PPhonons in lattices with rod-like particles
A. Sparavigna
Dipartimento di Fisica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino,Italy
The paper studies the modes of vibrations of a lattice with rod-like particles, in a continuum modelwhere the sites of the lattice are the connections among strings and rigid rods. In these structuresthen, translational and rotational degrees of freedom are strongly coupled. We will discuss inparticular two-dimensional lattices with auxetic-like behaviour. Auxetics are materials with a negativePoisson elastic parameter, meaning that they have a lateral extension, instead to shrink, when they arestretched. We assume as "auxetic-like" two-dimensional structures, structures which do not collapse,when stretched in one of the in-plane directions. The presence of rigid rod-like particles in the latticeprevents the shrinking of the membrane. Complete bandgaps between acoustic and optical modes areobserved in analogy with the behaviour of crystalline materials.PACS: 62.20, 63.20
Introduction
In this paper we discuss the vibrations of two-dimensional structures composed of flexible and rigidparts. The membrane contains rod-like particles connected with strings. In these structures,translational and rotational degrees of freedom are strongly coupled. As the models demonstrate, thesestructures are able to display complete band-gaps. Elastic materials can have phononic bandgaps -intervals of frequencies where no propagating phonons exist - in analogy with the photonic materials,known as photonic crystals, displaying bandgaps for light waves [1]. Phononic band-gaps can befound in crystalline materials [2,3] or in macroscopic periodic elastic media, the "phononic crystals"[4].Another family of interesting materials are those with the auxetic elastic behaviour, characterised by anegative Poisson elastic parameter [5]: this relevant property means that these materials have a lateralextension, instead to shrink, when they are stretched. Natural auxetic materials and structures occur inbiological systems, in skin and bone tissues and in crystalline membranes [6-8]. Man made auxeticmaterials have been produced from the nanoscale till the micro- and macroscales and cover the majorclass of materials, such as metals, ceramics, polymers and composites [9-14].We propose as "auxetic-like" two-dimensional structures, those structures which do not collapse,when stretched in one of the in-plane directions. The mesh is composed of strings and rigid parts: it isthe presence of rigid extended particles that prevents the shrinking of the membrane. The approach wefollow in the calculations is that recently proposed by Martinsson and Movchan [15]; these authorsstudied the phonon dispersions for membrane-like lattices with several geometries and non-uniformmass distribution along the strings, finding phononic band-gaps.
Flexible and rigid parts in the mesh.
To understand the behaviour of a mechanical two-dimensional mesh including rigid extendedparticles, let us start from a one-dimensional model, a chain composed of rigid rod-like particles andstring connections, where ' L is the length of the rigid mass M , and L the length of the stringconnecting two massive elements (see Fig.1). The vibrations of the chain we are considering are thoseperpendicular to the chain. In fact there are two possible directions, perpendicular to the chain: weconsider just one direction because they are degenerate.he unit cell of the lattice has a position given by the vector ( ) ' LL + . For simplicity, ' LL = . Thepositions of the lattice sites (0) are denoted by the lattice indices ,...2,1, ++ iii , and the sites of thebasis are denoted by B . The mass per unit length of the rigid rod is ' ρ . Ropes have a linear density ρ . Due to the geometry, o T is the equilibrium axial force in each string line part.Let us investigate the harmonic vibrations of the chain supposed to be infinite with displacements oflines and nodes along one of the transversal directions. bi u , is the displacement of one of the twonodes in the basis of the lattice cell at the reticular position i from the equilibrium. b can have thentwo possible determinations and B . With Bji w , it is called the displacement of a stringconnecting a node i in the lattice cell with the nearest neighbour node j . A linear co-ordinate ζ isranging from zero to the length Bji L of the string. For the strings, the equation in the case oftransverse vibrations is the usual wave equation, with a phase velocity ρ= o Tv . Solving theequation of motion for the string line units, we have as in Ref.15: ( ) ( )( ) ( ) κζκ κ−+κζ=ζ sinsin coscos)( ,0,,0, L Luuuw
BijBiBji (1)and in the case of a time-harmonic oscillation of frequency ω , v ω=κ .The dynamic force the string exerts at node 0 equals: ( )( ) L uLuvTddwTf iBioBjio κ −κω=ζ−= =ζ sincos (2)Assuming the rod-like particles in the chain as rigid, we can write the equations for the motion of thecentre of mass and for rotation around it in a plane perpendicular to the chain. In the case of smalldisplacements of the lattice nodes, equations are: ( ) ζ+ζ=+ ddwddwMTdt uud BijBijoiBi (3) ( ) ( ) iBioBijBijoiBi uuILTddwddwITLdt uud −− ζ−ζ=− (4)where index j denotes the lattice site connected with the site i on the right, and index ' j the siteconnected with i on the left of the rigid particle. Let us remember that ζζ ddwddw BijBij , areproportional to the force components perpendicular to the chain, as from Eq.(2), and then pulling therigid body. The force component parallel to the chain is simply considered equal to o T . In Eq.(4), thefirst contribution on the r.h.s. represents the momentum of the dynamic forces f on the rigid mass;the second contribution represents the momentum of axial tension o T on the rod-like particle. In Fig.1,a diagram shows the forces on the mass. Fig.1(b) illustrates that the axial forces produce a torque onthe rod-like particle when it is not in the equilibrium position.f we are looking for Bloch waves with wavevector k, it is possible to write for each lattice site: ( ) )2exp(;2exp ,00, ikLtiuuikLtiuu BBii −ω=−ω= (5)and then the dispersion relations for the frequency ω can be easily obtained from the dynamicequations (3) and (4) of the rods. To simplify the solution, a Bogoliubov rotation of B uu , ispossible. The Bolgoliubov transformation is the following: θ−η=θ+η= iuiu B ; (6)and then we have explicitly the translation and the rotation degrees of freedom coupled in the twoequations: ( )( ) ( ){ }( )( ) ( ){ } kLSCkLSvTLI kLCkLvSTM oo sincos2 sincos
22 2 ωη+θ−+θω−=θω− ωθ+−ηω=ηω− (7)where ( ) LC κ= cos ; ( ) LS κ= sin . For ( ) =κ L , for any κ standing wave, modes existcorresponding to internal vibration of the strings, with no associated nodal displacements. Let usconsider the reduced frequency o ωω=Ω where vMT oo =ω . The dispersion relations of thechain as a function of the wavenumber k are shown in Fig.2 for different values of the ratio MLI .It is interesting to note the existence of a gap in the phonon dispersion between acoustic and opticalmodes . Honeycomb lattices .A two-dimensional model will be a planar membrane, for instance with a honeycomb structure, asshown in Fig.2 on the left: the lattice will be described using a unit cell with a convenient set ofvectors ( , ll ), giving the lattice reticular positions. In the case of the honeycomb structure, the latticehas two nodes per unit cell. A mechanical model of the lattice can be made with rigid rod-likeparticles, with length L and mass per unit length ' ρ , and ropes with length L and linear density ρ .The honeycomb lattice shown in Fig.2 on the right, has rigid connections between sites substitutingall the bonds parallel to one of the lattice directions.It is straightforward to investigate the harmonic vibrations of a two-dimensional mesh, if it is supposedto be infinite with displacements of lines and nodes in the direction perpendicular to its plane. As inthe case of the one-dimensional lattice, bi u , is the displacement of one of the nodes in the basis of thelattice cell at the reticular position. The same for ', bbji w , the displacement of a string connecting anode in the lattice cell with the nearest neighbour node. If we are looking for Bloch waves withwavevector k , it is possible to write for each lattice cell the displacements as: ( ) ( ) exp;exp lklk ⋅−ω=⋅−ω= itiuuitiuu BBii (8)If the basis has two sites for instance, the dispersion relations for the frequency ω are then obtainedsubstituting these waves in the dynamics of rods, solving the following equations: )( ) ( ) ∑ − ∑− ζ−ζ=−ω− ∑ ζ+ζ=+ω− ', 0,,', ||,0,'0,20,,2 ', 0,'0,0,,2 jj iBijj oBijBijoiBi jj BijBijoiBi uuTILddwddwITLuu ddwddwMTuu (9)where 'j,j are indices for the nearest neighbour sites, as in the one-dimensional chain. The secondterm in the r.h.s. of the second equation is containing the components ||, o T of the forces, parallel toaxial direction of the rod in its equilibrium configuration. For the honeycomb cell, ( ) ||, π= oo TT . In the case of the honeycomb lattice, there are two forces applied to the points 0 or B of the rod: thecontribution of the components of these two forces is positive and then giving a resulting torquestabilising the rod-like oscillators. Equation (9) becomes: ( )( ) ( ) ∑ −− ζ−ζ=−ω− ∑ ζ+ζ=+ω− ', 0,,0,'0,20,,2 ', 0,'0,0,,2 jj iBioBijBijoiBi jj BijBijoiBi uuILTddwddwITLuu ddwddwMTuu (10)The reduced frequency o ωω=Ω (where vMT oo =ω ) of the honeycomb two-dimensional latticecan then be evaluated for different values of the ratio MLI as a function of the wavevector k , inthe directions OX, OC and OY, where O is the centre of the Brillouin Zone. The dispersions areshown in the Figure 4. Re-entrant honeycomb lattices .A two-dimensional model for an auxetic mechanical system is that proposed in Ref.[4] and shown onthe left of Fig.5: the model was introduced to easily explain the behaviour of a material with anegative Poisson elastic parameter. The cell is a re-entrant honeycomb cell. The auxetic model can beviewed as a structure composed of rigid parts of length L and ' L : when the lattice is stretched, itexpands instead to shrink. A model composed by ropes with length L and rod-like particles ' L as inFig.5 on the left is not suitable for discussing the vibrations, as we immediately explain.Let us imagine to consider a two-dimensional honeycomb lattice where it is inserted in the honeycombcell the rigid unit ' L , as shown on the right part of Fig.5. On points 0 or B of the rigid masses, threeropes and then three forces are acting. A force is parallel to the rod, then oo TT = ||, ; the other two havecomponents ( ) ||, π= oo TT , but negative with respect to the rod axis. If just these two forceswere present in the model, as in a structure like Fig.5 on the left, the resulting torque in Eq.(9) wouldbe destabilizing the acoustic oscillations of the membrane, in the case of a wavevector k in the Xdirection.For the model in Fig.5 on the right, the dispersion relations for the frequency ω can be easily obtainedfrom the dynamics of the rod-like particles. Fig.6 shows the reduced frequency as a function of thewavevector k , in the directions OX, OC and OY, where O is the centre of the Brillouin Zone. Inthis image, the auxetic lattice has all the ropes subjected to the same tension o T . The reducedrequency is o ωω=Ω , where vMT oo =ω , as for the honeycomb lattice. The dispersion relationsdepend on the value of the ratio 'MLI . Note that 'L is the length of the rods, different from thelength of the ropes. As previously told, the true auxetic network shown in Fig.5 on the left must besubstituted with the auxetic lattice of Fig.5 on the right. We still call this model auxetic, because it isdifferent from the honeycomb in the forces distribution.Moreover, in this model, the strings parallel to the rigid masses can have a different tension o T ξ .These ropes have a sound speed ρξ= o Tv , different from the sound speed ρ= o Tv of theother ropes. The reduced frequency we use is o ωω=Ω , with vMT oo =ω . The Eq.(9) must berewritten in the following form: ( )( ) ( )( )∑∑ ∑∑ −− −+ +−+ −=−− ++ +=+− 'k,k 0,iB,ioB0,'ik0B,iko2 'j,j 0,iB,ioB0,'ij0B,ijo20,iB,i2 'k,k B0,'ik0B,iko'j,j B0,'ij0B,ijo0,iB,i2 uuI2T'LddwddwI2 T'L uuI2T'LddwddwI2 T'Luu ddwddwMTddwddwMTuu ξζζξ ζζω ζζξζζω (11)where indices ', jj are used when sites are connected by ropes with tension o T and ', kk when sitesare connected with ropes with tension o T ξ . Note the different sign from Eq.10. An approach to solvesystem (11) with the Bogoliubov transformation was previously proposed [16].The Fig.6 shows the phonon dispersions of the auxetic lattice, for different values of ratio MLI and for =ξ . When parameter ξ increases over a certain value, a complete bandgap between theacoustic and the optical mode appears and this is shown in Fig.7. Of course, this approach withdifferent axial tensions is possible for the honeycomb model too.In both models, honeycomb and auxetic, the rod-like masses are viewed by the waves in the Y-direction as point-like ones. Auxetic membranes.
We have seen that a complete bandgap is easily obtained in two-dimensions, adjusting latticeparameters and interactions, that is changing elastic properties or densities of ropes. Of course,different and more complex auxetics must be proposed and studied, to exhaustively understand thebehaviour of these structures. If we consider as "auxetic-like" two-dimensional structures, thosestructures which do not collapse, when stretched along one of the in-plane directions, severalmembranes can be proposed, but it is necessary to insert some rigid parts in their mesh.Let us consider for instance the square lattice in the upper part of Fig.8. The thick lines represents therod-like particles, which have different orientations in the plane of the lattice. Then the lattice unitcontains two rigid rods. In the lower part of the figure, the phonon dispersions are shown. If themasses in the lattice unit are different, a complete band-gap appears, in agreement with the behaviourof crystalline systems [3] and mechanical systems with point-like masses proposed in Ref.[15].More complex two-dimensional structures can be proposed as that shown in Fig.9. The structureresembles the model for auxetics proposed in [17] and the membrane studied in [18]. This lastreference discusses the in-plane vibrations of rectangular rigid particles connected by harmonicelements.f course, different approaches to the problem of vibrations of auxetic structures are possible. Forinstance, a solution based on finite elements was used in Ref. [19] to solve a macroscopic mechanicalsystem. These studies are in fact very important for applications. The aim of this paper is instead theinvestigation of the role played by rod-like particles in the lattice vibrations and if it is possible tocreate a band-gap with proper mass differences or interaction anisotropy in the unit cell of the lattice.
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Figure captions
Fig.1: The chain with rigid particles in the upper part ( L is the length of the rods and strings). In (a),the dynamic forces acting on the rigid particle and in (b) the axial tension of ropes.Fig.2: Reduced frequency of the phonon dispersions as a function of the wavelength for differentvalues of the ratio MLI (0.1 red, 1 green and 10 blue). Note the behaviour of the optical mode forhigh values of the ratio MLI .
The gap between acoustic and optical modes is quite pronounced forthe green curve.Fig.3: The honeycomb structure. On the right, the primitive lattice points (black dots) and the points ofthe basis (white dots), and a set of two lattice vectors convenient for calculations. Thick lines arerepresenting the rigid rods. The three directions OX, OC and OY along which evaluate thedispersion relations are also displayed.Fig.4: The phonon dispersions for the honeycomb lattice for three different ratios MLI (a=0.5,b=1, c=5). The group velocity is strongly dependent on the direction. For the Y direction, the acousticmode does not change when the ratio MLI changes. In fact, the rod-like masses are viewed by thewaves in this direction as point-like masses.ig.5: The auxetic structure. For the auxetic lattice the Poisson coefficient is negative: if the lattice isstretched, it expands instead to shrink. L and ' L are the lengths of the rigid rods in the auxetic mesh.On the right, the mesh used to investigate vibrations. The figure shows the primitive lattice points(black dots) and the points of the basis (white dots), and a set of two lattice vectors convenient forcalculations. Thick lines are representing the rigid rods, thin lines the ropes. Note that at each point 0or B are attached three ropes. The directions OX, OC and OY along which the dispersion relationsare evaluated are also displayed.Fig.6: The phonon dispersions for the auxetic lattice for three different ratios ' MLI (a=0.5, b=1,c=5). Parameter =ξ . For the Y direction, the acoustic mode does not change when the ratio ' MLI changes.Fig.7: The phonon dispersions for the auxetic lattice with ratio = MLI for =ξ (a), and =ξ (b) reported for comparison. Note the complete band-gap for (a). The same result is possible in thehoneycomb lattice.Fig.8: The phonon dispersions for the square lattice depicted in the upper part. If the rods parallel to Xdirection have a different mass from the rods parallel to Y, then a complete band-gap appears (massratio 2, red lines).Fig.9: Phonon reduced frequency for the lattice depicted in the upper part. If the rods parallel to Xdirection have a different mass from the rods parallel to Y, then a complete band-gap appears (massratio 4).(b) reported for comparison. Note the complete band-gap for (a). The same result is possible in thehoneycomb lattice.Fig.8: The phonon dispersions for the square lattice depicted in the upper part. If the rods parallel to Xdirection have a different mass from the rods parallel to Y, then a complete band-gap appears (massratio 2, red lines).Fig.9: Phonon reduced frequency for the lattice depicted in the upper part. If the rods parallel to Xdirection have a different mass from the rods parallel to Y, then a complete band-gap appears (massratio 4).