a r X i v :0704.2171v 2 [c o n d -m a t .s t a t -m e c h ] 5 O c t 2007Fluctuation-induced interactions between dielectrics in general
geometries
S.Pasquali, A.C.Maggs
Laboratoire de Physico-Chime Th′e orique,Gulliver CNRS-ESPCI 7083,10rue Vauquelin,75231Paris Cedex 05,France.Abstract We study thermal Casimir and quantum non-retarded Lifshitz interactions between dielectrics in general geometries.We map the calculation of the classical partition function onto a determinant which we discretize and evaluate with the help of Cholesky factorization.The quantum partition function is treated by path integral quantization of a set of interacting dipoles and reduces to a product of determinants.We compare the approximations of pairwise additivity and proximity force with our numerical methods.We propose a “factorization approximation”which gives rather good numerical results in the geometries that we study.PACS numbers:05.10.-a 41.20.Cv
I.INTRODUCTION
Much is known about?uctuation-induced interactions between bodies.One distinguishes two parts to the interaction[1,2,3]:The?rst is quantum in nature,and corresponds to a multi-center generalization of London dispersion forces.The second,the so called thermal Casimir e?ect,has as its origin the thermal excitation of polarization modes in dielectrics,giving rise to temperature dependent forces.Together they form the multi-body generalization of the long-ranged part of the van der Waals interaction.An alternative,but equivalent vision comes from associating these forces to the energy and entropy of?uctuating electrodynamic?elds.A number of sophisticated theoretical techniques have been applied to the calculation of this interaction,but only the simplest of geometries are analytically tractable to exact solution,for instance planar surfaces[4],spheres or cylinders.
More complicated physical situations are generally studied with perturbation theory,or in exceptional cases with non-perturbative methods adapted to individual geometries[5]. Such methods,that keep into account the totality of the interaction,are usually developed on the basis of Lifshitz theory for dielectrics.Analytic methods are typically based on an approximation around one of the few solvable problems.These range from the simple proximity force approximation,to a semi-classical interactions approximation[6],to the multipole expansion[7],to the more recent and sophisticated ray optics approximation[8]. The main problem with these methods is they are limited when the system under study departs too far from the unperturbed system.Given the limitations of all analytic methods, which are in practice only capable of studying very regular geometries,numerical approaches have been developed.These methods have been limited both by storage space required and by long computational times,making it possible to study only two-dimensional systems, or three-dimensional systems translationally invariant in one direction[9,10].It is only very recently that more powerful methods have been developed based on the numerical calculation of the Maxwell stress tensor[11].The numerical approach we present in this paper is complementary to this last method,but it is based on somewhat di?erent theoretical grounds.We will explain in the discussion how the two methods compare in detail.
We note,as an aside,that so called“atomistic modeling”of materials,as performed with most molecular dynamics codes,with the aim of understanding micro-mechanical response, conformations of macromolecules or interfaces,often uses assumptions such as pairwise ad-
ditivity of interactions[12]parameterized with phenomenological Lennard-Jones potentials. These methods neglect retardation,screening of the classical interaction by ions and multi-body e?ects.It is of clear interest to develop methods which will enable one to have a better quantitative understanding of such e?ects in both soft and hard condensed matter physics.More sophisticated(and even more costly)quantum simulation methods based on local density functionals exsist,but are known to miss long-ranged dispersion interactions completely[13].
In this paper we want to focus on important geometries which are clearly beyond per-turbative study.One simple example is the case of a tip near a structured surface,where the curvature of the surface makes most approximation methods ine?ective,unless the di-electric contrast is very weak[14].This paper has as its principle aim the generalization of a recent paper[15]which used direct diagonalization of a large matrix in order to calcu-late the free energy of?uctuating dielectrics in the classical(thermal Casimir)limit.The methods in[15]were rather limited,only very small three dimensional systems could be studied.Quantum e?ects were neglected entirely.In this paper we use more sophisticated factorization techniques which allow us to treat?ner discretizations of physical systems.We will,in addition,show how to introduce quantum mechanics into our formalism and use it to study the full,non-retarded interaction between two dielectric bodies.In order to com-pare with our numerical method,we analyze the performance of two simple approximation methods:pairwise additivity,and the proximity force approximation.We also introduce an approximation based on the factorization of the geometric and material properties that works surprisingly well for the geometries that we study.
We begin by formulating the theory of classical?uctuating dielectrics.In this we prefer a formulation of the classical interaction in terms of the true microscopic?uctuating?eld, the polarization,rather than the electric?eld and electric displacement.We then show how to e?ciently factorize the resulting quadratic forms and apply our formalism to calculate the free energy of interaction between a tip and an indentation.We then generalize our microscopic energy functional in order to consider the quantum?uctuations of a dielectric and evaluate the non-retarded interaction between a tip and a structured surface.
II.CLASSICAL FLUCTUATING DIELECTRICS
We begin by evaluating the classical thermal interaction,since most of the technical di?culties of discretization and factorization are already present,before showing how to treat the quantum case.We start with the energy of a heterogeneous dielectric system[16] written in terms of the polarization density p.We note that formulation in terms of the polarization?eld is also much more convenient for the generalization to scale dependent dielectric e?ects[17],which are particularly important in water based systems.
The energy of a linear dielectric has two contributions[16],?rstly a long-ranged Coulomb energy for the induced charged densityρi=??·p and secondly a local contribution which depends on the local electric susceptibilityχ0(r).
U p= ?·p(r)?·p(r′)2χ0(r)d3r.(1) We note that the dielectric constant?(r)=1+χ0(r);we use units where?0=1.The partition function of the?uctuating dipoles
Z= D p exp(?βU p),(2) can be simpli?ed by re-writing the Coulomb potential,1/4πr as an integral over a potential φ.We then?nd the e?ective energy
U p,φ= (?φ)22χ0(r) d3r.(3) One now performs the Gaussian integral over p,to?nd the partition function expressed as an integral overφ.To do this it is useful to integrate by parts replacingφ?·p with?p·?φ; boundary terms vanish in periodic systems or those in which?elds decay at in?nity.
In order to perform numerical calculations,we derive a discretized theory.We discretize by placing scalar quantities such asφon V=L3nodes of a cubic lattice.We choose a length scale such that the lattice spacing is unity.The lattice spacing should be much larger than the atomic scale,so that a formulation in terms of continuum dielectric properties is possible;the lattice should however be su?ciently?ne to resolve features of physical interest, such as points or rough surfaces.Vector?elds,such as?φ,are associated with the3V links,?andχ0are also associated with the links.We?nd
Z(?)= lχ1/20,l Dφexp ?β l?l
where l is a link;the discretization of the derivative?l evaluates the di?erence between variables on the corresponding nodes.The integral in eq.(4)is over all modes excluding the uniform mode,φ=const which has eigenvalue zero[15].We write the quadratic form appearing in eq.(4)asφMφ/2with a symmetric matrix M.The interesting,long ranged, part of the free energy of interaction comes from
k B T
F=
determinant without forming intermediate dense matrices,saving computer memory and accelerating calculations.We evaluate the determinant by?rstly modifying the matrix in order to render it positive de?nite,so that we are not troubled by the zero eigenvalue.We do this by adding V to a single arbitrary diagonal element of the matrix;this can be shown to be equivalent to neglecting the zero mode of the determinant[15].With the matrix,M now positive de?nite we write it as a product of Cholesky factors M=R T R where R is an upper triangular matrix.Because R is triangular the determinant is given by the product of the diagonal elements.From the factor R we?nd det M=(det R)2.
This factorization has a number of remarkable properties that make it far more powerful than diagonalization.A good choice for the ordering that is used for evaluating the Cholesky factors renders the method particularly interesting:Nested dissection numbers the nodes in a non-consecutive manner by recursively cutting a graph into equal pieces[18].For this ordering of our matrix the Cholesky factor(in three dimensions)can be generated with a storage of S～V4/3,and with arithmetic e?ortτe～V2[19].We used the software package Taucs[20]to perform the Cholesky factoring.We?nd that a system of V=643can be factored on a3.2GHz64-bit Xeon workstation in approximately300seconds,using3GB of memory.
IV.INTERACTIONS BETWEEN A TIP AND INDENTATION
We now consider the interactions between a tip and a surface indentation in a system in which the thermal contribution is dominant.Experimentally this can occur either with tips in water which are made with materials which have similar optical properties(so that the high frequency contributions to the Lifshitz energy cancel out),or in systems that are su?ciently separated such that the quantum interactions have died out requiring separations larger than c/2k B T～5μm at room temperature[3,21].
We evaluated the free energy of interaction for a system composed of a three dimensional rounded tip close to an indentation in a surface discretized to a lattice of dimension L=803, Figure1.We evaluated the free energies with two vertical displacements of l=6and l=20.The physically interesting case for optically matched systems has a contrast ratio of approximately r?=?1/?2=50between the two media.However in order to study the evolution of the interactions with contrast we worked with a minimum ratio of r?=1.05to
a maximum of r?=100.
FIG.1:Section of the tip-indentation system taken on a plane going through the center of the tip. The system is discretized to a lattice with L=80.The tip and the indentation are taken to be half ellipsoids with vertical semi-axes a=18and horizontal semi-axes b=15and b=19respectively. In the?gure l=6.
In the absence of a general method to compute dispersion forces in arbitrary geometries, two approximations are commonly used.The?rst is the proximity force approximation according to which the interaction between surfaces of any geometry is found by assuming that the surfaces are locally?at.One then sums an e?ective,local free energy of the form
A
U(h,A)=
FIG.2:Minimal distance map for l=6(top)and l=20(bottom).Black corresponds to the minimal distance,while white corresponds to the maximal distance.As the tip moves up the geometry of the closest approach changes from a central disk to a ring from the rim of the indentation.Mesh corresponds to the discretization lattice.
the?nal free energy.The maps for the two chosen gaps are shown in Figure2.We?nd that the free energies computed with the proximity force approximation can deviate to up to40%from the full numerical evaluation.In particular the proximity force approximation performs the worse for high dielectric contrast(r?=50)and when the two surfaces are close together.Results for this system are shown in Figure3.When the l=20proximity force approximation performs better with deviations from the full evaluation of8%for large contrast.
The second method,widely used in computer modeling,is an additivity assumption according to which the overall interaction is found by summing pairwise interactions of ele-mentary components.A typical example of such approximation performs an integration of the long-ranged tail of Lennard-Jones1/r6potentials between in?nitesimal constituents of the interacting macroscopic objects[22].More generally,if one assumes a pairwise interac-tion between elements of a dielectric of the form V(r)=αv(r)whereαcharacterizes the
FIG.3:Ratio between the free energies computed via proximity force approximation and our numerical results as a function of the dielectric contrast between the two materials for l=6.
strength of the potential,then the long-ranged part of the interaction energy between two macroscopic bodies can be written as
U(R,α)=αG(R)(7)
where the coordinates R are the relative positions of the bodies and the function G encodes all the geometric information.
We notice that both eq.(6)and eq.(7)display a factorization property(they are expressed as a product of material and of geometric terms)so that if we for instance consider the ratio
R12=U(R1,α)/U(R2,α)(8)
for two di?erent geometries we?nd a result independent of the material property.Since this factorization property is true in two very di?erent limiting approximations it seems interesting to study the degree to which it remains valid over a wide range of dielectric contrasts in a geometry which is far from planar.
Surprisingly,Figure4,we?nd that the ratio of free energies computed at di?erent dis-tances exhibits only a moderate variation with the dielectric contrast.The robustness of this result has been tested by changing the system size and the accuracy of the tip/indentation discretization.Our study suggests a way of inferring free energy values at high dielectric
FIG.4:Test of the factorization property,eq.(8).The ratio of interaction energies for two separations of the tip-indentation system of Figure1is plotted as a function of r?.The ratio varies by just13%over the whole range of dielectric contrasts,despite the great variation in geometry displayed in Figure2.
contrast once the results at low contrast are known.It is su?cient to determine the inter-action at low?to obtain the whole set of measurements for the desired system.Our results show that for classical,thermal interaction,the factorization method we propose,works better than proximity force approximation when large surface deformations are present and for high dielectric contrasts,both instances where proximity force approximation performs poorly.This approach can be combined with analytic approximations that compute free energies for arbitrary geometries in a small dielectric contrast expansion[14].
V.NON-RETARDED QUANTUM INTERACTIONS
The quantum interaction between two materials is classi?ed as non-retarded when the interaction is instantaneous,or retarded when one must take into account the?nite propa-gation speed of electromagnetic radiation.In the?rst case the decay of interactions between two atoms is1/r6;in the second the interaction falls as1/r7.The characteristic crossover distance between these two forms is determined by the wavelength of typical spectral fea-tures that dominate dielectric response.This is often a feature in the ultraviolet,leading to
a cross-over length of tens of nanometers for most materials.
We now generalize our treatment of the interaction to the short distance,non-retarded regime.The systems for which this e?ect is dominant are limited to the nano-scale,however we will show that this case is particularly simple to treat with the methods developed above. We leave the generalization to retarded interactions,which require the correct treatment of the propagating electromagnetic?eld,for a future publication.
A.Quantization
Since in the energy eq.(1)the?eld p represents a microscopic polarization vector we can study its dynamics by adding the kinetic energy.
T=ρ(r)˙p2/2(9) whereρ(r)is a mass density.
The thermodynamics of a quantum system are particularly simple to treat with the method of path integral quantization[23].The potential and kinetic energies are combined in an e?ective statistical weight for an ensemble of N identical replicas of the original system; these replicas are coupled in the time/temperature direction by harmonic springs.The exact quantum partition function is then generated in the limit of large N.
The e?ective action at each time slice,m is
U m= d3rρ(r)(p m?p m+1)2
χ(ω)=
1
2τ2
(2?2cosω)
with Fourier frequencyω=2πn/N.
We again perform the transformation from eq.(1)to eq.(3)by introducing an integral over the potential.After integrating over p the partition function of the quantum system is given by a product
Z=N?1
n=0Z(?(n))(11)
where in the Z(?)of eq.(4),the dielectric function is replaced by
χ0
?(n)=1+
χ0(r)ρ(r).If we introduce the Matsubara frequenciesωn=2nπ/βwe?nd that for N large eq.(12)simpli?es to the single pole approximation
χ0
?(n)=1+
N=240for the evaluation of the free energy landscape described below.For this value F∞/F240=0.987.Higher accuracy and faster convergence(in1/N4)are possible if we work with two values of N and extrapolate to F∞.
FIG.5:Convergence of free energy di?erences with copy number N.We evaluated the free energy di?erence of two parallel plates separated by l=2and l=10for N varying from80to260. The free energy is plotted as a function of N?2.From values of N above180we extrapolate F∞=2.051×10?3.L=48.
We then evaluated the free energy of a system composed of a sharp point over a surface with regular wells as a function of the horizontal position.Given that typical small force microscopy tip sizes are of the order of10nm?50nm forces measured experimentally by this technique will generally fall into the crossover between the retarded and non-retarded regimes.However,there do exist extreme cases of tips of radii of2nm(super sharp silicon tips),where this case could be relevant,even if we are beginning to be close to the atomic scale where a continuum dielectric description is more di?cult to justify.
We scanned a region of12×12lattice points,covering the area of one well and its surroundings.Results of the free energy landscape are shown in Figure6.The free energy of each point F(x,y)is evaluated for N=240,for which,due to the symmetry?(n)=?(N?n), only N/2+1=121determinants are needed.Firstly,the matrices(122×121～17,500) were built using Matlab,and stored on disk.This took approximately two days on a single workstation.Then,the free energy evaluation was performed on a cluster of nine processors,
and took approximately2days to complete,with the evaluation of the interaction at each position taking about3hours.The energy landscape we obtained re?ects the underlying well pro?le,but with smoothed features.
FIG.6:Quantum free energy landscape for a system composed of a sharp,conical,tip near a surface with regular wells,side and depth l=6,separated by6lattice spaces in both x and y direction.Box size L=48.A portion of the system of dimensions12×12is scanned and the result reproduced periodically.Top surface:the free energy landscape at one lattice space above the top of the wells.Lower surface indicates the positions of the wells.
VI.DISCUSSION
Quite?ne and useful discretization of?uctuating dielectrics can be studied using methods based on Cholesky factorization of a sparse determinant.One can study the full multibody dispersion interaction at nanometric length scales,a scale suitable for the study of macro-molecules and nano-particles,in non-trivial three dimensional geometries.Conventional methods for interpreting such systems use extensive modeling with force?elds that contain many simplifying assumptions which can now be checked(in the near?eld regime)against explicit numerical results.
We now compare our approach with recent numerical work[11].The authors work with
the full discretized Maxwell equations which enables them to consider the full retarded interaction between bodies;they calculate the stress tensor rather than the free energy that we chose to evaluate.The method requires the calculation of Green functions which are then integrated over a closed surface surrounding the body of interest.The authors propose several di?erent methods of solving for the Green functions,including fast multipole and multigrid methods,which they argue can solve the problem in with memory S～V and time τe～V2?1/d.They work with modest system sizes of20×40in simpli?ed2+1dimensional geometries,suitable for studing grooved surfaces.
In practice they used a conjugate method which requires a number of iterations which increase with the systems size as V1/d[26],so that their actual implementation scales like our own asτe～V2.It could have been useful to compare real computing times,and not only theoretically derived asymptotic behaviors,but unfortunately the authors do not report these data.Any consideration of the choice of algorithm must take into account the prefactors in these laws;fast multipole methods have been abandoned in applications such as molecular dynamics due to the enormous prefactors in the(apparently favorable) asymptotic scaling.
For system sizes comparable to those used in this paper one?nds that the number of ?oating point operations needed to perform the Cholesky factorization is comparable to N flop=6.5V2[27].Multigrid methods while being asymptotically fast can require hundreds of iterations in order to converge[28].It thus seems possible that they perform less well than the methods of the present paper for moderate system sizes.It would be most interesting to study the crossover point in the e?ciency of the various methods.
When we restrict ourselves to systems in2+1dimensions the Cholesky factorization we use also improves in performance.Storage requirement in this case is only S～N log(N), and computation time isτe～N4/3.The interaction of grooved surfaces is studied by Fourier transforming in the uniform direction before performing the factorization[15].In such a system evaluation of the interaction of a system discretized to a lattice of dimensions 5003can be performed in30minutes on a3.2GHz64-bit Xenon workstation.
The main physical problems that are still not possible to study with this method have length scales in the range20nm?10μm where the retarded interaction dominates.This requires a di?erent discretization strategy based on the full Maxwell equations and will be considered in an future paper[29].
We wish to thank F.Nitti for many useful discussions.Work?nanced in part by the Volkswagenstiftung.
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T h e r e?b e?句型用法归纳 一、1.?定义：There?be句型表示某处存在某物或某人。? 2.?结构：(1)?There?is?+?单数可数名词????? ?? ?(2)?There?are?+?复数可数名词? 动词要与主语（某人或某物）的数保持一致。 ?eg. ?①?There?is?a?bird?in?the?tree.???树上有一只鸟。 ②?There are?two birds?in?the?tree. ?树上有两只鸟。 4.当主语是两个或两个以上的名词时，谓语动词要与跟它最近的那个名词一致。（就 近原则） ?①?There?is?a?teacher?and?many?students?in?our?classroom.??我们教室里有一位老师和许多学生。（就近原则） ②??There?are?many?students?and?a?teacher?in?our?classroom.?我们教室里有许多学生 和一位老师。（就近原则） ?5.?There?be句型与have的区别：? There?be?句型和have都表示“有”的含义。区别如下：There?be表示“某处存在某物或某人”；have表示“某人拥有某物/某人”，它表示所有、拥有关系。 eg.?①He?has?two?potatoes.???他有两个儿子。? ??②There?are?two?potatoes under the bed.????床的下面有两个土豆。 二、一般疑问句?? ????There?be句型的一般疑问句变化是把be动词放到句首（首字母大写），再在句尾加上问号。?? eg.??There?is?a book on the desk. ?→?Is?there?a book on the desk??肯定回答：Yes, there is. /No, there isn’t. There are two books on the desk. 改成一般疑问句 肯定回答： 否定回答： 三、练习 ??1..用is或are填空? (1).? There?_____?a?book?and?two?pens?on?the?desk.?? (2).? There?____?some?water in?the?picture.?? (3).? There ?_____?some?cards?in?Jim’s?bag.? (4).? There _____?an eraser in the pencil box. (5).? There?_____?one?pupil in?our?school. ?(6).? There?_____ three footballs and a cap on the chair. 2.把下列的句子改成一般疑问句并作回答。 1. There is an orange in the box. are four hamburgers on the floor. is one strawberry and twelve pears under the chair. 七、阅读理解，选择正确答案（每空2?分，共8分） ?Lucy?is?my?friend.?She?lives?in?a?small?house?near?a?park?in?Canada.?I?am?ten,?L ucy?is?ten?,?too.?Lucy?tell?me?more?about?Canada.?I?know?Canada?is?such?a?big?coun try.?So?I?want?to?learn?English.?And?I?tell?Lucy?more?about?China.?Lucy?likes?China?
There be 句型的几种特殊结构及其用法 There be句型是一种应用十分广泛和频繁的句型。但是对there be句型的多变的特点及其特殊结构的复杂性，并不是每个考生都熟悉了解。在大学英语四级考试题中也常常遇到这一句型结构的试题。 例如： Ann never dreams of _________ for her to be sent abroad very soon．（CET-4，1998．6-43） A）there to be a chance B）there being a chance C）there be a chance D）being a chance 该题的答案为B）。 动词 dream of 要求跟 V-ing 分词结构作宾语，更准确地讲是介词of后要求跟 V-ing 分词结构作宾语，there being a chance意为"有一个机会"；A）和C）两项均不符合句子结构的要求，所以不是答案选项，而D）项 being a chance 虽然是 V-ing 分词结构，但语义不通，故不能雪 又如： No one had told Smith about ____ a lecture the following day．（CET-4，2001．1-66） A）there be B）there would be C）there was D）there being 该题的答案为D）。介词 about 后要求跟 V-ing 分词结构作宾语，there being（a lecture）意为"有（一个讲座）"，而A）、B）和C）项内容均不符合结构要求，故不能雪 再如： It is fairly common in Africa for there to be an ensemble of expert musicians surrounded by others who join in by clapping，singing，or somehow adding to the totality of musical sound．（CET-4，1995．1） 该句子中出现了for there to be 的结构，如果按照上一题的解题思路去理解：介词后要求跟 V-ing 分词结构作宾语，那么，这一结构似乎是错的；但是，实际上此结构没有错，此处只能够用 for there to be，而不能用 there being。为什么呢？这就是本文想要解答的问题：there be 句型的特殊结构及其用法。笔者将 there be 句型的几种特殊结构及其用法进行归纳、总结如下，供读者参考。 一、there be句型与各种情态动词连用。 例如： There must be something wrong here． There might still be some vacant seats in the rear． There ought to be something with which to fill your stocking． 二、there be句型中的谓语动词be被be likely to be，happen to be，seem（to be）， occur等代替，用来描写事物。例如： ] There are likely to be more difficulties than they have been prepared for． There happened to be nobody in the room． There doesn’t seem to be much hope of our beating that team ． There seems no doubt that the general character of the landscape，the relative length of day and night，and the climate must all play a big part in determining what kind of people we are． There have occurred many great changes since we met last． 三、there be 句型中的谓语动词be被一些不及物动词代替，如 live，stand，exist，remain等，用来表示"静止、存在、有"。例如： There lives a family of five in the village． There remains nothing more to be done． There stands the Monument to the People’s Heroes at the center of the Tian’anmen Square． There exist different opinions on this question． 四、there be 句型中的谓语动词be被一些不及物动词代替，如 come，spring up，appear，emerge，arrive，enter，follow, 等，用来表示"突然出现"。例如：
一、There be 句型 1. 定义：There be句型表示某处存在某物或某人。 2. 结构：(1) There is +单数可数名词/不可数名词+ 地点状语. (2) There are +复数名词+地点状语. There be结构中的动词be的确定 1. there be 结构中的谓语动词be在人称和数上应与其后的主语保持一致。主语是不可数名词或单数可数名词时用is,是复数时用are。如: There is a flower in the bottle. 瓶里有一朵花。 There is some money in the purse. 钱包里有些钱。 2. 若句子中有几个并列的主语时,be的形式要与离其最近的一个主语在人称和数上保持一致。如: There is a boy, a girl and two women in the house. 房子里有一个男孩,一个女孩和两个妇女。 There are ten students and a teacher in the office. 办公室里有十个学生和一个教师。 另外,在陈述句中为了强调地点,也可将介词短语提置句首。如: In the tree there are five birds. 树上有五只鸟。 二、 There be 结构的句型转换 1. 否定句: there be的否定式通常在be后加not构成(在口语中be时常与not缩写在一起,is not=is n’t are not=are n’t)。如果句中有some,一般要变成any。如:
There are some children in the picture. →There aren't any children in the picture. 2.一般疑问句及其回答:把be提到there前,首字母大写,句末用问号即可(句中的some一样要改为any)。其肯定回答是Yes, there is / are;否定回答为No, there isn't / aren't。如: -Are there two cats in the tree? -Yes, there are. (No, there aren't.) 3. 特殊疑问句及其回答:①提问句子的主语(包括主语前的修饰语)时,句型一律用"what is + 地点介词短语?"(无论主语是单数还是复数都用is)。如: There are some birds in the tree. →What's in the tree? ②就there be后面的地点状语进行提问时,句型用"where is / are + 主语?"如: There is a car in the street. →Where is the car? ③提问可数名词(主语)前的数量时,用how many,句型结构为"how many + 复数名词+ are there + 其它?"(主语无论是单数还是复数,be通常要用are)。如: The re is a cat under the bed. →How many cats are there under the bed? 三、There be句型与have、has的区别： (1) There be 句型和have, has都表示“有”的含
There be 句型用法总结 There be 结构是英语中陈述事物客观存的常用句型，表示“有”，其确切含义是“存在”there 作为引导词，本身没有意义，用动词be的某些形式作为谓语动词，它的主语是用一些表示泛指或不定特指的名词词组，动词be和 主语的数必须一致。句子最后通常为表示地点和时间的状语。因此要表达“某个地方或某个时间存在什么事物或人”的时候常用“There be + 名词+ 地点（时间）这一句型。例如： There is a great Italian deli across the street. 穿过街道，有一家大的意大利熟食店。There are some students in the dormitory. 在宿舍里有一些学生。 一、There be 结构中的主谓一致 1.当动词be后所接的名词是单数可数名词或不可数名词时，be 应该取单数is；当其后所接的名词是复数的可数名词时，be用复数are。 There's a man at the door.门口有个人。 There is some apple juice in the bottle. 瓶子里有些苹果汁。 There are some strangers in the street.大街上有一些陌生人。 2.如果There be 后面是几个并列名词做主语时，动词be的形式和最靠近它的那个名词保持数的一致。 There is an ashtray and two bottles on the shelf. 架子上有一只烟灰缸和两个瓶子。There are two bottles and an ashtray on the shelf. 架子上有两个瓶子和一个烟灰缸。 二、There be 结构中的时态 1.There be 句型中动词be可以有一般现在时、一般过去时、将来时和完成时。 There is no harm in trying.不妨一试。 There were fabulous wildflowers in the hills last spring. 去年春天，山中有极美的野花。There will be a fine day tomorrow. 明天将是一个晴天。 There have been several private schools in our area this year. 今年，我们这里已经有好几所私立学校了。 2.There be 句型可以和各种助动词、情态动词连用。 There may be a cigarette in that box. 那只盒子里或许有支香烟。 There must be some cakes on the table. 桌子上一定有些蛋糕。 There used to be a hospital there before the war. 战前，那里曾经有家医院。 3.There be 句型也可以和这样一些的谓语动词连用：be going to 、seem to 、appear to 、used to、be likely to 、happen to …. There seem to be a few trees between me and the green. 在我与草坪之间好像有一些树。 There is gong to be a meeting tonight. 今天晚上有个会议。 There is likely to be a storm 可能有一场暴雨。 There happened to be a bus nearby. 碰巧附近有辆公交车。 There appears to have been a nasty accident. 似乎发生了一起严重事故。 4.there be 结构中除可以用be 外，还可以用其它动词。例如： There came a scent of lime-blossom. 飘来一阵菩提树的花香。 Once upon a time there lived a king in China. 从前中国有一个国王。
There be句型用法归纳 、1.定义：There be句型表示某处存在某物或某人 2. 结构：⑴There is +单数可数名词 （2） There are +复数可数名词 3. be动词要与主语（某人或某物）的数保持一致。 eg. ① There is a bird in the tree. 树上有一只鸟。 ② There are two birds in the tree. 树上有两只鸟。 4. 当主语是两个或两个以上的名词时，谓语动词要与跟它最近的那个名词一致。 （就近原则） ①There is a teacher and manystudents in our classroom. 我们教室里有一 位老师和许多学生。（就近原则） ②There are manystudents and a teacher in our classroom.我们教室里有许多学 生和一位老师。（就近原则） 5. There be句型与have的区别： There be句型和have都表示“有”的含义。区别如下：There be表示“某处存在某物或某人” ；have表示“某人拥有某物/某人”，它表示所有、拥有关系。eg.①Hehastwo potatoes. 他有两个儿子。 ②There are two potatoes under the bed. 床的下面有两个土豆。 二、一般疑问句 There be句型的一般疑问句变化是把be动词放到句首（首字母大写），再在句尾加上问号。 eg. There is a book on the desk. f Is there a book on the desk? 肯定回答：Yes, there is. /No, there isn '. There are two books on the desk. 改成一般疑问句 ______________________________ 肯定回答： _____________________ 否定回答： ___________________ 三、练习
T h e r e b e句型用法归纳 1.定义：Therebe句型表示某处存在某物或某人。 2.Therebe句型结构中的is/are的选择： (1)Thereis+单数可数名词/不可数名词+地点/时间状语. (2)Thereare+复数名词+地点/时间状语. there是引导词，在句中不充当任何成分，翻译时也不必译出。句子的主语是某人或某物，谓语动词be要与某人或某物的数保持一致。当be后是两个或两个以上的名词时，谓语动词要与跟它最近的第一个名词一致即采用就近原则。 eg.①Thereisabirdinthetree.树上有一只鸟。 ②Thereisateacherandmanystudentsinourclassroom. ③Therearetwoboysandagirlunderthetree. 3.句式转换： (1)肯定句：Thereis/are+名词/sb.+地点/时间状语 (2)否定句：Thereis/are+not+名词/sb.+地点/时间状语 Therebe句型的否定式的构成和含有be动词的其它句型一样，在be后加上not即可。例如：Therearesomepicturesonthewall.→Therearen'tanypicturesonthewall.Thereisabikebehindthetree.→Thereisn'tabikebehindthetree. (3)：一般疑问句:Is/Arethere+名词/sb.+地点/时间状语? Therebe句型的一般疑问句变化是把be动词调整到句首，再在句尾加上问号即可，此为"调整法"。 但同时要注意：当肯定句中有some时，要将其改为any（否定变化也一样）。看看下面两句是如何"改头换面"的吧： ThereissomewateronMars.→IsthereanywateronMars? Therearesomefishinthewater.→Arethereanyfishinthewater? (4)：特殊疑问句 Therebe句型的特殊疑问句形式有以下两种变化： ①对名词/sb.提问：用"Who/What+is+介词短语?" 注意：无论原句的名词是单数还是复数，对之提问时一般都用be的单数形式(回答时却要根据实际情况来决定)。如： Therearemanythingsoverthere.→What'soverthere? Thereisalittlegirlintheroom.→Whoisintheroom? ②对地点状语提问：疑问词+is/are+名词/sb.? 例如： Thereisacomputeronthedesk.→Whereisthecomputer? Therearefourchildrenontheplayground.→Wherearethefourchildren? 4.therebe结构的时态 therebe结构有不同的时态，而且可以和各种助动词或情态动词连用。如：Therewasasportmeetingintheplaygroundyesterday. Therewillbe(=Thereisgoingtobe)anewfilmshowonMonday. Thereistobeaconcertattheschoolhall.学校礼堂有场音乐会。Therehavebeenalotofaccidentsroundhere.这里已经发生多起事故了。Hetoldmethattherehadbeenanargumentbetweenthem.他告诉我们之间发生了一场争论。TherewillhavebeenadefiniteresultbyFriday.到星期五前就已经有明确的结果Theremustbeamistakesomewhere.一定在什么地方有错误。Theremusthavebeenarainlastnight,forthegroundiswet.昨晚一定下了雨，因为地是湿的。 5.therebe结构的变体
There be句型的基本用法是表示“某地(或某时)存在有某人(或某物)，而并非某地(某人、某物或某时)拥有什么东西”，其形式为“There be＋代词或名词(短语)＋地点/时间状语”。(其实质句式为倒装句)这里there是引导词，没有词义，be是谓语动词，代词或名词(短语)是主语。be要与主语保持人称和数的一致。否定句是在be后加not；一般疑问句是将be放在句首；反意疑问句中的简短问句是由“be(或其否定式)＋there”构成。例如： 1.There is a desk and two chairs in the room. (紧挨着be动词的主语是a desk，是单数，故be的形式要用is) 2.There aren't two chairs and a desk in the room.(否定句) 3.Is there anything wrong with your ears？(Yes，there is/No，there isn't.) 4.There wasn't a meeting yesterday，was there？(反意疑问句) 除此之外，还有一个重要句式“有某人在做某事”，要用“There be ＋sb.＋doing sth.＋地点/时间状语”。例如： There are several children swimming in the river.河里有几个孩子在游泳。 不难看出，各种时态的变化是通过be动词的变化来体现的。我们在使用过程中，首要的问题是弄清楚There be与have所表示的意义。There be 句型表示“存在”关系，have表示“所属”关系，两者不能混合在一个句子中。 例如：我们要说“明天有一个班会。” (1)There will have a class meeting tomorrow.(×） (2)There is going to/will be a class meeting tomorrow.(√) 有时候既表示“存在”又表示“所属”时，两种都可以用。 例如：Class Three have a map of China on the wall.(地图为三班学生所有。)
T h e r e b e句型用法归纳标准化文件发布号：（9312-EUATWW-MWUB-WUNN-INNUL-DDQTY-KII
There be 句型用法归纳 1. 定义：There be句型表示某处存在某物或某人。 2. There be句型结构中的is/are的选择： (1) There is +单数可数名词/不可数名词+ 地点/时间状语. (2) There are +复数名词+地点/时间状语. there是引导词，在句中不充当任何成分，翻译时也不必译出。句子的主语是某人或某物，谓语动词be要与某人或某物的数保持一致。当be后是两个或两个以上的名词时，谓语动词要与跟它最近的第一个名词一致即采用就近原则。 eg. ① There is a bird in the tree. 树上有一只鸟。 ② There is a teacher and many students in our classroom. ③ There are two boys and a girl under the tree. 3.句式转换： (1)肯定句：There is/are +名词/sb. + 地点/时间状语 (2)否定句：There is/are +not +名词/sb. + 地点/时间状语 There be句型的否定式的构成和含有be动词的其它句型一样，在be后加上not即可。例如： There are some pictures on the wall. →There aren't any pictures on the wall. There is a bike behind the tree. → There isn't a bike behind the tree. (3)：一般疑问句: Is / Are there+名词/sb. + 地点/时间状语 There be句型的一般疑问句变化是把be动词调整到句首，再在句尾加上问号即可，此为"调整法"。但同时要注意：当肯定句中有some时，要将 其改为any（否定变化也一样）。看看下面两句是如何"改头换面"的吧： There is some water on Mars. → Is there any water on Mars There are some fish in the water. →Are there any fish in the water (4)：特殊疑问句 There be句型的特殊疑问句形式有以下两种变化： ①对名词/sb.提问：用"Who/What + is +介词短语 " 注意：无论原句的名词是单数还是复数，对之提问时一般都用be的单数形式(回答时却要根据实际情况来决定)。如： There are many things over there. →What's over there There is a little girl in the room.→Who is in the room ②对地点状语提问：疑问词+ is / are+名词/sb. 例如： There is a computer on the desk. → Where is the computer There are four children on the playground. →Where are the four children 4. there be结构的时态 there be结构有不同的时态，而且可以和各种助动词或情态动词连用。如： There was a sport meeting in the playground yesterday. There will be (=There is going to be) a new film show on Monday. There is to be a concert at the school hall. 学校礼堂有场音乐会。 There have been a lot of accidents round here. 这里已经发生多起事故了。
There be 句型课程讲解（一） 一、There be 句型的用法： 表示某个地方存在某物或某人，可以翻译成有的意思。Be动词的单复数必须依主语的变化而变化。如：房间里有一张桌子。There is a table in the room. 区别: 表达一个人拥有某样东西时则用have/has,如:我有一张桌子。I have a table. There be 结构和have的区别与联系 1.区别点：there be 意为存在，强调某地有某物，不表示所属关系；have 表示所有关系， 强调某地有某物或某人，这是其基本用法。如： There are some trees in front of the house. 房前有些树。 Tom has many friends in China. 汤姆在中国有许多朋友。 2.相同点：在表示结构上的含有时，既可以用there be 句型，也可以用have（has）来表示。如：中国有许多长河。 There are many long rivers in China. China has many long rivers. 三月份有多少天？ How many days are there in March? How many days has March?
二、There be 句型的结构： There is+第三人称单数可数/不可数主语+地点状语（介词短语） 例：There is a ruler on the desk. 书桌上有一把尺。 There is some water in the bottle. 瓶子里有一些水。 There are+复数主语+地点状语（介词短语） 例：There are four apples on the tree. 树上有四个苹果。 There are many flowers in the park. 公园里有许多花。 There is+第三人称单数可数+V-ing+地点状语（介词短语） 例：There is a bird singing in the tree. 树上有一只鸟正在唱歌。 There is a baby sleeping in the room. 房间里有一个宝宝正在睡觉。 There are+复数主语+V-ing+地点状语（介词短语） 例：There are some birds singing in the tree. 树上有一些鸟正在唱歌。 There are two boys running on the street. 街上有两个男孩正在奔跑。 There be 句型的疑问句及回答（二） 一、There be 句型的疑问 1. 在“There is/are...”的疑问句中,要把Be动词放在there之前,并将句号变问号。难点:句中出 现的some要改成any。 2. 肯定回答为：Yes, there is/are. 3. 否定回答为: No, there isn’t/aren’t. 结构分析：
There be句型结构及其用法 一、there be句型与情态动词连用。There must be something wrong here．? There ought to be something with which to fill your stocking． 二、there be句型中的谓语动词be被be likely to be，happen to be， seem/appear（to be），等代替，用来描写事物。例如： There are likely to be more difficulties than they have been prepared for．There happened to be nobody in the room． There doesn't seem to be much hope of our beating that team ． There appears to be no substitute for this stuff yet． 三、there be 句型中的谓语动词be被 一些不及物动词代替，如live，
stand，exist，remain等，用来表 示"静止、存在、有"。例如：There lives a family of five in the village． There remains nothing more to be done． There stands the Monument to the People's Heroes at the center of the Tian'anmen Square． There exist different opinions on this question． 四、there be 句型中的谓语动词be被一些不及物动词代替，如come，arrive，enter，follow, 等，用来表示"突然出现"。例如： There came a company of actors and actresses． There followed a spirited discussion after class． 五、“there to be ＋名/代词"结构，表 示"有"或"存在（某种情况）这种不
T h e r e b e句型用法总结文档编制序号：[KKIDT-LLE0828-LLETD298-POI08]
There be 句型用法总结 There be 结构是英语中陈述事物客观存的常用句型，表示“有”，其确切含义是“存 在”there 作为引导词，本身没有意义，用动词be的某些形式作为谓语动词，它的主语是用一些表示泛指或不定特指的名词词组，动词be和 主语的数必须一致。句子最后通常为表示地点和时间的状语。因此要表达“某个地方或某个时间存在什么事物或人”的时候常用“There be + 名词+ 地点（时间）这一句型。例如： There is a great Italian deli across the street. 穿过街道，有一家大的意大利熟食店。There are some students in the dormitory. 在宿舍里有一些学生。 一、There be 结构中的主谓一致 1.当动词be后所接的名词是单数可数名词或不可数名词时，be 应该取单数is；当其后所接的名词是复数的可数名词时，be用复数are。 There's a man at the door.门口有个人。 There is some apple juice in the bottle. 瓶子里有些苹果汁。 There are some strangers in the street.大街上有一些陌生人。 2.如果There be 后面是几个并列名词做主语时，动词be的形式和最靠近它的那个名词保持数的一致。 There is an ashtray and two bottles on the shelf. 架子上有一只烟灰缸和两个瓶子。There are two bottles and an ashtray on the shelf. 架子上有两个瓶子和一个烟灰缸。 二、There be 结构中的时态 be 句型中动词be可以有一般现在时、一般过去时、将来时和完成时。 There is no harm in trying.不妨一试。
There be 句型的用法 ． there be 句型基本认识 1、定义：There be 句型表示某处有某物或某人。there 是引导词，在句中不充 当任何成分，翻译时也不必译出。 这种句型之所以要用 there 作形式主语而把 “实义主语” 置于动词之后正是 为了让这种“实义主语”称为信息中心，以达到引出新话题的目的。 2、句型构成:There 十连系动词be+主语（人/物）+地点（介词短语或副词） （1） There is +单数可数名词 / 不可数名词 + 地点状语 . （2） There are +复数名词 +地点状语 . :①当主语是可数名词单数或不可数名词时， be 动词用is ； ② 当主语是可数名词复数时，其 be 动词就用are ； ③ 当主语是两个或两个以上的名词时， （遵循就近原则） be 动词要根据离它最近的名词的数来确定 be 的形式。 ① There is a bird in the tree. 树上有一只鸟。 ②There are two children in the room. 房间里有两个孩子。 ③ There is a teacher and many stude nts in our classroom. 我 们 教室里有一位老师和许多学生。 There are two boys and a girl under the tree. 树下有两个男 孩，一个女孩。 一）如何改成否定句：直接在 be 后加上 not 或 no 即可。 例如： ① There are some pictures on the wall. f There aren't any pi ctures on the wall. =There are no pi ctures on the wall. ② There is a bike behind the tree. There isn't a bike behind the tree. =There is no bike behind the tree. 名词及主谓一致 对应例子： there be 句型的各种句型转化
There be 句型用法归纳 一、1. 定义：There be句型表示某处存在某物或某人。 2. 结构：(1) There is + 单数可数名词 (2) There are + 复数可数名词 3.be动词要与主语（某人或某物）的数保持一致。 eg. ① There is a bird in the tree. 树上有一只鸟。 ② There are two birds in the tree. 树上有两只鸟。 4.当主语是两个或两个以上的名词时，谓语动词要与跟它最近的那个名词一 致。（就近原则） ① There is a teacher and many students in our classroom. 我们教室里有一位老师 和许多学生。（就近原则） ② There are many students and a teacher in our classroom. 我们教室里有许多学 生和一位老师。（就近原则） 5. There be句型与have的区别： There be 句型和have都表示“有”的含义。区别如下：There be表示“某处存在某物或某人”；have表示“某人拥有某物/某人”，它表示所有、拥有关系。eg. ①He has two potatoes. 他有两个儿子。 ②There are two potatoes under the bed. 床的下面有两个土豆。 二、一般疑问句 There be句型的一般疑问句变化是把be动词放到句首（首字母大写），再在句尾加上问号。 eg. There is a book on the desk. → Is there a book on the desk? 肯定回答：Yes, there is. /No, there isn’t. There are two books on the desk. 改成一般疑问句 肯定回答： 否定回答： 三、练习 1..用is或are填空 (1). There _____ a book and two pens on the desk. (2). There ____ some water in the picture. (3). There _____ some cards in Jim’s bag. (4). There _____ an eraser in the pencil box. (5). There _____ one pupil in our school. (6). There _____ three footballs and a cap on the chair.
There be句型的用法 一、构成：There be ...句型表示的是“某处有(存在)某人或某物”，其结构为There be(is，are，was, were )+名词+地点状语。例如：There are fifty-two students in our class. There is a pencil in my pencil-case. There was an old house by the river five years ago. 二、各种句式： 否定句：There be句型否定句式的构成和含有be动词的其它句型一样，在be后加上“not”。也可用“no”来表示。即：no + n.（名词）= not a\an\any + n.（名词）。注意：no + n.（可数名词单数）= not a\an + n.（可数名词单数）；no + n.（可数名词复数）= not any + n.（可数名词复数）；no + n.（不可数名词）= not any + n.（不可数名词）。例如： There is an orange in her bag. →There isn’t an orange in her bag. →There is no orange in her bag. There are some oranges in her bag. →There aren’t any oranges in her bag. →There are no oranges in her bag. There is some juice in the bottle. →There isn’t any juice in the bottle. →There is no juice in the bottle.