Structural change in Bi4SixGe1-x3O12 glasses during crystallization

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Multi-scale structural similarity for image quality assesment

MULTI-SCALE STRUCTURAL SIMILARITY FOR IMAGE QUALITY ASSESSMENT Zhou Wang1,Eero P.Simoncelli1and Alan C.Bovik2(Invited Paper)1Center for Neural Sci.and Courant Inst.of Math.Sci.,New York Univ.,New York,NY10003 2Dept.of Electrical and Computer Engineering,Univ.of Texas at Austin,Austin,TX78712 Email:zhouwang@,eero.simoncelli@,bovik@ABSTRACTThe structural similarity image quality paradigm is based on the assumption that the human visual system is highly adapted for extracting structural information from the scene,and therefore a measure of structural similarity can provide a good approxima-tion to perceived image quality.This paper proposes a multi-scale structural similarity method,which supplies moreﬂexibility than previous single-scale methods in incorporating the variations of viewing conditions.We develop an image synthesis method to calibrate the parameters that deﬁne the relative importance of dif-ferent scales.Experimental comparisons demonstrate the effec-tiveness of the proposed method.1.INTRODUCTIONObjective image quality assessment research aims to design qual-ity measures that can automatically predict perceived image qual-ity.These quality measures play important roles in a broad range of applications such as image acquisition,compression,commu-nication,restoration,enhancement,analysis,display,printing and watermarking.The most widely used full-reference image quality and distortion assessment algorithms are peak signal-to-noise ra-tio(PSNR)and mean squared error(MSE),which do not correlate well with perceived quality(e.g.,[1]–[6]).Traditional perceptual image quality assessment methods are based on a bottom-up approach which attempts to simulate the functionality of the relevant early human visual system(HVS) components.These methods usually involve1)a preprocessing process that may include image alignment,point-wise nonlinear transform,low-passﬁltering that simulates eye optics,and color space transformation,2)a channel decomposition process that trans-forms the image signals into different spatial frequency as well as orientation selective subbands,3)an error normalization process that weights the error signal in each subband by incorporating the variation of visual sensitivity in different subbands,and the vari-ation of visual error sensitivity caused by intra-or inter-channel neighboring transform coefﬁcients,and4)an error pooling pro-cess that combines the error signals in different subbands into a single quality/distortion value.While these bottom-up approaches can conveniently make use of many known psychophysical fea-tures of the HVS,it is important to recognize their limitations.In particular,the HVS is a complex and highly non-linear system and the complexity of natural images is also very signiﬁcant,but most models of early vision are based on linear or quasi-linear oper-ators that have been characterized using restricted and simplistic stimuli.Thus,these approaches must rely on a number of strong assumptions and generalizations[4],[5].Furthermore,as the num-ber of HVS features has increased,the resulting quality assessment systems have become too complicated to work with in real-world applications,especially for algorithm optimization purposes.Structural similarity provides an alternative and complemen-tary approach to the problem of image quality assessment[3]–[6].It is based on a top-down assumption that the HVS is highly adapted for extracting structural information from the scene,and therefore a measure of structural similarity should be a good ap-proximation of perceived image quality.It has been shown that a simple implementation of this methodology,namely the struc-tural similarity(SSIM)index[5],can outperform state-of-the-art perceptual image quality metrics.However,the SSIM index al-gorithm introduced in[5]is a single-scale approach.We consider this a drawback of the method because the right scale depends on viewing conditions(e.g.,display resolution and viewing distance). In this paper,we propose a multi-scale structural similarity method and introduce a novel image synthesis-based approach to calibrate the parameters that weight the relative importance between differ-ent scales.2.SINGLE-SCALE STRUCTURAL SIMILARITYLet x={x i|i=1,2,···,N}and y={y i|i=1,2,···,N}be two discrete non-negative signals that have been aligned with each other(e.g.,two image patches extracted from the same spatial lo-cation from two images being compared,respectively),and letµx,σ2x andσxy be the mean of x,the variance of x,and the covariance of x and y,respectively.Approximately,µx andσx can be viewed as estimates of the luminance and contrast of x,andσxy measures the the tendency of x and y to vary together,thus an indication of structural similarity.In[5],the luminance,contrast and structure comparison measures were given as follows:l(x,y)=2µxµy+C1µ2x+µ2y+C1,(1)c(x,y)=2σxσy+C2σ2x+σ2y+C2,(2)s(x,y)=σxy+C3σxσy+C3,(3) where C1,C2and C3are small constants given byC1=(K1L)2,C2=(K2L)2and C3=C2/2,(4)Fig.1.Multi-scale structural similarity measurement system.L:low-passﬁltering;2↓:downsampling by2. respectively.L is the dynamic range of the pixel values(L=255for8bits/pixel gray scale images),and K1 1and K2 1aretwo scalar constants.The general form of the Structural SIMilarity(SSIM)index between signal x and y is deﬁned as:SSIM(x,y)=[l(x,y)]α·[c(x,y)]β·[s(x,y)]γ,(5)whereα,βandγare parameters to deﬁne the relative importanceof the three components.Speciﬁcally,we setα=β=γ=1,andthe resulting SSIM index is given bySSIM(x,y)=(2µxµy+C1)(2σxy+C2)(µ2x+µ2y+C1)(σ2x+σ2y+C2),(6)which satisﬁes the following conditions:1.symmetry:SSIM(x,y)=SSIM(y,x);2.boundedness:SSIM(x,y)≤1;3.unique maximum:SSIM(x,y)=1if and only if x=y.The universal image quality index proposed in[3]corresponds to the case of C1=C2=0,therefore is a special case of(6).The drawback of such a parameter setting is that when the denominator of Eq.(6)is close to0,the resulting measurement becomes unsta-ble.This problem has been solved successfully in[5]by adding the two small constants C1and C2(calculated by setting K1=0.01 and K2=0.03,respectively,in Eq.(4)).We apply the SSIM indexing algorithm for image quality as-sessment using a sliding window approach.The window moves pixel-by-pixel across the whole image space.At each step,the SSIM index is calculated within the local window.If one of the image being compared is considered to have perfect quality,then the resulting SSIM index map can be viewed as the quality map of the other(distorted)image.Instead of using an8×8square window as in[3],a smooth windowing approach is used for local statistics to avoid“blocking artifacts”in the quality map[5].Fi-nally,a mean SSIM index of the quality map is used to evaluate the overall image quality.3.MULTI-SCALE STRUCTURAL SIMILARITY3.1.Multi-scale SSIM indexThe perceivability of image details depends the sampling density of the image signal,the distance from the image plane to the ob-server,and the perceptual capability of the observer’s visual sys-tem.In practice,the subjective evaluation of a given image varies when these factors vary.A single-scale method as described in the previous section may be appropriate only for speciﬁc settings.Multi-scale method is a convenient way to incorporate image de-tails at different resolutions.We propose a multi-scale SSIM method for image quality as-sessment whose system diagram is illustrated in Fig. 1.Taking the reference and distorted image signals as the input,the system iteratively applies a low-passﬁlter and downsamples theﬁltered image by a factor of2.We index the original image as Scale1, and the highest scale as Scale M,which is obtained after M−1 iterations.At the j-th scale,the contrast comparison(2)and the structure comparison(3)are calculated and denoted as c j(x,y) and s j(x,y),respectively.The luminance comparison(1)is com-puted only at Scale M and is denoted as l M(x,y).The overall SSIM evaluation is obtained by combining the measurement at dif-ferent scales usingSSIM(x,y)=[l M(x,y)]αM·Mj=1[c j(x,y)]βj[s j(x,y)]γj.(7)Similar to(5),the exponentsαM,βj andγj are used to ad-just the relative importance of different components.This multi-scale SSIM index deﬁnition satisﬁes the three conditions given in the last section.It also includes the single-scale method as a spe-cial case.In particular,a single-scale implementation for Scale M applies the iterativeﬁltering and downsampling procedure up to Scale M and only the exponentsαM,βM andγM are given non-zero values.To simplify parameter selection,we letαj=βj=γj forall j’s.In addition,we normalize the cross-scale settings such thatMj=1γj=1.This makes different parameter settings(including all single-scale and multi-scale settings)comparable.The remain-ing job is to determine the relative values across different scales. Conceptually,this should be related to the contrast sensitivity func-tion(CSF)of the HVS[7],which states that the human visual sen-sitivity peaks at middle frequencies(around4cycles per degree of visual angle)and decreases along both high-and low-frequency directions.However,CSF cannot be directly used to derive the parameters in our system because it is typically measured at the visibility threshold level using simpliﬁed stimuli(sinusoids),but our purpose is to compare the quality of complex structured im-ages at visible distortion levels.3.2.Cross-scale calibrationWe use an image synthesis approach to calibrate the relative impor-tance of different scales.In previous work,the idea of synthesizing images for subjective testing has been employed by the“synthesis-by-analysis”methods of assessing statistical texture models,inwhich the model is used to generate a texture with statistics match-ing an original texture,and a human subject then judges the sim-ilarity of the two textures [8]–[11].A similar approach has also been qualitatively used in demonstrating quality metrics in [5],[12],though quantitative subjective tests were not conducted.These synthesis methods provide a powerful and efﬁcient means of test-ing a model,and have the added beneﬁt that the resulting images suggest improvements that might be made to the model[11].M )distortion level (MSE)12345Fig.2.Demonstration of image synthesis approach for cross-scale calibration.Images in the same row have the same MSE.Images in the same column have distortions only in one speciﬁc scale.Each subject was asked to select a set of images (one from each scale),having equal quality.As an example,one subject chose the marked images.For a given original 8bits/pixel gray scale test image,we syn-thesize a table of distorted images (as exempliﬁed by Fig.2),where each entry in the table is an image that is associated witha speciﬁc distortion level (deﬁned by MSE)and a speciﬁc scale.Each of the distorted image is created using an iterative procedure,where the initial image is generated by randomly adding white Gaussian noise to the original image and the iterative process em-ploys a constrained gradient descent algorithm to search for the worst images in terms of SSIM measure while constraining MSE to be ﬁxed and restricting the distortions to occur only in the spec-iﬁed scale.We use 5scales and 12distortion levels (range from 23to 214)in our experiment,resulting in a total of 60images,as demonstrated in Fig.2.Although the images at each row has the same MSE with respect to the original image,their visual quality is signiﬁcantly different.Thus the distortions at different scales are of very different importance in terms of perceived image quality.We employ 10original 64×64images with different types of con-tent (human faces,natural scenes,plants,man-made objects,etc.)in our experiment to create 10sets of distorted images (a total of 600distorted images).We gathered data for 8subjects,including one of the authors.The other subjects have general knowledge of human vision but did not know the detailed purpose of the study.Each subject was shown the 10sets of test images,one set at a time.The viewing dis-tance was ﬁxed to 32pixels per degree of visual angle.The subject was asked to compare the quality of the images across scales and detect one image from each of the ﬁve scales (shown as columns in Fig.2)that the subject believes having the same quality.For example,one subject chose the images marked in Fig.2to have equal quality.The positions of the selected images in each scale were recorded and averaged over all test images and all subjects.In general,the subjects agreed with each other on each image more than they agreed with themselves across different images.These test results were normalized (sum to one)and used to calculate the exponents in Eq.(7).The resulting parameters we obtained are β1=γ1=0.0448,β2=γ2=0.2856,β3=γ3=0.3001,β4=γ4=0.2363,and α5=β5=γ5=0.1333,respectively.4.TEST RESULTSWe test a number of image quality assessment algorithms using the LIVE database (available at [13]),which includes 344JPEG and JPEG2000compressed images (typically 768×512or similar size).The bit rate ranges from 0.028to 3.150bits/pixel,which allows the test images to cover a wide quality range,from in-distinguishable from the original image to highly distorted.The mean opinion score (MOS)of each image is obtained by averag-ing 13∼25subjective scores given by a group of human observers.Eight image quality assessment models are being compared,in-cluding PSNR,the Sarnoff model (JNDmetrix 8.0[14]),single-scale SSIM index with M equals 1to 5,and the proposed multi-scale SSIM index approach.The scatter plots of MOS versus model predictions are shown in Fig.3,where each point represents one test image,with its vertical and horizontal axes representing its MOS and the given objective quality score,respectively.To provide quantitative per-formance evaluation,we use the logistic function adopted in the video quality experts group (VQEG)Phase I FR-TV test [15]to provide a non-linear mapping between the objective and subjective scores.After the non-linear mapping,the linear correlation coef-ﬁcient (CC),the mean absolute error (MAE),and the root mean squared error (RMS)between the subjective and objective scores are calculated as measures of prediction accuracy .The prediction consistency is quantiﬁed using the outlier ratio (OR),which is de-Table1.Performance comparison of image quality assessment models on LIVE JPEG/JPEG2000database[13].SS-SSIM: single-scale SSIM;MS-SSIM:multi-scale SSIM;CC:non-linear regression correlation coefﬁcient;ROCC:Spearman rank-order correlation coefﬁcient;MAE:mean absolute error;RMS:root mean squared error;OR:outlier ratio.Model CC ROCC MAE RMS OR(%)PSNR0.9050.901 6.538.4515.7Sarnoff0.9560.947 4.66 5.81 3.20 SS-SSIM(M=1)0.9490.945 4.96 6.25 6.98 SS-SSIM(M=2)0.9630.959 4.21 5.38 2.62 SS-SSIM(M=3)0.9580.956 4.53 5.67 2.91 SS-SSIM(M=4)0.9480.946 4.99 6.31 5.81 SS-SSIM(M=5)0.9380.936 5.55 6.887.85 MS-SSIM0.9690.966 3.86 4.91 1.16ﬁned as the percentage of the number of predictions outside the range of±2times of the standard deviations.Finally,the predic-tion monotonicity is measured using the Spearman rank-order cor-relation coefﬁcient(ROCC).Readers can refer to[15]for a more detailed descriptions of these measures.The evaluation results for all the models being compared are given in Table1.From both the scatter plots and the quantitative evaluation re-sults,we see that the performance of single-scale SSIM model varies with scales and the best performance is given by the case of M=2.It can also be observed that the single-scale model tends to supply higher scores with the increase of scales.This is not surprising because image coding techniques such as JPEG and JPEG2000usually compressﬁne-scale details to a much higher degree than coarse-scale structures,and thus the distorted image “looks”more similar to the original image if evaluated at larger scales.Finally,for every one of the objective evaluation criteria, multi-scale SSIM model outperforms all the other models,includ-ing the best single-scale SSIM model,suggesting a meaningful balance between scales.5.DISCUSSIONSWe propose a multi-scale structural similarity approach for image quality assessment,which provides moreﬂexibility than single-scale approach in incorporating the variations of image resolution and viewing conditions.Experiments show that with an appropri-ate parameter settings,the multi-scale method outperforms the best single-scale SSIM model as well as state-of-the-art image quality metrics.In the development of top-down image quality models(such as structural similarity based algorithms),one of the most challeng-ing problems is to calibrate the model parameters,which are rather “abstract”and cannot be directly derived from simple-stimulus subjective experiments as in the bottom-up models.In this pa-per,we used an image synthesis approach to calibrate the param-eters that deﬁne the relative importance between scales.The im-provement from single-scale to multi-scale methods observed in our tests suggests the usefulness of this novel approach.However, this approach is still rather crude.We are working on developing it into a more systematic approach that can potentially be employed in a much broader range of applications.6.REFERENCES[1] A.M.Eskicioglu and P.S.Fisher,“Image quality mea-sures and their performance,”IEEE munications, vol.43,pp.2959–2965,Dec.1995.[2]T.N.Pappas and R.J.Safranek,“Perceptual criteria for im-age quality evaluation,”in Handbook of Image and Video Proc.(A.Bovik,ed.),Academic Press,2000.[3]Z.Wang and A.C.Bovik,“A universal image quality in-dex,”IEEE Signal Processing Letters,vol.9,pp.81–84,Mar.2002.[4]Z.Wang,H.R.Sheikh,and A.C.Bovik,“Objective videoquality assessment,”in The Handbook of Video Databases: Design and Applications(B.Furht and O.Marques,eds.), pp.1041–1078,CRC Press,Sept.2003.[5]Z.Wang,A.C.Bovik,H.R.Sheikh,and E.P.Simon-celli,“Image quality assessment:From error measurement to structural similarity,”IEEE Trans.Image Processing,vol.13, Jan.2004.[6]Z.Wang,L.Lu,and A.C.Bovik,“Video quality assessmentbased on structural distortion measurement,”Signal Process-ing:Image Communication,special issue on objective video quality metrics,vol.19,Jan.2004.[7] B.A.Wandell,Foundations of Vision.Sinauer Associates,Inc.,1995.[8]O.D.Faugeras and W.K.Pratt,“Decorrelation methods oftexture feature extraction,”IEEE Pat.Anal.Mach.Intell., vol.2,no.4,pp.323–332,1980.[9] A.Gagalowicz,“A new method for textureﬁelds synthesis:Some applications to the study of human vision,”IEEE Pat.Anal.Mach.Intell.,vol.3,no.5,pp.520–533,1981. [10] D.Heeger and J.Bergen,“Pyramid-based texture analy-sis/synthesis,”in Proc.ACM SIGGRAPH,pp.229–238,As-sociation for Computing Machinery,August1995.[11]J.Portilla and E.P.Simoncelli,“A parametric texture modelbased on joint statistics of complex wavelet coefﬁcients,”Int’l J Computer Vision,vol.40,pp.49–71,Dec2000. [12]P.C.Teo and D.J.Heeger,“Perceptual image distortion,”inProc.SPIE,vol.2179,pp.127–141,1994.[13]H.R.Sheikh,Z.Wang, A. C.Bovik,and L.K.Cormack,“Image and video quality assessment re-search at LIVE,”/ research/quality/.[14]Sarnoff Corporation,“JNDmetrix Technology,”http:///products_services/video_vision/jndmetrix/.[15]VQEG,“Final report from the video quality experts groupon the validation of objective models of video quality assess-ment,”Mar.2000./.PSNRM O SSarnoffM O S(a)(b)Single−scale SSIM (M=1)M O SSingle−scale SSIM (M=2)M O S(c)(d)Single−scale SSIM (M=3)M O SSingle−scale SSIM (M=4)M O S(e)(f)Single−scale SSIM (M=5)M O SMulti−scale SSIMM O S(g)(h)Fig.3.Scatter plots of MOS versus model predictions.Each sample point represents one test image in the LIVE JPEG/JPEG2000image database [13].(a)PSNR;(b)Sarnoff model;(c)-(g)single-scale SSIM method for M =1,2,3,4and 5,respectively;(h)multi-scale SSIM method.。

多原子分子的结构和性质习题解答

+ −
四面体空隙四面体空穴
八面体空隙
−
密堆积，而 P
5+
分别填入四面体空隙和八面体空隙之中，有利于密堆积结构。所以由于堆积
7
乐山师范学院化学学院
密度因素，使 P 原子改变成键方式， [ PCl4 ] 中的 P 采用 sp 杂化， [ PCl6 ] 中的 P 采用 d sp
+ 3 − 2
3
杂化。【5.10】N2H2 有两种同分异构体，是哪两种？为什么 C2H2 只有一种同分异构体？解：N2H2 分子中 N=N 为双键，不能自由旋转（因双键中 π 轨道叠加有方向性），故有顺式和反式两种异构体，它们的结构式如下：
6
八面体
4
2
Cl Cl
I
Cl
正方形
F
IF4+
F
7+4-1=10
5
三角双锥
4
1
I F F
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跷跷板型
SbF
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F
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5
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4
1
Sb F F
跷跷板型
5
乐山师范学院化学学院
F O
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8+0+2=10
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4
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6
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图 5.10
【5.11】试证明含 C、H、O、N 的有机分子，若相对分子质量为奇数。则分子中含有 N 原子数必为奇数；若相对分子含量为偶数，则含 N 原子数亦为偶数。证明：本题所涉及的是分子中各原子相互化合时的数量关系，其实质是共价键的饱和性。这些数量关系对于确定有机化合物的结构式很有用。分子中各个原子周围化学键数目的总和为偶数（因为一个键总是连接两个原子，计算时算两个键，n 重键计作 n 个单键）。由此可推得，具有奇数个单键的原子的数目之和必为偶数，即奇数价元素的原子数之和必为偶数。在含 C，H，O，N 的有机物分子中，C 和 O

Structural_Chemistry_Chapter_1-3

ν=
En j − Eni h
h h 2 2 (r + 3) − (r + 2) = 2 (2r + 5) = 2 8ml 8ml
[
]
第一章量子力学基础
8ml 2 c 3.30l 2 pm λ= = = ν h(2r + 5) 2r + 5 c
势箱总长：l =（248r+565）pm
3.30(248r + 565) λ= pm 2r + 5
h 2 d 2ψ − 2 = Eψ 2 8π m dx
d 2ψ 8π 2 mE + ψ =0 2 2 dx h
− 二阶常系数线性齐次方程，通解为：
⎡ ⎛ 8π 2 mE ⎞ 1 2 ⎤ ⎡⎛ 8π 2 mE ⎞ 1 2 ⎤ ψ ( x ) = c1 cos ⎢ ⎜ ⎟ x⎥ + c2 sin ⎢⎜ ⎟ x⎥ 2 2 ⎢⎝ h ⎢⎝ h ⎥ ⎠ ⎥ ⎠ ⎣ ⎦ ⎣ ⎦
第一章量子力学基础
− 根据品优函数的连续性和单值性以及边界条件：当x=0时， ψ (0) = c1 cos(0) + c2 sin (0) = 0
∴c1 = 0
⎡⎛ 8π 2 mE ⎞ 12 ⎤ ⎡⎛ 8π 2 mE ⎞ 12 ⎤ ⎥ ⎢ ⎥ 当x=l 时， ψ (l ) = 0 ⋅ cos⎢⎜ ⎜ h 2 ⎟ ⋅ l ⎥ + c2 sin ⎢⎜ h 2 ⎟ l ⎥ = 0 ⎜ ⎟ ⎟ ⎢⎝ ⎠ ⎠ ⎦ ⎣ ⎦ ⎣⎝
⎛ ∂2 ∂2 ∂2 ⎞ − ⎜ 2 + 2 + 2 ⎟ψ ( x, y, z ) = Eψ ( x, y, z ) 2m ⎝ ∂x ∂y ∂z ⎠
2
第一章量子力学基础

英文版原子物理课件

Shanxi University Atomic Physics
1.1 Introduction
The origins of atomic physics :quantum mechanics Bohr model of the H This introductory chapter surveys some of the early ideas: Spectrum of atomic H and Bohr Theory Einstein's treatment of interaction of atom with light the Zeeman effect Rutherford scattering And so on
Shanxi University Atomic Physics
1.2 Spectrum of atomic hydrogen_3
Wavenumbers may seem rather old-fashioned but they are very useful in atomic physics
the characteristic spectrum for atoms is composed of discrete lines that are the ‘fingerprint' of the element.
In 1888, the Swedish professor J. Rydberg found that the spectral lines in hydrogen obey the following mathematical formula:
Shanxi University Atomic Physics
Lyman series: n’ = 2; 3; 4; … n = 1. Balmer (n = 2), Paschen series: (n = 3), Brackett (n = 4) and Pfund (n = 5)

克莱森重排综述

Claisen Rearrangement over the Past Nine DecadesAna M.Martı´n Castro*Departamento de Quı´mica Orga´nica,Universidad Auto´noma de Madrid,Cantoblanco,28049Madrid,SpainReceived February18,2003Contents1.Introduction29392.Definition and Historic Overview of the ClaisenRearrangement29403.Related[3,3]Sigmatropic Rearrangements29413.1.Carroll Rearrangement29413.2.Eschenmoser Rearrangement29413.3.Johnson Rearrangement29423.4.Ireland−Claisen Rearrangement29423.5.Reformatsky−Claisen Rearrangement29423.6.Thio−Claisen Rearrangement29433.7.Aza−Claisen Rearrangement29433.8.Chelate Claisen Rearrangement29433.9.Diosphenol−Claisen Rearrangement29443.10.Metallo−Claisen Rearrangement29443.11.Retro-Claisen Rearrangement29444.Mechanistic and Kinetic Aspects29454.1.General Remarks29454.2.Factors Affecting the Reaction Rate29454.2.1.Influence of the Substituents29454.2.2.Influence of Charged Intermediates29474.2.3.Catalyzed Claisen Rearrangements29494.2.4.Other Parameters29545.Enzymatic Claisen Rearrangement29566.Stereoselective Claisen Rearrangement29566.1.General Aspects29566.2.Intraannular Diastereoselectivity29566.2.1.Transition-State Geometry29566.2.2.Vinyl Double-Bond Geometry29586.2.3.Allyl Double-Bond Geometry29606.2.4.Configuration at C-429616.3.Diastereoselective Synthesis of AchiralProducts29636.3.1.Diastereoselective Synthesis ofCycloalkane Derivatives29636.3.2.E/Z Selectivity29636.4.Diastereoselective Synthesis of ChiralProducts29636.4.1.Chiral Auxiliary at the Allyl Fragment29646.4.2.Chiral Auxiliary at the Vinyl Fragment29656.5.Enantioselective Claisen Rearrangement29686.5.1.Chiral Catalysts29686.5.2.Chiral Solvents29727.Application of Claisen Rearrangement Productsto the Synthesis of Organic Building Blocks29737.1.Heterocyclic Compounds29737.2.Carbocyclic Skeletons29807.3.Dienes29827.4.Condensed Aromatic Structures29847.5.Carboxylic Acid Derivatives29847.6.Quaternary Carbons29887.7.Polysubstituted Alkenes29897.8.Sugar Derivatives29898.Application of Claisen Rearrangement to theSynthesis of Natural Products29909.Other Applications299810.Conclusion299811.Acknowledgments299912.References2999 1.IntroductionThe discovery of the Claisen rearrangement almost a century ago1offered a potentially useful synthetic tool to the organic chemist.Over the decades this usefulness has been realized and the reaction has drawn the attention of numerous research groups, which has been reflected in the number of papers on the topic published in the literature.2In the1970s and1980s several general reviews appeared on the title reaction or related processes.2a-g However,in recent years only specific issues related to this type of rearrangement have been addressed,2k-m the stud-ies on the stereochemical aspects of the reaction deserving special mention.This review provides a general overview covering the most relevant topics related to the Claisen rearrangement,starting from its first publication by Ludwig Claisen in1912as a new[3,3]reorganization of allyl aryl(or vinyl)ethers up to its most recent applications in different organic chemistry fields.First,a brief description of the reaction along with its historic profile are given.This leads to the presentation of other[3,3]rearrange-ments closely related to the title reaction,which are of relevant interest for having been largely exploited as synthetic methods.Next,mechanistic and kinetic aspects are discussed with attention focused on the main factors affecting the reaction rate,basically the presence of different substituents at the1,5-hetero-diene skeleton,the use of catalysts,and changes in the physical parameters affecting the reaction.The following section in the review briefly deals with the enzymatic version of the Claisen rearrangement, which is of relevant interest in metabolic routes.A significant section of the review is constituted by the study of the stereoselective version of the rearrangement.After a presentation of some general*E-mail:martin.castro@uam.es.2939 Chem.Rev.2004,104,2939−300210.1021/cr020703u CCC:$48.50©2004American Chemical SocietyPublished on Web04/23/2004aspects regarding chirality transfer in these pro-cesses,different strategies to control intraannular diastereoselectivity as well as methods to perform diastereoselective synthesis of both achiral and chiral compounds are examined.The enantioselective Clais-en rearrangement is also thoroughly considered.Finally,in the last sections of the review a selection of the most outstanding applications of the reaction is presented.A number of examples illustrating the use of the Claisen rearrangement in the preparation of a wide range of synthetically interesting building blocks and natural or biologically active compounds are considered.Some other potentially interesting applications of the rearrangement in further fields of organic chemistry are also presented.2.Definition and Historic Overview of the Claisen RearrangementThe [3,3]sigmatropic rearrangement of allyl vinyl ethers,which allows the preparation of γ,δ-unsatur-ated carbonyl compounds,is worthy of study due to its special synthetic relevance as well as the large number of theoretical studies generated.This reac-tion,first reported by Ludwig Claisen in 1912,1was originally described as “the thermal isomerization of an allyl vinyl ether 1s or of its nitrogen or sulfur containing analogue derivatives s to afford a bifunc-tionalized molecule 2”(Scheme 1)in a [π2s +σ2s +π2s]process.This first paper essentially described the transfor-mation of allyl phenyl ether into C -allyl phenol.However,it also dealt with the formation,startingfrom O -allylated ethyl acetoacetate (3),of its C -allyl isomer 4after distillation in the presence of NH 4Cl (Scheme 2)in a process which adopted the generaldenomination of Claisen rearrangement .As we shall see,this is a reaction exhibiting all the essential properties required by a synthetic procedure to be considered as efficient:it can be chemo-,regio-,diastereo-,and enantioselective,3can be performed under mild conditions,and affords potentially useful polyfunctionalized molecules.The synthetic potential of the process encouraged,in the following years after its first publication,a number of research groups to make significant efforts to find the experimental conditions which would allow the generalization of the reaction to a wide variety of substrates.To verify that the conditions initially reported by Claisen to perform the rear-rangement on aromatic substrates 4could be success-fully applied to aliphatic skeletons,independently Bergman and Corte in 19355a and Lauer and Kilburn in 19375b studied the rearrangement in the presence of NH 4Cl of ethyl cinnamyl oxycrotonate (5).This substrate was generated either by reaction of cin-namyl alcohol and ethyl 3-ethoxy-2-crotonate 5a or from sodium cinnamylate and ethyl -chlorocrotonate 5b (Scheme 3).The formation of the rearranged productafforded a formal way of S N 2′C -alkylation of cin-namyl halides with the anion derived from aceto-acetates.The interest of this new rearrangement prompted the development of different methods for the prepa-ration of the starting materials.Hurd and Pollack 6described the synthesis of allyl vinyl ethers by acidic or basic elimination as well as the rearrangement of such compounds into the corresponding γ,δ-unsatur-ated carbonyl compounds (Scheme 4).However,this method did not provide general access to allyl vinylethers.Ana M.Martin Castro was born in Madrid,Spain,in 1964.She received her B.S.(1988)and Ph.D.(1994)degrees under the supervision of Professors J.L.Garcia Ruano and J.H.Rodriguez Ramos at the Universidad Auto ´noma of Madrid.As a postdoctoral fellow she joined the groups of Professor P.R.Raithby at the University of Cambridge (U.K.,1997)and Professor V.K.Aggarwal at the University of Sheffield (U.K.,1998−2000).She retuned to Professor Garcia Ruano’s group as an Assistant Professor.Her present research interests include the develop-ment of novel asymmetric methodologies assisted by a chiral sulfinyl group,particularly hydrocyanation processes and Diels −Alder and 1,3-dipolar cycloadditions.Scheme1Scheme2Scheme3Scheme42940Chemical Reviews,2004,Vol.104,No.6Martı´n CastroSeveral years later this procedure for the synthesis of allyl vinyl ethers was improved by the interchange of alcohols with alkyl vinyl ethers catalyzed by Hg(OAc)2.7Those compounds once again proved to be excellent substrates to undergo a [3,3]rearrange-ment (Scheme 5).This mercury-catalyzed reactionhas become one of the typical methods of preparation of allyl vinyl ethers despite that the yields of these reactions are often low.The development of the aliphatic Claisen rear-rangement was simultaneous with the study of the aromatic version of the reaction.2a,b,8Thus,in the Claisen rearrangement of an allyl aryl ether,the first [3,3]step affords an ortho dienone which usually enolizes into an o -allylphenol.It is the reaction known as the ortho Claisen rearrangement (Scheme 6).When the rearrangement takes place on an orthoposition bearing a substituent,a second [3,3]rear-rangement (Cope rearrangement)takes place fol-lowed by enolization.This reaction,usually called the para Claisen rearrangement ,leads to the correspond-ing p -allylphenol.The product resulting from the ortho Claisen rear-rangement is usually obtained from the reaction,although the para process can compete even when both ortho positions are unoccupied.2b The proposed mechanism for the aromatic Claisen rearrangement has been corroborated by the product resulting from the rearrangements of allyl phenyl ether and allyl 2,6-dimethylphenyl ether,both compounds 14C-labeled on the γcarbon of the allyl chain 8a (see Scheme 6).The result of the ortho rearrangement shows that the rearrangement with inversion at the allyl group is the only reaction taking place.In the case of the para rearrangement,the migration only proceeds without inversion of the allyl group.This means that during the course of the reaction such a group is never free enough to undergo resonance.8b3.Related [3,3]Sigmatropic RearrangementsThe interest generated by the Claisen rearrange-ment prompted the development of a considerable number of different versions of [3,3]sigmatropicrearrangement.Some of the most noteworthy ones will be considered next.3.1.Carroll RearrangementThe Carroll reaction,initially described in 1940,9is a thermal rearrangement of allylic -ketoesters followed by decarboxylation 10to yield γ,δ-unsaturated ketones (Scheme 7).This reaction has not beenwidely developed due to the drastic conditions (tem-peratures of 130-220°C after in situ preparation of the -ketoester)which are required to perform the transformation.After the publication of these results,it was re-ported that dianions derived from allylic acetoace-tates,prepared by treatment of acetoacetates with 2equiv of LDA,rearranged under milder thermal con-ditions to give easily isolated -keto acids (Scheme 8).11A dependence of the reaction rate on the substitu-tion pattern on the allylic fragment (R,R ′)H,alkyl,aryl)has also been detected.Thus,acetoacetates derived from primary alcohols rearrange more slowly than those derived from secondary and tertiary alcohols.3.2.Eschenmoser RearrangementIn 1964Eschenmoser,12based on observations previously reported by Meerwein 13on the interchange of amide acetals with allylic alcohols,described the [3,3]rearrangement of N,O -ketene acetals to yield γ,δ-unsaturated amides (Scheme 9).Scheme5Scheme6Scheme7Scheme8Scheme9Claisen Rearrangement over the Past Nine Decades Chemical Reviews,2004,Vol.104,No.62941This reaction allows the formation of a carbon -carbon bond at the position to a nitrogen atom,which is of great applicability in alkaloid synthesis,although it has the inconvenience of the difficulty inherent to the preparation of more elaborated N,O -ketene acetals,which usually requires elevated tem-peratures leading,in some cases,to decomposition of the resulting amides.Several years later an Eschenmoser rearrangement by reaction of lithium allyl alkoxides with acyclic 14and cyclic 15salts of N,N -dialkylalkoxymethylene-iminium was reported to proceed in excellent yields (Scheme 10).The high temperatures reported for theEschenmoser rearrangement are usually required for the alcohol exchange reaction,not for the actual rearrangement.Therefore,the mild conditions em-ployed for the preparation of N,O -ketene acetals such as those depicted in Scheme 10increased the syn-thetic interest of the method.Similar results are obtained by the ynamine -Claisen rearrangement,16also known as Ficini -Claisen rearrangement,by reaction of an allylic alco-hol with 1-(diethylamino)propyne (Scheme 11).Thisis a process whose stereochemical course may be modified by the reaction conditions,as discussed in section 6.2.2.When the N,O -ketene acetal is obtained by adding the alcohol slowly to a refluxing solution of the ynamine in xylene,the rearrangement takes place via the kinetically formed (E )-isomer.In the presence of a Lewis acid,equilibration to the ther-modynamically favored (Z )-stereoisomer occurs before rearrangement.Transformation of the kinetically favored (E )-N,O -ketene acetal to the threo γ,δ-un-saturated amide can be considered as complementary to the Eschenmoser rearrangement,which evolves through the (Z )-isomer affording the erythro product.3.3.Johnson RearrangementFirst reported in 1970,17the Johnson rearrange-ment,which may afford trans -trisubstituted alkenes,was originally described as the process consisting of the heating of an allylic alcohol (6)with an excess of ethyl orthoacetate in the presence of trace amounts of a weak acid (typically propionic acid).The initially formed mixed ortho ester (7)loses ethanol to generate the ketene acetal 8,which undergoes rearrangement leading to a γ,δ-unsaturated ester (9)(Scheme 12).Subsequently the Claisen rearrangement of ortho esters was shown to be compatible with the presence of a heteroatomic substituent directly bonded to the allyl vinyl ether moiety.One of the few reported ex-amples of Johnson rearrangement with heteroatomic substitution (OCH 3)is the reaction of allylic alcohols with methyl methoxyorthoacetate that gives methyl R -methoxy-γ,δ-unsaturated esters in a process that occurs under acidic conditions (Scheme 13).183.4.Ireland −Claisen RearrangementIn 1972Ireland reported the rearrangement of allyl trimethylsilyl ketene acetals,19prepared by reaction of allylic ester enolates with trimethylsilyl chloride,to yield γ,δ-unsaturated carboxylic acids (Scheme 14).As compared with other reported rearrangements,this reaction proceeds under mild basic or neutral conditions.These conditions have allowed the preparation of polyfunctionalized structures,such as the vinylstan-nanes represented in Scheme 15.20This exampleprovides a method of functionalizing the newly formed double bond due to the high synthetic versa-tility of organotin compounds.3.5.Reformatsky −Claisen RearrangementWe have so far seen that allylic ester enolates rearrange quite easily.[3,3]Sigmatropic rearrange-Scheme10Scheme11Scheme12Scheme13Scheme14Scheme152942Chemical Reviews,2004,Vol.104,No.6Martı´n Castroment of zinc enolates,known as the Reformatsky -Claisen rearrangement,has also been reported.21These zinc enolates (10),generated by Reformatsky reaction of R -haloesters (11)with zinc dust,lead to the corresponding γ,δ-unsaturated zinc carboxylates (12)(Scheme 16)in good yields under neither acidic nor basic conditions.Similar reaction conditions,in the presence of trimethylsilyl chloride,allowed the synthesis of 2,2-difluoro-4-pentenoic acid starting from allyl chlorodi-fluoroacetate 22(Scheme 17).This silicon-inducedReformatsky -Claisen reaction did not occur in the absence of chlorotrimethylsilane.This indicates that the ketene acetal depicted in Scheme 17is most likely a reaction intermediate.3.6.Thio −Claisen RearrangementThermolysis of allyl phenyl sulfides (13)23leading to a [3,3]sigmatropic rearrangement contrasts with the classic Claisen rearrangement:it requires higher temperature to produce the corresponding thiols,intermediates which are not easily isolated and usually evolve into the corresponding diallyl deriva-tives due to a S N 2displacement by the intermediate thiolate on the starting sulfide (Scheme 18).24In contrast,the aliphatic version of the thio -Claisen rearrangement may proceed under milder conditions than those reported for oxygenated sub-strates (Scheme 19).25Nevertheless,the low applicability of this method-ology is a consequence of the instability of the prod-ucts.This prompted the development of conditions to trap and transform them into more stable com-pounds such as,for example,the hydrolysis of the intermediate thioaldehyde into the corresponding aldehyde (Scheme 20).263.7.Aza −Claisen RearrangementThe [3,3]sigmatropic rearrangement of N -allyl-N -arylamines,known as the aza -Claisen rearrange-ment (Scheme 21),27usually requires more drasticconditions than those required for the classic Claisen rearrangement of oxygenated substrates (this rear-rangement occurs at 200-350°C).In addition,it affords the corresponding anilines along with undes-ired byproducts.Similar energetic conditions are needed for the aliphatic aza -Claisen rearrangement to take place (Scheme 22).2a The thermal process requires highertemperatures than those needed for oxygen sub-strates.In a number of cases the reaction only evolves under Lewis-acid catalysis.3.8.Chelate Claisen RearrangementChelated enolates derived from amino acid esters undergo Claisen rearrangement upon standing at room temperature to produce γ,δ-unsaturated amino acids (Scheme 23).28Starting from E -allylic esters,syn products are obtained in a diastereoselective fashion.These reaction conditions are based on the fact that,in general,chelation sharply increases the thermal stability with no negative influence on the reactivity of those enolates.As a consequence of the rigid geometry of the enolate and the predictable geometry of the transition state,any transformationScheme20Scheme21Scheme22Scheme23Scheme16Scheme17Scheme18Scheme19Claisen Rearrangement over the Past Nine Decades Chemical Reviews,2004,Vol.104,No.62943will tend to take place with a very high stereoselec-tivity (see section 6.2.2).3.9.Diosphenol −Claisen RearrangementThis variety of Claisen rearrangement uses allyl ethers (14)derived from diosphenol,with an endocy-clic vinyl double bond,to give rise to a bond between a functionalized carbon moiety and a sterically hindered carbon which is part of a cyclic structure (Scheme 24).29The resulting bisketone usually tau-tomerizes into the ketoenol derivative.3.10.Metallo −Claisen RearrangementSeveral studies focused on the development of synthetic applications of gem -dimetallic compounds,30which were prepared by carbometalation of an al-kenyl organometallic magnesium,lithium,or alumi-num derivative with an allyl zinc bromide.The initially accepted pathway for the carbometalation consists first in the formation of an allyl vinyl zinc compound (15)(Scheme 25),which next undergoes a[3,3]rearrangement s a process known as the met-allo -Claisen rearrangement 30a s to afford the 1,1-bimetallic species 16.However,two mechanistic rationales,a metallo -ene reaction and a metallo -Claisen rearrangement,account for the resulting products as shown in Scheme 25.Density functional (B3LYP)studies on this reaction have demonstrated that the process is an endothermic Lewis-acid-as-sisted metallo -Claisen rearrangement with some character of metallo -ene reaction of the vinylmag-nesium species (MX n )MgCl in Scheme 25).31The high diastereoselectivity of the reaction has been explained by the short length of the forming C -C bond in the late transition state of the metallo -Claisen process.These organic gem -dimetallic compounds are able to react successively with two different electrophilesto produce various gem -difunctionalized structures 30b -d(Scheme 26).Several years later this methodology was expanded to the rearrangement of allyl allenyl derivatives to give access to gem -dimetalated dienes 30e,f (Scheme 27).As mentioned,recently a series of theoretical cal-culations has supported a mechanism in which the reaction of an allyl zinc bromide with a vinylmagne-sium bromide initially proceeds through a fast trans-metalation process to generate an allyl vinyl zinc intermediate A ,which undergoes a MgBr 2-assisted metallo -Claisen rearrangement through transition state B ,which generates the 1,1-dimetallic species C .The reaction product D results from a final oligomerization step 31(Scheme 28).An equilibriummixture of (E/Z )-allyl zinc bromide affords a single diastereomer resulting from reaction of the minor (Z )-allyl isomer.This result is explained by comparison of the relative energies of the diastereomeric transi-tion states since the pathway through the (Z )-isomer is favored over that evolving through the (E )reagent.3.11.Retro-Claisen RearrangementThe Claisen rearrangement,as in any other [3,3]sigmatropic rearrangement,takes place under ther-modynamic control.This reaction is irreversible toward the formation of the carbonyl compounds (Scheme 29)due to their higher thermodynamic stability.Scheme26Scheme27Scheme28Scheme29Scheme24Scheme252944Chemical Reviews,2004,Vol.104,No.6Martı´n CastroHowever,some structural features have been iden-tified as being responsible for inversion of the normal situation,favoring the transformation of the carbonyl compound into the vinyl ether.In this sense,both the presence of any substituent at a bridgehead position and vicinal quaternary carbons in the car-bonyl compound shifts the equilibrium toward the retro-Claisen isomer as a result of a relief in torsional strain (Scheme 30).32This effect is particularlymarked in the presence of a catalytic amount of Lewis acid (BF 3.OEt 2).This retro-Claisen process is general for a number of substrates containing contiguous quaternary centers whenever the R -carbonyl sub-stituent is not an electron-releasing group.A similar thermolability has been detected for vinylcyclopropane carboxaldehydes,which evolve into 2,5-dihydrooxepines by retro-Claisen rearrangement,as indicated in Scheme 31a.In this example thethermodynamic stability of the carbonyl group,which is favored at equilibrium,compensates for the un-stabilizing strain of the cyclopropane ring.33a The ready retro-Claisen reaction of this type of vinylcy-clopropanes is evidenced by a number of examples reported in the literature (see,for example,Scheme 31b 33b ).4.Mechanistic and Kinetic Aspects 4.1.General RemarksThe term “Claisen rearrangement”was originally applied to rearrangements of allyl aryl ethers to afford ortho-and occasionally para -substituted phe-nols.Afterward it expanded to analogue rearrange-ments of allyl vinyl ethers into unsaturated carbonyl compounds,which were classified as [3,3]sigmatropic rearrangements.34Initially a synchronic evolution for these reactions through aromatic transition states was accepted,35,36formed by a combination of σand πoverlap of 2p atomic orbitals of the carbon atoms of both allyl fragments.It was concluded that,out of the two feasible geometries for the transition state,the reaction proceeded through chairlike intermedi-ates (17)instead of boatlike intermediates (18)(Figure 1).37Both transition states (17and 18)arethe only ones corresponding to supra-supra processes and,therefore,allowed by Woodward -Hoffmann rules.34The intramolecular cyclic character of the rear-rangement is generally accepted.However,the re-search to understand the precise nature and the geometry of the transition state continues.A large number of theoretical calculations aiming to pre-dict the structures of the transition states involved in the Claisen rearrangement have been reported.38-41Most of them accept a concerted rearrangement through a chairlike transition state.However,re-cently Houk used quantum mechanical calculations to rationalize the stereoselectivity of the Ireland -Claisen rearrangement of cyclohexenyl silyl enol ethers from the chair or boat preferences in the transition state which derived from the substituents on the cyclohexenyl ring.38c In addition,there is no general agreement about the structure of this transi-tion state (Figure 2).42Over the decades several experimental studies based on kinetic isotopic effects have been reported in order to determine the geometry of the transition state of aliphatic and aromatic Claisen rearrange-ments.42-44However,this has not proved to be an easy task,and despite the numerous papers dealing with the Claisen rearrangement in different fields related to organic chemistry,there is no general agreement about such a geometry from theoretical predictions.The difficulty of describing such a ge-ometry still persists.4.2.Factors Affecting the Reaction RateThe most frequently reported Claisen rearrange-ment s thermal isomerization of allyl vinyl ethers s is a process that requires high temperatures and proceeds quite slowly at atmospheric pressure.To transform this reaction into a synthetically useful procedure,numerous attempts to find milder experi-mental conditions have been reported.The introduc-tion of different substituents in the carbon skeleton of the substrate as well as variations of the catalyst are worth mentioning.4.2.1.Influence of the SubstituentsIn the last 20years a considerable number of studies to determine the inductive or mesomeric effects of electron-withdrawing or electron-donating substituents located at different positions of the carbon skeleton have been mentioned.These effects are qualitatively described in Scheme 32.Carpenter studied the effect of the cyano group at different positions of allyl vinyl ethers.Such an effect was interpreted as basically electronic.45In the case of substrates substituted at positions C-2(k rel 111),C-4(k rel 270),and C-5(k rel 15.6),an acceleration of the rearrangement was detected,whereas substitu-Scheme30Scheme31Figure1.Figure 2.Claisen Rearrangement over the Past Nine Decades Chemical Reviews,2004,Vol.104,No.62945tion at C-1and C-6resulted in a decrease in the reaction rate.These observations were rationalized from Hu ¨ckel molecular orbital (HMO)theory,which allowed evaluation of the effect of a substituent in the transition state and in the ground state.45b The comparison of the difference of πenergy of HMO (∆E π)between the ground state and the transition state with the value of ∆E πfor the unsubstituted analogue compound allowed them to predict the sign and magnitude of the effect of the substituent in the activation enthalpy of the reaction.In this model the electron-withdrawing and -donating substituents are represented as carbocations and carbanions,respec-tively.From the delocalized model of the transition state,some qualitative predictions about the effects of the substituents in the Claisen rearrangement could be made.The main inconvenience of this model is that the cyano group is not only an electron-withdrawing group,but also a radical-stabilizing group,so that the acceleration resulting from the presence of a cyano group at C-2and C-4may not be a consequence of its electron-withdrawing character.To differentiate the inductive electron-withdrawing character and the result of a combination of inductive and mesomeric electron-withdrawing effects,the behavior of allyl vinyl ethers bearing a trifluorom-ethyl group at C-2and C-4was studied.46A CF 3group is an electron-withdrawing substituent with an inductive character but with no mesomeric one;it is not able to stabilize radicals.Hence,the Claisen rearrangement of allyl vinyl ethers bearing a CF 3group at C-2suffered an accelerating factor of 73in relation with the unsubstituted substrate,in com-parison with the value of k rel 111observed for cyano derivatives at C-2.45a In its turn,a CF 3group at C-4exerted no influence in the reaction rate.These results allowed Gajewski 46to suggest that the electron-withdrawing character of the substituent at C-4is not that responsible for the increase in the rate but its ability to stabilize radicals,which is reflected in the stabilization of the transition state.Similarly,quite recently it has been reported that a CF 3groupat C-1does not modify the reaction rate,whereas the effect of a fluorine atom at the same position will depend on the influence of an alkyl substituent R at C-2(Figure 3).47Different theoretical models predicting the effects of several substituents in the Claisen rearrangement rate have been proposed.The model suggested by Gajewski 48assumes that the structure of the transi-tion state adopts the features of the substrate or product depending on the exothermic properties of the reaction.In addition,it will have an associative or dissociative character according to the way that the substituents can stabilize such a character.Also,recently some theoretical calculations on the effects of cyano,amino,and trifluoromethyl substituents on the rate,whose results are coincident with those attained from experimental studies,have been re-ported.49The effect of alkoxy groups has been largely studied by Curran.50,51An electron-donating substituent (alkoxy group)at C-6sharply accelerates the Claisen rearrangement.50This effect seems to contradict Carpenter’s model,45which predicts a deceleration in the presence of a donating substituent at C-6,despite the loss of resonance energy from the ground state to the transition state.To compensate for this effect,the model proposes a πf σ*stabilization by a “vinylogous anomeric effect”of the O 3-C 4bond of the vinyl ether as responsible for the Claisen rearrange-ment acceleration by donating substituents at C-6.The transition state of the Claisen rearrangement (eq 2in Scheme 33)can be understood in a similar wayto the “double bond -no bond”resonance explaining the vinylogous anomeric effect (eq 1in Scheme 33).The process goes through an early transition state where the bond breaking is more advanced than the bond formation.45As can be seen,an oxygenated substituent at C-6must decrease the energy of the transition state s and,therefore,accelerate the reac-tion s as it makes the breaking of the weakened O 3-C 4bond easier.A similar acceleration was detected with the pres-ence of an alkoxy group at C-4.In addition,the rates of the rearrangements of 4-and 6-alkoxyallyl enol ethers were quite sensitive to the solvent polarity and considerably increased in hydrogen bonding sol-vents 51s it could not be detected from unsubstituted substrates.These results were attributed to anScheme32Figure 3.Scheme332946Chemical Reviews,2004,Vol.104,No.6Martı´n Castro。

Structural_Change_in_a_Multi-Sector_Model_of_Growth

Structural Change in a Multi-Sector Modelof GrowthL Rachel Ngai Christopher A Pissarides∗November2004(this revision,May2006,forthcoming in theAmerican Economic Review)AbstractWe study a multi-sector model of growth with diﬀerences in TFP growth rates across sectors and derive suﬃcient conditions for thecoexistence of structural change,characterized by sectoral labor re-allocation,and balanced aggregate growth.The conditions are weakrestrictions on the utility and production functions.Along the bal-anced growth path,labor employed in the production of consump-tion goods gradually moves to the sector with the lowest TFP growthrate,until in the limit it is the only sector with nontrivial employ-ment of this kind.The employment shares of intermediate and cap-ital goods remain constant during the reallocation process.(JELO41,O14,E29)Economic growth takes place at uneven rates across diﬀerent sectors of the economy.This paper has two objectives related to this fact:(a)to derive the implications of diﬀerent sectoral total factor productivity(TFP) growth rates for structural change,the name given to the shifts in industrial∗Ngai:Centre for Economic Performance,London School of Economics and CEPR (e-mail:l.ngai@).Pissarides:Centre for Economic Performance,London School of Economics,CEPR and IZA(e-mail:c.pissarides@).We have beneﬁted from comments received at several presentations(the CEPR ESSIM2004meetings,the SED 2004annual conference,the NBER2004Summer Institute,the2004Canadian Macro-economic Study Group and several universities),and from Fernando Alvarez,Francesco Caselli,Antonio Ciccone,Nobu Kiyotaki,Robert Lucas,Nick Oulton,Danny Quah, Sergio Rebelo,Robert Shimer,Nancy Stokey,Richard Rogerson,Jaume Ventura and two anonymous referees.Funding from the CEP,a designated ESRC Research Centre, is acknowledged.employment shares that take place over long periods of time,and(b)to show that even with ongoing structural change,the economy’s aggregate ratios can be constant.We refer to the latter as aggregate balanced growth. The restrictions needed to yield structural change consistent with the facts and aggregate balanced growth are weak restrictions on functional forms that are frequently imposed by macroeconomists in related contexts.We obtain our results in a baseline model of many consumption goods and a single capital good,supplied by a sector that we label manufacturing. Our baseline results are consistent with the existence of intermediate goods and many capital goods under some reasonable restrictions.Production functions in our model are identical in all sectors except for their rates of TFP growth and each sector produces a diﬀerentiated good that enters a constant elasticity of substitution(CES)utility function.We show that a low(below one)elasticity of substitution acrossﬁnal goods leads to shifts of employment shares to sectors with low TFP growth.In the limit the employment share used to produce consumption goods vanishes from all sectors except for the one with the smallest TFP growth rate,but the employment shares used to produce capital goods and intermediate goods converge to non-trivial stationary values.If the utility function in addition has unit inter-temporal elasticity of substitution,during structural change the aggregate capital-output ratio is constant and the aggregate economy is on a balanced growth path.Our results contrast with the results of Cristina Echevarria(1997),John Laitner(2000),Francesco Caselli and Wilbur Coleman II(2001)and Dou-glas Gollin,Stephen Parente and Richard Rogerson(2002)who derived structural change in a two-or three-sector economy with non-homothetic preferences.Our results also contrast with the results of Piyabha Kongsamut, Sergio Rebelo and Danyang Xie(2001)and Reto Foellmi and Josef Zweimuller (2005),who derived simultaneous constant aggregate growth and struc-tural change.Kongsamut et al.(2001)obtain their results by imposing a restriction that maps some of the parameters of their Stone-Geary utility function onto the parameters of the production functions,abandoning one of the most useful conventions of modern macroeconomics,the complete in-dependence of preferences and technologies.Foellmi and Zweimuller(2005) obtain their results by assuming endogenous growth driven by the intro-duction of new goods into a hierarchic utility function.Our restrictions are quantitative restrictions on a conventional CES utility function that main-tains the independence of the parameters of preferences and technologies.Our results conﬁrm William J Baumol’s(1967)claims about structural change.Baumol divided the economy into two sectors,a“progressive”one that uses new technology and a“stagnant”one that uses labor as the only input.He then claimed that the production costs and prices of the stagnant sector should rise indeﬁnitely,a process known as“Baumol’s cost disease,”and labor should move in the direction of the stagnant sector.1 In the more recent empirical literature two competing explanations (which can coexist)have been put forward for structural change.Our ex-planation,which is sometimes termed“technological”because it attributes structural change to diﬀerent rates of sectoral TFP growth,and a utility-based explanation,which requires diﬀerent income elasticities for diﬀerent goods and can yield structural change even with equal TFP growth in all sectors.Baumol,Sue Anne Batey Blackman and Edward N.Wolﬀ(1985) provide empirical evidence at the2-digit industry level,consistent with our model.Irving B.Kravis,Alan W.Heston and Robert Summers(1983)also present evidence that favors the technological explanation,at least when the comparison is between manufacturing and services.Two features of their data that are satisﬁed by the technological explanation proposed in this paper are(a)relative prices reﬂect diﬀerences in TFP growth rates and(b)real consumption shares vary a lot less over time than nominal consumption shares.2Our model is also consistent with the observed pos-itive correlation between employment growth and relative price inﬂation across two-digit sectors3and with historical OECD evidence presented by Simon Kuznets(1966)and Angus Maddison(1980)for one-digit sectors.41Baumol controversially also claimed that as more weight is shifted to the stagnant sector,the economy’s growth rate will be on a declining trend and eventually converge to zero.This claim contrasts with ourﬁnding that the economy is on a balanced-growth path.We get our result because we include capital in our analysis,ironically left out of the analysis by Baumol(1967,p.417)“primarily for ease of exposition...that is [in]essential to the argument”.2See Rodney E.Falvey and Norman Gemmell(1996)for an update of some of their results.Falvey and Gemmellﬁnd a unit income elasticity and a small(negative)price elasticity for services in a cross-section of countries,consistent with our results.3These correlations are shown in the working paper version of this paper,L.Rachel Ngai and Christopher A.Pissarides(2004).4Kuznets(1966)documented structural change for13OECD countries and the USSR between1800and1960and Maddison(1980)documented the same pattern for16OECD countries from1870to1987.They both found a pattern with the same general features as the predictions that we obtain when the ranking of the average historical TFP growth rates is agriculture followed by manufacturing followed by services.Section1describes our model of growth with many sectors and sec-tions2and3respectively derive the conditions for structural change and balanced aggregate growth.In sections4and5we study two extensions of our baseline model,one where consumption goods can also be used as intermediate inputs and one where there are many capital goods.The Appendix discusses the implications of one more extension,diﬀerences in capital intensities across sectors,and contains proofs of the main results.1An economy with many sectorsThe baseline economy consists of an arbitrary number of m sectors.Sectors i=1,...,m−1produce only consumption goods.The last sector,which is denoted by m and labeled manufacturing,produces both aﬁnal consump-tion good and the economy’s capital stock.We derive the equilibrium as the solution to a social planning problem.The objective function isU=Z∞0e−ρt v(c1,..,c m)dt,(1) whereρ>0,c i≥0are per-capita consumption levels and the instanta-neous utility function v(.)is concave and satisﬁes the Inada conditions. The constraints of the problem are as follows.The labor force is exogenous and growing at rateνand the aggregate capital stock is endogenous and deﬁnes the state of the economy.Sectoral allocations are controls that satisfyP m i=1n i=1;P m i=1n i k i=k,(2)where n i≥0is the employment share and k i≥0is the capital-labor ratio in sector i,and k≥0is the aggregate capital-labor ratio.There is free mobility for both factors.All production in sectors i=1,...,m−1is consumed but in sector m production may be either consumed or invested.Therefore:c i=F i(n i k i,n i)∀i=m(3)˙k=F m(nk m,n m)−c m−(δ+ν)k(4)mwhereδ>0is the depreciation rate.Production function F i(.,.)has constant return to scale,positive and diminishing returns to inputs,and satisﬁes the Inada conditions.The social planner chooses the allocation of factors n i and k i across m sectors through a set of static e ﬃciency conditions,v i /v m =F m K /F i K =F m N /F i N ∀i.(5)The allocation of output to consumption and capital is chosen through a dynamic e ﬃciency condition,−˙v m /v m =F m K −(δ+ρ+ν).(6)where F i Nand F i K are the marginal products of labor and capital in sector i.5By (5),the rates of return to capital and labor are equal across sectors.In order to focus on the implications of di ﬀerent rates of TFP growth across sectors we assume production functions are identical in all sectors except for their rates of TFP growth:F i =A i F (n i k i ,n i );˙A i /A i =γi ;∀i,(7)With these production functions,we show in the Appendix that static e ﬃciency and the resource constraints (2)implyk i =k ;p i /p m =v i /v m =A m /A i ;∀i,(8)where p i is the price of good i in the decentralized economy.The utility function has constant elasticities both across goods and over time:v (c 1,...,c m )=φ(.)1−θ−11−θ;φ(.)=³P m i =1ωi c (ε−1)/εi´ε/(ε−1)(9)where θ,ε,ωi >0and Σωi =1.Of course,if θ=1,v (.)=ln φ(.)and if ε=1,ln φ(.)=P m i =1ωi ln c i .In the decentralized economy demand functions have constant price elasticity −εand unit income elasticity.With this utility function,(8)yields:p i c i m m =µωi m ¶εµA m i ¶1−ε≡x i ∀i.(10)The new variable x i is the ratio of consumption expenditure on good i to consumption expenditure on the manufacturing good and will prove useful in the subsequent analysis.The intuition behind this formula is in terms 5The corresponding transversality condition is lim t −→∞k exp ³−R t 0(F m k −δ−ν)dτ´=0.of price elasticities,given that all goods have unit income elasticity.The ratio of consumption expenditures is a weighted average of the ratio of the weights of each good in the utility function and of their relative prices.A higher price ratio p i/p m raises the ratio of expenditure on good i to good m by one minus their common price elasticity.We also deﬁne aggregate consumption expenditure and output per capita in terms of manufacturing:c≡P m i=1p i p m c i;y≡P m i=1p i p m F i(11) Using static eﬃciency we derive:c=c m X;y=A m F(k,1)(12)where X≡P m i=1x i.2Structural changeWe deﬁne structural change as the state in which at least some of the labor shares are changing over time,i.e.,˙n i=0for at least some i.We derive in the Appendix(Lemma A2)the employment shares:x iµc¶∀i=m,(13)n i=x mµc¶+µ1−c¶.(14)n m=Theﬁrst term in the right side of(14)parallels the term in(13)and so represents the employment needed to satisfy the consumption demand for the manufacturing good.The second bracketed term is equal to the sav-ings rate and represents the manufacturing employment needed to satisfy investment demand.Conditions(13)and(14)drive our structural change results.To see the intuition behind them,note that by aggregation over all i,we obtain that in our economy the employment share used to produce consumption goods is equal to c/y,and the employment share used to produce capital goods is 1−c/y.Conditions(13)and(14)state that the same holds for each sector i.From(10)and(12),the consumption expenditure share of each sector is p i c i/p m c=x i/X.So the employment share of consumption good i is the consumption share of good i multiplied by the employment share of totalconsumption.Equivalently,the employment share of consumption good i is the average propensity to consume good i:n i=p i c i/p m y.Condition(13)has the important implication that the growth rate of two sectors’relative employment depends only on the diﬀerence between the sectors’TFP growth rates and the elasticity of substitution betweengoods:˙n in i−˙n jn j=(1−ε)¡γj−γi¢∀i,j=m.(15)But(8)implies that the growth rate of relative prices is:˙p i p i−˙p jp j=γj−γi∀i(16)and so,˙n in i−˙n jn j=(1−ε)µ˙p i p i−˙p j p j¶∀i,j=m(17)Proposition1The rate of change of the relative price of good i to good j is equal to the diﬀerence between the TFP growth rates of sector j and sector i.In sectors producing only consumption goods,relative employment shares grow in proportion to relative prices,with the factor of proportionality given by one minus the elasticity of substitution between goods.6The dynamics of the individual employment shares satisfy:˙n i n i =c˙/yc/y+(1−ε)(¯γ−γi);∀i=m(18)˙n mn m="c˙/y c/y+(1−ε)(¯γ−γm)#(c/y)(x m/X)n m(19)+Ã−c˙/y1−c/y!µ1−c/y n m¶where¯γ≡P m i=1(x i/X)γi is a weighted average of TFP growth rates,with the weight given by each good’s consumption share.Equation(18)gives the growth rate in the employment share of each consumption good as a linear function of its own TFP growth rate.The intercept and slope of this function are common across sectors but although the slope is a constant,the intercept is in general a function of time because both c/y and¯γare in general functions of time.Manufacturing,however, does not conform to this rule,because its employment share is a weighted 6All derivations and proofs,unless trivial,are collected in the Appendix.average of two components,one for the production of the consumption good,which conforms to the rule,and one for the production of capital goods,which behaves diﬀerently.The properties of structural change follow immediately from(18)and (19).Considerﬁrst the case of equality in sectoral TFP growth rates,i.e., letγi=γm∀i.Our economy in this case is one of balanced TFP growth, with relative prices remaining constant but with many diﬀerentiated goods. Because of the constancy of relative prices all consumption goods can be aggregated into one,so we eﬀectively have a two-sector economy,one sector producing consumption goods and one producing capital goods.Structural change can still take place in this economy but only between the aggregate of the consumption sectors and the capital sector,and only if c/y changes over time.If c/y is increasing over time,the investment rate is falling and labor is moving out of the manufacturing sector and into the consumption sectors.Conversely,if c/y is falling over time labor is moving out of the consumption sectors and into manufacturing.In both cases,however,the relative employment shares in consumption sectors are constant.If c/y is constant over time,structural change requiresε=1and diﬀer-ent rates of sectoral TFP growth rates.It follows immediately from(16), (18)and(19)that if c˙/y=0,ε=1implies constant employment shares but changing prices.With constant employment shares faster-growing sec-tors produce relatively more output over time.Price changes in this case are such that consumption demands exactly match all the output changes due to the diﬀerent TFP growth rates.But ifε=1,prices still change as before but consumption demands are either too inelastic(in the case ε<1)to match all the output change,or are too elastic(ε>1)to be satisﬁed merely by the change in output due to TFP growth.So ifε<1 employment has to move into the slow-growing sectors and ifε>1it has to move into the fast-growing sectors.Proposition2Ifγi=γm∀i=m,a necessary and suﬃcient condition for structural change is˙c/c=˙y/y.The structural change in this case is between the aggregate of consumption sectors and the manufacturing sector.If˙c/c=˙y/y,necessary and suﬃcient conditions for structural change areε=1and∃i∈{1,..,m−1}s.t.γi=γm.The structural change in this case is between all sector pairs with diﬀerent TFP growth rates.Ifε<1 employment moves from the sector with the higher TFP growth rate to the sector with the lower TFP growth rate;conversely ifε>1.Proposition2forε<1conﬁrms the structural change facts identiﬁed by Baumol et al.(1985).When demand is price inelastic,the sectors with the low productivity growth rate attract a bigger share of labor,despite the rise in their price.From the static eﬃciency results in(8)and(12) weﬁnd that the nominal output shares(deﬁned as p i F i/p m y)are equal to the employment shares in all sectors,and by(10)the nominal consumption shares are given by x i/X,so the results obtained for employment shares also hold for nominal consumption and output shares.But real consumption growth satisﬁes˙c i/c i−˙c j/c j=ε¡γi−γj¢;∀i,j,(20) an expression also satisﬁed by real output shares∀i,j=m.A comparison of(15)with(20)reveals that a smallεcan reconcile the small changes in the relative real consumption shares with the large changes in relative nominal consumption shares found by Kravis et al. (1983).The authors concluded that theirﬁnding is evidence in favor of the technological explanation of structural change.More recently Daniel E. Sichel(1997)found the same pattern for relative output shares,and Falvey and Gemmell(1996)found that the real consumption share of services(a sector with low TFP growth rate)falls very gradually with income,both of which are consistent with our model whenε<1.3Aggregate growthWe now study the aggregate growth path of this economy,with the objec-tive ofﬁnding a suﬃcient set of conditions that satisfy structural change as derived in the preceding section,and in addition satisfy Kaldor’s stylized facts of aggregate growth.Recall that for the analysis of structural change we imposed a Hicks-neutral technology.It is well-known that with this type of technology,the economy can be on a steady state only if the production function is Cobb-Douglas.We therefore let F(n i k i,n i)=kαi n i,α∈(0,1).7 With TFP in each sector growing at some rateγi,the aggregate economy will also grow at some rate related to theγi s.The following Proposition derives the evolution of the aggregate economy:7Daron Acemoglu and Veronica Guerrieri(2005)examined the implications of dif-ferent capital intensities for economic growth and structural change.They show that capital deepening can cause both structural change and unbalanced growth.We exam-ine in the Appendix the implications of diﬀerent capital shares and aﬁxed factor for our model.Proposition 3Given any initial k (0),the equilibrium of the aggregate economy is a path for the pair {c,k }that satis ﬁes the following two dif-ferential equations:˙k =A m k α−1−c −(δ+ν),(21)θ˙c c=(θ−1)(γm −¯γ)+αA m k α−1−(δ+ρ+ν).(22)We de ﬁne an aggregate balanced growth path such that aggregate out-put,consumption and capital grow at the same rate.It follows from Propo-sition 3that a necessary condition for the existence of an aggregate bal-anced growth path is that the expression (θ−1)(γm −¯γ)be a constant.To show this,let:(θ−1)(γm −¯γ)≡ψconstant.(23)De ﬁne aggregate consumption and the capital-labor ratio in terms of e ﬃ-ciency units,c e ≡cA −1/(1−α)m and k e ≡kA −1/(1−α)m and let g m ≡γm /(1−α),the rate of labor-augmenting technological growth in the capital-producing sector.The dynamic equations (21)and (22)become˙c e /c e =£αk α−1e +ψ−(δ+ν+ρ)¤/θ−g m (24)˙k e =k αe−c e −(g m +δ+ν)k e .(25)Equations (24)and (25)parallel the two di ﬀerential equations in the con-trol and state of the one-sector Ramsey economy,making the aggregate equilibrium of our many-sector economy identical to the equilibrium of the one-sector Ramsey economy when ψ=0,and trivially di ﬀerent from it otherwise.Both models have a saddlepath equilibrium and stationary so-lutions ³ˆc e ,ˆke ´that imply balanced growth in the three aggregates.The capital-labor ratio is growing at the rate of growth of labor-augmenting technological progress in the sector that produces capital goods,g m .Ag-gregate consumption and output de ﬂated by the price of manufacturing goods are also growing at the same rate.Proposition 2and the requirement that ψbe constant yield the impor-tant Proposition:Proposition 4Necessary and su ﬃcient conditions for the existence of an aggregate balanced growth path with structural change are:θ=1,ε=1;and ∃i ∈{1,..,m −1}s.t.γi =γm .Recalling the deﬁnition of¯γfollowing equation(19),Proposition3im-plies that the contribution of each consumption sector i to aggregate equi-librium is through its weight x i in¯γ.Because each x i depends on the sector’srelative TFP level,the weights here are functions of time.So¯γcannot beconstant during structural change and the only way thatψcan be constantis throughθ=1,which yieldsψ=0.In this case our aggregate economyin c and k becomes formally identical to the one-sector Ramsey economywith growth rateγm.There are two other conditions that give a constantψand so yield balanced aggregate growth:γi=γm∀i orε=1.But as Proposition2demonstrates neither condition permits structural change onthe balanced growth path,where c/y is constant.Proposition4requires the utility function to be logarithmic in the con-sumption compositeφ,which implies an intertemporal elasticity of substi-tution equal to one,but be non-logarithmic across goods,which is neededto yield non-unit price elasticities.A noteworthy implication of Proposi-tion4is that balanced aggregate growth does not require constant ratesof growth of TFP in any sector other than manufacturing.Because bothcapital and labor are perfectly mobile across sectors,changes in the TFPgrowth rates of consumption-producing sectors are reﬂected in immediateprice changes and reallocations of capital and labor across sectors,withouteﬀect on the aggregate growth path.To give intuition for the logarithmic intertemporal utility function werecall that balanced aggregate growth requires that aggregate consumptionbe a constant fraction of aggregate wealth.With our homothetic utilityfunction this can be satisﬁed either when the interest rate is constant orwhen consumption is independent of the interest rate.The relevant interestrate here is the rate of return to capital in consumption units,which isgiven by the net marginal product of capital,αy/k−δ,minus the changein the relative price of the consumption composite,γm−¯γ.The latter is not constant during structural change.In the caseε<1,¯γis falling over time(see Lemma A3in the Appendix for proof),and so the real interest rate is also falling,and converging toαy/k−δ.With a non-constant interest rate the consumption-wealth ratio is constant only if consumption is independent of the interest rate,which requires a logarithmic utility function.88After re-examining the evidence,Robert Barro and Xavier Sala-i-Martin(2004,p.13)concluded,consistent with our model,“it seems likely that Kaldor’s hypothesis of a roughly stable real rate of return should be replaced by a tendency for returns to fallUnder the condition of Proposition4there is a steady-state character-ized by aggregate balanced growth,in the sense that in this steady state the aggregate ratios are constant.In order to achieve this balance,the aggregates c and y are divided by manufacturing price,to conform to the aggregate k.If some other price index is used as deﬂator,the rate of growth of the aggregates is constant only if the rate of growth of the price index is constant,but of course the aggregate ratios are still constant.The pub-lished aggregate series studied by macroeconomists usually use an average price as deﬂator which does not haveﬁxed weights.If the price index used to deﬂate national statistics is some˜p,the published real aggregate income is y/˜p.If the weights used to construct˜p are the sector shares,˜p changes during structural change.But because sector shares do not change rapidly over time,visually there is virtually nothing to distinguish the“stylized fact”of constant growth in reported per capita GDP with another“styl-ized fact”of constant growth in our per capita output measure.9 Next,we summarize the dynamics of employment shares along the ag-gregate balanced growth path.Proposition5Let sector l denote the sector with the smallest TFP growth rate whenε<1or the sector with the biggest TFP growth rate whenε>1. On the aggregate balanced growth path,n l increases monotonically.Employ-ment in the other sectors is either hump-shaped or declines monotonically. Asymptotically,the economy converges to an economy withn∗m=ˆσ=αµδ+ν+g mδ+ν+ρ+g m¶;n∗l=1−ˆσwhereˆσis the savings rate along the aggregate balanced growth path.Proposition5follows immediately from(18)-(19)and Lemma A3.Con-sider the caseε<1,the one forε>1following by a corresponding ar-gument.Forε<1,sector i expands if and only if its TFP growth rate is smaller than¯γ,and contracts if and only if its growth rate exceeds it. But ifε<1,the weighted average¯γis decreasing over time(see Lemma A3in the Appendix).Therefore,the set of expanding sectors is shrinking over some range as an economy develops.”In our model it is converging from above to a positive value.9Nicholas Kaldor(1961,p.178)spoke of a“steady trend rate”of growth in the“ag-gregate volume of production.”In Ngai and Pissarides(2004,Fig.4)we plot our series of per capita real incomes and the published chain-weighted series for the United States since1929,and show that they are virtually indistinguishable from each other.over time,as more sectors’TFP growth rates exceed¯γ.This feature of the model implies that sectors with TFP growth rates below the initial¯γexhibit a hump-shaped employment share,an implication that we believe is unique to our model.These employment sharesﬁrst rise but once¯γdrops down to their ownγi they fall.10In contrast to each sector’s employment share,once the economy is on the aggregate balanced growth path output and consumption in each consumption sector grow according to˙F i=˙Aii+α˙ki+˙n ii=εγi+αg m+(1−ε)¯γ.(26)Ifε61the rate of growth of consumption and output in each sector is positive(providedγi≥0),and so sectors never vanish,even though their employment shares in the limit may vanish.Ifε>1the rate of growth of output may be negative in some low-growth sectors,and since by Lemma A3¯γis rising over time in this case,their rate of growth remains indeﬁnitely negative until they vanish.Finally,we examine brieﬂy the implications ofθ=1.Whenθ=1 balanced aggregate growth cannot coexist with structural change,because the termψ=(θ−1)(γm−¯γ)in the Euler condition(24)is a function of time.But as shown in the Appendix Lemma A3,¯γis monotonic.As t→∞,ψconverges to the constant(θ−1)(γm−γl),whereγl is the TFP growth rate in the limiting sector(the slowest or fastest growing consumption sector depending on whetherε<or>1).Therefore,the economy withθ=1converges to an asymptotic steady state with the same growth rate as the economy withθ=1.What characterizes the dynamic path of the aggregate economy when θ=1?By diﬀerentiation and using Lemma A3,we obtain˙ψ=(θ−1)(1−ε)P m i=1(x i/X)(γi−¯γ)2(27) which is of second-order compared with the growth in employment shares in(15),given that theγs are usually small numbers centered around0.02. Therefore,the rate of growth of the economy during the adjustment to the asymptotic steady state withθ=1is very close to the constant growth 10Maddison(1980,p.48)in his study of historical OECD data found a“shallow bell shape”for manufacturing employment for each of the16OECD countries,which can be reproduced by our model if the manufacturing TFP growth rate takes values between the TFP growth rates of agriculture and services.。

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strucchange_An_R_Package_for_Testing_for_Structural_Change_in_Linear_Regression_Models

against the alternative that the coefﬁcient vector varies over time, with certain tests being more or less suitable (i.e., having good or poor power) for certain patterns of deviation from the null hypothesis. It is assumed that the regressors are nonstochastic with ||xi || = O(1) and that 1 n
Abstract This paper introduces ideas and methods for testing for structural change in linear regression models and presents how these have been realized in an R package called strucchange1 . It features tests from the generalized ﬂuctuation test framework as well as from the F test (Chow test) framework. Extending standard signiﬁcance tests it contains methods to ﬁt, plot and test empirical ﬂuctuation processes (like CUSUM, MOSUM and estimates-based processes) on the one hand and to compute, plot and test sequences of F statistics with the supF , aveF and expF test on the other. Thus, it makes powerful tools available to display information about structural changes in regression relationships and to assess their signiﬁcance. Furthermore it is described how incoming data can be monitored online.

Asymptotically Exact Confidence Intervals of CUSUM and CUSUMSQ Tests A Numerical Derivation

ABSTRACT: In testing a structural change, the approximated conﬁdence intervals are conventionally used for CUSUM and CUSUMSQ tests. This paper numerically derives the asymptotically exact conﬁdence intervals of CUSUM and CUSUMSQ tests. It can be easily extended to nonnormal and/or nonlinear models. KEY WORDS: CUSUM test, CUSUMSQ test, Monte-Carlo simulation, Asymptotically exact conﬁdence interval
2
OVERVIEW OF CUSUM AND CUSUMSQ TESTS
yt = xt β + ut , ut ∼ N (0, σ 2 ), t = 1, · · · , T,
Consider the following regression model:
where β is a k × 1 unknown parameter vector. yt is a dependent variable while xt is a 1 × k vector of independent variables. ut is assumed to be normally distributed with mean zero and variance σ 2 . Deﬁne Xt and Yt as follows: y1 x1 y2 x2 , . Yt = Xt = . . . . . . xt yt The null hypothesis of no structural change for the model is speciﬁed as:

基于复杂网络理论的大型换热网络节点重要性评价

2017年第36卷第5期 CHEMICAL INDUSTRY AND ENGINEERING PROGRESS·1581·化工进展基于复杂网络理论的大型换热网络节点重要性评价王政1，孙锦程1，刘晓强1，姜英1，贾小平2，王芳2（1青岛科技大学化工学院，山东青岛 266042；2青岛科技大学环境与安全工程学院，山东青岛 266042）摘要：鉴于换热网络大型化和流股间复杂关系，使得换热网络换热器节点重要性的研究显得越来越重要，对其控制和安全运行的工程实践方面具有指导意义。

本文以大型换热网络为研究对象，将换热器抽象为节点，换热器之间的干扰传递抽象为边，构造网络拓扑结构。

在复杂网络理论的基础上，提出了评价大型换热网络节点重要性的策略和模型。

首先，从网络的点度中心性、中间中心性、接近中心性和特征向量中心性等网络拓扑结构属性出发，依据多属性决策方法对网络节点重要性进行综合评价；其次，考虑换热网络的方向性，基于PageRank 算法对该网络进行节点重要性评价研究。

综合两个算法的计算结果得出最终结论。

案例分析表明：该研究方法是有效的，可从不同的角度全面评价换热网络的节点重要性，丰富了换热器节点重要性评价的相关理论。

关键词：换热网络；复杂网络；节点重要性；多属性决策；PageRank 算法中图分类号：X92 文献标志码：A 文章编号：1000–6613（2017）05–1581–08 DOI ：10.16085/j.issn.1000-6613.2017.05.004Evaluation of the node importance for large heat exchanger networkbased on complex network theoryWANG Zheng 1，SUN Jincheng 1，LIU Xiaoqiang 1，JIANG Ying 1，JIA Xiaoping 2，WANG Fang 2（1College of Chemical Engineering ，Qingdao University of Science and Technology ，Qingdao 266042，Shandong ，China ；2College of Environment and Safety Engineering ，Qingdao University of Science and Technology ，Qingdao266042，Shandong ，China ）Abstract ：Because of the complexity of large-scale heat exchanger network ，it is important to investigate the importance of heat exchanger nodes in heat exchanger network. It can provide guidance for the control and safe operation of heat exchanger networks ，as well as engineering practices. In this paper ，the network topology structure of large-scale heat exchanger network was constructed by treating heat exchangers as nodes and treating the transfer of interference between heat exchangers as edges. Based on the complex network theory ，the strategies and models for evaluating the node importance of the heat exchanger network were proposed. Firstly ，the importance of nodes were evaluated by the multi-attribute decision method based on the degree centrality, betweenness ，closeness and eigenvector centralities. Next ，considering the direction of case heat exchanger network ，PageRank algorithm was used to evaluate the importance of nodes. Considering the results from these two algorithms ，the final results were obtained. The case analysis showed that the strategy is effective and it can evaluate the node importance from different views ，which will enrich the node importance evaluation theory for heat exchanger network.Key words ：heat exchanger network ；complex network ；node importance ；multi-attribute decision ；PageRank algorithm第一作者及联系人：王政（1968—），男，博士，副教授，硕士生导师，主要研究过程系统工程。

Structural Studies of the Ferroelectric Phase Transition in Bi4Ti3O12

Structural Studies of the Ferroelectric Phase Transitionin Bi4Ti3O12Qingdi Zhou and Brendan J.Kennedy*School of Chemistry,The University of Sydney,Sydney,NSW2006AustraliaChristopher J.HowardAustralian Nuclear Science and Technology Organization,Private Mail Bag1,Menai, NSW2234,Australia,and School of Physics,The University of Sydney,Sydney,NSW2006AustraliaReceived July3,2003.Revised Manuscript Received October20,2003A variable-temperature synchrotron X-ray diffraction study of the phase transitions in the ferroelectric n)3Aurivillius oxide Bi4Ti3O12is described.At room temperature the structure of Bi4Ti3O12is orthorhombic in space group B2eb and this continuously transforms to the high-temperature tetragonal I4/mmm structure via an intermediate orthorhombic phase.The possible space groups of this intermediate orthorhombic phase have been identified by using group theory.IntroductionThe Aurivillius oxides are represented by the general formula(Bi2O2)2+(A n-1B n O3n+1)2-where B is a diamag-netic transition metal such as Ti4+or Nb5+and A is an alkali or alkaline earth cation.1The structure of the Aurivillius oxides consists of arrays of Bi2O2and per-ovskite-like A n-1B n O3n+1layers.The ferroelectric prop-erties of such oxides have been known for around50 years,2,3yet the structural origins of their ferroelectricity have only recently been established.4,5Following the pioneering work of Scott and co-work-ers,6the possibility of employing these Aurivillius oxides in ferroelectric memory devices has been extensively studied.A serious barrier to their practical utilization is their poor thermal stability.7-10Formation of thin-film ferroelectric devices involves a sintering step at high temperatures and this invariably degrades the performance of the simpler Aurivillius oxides.Conse-quently,a number of studies of the high-temperature behavior and structures of Aurivillius oxides have been reported.11-14In comparison to the n)2oxides based on SrBi2Ta2O9very little is known about the high-temperature properties of the n)3oxides such as Bi4-Ti3O12,yet such oxides are reported to have ferroelectric properties superior to those of the better-studied n)2 oxides.15Although it has been reported5that Bi4Ti3O12is monoclinic at room temperature,very-high-resolution powder diffraction data suggest that powder samples of Bi4Ti3O12are actually orthorhombic at room temper-ature.16Heating Bi4Ti3O12above670°C is reported to result in an apparently first-order phase transition to a paraelectric tetragonal phase.16,17A significant volume change between the high-temperature and low-temper-ature phases is likely to be detrimental to the stability of any thin films annealed or sintered above the Curie temperature.Conversely,continuous transitions are less likely to adversely influence the properties of the thin films.Hervoches and Lightfoot have demonstrated, using powder neutron diffraction methods,that for Bi4-Ti3O12the high-temperature paraelectric phase is in space group I4/mmm and that the room-temperature orthorhombic phase is in B2eb.16The main structural basis for the ferroelectricity in Bi4Ti3O12is the displacement of the Bi atoms within the perovskite-like layers,along the crystallographic a-axis with respect to the chains of corner-sharing TiO6 octahedra.This corresponds to a[110]displacement referred to the parent I4/mmm.The TiO6octahedra are tilted relative to each other and the tilt system can be*To whom correspondence should be addressed.Phone:61-2-9351-2742.Fax:61-2-9351-3329.E-mail: b.kennedy@.au.(1)Aurivillius,B.Arkiv Kemi1949,1,463.(2)Subbarao,E.C.J.Phys.Chem.Solids1962,23,665.(3)Smolenskii,G.A.;Isupov,V.A.;Agranovskaya,A.I.Sov.Phys. Solid State1961,3,651.(4)Rae,A.D.;Thompson,J.G.;Withers,R.Acta Crystallogr.,Sect. B:Struct.Sci.1992,48,418.(5)Rae,A.D.;Thompson,J.G.;Withers,R.;Willis,A.C.Acta Crystallogr.,Sect.B:Struct.Sci.1990,46,474.(6)Paz de Araujo,C.A.;Cuchlaro,J.D.;McMillan,L.D.;Scott, M.;Scott,J.F.Nature(London)1995,374,627.(7)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Kamiyama,T.;Asano,H.; Izumi,F.Appl.Phys.Lett.2000,77,2749.(8)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Asano,H.;Kamiyama,T.; Izumi,F.;Hiroi,Z.Appl.Phys.Lett.2001,79,2791.(9)Boulle, A.;Legrand, C.;Guinebretie`re,R.;Mercurio,J.P.; Dauger,A.Thin Solid Films2002,391,42.(10)Aizawa,K.;Tokumitsu,E.;Okamoto,K.;Ishiwara,H.Appl. Phys.Lett.2000,79,2791.(11)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.; Shimakawa,Y.Integr.Ferroelectr.2002,44,101-112.(12)Hervoches,C.H.;Irvine,J.T.S.;Lightfoot,P.Phys.Rev.B 2002,64,100102(R).(13)Liu,J.;Zou,G.;Yang,H.;Cui,Q.Solid State Commun.1994, 90,365.(14)Macquart,R.;Kennedy,B.J.;Vogt,T.;Howard,C.J.Phys. Rev B2002,66,212102.(15)Park,B.H.;Kang,B.S.;Bu,S.D.;Noh,T.W.;Lee,J.;Jo,W. Nature(London)1999,401,682.(16)Hervoches,C.H.;Lightfoot,P.Chem.Mater.1999,11,3359.(17)Hirata,T.;Yokokawa,T.Solid State Commun.1997,104,673.5025Chem.Mater.2003,15,5025-502810.1021/cm034580l CCC:$25.00©2003American Chemical SocietyPublished on Web11/21/2003described as a-a-c0in Glazer’s notation.18These two structural features act in concert to lower the symmetry from tetragonal to orthorhombic,however,they are not linked,and there is no reason to suppose that both modes will condense at precisely the same temperature. Rather,it is probable that these modes will condense successively and the two end member phases will be linked by an intermediate ing powder neutron diffraction methods,Macquart and Lightfoot have in-dependently shown that the A21am to I4/mmm transi-tion in the n)2Aurivillius phases SrBi2Ta2O911and Sr0.85Bi2.1Ta2O912proceeds via an intermediate paraelec-tric Amam phase in each case.This Amam phase is also seen in PbBi2M2O9(M)Nb,Ta).19That is,upon cooling from the I4/mmm phase,the tilting of the octahedra occurs before the cation displacement.The same se-quence is reported to occur in the n)4Aurivillius oxide SrBi4Ti4O15.20The aim of the present work is to establish whether the B2eb to I4/mmm transition in Bi4Ti3O12is first order as proposed by Hirata and Yokakowa17or if it actually occurs continuously via an intermediate phase as seen in SrBi2Ta2O9.11To establish this we have investigated the temperature dependence of the structure of Bi4-Ti3O12from room temperature to800°C using high-resolution synchrotron radiation.Near the Curie tem-perature fine temperature intervals have been used to detect the intermediate phase which exists over a very limited temperature range.Experimental SectionThe crystalline sample of Bi4Ti3O12was prepared by the solid-state reaction of stoichiometric quantities of Bi2O3 (99.999%,Aldrich)and TiO2(99.9%,Aldrich).The heating sequence used was700°C/24h and850°C/48h,with intermediate regrinding.The sample was slowly cooled to room temperature in the furnace.The sample purity was established by powder X-ray dif-fraction measurements using Cu K R radiation on a Shimadzu D-6000Diffractometer.Room-and variable-temperature syn-chrotron X-ray powder diffraction patterns were collected on a high-resolution Debye Scherrer diffractometer at beamline 20B,the Australian National Beamline Facility,at the Photon Factory,Japan.21The sample was finely ground and loaded into a0.3-mm quartz capillary that was rotated during the measurements.All measurements were performed under vacuum to minimize air scatter.Data were recorded using two Fuji image plates.Each image plate was20×40cm and each covered40°in2θ.A thin strip(ca.0.5cm wide)was used to record each diffraction pattern so that up to30patterns could be recorded before reading the image plates.The data were collected at a wavelength of0.75Å(calibrated with a NIST Si 640c standard)over the2θrange of5-75°with step size of 0.01°.The patterns were collected in the temperature range of100-800°C in25°C steps or600-803°C in7°C steps, and with30-min counting time at each temperature.Struc-tures were refined by the Rietveld method using the program Rietica.22The positions of the cations were well described in these analyses,however the estimated standard deviations (esds)of the Ti-O bond distances(typically around0.02Å) preclude any detailed discussion of the temperature depen-dence of the bond distances.Results and DiscussionThe published room-temperature structure for Bi4-Ti3O12was used as a starting model in our Rietveldrefinements,and the structural refinement proceededwithout event.The temperature dependence of thelattice parameters and volumes are illustrated in Figure1.All the lattice parameters show a smooth increasedue to thermal expansion as the sample is heated toca.500°C.Above this temperature the cell continuesto expand along both the b-and c-directions,howeverthe a-parameter is essentially constant.This behavioris very similar to that displayed by a number of simplerABO3perovskites23-25and is apparently related to thegradual reduction in distortion resulting from a reduc-tion in the magnitude of the tilting of the BO6octahedraas the temperature is increased.Near675°C there is arapid decrease in the a-parameter although as is clearlyevident from Figure1this is not a discontinuousdecrease but rather a rapid progressive drop.At thesame temperature the c-axis expands rapidly,Figure1.The transition to the tetragonal phase is clearlyevident in the200/020and317/137reflections(near2θ≈15.8and26.7°,respectively,whereλ)0.75Å).As illustrated in Figure2the diffraction pattern recordedat670°C shows obvious splitting of the200/020and317/137reflections that is clearly indicative of orthor-hombic symmetry.This splitting remains clearly visibleto the eye until677°C and can be discerned by profileanalysis at684°C.At691°C no splitting or diagnosticasymmetry of these or other peaks is apparent and itis concluded that the structure is tetragonal.That is,the transition to the tetragonal structure occurs near690°C,which is around15°above the reported ferro-electric Curie temperature for Bi4Ti3O12.(Some care isrequired when comparing the transition temperaturesreported in various studies because of both samplevariation(induced by the different heating regimesused)and possible variations in the high-temperaturethermometry.)Above691°C the structure has been refined in thetetragonal space group I4/mmm and is as described byHervoches and Lightfoot.16That the paraelectric phaseis tetragonal in I4/mmm well above the ferroelectricCurie temperature was confirmed from the high-resolu-tion powder neutron diffraction data by Hervoches andLightfoot from both the cell metric and the absence ofany superlattice reflections indicative of TiO6tilting. The thermal expansion of the c-axis both above and below the T c is relatively linear and can be well-fitted to a simple linear equation c)5.277×10-4T+32.725 for T<677°C,and c)5.116×10-4T+32.725for T> 691°C.Examining the temperature dependence of the long c-axis we observed a large difference between the(18)Glazer,A.M.Acta Crystallogr.B1972,28,3384.(19)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.J. Phys.:Condens.Matter2002,14,7955.(20)Hervoches,C.H.;Snedden,A.;Riggs,R.;Kilcoyne,S.H.; Manuel,P.;Lightfoot,P.J.Solid State Chem.2002,164,280.(21)Sabine,T.M.;Kennedy,B.J.;Garrett,R.F.;Foran,G.J.; Cookson,D.J.J.Appl.Crystallogr.1995,28,513.(22)Howard C.J.;Hunter,B.A.A Computer Program for Rietveld Analysis of X-ray and Neutron Powder Diffraction Patterns;Lucas Heights Research Laboratories:New South Wales,Australia,1998; pp1-27.(23)Howard,C.J.;Knight,K.S.;Kisi,E.H.;Kennedy,B.J.J. Phys.:Condens.Matter2000,12,L677.(24)Kennedy,B.J.;Howard,C.J.;Thorogood,G.J.;Hester,J.R. J.Solid State Chem.2001,161,106.(25)Kennedy,B.J.;Howard,C.J.;Chakoumakos,B.C.J.Phys. C:Condens.Matter1999,11,1479.5026Chem.Mater.,Vol.15,No.26,2003Zhou et al.values of the low-temperature orthorhombic,ferroelec-tric phase and those for the high-temperature tetragonal and paraelectric phase.Clearly the value for the c -parameter at 684°C does not fall into either series.A similar conclusion can be made for both the a -and b -parameters.These temperature-dependent changes in the lattice parameters are more reminiscent of the continuous-phase transitions observed in oxides such as Bi 2PbNb 2O 919than of the more subtle changes observed in the high-temperature second-order incommensurate-to-commensurate phase transition observed in Bi 2-MoO 6.26A feature of many first-order phase transitions is the coexistence of a two phase region.Attempts to fit the pattern at 684°C to a two-phase orthorhombic/tetragonal model with lattice parameters obtained by appropriate linear extrapolation were unsuccessful.It was concluded that a single phase was present at this temperature and this was neither the orthorhombic in B 2eb nor the tetragonal I 4/mmm .(26)Buttrey,D.J.;Vogt,T.;White,B.D.J.Solid State Chem.2000,155,206.(27)/∼stokesh/isotropy.html.Figure 1.Temperature dependence of the lattice parameters and volume for Bi 4Ti 3O 12obtained from Rietveld analysis of variable-temperature synchrotron diffraction data.For ease of comparison the values in the 2× 2×1orthorhombic cells have been reduced to the equivalent tetragonal values.Figure 2.Portions of the Rietveld fits for Bi 4Ti 3O 12showing the temperature dependence of the splitting of the tetragonal 110reflection into the orthorhombic 200/020pair near 2θ)15.8°and of the 127reflection into the 317/137pair near 2θ)26.8°.The 0012reflection is apparent near 2θ)15.6°.Ferroelectric Phase Transition in Bi 4Ti 3O 12Chem.Mater.,Vol.15,No.26,20035027In summary,we observe an orthorhombic structure at 684°C whose lattice parameter clearly distinguishes it from the low-temperature orthorhombic and high-temperature tetragonal phases.(In fact,even if we could not distinguish this as a separate phase,the observed continuity of transition,together with the group theo-retical arguments to follow,would indicate that such an intermediate orthorhombic phase is involved in the transition.)If we assume that all three phases have commensurate structures then it should be possible to identify the space group of the intermediate phase from a group theoretical analysis.To do this we used the program ISOTROPY.27This analysis confirmed that a direct B 2eb to I 4/mmm transition could not be continu-ous.The two modes responsible for the transition were identified as Γ5-describing the cation displacement and X 3+associated with the tilting of the TiO 6octahedra.28Whichever of these modes condenses first,an interme-diate orthorhombic structure based on a 2× 2×1superstructure of the parent I 4/mmm structure is involved,as illustrated in Figure 3,and the successive phase transitions through either of these intermediates are allowed to be continuous.If the initial distortion is cation displacement via the Γ5-mode then a ferroelec-tric orthorhombic structure that lacks any tilting of the TiO 6octahedra is expected.The resulting structure in Fmm 2can continuously transform to the observed room-temperature orthorhombic B 2eb phase through tilting of the TiO 6octahedra via the X 3+mode.(This descrip-tion refers still to the parent structure in I 4/mmm .)Alternatively,the initial distortion may be the introduc-tion of tilting of the TiO 6octahedra resulting in a Cmca phase followed by cation displacement.The available synchrotron diffraction data do not allow us to unequivoc-ally distinguish between these two possibilities;how-ever,by analogy with SrBi 2Ta 2O 9,we favor the latter possibility.11The confirmation of this will require a high-resolution neutron diffraction study.It is illuminating to compare our results with those of Hirata and Yokokawa.17First,Figure 2of their report suggests they recorded very little data in the ferroelec-tric phase with only four temperature points obvious.A somewhat greater number of temperatures were examined (nine)in the paraelectric phase,apparently at 20°C intervals.Crucially,no data appear to have been collected between 475and 675°C,the latter corresponding to the reported ferroelectric Curie tem-perature for Bi 4Ti 3O 12.There is then a relatively large jump in the temperatures used by Hirata and Yokokawa to above 695°C.In our present study we have used relatively coarse temperature increments (25°C)to monitor the general form of the phase transition,but then we have used much finer intervals (7°C)to probe the nature (first order or continuous)of the ferroelectric to paraelectric transition.We conclude that as a result of the relatively coarse temperature intervals used by Hirata and Yokokawa it was not possible for those authors to establish whether the ferroelectric to paraelec-tric transition in Bi 4Ti 3O 12was first order or continuous.The temperature dependence of the lattice parameters established using powder neutron diffraction data de-scribed by Lightfoot and Hervoches 16,29is very similar to that observed here for the data collected in 25°C intervals.On the basis of a similar density of data,Lightfoot and Hervoches 29concluded that the transition is first order.In comparison,our data collected in 7°C intervals using high-resolution synchrotron X-ray meth-ods strongly suggests the transition is continuous,albeit involving an intermediate phase.In conclusion we have identified the existence of an intermediate orthorhombic phase in the solid-state phase transition of the ferroelectric n )3Aurivillius phases Bi 4Ti 3O 12.Two possible orthorhombic phases were iden-tified using group theory,Fmm 2and Cmca ,depending on the sequence in which the two modes responsible for the lowering of symmetry condense.By analogy with the n )2oxide SrBi 2Ta 2O 9,the most likely sequence of transitions is B 2cb 670°C fCmca 695°CfI 4/mmm .Clearly confirming the existence of and establishing the precise structure of the proposed intermediate phase is of considerable interest,and efforts aimed at this are in progress.Acknowledgment.This work performed at the Australian National Beamline Facility was supported by the Australian Synchrotron Research Program,which is funded by the Commonwealth of Australia under the Major National Research Facilities program.B.J.K.acknowledges the support of the Australian Research Council.The assistance of Dr.James Hester at the ANBF is gratefully acknowledged.We thank Dr.P.Lightfoot for bringing ref 29to our attention while this manuscript was under review.CM034580L(28)Miller,S.C.;Love,W.F.Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups ;Pruett Press:Boulder,CO,1967.(29)Hervoches,C.H.;Lightfoot,P.Proceedings of CIMTEC,10th International Ceramics Congress ,Florence,Italy,July 2002.Figure 3.Schematic diagram showing the group -subgroup relationships for the n )3Aurivillius oxides.The solid lines show the transitions that are allowed to be continuous.The tilt system for the intermediate phases is given.5028Chem.Mater.,Vol.15,No.26,2003Zhou etal.。

A thermal study on the structural changes of bimetallic ZrO2-modified TiO2 nanotubes synthesized

IOP P UBLISHING N ANOTECHNOLOGY Nanotechnology20(2009)195601(9pp)doi:10.1088/0957-4484/20/19/195601A thermal study on the structural changes of bimetallic ZrO2-modiﬁed TiO2 nanotubes synthesized using supercritical CO2R A Lucky and P A Charpentier1Department of Chemical and Biochemical Engineering,University of Western Ontario,London,ON,N6A5B9,CanadaE-mail:pcharpentier@eng.uwo.caReceived4November2008,inﬁnal form19February2009Published21April2009Online at /Nano/20/195601AbstractIn this study the thermal behavior of bimetallic ZrO2–TiO2(10/90mol/mol)nanotubes arediscussed which were synthesized via a sol–gel process in supercritical carbon dioxide(scCO2).The effects of calcination temperature on the morphology,phase structure,mean crystallite size,speciﬁc surface area and pore volume of the nanotubes were investigated by using a variety ofphysiochemical techniques.We report that SEM and TEM images showed that the nanotubularstructure was preserved at up to800◦C calcination temperature.When exposed to highertemperatures(900–1000◦C)the ZrO2–TiO2tubes deformed and the crystallites fused together,forming larger crystallites,and a bimetallic ZrTiO4species was detected.These results werefurther examined using TGA,FTIR,XRD and HRTEM analysis.The BET textural propertiesdemonstrated that the presence of a small amount of Zr in the TiO2matrix inhibited the graingrowth,stabilized the anatase phase and increased the thermal stability.(Someﬁgures in this article are in colour only in the electronic version)1.IntroductionConsiderable effort is being devoted to the preparation of one-dimensional(1D)oxide nanostructures due to their poten-tial applications in a diversity of technologies including catal-ysis,high efﬁciency solar cells,coatings and sensors[1–3].In particular,titania(TiO2)nanotubes are receiving considerable attention due to titania’s unique optoelectronic,photochemical and dielectric properties,along with being a low cost material for potential commercial employment[4–7].Having such unique properties,TiO2nanomaterials with deﬁned structures are highly desirable for electron-transport materials in dye-sensitized solar cells,as photocatalysts,photoconductive agents and nanoﬁllers in polymer composites[7,8].The performance of these titania nanomaterials widely relies on their crystallinity,crystallite size,crystal structure,speciﬁc surface area and thermal stability[9,10].As examples,the 1Author to whom any correspondence should be addressed.role of anatase and rutile crystal phases in TiO2nanostructures is still under active investigation for photocatalysis,where the interface between these nanostructures is believed to be the catalytic active site[11].The rutile crystal structure is important for stronger materials for use in orthopedic applications such as bone cements[12].For preparing these oxide nanostructures,the sol–gel method is becoming the standard method as it provides a uniform phase distribution,high purity,low temperature processing,and better size and morphology control[13]. However,the properties of sol–gel-synthesized binary metal oxides strongly depend on the synthesis conditions,such as the type of alkoxide(s),temperature,catalyst,solvent and solvent removal process[14,15].In the past decade,direct sol–gel reactions in supercritical carbon dioxide(scCO2)have attracted much attention for synthesizing oxide nanomaterials[16–19]. This approach has many advantages over the conventional sol–gel process operated in an organic solvent,as the resulting ma-terials maintain nanofeatures and a high surface area after CO2drying and venting[20].Low viscosity,‘zero’surface tension and high diffusivity of scCO2are favorable physical properties of the solvent for synthesizing potential superior ultraﬁne and uniform nanomaterials.As well,CO2is inexpensive, inﬂammable and considered a‘green’reaction medium[21].Previously,we discovered that bimetallic Zr-modiﬁed TiO2(Zr–TiO2)nanotubes could be synthesized in scCO2,and our preliminary results indicated that the nanotubes gave a high surface area,up to430m2g−1,as prepared[22].According to several studies,a small amount of transition metal doping has been found very effective to improve the thermal stability and activity of TiO2,particularly by using zirconia[23–26].In addition to the synthesis conditions,the calcination conditions are very important to the crystal structures of the metal oxide nanomaterials obtained,and subsequently,their potential end use applications.Spijksma et al[27]synthesized titania–zirconia microporous composite membranes using a1:1molar ratio by using the sol–gel process.The crystallization temperature for these materials was750◦C;however,after calcining at800◦C,an orthorhombic ZrTi2O6structure was formed,commonly known as srilankite.Whereas Zou et al [28]synthesized binary oxides by hydrolysis of titanium and zirconium nitrate solutions at various ratios.After calcining at 800◦C,the binary oxide showed the presence of the ZrTiO4 crystal phase and very low surface areas.Kitiyanan et al [23]synthesized5mol%zinconia-modiﬁed TiO2using a sol–gel process.They showed that this small amount of Zr stabilized the anatase phase up to800◦C,but that the anatase phase completely transformed into the rutile phase at 1000◦C.However,the materials calcined at these very high temperatures showed very low surface areas.As the nanotubular structure of the bimetallic ZrO2/TiO2 nanotubes has many potentially interesting applications, however,their structure and crystal morphology changes with calcination temperature have not been investigated.Hence, this study focused on the thermal behavior of the synthesized Zr–TiO2(10/90mol/mol)nanotubes prepared via an acetic acid modiﬁed sol–gel process in scCO2.The synthesized materials were calcined at different temperatures and the effects of calcination temperature on the morphology,phase structure,mean crystallite size,speciﬁc surface area and pore volume were investigated using a variety of physicochemical characterization techniques.2.Experimental details2.1.MaterialsReagent grade titanium(IV)isopropoxide(TIP,97%,Aldrich), zirconium(IV)propoxide(ZPO,70%,Aldrich),acetic acid(99.7%,Aldrich)and instrument grade carbon dioxide (99.99%,BOC)were used without further puriﬁcation.2.2.Nanotube synthesisThe procedure previously reported[22]was used for synthesis of10%ZrO2–90%TiO2nanotubes.2.3.CharacterizationScanning electron microscopy(SEM)measurements were usedto determine the size and morphology of nanomaterials usinga LEO1530scanning electron microscope.Transmissionelectron microscopy(TEM)and HRTEM images wereobtained using a Philips CM10and JEOL2010f.Thespecimens were dispersed in methanol and placed on a coppergrid covered with a holey carbonﬁlm.Thermo-gravimetricanalysis(TGA)was performed under nitrogen atmosphere ona TA Instruments TA-Q500at a heating rate of10◦C min−1 from room temperature to1000◦C.IR spectra were recordedon a Bruker IFS55Spectrometer in the range500–4000cm−1.Each spectrum was recorded at4cm−1resolution using500scans.Sample pellets were obtained from the powder calcinedat various temperatures by mixing with a small amount of KBr,then analyzing in transmission mode.Bulk composition wasdetermined using energy-dispersive x-ray spectroscopy(EDX)attached to a LEO1530scanning electron microscope(SEM).X-ray diffraction(XRD)was performed utilizing a Rigakuemploying Cu Kα1+Kα2=1.54184˚A radiation with a power of40kV–35mA for the crystalline analysis.The broad-scan analysis was typically conducted within the2θrangeof10◦–80◦.The samples were further analyzed using a RenishawModel2000Raman spectrometer equipped with a633nmlaser.The power at the sample varied between0.2and0.5mWwith the beam defocused to an area of approximately5–10μmin diameter.The textural characterization,such as surface area,pore volume and pore size distribution of the aerogels andthe oxides,was obtained by N2physisorption at77K usinga Micromeritics ASAP2010.Prior to the N2physisorption,the samples were degassed at200◦C under vacuum.Fromthe N2adsorption isotherms,the speciﬁc surface area wascalculated.The mesopore volume(V BJH),the average pore diameters(d p)and the pore size distributions were estimated by the Barret–Joyner–Halenda(BJH)method applied to the desorption branch of the isotherm.3.Results and discussion3.1.Electron microscopy(SEM/TEM)The effects of calcination temperature on the morphology of the Zr–TiO2nanotubes’shape and size were characterized by SEM and TEM analysis.In the SEM analysis for the as-prepared materials,it can be seen inﬁgure1(a)that the aerogel powders were composed of a one-dimensional structure,with the nanotubes having a diameter of50–140nm and a length of several micrometers.Throughout the course of heat treatment, phase changes(amorphous to anatase to rutile)and sintering phenomena of the nanotubes were revealed by SEM and TEM investigations.The SEM image inﬁgure1(b)shows that the material calcined at500◦C has a similar structure,although very small holes are visible in the TEM image(ﬁgure2(b))on the walls of the nanotubes.The morphology of the calcined nanotubes at800◦C is still preserved,as shown inﬁgure1(c). As the temperature was increased further to1000◦C,the initial nanotubes disappeared and were replaced by nanometer-sizedFigure1.SEM:Zr–TiO2nanotubes calcined at:(a)as-prepared,(b)500,(c)800and(d)1000◦C(bar represents200nm;all the samples were examined after platinum coating).aggregated particles in the50–100nm size range,as shown in ﬁgure1(d).Along with SEM,TEM images gave more detailed morphological information on the tubular structure.The TEM image inﬁgure2(a)indicates that the as-prepared nanotubes possessed uniform inner and outer diameters,having thicknesses approx.14–50nm along their length,depending on synthesis conditions.Upon heat treatment to500◦C, the chemically bonded organic layer was removed from the synthesized nanotubes,resulting in small holes on the tube wall,which was conﬁrmed by the TEM image in ﬁgure2(b),although the internal structure was still maintained at this temperature.The SEM images showed that the outer morphology was preserved at800◦C,although at900◦C the TEM images shown that the inner hole has almost vanished with this additional heat energy(ﬁgure2(c)).The TEM images along with SEM images reveal that the nanotubes were deformed and the crystallites were fused together when the calcination temperature was increased to1000◦C (ﬁgure2(d)).3.2.Decomposition behavior(TGA/FTIR)Thermo-gravimetric analysis(TG–DTG)analysis was carried out to study the thermal decomposition behavior of the synthesized Zr–TiO2nanotubes.Figure3shows three main peaks in the TG–DTG analysis,which are in the ranges of 20–120,120–250and250–500◦C.Theﬁrst stage with peak maxima at37◦C gave only6wt%loss,which we attribute to the removal of residual solvent present in the synthesized materials.The second peak with its maximum at200◦C is attributed to the removal of bounded water and chemically bonded organic material,with approx.19wt%lost at this stage. The third peak,with its maximum at341◦C,is broad and is attributed to the removal of any bonded/coordinated organic material and−OH groups,with approximately an additional 21wt%loss at this temperature.The weight loss over500◦C was extremely small(0.14%)and attributed to removal of bounded−OH groups.The total weight loss measured from the TG curve was46wt%.Elemental analysis(EDX)was also performed(see table1)to investigate the change of composition with calci-nation temperature.It showed that the as-prepared nanotubes contained approx.30%carbon,with this value decreasing upon increasing calcination temperature.At300◦C,the carbon content was about18%,while when the temperature was increased to500◦C,all carbon-containing organic material was removed,consistent with the TG–DTG results.Materials calcined at higher temperatures had only metal,oxygen and a ratio of oxygen to metal atom change with temperature. Surfaces of metal oxides consist of unsaturated metal and oxide ions,and are usually terminated by−OH groups.Figure 2.TEM:Zr–TiO 2nanotubes calcined at (a)as-prepared,(b)500,(c)900and (d)1000◦C.Figure 3.Weight loss of nanotubes as a function of temperature.The −OH groups are formed by dissociative adsorption of H 2O molecules to reduce the coordinative unsaturation of the surface sites.It is very difﬁcult to analyze the amount of oxygen bonded with metal atoms only by the EDX method,as the amount of H present in the materials cannot be determined by EDX due to the low atomic weight of H.Infrared spectroscopy is an excellent method to study the behavior and properties of metal oxides [28].The powder ATR-FTIR spectra of the Zr–TiO 2nanotubes calcined at different temperatures in air are given in ﬁgure 4.TheTable positional change with calcination temperature determined by EDX.Sample(Cal.temp ◦C)Carbon (at.%)Oxygen (at.%)Ti (at.%)Zr (at.%)As-prepared 30.6±258.5±19.9±0.51.3±0.2T-30018.5±362.7±116.5±12.3±0.5T-5000±379.6±317.9±0.52.5±0.5spectra (ﬁgure 4(a))for the as-prepared nanotubes shows a broad peak at 3400cm −1assigned to the −OH group of absorbed water [29].The peaks at 1548and 1452cm −1are due to symmetric and asymmetric stretching of the zirconium titanium acetate complex,respectively [30].This metal acetate complex conﬁrms that the acetic acid formed bridging complexes with the metal ions,helping to stabilize the structures during their synthesis and self-assembly into nanotubular structures in scCO 2.The −CH 3group contributes the small peak at 1343cm −1,while the two small peaks at 1037and 1024cm −1correspond to the ending and bridging −OPr groups,respectively [31],indicating that unhydrolyzed −OPr groups were present in the as-prepared materials [32].The oxo bonds can be observed by the bands present below 657cm −1[30].Calcination at 400◦C signiﬁcantly diminishes the intensity of the C–H stretching at 2800–3000cm −1and the zirconium titanium acetate complex band at 1548andFigure4.The powder ATR-FTIR spectra of Zr–TiO2nanotubes calcined at different temperatures.1452cm−1,spectra given inﬁgure4(b).This indicates thatthe calcination at400◦C removes any organic material presentin the as-prepared nanotubes.No trace of IR bands fromthe organic groups was detected upon further heat treatment(ﬁgure4(c))and the broad peak at3400cm−1signiﬁcantlydecreased.The nanotubes calcined at1000◦C showed onlya small band at3400cm−1(ﬁgure4(d)),indicating only asmall amount of−OH groups were still present at this hightemperature.These results are consistent with the TG–DTGmeasurements and the electron microscopy results.3.3.XRD and HRTEMIn order to examine the phase structure and crystallite size,XRD and HRTEM were used to investigate the effects of thecalcination temperature on the crystal size and phase structure.During heat treatment,the as-prepared materials transferredfrom the amorphous to anatase to rutile phases.The XRDpatterns(ﬁgure5)of all the calcined samples indicate that theZr–TiO2nanotubes consist of anatase crystal,with no rutile phase being present up to700◦C.The as-prepared materials were amorphous,while whenincreasing the calcination temperature up to400◦C,thematerial reorganized itself and the anatase particles beganto grow,resulting in crystalline material.There wasno distinct Zr peak,indicating no phase separation,andthat the Zr was integrated within the anatase crystalstructure for this composition.Previous experimentsshowed that increasing concentrations of Zr alkoxidewere incorporated homogeneously into the nanotubularstructure[22].As well,we previously prepared a crystalof Zr2Ti4(μ3–O)4(OPr)4(μ–OPr)2(μ–OAc)10using lower concentrations of acetic acid[33],showing that the Zr is partof the crystal hexamer structure.By increasing the calcinationtemperature from400to800◦C,the peak intensities increasedas well as the width of the peaks becoming narrower,indicatingan improvement of the anatase phase and simultaneously thegrowth of anatase crystallites.The XRD patterns for theobserved nanotubes indicate that no rutile phase appeared upto calcination temperatures of700◦C.It also shows thatheat Figure5.The powder XRD spectra of Zr–TiO2nanotubes calcined at different temperatures.(A–anatase,R–rutile).Table2.Crystal size and crystal structure at different calcination temperatures.Sample(Cal.temp◦C)Crystallite size(nm)Crystal structure As-prepared—AmorphousT-4009.8AnataseT-50012.5AnataseT-60015.9AnataseT-70019.5AnataseT-80021.8Anatase(88%)48.1Rutile(12%)T-90027.8Anatase(42%)69.5Rutile(58%)T-100028.3Anatase(6%)90.8Rutile(94%) treatment at800◦C forms a very small peak of the rutile phase, and further heat treatment increased the amount of rutile phase, and a new peak appeared at2θ=30.4◦,which is assigned to zirconium titanium oxide(ZrTiO4)[28].The crystallite sizes of the calcined samples are summarized in table2and were estimated from these XRD patterns using Scherrer’s equation(equation(1)):D=0.9λβcosθ(1) where D is the average nanocrystallite size(nm),λis the x-ray wavelength(1.541˚A),βis the full width at half-maximum intensity(in radians)andθis half of the diffraction peak angle.For the nanotubes calcined at400◦C,crystallite sizes of approx.9.8nm were calculated,while further heat treatment increased the crystallite size moderately.Nanotubes calcined at800◦C gave crystallite sizes up to21.8nm,resulting in smaller crystallite materials,indicating that a small amount of zirconia inhibited grain growth during heat treatment[26]. The rutile crystallite size was calculated by Scherrer’s equation using rutile(110).The obtained crystallite size was>90nm at1000◦C calcination temperature,smaller than the value reported in the literature for rutile crystallites[29],likely due to the constrained geometry of the nanotubular structure.The TEM images of the1000◦C calcined nanotubes previously showed that the crystallites were fused together,formingFigure6.HRTEM:Zr–TiO2nanotubes calcined at(a)as-prepared,(b)500and(c)1000◦C.Bar represents10nm. larger crystallites.It is known that rutile and anataseshare two and four polyhedra edges,respectively,althoughboth are tetragonal[34].Due to changes in the crystalstructure,the nanotubes’morphology deformed at highercalcination temperatures.The phase compositions of the calcined samples are alsoreported in table2and were calculated using the integratedintensities of anatase(101)and rutile(110)peaks by theequation developed by Spurr and Myers[35]:X rutile=11+K(I a/I r)(2)where I a and I r are the integrated peak intensities of the anatase and rutile phases,respectively,and the empirical constant K was taken as0.79according to Spurr and Myers.From table2 we see that the material calcined at800◦C contained only12% rutile.Further heat treatment caused a dramatic increase in both the composition and size of the rutile particles,where at 900◦C,58%of the material was converted into rutile,which increased to approx.94%at1000◦C.Rutile is the most stable crystalline phase of TiO2and the phase transformation(anatase to rutile)depends on both the size and dopant present in the system[26].Sui et al reported that anatase-type TiO2nanostructures transformed into rutile phase(56%)after calcination at600◦C[18],whereas only 12%of the nanomaterial was converted into rutile at800◦C in the present study.Hence,consistent with the literature for non-nanotubular structures[23],the anatase phase of the bimetallic nanotubes can be stabilized by modifying titania with a small amount of ZrO2.However,the mechanism by which the zirconia stabilizes the TiO2anatase phase at higher temperatures is unclear.As shown in the XRD data,no distinct zirconium peak was observed,indicating that zirconia was well integrated into the anatase structure.This suggests that particle agglomeration was not favored and the particles grew by the Oswald ripening process during heat treatment[26].Due to the presence of zirconia,the Oswald ripening process was restricted,reducing the crystal growth rate and increasing the phase transformation temperature.Once individual crystallites reach a threshold size,a spontaneous phase change can occur.The rapid growth of rutile particles formed during heat treatment suggests that the growth mechanism consists of particle agglomeration or grain coalescence by grain boundary diffusion[36].There may be a threshold size limit for this transformation,below which no transformation occurs,which was>27nm for this study, similar to that reported for SiO2and ZrO2doping in a titania matrix[26,36].For this reason,no rutile phase appeared up to 700◦C.When calcining at800◦C,the crystallites size became >27nm and the anatase phase started transforming rapidly into the rutile phase.In addition to the XRD data,detailed information on the structural transformations and crystal growth can be obtained using HRTEM.The HRTEM micrographs of the as-preparedFigure7.The Raman spectra of Zr–TiO2nanotubes calcined at: (a)500,(b)600,(c)700,(d)800,(e)900and(f)1000◦C.(A–anatase,R–rutile).nanotubes were amorphous(ﬁgure6(a))with no ordered structure.The lattice image of nanotubes calcined at500◦C is given inﬁgure6(b),showing a grain size of approx.12nm width with a d spacing0.35nm,very close to the lattice spacing of the(101)planes of the anatase phase.However, all grains were not the same in terms of size and shape. Some were long with signiﬁcant lattice mismatch and grain boundaries.All these defects prevent rapid grain growth.The HRTEM image for the material calcined at1000◦C is given inﬁgure6(c),where little amorphous phase is observed.The crystallites were very large having a d spacing of0.245nm, which value is very close to the rutile(110)plane of titania. These observations also support the XRD and TEM analysis.3.4.RamanTo further verify these results,Raman spectra for the bimetallic nanotube samples were measured for several different calcination temperatures,as shown inﬁgure7.The spectrum for the sample calcined at500◦C(ﬁgure7(a)) shows Raman peaks at142,395,517and639cm−1,which can be assigned to the E g,B1g,B1g/A1g and E g modes of the anatase phase of titania,respectively,which agrees with published values[37].With increasing calcination temperature inﬁgures7(b)–(d),the intensity of the anatase phase increased, indicating a larger particle size being present,with the anatase peak shifting to lower frequencies.Lottici et al[38]explained this effect as the size-induced pressure effect on the vibrational modes,with the smaller the crystallite size,the higher the pressure and Raman frequencies.After calcining at900◦C (ﬁgure7(e)),three new peaks at230,442and612cm−1 appeared,which match the literature values for the rutile phase[37,38].Upon calcining at1000◦C(ﬁgure7(f)), all anatase-related peaks vanished and only the rutile-related peaks remain,indicating complete anatase to rutile phase transformation.The Raman results agree with the previous characterizationresults.Figure8.Ln of anatase and rutile crystallite size in nm as a function of the reciprocal of absolute temperature according to equation(3). (—anatase,•/◦—rutile).3.5.Activation energy of phase transformationsThe activation energy(kJ mol−1)of phase transformations can be calculated from the slope of a plot of ln rutile weight fraction versus the reciprocal of annealing temperature from XRD spectra according to Burns et al[39].This relationship is given asE a=−∂ln(X r)∂(1/T)R(3) where T is the temperature in kelvin,R is the universal gas constant(8.314J mol−1K−1)and X r is the weight fraction of the rutile phase as determined using equation(2).Figure8 shows the plot of the logarithm of the average crystallite size versus the reciprocal of the calcination temperature(solid lines),according to equation(3).A linear relationship is observed and the activation energy for the crystal growth of the Zr–TiO2nanotubes was calculated as12.8kJ mol−1and 36.4kJ mol−1for anatase and rutile phases,respectively.The activation energy for the phase transformation from anatase to rutile was also calculated,as shown by the dashed line in ﬁgure8,with a value of171kJ mol−1.These values are higher than those reported for nanocrystalline pure titania[39,40], which is a beneﬁcial effect of doping.In explanation,during the heat treatment,there are two competitive processes:grain growth and A→R phase transformation in the nanocrystalline materials.Both processes are easier for small grain-sized materials because the activation energy for growth and phase transformation is lower[41].3.6.BET analysisThe textural properties,i.e.the surface area,pore volume and pore size distributions of the as-prepared and calcined bimetallic nanotubes were characterized by nitrogen adsorp-tion studies.Figure9shows the nitrogen adsorption isotherms for both the as-prepared and calcined materials,which exhibit H3hysteresis loops(to800◦C),typical for mesoporous materials.The isotherm for the bimetallic metal oxide nanotubes calcined at1000◦C changes to a type I isotherm, typical for a microporous material[42].The lower limit ofFigure9.N2adsorption/desorption isotherm of the Zr–TiO2 nanotubes calcined at differenttemperatures.Figure10.BJH pore size distribution of the Zr–TiO2nanotubes calcined at different temperatures.the relative pressure for the hysteresis loop is characteristic of a given adsorbate at a given temperature[43].It can be seen fromﬁgure9for both the as-prepared nanotubes,and for those calcined at300and400◦C that the lower pressure limit of the hysteresis loop is at P/P0=0.4.Calcinations at higher temperatures increase this value,e.g.at600◦C,P/P0increasesto approx.0.55,while after800◦C this value increases to0.6, indicating that the pores are becoming larger as the materials are calcined at higher temperatures.To evaluate the pore size distribution,the as-prepared and calcined materials were plotted as shown inﬁgure10. The average pore diameter for the as-prepared nanotubes is approx.3nm,whereas upon calcination the pore size became gradually larger and the pore size distribution shifted,forming larger pores at the expense of the smaller ones.At600◦C the pore size is approx.9nm while,when the nanomaterials were calcined at800◦C,the pore size became more than double at approx.19nm.Calcining the materials at1000◦C collapsed the small pores,resulting in only larger pores.The surface properties of the Zr–TiO2nanotubes calcined at different temperatures are summarized inﬁgure11.The as-prepared Zr–TiO2nanotubes,having a surface area of 430m2g−1,gradually decrease through calcination.Due to the sintering phenomena,the small pores collapse,reducingthe Figure11.Surface area and pore volume of the Zr–TiO2nanotubes calcined at different temperatures as a function of calcination temperature.pore volume and surface area.The transformations into anatase and rutile crystalline phases will also help to reduce the surface area.Interestingly,literature values show that pure titania has almost zero surface area at this high temperature[44,45]. Hence,the presence of10%zirconia increased the thermal stability and reduced grain growth rates during the course of heat treatment,resulting in moderate(23m2g−1)surface areas at very high temperature.4.ConclusionsThe thermal behavior of the Zr–TiO2nanotubes synthesized by an acid-modiﬁed sol–gel process in scCO2has been investigated in detail using SEM,TEM,TG,EDX,FTIR, XRD,HRTEM,Raman and BET analysis.SEM and TEM analysis conﬁrmed that the morphology of the nanotube structure was preserved at up to800◦C,whereas further heat treatment deformed the tubes.FTIR and EDX analysis showed that different organic residues were removed,depending on the calcination temperature.Along with HRTEM and Raman, XRD results showed anatase nanocrystallites were formed after calcining at400◦C,while no rutile phase appeared until calcining at700◦C,with further heat treatment resulting in a rutile phase transformation and ZrTiO4being formed. The activation energy for anatase and rutile crystal growth was calculated and the values were12.8and36.4kJ mol−1, respectively.The activation energy for phase transformation was determined to be171kJ mol−1,higher than that of pure titania nanomaterials.The as-prepared nanotubes had a430m2g−1speciﬁc surface area(SSA),whereas after calcining at1000◦C the SSA was reduced to23m2g−1. Hence,the ZrO2present in the titania matrix increased thermal stability,reducing grain growth resulting in smaller crystallites, and hence preserving the morphology and surface area at high temperatures.AcknowledgmentsThe authors would like to thank Nancy Bell from the UWO Nanotechnology Centre for the SEM analysis,Ronald。

数值分析原理习题答案

数值分析原理习题答案【篇一：数值分析习题】学号班级习题主要考察点：有效数字的计算、计算方法的比较选择、误差和误差限的计算。

1 若误差限为0.5?10,那么近似数0.003400有几位有效数字？（有效数字的计算） 2 ??3.14159?具有4位有效数字的近似值是多少？（有效数字的计算）3 已知a?1.2031，b?0.978是经过四舍五入后得到的近似值，问a?b，a?b有几位有效数字？（有效数字的计算）4 设x?0，x的相对误差为?，求lnx的误差和相对误差？（误差的计算）**5测得某圆柱体高度h的值为h?20cm，底面半径r的值为r?5cm，已知?5|h?h*|?0.2cm，|r?r*|?0.1cm，求圆柱体体积v??rh的绝对误差限与相对误差限。

（误差限的计算）6 设x的相对误差为a%,求y?xn的相对误差。

（函数误差的计算） 7计算球的体积，为了使体积的相对误差限为1%，问度量半径r时允许的相对误差限为多大？（函数误差的计算）128 设in?e?1nxx?edx，求证： 0（1）in?1?nin?1(n?0,1,2?)（2）利用（1）中的公式正向递推计算时误差逐步增大；反向递推计算时误差逐步减小。

（计算方法的比较选择）第二章插值法姓名学号班级习题主要考察点：拉格朗日插值法的构造，均差的计算，牛顿插值和埃尔米特插值构造，插值余项的计算和应用。

1 已知f(?1)?2,f(1)?1,f(2)?1，求f(x)的拉氏插值多项式。

（拉格朗日插值）2 已知y?x,x0?4,x1?9，用线性插值求7的近似值。

（拉格朗日线性插值） 3 若xj(j?0,1,...n)为互异节点，且有lj(x)?试证明(x?x0)(x?x1)?(x?xj?1)(x?xj?1)?(x?xn)(xj?x0)(xj?x1)?(xj?xj?1)(x j?xj?1)?(xj?xn)?xlj?0nkjj（拉格朗日插值基函数的性质） (x)?xk(k?0,1,...n)。

单向凝固中温度梯度和凝固速度对铝硅合金共晶生长形态的影响

··凝固过程中，固-液界面的生长形貌是影响合金组织性能以及缺陷形成的重要因素。

Al-Si 合金具有优异的铸造性能，良好的力学性能及其他物理化学性能，是研究和应用最为广泛的铸造铝合金，占铝铸件产量的85%~90%，且适用于各种铸造方法。

大多数的铸造缺陷往往出现在合金凝固的最后阶段，而铝硅共晶体是铝硅合金凝固最后阶段的主要组织，铝硅共晶体的生长形态是影响合金性能的重要因素。

20世纪60年代，Kim 和Heine [1]对变质和未变质共晶铝硅合金的凝固模式进行了研究，发现变质合金中共晶体倾向于从铸件表层向中心生长，而未变质合金中共晶体在熔体中沿任意方向生长。

A.K.Dahle 等[2-7]对铝硅合金共晶凝固形核与生长进行了大量研究，发现未变质A356合金中有独立形核的共晶团且大部分α-Al 共晶体和树枝状α-Al 晶体取向一致，说明先析出α-Al 相是共晶体有效形核核心；另外还发现未变质工业铝硅合金中共晶体的形核是充分的，而未变质高纯铝硅合金中只有很少共晶团形成。

加入Sr 后，工业铝硅合金中形核率降低，而对高纯铝硅合金无明显影响。

蔡惠民和孙伟成等[8-10]对共晶凝固的机制及组织形态进行了研究，发现凝固条件不同得到的硅晶体形貌差别很大。

然而，他们的研究重点都是共晶组织的微观形成机制，探讨共晶组织中α-Al 相和Si 相的生长形态和相互关系，而没有当成一个整体来研究共晶体的生长形态。

铝硅合金凝固后期往往是共晶体的凝固过程，也是合金中微孔大量形成的阶段，本文研究了不同凝固条件下共晶体的生长形态，这将对研究铝合金中微孔的形成有重大的理论意义和应用价值。

由于凝固过程中固液界面形貌演化是一个涉及热量、质量和动量传输，以及界面动力学和毛细作用效应的自由边界问题，这一问题的复杂性造成目前在试验研究和理论分析上存在许多障碍，故在一黄婉如，廖恒成，吴申庆，孙国雄（东南大学材料科学与工程学院，江苏南京211189）摘要：对Al-13Si-0.2Sr-0.35Mg 合金进行了一系列单向凝固实验，研究了界面前沿温度梯度G L 和生长速度R 对铝硅合金共晶生长形态的影响。

jacs fe

† ‡
Sharp Laboratories of America, Camas, Washington 98607, United States The University of Texas at Austin, Austin, Texas 78712, United States § Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States ∥ Corporate Research & Development Division (CRDD), Sharp Corporation, Tenri 632-8567, Japan ⊥ Department of Physics, New York University, New York, New York 10003, United States # Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, United States
S Supporting Information *
ABSTRACT: A novel air-stable sodium iron hexacyanoferrate (R-Na1.92Fe[Fe(CN)6]) with rhombohedral structure is demonstrated to be a scalable, low-cost cathode material for sodiumion batteries exhibiting high capacity, long cycle life, and good rate capability. The cycling mechanism of the iron redox is clariﬁed and understood through synchrotron-based soft X-ray absorption spectroscopy, which also reveals the correlation between the physical properties and the cell performance of this novel material. More importantly, successful preparation of a dehydrated iron hexacyanoferrate with high sodium-ion concentration enables the fabrication of a discharged sodiumion battery with a non-sodium metal anode, and the manufacturing feasibility of low cost sodium-ion batteries with existing lithium-ion battery infrastructures has been tested.

SIXSIGMA-项目移转与文化变革(ppt 21)

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•黑帶負責訂出時程與引導會議進行 •會議長度約1 – 2 小時 : 屬工作會議!! •依據 DMAIC階段解討專案 •主要學習心得與控制計劃細節 •財務結果與計算之覆核 •參加人員包括, 專案成員, 流程控制者, 流程監控者與主要影響範圍內的主要成員 •大黑帶必須出席 •專案成果正式生效程序是必須的
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• 有效控制計劃的關鍵- 確認不足部分 • 每個重要的X值均持續地被改善 • 著重於以先買的方式來改善流程(在失敗發生前
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Structural change in Bi 4(Si x Ge 12x )3O 12glassesduring crystallizationJ.H.Cho a,b ,S.J.Kim a ,Y.S.Yang a,*aResearch Center for Dielectric and Advanced Matter Physics,Department of Physics,Pusan National University,Pusan 609-735,South KoreabInoue Superliquid Glass Project,ERATO,Japan Science and Technology Corporation (JST),Sendai 982-0807,JapanReceived 23April 2001;accepted 4June 2001by H.TakayamaAbstractStructural changes in Bi 4(Si x Ge 12x )3O 12glasses during crystallization,where x 0;0.33,0.5,0.66and 1,were investigated systematically by X-ray diffraction (XRD),differential scanning calorimetry (DSC),Raman spectroscopy and scanning elec-tron microscopy (SEM).The results showed that the glass transition temperature T g ,crystallization temperature T c and maximum crystallization temperature T p ,observed on DSC,increased as the Si content increases.The presence of two crystal-line phases,metastable Bi 2(Si,Ge)O 5and stable Bi 4(Si,Ge)3O 12,were detected by XRD.For x 0:5;it was possible to detect only the metastable crystalline phase at the early stage of crystallization.For the rest of the compositions,the metastable and stable crystals were almost formed simultaneously at this early stage.SEM morphology showed that the crystallization was dominated by crystal growth without prominent nucleation.It was concluded that the structural change occurs as follows:amorphous !amorphous'1Bi 2(Ge,Si)O 5crystal !Bi 4(Ge,Si)O 12crystal.q 2001Elsevier Science Ltd.All rights reserved.PACS :61.43.Er;61.50.2f;81.05.KfKeywords:A.Disordered systems;C.X-Ray scattering;D.Phase transitions1.IntroductionGlasses containing heavy metal oxide cations were proposed as candidates for the scintillation detector because of their high stopping power and medical imagery that has long been known.Moreover,the large polarizations of heavy metal oxide glasses are shown to be suitable for non-linear optical instruments [1].Bismuth germanate Bi 4Ge 3O 12(BGO)and bismuth silicate Bi 4Si 3O 12(BSO)are synthetic crystallines having an eulytite structure.These crystals are attractive because of their electro-optical [2],electro-mechanical [3]and intense luminescence proper-ties,the short wavelength of radiation [4],laser materials when rare earth elements are doped,and the fact that they are non-hygroscopic.In particular,BGO has received a lot of attention because of its uses as a scintillator,in X-ray and positron emission tomography,and telescopes for hard X-and g -rays in astrophysics and nuclear physics.There has been great effort to obtain high quality crystal for use in such applications and to understand the lumi-nescence mechanism [5±7].In order to produce these crystals successfully,a detailed fundamental knowledge of their growth behavior is necessary.But studying glass systems with properties similar to these crystals and obtain-ing the polycrystalline materials via glasses is easier than preparing single crystals.For BGO,there have been iso-thermal and non-isothermal experiments for crystallization in the molten and glass states [8,9],and studies of thermal and structural properties in the molten state [10,11].But even though there have been theoretical studies for the optical and solidi®cation properties of BSO [7,12],its crystallization process is not well known.In this work,the authors report a systematic study of structural changes on the crystallization of 2Bi 2O 3-3[x SiO 2-(12x )GeO 2]glasses.2.ExperimentalCrystalline powders of 2Bi 2O 3-3[x SiO 2-(12x )GeO 2],with x 0,0.33,0.5,0.66and 1were well mixed andSolid State Communications 119(2001)465±4700038-1098/01/$-see front matter q 2001Elsevier Science Ltd.All rightsreserved.PII:S0038-1098(01)00259-9PERGAMON /locate/ssc*Corresponding author.Tel.:182-51-510-2958;fax:182-51-516-5682.E-mail address:ysyang@hyowon.pusan.ac.kr (Y.S.Yang).annealed for 20h at 1123K.The mixture was ground and the same annealing process was performed for 6h.The powder samples were melted in a cylindrical electric furnace at 1423K in air.The melted sample was quenched through a twin roller.The estimated cooling rate was about 105K s 21.The glass samples identi®ed by X-ray diffraction (XRD)and differential scanning calorimetry (DSC)were transparent and had a thickness of 25m m.The glass tran-sition temperature T g ,crystallization temperature T c ,the maximum crystallization temperature T p and the thermal characteristics during crystallization were obtained from DSC measurements by applying a heating rate of 5K min 21.XRD,DSC,SEM and Raman spectroscopy measurements were conducted to investigate the crystalli-zation processes.3.Results and discussionIn this study,we simplify the notations according to the Ge/Si atomic mole ratio and thus they are:Bi 4Ge 3O 12 BGO,Bi 4Ge 2SiO 12 BGSO (2:1),Bi 4Ge 1.5Si 1.5O 12 BGSO (1:1),Bi 4GeSi 2O 12 BGSO (1:2),Bi 4Si 3O 12 BSO and Bi 4(Si x Ge 12x )3O 12 BGSO.T g and T c were determined as the point where the DSC curves depart from the hori-zontal baselines and meet with an extrapolation at in¯ection of thermal chord.The composition dependence of the thermal histogram of the glasses in DSC at a heating rate of 5K min 21and their characteristic temperatures are shown in Fig.1and Table 1.As can be seen in the ®gure and table,T g ,T c ,T p and the magnitude of T c 2T g ,which is one of the measures of stabi-lity of the glass,increase as Si content increase.This is due to increasing connectivity in the network with increase in SiO 2content and decreasing polymerization with increase in GeO 2content.The glass-forming ability is often given as D (T c 2T g )/(T m 2T c ),where T m is melting temperature and the better glass-forming ability is obtained by increasing the value of D .The melting temperatures of BGO and BSO are ,1050and ,1170K,respectively,and the correspond-ing D values are ,0.25and ,0.40for BGO and BSO.Therefore it can be concluded that the glass-forming abilityJ.H.Cho et al./Solid State Communications 119(2001)465±470position dependence of thermal histogram of the BGSO glasses in DSC at a heating rate of 5K min 21.Table 1Characteristic temperatures obtained from DSC curve in Fig.1T g (K)T c (K)T p (K)BGO704772794BGSO (2:1)708790818BGSO (1:1)709802835BGSO (1:2)711812845BSO721849868continues to increase with decrease in x.In the®gure,the DSC curves are asymmetric for all compositions.This feature often appears due to self heating and is apparent for materials with poor thermal and electric conductivities, such as dielectrics.In order to identify the nature and origin of the small shoulder and main peak in DSC curve in Fig.1,the authors carried out the isothermal heat treatment in the electric furnace and nonisothermal heat treatment in DSC.Then XRD measurements were performed for the annealed samples.In the case of isothermal heat treatment in the electric furnace,the sample environment changed from a given annealing temperature to room temperature within a second.This should be suf®ciently fast for only very little structural change to take place during quenching.For the heat treatment in DSC,the initial cooling rate was about40±50K min21.Some structural change could occur during quenching.The quenching point used in the DSC curve was a few degrees lower than the ideal quenching tempera-ture to avoid the effect of a relatively low quenching speed. Fig.2(a)shows the XRD patterns taken at room tempera-ture for quenched sample and samples quenched rapidly after isothermal heat treatment in the electric furnace at 773and1073K for1h.The XRD pattern of the quenched sample represents the glass state.The sample heated at 1073K exhibits the stable BGO crystalline phase(label s). Meanwhile,the XRD pattern of the sample quenched at 773K shows not only the scattering from stable BGO crystal but also some extra peaks,identi®ed as scattering from the orthorhombic Bi2GeO5crystalline phase(label m).These results indicate that the single DSC peak of BGO in Fig.1consists of two crystallization processes.In order to make the intermediate crystallization process clear,we measured XRD with the BGSO(1:1)sample quenched after heat treatment in DSC with a heating rate of0.5K min21.This low heating rate was applied in order to minimize the overshoot at the quenching temperature.The quenching point was783KÐjust below the small exo-thermic shoulder in the DSC curve,which is similar to that shown in Fig.1.As can be seen in Fig.2(b),the metastable crystalline phase of Bi2(Ge,Si)O5is dominant.With further heatingJ.H.Cho et al./Solid State Communications119(2001)465±470467Fig.2.XRD patterns taken at room temperature of(a)BGO for as quenched sample and samples after isothermal heat treatment in the electric furnace at773and1073K for1h;(b)and(c)are the XRD patterns for BGSO(1:1)and BSO after heat treatment in DSC with a heating rate of 0.5K min21at given conditions.The notation m and s indicate the metastable Bi2(Ge,Si)O5and stable Bi4(Ge,Si)3O12crystalline phases, respectively.to 973K in DSC,the glass and metastable phases com-pletely transform into the stable BGSO crystal.It is not clear whether the BGSO (1:1)glass transforms into the metastable and stable phases in sequence,but it is worth-while only having the intermediate crystalline phase.Fig.2(c)shows XRD patterns obtained after performing the same experimental procedure in Fig.2(b),and the results are similar to those shown in Fig.2(a).Fig.3shows XRD patterns of BGO taken at room temperature after heat treatment in the electric furnace at 733K for a given time.This measurement was taken to identify the initial sequence of production in the metastable and stable crystalline phases.At an early stage,the scatter-ings from the metastable Bi 2GeO 5and stable BGO crystal-line phases occur.With increase in annealing time,intensities from both the crystalline phases continue to increase.This indicates that the BGO glass almost simul-taneously transforms into two crystalline states at the early stage.The metastable Bi 2GeO 5crystal eventually transforms into the ®nal BGO crystal at a later stage as can be seen in Fig.2.These results clearly show that the structural change of BGO glass on heating is as follows:glass !glass 01metastable Bi 2GeO 5crystal !stable BGO crystal.Fig.4shows SEM morphology taken with the same sample used for XRD measurement in Fig. 3.After 40min,small nuclei appear and then continue to grow as heat treatment time increases.The ®gure shows that there is no pronounced increase in the number density of nuclei,but the growth of crystal is dominant.XRD patterns in Fig.3show the coexistence of the metastable Bi 2GeO 5and stable BGO crystalline phases.It should be noted that the bright and dark (a few crystals penetrate into large ones)parts of the SEM ®gure correspond to the metastable and stable crystalline phases,respectively.Fig.5shows the Raman spectra for samples annealed at 733K for 80,160and 320min in the electric furnace.Small peaks from crystals which are on top of a broad band that is characteristic of glass that appear after 80min.The peakJ.H.Cho et al./Solid State Communications 119(2001)465±470468Fig.3.XRD patterns of BGO taken at room temperature after heat treatment in the electric furnace at 733K for given time.intensity continues to increase as annealing time increases. Comparing with the XRD data,it is concluded that these Raman peaks are from the metastable and stable crystals.It is known that the low frequency region is nearly inde-pendent of x and the most evident change appears for the peak at,360cm21in BGO,which shift to,390cm21in BSO.Other distinguishable behavior is found in the high frequency region,where the energies corresponding to ,720and,820cm21in BGO shift to,820and ,880cm21in BSO.In the eulytite crystals,the Raman spectra is often interpreted in terms of the internal vibrations of the tetrahedral groups(GeO4,SiO4)and the external vibrations corresponding to the motion of the total tetra-hedral groups against Bi31ions.The internal motions are assigned to Raman modes with frequencies greater than ,360cm21.The authors have shown the Raman mode for BGO onlyJ.H.Cho et al./Solid State Communications119(2001)465±470469Fig.4.SEM morphologies of BGO taken with the same heat treatment conditions used in Fig.3.Fig.5.Raman spectra of BGO taken with the same heat treatment conditions used in Fig.position dependence of lattice spacing obtained from XRD for the stable cubic phase.The mole fraction x is from Bi4(Si x Ge12x)3O12.and found a good agreement with other studies.They also measured the Raman spectrum of BSO and found a pronounced Raman shift towards a higher frequency for the SiO4vibration mode compared with that of GeO4 mode[13].Fig.6shows the change in lattice spacing on various compositions for(211),(310)and(321)scattering planes obtained from XRD.The known lattice parameters for polycrystals of BSO and BGO are1.0288and1.0513nm, respectively[14].The present results give the values as 1.0287for BSO and1.0517nm for BGO,which are very close to those of Fisher and Waldner[14].The continuous increase of lattice parameters increase as the Ge/Si ratio increases is attributed to the large atomic size of Ge compared with the value of Si,and there are no appreciable anomalies for the stable cubic phase.It has been known that BGO and BSO crystals exhibit very different optical behavior[12].But no studies have been reported for the structural properties when the mixture of disordered BGO and BSO transform into the ordered crystalline phase.Our results can provide some information for further studies of the combined BGO and BSO systems.4.SummaryStructural properties during crystallization of amorphous Bi4(Si x Ge12x)3O12have been studied.The DSC results show two step crystallization for all compositions on heating.The ®rst exothermic shoulder in the DSC curve is due to the precipitation of Bi2(Ge,Si)O5from the amorphous and the main peak is caused by the crystallization of the Bi4(Si x-Ge12x)3O12stable phases.For the crystallization of BGO and BSO,both metastable and stable crystalline phases appear at the same time.Mean-while,for BGSO(1:1),it is possible to have an XRD pattern corresponding to only metastable crystalline structure. Concerning the crystallization behavior,two processes can be considered.First amorphous!crystal Bi2(Ge, Si)O5!crystal Bi4(Ge,Si)3O12.Secondly,amorphous! amorphous01crystal Bi2(Ge,Si)O5!crystal Bi4(Ge, Si)3O12.The authors do not rule out the®rst crystallization process.But,in spite of their efforts,there is no evidence that an amorphous phase completely crystallizes into the metastable Bi2(Ge,Si)O5phase and Bi4(Ge,Si)O12crystal-lization subsequently follows.Therefore,it is concluded that the crystallization process follows the second process in most cases.The BGO sample annealed at733K for80min shows a broad peak originating from glass and the small peaks origi-nating from crystals in Raman spectroscopy.The well-de®ned modes from the internal and external vibrations of GeO4are obtained for the fully crystallized BGO.The intensities of these modes increase as crystal volume fraction increases.It is found that the estimated lattice parameters of1.0287 and1.0517nm for BSO and BGO crystals are in agreement with the values reported previously.The lattice parameters continue to increase as Ge/Si ratio increases.AcknowledgementsThis work was supported by the Ministry of Information and Communication of Korea(Support Project of University Foundation Research(2000)supervised by IITA).References[1]N.Sugimoto,H.Kanbara,S.Fujiwara,K.Tanaka,Opt.Lett.21(1996)1637.[2]J.Link,J.Fontanella,J.Appl.Phys.51(1980)4352.[3]H.Schweppe,IEEE Trans.Sonics Ultras.SU-16(1969)219.[4]R.Moncorge,B.Jacquier,G.Boulon,J.Lumines.14(1976)337.[5]G.Boulon,B.Moine,J-C.Bourcet,Phys.Rev.B22(1980)1163.[6]A.Horowitz,G.Kramer,J.Cryst.Growth79(1986)296.[7]C.Lin,S.Motakef,J.Cryst.Growth128(1993)834.[8]L.Dimesso,G.Gnappi,A.Montenero,P.Fabeni,G.P.Pazzi,J.Mater.Sci.26(1991)4215.[9]G.Corsmit,M.A.van Driel,R.J.Elsenaar,W.van de Guchte,J.Cryst.Growth75(1986)551.[10]K.Shigematsu,Y.Anzai,K.Omote,S.Kimura,J.Cryst.Growth137(1994)509.[11]S.Takeda,K.Sugiyama,Y.Waseda,Jpn.J.Appl.Phys.32(1993)5633.[12]J.F.Rivas-Silva,M.Berrondo,J.Phys.Chem.Sol.59(1998)1627.[13]P.Beneventi,D.Bersani,P.P.Lottici,L.Kovacs,Sol.Stat.Commun.93(1995)143.[14]P.Fisher,F.Waldner,mun.44(1982)657.J.H.Cho et al./Solid State Communications119(2001)465±470 470。