非牛顿流体电学：综述外文文献翻译@外文翻译@中英文翻译

1 附录 A 外文翻译 译文： 非牛顿流体电学：综述 3.在非牛顿流体电泳 在第二节讨论了关于电渗流带电表面，如果我们通过想象改变参考系统，带电表面的流体应该是静止的，然后将带电面以速度大小相等但与以前面讨论的亥姆霍兹Smoluchowski 的速度方向相反移动。这种情况下有效地代表了电泳具有很薄的 EDL 的粒子在一个无限大的非运动牛顿流体范围 17,18,26,34 。显然，先前讨论电渗的亥姆霍兹 Smoluchowski 速度当然也可适用于分析在无限大非牛顿流体域具有薄 EDL 颗粒的电泳速度，仅仅与它的符号相反，并改 变了充电通道壁与带电粒子的潜力。 事实上，支付给非牛顿液体粒子电泳最早的关注可以追溯到 30年前 Somlyody 68 提起的一项有关采用非牛顿液体以提供优越的阈值特性的电泳显示器的专利。在 1985年， Vidybida 和 Serikov 69 提出关于球形颗粒的非牛顿电泳研究第一个理论解决方案。他们展示了一个粒子在非牛顿净电泳运动流体可通过以交替的电场来诱导一个有趣的且违反直觉的效果。最近才被 Hsu 课题组填补这方面 20 年的研究空白。在 2003 年，Lee70等人通过一个球形腔的低 zeta电位假设 封闭 andweak施加电场分析了电泳刚性球形颗粒在非牛顿的 Carreau 流体的运动。他们特别重视电泳球形粒子位于中心的空腔特征。之后，该分析被扩展来研究电泳位于内侧的球面的任意位置的球形颗粒的腔体71 。除了单个粒子电泳外， Hsu72等人假设粒子分散潜力在卡罗流体 zeta 进行了集中的电泳调查分析，并分析了由 Lee73完成的其它任意潜力。为了研究在边界上非牛顿流体电泳的影响， Lee74等人分析了电泳球状 粒子在卡罗体液从带电荷到不带电荷的平面表 面，发现平面表面的存在增强了剪切变稀效果，对电 泳迁移率产生影响。类似的分析后来由 Hsu 等 75进行了扩展。为了更紧密地模拟真实的应用环境， Hsu 等人76分析了球形粒子的电泳由一个圆柱形的微细界卡罗流体低 zeta 电位到弱外加电场的条件。许多实际电泳应用涉及生物粒子或是棒状颗粒，比如蛋白和 DNA。为此， Yeh和 Hsu77在延续以往的研究上分析了球状 粒子在非牛顿流体的电泳 沿圆筒形通道的圆柱形颗粒的情况。从 Hsu 课题组研究提出的一般结论是，剪切降粘流体或更薄的 EDL 的周围的粒子可提高电泳迁移率。这与流体流变学和 EDL 厚度的依赖性达成了一致性。最近，海尔哈纳等 78证明了均匀带电粒子在非牛顿流体的电泳速度和 EDL 取决于该粒子的形状和大小。这种行为是完全相反的电泳牛顿流体。此外，经鉴定，外界的应力取决于该 EDL （即散装电中性非牛顿流体内部）。有趣的是，这样的尺寸和电泳的非牛顿流体的几何形状作为定性地重合和报道的文献。 总之，可以断定的是粒子在非牛顿流体秤 nonlinearlywith 外部施加场的粒子的2 电位和电泳速度，以及类似的电渗的非牛顿流体。另一个显着特点是，利用剪切变稀的液体会提高颗粒的电泳迁移率，从而导致粒子在电场中快速运动。 4.非牛 顿流体的潜在作用 早在 20 世纪 60 年代， Raza 和 Marsden79,80报告了他们通过派热克斯管和相关联的流势水性泡沫体的实验获得的压力驱动流的测量，第一次尝试研究非牛顿流体的分流作用。他们观察到非离子型发泡剂具有非常高的流势（如 50 V）。这样大的流动电位规模产生抵抗压力驱动显著作用的电渗流，从而被认为是 foamflow 在多孔介质的堵塞的主要原因。把实验的结果关联起来，该泡沫被认为是一种非牛顿幂律流体和理论模型，然后具有制定描述跨越圆管流的潜在作用。在接下来的几十年里，这种影响显然已经引起了关注。 最近， Bharti 等人 81从理论上研究幂律液体的筒形的压力驱动流微通道电粘性效应。他们估计数字如图减小，增加了流势场 whichwas 流体行为指数。此外，人们发现，由于非牛顿流体的潜在影响，剪切稀化流体比剪切增稠流体的作用更显著。利用更一般的卡罗液体模型， Bharti 等人 82通过考虑在一圆柱形的压力驱动流微管具有收缩 - 扩张结构进行了一项类似的调查。赵存璐和杨春 83 根据流动电位的微影响分析了幂律流体在狭缝中的压力驱动流。在该分析中，对于在不完整的条件下获得对任意流体行为指数的流体的潜力和速度场 做出了解析。 Vasu 和 De 84 研究了类似的问题。通过确切几个特殊值，获得潜在的流体解决方案的流动特性指数（如 n = 1 时， 1/2 ，1/3）和数值解从而求出流动行为指数任意值。此外，他们还对参数进行了研究，这些参数是用来评估所施加的压力的影响梯度，流体行为指数， EDL 厚度和 Zeta 电位的表观粘度，流动电位和摩擦系数。图 5所示的是压力梯度和流体行为指数的关系。随着流动电位升高压力梯度和流体行为指数降低，并且流动电位对流体行为指数的影响比对压力梯度的作用更显著。 图 5 3 Tang 等人 85用格子玻尔兹曼方法研究流动电位在微压驱动的非牛顿流体流动的影响。 Lattice Boltzmann 方法具有高效用于表征非牛顿流体流场与 sheardependent粘度的二阶精度。 Lattice Boltzmann 方法是进一步利用由同一组用来研究流动电位在微孔结构的压力驱动流 86的影响。除了这些，非牛顿流体潜在作用最近还用来探讨了非弹性流体转换工作机械电力 83,87。 Itwas 表示，相比转换效率相同的操作条件下的解 决方案，非牛顿型聚合物溶液的剪切变稀的性质可以大幅提高能源与牛顿电解质。最近，使用非牛顿流的潜在影响发电是由非牛顿粘弹性流体 88来实现的。 上述分析普遍预测，由于非牛顿流体的潜在效果，剪切变稀流体比剪切增稠液体在一个相同的施加压力梯度下，诱发更大的流势，并相应地大大减少流体流量。这些发现可以提供更多的关于非牛顿流体的物理洞察的特点，由此更有效地控制微流体的非牛顿流体流动的设备。 5.其他电学上非牛顿效应 在本节中，将讨论包括（） viscoelectric 效果，（）离子拥挤引起的增厚，（）胶体的 电流变效应（）电粘性效应悬浮液的电学非牛顿效应。在非牛顿流体流变中，第一和第三种情况是源于电场动态粘度的依赖性，而第二种情况指的是在 EDL 中由于高度填充的离子粘度增加，同时，第四种情况是由于 EDL 中粒子的流动而引起的变形。非牛顿的这四个效应均说明了动态粘度对剪切速率有影响。 5.1 电学上的 Viscoelectric 效果 在非牛顿液体中的粘度与外部电场的强度变化的现象被称为 viscoelectric 效果。在第一份该实验报告中， Andrade 和 Dodd89,90通常用公式描述这种现象为 20= + f EE（ ） （ 1 ） （ 12） 其中0 是在没有外部电气的液体粘度， E 是局部电场的强度，并且 f 表示的是viscoelectric 系数。对于氯仿，氯苯和乙酸戊酯三种液体， viscoelectric 系数 89,90分别测得为 161.89 10 ， 162.12 10 和 162.74 10 22/Vm 。根据公式（ 12），因为附近高电荷的固体表面，所述粘度可以显著提高垂直于 EDL 内表面上的电场。 Lyklema，Overbeek 91和 Lyklema 92通过实验确定 viscoelectric 系数的值是 1.02 10-15 22/mV 并表明 viscoelectric 系数将改变传统的亥姆霍兹 Smoluchowski 速度。在这里，我们展示的一个简单的推导受到 viscoelectric 影响的亥姆霍兹 - Smoluchowski 速度。当一个平坦的表面电位被认为是的壁，超过这个表面可推导出亥姆霍兹 - Smoluchowski 速度 91,92 0 0sduE (13) 4 其中，为取为恒定的溶液的介电常数，0E是与表面相切的外部电场，并且 为里面的EDL 的分布电位。为 了得出一个分析公式，我们将整合公式（ 13）。在这里 ,将 和 明确相关，方程（ 12）所述的 EDL 内溶液的动态粘度可以表示为如下 20 1 ( ) df dy （ 14） 其中 d /dy 是双层的场强（其中 y代表坐标垂直于平的表面），并且可以从 Gouy Chapman推导出 EDL潜在的查普曼解决方案 22( | d / d y | ) s i n h z e / 2 T BCk。则方程（ 13）可以转化为 00 20 ze1 s i n h ( ) 2sBE dufC kT （ 15） 其中 8/BC n k T （所有的符号与公式（ 1）具有相同的定义），和 / (1 / C )fC f 可以被解释为无量纲的 viscoelectric 系数。应当指出的是，因为通常0|E | |d /dy|，贡献的外加电 场公式（ 14）是可忽略的 。最后，得到用于亥姆霍兹 Smoluchowski 一个确切的配方速度为 00zea r c t a n 1 t a n h ( ) 221BBsfCE k T k Tu ze fC (16) 当 1fC 根据公式（ 16）可以推导出 002 t a n h ( )2Bs BE kT zeu z e k T (17) 显然，亥姆霍兹 - Smoluchowski 速度现在变成非线性的关系，因为上壁的 Zeta 电位存在 viscoelectric 效果。当 viscoelectric 效果是不存在时，即， F = 0，自然降低到常规形式亥姆霍兹 Smoluchowski 速度，由方程（ 8）给出公式（ 16）。对于非常大的 Zeta 势，即 |ze /2 k T | 1B ，亥姆霍兹 - Smoluchowski 速度在方程（ 16）达到渐近值 00a r c t a n 121BsfCE kTuze fC （ 18） 在第 2节中讨论过平坦的表面的类似的非牛顿流体动力的电渗 ，我们还定义了一个有效的 Zeta 电位 5 a r c t a n 1 t a n h ( ) 221BBe f fzefCk T k Tze fC （ 19） 公式（ 16）然后可以在常规的形式重写亥姆霍兹 Smoluchowski 速度。方程（ 8）。预测结果从方程（ 19）如图 6 所示，这是值得注意的是在 Zeta 接近其渐近值，由于viscoelectric 效果 Zeta 电位存在有效电势（例如， 0fC ）。这行为是让人联想到第2 节中所讨论的由于 shearthickening 作用的液体的渐近行为，但剪切增稠在这种情 况下，根据公式（ 14），在第 2 诱导的剪切速率相关的粘度与有效的 zeta 电位的渐近饱和相关。图 6表示该 viscoelectric 效果大大增强的 EDL 的内部分的粘度，使得内该 EDL的部分看起来像是固定不变的。 图 6 Lyklema 91,92 假设这样的增强粘度的纯粹是归因于电动 fielddependent 溶剂粘度（所建议的方程（ 14），并且是不依赖于当地的净电荷密度。然而，这种假设后来被证明是与最近的实验冲突和流体力学滑移和 在电学理论研究纳米通道（见参考文献第4.1.2 详细的讨论 93） Bazant 等 93进一步指出，高电荷表面会导致里面的 EDL 抗衡的拥挤（在讨论中的第 5.2 节），从而导致表面附近的表观粘度显著从单纯的本体溶剂偏离。 除了增加的溶剂的粘度，电场也减少了，因为饱和的溶剂的介电常数作用 94,95 。考虑到在 EDL 中，高电场强度的溶剂的介电常数可以根据需要修改 20= 1 B ( ) ddy （ 20） 6 其中 0 是零电场下的介电 常数溶剂强度和系数 B 描述了介电还原的强度，并估计为 4 10-18 平方米 / V 为室温水。下的 viscoelectric 效应的共同作用和介电减少，可以遵循推导公式（ 16）的过程相似，以得到爱茉莉亥姆霍兹 - 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） 其中明确包括公式（ 16）作为一种特殊情况，当 B = 0 。 5.2 离子拥挤引起的剪切增稠 在稀的电解质溶液，离子物质的浓度和的电势中的 EDL 的一部分漫过充电的表面是泊松 - 玻尔兹曼方程 有关。在经典线性电动现象，横跨 EDL 的电位降（或所谓的电势）通常可比的热电压（ / (ze)BKT ）。然而，实际应用中可能遇到情况与大 Zeta 电位。一个例子是，感应充电的电动订单 100 / (ze)BKT一个典型的驱动电压下的现象涉及大引起的 Zeta 电位显著超过热电压 93。研究发现，在这种情况下，泊松 - 玻尔兹曼通过理论预测的离谱分解高浓度在固体表面上的反离子的。这是由于到嵌入在经典的点状离子的假设泊松 - 玻耳兹曼方程。然而，离 子具有有限大小的已经证明对 EDL 充电重要意义 96和 ACEO 抽 97。对于大量带电固体表面时，在电解质溶液中的反离子变成高度内包装该 EDL 。因此，传统的泊松 - 波尔兹曼方程将变为无效，并在 EDL 当地粘度也大大增加。更先进的模型需要修改泊松玻耳兹曼方程和 EDL 内的粘度。特等等。 98解决了这个问题，他们假设的拥挤在 EDLmodifies 抗衡传统的泊松 - 波尔兹曼理论而变稠流体（等效增加了当地的粘度）。为了澄清离子拥挤在里面的离子分布的影响该 EDL，他们所采用的最简单的改进泊松 - 玻尔兹曼 Bikerman 等 99,100的理论。的此外，比介电常数粘度也假定发散的抗衡变得非常打包。一个简单的模型是当时配制来描述剪切增稠液体 98 300|(1 )e aze （ 22） 其中e是本地净电荷密度， a 是离子的有效直径，0和0分别是该溶液的介电常数和动力粘度。 对比公式（ 22），以及修改后的泊松 -波尔兹曼理论 Bikerman ，可以推导出有效的电位为 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为电解液的体积摩尔浓度和AN为阿伏加德罗常数），表 示散装溶剂化的离子的体积分数，也可以看作是离子拥挤强度的特性。代入离子的极限（ 0 ）可以很容易地获得结果，eff 。公式（ 23）是用一个相当简单的附加参数介入实际离子尺寸。因此，它无法满足一些实验数据。随着越来越多的改进的泊松 - 玻尔兹曼理论和更复杂的带电引起的剪切增稠的相关性理论出现，引入一个以上的参数，用来适应所观察到的离子尺寸效应。对于这方面的更详细的研究分析，可以参考文献 93。图7 呈现由公式（ 23）演算的反映离子拥挤的不同强度的结果。有效的 Zeta 电位从变成实际的较大值到一个渐近值饱和时对 电势的有限离子尺寸的影响是很重要的。对于较大的值（更显著离子拥挤强度），有效的电位达到比较低的实际 zeta 电位的渐近值。从该图中得出的结论是，溶液中的化学反应（ A， Z 和0c ）对有效的电位的影响，与在eff下离子的极限行为（ 0 ）是不同的。 上述离子拥挤引起的稳健性剪切增稠的概念是由它与之前验证预测增长体积粘度随散装电解质浓度 101-103是一致的。从图 7 看出，离子拥挤引起的剪切增稠可以解释亥姆霍兹 - Smoluchowski 速度通过 z 和0c对溶液化学反应的影响。此外， alongwithmodified 泊松 - 玻尔兹曼理论，也可以适用于模拟在纳米结构中电解液的行为，其中离子拥挤的解决方案具有是巨大的意义。 8 译文原文： 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