Nucleation and Growth of Crystals

Nucleation and Growth Rates Control Rc

• Nucleation, the first step…• First process is for microscopic clusters (nuclei) of atoms or ions to form

– Nuclei possess the beginnings of the structure of the crystal– Only limited diffusion is necessary– Thermodynamic driving force for crystallization must be present

NUCLEATION water as example

• initiation of freezing • formation of small nuclei -center of crystal

s • homogeneous or heterogeneous • homogeneous -water -random accumulati

on of water molecules • heterogeneous -small particles present in

the solution act as nuclei

Crystal Growth water as example

• can only occur after nuclei are formed • exceed the crystal size • function of :

– rate at which the water molecules reacts at the crystal surface

– diffusion rate of water molecules from the unfrozen solution to the crystal surface

– rate heat is removed

Growth of crystals from nuclei• Growth processes then enlarge existing nuclei• Smallest nuclei often redissolve• Larger nuclei can get larger• Thermodynamics favors the • formation of larger nuclei

Nucleation and Growth Rates – Poor Glass Formers

Tm

T

Strong overlap of growth and nucleation rates•Nucleation rate is high•Growth rate is high•Both are high at the same temperature

Rate

Growth Rate

Nucleation Rate

Nucleation and Growth Rates – Good Glass Formers

Tm

T

No overlap of growth and nucleation rates• Nucleation rate is small• Growth rate is smallAt any one temperature one of the two is zero

Rate

Growth Rate

Nucleation Rate

Fluid Processing

• Molten glass is processed to maximize clarity or durability.

• Pure raw materials are often produced to exacting sizes through precipitation from aqueous solutions.

• Large single crystals are grown from pure molten solids for specialty applications.

Liquid

Critical Cooling Rate

• Rapidly cooled liquids skip crystallization and form random amorphous solids.

• Calculating this rate involves minimizing both nucleation and grain growth.

Glass

What This Means

• Clear glasses are processed to reduce the number and size of crystals.

• Crystalline ceramics are processed so that properties are optimized.

Nucleation Rate – Thermodynamic barrier W*

• At r*, (W(r)/ r)r=r* = 0

• r* = -2/ Gcryst(T)

• W(r*) W* = 16 3/3(Gcryst(T))2

r

Wto

t

WS = 4r2, is the surface energy

WB = 4/3r3Gcrsyt(T),

the Gibb’s Free-Energy of Crystallization

Wtot = WS + WB

W*

r*

+

-

0

Nucleation Rate I(T)

• I = nexp(-N 16 3/3(Gcrsyt(T))2 /RT)exp(-ED/RT)

Gcryst(T) = Hcryst(Tm )(1 – T/Tm)Gcryst(T)

0

+

-

Tm

Approx. for : ~ 1/3Hcryst/N1/3Vm

2/3

RT

E

T

TH

RTVnI Dmcryst

m

exp81

16exp

2

2

Liquid is Stable

Crystal is Stable

Growth Rates - • Crystal Growth requires diffusion to the nuclei surface• Crystallization onto the exposed crystal lattice

Gcryst

ED

lc = exp(-ED/RT)cl = exp(-(ED- Gcryst) /RT)net = lc - cl = exp(-ED/RT) - exp(-(ED- Gcryst) /RT) = a net = a exp(-ED/RT) x

(1 – exp(Gcryst) /RT)

Growth Rates -

• Diffusion coefficient, D

• D = a2 exp(-ED/RT) = fRT/3Na(T)

• Hence: = fRT/3Na2(T)(1 – exp(Hm/RT(T/Tm)

• where (T) = 0exp(ED/RT)

Precipitation

The objective of precipitation is to remove salts, metals, or other contaminants present in liquid waste streams. Most often, this deals with the removal of metals at varying pH levels. Generally, the size of a precipitated particle increases if the reaction is allowed to occur with previously precipitated particle.

1. Process Description

Precipitation has generally been shown to occur in three steps: (a) nucleation; (b) crystal growth; and (3) agglomeration and the ripening of the solids.

(a) Nucleation:a nucleus is a fine particle on which the spontaneous formation or precipitation of a solid phase can take place in a supersaturated solution.

Homogeneous nucleation occurs when the nuclei is formed from component ions of the precipitate; if foreign particles are the nuclei, heterogeneous nucleation occurs.

PRECIPITATION

(b) Crystal growth: crystals form by the deposition of the precipitate constituent ions onto nuclei.

Crystal growth rate can be expressed as:

whereC* = saturation concentration (mole/L)C = actual concentration of limiting ion (mole/L)k = rate constant (Ln / time mg)S = surface area available for precipitation (mg/L of a given particle size)n = constant

When the diffusion rate of ions to the surface of the crystal controls the crystal growth rate, the exponent n has a value of unity; when other processes such as the reaction rate at the crystal surface are rate limiting, n may have a value other than unity.

dC

dtkS C C n ( * )

PRECIPITATION

Protein Crystals day 6 day 10

Day 13

Day 16

(c) Agglomeration and ripening

: conversion of small particles into larger particles is enhance by agglomeration of particles to form larger particles, which is the continual growth until equilibrium is reached. The changes in crystal structure that take place over time are often called aging.

A phenomenon called ripening may also take place whereby the crystal size of the precipitate increases.

PRECIPITATION

Definition of Ostwald ripening

• Many small crystals form in a system initially but slowly disappear except for a few that grow larger, at the expense of the small crystals. The smaller crystals act as "nutrients" for the bigger crystals. As the larger crystals grow, the area around them is depleted of smaller crystals.

“LEEM (Low-energy electron microscopy) images of ripening of single atomic layer height islands on Si(001) at various times after the temperature was increased to 670˚ C: (a) 10 s, (b) 50 s, (c) 400 s,

and (d) 1300 s.

Explanation for the occurrence of Ostwald ripening

• This is a spontaneous process that occurs because larger crystals are more energetically favored than smaller crystals. (This might be hard to believe seeing as how it seems far more common to get many small crystals than a few large ones, but there is a believable explanation.) . While the formation of many small crystals is kinetically favored, (i.e. they nucleate more easily) large crystals are thermodynamically favored. Thus, from a standpoint of kinetics, it is easier to nucleate many small crystals. However, small crystals have a larger surface area to volume ratio than large crystals. Molecules on the surface are energetically less stable than the ones already well ordered and packed in the interior. (Think of packing your vacation clothes in a suitcase. Which ones are more energetic? The ones in the middle or the ones you are packing in on top, trying to get them to fit?) Large crystals, with their greater volume to surface area ratio, represent a lower energy state. Thus, many small crystals will attain a lower energy state if transformed into large crystals and this is what we see in Ostwald ripening.

• So why doesn't Ostwald ripening happen all the time? One reason is that the nucleation of many small crystals reduces the amount of supersaturation and thus, the thermodynamically favored large crystals never get a chance to appear.