Nucleation

Last updated
When sugar is supersaturated in water, nucleation will occur, allowing sugar molecules to stick together and form large crystal structures. Rock-Candy-Closeup.jpg
When sugar is supersaturated in water, nucleation will occur, allowing sugar molecules to stick together and form large crystal structures.

Nucleation is the first step in the formation of either a new thermodynamic phase or a new structure via self-assembly or self-organization. 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 water is cooled (at atmospheric pressure) below 0 °C, it will tend to freeze into ice, but volumes of water cooled only a few degrees below 0 °C often stay completely free of ice for long periods. At these conditions, nucleation of ice is either slow or does not occur at all. However, at lower temperatures ice crystals appear after little or no delay. At these conditions ice nucleation is fast. [1] [2] Nucleation is commonly how first-order phase transitions start, and then 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.

Contents

Nucleation is often found to be very sensitive to impurities 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 nucleation sites on surfaces in the system. [1] Homogeneous nucleation occurs away from a surface.

Characteristics

Nucleation at a surface (black) in the 2D Ising model. Up spins (particles in lattice-gas terminology) shown in red, down spins shown in white.

Nucleation is usually a stochastic (random) process, so even in two identical systems nucleation will occur at different times. [1] [2] [3] 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 classical nucleation theory. However, the CNT fails in describing experimental results of vapour to liquid nucleation even for model substances like argon by several orders of magnitude. [4]

For nucleation of a new thermodynamic phase, such as the formation of ice in water below 0 °C, if the system is not evolving with time and nucleation occurs in one step, then the probability that nucleation has not occurred should undergo exponential decay. This is seen for example in the nucleation of ice in supercooled small water droplets. [5] The decay rate of the exponential gives the nucleation rate. Classical nucleation theory 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 supersaturated. [1] [2]

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 amyloid aggregates associated with Alzheimer's disease also starts with nucleation. [6] Energy consuming self-organising systems such as the microtubules in cells also show nucleation and growth.

Heterogeneous nucleation often dominates homogeneous nucleation

Three nuclei on a surface, illustrating decreasing contact angles. The contact angle 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 classical nucleation theory 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 barrier. Surface tension.svg
Three nuclei on a surface, illustrating decreasing contact angles. The contact angle 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 classical nucleation theory 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 barrier.

Heterogeneous nucleation, nucleation with the nucleus at a surface, is much more common than homogeneous nucleation. [1] [3] 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 −35 °C, [1] [3] [5] whereas water that contains impurities may freeze at −5 °C or warmer. [1]

This observation that heterogeneous nucleation can occur when the rate of homogeneous nucleation is essentially zero, is often understood using classical nucleation theory. This predicts that the nucleation slows exponentially with the height of a free energy barrier 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 . This reduction in surface area of the nucleus reduces the height of the barrier to nucleation and so speeds nucleation up exponentially. [2]

Nucleation can also start at the surface of a liquid. For example, computer simulations of gold nanoparticles show that the crystal phase nucleates at the liquid-gold surface. [7]

Computer simulation studies of simple models

Classical nucleation theory 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 always clear that we can treat something so small as a volume plus a surface. Also nucleation is an inherently out of thermodynamic equilibrium phenomenon so it is not always obvious that its rate can be estimated using equilibrium properties.

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 thermal motion, and is a simple model of some colloids. For the crystallization of hard spheres the classical theory is a very reasonable approximate theory. [8] 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.

The spinodal region

Phase-transition processes can also be explained in terms of spinodal decomposition, 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. [9] 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 Cahn–Hilliard equation.

The nucleation of crystals

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. 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 °C. [1]

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. [3] Visualization of the initial stage of crystal nucleation of sodium chloride was achieved through atomic-resolution real-time video imaging. [10] Calcium carbonate crystal nucleation depends not only on degree of supersaturation but also the ratio of calcium to carbonate ions in aqueous solutions. [11] In larger volumes many nucleation events will occur. A simple model for crystallisation in that case, that combines nucleation and growth is the KJMA or Avrami model.

Primary and secondary nucleation

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 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. [12]

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 their mechanisms are very different, and secondary nucleation relies on crystals already being present.

Experimental observations on the nucleation times for the crystallisation of small volumes

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 growth of the nucleated phase. These problems can be overcome by working with small droplets. As nucleation is stochastic, many droplets are needed so that statistics for the nucleation events can be obtained.

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 data. Fraction of a set of supercooled liquid tin droplets that have not frozen, as a function of time(png).png
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 data.

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. [13]

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 Gompertz function to the data. This is a simplified version of the model Pound and La Mer used to model their data. [13] 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 also assumes that these particles are Poisson distributed 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 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.

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.

Ice

The freezing of small water droplets to ice is an important process, particularly in the formation and dynamics of clouds. [1] Water (at atmospheric pressure) does not freeze at 0 °C, but rather at temperatures that tend to decrease as the volume of the water decreases and as the water impurity increases. [1]

Survival curve for water droplets 34.5 mm in diameter. Blue circles are data, and the red curve is a fit of a Gumbel distribution. Survival curve 34.5 micrometre water droplets 1950 NACA Dorsch & Hacker ice nucleation.png
Survival curve for water droplets 34.5 μm in diameter. Blue circles are data, and the red curve is a fit of a Gumbel distribution.

Thus small droplets of water, as found in clouds, may remain liquid far below 0 °C.

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 droplets freezes is close to -19 °C, while the last droplet to freeze does so at almost -35 °C. [14]

Examples

Examples of the nucleation of fluids (gases and liquids)

Nucleation of carbon dioxide bubbles around a finger Nucleation finger.jpg
Nucleation of carbon dioxide bubbles around a finger

Examples of the nucleation of crystals

Nucleation in solids

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 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.

Nucleation of failures in networks

It was found that in interdependent spatial networks (like infrastructures), a localized failure above a critical radius can propagate like nucleation and the system will collapse. [18] [19]

Related Research Articles

Frost Coating or deposit of ice

Frost is a thin layer of ice on a solid surface, which forms from water vapor in an above-freezing atmosphere coming in contact with a solid surface whose temperature is below freezing, and resulting in a phase change from water vapor to ice as the water vapor reaches the freezing point. In temperate climates, it most commonly appears on surfaces near the ground as fragile white crystals; in cold climates, it occurs in a greater variety of forms. The propagation of crystal formation occurs by the process of nucleation.

Ice Frozen water: the solid state of water

Ice is water frozen into a solid state. Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaque bluish-white color.

Freezing phase transition in which a liquid turns into a solid due to a decrease in thermal energy

Freezing is a phase transition where a liquid turns into a solid when its temperature is lowered below its freezing point. In accordance with the internationally established definition, freezing means the solidification phase change of a liquid or the liquid content of a substance, usually due to cooling.

Supercooling Lowering the temperature of a liquid or gas below freezing without its becoming a solid

Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid. It achieves this in the absence of a seed crystal or nucleus around which a crystal structure can form. The supercooling of water can be achieved without any special techniques other than chemical demineralization, down to −48.3 °C (−55 °F). Droplets of supercooled water often exist in stratus and cumulus clouds. An aircraft flying through such a cloud sees an abrupt crystallization of these droplets, which can result in the formation of ice on the aircraft's wings or blockage of its instruments and probes.

In physics and chemistry, flash freezing is the process whereby objects are frozen in just a few hours by subjecting them to cryogenic temperatures, or through direct contact with liquid nitrogen at −196 °C (−320.8 °F). It is commonly used in the food industry.

Cloud physics Study of the physical processes in atmospheric clouds

Cloud physics is the study of the physical processes that lead to the formation, growth and precipitation of atmospheric clouds. These aerosols are found in the troposphere, stratosphere, and mesosphere, which collectively make up the greatest part of the homosphere. Clouds consist of microscopic droplets of liquid water, tiny crystals of ice, or both. Cloud droplets initially form by the condensation of water vapor onto condensation nuclei when the supersaturation of air exceeds a critical value according to Köhler theory. Cloud condensation nuclei are necessary for cloud droplets formation because of the Kelvin effect, which describes the change in saturation vapor pressure due to a curved surface. At small radii, the amount of supersaturation needed for condensation to occur is so large, that it does not happen naturally. Raoult's law describes how the vapor pressure is dependent on the amount of solute in a solution. At high concentrations, when the cloud droplets are small, the supersaturation required is smaller than without the presence of a nucleus.

Crystallization Process by which a solid with a highly organised atomic or molecular structure forms

Crystallization or crystallisation is the process by which a solid forms, where the atoms or molecules are highly organized into a structure known as a crystal. Some of the ways by which crystals form are precipitating from a solution, freezing, or more rarely deposition directly from a gas. Attributes of the resulting crystal depend largely on factors such as temperature, air pressure, and in the case of liquid crystals, time of fluid evaporation.

Rime ice

Rime ice forms when supercooled water liquid droplets freeze onto surfaces. Meteorologists distinguish between three basic types of ice forming on vertical and horizontal surfaces by deposition of supercooled water droplets. There are also intermediate formations.

Impurities are chemical substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound. Firstly, a pure chemical should appear thermodynamically in at least one chemical phase and can also be characterized by its one-component-phase diagram. Secondly, practically speaking, a pure chemical should prove to be homogeneous. The perfect pure chemical will pass all attempts and tests of further separation and purification. Thirdly, and here we focus on the common chemical definition, it should not contain any trace of any other kind of chemical species. In reality, there are no absolutely 100% pure chemical compounds, as there is always some minute contamination. Indeed, as detection limits in analytical chemistry decrease, the number of impurities detected tends to increase.

The Wegener–Bergeron–Findeisen process, is a process of ice crystal growth that occurs in mixed phase clouds in regions where the ambient vapor pressure falls between the saturation vapor pressure over water and the lower saturation vapor pressure over ice. This is a subsaturated environment for liquid water but a supersaturated environment for ice resulting in rapid evaporation of liquid water and rapid ice crystal growth through vapor deposition. If the number density of ice is small compared to liquid water, the ice crystals can grow large enough to fall out of the cloud, melting into rain drops if lower level temperatures are warm enough.

Critical radius is the minimum particle size from which an aggregate is thermodynamically stable. In other words, it is the lowest radius formed by atoms or molecules clustering together before a new phase inclusion is viable and begins to grow. Formation of such stable nuclei is called nucleation.

Atmospheric icing

Atmospheric icing occurs in the atmosphere when water droplets freeze on objects they come in contact with. Icing conditions can be particularly dangerous to aircraft, as the built-up ice changes the aerodynamics of the flight surfaces, which can increase the risk of a stall. For this reason, on-board ice protection systems have been developed, and aircraft are often deiced prior to take-off in icy environments.

Crystal growth Major stage of a crystallization process

A crystal is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. Crystal growth is a major stage of a crystallization process, and consists in the addition of new atoms, ions, or polymer strings into the characteristic arrangement of the crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous nucleation, unless a "seed" crystal, purposely added to start the growth, was already present.

Ostwald ripening Process by which small crystals dissolve in solution for the benefit of larger crystals

Ostwald ripening is a phenomenon observed in solid solutions or liquid sols that describes the change of an inhomogeneous structure over time, i.e., small crystals or sol particles dissolve, and redeposit onto larger crystals or sol particles.

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

Insect winter ecology describes the overwinter survival strategies of insects, which are in many respects more similar to those of plants than to many other animals, such as mammals and birds. Unlike those animals, which can generate their own heat internally (endothermic), insects must rely on external sources to provide their heat (ectothermic). Thus, insects persisting in winter weather must tolerate freezing or rely on other mechanisms to avoid freezing. Loss of enzymatic function and eventual freezing due to low temperatures daily threatens the livelihood of these organisms during winter. Not surprisingly, insects have evolved a number of strategies to deal with the rigors of winter temperatures in places where they would otherwise not survive.

Snowflake Single ice crystal or an aggregation of ice crystals which falls through the Earths atmosphere

A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, then falls through the Earth's atmosphere as snow. Each flake nucleates around a dust particle in supersaturated air masses by attracting supercooled cloud water droplets, which freeze and accrete in crystal form. Complex shapes emerge as the flake moves through differing temperature and humidity zones in the atmosphere, such that individual snowflakes differ in detail from one another, but may be categorized in eight broad classifications and at least 80 individual variants. The main constituent shapes for ice crystals, from which combinations may occur, are needle, column, plate, and rime. Snow appears white in color despite being made of clear ice. This is due to diffuse reflection of the whole spectrum of light by the small crystal facets of the snowflakes.

Ice nucleus

An ice nucleus, also known as an ice nucleating particle (INP), is a particle which acts as the nucleus for the formation of an ice crystal in the atmosphere.

In thermodynamics, explosive boiling or phase explosion is a method whereby a superheated metastable liquid undergoes an explosive liquid-vapor phase transition into a stable two-phase state because of a massive homogeneous nucleation of vapor bubbles. This concept was pioneered by M. M. Martynyuk in 1976 and then later advanced by Fucke and Seydel.

Classical nucleation theory (CNT) is the most common theoretical model used to quantitatively study the kinetics of nucleation.

References

  1. 1 2 3 4 5 6 7 8 9 10 11 12 H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, Kluwer (1997).
  2. 1 2 3 4 Sear, R.P. (2007). "Nucleation: theory and applications to protein solutions and colloidal suspensions" (PDF). Journal of Physics: Condensed Matter. 19 (3): 033101. Bibcode:2007JPCM...19c3101S. CiteSeerX   10.1.1.605.2550 . doi:10.1088/0953-8984/19/3/033101.
  3. 1 2 3 4 Sear, Richard P. (2014). "Quantitative Studies of Crystal Nucleation at Constant Supersaturation: Experimental Data and Models". CrystEngComm. 16 (29): 6506–6522. doi: 10.1039/C4CE00344F .
  4. A. Fladerer, R. Strey: "Homogeneous nucleation and droplet growth in supersaturated argon vapor: The cryogenic nucleation pulse chamber". The Journal of Chemical Physics 124(16), 164710 (2006). doi : 10.1063/1.2186327.
  5. 1 2 Duft, D.; Leisner (2004). "Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets". Atmospheric Chemistry and Physics. 4 (7): 1997. Bibcode:2004ACP.....4.1997D. doi: 10.5194/acp-4-1997-2004 .
  6. Gillam, J.E.; MacPhee, C.E. (2013). "Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth". Journal of Physics: Condensed Matter. 25 (37): 373101. Bibcode:2013JPCM...25K3101G. doi:10.1088/0953-8984/25/37/373101. PMID   23941964.
  7. Mendez-Villuendas, Eduardo; Bowles, Richard (2007). "Surface Nucleation in the Freezing of Gold Nanoparticles". Physical Review Letters. 98 (18): 185503. arXiv: cond-mat/0702605 . Bibcode:2007PhRvL..98r5503M. doi:10.1103/PhysRevLett.98.185503. PMID   17501584. S2CID   7037979.
  8. Auer, S.; D. Frenkel (2004). "Numerical prediction of absolute crystallization rates in hard-sphere colloids" (PDF). The Journal of Chemical Physics. 120 (6): 3015–29. Bibcode:2004JChPh.120.3015A. doi:10.1063/1.1638740. hdl:1874/12074. PMID   15268449.
  9. Mendez-Villuendas, Eduardo; Saika-Voivod, Ivan; Bowles, Richard K. (2007). "A limit of stability in supercooled liquid clusters". The Journal of Chemical Physics. 127 (15): 154703. arXiv: 0705.2051 . Bibcode:2007JChPh.127o4703M. doi:10.1063/1.2779875. PMID   17949187. S2CID   9762506.
  10. Nakamuro, Takayuki; Sakakibara, Masaya; Nada, Hiroki; Harano, Koji; Nakamura, Eiichi (2021). "Capturing the Moment of Emergence of Crystal Nucleus from Disorder". Journal of the American Chemical Society. 143 (4): 1763–1767. doi: 10.1021/jacs.0c12100 . PMID   33475359.
  11. 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", Crystal Growth & Design, 21 (3): 1576–1590, doi: 10.1021/acs.cgd.0c01403 , PMC   7976603 , PMID   33762898 CS1 maint: date and year (link)
  12. Botsaris, GD (1976). "Secondary Nucleation — A Review". In Mullin, J (ed.). Industrial Crystallization . Springer. pp.  3–22. doi:10.1007/978-1-4615-7258-9_1. ISBN   978-1-4615-7260-2.
  13. 1 2 Pound, Guy M.; V. K. La Mer (1952). "Kinetics of Crystalline Nucleus Formation in Supercooled Liquid Tin". Journal of the American Chemical Society. 74 (9): 2323. doi:10.1021/ja01129a044.
  14. Dorsch, Robert G; Hacker, Paul T (1950). "Photomicrographic Investigation of Spontaneous Freezing Temperatures of Supercooled Water Droplets". NACA Technical Note. 2142.
  15. Kelton, Ken; Greer, Alan Lindsay (2010). Nucleation in Condensed Matter: Applications in Materials and Biology . Amsterdam: Elsevier Science & Technology. ISBN   9780080421476.
  16. Palmans, Roger; Frank, Arthur J. (1991). "A molecular water-reduction catalyst: Surface derivatization of titania colloids and suspensions with a platinum complex". The Journal of Physical Chemistry. 95 (23): 9438. doi:10.1021/j100176a075.
  17. Rajh, Tijana; Micic, Olga I.; Nozik, Arthur J. (1993). "Synthesis and characterization of surface-modified colloidal cadmium telluride quantum dots". The Journal of Physical Chemistry. 97 (46): 11999. doi:10.1021/j100148a026.
  18. Y. Berezin, A. Bashan, M.M. Danziger, D. Li, S. Havlin (2015). "Localized attacks on spatially embedded networks with dependencies". Scientific Reports. 5: 8934. Bibcode:2015NatSR...5E8934B. doi:10.1038/srep08934. PMC   4355725 . PMID   25757572.CS1 maint: multiple names: authors list (link)
  19. D Vaknin, MM Danziger, S Havlin (2017). "Spreading of localized attacks in spatial multiplex networks". New J. Phys. 19 (7): 073037. arXiv: 1704.00267 . Bibcode:2017NJPh...19g3037V. doi:10.1088/1367-2630/aa7b09. S2CID   9121930.CS1 maint: multiple names: authors list (link)