非牛頓流體電學(xué)：綜述外文文獻(xiàn)翻譯@外文翻譯@中英文翻譯

1 附錄 A 外文翻譯 譯文： 非牛頓流體電學(xué)：綜述 3.在非牛頓流體電泳 在第二節(jié)討論了關(guān)于電滲流帶電表面，如果我們通過想象改變參考系統(tǒng)，帶電表面的流體應(yīng)該是靜止的，然后將帶電面以速度大小相等但與以前面討論的亥姆霍茲Smoluchowski 的速度方向相反移動。這種情況下有效地代表了電泳具有很薄的 EDL 的粒子在一個無限大的非運動牛頓流體范圍 17,18,26,34 。顯然，先前討論電滲的亥姆霍茲 Smoluchowski 速度當(dāng)然也可適用于分析在無限大非牛頓流體域具有薄 EDL 顆粒的電泳速度，僅僅與它的符號相反，并改 變了充電通道壁與帶電粒子的潛力。 事實上，支付給非牛頓液體粒子電泳最早的關(guān)注可以追溯到 30年前 Somlyody 68 提起的一項有關(guān)采用非牛頓液體以提供優(yōu)越的閾值特性的電泳顯示器的專利。在 1985年， Vidybida 和 Serikov 69 提出關(guān)于球形顆粒的非牛頓電泳研究第一個理論解決方案。他們展示了一個粒子在非牛頓凈電泳運動流體可通過以交替的電場來誘導(dǎo)一個有趣的且違反直覺的效果。最近才被 Hsu 課題組填補這方面 20 年的研究空白。在 2003 年，Lee70等人通過一個球形腔的低 zeta電位假設(shè) 封閉 andweak施加電場分析了電泳剛性球形顆粒在非牛頓的 Carreau 流體的運動。他們特別重視電泳球形粒子位于中心的空腔特征。之后，該分析被擴(kuò)展來研究電泳位于內(nèi)側(cè)的球面的任意位置的球形顆粒的腔體71 。除了單個粒子電泳外， Hsu72等人假設(shè)粒子分散潛力在卡羅流體 zeta 進(jìn)行了集中的電泳調(diào)查分析，并分析了由 Lee73完成的其它任意潛力。為了研究在邊界上非牛頓流體電泳的影響， Lee74等人分析了電泳球狀 粒子在卡羅體液從帶電荷到不帶電荷的平面表 面，發(fā)現(xiàn)平面表面的存在增強了剪切變稀效果，對電 泳遷移率產(chǎn)生影響。類似的分析后來由 Hsu 等 75進(jìn)行了擴(kuò)展。為了更緊密地模擬真實的應(yīng)用環(huán)境， Hsu 等人76分析了球形粒子的電泳由一個圓柱形的微細(xì)界卡羅流體低 zeta 電位到弱外加電場的條件。許多實際電泳應(yīng)用涉及生物粒子或是棒狀顆粒，比如蛋白和 DNA。為此， Yeh和 Hsu77在延續(xù)以往的研究上分析了球狀 粒子在非牛頓流體的電泳 沿圓筒形通道的圓柱形顆粒的情況。從 Hsu 課題組研究提出的一般結(jié)論是，剪切降粘流體或更薄的 EDL 的周圍的粒子可提高電泳遷移率。這與流體流變學(xué)和 EDL 厚度的依賴性達(dá)成了一致性。最近，海爾哈納等 78證明了均勻帶電粒子在非牛頓流體的電泳速度和 EDL 取決于該粒子的形狀和大小。這種行為是完全相反的電泳牛頓流體。此外，經(jīng)鑒定，外界的應(yīng)力取決于該 EDL （即散裝電中性非牛頓流體內(nèi)部）。有趣的是，這樣的尺寸和電泳的非牛頓流體的幾何形狀作為定性地重合和報道的文獻(xiàn)。 總之，可以斷定的是粒子在非牛頓流體秤 nonlinearlywith 外部施加場的粒子的2 電位和電泳速度，以及類似的電滲的非牛頓流體。另一個顯著特點是，利用剪切變稀的液體會提高顆粒的電泳遷移率，從而導(dǎo)致粒子在電場中快速運動。 4.非牛 頓流體的潛在作用 早在 20 世紀(jì) 60 年代， Raza 和 Marsden79,80報告了他們通過派熱克斯管和相關(guān)聯(lián)的流勢水性泡沫體的實驗獲得的壓力驅(qū)動流的測量，第一次嘗試研究非牛頓流體的分流作用。他們觀察到非離子型發(fā)泡劑具有非常高的流勢（如 50 V）。這樣大的流動電位規(guī)模產(chǎn)生抵抗壓力驅(qū)動顯著作用的電滲流，從而被認(rèn)為是 foamflow 在多孔介質(zhì)的堵塞的主要原因。把實驗的結(jié)果關(guān)聯(lián)起來，該泡沫被認(rèn)為是一種非牛頓冪律流體和理論模型，然后具有制定描述跨越圓管流的潛在作用。在接下來的幾十年里，這種影響顯然已經(jīng)引起了關(guān)注。 最近， Bharti 等人 81從理論上研究冪律液體的筒形的壓力驅(qū)動流微通道電粘性效應(yīng)。他們估計數(shù)字如圖減小，增加了流勢場 whichwas 流體行為指數(shù)。此外，人們發(fā)現(xiàn)，由于非牛頓流體的潛在影響，剪切稀化流體比剪切增稠流體的作用更顯著。利用更一般的卡羅液體模型， Bharti 等人 82通過考慮在一圓柱形的壓力驅(qū)動流微管具有收縮 - 擴(kuò)張結(jié)構(gòu)進(jìn)行了一項類似的調(diào)查。趙存璐和楊春 83 根據(jù)流動電位的微影響分析了冪律流體在狹縫中的壓力驅(qū)動流。在該分析中，對于在不完整的條件下獲得對任意流體行為指數(shù)的流體的潛力和速度場 做出了解析。 Vasu 和 De 84 研究了類似的問題。通過確切幾個特殊值，獲得潛在的流體解決方案的流動特性指數(shù)（如 n = 1 時， 1/2 ，1/3）和數(shù)值解從而求出流動行為指數(shù)任意值。此外，他們還對參數(shù)進(jìn)行了研究，這些參數(shù)是用來評估所施加的壓力的影響梯度，流體行為指數(shù)， EDL 厚度和 Zeta 電位的表觀粘度，流動電位和摩擦系數(shù)。圖 5所示的是壓力梯度和流體行為指數(shù)的關(guān)系。隨著流動電位升高壓力梯度和流體行為指數(shù)降低，并且流動電位對流體行為指數(shù)的影響比對壓力梯度的作用更顯著。 圖 5 3 Tang 等人 85用格子玻爾茲曼方法研究流動電位在微壓驅(qū)動的非牛頓流體流動的影響。 Lattice Boltzmann 方法具有高效用于表征非牛頓流體流場與 sheardependent粘度的二階精度。 Lattice Boltzmann 方法是進(jìn)一步利用由同一組用來研究流動電位在微孔結(jié)構(gòu)的壓力驅(qū)動流 86的影響。除了這些，非牛頓流體潛在作用最近還用來探討了非彈性流體轉(zhuǎn)換工作機(jī)械電力 83,87。 Itwas 表示，相比轉(zhuǎn)換效率相同的操作條件下的解 決方案，非牛頓型聚合物溶液的剪切變稀的性質(zhì)可以大幅提高能源與牛頓電解質(zhì)。最近，使用非牛頓流的潛在影響發(fā)電是由非牛頓粘彈性流體 88來實現(xiàn)的。 上述分析普遍預(yù)測，由于非牛頓流體的潛在效果，剪切變稀流體比剪切增稠液體在一個相同的施加壓力梯度下，誘發(fā)更大的流勢，并相應(yīng)地大大減少流體流量。這些發(fā)現(xiàn)可以提供更多的關(guān)于非牛頓流體的物理洞察的特點，由此更有效地控制微流體的非牛頓流體流動的設(shè)備。 5.其他電學(xué)上非牛頓效應(yīng) 在本節(jié)中，將討論包括（） viscoelectric 效果，（）離子擁擠引起的增厚，（）膠體的 電流變效應(yīng)（）電粘性效應(yīng)懸浮液的電學(xué)非牛頓效應(yīng)。在非牛頓流體流變中，第一和第三種情況是源于電場動態(tài)粘度的依賴性，而第二種情況指的是在 EDL 中由于高度填充的離子粘度增加，同時，第四種情況是由于 EDL 中粒子的流動而引起的變形。非牛頓的這四個效應(yīng)均說明了動態(tài)粘度對剪切速率有影響。 5.1 電學(xué)上的 Viscoelectric 效果 在非牛頓液體中的粘度與外部電場的強度變化的現(xiàn)象被稱為 viscoelectric 效果。在第一份該實驗報告中， Andrade 和 Dodd89,90通常用公式描述這種現(xiàn)象為 20= + f EE（ ） （ 1 ） （ 12） 其中0 是在沒有外部電氣的液體粘度， E 是局部電場的強度，并且 f 表示的是viscoelectric 系數(shù)。對于氯仿，氯苯和乙酸戊酯三種液體， viscoelectric 系數(shù) 89,90分別測得為 161.89 10 ， 162.12 10 和 162.74 10 22/Vm 。根據(jù)公式（ 12），因為附近高電荷的固體表面，所述粘度可以顯著提高垂直于 EDL 內(nèi)表面上的電場。 Lyklema，Overbeek 91和 Lyklema 92通過實驗確定 viscoelectric 系數(shù)的值是 1.02 10-15 22/mV 并表明 viscoelectric 系數(shù)將改變傳統(tǒng)的亥姆霍茲 Smoluchowski 速度。在這里，我們展示的一個簡單的推導(dǎo)受到 viscoelectric 影響的亥姆霍茲 - Smoluchowski 速度。當(dāng)一個平坦的表面電位被認(rèn)為是的壁，超過這個表面可推導(dǎo)出亥姆霍茲 - Smoluchowski 速度 91,92 0 0sduE (13) 4 其中，為取為恒定的溶液的介電常數(shù)，0E是與表面相切的外部電場，并且 為里面的EDL 的分布電位。為 了得出一個分析公式，我們將整合公式（ 13）。在這里 ,將 和 明確相關(guān)，方程（ 12）所述的 EDL 內(nèi)溶液的動態(tài)粘度可以表示為如下 20 1 ( ) df dy （ 14） 其中 d /dy 是雙層的場強（其中 y代表坐標(biāo)垂直于平的表面），并且可以從 Gouy Chapman推導(dǎo)出 EDL潛在的查普曼解決方案 22( | d / d y | ) s i n h z e / 2 T BCk。則方程（ 13）可以轉(zhuǎn)化為 00 20 ze1 s i n h ( ) 2sBE dufC kT （ 15） 其中 8/BC n k T （所有的符號與公式（ 1）具有相同的定義），和 / (1 / C )fC f 可以被解釋為無量綱的 viscoelectric 系數(shù)。應(yīng)當(dāng)指出的是，因為通常0|E | |d /dy|，貢獻(xiàn)的外加電 場公式（ 14）是可忽略的 。最后，得到用于亥姆霍茲 Smoluchowski 一個確切的配方速度為 00zea r c t a n 1 t a n h ( ) 221BBsfCE k T k Tu ze fC (16) 當(dāng) 1fC 根據(jù)公式（ 16）可以推導(dǎo)出 002 t a n h ( )2Bs BE kT zeu z e k T (17) 顯然，亥姆霍茲 - Smoluchowski 速度現(xiàn)在變成非線性的關(guān)系，因為上壁的 Zeta 電位存在 viscoelectric 效果。當(dāng) viscoelectric 效果是不存在時，即， F = 0，自然降低到常規(guī)形式亥姆霍茲 Smoluchowski 速度，由方程（ 8）給出公式（ 16）。對于非常大的 Zeta 勢，即 |ze /2 k T | 1B ，亥姆霍茲 - Smoluchowski 速度在方程（ 16）達(dá)到漸近值 00a r c t a n 121BsfCE kTuze fC （ 18） 在第 2節(jié)中討論過平坦的表面的類似的非牛頓流體動力的電滲 ，我們還定義了一個有效的 Zeta 電位 5 a r c t a n 1 t a n h ( ) 221BBe f fzefCk T k Tze fC （ 19） 公式（ 16）然后可以在常規(guī)的形式重寫亥姆霍茲 Smoluchowski 速度。方程（ 8）。預(yù)測結(jié)果從方程（ 19）如圖 6 所示，這是值得注意的是在 Zeta 接近其漸近值，由于viscoelectric 效果 Zeta 電位存在有效電勢（例如， 0fC ）。這行為是讓人聯(lián)想到第2 節(jié)中所討論的由于 shearthickening 作用的液體的漸近行為，但剪切增稠在這種情 況下，根據(jù)公式（ 14），在第 2 誘導(dǎo)的剪切速率相關(guān)的粘度與有效的 zeta 電位的漸近飽和相關(guān)。圖 6表示該 viscoelectric 效果大大增強的 EDL 的內(nèi)部分的粘度，使得內(nèi)該 EDL的部分看起來像是固定不變的。 圖 6 Lyklema 91,92 假設(shè)這樣的增強粘度的純粹是歸因于電動 fielddependent 溶劑粘度（所建議的方程（ 14），并且是不依賴于當(dāng)?shù)氐膬綦姾擅芏?。然而，這種假設(shè)后來被證明是與最近的實驗沖突和流體力學(xué)滑移和 在電學(xué)理論研究納米通道（見參考文獻(xiàn)第4.1.2 詳細(xì)的討論 93） Bazant 等 93進(jìn)一步指出，高電荷表面會導(dǎo)致里面的 EDL 抗衡的擁擠（在討論中的第 5.2 節(jié)），從而導(dǎo)致表面附近的表觀粘度顯著從單純的本體溶劑偏離。 除了增加的溶劑的粘度，電場也減少了，因為飽和的溶劑的介電常數(shù)作用 94,95 。考慮到在 EDL 中，高電場強度的溶劑的介電常數(shù)可以根據(jù)需要修改 20= 1 B ( ) ddy （ 20） 6 其中 0 是零電場下的介電 常數(shù)溶劑強度和系數(shù) B 描述了介電還原的強度，并估計為 4 10-18 平方米 / V 為室溫水。下的 viscoelectric 效應(yīng)的共同作用和介電減少，可以遵循推導(dǎo)公式（ 16）的過程相似，以得到愛茉莉亥姆霍茲 - Smoluchowski 速度一般版本，如 000f 1 2 ( B f ) a r c t a n f 1 t a n h ( ) 2u f1B B Bsz e z eB C CE k T k T k Tze fC （ 21） 其中明確包括公式（ 16）作為一種特殊情況，當(dāng) B = 0 。 5.2 離子擁擠引起的剪切增稠 在稀的電解質(zhì)溶液，離子物質(zhì)的濃度和的電勢中的 EDL 的一部分漫過充電的表面是泊松 - 玻爾茲曼方程 有關(guān)。在經(jīng)典線性電動現(xiàn)象，橫跨 EDL 的電位降（或所謂的電勢）通?？杀鹊臒犭妷海?/ (ze)BKT ）。然而，實際應(yīng)用中可能遇到情況與大 Zeta 電位。一個例子是，感應(yīng)充電的電動訂單 100 / (ze)BKT一個典型的驅(qū)動電壓下的現(xiàn)象涉及大引起的 Zeta 電位顯著超過熱電壓 93。研究發(fā)現(xiàn)，在這種情況下，泊松 - 玻爾茲曼通過理論預(yù)測的離譜分解高濃度在固體表面上的反離子的。這是由于到嵌入在經(jīng)典的點狀離子的假設(shè)泊松 - 玻耳茲曼方程。然而，離 子具有有限大小的已經(jīng)證明對 EDL 充電重要意義 96和 ACEO 抽 97。對于大量帶電固體表面時，在電解質(zhì)溶液中的反離子變成高度內(nèi)包裝該 EDL 。因此，傳統(tǒng)的泊松 - 波爾茲曼方程將變?yōu)闊o效，并在 EDL 當(dāng)?shù)卣扯纫泊蟠笤黾?。更先進(jìn)的模型需要修改泊松玻耳茲曼方程和 EDL 內(nèi)的粘度。特等等。 98解決了這個問題，他們假設(shè)的擁擠在 EDLmodifies 抗衡傳統(tǒng)的泊松 - 波爾茲曼理論而變稠流體（等效增加了當(dāng)?shù)氐恼扯龋?。為了澄清離子擁擠在里面的離子分布的影響該 EDL，他們所采用的最簡單的改進(jìn)泊松 - 玻爾茲曼 Bikerman 等 99,100的理論。的此外，比介電常數(shù)粘度也假定發(fā)散的抗衡變得非常打包。一個簡單的模型是當(dāng)時配制來描述剪切增稠液體 98 300|(1 )e aze （ 22） 其中e是本地凈電荷密度， a 是離子的有效直徑，0和0分別是該溶液的介電常數(shù)和動力粘度。 對比公式（ 22），以及修改后的泊松 -波爾茲曼理論 Bikerman ，可以推導(dǎo)出有效的電位為 2s g n ( ) l n 1 2 s i n h ( ) 2Be f fBkT zez e k T (23) 7 其中 302 Aa c N （0c為電解液的體積摩爾濃度和AN為阿伏加德羅常數(shù)），表 示散裝溶劑化的離子的體積分?jǐn)?shù)，也可以看作是離子擁擠強度的特性。代入離子的極限（ 0 ）可以很容易地獲得結(jié)果，eff 。公式（ 23）是用一個相當(dāng)簡單的附加參數(shù)介入實際離子尺寸。因此，它無法滿足一些實驗數(shù)據(jù)。隨著越來越多的改進(jìn)的泊松 - 玻爾茲曼理論和更復(fù)雜的帶電引起的剪切增稠的相關(guān)性理論出現(xiàn)，引入一個以上的參數(shù)，用來適應(yīng)所觀察到的離子尺寸效應(yīng)。對于這方面的更詳細(xì)的研究分析，可以參考文獻(xiàn) 93。圖7 呈現(xiàn)由公式（ 23）演算的反映離子擁擠的不同強度的結(jié)果。有效的 Zeta 電位從變成實際的較大值到一個漸近值飽和時對 電勢的有限離子尺寸的影響是很重要的。對于較大的值（更顯著離子擁擠強度），有效的電位達(dá)到比較低的實際 zeta 電位的漸近值。從該圖中得出的結(jié)論是，溶液中的化學(xué)反應(yīng)（ A， Z 和0c ）對有效的電位的影響，與在eff下離子的極限行為（ 0 ）是不同的。 上述離子擁擠引起的穩(wěn)健性剪切增稠的概念是由它與之前驗證預(yù)測增長體積粘度隨散裝電解質(zhì)濃度 101-103是一致的。從圖 7 看出，離子擁擠引起的剪切增稠可以解釋亥姆霍茲 - Smoluchowski 速度通過 z 和0c對溶液化學(xué)反應(yīng)的影響。此外， alongwithmodified 泊松 - 玻爾茲曼理論，也可以適用于模擬在納米結(jié)構(gòu)中電解液的行為，其中離子擁擠的解決方案具有是巨大的意義。 8 譯文原文： Electrokinetics of non-Newtonian fluids: A review 3. Electrophoresis in non-Newtonian fluids. For the electroosmotic flow over a charged surface discussed in Section 2, if we change the system of reference by imagining that the fluid from the charged surface is stationary, and then the charged surface is expected to move with a velocity equal in magnitude but opposition in direction to the previously discussed HelmholtzSmoluchowski velocity. This scenario effectively represents the electrophoretic motion of a particle with thin EDL in an infinitely large non-Newtonian fluid domain 17,18,26,34. Apparently, the previously discussed Helmholtz Smoluchowski velocity of electroosmosis can be naturally applicable to analyzing the electrophoretic velocity of a particle with thin EDL in unbounded non-Newtonian fluid domains, only with the reversion of its sign and the replacement of the zeta potential of charged channel wall with that of charged particle. Actually, the earliest attention paid to electrophoresis of particles in non-Newtonian liquids could be traced back to Somlyody 68 who filed a patent 30 years ago about electrophoretic display which utilizes a non-Newtonian liquid to provide superior threshold characteristics. In 1985, Vidybida and Serikov 69 presented probably the first theoretical study of the electrophoresis of a spherical particle in a non-Newtonian solution. They demonstrated an interesting and counterintuitive effect that the net electrophoretic motion of a particle in non-Newtonian fluids can be induced by an alternating electric field. Then this area of research was left blank for nearly 20 years, and was recently renewed by Hsus group. In 2003, Lee et al. 70 analyzed the electrophoretic motion of a rigid spherical particle in non-Newtonian Carreau fluids enclosed by a spherical cavity with assumptions of low zeta potential andweak applied electric field. They specially paid attention to the electrophoretic characteristics of a spherical particle located at the center of the cavity. Later, the analysis was extended to investigate electrophoresis of spherical particles located at arbitrary position inside the spherical cavity 71. In addition to single particle electrophoresis, Hsu et al. 72 conducted an investigation of the electrophoresis of a concentrated particle dispersion in a Carreau fluid with assumptions of low zeta potentials, and the 9 analysis with arbitrary potentials was done by Lee et al. 73. To investigate the effect of boundary on electrophoresis in non-Newtonian fluids, Lee et al. 74 considered the electrophoresis of a spherical particle in a Carreau fluids normal to a uncharged planar surface, and found that the presence of planar surface enhances the shear-thinning effect and thus the electrophoretic mobility. A similar analysis by Hsu et al. 75was later carried out to investigate the electrophoresis of a spherical particle in a Carreau fluid normal to a large charged disk. In order to more closely simulate the real applications, Hsu et al. 76 analyzed the electrophoresis of a spherical particle in Carreau fluids bounded by a cylindrical microcapillary under the conditions of low zeta potential and weak applied electric field. Many practical electrophoretic applications involve biological particles that are more reasonably represented by rod-like particles, such as protein, and DNA. To this end, Yeh and Hsu 77 extended previous studies on the electrophoresis of a spherical particle in non-Newtonian fluids along the axis of a cylindrical channel to the case of a cylindrical particle (a finite rod). The general conclusion from the studies by Hsus group is that the electrophoretic mobility of a particle is enhanced with a shearthinning fluid and/or a thinner EDL surrounding the particle. This is in consistency with the dependence of electroosmotic velocity on the fluid rheology and EDL thickness presented and reviewed in the previous section. Very recently, Khair et al. 78 demonstrated that the electrophoretic velocity of a uniformly charged particle with a thin EDL in non-Newtonian fluids explicitly depends on shape and size of the particle. This behavior is quite contrary to the electrophoresis in Newtonian fluids. Moreover, it was identified that the stresses outside the EDL (i.e., inside the bulk electroneutral non-Newtonian fluid) are responsible for such complicated dependence. Interestingly, such size and geometry dependence of electrophoresis in non-Newtonian fluids qualitatively coincides with electroosmosis of non-Newtonian fluids as reported in Ref. 20. In summary, it can be concluded that the electrophoretic velocity of particle in a non-Newtonian fluid scales nonlinearlywith the external applied field and the particle zeta potential, resembling the electroosmotic flows of non-Newtonian fluids. Another notable feature is that use of shear-thinning liquids would enhance the electrophoretic mobility of particles and thus leads to fast motion of particles under electric fields. 10 4. Streaming potential effect of non-Newtonian fluids The first attempt made to investigate the streaming effect of non-Newtonian fluids was as early as 1960s. Raza and Marsden 79,80 reported their experimental measurements of the pressure-driven flow of aqueous foams through Pyrex tubes and the associated streaming potential.They observed remarkably high streaming potentials (e.g., 50 V) for nonionic foaming agents. The streaming potential of such large magnitude generates significant electroosmotic flow which resists the pressure-driven flow, thereby being regarded as the main reason for the blockage of foamflow in porous mediums. In order to correlate the experimental results, the foam was assumed to be a non-Newtonian power-law fluid and a theoretical model was then formulated to describe the streaming potential across circular tubes. The calculated streaming potential favorably agrees with the measured streaming potential. In next several decades, this effect apparently has been left unnoticed in the literature. Recently, Bharti et al. 81 theoretically investigated the pressure-driven flow of a power-law liquid in a cylindrical microchannel with electroviscous effects. They numerically estimated the streaming potential field whichwas shown to decrease as increasing the fluid behavior index. Furthermore, it was found that due to the streaming potential effects the flow reduction for shear-thinning fluids is more significant than that for shear-thickening fluids. Utilizing a more general Carreau liquid model, Bharti et al. 82 carried out a similar investigation by considering a pressure-driven flow in a cylindrical micropipe with a contraction-expansion structure. Zhao and Yang83 analyzed the pressure-driven flow of power-law fluids in slit microchannels under the effect of streaming potential. In this analysis, analytical solutions for the streaming potential and velocity field were obtained for arbitrary fluid behavior indices in terms of incomplete gamma functions. Vasu and De 84 studied the similar problem. Exact solutions of streaming potential were obtained for several special values of flow behavior index (such as n = 1, 1/2, 1/3), and numerical solutions were sought for arbitrary values of flow behavior index. Additionally, parametric studies were carried out to assess effects of applied pressure gradient, fluid behavior index, EDL thickness and zeta potential on the apparent viscosity, streaming potential and friction coefficient. Fig. 5 shows the effects of pressure-gradient and the fluid behavior index on the streaming potential. The 11 streaming potential increases with increasing pressure gradient and/or decreasing fluid behavior index, and the dependence of the streaming potential on the fluid behavior index becomes more significant for a larger pressure gradient. Tang et al. 85 numerically investigated the streaming potential effect on pressure-driven non-Newtonian fluid flow in microchannels using the Lattice Boltzmann method. The proposed Lattice Boltzmann method with second-order accuracy was claimed highly efficient for characterizing flow fields of non-Newtonian fluids with sheardependent viscosities. The Lattice Boltzmann method was further utilized by the same group to investigate the streaming potential effect on pressure-driven flows in microporous structures 86. Aside from fundamental interests, streaming potential effect of non-Newtonian inelastic fluids was also recently explored to convert mechanical work to electricity 83,87. Itwas shown that non-Newtonian polymeric solutions of shear-thinning nature can substantially increase the energy conversion efficiency in comparison with Newtonian electrolyte solutions under the same operating conditions. More recently, the use of streaming potential effect for electricity generation was achieved by using non-Newtonian viscoelastic fluids 88. The above analyses generally predict that shear-thinning fluids induce larger streaming potentials than shear-thickening fluids does under a same applied pressure gradient, and correspondingly would experience more significant flow reduction due to streaming potential effect. These findings can provide more physical insight into the characteristics of non-Newtonian fluid flows inmicrochannels, thereby giving rise to better control of the non-Newtonian fluid flows in microfluidic devices. 5. Other non-Newtonian effects in electrokinetics In this section, non-Newtonian effects in electrokinetics including (i) viscoelectric effect, (ii) ion-crowding induced thickening, (iii) electrorheological effect, and (iv) electroviscous effect of colloidal suspensions will be discussed. Non-Newtonian rheology in the first and third cases stems from the electric field dependent dynamic viscosity, and that in the second case refers to the increased viscosity 12 in EDLs due to highly packed ions, and that in the fourth case is due to the flow-induced deformation of EDLs. The origins of these four non-Newtonian effects are different fromthose reviewed in previous three sections which are due to the dependence of dynamic viscosity on the rate of shear. 5.1 Viscoelectric effect in electrokinetics The phenomenon in which the viscosity of liquid varies with the strength of external electric field is referred to as viscoelectric effect which was first experimentally reported by Andrade and Dodd 89,90 and commonly described as 20= + f EE（ ） （ 1 ） （ 12） where 0 is the usual liquid viscosity in absence of external electric fields, E is the strength of local electric field, and f represents the so-called viscoelectric coefficient. For three liquids of chloroform, chlorobenzene and amyl acetate, the viscoelectric coefficients were measured to be 161.89 10 ， 162.12 10 and 162.74 10 22/Vm , respectively 89,90. As suggested by Eq. (12), the local viscosity near highly charged solid surfaces can increase significantly because of 13 the very strong electric field normal to the surface inside the EDL. Lyklema and Overbeek 91, and Lyklema 92 determined experimentally the value of f for water to be 1.02 1015 22/mV and showed that viscoelectric effect would modify the conventional HelmholtzSmoluchowski velocity. Here, we show a simple derivation of the HelmholtzSmoluchowski velocity subjected to viscoelectric effect. When a flat surface with a wall zeta potential of is considered, the HelmholtzSmoluchowski velocity over this surface can be derived as 0 0sduE (13) where is the electric permittivity of solution taken to be a constant at the moment, E0 is the external electric field tangential to the surface, and is the potential distribution inside EDL. In order to obtain an analytical formula for us, an integration of Eq. (13) needs to be carried out. To do so, and should be explicitly correlated. With aid of Eq. (12), the dynamic viscosity of solution inside the EDL can be expressed as 20 1 ( ) df dy （ 14） where d/dy is the double layer field strength (wherein y represents the coordinate normal to the flat surface) and can be derived from the Gouy Chapman solution of EDL potential as (d/dy)2 = C sin 2h ze /2k TB.Then Eq. (13) can be transformed as 00 20 ze1 s i n h ( ) 2sBE dufC kT （ 15） 14 where C = 8nkBT/ (all the symbols have the same definitions as those in Eq. (1), and fC = f/(1/C) can be interpreted as the dimensionless viscoelectric coefficient. It should be noted that the contribution of externally applied electric field to Eq. (14) is negligible since usually |E0| |d/dy|. Finally, one is able to integrate Eq. (15) to obtain an exact formula for the HelmholtzSmoluchowski velocity as 00zea r c t a n 1 t a n h ( ) 221BBsfCE k T k Tu ze fC (16) When f C 1, one can readily show that Eq. (16) reduces to 002 t a n h ( )2Bs BE kT zeu z e k T (17) Clearly the HelmholtzSmoluchowski velocity now becomes nonlinearly dependent on the wall zeta potential because of the viscoelectric effect. When the viscoelectric effect is absent, i.e., f = 0, Eq. (16) naturally reduces to the conventional form of the HelmholtzSmoluchowski velocity given by Eq. (8). For extremely large zeta potentials, i.e., |ze /2 k T | 1B, the HelmholtzSmoluchowski velocity in Eq. (16) reaches an asymptotic value 00a r c t a n 121BsfCE kTuze fC （ 18） Similar to the electroosmosis of non-Newtonian power fluids over flat surface discussed in Section 2, we also define an effective zeta potential a r c t a n 1 t a n h ( ) 221BBe f fzefCk T k Tze fC （ 19） Eq. (16) then can be rewritten in the conventional form of HelmholtzSmoluchowski velocity as in Eq. (8). The results predicted from Eq. (19) are shown in Fig. 6. It is interesting to note the effective zeta potential approaches its asymptotic values at large actual zeta potentials when the viscoelectric effect is present (e.g., fC 0). This behavior is reminiscent of the asymptotic behavior due to the shearthickening behavior of liquids discussed in Section 2. However, the shear-thickening in this case is induced by the EDL electric field as suggested by Eq. (14), while that in Section 2 is induced by the 15 shear-rate dependent viscosity. The asymptotic saturation of effective zeta potential shown in Fig. 6 indicates that the viscoelectric effect tremendously enhances the viscosity of the inner part of the EDL so that the inner part of the EDL looks like immobilized. Lyklema 91,92 assumed that such enhancement of viscosity is attributed purely to the electric fielddependent viscosity of solvent (as suggested by Eq. (14), and is not dependent on the local net charge density. However, this assumption was later shown to be in conflict with recent experimental and theoretical investigations on hydrodynamic slip and electrokinetics in nanochannels (see detailed discussion in Section 4.1.2 of Ref. 93).Bazant et al. 93 further argued that the highly charged surface would result in the crowding of counterions inside EDL (to be discussed in Section 5.2), leading to the apparent viscosity near the surface to significantly deviate from the pure bulk solvent. In addition to increasing the viscosity of solvent, the electric field also reduces the dielectric constant of solvent because of the saturation effect 94,95. Considering the high electric field strength in the EDL, the dielectric constant of solvent can be modified according to 20= 1 B ( ) ddy （ 20） 16 where 0 is the solvent permittivity under the zero electric field strength, and the coefficient B describes the strength of dielectric reduction, and is estimated to be 8 2 24 1 0 /mV for the room-temperature water. Under the joint action of the viscoelectric effect and the dielectric reduction, one can follow a similar procedure of deriving Eq. (16) to obtain amore general version of the HelmholtzSmoluchowski velocity as 000f 1 2 ( B f ) a r c t a n f 1 t a n h ( ) 2u f1B B Bsz e z eB C CE k T k T k Tze fC （ 21） which clearly includes Eq. (16) as a special case when B = 0. 5.2 Ion-crowding induced shear-thickening In dilute electrolyte solutions, the concentration of ionic species and the electric potential in the diffuse part of EDL over a charged surface is related by the PoissonBoltzmann equation. In classical linear electrokinetic phenomena, the potential drop across the EDL (or so-called zeta potential) is typically comparable to the thermal voltage ( / (ze)BkT). However, practical applications may encounter the situations with large zeta potentials. One example is that the induced-charge electrokinetic phenomena under a typical driving voltage of order 100 / (ze)BkT involves large induced zeta potentials significantly exceeding the thermal voltage 93. It is found that under such circumstances the PoissonBoltzmann theory breaks down by predicting ridiculously high concentrations of counterions on the solid surface. This is attributed to the assumption of point-like ions embedded in the classic PoissonBoltzmann equation. However, ions have finite sizes which were already shown to have important implications on EDL charging 96 and ACEO pumping 97. For heavily charged solid surfaces, the counterions in the electrolyte solution become highly packed inside the EDL. Therefore, the conventional PoissonBoltzmann equation becomes invalidated, and the local viscosity in the EDL also drastically increases. More advanced models are required to modify the PoissonBoltzmann equation and the viscosity inside EDL. Bazant et al. 98 addressed this issue and they hypothesized that the crowding of counterions in a EDLmodifies the conventional PoissonBoltzmann theory and thickens the fluids (equivalently increases the local viscosity).To clarify the effects of ionic crowding on the ion distribution inside the EDL, they employed the simplest 17 modified PoissonBoltzmann theory of Bikerman and others 99,100. Furthermore, the rati