Outside region
IV. 電泳動度
在這一節中我們考慮的例子，即求解非線性之Poison主控方程式並考慮在高 表面電位及極化效應的影響。圖4-8 為在考慮沒有極化效應在不同無因次表面電 位φr下，無因次的電泳速度(mobility)U*/E*對κa作圖(其中φr→0 的這一條線為其於 線性化Boltzmann 分佈所得之結果)。從圖中可以看出電泳速度在κa小的時候，
即代表電雙層厚，此時的電泳速度和表面電位無關為一固定值。在κa逐漸的變 大，此時電雙層漸漸的縮小，因此電泳速度開始上升。另外由圖中可以看出表面 電位的上升，會使電泳速度上升但不明顯。可見增加表面電位所增加之電力並不 甚大。同樣我們再將粒子的受力表示出來，圖4-9 為圖 4-8 相同條件下粒子所受 總力(a)為第一個子問題、(b)為第二個子問題。首先我們看圖(a)，它表現出和低 表面電位之相同行為。由於此時以流力為主和電力較無關係，表面電位的影響並 不是相當重要。而圖(b)為可以看到當表面電之較大時則其所受之力較大，主要 是因原為表面電位的上升而增加了電位勢。由於以上的受力情形，應可了解電泳 速度的變化原因。
圖 4-10 為考慮極化效應的影響在不同無因次表面電位下，無因次電泳速度 (mobility)U*/E*對κa作圖，其中(a)η0=1.5、(b)η0=2.0。由圖中可以看到在κa小的時 候，其現象和圖4-8 相同。粒子受到平板相當的影響，電泳速度無明顯之改變。
但在隨著κa上升，圖中表現出和圖 4-8 截然不同的趨勢。在本圖中可以看到表面 電位的改變對於電泳速度的影響相當的明顯。當電位上升時，電泳速度則有相當 程度的下降。此現象最主要的因素即是加入了極化效應的原故。因電雙層的極化 效應會使得帶電膠體粒子周圍之流動產生影響，進一步使得電雙層變形而誘發一 內部電場。此一誘發電場會減弱外加電場所施之力量，並隨著表面電位變大其電 位勢則誘發電場越強，故電泳速度則較小。
我們亦可由其受力來看，圖 4-11 為η0=1.5 時其粒子對κa作圖的所受總力，
我們可以看到無論是圖 4-11 或是圖 4-12 問題一所受之總力f1隨著表面電位上升 而下降，其變化雖不大但相對於無極化效應之圖4-9 來看已比較明顯。可見極化 效應使得離子運動也會減低粒子運動之阻力，但相對於電力的成長仍屬不明顯。
而問題二所受之總力f2即可以看到極化效應所帶來的影響，由圖中可以明顯的看 出在極化效應的影響下，越高的表面電位所誘發的內電場越大下而抵消了大部分 的電力，使得總力下降。圖4-13~圖 4-15 則是再將上述二圖更詳細地分成三種不 同的力。可以看到了圖4-15 和圖 4-16 的問題二中因外加電場所引起的電力，一 般而言在無極化效應時表面電位越大則所感應到的電力越大。但因極化效應的加 入則反而使得電力下降。
再來我們看看不同粒子和平板距離對於電泳速度的影響。圖 4-17 表示電泳 速度在不同的表面電位下對η0作圖，圖中明顯可以看出在κa=1.0 時，只要η0<1.0 時，則此時的電泳速度己和表面電位無關。只因膠體粒子已相當靠近平板，而受 到其平板給予相當大的阻力。
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ρ
z
0 1 2 3 4
0 1 2 3
(a)
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ρ
z
0 1 2 3 4
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(b)
Fig. 4-2 The stream function profile in problem1 for the case
η
0 =1.0 (a)κ
a=0.01(b)
κ
a=3.98ρ
z
0 1 2 3 4
0 1 2 3
(a)
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z
0 1 2 3 4
0 1 2 3
(b)
Fig. 4-3 The stream function profile in problem2 for the case
η
0 =1.0 (a)κ
a=0.01 (b)κ
a=3.982
Level FI8 14 -1.80514E-06 13 -5.03312E-06 12 -9.50305E-06 11 -1.42321E-05 10 -2.11665E-05 9 -2.72139E-05 8 -3.36898E-05 7 -4.04507E-05 6 -4.68358E-05 5 -5.06068E-05 4 -5.45727E-05 3 -5.80261E-05 2 -6.34793E-05 1 -0.00024122
(a)
Level FI8 15 0.000353533 14 -0.000509525 13 -0.00137258 12 -0.00223564 11 -0.0030987 10 -0.00396175 9 -0.00482481 8 -0.00568787 7 -0.00655093 6 -0.00741398 5 -0.00827704 4 -0.0091401 3 -0.0100032 2 -0.0108662 1 -0.0117293
(b)
Fig 4-4The potential function profile was related applied field in problem 1(φ21*) for the case η0=1.0, φr=1.0 and (a) κa=0.01 (b)κa=6.3
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Level FI8 14 -7.17501E-05 13 -0.000227267 12 -0.000405707 11 -0.000586421 10 -0.000734116 9 -0.000877319 8 -0.00102589 7 -0.0011779 6 -0.00126322 5 -0.00134897 4 -0.00140094 3 -0.00144349 2 -0.00150004 1 -0.00390118
(a)
Level FI8 15 -0.0730326 14 -0.146069 13 -0.219105 12 -0.292141 11 -0.365177 10 -0.438213 9 -0.511249 8 -0.584285 7 -0.657321 6 -0.730357 5 -0.803393 4 -0.876429 3 -0.949465 2 -1.0225 1 -1.09554
(b)
Fig 4-5 The potential function profile was related applied field in problem 1(φ21*) for the case η0=1.0, φr=4.0 and (a) κa=0.01 (b)κa=6.3
1
Level FI4 15 -0.135 14 -0.377186 13 -0.619371 12 -0.861557 11 -1.10374 10 -1.34593 9 -1.58811 8 -1.8303 7 -2.07249 6 -2.31467 5 -2.55686 4 -2.79904 3 -3.04123 2 -3.28341 1 -3.5256
(a)
Level FI4 15 -0.135 14 -0.377186 13 -0.619371 12 -0.861557 11 -1.10374 10 -1.34593 9 -1.58811 8 -1.8303 7 -2.07249 6 -2.31467 5 -2.55686 4 -2.79904 3 -3.04123 2 -3.28341 1 -3.5256
(b)
Fig 4-6 The potential function profile was related applied field in problem 2(φ22*) for the case η0=1.0, φr=1.0 and (a) κa=0.01 (b)κa=6.3
1
Level FI4 15 -0.135 14 -0.377186 13 -0.619371 12 -0.861557 11 -1.10374 10 -1.34593 9 -1.58811 8 -1.8303 7 -2.07249 6 -2.31467 5 -2.55686 4 -2.79904 3 -3.04123 2 -3.28341 1 -3.5256
(a)
Level FI4 15 -0.135 14 -0.377186 13 -0.619371 12 -0.861557 11 -1.10374 10 -1.34593 9 -1.58811 8 -1.8303 7 -2.07249 6 -2.31467 5 -2.55686 4 -2.79904 3 -3.04123 2 -3.28341 1 -3.5256
(b)
Fig 4-7 The potential function profile was related applied field in problem 2(φ22*) for the case η0=1.0, φr=4.0 and (a) κa=0.01 (b)κa=6.3
κa
U* /E*
10-2 10-1 100
0.55 0.6 0.65 0.7 0.75 0.8
φr→0 1.0
2.0 4.0 3.0
(a)
Fig.4-8 Variation of scaled electrophortic mobility(U*/E*) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and without polarization effect. Key α=1.0, η0=1.5
κa f1
10-2 10-1 100
-8.65 -8.6 -8.55 -8.5
φr→0 1.0 4.0
(a)
κa f2
10-2 10-1 100
4.5 5 5.5 6
6.5 φr→0
1.0
4.0
(b)
2.0 3.0
Fig.4-9 (a)Variation of total force in problem 1(f1) (b) Variation of total force in problem 2(f2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and without polarization effect. Key α=1.0, η0=2.0
κa U* /E*
10-2 10-1 100
0.4 0.45 0.5 0.55 0.6 0.65 0.7
0.75 1.0
2.0
3.0
4.0 φr→0
(a)
κa U* /E*
10-2 10-1 100
0.5 0.55 0.6 0.65 0.7 0.75
0.8 1.0
2.0
3.0
4.0
φr→0
(b)
Fig.4-10 Variation of scaled electrophortic mobility(U*/E*) as a function of inverse double layer thickness(κa) at various scaled surface potential φr. (a)η0=1.5 (b)η0=2.0 for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0,Pe1=0.01,Pe2=0.01
κa f1
10-2 10-1 100 101
-11.6 -11.5 -11.4 -11.3 -11.2 -11.1 -11 -10.9 -10.8 -10.7 -10.6 -10.5 -10.4
1.0 2.0 3.0 4.0
φr→0
(a)
κa f2
10-2 10-1 100 101
4.5 5 5.5 6 6.5 7 7.5 8 8.5
1.0
2.0
3.0
4.0 φr→0
(b)
Fig.4-11 (a)Variation of total force in problem 1(f1) (b) Variation of total force in problem 2(f2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=1.5,Pe1=0.01,Pe2=0.01
κa f1
10-2 10-1 100
-8.65 -8.6 -8.55 -8.5 -8.45 -8.4 -8.35 -8.3 -8.25 -8.2
1.0 2.0 3.0 4.0
φr→0
(a)
κa f2
10-2 10-1 100
4.5 5 5.5 6 6.5
1.0
2.0
3.0
4.0
φr→0
(b)
Fig.4-12 (a)Variation of total force in problem 1(f1) (b) Variation of total force in problem 2(f2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=2.0,Pe1=0.01,Pe2=0.01
κa DF1
10-2 10-1 100 101
-30 -25 -20 -15 -10
1.0 2.0
3.0
4.0
φr→0
(a)
κa DF2
10-2 10-1 100 101
-200 -150 -100 -50 0
1.0 2.0 3.0 4.0
φr→0
(a)
Fig.4-13 (a)Variation of drag force in problem 1(DF1) (b) Variation of drag force in problem 2(DF2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=1.5,Pe1=0.01,Pe2=0.01
κa DFE1
10-2 10-1 100 101
0 2 4 6 8 10 12 14 16 18 20
1.0 2.0 3.0 4.0
φr→0
(a)
κa DFE2
10-2 10-1 100 101
0 50 100
150 1.0
2.0
4.03.0
φr→0
(b)
Fig.4-14 (a)Variation of electrical force induced by the imbalance charge density distribution in problem 1(DEF1) (b) Variation of electrical force induced by the imbalance charge density distribution in problem 2(DEF2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=1.5,Pe1=0.01,Pe2=0.01
κa E1
10-2 10-1 100 101
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
1.0 2.0
3.0 4.0
φr→0
(a)
κa E2
10-2 10-1 100 101
5 10 15 20 25 30 35 40
1.0 2.0
3.0
4.0
φr→0
(b)
Fig.4-15 (a)Variation of electrical force in problem 1(E1) (b) Variation of electrical force in problem 2(E2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=1.5,Pe1=0.01,Pe2=0.01
κa E1
10-2 10-1 100
0 0.25 0.5 0.75 1 1.25 1.5 1.75
1.0 2.0 3.0 4.0
φr→0
(a)
κa E2
10-2 10-1 100
5 10 15 20 25 30 35
1.0
2.0
3.0
4.0 φr→0
(b)
Fig.4-16 (a)Variation of electrical force in problem 1(E1) (b) Variation of electrical force in problem 2(E2) as a function of inverse double layer thickness(κa) at various scaled surface potential φr for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0, η0=2.0,Pe1=0.01,Pe2=0.01
η0
U* /E*
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 φr→0.0
2.0 3.0 4.0 1.0
Fig.4-17 Variation of scaled electrophortic mobility(U*/E*) as a function of η0 at various scaled surface potential φr. for the case when the surface potential remain constant at unity and with polarization effect. Key α=1.0,Pe1=0.01,Pe2=0.01,κa=1.0
第五章
球對平板在任意外加電場下之結果與討論
先前我們有提到，求解粒子的電泳速度需須先求解數條主控方程式。首先是 平衡電位方程式，再來是其它的外加擾動方程式。由於外加擾動相對於平衡電位 不可忽略時，我們必須直接處理幾條複雜的方程式。在這裡我們是採用牛頓法處 理這些非線性的主控方程式。了解問題的求解方法之後，則以下就分別探討外加 電場強度(即變數E)、距離平板的距離(即變數η0)在不同電雙層厚度下對膠體粒子 的電泳有什麼影響。