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| '''Critical radius''' is the minimum size that must be formed by [[atoms]] or molecules clustering together (in a gas, liquid or solid matrix) before a new-phase inclusion (a bubble, a droplet, or a solid particle) is stable and begins to grow. Formation of such stable "nuclei" is called [[nucleation]].
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| In precipitation models this is generally a prelude to models of the growth process itself. Sometimes precipitation is rate-limited by the nucleation process. This happens for example before one takes a cup of superheated water from a microwave and, when jiggling it against dust particles on the wall of the cup, enables "heterogeneous" nucleation that then rapidly converts much of that water into steam.
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| If the change in phase forms a crystalline solid in a liquid matrix, the atoms might then form a [[Dendrite (crystal)|dendrite]]. The [[crystal]] growth continues in three dimensions, the atoms attaching themselves in certain preferred directions, usually along the axes of a crystal, forming a characteristic tree-like structure of a dendrite.
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| Example: the critical radius for spheric-like dendride in an ideal system can be determined from its Gibbs free energy
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| <math>G=\frac{4 \pi}{3} r^3 G_v + 4 \pi r^2 \gamma</math> | |
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| where <math>G_v</math> is the Gibbs volume energy and <math>\gamma</math> is the interfacial energy. The critical radius <math>r_c</math> is found by setting the derivative of <math>G</math> equal to zero
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| <math>\frac{dG}{dr}=4\pi r_c^2 G_v+ 8 \pi r_c \gamma = 0</math>
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| yielding
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| <math>r_c = -\frac{2\gamma}{G_v}</math>,
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| where <math>\gamma</math> is the surface energy, and <math>G_v</math> is Gibbs energy per volume.
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| ==See also==
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| * [[Nucleation]]
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| * [[Homogeneous nucleation]]
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| * [[Heterogeneous nucleation]]
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| * [[Ostwald ripening]]
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| ==References==
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| * N.H.Fletcher, Size Effect in Heterogeneous Nucleation, J.Chem.Phys.29, 1958, 572.
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| [[Category:Critical phenomena]]
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| [[Category:Phase transitions]]
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| {{Physics-stub}}
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