# 液相法原理

```Che5700 陶瓷粉末處理

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decomposition of solutions EDS; 乳化乾燥(emulsion
drying), 冷凍乾燥(freeze drying)

precipitation)

Che5700 陶瓷粉末處理

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Precursor 必須先溶入溶劑中 (水或有機溶劑)  一

powder

emulsion precipitation, hydrothermal precipitation,
hydrolytic precipitation (實際指sol-gel, alkoxide被水

UO2 核電廠燃

UO2F2 +
(NH4)2CO3 
(NH4)4UO2(C
O3) 3 + 2
NH4F

- Burman等
•上圖中, a,b,c 分別代

•下圖則反應煆燒後粒

* 取自JS Reed; 沉澱法都經由兩種化合物反應而得, 粒子的

Che5700 陶瓷粉末處理

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supersaturation: C = C – C or m, x
Supersaturation ratio:  = C/C
Relative supersaturation: C = C/C; x + 1 = x/x
Dimensionless growth affinity:  = /RT

Debye-Huckel equation
   i (i  i ) & i  i  RT ln ai
eq
o
ai   i xi & xi   i x
  {1  ln(  /   ) / ln(x / xeq )}RT ln(1   x )
eq
=RT ln;  = (i/Ksp) 1/;  = i
Ksp 代表平衡的離子濃度乘積; i 則代表目前的離子濃度

Solubility
 Thermodynamic data: 主要受溫度影響, 當然溶液

a
G  ( 2  1 )   RT ln( )
ao
C
a
S

Co ao
Solubility (2)
T & pH對溶解度之影響; DCP
= dicalclium phosphate; HAP
= hydroxyapatite;
System of: Ca(OH)2-H3PO4 –
KOH – HNO3 – CO2 – H2O;
Ca/P = 1
2

A  2 B  AB2( s )
2
 2
o
K sp  [ A ]o [ B ]
2
 2
S  [ A ][B ] / K sp
•溫度差也可以作

•溶解度多半隨溫

Che5700 陶瓷粉末處理
Nucleation
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nucleation, secondary nucleation等幾種現象

energy of formation of new crystals  =  Ac –
( - ) Mc [Ac: crystal surface area; Mc: crystal
mass)  當粒子粒徑到達critical nucleus時  d 
/d(d) = 0  得到critical nucleus size d* = 4 Vm
/(RT ln)
 最終推導出rate of nucleation為:
Bo = C exp(- */kT) & * = 32 b 3 Vm2/(RT
ln)2
S=
Che5700 陶瓷粉末處理
Kinetic Expression of Nucleation
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A1代表monomer  所以其動力學公式如下:
Ci = condensation rate消失; Ei = evaporation rate長大
Under steady state d fi/dt = 0, and B.C. f1 =n1 = constant; fG =
0 or constant (G: 表示某種大小, 例如critical nucleus size)
dfi / dt  Ci 1 fi 1  Ei fi  Ci fi  Ei 1 fi 1  I (i 1, t )  I (i.t )
I  ao P / 2mkT(ao / 9kT )1/ 2 exp(4ao  3 / 27k 3T 3 (ln P / Pe )2 )
3
I   Z (i*)C (i*)n(i*)
Zeldovich factor
Che5700 陶瓷粉末處理
Solute Clustering & Nucleation
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stand for a long time, 則不同尺寸的cluster會依其重量分

Typical cluster size: 4-10nm, ~ 103 molecules
Che5700 陶瓷粉末處理
Heterogeneous Nucleation

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In a sense, 共沉澱也是這一種效應
Epitaxial growth 指成核物種與雜質微粒表面的晶格

nucleus
size因而有

Che5700 陶瓷粉末處理
More on Nucleation

Ring, 1996;
data for
BaSO4
Che5700 陶瓷粉末處理
Secondary Nucleation

 原因包括:
Initial breeding
Needle breeding
Contact breeding
Fluid shear 等效應造成
 影響因素: 過飽和程度, 攪拌情形, 懸浮粒子碰撞情形
(碰撞頻率, 碰撞能量, 材質特色)等
 經驗式顯示: 二次成核速率 Bo ~ (S-1)b MTj (rpm)h;

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A Model on Secondary Nuleation
Taken from Botsaris, et. al. Chem. Eng. Sci., 52(20),
3429-3440, 1996;
 他們提出的概念: 過飽和溶液中, 原本就有embryos (may
be viewed as a result of coagulation between clusters), 彼

secondary nuclei, 也許有些與seed聚集, 讓seed 長大
Theory of rapid coagulation: - dn/dt = 8D r n2 = (4kT/3)
n2 (by Smoluchowski) (粒子運動靠Brownian motion; n 粒

Botsaris計算結果: 假設seed 附近的secondary nucleation rate;
curve 7: 假設cluster g = 622; At = seed surface area = 1.67
cm2/cm3; 對象KCl-H2O system; curve 6前半段: contact
nucleation; 後半段與Botsaris theory相仿
LH left-hand

Che5700 陶瓷粉末處理
Induction Times
•

• 觀察晶體的技巧: turbidity, visual observation, conductivity, 或

• 具體而言, 包括三個部分:
 ti = tr + tn + tg
 ti : induction time;
 tr: time required for attainment of stationary embryo
distribution (relaxation time)
 tn: time for the formation of nucleus
 tg: time for nucleus to grow into detectable crystals
* 可能造成成核障礙之一: 如dehydration reaction of ions
Che5700 陶瓷粉末處理
More on Induction Times
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If tn 為主, 表示成核過程為主, tn ~ 1/Bo  then ln(tn)
or ln(ti) vs ln()-2 should be linear
If tg 為主, ti 通常變得很長, 晶體成長可能受surface
nucleation控制;  ln(ti) vs ln()-1 will be linear

Che5700 陶瓷粉末處理
Crystal Growth
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(L small), 則其溶解度越高

ln(X / X eq )  2M / 3LRT
•成長過程可以細分為以下幾各步驟:
(dehydration); Surface diffusion; Integration at kink site
•名詞: ledge, step and kink
xL/ xeq = 也

supersaturation
F 代表表面能surface
energy

Che5700 陶瓷粉末處理
Growth Rates

 Birth & Spread mechanism (2D nucleation
model):
 growth rate ~ (step height) x (step velocity)
2/3 x (#critical nuclei formed/area-time) 1/3
 G = A i 5/6 exp(-B/i)
 綜合的經驗式則多寫為: G = k n
 注意所表達的過飽和, 可以代表bulk內的數

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linear,
parabolic,
and
exponential
law幾種, 描

Che5700 陶瓷粉末處理
More on Growth Rates
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Growth rate dispersion: 因為粒子在容器內停留時間

Heat production: 介面溫度改變會影響到溶解度 

In general: linear growth rate = 質傳或吸附效應
parabolic rate = spiral steps
exponential rate = polynuclear surface
control
(H / M )GC  hi (Ti  T )

(screw
dislocation) 密

Summary on Particle
Formation
Reaction  formation of some “species” 
supersaturation  (induction times)
 Nucleation (home-, hetereo- ..) (critical nucleus
size, nucleation rate)
 growth (growth rate, crystal habit, …)
 agglomeration
 final particle size distribution and morphology
Crystal Habit
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Equilibrium shape versus growth shape

Equilibrium shape: (下列公式 Wulff theorem)
 表面能大者, 其表面積小, I.e. easy to
disappear
 S / L (1) A(1)   S / L (2) A(2)  ...  const.

equilibrium
shape;
Elimination of high
energy surface via
growth

1996; 藉由雜

Che5700 陶瓷粉末處理
Ostwald Ripening
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coarsening of large particles at the expense
of small ones

Cs (r )  C  C [exp(a / r ) 1]
a  2VM /RT

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