颗粒气泡间相互作用力及液膜薄化破裂动力学是揭示浮选黏附机理的核心问题，也是近年来浮选胶体化学领域的研究热点。从热力学角度讲，水相中气泡会在固体表面形成有限接触角，其中气液固三相润湿周边铺展状态可由Young方程[1]描述:
γSG=γSL+γLGcos θ
(1)
式中，γSG为固气界面自由能;γSL为固液界面自由能;γLG为气液界面自由能;θ为接触角。
浮选气泡与固体颗粒发生黏附过程中的吉布斯自由能变化ΔG可由Dupre方程[1]来表示，联合Young方程可得
ΔG=γLG(1-cos θ)
(2)
由式(2)可知，一旦固体表面接触角大于0，气泡可在其表面成功黏附并达到热力学稳定状态。然而，在实际浮选中，只有疏水性颗粒才能被气泡捕获，造成此矛盾的原因是热力学方程没有考虑颗粒气泡黏附的动力学特性，黏附过程中的作用力及排液动力学不容忽视。基于此，对当前颗粒气泡间相互作用力及液膜排液动力学模型理论研究进展进行了系统综述。
1 颗粒气泡间相互作用力
1.1 DLVO力
根据经典DLVO理论，颗粒气泡黏附应由范德华力及双电层力共同支配[2-3]。当排斥力主导时，颗粒气泡间形成稳定液膜;当吸引力主导时，液膜会发生薄化破裂。对于颗粒气泡体系，单位面积的范德华力Πd通常可以由Hamaker方法或Lifshitz连续性理论表示。在不考虑延迟效应条件下有
(3)
其中，A132为以水相为中间介质的颗粒气泡间Hamaker常数，下标1代表颗粒，2代表气泡，3代表水相介质;h为分离距离。
A132可以由物质介电常数及光学折射率来确定，可近似为
(4)
式中，A11为颗粒在真空介质中的Hamaker常数;A22为气泡在真空介质中的Hamaker常数;A33为水在真空介质中的Hamaker常数。
对于单位面积的双电层力Πe，可以由恒定表面电势条件下的Hogg-Healy-Fuerstenau近似公式[3-4]计算:
(5)
式中，ε为水的介电常数;ε0为真空介电常数;κ-1为德拜长度;ψ1和ψ2为颗粒及气泡表面电势。在计算中通常使用Zeta电位来替代表面电势。
一般来说，浮选颗粒气泡表面均呈电负性，体系Hamaker常数始终为正值，因此范德华及双电层力均表现为排斥作用，颗粒气泡间液膜应一直处于稳定状态。由此可见，经典DLVO理论可以诠释亲水性颗粒的浮选行为但并不能有效的解释疏水性颗粒气泡间的黏附机理，体系必须存在着第3种力(疏水力)以克服颗粒气泡间DLVO斥力诱发黏附。
1.2 疏水力及其来源机制
LASKOWSKI 和KITCHENER[5]首次发现了甲基化石英表面润湿膜的自发薄化破裂现象，并猜测疏水化石英基板与气泡间可能存在一种吸引力克服排斥性的范德华及双电层力。BLAKE 和KITCHENER[6]进一步测量了气泡与不同接触角的石英基板间的液膜临界破裂厚度，发现疏水性石英表面的润湿膜破裂厚度在60～220 nm变化，指出与传统DLVO力相比该吸引力具有更长的作用范围。1982年，PASHLEY和ISRAELACHVILI[7-8]首次实现了疏水力的定量表示，他们利用表面力仪(SFA)测量了十六烷基三甲基溴化铵(CTAB)存在体系中云母间的相互作用力，发现疏水力以1 nm的衰变长度按单指数模型衰减。自此，众多学者相继利用SFA，原子力显微镜(AFM)及薄膜压力平衡技术(TFB)对疏水力进行了测量[9-28]。然而，目前关于疏水力的来源机制及大小仍是一个充满争议的课题，疏水样品的制备方法及力测试手段均对结果有着很大影响，难以找到一个通用机制可以适用并概括所有的疏水力测试结果[29]。
21世纪初期，CHRISTENSON和CLAESSON[27]、MEYER等[30]和XING等[29]详细探讨了疏水力的来源机制，其中较具代表性的来源机制有:固水界面水分子重排熵效应、局部电荷波动、纳米气泡桥接及亚稳态液膜分离诱导空化作用，如图1所示。
图1 疏水力来源机制[30]Fig.1 Possible mechanisms for hydrophobic force[30]
固水界面水分子重排可能是诱发短程疏水力的主要原因。水分子难以与疏水固体表面分子发生氢键作用，疏水界面的存在会打散体相水分子氢键笼形网状结构。因此，水分子趋向于恢复原有笼形配置使得疏水界面会受到“排挤”作用产生疏水力。MEZGER等[31]利用高能X射线反射技术对十八烷基三氯硅烷(OTS)覆盖的二氧化硅-水界面的水分子结构和密度云图进行了检测(图2)，X射线穿过系列复合折射透镜(CRL)完成聚焦，发现界面存在0.5 nm左右的水分子排空区，这也直接说明了疏水性OTS与水分子间存在一定的排斥作用，界面水分子处于热力学不稳定状态。ERIKSSON等[32]则推导了基于水分子结构有序性的疏水力模型，随着分离距离的减小水分子有序性系数增加，模型能够很好的预测疏水力行为。HAMMER 等[33]指出疏水表面对界面水分子氢键排列顺序的影响仅在几个水分子层内起作用(0～1 nm)，这意味着水分子重排只能诱发短程疏水力。TABOR 等[34]使用AFM对油滴间相互作用力进行了测试，通过配制与油滴折射率相等的溶剂来有效屏蔽范德华力，试验疏水力按0.3 nm(水分直径)的衰减长度衰减，进一步验证了基于水分子重排的短程疏水力假说，然而仍然存在部分文献结果与该假说相矛盾[12-13,35]。如熵起源力应受系统温度的影响，ISHIDA等[19]发现短程疏水力并不具有温度响应行为。
图2 MEZGER等实验系统及测试结果[31]
Fig.2 Experimental setup and results of MEZGER etc[31]
基于双分子层荷电补丁的静电作用是疏水力的另一种可能来源机制[36]，但此假说主要针对表面活性体系。具体来说，覆盖在云母表面的表面活性剂单分子层逐渐向不完整的双层膜结构转变并最终形成荷正电双层膜与电负性裸露云母微区相间的新表面，如图1(c)所示。在云母相互靠近过程中，上下表面间电荷相反微区的对齐排列最终导致长程静电引力作用。MIKLAVCIC等[37-38]曾对恒定电势及恒定电量边界条件下非均质表面间的双电层力数学表达式进行了推导，表面电荷的非均匀排列可产生较强的引力作用。然而，表面活性剂系统中的短程吸引力来源仍然无法得到合理的解释。
CHRISTENSEN和CLAESSON[26]认为疏水界面间亚稳态液膜空化作用是产生长程疏水引力的原因。利用SFA对碳氢化合物及氟碳化合物表面间的表面力进行了测量，通过监视中间介质的光学折射率变化发现了空化气泡的生成。但空化是在界面接触前还是在接触后产生仍存在一些争议，YAKUBOV等[39]及AZADI 等[40]发现AFM力曲线中疏水颗粒间的跳入黏附距离随着接近次数的增加逐渐增加至最大，如图3所示。这说明第1次颗粒间接触后的分离压降诱导了空化气泡的形成，再次接触时由于界面空化气泡的存在产生长程疏水力。FAGHIHNEJAD和ZENG[41]则使用SFA-正置光学显微镜连用系统对不同电解质浓度下的聚苯乙烯表面间的相互力进行了测试，在样品接近过程中观察到了空化气泡的形成。
图3 去离子水中不同接近次数下的疏水化石英小球与硅板
之间的力曲线[40]
Fig.3 Measured force curves for consecutive approaches in deionized water between silicon wafer and silica spheres after hydrophobic treatment[40]
纳米气泡桥接也是较具代表性的长程疏水力来源机制之一，并在21世纪开始受到了学者们的广泛关注。PARKER等[17]首次指出疏水固体表面的亚微米气泡是导致阶梯状AFM力曲线的原因。ISHIDA等[42]和LOU等[43]使用轻敲模式AFM实现了界面纳米气泡成像，采用非侵入性光学检测手段[44-46]也相继发现了纳米气泡使得其存在性已被学者们广为接受[23,47-51]。如果长程疏水力来源于纳米气泡桥接，那么水相中溶解气体及纳米气泡性质会对力学行为产生影响。TYRRELL和ATTARD[52]发现疏水石英胶体探针与玻璃基板间的黏附距离与表面纳米气泡的高度相当。一些研究也表明脱气处理会显著减小疏水力作用程[41,53-57]，但脱气处理后仍然可以检测到一种明显强于范德华力的短程疏水力。此外，学者们发现毛细力数学模型可以很好的拟合长程疏水吸引力，间接证明了疏水力的纳米气泡起源[47,58-60]。MISHCHUK等[61-62]认为纳米气泡的存在会显著改变中间介质的介电常数，进而影响范德华力及双电层力，仅需对传统DLVO力进行适当修正即可对疏水力做出合理的解释。
FAGHIHNEJAD和ZENG[41]提出了一种三区段疏水力模型，不同的区域内的疏水力形成机制不同，如图4所示。包括:由微米及亚微米气泡桥接或荷电补丁引起的长程引力区(20 nm到几百纳米)，纳米气泡桥接或Hamaker常数强化引起的中等作用区(几纳米到20 nm)及固液界面水分子结构效应造成的短程引力区。亚微米气泡及纳米气泡桥接从本质上讲同属于毛细作用力，同时考虑到荷电补丁假说的应用体系局限性，疏水力的来源机制可以更广义的概括为基于亚微米/纳米气泡桥接引发的长程力和基于固液界面水分子重排引发的短程力。
图4 三区段疏水力模型[41]
Fig.4 Three-region hydrophobic interaction model[41]
疏水力通常可以使用指数衰减或幂式衰减模型来表示。其中单位面积疏水力的双指数模型[2-3,63]表达式为
(6)
式中，Πh为单位面积疏水力;C1为长程疏水力常数;D1为长程疏水力衰变长度;C2为短程疏水力常数;D2为短程疏水力衰变长度;h为分离距离。
目前大部分疏水力来源机制及力行为研究均聚焦于固-固系统，但疏水力作为颗粒气泡间液膜薄化破裂的驱动力已经达成了共识[29]。目前的主要挑战是如何定量颗粒气泡间疏水作用力。一方面，由于其强烈的吸引性作用，传统力测试技术如AFM及SFA很难在准静态模式下操作，当吸引力梯度超过仪器的弹簧的弹性系数时，跳入黏附过程中的有效力信息难以提取。另一方面，由于气液界面的变形效应，颗粒气泡间绝对分离距离的确定存在困难。
2 颗粒气泡间流体动力学排液模型
学者们通常使用雷诺润滑理论来描述颗粒气泡间液膜薄化动力学行为，其中流体力学边界条件对液膜排液动力学方程有着显著影响。对于气液界面，研究发现极微量的表面活性剂可使得气液界面变成无滑移界面[64]。对于固液界面，一般认为亲水性表面为无滑移界面[65];而疏水性表面则遵守Navier滑移准则[66]，但仍有很多学者将疏水界面看做无滑移界面处理[67-68]。目前，经常使用的代表性排液模型包括:Stefan-Reynolds平坦膜模型，Taylor模型及Stokes-Reynolds-Young-Laplace(SRYL)模型。
Stefan-Reynolds模型是基于平行液膜假设的条件下获得的。在无滑移边界条件下，平坦液膜排液动力学方程可表示为
(7)
式中，t为时间;ΔP为液膜及液体体相压力差;μ为液体黏度;Rf为液膜半径。
众多排液试验发现在颗粒气泡接近过程中气液界面经常存在涟漪变形[69]，Stefan-Reynolds模型未考虑涟漪形成后的流体阻力，因此会对排液速率产生过高估计。MANICA等[70]认为Stefan-Reynolds模型因其内在的不连续性不能准确描述液膜薄化动力学。
Taylor模型最早用于描述固体小球接近接近固体基板过程中的液膜薄化行为。在无滑移边界条件下有[71-73]:
(8)
式中，Fh为流体阻力;R为小球半径。
当固体小球被气泡所替代，并进一步考虑范德华力及静电力作用时[72]，则有:
(9)
式中，ρ为水的密度;g为重力加速度。
Taylor模型的局限性同样在于没有考虑气泡接近过程中的变形效应，气泡内部的拉普拉斯压力难以克服排液过程中产生的流体阻力。对于Stefan-Reynolds和Taylor模型，均需要借助薄膜干涉技术观测液膜厚度的时空演化，通过拟合排液试验数据来获得颗粒气泡间相互作用力。
SRYL模型则在描述液膜薄化速率的同时，兼顾了气泡表面在流体力和表面力等外力作用下的变形行为。在给定起始与边界条件下，SRYL模型通过数值迭代法与液膜排液试验测试结果对比，可以计算出颗粒气泡间的相互作用力信息;也可通过与相互作用力试验结果对比获得液膜排液数据。IVANOV等[74]首次构建了SRYL控制方程。CHAN等[75]在此基础上将表面力项加入到了传统SRYL模型中。至此，SRYL模型使得力及液膜排液动力学真正意义上的实现了统一，其描述了颗粒气泡间作用力、界面变形及液膜排液3者间的动态耦合关系。模型控制方程主体为2部分:1部分是描述液膜排液的SR方程;另1部分是描述界面变形的YL方程。无滑移边界条件下，SR排液方程和YL变形方程可以表示为
(10)
(11)
式中，
为等效半径;p(r,t)为液膜内部与外围体相的流体压力差;Π(r,t)为单位面积的总表面力，含范德华力，静电力及疏水力。
由YL方程可知，当流体力和表面力为排斥力时，气液界面为了维持力平衡状态将向气泡圆心方向变形，当流体压力和分离压力之和大于气泡内部拉普拉斯压力时，此时界面会形成浅凹变形。相反，当流体压力和分离压力之和为吸引力，气液界面将背离圆心方向形成凸起。
根据Derjaguin近似原理对流体压力和单位面积表面力在[0,∞]区间内积分可得颗粒气泡间的动态相互作用力,即
F(t)=2π
[p(r,t)+Π(r,t)]rdr
(12)
其中，F(t)为颗粒气泡间相互作用力，在AFM体系下，探针所测得的力即为F(t)。
使用Matlab ODE15S标准软件包可完成对上述SRYL偏微分方程的数值求解，需要注意的是求解前需要使用毛细数对各变量进行无量纲化处理。
3 结 论
(1)对于颗粒气泡间相互作用力，疏水引力可克服颗粒气泡间范德华力和静电斥力，诱发黏附。不同作用程范围内疏水力的来源机制不同:长程疏水力(>20 nm)主要源于固液界面亚微米/纳米气泡桥接，而短程疏水力(<20 nm)则主要源于固液界面水分子重排效应。由于疏水力强烈的吸引性和气液界面的变形效应，颗粒气泡间疏水力的定量表征仍存在较大的挑战。
(2)对于颗粒气泡间液膜排液动力学模型，最具代表性的有Stefan-Reynolds平坦膜模型，Taylor模型和SRYL模型。Stefan-Reynolds及Taylor模型并未考虑排液过程中气泡表面曲率的变化，其应用存在着较大的局限性。SRYL模型则在描述液膜薄化速率的同时，兼顾了气泡表面在流体力和表面力等外力作用下的变形行为。在给定起始与边界条件下，SRYL模型通过数值迭代法与液膜排液试验测试结果对比，可以计算出颗粒气泡间的相互作用力信息;也可通过与相互作用力试验结果对比获得液膜排液数据。在今后的研究中，应重点将SRYL模型与试验测试相结合，对颗粒气泡间疏水力进行定量表征，揭示浮选黏附机理。
[1] BUTT H J,GRAF K,KAPPL M.Physics and Chemistry of Interfaces[M].Weinheim:Wiley-VCH Verlag GmbH & Co.KGaA,2003.
[2] ISRAELACHVILI J.Intermolecular and Surface Forces[M].Third Edition.USA:Elsevier Pte Ltd,2012.
[3] BUTT H J,KAPPL M.Surface and Interfacial Forces[M].Weinheim:Wiley-VCH Verlag GmbH & Co.KGaA,2010.
[4] GUI X H,XING Y W,RONG G Q,et al.Interaction forces between coal and kaolinite particles measured by atomic force microscopy[J].Powder Technology 2016,301:349-355.
[5] LASKOWSKI J,KITCHENER J A.The hydrophilic-hydrophobic tr-ansition on silica[J].Journal of Colloid and Interface Science,1969,29(4):670-679.
[6] BLAKE T D,KITCHENER J A.Stability of aqueous films on hydrophobic methylated silica[J].Journal of the Chemical Society,Faraday Transactions 1:Physical Chemistry in Condensed Phases,1972,68(8):1435-1442.
[7] ISRAELACHVILI J,PASHLEY R.The hydrophobic interaction is long-range,decaying exponentially with distance[J].Nature,1982,300(5890):341-342.
[8] ISRAELACHVILI J,PASHLEY R M.Measurement of the hydrophobic interaction between two hydrophobic surfaces in aqueous electrolyte solutions[J].Journal of Colloid and Interface Science,1984,98(2):500-514.
[9] RABINOVICH Y I,YOON R H.Use of atomic-force microscope for the measurements of hydrophobic forces between silanated silica plate and glass sphere[J].Langmuir,1994,10(6):1903-1909.
[10] CLAESSON P M,BLOM C E,HERDER P C,et al.Interactions between water-stable hydrophobic Langmuir-Blodgett monolayers on mica[J].Journal of Colloid and Interface Science,1986,114(1):234-242.
[11] CHRISTENSON H K,CLAESSON P M,BERG J,et al.Forces between fluorocarbon surfactant monolayers-salt effects on the hydrophobic interaction[J].Journal of Physical Chemistry,1989,93(4):1472-1478.
[12] CHRISTENSON H K,PARKER J L,YAMINSKY V V.Interactions between hydrophobic surfaces-dependence on temperature and alkyl chain-length-Comment[J].Langmuir,1992,8(8):2080-2080.
[13] TSAO Y H,YANG S X,EVANS D F,et al.Interactions between hydrophobic surfaces-dependence on temperature and alkyl chain-length[J].Langmuir,1991,7(12):3154-3159.
[14] TSAO Y H,EVANS D F,WENNERSTROM H.Long-range attractive force between hydrophobic surfaces observed by atomic force microscopy[J].Science,1993,262(5133):547-550.
[15] YAMINSKY V,JONES C,YAMINSKY F,et al.Onset of hydrophobic attraction at low surfactant concentrations[J].Langmuir,1996,12(15):3531-3535.
[16] WANG J L,YOON R H.AFM forces measured between gold surfaces coated with self-assembled monolayers of 1-hexadecanethiol[J].Langmuir,2008,24(15):7889-7896.
[17] PARKER J L,CLAESSON P M,ATTARD P.Bubbles,cavities,and the long-ranged attraction between hydrophobic surfaces[J].The Journal of Physical Chemistry,1994,98(34):8468-8480.
[18] ZHANG J H,YOON R H,ERIKSSON J C.AFM surface force measurements conducted with silica in CnTACl solutions:Effect of chain length on hydrophobic force[J].Colloids and Surfaces A:Physicochemical and Engineering Aspects,2007,300(3):335-345.
[19] ISHIDA N,KUSAKA Y,USHIJIMA H.Hydrophobic attraction between silanated silica surfaces in the absence of bridging bubbles[J].Langmuir,2012,28(39):13952-13959.
[20] ERIKSSON J C,LJUNGGREN S.Alleged formation of bridging cavities bubbles between planar hydrophobic surfaces-Comment[J].Langmuir,1995,11(6):2325-2328.
[21] CARAMBASSIS A,JONKER L C,ATTARD P,et al.Forces measured between hydrophobic surfaces due to a submicroscopic bridging bubble[J].Physical Review Letters,1998,80(24):5357-5360.
[22] ATTARD P.Bridging bubbles between hydrophobic surfaces[J].Langmuir,1996,12(6):1693-1695.
[23] ATTARD P.Nanobubbles and the hydrophobic attraction[J].Advances in Colloid and Interface Science,2003,104:75-91.
[24] WANG L G,YOON R H.Hydrophobic forces in the foam films stabilized by sodium dodecyl sulfate:Effect of electrolyte[J].Langmuir,2004,20(26):11457-11464.
[25] CRAIG V S J.An historical review of surface force measurement techniques[J].Colloids and Surfaces A:Physicochemical and Engineering Aspects,1997,129:75-93.
[26] CHRISTENSEN H,CLAESSON P.Cavitation and the interaction between macroscopic surfaces[J].Science,1988,239:390-392.
[27] CHRISTENSON H K,CLAESSON P M.Direct measurements of the force between hydrophobic surfaces in water[J].Advances in Colloid and Interface Science,2001,91(3):391-436.
[28] PASHLEY R M,MCGUIGGAN P M,NINHAM B W,et al.Attractive forces between uncharged hydrophobic surfaces-Direct measurements in aqueous-solution[J].Science,1985,229(4718):1088-1089.
[29] XING Y W,GUI X H,CAO Y J.The hydrophobic force for bubble-particle attachment in flotation-a brief review[J].Physical Chemistry Chemical Physics,2017,19(36):24421-24435.
[30] MEYER E E,ROSENBERG K J,ISRAELACHVILI J.Recent progress in understanding hydrophobic interactions[J].Proceedings of the National Academy of Sciences,2006,103(43):15739-15746.
[31] MEZGER M,REICHERT H,SCHODER S,et al.High-resolution in situ x-ray study of the hydrophobic gap at the water-octadecyl-trichlorosilane interface[J].Proceedings of the National Academy of Sciences,2006,103(49):18401-18404.
[32] ERIKSSON J C,LJUNGGREN S,CLAESSON P M.A phenomenological theory of long-range hydrophobic attraction forces based on a square-gradient variational approach[J].Journal of the Chemical Society Faraday Transactions,1989,85:163-176.
[33] HAMMER M U,ANDERSON T H,CHAIMOVICH A,et al.The search for the hydrophobic force law[J].Faraday Discussions,2010,146:299-308.
[34] TABOR R F,WU C,GRIESER F,et al.Measurement of the hydrophobic force in a soft matter system[J].Journal of Physical Chemistry Letters,2013,4(22):3872-3877.
[35] MASTROPIETRO D J,DUCKER W A.Forces between hydrophobic solids in concentrated aqueous salt solution[J].Physical Review Letters,2012,108(10):1061011-1061015.
[36] MEYER E E,LIN Q,HASSENKAM T,et al.Origin of the long-range attraction between surfactant-coated surfaces[J].Proceedings of the National Academy of Sciences,2005,102(19):6839-6842.
[37] MIKLAVIC S J,CHAN D Y C,WHITE L R,et al.Double-layer forces between heterogeneous charged surfaces[J].Journal of Physical Chemistry,1994;98(36):9022-9032.
[38] MIKLAVCIC S J.Double-layer forces between heterogeneous cha-rged surfaces-the effect of net surface-charge[J].Journal of Chemical Physics,1995,103(11):4794-4806.
[39] YAKUBOV G E,BUTT H J,VINOGRADOVA O I.Interaction forces between hydrophobic surfaces.Attractive jump as an indication of formation of “stable” submicrocavities[J].The Journal of Physical Chemistry B,2000,104(15):3407-3410.
[40] AZADI M,NGUYEN A V,YAKUBOV G E.Attractive forces between hydrophobic solid surfaces measured by AFM on the first approach in salt solutions and in the presence of dissolved gases[J].Langmuir,2015,31(6):1941-1949.
[41] FAGHIHNEJAD A,ZENG H B.Hydrophobic interactions between polymer surfaces:Using polystyrene as a model system[J].Soft Matter,2012,8(9):2746-2759.
[42] ISHIDA N,INOUE T,MIYAHARA M,et al.Nano bubbles on a hydrophobic surface in water observed by tapping-mode atomic force microscopy[J].Langmuir,2000,16(16):6377-6380.
[43] LOU S T,OUYANG Z Q,ZHANG Y,et al.Nanobubbles on solid surface imaged by atomic force microscopy[J].Journal of Vacuum Science & Technology B Microelectronics & Nanometer Structures Processing Measurement & Phenomena,2000,18(5):2573-2575.
[44] HAIN N,WESNER D,DRUZHININ S I,et al.Surface nanobubbles studied by time-resolved fluorescence microscopy methods combined with AFM:The impact of surface treatment on nanobubble nucleation[J].Langmuir,2016,32(43):11155-11163.
[45] KARPITSCHKA S,DIETRICH E,SEDDON J R,et al.Nonintrusive optical visualization of surface nanobubbles[J].Physical Review Letters,2012,109(6):0661021-0661025.
[46] SEO D,GERMAN S R,MEGA T L,et al.Phase state of interfacial nanobubbles[J].The Journal of Physical Chemistry C,2015,119(25):14262-14266.
[47] HAMPTON M A,NGUYEN A V.Nanobubbles and the nanobubble bridging capillary force[J].Advances in Colloid and Interface Science,2010,154(1-2):30-55.
[48] CRAIG V S J.Very small bubbles at surfaces-the nanobubble puzzle[J].Soft Matter,2011,7:40-48.
[49] ALHESHIBRI M,QIAN J,JEHANNIN M,et al.A history of nanobubbles[J].Langmuir,2016,32(43):11086-11100.
[50] PENG H,BIRKETT G R,NGUYEN A V.Progress on the surface nanobubble story:What is in the bubble? Why does it exist?[J].Advances in Colloid and Interface Science,2015,222:573-580.
[51] ZHANG X,LOHSE D.Perspectives on surface nanobubbles[J].Biomicrofluidics,2014,8(4):0413011.
[52] TYRRELL J W G,ATTARD P.Atomic force microscope images of nanobubbles on a hydrophobic surface and corresponding force-separation data[J].Langmuir,2002,18(1):160-167.
[53] CONSIDINE R F,HAYES R A,HORN R G.Forces measured between latex spheres in aqueous electrolyte:Non-DLVO behavior and sensitivity to dissolved gas[J].Langmuir,1999,15(5):1657-1659.
[54] CRAIG V S J,NINHAM B W,PASHLEY R M.Direct measurement of hydrophobic forces:A study of dissolved gas,approach rate,and neutron irradiation[J].Langmuir,1999,15(4):1562-1569.
[55] HAYES R.The influence of dissolved gas on the interactions between surfaces of different hydrophobicity in aqueous media Part I.Measurement of interaction forces[J].Physical Chemistry Chemical Physics,1999,1(11):2793-2798.
[56] MEAGHER L,CRAIG V S J.Effect of dissolved-gas and salt on the hydrophobic force between polypropylene surfaces[J].Langmuir,1994,10(8):2736-2742.
[57] STEVENS H,CONSIDINE R F,DRUMMOND C J,et al.Effects of degassing on the long-range attractive force between hydrophobic surfaces in water[J].Langmuir,2005,21(14):6399-6405.
[58] ATTARD P.Thermodynamic analysis of bridging bubbles and a quantitative comparison with the measured hydrophobic attraction[J].Langmuir,2000,16(10):4455-4466.
[59] HAMPTON M A,NGUYEN A V.Systematically altering the hydrophobic nanobubble bridging capillary force from attractive to repulsive[J].Journal of Colloid and Interface Science,2009,333(2):800-806.
[60] HAMPTON M A,DONOSE B C,NGUYEN A V.Effect of alcohol-water exchange and surface scanning on nanobubbles and the attraction between hydrophobic surfaces[J].Journal of Colloid and Interface Science,2008,325(1):267-274.
[61] MISHCHUK N A.The model of hydrophobic attraction in the framework of classical DLVO forces[J].Advances in Colloid and Interface Science,2011,168(1-2):149-166.
[62] MISHCHUK N,RALSTON J,FORNASIERO D.Influence of very small bubbles on particle/bubble heterocoagulation[J].Journal of Colloid and Interface Science,2006,301(1):168-175.
[63] LIN Q,MEYER E E,TADMOR M,et al.Measurement of the long-and short-range hydrophobic attraction between surfactant-coated surfaces[J].Langmuir,2005,21(1):251-255.
[64] PARKINSON L,RALSTON J.Dynamic aspects of small bubble and hydrophilic solid encounters[J].Advances in Colloid and Interface Science,2011,168(1-2):198-209.
[65] HONIG C D F,DUCKER W A.No-slip hydrodynamic boundary condition for hydrophilic particles[J].Physical Review Letters 2007,98(2):0283051-0283054.
[66] VINOGRADOVA O I.Drainage of a thin liquid film confined between hydrophobic surfaces[J].Langmuir,1995,11(6):2213-2220.
[67] SHI C,CUI X,XIE L,et al.Measuring forces and spatiotemporal evolution of thin water films between an air bubble and solid surfaces of different hydrophobicity[J].ACS Nano,2015,9(1):95-104.
[68] PAN L,JUNG S,YOON R H.A fundamental study on the role of collector in the kinetics of bubble-particle interaction[J].International Journal of Mineral Processing,2012,106:37-41.
[69] KLASEBOER E,CHEVAILLIER J P,GOURDON C,et al.Film drainage between colliding drops at constant approach velocity:Experiments and modeling[J].Journal of Colloid and Interface Science,2000,229(1):274-285.
[70] MANICA R,KLASEBOER E,CHAN D Y C.Dynamic interactions between drops-a critical assessment[J].Soft Matter,2008,4(8):1613-1616.
[71] HORN R G,VINOGRADOVA O I,MACKAY M E,et al.Hydrodynamic slippage inferred from thin film drainage measurements in a solution of nonadsorbing polymer[J].Journal of Chemical Physics,2000,112(14):6424-6433.
[72] PARKINSON L,RALSTON J.The interaction between a very small rising bubble and a hydrophilic titania surface[J].Journal of Physical Chemistry C,2010,114(5):2273-2281.
[73] HARDY W,BIRCUMSHAW I.Bakerian lecture,boundary lubrication,plane surfaces and the limitations of Amontons’ law[J].Proceedings of the Royal Society of London Series A,Containing Papers of a Mathematical and Physical Character,1925,108(745):1-27.
[74] IVANOV I B,DIMITROV D S,SOMASUNDARAN P,et al.Thi-nning of films with deformable surfaces-diffusion controlled surfactant transfer[J].Chemical Engineering Science,1985,40(1):137-150.
[75] CHAN D Y,KLASEBOER E,MANICA R.Theory of non-equilibrium force measurements involving deformable drops and bubbles[J].Advances in Colloid and Interface Science,2011,165(2):70-90.
