Liquid

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, and plasma), and is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena. Water is, by far, the most common liquid on Earth.

For other uses, see Liquid (disambiguation).

The formation of a spherical droplet of liquid water minimizes the surface area, which is the natural result of surface tension in liquids.

The density of a liquid is usually close to that of a solid, and much higher than in a gas. Therefore, liquid and solid are both termed condensed matter. On the other hand, as liquids and gases share the ability to flow, they are both called fluids. Although liquid water is abundant on Earth, this state of matter is actually the least common in the known universe, because liquids require a relatively narrow temperature/pressure range to exist. Most known matter in the universe is in gaseous form (with traces of detectable solid matter) as interstellar clouds or in plasma from within stars.

Introduction

Thermal image of a sink full of hot water with cold water being added, showing how the hot and the cold water flow into each other.

Liquid is one of the four primary states of matter, with the others being solid, gas and plasma. A liquid is a fluid. Unlike a solid, the molecules in a liquid have a much greater freedom to move. The forces that bind the molecules together in a solid are only temporary in a liquid, allowing a liquid to flow while a solid remains rigid.

A liquid, like a gas, displays the properties of a fluid. A liquid can flow, assume the shape of a container, and, if placed in a sealed container, will distribute applied pressure evenly to every surface in the container. If liquid is placed in a bag, it can be squeezed into any shape. Unlike a gas, a liquid is nearly incompressible, meaning that it occupies nearly a constant volume over a wide range of pressures; it does not generally expand to fill available space in a container but forms its own surface, and it may not always mix readily with another liquid. These properties make a liquid suitable for applications such as hydraulics.

Liquid particles are bound firmly but not rigidly. They are able to move around one another freely, resulting in a limited degree of particle mobility. As the temperature increases, the increased vibrations of the molecules causes distances between the molecules to increase. When a liquid reaches its boiling point, the cohesive forces that bind the molecules closely together break, and the liquid changes to its gaseous state (unless superheating occurs). If the temperature is decreased, the distances between the molecules become smaller. When the liquid reaches its freezing point the molecules will usually lock into a very specific order, called crystallizing, and the bonds between them become more rigid, changing the liquid into its solid state (unless supercooling occurs).

Examples

Only two elements are liquid at standard conditions for temperature and pressure: mercury and bromine. Four more elements have melting points slightly above room temperature: francium, caesium, gallium and rubidium.[1] Metal alloys that are liquid at room temperature include NaK, a sodium-potassium metal alloy, galinstan, a fusible alloy liquid, and some amalgams (alloys involving mercury).

Pure substances that are liquid under normal conditions include water, ethanol and many other organic solvents. Liquid water is of vital importance in chemistry and biology; it is believed to be a necessity for the existence of life.

Inorganic liquids include water, magma, inorganic nonaqueous solvents and many acids.

Important everyday liquids include aqueous solutions like household bleach, other mixtures of different substances such as mineral oil and gasoline, emulsions like vinaigrette or mayonnaise, suspensions like blood, and colloids like paint and milk.

Many gases can be liquefied by cooling, producing liquids such as liquid oxygen, liquid nitrogen, liquid hydrogen and liquid helium. Not all gases can be liquified at atmospheric pressure, however. Carbon dioxide, for example, can only be liquified at pressures above 5.1 atm.[2]

Some materials cannot be classified within the classical three states of matter; they possess solid-like and liquid-like properties. Examples include liquid crystals, used in LCD displays, and biological membranes.

Applications

A lava lamp contains two immiscible liquids (a molten wax and a watery solution) which add movement due to convection. In addition to the top surface, surfaces also form between the liquids, requiring a tension breaker to recombine the wax droplets at the bottom.

Liquids have a variety of uses, as lubricants, solvents, and coolants. In hydraulic systems, liquid is used to transmit power.

In tribology, liquids are studied for their properties as lubricants. Lubricants such as oil are chosen for viscosity and flow characteristics that are suitable throughout the operating temperature range of the component. Oils are often used in engines, gear boxes, metalworking, and hydraulic systems for their good lubrication properties.[3]

Many liquids are used as solvents, to dissolve other liquids or solids. Solutions are found in a wide variety of applications, including paints, sealants, and adhesives. Naphtha and acetone are used frequently in industry to clean oil, grease, and tar from parts and machinery. Body fluids are water based solutions.

Surfactants are commonly found in soaps and detergents. Solvents like alcohol are often used as antimicrobials. They are found in cosmetics, inks, and liquid dye lasers. They are used in the food industry, in processes such as the extraction of vegetable oil.[4]

Liquids tend to have better thermal conductivity than gases, and the ability to flow makes a liquid suitable for removing excess heat from mechanical components. The heat can be removed by channeling the liquid through a heat exchanger, such as a radiator, or the heat can be removed with the liquid during evaporation.[5] Water or glycol coolants are used to keep engines from overheating.[6] The coolants used in nuclear reactors include water or liquid metals, such as sodium or bismuth.[7] Liquid propellant films are used to cool the thrust chambers of rockets.[8] In machining, water and oils are used to remove the excess heat generated, which can quickly ruin both the work piece and the tooling. During perspiration, sweat removes heat from the human body by evaporating. In the heating, ventilation, and air-conditioning industry (HVAC), liquids such as water are used to transfer heat from one area to another.[9]

Similarly, liquids are often used in cooking for their better heat-transfer properties. In addition to better conductivity, because warmer fluids expand and rise while cooler areas contract and sink, liquids with low kinematic viscosity tend to transfer heat through convection at a fairly constant temperature, making a liquid suitable for blanching, boiling, or frying. Even higher rates of heat transfer can be achieved by condensing a gas into a liquid. At the liquid's boiling point, all of the heat energy is used to cause the phase change from a liquid to a gas, without an accompanying increase in temperature, and is stored as chemical potential energy. When the gas condenses back into a liquid this excess heat-energy is released at a constant temperature. This phenomenon is used in processes such as steaming. Since liquids often have different boiling points, mixtures or solutions of liquids or gases can typically be separated by distillation, using heat, cold, vacuum, pressure, or other means. Distillation can be found in everything from the production of alcoholic beverages, to oil refineries, to the cryogenic distillation of gases such as argon, oxygen, nitrogen, neon, or xenon by liquefaction (cooling them below their individual boiling points).[10]

Liquid is the primary component of hydraulic systems, which take advantage of Pascal's law to provide fluid power. Devices such as pumps and waterwheels have been used to change liquid motion into mechanical work since ancient times. Oils are forced through hydraulic pumps, which transmit this force to hydraulic cylinders. Hydraulics can be found in many applications, such as automotive brakes and transmissions, heavy equipment, and airplane control systems. Various hydraulic presses are used extensively in repair and manufacturing, for lifting, pressing, clamping and forming.[11]

Liquids are sometimes used in measuring devices. A thermometer often uses the thermal expansion of liquids, such as mercury, combined with their ability to flow to indicate temperature. A manometer uses the weight of the liquid to indicate air pressure.[12]

Mechanical properties

Volume

Quantities of liquids are measured in units of volume. These include the SI unit cubic metre (m³) and its divisions, in particular the cubic decimeter, more commonly called the litre (1 dm³ = 1 L = 0.001 m³), and the cubic centimetre, also called millilitre (1 cm³ = 1 mL = 0.001 L = 10⁻⁶ m³).[13]

The volume of a quantity of liquid is fixed by its temperature and pressure. Liquids generally expand when heated, and contract when cooled. Water between 0 °C and 4 °C is a notable exception.[14]

On the other hand, liquids have little compressibility. Water, for example, will compress by only 46.4 parts per million for every unit increase in atmospheric pressure (bar).[15] At around 4000 bar (400 megapascals or 58,000 psi) of pressure at room temperature water experiences only an 11% decrease in volume.[16] Incompressibility makes liquids suitable for transmitting hydraulic power, because a change in pressure at one point in a liquid is transmitted undiminished to every other part of the liquid and very little energy is lost in the form of compression.[17]

However, the negligible compressibility does lead to other phenomena. The banging of pipes, called water hammer, occurs when a valve is suddenly closed, creating a huge pressure-spike at the valve that travels backward through the system at just under the speed of sound. Another phenomenon caused by liquid's incompressibility is cavitation. Because liquids have little elasticity they can literally be pulled apart in areas of high turbulence or dramatic change in direction, such as the trailing edge of a boat propeller or a sharp corner in a pipe. A liquid in an area of low pressure (vacuum) vaporizes and forms bubbles, which then collapse as they enter high pressure areas. This causes liquid to fill the cavities left by the bubbles with tremendous localized force, eroding any adjacent solid surface.[18]

Pressure and buoyancy

In a gravitational field, liquids exert pressure on the sides of a container as well as on anything within the liquid itself. This pressure is transmitted in all directions and increases with depth. If a liquid is at rest in a uniform gravitational field, the pressure ${\displaystyle p}$ at depth ${\displaystyle z}$ is given by[19]

${\displaystyle p=p_{0}+\rho gz\,}$

where:

${\displaystyle p_{0}\,}$ is the pressure at the surface

${\displaystyle \rho \,}$ is the density of the liquid, assumed uniform with depth

${\displaystyle g\,}$ is the gravitational acceleration

For a body of water open to the air, ${\displaystyle p_{0}}$ would be the atmospheric pressure.

Static liquids in uniform gravitational fields also exhibit the phenomenon of buoyancy, where objects immersed in the liquid experience a net force due to the pressure variation with depth. The magnitude of the force is equal to the weight of the liquid displaced by the object, and the direction of the force depends on the average density of the immersed object. If the density is smaller than that of the liquid, the buoyant force points upward and the object floats, whereas if the density is larger, the buoyant force points downward and the object sinks. This is known as Archimedes' principle.[20]

Surfaces

Main articles: Surface tension and Surface science

Surface waves in water

Unless the volume of a liquid exactly matches the volume of its container, one or more surfaces are observed. The presence of a surface introduces new phenomena which are not present in a bulk liquid. This is because a molecule at a surface possesses bonds with other liquid molecules only on the inner side of the surface, which implies a net force pulling surface molecules inward. Equivalently, this force can be described in terms of energy: there is a fixed amount of energy associated with forming a surface of a given area. This quantity is a material property called the surface tension, in units of energy per unit area (SI units: J/m²). Liquids with strong intermolecular forces tend to have large surface tensions.[21]

A practical implication of surface tension is that liquids tend to minimize their surface area, forming spherical drops and bubbles unless other constraints are present. Surface tension is responsible for a range of other phenomena as well, including surface waves, capillary action, wetting, and ripples. In liquids under nanoscale confinement, surface effects can play a dominating role since – compared with a macroscopic sample of liquid – a much greater fraction of molecules are located near a surface.

The surface tension of a liquid directly affects its wettability. Most common liquids have tensions ranging in the tens of mJ/m², so droplets of oil, water, or glue can easily merge and adhere to other surfaces, whereas liquid metals such as mercury may have tensions ranging in the hundreds of mJ/m², thus droplets do not combine easily and surfaces may only wet under specific conditions.

The surface tensions of common liquids occupy a relatively narrow range of values, which contrasts strongly with the enormous variation seen in other mechanical properties, such as viscosity.[22]

Flow

A simulation of viscosity. The fluid on the left has a lower viscosity and Newtonian behavior while the liquid on the right has higher viscosity and non-Newtonian behavior.

Main articles: Fluid mechanics and Fluid dynamics

An important physical property characterizing the flow of liquids is viscosity. Intuitively, viscosity describes the resistance of a liquid to flow.

More technically, viscosity measures the resistance of a liquid to deformation at a given rate, such as when it is being sheared at finite velocity.[23] A specific example is a liquid flowing through a pipe: in this case the liquid undergoes shear deformation since it flows more slowly near the walls of the pipe than near the center. As a result, it exhibits viscous resistance to flow. In order to maintain flow, an external force must be applied, such as a pressure difference between the ends of the pipe.

The viscosity of liquids decreases with increasing temperature.[24][25]

Precise control of viscosity is important in many applications, particularly the lubrication industry. One way to achieve such control is by blending two or more liquids of differing viscosities in precise ratios.[26] In addition, various additives exist which can modulate the temperature-dependence of the viscosity of lubricating oils. This capability is important since machinery often operate over a range of temperatures (see also viscosity index).[27]

The viscous behavior of a liquid can be either Newtonian or non-Newtonian. A Newtonian liquid exhibits a linear strain/stress curve, meaning its viscosity is independent of time, shear rate, or shear-rate history. Examples of Newtonian liquids include water, glycerin, motor oil, honey, or mercury. A non-Newtonian liquid is one where the viscosity is not independent of these factors and either thickens (increases in viscosity) or thins (decreases in viscosity) under shear. Examples of non-Newtonian liquids include ketchup, mayonnaise, hair gels, play dough, or starch solutions.[28]

Elasticity under confinement

Confined liquids may exhibit different mechanical properties compared to bulk liquids. For example, liquids under sub-millimeter confinement (e.g. in the gap between rigid walls) exhibit a solid-like mechanical response and possess a surprisingly large low-frequency elastic shear modulus, which scales with the inverse cubic power of the confinement length.[29]

Sound propagation

Main article: speed of sound § Speed of sound in liquids

The speed of sound in a liquid is given by ${\displaystyle c={\sqrt {K/\rho }}}$ where ${\displaystyle K}$ is the bulk modulus of the liquid and ${\displaystyle \rho }$ the density. As an example, water has a bulk modulus of about 2.2 GPa and a density of 1000 kg/m³, which gives c = 1.5 km/s.[30]

Thermodynamics

Phase transitions

Main articles: boiling, boiling point, melting, and melting point

A typical phase diagram. The dotted line gives the anomalous behaviour of water. The green lines show how the freezing point can vary with pressure, and the blue line shows how the boiling point can vary with pressure. The red line shows the boundary where sublimation or deposition can occur.

At a temperature below the boiling point, any matter in liquid form will evaporate until the condensation of gas above reach an equilibrium. At this point the gas will condense at the same rate as the liquid evaporates. Thus, a liquid cannot exist permanently if the evaporated liquid is continually removed. A liquid at its boiling point will evaporate more quickly than the gas can condense at the current pressure. A liquid at or above its boiling point will normally boil, though superheating can prevent this in certain circumstances.

At a temperature below the freezing point, a liquid will tend to crystallize, changing to its solid form. Unlike the transition to gas, there is no equilibrium at this transition under constant pressure, so unless supercooling occurs, the liquid will eventually completely crystallize. Note that this is only true under constant pressure, so e.g. water and ice in a closed, strong container might reach an equilibrium where both phases coexist. For the opposite transition from solid to liquid, see melting.

Liquids in space

The phase diagram explains why liquids do not exist in space or any other vacuum. Since the pressure is zero (except on surfaces or interiors of planets and moons) water and other liquids exposed to space will either immediately boil or freeze depending on the temperature. In regions of space near the earth, water will freeze if the sun is not shining directly on it and vapourize (sublime) as soon as it is in sunlight. If water exists as ice on the moon, it can only exist in shadowed holes where the sun never shines and where the surrounding rock doesn't heat it up too much. At some point near the orbit of Saturn, the light from the sun is too faint to sublime ice to water vapour. This is evident from the longevity of the ice that composes Saturn's rings.

Solutions

Main article: solution

Liquids can form solutions with gases, solids, and other liquids.

Two liquids are said to be miscible if they can form a solution in any proportion; otherwise they are immiscible. As an example, water and ethanol (drinking alcohol) are miscible whereas water and gasoline are immiscible.[31] In some cases a mixture of otherwise immiscible liquids can be stabilized to form an emulsion, where one liquid is dispersed throughout the other as microscopic droplets. Usually this requires the presence of a surfactant in order to stabilize the droplets. A familiar example of an emulsion is mayonnaise, which consists of a mixture of water and oil that is stabilized by lecithin, a substance found in egg yolks.[32]

Microscopic description

The molecules which compose liquids are disordered and strongly interacting, which makes liquids difficult to describe rigorously at the molecular level. This stands in contrast with the other two common phases of matter, gases and solids. Although gases are disordered, they are sufficiently dilute that many-body interactions can be ignored, and molecular interactions can instead be modeled in terms of well-defined binary collision events. Conversely, although solids are dense and strongly interacting, their regular structure at the molecular level (e.g. a crystalline lattice) allows for significant theoretical simplifications. For these reasons, the microscopic theory of liquids is less developed than that of gases and solids.[33]

Static structure factor

Main article: structure of liquids and glasses

Structure of a classical monatomic liquid. Atoms have many nearest neighbors in contact, yet no long-range order is present.

In a liquid, atoms do not form a crystalline lattice, nor do they show any other form of long-range order. This is evidenced by the absence of Bragg peaks in X-ray and neutron diffraction. Under normal conditions, the diffraction pattern has circular symmetry, expressing the isotropy of the liquid. In radial direction, the diffraction intensity smoothly oscillates. This is usually described by the static structure factor S(q), with wavenumber q=(4π/λ)sinθ given by the wavelength λ of the probe (photon or neutron) and the Bragg angle θ. The oscillations of S(q) express the near order of the liquid, i.e. the correlations between an atom and a few shells of nearest, second nearest, ... neighbors.

A more intuitive description of these correlations is given by the radial distribution function g(r), which is basically the Fourier transform of S(q). It represents a spatial average of a temporal snapshot of pair correlations in the liquid.

Radial distribution function of the Lennard-Jones model fluid.

Sound dispersion and structural relaxation

The above expression for the sound velocity ${\displaystyle c={\sqrt {K/\rho }}}$ contains the bulk modulus K. If K is frequency independent then the liquid behaves as a linear medium, so that sound propagates without dissipation and without mode coupling. In reality, any liquid shows some dispersion: with increasing frequency, K crosses over from the low-frequency, liquid-like limit ${\displaystyle K_{0}}$ to the high-frequency, solid-like limit ${\displaystyle K_{\infty }}$. In normal liquids, most of this cross over takes place at frequencies between GHz and THz, sometimes called hypersound.

At sub-GHz frequencies, a normal liquid cannot sustain shear waves: the zero-frequency limit of the shear modulus is ${\displaystyle G_{0}=0}$. This is sometimes seen as the defining property of a liquid.[34][35] However, just as the bulk modulus K, the shear modulus G is frequency dependent, and at hypersound frequencies it shows a similar cross over from the liquid-like limit ${\displaystyle G_{0}}$ to a solid-like, non-zero limit ${\displaystyle G_{\infty }}$.

According to the Kramers-Kronig relation, the dispersion in the sound velocity (given by the real part of K or G) goes along with a maximum in the sound attenuation (dissipation, given by the imaginary part of K or G). According to linear response theory, the Fourier transform of K or G describes how the system returns to equilibrium after an external perturbation; for this reason, the dispersion step in the GHz..THz region is also called structural relaxation. According to the fluctuation-dissipation theorem, relaxation towards equilibrium is intimately connected to fluctuations in equilibrium. The density fluctuations associated with sound waves can be experimentally observed by Brillouin scattering.

On supercooling a liquid towards the glass transition, the crossover from liquid-like to solid-like response moves from GHz to MHz, kHz, Hz, ...; equivalently, the characteristic time of structural relaxation increases from ns to μs, ms, s, ... This is the microscopic explanation for the above-mentioned viscoelastic behaviour of glass-forming liquids.

Effects of association

The mechanisms of atomic/molecular diffusion (or particle displacement) in solids are closely related to the mechanisms of viscous flow and solidification in liquid materials. Descriptions of viscosity in terms of molecular "free space" within the liquid[36] were modified as needed in order to account for liquids whose molecules are known to be "associated" in the liquid state at ordinary temperatures. When various molecules combine together to form an associated molecule, they enclose within a semi-rigid system a certain amount of space which before was available as free space for mobile molecules. Thus, increase in viscosity upon cooling due to the tendency of most substances to become associated on cooling.[37]

Similar arguments could be used to describe the effects of pressure on viscosity, where it may be assumed that the viscosity is chiefly a function of the volume for liquids with a finite compressibility. An increasing viscosity with rise of pressure is therefore expected. In addition, if the volume is expanded by heat but reduced again by pressure, the viscosity remains the same.

The local tendency to orientation of molecules in small groups lends the liquid (as referred to previously) a certain degree of association. This association results in a considerable "internal pressure" within a liquid, which is due almost entirely to those molecules which, on account of their temporary low velocities (following the Maxwell distribution) have coalesced with other molecules. The internal pressure between several such molecules might correspond to that between a group of molecules in the solid form.

References

1. Theodore Gray, The Elements: A Visual Exploration of Every Known Atom in the Universe New York: Workman Publishing, 2009 p. 127 ISBN 1-57912-814-9
2. Silberberg, Martin S. (2009), Chemistry: The Molecular Nature of Matter and Change, McGraw-Hill Higher Education, pp. 448–449, ISBN 978-0-07-304859-8
3. Theo Mang, Wilfried Dressel ’’Lubricants and lubrication’’, Wiley-VCH 2007 ISBN 3-527-31497-0
4. George Wypych ’’Handbook of solvents’’ William Andrew Publishing 2001 pp. 847–881 ISBN 1-895198-24-0
5. N. B. Vargaftik ’’Handbook of thermal conductivity of liquids and gases’’ CRC Press 1994 ISBN 0-8493-9345-0
6. Jack Erjavec ’’Automotive technology: a systems approach’’ Delmar Learning 2000 p. 309 ISBN 1-4018-4831-1
7. Gerald Wendt ’’The prospects of nuclear power and technology’’ D. Van Nostrand Company 1957 p. 266
8. ’’Modern engineering for design of liquid-propellant rocket engines’’ by Dieter K. Huzel, David H. Huang – American Institute of Aeronautics and Astronautics 1992 p. 99 ISBN 1-56347-013-6
9. Thomas E Mull ’’HVAC principles and applications manual’’ McGraw-Hill 1997 ISBN 0-07-044451-X
10. Unit Operations in Food Processing by R. L. Earle -- Pergamon Press 1983 Page 56--62, 138--141
11. R. Keith Mobley Fluid power dynamics Butterworth-Heinemann 2000 p. vii ISBN 0-7506-7174-2
12. Bela G. Liptak ’’Instrument engineers’ handbook: process control’’ CRC Press 1999 p. 807 ISBN 0-8493-1081-4
13. Knight, Randall D. (2008), Physics for Scientists and Engineers: A Strategic Approach (With Modern Physics), Addison-Wesley, p. 443, ISBN 978-0-8053-2736-6
14. Silberberg, Martin S. (2009), Chemistry: The Molecular Nature of Matter and Change, McGraw-Hill Higher Education, p. 461, ISBN 978-0-07-304859-8
15. "Compressibility of Liquids". hyperphysics.phy-astr.gsu.edu. Archived from the original on 7 December 2017. Retrieved 8 May 2018.
16. Intelligent Energy Field Manufacturing: Interdisciplinary Process Innovations By Wenwu Zhang -- CRC Press 2011 Page 144
17. Knight (2008) p. 454
18. Fluid Mechanics and Hydraulic Machines by S. C. Gupta -- Dorling-Kindersley 2006 Page 85
19. Knight (2008) p. 448
20. Knight (2008) pp. 455-459
21. Silberberg, Martin S. (2009), Chemistry: The Molecular Nature of Matter and Change, McGraw-Hill Higher Education, p. 457, ISBN 978-0-07-304859-8
22. Edward Yu. Bormashenko (5 November 2018). Wetting of Real Surfaces. De Gruyter. pp. 3–5. ISBN 978-3-11-058314-4.
23. Landau, L.D.; Lifshitz, E.M. (1987), Fluid Mechanics (2nd ed.), Pergamon Press, pp. 44–45, ISBN 978-0-08-033933-7
24. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007), Transport Phenomena (2nd ed.), John Wiley & Sons, Inc., p. 21, ISBN 978-0-470-11539-8
25. Krausser, J.; Samwer, K.; Zaccone, A. (2015). "Interatomic repulsion softness directly controls the fragility of supercooled metallic melts". Proceedings of the National Academy of Sciences of the USA. 112 (45): 13762. doi:10.1073/pnas.1503741112.
26. Zhmud, Boris (2014), "Viscosity Blending Equations" (PDF), Lube-Tech, 93
27. "Viscosity Index". UK: Anton Paar. Retrieved 29 August 2018.
28. Honey in Traditional and Modern Medicine by Laid Boukraa -- CRC Press 2014 Page 22--24
29. Zaccone, A.; Trachenko, K. (2020). "Explaining the low-frequency shear elasticity of confined liquids". Proceedings of the National Academy of Sciences of the USA. doi:10.1073/pnas.2010787117.
30. Taylor, John R. (2005), Classical Mechanics, University Science Books, pp. 727–729, ISBN 978-1-891389-22-1
31. Silberberg, pp. 188 and 502
32. Miodownik, Mark (2019), Liquid rules: The Delightful and Dangerous Substances that Flow Through Our Lives, Houghton Mifflin Harcourt, p. 124, ISBN 978-0-544-85019-4
33. Fisher, I.Z. (1964), Statistical Theory of Liquids, The University of Chicago Press, pp. 1–11
34. Born, Max (1940). "On the stability of crystal lattices". Mathematical Proceedings. Cambridge Philosophical Society. 36 (2): 160–172. Bibcode:1940PCPS...36..160B. doi:10.1017/S0305004100017138.
35. Born, Max (1939). "Thermodynamics of Crystals and Melting". Journal of Chemical Physics. 7 (8): 591–604. Bibcode:1939JChPh...7..591B. doi:10.1063/1.1750497. Archived from the original on 2016-05-15.
36. D.B. Macleod (1923). "On a relation between the viscosity of a liquid and its coefficient of expansion". Trans. Faraday Soc. 19: 6. doi:10.1039/tf9231900006.
37. G.W. Stewart (1930). "The Cybotactic (Molecular Group) Condition in Liquids; the Association of Molecules". Phys. Rev. 35 (7): 726. Bibcode:1930PhRv...35..726S. doi:10.1103/PhysRev.35.726.

Phase transitions of matter ()

		To
		Solid	Liquid	Gas	Plasma
From	Solid		Melting	Sublimation
	Liquid	Freezing		Vaporization
	Gas	Deposition	Condensation		Ionization
	Plasma			Recombination