Cloaking with optimized homogeneous anisotropic layers

合集下载

211251896_枳椇果梗多糖的提取工艺优化及其抗氧化性

刘旭东，张玉超，朱思洁，等. 枳椇果梗多糖的提取工艺优化及其抗氧化性[J]. 食品工业科技，2023，44（11）：230−237. doi:10.13386/j.issn1002-0306.2022090032LIU Xudong, ZHANG Yuchao, ZHU Sijie, et al. Optimization of Extraction Process of Polysaccharides from Hovenia dulcis Fruit Pedicels and Its Antioxidant Activity[J]. Science and Technology of Food Industry, 2023, 44(11): 230−237. (in Chinese with English abstract). doi: 10.13386/j.issn1002-0306.2022090032· 工艺技术 ·枳椇果梗多糖的提取工艺优化及其抗氧化性刘旭东1,2，张玉超3, *，朱思洁1,2，宋　亚1,2，张智淮1，陈光祥1，姜斯媛1（1.茅台学院食品科学与工程系，贵州仁怀 564507；2.贵州省保健酒酿造技术工程研究中心，贵州仁怀 564507；3.茅台学院酿酒工程系，贵州仁怀 564507）摘　要：为获得枳椇果梗多糖，并进一步评价其自由基清除及抑制生物大分子（蛋白质、脂质、DNA ）氧化的能力。

以枳椇果梗为试验材料，在单因素实验的基础上，结合正交试验及方差分析优化枳椇果梗多糖热水提取工艺条件；对所提多糖清除DPPH 、ABTS +自由基能力进行测定；并利用Cu 2+/H 2O 2、FeSO 4、APPH 分别诱导牛血清蛋白、亚油酸、鲱鱼精子DNA 氧化，构建体外蛋白质、脂质、DNA 氧化模型，对所提多糖体外抑制生物大分子氧化能力进行评价。

结果表明：醇沉体积分数对枳椇果梗多糖的得率有显著性（P <0.05）的影响，最佳的热水提取工艺条件为：料液比1:25 g/mL ，提取温度85 ℃，提取时间1 h ，醇沉体积分数80%，此时多糖得率为3.06%±0.181%；且随着浓度的增大，所提多糖对自由基的清除和生物大分子的氧化抑制效果也逐渐提高，对DPPH 自由基、ABTS +自由基清除IC 50分别为1.687、1.824 mg/mL ，对牛血清蛋白羰基化、亚油酸过氧化和鲱鱼精子DNA 氧化抑制IC 50分别为：13.84、10.88、74.70 mg/mL 。

Optics Express 18, 12646-12652(2010)

Electrically controlled multifrequencyferroelectric cloakPeining Li, Youwen Liu*, and Yunji MengCollege of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China*ywliu@Abstract:We have suggested an electrically controlled multifrequencycloak with a single shell of ferroelectric material for the first time to the bestof our knowledge. The theoretical and simulated results have demonstratedthat this cloak with high-index ferroelectrics can reduce the total scatteringcross section of the cloaked system at multiple frequencies. These cloakingfrequencies of our cloak can be externally controlled since the dielectricconstant of ferroelectrics is well tuned with the applied electric field. It mayprovide a potential way to design a tunable multifrequency cloak withconsiderable flexibility.©2010 Optical Society of AmericaOCIS codes: (260.2110) Electromagnetic optics; (290.5839) Scattering, invisibility; (160.2260)Ferroelectrics.References and links1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782(2006).2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterialelectromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).4. G. W. Milton, and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonances,”Proc. R. Soc. A 462(2074), 3027–3059 (2006).5. A. Alù, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat.Nonlin. Soft Matter Phys. 72(1), 016623 (2005).6. M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. EStat. Nonlin. Soft Matter Phys. 75(3), 036603 (2007).7. F. Bilotti, S. Tricarico, and L. Vegni, “Electromagnetic cloaking devices for TE and TM polarizations,” N. J.Phys. 10(11), 115035 (2008).8. B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking atmicrowave frequencies with metamaterials,” Phys. Rev. Lett. 103(15), 153901 (2009).9. P. Alitalo, O. Luukkonen, L. Jylhä, J. Venermo, and S. A. Tretyakov, “Transmission-Line networks cloakingobjects from electromagnetic fields,” IEEE Trans. Antenn. Propag. 56(2), 416–424 (2008).10. Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at adistance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).11. Y. Gao, J. P. Huang, and K. W. Yu, “Multifrequency cloak with multishell by using transformation medium,” J.Appl. Phys. 105(12), 124505 (2009).12. A. Alù, and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev.Lett. 100(11), 113901 (2008).13. A. E. Serebryannikov and E. Ozbay, “Multifrequency invisibility and masking of cylindrical dielectric objectsusing double-positive and double-negative metamaterials,” J. Opt. A, Pure Appl. Opt. 11(11), 114020 (2009). 14. A. E. Serebryannikov, P. V. Usik, and E. Ozbay, “Non-ideal cloaking based on Fabry-Perot resonances in single-layer high-index,” Opt. Express 17(19), 16869–16876 (2009).15. D. P. Gaillot, C. Croënne, and D. Lippens, “An all-dielectric route for terahertz cloaking,” Opt. Express 16(6),3986–3992 (2008).16. A. K. Tagantsev, V. O. Sherman, K. F. Astafiev, J. Venkatesh, and N. Setter, “Ferroelectric materials formicrowave tunable applications,” J. Electroceram. 11(1/2), 5–66 (2003).17. K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-typemetamaterials at optical frequencies,” Phys. Rev. Lett. 102(13), 133901 (2009).18. G. A. Smolensky, Ferroelectrics and Related Materials, (New York: Academic Press 1981).19. O. Vendik, Ferroelectrics at Microwave technology, (Moscow: Sov. Radio 1979).#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 126461. IntroductionCloaking an object with metamaterials and artificial structures has recently attracted a great deal of attention because of potential applications in various scientific fields. Different approaches have been proposed to suppress the scattering from a given object, involving coordinate transformation techniques [1–3], anomalous localized resonances [4], scattering cancellation [5–8] and other novel concepts [9,10]. These techniques are usually designed to work at a single operating frequency.Obviously, extending these schemes to multifrequency (MF) operation would invest them with considerable practicability. Recently, several possibilities have been put forward in order to achieve the MF cloaking [11–14]. Gao et al. have reported a MF cloak with multiple shells based on coordinate transformation method [11]. Alu and Engheta [12], extended their scattering cancellation cloaking (SCC) theory to multi-layered geometries and suggested that suitably designed multiple homogeneous and isotropic plasmonic layers may drastically reduce the total scattering cross section simultaneously at several distinct frequencies. Serebryannikov et al. have proposed a MF cloak with a single cloaking shell with high-index media based on the Fabry-Perot type radial resonances [13,14]. This latter cloak with high-index medium would be very feasible to implement in reality, since proper materials for the cloaking shell are easy to obtain, for instance, polar dielectrics at THz frequencies and ferroelectrics at microwave and THz frequencies [15], as well as Drude-Lorentz composites in a wide frequency range. Among these candidates, ferroelectrics may be a good one whose relative dielectric constant is effectively upon 100 at microwave frequencies and THz [16].The dielectric response of ferroelectric materials is well tuned by temperature or external DC (or low-frequency) electric field. This tunable property has been proven successful in describing the performance of tunable microwave and THz devices [16]. Here we extend the high-index MF cloak to the frequency-tunable operation by taking into account the fact that the dielectric constant of ferroelectric materials can be well tuned external DC electric field. Firstly, we analytically study the cloaking mechanism of the high-index concentric shell based on the well-known Mie scattering theory and obtain the condition that frequencies of the minima of the scattering cross section satisfy, which is consistent with results based on the conventional Fabry-Perot resonators [13,14]. In the following section, we brief introduce the electrical tunability of the dielectric constant of ferroelectrics and present a cloaking scheme with bulk Ba0.5Sr0.5TiO3(BST-0.5) material. In Section 4, we show that the cloaking frequencies of this cloak with a shell of BST-0.5 can be effective controlled by external electric field, and demonstrate the cloaking effect by numerical simulations based on the finite element method. The final section summarizes our results.2. Basic theory and the minimum scattering conditionIn [13,14] it was shown that, under suitable conditions, it is possible to drastically reduce the scattering cross section of cylindrical objects using single-layer high-index shell. This phenomenon can be heuristically understood by using the analogy with between the zero reflection regime in the planar resonators and near-zero scattering cross section regime in the cylindrical resonators. Although this used analogy is quite justified, since the location of the corresponding frequencies can be estimated with a high accuracy by using a mode of planar resonators, an analytical proof is still necessary to better understand the physical mechanism of this high-index cloaking. Below we would deal with this problem. In this work, the ﬁelds are assumed to be TM polarized (electric ﬁeld parallel to the axis of the cylindrical object) but similar steps could be carried out in the TE polarization. Moreover, all the materials are assumed to be nonmagnetic.We start by considering the dielectric obstacle is an infinitely long cylinder along z with radius a and the symmetry of z-axis. The dielectric constant of the obstacle is assumed to be ε. The object is covered with a concentric shell with radius a c and high dielectric constant εc. This combined system is surrounded by air as shown in the inset of Fig. 1. The e-jωt time #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12647convention is assumed throughout. A plane wave with the unit amplitude 0ik x i z E z e ∧=is incident on the system along x direction depicted in Fig. 1, where k 0 is the wave number in free space. The cylinder symmetry of the problem allows us to analytically solve the electric ﬁelds inside and outside our structure from the Mie scattering theory. By applying the continuity at the inner (r = a ) and outer (r = a c ) interface of the cloak, the scatteringcoefficients TM n c can be determined ,TM TMn n TM TM n n U c U iV =−+ (1)where TM nU , TM n V are available in the literature [6]. The total SCS of the combined system denoted as Q s , is given by 2,004(2).TM s n n n Q c k δ∞=−∑ (2)where δ is the Kronecker delta.It is expected that some interconnects would exist between the approach discussed in this work and the theory of scattering cancellation in plasmonic cloaking [5,6]. The purpose of this high-index cloaking shell is also to cancel the electric dipole moment of the system, since which contributes the most to the scattering properties for a relative small object [6,17].Therefore, the locations of minimum 0TM c are the case of interest in this work, which alsocorrespond to the minimum of the total scattering. Considering a limit of a relatively smallshell (λ>>a c ) with high dielectric constant (εc >>1), the minimum scattering condition 0TM U =0, which ensures 0TM c = 0, may be obtained analytically. This is because the expression TM n Ufor 0thorder is reduced to the following expression,2021/4)/4)0/23/4)3/4)0/4)/4)103/4)3/4)/2c c c c TM c c c c c c c c o c k a k a k a k k a k k a U k a k a k k a k k a k a ππππππππ−−−−−−−=−−−−−−− 2204sin[()]().2c c c k k a a o k a π=−+ (3)If we set 0TM U = 0, the following minimum scattering condition is obtained,λ= (4) where m = 1, 2, 3…, and the max of integer m ensures λ>>a c .The above condition is also expressed as 2c c a a m λ−=,where c λ= is thewavelength in the shell. This equation exactly coincides with the model of multiple zeros of the refection coefficient of the conventional planar Fabry-Perot resonators, and well supports #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010(C) 2010 OSA 7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12648the cloaking theory based on Fabry-Perot type radial resonances [13,14]. As is shown in [13,14], it is easy to estimate the cloaking frequencies with a high accuracy by using above equation. It is an obvious advantage of this method in designing cloaking frequencies over other recently proposed techniques.Let’s come back to the main purpose of this work. It can be easily deduced from Eq. (4) that externally changing the dielectric constant of the shell would naturally lead to externally control cloaking frequencies when the dimensions have been determined. From this important feature, the ferroelectrics provide a possibility of designing a tunable MF cloak, since whose dielectric constant of is well tuned by the applied electric field.3. The electrical tunability of the dielectric constant of ferroelectricsThe main attraction of ferroelectric materials is the strong dependence of their high dielectric permittivity on the applied bias electric field. This dependence would lead to considerably tunable application in our cloaking model. Typical representative ferroelectric material is Ba x Sr1-x TiO3 (BST) that can be synthesized in polycrystalline, ceramic layer, and bulk forms the real part of the relative dielectric constant exceeds several hundred depending on the barium concentration, whereas the loss tangent can be less than 10−2. For this material, the aforementioned dependence has an approximated form [16],(V/ µm),0.7≈<(5)E xwhere n= ε(0)/εr defined as the ratio of the dielectric permittivity of the material at zero electric field to its permittivity at some non-zero electric field. In this work, we consider that the cloaking shell consists of bulk ferroelectric BST with barium concentration x = 0.5, and ε(0) = 200 [15]. The electric field dependence for this material is shown in the Fig. 1.Fig. 1. The dependence of the dielectric permittivity of BST(x= 0.5) on the applied electricfield. The inset is schematic diagram of a cylindrical object covered with a ferroelectric shell.The tunability is obvious that the dielectric constant decreases from 200 to 150 when the bias field varies from 0 to 77V/µm. The dielectric constant approximately varies linearly with the applied electric field in the range of 20-80V/µm. Evidently, the cloaking frequencies would be externally controlled by applying this bias-field-dependent property to the high-index cloaking.4. The performance of the tunable cloaking with a BST shellIn this section, we present the results for tuning cloaking frequencies of the cloak with ferroelectric shell through the biased electric field. As an example, the dielectric obstacle is characterized by given material parameters ε= 6, which is covered with a BST-0.5 layer discussed in Sec. 3. And the structure of the system is set as a c/a = 2. In Fig. 2(a), we report the contour plot of the variation of Q s of our cloaked system as a function of frequency and the external field on the BST-0.5 shell. The orange regions represent the small Q s regions. It is obvious that there are two cloaking frequencies for the minimum Q s in the range of interest when the applied field is absent: a c/λ = 0.07 and 0.14, corresponding to mλ = 2(a c−a)εc1/2 at m#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12649= 1 and 2, respectively. The higher dielectric constant of the shell, the more the cloaking frequencies in the same frequency range. This is because the cloaking frequency interval is inversely proportional to εc1/2. It is clear that these cloaking frequencies change with the variations of the external field. When the electric field of 70 V/µm is applied (εc = 155.8 under this field), cloaking frequencies vary to the high frequencies, 0.08 for m = 1and 0.16 for m = 2. The cloaking frequency for large m model are more sensitive to the applied electric field, since the variance ratio ∆E/∆(a c/λ) is proportional with m.Fig. 2. (a) Contour plot of the variation of total scattering cross section Q s as a function offrequency and the applied electric field. (b)The total scattering cross section Q s as a function offrequency for three cases: No cover (green), with cover and zero applied electric field (red),with cover and an applied electric field of 70 V/µm (blue).For implementing such a cloak in reality, the loss of shell material should be under consideration. For bulk BST-0.5, the loss tangent is tanδ = 0.01 [16] for both in the absence of a bias field and under it. This choice relies on the fact that the increase of the loss induced by the bias field is not substantial in this BST with intermediate concentrations of barium [16]. Figure 2(b) reports the total scattering cross section Q s for three cases of uncovered cylinder object, the cloaked system in the absence and in existence of the applied electric field. It can be seen that the reduction effect of scattering is still very obvious. The total scattering cross section have been reduced to about 27% (at a c/λ = 0.07) and 11% (at a c/λ = 0.14) with respect to the uncloaked scenario without the applied field, and to about 16% (at a c/λ = 0.08) and 8% (at a c/λ = 0.16) with an applied field E = 70 V/µm. Actually, as is shown in [14], the presence of the loss only affects the minima locations in a very minor way, and affects the corresponding Q s values signiﬁcantly. For the lossless case, the total scattering cross section can be reduced to about 8% at a c/λ = 0.07 with respect to the uncloaked scenario without the applied field.To demonstrate the cancelling of scattering and frequency-tuning effect of the cloak, we perform the numerical simulations of the propagation of plane electromagnetic wave by the finite element method. Figure 3 show the modulus of the axial electric field in the three cases of uncovered cylinder object (corresponding to green line in Fig. 2(b)), cloaked system without the applied electric field (corresponding to red line) and under an applied field (corresponding to blue line) at two frequencies of a c/λ = 0.14 and a c/λ = 0.16 (the mode m = 2). In all the cases the structure is excited by a uniform plane wave impinging from the left of the figure with electric field amplitude equal to 1 V/m. For the case of the cloaked system without the applied field, it can be seen that the field in the air region is nearly uniform at the cloaking frequency a c/λ= 0.14, showing that the scattering field is very weak, whilst the scattering field is very strong in the case of uncover cylinder object. This confirms that the scattering is effectively suppressed by the cloaking shell. If an electric field of E = 70 V/µm is applied in the shell, the cloaking frequency becomes a c/λ= 0.16, where the field is nearly uniform in the air region, while the scattering interaction is very strong at the non-cloaking frequencies a c/λ = 0.14. The frequency-tuning effect is very obvious when the electric field is applied on the shell. The simulated results based on the finite element method demonstrate the #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12650minimum scattering condition and the electrically frequency-tunable cloaking from the Mie scattering theory.Fig. 3. The modulus of the axial electric field for the three cases in Fig. 2(b) with reasonableloss at two frequencies: (a) a c/λ = 0.14, (b) a c/λ = 0.16.Figure 4 further shows simulated modulus of the total magnetic field in the orthogonal plane of polarization (x-y plane), which is dominated by high-order multipoles [12]. Although the cloak has been originally designed to cancel the electric dipole moment, it is clear that the higher-order moments are also reduced at cloaking frequencies by this BST-0.5 shell. And the frequency-tuning effect is valid for the higher-order moments in the same way. This polarization independence is another advantage of this cloak.Fig. 4. The modulus of the total magnetic field in the orthogonal plane of polarization for thethree cases in Fig. 2 (b) with reasonable loss at two frequencies: (a) a c/λ = 0.14, (b) a c/λ =0.16.5. ConclusionsIn conclusion, we have theoretically proposed the possibility of designing an electrically controlled multifrequency cloak with a single shell of ferroelectric materials. The proposed scheme depends on the tuning of the dielectric permittivity of ferroelectric material with the applied field. The calculated and simulated results have demonstrated that such cloak can drastically reduce the total scattering cross section at multiple frequencies and these cloaking frequencies can be controlled by the applied field. We also analytically obtain the minimum scattering condition based on the Mie scattering theory, which may provide better understanding of the high-index cloak.#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12651AcknowledgmentsThis work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Specialized Research Fund for the Doctoral Program of Higher Education (200802871028), and the Natural Science Foundation of Jiangsu Province (BK2009366).#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12652。

F离子传感

Analytica Chimica Acta 539(2005)311–316A colorimetric method for ﬂuoride determination in aqueous samplesbased on the hydroxyl deprotection reaction of a cyanine dyeChang-Qing Zhu a ,Jin-Long Chen b ,Hong Zheng a ,Yu-Qin Wu b ,Jin-Gou Xu a ,∗aThe Key Laboratory of Analytical Science of MOE,Department of Chemistry,Xiamen University,Xiamen 361005,Chinab College of Chemistry and Materials Science,Anhui Normal Unversity,Wuhu 241000,ChinaReceived 6January 2005;received in revised form 3March 2005Available online 30March 2005AbstractA new highly sensitive and selective colorimetric method for ﬂuoride determination in water is described.The novel reagent for this method is a cyanine dye (C 1)on which the hydroxyl group has been protected by reaction with tert -butyldimethylsilane (TBS)to form the silanated dye,C 2.C 2is selectively attacked by ﬂuoride ions to reform C 1.C 1has an absorption maximum at 600nm with a molar absorptivity of about 200,000.Under optimized conditions,absorbance at 600nm is proportional to ﬂuoride concentration up to about 1×10−4mol/L with a detection limit of 1.0×10−7mol/L.Because of the speciﬁc afﬁnity of ﬂuoride for the TBS functional group on C 2,there is little interference by other ions.The method has been successfully applied for ﬂuoride determinations in local rainwater samples.Results determined by the proposed method agree favorably with those determined by a ﬂuoride ion selective electrode method.©2005Elsevier B.V .All rights reserved.Keywords:Fluoride ion;Cyanine;Hydroxyl deprotection reaction;Colorimetric method1.IntroductionThe effect of ﬂuoride on both humans and animals has a dual role as an essential element and at high levels as a toxic substance [1].Excess ﬂuoride ion results in ﬂuorosis and renal,gastrointestinal and immunological toxicity [2].Also,plants and fruits such as apricots,plums grapes,tulips,iris and sweet corn are sensitive to ﬂuoride [3].Hydrogen ﬂuoride is often found in rainwater [4]as a result of various industrial processes,such as phosphate fertilizer production,aluminium production and metal smelting.Therefore,an ac-curate and rapid method for the determination of ﬂuoride in rainwater samples is important for assessment of air and water quality.Several analytical techniques have been developed and described for ﬂuoride determinations in water including spec-trophotometry [5–7],ﬂuorometry [3,8–10],potentiometry∗Corresponding author.Tel.:+865922180307;fax:+865922188054.E-mail address:jgxu@ (J.-G.Xu).(ion-selective electrodes (ISE)[11–13],ion chromatography (IC)[14],gas chromatography (GC)[15],capillary zone electrophoresis (CZE)[16]and radioanalysis [17].Among these,ISE methods are most widely applied because they are easy to use,selective and provide a wide dynamic range [11].Determination of low-concentration ﬂuoride by ISE can be difﬁcult due to long equilibration times,electrode drift and dissolution of the lanthanum ﬂuoride membrane crystal [18,19].Compared to ISE ﬂuoride determination methods,chromatographic methods are time consuming [15]and require more expensive instrumentation.Several spectrometric methods [20–22]are widely accepted for the determination of trace ﬂuoride in natural waters due to their simplicity,low cost and reliability.In most of these spectro-metric methods,ﬂuoride ions displace organic ligands from colored complexes of metal ions such as zirconium,thorium,titanium,aluminium and ferric iron [20–22].Displacement of the metal ion or organic ligand by ﬂuoride therefore causes the absorbance of the reaction mixture to decrease.Often,however,Beer’s law is not obeyed in such methods0003-2670/$–see front matter ©2005Elsevier B.V .All rights reserved.doi:10.1016/j.aca.2005.03.002312 C.-Q.Zhu et al./Analytica Chimica Acta539(2005)311–316due to the diversity of co-existing complexes ofﬂuoride with the metal ion in the solution.Furthermore,many anions or cations compete withﬂuoride or the central metal ion for coordination[7],which results in serious interference.Recently,Descalzo et al.[23],Kim and Swager[24]re-ported a new reaction system for the selective detection of ﬂuoride ion that exploits the unique chemical reactivity of ﬂuoride ions with silicon.Long reaction times and organic solvent requirements of these reaction systems[24]greatly limited their utility for water analysis applications.In this paper,we describe using a new reagent forﬂuoride determi-nation in aqueous solution,based on the selective attack of ﬂuoride on the Si–O bond of the cyanine dye,C2(1-ethyl-4-(p-tert-butyldimethylsilane ether styryl)quinolinium iodide). C2is conveniently obtained in98%yield by reaction of cya-nine dye,C1(1-ethyl-4-(p-hydroxystyryl)quinolinium io-dide)with tert-butyldimethylsilane(TBS)as shown in Fig.1. The large molar absorptivity of C1(≈200,000L cm mol−1) [25]makes it an excellent chromophoric reporter.When C2is incubated withﬂuoride ions in7:3v/v THF:water solutions, the characteristic absorption band of C1at600nm gradually appears.Under optimized conditions,a linear relationship be-tween the absorbance at600nm andﬂuoride ion concentra-tion is observed.This analytical approach provides a highly sensitive and selective colorimetric method for the determi-nation ofﬂuoride ion in aqueous solution that was applied to the detection ofﬂuoride ion in local rainwater samples.In-terference by foreign anions is negligible and results by the proposed method compare favorably with results by standard F−ISE methods.2.Experimental2.1.ApparatusA Hitachi U-3010spectrophotometer(Tokyo,Japan)was used for recording absorption spectra and making absorption measurements.The pH was measured with a Model pHs-3c meter(Shanghai,China).A201ﬂuoride Ion Selective Elec-trode(Jiangshu Electroanalysis Apparatus Factory,China) was used for the detection ofﬂuoride in rainwater samples.2.2.ReagentsTheﬂuoride standard solutions were prepared by a se-rial dilution of a10−2mol/L sodiumﬂuoride stock solu-tion.Cyanine dye C1was synthesized and puriﬁed accord-ing to Blazsek-Bodo et al.[26,27]and silanated with TBS as described by Sartori et al.[28](Fig.1),to form the indicator,C2in98%yield.C2was dissolved in chloro-form solution to make a1.7×10−3mol/L stock solution. The purity of C2was conﬁrmed by1H NMR and HRMS spectra.All chemicals were analytical reagent grade and used as received.Doubly deionized water was used for all experiments.2.3.Pretreatment of samplesTwo local rainwater samples were obtained after a pre-treatment according to the literature[29];two tap water sam-ples were diluted by10-fold with doubly deionized water for comparative experiment.2.4.Standard procedure for the detection ofﬂuorideAdd sequentially to5.0mL volumetricﬂasks containing 3.5mL of THF solution,0.1mL of C2working solution, 1.3mL of standard sodiumﬂuoride solution or sample and 0.1mL of NaH2PO4–NaOH buffer solution(pH9.0).Mix the contents of theﬂask thoroughly.After10min,measure the absorbance at600nm using THF–water(7:3,v/v)solution as the reference.After determinations are complete,recycle THF by simple collection and distillation to reduce the pos-sible pollution.3.Results and discussion3.1.Spectral characteristicsThe visible absorption spectra of C1in pH9.0buffer dis-plays two bands as shown in Fig.2b.At low pH,however, the spectra of C1display only one visible band at440nm as shown in Fig.3that corrsesponds to the absorption ofthe Fig.1.Protection and deprotection of cyanine’s hydroxyl.C.-Q.Zhu et al./Analytica Chimica Acta 539(2005)311–316313Fig.2.Absorption change of C 2upon addition of NaF:(a)a serial visual color change,the ﬂuoride concentrations (from the left to the right)are:0.0,2.8×10−6,6.6×10−6,9.4×10−6,2.8×10−5,4.7×10−5,6.6×10−5and 9.4×10−5mol/L.(b)UV–vis absorption spectra in THF–water (70:30,v/v)solvent.Concentrations of ﬂuoride are:0.0,4.8×10−6,9.4×10−6,2.8×10−5,4.8×10−5,6.6×10−5and 9.4×10−5mol/L.Other conditions are the same as those described in theprocedure.Fig.3.UV–vis absorption change of the cyanine with pH before (C 1)and after (C 2)its hydroxy was protected.Solvent:THF–water (70:30,v/v);con-centration of C 1and C 2:3.4×10−5mol/L;NaH 2PO 4–NaOH buffer:pH 6.0–9.7.species with non-ionized hydroxyl group.The band at 440nm was assigned to an intramolecular charge transfer transition [30].Its intensity decreases as pH increases above 7.The in-tensity of the band at 600nm increases as the pH increases above pH 7.The behavior can be explained as follows.With the increase of pH the hydroxyl proton begins to ionize,which increases the proportion of the ionized form in solution due to making of the energy barrier of the intramolecular CT smaller.The absorption band at 600nm of C 2at pH 9.0disap-peared,with a little blue shifting of the main absorption band at 440nm (Fig.2b)due to the increase of the energy bar-rier of the intramolecular CT after the hydroxyl group of C 1was protected.At the same time,the dependence to pH dis-appeared since only one cyanine form existed.As shown in Fig.3,C 2did not show the absorption at about 600nm even at the pH up to 9.7.When C 2was incubated with various amounts of ﬂuoride ion,a dramatic color change from yellow to green was ob-served (Fig.2a)and the characteristic absorption of initial cyanine C 1at about 600nm gradually returned to the solu-tion (Fig.2b).Under the optimum conditions,there was a good linear relationship between the absorbance at 600nm and the concentration of ﬂuoride ion.3.2.Effects of reaction medium and pHTHF greatly increases the rate of ﬂuoride reaction with C 2.Experiments showed that 70%THF was optimal for speciﬁc reaction of ﬂuoride with C 2.Because C 2was sta-ble and soluble in chloroform at room temperature,chlo-roform was used to prepare the stock solution of C 2and this small amount of chloroform (<3%)is unavoid-ably introduced into the detection system.Experiments showed that its effect on analytical results was negligi-ble.Two buffer systems,Tris–HCl and NaH 2PO 4–NaOH buffer,were investigated.The latter showed a very low blank absorbance at 600nm (see Fig.2b).Therefore,0.1mol/L pH 9.0NaH 2PO 4–NaOH buffer solution was chosen in order to have higher sensitivity and lower background.3.3.Probe C 2concentration and incubation timeThe effect of concentration of C 2on the determination of ﬂuoride was investigated.When concentrations of other reagents were kept constant,the absorbance difference be-tween the absence and the presence of ﬂuoride ion in-creased with increasing the amount of C 2.However,when the concentration of C 2exceeded 3.4×10−5mol/L,the background absorbance (without ﬂuoride ion)increased ac-cordingly.Thus,C 2concentration of 3.4×10−5mol/L was recommended.Under these optimized conditions,complete cleavage of the Si–O bond (or the association of ﬂuoride with the silicon atom)on C 2required about 1.5h (Fig.4).Although analyti-314 C.-Q.Zhu et al./Analytica Chimica Acta539(2005)311–316Fig.4.Effect of reaction time.Concentrations ofﬂuoride(from the bottom to the top):0.0,4.8×10−6and4.8×10−5mol/L.Other conditions are the same as those described in the procedure.cal sensitivity could have been increased with longer reaction times,an incubation time of10min was chosen for this work as a practical matter.Our experiments demonstrate that accu-rate results could be obtained at this relatively short reaction time.3.4.Interference of foreign ionsThe effects of foreign ions on the determination of 1.6×10−6mol/Lﬂuoride in deionized water were studied. Foreign ion concentrations that result in relative errors less than±5%are listed in Table1.From Table1,it can be seen that the tested ions have very little interference with the deter-mination.The relatively high concentrations of Cl−,Br−,I−, SO42−,SCN−and PO43−that can be tolerated are explained Table1Tolerance of foreign substancesForeign ions Concentration(10−6mol/L)Relative error caused(%)Cl−280−5.0Br−125−1.7I−78−1.0SO42−80+2.5SCN−90+3.5PO43−200+1.0Fe3+50−3.0NH4+200−2.0Cr3+50+3.5Co2+70+2.5Ni2+45+3.5Ca2+80−1.5Ba2+100+2.8Cu2+100−3.5Al3+40−2.0Mn2+60+4.0 Concentration ofﬂuoride:1.6×10−6mol/L.Other conditions are the same as those described in theprocedure.Fig.5.Calibration graphs for different incubation time(10min and30min). Absorbance was measured at600nm;solvent:THF–water(7:3,v/v).Other conditions are the same as those described in the procedure.by the speciﬁcity with whichﬂuoride reacts with the silicon atom on C2.3.5.Calibration graphs and analysis of samplesThe calibration graphs for two kinds of incubation time, i.e.10min and30min,were constructed(Fig.5)under the optimum conditions.From Figs.4and5,it can be seen that reasonably prolonging the reaction time can improve the an-alytical sensitivity.The limit of detection(LOD)for the proposed method was1.0×10−7mol/L given by the equation,LOD=KS0/S, where K is a numerical factor chosen according to the con-ﬁdence level desired,S0the standard deviation of the blank measurements(n=6)and S the sensitivity of the calibration graph.Here a value of3for K was used.And the relative standard deviation(n=6)was3.3%for the determination of 1.6×10−6mol/L NaF.The proposed method was applied to the determination of ﬂuoride in rainwater samples.Two local rainwater samples were collected andﬁltered according to the literature[29]. The determination results by this method and comparison with that obtained by the F−ISE standard method are shown in Table2.All the determination results shown in Table2were the mean of six measurements.The recovery test was carried out by adding1.0×10−6mol/Lﬂuoride standard solution in the sample solutions.The recoveries were found to be 98–102%.At the same time,the data shown in Table2,which are very close to that obtained by the F−ISE standard method, are also in agreement with that provided by local environment monitor station.Furthermore,two tap water samples diluted by10-fold with doubly deionized water were detected by the proposed method and F−ISE method,the results are shown in Table3.From the data of Tables2and3,it can be seen that the proposed method is sensitive and reliable.C.-Q.Zhu et al./Analytica Chimica Acta 539(2005)311–316315Table 2Analytical results of rainwater samples Rainwater samples no.This method F −,found a (n =6)R.S.D.(%)F −,added (10−6mol/L)F −,found (10−6mol/L)Recovery (%)F −ISE method F −,found (10−6mol/L)t -test b 1 3.78±0.102.51.04.801023.851.8123.55±0.113.01.04.53983.501.15aThe analytical result is expressed as ¯x ±(t p,n −1S/√n ),where t p ,n −1=t 0.95,5=2.57,¯x and S are the mean and standard deviation,respectively.The unit for the concentration of ﬂuoride in rainwater is expressed in 10−6mol/L.b t0.95,5=2.57;t 0.95,5=2.02.Table 3Comparative determination of F −in tap water samples SampleThis method F −,found a (10−6mol/L)F −ISE method F −,found (10−6mol/L)F −found aF −added F −found b Recovery (%)F −found a F −added F −found b Recovery (%)Tap water11.02.251041.0Undetected –1.214.05.2099Undetected 4.0 4.25106Tap water21.02.08981.0Undetected –1.104.05.15101Undetected4.04.16104a The ﬂuoride in tap water with 10-fold dilution.Mean of six determinations.bThe total ﬂuoride in 10-fold diluted tap water with the addition of ﬂuoride.Mean of six determinations.4.ConclusionsA novel colorimetric method for the determination of ﬂuoride ion has been developed based on the selective at-tack of ﬂuoride on a silanated cyanine dye.The method has several merits that include:(i)high sensitivity and speciﬁcity;(ii)semi-quantitative determinations by opti-cal comparison is possible;and (iii)synthesis of C 1and C 2are straightforward and spectrophotometric or optical comparison detection methods are convenient and economi-cal.AcknowledgementsThis work was supported by the National Natural Science Foundation of China (NNSFC,No.29775021)and the Ed-ucation Commission Natural Science Foundation of Anhui Province (No.2001KJ114ZD);all the authors wish to ex-press their gratitude.References[1]The Impact of Fluoride on Health J.Am.Diet.Assoc.100(2000)1208–1213;The Impact of Fluoride on Health J.Am.Diet.Assoc.101(2001)126–131.[2]D.Purves,Trace Element Contamination of the Environment,Else-vier,Amsterdam,1977,pp.79–82(Chapter 3).[3]D.H.Chen,M.D.Luque de Castro,M.Valcarcel,Anal.Chim.Acta230(1990)137.[4]A.G.T.Marc,Hoop Van den,F.M.J.Rob,J.J.Cleven,Staden Van,J.Neele,J.Chromatogr.A 739(1996)241.[5]Standard Methods for the Examination of Water and Wastewater,Method 4500FD,20th ed.,American Public Health Association,Washington,1998,pp.4–62.[6]EPA Methods for Chemical Analysis of Water and Wastes,Method340.1,US Environment Protection Agency,Washington,1978.[7]E.Bellack,P.Schouboe,Anal.Chem.30(1958)2032.[8]M.Garrido,A.G.Lista,M.Palomeque,B.S.Fern´a ndez Band,Talanta58(2002)849.[9]T.L.Har,T.S.West,Anal.Chem.43(1971)136.[10]J.Nishimoto,T.Yamada,M.Tabata,Anal.Chim.Acta 428(2001)201.[11]M.S.Frant,J.W.Ross Jr.,Science 154(1966)1553.[12]D1179–99Standard Test Methods for Fluoride Ion in Water,ASTMBook of Standards,vol.11.01,ASTM International,West Con-shohocken,PA,USA,2003.[13]H.Hara,Chun-ching Huang,Anal.Chim.Acta 338(1997)141.[14]L.N.Moskvin, A.N.Katruzov,T.G.J.Nikitina,Anal.Chem.53(1998)173.[15]M.Yuwono,S.Ebel,Arch.Pharm.Pharm.Med.Chem.330(1997)348–352.[16]R.Bodor,V .Madajova,D.Kaniansky,M.Masar,M.Johnck,B.J.Stanislawski,J.Chromatogr.A 916(2001)155.[17]D.I.Paik,D.S.Ma,D.Y .Park,H.S.Moon,Y .I.Chang,J.B.Kim,J.Radioanal.Chem.217(1997)221–223.[18]R.C.Hawkings,L.P.V .Corriveau,S.A.Kushneriuk,P.Y .Wong,Anal.Chim.Acta 102(1978)61.[19]C.Fuchs, D.Dom, C.A.Fuchs,H.V .Henning, C.Mcintosh, F.Scheler,Clin.Chim.Acta 60(1975)157–167.[20]M.A.Faraj-Zadeh, E.G.Kalhor,Microchim.Acta 137(2001)169.[21]Z.J.Huang,Y .L.Yang,G.Y .Yang,J.Y .Yin,Fenxi Huaxue 27(1999)1363.[22]I.Nemcova,B.Evtimova,J.Hrachovska,Dokl.Bolg.Akad.Nauk.43(1990)45.[23]A.B.Descalzo,D.Jim´e nez,J.E.Haskouri,D.Beltr´a n,P.Amor´o s,M.D.Marcos,R.Mart´ınez-M´a ˜nez,J.Soto,mun.(2002)562.[24]T.-H.Kim,T.M.Swager,Angew.Chem.Int.Ed.42(2003)4803.[25]G.Patonay,M.D.Antonie,Anal.Chem.63(1991)321.316 C.-Q.Zhu et al./Analytica Chimica Acta539(2005)311–316[26]A.Blazsek-Bodo,M.Vernes,Proceedings of the2nd conference onApplied Physical Chemistry,vol.1,1971,p25.[27]A.Blazsek-Bodo,M.Vernes,A.Szurkos,Farmacia(Bucharest)22(1974)345.[28]G.Sartori,R.Ballini,F.Bigi,G.Bosica,R.Maggi,P.Righi,Chem.Rev.104(2004)199.[29]Chinese Standard GB5750-85,Standard examination methods fordrinking water:determinationﬂuoride of environment water by F−ISE method,PR China Sanitary Department,1986,pp.46–50.[30]M.R.Mahmoud,Z.H.Khalil,R.M.Issa,Acta Chim.Acad.Sci.Hung.87(1975)121.。

Singularity of the density of states in the two-dimensional Hubbard model from finite size

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from ﬁnitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepﬁsh 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A ﬁnite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external ﬁeld”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some ﬁnite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at ﬁnite “exter-nal ﬁeld”was found in ﬁnite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at ﬁnite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a ﬁxed particle number.Hence the chem-ical potential can be thought of as an external ﬁeld for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at ﬁnite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at ﬁnite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed aﬀected,which in turn may lead to an extension of the Lifshitz-type singularities into the ﬁnite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tiﬁcation of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative diﬀerences from the 42lattice.In Section 4we analyse the ﬁnite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-ﬁlling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are deﬁned asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the conﬁgurations of Ising ﬁelds isdenoted by an integral.Since ˜z (µ)is not positive deﬁnite for Re(µ)=0we weight the ensemble of conﬁgurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice conﬁgurations reﬂect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given conﬁguration.In this regime the adiabatic approximation,which is the basis of the ﬁnite-temperature algorithm,is valid.The situa-tion diﬀers strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)conﬁgurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSﬁelds the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonﬁelds presented in Ref.[1].The auxilary HS ﬁelds do not have a kinetic energy term in the bosonic action which would suppress their rapidﬂuctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-ﬁlling,results in high uncontrolledﬂuctuations of the expansion coeﬃcients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coeﬃcients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coeﬃcients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of theﬁrst coeﬃcients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coeﬃcient)forβ=5−7.5for diﬀer-entµ0.The coeﬃcients are obtained using the bootstrap method on over10000conﬁgurations forβ=5increasing to over30000forβ=7.5.In spite of diﬀerent values of the average sign in these simulations,the coeﬃcients of the expansion(16)indicate good correspondence between coeﬃcients obtained with diﬀerent values of the update chemical potentialµ0:the normalized coeﬃcients taken from diﬀerentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeﬃ-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with diﬀerent values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identiﬁed by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect toﬁnd that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in theseﬁgures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for theﬁnite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Diﬀerentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which deﬁnes the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for diﬀerent values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the ﬁnite size eﬀects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at ﬁnite temperature.However,for suﬃciently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of theﬁnite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreﬂected in the diﬀerence between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourﬁnite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove signiﬁcant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearﬁts for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reﬂecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coeﬃcients and the Yang-Lee zeros,the smallnumber of zeros obtained with suﬃcient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaborateﬁttings of the scal-ing behaviour.Theﬁnite-size eﬀects may still be signiﬁ-cant,especially as the simulation temperature decreases,thus aﬀecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atﬁnite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepﬁsh,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creﬃeld,E.G.Klepﬁsh,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coeﬃcients for ex-pansion(14)at diﬀerent update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topﬁgure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the theﬁt of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

SLiCE

SLiCE:a novel bacterial cell extract-based DNA cloning methodYongwei Zhang*,Uwe Werling and Winfried Edelmann*Department of Cell Biology,Albert Einstein College of Medicine,Bronx,NY 10461,USAReceived July 28,2011;Revised December 13,2011;Accepted December 15,2011ABSTRACTWe describe a novel cloning method termed SLiCE (Seamless Ligation Cloning Extract)that utilizes easy to generate bacterial cell extracts to assemble multiple DNA fragments into recombinant DNA molecules in a single in vitro recombination reaction.SLiCE overcomes the sequence limitations of traditional cloning methods,facilitates seamless cloning by recombining short end homologies (!15bp)with or without flanking heterologous se-quences and provides an effective strategy for dir-ectional subcloning of DNA fragments from Bacteria Artificial Chromosomes (BACs)or other sources.SLiCE is highly cost effective as a number of standard laboratory bacterial strains can serve as sources for SLiCE extract.In addition,the cloning efficiencies and capabilities of these strains can be greatly improved by simple genetic modifica-tions.As an example,we modified the DH10B Escherichia coli strain to express an optimized j prophage Red recombination system.This strain,termed PPY,facilitates SLiCE with very high efficiencies and demonstrates the versatility of the method.INTRODUCTIONThe generation of recombinant DNA molecules is an essential tool in modern molecular biology.The conven-tional DNA cloning strategies that have been used for several decades typically involve the use of type II restric-tion enzymes to generate appropriate DNA fragments,the modiﬁcation of DNA ends to generate blunt or sticky ends and the ligation of the DNA fragments to generate plasmid or other type DNA vectors (1–3).However,these proced-ures depend on the presence of appropriate restriction sites to generate both vector and insert molecules and often leave unwanted sequences at the junction sites.In addition,the restriction enzymes and modifying enzymes required for these manipulations are often expen-sive making these procedures costly especially in high throughput settings.To circumvent these limitations,we developed a new restriction site independent cloning method that does not leave any unwanted sequences at the junction sites (seamless)and is based on in vitro recombination between short regions of homologies (15–52bp)in bacterial cell extracts termed SLiCE (Seamless Ligation Cloning Extract).SLiCE allows for efﬁcient restriction site independent cloning of DNA frag-ments generated by restriction digestion or PCR ampliﬁ-cation into linearized vectors.In addition,SLiCE does not require the use of enzymes for the modiﬁcation of vector and insert end sequences (such as Klenow or T4DNA polymerase)or ligases.The SLiCE method can be used for virtually any type of cloning approach including the simple subcloning of PCR or restriction fragments,the generation of tagged expression vectors,the construction of more complex vectors such as gene targeting vectors or the directional subcloning of larger DNA fragments from more complex vectors such as bacterial artiﬁcial chromo-somes (BACs).In addition,SLiCE allows the assembly of multiple DNA fragments in one cloning step,which may make it an ideal method for the assembly of multiple DNA fragments during gene synthesis applications.The SLiCE method is based on bacterial extracts that can be derived from a variety of common RecA ÀEscherichia coli laboratory strains such as DH10B and JM109.These strains can also be further optimized by simple genetic modiﬁcations to improve SLiCE cloning efﬁciencies and capabilities making SLiCE highly versa-tile.For example,we established a DH10B-derived strain,termed PPY that was engineered to contain an optimized prophage Red recombination system (4–6).We found that extracts derived from this strain provide the highest cloning efﬁciencies thus far and that it can be used for all cloning approaches that are routinely used in the laboratory.The SLiCE method overcomes many problems related to conventional cloning procedures and provides a highly cost-effective approach for the*To whom correspondence should be addressed.Tel:+17186781087;Fax:+17186781019;Email:yongwei.zhang@ Correspondence may also be addressed to Winfried Edelmann.Tel:+17186781086;Fax:+17186781019;Email:winfried.edelmann@Published online 12January 2012Nucleic Acids Research,2012,Vol.40,No.8e55doi:10.1093/nar/gkr1288ßThe Author(s)2012.Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (/licenses/by-nc/3.0),which permits unrestricted non-commercial use,distribution,and reproduction in any medium,provided the original work is properly cited.by guest on September 10, 2013/Downloaded fromgeneration of recombinant DNA molecules in a seamlessand restriction site independent manner.In this report,wedescribe the SLiCE method,its principal features andapplications.MATERIALS AND METHODSBacteria strainsThe following laboratory E.coli strains were used:DH10B(Invitrogen),JM109(Promega),BL21(DE3)(Invitrogen),BLR(DE3)(Novagen)and ER2566(NEB).The DH10B derived E.coli strain PPY was constructedby Suicide Plasmid Based Genome Modiﬁcation(7)using plasmid pGT1(PPY genotype:FÀend A1rec A1gal E15gal K16nup G rps LÁlac X74F80lac ZÁM15ara D139Á(ara,leu)7697mcr AÁ(mrr-hsd RMS-mcr BC)cyn X::[ara C pBAD-red EM7-red Tn5-gam]lÀ).The competent cells used for transformation of SLiCEgenerated recombination products were ElectroMAX DH10B TM cells and MAX EfﬁciencyÕDH10B TM compe-tent cells(Invitrogen).PlasmidsThe plasmid pBL was constructed by insertion of a70-bpchemically synthesized multiple cloning site into the2.5-kbPCR-generated plasmid backbone of pBluescript II KS(+)(Stratagene)and deletion of the LacZ ORF by conven-tional cloning.The plasmid pBL-DL was constructed byinsertion of a1-kb PCR fragment from pGEMÕ-luc(Promega)into the NotI/SalI sites of pBL by SLiCE. The suicide plasmid pGT1was constructed by SLiCE-mediated insertion of a830-bp PCR-ampliﬁed fragmentspanning the30region of the E.coli DH10B cyn X geneand an ara C-pBAD-red /EM7-red /Tn5-gam expressioncassette isolated from plasmid pBAD24(8)and lambdaphage DNA(NEB)into the Sma I site of plasmid pEL04(9).pGT1also contains a temperature-sensitive replicon and a chloramphenicol selection marker.Preparation of SLiCE extractEscherichia coli strains were grown at37 C in100ml2X YT medium until they reached OD600&5.3(OD600readings were calculated by diluting the sample toenable photometric measurement in the linear rangebetween0.1and0.5OD600).PPY was subsequentlyincubated for2h in2X YT medium containing0.2% L-arabinose to express prophage protein Red .The cells were harvested by centrifugation at5000g for 20minutes at4 C.The cells from96ml of originalculture(wet weight&0.92g)were washed1time with200ml ddH2O and resuspended in1.2ml CelLytic TM BCell Lysis Reagent(Sigma).The resuspended cells wereincubated at room temperature for10minutes to allowlysis to occur.The cell lysates were centrifuged at20000g for2min at room temperature to pellet the insol-uble material.The resulting supernatants were carefully removed from the cell debris into a low binding1.5ml tube(Protein LoBind Tube1.5ml,Eppendorf).The cell extracts were mixed with equal volume of100%glycerol,aliquoted into40–60ul portions in low binding0.5mltubes(Protein LoBind Tube0.5ml,Eppendorf),andstored atÀ20 C for about2months without signiﬁcantloss of activity.For long-term storage,the aliquoted cellextracts were stored atÀ80 C in50%glycerol,which canbe thawed on wet ice and refrozen up to10times withoutsigniﬁcant loss of activity.SLiCE reaction and transformationThe vectors used for SLiCE were linearized by restriction digestion or PCR ampliﬁcation.The cloning inserts werePCR ampliﬁed using primers containing50-end homo-logies to the vector or to other inserts for coassembly.Vector or insert DNAs that were generated by PCR amp-liﬁcation using plasmid DNA as templates were treatedwith DpnI prior to puriﬁcation to remove residualplasmid template DNA.The linearized vectors and PCRinserts were subjected to gel electrophoresis and puriﬁedusing the QIAEX II gel extraction kit.For SLiCE cloningof BAC DNA,the restriction digested BAC DNA waspuriﬁed by phenol/chloroform extraction.The standard SLiCE reaction mix contained the follow-ing ingredients:50–200ng linear vector,appropriateamount of insert DNA in a1:1to10:1molar ratio ofinsert to vector,1m l10X SLiCE buffer(500mM Tris–HCl(pH7.5at25 C),100mM MgCl2,10mM ATP,10mMDTT),1m l SLiCE extract and ddH2O to a total volumeof10m l.The SLiCE reaction was incubated at37 C for1hand subsequently1m l of the SLiCE reaction was electroporated into20m l ElectroMAX DH10B TM cells (Invitrogen)or chemically transformed into100m l MAXEfﬁciencyÕDH10B TM competent cells(Invitrogen)fol-lowing the manufacturer’s instructions.The transform-ation efﬁciencies of ElectroMAX DH10B TM cells (Invitrogen)and MAX EfﬁciencyÕDH10B TM competentcells(Invitrogen)were$1Â1010and1Â109transfor-mants/m g of pUC19DNA,respectively.The transformedcells were plated on ampicillin/Xgal agar plates or agarplates containing appropriate antibiotics.RESULTSComparison of E.coli K12strains for SLiCESLiCE is a cloning method that is based on in vitro recom-bination in bacterial extract.SLiCE is a simple and efﬁ-cient procedure with the entire process consisting of threesteps(Figure1a):(i)The preparation of linear vector andinsert fragments containing short end homologies introduced by PCR with primers having appropriate50 extension sequences;(ii)the SLiCE in vitro reaction and(iii)the standard transformation(electroporation or chemical transformation)of recombination products intosuitable host bacteria.In this article,all bacterial trans-formations were performed by electroporation using ElectroMAX DH10B TM cells(Invitrogen)unless other-wise noted.To determine the most efﬁcient bacterial strains forSLiCE,ﬁve standard laboratory E.coli K12strains weretested including DH10B,JM109,ER2566,BL21(DE3)and BLR(DE3).The main criteria for their selectionby guest on September 10, 2013/Downloaded fromincluded their genetic status with regard to the RecA hom-ologous recombination protein and the presence of restric-tion systems that could interfere with the stability of exogenous DNA (Supplementary Table S1).To determine the suitability of extracts derived from these strains for SLiCE,a simple cloning strategy was devised.A plasmid vector termed pBL was linearized by restriction digestion and incubated together with a 500-bp PCR-ampliﬁed LacZ a in the different bacterial extracts.To facilitate recombination,the LacZ a fragment con-tained 42-bp end sequences that were homologous to the end sequences of pBL.According to the experimental design positive recombinant clones could be identiﬁed by blue/white selection after transformation and growth of bacteria on ampicillin/Xgal agar plates.This analysis showed that extracts from the two RecA Àstrains DH10B and JM109yielded the highest cloning efﬁciencies,indicating that SLiCE is facilitated by a RecA independent mechanism (Figure 1c).Due to the high cloning efﬁciency,cell extract from the DH10B strain was further used for the analysis of ing the pBL–LacZ a cloning strategy described above,the inﬂuence of several parameters on SLiCE cloning ef-ﬁciency was tested including the lengths of end homologies,the vector/insert ratio,the overall DNA con-centration and the transformation methods.To determine the effect of end homology length on SLiCE,homologies ranging from 0to 100bp (as counted from the 30-ends of the vector)were tested (Table 1).This analysis showed that inserts without end homology or 10-bp end homology did not yield any recombinant colonies.In contrast,15bp of end homology already yielded an appreciable number of recombinant colonies at a cloning efﬁciency of 75colony forming unit (CFU)/ng of vector,while 30bp of end homology provided very robust cloning efﬁciencies (920CFU/ng of vector).A further increase in end homology length resulted in even higher cloning efﬁciencies with an end homology length of 52bp pro-viding the highest efﬁciency (21965CFU/ng of vector).However,in contrast to in vivo homologous recombin-ation cloning,the cloning efﬁciency dropped signiﬁcantly when the end homology length was further increased (Table 1)indicating that SLiCE promotes in vitro recom-bination by a different pathway.Control reactions that contained the same vector/insert combinations with increasing end homologies but without SLiCE extract did not yield any recombinant colonies.Next,SLiCE was performed with varying vector/insert molar ratios at a vector concentration of 10ng/m l and 42bp of end homology.These studies showed that vector/insert ratios of 1:1,1:2,1:6and 1:10yielded 1335,2330,11120and 12120CFU/ng of vector,respectively,demonstrating that increased insert ratios could yield higher cloning efﬁpared to the standard vector concentra-tion of 10ng/m l,SLiCE using low concentrations of vector (1ng/m l)and a vector/insert at ratio of 1:1led to a 200-fold decrease in cloning efﬁciency which is likely due to the instability of vector and insert DNA at these low concentration in the SLiCE reaction mix.All the data above were derived by electroporation of the SLiCE reaction products.We also performed chemical transform-ation of SLiCE reaction products with 42bp of end homology using MAX Efﬁciency ÕDH10B TM competent cells (Invitrogen),which yielded about a 10-fold lower cloning efﬁciency than electroporation (1063CFU versus 10480CFU/ng of vector,chemical transformation versus electroporation).Besides promoting recombination between homologous sequences at the ends of vector and inserts,SLiCE is also capable of facilitating recombination between DNA frag-ments that contain ﬂanking heterologous sequences and of deleting the extra ﬂanking sequences to generate precise junctions at the recombination sites.This feature of SLiCE provides a highly useful cloning tool,especially in those cases where the absence of suitable restriction sites in a vector prevent the seamless cloning of aninsertFigure 1.SliCE cloning.(a )Outline of SLiCE Cloning.(b )Schematic illustrating seamless SLiCE cloning with ﬂanking heterologous se-quences.(c )Comparison of SLiCE efﬁciency of E.coli K 12strains.(d )BsaAI/SapI restriction analysis of the recombinants derived from SLiCE cloning.Plasmid DNAs from 12independent ampicillin-resistant blue colonies (lanes 1–12)were digested with BsaAI/SapI.The digestion products were separated on a 1%agarose gel and visualized after ethidium bromide staining.Recombinant plasmids contain one BsaAI site and one SapI site yielding diagnostic 2.1-and 0.8-kb restriction fragments.by guest on September 10, 2013/Downloaded fromfragment into a desired vector region (Figure 1b).To test this feature,DH10B SLiCE reactions were performed with vector pBL-DL that was designed to provide heterologous ﬂanking sequences at the cloning sites.pBL-DL was linearized with appropriate restriction enzymes to generate ﬂanking heterologous sequences on one side (319,738and 998bp)or on both sides (45bp plus 23bp and 319bp plus 738bp)and was together with a LacZ a fragment of 500bp with 42bp of end homologies (Figure 1b)subjected to SLiCE followed by blue/white selection.Our results showed that DH10B SLiCE can ef-ﬁciently remove 45bp plus 23bp on both sides of ﬂanking heterologous sequences but it cannot facilitate DNA cloning with longer ﬂanking heterologous sequences of 319,738,998bp on one side or 319bp plus 738bp on both sides (Supplementary Table S2).Generation of a modiﬁed DH10B strain for the optimization of SLiCEIn vivo homologous recombination in E.coli can be facilitated by three different recombination pathways:the RecA dependent pathway,a RecA independent pathway of unknown nature and a RecA independent pathway that utilizes prophage Red/ET recombination systems (4–6,10–14).The studies above indicate that a RecA independent pathway catalyzes SLiCE.To optimize SLiCE and acquire even more efﬁcient strains as a source for cell extracts,we modiﬁed the DH10B genome using a suicide plasmid based strategy to insert an optimized prophage Red recombination system into the bacterial genome.Speciﬁcally,the genome of DH10B bacteria were modiﬁed to constitutively express the phage red b and gam genes under the control of the EM7and Tn5promoters,respectively,and also the red a gene under the control of an arabinose-inducible pBAD promoter (araC-pBAD )(8).The modiﬁed DH10B strain was termed PPY and tested for SLiCE.Extracts derived from PPY bacteria yielded signiﬁcantly higher cloningefﬁciencies and a more robust seamless cloning activity in the presence of heterologous ﬂanking sequences than the DH10B extracts (see below)and were used in the fol-lowing series of experiments for the analysis of optimized SLiCE capabilities.Efﬁciency and ﬁdelity of PPY SLiCEIn the ﬁrst series of experiments we investigated the efﬁ-ciency and ﬁdelity of the improved PPY SLiCE extract.First,we examined the inﬂuence of end homology length on PPY SLiCE mediated cloning in more ing the pBL–LacZ a cloning strategy end homologies varying from 0to 100bp were examined (Table 1).Vector and insert fragments with no end homology or a short homology of 10bp did not yield any recombinant colonies.Similar to DH10B SLiCE without the Red system,the minimum length of homology required for efﬁcient cloning was 15bp,however,the PPY SLiCE extract yielded a dramatic increase in the number of blue recombinant colonies (4640CFU versus 75CFU per ng of vector,PPY SLiCE versus DH10B SLiCE)with a high cloning accuracy.Similar to DH10B SLiCE cloning,the cloning efﬁciency for PPY SLiCE increased with homology length in a range up to 52bp but dropped signiﬁcantly when the end homologies were further increased (Table 1).PPY SLiCE-mediated cloning was also performed using another vector/insert combination (p3XFLAG-CMV-7.1vector (Invitrogen)and a 800-bp PCR insert with end homologies ranging from 0to 42bp).These studies yielded similar results (data not shown).The recombinant colonies derived from PPY SLiCE cloning were further analyzed by colony PCR,restriction digestion and DNA sequencing analyses.More than 300blue colonies were screened using colony PCR (data not shown)and some of the colonies were veriﬁed by restric-tion digestion (Figure 1d).All of the analyzed clones con-tained the correct insert.The vector/insert junctions ofTable 1.Inﬂuence of End Homology Length on SLiCE Cloning Homology length (bp)Cloning efﬁciencyCloning accuracyDH10B SLiCEPPY SLiCE DH10B SLiCE PPY SLiCE Vector only 00––000––1000––1575464090%99%20803450088%99%3092012400098%99%421048063200099%99%522196576600099%99%68179516200099%99%78332511925099%99%8818906800099%99%10033203250099%99%Cloning efﬁciencies using different lengths of end homologies are given as CFUs of blue colonies per nanogram of vector.Cloning accuracies are given as the percentage of blue colonies among the total number of all amp r colonies (blue and white).The 2.5-kb vector pBL was linearized by NotI/SalI digestion and the 500–700-bp LacZ a fragments were prepared by PCR.Experiments were performed using 10ng/m l of vector and the corresponding amount of insert DNA at a 1:6molar ratio of vector:insert.The blue colonies contain recombinant plasmid and the white colonies contain non-recombinant vector background.by guest on September 10, 2013/Downloaded from30recombinant clones were sequenced and all of the clones contained the correct cloning junctions indicating that SLiCE fuses vector and insert in a precise manner. The small number of white background colonies that were observed in these test experiments could be traced back to spurious amounts of undigested pBL vector during linearization.We next examined theﬁdelity of PPY SLiCE without prior selection of positive recombinant clones.For these studies we used a NotI/XbaI linearized5.2-kb plasmid vector and a1.4-kb PCR-ampliﬁed insert with30bp of end homologies.The entire insert and the junction regions of20positive recombinants were sequenced. Eighteen recombinants contained completely correct se-quences and2recombinants presented one mutant nucleo-tide located in the PCR insert,which is consistent with the error rate of the DNA polymerase that was used for PCR ampliﬁcation(Fastart Fidelity PCR system,Roche)at 2.4Â10À6/bp/cycle and30cycle ampliﬁcation.The error rate and location of mutations indicate that these muta-tions were caused by PCR ampliﬁcation of inserts and that SLiCE did not introduce further mutations.We next examined the effect of various molar ratios of vector and insert and the overall DNA concentration on PPY SLiCE.The results were again similar to DH10B SLiCE mediated cloning.PPY SLiCE with pBL–LacZ a vector/inserts at molar ratios of1:1,1:2,1:6and1:10 with20-bp end homologies yielded19240,27320,34500 and65350CFU/ng of vector,respectively,showing that increasing the amount of insert yields slightly higher cloning efﬁciencies for PPY SLiCE.We also observed that PPY SLiCE at a low concentration of vector and insert(1ng/m l)at a vector/insert ratio of1:1also resulted in a10-fold reduced cloning efﬁciency.We also determined the sequence dependence of PPY SLiCE in more detail.For this,weﬁrst generated four LacZ a fragment and vector combinations containing 20-bp end homologies that each differed in GC content (ranging from40%to80%)and sequence.These vector/ insert combinations were subjected to PPY SLiCE.We did notﬁnd a signiﬁcant difference in the cloning efﬁciencies indicating that SLiCE is indeed sequence inde-pendent within the range of sequence diversity tested (Supplementary Table S3).Next,we generated four frag-ments with20-bp end homologies containing a mismatch at different positions.We found that a single mismatched nucleotide at the end of the20-bp homology produced one type of recombination product and the heterologous nu-cleotide was removed during SLiCE.A single mismatched nucleotide within the middle of the20-bp homology produced two types of recombination products in which either one of the mismatched nucleotides was retained.In addition,we observed that the presence of a single mismatch only led to a slight reduction in cloning efﬁ-ciency(Supplementary Table S4).To determine the effect of insert length on PPY SLiCE cloning efﬁciencies we attempted to assemble vectors ranging in size from2to15kb and containing inserts ranging from80bp to21kb in size.We found that the cloning of larger fragments by SLiCE occurred at robust but somewhat reduced efﬁciencies.For example,the assembly of an11-kb recombinant plasmid containingan8-kb insert was achieved at a cloning efﬁciency of140CFU/ng of vector.The restriction analysis of24clones revealed that22contained the expected recombinant plasmid.At present,we have successfully used SLiCE-mediated cloning to generate more than100recombinant plasmids employing various cloning strategies and many different vector/insert combinations.Our results indicate that SLiCE can be considered a universal cloning method forthe generation of recombinant DNA at highﬁdelity. Furthermore,the nature of vector and insert ends suchas blunt ends or30or50sequence overhangs did not inﬂu-ence SLiCE efﬁciency or accuracy.However,the use of vectors with complementary50or30overhanging ends for SLiCE increased the formation of empty vector back-ground colonies,which is probably due to annealing ofthe single-stranded ends in the bacterial extracts or inthe transformed host cells.PPY SLiCE withﬂanking heterologous sequencesThe seamless cloning activity of PPY SLiCE was examined using the same pBL-DL–LacZ a cloning strategy as for DH10B SLiCE withﬂanking heterologous sequences at one side(2,319,738and998bp)or on bothsides(45bp plus23bp and319bp plus738bp)and with various end homologies.In comparison to DH10B extracts,PPY extracts pre-sented a much stronger seamless activity,which can remarkably increase the efﬁciency of DNA cloning espe-cially with shorterﬂanking heterologies(45bp plus23bpon both sides;7600CFU versus1265CFU per ng of vector,PPY extract versus DH10B extract).In addition,PPY SLiCE can efﬁciently remove longerﬂanking heter-ologous sequences of up to998bp on one side or up to319bp plus738bp on both sides(Table2).In general, vectors with shorterﬂanking sequences orﬂanking se-quences on only one side yielded higher cloning efﬁciencies compared to vectors with longer and/or double-sidedﬂanking heterologous sequences(Table2).Similar to SLiCE cloning withoutﬂanking heterologous sequences, longer end homologies between vector and insert resultedin higher cloning efﬁciencies(Table2).SLiCE cloning with multiple fragmentsThe high cloning efﬁciency andﬁdelity of PPY SLiCE suggested it might be possible to generate more complex recombinant plasmids using multiple inserts in a single cloning reaction.To test this idea we designed two differ-ent SLiCE strategies for the cloning of multiple insert fragments.In theﬁrst strategy,we attempted to clone multiple inserts into one vector in one SLiCE reaction to generate a single recombinant DNA molecule derivedfrom multiple fragments that was termed multiple-way SLiCE cloning(Figure2a).The second strategy was designed to clone several different inserts carrying thesame end homology into a vector in one SLiCE reactionin parallel.This strategy creates multiple different recom-binant DNA molecules and was termed SLiCE batch cloning(Figure2b).by guest on September 10, 2013/Downloaded fromTo examine the potential of SLiCE for multiple-way cloning,three-,four-and seven-way SLiCE cloning was performed using the pBL vector and three different sets of PCR ampliﬁed inserts with 42bp of end homology that can assemble into a single 1.9-kb DNA fragment express-ing LacZ a activity (Figure 2c and Table 3).Our studies showed that PPY SLiCE mediated three-,four-and seven-way cloning occurred at signiﬁcant efﬁciencies and high accuracies (Table 3).DNA sequencing analysis showed that all of the multiple fragments were precisely joined by SLiCE-mediated multiple-way cloning.We next examined multiple-way SLiCE cloning using other vectors and inserts.A three-way SLiCE using a 4.7-kb vector (p3XFLAG-CMV-7.1,Invitrogen)and two 250-bp inserts with 24bp of end homologies produced about 500CFU/ng of vector with a 10-fold stimulation over non-recombinant background colonies.In another three-way experiment we successfully assembled a 3-kb vector and two 2.6-and 2.5-kb inserts using 42bp of end homology with 80%accuracy and a cloning efﬁciency of 60CFU/ng of vector.To determine the ability of multiple-way SLiCE cloning to assemble highly complex vector/insert combinations,a four-way cloning strategy (42-bp end homology,2.5kb of vector,inserts of 500bp,1.4and 2.5kb)and a seven-way strategy with shorter end homology (24-bp homology,2.5kb of vector and six inserts totaling 2kb)were performed.For both multiple-way cloning strategies the cloning efﬁciencies were reduced but still provided at least 20CFU/10ng of vector.For SLiCE batch cloning,two sets of experiments were performed.Six PCR inserts varying from 300bp to 1kb with 30bp of end homology were mixed together with a linearized 6.7-kb prokaryotic expression vector (PTXB1,NEB)and incubated in PPY SLiCE extract (Figure 2b).About 340CFU/ng of vector were obtained.The analysis of 32colonies showed that all six possible recombinant vector/insert combinations were obtained (Figure 2d and e).Another experiment using a 4.7-kb mammalian expression vector (p3XFLAG-CMV-7.1,Invitrogen)andthree inserts of 1,1.5and 2.5kb with 24-bp homology yielded similar results (data not shown).SLiCE cloning of genomic fragments from BAC clones It is often challenging to subclone a genomic DNA fragment from larger DNA vectors such as BACs into a plasmid vector.Due to the high cloning efﬁciency of PPY SLiCE,we tested whether SLiCE could also facilitate this type of cloning.A SLiCE cloning strategy was designed to subclone individual genomic DNA fragments from BAC vectors (Figure 3a).Speciﬁcally,BAC DNA isolated from clone RP23-303G13(CHORI),165kb in size and contain-ing 66Bgl II and 19Eco RV restriction sites,was digested with either Bgl II or Eco RV to generate a complex pool of DNA fragments.The digested BAC DNA was phenol/chloroform puriﬁed and subjected to PPY SLiCE cloning with PCR-generated pBluescript II KS(+)(Stratagene)derived vectors that contained end homologies to different Bgl II or Eco RV BAC restriction fragments.We attempted SLiCE cloning of several Bgl II BAC fragments of differ-ent sizes (830bp,3.7,6.7,8.7and 14kb)with 42or 52bp of end homology.In addition,SLiCE cloning was also performed for several Eco RV BAC fragments of larger sizes (5.3,6.3,12.2and 21kb)(Table 4).In all cases we were able to obtain recombinant clones carrying the dif-ferent BAC fragments with high or acceptable cloning efﬁciencies (Table 4and Figure 3b and c),indicating that SLiCE cloning is an effective strategy for the direc-tional subcloning of small or large BAC genomic fragments.DISCUSSIONThe observation that bacterial cell extracts can efﬁciently recombine DNA molecules using short-end homolo-gies was a serendipitous discovery in our laboratory.After initial characterization and further optimization,we were able to establish a novel restriction siteTable 2.PPY SLICE with ﬂanking heterologous sequences Homology length (bp)Flanking heterology length (bp)Vector length (bp)Cloning efﬁcencyCloning accuracy (%)Side 1Side 220202500100009942319028032270993031902803125098427380322212328730738032224327742998034825709442452325527600813045232552128863204523255271059423197383541598Cloning efﬁciencies are given as CFUs of blue colonies per nanogram of vector.Cloning accuracies are given as the percentage of blue colonies among the total number of all amp r colonies (blue and white).Vectors containing different end heterologies were derived from plasmid pBL-DL by digesting with various restriction cZ inserts of 500-bp size containing the indicated end homologies were generated by PCR.The experiments were performed using 10–40ng/m l vector DNA and the corresponding amount of insert DNA at a 1:6molar ratio of vector:insert in a 10m l reaction volume.The blue colonies contain recombinant plasmid and the white colonies contain non-recombinant vector background.by guest on September 10, 2013/Downloaded from。

异性材料方解石制备的隐身斗篷macroscopic invisibility cloaking of visible light

nature communications | 2:176 | DOI: 10.1038/ncomms1176 | /naturecommunications

© 2011 Macmillan Publishers Limited. All rights reserved.
nature communications | DOI: 10.1038/ncomms1176
T
height H2 and filled with an isotropic material of permittivity ε and µ (µ = 1; blue region in Fig. 1a) is mapped to a quadrilateral region in the physical space with anisotropic electromagnetic properties ε′ and µ′ (brown region in Fig. 1b). Thus, the cloaked region is defined by the small grey triangle of height H1 and half-width d. Mathematically, the transformation is defined by
ARTICLE
Received 15 Sep 2010 | Accepted 4 Jan 2011 | Published 1 Feb 2011
DOI: 10.1038/ncomms1176
Macroscopic invisibility cloaking of visible light
Xianzhong Chen1, Yu Luo2, Jingjing Zhang3, Kyle Jiang4, John B. Pendry2 & Shuang Zhang1

The Role of the Exchange-Correlation Potential in ab initio Electron Transport Calculations

a r X i v :c o n d -m a t /0609637v 2 [c o n d -m a t .m e s -h a l l ] 6 M a r 2007The Role of the Exchange-Correlation Potential in ab initioElectron Transport CalculationsSan-Huang Ke,1Harold U.Baranger,2and Weitao Yang,11Department of Chemistry,Duke University,Durham,NC 27708-03542Department of Physics,Duke University,Durham,NC 27708-0305(Dated:March 5,2007)The eﬀect of the exchange-correlation potential in ab initio electron transport calculations is investigated by constructing optimized eﬀective potentials (OEP)using diﬀerent energy functionals or the electron density from second-order perturbation theory.We calculate electron transmission through two atomic chain systems,one with charge transfer and one without.Dramatic eﬀects are caused by two factors:changes in the energy gap and the self-interaction error.The error in conductance caused by the former is about one order of magnitude while that caused by the latter ranges from several times to two orders of magnitude,depending on the coupling strength and charge transfer.The implications for accurate quantum transport calculations are discussed.PACS numbers:73.40.Cg,72.10.-d,85.65.+hThe calculation of electron transport through sin-gle molecules directly from quantum mechanics is cur-rently being intensively investigated for both funda-mental physics and applications in molecular electron-ics [1].In such a calculation,properties of the par-ticular molecule must be incorporated into an accu-rate transport model.A frequently used theoretical ap-proach is the single-particle Green function (GF)method [2]combined with a density functional theory (DFT)[3]electronic structure calculation.In this approach [4,5,6,7,8],the atomic structure of the entire lead-molecule-lead system is taken into account explicitly [7,8].Despite its advantages and high eﬃciency for large systems,several aspects of this approach remain prob-lematic [9,10,11,12,13,14].Here we address one aspect:we show that an improved description of electron-electron exchange and correlation within Kohn-Sham DFT dra-matically changes the predicted conductance.In the standard GF+DFT approach,all electron-electron interaction eﬀects are incorporated through the self-consistent DFT calculation,while the transmis-sion calculation is simply single-particle.Consequently,as emphasized by others [11,13],exchange-correlation corrections to the expression for the current are ne-glected.But even before considering those corrections,self-interaction error (SIE)is a potentially serious prob-lem within the GF+DFT method itself [3,12,14]:it leads to an overly extended charge distribution and,therefore,inaccurate molecule-lead charge transfer,espe-cially for weakly coupled systems.In ab initio transport calculations,SIE will directly aﬀect the position of the chemical potential in the molecular HOMO-LUMO gap (“gap”for short),as well as the broadening of the HOMO and LUMO orbitals,possibly producing large errors in the conductance [12].It is thus critical to improve the xc potential so as to eliminate the eﬀects of SIE.Another well-known problem with DFT is that the predicted gap is too small,often leading to a signiﬁcant overestimation of the conductance.The solution to this problem relies on a quasiparticle calculation or the construction of a SIE-free functional with a nonlocal xc potential.In this paper,we focus on eliminating the SIE and revealing the magnitude of the errors caused by the two problems.One way to eliminate SIE is to use Hartree-Fock (HF)theory:the exact treatment of exchange eliminates SIE.However,because HF involves a single determinant and lacks dynamical screening,the LUMO orbital is not phys-ically meaningful and the gap is too large for extended systems and large molecules.Hybrid functionals,like B3LYP [15,16],are a possible compromise:these mix the HF exchange potential with the local eﬀective potential obtained from the local density approximation (LDA)or generalized gradient approximation.Although B3LYP is a signiﬁcant improvement over LDA for almost all molec-ular systems,SIE still remains [17].The optimized eﬀective potential (OEP)approach is a direction for improving DFT calculations [18],in which the (local)eﬀective potential is expressed as an implicit density functional in terms of the Kohn-Sham orbitals.OEP enables one to construct a local xc potential from any energy functional,such as the HF exact exchange (EXX)or B3LYP functionals,or from an electron den-sity obtained from a more accurate theory [19],such as second-order many-body perturbation theory (MP2).Most OEP calculations to date use the HF energy func-tional (EXX-OEP).This simplest exchange-only OEP approach improves systematically the electronic struc-ture of various semiconductors [20]:the band gaps are signiﬁcantly improved over those of both LDA and HF,although the underlying reason is still open [21,22].In this paper,we implement the OEP approach in DFT-based ab initio transport calculations and inves-tigate,for the ﬁrst time,the eﬀect of diﬀerent xc potentials—LDA,HF,EXX-OEP,B3LYP,B3LYP-OEP,and MP2-OEP.Our purpose in using OEP is to con-struct a local xc potential which is SIE free (EXX-OEPFIG.3:(color online)Electric ﬁeld induced electron transfer (change in Q )between two H 8clusters separated by 8˚A (as shown in the inset).For such a large separation,the electron transfer should be an integer.For HF,this is the case;how-ever,LDA and B3LYP calculations show a substantial SIE.the peaks in T (E )become sharper and the conductance decreases.Note that the spread in conductance values becomes larger:now the maximum diﬀerence (between LDA and EXX-OEP)is about a factor of 10.To show the eﬀect from the gap,we ﬁrst examine the real transport gap of the H 16molecule by calcu-lating the ionization potential (I )and electron aﬃnity (A )using the delta self-consistent ﬁeld method (∆SCF)and the outer valence Green function method (OVGF)[24].The result for I −A is ∆SCF(HF,LDA,B3LYP)=4.7eV,6.0eV,5.9eV,and OVGF=6.6eV.Note that all the results are substantially smaller than the 8eV HF gap,indicating that it is too large.In particular,the large diﬀerence between ∆SCF(HF)and HF shows that HF does not give a good description for this long-chain molecule because of the lack of screening/correlation,de-spite the fact that it works well for very small molecules (the screening is very weak there,see the database at /cccbdb).On the other hand,I −A is signiﬁcantly larger than all the DFT gaps (2∼3eV)in the second class,indicating that they are too small (EXX-OEP does not work well here).A rough estima-tion of the eﬀect of the gap is the diﬀerence in conduc-tance between EXX-OEP and HF,both of which lack correlation and are SIE free.In Fig.2,this diﬀerence is about one order of magnitude for both strong and weak coupling;because the HF gap is too large,this rough estimation is probably an overestimate.So far we have discussed the SIE and gap issues for the system without molecule-lead charge transfer,where SIE causes overly broadened HOMO and LUMO states.For systems with charge transfer,SIE is a more signiﬁcant problem because it may lead to too much charge trans-fer,particularly in weakly coupled systems.To directly demonstrate this,we calculate,by using Mulliken popu-lation analysis,the charge transfer between two weakly coupled H atomic chains induced by a strong electric ﬁeld (see Fig.3).Each chain contains 8H atoms separated byFIG.with (a)2.8energy the the Li cluster to the H-chain are listed.The SIE-infected functionals place the chemical potential near a molecular resonance,while the SIE-free functionals place it near the middle of the gap.1˚A ,and the separation between the two chains is 8˚A.Because of the very large separation,the physical elec-tron transfer must be an integer.In HF,the electron transfer is indeed always an integer,showing that it is SIE free.For LDA,the result is almost linear in the elec-tric ﬁeld (full SIE),while B3LYP signiﬁcantly improves upon LDA but is still not accurate (partial SIE).The transmission through system B,in which there may be substantial charge transfer,is shown in Fig.4for two values of the molecule-lead separation.Molecule-lead charge transfer determines the position of the chemical potential (ﬁxed in the lead)in the molecular gap,and therefore the resulting conductance.The charge transfer and conductance are listed in the ﬁgure.Note the strik-ingly diﬀerent behavior of the two groups of function-als:for functionals with SIE (LDA,B3LYP,and B3LYP-OEP),the chemical potential enters the HOMO reso-nance because the charge transfer is large,while for func-tionals without SIE (HF,EXX-OEP,and MP2-OEP),the chemical potential is at the middle of the gap because the charge transfer is near zero.Consequently,the conduc-tance given by these two groups of functionals are very4diﬀerent,up to three orders of magnitude.For the smaller separation,2.8˚A in Fig.4(a),the func-tionals with SIE give a charge transfer of about0.5e,and the resulting conductance is around0.8G0.When the separation is increased to4.0˚A[panel(b)],the peaks be-come sharper,and the conductance in all cases decreases by more than an order of magnitude.In contrast,the charge transfer resulting from the SIE functionals de-creases only slightly,showing clearly that it is an arti-fact of SIE.Despite the quantitative diﬀerences between the stronger and weaker coupling,the broad features in the two cases are the same:the biggest step in conduc-tance(a factor of∼30)is between the SIE functionals and MP2-OEP followed by two smaller decreases,ﬁrst from MP2-OEP to EXX-OEP and then further to HF, each by about a factor of10.Here the eﬀect from the gap is also about one order of magnitude,from comparing EXX-OEP and HF as for system A.While it is clear,in terms of SIE,that the EXX-OEP and MP2-OEP calculations improve signiﬁcantly the standard GF+DFT calculation,it is not obvious which one of the SIE-free functionals–HF,EXX-OEP, or MP2-OEP–gives a conductance closest to the truth. MP2-OEP provides a near-exact local eﬀective potential for Kohn-Sham DFT,but itsﬁnite xc potential disconti-nuity is not included in the gap.As a result,its gap is too small.EXX-OEP gives a slightly larger gap which,how-ever,is still too small,and correlation is absent.HF,on the other hand,yields too large a gap,and correlation is also absent.Therefore,in terms of transport,EXX-OEP and MP2-OEP probably overestimates the conductance while HF probably underestimates it.The error seems to be about a factor of10.Finally,we relate the present calculation to the more rigorous time-dependent DFT formalism(TDDFT)[25, 26].In principle,unlike DFT,TDDFT can treat the elec-tronic structure of excited states[27].A Landauer-like form for the steady-state current can be derived from TDDFT[26]:in the linear-response regime(zero bias), the current is a Kohn-Sham term plus a correction from dynamical xc eﬀects.The eﬀective potential in our cal-culation can be regarded as the long time limit of that in the Kohn-Sham term,which is the major part of the current.The missing dynamical xc eﬀect is an open issue studied in[11,13].Our results on the eﬀects of the xc potential are helpful for improving TDDFT calculations within the adiabatic approximation.In summary,by implementing the OEP approach in an ab initio transport calculation,we have systematically investigated the eﬀect of diﬀerent local and nonlocal xc potentials.Dramatic eﬀects,up to orders of magnitude, originate from two factors–the SIE and the energy gap. The former will dominate for systems with charge trans-fer and can be eliminated by using a SIE-free OEP po-tential,while the latter is diﬃcult to treat within DFT and leads to a typical overestimation of the conductance by about a factor of10.Possible solutions are either to perform transport calculations using quasiparticle states, like those in GW approximation,or to develop SIE-free energy functionals without the discontinuity problem. We thank Aron Cohen,Kieron Burke,and Qin Wu for valuable conversations.This work was supported in part by the NSF(DMR-0506953).[1]J.R.Heath and M.A.Ratner,Physics Today56,43(2003).[2]S.Datta,Electronic Transport in Mesoscopic Systems(Cambridge University Press,Cambridge,1995).[3]R.Parr and W.Yang,Density-Functional Theory ofAtoms and Molecules(Oxford University Press,New York,1989).[4]J.Taylor,H.Guo,and J.Wang,Phys.Rev.B63,245407(2001).[5]P.S.Damle,A.W.Ghosh,and S.Datta,Phys.Rev.B64,201403(R)(2001).[6]Y.Xue,S.Datta,and M.Ratner,Chem.Phys.281,151(2002).[7]M.Brandbyge,J.-L.Mozos,P.Ordej´o n,J.Taylor,andK.Stokbro,Phys.Rev.B65,165401(2002).[8]S.-H.Ke,H.Baranger,and W.Yang,Phys.Rev.B70,085410(2004).[9]F.Evers,F.Weigend,and M.Koentopp,Phys.Rev.B69,235411(2004).[10]K.Burke,R.Car,and R.Gebauer,Phys.Rev.Lett.94,146803(2005).[11]N.Sai,M.Zwolak,G.Vignale,and M.DiVentra,Phys.Rev.Lett.94,186810(2005).[12]C.Toher,A.Filippetti,S.Sanvito,and K.Burke,Phys.Rev.Lett.95,146402(2005).[13]M.Koentopp,K.Burke,and F.Evers,Phys.Rev.B73,121403(R)(2006).[14]B.Muralidharan,A.W.Ghosh,and S.Datta,Phys.Rev.B73,155410(2006).[15]A.D.Becke,J.Chem.Phys.98,5648(1993).[16]C.Lee,W.Yang,and R.Parr,Phys.Rev.B37,785(1988).[17]Y.Zhang and W.Yang,J.Chem.Phys.109,2604(1998).[18]W.Yang and Q.Wu,Phys.Rev.Lett.89,143002(2002),and references therein.[19]Q.Wu and W.Yang,J.Chem.Phys.118,2498(2003),and references therein.[20]M.St¨a dele,M.Moukara,J.A.Majewski,P.Vogl,andA.G¨o rling,Phys.Rev.B59,10031(1999).[21]S.Sharma,J.K.Dewhurst,and C.Ambrosch-Draxl,Phys.Rev.Lett.95,136402(2005).[22]M.Gr uning,A.Marini,and A.Rubio,J.Chem.Phys.124,154108(2006).[23]NWChem,A Computational Chemistry Package for Par-allel Computers,Paciﬁc Northwest National Lab,Rich-land,Washington,USA(2003).[24]Ortiz,J.Chem.Phys.89,6348(1988).[25]G.Stefanucci and C.-O.Almbladh,Europhys.Lett.67,14(2004).[26]G.Stefanucci,S.Kurth,and E.K.U.Gross,cond-mat/0607333(2006).[27]E.Runge and E.K.U.Gross,Phys.Rev.Lett.52,997(1984).。

Synthesis and surface activities of organic solvent-soluble luorinated surfactants

Synthesis and surface activities of organic solvent-soluble ﬂuorinated surfactantsGuo-Li Li a ,b ,Li-Qiang Zheng a ,*,Jin-Xin Xiao c ,**aKey Laboratory for Colloid and Interface Chemistry of Education Ministry,Shandong University,Jinan 250100,PR China bSchool of Light Chemical and Environmental Engineering,Shandong Institute of Light Industry,Jinan 250353,PR China cBeijing FLUOBON Surfactant Institute,Beijing 100080,PR China1.IntroductionFluorinated surfactants,containing ﬂuorocarbon chains as hydrophobic groups,have a number of special properties such as chemical inertness,thermal stability,high surface activity and water and oil repellence that offer advantages over hydrocarbon surfactants [1].Among all surfactants,ﬂuorinated surfactants are most effective in reducing the surface tension of aqueous solutions [2].Their outstanding chemical and thermal stability expands their applications to extreme conditions which are too severe for hydrocarbon surfactants [3].Utilization of surfactants is most frequently implemented in aqueous systems.However,there are many cases of non-aqueous systems,such as the coating industry,paint industry,petroleum exploitation,and extraction process and micelle catalysis.Con-ventional surfactants used in aqueous systems that combine two basically different parts,at least one hydrophobic and one hydrophilic group,are not as effective in organic solvents as in water.It is known that conventional surfactants orient with the hydrophilic group away from organic solvent surface,whichresults in a higher surface energy and an increase of surface tension of the organic solvent [4].Therefore,structural requirements for a surfactant in organic solvents should be different from those of conventional surfactants.Previous investigations have shown that ﬂuorinated surfactants consisting of solvophobic and solvophilic segments in the molecule could be adsorbed at organic solvent/air interfaces as monomole-cular ﬁlms,thus depressing the surface tension of the organic solvent.Such investigations have thrown light on the study of ﬂuorinated surfactants in organic solvents [5–12].Several ﬂuori-nated surfactants such as F[CF(CF 3)CF 2O]n CF(CF 3)–COAr,(Ar =aryl group,n =1–4)[13],F[CF(CF 3)CF 2O]n CF(CF 3)–C 6H 5(n =1,2)[14],semiﬂuorinated diblock copolymers based on methyl methacrylate and 1H ,1H ,2H ,2H perﬂuoroalkyl methacrylate [15],ﬂuoroalkyl end-capped diacetone (N -1,1-dimethyl-3-oxobutylacrylamide)oli-gomers [16],ﬂuoroalkyl end-capped cooligomers containing poly-dimethylsiloxane and polyoxyethylene segments [17]were found to be effective in reducing the surface tension of m -xylene,while their surface tension curves exhibited a clear break point like the critical micelle concentration of ordinary surfactants in aqueous solutions.These ﬁndings suggest that these ﬂuorinated surfactants can form self-assembled molecular aggregates in aromatic solvents.Also some semiﬂuorinated alkanes which possess no charged or polar groups were found to aggregate in either hydrocarbon or ﬂuorocarbon solvents when the incompatibility between the semiﬂuorinated alkanes and solvents is sufﬁciently strong [18–21].Journal of Fluorine Chemistry 130(2009)674–681A R T I C L E I N F O Article history:Received 11February 2009Received in revised form 5May 2009Accepted 7May 2009Available online 18May 2009Keywords:Fluorinated surfactants FluorocarbonOrganic solvent-soluble surfactant Surface activity AdsorptionA B S T R A C TA variety of ﬂuorinated surfactants soluble in organic solvent were prepared,including C 8F 17SO 2NHC n H 2n +1(n =2,4,6,8,10),C 8F 17SO 2NHR (R =C 6H 11,C 6H 5),C 8F 17SO 2N(C n H 2n +1)2(n =1,2,3,4)and C 8F 17SO 2NH(CH 2)n NHO 2SC 8F 17(n =6,10).Their surface activities in various organic solvents were determined by surface tension measurement.The results showed that these ﬂuorinated surfactants can reduce the surface tension of both polar and non-polar organic solvents.In general,organic solvents with strong polarity or long alkyl chain are beneﬁcial to increase the surface activity of these polar ﬂuorinated surfactants.By comparing ﬂuorinated surfactants with the same ﬂuorocarbon segment and connecting group,C 8F 17SO 2N(C n H 2n +1)2(n =1,2,3,4)showed lower surface activity in organic solvents than C 8F 17SO 2NHC n H 2n +1(n =2,4,6,8)with an equal carbon number of the solvophilic group.Through surface tension vs.concentration curves given for N -octyl perﬂuorooctanesulfonamide in various organic solvents,a break point like the critical micelle concentration of ordinary surfactants in aqueous solutions was observed,and the effect of the different types of organic solvents on adsorption and aggregation behavior was also studied.ß2009Elsevier B.V.All rights reserved.*Corresponding author.Tel.:+8653188366062;fax:+8653188564750.**Corresponding author.Tel.:+861062561871;fax:+861062561871.E-mail addresses:lqzheng@ (L.-Q.Zheng),xiaojinxin@ (J.-X.Xiao).Contents lists available at ScienceDirectJournal of Fluorine Chemistryj o ur n a l h o m e p a g e :w w w.e l se v i e r.c om /l oc a t e /f l uo r0022-1139/$–see front matter ß2009Elsevier B.V.All rights reserved.doi:10.1016/j.jﬂuchem.2009.05.006Surface activities ofﬂuorinated surfactants at the air/organic solvent interface have been studied since the1950s[5].However, rather little work has been reported in recent decades[22–27].N-Alkyl perﬂuorooctanesulfonamides have been widely used in fabrics and papers,ﬁre retardants,anticorrosion agents,and many other commercial formulations[5,28].To our knowledge,there is little information available in the literature about the surface activity of derivations of N-alkyl perﬂuoralkylsulfonamides.In this work,we synthesized a series of N-alkyl perﬂuoralk-ylsulfonamides as organic solvent-solubleﬂuorinated surfactants in which straight,branched and cyclic alkyls and phenyl acted as solvophilic segment and single or doubleﬂuorocarbon chains as solvophobic segment.Surface activities of these surfactants in organic solvents including aliphatic,aromatic and non-protonic polar solvents were determined by surface tension measurement. Also,the relationship between surface activity and molecular structure of the organic solvent-solubleﬂuorinated surfactants as well as the type and structure of organic solvents was discussed. The surface adsorption state and surface adsorbed layer of ﬂuorinated surfactants were also discussed.2.Results and discussion2.1.Surfactant structureThe nature of common surfactants is based on the chemical antipathy of the surfactant head and tail,and on their opposite sympathy for water molecules.The situation is different in organic solvents.Surfactants which operate in organic solvent are generally composed of a solvophobic group and a solvophilic group within the same molecule.The hydrocarbon tail,hydro-phobic in water,may be the solvophilic group providing solubility in organic solvents and theﬂuorinated alkyl tail may be the solvophobic group.Such surfactants are expected to adsorb at the organic solvent/air surface to form a monolayer in which the ﬂuorocarbon segment of the surfactant tilts away from the surface. Thus their surface tension is expected to be reduced to a low value as a result of low cohesive energy density of perﬂuorocarbons[29]. In this work,perﬂuorooctanesulfonylﬂuoride was used as the raw material in view of its cheapness.Three series ofﬂuorinated surfactants were designed as following:(1)R F–Q–R H.where R F=C8F17,Q=SO2NH,R H=(CH2)n H(n=2,4,6,8,10),C6H11,C6H5.(2)R F–Q–(R H)2.where R F=C8F17,Q=SO2N,R H=(CH2)n H(n=1,2,3,4).(3)R F–Q–R H–Q–R F.where R F=C8F17,Q=SO2NH,R H=(CH2)n(n=6,10).2.2.The choice of organic solventCompared with aqueous solutions,various organic solvents used to dissolve surfactant may enlarge the research system and present diversiﬁed properties.Considering the extensive applica-tion and the solvent cost,a wide variety of organic solvents including alkanes(n-dodecane,n-tetradecane,n-hexadecane and liquid parafﬁn),cyclanes(cyclohexane),aromatic hydrocarbons (toluene and m-xylene),polar solvents(ethyl acetate,2-butanone, nitromethane,DMF,and DMSO)were selected for further investigation.2.3.Surface activity of surfactants in organic solventSurface activity results of each synthesizedﬂuorinated surfac-tant in saturated solutions of various solvents are presented in Table10(see supporting information)and Table1.In this work,the effectiveness of theﬂuorinated surfactants was expressed by the maximum reduction of surface tension of the organic solvents,D g.D g¼gsolventÀg solution(1)where g solvent is the surface tension of pure solvent,and g solution is the surface tension of the saturated solution.Most of surfactant solutions became saturated at concentration lower than0.05M. The lower the total concentration at which a small amount of precipitate was observed to appear in solution,the lower the solubility was.And at a same total concentration,more precipitate existed in solution,the lower solubility of surfactant was. However,for some solvents in which surfactants were too soluble, the effectiveness of surfactants was calculated on the basis of a 0.05M solution(footnote a).For example,the solution of surfactants in DMF remained homogeneous even at0.2M.The results in Table10(see supporting information)should be quite comparable because only a small concentration of an effective surfactant can give the maximum possible surface tension lowering of solvents[11].Besides,the surface tension of a solution containing0.05M of surfactant amounts to the value which is very close to the surface tension of a saturated solution.From Table10(see supporting information),it can be seen that the synthesizedﬂuorinated surfactants can reduce the surface tension of both polar and non-polar organic solvents.In all cases, the surface tension depression is smaller than21mN/m.However, commonﬂuorinated surfactants can lower the surface tension of water by more than50mN/m.Surfactants are not as effective in organic solvents as in water because of two reasons.First,the initial surface tension of organic solvents,19–30mN/m for most of hydrocarbon solvents,is much lower than that of pure waterTable1Surface tension of saturated solution of surfactant containing doubleﬂuorocarbon chains in various organic solvents at258C.Surfactant R H Solvent g solution(mN mÀ1)D gC8F17SO2NH–R H–NHO2SC8F17(CH2)6Toluene24.6 2.1m-xylene24.3 2.7DMSO24.017.5Nitromethane23.710.8DMF22.012.62-butanone19.5 2.8Ethyl acetate20.9 1.4(CH2)10Toluene24.7 2.0m-xylene24.9 2.1DMSO26.6a14.9Nitromethane24.110.4DMF28.4a 6.22-butanone20.9 1.4R H,the solvophilic group ofﬂuorinated surfactant.a Minimum surface tension at concentration0.05mol lÀ1.G.-L.Li et al./Journal of Fluorine Chemistry130(2009)674–681675(72mN/m,258C).Second,the free energy cost of transferring a –CF 2–group from the pure ﬂuorocarbon to an alkane solution is approximately 1.4kJ/mol,whereas the cost of transferring a –CF 2–from ﬂuorocarbon to water is 6.5kJ/mol [30].As expected,there is a close relationship between the structure of the solvophobic and solvophilic constituents of surfactant mole-cules,the property of organic solvents,and the surface activity of surfactants.That is why a ﬂuorinated surfactant cannot have the same optimum structure for all organic solvents [14].In order to clarify the effect of surfactant structure on the decrease of surface tension of various organic solvents,Figs.1–3were drawn according to corresponding results in Table 10(see supporting information).Fig.1.Relationship between the decrease of surface tension of organic solvents and the carbon atom numbers of single alkyl chain of C 8F 17SO 2NHC n H 2n +1at 258C.(A)alkanes;(B)cyclohexane and aromatic solvents;(C)polar solvents.D g =g solvent Àg solution indicates the effectiveness of the surfactant in lowering the surface tension of the solvent;n ,the carbon atom numbers of single alkyl of C 8F 17SO 2NHC n H 2n +1.Fig.2.Relationship between the decrease of surface tension of organic solvents and the carbon atom numbers of double alkyl chains of C 8F 17SO 2N (C n H 2n +1)2at 258C.(A)alkanes;(B)cyclohexane and aromatic solvents;(C)polar solvents.D g =g s olvent Àg solution indicates the effectiveness of the surfactant in lowering the surface tension of the solvent;n ,the carbon atom numbers of solvophilic segment of C 8F 17SO 2N (C n H 2n +1)2.G.-L.Li et al./Journal of Fluorine Chemistry 130(2009)674–6816762.3.1.Effect of the length of the solvophilic group2.3.1.1.Single alkyl solvophilic group structure.Fig.1shows the relationship between the decreases of surface tension of organic solvents (D g )and the carbon number (n )of C 8F 17SO 2NHC n H 2n +1.Itcan be seen from Fig.1A that in n-alkane (dodecane,tetradecane and hexadecane)solutions,surface activity increases with increasing alkyl chain length of the surfactants.Only those C 8F 17SO 2NHC n H 2n +1with enough long carbon chain and sufﬁcient solubility can exhibit surface activity in n-alkane solutions.For example,in n-hexadecane,such a surfactant has the highest surface tension reduction when the solvophilic group is n-decyl (D g =6.2mN/m).In liquid parafﬁn,the surfactants with a long alkyl chain,such as N -hexyl,N -octyl and N -decyl perﬂuoroocta-nesulfonamide exhibit surface activity whereas surfactants with a short alkyl chain cannot decrease the surface tension of liquid parafﬁn.The differences in surface activity of surfactants are based on the high solubility of N -alkyl perﬂuorooctanesulfonamide with a long alkyl chain,leading to the adsorption of a high concentration of perﬂuorocarbon chains at the organic solvent/air interface.Thus,it is not surprising to ﬁnd that N -ethyl perﬂuorooctanesulfonamide has little surface activity due to the short alkyl chain.However,in n -tetradecane and n-dodecane,the highest surface tension decrease can be obtained by N -octyl perﬂuorooctanesulfonamide to 5.7mN/m and 3.9mN/m,respectively.A possible explanation is that a long alkyl chain favors sufﬁcient solubility,yet a long alkyl chain arranged in zigzag conformation is easier to bend than a short alkyl chain.Because of the increasing steric hindrance between the bending long alkyl chains,a long alkyl chain maybe unfavorable for high packing density of ﬂuorocarbon chains at the interface.Moreover,it can also be inferred from Fig.1A that C 8F 17SO 2NHC n H 2n +1show higher surface activity in long chain n-alkane solvents than that in short chain n -alkane solvents.For example,the D g of N -octyl perﬂuorooctanesulfonamide,one surfactant of C 8F 17SO 2NHC n H 2n +1is 5.5mN/m in n -hexadecane which is higher than the D g in n -dodecane.The reason for this trend is quite obvious.Long chain alkane solvents have the contribution to maximize the mutual immiscibility and incompat-ibility between the ﬂuorocarbon chain and solvent,to promote ﬂuorocarbon chain escaping from the bulk of organic solvent,and thus to increase adsorption [21].In general,the surface tension decrease of n-alkane depends mostly on the length of the solvophilic group of C 8F 17SO 2NHC n H 2n +1,which exhibits a great decrease when the length of the single alkyl is long enough.In cyclohexane,as can be seen from Fig.1B,the values of D g for most of C 8F 17SO 2NHC n H 2n +1(n =2,6,8,10)are all small (near 2mN/m),so the surface activities do not obviously change with an increase of the alkyl chain because of limited solubility.In particular,D g of C 8F 17SO 2NHC n H 2n +1(n =4)is a little higher (3mN/m)than the other four surfactants,and its solubility is a little higher too.In the case of aromatic solvents,such as toluene and m -xylene,D g of C 8F 17SO 2NHC n H 2n +1(n =2,4,6,8,10)increases,then decreases a bit and increases again with an increase of alkyl chain length.This trend might be ascribed to different solubility of surfactants,and N -butyl perﬂuorooctanesulfonamide has a higher solubility than the other four surfactants.As discussed above,a long alkyl chain in solvophilic group favors sufﬁcient solubility in n -alkane solvents with ten or more carbon atoms,such as n-dodecane,n-tetradecane and n -hexadecane.But for cyclohex-ane and aromatic solvents,the number of carbon atoms is six.Thus compared to N -butyl perﬂuorooctanesulfonamide,other similar surfactants with shorter (n =2)or longer alkyl chains (n >6)as the solvophilic group have too low a solubility to impart the desired orientation of the surfactant molecule at the interface,resulting in low surface activity.As shown in Fig.1C,surfactant with ethyl solvophilic group exhibits the highest surface tension reduction of DMSO.Moreover,the surface activity in DMSO gradually decreases with the increase of the length of alkyl chain.This variation of surface activity may be affected by the result of change of solubility.In strong polar DMSO solvent,the longer the alkyl chain,the lower the solubilityofFig.3.Relationship between the decrease of surface tension of organic solvents and the structure of solvophilic group of C 8F 17SO 2NH–R H (R H =n-hexyl,cyclohexyl,phenyl)at 258C.(A)alkanes;(B)cyclohexane and aromatic solvents;(C)polar solvents.D g =g solvent Àg solution indicate the effectiveness of the surfactant in lowering the surface tension of the solvent;n -Hexyl,cyclohexyl and phenyl,the solvophilic segment of C 8F 17SO 2NH–R H .G.-L.Li et al./Journal of Fluorine Chemistry 130(2009)674–681677surfactants.During the preparation of solutions,it was found that the solution of N -butyl perﬂuorooctanesulfonamide or N -ethyl perﬂuorooctanesulfonamide in DMSO was homogeneous at 0.05M,but the solution of other longer alkyl perﬂuorooctane-sulfonamides at the same concentration became supersaturated.These phenomena may support an explanation that the solubility of N -alkyl perﬂuorooctanesulfonamides with longer alkyl chain is too small to obtain high surface activity in DMSO solution.Known as the ‘‘all-purpose dissolvent’’,DMF has excellent dissolving ability for C 8F 17SO 2NHC n H 2n +1.The solution of these surfactants in DMF was still homogeneous even at 0.2M.But the excessive solvophilic property of surfactants increases the con-centration to obtain minimum surface tension of organic solvents,and may prevent the close packing of ﬂuorocarbon chains adsorbed at the interface.Consequently,it is well understood that D g of all surfactants in DMF is less than that in DMSO.Moreover,surfactant with the n-decyl solvophilic group exhibits the highest surface tension reduction of DMF.2.3.1.2.Double alkyl solvophilic groups structure.Fig.2shows the relationship between the decrease of surface tension of organic solvents (D g )and the carbon atom numbers (n )of C 8F 17SO 2N(C n H 2n +1)2.Among the four surfactants with double alkyl as solvophilic groups,only N ,N -diethyl perﬂuorooctanesulfonamide can show surface activity in n-alkane and liquid parafﬁn solutions (see Fig.2A).These results are not surprising because N ,N -diethyl perﬂuorooctanesulfonamide has not only sufﬁcient solubility but also low steric hindrance.But surfactant with the dimethyl solvophilic groups has such poor solubility in long chain alkane solvents that the molecule cannot effectively be inserted into the bulk of solvents,resulting in a limited surface activity.For surfactants with dibutyl or dipropyl solvophilic groups,low surface activity may be caused by higher steric hindrance,and thus lower density of ﬂuorocarbon chains packed at the interface.In toluene and m -xylene solutions,it can be seen from Fig.2B that the surface activity sequence of C 8F 17SO 2N(C n H 2n +1)2is N ,N -dipropyl perﬂuoroctanesulfonamide,N ,N -dibutyl perﬂuoroocta-nesulfonamide,N ,N -diethyl perﬂuorooctanesulfonamide and N ,N -dimethyl perﬂuorooctanesulfonamide in terms of D g value from high to low.This is because the solubility of the surfactants is controlled by the structural similarity between the solvopho-bic portion of the solute molecule and the solvent molecules.Among the surfactants with the dialkyl solvophilic groups,the dipropyl structure is similar to the six-membered ring of the solvents,so it shows higher solubility.In cyclohexane,poor solubility of N ,N -dimethyl perﬂuorooctanesulfonamide limits the maximum surface tension decrease,while surfactants with the dipropyl or dibutyl solvophilic groups have relatively higher surface activity.As shown in Fig.2C,in strong polar organic solvents such as nitromethane,DMSO and DMF,N ,N -diethyl perﬂuorooctanesulfo-namide also exhibits higher surface activity than that of other surfactants with the dialkyl solvophilic groups,which might also be explained in the same way as in n-alkane and liquid parafﬁn solutions.parison of single,double and cyclic alkyl and phenyl structure of the solvophilic segment in the surfactant moleculeIt can be seen from Figs.1and 2the single alkyl solvophilic group is more effective in lowering surface tension of organic solvents discussed in this paper than the double alkyl structure when the total carbon number of the solvophilic segment are equal.That is,the D g is n-octyl >dibutyl,n-hexyl >dipropyl,n-propyl >diethyl (except for DMF solution)and n-ethyl >dimethyl.This can be explained in part by the branching chain effect of the solvophilic segment which increases the steric hindrance between the surfactant molecules,leading to a lower density of ﬂuorocarbon chains at the organic solvent/air interface.That is why only diethyl as the solvophilic segment among C 8F 17SO 2N(C n H 2n +1)2shows surface activity in n-alkane solutions.Fig.3shows the surface activity of C 8F 17SO 2NH–R H (R H =n-hexyl,cyclohexyl,phenyl)in different organic solvents.It can be found that N -cyclohexyl perﬂuorooctanesulfonamide exhibits the highest surface tension reduction of cyclohexane solution owing to the similar polarity of solvophilic group with solvent molecule (see Fig.3B).Likewise,N -phenyl perﬂuorooctanesulfo-namide has the best surface activity in toluene and m -xylene solution (see Fig.3B),and N-hexyl perﬂuorooctanesulfonamide can effectively decrease the surface tension of n-dodecane,n-tetradecane and n-hexadecane (see Fig.3A).It can also be found that N -cyclohexyl perﬂuorooctanesulfonamide is only effective in decreasing the surface tension of n-hexadecane.As for surfac-tants with phenyl or dipropyl as the solvophilic segment,they are all not effective in decreasing the surface tension of n-alkane solvents.This different ability in lowering the surface tension of n -alkane solvents might be not only caused by insufﬁcient solubility,but also due to the increased steric hindrance of cyclohexyl,phenyl and dipropyl groups in surfactant molecules as shown in Fig.4.parison of single ﬂuorocarbon chain and double ﬂuorocarbon chains of the solvophobic segmentAs is known,long ﬂuoroalkylated compounds exhibit a strong repellent property against water or hydrocarbons.Therefore,compounds containing two ﬂuorinated chains listed in Table 1were designed and the surface activity was examined in a series of organic solvents.Effect of the number of solvophobic groups on the decrease of surface tension of organic solvents can be obtained by comparing results in Figs.1–3and Table 1.Results show that the most effective surfactant that lowers the surface tension of nitromethane and DMF is C 8F 17SO 2NHC 6H 12NHO 2SC 8F 17among all synthesized surfactants.Especially,in 2-butanone and ethyl acetate,surfactants with a single ﬂuorocarbon chain as the solvophobic group cannot play a role due to excessive solubility in these solvents.However,the D g values of C 8F 17SO 2NHC 6H 12N-HO 2SC 8F 17are both higher than 1mN/m,which suggests that this ﬂuorocarbon compound can function as a surfactant in these solvents.Consequently,the use of ﬂuorinated surfactants in industrial applications may be widened.The discrepancy between the ability of surfactants containing different ﬂuorinated chains may reﬂect the differences in adsorption states.It is believedthatFig.4.Schematic illustration of the adsorption of C 8F 17SO 2NH(CH 2)5CH 3,C 8F 17SO 2N(CH 2CH 2CH 3)2and C 8F 17SO 2NHC 6H 11in n -alkane solvent.Bent curves depict alkyl groups,shaded blocks represent ﬂuorinated chains,and dark dots indicate the connecting group.G.-L.Li et al./Journal of Fluorine Chemistry 130(2009)674–681678an increase of the solvophobic part in surfactant molecules may decrease the solubility and lead to closer packing at the surface.2.4.Adsorption at interface and the adsorbing state of surfactant moleculesThe orientation and packing ofﬂuorinated surfactant molecules at the organic solvent/air interface prominently depend upon the molecular structure,solubility and extent of association of the solute and solvent molecules.As described above,N-octyl perﬂuorooctanesulfonamide exhibits remarkably decrease the surface tension of various organic solvents,so special attention was paid to the adsorption of N-octyl perﬂuorooctanesulfonamide in some representative organic solvents including n-tetradecane, n-hexadecane,DMF,DMSO and nitromethane.As shown in Fig.5, the surface tension curves(g-log c curves)of N-octyl perﬂuor-ooctanesulfonamide solutions inﬁve organic solvents at258C,are similar to those of common surfactants in aqueous solution.The initial decrease of the surface tension is followed by an abrupt change in the slope of the surface tension curve.After the break point,the surface tension of the solutions no longer changes, suggesting the aggregation of surfactant molecules in the organic solvent.The intersection in the surface tension curves can be deﬁned as critical aggregate concentrations(cac),and the corresponding surface tension is deﬁned as g cac.The main difference between these curves and the analogous curves for common surfactants in aqueous solutions is that the initial surface tensions of the pure organic solvents are much lower than that of pure water.The values of cac and g cac for C8F17SO2NHC8H17in various organic solvents at258C are shown in Table2.It is evident that the cac values increase and then decrease with an increase of polarity of organic solvents,which agrees with the idea that excessive solubility increases the concentration to obtain minimum surface tension.To investigate the adsorbing state of surfactant molecules at the organic solvent/air interface,the maximum surface excess con-centration,G max,and the area occupied by a single surfactant molecule at the organic solvent/air interface,A min,were both estimated from the Gibbs adsorption isotherm of nonionic surfactants[11]and also shown in Table2.G max¼À1RT@g@ln CT(2)A min¼1014G max o(3)where C is the concentration of surfactant,(@g/@ln C)T is the slope in the surface tension isotherm when the concentration is near the cac.R is the gas constant(8.314J molÀ1KÀ1),T is the absolute temperature and N o is the Avogadro constant.The maximum surface excess concentration(G max)reﬂects the degree of packing and orientation of the adsorbed surfactant molecules at the interface.As shown in Table2,the values of G max in n-tetradecane,n-hexadecane,nitromethane and DMSO is higher than that in DMF,therefore,the area occupied by a single surfactant molecule at organic solvent/air interface(A min)of these four systems is found to be smaller than that at DMF/air interface.A larger G max means that there are more surfactant molecules adsorbed on the surface of the solution,which also means a lower surface tension.The lowest value of A min obtained from n-tetradecane system is0.44nm2,which suggests adsorption of surfactant with the solvophobicﬂuorocarbon chain oriented away from the liquid in a more tilted position,approaching to the cross sectional area of theﬂuorocarbon chain(28.3A˚2)[31].The sulfonamide group of the surfactant molecule coexisting at the organic solvent/air interface increases the steric hindrance betweenﬂuorocarbon chains,thus inevitably increases the area occupied by a single surfactant molecule at the interface.And it is reasonable for the high A min data calculated from N-octyl perﬂuorooctanesulfonamide in n-hexadecane,nitromethane and DMSO.For example,the A min data(0.87nm2)of N-octyl perﬂuorooctanesulfonamide in nitromethane is basically coin-cident with the literature data(0.84nm2)of perﬂuorinatedoctyl ethanesulfonate in nitromethane[11].However,a high A min data suggest incompact adsorption ofﬂuorocarbon chains at interface or incomplete surface coverage by aﬂuorocarbonﬁlm,even though theﬂuorinated chains are oriented away from the liquid surface.When N-octyl perﬂuorooctanesulfonamide dissolves in DMF,it is evident that G max values are the minimum,cac is the maximum,and A min is the maximum(even up to1.33nm2).These results may be due to the excessive solubility of the surfactant in DMF to prevent the close packing ofﬂuorocarbon chains adsorbed at the interface.Therefore,the adsorbed molecules fail to form close-packed condensed monolayer even at cac.At the same time, theﬂuorinated groups probably lieﬂat on the surface when they are adsorbed.For conventional surfactants in water,the driving force for micellization and adsorption derives from the unfavorable contact between water and the hydrocarbon chain,called the hydrophobic effect of surfactant.The magnitude of this hydrophobic effect can be quantiﬁed in terms of the standard free energy of transfer of the hydrocarbon chain from bulk alkane to water.In organic solvents, the adsorption and aggregation of an organic solvent-soluble surfactant can be attributed to the solvophobic driving force of the ﬂuorocarbon chain.And the magnitude of the driving force can be estimated from the approximate Eq.(4)[30,32].D G uagg¼ÀRT ln S(4) where S is the solubility of the surfactant in organic solvents in mole fraction units and D G u agg is the standard free energy of aggregation.As illustrated in Table2,the values of D G u agg in n-tetradecane,n-hexadecane,nitromethane and DMSO are higher than that in DMF,indicative of higher solvophobic driving force of theﬂuorocarbon chain in n-tetradecane,n-hexadecane,nitro-methane and DMSO.However,the driving force of aggregation and adsorption in DMF are lower than n-alkanes in spite of anincrease Fig.5.Surface tension curves of C8F17SO2NHC8H17in various organic solvents at 258C.G.-L.Li et al./Journal of Fluorine Chemistry130(2009)674–681679。

s1-Conversion of Biomass into Chemicals over Metal Catalysts

Chemical Reviews focused on the design and mechanism of action of multifunctional catalysts.16 Some of the challenges to improve the activity, selectivity, and stability of metal catalysts have been pinpointed.16i,17 The addition of a second metal acting as a promoter of activity, selectivity, and stability was reviewed.1f,16i,18 Theoretical modeling has been applied to understand the reaction mechanism of biomass-derived molecules on the surface of metal particles.16c,19 Progress has been achieved in the development of multifunctional catalysts allowing process intensiﬁcation,20 in the combination of homogeneous and heterogeneous catalytic processes,21 and in cascade catalysis combining enzymatic and chemo-catalytic steps.16b,22 The inﬂuence of organic impurities contained in biosourced raw materials on catalyst deactivation,18,23 the eﬀect of porosity, hydrophilic, and acidic properties of supports,24 and the role played by water25 were considered. The importance of new reaction media and activation methods employed to improve biomass conversion and selectivity such as ionic liquids,26 molten salt hydrates,27 supercritical ﬂuids,28 microwave activation,29 and ultrasonication30 has been pinpointed. This Review puts focus on the catalytic conversion of biosourced feedstocks into chemicals in the presence of monometallic, multimetallic, and multifunctional catalysts. The production of hydrocarbons, biofuels, and fuel additives was not considered. The performances of metal catalysts will be examined in the conversion of carbohydrates, triglycerides, and terpenes, and some applications will be mentioned on amino acids derived from proteins and phenolic compounds derived from lignin. Starting feedstocks will be either pure platform molecules obtained from carbohydrates by chemical or enzymatic processes or more complex mixtures of molecular species such as those present in plant oils or polysaccharides. Metal-catalyzed reactions including hydrogenation, dehydrogenation, dehydroxylation/hydrogenolysis, and oxidation reactions will be considered possibly in combination with acid/ base-catalyzed reactions such as hydrolysis and dehydration reactions to achieve a multistep conversion in one-pot process. Selected examples of biomass conversion into chemicals, either those already produced by traditional synthesis routes or those without synthetic counterpart, will be highlighted. In addition to pure chemicals, metal-catalyzed reactions leading to a mixture of chemicals that could be employed in the manufacture of high tonnage end-products such as paper additives, paints, resins, foams, surfactants, lubricants, and plasticizers will be considered.1e,31 Scheme 1. Glucose Hydrogenation to Sorbitol

任意形状声学黑洞的简化设计方法

任意形状声学黑洞的简化设计方法许卫锴;张蒙;王伟【摘要】提出一种简化的任意形状声学黑洞设计方法.利用比例映射变换,可以消除传统声学变换方法所导致的材料参数的高度各向异性,降低声学黑洞吸收体的制作成本.数值结果表明,该模型可以实现复杂形状下声波的显著吸收,证明了任意形状声学黑洞的有效性.该工作将对声学能量吸收和噪声控制提供一定的理论依据.【期刊名称】《沈阳航空航天大学学报》【年(卷),期】2017(034)006【总页数】5页(P22-26)【关键词】声学黑洞;任意形状;比例映射;声学变换【作者】许卫锴;张蒙;王伟【作者单位】沈阳航空航天大学航空航天工程学部(院),沈阳110136;沈阳航空航天大学航空航天工程学部(院),沈阳110136;沈阳建筑大学土木工程学院,沈阳110168【正文语种】中文【中图分类】O429;TB535波的吸收问题一直以来都是人们研究的重点，并在多个学科都得到了广泛地关注，如光学、微波、声学、热学等领域。

由于其惊人的吞噬能力，很多学者用“黑洞”的概念来形容完美的吸收体，并进行了广泛的研究。

然而对于传统的黑洞设计来说，无论是吸波材料或是吸波结构都存在着各自的局限性，或者受限于特殊的入射角度[1-2],或者只具有非常窄的带宽[3]。

因此，研究宽频带、高吸收的黑洞结构成为一个重要的课题。

近年来，一种基于“有效势(Effective potential)”概念的黑洞结构得到了研究人员的关注。

通过合理设计材料的参数分布，可以实现折射率随位置变化的函数式分布[4]，从而形成类似于势力场的空间，最终达到控制波的传播路径的目的。

第一个光学黑洞由Narimanov等人提出[5]，并很快得到了实验验证[6-8]。

而利用光波与声波方程的相似性，与光学黑洞类似的声学黑洞也得到了关注，并由理论证实[9]和实验验证[10]。

随后，声学黑洞得到了大量的研究，例如Song等人利用超材料实现声相干完美吸收器[11]，Naify[12]和Elliott[13]利用梯度材料设计了全方位的水下完美吸收体，Cheng等人利用均匀的各向异性超材料设计了一个二维宽频带的全方位声学吸收体并完成实验验证[14]，Hu等人则利用坐标变换方法设计了弹性波的黑洞结构[15]。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

Cloaking with optimized homogeneous anisotropic layersBogdan-Ioan Popa*and Steven A.Cummer†Department of Electrical and Computer Engineering,Duke University,North Carolina27708,USA͑Received7November2008;published5February2009͒We present a method to reduce the scattering from arbitrary objects by surrounding them with shells composed of several layers of homogeneous anisotropic materials.An optimization procedure is used toﬁnd the material parameters for each layer,the starting point of which is a discretized approximation of a coordinate transformation cloaking shell.We show that an optimized,three-layer shell can reduce the maximum scattering of an object by as much as15dB more than a100-layer realization of a coordinate transformation cloaking shell.Moreover,using an optimization procedure can yield high-performance cloaking shell solutions that also meet external constraints,such as the maximum value of permittivity or permeability.This design approach can substantially simplify the fabrication of moderate-size cloaking shells.DOI:10.1103/PhysRevA.79.023806PACS number͑s͒:42.25.Fx,41.20.Jb,42.25.GySigniﬁcant research has recently focused on developing new methods to minimize the interaction between given ob-jects and electromagnetic waves.Pendry et al.͓1͔showed that carefully designed inhomogeneous and anisotropic shells can prevent electromagnetic radiation from penetratinginside them and,more importantly,cancel the scattering offthese shells,making them and their interior effectively trans-parent to electromagnetic waves.What is appealing aboutthis coordinate transformation approach is its generality:itcan be used to conceal objects of any size and shape.Even though numerical simulations͓2͔and further theo-retical analysis͓3͔conﬁrmed the efﬁcacy of this method,experimental demonstrations have proven more challenging.One effort͓4͔involved a cylindrical shell surrounding ametal cylinder and demonstrated the basic physics of suchstructures,namely,that waves can be steered around thestructure.However,in this work approximations were made ͓2͔that signiﬁcantly reduced the scatter reduction perfor-mance of the shell.Other work has derived different approxi-mations to ideal cloaking shell parameters that also sacriﬁceperformance for fabrication simplicity͓5–7͔.The difﬁculty of fabricating cloaking shells speciﬁed bycoordinate transformation theory stems from the require-ments on the material that composes it:the shell has to beanisotropic with its permittivity and permeability varyingcontinuously with position over a broad range of values.Itsphysical implementation will always require some form ofdiscretization of these continuous proﬁles.For example,Schurig et al.͓4͔used a ten-layer stepwise approximation ofthe ideal parameters in their experiment.Beginning with the notion that anisotropy appears to bethe most important ingredient of cloaking shells that are notelectrically small,one might ask whether cloaking shellscomposed of anisotropic layers can be designed through an-other approach.Here we show that scatter reducing shellscomposed of a relatively small number of homogeneous lay-ers can be designed through an optimization procedure thatuses a coordinate transformation shell as the initial condition.The performance of fewer thanﬁve optimized layers can equal or even exceed the performance of a100-layer discrete approximation of the smoothly homogeneous shell designed through coordinate transformations.For simplicity,we will focus on the two-dimensional cy-lindrical shell for electromagnetics,but the analysis pre-sented here can be applied for other geometries and wave types as well,such as acoustics͓8͔.Figure1shows a perfect electric conductor͑PEC͒cylindrical object of radius a sur-rounded by a shell of outer radius b.It has been shown͓2͔that one set of relative material parameters that completely cancel the scattering from the structure is͑in cylindrical co-ordinates͒,⑀r͑r͒=␮r͑r͒=r−ar,⑀␾͑r͒=␮␾͑r͒=rr−a,⑀z͑r͒=␮z͑r͒=ͩb b−aͪ2r−a r,͑1͒where z is the invariant direction,and r and␾are the radial and azimuthal coordinates,respectively.We consider the TE͑transverse electric͒polarization for which only⑀z,␮r,and␮␾are relevant.We assume that the cloak can be assembled from M concentric layers of homo-*bap7@ †cummer@kIncidentplane wavePECFIG.1.Cylindrical perfect electric conductor surrounded by amultilayer shell and illuminated with a plane wave.The input andoutput radii of the shell are a and b,respectively.PHYSICAL REVIEW A79,023806͑2009͒geneous materials as shown in Fig.1.Each layer is charac-terized by permittivity and permeability tensors that are con-stant with position inside the layer.Our goal is to ﬁnd a set of parameters that minimizes the scattering off the cloak.One choice is to use the stepwise approximation of Eqs.͑1͒as in Ref.͓4͔.Using this approach and TE polarization,as was pointed out before ͓9,10͔,the boundary conditions at the inner interface of the cloak with the PEC object induce sig-niﬁcant scattering.We therefore expect that other choices of material parameters may improve cloaking performance.The exp ͑+j ␻t ͒time convention is assumed throughout.Consider a plane wave having the electric ﬁeld E inc =zˆexp ͑−jk 0r cos ␾͒incident on the object and shell depicted in Fig.1.The symmetry of the problem allows us to compute the ﬁelds inside and outside our structure analytically,by employing the procedure outlined in ͓11͔.Thus,the incident plane wave can be expanded into a sum of Bessel functions of the ﬁrst kind asE inc =J 0͑k 0r ͒+2͚n =1ϱj n J n ͑k 0r ͒cos ͑n ␾͒͑2͒while the scattered ﬁeld in the r Ͼb region can be written in terms of Hankel functions of the second kind asE sc =͚n =0ϱA n H n ͑2͒͑k 0r ͒cos ͑n ␾͒.͑3͒The ﬁelds in layer m inside the shell are given byE m =͚n =0ϱ͓B mn J ␯͑k m r ͒+C mn Y ␯͑k m r ͔͒cos ͑n ␾͒,͑4͒where m =1,M ,Y ␯is the Bessel function of the second kind,k m =␻ͱ⑀z ,m ␮␾,m ,and ␯=n ͱ␮␾,m /␮r ,m .The coefﬁcients A n ,B mn ,and C mn can be found by imposing the continuity of tangential E and H ﬁelds across the boundaries of each layer.Once we know the ﬁelds inside and outside our structure,we can compute the ﬁgure of merit used throughout this paper:the radar cross section per unit length also known as scattering width ͑SW ͒,which is deﬁned as ␴͑␾͒=2␲R ͉E sc ͑␾,R ͒/E inc ͉2,where R is the distance from the ob-ject where the far-ﬁeld scattered ﬁeld E sc is evaluated.Since in the far-ﬁeld region E sc is inversely proportional to R ,␴is independent of R as long as R is big enough.The question is whether we can derive a set of material parameters for the layered shell that would reduce the scat-tering more than if we simply discretize the proﬁles given byEqs.͑1͒.The answer is afﬁrmative and we outline the pro-cedure for the general case of an M -layer shell,after which we apply the method to a speciﬁc example.To maximally reduce object visibility,one should mini-mize max ␾␴͑␾͒.However,since the forward scattering of an object that is not electrically small is usually largest ͓12͔,we solve the simpler problem of optimizing for the permit-tivity and permeability tensors that minimize the forward scattering ͑i.e.,the shadow ͒.As we will see next,this heu-ristic gives a strongly reduced SW in all directions.Math-emathically,we want to ﬁnd the ⑀and ␮values that mini-mize the function ␴͑␾=0,⑀z͑1¯M ͒,␮r ͑1¯M ͒,␮␾͑1¯M ͒͒,where the superscripts signify that we have one set of material pa-rameters for each one of the M layers.This is a classical optimization problem that can be solved with a variety of algorithms.In this paper,we use the Broyden-Fletcher-Goldfarb-Shanno ͑BFGS ͒method,which is already imple-mented in software tools such as MATHEMATICA and MATLAB .The BFGS method provides a local minimum for f around aspeciﬁed starting point,X 0=͑⑀z ,0͑1¯M ͒,␮r ,0͑1¯M ͒,␮␾,0͑1¯M ͒͒,whose choice is very important for the success of this algorithm.Since it already gives good results,we choose X 0to be thediscretized version of Eqs.͑1͒,namely,⑀z ,0͑i ͒=⑀z ͑r =͓R i +R i +1͔/2͒͑similar expressions for ␮r ,0͑i ͒and ␮␾,0͑i ͒͒,where R i and R i +1are the inner and,respectively,outer boundaries of layer i ,R 1=a ,and R M +1=b .The local minimum in ␴͑␾=0͒gives the material parameters for each of the M layers.We illustrate the procedure for an object that has the di-mensions speciﬁed in ͓2͔,namely,a =10cm,which makes it 1.33wavelengths at the working frequency of 2GHz.We choose to design a thin shell of 1.3cm,i.e.,b =11.3cm.For this size,three layers provide enough degrees of freedom ͑nine in total,i.e.,⑀z ,␮r ,and ␮␾for each layer ͒to signiﬁ-cantly reduce the scattering width of our object.Table I shows the starting point and material parameters found by the optimization algorithm,while Fig.2shows the scattering width normalized to the object diameter versus angle for our design ͑curve d ͒and,for comparison,for three other shells whose material parameter proﬁles are the three-,nine-,and 100-level staircase approximation of Eqs.͑1͒.This example demonstrates that the optimization method presented here not only enables the design of shells com-posed of a reduced number of layers,but also can signiﬁ-cantly improve their performance.In this case,the maximum scattered ﬁeld is approximately 15dB smaller for the opti-mized three-layer cloak than for the 100-layer discretization of the coordinate transformation cloak.Figure 3illustrates the contribution of these scattered ﬁelds to the total ﬁelds:TABLE I.Optimized material parameters for the three-layer cloak.Initial guessOptimized parametersLayer ⑀z ␮r ␮␾⑀z ␮r ␮␾1 1.600.02147.15 3.320.00647.212 4.610.06116.38 6.060.03916.8837.400.09810.237.990.1010.63BOGDAN-IOAN POPA AND STEVEN A.CUMMER PHYSICAL REVIEW A 79,023806͑2009͒the optimized version perturbs the incident ﬁeld considerably less than the staircase approximation to the analytical cloak.We also notice that,even though a ␮␾of 47in the innermost layer of the shell is difﬁcult to achieve with current materi-als,it is still signiﬁcantly lower than what would be required for the nine-layer ͑␮␾=139͒or 100-layer ͑␮␾=1539͒approximations of the analytical cloak.From a sensitivity point of view,random perturbations of up to 1%applied simultaneously to all the optimized ⑀and ␮parameters given in Table I change the scattering width by approximately Ϯ5dB for most angles.This makes the opti-mized shell fairly sensitive to changes in the desired param-eters,but even with these perturbations the performance of the optimized shell is signiﬁcantly better than that of the discretized analytical shell.We also emphasize the impor-tance of choosing a good starting point in the BFGS optimi-zation algorithm.If,for instance,we choose all the initial permittivity and permeability components to be unity,the ﬁnal values given by the iterative algorithm result in a shell that,even though it reduces the overall scattering compared to the bare cylinder,has a maximum scattering cross section that is 30dB bigger than the optimized cloak speciﬁed in Table I .The optimization procedure presented above shows that for an object of 1.33wavelengths in diameter,a three-layer optimized cloak is more than 100times better than a three-layer approximation of the analytical cloak.If we double the size of the object ͑i.e.,2.66wavelengths in diameter ͒,the improvement provided by optimization of three layers is only a factor of 7.With another factor of 2͑i.e.,5.32wave-lengths ͒optimization yields a factor of 2improvement.This size dependence of the optimization improvement is ex-pected;as the object becomes larger,we need to minimize increasingly more coefﬁcients in the Hankel function expan-sion of the scattered ﬁeld ͓see Eq.͑3͔͒.Three layers do not provide enough degrees of freedom to dramatically improve the cloaking of a ﬁve-wavelength-wide object.However,by increasing the number of layers,we expect to obtain signiﬁ-cant improvements through optimization for larger objects.Optimization adds a great deal of ﬂexibility to the con-straints that can be imposed on cloaking shells.Equation ͑1͒shows that for thin shells,the required values of ␮␾inside the innermost layers have to be large ͑in the previous ex-ample,it is ␮␾=47inside the innermost layer ͒,which can be an obstacle to fabrication.These extreme values can be re-duced through optimization and still yield signiﬁcant im-provements over the discretized analytical shell.We demon-strate this in the following example by designing a nonmagnetic cloak that can potentially be implemented at optical frequencies.3020100-10-20-30-2-1012Position (cm)P o s i t i o n (c m )Electric field (arbitrary units)Angle (deg)N o r m a l i z e d s c a t t e r i n gw i d t h (d B )FIG.4.͑Color online ͒Top:The scattering width vs angle for the three-layer approximation of the analytical cloak ͑dash-dotted ͒,nonmagnetic cloak ͑dashed ͒,and three-layer optimized nonmag-netic cloak ͑solid ͒.Bottom:The total ﬁelds inside and around the nonmagnetic optimized cloak.Angle (deg)N o r m a l i z e d sc a t t e ri n g w i dt h (d B )FIG.2.͑Color online ͒The scattering width normalized to the object diameter for the three-layer ͑a ͒,nine-layer ͑b ͒,and 100-layer ͑c ͒approximations of the analytical cloak compared to the opti-mized cloak ͑d ͒.Position (cm)P o s i t i o n (c m )40200-20-11Electric field (arbitrary units)FIG.3.͑Color online ͒Total electric ﬁeld for the nine-layer ap-proximation of the analytical cloak ͑top ͒and the optimized three-layer cloak ͑bottom ͒.CLOAKING WITH OPTIMIZED HOMOGENEOUS …PHYSICAL REVIEW A 79,023806͑2009͒From a practical perspective,even though it requires only three layers,the optimized shell described above is hard to fabricate:one needs to control both⑀and␮;in addition, these parameters need to be quite big͑for example,␮␾needs to be47in the innermost layer͒,which is hard to achieve at high frequencies.For this reason,there is signiﬁcant interest in designing cloaking shells that have certain properties that make them easier to fabricate.Some approximations to ideal parameters have been devised͓2,5͔to facilitate physical re-alizability,but most of these sacriﬁce performance to a high degree.One such set of parameters that avoids using the harder to fabricate magnetic materials and can be used to create shells able to conceal objects detected using TM͑transverse mag-netic͒polarized waves͑only␮z,⑀r,and⑀␾are relevant for this polarization͒is͓5͔⑀r=ͩb b−a ͪ2ͩr−a rͪ2,⑀␾=ͩb b−aͪ2,␮z=1.͑5͒Considering the same object as before͑i.e.,a=10cm͒, Fig.4͑top͒shows the scattering width versus angle,␴͑␾͒, for a relatively thick cloaking shell of outer radius b =17cm made of a medium whose parameters are given by the above equations͑dashed curve͒compared to that of the three-layer approximation of the analytical cloak͑dash-dotted curve͒.We chose a slightly thicker shell than before in order to obtain easier to achieve material parameters inside it.Neither of these approximations achieves especially good performance.Optimization can dramatically improve the performance of a three-layer nonmagnetic cloak.The initial guess was given by Eqs.͑5͒,and we added the additional constraint to simplify fabrication that no component of permeability or permittivity could be larger than10.The resulting optimized cloak material parameters are given in Table II and the com-puted scattering width is shown in Fig.4͑top͒.This opti-mized three-layer cloak gives an approximately15dB reduc-tion in the scattering width over the cloak described by Eqs.͑5͒,and Fig.4͑bottom͒illustrates the total electricﬁeld dis-tribution around the shell and shows its good performance. We note that,as expected,stricter requirements on⑀increase␴and consequently decrease to some extent the performance of the shell.For example,if we require⑀ഛ5,then max␾␴reaches0dB,and for⑀ഛ8,max␾␴Ϸ−5dB.In conclusion,we present a practical optimization ap-proach used to design layered shells of anisotropic materials that signiﬁcantly reduce the scattering width of an object. This approach yields better performance and can also pro-vide simpler material parameters compared to cloaking shells designed with coordinate transformation theory.As an ex-ample,weﬁnd that an optimized three-layer shell can per-form much better than a100-layer approximation to the ana-lytical cloak.We also used this approach to design a three-layer nonmagnetic cloak made of relatively low permittivity anisotropic materials realizable even at optical frequencies that behaves signiﬁcantly better than comparable designs.͓1͔J.B.Pendry,D.Schurig,and D.R.Smith,Science312,1780͑2006͒.͓2͔S.A.Cummer,B.-I.Popa,D.Schurig,D.R.Smith,and J.B.Pendry,Phys.Rev.E74,036621͑2006͒.͓3͔H.Chen,B.-I.Wu,B.Zhang,and J.A.Kong,Phys.Rev.Lett.99,063903͑2007͒.͓4͔D.Schurig et al.,Science314,977͑2006͒.͓5͔W.Cai et al.,Appl.Phys.Lett.91,111105͑2007͒.͓6͔Y.Huang,Y.Feng,and T.Jiang,Opt.Express15,11133͑2007͒.͓7͔H.Ma et al.,Opt.Express16,15449͑2008͒.͓8͔S.A.Cummer,B.I.Popa,D.Schurig,D.R.Smith,J.Pendry, M.Rahm,and A.Starr,Phys.Rev.Lett.100,024301͑2008͒.͓9͔Z.Ruan,M.Yan,C.W.Neff,and M.Qiu,Phys.Rev.Lett.99, 113903͑2007͒.͓10͔A.Greenleaf et al.,Opt.Express15,12717͑2007͒.͓11͔C. A.Balanis,Advanced Engineering Electromagnetics ͑Wiley,New York,1989͒.͓12͔J.I.Glaser,IEEE Trans.Aerosp.Electron.Syst.AES-21,70͑1985͒.TABLE II.Optimized material parameters for the three-layer nonmagnetic cloak.Initial guess Optimized parametersLayer␮z⑀r⑀␾␮z⑀r⑀␾110.064 5.9010.10 4.49210.40 5.901 1.389.99310.80 5.9010.700.89BOGDAN-IOAN POPA AND STEVEN A.CUMMER PHYSICAL REVIEW A79,023806͑2009͒。