Carbon quantum dot

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Carbon_quantum_dot 

Carbon quantum dots also commonly called carbon nano dots or simply carbon dots (abbreviated as CQDs, C-dots or CDs) are carbon nanoparticles which are less than 10 nm in size and have some form of surface passivation.

History

CQDs were first discovered by Xu et al. in 2004 accidentally during the purification of single-walled carbon nanotubes. This discovery triggered extensive studies to exploit the fluorescence properties of CQDs.

As a new class of fluorescent carbon nanomaterials, CQDs possess the attractive properties of high stability, good conductivity, low toxicity, environmental friendliness, simple synthetic routes as well as comparable optical properties to quantum dots. Carbon quantum dots have been extensively investigated especially due to their strong and tunable fluorescence emission properties, which enable their applications in biomedicine, optronics, catalysis, and sensing. In most cases CQDs emits the light in a band of about several hundred nanometers in visible or near-infrared range, however it was also reported on broadband CQDs covering the spectrum from 800 to 1600 nm.

Carbon dots prepared from different precursors: urea, alanine and sucrose (made by Paliienko Konstantin)

The fundamental mechanisms responsible of the fluorescence capability of CQDs are very debated. Some authors have provided evidence of size-dependent fluorescence properties, suggesting that the emission arises from electronic transitions with the core of the dots, influenced by quantum confinement effects, whereas other works, including single particle measurements, have rather attributed the fluorescence to recombination of surface-trapped charges, or proposed a form of coupling between core and surface electronic states. The excitation-dependent fluorescence of CQDs, leading to their characteristic emission tunability, has been mostly linked to the inhomogeneous distribution of their emission characteristics, due to polydispersity, although some works have explained it as a violation of Kasha's rule arising from an unusually slow solvent relaxation.

Properties

The structures and components of CQDs determine their diverse properties. Many carboxyl moieties on the CQD surface impart excellent solubility in water and biocompatibility. Such surface moieties enable CQDs to serve as proton conducting nanoparticles. CQDs are also suitable for chemical modification and surface passivation with various organic, polymeric, inorganic or biological materials. By surface passivation, the fluorescence properties as well as physical properties of CQDs are enhanced. Recently, it has been discovered that amine and hydroxamic acid functionalized CD can produce tricolor (green, yellow and red) emission when introduced with different pH environment and this tricolor emission can be preserved in ORMOSIL film matrix.

Synthesis

Synthetic methods for CQDs are roughly divided into two categories, "top-down" and "bottom-up" routes. These can be achieved via chemical, electrochemical or physical techniques. The CQDs obtained could be optimized during preparation or post-treatment. Modification of CQDs is also very important to get good surface properties which are essential for solubility and selected applications.

Synthetic methods

"Top-down" synthetic route refers to breaking down larger carbon structures such as graphite, carbon nanotubes, and nanodiamonds into CQDs using laser ablation, arc discharge, and electrochemical techniques. For example, Zhou et al. first applied electrochemical method into synthesis of CQDs. They grew multi-walled carbon nanotubes on a carbon paper, then they inserted the carbon paper into an electrochemical cell containing supporting electrolyte including degassed acetonitrile and 0.1 M tetrabutyl ammonium perchlorate. Later, they applied this method in cutting CNTs or assembling CNTs into functional patterns which demonstrated the versatile callability of this method in carbon nanostructure manipulations.

"Bottom-up" synthetic route involves synthesizing CQDs from small precursors such as carbohydrates, citrate, and polymer-silica nanocomposites through hydrothermal/solvothermal treatment, supported synthetic, and microwave synthetic routes. For instance, Zhu et al. described a simple method of preparing CQDs by heating a solution of poly(ethylene glycol) (PEG) and saccharide in 500 W microwave oven for 2 to 10 min. By varying the molar ratio of citric acid and urea (two common precursor molecules) of the mixture that is subjected to pyrolysis, a number of distinct fluorescent materials in both liquid and solid state can be synthesised, predominantly comprising Carbon dots with embedded fluorophores. Also a laser-induced thermal shock method is exploited for synthesis ultra-broadband QCDs. Recently, green synthetic approaches have also been employed for fabrication of CQDs. Care must be taken to separate the "bottom-up" carbon dots from fluorescent byproducts such as small molecules or polyester condensates by using multiple dialysis and chromatography separation methods.

Size control

In addition to post-treatment, controlling the size of CQDs during the preparing process is also widely used. For instance, Zhu et al. reported hydrophilic CQDs through impregnation of citric acid precursor. After pyrolyzing CQDs at 300 °C for 2 hours in air, then removing silica, followed by dialysis, they prepared CQDs with a uniform size of 1.5–2.5 nm which showed low toxicity, excellent luminescence, good photostability, and up-conversion properties.

Modification

Being a new type of fluorescent nanoparticles, applications of CQD lie in the field of bioimaging and biosensing due to their biological and environmental friendly composition and excellent biocompatibility. In order to survive the competition with conventional semiconductor quantum dots, a high quantum yield should be achieved. Although a good example of CQDs with ~80% quantum yield was synthesized, most of the quantum dots synthesized have a quantum yield below 10% so far. Surface-passivation and doping methods for modifications are usually applied for improving quantum yield.

To prevent surfaces of CQDs from being polluted by their environment, surface passivation is performed to alleviate the detrimental influence of surface contamination on their optical properties. A thin insulating layer is formed to achieve surface passivation via the attachment of polymeric materials on CQDs surface treated by acid.

In addition to surface passivation, doping is also a common method used to tune the properties of CQDs. Various doping methods with elements such as N, S, P have been demonstrated for tuning the properties of CQDs, among which N doping is the most common way due to its great ability in improving the photo luminescence emissions. The mechanisms by which Nitrogen doping enhances the fluorescence quantum yield of CQDs, as well as the structure of heavily N-doped CDs, are very debated issues in the literature. Zhou et al. applied XANES and XEOL in investigating the electronic structure and luminescence mechanism in their electrochemically produced carbon QDS and found that N doping is almost certainly responsible for the blue luminescence. Synthesis of new nanocomposites based on CDs have been reported with unusual properties. For example, a nanocomposite has been designed by using of CDs and magnetic Fe₃O₄ nanoparticles as precursors with nanozyme activity.

Post synthesis electrochemical etching results in dramatic changes in GQDs size and fluorescence intensity.

Applications

CQDs with unique properties have great potential in biomedicine, optronics, catalysis and sensors

Bioimaging

CQDs can be used for bioimaging due to their fluorescence emissions and biocompatibility. By injecting solvents containing CQDs into a living body, images in vivo can be obtained for detection or diagnosis purposes. One example is that organic dye-conjugated CQDs could be used as an effective fluorescent probes for H₂S. The presence of H₂S could tune the blue emission of the organic dye-conjugated CQDs to green. So by using a fluorescence microscope, the organic dye-conjugated CQDs were able to visualize changes in physiologically relevant levels of H₂S. Another example can be dual-mode bioimaging using their highly accessible surface functional groups to conjugate them via EDC-NHS chemistry. Saladino et al. demonstrated the concept using MW-assisted synthesized nitrogen-doped excitation-independent CQDs. These were conjugated with rhodium nanoparticles – X-ray fluorescence contrast agents – leading to dual-mode nanohybrids with both optical and X-ray fluorescent properties. Moreover, the conjugation process not only accounts for dual-mode bioimaging but also passivates the rhodium nanoparticle surface, resulting in reduced cytotoxicity.

Sensing

CQDs were also applied in biosensing as biosensor carriers for their flexibility in modification, high solubility in water, nontoxicity, good photostability, and excellent biocompatibility. The biosensors based on CQD and CQs-based materials could be used for visual monitoring of cellular copper, glucose, pH, trace levels of H₂O₂ and nucleic acid. A general example is about nucleic acid lateral flow assays. The discriminating tags on the amplicons are recognized by their respective antibodies and fluorescence signals provided by the attached CQDs. More generally, the fluorescence of CQDs efficiently responds to pH, local polarity, and to the presence of metal ions in solution, which further expands their potential for nanosensing applications, for instance in the analysis of pollutants.

Drug delivery

The nontoxicity and biocompatibility of CQDs enable them with broad applications in biomedicine as drug carriers, fluorescent tracers as well as controlling drug release. This is exemplified by the use of CQDs as photosensitizers in photodynamic therapy to destroy cancer cells.

Catalysis

The flexibility of functionalization with various groups CQDs makes them possible to absorb lights of different wavelengths, which offers good opportunities for applications in photocatalysis. CQDs-modified P25 TiO₂ composites exhibited improved photocatalytic H2 evolution under irradiation with UV-Vis. The CQDs serve as a reservoir for electrons to improve the efficiency of separating of the electron-hole pairs of P25. In the recent times, metal-free CQDs have been found to improve the kinetics of hydrogen evolution reaction (HER), making CQDs a sustainable choice for catalysis.

Optronics

CQDs possess the potential in serving as materials for dye-sensitized solar cells, organic solar cells, supercapacitor, and light emitting devices. CQDs can be used as photosensitizer in dye-sensitized solar cells and the photoelectric conversion efficiency is significantly enhanced. CQD incorporated hybrid silica based sol can be used as transparent Fluorescent paint,

Rocket fuels

Recently, CQDs have been employed in hybrid rocket fuels.

Fingerprint recovery

CQDs are used for the enhancement of latent fingerprints.

Electrocatalyst

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Electrocatalyst

A platinum cathode electrocatalyst's stability being measured by chemist Xiaoping Wang

An electrocatalyst is a catalyst that participates in electrochemical reactions. Electrocatalysts are a specific form of catalysts that function at electrode surfaces or, most commonly, may be the electrode surface itself. An electrocatalyst can be heterogeneous such as a platinized electrode. Homogeneous electrocatalysts, which are soluble, assist in transferring electrons between the electrode and reactants, and/or facilitate an intermediate chemical transformation described by an overall half reaction. Major challenges in electrocatalysts focus on fuel cells.

Background and theory

An electrocatalyst lowers the activation energy required for an electrochemical reaction. Some electrocatalysts change the potential at which oxidation and reduction processes occur. In other cases, an electrocatalyst can impart selectivity by favoring specific chemical interaction at an electrode surface. Given that electrochemical reactions occur when electrons are passed from one chemical species to another, favorable interactions at an electrode surface increase the likelihood of electrochemical transformations occurring, thus reducing the potential required to achieve these transformations.

Electrocatalysts can be evaluated according to activity, stability, and selectivity. The activity of electrocatalysts can be assessed quantitatively by the current density is generated, and therefore how fast a reaction is taking place, for a given applied potential. This relationship is described with the Tafel equation. In assessing the stability of electrocatalysts, the a key parameter is turnover number (TON). The selectivity of electrocatalysts refers to the product distribution. Selectivity can be quantitatively assessed through a selectivity coefficient, which compares the response of the material to the desired analyte or substrate with the response to other interferents.

In many electrochemical systems, including galvanic cells, fuel cells and various forms of electrolytic cells, a drawback is that they can suffer from high activation barriers. The energy diverted to overcome these activation barriers is transformed into heat. In most exothermic combustion reactions this heat would simply propagate the reaction catalytically. In a redox reaction, this heat is a useless byproduct lost to the system. The extra energy required to overcome kinetic barriers is usually described in terms of low faradaic efficiency and high overpotentials. In these systems, each of the two electrodes and its associated half-cell would require its own specialized electrocatalyst.

Half-reactions involving multiple steps, multiple electron transfers, and the evolution or consumption of gases in their overall chemical transformations, will often have considerable kinetic barriers. Furthermore, there is often more than one possible reaction at the surface of an electrode. For example, during the electrolysis of water, the anode can oxidize water through a two electron process to hydrogen peroxide or a four electron process to oxygen. The presence of an electrocatalyst could facilitate either of the reaction pathways.

Types of electrocatalyst materials, including homogeneous and heterogeneous electrocatalysts.

Homogeneous electrocatalysts

A homogeneous electrocatalyst is one that is present in the same phase of matter as the reactants, for example, a water-soluble coordination complex catalyzing an electrochemical conversion in solution. This technology is not practiced commercially, but is of research interest.

Synthetic coordination complexes

Many coordination complexes catalyze electrochemical reactions, although few have achieved commercial success. Well investigated processes include the hydrogen evolution reaction.

Electrification of catalytic processes

There is much interest in replacing traditional chemical catalysis with electrocatalysis. In such a scheme electrons supplied by an electrode are reagents. The topic is a theme within the area of green energy, because the electrons can be sourced from renewable resources. Several conversions that use of hydrogen gas could be transformed into electrochemical processes that use protons. This technology remains economically noncompetitive.

Another example is found in the area of nitrogen fixation. The traditional Haber-Bosch process produces ammonia by hydrogenation of nitrogen gas:

N₂ + 3 H₂ → 2 NH₃

In the electrified version, the hydrogen is provided in the form of protons and electrons:

N₂ + 6 H⁺ + 6 e⁻ → 2 NH₃

The ammonia represents an energy source since it is combustable. In this way electrification can be seen as a means for energy storage.

Another process attracting much effort is the electrochemical reduction of carbon dioxide.

Enzymes

Some enzymes can function as electrocatalysts. Nitrogenase, an enzyme that contains a MoFe cluster, can be leveraged to fix atmospheric nitrogen, i.e. convert nitrogen gas into molecules such as ammonia. Immobilizing the protein onto an electrode surface and employing an electron mediator greatly improves the efficiency of this process. The effectiveness of bioelectrocatalysts generally depends on the ease of electron transport between the active site of the enzyme and the electrode surface. Other enzymes provide insight for the development of synthetic catalysts. For example, formate dehydrogenase, a nickel-containing enzyme, has inspired the development of synthetic complexes with similar molecular structures for use in CO₂ reduction. Microbial fuel cells are another way that biological systems can be leveraged for electrocatalytic applications. Microbial-based systems leverage the metabolic pathways of an entire organism, rather than the activity of a specific enzyme, meaning that they can catalyze a broad range of chemical reactions. Microbial fuel cells can derive current from the oxidation of substrates such as glucose, and be leveraged for processes such as CO₂ reduction.

Heterogeneous electrocatalysts

A heterogeneous electrocatalyst is one that is present in a different phase of matter from the reactants, for example, a solid surface catalyzing a reaction in solution. Different types of heterogeneous electrocatalyst materials are shown above in green. Since heterogeneous electrocatalytic reactions need an electron transfer between the solid catalyst (typically a metal) and the electrolyte, which can be a liquid solution but also a polymer or a ceramic capable of ionic conduction, the reaction kinetics depend on both the catalyst and the electrolyte as well as on the interface between them. The nature of the electrocatalyst surface determines some properties of the reaction including rate and selectivity.

Bulk materials

Electrocatalysis can occur at the surface of some bulk materials, such as platinum metal. Bulk metal surfaces of gold have been employed for the decomposition methanol for hydrogen production. Water electrolysis is conventionally conducted at inert bulk metal electrodes such as platinum or iridium. The activity of an electrocatalyst can be tuned with a chemical modification, commonly obtained by alloying two or more metals. This is due to a change in the electronic structure, especially in the d band which is considered to be responsible for the catalytic properties of noble metals.

Nanomaterials

Nanoparticles

A variety of nanoparticle materials have been demonstrated to promote various electrochemical reactions, although none have been commercialized. These catalysts can be tuned with respect to their size and shape, as well as the surface strain.

Electronic density difference of a Cl atom adsorbed on a Cu(111) surface obtained with a DFT simulation.

Also, higher reaction rates can be achieved by precisely controlling the arrangement of surface atoms: indeed, in nanometric systems, the number of available reaction sites is a better parameter than the exposed surface area in order to estimate electrocatalytic activity. Sites are the positions where the reaction could take place; the likelihood of a reaction to occur in a certain site depends on the electronic structure of the catalyst, which determines the adsorption energy of the reactants together with many other variables not yet fully clarified.

According to the TSK model, the catalyst surface atoms can be classified as terrace, step or kink atoms according to their position, each characterized by a different coordination number. In principle, atoms with lower coordination number (kinks and defects) tend to be more reactive and therefore adsorb the reactants more easily: this may promote kinetics but could also depress it if the adsorbing species isn't the reactant, thus inactivating the catalyst. Advances in nanotechnology make it possible to surface engineer the catalyst so that just some desired crystal planes are exposed to reactants, maximizing the number of effective reaction sites for the desired reaction.

To date, a generalized surface dependence mechanism cannot be formulated since every surface effect is strongly reaction-specific. A few classifications of reactions based on their surface dependence have been proposed but there are still many exceptions that do not fall into them.

Particle size effect

An example of a particle-size effect: the number of reaction sites of different kinds depends on the size of the particle. In this four FCC nanoparticles model, the kink site between (111) and (100) planes (coordination number 6, represented by golden spheres) is 24 for all of the four different nanoparticles, while the number of other surface sites varies.

The interest in reducing as much as possible the costs of the catalyst for electrochemical processes led to the use of fine catalyst powders since the specific surface area increases as the average particle size decreases. For instance, most common PEM fuel cells and electrolyzers design is based on a polymeric membrane charged in platinum nanoparticles as an electrocatalyst (the so-called platinum black).

Although the surface area to volume ratio is commonly considered to be the main parameter relating electrocatalyst size with its activity, to understand the particle-size effect, several more phenomena need to be taken into account:

• Equilibrium shape: for any given size of a nanoparticle there is an equilibrium shape which exactly determines its crystal planes
• Reaction sites relative number: a given size for a nanoparticle corresponds to a certain number of surface atoms and only some of them host a reaction site
• Electronic structure: below a certain size, the work function of a nanoparticle changes and its band structure fades away
• Defects: the crystal lattice of a small nanoparticle is perfect; thus, reactions enhanced by defects as reaction sites get slowed down as the particle size decreases
• Stability: small nanoparticles have the tendency to lose mass due to the diffusion of their atoms towards bigger particles, according to the Ostwald ripening phenomenon
• Capping agents: in order to stabilize nanoparticles it is necessary a capping layer, therefore part of their surface is unavailable for reactants
• Support: nanoparticles are often fixed onto a support in order to stay in place, therefore part of their surface is unavailable for reactants

Carbon-based materials

Carbon nanotubes and graphene-based materials can be used as electrocatalysts. The carbon surfaces of graphene and carbon nanotubes are well suited to the adsorption of many chemical species, which can promote certain electrocatalytic reactions. In addition, their conductivity means they are good electrode materials. Carbon nanotubes have a very high surface area, maximizing surface sites at which electrochemical transformations can occur. Graphene can also serve as a platform for constructing composites with other kinds of nanomaterials such as single atom catalysts. Because of their conductivity, carbon-based materials can potentially replace metal electrodes to perform metal-free electrocatalysis.

Framework materials

Metal—organic frameworks (MOFs), especially conductive frameworks, can be used as electrocatalysts for processes such as CO₂ reduction and water splitting. MOFs provide potential active sites at both metal centers and organic ligand sites. They can also be functionalized, or encapsulate other materials such as nanoparticles. MOFs can also be combined with carbon-based materials to form electrocatalysts. Covalent organic frameworks (COFs), particularly those that contain metals, can also serve as electrocatalysts. COFs constructed from cobalt porphyrins demonstrated the ability to reduce carbon dioxide to carbon monoxide.

However, many MOFs are known unstable in chemical and electrochemical conditions, making it difficult to tell if MOFs are actually catalysts or precatalysts. The real active sites of MOFs during electrocatalysis need to be analyzed comprehensively.

Research on electrocatalysis

Some transition metal complexes that exhibit some activity as homogeneous electrocatalysts.

Water splitting / Hydrogen evolution

Main article: Electrolysis of water

A schematic of a hydrogen fuel cell. To supply hydrogen, electrocatalytic water splitting is commonly employed.

Hydrogen and oxygen can be combined through by the use of a fuel cell. In this process, the reaction is broken into two half reactions which occur at separate electrodes. In this situation the reactant's energy is directly converted to electricity. Useful energy can be obtained from the thermal heat of this reaction through an internal combustion engine with an upper efficiency of 60% (for compression ratio of 10 and specific heat ratio of 1.4) based on the Otto thermodynamic cycle. It is also possible to combine the hydrogen and oxygen through redox mechanism as in the case of a fuel cell. In this process, the reaction is broken into two half-reactions which occur at separate electrodes. In this situation the reactant's energy is directly converted to electricity.

The standard reduction potential of hydrogen is defined as 0V, and frequently referred to as the standard hydrogen electrode (SHE).

Half Reaction	Reduction Potential Eored (V)
2H+ + 2e− → H2 (g)	≡ 0
O2(g) + 4H+ + 4e− → 2H2O(l)	+1.23

HER can be promoted by many catalysts.

Carbon dioxide reduction

Main article: Electrochemical reduction of carbon dioxide

Electrocatalysis for CO₂ reduction is not practiced commercially but remains a topic of research. The reduction of CO₂ into useable products is a potential way to combat climate change. Electrocatalysts can promote the reduction of carbon dioxide into methanol and other useful fuel and stock chemicals. The most valuable reduction products of CO₂ are those that have a higher energy content, meaning that they can be reused as fuels. Thus, catalyst development focuses on the production of products such as methane and methanol. Homogeneous catalysts, such as enzymes and synthetic coordination complexes have been employed for this purpose. A variety of nanomaterials have also been studied for CO₂ reduction, including carbon-based materials and framework materials.

Ethanol-powered fuel cells

Aqueous solutions of methanol can decompose into CO₂ hydrogen gas, and water. Although this process is thermodynamically favored, the activation barrier is extremely high, so in practice this reaction is not typically observed. However, electrocatalysts can speed up this reaction greatly, making methanol a possible route to hydrogen storage for fuel cells. Electrocatalysts such as gold, platinum, and various carbon-based materials have been shown to effectively catalyze this process. An electrocatalyst of platinum and rhodium on carbon backed tin-dioxide nanoparticles can break carbon bonds at room temperature with only carbon dioxide as a by-product, so that ethanol can be oxidized into the necessary hydrogen ions and electrons required to create electricity.

Chemical synthesis

Electrocatalysts are used to promote certain chemical reactions to obtain synthetic products. Graphene and graphene oxides have shown promise as electrocatalytic materials for synthesis. Electrocatalytic methods also have potential for polymer synthesis. Electrocatalytic synthesis reactions can be performed under a constant current, constant potential, or constant cell-voltage conditions, depending on the scale and purpose of the reaction.

Advanced oxidation processes in water treatment

Water treatment systems often require the degradation of hazardous compounds. These treatment processes are dubbed Advanced oxidation processes, and are key in destroying byproducts from disinfection, pesticides, and other hazardous compound. There is an emerging effort to enable these processes to destroy more tenacious compounds, especially PFAS

Fluid solution

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Fluid_solution
 

In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid.

In astrophysics, fluid solutions are often employed as stellar models, since a perfect gas can be thought of as a special case of a perfect fluid. In cosmology, fluid solutions are often used as cosmological models.

Mathematical definition

The stress–energy tensor of a relativistic fluid can be written in the form

${\displaystyle T^{ab}=\mu u^a{u}^b+p{h}^{ab}+\left(u^a{q}^b+q^a{u}^b\right)+{\pi }^{ab}}$

Here

• the world lines of the fluid elements are the integral curves of the velocity vector ${\displaystyle u^a}$,
• the projection tensor ${\displaystyle h_{ab}=g_{ab}+u_a{u}_b}$ projects other tensors onto hyperplane elements orthogonal to ${\displaystyle u^a}$,
• the matter density is given by the scalar function ${\displaystyle \mu }$,
• the pressure is given by the scalar function ${\displaystyle p}$,
• the heat flux vector is given by ${\displaystyle q^a}$,
• the viscous shear tensor is given by ${\displaystyle {\pi }^{ab}}$.

The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that

${\displaystyle q_a{u}^a=0,{\pi }_{ab}{u}^b=0}$

This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively three and five linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor.

Special cases

Several special cases of fluid solutions are noteworthy (here speed of light c = 1):

• A perfect fluid has vanishing viscous shear and vanishing heat flux:

${\displaystyle T^{ab}=(\mu +p)u^a{u}^b+p{g}^{ab},}$

• A dust is a pressureless perfect fluid:

${\displaystyle T^{ab}=\mu u^a{u}^b,}$

• A radiation fluid is a perfect fluid with ${\displaystyle \mu =3p}$:

${\displaystyle T^{ab}=p\left(4u^a{u}^b+g^{ab}\right).}$

The last two are often used as cosmological models for (respectively) matter-dominated and radiation-dominated epochs. Notice that while in general it requires ten functions to specify a fluid, a perfect fluid requires only two, and dusts and radiation fluids each require only one function. It is much easier to find such solutions than it is to find a general fluid solution.

Among the perfect fluids other than dusts or radiation fluids, by far the most important special case is that of the static spherically symmetric perfect fluid solutions. These can always be matched to a Schwarzschild vacuum across a spherical surface, so they can be used as interior solutions in a stellar model. In such models, the sphere ${\displaystyle r=r[0]}$ where the fluid interior is matched to the vacuum exterior is the surface of the star, and the pressure must vanish in the limit as the radius approaches ${\displaystyle r_0}$. However, the density can be nonzero in the limit from below, while of course it is zero in the limit from above. In recent years, several surprisingly simple schemes have been given for obtaining all these solutions.

Einstein tensor

The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer.

In the special case of a perfect fluid, an adapted frame

${\displaystyle {\overrightarrow{e}}_0,{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3}$

(the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein tensor takes the simple form

${\displaystyle G^{\hat{a}\hat{b}}=8\pi \left[\begin{array}{cccc}\mu & 0& 0& 0\\ 0& p& 0& 0\\ 0& 0& p& 0\\ 0& 0& 0& p\end{array}\right]}$

where ${\displaystyle \mu }$ is the energy density and ${\displaystyle p}$ is the pressure of the fluid. Here, the timelike unit vector field ${\displaystyle {\overrightarrow{e}}_0}$ is everywhere tangent to the world lines of observers who are comoving with the fluid elements, so the density and pressure just mentioned are those measured by comoving observers. These are the same quantities which appear in the general coordinate basis expression given in the preceding section; to see this, just put ${\displaystyle \overrightarrow{u}={\overrightarrow{e}}_0}$. From the form of the physical components, it is easy to see that the isotropy group of any perfect fluid is isomorphic to the three dimensional Lie group SO(3), the ordinary rotation group.

The fact that these results are exactly the same for curved spacetimes as for hydrodynamics in flat Minkowski spacetime is an expression of the equivalence principle.

Eigenvalues

The characteristic polynomial of the Einstein tensor in a perfect fluid must have the form

${\displaystyle \chi (\lambda )=\left(\lambda -8\pi \mu \right){\left(\lambda -8\pi p\right)}^3}$

where ${\displaystyle \mu ,p}$ are again the density and pressure of the fluid as measured by observers comoving with the fluid elements. (Notice that these quantities can vary within the fluid.) Writing this out and applying Gröbner basis methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following two algebraically independent (and invariant) conditions:

${\displaystyle 12a_4+a_2^2-3a_1{a}_3=0}$

${\displaystyle a_1{a}_2{a}_3-9a_3^2-9a_1^2{a}_4+32a_2{a}_4=0}$

But according to Newton's identities, the traces of the powers of the Einstein tensor are related to these coefficients as follows:

${\displaystyle {G^a}_a=t_1=a_1}$

${\displaystyle {G^a}_b{G^b}_a=t_2=a_1^2-2a_2}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_a=t_3=a_1^3-3a_1{a}_2+3a_3}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_d{G^d}_a=t_4=a_1^4-4a_1^2{a}_2+4a_1{a}_3+2a_2^2-a_4}$

so we can rewrite the above two quantities entirely in terms of the traces of the powers. These are obviously scalar invariants, and they must vanish identically in the case of a perfect fluid solution:

${\displaystyle t_2^3+4t_3^2+t_1^2{t}_4-4t_2{t}_4-2t_1{t}_2{t}_3=0}$

${\displaystyle t_1^4+7t_2^2-8t_1^2{t}_2+12t_1{t}_3-12t_4=0}$

Notice that this assumes nothing about any possible equation of state relating the pressure and density of the fluid; we assume only that we have one simple and one triple eigenvalue.

In the case of a dust solution (vanishing pressure), these conditions simplify considerably:

${\displaystyle a_2=a_3=a_4=0}$

or

${\displaystyle t_2=t_1^2,t_3=t_1^3,t_4=t_1^4}$

In tensor gymnastics notation, this can be written using the Ricci scalar as:

${\displaystyle {G^a}_a=-R}$

${\displaystyle {G^a}_b{G^b}_a=R^2}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_a=-R^3}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_d{G^d}_a=-R^4}$

In the case of a radiation fluid, the criteria become

${\displaystyle a_1=0,27a_3^2+8a_2^3=0,12a_4+a_2^2=0}$

or

${\displaystyle t_1=0,7t_3^2-t_2{t}_4=0,12t_4-7t_2^2=0}$

In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a timelike eigenvector, since there are Lorentzian manifolds, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a spacelike eigenvector, and these cannot represent radiation fluids.

The coefficients of the characteristic will often appear very complicated, and the traces are not much better; when looking for solutions it is almost always better to compute components of the Einstein tensor with respect to a suitably adapted frame and then to kill appropriate combinations of components directly. However, when no adapted frame is evident, these eigenvalue criteria can be sometimes be useful, especially when employed in conjunction with other considerations.

These criteria can often be useful for spot checking alleged perfect fluid solutions, in which case the coefficients of the characteristic are often much simpler than they would be for a simpler imperfect fluid.

Examples

Noteworthy individual dust solutions are listed in the article on dust solutions. Noteworthy perfect fluid solutions which feature positive pressure include various radiation fluid models from cosmology, including

• FRW radiation fluids, often referred to as the radiation-dominated FRW models.

In addition to the family of static spherically symmetric perfect fluids, noteworthy rotating fluid solutions include

• Wahlquist fluid, which has similar symmetries to the Kerr vacuum, leading to initial hopes (since dashed) that it might provide the interior solution for a simple model of a rotating star.