layers.
A cholesteric (or twisted nematic) liquid crystal is chiral: the molecules have left or right handedness. When the molecules
align in layers, this causes the director orientation to rotate slightly between the layers, eventually bringing the molecules
back into the original orientation. The distance required to achieve this is known as the *pitch* of the twisted
nematic, as seen in the diagram above. The pitch is not equal to the distance marked x, because only 180º of rotation
occurs over this length, so the molecules are aligned antiparallel to their starting orientation.
.
When viewed between crossed polars, thin films (approximately 10μm thick) of liquid crystals
exhibit *schlieren textures*, as seen in the micrograph below, which shows a nematic liquid crystalline polymer.Micrograph of nematic liquid crystalline polymer, courtesy of Professor TW Clyne
and the (click on image to view larger version, or )
The black brushes are regions where the director is either parallel or perpendicular to the plane of polarisation of the
incident radiation, and the points at which the brushes meet are known as disclinations.
If the temperature of a liquid crystal is raised, the constituent molecules have more energy, and are able to move and
rotate more, so the liquid crystal becomes less ordered. As a result, the magnitude of the anisotropy of the bulk properties
of the liquid crystal decreases, usually eventually resulting in an isotropic fluid.
Liquid crystals are used in many different applications, for example the displays on calculators, digital watches and mobile
phones.
Summary
=
In some materials a property will be the same, irrespective of the direction in which it is measured, but this is not always
the case. On completion of this TLP, you should now understand the concept of anisotropy, and be able to appreciate that
a response can be non-parallel to the applied stimulus. Anisotropy in a range of properties has been discussed, including
electrical and thermal conductivity, diffusion, dielectric permittivity, and optical properties. You should also now be
familiar with the use of representation surfaces for a range of anisotropic properties, including the basis behind their
mathematical description.
Anisotropic properties are exploited in many applications. In polarised-light
microscopy, a quartz wedge can be used to determine birefringence and optical
sign. Liquid crystals have electronic uses such as displays, and the liquid crystalline
state has advantages in the processing of polymers (such as Kevlar). The anisotropic
thermal conductivity in polymer thin films has use in microelectronic devices,
for example, solid-state transducers.
Anisotropic properties described by higher than second rank tensors (not discussed here) can also have useful applications.
Examples include:
* Piezoelectricity (relating an applied stress to the induced polarisation)
* The electro-optic effect (when a field causes a change in the dielectric impermeability)
* Elastic compliance and elastic stiffness (relating stress and strain)
* Piezo-optical effect (when a stress induces a change in refractive index)
* Electrostriction (strain arising from an electric field)
Non-tensor properties can also demonstrate anisotropy; for example, yield
stress can vary with direction of applied stress.
Questions
=
1. Which of these properties of a crystal may be anisotropic? | | | | |
| - | - | - | - |
| Yes
| No
|
a |
Density |
| Yes
| No
|
b |
Young's modulus |
| Yes
| No
|
c |
Surface energy |
| Yes
| No
|
d |
Refractive index |
| Yes
| No
|
e |
Electrical conductivity |
| Yes
| No
|
f |
Thermal conductivity |
| Yes
| No
|
g |
Heat capacity |
| Yes
| No
|
h |
Melting point |
| Yes
| No
|
i |
Coefficient of thermal expansion |
2. Two similar transparent uniaxial crystals show the same (principal) extraordinary refractive index, *n*e. However one is optically positive, and the other is optically negative. In which will the light travel faster along the optic axis?
3. Which of these could not induce anisotropy in an initially isotropic material? | | | |
| - | - | - |
| |
a |
Application of a stress |
| |
b |
Application of an electric field |
| |
c |
Application of a magnetic field |
| |
d |
Application of a high temperature |
4. Below 0°C a particular material has a crystal structure that gives rise to anisotropic thermal conductivity. At room temperature the thermal conductivity of a sample of this material is found to be isotropic. In what circumstances would the following hypotheses explain this observation? | | | |
| - | - | - |
| |
a |
The sample is polycrystalline |
| |
b |
The sample has undergone a phase transition when brought up to room temperature |
| |
c |
The principal values of thermal conductivity have changed with temperature |
5. Which of these materials will show isotropy in its mechanical properties? | | | |
| - | - | - |
| |
a |
Wood |
| |
b |
Carbon fibre reinforced polymer |
| |
c |
Window glass |
| |
d |
Extruded polyethylene |
6. An olivine crystal has the following diffusion constants for Ni at a certain temperature: Dx = 6 x 10-8 m2s-1,
Dy = 4 x 10-18 m2s-1
Dz = 120 x 10-18 m2s-1. The unit cell dimensions are as follows: a = 0.5 nm
b = 1.0 nm
c = 0.6 nm By considering the representation surface, what will the diffusion constant be when the concentration gradient lies along the [101] direction?
7. A certain orthorhombic crystal has the following principal values of thermal conductivity: *k*x = 6.25 Wm-1K-1
*k*y = 1.00 Wm-1K-1
*k*z = 1.75 Wm-1K-1 (where the subscripts represent conductivity parallel to the *x*, *y*, and *z* axes respectively). The unit cell dimensions are: a = 0.8 nm, b = 0.6 nm, c = 1.0 nm.Calculate the thermal conductivity along the [111] direction.
8. Explain how anisotropy is involved in the operation of the following devices: 1. a liquid crystal display
2. a fuel cell that uses a sold state electrolyte
3. a pyroelectric intruder alarm
9. Suggest ways of distinguishing between the answers to Question 4.
Going further
=
### Books
* A. Putnis, *Introduction to Mineral Sciences*, CUP, 1992 (specifically
Chapter 2, "Anisotropy and physical properties").
* R.E. Newnham, *Structure-Property Relations Relations (Crystal chemistry
of non-metallic materials)*, Springer-Verlag, 1975
* D.R. Lovett, *Tensor Properties of Crystals*, IOP, 1999
* P.J. Collings and M. Hird, *Introduction to Liquid Crystals: Chemistry
and Physics*, Taylor & Francis, 1997
More advanced and detailed books:
* C. Kittel, *Introduction to Solid State Physics*, 7th Edition 1995
* J.F. Nye, *Physical Properties of Crystals: Their Representation by Tensors
and Matrices*, Oxford, 2nd Edition 1985
* E. Hecht, *Optics*, Addison-Wesley, 4th Edition 2001
### Websites
* An award-winnning website based at Case Western Reserve University in
the USA, with a and .
* A TLP covering many features of birefringence under polarised light.
* Summary of phase transitions and formation of domains in perovskites.
* This comprehensive introduction to optical birefringence is part of the excellent
award-winning website based at Florida State University in the USA.
* Also part of the website, this is an interactive Java tutorial.
For some light relief, take a look at:
*
|
Aims
On completion of this TLP, you should:
* Be familiar with the concept and mechanism of aqueous corrosion
* Know what factors affect the rate of aqueous corrosion
* Be familiar with the use of Tafel plots to predict aqueous corrosion rates
Before you start
This TLP is largely self-contained, though some sections require knowledge of the Pourbaix diagram. Details of the thermodynamics of aqueous corrosion and the Pourbaix diagram can be found in the TLP entitled .
Introduction
We are reliant on metallic structures to support our everyday activities, be it getting to work, transporting goods around the world, or storing and preserving food. Metals are everywhere.
However, from the moment most metals come into contact with water, they are subject to sustained and continuous attack which can lead to the metal corroding and failing to do its job.
It is therefore important to understand when corrosion will occur, how fast it will proceed and what can be done to slow down or stop it. Whether or not corrosion occurs at all is dependent on thermodynamics and is covered in the .
How to predict and control the rate of corrosion is covered in this TLP.
What's Going On? The Mechanism of Aqueous Corrosion
=
Corrosion involves two separate processes or **half-reactions**, oxidation and reduction. Oxidation is the reaction that consumes metal atoms when they corrode, releasing electrons. These electrons are used up in the reduction reaction.
When a metal corrodes in solution, the two halves of the reaction can be separated by large distances. This is unlike oxidation in air, when one reaction occurs at the surface of the film and the other at the surface of the metal, meaning that the reaction sites are always close to each other. In fact, in aqueous solution the reaction separation can be very large and as long as there is both electronic and electrolytic contact between the anodic and cathodic sites, corrosion will occur regardless of separation of the half-reactions.
When a Metal Corrodes - the Electrical Double Layer
=
An electrical double layer is the name given to any region between two different phases when charge is separated across the interface between them.
In aqueous corrosion, this is the region between a corroding metal and the bulk of the aqueous environment (“free solution”). In the double layer, the water molecules of the solution align themselves with the electric field generated by applying a potential to the metal. In the Helmholtz model, there is a layer of aligned molecules (or ions), which is one particle thick and then immediately next to that, free solution. In later models (proposed by Louis Georges Gouy, David Leonard Chapman and Otto Stern) the layer is not well defined, and the orientation becomes gradually less noticeable further from the metal surface. However, for the purposes of determining the rate of corrosion, the Helmholtz model will suffice.
To corrode, an ion in the metallic lattice must pass through the double layer and enter free solution. The double layer presents a potential barrier to the passage of ions and so has an acute effect on corrosion kinetics.
Like all chemical processes, the kinetics involved in corrosion obey the Arrhenius relationship:
\[k = {k\_0}\;\exp \left( {\frac{{ - \Delta G}}{{RT}}} \right)\]
where k is the rate of reaction, k0 is a fundamental rate constant, ΔG is the activation energy. R and T have their usual meanings of the ideal gas constant (8.3145 J K−1 mol‑1) and temperature (in Kelvin) respectively.
The chemical nature of corrosion suggests that it is driven by a change in Gibbs Free Energy, ΔG but the electrical nature of corrosion leads to the conclusion that a voltage drives the reaction. Since both quantities can be considered as the driving force, they must be equivalent and, indeed they are related through the expression ΔG = −z F E , where z is the stoichiometric number of electrons in the reaction, F is Faradays constant, 96485 C mol−1 and E is the voltage driving the reaction.
Note the minus sign, used to correct for the conventions that a chemical reaction only proceeds if ΔG is negative but an electrical reaction only occurs if E is positive.
Since the absolute driving force of an applied voltage depends on what reaction is occurring, potentials are usually defined as the difference between applied voltage and the equilibrium potential of the reaction. The difference between applied potential and equilibrium potential is defined as the ***overpotential***, *η.*
η = E − Ee
It is worth noting that the “equilibrium potential” is not necessarily the standard electrode potential of the reaction, as this has the added requirement that all reagents are in standard states. The equilibrium referred to is, in fact the “equilibrium electrode potential”, Ee, which is specific to every electrode individually.
If an electrode is at its equilibrium potential, both forwards and backwards reactions occur at the same rate, so no net reaction will occur. Net reactions only occur when the potential is moved away from equilibrium.
The Energy Landscape
Under equilibrium conditions, the energy landscape is symmetrical when free energy is plotted against distance from metallic surface:
The fraction of the width of the double layer that must be crossed to reach the excited state is known as the symmetry factor, α.
However, when overpotential is applied, the energy is changed on the free solution side of the plot by an amount -*z*F*η*. The overpotential is distributed so that a fraction, α lies across the barrier in the forward direction and (1 − α) lies across the barrier in the backward direction.
The overall effect of the overpotential is to **lower** activation energy for the forward reaction by α z Fη. Thus the Arrhenius relation now becomes:
\[{k^′} = {k\_0}\;\exp \left( {\frac{{ - (\Delta {G^0} - \alpha zF\eta )}}{{RT}}} \right)\]
\[{k^′} = k\;\exp \left( {\frac{{\alpha zF\eta }}{{RT}}} \right)\]
where k is the new rate, k is the rate without the overpotential.
Kinetics of Corrosion - the Tafel Equation
Tafel equation
Armed with the new Arrhenius expression and the generalised reaction:
M ⇌ Mz+ + ze,
where M is a metal that forms Mz+ ions in solution, we can now derive an equation describing corrosion kinetics.
Consider the rate of the anodic (oxidation, corrosion) reaction, ka
\[{k\_{\rm{a}}} = k\_{\rm{a}}^{'}\;\exp \left( {\frac{{ - \Delta G^0}}{{RT}}} \right)\]
Since the reaction involves the release of electrons, its progress can be expressed as a current density, i (current per unit area).
The **exchange** current density, i0 is defined as the current flowing in both directions per unit area when an electrode reaction is at equilibrium (and, hence, at its equilibrium potential).
If i0 is small, then little current flows and the reactions at dynamic equilibrium are generally slow. Likewise, a high i0 gives a fast reaction. The metal itself affects the value of i0, even if the reaction does not involve the metal directly.
\[{i\_0} = z{\rm{F}}{k\_{\rm{a}}} = z{\rm{F}}k\_{\rm{a}}^{'}\;\exp \left( {\frac{{ - \Delta G^0}}{{RT}}} \right)\]
If overpotential is applied, the activation energy is changed, as described on the previous page:
\[{i\_{\rm{a}}} = i\_{0}\;\exp \left( {\frac{{\alpha z{\rm{F}}\eta }}{{RT}}} \right)\]
This is one form of the **Tafel equation**.
The Tafel equation can also be written in several equivalent ways, as shown .
The quantity \(\frac{{2.303\,RT}}{{\alpha z{\rm{F}}}}\) is given the symbol *b*a and is known as the anodic Tafel slope. It has units of volts per decade of current.
Similarly, if the cathodic reaction were to be considered, the quantity would be \(\frac{{ - 2.303\,RT}}{{(1 - \alpha )z{\rm{F}}}}\) since (1 − α) is applicable instead of α and E - Ee is negative. This quantity is the cathodic Tafel slope, bc. .
The usual form of Tafels equation is η = *a* + *b*a log *i*a where \(a = \frac{{ - 2.303\,RT}}{{\alpha z{\rm{F}}}}\;\log {i\_0}\)
Through consideration of the reaction as both a chemical and electrical process and manipulation of algebra, we have found that the applied potential is proportional to the log of the resulting corrosion current. This is certainly different to Ohmic behaviour where applied potential is directly proportional to the resulting current.
The Tafel Plot
Using the Tafel equation, useful plots can be drawn to help find corrosion rates.
In a plot of i vs. E, for a single electrode, the following is seen:
Considering the sum of the currents and **then** ignoring the signs and **then** taking log of current gives a plot known as a Tafel plot, which is described in the animation below for a single electrode:
As can be seen, at Ee, the net current flow is 0, as must be the case for equilibrium (the anodic and cathodic currents are equal and opposite). The straight-line sections have gradients related to the Tafel slopes – anodic, ba and cathodic bc. (If we had plotted E on the vertical axis and log i horizontally the gradients would be equal to ba and bc. )
There are several important points to note:
* *i*a and *i*c never reach zero individually. However, the resultant net current flow will be zero if anodic and cathodic currents are equal in magnitude.
* This derivation applies both to dissolution (corrosion) of metals and deposition (electroplating) of metals.
* This derivation also applies to hydrogen evolution and oxygen reduction, even though they dont involve metal ions – they can still **activation controlled**.
* The signs may be dropped since they serve only to define direction of current flow. Had current been defined the opposite way around, all signs would be reversed, so dropping signs (to allow logs to be taken) is not unreasonable.
* α is usually 0.5 for a single step reaction.
* Multiple step reactions can have different steps with different stoichiometric numbers of electrons (different z). In this case, the value of z for the rate determining step should be used, **not the overall stoichiometric number**.
* However, the overall stoichiometric value of z **is** used to relate current density to rate constant and in the Nernst Equation. If these are different, it is standard to rename the value for the RDS (the value inside the exponential) as n.
N.B. There is no universally adopted standard to plot log (i) on the y-axis and E on the x-axis. In fact, it is more common to see polarisation curves plotted as E vs. log (i). In this TLP graphs will be plotted as log (i) vs. E.
Tafel plots can be linked to Pourbaix diagrams:Corrosion occurs when two electrodes with different equilibrium potentials are in both electronic and electrolytic contact. We can use Tafel plots to predict corrosion rates as explained in the animation below.
Diffusion Limited Corrosion
=
So far all reactions have been assumed to proceed (if they are thermodynamically possible) at the rate predicted by the Tafel analysis. In reality, reactions are often limited by other factors and dont achieve this maximum rate. One such factor is the availability of oxygen in solution.
In aqueous solutions that contain dissolved oxygen, an important cathodic reaction is the oxygen reduction reaction:
O2 + 4 H+ + 4 e- → 2 H2O
The reaction takes place at the surface of the metal and so oxygen must be present at that site. If the reaction occurs quickly enough, the concentration of oxygen at the surface cannot be maintained at the same level as that in the bulk of the solution. In this case the rate of oxygen diffusion may become a limiting factor. With less oxygen available, the cathodic reaction slows down and so must the anodic reaction to conserve electrons (electrons can only be used up at the same rate as they are released as charge must always be conserved).
\* can be used to find the maximum rate of oxygen diffusion. Since each oxygen molecule consumes 4 electrons, according to the reaction above, this maximum rate of diffusion corresponds to a maximum current density that the oxygen reduction reaction can sustain and, hence, a maximum corrosion rate for the anode (since electrons must be used at the cathode at the same rate as they are released at the anode).
Since the corrosion current is limited, the cathodic arm of the Tafel plot is flattened:
Oxygen reduction is not the only process that deviates from the Tafel analysis. The hydrogen evolution reaction can be limited by the rate at which molecules desorb from the cathode surface. This is usually the rate-determining factor for hydrogen evolution on iron, copper, platinum and other metals. Relatively few metals behave as predicted by the Tafel analysis, examples being cadmium, mercury and lead.
Passivation
=
Another effect that limits the rate of corrosion is **passivation**. If the potential of an electrode is raised above some **passivation potential**, a passive product may become favourable forming a layer on the surface of the anode. In this case, the rate of corrosion can be much reduced. This is characterised by the value of log (i) peaking at a **critical current density**, before falling to some lower value.
In other words, the anodic arm of the Tafel plot reaches a peak and falls away to a roughly horizontal region:
It is possible to deliberately drive the reaction into the regime in which a passive layer forms. This technique is used in the process known as **anodising** in which thick oxide layers are developed on aluminium components.
The Tafel plot below shows two electrodes. The cathodic branch of the electrode with the higher equilibrium potential (shown in blue) is diffusion limited. The anodic branch of the electrode with the lower equilibrium potential (shown in red) is passivated. The resulting intersection is often a good representation of real corrosion scenarios, where the metal can passivate and there is a limited oxygen supply for the cathode.
Predicting Corrosion Rates
Armed with the Tafel equation and Tafel plots, it is now possible to predict whether a particular setup will result in corrosion and if so how fast the corrosion will be.
In order for corrosion to occur, there must be a suitable anodic reaction and an appropriate cathodic reaction. This is manifested as an intersection of a cathodic branch and an anodic branch on a Tafel plot. The point of intersection gives the corrosion potential and the corrosion current (or, more accurately the log of the corrosion current density).
The rate of corrosion is governed by all the factors discussed previously. When all the effects are taken into account, Tafel plots get quite complicated and some interesting effects occur:
Faradays law allows the current density to be expressed as the mass of material lost per unit time.
The calculation involves a few simple steps. For a corrosion reaction:
1. The current is converted into a rate of electron consumption using the electronic charge constant.
2. The number of electrons is divided by the stoichiometric number of electrons in the corrosion reaction, giving the number of metal atoms lost per unit time.
3. This answer is then divided by Avogadros number to give the number of moles of metal atoms lost per unit time.
4. The number of moles is then converted to mass lost per unit time, using the molar mass.
5. The mass is then converted to a volume using the density.
6. The volume is then converted to a thickness lost per unit time by dividing by the area that the current passes over. If a current density was given, this step has already been done.
Overall, the thickness of metal lost per unit time is given by the formula:
$$t = {{i{m\_{\rm{M}}}} \over {\rho ez{N\_{\rm{A}}}}}$$
where t = thickness (m), i = current density (A m-2), mM = molar mass (kg mol-1), e = electronic charge (C), z = stoichiometric number of electrons in oxidation reaction, NA is Avogadros number.
It is also possible to have a situation where corrosion does not occur for thermodynamic reasons, for example if there was a driving force for the reverse of the corrosion reaction to occur due to an applied potential. This would result in deposition (electroplating) if there were metal ions in solution available to be reduced. If deposition is being carried out commercially, for example to electroplate silver onto stainless steel cutlery, the rate must be maximised to make production as cost effective as possible. However, care must be taken to avoid the hydrogen evolution reaction starting at the cathode in addition to the metal ion deposition.
We can now draw Tafel plots and use them to determine corrosion current densities and corrosion rates. Below is an interactive graph that allows the corrosion rates of several metals to be investigated. Notice how, in this idealised situation, i.e. with no diffusion limits, the corrosion rate in aerated water can be extremely high. This shows how important a consideration the diffusion layer is.
Corrosion Control
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There are several ways kinetics can be employed to reduce or prevent corrosion.
Barriers and coatings
-
A barrier can be employed to prevent the electrolyte coming into contact with the metal.
Tin is the usual barrier, as it does not react in most aqueous solutions. It is used in food cans and has the useful property that if the barrier fails, then the steel in the can corrodes. If zinc were used instead it would begin to act as a sacrificial anode should the barrier fail. This would protect the steel but the distinct disadvantage is that hydrogen is evolved at the steel cathode and if it were to build up inside a closed container an explosive situation could arise.
Other inert metals may be used as barriers, as can polymers, ceramics and paint.
Anodic protection
-
Anodic protection involves raising the potential of the metal in order to develop
a passivating layer (such that protection is due to inhibited kinetics).
Sodium carbonate, a base, acts to remove the acidity in the solution and drives
the reaction towards the right of the Pourbaix diagram. Above a certain
pH, metals tend to form a passive layer as a passive species is stable under
these conditions.
Potassium chromate works by providing a source of chromate ions that penetrate the surface of the metal, forming a stable, passive chromium oxide layer on the surface of the anode and thus prevents corrosion by forming a passivation layer in a similar way to that seen in stainless steels.
Cathodic protection
-
Cathodic protection involves lowering the potential of the metal in order to
make it more thermodynamically stable. This may be done using an impressed current
(a supply of electrons from an external source) or with a sacrificial anode,
as shown below:
Summary
=
Corrosion is a problem facing us every day and in almost every activity. Corrosion wastes material and energy, and could prevent objects from doing the job they were made to do, possibly with dangerous consequences.
The rate at which corrosion occurs depends on the kinetics of the reactions taking place and so the electrical double layer is important.
Applying an overpotential to an electrode drives the reaction in one direction and away from equilibrium. Tafels law governs the new rate and as long as the reaction kinetics are activation controlled, the overpotential is proportional to the log of the corrosion current.
Other factors may limit the maximum rate of corrosion, with oxygen depletion limiting the speed of the cathodic reaction to the rate at which oxygen can be supplied from the bulk. The anodic reaction may be limited by passivation, if a sufficiently large overpotential is applied to form a passive layer. Passive layers separate the metal from the electrolyte and slow the corrosion reaction.
Faradays law can give meaningful results from the predicted corrosion current, i.e. giving the mass loss per unit time.
Corrosion can be slowed by either adding an inhibitor to remove hydrogen ions and move to a passivating region of the Pourbaix diagram, by adding an inhibitor to form a passive layer on the anode, or by adding an inert barrier to the surface of the anode.
Questions
=
### Quick questions*You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!*1. Which of the following half-reactions represent a general corrosion process? | | | |
| - | - | - |
| |
a |
M + 2e → M2+ and H2 → 2H+ + 2e |
| |
b |
M → M2+ + 2e and H2 → 2H+ + 2e |
| |
c |
M2+ + 2e → M and H2 → 2H+ + 2e |
| |
d |
M → M2+ + 2e and 2H+ + 2e → H2 |
2. Which of the following are possible cathodic reactions that accompany corrosion | | | |
| - | - | - |
| |
a |
2H+ + 2e → H2 |
| |
b |
O2 + 4H+ + 4e → 2H2O |
| |
c |
4OH- → O2 + 2H2O + 4e |
| |
d |
Mnz+ + ze → M |
3. Which of the following is **not** a form of Tafel's equation? | | | |
| - | - | - |
| |
a | |
| |
b | |
| |
c | |
| |
d | |
4. Using the electrochemical series, which of the following metals can be used as a sacrificial anode for steel under standard conditions? Click on this link to | | | |
| - | - | - |
| |
a |
Nickel |
| |
b |
Zinc |
| |
c |
Magnesium |
| |
d |
Tin |
5. Look at the following Tafel plot. What is the critical current density?
 | | | |
| - | - | - |
| |
a |
15 μA m-2 |
| |
b |
1.5 A m-2 |
| |
c |
32 A m-2 |
| |
d |
0.3 μA m-2 |### Deeper questions*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*6. Draw the Tafel plot of the following information on graph paper and find:
a) The corrosion potential
b) The corrosion current density
c) How long would a 3 mm thick component survive in this scenario (use Faraday's law)
Both reactions have ba = -bc = 0.12 V / decade.
One half-reaction has an equilibrium potential -0.25 V (SHE) and an exchange current density of 10 μA m-2. This reaction has a passivation potential 0.15 V (SHE) and passive current density 10 mA m-2.
The other half-reaction has equilibrium potential 0.8 V (SHE) and exchange current density 0.1 mA m-2.
The metal corroding forms 2+ ions, has a molar mass of 35 g mol-1 and has a density of 6400 kg m-3.
7. a) Write balanced half-reactions and the overall reaction for an iron water pipe corroding in fully aerated water under standard conditions, and with the water flowing at such a rate to maintain a diffusion layer 1 μm thick.
b) Derive the anodic and cathodic Tafel slopes for the two halves of the reaction if α = 0.5
Using the fact that the iron does not passivate in the potential range being considered, some of the data in and the other information and answers above, draw the Tafel plot on graph paper, and then calculate
c) The corrosion potential
d) The corrosion current density
e) Whether or not the pipe will survive 1 year if it has walls 5 mm thick
f) What happens if the pipe is part of a sealed system such as central heating?
g) Why else might your answer to c) be flawed?
Going further
=
### Books
J M West, *Basic oxidation and corrosion*, Ellis Harwood (1936)
K R Trethewey and J Chamberlain, *Corrosion for students of science and engineering*, Longman (1988)
### Websites
A really good website with definitions and explanations
Another website that covers most things in this TLP. A bit more detailed but needs more prior knowledge
|
Aims
On completion of this TLP you should:
* be able to identify the materials used in common objects using suitable techniques;
* be able to identify the processes used to produce a particular shape or microstructure within components of an article;
* be able to understand why materials and processes might be chosen to produce a given article.
Before you start
This TLP brings together considerations from a range of other topics, and so many other TLPs might be helpful in understanding the properties and techniques referred to. There will be links to the relevant TLPs at appropriate points.
Introduction
Most manufactured items contain a large number of individual components, often using a surprisingly wide variety of materials. Examining a manufactured item (article) can help to understand what materials are commonly used in manufacturing and why.
It is important to keep in mind economic factors as well as the properties of materials and the specific purpose of each component. This means it can be quite a complex procedure to identify, with confidence, the materials within an article. It is usually done by considering a range of all appropriate factors and making use of some of a broad range of identification techniques.
This TLP suggests some factors that may be considered in attempting to identify the materials used in various articles. It will also give some examples of where these factors may be important.
The other DoITPoMS TLPs provide a wide variety of information about the functional and mechanical behaviour of materials, as well as a range of information on techniques for examining materials. This TLP will not aim to describe all of these, but will link to them.
Dismantling the Article
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The first step to determining the materials and reasons for their use in an article is to take it apart. This is an important stage because it is a chance to see where components fit, and what their purposes might be.
An exploded diagram of components and where they fit in relation to the rest of the article is often extremely helpful. Once the article has been dismantled it may not be possible to reassemble it, and knowing the position of a component in the article is vital to understanding what its function might be. The exploded diagram helps to sort out what the purpose of components are, and therefore what might be required of their properties.
The diagram should be clear and well labelled, including scale bars.
Important properties to consider for each component could include:
* **Mechanical properties:** such as strength or toughness
* **Electrical properties**: conducting or insulating
* **Aesthetic properties**: is the component on display, and would the appearance be important?
* **Corrosion resistance**: the component will probably need to last for at least the lifetime of the product
* **Density of component**: it may be important that the article is lightweight
* **Specific functions**: some materials are chosen because they fulfil a specific function within the article (for example or materials)
It is also important to consider economic factors, such as the costs of the raw materials and processing. The cost of a material can influence choice as much as its physical properties; for example diamond is an extremely hard material, but it is not widely used due to cost.
Each component will probably be reasonably easily identifiable as belonging to one of four classes of materials:
* Metals
* Polymers
* Ceramics
* Composites
Classes of Materials
It can often be quite straightforward to tell materials classes apart by look or feel. Metals are usually more reflective or 'metallic' looking, ceramics are commonly matte and polymers may be shiny or matte, but are typically less dense than either metals or ceramics. Composites may be harder to immediately identify, but the surface may appear non-uniform and/or sectioning the sample may reveal fibres or particles.
It is useful to note that taking a cross-section can often be helpful in identifying materials used in components as the internal material and/or microstructure may differ from that at the edge.
Some materials may not be quite so easily identified simply from appearance and texture, but considering the factors mentioned can help to narrow down the possibilities. The table below gives a rough guideline to the kinds of properties you would expect from each class, and a few examples:
| | | |
| - | - | - |
| **Class** | **Common Properties** | **Examples** |
| Metal | Hard, ductile and conduct heat and electricity | Copper (wires), stainless steel (cutlery) |
| Polymer | Widely variable, often soft and flexible | Polystyrene (cups), polycarbonate (CDs), polyethylene (plastic bags) |
| Ceramic | Hard, brittle, resistant to corrosion, electrically non-conductive | Concrete (buildings), PZT (piezoelectric used in lighters and ultrasonic transducers), porcelain (vases, teacups) |
The tree-diagram below shows an overview of a variety of materials that might be encountered:
Coatings
Many components may have some kind of coating; a covering of another material designed to improve the surface qualities of the item. The improvement could be for many reasons including: corrosion resistance, appearance, adhesion, wear resistance and scratch resistance.
Different kinds of coating will have different processing methods. It is often possible to deduce the method from the composition (of the coating and the bulk of the component) and the shape of the component.
**Common coating methods include:** ***Hot dip coating*** – a method used for coating metals (commonly ferrous alloys) with a low melting point alloy. The component is dipped in a bath of the molten coating alloy. For example zinc is often hot dipped onto steel (called ‘galvanising). This also offers sacrificial corrosion protection and gives a distinctive ‘spangled appearance (which can be prevented by including particles to encourage nucleation in the electrolyte).
Steel coated with zinc
***Electroplating*** – reduction of cations in an electrolytic solution onto conducting components. For example silver plated cutlery.
***Anodising*** – commonly used for aluminium components, an electrochemical cell is set up which drives the oxidation of the metal, increasing the thickness of the protective oxide layer.
***Vacuum deposition*** – also known as PVD – physical vapour deposition. For example, ‘evaporation involves the heating of the coating metal in a vacuum, so that it evaporates and is deposited onto the surface of the component that is positioned above. This process is used to make mirrors, depositing a thin layer of metal, usually aluminium.
***Thermal Spraying*** - powder particles are fed into a high temperature torch (combustion or plasma), where they melt and are accelerated against the substrate. It is mainly used to produce relatively thick ceramic and metallic layers.
***Enamelling*** - a powder is distributed on a surface, which is then heated so that the powder melts and bonds to the substrate. The resultant layer is usually glassy. Originally developed in ancient Egypt, and extensively used for jewelry, it is also widely employed for cooking utensils and various domestic items, especially those subjected to high temperature.
One example of a coated metal is shown below; a drawing pin, which appears to be brass, was found to be magnetic and so the surface was abraded, revealing a grey metal within – steel. Steel, which has very good mechanical properties, is covered with brass for aesthetic reasons and also protects the surface from corrosion.
| | |
| - | - |
| | |
| Magnetic drawing pin | Abraded pin revealing steel |
Metals
Metals are extremely widely used in manufacturing, often for mechanical or electrical properties. They are often easily shaped and have good mechanical properties so they may be used in ‘structural elements of an article, and they also have good conduction (electrical and thermal) and so may also be used, for example, in electrical wiring. Familiar metals can often be reasonably well identified by eye (see examples below), but there are more complex methods of metal identification available too.
Techniques for the Identification of Metals
-
One very easy test for a metal is to see if the component is magnetic, this narrows down the possible materials to those that are ferromagnetic (most commonly iron or nickel).
Simple corrosion tests involving immersing a scratched sample (to remove any coating) in water (or some other electrolyte) can be helpful. Leaving a sample in water overnight might reveal rusting. For example a scratched zinc coated steel sample would not rust due to the zinc offering sacrificial protection. However, a scratched tin coated steel sample would rust, because the tin is supposed to act as a barrier between the steel and air.This is especially useful for ferrous alloys as corrosion resistance is very often a concern for these (and they are very common).
**Optical Microscopy**
This involves looking at mounted, polished and etched samples under a light microscope. It reveals the microstructure of the sample; this can give information on both the composition and processing of the component.
See TLP for more information on how to go about this.
| | |
| - | - |
| Image showing Al-Cu Eutectic composition | Image of cold rolled zinc showing deformation twins |
| Al-Cu Eutectic composition
This is an Al-Cu alloy showing a very clear eutectic lamellar microstructure.
(See for more information) | Cold rolled zinc showing deformation twins
This is zinc, it has been cold rolled as can be seen from lenticular deformation twins
|
See the for further examples.
The benefits of this method are that optical micrographs can reveal a large amount of information about a metallographic sample, and it is possible to find known examples (see above links) to compare your work to. After an initial examination by eye and consideration of properties, optical microscopy is an important step in the characterisation of metals. It can reveal many things that the initial examination does not.
Scanning Electron Microscopy (SEM)
Scanning Electron Microscopy uses a focussed beam of high-energy electrons to form images of samples. Electron Microscopy is not limited by the wavelength of light, so very closely spaced features can be resolved, so this method gives very clear high magnification images when set up correctly. It also gives a large depth of field, so rough surfaces can still be in focus.
The SEM can give very good high magnification images, again revealing more than optical microscopy could. One limitation is that the sample must be electrically conducting; the mounting polymer and the sample must both conduct electricity.**Energy Dispersive X-ray Spectroscopy (EDS)**
This is a technique often used in conjunction with the SEM, with an electron beam of ~20 keV. The beam strikes the sample resulting in X-rays being emitted; the X-rays are collected and the intensities and energies examined. The results can determine the atomic composition of the sample at the point of beam-sample interaction. Examples: It can reveal the composition of a very thin coating; for example by taking a linescan. The diagram below shows results for a linescan taken across a coating, showing a layer of copper and a finer layer of nickel on an iron alloy (all other elements were ignored). This may not be easy to tell from optical microscopy alone.
EDS can be a very helpful method of characterisation, but it is not always absolutely reliable. Elements of low atomic number (less than about 11, i.e. below sodium) are difficult to detect by EDS. There is often contamination of elements like carbon from the environment. It is important to use common sense when interpreting the EDS results; most of the time is it unlikely that very heavy elements like uranium are actually present. Results may be very good qualitatively, but care must be taken when trying to obtain and interpret quantitative results. The quantitative analysis results depend upon things like the set-up of the SEM and the geometry of the sample.
Examples of Fabrication Processes
-
The ease with which metals are shaped leads to a wide range of processing techniques for different end products; any coatings will also be applied by one from a range of processes. It may be possible to deduce the method of processing from the shape and the properties of the metal. The microstructure may also give further clues:
| | | |
| - | - | - |
| **Process** | **Description** | **Features** |
| Deformation | Includes a variety of techniques including forging, extruding and drawing, see the TLP for more information | May expect to see directionality or squashed grains in places that have been stressed (see the animation) |
| Machining | Includes, for example, laser cutting and water-jet cutting as well as more conventional methods like sawing or grinding. | These methods would not result in larger scale microstructural directionality, but may show localised deformation.
Can give a very good finish |
| Casting | The molten metal is set in a cast of some kind, see the TLP for further details on the different kinds of casting. | There is a lot of variety of microstructure from cast products, which ranges from single crystal components to those with clear chill, columnar and equiaxed zones. |
| Carburisation – a surface heat treatment | A surface treatment in which carbon is diffused into the surface of a steel object above the ferrite-austenite transition temperature (~ 740 °C). This is done by heating the steel in a C-rich atmosphere, (e.g. in CO gas). The result is a hard, high carbon surface several hundreds of microns thick, surrounding a tough, low carbon interior. To improve the hardness, the surface may be quenched, which helps the production of martensite. | See micrograph below |
Micrograph #271 An example of carburisation; see the for further information.
Polymers
Polymers are very widely used in many areas today. They have a range of properties that can often be controlled by additives, blending or copolymerisation. Many structures and chemical compositions are seen in polymers, but they can be separated into three main groups:
**Thermoplastics:**
These are the most widely used polymers due to the ease of processing (especially for injection molding). Thermoplastics can, once they have been set (solidified) for the first time, be re-melted and remoulded (unlike thermosets). Some examples of thermoplastics are: polyethylene, polystyrene and PET.
**Thermosets:** These differ from thermoplastics in that they do not re-melt after they have been set (or cured). This is due to the long polymer chains forming cross links on curing. One example is melamine formaldehyde, which is used in domestic electrical plugs.
**Elastomers:** These polymers have a glass transition temperature below room temperature (see TLP). Rubbers are examples of commonly used elastomers (for more information see the TLP)
Techniques for identifying polymers
-
### **Polymer tests**
The polymer tests are a simple way to identify polymers, or at least to narrow down the possibilities. Some of the steps rely on recognising smells, which can be difficult, and it is important to remember that some tests (for example transparency) can be unhelpful due to additives like dyes.
The goes through a series of simple tests, which should be carried out on a small sample of the polymer. Below is an interactive version of the identification chart. It is important to connect the results of the test to the function and cost of the item.
### Infra Red (IR)
In an IR spectrometer IR radiation excites covalent bonds, causing them to vibrate at their resonant frequency. This frequency depends on the exact nature of the bonds (e.g. single/double and the atomic masses of the elements involved). The output is a graph of intensities at different wavelengths (and therefore energies) of infrared radiation. This plot shows the transmitted intensities, so at resonant frequencies, where the energy is absorbed, there is a peak. This allows the bonds to be identified and therefore the sample identified.
Here is a collection of IR spectra for some common polymers:
IR spectrometry is often a very quick method of polymer identification (depending on the equipment available). Preparing a sample for IR spectroscopy may be very simple. A small sample of the polymer with any coatings removed should be placed in the IR machine, and analysed. If it is likely that the plastic contains plasticizers and colours, placing the polymer in ether for an hour and then fully drying it may remove them prior to carrying out IR spectroscopy on the sample. (Test this with a small piece of your sample polymer first though, as some polymers are soluble in ether).
### Differential Scanning Calorimetry (DSC)
DSC measures specific heat capacity and how it varies with temperature. As a polymer is heated through its glass transition point, it experiences a sudden change in heat capacity, as chain rotation allows it to take up more energy. This means that DSC allows us to identify the glass transition temperature of a polymer. It can also aid the interpretation of the type of a copolymer (e.g. block copolymer, random copolymer, graft copolymer).
See for an explanation of this technique.
Examples of processes
-
Polymers are usually processed by moulding methods:
| | | |
| - | - | - |
| Process | Description | Features |
| Injection moulding | Polymer granules are melted and forced into a mould. This is extremely widely used to mass produce small, precise polymer components. | It gives a good finish and the injection points where excess material has been cut off are often visible. In transparent polymers a residual stress field may be visible under crossed polars (see the TLP). |
| Blow moulding | Cylinders of polymer are inserted into a die and hot air is forced in, pushing the polymer out to the walls of the die. | Gives hollow components, such as bottles or containers. It is only used for thermoplastics. |
For further examples see on .
### Additives and Blends:
Polymers very often have some form of additive, even if it is simply to add colour. These may or may not impair the ability to identify the polymer. When identifying any material it is important to think about the cost and properties, but blending or additives can change the properties of a polymer.
One very common example of a polymer commonly found both with and without additives is polyvinylchloride (PVC). This polymer is used in its rigid, un-plasticized form in plastic guttering and water and gas piping, but is also often found with added plasticizers in a variety of applications from clothing to coating electrical wires.
Ceramics and Composites
=
Ceramics
Ceramics cover a very wide range of materials from structural materials like concrete to technical ceramics like PZT – a . Usually they are defined as solids with a mixture of metallic or semi-metallic and non-metallic elements (often, although not always, oxygen), that are quite hard, non-conducting and corrosion-resistant.
**Techniques for identifying ceramics**
It is effectively impossible to identify ceramics by eye. Optical microscopy will allows the examination of the microstructure to identify the method of processing, however, it does not allow the identification of different phases.
The most useful technique for finding the composition of a ceramic is energy dispersive x-ray spectroscopy (EDS). Note that for non-conducting ceramics the surface of the sample must be covered with a metallic coating (often gold) to prevent charge build-up.
Here is an example EDS for PZT – a piezoelectric ceramic: Pb[Zr*x*Ti1-*x*]O3, this data gives the formula to be: Pb0.7[Zr0.49Ti0.44]O3. For the piezoelectric ceramic we would expect to have *x* ~ 0.52.
| | |
| - | - |
| **EDS data for PZT** | |
| **Element** | **Weight%** | **Atomic%** |
| O K | 17.26 | 63.49 |
| Ti K | 7.58 | 9.32 |
| Zr L | 16.15 | 10.42 |
| Pb M | 59.01 | 16.77 |
| Totals | 100.00 | 100.00 |
Another appropriate method is . This allows you to detect the phase or phases present as well as measuring lattice parameter(s) in order to specify precise compositions.
**Processing techniques for ceramics**
Ceramics are mostly made by powder processing techniques, for example
sintering. It may be possible to identify the kind of processing from directionality or porosity in the sample.
Composites
Composites are often used in applications that require specific ‘conflicting properties such as a high strength and high toughness. The properties may be conflicting because having a high yield stress sometimes relies on trapping and tangling dislocations, but these reduce the ductility and toughness of the material. Composites often consist of a matrix and fibres or particles that affect the properties (see the TLP on the ).
Usually for composites, once they have been identified as such, it is better to treat each part of the composite as a separate material, and then subsequently look at costs of manufacture and processing.
One important distinction to make is the structure of the two parts that make up the composite – i.e. is it a matrix with long, aligned fibres? Or a matrix with particles? etc
Example Article
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The best way to understand the concepts in this TLP is to try analysing something. Here is a 'virtual' article, which can be clicked through:
Summary
=
Many articles can be analysed using the relatively simple techniques described here. This can help with determining the types of material used for different components, their composition, and also processing history. The examination of an article can start to put into use the methods and theory of materials science. Looking at the mechanical, thermal and aesthetic properties of materials can help materials scientists design similar items.
The range of techniques available today is very large, but often a reasonable amount of understanding can be gained from fairly simple techniques and using some common sense. It is essential not to forget the importance of stepping back from results and considering whether or not they are logical; do they fulfil the requirements in terms of mechanical, thermal, aesthetic and economic properties?
Questions
=
### Quick questions*You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!*1. Which of these methods of characterisation would be helpful in identifying a ceramic? | | | |
| - | - | - |
| |
a |
Infra red spectroscopy |
| |
b |
Differential Scanning Calorimetry |
| |
c |
Energy Dispersive X-ray Spectroscopy |
| |
d |
Scanning Electron Microscopy |### Deeper questions*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*2. Match the material to the properties, try to think of reasons why the material may have been chosen.
Material 1 has a shiny 'spangled' appearance due to large dendritic grains on the surface, it is a structural component within the article and must withstand relatively high stresses, and may experience wear in service, it must not corrode in warm air and is in a low cost article. | | | |
| - | - | - |
| |
a |
Tinned Steel |
| |
b |
Stainless Steel |
| |
c |
Aluminium |
| |
d |
Galvanised Steel |
3. Match the material to the properties, try to think of reasons why the material may have been chosen.
Material 2 is a brightly coloured, low-density component, it must be tough, rigid and non-toxic, the recycling mark is number 7. | | | |
| - | - | - |
| |
a |
Polyethylene (PE) |
| |
b |
Acrylonitrile-butadiene-styrene (ABS) |
| |
c |
Polyethylene terephthalate (PET) |
| |
d |
Polytetrafluoroethylene (PTFE) |
4. Match the material to the properties, try to think of reasons why the material may have been chosen.
Material 3 is an electrical component, it forms a contact that is subjected to a reasonable amount of wear, it must not corrode in air. | | | |
| - | - | - |
| |
a |
Low carbon steel |
| |
b |
Copper |
| |
c |
Brass: Copper/Zinc alloy |
| |
d |
Stainless steel |
Going further
=
### Websites
Information
on selection, design and processingSome simple ways to identify metals
Some simple ways to identify polymers
This TLP covers the fundamentals of metal forming processesMacrogalleria's explanation of using DSC for sudying thermal transitions of polymers "Probably" the Web's largest plastics encyclopedia, including plastics processes
Metallography advice, including sample preparation methods and choice of standard etchants
|
Aims
On completion of this TLP you should:
* know the differences between single crystal, polycrystalline and amorphous solids
* be able to identify the characteristic features of single crystals and polycrystals
* understand the nature of crystal defects
* appreciate the use of polarised light to examine optical properties
Introduction
The fundamental difference between single crystal, polycrystalline
and amorphous solids is the length scale over which the atoms are related
to one another by translational symmetry ('periodicity'
or 'long-range order'). Single crystals have infinite periodicity, polycrystals have local periodicity, and amorphous solids
(and liquids) have no long-range order.
* An *ideal single crystal* has an atomic structure that repeats periodically
across its whole volume. Even at infinite length scales, each atom is related
to every other equivalent atom in the structure by translational symmetry.
* A *polycrystalline solid* or *polycrystal* is comprised of many
individual *grains* or *crystallites*. Each grain can be thought
of as a single crystal, within which the atomic structure has long-range order.
In an *isotropic* polycrystalline solid, there is *no relationship*
between neighbouring grains. Therefore, on a large enough length scale, there
is no periodicity across a polycrystalline sample.
* *Amorphous* materials, like window glass, have no long-range order
at all, so they have no translational symmetry. The structure of an amorphous
solid (and indeed a liquid) is not truly random - the distances between atoms
in the structure are well defined and similar to those in the crystal. This
is why liquids and crystals have similar densities - both have *short-range
order* that fixes the distances between atoms, but only crystals have long-range
order.
The range of crystalline order distinguishes single crystals, polycrystals and amorphous solids. The figure shows how the *periodicity* of the atomic structure of each type of material compares.
Many characteristic properties of materials, such as mechanical, optical, magnetic and electronic behaviour, can be attributed
to the difference in structure between these three classes of solid.
Single crystals: Shape and anisotropy
=
A single crystal often has distinctive plane faces and some symmetry. The actual
shape of the crystal will be determined by the availability of crystallising
material, and by interference with other crystals, but the angles between the
faces will be characteristic of the material and will define an ideal shape.
Single crystals showing these characteristic shapes can be grown from salt solutions
such as alum and copper sulphate.
Gemstones are often single crystals. They tend to be cut artificially to obtain
aesthetically pleasing refractive and reflective properties. This generally
requires cutting along crystallographic planes. This is known as cleaving the
crystal. A familiar example is diamond, from which decorative stones can be
cleaved in different ways to produce a wide range of effects.
To see a variety of symmetrical naturally formed minerals, visit the website.
Consider the following three-dimensional shapes:
| | |
| - | - |
| Diagram of a cube | Cube: 6 identical squares |
| Diagram of a tetrahedron | Tetrahedron: 4 identical equilateral triangles |
| Diagram of an octahedron | Octahedron: 8 identical equilateral triangles |
| Diagram of a rhomohedron | Rhombohedron: 6 identical parallelograms with
sides of equal length |
You can make your own cube, octahedron and tetrahedron by printing the following
pages and following the instructions on them.
*
*
These three shapes are the most important in materials science, and you should
be very familiar with them!
The symmetry exhibited by real single crystals is determined by the crystal
structure of the material. Many have shapes composed of less regular polyhedra,
such as prisms and pyramids.
| | |
| - | - |
| Diagram of a hexagonal prism | Hexagonal prism: 2 hexagons and 6 rectangles |
| Diagram of a square-based pyramid | Square-based pyramid: 4 triangles and a square |
Not all single crystal specimens exhibit distinctive polyhedral shapes. Metals,
for example, often have crystals of no particular shape at all.
| | | |
| - | - | - |
| | | |
These quartz specimens show a range of shapes
typically exhibited by crystals. (Click on an image to see a larger version.)
Most single crystals show anisotropy in certain properties, such as
optical and mechanical properties. An amorphous substance, such as window glass, tends to be isotropic.
This difference may make it possible to distinguish between a glass and a crystal. The characteristic shape of some single
crystals is a clue that the properties of the material might be directionally dependent. The properties of polycrystalline
samples can be completely isotropic or strongly anisotropic depending on the nature of the material and the way in which
it was formed.
Single crystals: Mechanical properties
Gypsum can be cleaved along particular crystallographic planes using
a razor blade. The bonding perpendicular to these cleavage planes
is weaker than that in other directions, and hence the crystal breaks preferentially along these planes. Quartz and diamond
do not have such distinct cleavage planes, and so cleaving these crystals requires much more effort and care.
There are distinct planes in the gypsum structure, with no bonding between them. These are the cleavage planes. It is much
more difficult to cleave gypsum along planes other than these. In contrast, all of the planes in the quartz structure are
interconnected and the material is much more difficult to cleave in any direction. This is a demonstration of a way in which
the crystal structure of a material can influence its mechanical properties.Certain crystals, such as gypsum, can be cleaved with a razor blade along particular crystallographically-determined planes. (Click on image to view larger version.)
Glass is impossible to cleave. As an amorphous substance, glass has no crystallographic
planes and therefore can have no easy-cleavage directions. Glassy materials
are often found to be mechanically harder than their crystalline equivalents.
This is an example of how mechanical properties of crystals and amorphous substances
differ.
Single crystals: Optical properties
=
Quartz crystals are birefringent, so they exhibit optical anisotropy. Consider plane polarised light passing through a birefringent crystal. Inside the crystal, the light is split into two rays travelling along *permitted vibration directions* (p.v.d.s). The two rays are subject to different refractive indices, so the light travelling along each p.v.d. reaches the opposite side of the crystal at a different time. When the two rays recombine, there is a phase difference between the two rays that causes the *polarisation state* of the transmitted light to be different from that of the incident light.
Optical anisotropy in thin samples can be observed by placing the sample between crossed polarising filters in a light box. The bottom filter, between the light source and the sample, is called the polariser. The top filter, between the sample and the observer, is called the analyser. The polariser and analyser have polarising directions perpendicular to one another.
The apparatus used for examining optical anisotropy consists of a white-light source, two polarising filters
and a frame to hold them apart so creating a working space.
When no sample is in place the light that reaches the analyser is polarised at 90° to the analyser's polarisation direction, so no light is transmitted to the observer. When a quartz sample (with favourable orientation, see later) is placed between the filters, the crystal changes the polarisation state of the light that is transmitted through it. When this light reaches the analyser, some component of it lies parallel to the polarisation direction of the analyser, and therefore some light is transmitted to the observer.
If a quartz slice shows optical anisotropy, the intensity of light transmitted through the analyser varies as a function of the angle of rotation of the quartz sample in the plane of the filters. At certain orientations, no light is transmitted. These 'extinction positions' are found at 90° intervals.
Your browser does not support the video tag.
Video animation of anisotropic quartz rotated between crossed polarisersWhen the same experiment is done using a piece of glass, it is found that light is not transmitted for *any* orientation. This is because the glass is *optically isotropic*, and does not change the polarisation state of the light passing though it.
In quartz, there is one direction of propagation for which no birefringence is observed. If a sample is cut so that the incident light is parallel to this direction, the sample behaves as if it is optically isotropic and no light is transmitted. The crystallographic direction that exhibits this property is known as the *optic axis*.
When the quartz sample is cut so that the incident light is parallel to the optic axis, no light is transmitted in any orientation.
This experiment demonstrates that some single crystals, such as quartz, show anisotropic optical properties. The phenomenon depends on the crystallographic orientation of the crystal with respect to the incident light. Amorphous materials like glass have no 'distinct' crystal directions, so anisotropic properties are generally not observed.
Polycrystals
Single crystals form only in special conditions. The normal solid form of an element or compound is *polycrystalline*.
As the name suggests, a polycrystalline solid or *polycrystal* is made up of many crystals. The properties of a polycrystal
are notably different from those of a single crystal. The individual component crystallites
are often referred to as grains and the junctions between these grains are
known as grain boundaries.
The size of a grain varies according to the conditions under which it formed. Galvanised steel has a zinc coating with
visibly large grains. Other materials have much finer grains, and require the use of optical microscopy.
In galvanised steel, the grains are big enough to be seen unaided. The plate measures 5 cm across.In many other metals, such as this hypoeutectoid iron-carbon alloy, the grains may only be seen under a
microscope. (Click on image to see larger version.)
These photographs show a polycrystalline sample of quartz mixed with feldspar in which the grains all have optically anisotropic
properties. Between the crossed polarisers, each grain allows transmission of light at a slightly different point in the rotation.
This gives the strange effect seen here.
This polycrystal contains randomly oriented grains that allow transmission at different angles. Consequently
different regions of the polycrystal are seen in these two photographs. (Click on the images to view larger versions.)
The three-dimensional shape of grains in a polycrystal is similar to the shape of individual soap bubbles made by blowing
air into a soap solution contained in a transparent box.
The surface between bubbles is a high-energy feature. If the area of the surface is decreased, the overall energy of the
system decreases, so *reduction of surface area is a spontaneous process*. If all the bubbles were the same size, the
resulting structure would be a regular *close-packed* array, with 120° angles between the surfaces of neighbouring
bubbles.
In practise, *bubble growth* can occur because the surface area of a few large bubbles is lower than that of many
small bubbles. Large bubbles tend to grow, and small bubbles tend to shrink. The bubbles are therefore different sizes so
there are large deviations from the close-packed structure. On average, however, three bubbles meet at a junction, and the
angle between the bubble surfaces is usually within a few tens of degrees of 120°.
The *curvature* of the surfaces is also important. Surfaces with a smaller radius of curvature have a higher energy
than those with a larger radius of curvature. As a result, some small bubbles cannot shrink and disappear, even though the
surface area would decrease if they did so. This is because the curvature of the boundaries, and the associated energy, would
be too high.
In a real polycrystal, the grain boundaries are high-energy features, similar to the surfaces between bubbles. The soap
froth is a very good model for the grain structure of a simple polycrystalline material, and many similar features can be
observed in the two systems. The soap bubbles are analogous to the grains, and the surfaces of the bubbles are analogous to
the grain boundaries. Compare the photographs of the soap bubbles with the micrograph of a polycrystalline material that has
been etched to reveal the grain boundaries.
| | | |
| - | - | - |
| | | |
The packing of soap bubbles is somewhat similar to the packing of crystals - both systems seek to minimise
their surface area. Note the angles at the junctions of grain boundaries. (Click on images to view larger versions.)The grains of this hypereutectoid iron-carbon alloy are packed in a similar way to the bubbles in the previous
photographs. (Click on image to view larger version.)
Grain boundaries in a polycrystalline solid can *relax* (move in such a way to decrease the total energy of the system)
when atomic rearrangement by diffusion is possible. In real materials, many other effects can influence the observed grain
structure.
Defects
=
Within a single crystal or grain, the crystal structure is not perfect. The
structure contains defects such as vacancies, where an atom is missing altogether, and dislocations, where the perfection of the structure is disrupted along a line. Grain boundaries
in polycrystals can be considered as two-dimensional defects in the perfect crystal
lattice. Crystal defects are important in determining many material properties,
such as the rate of atomic diffusion
and mechanical strength.
We can use a "shot model" to get a picture of crystal defects. The
model consists of many small ball bearings trapped in a single layer between two
transparent plates. They tend to behave like the atoms in a crystal, and can show
the same kind of defects.
When the shot model is held horizontally, so that the balls flow freely, the
resulting structure is similar to a liquid.Shot model held horizontally. The balls form a
liquid-like structure. (Click on image to view larger version.)
As the model is tilted towards the vertical, the balls pack closely together.
This represents crystallisation. One or two balls may be suspended above the main
body by electrostatic forces: this is comparable to the vapour found above the
crystal.
In some places the balls form close-packed regions. Tapping of the model causes
minor rearrangements of the balls, especially at the top of the "solid"
region. This is similar to diffusion, in which case the tapping is analogous to
thermal activation. Occasionally, the "diffusion" process may cause
two grains to join together, or for some grains to "grow". The following
image sequence shows the behaviour of the shot model as it is rearranged by tapping,
starting from a polycrystalline state with many small grains and ending with much
larger grains. Note the presence of vacancies in the structure.
Your browser does not support the video tag.
Grain growth in the shot modelWith great care, it may be possible to create a single crystal, as *all*
the balls form a single pattern. Note that diffusion occurs mainly near the top
of the balls: those towards the middle and bottom do not easily move, as the photographs
show.
Even in a single crystal, or large-grained sample, there are still vacancies,
as the shot model shows. The reason for this involves *entropy*: at all finite
temperatures, there will be some disorder in the crystal.
The balls within a grain arrange themselves into close-packed planes. In metals,
close-packing of atoms is a very common structure. This pattern is typical of
*hexagonal close-packed* and *cubic close-packed* lattices. Note that
in this 2-D model, each ball touches six others. In a 3-D crystal, such as a real
one, each ball would also touch three on the plane above, and three on the plane
below.
A close-packed plane.
In the shot model, the balls are normally arranged in to a polycrystalline
form, shown schematically below:
A polycrystal will typically have crystalline regions (grains)
bounded by disordered grain boundaries. These boundaries are marked in the picture
on the right.
Note that the packing of atoms at the grain boundaries is disordered compared
to the grains. At a grain boundary, the normal crystal structure is obviously
disturbed, so the boundaries are regions of high energy. The ideal low energy
state would be a single crystal.
Polycrystals form from a melt because crystallisation starts from a number
of centres or nuclei. These developing crystals grow until they meet. Since they are not usually aligned
before meeting, the grains need not necessarily be able to fit together as a single
crystal, hence the polycrystalline structure.
After crystallisation the solid tends to reduce the boundary area, and hence
the internal energy, by *grain growth*. This can only happen by a process
of *atomic diffusion* within the solid. Such diffusion is more rapid at a
higher temperature since it is thermally activated.
Summary
=
The focus of this package is the difference between single crystals,
polycrystals and amorphous solids. This is explained in terms of the atomic
scale periodicity: single crystals are periodic across their entire volume;
polycrystals are periodic across individual grains; amorphous solids have
little to no periodicity at all.
The different atomic structures can have effects on the macroscopic
properties. A single crystal may exhibit anisotropy - we have seen
mechanical anisotropy of gypsum, and optical anisotrpy of quartz.
Polycrystals may also be anisotropic within each grain, as seen when the
polycrystalline quartz-feldspar mix was placed between the crossed
polarisers. Amorphous solids do not have anisotropic mechanical or optical
properties, since they are isotropic on the atomic scale.
Defects may exist in all structures, even single crystals. They include
vacancies and grain boundaries, where the regular repeating structure is
disrupted.
Questions
=
### Quick questions*You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!*1. Most single crystals contain: | | | |
| - | - | - |
| |
a |
No defects, since they must be perfect crystals. |
| |
b |
Exactly one defect, hence the term '*single* crystal'. |
| |
c |
Many defects. |
| |
d |
More defects than atoms, since every atom must generate at least one defect. |
2. If quartz had optical properties such that the refractive indices for all vibration directions were equal, the crossed polariser experiment would show: | | | |
| - | - | - |
| |
a |
No light transmitted for any orientation. |
| |
b |
Varying intensity as the orientation changes. |
| |
c |
Uniform non-zero intensity of transmitted light regardless of orientation. |
| |
d |
Circular dark patches to represent the symmetry of the optical properties. |
3. Bubbles in a box behave in a similar way to grains in a crystal in several ways, but not in all. Which of the following statements is TRUE? | | | |
| - | - | - |
| |
a |
The geometry of the places where bubbles meet one another is different from the geometry of the junctions between grains in a real polycrystal. |
| |
b |
The shape of the bubbles is different from the typical shape of a grain. |
| |
c |
The way a bubble deforms when a load is applied is different from the way a grain deforms. |
| |
d |
The three dimensional structure of bubbles in a box is unlike the three dimensional structure of a polycrystal. |
4. Which of the following is false? | | | |
| - | - | - |
| |
a |
Quartz crystals have optically anisotropic properties. |
| |
b |
Glass has no regular repeating crystalline structure. |
| |
c |
Certain crystals may cleave easily along certain planes, defined by the crystal structure. |
| |
d |
Crystal defects are not found in single crystals. |
5. What does the 'shot model' fail to show? | | | |
| - | - | - |
| |
a |
Polycrystallinity |
| |
b |
Crystalline defects |
| |
c |
The third dimension of the structure |
| |
d |
The difference between a vapour and a solid |### Deeper questions*The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.*6. Why is window glass transparent? | | | |
| - | - | - |
| |
a |
Because it has a single crystal structure and each sheet is cut with the optic axis normal to the plane of the window. |
| |
b |
Because it has an amorphous structure with large interatomic spacing. Light waves can pass between widely spaced atoms without any interaction with the solid structure. |
| |
c |
Because sheets of glass are cut thin enough for light to pass through without any significant absorption. |
| |
d |
Because of the electronic nature of the bonds between the atoms in the glass. |
7. A quantity of pure liquid aluminium is cooled slowly through its melting point. The solid is then left at room temperature for 100 years. What is the resulting structure? | | | |
| - | - | - |
| |
a |
A polycrystal with grains of identical chemical composition but different crystallographic orientation. |
| |
b |
A polycrystal consisting of finely spaced lamellae with alternating composition. |
| |
c |
A single crystal. |
| |
d |
An amorphous solid with good mechanical strength. |
8. *Self-diffusion* is the diffusion of a species within a body of material made from the same species. In general, self-diffusion in a polycrystalline solid can occur through the bulk of the grains (*lattice diffusion*) or along the grain boundaries (*grain boundary diffusion*). Which of the following statements gives the *best* description of the relative contribution of each process to the overall diffusion rate? | | | |
| - | - | - |
| |
a |
The contributions should be about the same in both cases. |
| |
b |
The contribution from lattice diffusion will always be greater than the contribution from grain boundary diffusion. |
| |
c |
The contribution from grain boundary diffusion will always be greater than the contribution from lattice diffusion. |
| |
d |
The relative contributions of the two processes depend upon the temperature of the material. |
9. Imagine a polycrystalline solid with cubic grains of edge length *D*. When *D* = 10 μm, what percentage of the volume of solid lies within a grain boundary, if the grain boundary width *d* is 1 nm? What must the grain size *D* be if 10% of the volume lies within a grain boundary? Comment on your answers.
10. Which of the following material properties could show anisotropy? *(answer yes or no for each)* | | | | |
| - | - | - | - |
| Yes
| No
|
a |
Density |
| Yes
| No
|
b |
Young's modulus |
| Yes
| No
|
c |
Electrical conductivity |
| Yes
| No
|
d |
Refractive index |### Open-ended questions*The following questions are not provided with answers, but intended to provide food for thought and points for further discussion with other students and teachers.*11. Think about some of the possible applications of materials showing optical anisotropy, like the quartz crystal.
12. How might you control the grain size of a material produced from a melt? How might the cooling rate and the chemical composition affect the results? Can you think of ways to change the grain structure *after* the material has solidified?
13. In this TLP, we have discussed *pure* materials. Real materials almost always contain some impurities. How might these impurities be incorporated into the crystal structure of a material? Consider the relative size of the impurity atoms and the host atoms. Are impurities always undesirable?
14. Graphite is sometimes used as a lubricant, and diamond can be used on the tips of cutting tools. In terms of the crystal structure, why might this be?
15. Why do the individual grains in a polycrystalline material, such as those in the photo of galvanised steel (on the page) appear to be different colours or shades, when the composition of every grain is approximately the same?
Going further
=
Most 'introductory' materials science textbooks will cover the basic material in this package. The following resources
cover the subjects in more detail than this teaching and learning package, and may prove useful to the interested student.
### Books
Introduction to Mineral Sciences by Putnis (Cambridge University Press, 1992)
Provides a mineral-based treatment of many of the topics introduced in this package. Of particular interest:
Chapter 1 on Periodicity and Symmetry
Chapter 2 on Anisotropy and Optical Properties, including the phenomenon of birefringence
Chapter 5 on Crystal Structures
Chapter 7 on Defects in Minerals
The Structure of Materials by Allen and Thomas (Wiley, 1999)
Gives a thorough mathematical treatment of the noncrystalline and crystalline states (chapters 2 and 3).
### Websites
* Contains java-based applets that allow the structure of common polyhedra and crystals to be explored.
* A library of 'crystal forms' - the shapes adopted by natural crystals. Contains Java applets.
* An excellent tutorial on birefringence. Contains Java applets.
### Other resources
The MATTER Project's 'Materials Science on CD-ROM' includes modules on:
Introduction to Crystallography (including Miller Indices etc.)
Introduction to Point Defects
Dislocations
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"\n Aims\n\n* In this TLP, it is aimed to consider the basic electrochemical principles involved in (...TRUNCATED) |
"\n Aims\n\nOn completion of this TLP package, you should: \n* Understand the stress distribution wi(...TRUNCATED) |
"\n Aims\n\nOn completion of this TLP you should:\n* Understand how nucleation and crystallization o(...TRUNCATED) |
"\n Aims\n\nOn completion of this TLP you should:\n* Appreciate that many materials in living system(...TRUNCATED) |
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