### 粒徑與形狀

```Che5700 陶瓷粉末處理

•名詞: powder, particle (primary, secondary), colloid,
agglomerate (soft, hard), aggregate, granule,
crystallite; 各具有略為不同的意義
•強調單一粒子或者一群粒子(particle system)
•Ideal powder: 單一粒徑分布, 0.1 – 1.0 m, 球形, 沒

consistency

Che5700 陶瓷粉末處

• 具有代表性representative!! 屬於統計學的理論
• 不同的粒子 (形狀, 大小, 密度等) 其運動行為也會不

•Golden Rules of Sampling: (a) 取樣的樣品應該處於運

powder should be taken for many short increments of
time in preference to part of the stream being taken
for the whole time.
•取樣的大小,有些學問!!
Grab sample;
cone & quarter;
Riffling 等三種實

From JS Reed,
2nd ed.
Che5700 陶瓷粉末處理
Sampling Accuracy
•Maximum sampling error: E = 2i/P where i = 本次取樣

•t = (i 2 + n 2) ½ where n = standard deviation from
measurement (total error: 前式中的i 用t取代); i.e.
sampling error + measurement error
•範例: 自10,000Kg樣品中取樣10g分析, 經由篩分後, 大於
44m的鐵雜質粒子(造成顏色)佔40ppm (其餘為silica粒子,

(1-P) w2) . (1- Ws/Wb)] ½  Ws =取樣重量; Wb = 全部重

= 4.0x10-6  所以本實驗物查主要來自量測, 而非取樣.
Two-Component Sampling Accuracy
If we count particles (instead of measuring weight),
then sampling error
 σi = [p (1-p)/Ns (1- Ns/Nb)] ½
where p = fraction of particles above a certain size
Ns = number of particles counted
Nb = number of particles in the bulk

Che5700 陶瓷粉末處理

•主要概念: 使用與球體對等的直徑為粒子的粒徑
equivalent diameter;
•對等的項目包括: 體積, 表面積, 沉降速度, 投影面積等

•  dv volume diameter V = /6 dv3 …測量粒子體積
•  ds surface diameter S =  ds2 ….測量粒子表面積
•  dvs surface volume diameter dsv = dv3/ds2 ..測量粒

•  Stoke’s diameter dStk 與球形粒子沉降速度相同
Che5700 陶瓷粉末處理

•  projected area A = /4 da2 …測量投影面積
•  Sieve diameter: 通過篩網的網口 (width of square
opening)
•  Martin diameter: mean chord length of projected
outline of particle
•  Feret’s diameter: mean value of distance between
pairs of parallel tangents to the projected outline of
particles
•利用投影面積者, 多半可以藉助影像分析的技術
* Martin diameter: 實際上可以選許多方向畫線, 然

From TA Ring, 1996.
Coulter counter: 每當

aperature時會因為排

Che5700 陶瓷粉末處理
Microscopic Method
• 各種粒徑測量技術的基準, 眼見為真!! OM, SEM, TEM
•運用標準粒子 (e.g. PS polystyrene monodispersed
particle 得自乳化聚合) 進行所有方法的校正工作.
•搭配影像分析軟體 image analysis: 可以迅速且大量的

•ASTM counting requirements: modal size class至少25

Che5700 陶瓷粉末處理

•Laser diffraction technique: 粒子一顆一顆的通過平行的雷

•DLS: dynamic light scattering (or photon correlation
spectroscopy, PCS): 因為粒子的Brownian motion 造成散射

d = (kB T)/(3L DT)

Forward scatter, side scatter, back scatter
Some Terminology
Rayleigh scattering: 粒子粒徑遠小於波長 d<<λ 
Rθ = Iθ r2/Io = 8 π4α2/λ4 (1+ cos2θ), where α =
polarisability; λ = wavelength of incident light
 粒子粒徑大於波長: 不透明, 只有繞射現象, Fraunhofer
diffraction
 Mie scattering: 任何粒子與光的作用 (一般 10λ – λ/10

PCS or DLS 基本上是量測粒子散射光的相對強度, 運動前

mathematics
Che5700 陶瓷粉末處理
Hydrodynamic Chromatography
•如同一般的層析技術, 本方法主要靠大小不同的粒子

•缺點: low sample recovery (也許被trap), 粒子與堆積

relative low resolution等
Rf = (time
of passage
of marker)/
(time of
passage of
colloid)
versus
colloid size

calibration
•由XRD 特性峰的半高寬 (full width at half maximum),

•Scherrer equation: d hkl = k /(o cos); k = constant

•波峰變寬的原因: 粒徑小, 結晶體內有strain (or
disorder), 儀器問題; 所以通常使用單晶作為校正
From JS Reed, 2nd ed. 好的儀器測量結果大致相仿, 但不

* From TA Ring, 1996; 量測誤差與所計粒子數目平方

Che5700 陶瓷粉末處理
Shape Factor
•Surface or volume shape factor: V = v d3; or S = s
d2;  就是形狀因子, 顯然與所選用的粒子粒徑也有關

• shpericity 球形度: = (surface area of a sphere
having same volume)/(actual surface area of particle)
 = (d NV/d NS) 2
• 類似的也可以定義: circularity = (perimeter of a
circle having same area)/(actual perimeter)
• aspect ratio: 對於細長的纖維, 可以使用此一因子表

dimension)
All particles
are hydrous
zinc oxide
(from
different
precipitation
conditions):
Shapes are
different
ψA/ψV  index of
angularity
(shape factor based
on area/shape factor
based on volume)
* 取自TA Ring, 1996; S/V = St/V . Dav ; 其中St/V 表

More Shape Factors
•Dynamic shape factor  = (d NV/ d st)2 ; 表示粒子運動

d st 分別表示 volume equivalent diameter based on
number & Stoke’s diameter;
•=
-½
(sphericity)
•這些數字提供一個簡單的方法量化形狀的效應, 可以用

Che5700 陶瓷粉末處理
Fractal Shapes
•對於有些像花耶菜的粒子, 使用fractal shapes 也許為

~ x 1-D , where D = fractal dimension of particle
• 例如在包含粒子的顯微照片中, 畫許多大小不同的圓圈,

•一般使用sol-gel法, or flame, plasma等法製備的陶瓷粒

•粒子特性會與fractal dimension有關, 例如  ~ R D-3; A
~ (ro)-1 R D; (ro = 基本粒子粒徑, R 聚集體粒徑)

Ring, 1996
Che5700 陶瓷粉末處理

•表達方式: (a) 圖形 – cumulative (oversize or
undersize); frequency – based on number, weight or
volume, etc. (b) 恰當的數學公式
•CNPF: cumulative distribution based on number,
percentage finer; CNPL (L = larger 比他大的部分)
•CMPF: (M for weight) 以重量為準的意思
Size 區分: linear or geometric 幾何級數, e.g. 2
½
Che5700 陶瓷粉末處理

•Normal distribution: f(x) = 1/(2) exp[-(x –
x)2/22] 公式中有兩各可調參數: x &  (平均粒徑與標

•Log-normal distribution: f(z) = 1/(z2) exp[-(z –
z)2/ 2z2] ; z = ln d 同樣也是兩個可調參數 or 改寫為
f(d) = 1/(lng 2) exp[- (ln(d/dg))2/ 2 (lng)2]
•以上兩個公式最常用。dg g 分別代表geometric mean
size & standard deviation; dg = d 84.13/d 50 = d 50/d
15.87
Che5700 陶瓷粉末處理
More Equations
•Rosin – Rammler distribution: f(x) = n b x (n-1) exp( b xn) ; n & b 同樣為可調參數, 換言之與粒子特性有關,

distribution = exp( - b xn) …簡單的數學公式
•Gaudin – Schulmann model: w(d) = a n d

(n-1)
; w(d)
•絕大部分的公式都採取兩個可調參數, 既然都希望描

From TA Ring,
1996; bimodal
distribution 本圖示

Che5700 陶瓷粉末處理

•可以使用 mean, modal (指

(指50%正好一半的地方)等

•Mean (or average)的計算,

Che5700 陶瓷粉末處理

•微粒間鍵結: electrostatic, magnetic, van der Waals,
capillary adhesion, hydrogen bonding, solid bridge (由

dissolution-evaporation等程序造成
•強度: 可以藉由compaction,ultrasonic vibration等法間

agglomerate porosity等資訊估算之
In general:  = o exp( - b ); /tho = 1 - 
From Am. Cer.
Soc. Bull., 65,
1591, 1986.

Soft
agglomerate
vs strong
agglomerate
Che5700 陶瓷粉末處理

• 當凝聚體已經形成後, 破壞或去除的方法:
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