2002:
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1751:(random) process, so even in two identical systems nucleation will occur at different times. A common mechanism is illustrated in the animation to the right. This shows nucleation of a new phase (shown in red) in an existing phase (white). In the existing phase microscopic fluctuations of the red phase appear and decay continuously, until an unusually large fluctuation of the new red phase is so large it is more favourable for it to grow than to shrink back to nothing. This nucleus of the red phase then grows and converts the system to this phase. The standard theory that describes this behaviour for the nucleation of a new thermodynamic phase is called
31:
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the transition to a new phase that does not rely on the new phase already being present, either because it is the very first nucleus of that phase to form, or because the nucleus forms far from any pre-existing piece of the new phase. Particularly in the study of crystallisation, secondary nucleation can be important. This is the formation of nuclei of a new crystal directly caused by pre-existing crystals.
1851:ΔG*. This barrier comes from the free energy penalty of forming the surface of the growing nucleus. For homogeneous nucleation the nucleus is approximated by a sphere, but as we can see in the schematic of macroscopic droplets to the right, droplets on surfaces are not complete spheres and so the area of the interface between the droplet and the surrounding fluid is less than a sphere's
1735:
2024:
this is assumed to be because, by chance, these droplets do not have even one impurity particle and so there is no heterogeneous nucleation. Homogeneous nucleation is assumed to be negligible on the timescale of this experiment. The remaining droplets freeze in a stochastic way, at rates 0.02/s if they have one impurity particle, 0.04/s if they have two, and so on.
1927:, where phase separation is delayed until the system enters the unstable region where a small perturbation in composition leads to a decrease in energy and, thus, spontaneous growth of the perturbation. This region of a phase diagram is known as the spinodal region and the phase separation process is known as spinodal decomposition and may be governed by the
2197:
Many of the materials we make and use are crystalline, but are made from liquids, e.g. crystalline iron made from liquid iron cast into a mold, so the nucleation of crystalline materials is widely studied in industry. It is used heavily in the chemical industry for cases such as in the preparation of
1902:
makes a number of assumptions, for example it treats a microscopic nucleus as if it is a macroscopic droplet with a well-defined surface whose free energy is estimated using an equilibrium property: the interfacial tension σ. For a nucleus that may be only of order ten molecules across it is not
2023:
among the liquid tin droplets. The fit values are that the nucleation rate due to a single impurity particle is 0.02/s, and the average number of impurity particles per droplet is 1.2. Note that about 30% of the tin droplets never freeze; the data plateau at a fraction of about 0.3. Within the model
2018:
to the data. This is a simplified version of the model Pound and La Mer used to model their data. The model assumes that nucleation occurs due to impurity particles in the liquid tin droplets, and it makes the simplifying assumption that all impurity particles produce nucleation at the same rate. It
1980:
The time until the appearance of the first crystal is also called primary nucleation time, to distinguish it from secondary nucleation times. Primary here refers to the first nucleus to form, while secondary nuclei are crystal nuclei produced from a preexisting crystal. Primary nucleation describes
1951:
In small volumes, such as in small droplets, only one nucleation event may be needed for crystallisation. In these small volumes, the time until the first crystal appears is usually defined to be the nucleation time. Calcium carbonate crystal nucleation depends not only on degree of supersaturation
1910:
However, modern computers are powerful enough to calculate essentially exact nucleation rates for simple models. These have been compared with the classical theory, for example for the case of nucleation of the crystal phase in the model of hard spheres. This is a model of perfectly hard spheres in
1984:
For example, if the crystals are in a solution and the system is subject to shearing forces, small crystal nuclei could be sheared off a growing crystal, thus increasing the number of crystals in the system. So both primary and secondary nucleation increase the number of crystals in the system but
1828:
Heterogeneous nucleation, nucleation with the nucleus at a surface, is much more common than homogeneous nucleation. For example, in the nucleation of ice from supercooled water droplets, purifying the water to remove all or almost all impurities results in water droplets that freeze below around
2005:
The black triangles are the fraction of a large set of small supercooled liquid tin droplets that are still liquid, i.e., where the crystal state has not nucleated, as a function of time. The data are from Pound and La Mer (1952). The red curve is a fit of a function of the
Gompertz form to these
2062:
An example of experimental data on the freezing of small water droplets is shown at the right. The plot shows the fraction of a large set of water droplets, that are still liquid water, i.e., have not yet frozen, as a function of temperature. Note that the highest temperature at which any of the
1993:
It is typically difficult to experimentally study the nucleation of crystals. The nucleus is microscopic, and thus too small to be directly observed. In large liquid volumes there are typically multiple nucleation events, and it is difficult to disentangle the effects of nucleation from those of
1971:
Although the existing theories including the classical nucleation theory explain well the steady nucleation state when the crystal nucleation rate is not time dependent, the initial non-steady state transient nucleation, and even more mysterious incubation period, require more attention of the
2214:
In addition to the nucleation and growth of crystals e.g. in non-crystalline glasses, the nucleation and growth of impurity precipitates in crystals at, and between, grain boundaries is quite important industrially. For example in metals solid-state nucleation and precipitate growth plays an
1915:. For the crystallization of hard spheres the classical theory is a very reasonable approximate theory. So for the simple models we can study, classical nucleation theory works quite well, but we do not know if it works equally well for (say) complex molecules crystallising out of solution.
1939:
In many cases, liquids and solutions can be cooled down or concentrated up to conditions where the liquid or solution is significantly less thermodynamically stable than the crystal, but where no crystals will form for minutes, hours, weeks or longer; this process is called
1717:
in the system. These impurities may be too small to be seen by the naked eye, but still can control the rate of nucleation. Because of this, it is often important to distinguish between heterogeneous nucleation and homogeneous nucleation. Heterogeneous nucleation occurs at
2138:
with respect to the pressure-dependent boiling point. More often, nucleation occurs on the heating surface, at nucleation sites. Typically, nucleation sites are tiny crevices where free gas-liquid surface is maintained or spots on the heating surface with lower
1734:
2010:
To the right is shown an example set of nucleation data. It is for the nucleation at constant temperature and hence supersaturation of the crystal phase in small droplets of supercooled liquid tin; this is the work of Pound and La Mer.
1778:
is a widely used approximate theory for estimating these rates, and how they vary with variables such as temperature. It correctly predicts that the time you have to wait for nucleation decreases extremely rapidly when
1944:. Nucleation of the crystal is then being prevented by a substantial barrier. This has consequences, for example cold high altitude clouds may contain large numbers of small liquid water droplets that are far below 0
1952:
but also the ratio of calcium to carbonate ions in aqueous solutions. In larger volumes many nucleation events will occur. A simple model for crystallisation in that case, that combines nucleation and growth is the
1823:
and hence results in faster nucleation on surfaces with smaller contact angles. Also, if instead of the surface being flat it curves towards fluid, then this also reduces the interfacial area and so the nucleation
1818:
the nucleus surface makes with the solid horizontal surface decreases from left to right. The surface area of the nucleus decreases as the contact angle decreases. This geometrical effect reduces the barrier in
2714:
Seepma; Ruiz
Hernandez; Nehrke; Soetaert; Philipse; Kuipers; Wolthers (January 28, 2021), ""Controlling CaCO3 particle size with {Ca2+}:{CO32-} ratios in aqueous environments" Crystal Growth & Design",
2027:
These data are just one example, but they illustrate common features of the nucleation of crystals in that there is clear evidence for heterogeneous nucleation, and that nucleation is clearly stochastic.
2014:
Nucleation occurs in different droplets at different times, hence the fraction is not a simple step function that drops sharply from one to zero at one particular time. The red curve is a fit of a
1320:
2190:°C is required to nucleate ice and for the water to freeze. For example, small droplets of very pure water can remain liquid down to below -30 °C although ice is the stable state below 0
1972:
scientific community. Chemical ordering of the undercooling liquid prior to crystal nucleation was suggested to be responsible for that feature by reducing the energy barrier for nucleation.
2104:. The excess vapor begins to nucleate and to form small water droplets which form a cloud. Nucleation of the droplets of liquid water is heterogeneous, occurring on particles referred to as
2215:
important role e.g. in modifying mechanical properties like ductility, while in semiconductors it plays an important role e.g. in trapping impurities during integrated circuit manufacture.
2143:
properties. Substantial superheating of a liquid can be achieved after the liquid is de-gassed and if the heating surfaces are clean, smooth and made of materials well wetted by the liquid.
1671:. Nucleation is typically defined to be the process that determines how long an observer has to wait before the new phase or self-organized structure appears. For example, if a volume of
2205:
catalyses the decomposition of water. It is an important factor in the semiconductor industry, as the band gap energy in semiconductors is influenced by the size of nanoclusters.
2036:
The freezing of small water droplets to ice is an important process, particularly in the formation and dynamics of clouds. Water (at atmospheric pressure) does not freeze at 0
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3001:
Palmans, Roger; Frank, Arthur J. (1991). "A molecular water-reduction catalyst: Surface derivatization of titania colloids and suspensions with a platinum complex".
990:
2040:°C, but rather at temperatures that tend to decrease as the volume of the water decreases and as the concentration of dissolved chemicals in the water increases.
489:
1755:. However, the CNT fails in describing experimental results of vapour to liquid nucleation even for model substances like argon by several orders of magnitude.
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1710:, and it is the start of the process of forming a new thermodynamic phase. In contrast, new phases at continuous phase transitions start to form immediately.
2150:
operate by providing many nucleation sites via high surface-area and sharp corners, speeding the release of bubbles and removing carbonation from the wine.
1786:
It is not just new phases such as liquids and crystals that form via nucleation followed by growth. The self-assembly process that forms objects like the
2001:
3028:
Rajh, Tijana; Micic, Olga I.; Nozik, Arthur J. (1993). "Synthesis and characterization of surface-modified colloidal cadmium telluride quantum dots".
1630:
1840:
This observation that heterogeneous nucleation can occur when the rate of homogeneous nucleation is essentially zero, is often understood using
2157:
offers another example. The surface of Mentos candy provides nucleation sites for the formation of carbon-dioxide bubbles from carbonated soda.
1331:
1699:). At these conditions, nucleation of ice is either slow or does not occur at all. However, at lower temperatures nucleation is fast, and ice
1219:
2808:
Wang, Zhi; Chen, Chunlin; Ketov, Sergey V.; Akagi, Kazuto; Tsarkov, Andrey A.; Ikuhara, Yuichi; Louzguine-Luzgin, Dmitri V. (October 2018).
2043:
1884:. This reduction in surface area of the nucleus reduces the height of the barrier to nucleation and so speeds nucleation up exponentially.
2409:
A. Fladerer, R. Strey: "Homogeneous nucleation and droplet growth in supersaturated argon vapor: The cryogenic nucleation pulse chamber".
453:
2762:
1309:
2810:"Local chemical ordering within the incubation period as a trigger for nanocrystallization of a highly supercooled Ti-based liquid"
2653:
Mendez-Villuendas, Eduardo; Saika-Voivod, Ivan; Bowles, Richard K. (2007). "A limit of stability in supercooled liquid clusters".
1342:
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Kreer, Markus (1993). "Classical Becker-Döring cluster equations: Rigorous results on metastability and long-time behaviour".
2903:
912:
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879:
446:
324:
262:
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Gillam, J.E.; MacPhee, C.E. (2013). "Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth".
1762:°C, if the system is not evolving with time and nucleation occurs in one step, then the probability that nucleation has
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889:
343:
1903:
always clear that we can treat something so small as a volume plus a surface. Also nucleation is an inherently out of
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2100:) and many small water droplets nucleate from the supersaturated air. The amount of water vapour that air can carry
1848:
1447:
918:
317:
1616:
2526:
Mendez-Villuendas, Eduardo; Bowles, Richard (2007). "Surface
Nucleation in the Freezing of Gold Nanoparticles".
1994:
growth of the nucleated phase. These problems can be overcome by working with small droplets. As nucleation is
1547:
79:
2047:
Survival curve for water droplets 34.5 μm in diameter. Blue circles are data, and the red curve is a fit of a
1967:
in water, nucleation will occur, allowing sugar molecules to stick together and form large crystal structures.
1442:
2154:
1988:
1522:
1295:
272:
2922:
Pound, Guy M.; V. K. La Mer (1952). "Kinetics of
Crystalline Nucleus Formation in Supercooled Liquid Tin".
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their mechanisms are very different, and secondary nucleation relies on crystals already being present.
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141:
75:
2341:"Quantitative Studies of Crystal Nucleation at Constant Supersaturation: Experimental Data and Models"
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metallic ultradispersed powders that can serve as catalysts. For example, platinum deposited onto TiO
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1904:
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2951:"Photomicrographic Investigation of Spontaneous Freezing Temperatures of Supercooled Water Droplets"
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phenomenon so it is not always obvious that its rate can be estimated using equilibrium properties.
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Fokin, Vladimir M.; Zanotto, Edgar D.; Yuritsyn, Nikolay S.; Schmelzer, Jürn W.P. (August 2006).
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2436:"Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets"
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is the process of adding artificial condensation nuclei to quicken the formation of clouds.
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Nucleation can also start at the surface of a liquid. For example, computer simulations of
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129:
1998:, many droplets are needed so that statistics for the nucleation events can be obtained.
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Experimental observations on the nucleation times for the crystallisation of small volumes
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For nucleation of a new thermodynamic phase, such as the formation of ice in water below 0
1743:. Up spins (particles in lattice-gas terminology) shown in red, down spins shown in white.
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2274:"Nucleation: theory and applications to protein solutions and colloidal suspensions"
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small water droplets. The decay rate of the exponential gives the nucleation rate.
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also starts with nucleation. Energy consuming self-organising systems such as the
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2589:"Numerical prediction of absolute crystallization rates in hard-sphere colloids"
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Thus small droplets of water, as found in clouds, may remain liquid far below 0
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on surfaces in the system. Homogeneous nucleation occurs away from a surface.
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Initial step in the phase transition or molecular self-assembly of a substance
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1844:. This predicts that the nucleation slows exponentially with the height of a
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894:
463:
424:
136:
2878:
Botsaris, GD (1976). "Secondary
Nucleation — A Review". In Mullin, J (ed.).
2841:"Homogeneous crystal nucleation in silicate glasses: A 40 years perspective"
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show that the crystal phase sometimes nucleates at the liquid-gold surface.
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process on Earth is the formation of ice. Liquid water does not freeze at 0
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1995:
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Three nuclei on a surface, illustrating decreasing contact angles. The
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2418:
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Nucleation in
Condensed Matter: Applications in Materials and Biology
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2186:°C unless there is ice already present; cooling significantly below 0
1959:
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nucleate shortly after the pressure is released from a container of
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1912:
1714:
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3057:"Mechanisms of Nucleation and Growth of Nanoparticles in Solution"
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1806:
Heterogeneous nucleation often dominates homogeneous nucleation
3055:
Thanh, Nguyen T. K.; Maclean, N.; Mahiddine, S. (2014-08-13).
2838:
1923:
Phase-transition processes can also be explained in terms of
1706:
Nucleation is a common mechanism which generates first-order
1672:
2761:
Kelton, K. F.; Greer, A. L.; Thompson, C. V. (1983-12-15).
2079:
1833:°C, whereas water that contains impurities may freeze at −5
367:
2168:
rely on nucleation, of bubbles and droplets, respectively.
2067:°C, while the last droplet to freeze does so at almost -35
1688:
1691:, but volumes of water cooled only a few degrees below 0
2525:
1770:. This is seen for example in the nucleation of ice in
1695:°C often stay completely free of ice for long periods (
2134:
if the pressure is reduced so that the liquid becomes
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Nucleation of carbon dioxide bubbles around a finger
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1975:
1651:is the first step in the formation of either a new
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2980:. Amsterdam: Elsevier Science & Technology.
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1911:thermal motion, and is a simple model of some
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2263:
2261:
1934:
1624:
3000:
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2871:
2325:
1895:Computer simulation studies of simple models
2971:
2763:"Transient nucleation in condensed systems"
2433:
2965:
2917:
2915:
2258:
1739:Nucleation at a surface (black) in the 2D
1631:
1617:
29:
2972:Kelton, Ken; Greer, Alan Lindsay (2010).
2949:Dorsch, Robert G; Hacker, Paul T (1950).
2738:
2728:
2666:
2539:
2459:
2356:
2300:
2172:
2942:
2924:Journal of the American Chemical Society
2877:
2253:Microphysics of Clouds and Precipitation
2083:
2080:Nucleation of fluids (gases and liquids)
2042:
2000:
1958:
1809:
1729:
2912:
2209:
3100:
2019:also assumes that these particles are
1918:
1713:Nucleation is often very sensitive to
2580:
2374:
2477:Journal of Physics: Condensed Matter
2429:
2427:
2338:
2281:Journal of Physics: Condensed Matter
2271:
13:
2251:H. R. Pruppacher and J. D. Klett,
1725:
838:
791:
706:
659:
571:
524:
344:Intensive and extensive properties
14:
3129:
3030:The Journal of Physical Chemistry
3003:The Journal of Physical Chemistry
2845:Journal of Non-Crystalline Solids
2440:Atmospheric Chemistry and Physics
2424:
2102:decreases with lower temperatures
1703:appear after little or no delay.
2865:10.1016/j.jnoncrysol.2006.02.074
2063:droplets freezes is close to -19
1976:Primary and secondary nucleation
1600:
1599:
919:Table of thermodynamic equations
3048:
3021:
2994:
2832:
2801:
2767:The Journal of Chemical Physics
2754:
2707:
2655:The Journal of Chemical Physics
2646:
2596:The Journal of Chemical Physics
2411:The Journal of Chemical Physics
2096:form when wet air cools (often
1395:Maxwell's thermodynamic surface
2519:
2497:10.1088/0953-8984/25/37/373101
2468:
2403:
1129:
1117:
1074:
1062:
1019:
1007:
979:
967:
1:
2587:Auer, S.; D. Frenkel (2004).
2558:10.1103/PhysRevLett.98.185503
2311:10.1088/0953-8984/19/3/033101
2218:
2155:Diet Coke and Mentos eruption
1296:Mechanical equivalent of heat
2826:10.1016/j.matdes.2018.07.013
908:Onsager reciprocal relations
7:
2896:10.1007/978-1-4615-7258-9_1
2717:Crystal Growth & Design
2074:
1900:Classical nucleation theory
1842:classical nucleation theory
1821:classical nucleation theory
1790:aggregates associated with
1776:Classical nucleation theory
1753:classical nucleation theory
1400:Entropy as energy dispersal
1211:"Perpetual motion" machines
1150:{\displaystyle G(T,p)=H-TS}
1095:{\displaystyle A(T,V)=U-TS}
1040:{\displaystyle H(S,p)=U+pV}
10:
3134:
2882:Industrial Crystallization
2434:Duft, D.; Leisner (2004).
1935:The nucleation of crystals
1877:{\displaystyle 4\pi r^{2}}
847:{\displaystyle \partial T}
800:{\displaystyle \partial V}
715:{\displaystyle \partial p}
668:{\displaystyle \partial V}
580:{\displaystyle \partial T}
533:{\displaystyle \partial S}
2339:Sear, Richard P. (2014).
2106:cloud condensation nuclei
1905:thermodynamic equilibrium
1321:An Inquiry Concerning the
2413:124(16), 164710 (2006).
2397:10.1002/andp.19935050408
1766:occurred should undergo
1747:Nucleation is usually a
1334:Heterogeneous Substances
751:{\displaystyle \alpha =}
619:{\displaystyle \beta =-}
2730:10.1021/acs.cgd.0c01403
2528:Physical Review Letters
2461:10.5194/acp-4-1997-2004
1679:) significantly below 0
2814:Materials & Design
2173:Nucleation of crystals
2130:can occur in the bulk
2089:
2052:
2031:
2007:
1968:
1929:Cahn–Hilliard equation
1925:spinodal decomposition
1878:
1825:
1744:
1667:within a substance or
1151:
1096:
1041:
986:
985:{\displaystyle U(S,V)}
848:
824:
801:
777:
752:
716:
692:
669:
645:
620:
581:
557:
534:
510:
485:
464:Specific heat capacity
68:Quantum thermodynamics
2886:. Springer. pp.
2098:because the air rises
2087:
2046:
2004:
1962:
1879:
1813:
1738:
1332:On the Equilibrium of
1152:
1097:
1050:Helmholtz free energy
1042:
987:
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802:
778:
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717:
693:
670:
646:
621:
582:
558:
535:
511:
486:
2851:(26–27): 2681–2714.
2210:Nucleation in solids
1954:KJMA or Avrami model
1855:
1683:°C, it will tend to
1677:atmospheric pressure
1345:Motive Power of Fire
1111:
1056:
1001:
961:
913:Bridgman's equations
890:Fundamental relation
835:
814:
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767:
739:
703:
682:
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568:
547:
521:
500:
472:
3042:10.1021/j100148a026
3015:10.1021/j100176a075
2955:NACA Technical Note
2936:10.1021/ja01129a044
2857:2006JNCS..352.2681F
2779:1983JChPh..79.6261K
2677:2007JChPh.127o4703M
2608:2004JChPh.120.3015A
2550:2007PhRvL..98r5503M
2489:2013JPCM...25K3101G
2452:2004ACP.....4.1997D
2389:1993AnP...505..398K
2293:2007JPCM...19c3101S
2272:Sear, R.P. (2007).
2049:Gumbel distribution
2021:Poisson distributed
1919:The spinodal region
1798:in cells also show
1792:Alzheimer's disease
1653:thermodynamic phase
1323:Source ... Friction
1255:Loschmidt's paradox
447:Material properties
325:Conjugate variables
2377:Annalen der Physik
2358:10.1039/C4CE00344F
2148:champagne stirrers
2090:
2053:
2008:
1969:
1889:gold nanoparticles
1874:
1826:
1745:
1587:Order and disorder
1343:Reflections on the
1250:Heat death paradox
1147:
1092:
1037:
982:
844:
820:
797:
773:
748:
712:
688:
665:
641:
616:
577:
553:
530:
506:
484:{\displaystyle c=}
481:
454:Property databases
430:Reduced properties
414:Chemical potential
378:Functions of state
301:Thermal efficiency
37:Carnot heat engine
3113:Materials science
3073:10.1021/cr400544s
3067:(15): 7610–7630.
2905:978-1-4615-7260-2
2773:(12): 6261–6276.
2685:10.1063/1.2779875
2616:10.1063/1.1638740
2419:10.1063/1.2186327
2351:(29): 6506–6522.
2016:Gompertz function
1768:exponential decay
1736:
1708:phase transitions
1665:self-organization
1641:
1640:
1582:Self-organization
1407:
1406:
1105:Gibbs free energy
903:Maxwell relations
861:
860:
857:
856:
823:{\displaystyle V}
776:{\displaystyle 1}
731:Thermal expansion
725:
724:
691:{\displaystyle V}
644:{\displaystyle 1}
590:
589:
556:{\displaystyle N}
509:{\displaystyle T}
437:
436:
353:Process functions
339:Property diagrams
318:System properties
308:
307:
273:Endoreversibility
165:Equation of state
3125:
3093:
3092:
3061:Chemical Reviews
3052:
3046:
3045:
3025:
3019:
3018:
2998:
2992:
2991:
2979:
2969:
2963:
2962:
2946:
2940:
2939:
2919:
2910:
2909:
2885:
2875:
2869:
2868:
2836:
2830:
2829:
2805:
2799:
2798:
2787:10.1063/1.445731
2758:
2752:
2751:
2742:
2732:
2723:(3): 1576–1590,
2711:
2705:
2704:
2670:
2650:
2644:
2643:
2593:
2584:
2578:
2577:
2543:
2541:cond-mat/0702605
2523:
2517:
2516:
2472:
2466:
2465:
2463:
2431:
2422:
2407:
2401:
2400:
2372:
2363:
2362:
2360:
2336:
2323:
2322:
2304:
2278:
2269:
2256:
2255:, Kluwer (1997).
2249:
2193:
2189:
2185:
2178:The most common
2070:
2066:
2058:
2039:
1947:
1883:
1881:
1880:
1875:
1873:
1872:
1836:
1832:
1761:
1737:
1720:nucleation sites
1694:
1682:
1633:
1626:
1619:
1603:
1602:
1310:Key publications
1291:
1290:("living force")
1240:Brownian ratchet
1235:Entropy and life
1230:Entropy and time
1181:
1180:
1156:
1154:
1153:
1148:
1101:
1099:
1098:
1093:
1046:
1044:
1043:
1038:
991:
989:
988:
983:
885:Clausius theorem
880:Carnot's theorem
853:
851:
850:
845:
829:
827:
826:
821:
806:
804:
803:
798:
782:
780:
779:
774:
761:
760:
757:
755:
754:
749:
721:
719:
718:
713:
697:
695:
694:
689:
674:
672:
671:
666:
650:
648:
647:
642:
629:
628:
625:
623:
622:
617:
586:
584:
583:
578:
562:
560:
559:
554:
539:
537:
536:
531:
515:
513:
512:
507:
494:
493:
490:
488:
487:
482:
460:
459:
333:
332:
152:
151:
33:
19:
18:
3133:
3132:
3128:
3127:
3126:
3124:
3123:
3122:
3098:
3097:
3096:
3053:
3049:
3026:
3022:
2999:
2995:
2988:
2970:
2966:
2947:
2943:
2920:
2913:
2906:
2876:
2872:
2837:
2833:
2806:
2802:
2759:
2755:
2712:
2708:
2651:
2647:
2591:
2585:
2581:
2524:
2520:
2473:
2469:
2432:
2425:
2408:
2404:
2373:
2366:
2337:
2326:
2302:10.1.1.605.2550
2276:
2270:
2259:
2250:
2225:
2221:
2212:
2201:
2191:
2187:
2183:
2180:crystallisation
2175:
2082:
2077:
2068:
2064:
2056:
2037:
2034:
1991:
1978:
1945:
1937:
1921:
1897:
1868:
1864:
1856:
1853:
1852:
1834:
1830:
1808:
1759:
1730:
1728:
1726:Characteristics
1692:
1680:
1637:
1592:
1591:
1567:
1559:
1558:
1557:
1417:
1409:
1408:
1387:
1373:
1348:
1344:
1337:
1333:
1326:
1322:
1289:
1282:
1264:
1245:Maxwell's demon
1207:
1178:
1177:
1161:
1160:
1159:
1112:
1109:
1108:
1107:
1057:
1054:
1053:
1052:
1002:
999:
998:
997:
962:
959:
958:
957:
955:Internal energy
950:
935:
925:
924:
899:
874:
864:
863:
862:
836:
833:
832:
815:
812:
811:
789:
786:
785:
768:
765:
764:
740:
737:
736:
704:
701:
700:
683:
680:
679:
657:
654:
653:
636:
633:
632:
605:
602:
601:
596:Compressibility
569:
566:
565:
548:
545:
544:
522:
519:
518:
501:
498:
497:
473:
470:
469:
449:
439:
438:
419:Particle number
372:
331:
320:
310:
309:
268:Irreversibility
180:State of matter
147:Isolated system
132:
122:
121:
120:
95:
85:
84:
80:Non-equilibrium
72:
47:
39:
17:
12:
11:
5:
3131:
3121:
3120:
3115:
3110:
3095:
3094:
3047:
3020:
2993:
2986:
2964:
2941:
2911:
2904:
2870:
2831:
2800:
2753:
2706:
2661:(15): 154703.
2645:
2602:(6): 3015–29.
2579:
2534:(18): 185503.
2518:
2483:(37): 373101.
2467:
2423:
2402:
2383:(4): 398–417.
2364:
2324:
2257:
2222:
2220:
2217:
2211:
2208:
2207:
2206:
2199:
2195:
2174:
2171:
2170:
2169:
2162:bubble chamber
2158:
2151:
2144:
2126:Nucleation in
2124:
2117:carbon dioxide
2113:
2081:
2078:
2076:
2073:
2033:
2030:
1990:
1987:
1977:
1974:
1965:supersaturated
1963:When sugar is
1936:
1933:
1920:
1917:
1896:
1893:
1871:
1867:
1863:
1860:
1837:°C or warmer.
1807:
1804:
1781:supersaturated
1727:
1724:
1675:is cooled (at
1645:thermodynamics
1639:
1638:
1636:
1635:
1628:
1621:
1613:
1610:
1609:
1608:
1607:
1594:
1593:
1590:
1589:
1584:
1579:
1574:
1568:
1565:
1564:
1561:
1560:
1556:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1419:
1418:
1415:
1414:
1411:
1410:
1405:
1404:
1403:
1402:
1397:
1389:
1388:
1386:
1385:
1382:
1378:
1375:
1374:
1372:
1371:
1366:
1364:Thermodynamics
1360:
1357:
1356:
1352:
1351:
1350:
1349:
1340:
1338:
1329:
1327:
1318:
1313:
1312:
1306:
1305:
1304:
1303:
1298:
1293:
1281:
1280:
1278:Caloric theory
1274:
1271:
1270:
1266:
1265:
1263:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1226:
1223:
1222:
1216:
1215:
1214:
1213:
1206:
1205:
1200:
1195:
1189:
1186:
1185:
1179:
1176:
1175:
1172:
1168:
1167:
1166:
1163:
1162:
1158:
1157:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1102:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1047:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
992:
981:
978:
975:
972:
969:
966:
951:
949:
948:
943:
937:
936:
931:
930:
927:
926:
923:
922:
915:
910:
905:
898:
897:
892:
887:
882:
876:
875:
870:
869:
866:
865:
859:
858:
855:
854:
843:
840:
830:
819:
808:
807:
796:
793:
783:
772:
758:
747:
744:
734:
727:
726:
723:
722:
711:
708:
698:
687:
676:
675:
664:
661:
651:
640:
626:
615:
612:
609:
599:
592:
591:
588:
587:
576:
573:
563:
552:
541:
540:
529:
526:
516:
505:
491:
480:
477:
467:
458:
457:
456:
450:
445:
444:
441:
440:
435:
434:
433:
432:
427:
422:
411:
400:
381:
380:
374:
373:
371:
370:
365:
359:
356:
355:
349:
348:
347:
346:
341:
322:
321:
316:
315:
312:
311:
306:
305:
304:
303:
298:
293:
285:
284:
278:
277:
276:
275:
270:
265:
260:
258:Free expansion
255:
250:
245:
240:
235:
230:
225:
220:
212:
211:
205:
204:
203:
202:
197:
195:Control volume
192:
187:
185:Phase (matter)
182:
177:
172:
167:
159:
158:
150:
149:
144:
139:
133:
128:
127:
124:
123:
119:
118:
113:
108:
103:
97:
96:
91:
90:
87:
86:
83:
82:
71:
70:
65:
60:
55:
49:
48:
45:
44:
41:
40:
35:The classical
34:
26:
25:
23:Thermodynamics
15:
9:
6:
4:
3:
2:
3130:
3119:
3116:
3114:
3111:
3109:
3106:
3105:
3103:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3051:
3043:
3039:
3036:(46): 11999.
3035:
3031:
3024:
3016:
3012:
3008:
3004:
2997:
2989:
2987:9780080421476
2983:
2978:
2977:
2968:
2960:
2956:
2952:
2945:
2937:
2933:
2929:
2925:
2918:
2916:
2907:
2901:
2897:
2893:
2889:
2884:
2883:
2874:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2835:
2827:
2823:
2819:
2815:
2811:
2804:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2757:
2750:
2746:
2741:
2736:
2731:
2726:
2722:
2718:
2710:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2669:
2664:
2660:
2656:
2649:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2601:
2597:
2590:
2583:
2575:
2571:
2567:
2563:
2559:
2555:
2551:
2547:
2542:
2537:
2533:
2529:
2522:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2471:
2462:
2457:
2453:
2449:
2445:
2441:
2437:
2430:
2428:
2420:
2416:
2412:
2406:
2398:
2394:
2390:
2386:
2382:
2378:
2371:
2369:
2359:
2354:
2350:
2346:
2342:
2335:
2333:
2331:
2329:
2320:
2316:
2312:
2308:
2303:
2298:
2294:
2290:
2287:(3): 033101.
2286:
2282:
2275:
2268:
2266:
2264:
2262:
2254:
2248:
2246:
2244:
2242:
2240:
2238:
2236:
2234:
2232:
2230:
2228:
2223:
2216:
2204:
2203:nanoparticles
2196:
2181:
2177:
2176:
2167:
2166:cloud chamber
2163:
2159:
2156:
2152:
2149:
2145:
2142:
2137:
2133:
2129:
2125:
2122:
2118:
2114:
2111:
2110:Cloud seeding
2107:
2103:
2099:
2095:
2092:
2091:
2086:
2072:
2060:
2050:
2045:
2041:
2029:
2025:
2022:
2017:
2012:
2003:
1999:
1997:
1986:
1982:
1973:
1966:
1961:
1957:
1955:
1949:
1943:
1932:
1930:
1926:
1916:
1914:
1908:
1906:
1901:
1892:
1890:
1885:
1869:
1865:
1861:
1858:
1850:
1847:
1843:
1838:
1822:
1817:
1816:contact angle
1812:
1803:
1801:
1797:
1793:
1789:
1784:
1782:
1777:
1773:
1769:
1765:
1756:
1754:
1750:
1742:
1723:
1721:
1716:
1711:
1709:
1704:
1702:
1698:
1690:
1686:
1678:
1674:
1670:
1666:
1662:
1661:self-assembly
1658:
1654:
1650:
1646:
1634:
1629:
1627:
1622:
1620:
1615:
1614:
1612:
1611:
1606:
1598:
1597:
1596:
1595:
1588:
1585:
1583:
1580:
1578:
1577:Self-assembly
1575:
1573:
1570:
1569:
1563:
1562:
1554:
1551:
1549:
1548:van der Waals
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1473:von Helmholtz
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1420:
1413:
1412:
1401:
1398:
1396:
1393:
1392:
1391:
1390:
1383:
1380:
1379:
1377:
1376:
1370:
1367:
1365:
1362:
1361:
1359:
1358:
1354:
1353:
1347:
1346:
1339:
1336:
1335:
1328:
1325:
1324:
1317:
1316:
1315:
1314:
1311:
1308:
1307:
1302:
1299:
1297:
1294:
1292:
1288:
1284:
1283:
1279:
1276:
1275:
1273:
1272:
1268:
1267:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1227:
1225:
1224:
1221:
1218:
1217:
1212:
1209:
1208:
1204:
1201:
1199:
1196:
1194:
1191:
1190:
1188:
1187:
1183:
1182:
1173:
1170:
1169:
1165:
1164:
1144:
1141:
1138:
1135:
1132:
1126:
1123:
1120:
1114:
1106:
1103:
1089:
1086:
1083:
1080:
1077:
1071:
1068:
1065:
1059:
1051:
1048:
1034:
1031:
1028:
1025:
1022:
1016:
1013:
1010:
1004:
996:
993:
976:
973:
970:
964:
956:
953:
952:
947:
944:
942:
939:
938:
934:
929:
928:
921:
920:
916:
914:
911:
909:
906:
904:
901:
900:
896:
895:Ideal gas law
893:
891:
888:
886:
883:
881:
878:
877:
873:
868:
867:
841:
831:
817:
810:
809:
794:
784:
770:
763:
762:
759:
745:
742:
735:
732:
729:
728:
709:
699:
685:
678:
677:
662:
652:
638:
631:
630:
627:
613:
610:
607:
600:
597:
594:
593:
574:
564:
550:
543:
542:
527:
517:
503:
496:
495:
492:
478:
475:
468:
465:
462:
461:
455:
452:
451:
448:
443:
442:
431:
428:
426:
425:Vapor quality
423:
421:
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1369:Heat engines
1341:
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1301:Motive power
1286:
946:Free entropy
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397:introduction
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291:Heat engines
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2136:superheated
2115:Bubbles of
1846:free energy
1772:supercooled
1741:Ising model
1260:Synergetics
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387:Temperature
248:Quasistatic
243:Isenthalpic
200:Instruments
190:Equilibrium
142:Open system
76:Equilibrium
58:Statistical
3102:Categories
2624:1874/12074
2219:References
2121:carbonated
1996:stochastic
1800:nucleation
1749:stochastic
1715:impurities
1649:nucleation
1572:Nucleation
1416:Scientists
1220:Philosophy
933:Potentials
296:Heat pumps
253:Polytropic
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2795:0021-9606
2668:0705.2051
2297:CiteSeerX
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1448:Clapeyron
1428:Boltzmann
1423:Bernoulli
1384:Education
1355:Timelines
1139:−
1084:−
872:Equations
839:∂
792:∂
743:α
707:∂
660:∂
614:−
608:β
572:∂
525:∂
233:Adiabatic
223:Isochoric
209:Processes
170:Ideal gas
53:Classical
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1824:barrier.
1701:crystals
1605:Category
1543:Thompson
1453:Clausius
1433:Bridgman
1287:Vis viva
1269:Theories
1203:Gas laws
995:Enthalpy
403:Pressure
218:Isobaric
175:Real gas
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3118:Physics
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2775:Bibcode
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2385:Bibcode
2319:4992555
2289:Bibcode
2141:wetting
2128:boiling
2123:liquid.
1849:barrier
1788:amyloid
1669:mixture
1528:Smeaton
1523:Rankine
1513:Onsager
1498:Maxwell
1493:Massieu
1198:Entropy
1193:General
1184:History
1174:Culture
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323:Note:
282:Cycles
111:Second
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