Hoffman nucleation theory

Last updated December 05, 2021

Hoffman nucleation theory is a theory developed by John D. Hoffman and coworkers in the 1970s and 80s that attempts to describe the crystallization of a polymer in terms of the kinetics and thermodynamics of polymer surface nucleation.^[1] The theory introduces a model where a surface of completely crystalline polymer is created and introduces surface energy parameters to describe the process. Hoffman nucleation theory is more of a starting point for polymer crystallization theory and is better known for its fundamental roles in the Hoffman–Weeks lamellar thickening and Lauritzen–Hoffman growth theory.

Polymer morphology

Polymers contain different morphologies on the molecular level which give rise to their macro properties. Long range disorder in the polymer chain is representative of amorphous solids, and the chain segments are considered amorphous. Long range polymer order is similar to crystalline material, and chain segments are considered crystalline.

The thermal characteristics of polymers are fundamentally different from those of most solid materials. Solid materials typically have one melting point, the T_m, above which the material loses internal molecular ordering and becomes a liquid. Polymers have both a melting temperature T_m and a glass transition temperature T_g. Above the T_m, the polymer chains lose their molecular ordering and exhibit reptation, or mobility. Below the T_m, but still above the T_g, the polymer chains lose some of their long-range mobility and can form either crystalline or amorphous regions. In this temperature range, as the temperature decreases, amorphous regions can transition into crystalline regions, causing the bulk material to become more crystalline over all. Below the T_g, molecular motion is stopped and the polymer chains are essentially frozen in place. In this temperature range, amorphous regions can no longer transition into crystalline regions, and the polymer as a whole has reached its maximum crystallinity.

Hoffman nucleation theory addresses the amorphous to crystalline polymer transition, and this transition can only occur in the temperature range between the T_m and T_g. The transition from an amorphous to a crystalline single polymer chain is related to the random thermal energy required to align and fold sections of the chain to form ordered regions titled lamellae, which are a subset of even bigger structures called spherulites. The crystallization of polymers can be brought about by several different methods, and is a complex topic in itself.

Nucleation

Nucleation is the formation and growth of a new phase with or without the presence of external surface. The presence of this surface results in heterogeneous nucleation whereas in its absence homogeneous nucleation occurs. Heterogeneous nucleation occurs in cases where there are pre-existing nuclei present, such as tiny dust particles suspended in a liquid or gas or reacting with a glass surface containing SiO₂ . For the process of Hoffman nucleation and its progression to Lauritzen–Hoffman growth theory, homogeneous nucleation is the main focus. Homogeneous nucleation occurs where no such contaminants are present and is less commonly seen. Homogeneous nucleation begins with small clusters of molecules forming from one phase to the next. As the clusters grow, they aggregate through the condensation of other molecules. The size continues to increase and ultimately form macroscopic droplets (or bubbles depending on the system).

Nucleation is often described mathematically through the change in Gibbs free energy of n moles of vapor at vapor pressure P that condenses into a drop. Also the nucleation barrier, in polymer crystallization, consists of both enthalpic and entropic components that must be over come. This barrier consists of selection processes taking place in different length and time scales which relates to the multiple regimes later on.^[2] This barrier is the free energy required to overcome in order to form nuclei. It is the formation of the nuclei from the bulk to a surface that is the interfacial free energy. The interfacial free energy is always a positive term and acts to destabilize the nucleus allowing the continuation of the growing polymer chain. The nucleation continues as a favorable reaction.

Thermodynamics of polymer crystallization

The Lauritzen–Hoffman plot (right) models the three different regimes when (logG) + U*/k(T-T₀) is plotted against (TΔT)⁻¹.^[3] It can be used to describe the rate at which secondary nucleation competes with lateral addition at the growth front among the different temperatures. This theory can be used to help understand the preferences of nucleation and growth based on the polymer's properties including its standard melting temperature.

Lamellar thickening (Hoffman–Weeks plot)

For many polymers, the change between the initial lamellar thickness at T_c is roughly the same as at T_m and can thus be modeled by the Gibbs–Thomson equation fairly well. However, since it implies that the lamellar thickness over the given supercooling range (T_m–T_c) is unchanged, and many homogeneous nucleation of polymers implies a change of thickness at the growth front, Hoffman and Weeks pursued a more accurate representation.^[4] In this regard, the Hoffman-Weeks plot was created and can be modeled through the equation

$T_{\text{m}}={T_{\text{c}} \over \beta }+(1-{1 \over \beta })T_{\text{m}}^{\circ }$

where β is representative of a thickening factor given by L = L₀ β and T_cand T_m are the crystallization and melting temperatures, respectively.

Applying this experimentally for a constant β allows for the determination of the equilibrium melting temperature, T_m° at the intersection of T_cand T_m.^[3]

Kinetics of polymer crystallization

The crystallization process of polymers does not always obey simple chemical rate equations. Polymers can crystallize through a variety of different regimes and unlike simple molecules, the polymer crystal lamellae have two very different surfaces. The two most prominent theories in polymer crystallization kinetics are the Avrami equation and Lauritzen–Hoffman growth theory.^[5]

Lauritzen–Hoffman growth theory

The Lauritzen–Hoffman growth theory breaks the kinetics of polymer crystallization into ultimately two rates. The model breaks down into the addition of monomers onto a growing surface. This initial step is generally associated with the nucleation of the polymer. From there, the kinetics become the rate which the polymer grows on the surface, or the lateral growth rate, in comparison with the growth rate onto the polymer extending the chain, the secondary nucleation rate. These two rates can result in three situations.^[6]

Three regimes of crystallization kinetics

For Regime I, the growth rate on the front laterally, referred to as g, is the rate-determining step (RDS) and exceeds the secondary nucleation rate, i. In this instance of g >> i, monolayers are formed one at a time so that if the substrate has a length of L_p and thickness, b, the overall linear growth can be described through the equation

$G_{\text{I}}=biL_{\text{p}}$

and the rate of nucleation in specific can further be described by

$G_{\text{I,n}}=e^{-(K_{\text{g}}/T\Delta T)}\,$

with K_g equal to

$K_{\text{g}}={4b\sigma _{\text{l}}\sigma _{\text{f}}T_{m}^{0} \over k\Delta h}$

where

σ_l is the lateral/lamellae surface free energy per unit area
σ_f is the fold surface free energy per unit area
T_m⁰ is the equilibrium melting temperature
k is equal to Boltzmann constant
Δh is equal to the change of enthalpy of fusion (or latent heat of fusion) per repeat unit at the standard temperature^[3]

This shows that in Region I, lateral nucleation along the front successfully dominates at temperatures close to the melting temperature, however at more extreme temperatures other forces such as diffusion can impact nucleation rates.

In Regime II, the lateral growth rate is either comparable or smaller than the nucleation rate g ≤ i, which causes secondary (or more) layers to form before the initial layer has been covered. This allows the linear growth rate to be modeled by

$G_{\text{II}}=b{\sqrt {ig}}$

Using the assumption that g and i are independent of time, the rate at which new layers are formed can be approximated and the rate of nucleation in regime II can be expressed as

$G_{\text{II,n}}=e^{-(K_{\text{g}}^{\prime }/T\Delta T)}\,$

with K_g^' equal to about 1/2 of the K_g from Regime I,

$K_{\text{g}}^{\prime }={2b\sigma _{\text{l}}\sigma _{\text{f}}T_{m}^{0} \over k\Delta h}$

Lastly, Regime III in the L-H model depicts the scenario where lateral growth is inconsequential to the overall rate, since the nucleation of multiple sites causes i >> g. This means that the growth rate can be modeled by the same equation as Regime I,

$G_{\text{III}}=biL_{\text{p}}=G_{\text{III}}^{\circ }e^{{-U^{*}/k(T-T_{\text{0}})}-(K_{\text{g}}/T\Delta T)}$

where G_III^° is the prefactor for Regime III and can be experimentally determined through applying the Lauritzen–Hoffman Plot.^[7]

Polyethylene crystallization kinetics

A reza's crystallization depends on the time it takes for layers of its chains to fold and orient themselves in the same direction. This time increases with a molecule's weight and branching.^[8] The table below shows that the growth rate is higher for Sclair 14B.1 than Sclair 2907 (20%), where 2907 is less highly branched than 14B.1.^[8] Here Gc is the crystal growth rate, or how quickly it orders itself depending on the layers, and t is the time it takes to order.

Polymer	Growth Temp (°C)	G_c (μm*min⁻¹)	t (ms)
Sclair 2907 (20%)	119	3.5-6.8	4.4-8.6
Sclair 14B.1	119	~0.2	~150

Further testing and applications

Many additional tests have since been run to apply and compare Hoffman's principles to reality. Among the experiments done, some of the more notable secondary nucleation tests are briefly explained in the table below.

Secondary nucleation testing	Experimental results observed
Potassium chloride (KCl)	Secondary nuclei form at a rate proportional to the degree of supercooling (above certain levels of agitation) and achieve the same amount of nucleation regardless of the shape of the parent crystal. This is due to the substantially greater effect of secondary nucleation over primary nucleation of the original crystal. This was proven through both temperature and shape dependent nuclei stimulated growth experiments to confirm that in cases of secondary nucleation only the degree and temperature of supercooling change the nucleation rate, whereas the parent crystal only serves to act as a catalytic initiator of the process.^[9]
Isotactic poly (vinylcyclohexane) (PVCH)	PVCH crystals were experimentally shown to increase their spread and lateral growth at high temperatures indicating that although they were feasibly unable to reach Regime III temperatures, extrapolations and hypotheses from the experiment point to confirmation of the expected behavior in each of the three regimes. Experiments concluded that additional growth mechanisms such as crystal twinning and twin boundary interactions can alter the traditional LH theory, but further research is needed to model each individual influence.^[10]
Zinc oxide (ZnO)	Zinc oxide crystals were proven to undergo secondary nucleation under an odd mixture of conditions including the addition of a diamine as well as surface etching. Overall, testing has shown that the morphology of the secondary crystals can fluctuate greatly depending on the amount of diamine added, due to its ability to deplete the substrate and hinder the growth prematurely.^[11]

Related Research Articles

Melting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which increases the substance's temperature to the melting point. At the melting point, the ordering of ions or molecules in the solid breaks down to a less ordered state, and the solid "melts" to become a liquid.

A polymer is a substance or material consisting of very large molecules, or macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass, relative to small molecule compounds, produces unique physical properties including toughness, high elasticity, viscoelasticity, and a tendency to form amorphous and semicrystalline structures rather than crystals.

Differential scanning calorimetry (DSC) is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Both the sample and reference are maintained at nearly the same temperature throughout the experiment. Generally, the temperature program for a DSC analysis is designed such that the sample holder temperature increases linearly as a function of time. The reference sample should have a well-defined heat capacity over the range of temperatures to be scanned.

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.

Polypropylene (PP), also known as polypropene, is a thermoplastic polymer used in a wide variety of applications. It is produced via chain-growth polymerization from the monomer propylene.

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.

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.

Nucleation Step of self-assembly, including crystallization

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 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. 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.

Amorphous ice is an amorphous solid form of water. Common ice is a crystalline material wherein the molecules are regularly arranged in a hexagonal lattice, whereas amorphous ice has a lack of long-range order in its molecular arrangement. Amorphous ice is produced either by rapid cooling of liquid water, or by compressing ordinary ice at low temperatures.

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 of 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.

Hot-melt adhesive Glue applied by heating

Hot melt adhesive (HMA), also known as hot glue, is a form of thermoplastic adhesive that is commonly sold as solid cylindrical sticks of various diameters designed to be applied using a hot glue gun. The gun uses a continuous-duty heating element to melt the plastic glue, which the user pushes through the gun either with a mechanical trigger mechanism on the gun, or with direct finger pressure. The glue squeezed out of the heated nozzle is initially hot enough to burn and even blister skin. The glue is sticky when hot, and solidifies in a few seconds to one minute. Hot melt adhesives can also be applied by dipping or spraying, and are popular with hobbyists and crafters both for affixing and as an inexpensive alternative to resin casting.

GeSbTe (germanium-antimony-tellurium or GST) is a phase-change material from the group of chalcogenide glasses used in rewritable optical discs and phase-change memory applications. Its recrystallization time is 20 nanoseconds, allowing bitrates of up to 35 Mbit/s to be written and direct overwrite capability up to 10⁶ cycles. It is suitable for land-groove recording formats. It is often used in rewritable DVDs. New phase-change memories are possible using n-doped GeSbTe semiconductor. The melting point of the alloy is about 600 °C (900 K) and the crystallization temperature is between 100 and 150 °C.

Germanium telluride (GeTe) is a chemical compound of germanium and tellurium and is a component of chalcogenide glasses. It shows semimetallic conduction and ferroelectric behaviour.

The Avrami equation describes how solids transform from one phase to another at constant temperature. It can specifically describe the kinetics of crystallisation, can be applied generally to other changes of phase in materials, like chemical reaction rates, and can even be meaningful in analyses of ecological systems.

In polymer physics, spherulites are spherical semicrystalline regions inside non-branched linear polymers. Their formation is associated with crystallization of polymers from the melt and is controlled by several parameters such as the number of nucleation sites, structure of the polymer molecules, cooling rate, etc. Depending on those parameters, spherulite diameter may vary in a wide range from a few micrometers to millimeters. Spherulites are composed of highly ordered lamellae, which result in higher density, hardness, but also brittleness when compared to disordered regions in a polymer. The lamellae are connected by amorphous regions which provide elasticity and impact resistance. Alignment of the polymer molecules within the lamellae results in birefringence producing a variety of colored patterns, including a Maltese cross, when spherulites are viewed between crossed polarizers in an optical microscope.

Environmental Stress Cracking (ESC) is one of the most common causes of unexpected brittle failure of thermoplastic polymers known at present. According to ASTM D883, stress cracking is defined as "an external or internal crack in a plastic caused by tensile stresses less than its short-term mechanical strength". This type of cracking typically involves brittle cracking, with little or no ductile drawing of the material from its adjacent failure surfaces. Environmental stress cracking may account for around 15-30% of all plastic component failures in service. This behavior is especially prevalent in glassy, amorphous thermoplastics. Amorphous polymers exhibit ESC because of their loose structure which makes it easier for the fluid to permeate into the polymer. Amorphous polymers are more prone to ESC at temperature higher than their glass transition temperature (T_g) due to the increased free volume. When T_g is approached, more fluid can permeate into the polymer chains.

The glass–liquid transition, or glass transition, is the gradual and reversible transition in amorphous materials from a hard and relatively brittle "glassy" state into a viscous or rubbery state as the temperature is increased. An amorphous solid that exhibits a glass transition is called a glass. The reverse transition, achieved by supercooling a viscous liquid into the glass state, is called vitrification.

Crystallization of polymers is a process associated with partial alignment of their molecular chains. These chains fold together and form ordered regions called lamellae, which compose larger spheroidal structures named spherulites. Polymers can crystallize upon cooling from melting, mechanical stretching or solvent evaporation. Crystallization affects optical, mechanical, thermal and chemical properties of the polymer. The degree of crystallinity is estimated by different analytical methods and it typically ranges between 10 and 80%, with crystallized polymers often called "semi-crystalline". The properties of semi-crystalline polymers are determined not only by the degree of crystallinity, but also by the size and orientation of the molecular chains.

Polymer fracture is the study of the fracture surface of an already failed material to determine the method of crack formation and extension in polymers both fiber reinforced and otherwise. Failure in polymer components can occur at relatively low stress levels, far below the tensile strength because of four major reasons: long term stress or creep rupture, cyclic stresses or fatigue, the presence of structural flaws and stress-cracking agents. Formations of submicroscopic cracks in polymers under load have been studied by x ray scattering techniques and the main regularities of crack formation under different loading conditions have been analyzed. The low strength of polymers compared to theoretically predicted values are mainly due to the many microscopic imperfections found in the material. These defects namely dislocations, crystalline boundaries, amorphous interlayers and block structure can all lead to the non-uniform distribution of mechanical stress.

References

↑ N. B. Hannay (1976). "7". Treatise on Solid State Chemistry. 3. Plenum Press. doi:10.1002/pol.1977.130150310.
↑ Cheng, Stephen; Lotz, Bernard (2005). "Enthalpic and entropic origins of nucleation barriers during polymer crystallization: the Hoffman–Lauritzen theory and beyond". Polymer. 46 (20): 8662–8681. doi:10.1016/j.polymer.2005.03.125.
1 2 3 Muthukumar, M (2004). "Nucleation in Polymer Crystallization". Advances in Chemical Physics. 128. ISBN 0-471-44528-2.
↑ Marand, Herve; Xu, Jiannong; Srinivas, Srivatsan (1998). "Determination of the Equilibrium Melting Temperature of Polymer Crystals: Linear and Nonlinear Hoffman-Weeks Extrapolations". Macromolecules. 31 (23): 8219–8229. Bibcode:1998MaMol..31.8219M. doi:10.1021/ma980747y.
↑ Paul C. Painter, Michael M. Coleman (1997). "8". Fundamentals of Polymer Science An Introductory Text, Second Edition. CRC Press.
↑ Snyder, Chad R.; Marand, Herve; Mansfield, Marc L. (1996). "Lateral Substrate Completion Rate in the Lauritzen-Hoffman Secondary Surface Nucleation Theory: Nature of the Friction Coefficient". Macromolecules. 29 (23): 7508–7513. Bibcode:1996MaMol..29.7508S. doi:10.1021/ma960589f.
↑ Snyder, Chad R.; Marand, Herve (1997). "Effect of Chain Transport in the Secondary Surface Nucleation Based Flux Theory and in the Lauritzen-Hoffman Crystal Growth Rate Formalism". Macromolecules. 30 (9): 2759–2766. Bibcode:1997MaMol..30.2759S. doi:10.1021/ma961633u.
1 2 el Maaty, M.I. Abo; Bassett, D. C. (4 October 2006). "On the time for fold surfaces to order during the crystallization of polyethylene from the melt and its dependence on molecular parameters". Polymer. 47 (21): 7469–7476. doi:10.1016/j.polymer.2006.08.015.
↑ Melia, T.P.; Moffitt, W.P. (1964). "Secondary Nucleation from Aqueous Solution". Industrial & Engineering Chemistry Fundamentals. 3 (4): 314–317. doi:10.1021/i160012a006.
↑ Alcazar, Daniel; Thierry, Annette; Schultz, Patrick; Kawaguchi, Akiyoshi; Cheng, Stephen Z. D.; Lotz, Bernard (2006). "Determination of the Extent of Lateral Spread and Density of Secondary Nucleation in Polymer Single Crystal Growth". Macromolecules. 39 (26): 9120–9131. Bibcode:2006MaMol..39.9120A. doi:10.1021/ma061697x.
↑ Sounart, Thomas L.; Liu, Jun; Voigt, James A.; Huo, Mae; Spoerke, Erik D.; McKenzie, Bonnie (2007). "Secondary Nucleation and Growth of ZnO". J. Am. Chem. Soc. 129 (51): 15786–15793. doi:10.1021/ja071209g.

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[1] N. B. Hannay (1976). "7". Treatise on Solid State Chemistry. 3. Plenum Press. doi:10.1002/pol.1977.130150310.

[2] Cheng, Stephen; Lotz, Bernard (2005). "Enthalpic and entropic origins of nucleation barriers during polymer crystallization: the Hoffman–Lauritzen theory and beyond". Polymer. 46 (20): 8662–8681. doi:10.1016/j.polymer.2005.03.125.

[LHtheory-3] 1 2 3 Muthukumar, M (2004). "Nucleation in Polymer Crystallization". Advances in Chemical Physics. 128. ISBN 0-471-44528-2.

[HWplot-4] Marand, Herve; Xu, Jiannong; Srinivas, Srivatsan (1998). "Determination of the Equilibrium Melting Temperature of Polymer Crystals: Linear and Nonlinear Hoffman-Weeks Extrapolations". Macromolecules. 31 (23): 8219–8229. Bibcode:1998MaMol..31.8219M. doi:10.1021/ma980747y.

[5] Paul C. Painter, Michael M. Coleman (1997). "8". Fundamentals of Polymer Science An Introductory Text, Second Edition. CRC Press.

[LHFriction-6] Snyder, Chad R.; Marand, Herve; Mansfield, Marc L. (1996). "Lateral Substrate Completion Rate in the Lauritzen-Hoffman Secondary Surface Nucleation Theory: Nature of the Friction Coefficient". Macromolecules. 29 (23): 7508–7513. Bibcode:1996MaMol..29.7508S. doi:10.1021/ma960589f.

[SSnucleation-7] Snyder, Chad R.; Marand, Herve (1997). "Effect of Chain Transport in the Secondary Surface Nucleation Based Flux Theory and in the Lauritzen-Hoffman Crystal Growth Rate Formalism". Macromolecules. 30 (9): 2759–2766. Bibcode:1997MaMol..30.2759S. doi:10.1021/ma961633u.

[ReferenceA-8] 1 2 el Maaty, M.I. Abo; Bassett, D. C. (4 October 2006). "On the time for fold surfaces to order during the crystallization of polyethylene from the melt and its dependence on molecular parameters". Polymer. 47 (21): 7469–7476. doi:10.1016/j.polymer.2006.08.015.

[KCLSupercooling-9] Melia, T.P.; Moffitt, W.P. (1964). "Secondary Nucleation from Aqueous Solution". Industrial & Engineering Chemistry Fundamentals. 3 (4): 314–317. doi:10.1021/i160012a006.

[Lateralgrowth-10] Alcazar, Daniel; Thierry, Annette; Schultz, Patrick; Kawaguchi, Akiyoshi; Cheng, Stephen Z. D.; Lotz, Bernard (2006). "Determination of the Extent of Lateral Spread and Density of Secondary Nucleation in Polymer Single Crystal Growth". Macromolecules. 39 (26): 9120–9131. Bibcode:2006MaMol..39.9120A. doi:10.1021/ma061697x.

[11] Sounart, Thomas L.; Liu, Jun; Voigt, James A.; Huo, Mae; Spoerke, Erik D.; McKenzie, Bonnie (2007). "Secondary Nucleation and Growth of ZnO". J. Am. Chem. Soc. 129 (51): 15786–15793. doi:10.1021/ja071209g.

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