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LettersSuper-Water-Repellent Fractal SurfacesT. Onda,*,S. Shibuichi,N. Satoh,and K. TsujiiKao Institute for Fundamental Research, Kao Corporation, 2606, Akabane, Ichikai-machi,Haga-gun, Tochigi, 321-34, Japan, and Tokyo Research Laboratories, Kao Corporation, 2-1-3,Bunka, Sumida-ku, Tokyo, 131, JapanReceived May 30, 1995. In Final Form: November 14, 1995XWettability of fractal surfaces has been studied both theoretically and experimentally. The contactangle of a liquid droplet placed on a fractal surface is expressed as a function of the fractal dimension,therangeoffractalbehavior,andthecontactingratioofthesurface. Theresultshowsthatfractalsurfacescanbesuperwaterrepellent(superwettable)whenthesurfacesarecomposedofhydrophobic(hydrophilic)materials. We also demonstrate a super-water-repellent fractal surface made of alkylketene dimer; awater droplet on this surface has a contact angle as large as 174.Ultrahydrophobic solid surfaces, which perfectly repelwater, would bring great convenience on our daily life.Conventionally,thewettabilityofsolidsurfaceshasbeencontrolledbychemicalmodificationsofthesurfaces,suchas fluorination.1The present Letter highlights the otherfactor determining the wettability, i.e., the geometricalstructureofsolidsurfaces.2-4Thereexists,asMandelbrothasemphasizedinhistextbook,5afascinatinggeometricalstructure called fractal, which is characterized by self-similarity and a noninteger dimension. Then a questionarsies: What is the wettability of the solid surface witha fractal structure? This Letter shows both theoreticallyand experimentally that fractal surfaces can be eithersuperrepellentorsuperwettabletoaliquid. Furthermore,we demonstrate a super-water-repellent fractal surfacemadeofalkylketenedimer;awaterdropletonthissurfacehas a contact angle as large as 174.The contact angle of a liquid placed on a flat solidsurface is given by Youngs equation2where R12, R13, and R23denote the interfacial tensions ofthe solid-liquid, the solid-gas, and the liquid-gasinterface, respectively. When a solid surface is rough,Youngs equation is modified into2where r is a coefficient giving the ratio of the actual areaof a rough surface to the projected area.A fractal surface is a kind of rough surface, so that thewettability of the fractal surface is basically described byeq 2.3The coefficient r of the fractal surface, however, isvery large and can even be infinite for a mathematicallyideal fractal surface. Therefore, the modification of thewettability due to surface roughness can be greatlyenhancedinthefractalsurface;3thatis,thefractalsurfacewill be superrepellent (superwettable) to a liquid when is greater (less) than 90.Strictly speaking, applicability of eq 2 is limited. Infact,eq2cannotgiveanycontactanglewhentheabsolutevalue of its right-hand side exceeds unity.A correctexpression for the contact angle can be derived conve-niently by introducing the effective interfacial tension ofKao Institute for Fundamental Research, Kao Corporation.Tokyo Research Laboratories, Kao Corporation.XAbstractpublishedinAdvanceACSAbstracts,March15,1996.(1) Watanabe, N.; Tei, Y. Kagaku 1991, 46, 477 in Japanese.(2) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; JohnWiley & Sons: New York, 1990; Chapter X, Section 4.(3) Hazlett, R. D. J. Colloid Interface Sci. 1990, 137, 527.(4) Good, R. J.; Mikhail, R. S. Powder Technol. 1981, 29, 53.(5) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman:San Francisco, CA, 1982.cos )R13- R12R23(1)cos r) rR13- R12R23() r cos )(2) Copyright 1996American Chemical SocietyMAY 1, 1996VOLUME 12, NUMBER 9S0743-7463(95)00418-5 CCC: $12.00 1996 American Chemical Societythe solid-liquid fractal interface, Rf12, and that of thesolid-gas fractal interface, Rf13. Then the contact anglefis given bySinceRf1j(j)2,3)maybeconsideredthetotalinterfacialenergy of the fractal surface per unit projected area, Rf1jcan be estimated, as a first approximation, by Rf1j)(L/l)D-2R1j. Here, D (2 e D 3) is the fractal dimensionof the surface; L and l are respectively the upper and thelower limit lengths of fractal behavior. Substitution ofthisRf1jintoeq3yieldseq2inwhich(L/l)D-2issubstitutedfor r.To be more precise in estimating Rf1j, we must takeaccount of the adsorption of a gas on the solid-liquidfractal interface and that of a liquid on the solid-gasfractal interface. Then Rf1jcan be expressed bywhere x (0e x e 1) is the fraction of the area of the fractalsurfacecoveredwiththeadsorbedgas,y(0eye1)isthatcovered with the adsorbed liquid, and S23is the area ofthe gas-liquid interface per unit projected area. Thesymbol “min” expresses the operation of minimizing thesucceedingquantitieswithrespecttoxory;thisoperationcorresponds to the fact that adsorption so occurs as tominimize the total interfacial energy. Typically, S23(z) (z) x, y) can be approximated by a cubic function of z, givenbyFigure1. Schematicillustrationforcosfvscostheoreticallypredicted.Figure2. SEMimagesofthefractalAKDsurface: (a,top)topview, (b, bottom) cross section.cos f)Rf13- Rf12R23(3)Figure 3. Water droplet on AKD surfaces: (a, top) fractalAKD surface (f) 174); (b, bottom) flat AKD surface ( )109). The diameter of the droplets is about 2 mm.Rf12)minxR13(Ll)D-2x + R12(Ll)D-2(1 - x) + R23S23(x)(4)Rf13)minyR13(Ll)D-2(1 - y) + R12(Ll)D-2y + R23S23(y)(5)S23(z) )z +(Ll)D-2- 1z(1 - z)(1 - )(1 - 3)1 - 2- z(6)2126Langmuir, Vol. 12, No. 9, 1996Letterswheredenotesthecontactingratioofthefractalsurface(the ratio of the area touching a flat plate placed on thefractal surface). Behavior of fdetermined by eqs 3-6 isshownschematicallyinFigure1,inwhichcosfisplottedas a function of cos ()(R13- R12)/R23). The obtainedcurve coincides with the line of cos f) (L/l)D-2cos inthe vicinity of ) 90, but deviates from it owing to theadsorption when approaches 0 or 180.To verify the theoretical prediction, we have experi-mentally examined the wettability of various kinds offractal surfaces. In this Letter, we report the results ofthe fractal surface made of alkylketene dimer (AKD),which is a kind of wax and one of the sizing agents forpapers.Alkylketenedimermostlyusedinourexperimentswassynthesized from stearoyl chloride with the use of tri-ethylamine as a catalyst and purified up to 98% with asilica gel column. The purity was checked by gel per-meation chromatography and capillary gas chromatog-raphy; the major impurity was dialkyl ketone, which wasproducedbythehydrolysisofAKD. AKDsolidfilmswitha thickness of about 100 m were grown on glass plates;a glass plate was dipped into melted AKD heated at 90C and was then cooled at room temperature in theambience of dry N2gas. AKD undergoes fractal growthwhen it solidifies, although the mechanism has not beenclarified yet. SEM images of the AKD solid surface areshown in Figure 2.To find the fractal dimension of the AKD surface, weapplied the box counting method to the cross section ofone of the AKD films. The AKD film peeled off from theglass plate was cleaved with the aid of a razor blade, andSEM images of its cross section (Figure 2b) were takenat several magnifications. Then the fractal dimension ofthe cross section, Dcross, has been measured by the boxcountingmethodandfoundtobe1.29intherangebetweenl)0.2mandL)34m(seetheinsetofFigure4);belowandabovetherange,Dcrossisfoundtobeunity. Thus,thefractal dimension of the AKD surface has been evaluatedas D = Dcross+ 1 ) 2.29.Figure 3a shows a water droplet placed on the fractalAKD surface. This surface, as expected, repels watercompletely and a contact angle as large as 174 has beenobtained. For comparison, we have also examined thewettabilityofamechanicallyflattenedAKDsurface,whichwas prepared by cutting a fractal AKD surface with arazor blade. The flat AKD surface, however, does notrepelwaterverymuch,showingacontactanglenotlargerthan 109 (Figure 3b). Comparison of parts a and b ofFigure3highlightstheimportanceofthefractaleffectonthe wettability.The relationship between the contact angle for thefractalAKDsurface,f,andthatfortheflatAKDsurface, has also been studied with the use of liquids of varioussurfacetensions. Astheliquids,weusedaqueoussolutionsof1,4-dioxaneatvariousconcentrations;becomessmalleras the concentration of 1,4-dioxane increases. The resultisshowninFigure4,inwhichcosfisplottedasafunctionof cos . The measurement of the contact angle has beenperformed several times at each concentration and thedispersion of the data is expressed by error bars. Thefractal AKD surface showed advancing and recedingcontact angles, whose difference was not negligibleparticularly when fwas close to 90, so that we havemeasured an average contact angle after bringing thewaterdropletintoanequilibriumstatebyvibratingit.InFigure 4, we draw also the line of cos f) (L/l)D-2cos with the value of (L/l)D-2determined by the box countingmeasurement: (L/l)D-2) (34/0.2)2.29-2= 4.43. The resultobtainedagreeswellwiththebehaviorpredictedinFigure1.Acknowledgment. WethankDr.J.Mino(KaoCorp.)and Professor T. Tanaka (MIT) for their valuable com-ments. We also thank Mr. Sodebayashi (Kao Corp.) fortaking photos of the liquid droplets.LA950418OFigure 4. cos fvs cos determined experimentally for theAKD surface. The line of cos f) (L/l)D-2cos deduced fromthe box counting measurement of the fractal AKD surface isalso drawn.In the inset, the result of the box countingmeasurement applied to the cross section of the AKD surfaceis shown.LettersLangmuir, Vol. 12, No. 9, 19962127 T. Onda,S. Shibuichi, N. Satoh, and K. Tsujii, Langmuir, 12( 9), 1996:2125-2128不规则疏水表面T. Onda,*, S. Shibuichi, N. Satoh, and K. TsujiiKao Institute for Fundamental Research, Kao Corporation, 2606, Akabane, Ichikai-machi,Haga-gun, Tochigi, 321-34, Japan, and Tokyo Research Laboratories, Kao Corporation, 2-1-3,Bunka, Sumida-ku, Tokyo, 131, JapanReceived May 30, 1995. In Final Form: November 14, 1995X对于不规则表面的润湿性，已经从理论上和实验上进行了研究。在不规则表面上的液滴接触角是表示不规则度，不规则行为的范围和表面接触比的函数。结果表明，当表面含有疏水物质时，它就会有疏水性。我们还示范了一个由烷基烯酮二聚物做成的表面，该表面上水珠的接触角是174o。几乎能完全防水的良好的疏水性固体表面将会给我们的日常生活带来很大的便利。按理说，固体表面的润湿性应该由表面的化学成分，比如说氟化物等决定1。目前的论文都认为其它的因数，比如固体表面的几何结构等决定了润湿性2-4。正如Mandelbrot在他的论文中强调的一样5，的确存在一种令人着迷的不规则表面结构，所谓的不规则，它有两个特点：自相似和大小非整数。那么，就有一个问题了：具有不规则结构的固体表面的润湿性是什么？本论文，从理论和实验上指出不规则表面对于液体来说既可能是超疏水性的也可能是超润湿性的。另外，我们还展示一个由烷基烯酮二聚物，水滴在这个表面上的接触角是174o。置于固体平面的液体的接触角可由杨氏方程式得出。其中12、13和14分别表示固液，固气，液气接触面的接触张力。当固体表面是粗糙的时候，杨氏不等式调整为其中r是粗糙表面的实际面积和突出面积的比值系数。不规则表面是一种粗糙表面，因此，不规则表面的润湿性基本上由等式2给出3。但是，不规则表面的系数r非常大，对于一个数学上的理想不规则表面来说，它甚至可能是无穷的。因此，在不规则表面上，对粗糙表面的润湿性的调整在很大程度上可能偏大了。就是说，当大于（小于）90o时，不规则表面就具有超疏水性（超润湿性）。严格地讲，等式2的应用是有限的。事实上，当等式右边的绝对值超过1的时候，就不能用等式2计算接触角了。引入不规则表面的固液有效界面张力12和固气有效界面张力13，就可以很方便地推导出接触角f的表达公式。其表达式为：f1j(j=2,3)可以看作不规则表面上单位突出面积的总界面能，所以f1j的大小是可以估计的。做为第一个近似数，f1j，其中，D(2D3是表面不规则度。L和I分别表示不规则范围的上下极限。将f1j带入等式2、3中，用(L/I)D-2代替式中的r。图1 理论上预测cosr和cos的关系示意图图2 实际AKD表面的SEM观测图像。a上图：俯视图；b下图：剖面图。为了更精确估计f1j，我们必须考虑固液交接面上水分和空气的吸收。则，f1j可以表示为：图3 AKD表面上的水滴。a上图：不规则AKD表面，r174o；b下图：平面AKD表面，109o。水滴的直径大约是2mm。其中，0x1，x是被吸收的气体覆盖的不规则表面面积的百分数，0y1，y是液体覆盖的不规则表面面积的百分数，S23是单位突出面积的气液交界面面积。符号“min”表示对其后关于x或者y的数量取最小值。做这样的运算是因为，这样的吸收是为了使总的交界面能量最小。很典型地，S23(z) (z=x,y)可以由一个z的立方表达式表示出来。图4 在AKF表面上，用实验得出的cosf和cos的关系，图中画有直线cosf=(L/I)D-2,它是从不规则AKD表面的箱式计算制中推导出来的。在插图中，还画了应用到AKD表面剖面的箱式计算制的结果。其中，是不规则表面的接触比例（放在不规则表面的平面的接触面积）。由等式36确立的f变化的示意图如图1所示。图中，cosf是cos的函数，cos（1312）/23。所得曲线和直线cosf=(L/I)D-2在90o附近一致，但是由于吸收了水和空气的缘故，在为0o和180o时偏离了。为了证明理论的推测，我们还必须从实验上检验各种不规则表面的润湿性。本文中，我们阐述了烷基烯酮二聚物制成的不规则表面的结果，该二聚物是一种腊，一种造纸的塑料。实验中使用的大多数烷基烯酮二聚物都是用三乙氨做催化剂，十八烷酰碌化物的合成物，并用硅胶提纯到98。纯度是通过胶渗透套色版和毛细管套色版检测的。主要的杂质是二氢酮，它主要是AKD水解形成的。在玻璃板上培植大约100m厚的AKD固体膜。将玻璃板浸入熔化的AKD中，加温至90摄氏度，然后放在干燥N2中，在室温下冷却。AKD在凝固时经过不规则变化，尽管其机理还没有得到充分阐释。AKD固体表面的SEM图如图2所示。为了找出AKD表面的不规则的直径大小，我们将箱式计算方法应用到AKD膜的横截面上。用刀片将从玻璃片上脱落的AKD薄膜分开。通过多次放大，获得它横截面的SEM图（如图2b所示）。然后用箱式计算方法测量截面的不规则直径Dcross，得出在I=0.2m和L34m的范围内，它的值是1.29（如图4插图所示），在这个范围之外，Dcross等于1。那么，我们估计AKD表面的不规则直径DDcross12.29。图3a表示的是一个置于AKD表面上的水滴。正如预计的一样，这个表面完全防水，并且得到的接触角达到174o。我们还检验了经机械压平的AKD表面的润湿性，该表面也是用刀片切割的。但是，这个平整的AKD表面疏水能力并不强，其接触角不大于109o(如图3b所示)。经过图3a和b的比较，我们可以看出不规则性对润湿性的影响。我们还用不同表面张力的液体研究了不规则AKD表面接触角f和平整表面接触角之间的关系。对于液体，我们用了不同浓度的1，4二氧杂环乙烷。随着1，4二氧杂环乙烷浓度增大角变小。结果如图4所示，图中cosf是cos的函数。每个浓度的接触角测量我们都进行了几次，测量值之间的误差用误差条表示。不规则AKD表面的接触角呈现出变大和变小的规律，这个规律很明显，特别是在f接近90o时，因此，我们振动水滴，使他进入一种平衡状态，这时测得的接触角做为平均接触角。在图4中，我们还画了直线cosf=(L/I)D-2，其中(L/I)D-2的值由箱式计算测量而得：(L/I)D-2（34/0.2）2.2924.43。所得结果和图1中预测的运动规律十分吻合。致 谢我们感谢J.Mino博士(KaoCorp.)和T.Tanaka教授（MIT），他们给了我们宝贵的意见。我们同样感谢Sodebayashi先生（Kao Corp.），感谢他为我们提供了液滴的图片。参考文献：(1) Watanabe, N.; Tei, Y. Kagaku 1991, 46, 477 in Japanese.(2) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; JohnWiley & Sons: New York, 1990; Chapter X, Section 4.(3) Hazlett, R. D. J. Colloid Interface Sci. 1990, 137, 527.(4) Good, R. J.; Mikhail, R. S. Powder Technol. 1981, 29, 53.(5) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman:San Francisco, CA, 1982.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?!? ? ? ? #$%& ? ? ? ( ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?)?$? ? ? ? ? $% ? ? ?*( ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?+? ? ? ? ? ?,? ? ? ?-? ? ? ? ? ?-? ? ? ? ? ? ?*?$? ? ? ? ? ? ? ? ? ? !? ? ? ? ? ? ? ? ? ? #$%./(&? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?,?,? ? #%01? ? ,? ? ?-&?2? ?3?4? ? ? #55& ? ? ? ?#5?67&?8? ? ? ? #9:57?$ 1& ? ?-? #?99!;&? +? 1? ?8? :?7A? $B ? C?B D $? 0C?%? :?/)*?/2+44+?*8 E ?8?*F?*4?*8? ? ?!? ?# ?$?#?% &? ? &? #?#? ? 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