1 Some Applications of the Lambert W Function to Physics
逆二次方定律和朗伯余弦定律及它们的应用
逆二次方定律和朗伯余弦定律及它们的应用逆二次方定律是指物体辐射的能量与物体表面到辐射源的距离的平方成反比。
The inverse square law refers to the fact that the energy radiated by an object is inversely proportional to the square of the distance from the object's surface to the radiation source.朗伯余弦定律描述了光线射向表面时被散射的强度与光线与表面法线的夹角的余弦成正比。
The Lambert cosine law describes the intensity of light scattered when a ray of light is directed towards a surface is directly proportional to the cosine of the angle between the ray of light and the surface normal.这些定律在光学、照明设计和遥感等领域有着重要应用。
These laws have important applications in optics,lighting design, and remote sensing.在照明设计中，人们使用逆二次方定律来确定灯具的布置位置以及光线照射范围。
In lighting design, the inverse square law is used to determine the placement of fixtures and the range of light coverage.朗伯余弦定律有助于我们理解光线在不同表面上的反射和散射现象。
The Lambert cosine law helps us understand the reflection and scattering of light on different surfaces.在遥感领域，科学家们利用逆二次方定律来分析遥感图像中的光谱数据。
固体材料的宏观光学性质 61页PPT文档
astronomynotes/light/emanim.gif
• Solids which bond ionically, show high absorption because ions of opposite charge move in opposite directions
in the same electric field hence we get effectively twice the interaction between the
• Finally we will mention applications, in particular optical fibres and lasers
Optic: 2
Nature of light
• Light is an electromagnetic wave:
with a velocity given by c = 1/(00) = 3 x 108 m/s
evidence for energy levels in atoms evidence for energy bands and band-gaps photoelectric effect
Optic: 4
General description of absorption
• Because of conservation of energy, we can say that I0 = IT + IA + IR
Modeling Illumination Variation With Spherical Harmonics
Ravi Ramamoorthi Columbia Universi
Introduction
Illumination can have a significant impact on the appearance of surfaces, as the patterns of shading, specularities and shadows change. For instance, some images of a face under different lighting conditions are shown in figure 1. Differences in lighting can often play a much greater role in image variability of human faces than differences between individual people. Lighting designers in movies can often set the mood of a scene with carefully chosen lighting. To achieve a sinister effect, for instance, one can use illumination from below the subject—a sharp contrast to most natural indoor or outdoor scenes where the dominant light sources are above the person. Characterizing the variability in appearance with lighting is a fundamental problem in many areas of computer vision, face modeling, and computer graphics. One of the great challenges of computer vision is to produce systems that can work in uncontrolled environments. To be robust, recognition systems must be able to work oudoors in a lighting-insensitive manner. In computer graphics, the challenge is to be able to efficiently create the visual appearance of a scene under realistic, possibly changing, illumination. At first glance, modeling the variation with lighting may seem intractable. For instance, a video projector can illuminate an object like a face with essentially any pattern. In this chapter, we will stay away from such extreme examples, making a set of assumptions that are approximately true in many common situations. One assumption we make is that illumination is distant. By this, we mean that the direction to, and intensity of, the light sources is approximately the same throughout the region of interest. This explicitly rules out cases like slide projectors. This is a reasonably good approximation in outdoor scenes, where the sky can be assumed to be far away. It is also fairly accurate in many indoor environments, where the light sources can be considered much further away relative to the size of the object. Even under the assumption of distant lighting, the variations may seem intractable. The illumination can come from any incident direction, and can be composed of multiple illuminants including localized light sources like sunlight and broad area distributions like skylight. In the general case, we would need to model the intensity from each of infinitely many incident lighting directions. Thus, the space we are dealing with appears to be infinite-dimensional. By contrast, a number of other causes of appearance variation are low-dimensional. For instance, appearance varies with viewing direction as well. However, unlike lighting, there can only be a single view direction. In general, variation because of pose and translation can be described using only six degrees of freedom. Given the daunting nature of this problem, it is not surprising that most previous analytic models have been restricted to the case of a single directional (distant) light source, usually without considering shadows. In computer graphics, this model is sufficient (but not necessarily efficient) for numerical Monte Carlo simulation of appearance, since one can simply treat multiple light sources separately and add up the results. In computer vision, such models can be reasonable approximations under some circumstances, such as controlled laboratory conditions. However, they do not suffice for modeling illumination variation in uncontrolled conditions like the outdoors. Fortunately, there is an obvious empirical method of analyzing illumination variation. One can simply record a number of images of an object under light sources from all different directions. In practice, this usually corresponds to moving a light source in a sphere around the object or person, while keeping camera and pose fixed. Because of the linearity of light transport—the image under multiple lights is the sum of that under the individual sources—an image under arbitrary distant illumination can be written as a linear combination of these source images. This observation in itself provides an attractive direct approach for relighting in computer graphics. Furthermore, instead of using the source images, we can try to find linear combinations or basis images that best explain the
IEEE485-1997(r2003)
Interpretations: Occasionally questions may arise regarding the meaning of portions of standards as they relate to specific applications. When the need for interpretations is brought to the attention of IEEE, the Institute will initiate action to prepare appropriate responses. Since IEEE Standards represent a consensus of all concerned interests, it is important to ensure that any interpretation has also received the concurrence of a balance of interests. For this reason, IEEE and the members of its societies and Standards Coordinating Committees are not able to provide an instant response to interpretation requests except in those cases where the matter has previously received formal consideration.
荧光光谱分析方法及原理
Emission lifetimes of absorption, fluorescence, and phosphorescence at the equilibrium internuclear distance of the ground state.
仪器结构
某些物质被一定波长的光照射时，会在一定时间内发射出波长比入射光长的光，如果这个时间比较短，这种光就称为荧光。荧光由一种能发荧光的矿物 萤石(fluospar)而得名。 我们这里要介绍的荧光，是指物质在吸收紫外光和可见光后发出的波长较长的紫外荧光或可见荧光。 除了紫外光和可见光可能激发荧光外，其它的光如红外光、X射线也可能激发出荧光，因此除紫外荧光或可见荧光外，还有红外荧光、X射线荧光等。
Schematic diagram of a double-beam (ratiometric) filter fluorometer.
Filter fluorometers are suitable for quantitative analysis applications in which spectral scanning and high resolution are not required. Filters transmit more light and cost less than monochromators, thereby providing better detection limits with less expensive instrumentationque for detecting biological materials
荧光光谱灵敏度高的原因
荧光辐射的波长比激发光波长长，测量到的荧光频率与入射光的频率不同； 荧光在各个方向上都有发射，因此可以在与入射光成直角的方向上检测； 这样，荧光不受来自激发光的本底的干扰，灵敏度大大高于紫外-可见吸收光谱，测量用的样品量很少，且测量方法简便。
的某些分量对任何产品个体都是相同...
淫辖辫学援本天学疆炎燕淹学位论文隆提出了开展激光ＩｃＦ研究的建议。
目前世界上主疆的发达国家已建成或正在建的ＩｃＦ激光蠢嚣惫糍：美蕃ｉ§８４年建成№ｖ８装置（惑藐量４５ＫＪ、稼宽０。
ｌ～３ｎｓ、ｌ§寒激巍），１９９《年瞳成Ｂｅａｌｌｌｌｅｔ装置（总能量６．４ＫＪ、脉宽３ｎｓ、１束激光），１９０５年建成ｏＩｆｌｅｇａ装置（总能量ｌＯ醛、艨宽０．Ｉ～ｌｎｓ、∞束激光），２∞８年拟建成ＮＩＦ装置（悲雏量１８００时、脉宽１５ｎｓ、１９２束激光）：日本１９８３年建成ＧｅｋｋｏｘＩＩ装置（总能量１５ＫＪ、脉宽Ｏ，ｌ～】ｎｓ、１２束激光）；箕国正在建Ｖｕｌｃａ硝￡置（总能量２ＫＪ、脉宽Ｏ．１～ｌｎｓ、８束激光）；法国１９８６年建成Ｐｈｅｂｕｓ寝置（憩能量８酊、赫宽０．１～３ｎｓ、２索激光），嚣翦正在建ＬＭＪ装嚣（蕊髓量２４∞酣、豫宽１５ｎｓ、２４０束激光）；俄罗斯正在建Ｉｓｋｒａ一６装置（总能量３００ＫＪ、脉宽１～３ｎｓ、１２８束激蓬）。
鞑器各国均已、凑ＩｅＦ激党装置传为霹家安全、戆深稠秘学磅究东乎熬重要椽悫。
９０年代，美国开始启动总投资约４０亿美元、为期１０年的图家点火装鼹ＮＩＦ（Ｎａｔｉｏｎａｌ】ｇｎｉｔｉｏｎＦａｃｉｌｉｔｙ，图１．卜１）科学工程ｎＮＩＦ是一座用于聚变和核武器研究的Ｅｂ激光驱动韵惯性约束核聚变设施，其原理怒闻对发射１９２柬分离弱紫外激光聚焦在较其微小的氘氚靶丸（图１．１—２）上，加热压缩到点火和燃烧，触发核聚变。
法国随即开始建避与ＮＩＦ娄稼蔑摸静Ｌ※ｊ激巍装蓑。
遮整装爱鼗卡偿予已建戏装萋躲藏模，耗资藏大，周期缀长，风险度筒，需要不断发展的先进技术米支撑，更面临着工程科学方面的许多课题，必然为疆关科学领域和熊源基础科学技术静发震宅４造条件。
圈１，ｌ一１ＮＩＦ装置实景ｌ簦｜图１１—２ＮｌＦ装置靶心外观我鬻王淦蓦教援虽帮苏焚季萼学家整不多霜露掇爨了舞鬟激光ｌｅＦ磺究豹建议，援壹到１９７７年，我国才开始ＩｃＦ激光装置建造工作，１９８３年建成了星光Ｉ装置（总能量Ｏ．１ＫＪ、脉宽ｌｎｓ、ｌ束激光），１９８６年建成神光ｌ装置（总能量１．６ＫＬｌ、脉宽Ｏ．１～ｌｎｓ、２京激光）。
WRFV3.1用户手册之中文版(非常难得之翻译版)
Chapter 1: Overview翻译:by 澳洲的牛牛 laiwf由于译者水平有限,其中一定存在翻译不妥的地方,希望大家能帮忙指正IntroductionThe Advanced Research WRF (ARW) modeling system has been in development for the past few years. The current release is Version 3, available since April 2008. The ARW is designed to be a flexible, state-of-the-art atmospheric simulation system that is portable and efficient on available parallel computing platforms. The ARW is suitable for use in a broad range of applications across scales ranging from meters to thousands of kilometers, including:•Idealized simulations (e.g. LES, convection, baroclinic waves)•Parameterization research•Data assimilation research•Forecast research•Real-time NWP•Coupled-model applications•Teaching简介Advanced Research WRF (ARW)模式系统在过去的数年中得到了发展。
最近公布了第三版，从2008年4月开始可供使用。
ARW是灵活的，最先进的大气模拟系统，它易移植，并且有效的应用于各种操作系统。
三氧化二钨
Reduction of NO x by H 2on Pt/WO 3/ZrO 2catalysts in oxygen-rich exhaustF.J.P.Schott,P.Balle,J.Adler,S.Kureti *Institut fu¨r Technische Chemie und Polymerchemie,Universita ¨t Karlsruhe,Kaiserstrasse 12,D-76128Karlsruhe,Germany 1.IntroductionNitrogen oxides (NO x )emitted by lean-burn engines con-tribute to various environmental problems,for instance forma-tion of acid rain as well as ozone.As a consequence,the emission limits have been worldwide tightened in the past.For the removal of NO x from oxygen-rich exhaust the selective catalytic reduction (SCR)using NH 3and NO x storage reduction catalyst (NSR)are currently the most favoured technologies.However,a serious constraint of these techniques is the minor deNO x performance below 2008C.Contrary,the catalytic reduction of NO x by H 2(H 2-deNO x )reveals an interesting potential for the low-temperature NO x abatement being particularly crucial for diesel passenger cars.In the driving cycle of the European Union the exhaust temperature is below 1508C for about 60%of cycle time.Thus,SCR and NSR do not cover the most part of the certification cycle and might therefore come under pressure when the exhaust limits will be markedly tightened in the future.This clearly indicates the need for a deNO x technique operating at low temperatures.Furthermore,low-temperature NO x reduction exhibits a potential for industrial applications as well, e.g.for fossil power plants,waste combustion plants,nitric acid production and air separation.First published in 1971Jones et al.show the effective NO x reduction by H 2in slight excess of O 2using a Pt/Al 2O 3catalyst [1].High NO x conversions are observed between 65and 2008C while indicating a high yield of nitrous oxide as well;at maximum deNO x the molar ratio of N 2/N 2O is shown to be unity.The mechanism of the reaction of NO with H 2on Pt/Al 2O 3involves the reduction of the active Pt sites by H 2followed by adsorption and dissociation of NO [2].The recombination of two N atoms leads to the formation of N 2,whereas the oxygen is retained onto the Pt surface.Contrary,N 2O is produced by combination of a N atom and NO being adsorbed on neighbouring Pt sites.In the last years some Pt H 2-deNO x catalysts are presented revealing considerable low-temperature activity even under strongly oxidising conditions [3–6].Wildermann reports on a very active Pt/Al 2O 3catalyst that shows maximum performance already at 708C [3].However,this material produces a huge proportion of N 2O being in line with the results from Jones et al.[1].For example,at peak NO x conversion the N 2O selectivity amounts to 80%.Moreover,Wildermann indicates the enhancement of activity and N 2production of Pt/Al 2O 3by using the promoter Mo (3.4wt.%)resulting in a N 2selectivity of 40%.The performance of this Pt/Mo/Al 2O 3catalyst is additionally enhanced by Co,whereas the N 2selectivity is slightly increased only.However,the activityApplied Catalysis B:Environmental 87(2009)18–29A R T I C L E I N F O Article history:Received 30June 2008Received in revised form 19August 2008Accepted 26August 2008Available online 31August 2008Keywords:NO x reduction H 2Pt WO 3ZrO 2Diesel exhaust Mechanism DRIFTSA B S T R A C TThis work addresses the low-temperature NO x abatement under oxygen-rich conditions using H 2as reductant.For this purpose Pt/ZrO 2and Pt/WO 3/ZrO 2catalysts are developed and characterised by temperature-programmed desorption of H 2(H 2-TPD),N 2physisorption (BET)and powder X-ray diffraction (PXRD).The most active catalyst is a Pt/WO 3/ZrO 2pattern with a Pt load of 0.3wt.%and a W content of 11wt.%.This material reveals high deNO x activity below 2008C and high overall N 2selectivity of about 90%.Additionally,the catalyst exhibits outstanding hydrothermal stability as well as resistance against SO x .Furthermore,the transfer from the powder level to real honeycomb systems leads to promising performance as well.Diffuse reflectance Fourier transform infrared spectroscopic studies,kinetic modelling of tempera-ture-programmed desorption of O 2(O 2-TPD)and NO x -TPD examinations indicate that the pronounced H 2-deNO x performance of the Pt/WO 3/ZrO 2catalyst is related to the electronic interaction of WO 3with the precious metal.The tungsten promoter increases the electron density on the Pt thus activating the sample for H 2-deNO x and N 2formation,respectively.Contrary,NO x surface species formed on the WO 3/ZrO 2support are not supposed to be involved in the H 2-deNO x reaction.ß2008Elsevier B.V.All rights reserved.*Corresponding author.Tel.:+497216088090;fax:+497216082816.E-mail address:kureti@ict.uni-karlsruhe.de (S.Kureti).Contents lists available at ScienceDirectApplied Catalysis B:Environmentalj 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 /a p c a t b0926-3373/$–see front matter ß2008Elsevier B.V.All rights reserved.doi:10.1016/j.apcatb.2008.08.021declines when CO exceeds0.15vol.%[7,8]being in accordance with Lambert and Macleod[9,10].Costa et al.report a Pt/La0.7Sr0.2Ce0.1FeO3catalyst with pronounced low-temperature activity as well as substantially increased N2selectivity up to80–90%[11,12].Detailed examina-tions performed with a related Pt/La0.5Sr0.2Ce0.51MnO3sample[12] indicate a different reaction mechanism as compared to that elucidated for Pt/Al2O3[2].Costa et al.postulate chemisorption of NO x on the support resulting in nitro and nitrato surface species, while H2adsorbs dissociatively on the Pt component.Then,the atomic hydrogen spills over to the support reducing the NO x surface complexes to release N2and H2O.Following Costa et al.this mechanism suppresses the formation of N2O.A very similar mechanism is postulated for a Pt/MgO–CeO2catalyst being very active as well[13,14].In contrast to platinum,Ru,Ir,Rh,Pd and Ag [3,15]as well as perovskite catalysts[11,16–18]reveal no or at least low performance in excess of O2.An exception is Pd/LaCoO3 showing considerable activity[19].Furthermore,it is worth mentioning that the H2-deNO x reaction is an important feature of the TWC technology as well as in regeneration of NSR catalystsreducing NO x under stoichiometric and rich conditions,respec-tively.The aim of this paper is the development of a H2-deNO x catalyst showing both pronounced low-temperature activity and minimum N2O production,whereas the present study mainly focuses on diesel exhaust.For this purpose a series of Pt/ZrO2and Pt/WO3/ ZrO2samples is systematically prepared,characterised andfinally tested for H2-deNO x.These systems are selected as a result of pre-studies in which different support materials have been screened [20,21],e.g.Al2O3,SiO2,ZrO2,TiO2and MgO carriers all modified with alkaline and alkaline earth metals,elements of the1st period of transition metals,Ce,La and Mo.For the evaluation of the technical potential of the most promising catalyst relevant conditions are varied,while a coated honeycomb is employed as well.Additionally,mechanistic examinations are performed to gain insight into the effectiveness of the best catalyst.2.Experimental2.1.Development of the ZrO2support and catalyst preparationPreliminary catalytic investigations show that the synthesis route,crystalline phase and BET surface area of the ZrO2substrate strongly affect the performance of the H2-deNO x catalysts[22].The best result is obtained with a self-prepared zirconia existing in the tetragonal phase.This material is synthesised by advancing the so-called hydrazine route[23,24]providing reliable product quality and sufficient mass to coat several full size honeycombs for automotive applications.For the synthesis a solution of234g ZrO(NO3)2(Fluka)in 1.6l distilled H2O is added to a boiling mixture of400ml N2H4ÁH2O(Fluka)and1.2l distilled H2O.The resulting blend is digested for12h under reflux,whereas shorter reaction time leads to unwanted monoclinic ZrO2as well(Fig.1). Afterfiltration and washing with H2O the solid is dried overnight at 1008C and calcined in air at7508C for6h.The yield of ZrO2is almost100%(85g).The introduction of Pt is carried out by incipient wetness method.In this impregnation a defined volume of Pt(NO3)2 (Chempur)solution is taken such that it is completely absorbed by the substrate.The adjusted Pt loads referring to the support range from0.1to2.0wt.%.After impregnation,the samples are dried overnight at1008C and are then activated by dosing a gas mixture of9vol.%H2and91vol.%N2.In the activation step the temperature is increased from20to3008C at the rate of1.0K minÀ1;the end temperature is held for30min.Finally,the samples are condi-tioned by heating in air at5008C for5h.For reference purposes a classical Pt/Al2O3catalyst is also prepared taking commercially available g-Al2O3balls(d=0.6mm,Sasol).The modification of ZrO2with tungsten is performed by incipient wetness method as well using a solution of (NH4)6H2W12O41(Fluka).Different loads of W(up to22wt.%) relating to the mass of ZrO2are established by varying the concentration of(NH4)6H2W12O41.After impregnation,the sample is dried overnight at1008C and is then impregnated by Pt as mentioned above.It is worth noting that in the H2activation tungsten oxide is not reduced[24].2.2.Characterisation of the catalystsCrystalline phase of the pure supports as well as catalysts is examined by powder X-ray diffraction(PXRD).The PXRD patterns are recorded at room temperature on a Siemens D501using Ni filtered Cu K a radiation.A2u step size of0.028is used with an integration time of4s.The diffractogram of the commercial Al2O3 carrier confirms the g-modification,while the prepared ZrO2is in the tetragonal phase as already demonstrated in Fig.1.Regardless of the load of W no reflexes of crystalline tungsten oxide are observed suggesting amorphous WO x domains,at least at W contents above6wt.%when tungsten oxide exists in sufficient abundance to be monitored.Additionally,signals of Pt are not found as well being associated with its low contents.The dispersion of Pt is studied by temperature-programmed desorption of H2(H2-TPD).In these analyses,same laboratory bench is used as in the catalytic investigations(Section2.4).Respective sample(1.5g)is charged into the quartz glass tube reactor(i.d. 8mm)and pre-treated in Arflow at6008C for15min.Subsequently, the catalyst is cooled to3008C and exposed to a gas mixture of 5vol.%H2and95vol.%Ar for30min to reduce the active Pt surface. The pattern is then rapidly cooled to308C in the H2/Arflow.After saturation,it isflushed with Ar and H2-TPD is started using Ar as carrier gas(100ml minÀ1,STP).In TPD the catalyst is heated to 6008C at the rate of20K minÀ1,whereupon temperature is recorded by a K type thermocouple(TC)fitted directly in front of the sample. Desorbing H2is continuously monitored by thermal conductivity detection(TCD,Shimadzu).For specific analysis of H2the reactor effluents pass a cold trap(À508C)removing H2O.The Pt dispersion (d Pt)is determined by supposing that one H adsorbs per active Pt site (Eq.(1))[25].The molar amount of desorbing H2ðn H2Þis obtained by integrating the corresponding TCD signal,whereas the total proportion of Pt(n Pt)is known from the impregnationprocedure. Fig.1.PXRD patterns of the prepared ZrO2support depending on the digestion time (a)5min,(b)1h,(c)4h,(d)7h,(e)12h and(f)17h;*monoclinic phase;+ tetragonal phase);final calcination is performed in air at7508C for6h;analytical parameters of PXRD are described in next section.F.J.P.Schott et al./Applied Catalysis B:Environmental87(2009)18–2919As derived from blank experiments,e.g.Pt free 11W/ZrO 2,the H 2desorption is specific for Pt.d Pt ¼0:5n H 2n Pt(1)Furthermore,the ZrO 2-based catalyst with a Pt load of 0.3wt.%and a W content of 11wt.%is exemplarily characterised by X-ray photoelectron spectroscopy.The spectrometer is a Phoibos 150MCD from Specs being equipped with a Mg anode (E (Mg K a )=1253eV).The spectrum shows a clear absorption at 36eV (W4f 7/2)being related to W 6+species [26].The BET surface area of the catalysts is investigated by multi-point Sorptomatic 1990using N 2as adsorbate.The BET data are listed in Table 1along with the loading,Pt dispersion and sample codes.It is worth mentioning that the declining BET surface area of the 0.3Pt/W/ZrO 2samples is mainly related to the increasing proportion of WO 3exhibiting negligible surface area and it is not due to the blocking of pores of the ZrO 2carrier.2.3.O 2-and NO x -TPD studiesO 2-TPD studies are performed to investigate the kinetics of the adsorption and desorption of O 2on selected catalysts.Oxygen is used as probe molecule since it represents a dominating species on the Pt surface under lean burn conditions [27].The procedure of O 2-TPD is similar to that described above for H 2-TPD.The catalyst (5.00g)is heated in Ar at 7508C for 15min,cooled to 508C and is then exposed to a mixture of 2vol.%O 2and 98vol.%Ar (99.996%,<6ppm O 2)until the sample is saturated.After flushing with Ar (500ml min À1,STP)TPD is started with a rate of 20K min À1.O 2is detected by CIMS (Airsense 500,V &F).As the axial and radial temperature gradients along the catalyst bed are below 10K heat transfer effects are to be neglected.Furthermore,Mears and Weisz Prater criteria [28],that amount to 10À5and 10À2,respectively,exclude transport limitation by film and pore diffusion.NO x -TPD examinations are conducted similar to H 2-TPD as well using a catalyst mass of 1.50g.After the pre-treatment performed at 5008C the sample is cooled to 1258C in Ar flow and is then exposed to a mixture of 2800ppm NO and 6vol.%O 2in Ar.In this treatment NO is partially converted into NO 2(15%)due to catalytic oxidation.After saturation,the dosage of NO and O 2is stopped and the reactor is flushed by Ar followed by starting the TPD with a rateof 20K min À1.Additionally,before beginning the TPD a mixture of (a)1500ppm H 2,6vol.%O 2,Ar (balance)or (b)1500ppm H 2in Ar is added for 10min to study the reaction of NO x surface species with H 2.Subsequently,it is purged again and TPD is finally started employing exclusively the CLD analyzer mentioned in the following section.2.4.H 2-deNO x studiesThe catalytic investigations are performed on a laboratory bench using a diesel model exhaust.Before the measurements,the samples are pressed to pellet with 40MPa,granulated and sieved in a mesh size of 125–250m m;an exception is the Pt/Al 2O 3pattern which is kept in form of balls.The samples (1.50g)are charged into the quartz glass tube reactor (i.d.8mm),fixed with quartz wool and pre-treated in Ar flow at 5008C for 15min to remove possible impurities and to provide reproducible conditions.Subsequently,the model exhaust is added and the temperature is decreased to 408C with a rate (b )of 1.0K min À1.Furthermore,some experi-ments are carried out under stationary conditions at selected temperatures.The standard feed (500ml min À1,STP)is composed of 500ppm NO,2000ppm H 2,6.0vol.%O 2and Ar as balance.To evaluate the best catalyst the concentration of H 2and O 2is varied and other relevant exhaust gas species,i.e.CO,H 2O and CO 2are dosed additionally.The feed is obtained by blending a special mixture of 2000ppm H 2and 6.0vol.%O 2in Ar with the other components (Air Liquide).The flow of each component is controlled by independent mass flow controllers (MKS Instru-ments),whereas water is supplied by a liquid pump (Kronlab).Temperature is measured by a K type TC located directly in front of and behind the catalyst ing the standard feed the temperature difference between inlet and outlet is below 10K and therefore only the inlet temperature is presented.The analysis of NO x is conducted by means of CLD (EL-ht,Eco Physics),while N 2O,CO and CO 2are monitored by NDIR spectroscopy (Uras 10E,Hartmann &Braun).N 2is detected by GC/TCD (RGC 202withpacked columns Haye Sep Q 60and mol sieve 5A˚,Siemens)resulting in a time resolution of 9min that corresponds to a temperature interval of 15K.Oxygen is analysed by using magnetomechanics (Magnos 6G,Hartmann &Braun).The NO x conversion (X (NO x ))refers to the formation of N 2and N 2O as defined by Eq.(2),whereas the temperature programmed H 2-deNO x data are found to be equal to stationary results.For the major part of the reaction the mass of N is balanced being associated with the production of N 2and N 2O thus excluding the genesis of NH 3.Nevertheless,below 808C H 2-deNO x is slightly interfered by NO x adsorption corresponding to less than 10%conversion.X ðNO x Þ¼2c ðN 2Þþ2c ðN 2O Þc ðNO x Þin(2)The selectivity of N 2(S (N 2))is defined by Eq.(3),whereupon a corresponding expression is used for the N 2O selectivity (S (N 2O)).Moreover,for the comparison of different catalysts the overall selectivity of N 2(S (N 2)overall )is taken as well (Eq.(4));T 1and T 2are the temperatures with NO x conversion of 20%.Selectivity data are exclusively presented for deNO x above 20%to minimise error propagation.S ðN 2Þ¼c ðN 2Þ22(3)S ðN 2Þoverall ¼ZT 2T 1S ðN 2Þd T21(4)Table 1Load of Pt and W,sample code,BET surface area and Pt dispersion of the H 2-deNO x catalysts Catalyst system m(Pt)(%)m(W)(%)Sample codeS BET(m 2g À1)d Pt (%)Pt/Al 2O 30.50.5Pt/Al 2O 3150522Pt/Al 2O 31489Pt/ZrO 20.10.1Pt/ZrO 299300.30.3Pt/ZrO 2100250.50.5Pt/ZrO 21003622Pt/ZrO 29811Pt/WO 3/ZrO 2a0.330.3Pt/3W/ZrO 297360.360.3Pt/6W/ZrO 2106500.3110.3Pt/11W/ZrO 268900.3220.3Pt/22W/ZrO 26295b 2112Pt/11W/ZrO 27011aThe loads of Pt and W refer to the ZrO 2support.bWhile the Pt dispersion of 0.3Pt/11W/ZrO 2is checked by HRTEM (particles <2nm),double H 2desorption is observed for 0.3Pt/22W/ZrO 2.Hence,different H/Pt stoichiometry is assumed being speculated to be 2.This change might be associated with Pt–W interactions dominating at high tungsten oxide coverages amounting to ca.45%for 22%W.F.J.P.Schott et al./Applied Catalysis B:Environmental 87(2009)18–29202.5.DRIFTS studiesThe DRIFT spectroscopic studies are performed with a Nicolet 5700FTIR spectrometer(Thermo Electron)being equipped with a MCT detector and DRIFTS optics(Thermo Mattson).The sample compartment is continuously purged with N2to avoid diffusion of air.The IR cell made of stainless steel contains a ZnSe window and is connected to a gas-handling system.The spectra are recorded in the range from1000to4000cmÀ1with an instrument resolution of 4cmÀ1.100scans are accumulated to a spectrum resulting in a time resolution of1min.Before the analysis,the catalyst powder is charged into the sample holder of the cell and is heated for30min at 5008C in N2or Arflow(500ml minÀ1,STP).In the studies using CO as probe molecule the sample is then exposed at2508C for30min to a mixture of2000ppm H2and6vol.%O2(N2balance);this is done to establish similar conditions as in catalytic studies(Section2.4). Subsequently,the catalyst is cooled in N2flow to258C and then a background spectrum is recorded.After this,the sample is treated for5min with a mixture of500ppm CO in N2followed by purging with N2.Finally,the sample spectrum is collected.The DRIFTS investigation of H2-deNO x is performed sequen-tially.Firstly,the catalyst is cooled from500to1258C under flowing Ar and then the background spectrum is taken.Subse-quently,the sample is exposed for10min to a mixture of 1000ppm NO and6vol.%O2(Ar balance)followed by purging with Ar and taking the spectrum.After this,the blend of40or2000ppm H2and6vol.%O2(Ar balance)is added while continuously collecting data.The H2concentration of40ppm is adjusted to definitely avoid hot spots on the catalyst.The DRIFT spectra are presented in terms of Kubelka Munk transformation defined as F(R)=(1ÀR)2/(2R)with R=R s/R r, whereas R s is the reflectance of the sample under reaction conditions and R r that under the Arflow.3.Results and discussion3.1.Performance of the Pt/Al2O3reference and the Pt/ZrO2catalystsA preliminary investigation performed in the absence of a catalyst shows no conversion of NO x excluding H2-deNO x by gas-phase reactions.In contrast to that,the0.5Pt/Al2O3referencereveals pronounced low-temperature activity,whereas the opera-tion window is rather narrow(50–1508C)and N2O forms as the major product;S(N2)overall amounts to20%(Fig.2).The perfor-mance of Pt/Al2O3is considered to be in fair agreement with literature addressing the same catalytic system[1,3,4].Fig.3illustrates the performance of0.3Pt/ZrO2indicating NO x conversion in the entire temperature range with two deNO x maxima at140and1708C.Furthermore,the peak NO x removal is higher as compared to0.5Pt/Al2O3.Another interesting feature is the markedly improved N2selectivity of0.3Pt/ZrO2,i.e.N2is formed as the major product showing an overall selectivity of55%.The Pt/ZrO2samples with Pt contents of0.1and0.5wt.%show similar peak NO x conversions of78and82%,respectively,whereas their operation range is restricted covering a range of ca.140K only.Furthermore,S(N2)overall is very similar to0.3Pt/ZrO2.Hence, the latter material is considered to be superior and is therefore adopted to the Pt/WO3/ZrO2system.The superiority of0.3Pt/ZrO2 is difficult to explain as Table1shows very similar physical–chemical properties for the Pt/ZrO2samples.3.2.Performance of the0.3Pt/W/ZrO2catalystsFor comparison of the0.3Pt/W/ZrO2samples revealing different loads of W the maximum NO x conversion(X(NO x)max)is used in addition to S(N2)overall.Fig.4shows that a little amount of tungsten is sufficient to increase deNO x as well as N2selectivity.The optimum content of W is11wt.%resulting in an overall N2 selectivity of85%and a peak NO x conversion of95%;assuming planar tungsten oxide the ZrO2coverage by WO3is estimated to be 0.22being significantly less than a monolayer.Furthermore,the results of XPS and PXRD suggest that the tungsten component exists in the form of amorphous WO3.For clarity the performance of the0.3Pt/11W/ZrO2catalyst is presented in Fig.5.The data point to a broad range of deNO x with two conversion peaks at90and2508C.As a consequence,0.3Pt/ 11W/ZrO2covers the low-as well as high-temperature regime thus representing a promising catalytic material.Nevertheless,it should be stated that the low-temperature activity is much more pronounced,while above2508C deNO x declines.The latter feature is attributed to increasing conversion of H2with O2being present in excess;a more detailed discussion of the educt selectivity is presented in Section3.4.The two NO x conversion maximums are ascribed to different active Pt sites as stated in Section 3.5.3, whereas no evidence for active NO x species located on the support is found as discussed in Section3.5.3as well.Contrary,the bare 11W/ZrO2support does not directly participate in deNO x as deduced from a measurement without Pt.However,0.3Pt/11W/ ZrO2still shows significant formation of N2O below1508C,e.g.at Fig.2.H2-deNO x performance of0.5Pt/Al2O3(X(NO x)—,S(N2)--,S(N2O)). Conditions:m=1.50g,c(NO)=500ppm,c(H2)=2000ppm,c(O2)=6.0vol.%,Ar balance,F=500ml minÀ1(STP),S.V.=22.000hÀ1,b=1.0K minÀ1.Fig.3.H2-deNO x performance of0.3Pt/ZrO2(X(NO x)—,S(N2)--,S(N2O)). Conditions:m=1.50g,c(NO)=500ppm,c(H2)=2000ppm,c(O2)=6.0vol.%,Ar balance,F=500ml minÀ1(STP),S.V.=22,000hÀ1,b=1.0K minÀ1.F.J.P.Schott et al./Applied Catalysis B:Environmental87(2009)18–2921the low-temperature deNO x peak S (N 2O)is about 45%.In contrast to that,N 2O is substantially suppressed above 2008C correspond-ing to a N 2selectivity of approximately 90%.Furthermore,it should be mentioned that no NH 3forms in H 2-deNO x on the 0.3Pt/11W/ZrO 2catalyst as referred from a NDIR analysis (Binos 1.1,Leybold-Heraeus).The suppression of NH 3formation is reported to be typical for NO x reduction by H 2on Pt catalysts under oxygen-rich conditions [1].The turnover frequency (TOF)being defined as the number of converted NO x molecules per Pt atom and time is 8.0Â10À3s À1for the deNO x peak at 90%and 5.0Â10À3s À1for the 2508C maximum.These specific H 2-deNO x data are very close to that of 0.1Pt/MgO–CeO 2showing a TOF of ca.8Â10À3s À1(908C)in a similar NO/H 2/O 2feed [14].A direct comparison of the activity range of both catalysts is problematic as data recorded under analogue condi-tions are not available in the study of 0.1Pt/MgO–CeO 2[14].3.3.Evaluation of the 0.3Pt/11W/ZrO 2catalyst3.3.1.Effect of O 2and CO on H 2-deNO x performanceThe previous section provides evidence that the content of O 2is a crucial parameter for the H 2-deNO x reaction.Hence,theeffect of O 2is systematically investigated,whereas for simplicity the performance of 0.3Pt/11W/ZrO 2is exemplarily illustrated at 1258C being representative for the low-temperature range (Fig.6).It is apparent that the O 2concentration does not drastically affect the low-temperature activity, e.g.deNO x is about 90%for 1.5vol.%O 2and ca.75%for 18vol.%.Moreover,the N 2selectivity does not change at all.On the contrary,the high-temperature activity declines markedly with growing O 2content being completely suppressed above 12vol.%O 2(Fig.7).This effect is referred to the increasing reaction of H 2with O 2(Section 3.4).However,diesel engines provide low temperatures in connection with rather high O 2concentrations and vice versa,e.g.at 1508C the O 2content is in the range of 10–15vol.%.Therefore,the decline in high-temperature deNO x observed for relatively high O 2contents is not a substantial issue for practical application.Furthermore,the effect of CO on H 2-deNO x is investigated as well,since this component is known to block active Pt sites at low temperatures which might affect the performance of 0.3Pt/11W/ZrO 2[27].For this examination two representative CO concentra-tions are supplied,i.e.a rather low (40ppm)and a rather high one (400ppm).Fig.8shows that the latter CO concentrationcausesFig.4.Effect of W load on the H 2-deNO x performance of the 0.3Pt/W/ZrO 2samples (X (NO x )max *,S (N 2)overall *).Conditions:m =1.50g,c (NO)=500ppm,c (H 2)=2000ppm,c (O 2)=6.0vol.%,Ar balance,F =500ml min À1(STP),S.V.=22.000h À1,b =1.0K min À1.Fig.5.H 2-deNO x performance of 0.3Pt/11W/ZrO 2(X (NO x )—,S (N 2)--,S (N 2O)).Conditions:m =1.50g,c (NO)=500ppm,c (H 2)=2000ppm,c (O 2)=6.0vol.%,Ar balance,F =500ml min À1(STP),S.V.=22.000h À1,b =1.0K min À1.Fig.6.Effect of O 2on the H 2-deNO x performance of 0.3Pt/11W/ZrO 2at 1258C (X (NO x )*,S (N 2)*).Conditions:m =1.50g,c (NO)=500ppm,c (H 2)=2000ppm,c (O 2)=1.5–18vol.%,Ar balance,F =500ml min À1(STP),S.V.=22.000h À1,b =1.0K min À1.Fig.7.H 2-deNO x performance of 0.3Pt/11W/ZrO 2with a O 2content of 12vol.%(X (NO x )—,S (N 2)--,S (N 2O)).Conditions:m =1.50g,c(NO)=500ppm,c (H 2)=2000ppm,c (O 2)=12vol.%,Ar balance,F =500ml min À1(STP),S.V.=22.000h À1,b =1.0K min À1.F.J.P.Schott et al./Applied Catalysis B:Environmental 87(2009)18–2922a drastic decline in catalytic activity,whereupon significant deNO x begins in parallel to the CO light-off (ca.1108C).This effect is in line with the literature which shows the appearance of free Pt sites being capable of reducing NO only when CO removal starts [29].Consequently,as deNO x is inhibited below 1108C no substantial quantity of N 2O is formed resulting in an overall N 2selectivity of about 90%.Additionally,it is worth mentioning that four maxima of NO x conversion appear pointing to a broad variety of active Pt sites.Furthermore,the CO concentration of 40ppm does not affect the performance of 0.3Pt/11W/ZrO 2at all.These results clearly show that high CO concentrations have to be avoided in practice to maintain H 2-deNO x .Indeed,this can be simply achieved by using a diesel oxidation catalyst (DOC)located in front the H 2-deNO x catalyst.DOC systems are a state-of-the-art technology and are applied in every diesel vehicle released in the industry countries oxidising CO and HC close to the engine outlet.3.3.2.Hydrothermal stability and resistance against SO xFor the use of 0.3Pt/11W/ZrO 2in diesel exhaust its hydrothermal resistance as well as chemical stability towards SO x is of particular concern.To pursue these requirements the catalyst (1.50g)is hydrothermally aged at 7808C for 15h adjusting a gas mixture of 2.5vol.%H 2O and 97.5vol.%Ar (500ml/min,STP),whereas exposure to SO x is carried out at 3508C for 24h while supplying a blend of 40ppm SO 2and synthetic air (500ml min À1,STP).The latter conditions are considered to be appropriate to form SO 3on the catalyst being a strong catalyst poison [30].Nevertheless,both aging procedures do not affect the catalytic performance at all evidencing high resistance of 0.3Pt/11W/ZrO 2against hydrothermal and sulphur exposure.parison of the reducing efficiency of H 2with C 3H 6and COTo assess the efficiency of H 2in the deNO x reaction on 0.3Pt/11W/ZrO 2additional reductants are used.For this purpose C 3H 6and CO are taken as they are potentially formed as major or side product in the on-board production of H 2from diesel,e.g.by catalytic cracking,catalytic partial oxidation or catalytic steam reforming [31];C 3H 6is taken as a model hydrocarbon.For an accurate comparison 10,000ppm H 2,1000ppm C 3H 6or 9000ppm CO are dosed to the standard feed.These concentra-tions demand very similar amount of oxygen for complete conversion.Fig.9shows that the presence of 10,000ppm H 2causes outstanding NO x conversion.Contrary,in the study withC 3H 6significant deNO x appears between 160and 3208C only with a peak conversion of ca.80%corresponding to a N 2selectivity of 45%.A minor deNO x activity is obtained with CO being in line with the results demonstrated in Section 3.3.1.As discussed there,the inhibition of deNO x is related to the blocking of active Pt sites by C 3H 6and CO suppressing substantial dissociation of NO [29].This effect is obviously stronger for CO,whereas it has to be taken into account that much more CO molecules are dosed as compared to C 3H 6.Additionally,Fig.9demonstrates that C 3H 6and CO mainly react with O 2,particularly above 3008C.Finally,from the present experiments it is concluded that among the tested reductants only H 2shows efficient NO x reduction on 0.3Pt/11W/ZrO 2.Fig.9.Effect of H 2,C 3H 6and CO on the deNO x performance of 0.3Pt/11W/ZrO 2(X (NO x )—,S (N 2)--,S (N 2O),X (C 3H 6)—,X (CO)—),The concentration of reducing agent 10,000ppm H 2,1000ppm C 3H 6or 9000ppm CO;remaining conditions are the same as described in Fig.5.Fig.8.Effect of CO on the H 2-deNO x performance of 0.3Pt/11W/ZrO 2(X (NO x )—,S (N 2)--,S (N 2O),X (CO)—).Conditions:m =1.50g,c (NO)=500ppm,c (CO)=400ppm,c (H 2)=2000ppm,c (O 2)=6.0vol.%,Ar balance,F =500ml min À1(STP),S.V.=22.000h À1,b =1.0K min À1.F.J.P.Schott et al./Applied Catalysis B:Environmental 87(2009)18–2923。
Thorpe-Ingold效应及其在有机成环反应中的应用
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结果， 无取代底物 1a 环化速率最慢， 单甲基取 代 1b 和单叔丁基取代 1c 的环化速率略有增加， 但
14711491thorpeingold效应及其在有机成环反应中的应用北京化工大学理学院有机化学系化工资源有效利用国家重点实验室北京100029本文首先介绍了thorpeingold效应及其相关理论和实验研究进展然后综述了近年来thorpeingold效应在形成三四五和六元环产物的环化反应中的应用
PＲOGＲESS IN CHEMIreaction
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ThorpeIngold 效应
ThorpeIngold 效 应， 又被称为偕二甲基效应 （ gemdimethyl effect ） 或角压缩 （ angle compression ） 效应， 是指通过分子链上的偕二取代促进环化 。 ［2 ］ 该效应最早由 Thorpe 和 Ingold 等提出 ， 即当 分子链中亚甲基上的氢被大体积烷基取代后 ， 该处 内角（ 键角） 被压缩， 分子链末端两个反应基团相互 靠近， 有利于分子内环化（ 图 1 ） 。
Waters公司产品说明：高效液相色谱(HPLC)及其相关技术的简要介绍说明书
Introduction to GPCColumns, Distributions,Sample Prep., Calibration,What’s NewPolymer Analysis TechniquesHigh Performance Liquid Chromatography (HPLC) (mainly Size Exclusion Chromatography, SEC)Mass Spectroscopy (MS)Thermal Analysis (TA)Rheometry, Nuclear Magnetic Resonance spectroscopy (NMR), Fourier Transform Infrared spectroscopy (FTIR)Introduction to GPC OutlineWhat is GPC?GAP of AdditivesWhat’s New?What is GPC?▪Gel Permeation Chromatography (GPC) separates samplemolecules based upon their relative size in solution▪Size Exclusion Chromatography (SEC)▪Gel Filtration Chromatography (GFC)▪GPC is an isocratic mode of separation▪GPC is well-suited for polymer analysis –provides a “molecularweight distribution”Particles arePorous, rigidpolymericmaterialsBig ones are not sloweddown because they aretoo big to go into thepores, so they elute firstWhat is GPC?▪The elution profile represents the molecular weight distribution based upon the relative content of different molecular weights…▪Based on size in solution Elution Volume (retention time)largest smallest Mn Mw Mz Mz+1B ig O nes C ome O ut F irst M o l e c u l a r W e i g h tSome definitionsMolecular weight averages are used to provide numerical differences between samples –Mn: Number average molecular weighto At this point in the curve the number of molecules in the sample to left is equal to the number of molecules to the right–Mw: Weight average molecular weighto At this point in the curve the weight of the molecules to the left is equal to the weight of the molecules to the right–Mz and Mz+1: these values are calculated based on molecular weight and abundance (obtained by ultracentrifugation and GPC software computation)o The values are used for “comparison” purposes•Known samples to unknown samplesThese averages are statistical moments calculated from the molecular weight distribution curveMolecular Weight AveragesGPC Delivers all MW information with one experiment GPC calculates the MW distributionof the polymer; this distribution canbe measured for:–Mn can affect a polymer’s brittleness,flow and compression properties.–Mw is related to strength properties,and impact resistance–Mz is related to elongation andflexibility, (Gumby -rubber)–Mz+1 is related to die swell,(extrusion parameter)Molecular Weight/Physical Property Correlations P r o p e r t y /P r o c e s s s P a r a m e t e r E f f e c t o f H i g h M W E f f e c t o f L o w M W I m p a c t S t r e n g t h M e l t V i s c o s i t y P r o c e s s i n g T e m p F l e x L i f e B r i t t l e n e s s D r a w a b i l i t y S o f t e n i n g T e m p S t r e s s -c r a c k R e s i s t a n c e M e l t F l o w Properties of PolymersGPC Process –Separates by Size in Solution Sample PolymerVo Vt“Bank” of3 GPC Columns Molecular Weight Distribution Chromatogram B igO nesC omeO utF irstWhat is happening inside a column?BOCOF Separation by size –Usually no chemical Retention mechanismBasic Column InformationColumns are put together in seriesto form a bank (>2)Always put the highest pore sizecolumn first (from the injector), andsmallest pore size column last–This reduces back pressure on themost fragile column (LMW column)Ramp the flow up slowly –0.1ml/minute –1.0 ml/minute insmall increments*Change over solvent at a0.1ml/minute flow rate over nightSome applications atHigh Temperature >150°CGPC Column TypesOrganic vs. Aqueous GPC columnsDifferent Pore SizesAnalytical vs. Preparative GPC columns Conventional vs. Solvent efficient, and High speed GPC columns.Care and Use of GPC ColumnsBasic GPC –typically a bank of 3 columns of different pore sizes to cover a broad MW range (as needed)Never use methanol or acetonitrile with Organic GPC columns because certain polar solvents will shrink the column, causing it to void–Note: Make sure to flush the complete system with solvent to be used before connecting the columns to the systemLife time of the columns can be as long as ~ 1 year or moreStore the columns in the solvent used with the column bank kept togetherMake sure that the end fittings on the column are tight to keep the column packing from drying out Be careful not to drop the columns, because they are fragileFilter sample solutionsPrevent air bubbles from getting into columnsGPC RulesPolymer is dissolved in a solvent at a low concentration (<0.10% w/v)Polymer solution passes through crosslinked, organic gel columns packed with controlled pore size particles.Larger molecules do not fit into pores (they are excluded) and elute firstVo : Exclusion volumeVt : Total volume or Permeation volumeExclusion principle assumes no adsorption of polymer molecules on packing materialEvery polymer will be eluted between Vo and VtDetectors in Polymer ChromatographyConcentration–Response ~ concentration, (C)–Refractometer; ∆N = (dn/dc) CStructure-selective–UV/Vis Detector (Also can use as concentration detector if sample has UV response), Lambert-Beer Law –IRMolecular weight sensitive–Response ~ C x f(M)–Light scattering: f(M) = M; M(C)–Viscometer: f(M) = [η] = kMα; [η]C•Where k, alpha are the Mark-Houwink constants–Mass Spec.: f(M) = 1/M; C/MRI’s were among the first detectors used in LC/GPC, (late 1960’s)Typically referred to as a “Universal Detector”Detects all dissolved solutes –“non -specific”Refractive index of any optical medium is defined as the ratio of the speed of light in a vacuum to the speed of light in the mediumDetection based on the refractive index of a given analyteMeasures the difference in RI from the eluent to the dissolved sample, (differential type)–The greater the dRI, the stronger the signal Sensitivity increase as RI difference increases Refractive Index (RI) Definition RSUV/Vis DetectionHigh SensitivityHigh SelectivityGain information on chemical compositionCan be used in gradient modeExcellent for most polymer additivesLinear response over a wide absorption range:–A = x l x C (Beer/Lambert)Sometimes UV/Vis detection is used for–Higher detection sensitivity.–Copolymer analysis (usually coupled with RI detector)–Under gradient elution condition.Polystyrene StandardsUV Detection at 260nmA U 0.000.010.020.030.040.050.060.070.080.090.100.110.120.130.140.150.16Minutes0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.57.06.0mm X 15cm RT MB-M Column (1)(Same Pore Size as Conventional Column Set)Flowrate: THF at 0.60 ml/min Injection Volume: 5 µl Detector: 2487 UV @260nm PS Standards3,840,0002,890,0001,260,000775,000422,000186,00042,80016,7005,5702,980890474Evaporative Light Scattering DetectorEvaporative Light Scattering Detection for polymer characterization–ELSD is a concentration detector.–ELSD is not affected by solvent changes–Most appropriate detector for Gradient Analysis of Polymers (GAP) or Gradient Polymer Elution Chromatography (GPEC)–Good alternative to dRI for compounds having a low dn/dcEvaporative Light Scattering Detection for additives–ELSD is a universal detector–Compounds without UV-chromophore groups will be detectedEvaporative Light Scattering DetectorPolymers are combined with low molecular compounds (0.1-3%)Additives have several key functions :–Protection : light stabilizers, antioxidants, anti-UV–Safety : flame retardants–Processing : plasticizers, slip agentsFull characterization of synthetic polymers involves detection of additivesMost of slip agents do not absorb UV light. ELS Detector is a good alternative to UV detectionAdditives MixtureComparison of Detection ModesCrodamide (oleamide) is not detected in UV mode5.0010.0015.0020.00T i n u v i n 312T i n u v i n PB H T L uw i n o x 44 B 25S u c c o n o x 16N a u g a r d 445S u c c o n o x 18T i n u v i n 328I r g a n o x 1330I r g a n o x 1076I r g a f o s 1680.005.0010.0015.0020.00T i n u v i n 312T i n u v i n PL u w i n o x 44B 25S u c c o n o x 16N a u g a r d 445S u c c o n o x 18C r o d a m i d eT i n u v i n 328I r g a n o x 1010I r g a n o x 1330I r g a n o x 1076I r g a f o s 168UV Detection (220 nm)ELS DetectionI r g a n o x 1010GPC Calibration -Narrow Standards A calibration curve is built with low dispersity (narrow) standards with known molecularweight; (ideal if same structure as unknown polymer)(PS, PMMA...).This calibration curve may be used toquantitate a polymer of different nature (PC, PMMA...). Then results are expressed in PSor PMMA equivalents, (or relative to PS orPMMA –incorrect for the polymer sample of interest).The chromatographic process is based on hydrodynamic volume (H)(size in solution), and not molecular weight.Two different polymers with the same MW will elute at different retention volumes.Vo Excluded V Total MW*RangeNoresolutionABOVEthis MW Resolutionrangeis created bydifferences inelution timeElution Time or Volume NoresolutionBELOWthis MW*MW is Log scaleSame “Pore Size” ColumnsVo Excluded V Total MW RangeNoresolution ABOVE this MWResolution range is created by differences in elution timeElution Time or VolumeNoresolution BELOW this MWVo Excluded V TotalMW RangeNoresolution ABOVE this MWResolution range is created by differences in elution timeElution Time or VolumeNoresolution BELOW this MW2 Columns of the SAME Type-SAME MW Range -More ResolutionDifferent “Pore Size” ColumnsVo ExcludedV TotalMW RangeNoresolution ABOVE this MWResolution range is created by differences in elution timeElution Time or VolumeNoresolution BELOW this MWVo Excluded V TotalNew MW RangeNoresolution ABOVE this MWResolution range is created by differences in elution timeElution Time or VolumeNoresolution BELOW this MW2 Columns of the Different Type -EXPANDED MW Range -More ResolutionActual Calibration Curves for the Different Pore SizeHT ColumnsH igh T emperatureHT 2 for Low MWHT 6 for High MWHT 6E for BlendedGPC Calibration 1 –Narrow StandardsCalibration:ually >10 standards usedbracketing MW range2.Standards may be injected as amixture3.Mixtures should be 1 order ofmagnitude different (1000, 10000,100000)To build a calibration curve:Narrow dispersity standards (PD<1.1)Elution volume at peak heightCurve : Log(M) = f(Ve)2.62.83.03.23.43.63.84.04.24.44.64.85.05.25.45.65.86.06.26.418192021222324252627Log Mol WtTime (min)11000001900009100Log(M)Elution volumeGPC Calibration –Narrow StandardsOrganic Polymers Aqueous PolymersPolystyrene Poly(ethylene oxides)Polybutadienes Poly(ethylene glycols)Poly(methylmethacrylates)PullulansPolyisoprenesNarrow Standards -Preparation ConsiderationsNarrow standards may be mixed together to develop a relative calibration curve –No more than 3 in one “cocktail” –be careful of concentrations–MW’s should be one decade apartNarrow standards should only be swirled gentlyNo need to filter narrow standardsAdd antioxidant if high temperature applicationSample Preparation ConsiderationsSample may be mixed to facilitate dissolution–Be careful of shear for high MW (>1M) samplesIn some cases, sample solution should be filtered–Presence of microgels, fillers, any other insolublesAllow enough time for complete dissolutionFor certain crystalline polymers, high temperature may be needed–Example: Isotactic polypropylene requires 2 hours at 170C in an external oven –The PP may then be run at 145CSample Concentration GuideThe more dilute the polymer solution, the better–This will prevent viscosity effects and non-reproducible retention–A dilute solution will allow the polymer to open up into its most relaxed conformation o No chain entanglemento No microgel formationInjection volume no more than 100 µL per columnFor very high MW polymers, flow rate may have to be lowered–HMW columns may be needed as well to prevent shearSample Concentration GuideMW < 1,000MW 1,000 –10,000MW 10,000 –100,000 MW 100,000 –500,000 MW 500,000 –1MMW >1M 0.20 –0.30% 0.15 –0.20% 0.10 –0.15% 0.05 –0.10% 0.01 –0.05% 0.005 –0.01%Above concentrations assumeno more than100 µL injection per columnGPC Column Operation Techniques Switch solvents by flushing to the column at0.1ml/min overnight in the new solventIncrease flow rates at 0.1 ml/minute/minute to the columns specified flow ratesRecommended to keep the flow through the columns at low flow, 50 ~ 100 m l/minute when idlingFor TCB at 140 -150 C -purge the toluene out of the columns at 0.1 mL/min. overnight at ~80C and thenramp up to temperature over 300 min after 3 column volumes of TCB have passed throughDetection–RI–Viscometry–UVPolymer DistributionsA polymer is a mixture of different size chain lengths of the same monomer. Tomeasure this distribution of sizes we use molecular weight averages.Slices are made to the chromatogram, where the height of each slice (H i), represents the population of molecules at that chain length or MW.iHlouvmeulEtionPolymer DistributionsMv High MWLow MWMw MnMv is derivedfrom ViscometryMP Mz Mz+1GPC and Flow Rate PrecisionMnMwVeLC Peak Identification (Narrow Standard)Based on retention timeMolecular Weight DeterminationGel permeation chromatography (GPC)Based on volume of solvent flowing through the column Essential for the flow rate to be absolutely constant To calibrate, plot Log MW of standards vs volume (time)Any flow variability will result in large MW errorsGPC and Flow Rate PrecisionPrecise solvent flow is essential for precise GPC…MwMnVe▪Comparisons of Molecular Weight Distributions of different samples can highlight even small MW shift information (good vs. bad PP’s below)▪Determine "good from bad" -e.g. QC of resin batches… these are typical formulators –use Mw, Mn, Mz and polydispersityPractical use –Fingerprinting the Polymerd w t /d (l o g M )0.000.200.400.60 3.504.00 4.505.00 5.506.00 6.50“Bad”“Good”Degradation of Polyethylene Terephthalate –HFIPAged PET Virgin PETd w t /d (l o g M )0.00.10.20.30.40.50.60.70.80.91.01.11.2Log Mol Wt2.83.03.23.43.63.84.04.24.44.64.85.05.25.45.6PET Exposed To Oils And Refrigerants For Several Days At Elevated TemperatureVirgin PETIntroduction to GPC OutlineWhat is GPC?GAP of AdditivesWhat’s New?Gradient Analysis of Polymers (GAP)In recent years there has been increased interest in using gradient HPLC techniques, such as Gradient Polymer Elution Chromatography (GPEC), with polymers for determining the compositional drift of copolymers, the composition of polymer blends, or for the analysis of polymer additives. Depending upon the gradient conditions and columns selected for analysis, separations may be obtained dependent on molecular weight or based upon precipitation, or adsorption mechanisms. The use of an Evaporative Light Scattering Detector (ELSD) allows one to perform solvent gradients with a universal mass detector and observe both UV absorbing and non-UV absorbing polymer samples without baseline disturbances from the solvent gradient. The addition of a Photodiode Array Detector (PDA) allows for compositional analysis across the molecular weight distribution of many copolymers, can be useful for the identification of components in a polymer blend, and also is invaluable for the quantitation of polymer additives and other small molecules in traditional reverse phase separations.Experimental ConditionsSystem:Waters Alliance 2690 Separations Module with column heater at 30 ºCDetector 1:Waters 996 Photodiode Array DetectorDetector 2:Alltech Model 500 ELSD with LTA Adapter (Drift Tube at 40º C, 1.75 Liters/min Nitrogen)Data System:Waters Millennium 32 Chromatography Manager Column:As listed in Figures, 30 ºCFlow Rate:1mL/minSamples:10 - 25 µl injections of 0.2 - 0.5% samplesGradient:Linear gradient, conditions and mobile phases as listed in Figures.GPC Analysis of a Polymer BlendGradient Analysis of a Polymer BlendStyrene-Acrylonitrile (25% Acrylonitrile)PolystyreneStyrene-ButadieneRubber (50% Styrene)% THF0102030405060708090100Minutes246810121416Symmetry Shield™ C8 Column (3.9x150mm)100% ACN to 100% THF over 20 minutes ELSD DetectionGradient Analysis of Narrow PS Standards% T H F102030405060708090100Time (Minutes)1234567891011Symmetry Shield C8 (3.9x150mm)100% ACN to 100% THF over 10 minutes ELSD DetectionStyrene Oligomers2890910043,900190,000355,000706,0001,090,0002,890,000Analysis of Styrene-Butadiene-Styrene Block% T H F102030405060708090100Minutes1011121314151617181920100% S t y r e n e40% S t y r e n e30% S t y r e n e22% S t y r e n e100% B u t a d i e n ePrototype DVB/Vinylpyrolidone Column (3.9x150mm)100% ACN to 100% THF over 20 minutes ELSD DetectionCalibration of %Styrene in SBR'sGradient Analysis of Low Molecular Weight WaxesTime (Minutes)68101214161820221112131415161718m V"Slack wax""C18 wax"Nova-Pak ®C18 (3.9x150mm)100% ACN to 100% THF over 30 minutes ELSD DetectionPolymer Additives –Tinuvins, (UV Stabilizers)4681012141618202224262814161820222426283032mVMinutesTinuvin 440Tinuvin 900Tinuvin 328ELSD DetectionPolymer Additives –Phthalate Plasticizers5.05.56.06.57.07.58.08.59.09.510.010.511.014161820222426283032343638mVMinutesDcHPDOP (DEHP)DIDPUDPELSD DetectionPolymer Additives -Slips and Antistats204060801001201401601802002201416182022242628303234mVMinutesOleamideErucamideStearic AcidELSD Detection。
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1 Abstract:Two standard physics problems are solved in terms of the Lambert function,in order to show the applicability of this recently defined function to physics.Otherapplications of the function are cited,but not described.The problems solved concern Wien’sdisplacement law and the fringingfields of a capacitor,the latter problem being representativeof some problems solved using conformal transformations.The physical content of thesolutions remains unchanged,but they gain a new elegance and convenience.1.IntroductionMany physicists have experienced,during their education,the surprise of seeing a known mathematical function appear in a new physical context.An example in elementary physics is one that arises when students arefirst taught about simple harmonic motion.We hope that some readers can remember their amazement on learning that the motion of objects bobbing on springs,or moving in circles,can be described using trigonometric functions.At the time,they might have wondered,“The functions sine and cosine are used for all of those triangle problems in geometry class;there are no triangles here”.A recent example of the same phenomenon is the discovery that a function called the Lambert function has applications in a number of areas of physics,even though it wasfirst defined by a computer scientist for the purpose of counting search trees.Several well-known problems in electrostatics and in quantum mechanics can be solved with greater facility using it.In addition to presenting these problems,we give references to other,less elementary,applications.Mathematical functions do not by themselves uncover new physics—rather they assist the physi-cist by facilitating numerical and algebraic computations.A physicist therefore demands several things of any new function before taking the time to learn about it.Thefirst feature it should have is that it is likely to have some general applicability.Abramowitz&Stegun[1]is full of functions that most of us would prefer not to know anything about;we only want to know about functions that will probably prove useful.Each person will draw the line between useful and not useful at a different place.The sec-ond feature is that there should be convenient access to numerical evaluation and to pertinent algebraic properties.A function that cannot be worked with easily is not much use.Many readers will be familiar with the conference scene in which a colleague asks you whether you know of a computer program to calculate values of the Chepooka function1.You probably reply that you yourself need a reference on the asymptotic properties of the Horrorshow function,at which point the two of you change the topic of conversation.Until the advent of the scientific calculator,even trigonometric functions and logarithms were not so easy to evaluate.The Lambert function is a function that meets the criteria just listed.Itfirst received a name in2Can.J.Phys.Vol., the early1980s,when the program Maple defined a function that was named simply.An historicalsearch,conducted while writing an account of this function[4],found work by the eighteenth centuryscientist mbert that foreshadowed the definition of the function;even though his work did not actually define the function,was named in his honour.The same search uncovered a fortuitous rea-son for calling the function,in that E.M.Wright,a mathematician known for his book with Hardyon pure mathematics,studied the complex values of the function,again without naming it.The function is not connected with the Lambert transform of a function,which has been defined independently[13].The definition of is that it is the function that solves the equation(1) where is a complex number.This equation always has an infinite number of solutions,most of them complex,and so is a multivalued function.The different possible solutions are labelled by an integervariable called the branch of.Thus the proper way to talk about the solutions of(1)is to say that they are,for any etc.There is always special interest in solutions that are purely real,and so we note immediately that when is a real number,equation(1)can have either two realsolutions,in which case they are and,or it can have only one real solution,this being [with now being complex],or no real solution.Even if is real,the branches other than are always complex.Admittedly,does not yet appear on any pocket calculator,but it is known to the computing systems Maple,Macsyma and Mathematica(in the case of Mathematica,the function is called).Therefore,as soon as a problem is solved in terms of,numerical values,plots,derivatives and integrals can be easily obtained.Thefirst physics problem to be solved explicitly in terms of was one in which the exchange forces between two nuclei within the hydrogen molecular ion()were calculated[11];this,however, is a long and difficult calculation(and it has already been published)so instead of describing it,we have taken two much simpler problems from standard physics textbooks,problems that many students meet in their physics education,and we have expressed the solutions in terms of.As mentioned above, the physical content does not change,only the ease of working.An additional point of interest is the fact that the electrostatic application helps to justify a mathematical decision concerning the definition of that was originally taken entirely on aesthetic(in a mathematical sense)grounds.2.Wien’s displacement lawThe spectral distribution of black body radiation is a function of the wavelength and absolute tem-perature,and is described by,defined such that is the power emitted in a wave-length interval per unit area from a black body at absolute temperature.The wavelength at which is a maximum obeys Wien’s displacement law,where is Wien’s displacement constant[3].This law was proposed by Wien in1893from general thermodynamic arguments.Once Planck’s spectral distribution law is known,Wien’s law can be deduced and the value of determined.The Planck Spectral distribution law iswhich,on the substitution,can be written concisely as the transcendental equation(2)Valluri,Jeffrey,Corless3 This equation has the trivial solution and the nontrivial oneTherefore Wien’s law is obtained with a new expression for Wien’s displacement constant:4Can.J.Phys.Vol.,Physical plane:(The -plane)–4–2024–2–112–3–2–10123–3–2–1123Starting solution:(The -plane)Fig.1.The top set of axes (the -plane)show the edge of a parallel-plate capacitor.The plates are the heavy lines.The equipotential lines of the fringing field are shown and are calculated as images of horizontal grid lines in the -plane.The bottom set of axes show the -plane,which is mapped to the -plane by .Valluri,Jeffrey,Corless5(5) Some further analysis(not given here)shows that the restriction implies that the branch index is also specified,once is known;moreover,we can give an analytic formula for this,in terms of,the unwinding number[5,7].The expression is(9)6Can.J.Phys.Vol.,Physical plane:Lines in -plane where values of potentialare required –6–4–20246–3–2–1123–3–2–10123–3–2–1123Lines where evaluationsare made in -plane These lines transformedusingThese lines transformedusing These lines transformedusing Fig.2.The top and bottom sets of axes again show the -and -planes.Now the transformation proceeds fromtop to bottom using the inverse mapping.Valluri,Jeffrey,Corless 7−1123−4−3−2−11xW−1/e Fig.3.The real values of the Lambert W function.The solid line shows and the dashed line .Many other algebraic properties have been found [4],but we quote only one more that might be useful in physical applications:an asymptotic formula for large (complex).(10)where is the principal branch of the (complex)natural logarithm (i.e.,the function implemented in software packages that support complex functions).Any physicist usingalso benefits from lengthy studies of the quickest way to evaluate the func-tion accurately.The basic strategy is to use the asymptotic formula to obtain a starting estimate for an iterative scheme similar to Newton iteration.Most physicists would jump to a Newton scheme if asked to evaluate the function without help,but the packages have optimized this strategy to ensure accuracy (always getting the branch correct,for example)and speed.The liberty in assigning the branches ofthat was referred to above can be illustrated using the values of given in table 1.The reader can notice that although are complex conjugates,it is not the case that are correspondingly complex conjugates.The branches ofcould be assigned so that this relation became true,and indeed,an early version of Maple did assign branches that way.Such a definition,however,would not satisfy the simple asymptotic relation (10).Our physics application has confirmed that the symmetries as displayed in the table are the best ones to have.5.Concluding remarksWe have discussed two standard problems of physics in which the Lambert function can be used.They are not the only two problems in whicharises.For example,Adler and Piran [2]used in their work on effective action models for a system of heavy antiquark and light scalar quark;they were working about the time when the function was named and before its properties had been set out,and so they did not benefit from the convenience they would now have.In their model,as well as in nonlinear quantum electrodynamics,the nonlinear dielectric constant has a logarithmic dependenceon the applied electric field,which meant thatcould be used to describe the electric displacement.Mann and Ohta [9,10]have used to elucidate the physics involved in their study of Lagrangians for two-dimensional gravity.8Can.J.Phys.Vol.,complex infinity。