F Dell'isola

S Vidoli

Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping

Keywords: modular truss beam, electric transmission line, piezoelectromechanical coupling, vibration control

We call piezoelectromechanical (PEM) truss beam a truss modular beam coupled with a transmission electrical line when the coupling is obtained by piezoelectric actuators which act as bars in the module and as capacitances in the electrical line.

The truss module length is assumed negligible with respect to the considered wave lengths. The transmission electrical line is assumed continuously distributed along the truss beam.

 we formulate a continuum model for PEM truss beams and we prove that there exists a critical value for the transmission electrical impedance in the neighborhood of which the electromechanical modal coupling is maximum and the possible electrical dissipation of mechanical energy is relevant.

Introduction

When formulating a mathematical model for a truss modular beam two different approaches are possible: i) in a re®ned description, the deformation of every element of the truss is accounted for, ii) in a continuum one, the kinematic of the truss is described by displacement and rotation ®elds de®ned on a curve.

The ®rst approach, when applied to technologically meaningful instances, often leads to oversized numerical problems and models not easy to handle. Therefore, the second one (see for instance [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF], [START_REF] Noor | Analysis of beam-like lattice trusses[END_REF], [START_REF] Abrate | Continuum modelling of latticed structures[END_REF], [START_REF] Vidoli | Strutture spaziali ¯essibili: modellazione e simulazione orientate al controllo[END_REF], [START_REF] Teresi | Analisi modale di travature reticolari modulari[END_REF]) has been proposed: the displacement and rotation ®elds describe a homogenized average displacement and rotation of the generic constituent module.

In this paper, using the method of virtual power [START_REF] Maugin | The method of virtual power in continuum mechanics: application to coupled ®elds[END_REF], we formulate a continuum description for a piezoelectromechanical truss beam which accounts for its modular structure, and use it to propose a new vibration damping technique. The typical module constituting the considered truss beam is composed: 1) by a pair of ¯at rigid diaphragms connected by a number of elastic bars (some of which are piezoelectric) constrained to remain straight and 2) by an electric net connected also with the piezoelectric actuators.

The electromechanical system we consider presents the following features: ± at least one of the bars of each module is a piezoelectric actuator, ± the electrical transmission line shows the same periodicity properties of the truss beam, so that every mechanical module corresponds to a modular electrical net, ± the piezoelectric actuators are connected to the electrical line so that their equivalent circuits are to be regarded as part of the modular electrical net.

One important assumption we apply in this paper concerns the electromechanical behavior of the piezoelectric actuators: following [START_REF] Dell' Isola | Saint-Venant Problem in Linear Piezoelasticity[END_REF], [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF], they are regarded as cylinders undergoing Saint-Venant deformations only. Then the results obtained in [START_REF] Dell' Isola | Saint-Venant Problem in Linear Piezoelasticity[END_REF], [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF] allow us to ®nd the relationship between the tabular Voigt piezoelectric constants and the relevant piezoelectric stiffnesses and impedances. A MATHEMATICA program, supplied in Appendix A, needed to be developed in order to use quoted results in the present situation.

The new structural system studied in this paper deserves, in our opinion, some interest. This is proved by the following results obtained for simple, but interesting, 2D and 3D module choices in which extension, bending and torsion are examined.

We have studied the effects of the variation of the transmission line impedance on the wave propagation speed and on the modal and frequency response of the structure; we ®nd that there exists a critical range of inductance in which the electromechanical coupling is relevant. In this range:

± the displacement-electric potential waves propagate simultaneously with two close propagation speeds, ± the wave energy is shared in comparable proportions in mechanical and electrical parts.

Unfortunately, the electromechanical coupling range for bending and the modular transmission line introduced in the present paper lies in the neighborhood of in®nity: the inductances necessary to insure electromechanical coupling are not technically feasible. As a consequence, the modal damping ratios for bending are typically 10 3 Ä 10 4 times smaller than in extension or in torsion, under comparable conditions. In a forthcoming paper, we will consider an electrical line whose evolution equation is a PDE in which fourth order space derivative appears, so obtaining a more ef®cient modal coupling.

Having used MATHEMATICA, we dispose (in the cases of extension and torsion) of the explicit symbolic formulas expressing wave speeds and the norms of the associated wave forms versus the electrical transmission impedance. This analytical treatment becomes dif®cult when studying the modal analysis under given boundary conditions or in the case of bending wave.

For this reason we have used a standard numerical procedure to ®nd:

± the bending wave dispersion formulae, ± the eigenfrequencies electrically insulated, the corresponding eigenfunctions and the frequency response operator H jx of the clamped-clamped PEM beam.

We remark, concerning the frequency response operator, that a partition of the set of impedance values is obtained, as proved by the plot of a norm of the H jx operator. In a range, the response operator yields for different forcing frequencies either a purely mechanical or a purely electrical response. Instead in a given interval, centered around what we call ``maximal exchange value'', the response operator yields only coupled electromechanical responses: the range of resonant forcing frequencies results to be also sensibly increased with respect to the purely mechanical one. This means that around the ``maximal exchange value'' for impedance it is possible to damp modular truss by dissipating electrical energy.

Our numerical computation shows that this phenomenon can be exploited in technologically relevant instances: indeed the available PZT materials and actuators seem to allow the damping of truss beams of technologically relevant size.

To our knowledge, a continuously distributed damping for a modular truss has never been proposed before (see [START_REF] Culshaw | Smart structures and materials[END_REF]). It seems to us that such damping technique is more ef®cient than those already proposed in the literature, characterized by the use of actuators supplying concentrated forces.

Identification procedure

In [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF] truss beams are modelled using a continuum description generalizing the standard Bernoulli-Navier theory. The main result of the cited paper, based on the identi®cation of the mechanical power on the re®ned level with that on the continuum one, allows for the determination of the constitutive equations for the contact and inertial actions explicitly in terms of the mechanical properties in the module. The identi®cation of the mechanical response of the truss modular beam, when modelled as a one-dimensional continuum, is then possible by means of an algorithmic procedure.

The same approach applied to the electrical transmission line leads to a standard homogenization procedure in which the electrical impedance is continuously distributed per unit length.

We adapt here the identi®cation procedure proposed in [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF] to piezoelectromechanical truss beams.

Refined model

We assume that: ± the bars are constrained to remain straight, therefore they can be only elongated, ± the electric potential is assumed linear in any piezoelectric bar (see [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF]).

With these hypotheses, the state of each module is completely determined by the placement of the diaphragms which are assumed to be rigid, and by the capacitive charges at the extremes of the piezoelectric bars.

Therefore, let us assume as ``kinematical descriptors'' characterizing the state of the considered system, the position of a point in the diaphragm X o , a triple of unit vectors G k representing the attitude of the diaphragm, and the set of the charges at the ends of the piezoelectric bars Q.

We shall label with a superscript minus ()) and plus (+) two consecutive diaphragms belonging to a module. So, respectively, on the reference and actual con®guration, the kinematical descriptors of a module are

X AE o ; G AE k ; Q AE x AE o ; g AE k ; q AE
so that the positions of the end points of the typical bar in the module are

x AE b x AE o a AE b g AE 2 b AE b g AE 3 ; 1
where a AE b ; b AE b are the time-independent coordinates of the bar labeled b with respect to the vectors g 2 and g 3 in the diaphragm plane.

Corresponding to the previous choices, being

e b : kx b À x À b k L b À 1
the elongation of the bar, the time-rate deformation ®eld in any bar is given by

_ e b x b À x À b _ x b À x À b L b kx b À x À b k ; 2
where L b :kX b ÀX À b kis the reference length of the bar. Their electrical counterparts will be

q b : q b À q À b
and the piezoelectric capacitive current

_ q b Á q b À q À b : 3
For any module, the expression of the electrical power associated to the set N p of its piezoelectric bars is

W refined e pPN p / p Á _ q p ; 4
where / p represents the electric potential difference between the end points of the piezoelectric bar p.

In a similar way, for any module we express the mechanical power of the contact actions n b Concerning the re®ned constitutive relations for the contact actions and for the electric potentials, we limit our attention to the linear case. Following [START_REF] Dell' Isola | Saint-Venant Problem in Linear Piezoelasticity[END_REF], [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF] the electro-mechanical behavior of the piezoelectric actuators is derived. We regard them as Saint-Venant cylinders, starting from the equations for the Cauchy stress tensor T and the electric displacement tensor D div T 0; div D 0 : 7

The linear constitutive equations for piezoelectric materials can be found in [START_REF] Eringen | Electrodynamics of Continua I[END_REF]. Equations are solved under the Saint-Venant hypothesis in order to determine the electromechanical stiffnesses of such actuators. As our interest is limited to purely extensional actuators, we quote here the linear constitutive relations expressing the contact action and the electric potential as functions of the electromechanical deformation ®eld n b k ee e b k eq q b ; 8

/ b k qe e b k qq q b : 9
The results of [START_REF] Dell' Isola | Saint-Venant Problem in Linear Piezoelasticity[END_REF], [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF] allow us to ®nd the relationship between the tabular Voigt piezoelectric constants and the more technologically useful actuator constants k ee ; k eq ; k qe ; k qq .AM ATHE-MATICA program devoted to transform the Voigt constants is given in Appendix A.

The constitutive relations for the purely electric elements (resistances and inductances) are written as

/ p k R _ q p ; / p k L q p : 10

Continuum model

The coarse model apt to describe our electromechanical system is a one-dimensional continuum endowed with structure: we assume as model for the mechanical part the standard Euclidean structure used in the theory of Bernoulli-Navier beam (see for a complete discussion [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF]), and for the electric part the standard model from the theory of the continuous electrical transmission line. Finally, the electromechanical coupling will be modelled assuming that every used constitutive equation depends on all the electro-mechanical state parameters listed below.

The parameters describing the state of every material point P of this continuum are chosen paralleling the choices made in the re®ned model. Respectively, in the reference and actual con®guration we have

P o 1; D k 1; C1 ; p o 1; t; d k 1; t; c1; t ;
where 1 and t are the spatial and temporal coordinates, p o represents the position of the point P; d k the directors orienting P and c1c 1 1;c 2 1;... the charges stored in the piezoelectric components in the 0; 1 interval.

The generalized displacement ®eld is described by the following relations:

u1; tp o 1;tÀP o 1; RPOrth, d k 1; tR1;tD k 1; d1;tc1;tÀC1 :
The velocity ®eld is de®ned deriving with respect to t the previous equations

w _ p o ; W _ RR T A _ d k Wd k ; i _ c : 12
Finally, the generalized deformation ®eld is chosen comparing, as usual, the actual and the reference con®gurations

e R T p H o À P H o ; E R T R H ; . c H À C H : 13
Let us now write the expressions of mechanical and electrical power: the actions which expend power on the time-rate generalized deformation ®eld are the Piola-Kirchhoff contact actions s; S and the electric potential ®eld u. On a module of reference length L we have

W coarse c L 0 s Á _ e À 1 2 S Á _ E d1 ; 14 W coarse e L 0 u Á _ .d1 ; 15
Finally, denoting by b; B the distributed inertial actions, the power expended is given by

W coarse c L 0 b Á w À 1 2 B Á Wd1 : 16
The mechanical balance equations in terms of the Piola-Kirchhoff stress ®elds are

Rs H b 0 ; 17 RSR T H p H o Rs B 0 : 18
Now, let us choose the typical module for the electrical circuit, i.e. show how the piezoelectric bars are connected to each other as well as with the resistance and inductance elements. Figure 1 represents the more general mesh used in the present paper. For a linear continuous transmission line characterized by the chosen electrical mesh, Kirchhoff circuital laws and constitutive equations read

u H j L c j R _ c V ; 19 _ c H _ v a _ v b ; 20
j qq v b j qe u H u ; 21 j l v a j r _ v a v u ; 22
where constitutive equations ( 21) and ( 22) are the continuum counterpart of ( 9) and [START_REF] Teresi | Analisi modale di travature reticolari modulari[END_REF]. For vanishing initial conditions we get

v a t 1 j l t 0 t H 0 exp À j r j l t H À s v usdsdt H : 23
Given a function f f 1; t, let us introduce the following integral operator

I a f : t 0 t H 0 exp Àat H Às f Á; sdsdt H ; 24
With the following two properties lim a3I I a f 0 ;

I a f H I a f H ; so that v a 1 j l Ijr j l v u :
With these assumptions

u H j R _ c j L c V ; 25
j qq c H j qe u H u j qq j l Ijr j l v u : 26
The previous equations are easily rearranged in terms of c to give

j qq c HH j qe u HH j L c j R _ c V j qq j l Ijr j l v H j R _ c j L c V ; 27
which for j r 3I Ijr j l 30 reduces to a wave equation for the charge vector density c.

We note that for j R 3I, Eq. ( 19) reduces to _ c 0 : 28

Then cÁ; tc 0 and _ c H 0, so that Eqs. ( 20), ( 21) and ( 22) become

_ v a À_ v b : _ v; j qq c H 0 À j qq v j qe u H u; j l v j r _ v v u ;
29 which reduce to the Kirchhoff laws for the RLC circuit separately connected to every piezoelectric bar (remark) that in this limit case c H 0 represents the initial distribution of capacitive charge:

j l v j r _ v j qq v À c H 0 Àj qe u H v
; q AE x À o p o À L 2 p H o ; x o p o L 2 p H o ; g À k d k À L 2 d H k ; g k d k L 2 d H k ; q À c À L 2 c H ; q c L 2 c H ;
31 denoting by L the reference length of the module.

In order to identify the continuum constitutive functions we assume that, for any pair of corresponding motions, the power expended by the actions prescribed within the ®ne model equals the power expended by the corresponding actions within the coarse model

W coarse c W refined c ; W coarse e W refined e ; W coarse in W refined in : 32
This identi®cation procedure was used by [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF] in a purely mechanical contest, and has shown its validity in the study of technical problems [START_REF] Vidoli | Strutture spaziali ¯essibili: modellazione e simulazione orientate al controllo[END_REF], [START_REF] Teresi | Analisi modale di travature reticolari modulari[END_REF]. Moreover, the results expounded in [START_REF] Maugin | The method of virtual power in continuum mechanics: application to coupled ®elds[END_REF] clearly indicate that, when dealing with the electromechanical coupling phenomena, the most useful tool is given by the method of virtual power: we are con®dent that the identities (32) allow among the possible continuum models the determination of that one which better approximates the discrete re®ned model.

By using explicitly the expressions (4), ( 5), ( 14), (15) for power, and by a massive use of computer aided symbolic calculation via Eqs. (31), we ®nally get the constitutive coarse relations for the contact actions s; S:

s bPN b s b ; s b : R T b x b À x À b kx b À x À b k n b ; S bPN b S b ; S b : 1 2 a À b a b D 2 b À b b b D 3 ÂÃ s b ; 33
and for the electrical and piezoelectrical ones:

j qe k qe ; j qq k qq L; j R k R L ; j L k L L : 34
Equations (34) represent a simple homogenization of the re®ned discrete parameters k:: in the coarse continuum ones j:: used in the classical theory of continuously distributed transmission lines. We refer to the Appendix B for a complete explicit expression of the non-linear inertial constitutive coarse relations.

3 Linearized equations: applications to particular truss modules Equations (33), (34) and (46 through 56) in Appendix B provide the linear expressions for the piezo electrical and purely electrical actions and the nonlinear ones for the contact and inertial actions. In order to start the analysis of the behavior of PEM beams, we have linearized these last ones (see in Appendix C for details of such a linearization) assuming that either j r 3Ior j R 3I.

Under the assumption j r 3I, we introduce the linearized expressions of s; S and b; B in the balance equations (17, 18), and rearrange the mechanical parameters in a function vector u1; t and the electrical ones in c1; t. For example, in the case of a 2D truss beam with one piezoelectric bar per module the function vector u1; t contains the axial displacement u, the transverse displacement v and the rotation ®elds # u1; t u1;t v1;t #1;t H d I e while c1; t is a scalar function representing the capacitive charge stored in segment [0; 1]o f the line. The linear equations of the considered electromechanical system may be written now as follows:

K 0 ee u K 1 ee u H K 2 ee u HH K eq c HH M 0 u M 1 u H M 2 u HH 0; K qq c HH K qe u HH K R _ c K L c V ; 35
where K :: :: ; M :: ; K R and K L , are, respectively, the stiffness, mass and impedance matrices, whose expression in term of the electromechanical properties of the module are known.

Similarly, under the assumption j R 3I, we have

K 0 ee u K 1 ee u H K 2 eq u HH K eq v H M 0 u M 1 u H M 2 u HH 0; K qq v K qe u HH K r _ v K l v v ; 36
in which the variable v plays the role of an internal variable (see [START_REF] Maugin | Material Inhomogeneities in Elasticity[END_REF]).

Description of the considered truss module

We have examined three different truss modules.

In the ®rst one, shown in Fig. 2, there is only one piezoelectric bar centered along the symmetry axis; in addition, all the geometrical and mechanical properties are symmetric with respect to this axis. This fact allows us to obtain the axial displacement and electric current equations uncoupled from the shear and bending ones. This is a particular case of Eqs. (35) in which the stiffness, mass and impedance matrices reduce to scalar numbers: wave analysis can be easily faced in a symbolic way. The bars containing the piezoelectric actuators are marked with bold lines; the normal lines indicate the aluminium bars.

The second module, with four PZT bars electrically connected in parallel, is designed to couple torsion with the electric transmission line, and can be studied exactly in the same way as the ®rst one.

The third module, with two symmetric piezoelectric actuators, is designed to establish coupling phenomena between the bending beam and the considered electric transmission line.

Table 1 resumes the geometrical and mechanical data of the described truss modules. In all the modules we assume as piezoelectric actuator a cylinder of PZT5 material whose constants are listed in Appendix A.

Analysis of numerical results

The numerical methods we use for the ®rst analysis of Eqs. (35) are based on standard FE and Runge-Kutta discretizations, implemented by the computation environment MATHEMATICA. The section is divided into three parts. In Sect. 4.2, we consider the ®nite clamped-clamped and electrically insulated electromechanical beam. The spectrum of the differential operator in (35) is discrete, and determined via the simultaneous diagonalization of the corresponding electromechanical rigidity and mass FEM matrices. Subsequently, we determine the set of eigenfunctions, each of which is composed by a purely mechanical and purely electrical part. Finally we determine, via a Fourier Transform of the FE discretized version of (35), its frequency response operator.

In Sect. 4.3, we introduce in the coupled transmission line a dissipative impedance. For all considered modules, we determine the modal damping ratio as a function of the line impedance. This leads us to consider damped vibration only for extensional and torsional problems. This is done projecting (35) on the ®rst two eigenfunctions found in the previous Section, thus ®nding a second order system of ODEs for the corresponding coef®cients. The Cauchy problem for this system of ODEs we solve using a standard Runge-Kutta integrator. Different values for impedance are considered. We ®nd the range for impedance in which a relevant energy exchange between the mechanical and the electrical components occurs and an ef®cient electrical damping is possible. The ratio x=k physically represents the wave speed; the ratio ju j j=jc j j between two components of the mechanical and electrical eigenvectors measures electromechanical coupling in the wave. We can analytically determine the explicit dispersion formula k kx for axial displacement-current and twist angle-current waves, arising when the modules in Fig. 2 and in Fig. 3 are considered. This can be done as in both cases in Eqs. (35) the stiffness, impedance and mass matrices are reduced to scalar numbers and the last two addends in Eqs (35) 1 are vanishing. With the obvious meaning of the symbols we have k x 2 j ee j L j qq m 0 AE j 2 ee j 2 L 4j eq j L j qe m 0 À 2j ee j L j qq m 0 j 2 qq m 2 0 q 2j ee j qq À j qe j eq v u u t : 38

The line inductance minimizing the difference between the two wave speeds and maximizing the electromechanical coupling is given by j L 2j eq j qe À j qq j ee AE 2j eq j qe À j qq j ee 2 À j 2 ee j 2 qq q j 2 ee Á m 0 : 39

In Fig. 5 are shown, for the ®rst truss module, the variation of the wave speeds when increasing the line inductance j L , and the electromechanical coupling measured comparing the elastic and capacitive energy of the different waves. We note that for both modules the existence of a critical value j L ± which we will call ``maximal exchange value'' ± for inductance is numerically proven. It maximizes the electromechanical coupling and minimizes the difference among the propagation speeds of those waves for which the electromechanical coupling is relevant. As one should expect, the maximal exchange value for inductance will play a relevant role also in the following modal analysis.

Unfortunately, the maximal exchange value for inductance, in the case of bending-current waves, is I for the present choice of the coupled transmission line; this is shown by the numerical determination of the dispersion formula and the corresponding study of electro-Fig. 5. Wave speeds and electromechanical coupling as functions of the line inductance (module 1: extensional problem) mechanical coupling. Our numerical results show that a relevant energy exchange occurs only in presence of too high impedance. In our opinion, this circumstance is due to the choice of an unsuitable spatial second order circuitry in the transmission line.

Modal analysis: a clamped-clamped electromechanical beam

We perform the modal analysis for Eqs. (35), adding to them the following boundary condition: u0; t0 uL B ;t0; c0;t0 cL B ;t0 ; 40 where L B is the total length of the beam (we assume L B 2 m). These boundary conditions physically mean that the beam is mechanically clamped and electrically insulated. We postpone to further investigations the study of more complex boundary conditions. In order to lighten the presentation of the obtained results, as from the qualitative point of view they are similar for the ®rst and the second modules, in the following we alternatively show plots concerning only one of them.

Eigenfrequencies and eigenfunctions

Although this does not mean any change in the numerical analysis, we assume, in this subsection, j R 0.

Numerical computation shows that it is possible to select pairs of eigenfunctions with electrical and mechanical modal shapes exchanged, as clearly shown by Fig. 6.

In Fig. 7 the locus of eigenfrequencies is plotted in the plane of frequency and line inductance; for j L j L the eigenvalues corresponding to the pairs of modal shapes determined in Fig. 6 are equal.

As one can also see from Fig. 5, in the neighborhood of j L the eigenfunctions show coupled electromechanical component; as a consequence, when j L 9 j L , for every eigenfunction the elastic deformation energy (relative to displacement modal form) and the electric capacitive energy (relative to charge modal form) are comparable; otherwise, for j L ( j L or K L ) j L , the eigenfunctions present either the mechanical or the electrical part sensibly greater than the other one.

Frequency response operator H jx

In this subsection, a dissipative line impedance is considered. A numerical evaluation of frequency response operator is obtained by a Fourier Transform of the FE discretized form of Eqs. (35). Let Hx be the FE matrix representing H jx . In Fig. 8 we show the plot of jdetHxj for the twist angle-current problem for six different values of j L .

For the case considered in Fig. 8, the units are chosen so that jdetH0j 1 and j L 9 330 H=m.

The whole set of numerical simulations we have performed allows us to state the following qualitative results: ± For j L ( j L and j L ) j L , the plot of jdetHj as a function of x shows two distinct maxima; one of them is always ®nite and corresponds to the electric eigenfrequency, the other one is always in®nite and corresponds to the mechanical eigenfrequency; this circumstance can be accounted for once one recalls the absence of any mechanical dissipation mechanism and the presence of an electric dissipative impedance. ± When j L 9 j L , the plot of jdet(H)j has only one maximum; the considerations developed in the previous section indicate that in this case there is a relevant electromechanical energy exchange and coupling; this means that, for every forcing action, the system response shows comparable electrical and mechanical components. ± When j L 9 j L the resonance frequency range increases its width with respect to the cases j L ( j L and j L ) j L ; apparently this could seem a drawback but the analysis of the following Sect. 4.3 shows that this is not the case. Indeed the resonance of the electrical signal allows for an ef®cient dissipation of a mechanical forcing action.

An example of piezoelectromechanical damping

In order to give a further test to the previous statements we implement a numerical simulation of a damping problem. More precisely, a time-evolution problem is studied: initial data are assumed to be given as a linear combination of the ®rst two eigenfunctions. Figure 9 clearly shows the electromechanical energy exchange in the case of non-dissipative impedance (dashed line) and the electrical damping of the mechanical vibration in the case of dissipative line (solid line).

In Fig. 10 are plotted the damping ratios of the ®rst four modes as a function of line resistance. The damping ration d is de®ned by the expansion:

xtx 0 e Àdt e jxt so that: x tx 0 =efor t d À1 . There is a critical value for resistance for which the mechanical damping ratio attains its maximum (see curves 1m and 2m).

In fact, Fig. 10 shows that for any mode there is a branching value jR beyond which modal shapes are characterized by either a mechanical or an electrical dominant component (respectively, curves 1m-2m or 1e-2e, where the ®rst digit refers to mode frequency); this can be physically interpreted as follows: even when the modal coupling controlled by j L is maximum, the choice of j R can avoid the ¯owing of mechanical into electrical energy. In Fig. 11 are plotted the damping ratios of the ®rst four bending-charge modes for module 3 in the same range of line resistance, they are sensibly smaller than those shown in Fig. 10; this is essentially due to the low modal coupling between the bending beam and the second order transmission line presented. In order to obtain an ef®cient electric coupling for the bending beam we will introduce a spatially fourth order transmission line.

Concluding remarks

A modelling procedure for smart truss beams has been introduced essentially based on an identi®cation in power; both the mechanical truss beam and electric transmission line have been described by continuum models in which PEM constitutive equations are expressed in terms of modular properties (generalizing [START_REF] Di Carlo | Continuum modelling of beam-like latticed truss: identi®cation of the constitutive functions for the contact and inertial[END_REF] with method suggested in [START_REF] Maugin | The method of virtual power in continuum mechanics: application to coupled ®elds[END_REF]).

The basic physical idea is to couple a truss beam with an electrical transmission line by means of piezoelectric elements which act respectively as bars and capacitances.

The piezoelectric stiffnesses and impedances have been derived by the tabular Voigt constants via a simple MATHEMATICA program (sketched in Appendix A), which implements results obtained in [START_REF] Dell' Isola | Saint-Venant Problem in Linear Piezoelasticity[END_REF] and [START_REF] Dell' Isola | Almansi-type boundary conditions for electric potential inducing ¯exure in linear piezoelectric beams[END_REF].

The continuum model leads to the equations of an internal variable continuum for k r 3I and of a microstructured continuum for k R 3I.

Three different modules have been studied in order to investigate coupling phenomena between capacitive charge and, respectively, axial displacement, twist angle and bending. Our aim was to answer the following question: what happens when changing the line impedance (i.e. j L and j R )?

We found an interesting result: there is a critical line inductance value that maximizes the electromechanical modal coupling; this means a relevant exchange of energy between the beam and the electrical line and the possibility of an ef®cient control on the mechanical phenomena acting on electric parameters. We only investigated a simple case of passive control in damping vibration, but several active control applications are conceivable, for instance acting on the terms V and v in Eqs. ( 35) and (36).

In particular, we obtain very high damping ratio in the case of extensional and torsional waves: distributing piezoelectric actuators along the beam and connecting them to form a transmission line proves to be a useful method for damping.

Unfortunately, in bending problems, the electromechanical coupling is a monotone increasing function of j L and in technologically meaningful range of inductance is low; this is also shown by the damping ratio plots of Fig. 11: this circumstance is essentially due to the absence in the functional spectrum of the electrical line operator of the bending eigenfunctions (i.e. electric line and truss beam have different modal shapes) and will lead us to consider a spatially fourth order transmission line electrically paralleling a bending beam.

Appendix A: Voigt constants

In [START_REF] Eringen | Electrodynamics of Continua I[END_REF]

6 where

 6 N d and N b are, respectively, the set of the bars constituting and connecting the diaphragms and d b ; d d are the densities per unit length of the bars.

Fig. 1 .

 1 Fig. 1. Typical modular electric mesh constituting the transmission line. The ®rst element from the left represents the capacitance of the PZT bar

Fig. 2 .

 2 Fig. 2. First truss module (bold lines for PZT bars)

Fig. 3 .Fig. 4 .

 34 Fig. 3. Second truss module (bold lines for PZT bars)

Fig. 6 .

 6 Fig. 6. The ®rst four eigenfunctions for module 1 ( j L 9 45 H/m; solid line for the displacement, dotted line for the charge eigenfunctions)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Eigenfrequencies locus in the inductance frequency plane (module 1)

Fig

  Fig.9a,b. Time evolutions for axial displacement-current ®elds. (j L j L ; dashed line for j R 0, solid line for j R T 0)

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Modal damping ratio vs. line resistance (module 2)

  the re®ned and the continuum model is established by assigning a map from the local placement in the coarse model p o ; d k ; c to the local placement of a module in the re®ned model x AE o ; g AE

	2.3
	Constitutive relations
	A relation between
	:	30

Table . 1

 . . Mechanical and geometrical properties of the aluminium and PZT bars

	Geometry	PZT bar	Al bar
	L 0:1m h 0:05 m	q PZT 7000 kg/m 3 A PZT 10 À4 m 2	q Al 2800 kg/m 3 A Al 2:510 À5 m 2
		l PZT 0:05 m	G Y;Al 70 GPa

2

  are de®ned the following matrices representing the linear constitutive relation for a transversely isotropic piezoelectric (TIP) material: There are ten independent material constants for such a material. The following MATHEMATICA program allows to determine the only four coarse parameters k ee ; k eq ; k qe ; k qq of Eqs. (8) and (9) that are most technologically relevant, once the matrices s, d and k are known.For the given PZT5 Voigt constants we obtain: k ee k eq k qe k qq 24:085 Á 10 9 Á A PZT À1:1537 Á 10 7 À4:4664 Á 10 8 Á l PZT 7:9394 Á 10 7 Á l PZT =A PZT :bd 1=LÀm d0 À 2m d1 À m d2 p À 1=La i m d0 a i m d1 a j m d1 a j m d2 d 2 À 1=Lb i m d0 b i m d1 b j m d1 b j m d2 d 3 L=4m d0 2m d1 m d2 p HH L=4a m d0 a i m d1 a j m d1 a j m d2 d HH L=4b i m d0 b i m d1 b j m d1 b j m d2 d HH 3 ;51 bd ÀL=4m d0 2m d1 m d2 p H ÀL=4a i m d0 a i m d1 a j m d1 a j m d2 d H

			kep B1/A2;
			kpe ÀB2/A2 lpzt; Ãlpzt=PZT cylinder lengthÃ
			kpp (1/A2) lpzt/Apzt;
		H	s 11 s 12 s 13 00	0	I
		f f	s 12 s 11 s 13 00	0	g g
	s	f f f f f	s 13 s 13 s 33 00 000s 44 00 0	g g g g g	;	41
		f d	0000s 44	0	g e
			000002 s 11 À s 12
		H	0000 d 15 0	I
	d	f d	000 d 15 00	g e ;		42
			d 31 d 31 d 33 00 0
		H	k 11 00	I
	k	f d	0 k 11 0	g e :			43
			00 k 33
	(Ã PZT5 Ã)
	s11 16:4; s12 À5:74; s13 À7:22; s33 47:5; s44 47:5; Ã 10^À 12Ã
	d15 584; d31 À172; d33 374; Ã 10^À 12Ã
	k11 1730; k33 1700;
	ÃExperimental data À3 Voigt Matrices Ã
	sigma Inverse[s] ;
	e d Á Transpose[sigma] ;
	eps eps0 k;
	Ã eps0 permittivity of free space 8:854 10^À 12 Ã
	Ã Voigt Matrices À3 TIP material representation Ã
	lambda sigma1; 1 sigma1; 2; mu sigma6; 6=2;
	alpha1 sigma4; 4; alpha2 sigma1; 3 ;
	alpha3 sigma3; 3=2 À alpha2 À alpha1;
	beta1 e3; 1 À e1; 5; beta2 Àe3; 1 ; beta3 Àe3; 3 ;
	gamma1 eps1; 1 ; gamma2 eps2; 2 ;
	( Ã TIP material representation À3 7 Ã )
	A1 2 alpha1 alpha2 alpha3Àalpha2^2=mu lambda;
	A2 gamma2 beta2 beta1=mu lambda;
	B1 beta3 Àbeta1 alpha2=mu lambda;
	B2 Àbeta3 beta2 alpha2=mu lambda;
	Ã 7À 3our actual definition Ã
	kee A1 À B1 B2=A2 Apzt; ÃApzt PZT cylinder section areaÃ

research contract of Agenzia Spaziale Italiana, issued under the scienti®c responsibility of Prof. A. Di Carlo.

Appendix B: Continuum inertial actions

The inertial actions identi®ed by the aforementioned procedure turn to be the sum of two parts

the actions for a generic bar or diaphragm are given in terms of these integral mass quantities and of the coordinates a; b de®ned in Eq. ( 1)

56

Appendix C: Linearization

The linearization of the continuum model inertial actions shown before is obtained with a simple MATHEMATICA symbolic program that provides to calculate (b, B) via the substitution p 3 P euD 1 vD 2 wD 3 ;


Susanne Jonsson

Maritha Hörsing

Investigation of sorption phenomena by solid phase extraction and liquid chromatography for the determination of some ether derivatives of tetrabromobisphenol A

Keywords: sorption, solid phase extraction, liquid chromatography

Observed sorption and chromatographic behaviour served as a starting point for examination of four ether derivatives of the flame retardant tetrabromobisphenol (TBBPA); dihydroxyethyl (DHEE), dimethyl (DMe), diallyl (DAE) and dibromopropyl (DBPE). To date, there is little or no information about these compounds in the scientific literature. The targets in the present study had calculated log K ow values of 6.0-10. A broad range of different types of solvents were used in order to elucidate the mechanisms of sorption onto silica (glass) and two solid phase extraction cartridges based on polystyrene divinylbenzene polymers, hydroxylated (Isolute ENV+) and vinylpyrrolidone (Oasis HLB) containing units. Also, three chromatographic columns (C18, C8, and dual C18-based, i.e. a polar ligand together with C18 on a silica surface) were evaluated. The following were observed: desorption of the targets from silica surface by 1-propanol showed the best results and the yields were further improved by silylation of the surface. Regardless of the experimental test conditions applied, the lowest recovery was seen for the DMe derivative, followed by TBBPA itself, which had calculated log K ow values in between the other targets. A washing solution containing up to 70% of methanol could be used without elution of the targets from the . The dihydroxethyl ether derivative of TBBPA is most probably not affected by environmental relevant pH due to a likely high pKa (ca 15). Also, this study shows that the polar character of the dual-phase column was not as pronounced as asserted by the manufacturer.

Introduction

Tetrabromobisphenol A (TBBPA) is the most extensively used brominated flame retardant (BFR), 1 with a global market of about 170 000 tonnes in 2004, representing more than 50% of the production of all BFRs. 2 The success of TBBPA arises from the fact that it is cheap, it is highly effective in fire prevention, and, most importantly, it is not considered to constitute a threat to either human health or environment. [START_REF]European Risk Assessment Report on 22'6,6'-tetrabromo-4,4'-isopropylidenendiphenol (tetrabromobisphenol-A). Part I, Environment and Part II, Human Health[END_REF] This profile is unique compared to the characteristics of other BFRs, such as polybrominated diphenyl ethers. Accordingly, TBBPA and oligomers and ether derivatives of TBBPA are incorporated into a broad range of plastic products and epoxy resins that are used in electrical and electronic equipment and installations worldwide. 2,[START_REF] De Wit | [END_REF] Together, the TBBPA derivatives utilized as flame retardants and TBBPA oligomers represent 18% of the total sales of TBBPA. 2 To our knowledge, few data are available on only one of the individual derivatives. That is unfortunate, since there are about 75 different BFRs, and thus the derivatives represent a substantial part of the compounds in use, even though each one hypothetically constitutes only 1% of the TBBPA family (i.e. 1700 tonnes). It should also be mentioned that the derivatives are relatively new as BFRs, and the proportions in use are expected to increase. 5 The two hydroxyl groups on the TBBPA molecule make it possible to reactively incorporate the compound into the backbone of the polymeric product, as is done in epoxy resins in printed circuit boards. [START_REF] Sinoharvest | Tetrabromobisphenol A (TBBPA) Application[END_REF] This process represents the most prevalent commercial application of TBBPA, although the compound is also employed to a lesser extent as a regular additive, for example in acrylonitrile butadiene styrene (ABS) plastics used in electronic housings. Moreover, the ether derivates are used as reactive or additive flame retardants, depending on whether or not they contain reactive groups (e.g. 1). However, the TBBPA oligomers are only added reactively and are part of the main constituents in the plastic resin.

Data on the various physico-chemical properties of the ether derivatives of TBBPA are scarce in the literature, although TBBPA itself has been studied extensively. Compared to all its derivatives, TBBPA has the lowest molecular weight with a log K ow of 4.5. 1 Thus, the TBBPA family is very hydrophobic, and, considering environmental partitioning, its members would preferably adsorb to solids and create TBBPA to concentrate in sediments. [START_REF] Watanabe | [END_REF]8,9 The same might be true for the ether derivatives, although to our knowledge there are no studies that support that conclusion. Field analyses have detected the dimethoxy derivative of TBBPA (TBBPA-DMe) in tissues of marine organisms, [START_REF] Watanabe | [END_REF] but reports are conflicting as to whether TBBPA-DMe is actually used as a flame retardant; for instance, Alaee et al. 10 and de Wit [START_REF] De Wit | [END_REF] have claimed that it is indeed the case, whereas the Bromine Science and Environmental Forum (BSEF) asserts that it is not (K.P. Rothenbacher, 2008, pers. comm.). Furthermore, it has been suggested that TBBPA-DMe is a product of microbe-mediated transformation of TBBPA. [START_REF] Watanabe | [END_REF] Sorption of hydrophobic constituents is usually ascribed to the organic (carbon) content of the matrix of interest, such as soil and sediment, but adsorption of hydrophobic compounds to mineral surfaces has been studied as well. 11 Non-ionic compounds may adsorb to silica-based constituents like the surface of laboratory glass, 12 which can lead to confusion when interpreting analytical results. 13 The latter problem is seldom discussed in the scientific literature, although Furtmann 14 and Sjödin 15 have considered these polar surface attraction phenomena in investigations of phthalates and organohalogens, respectively.

Liquid-liquid extraction using an organic solvent is the most commonly applied method for analysis of TBBPA in sediments or wastewater, 16,17 and solid phase extraction (SPE) has been employed to determine TBBPA in human plasma 9,18 and serum. 19 Although much analytical and environmental data on TBBPA have been accumulated and a paper concerning analysis of the derivative TBBPA bis(2,3-dibromopropylether) has been published 5 , little has been reported about the ether derivatives in the TBBPA family. Accordingly, we chose to investigate aspects of the analysis of some of those compounds and the problems associated with their adsorption on polar and non-polar surfaces, specifically silica (glass) and two types of polystyrene-divinyl benzene (PS-DVB) polymers used for solid phase extractions. We employed different solvents in the extractions, because our aim was to study and compare the derivatives, not to optimize analytical protocols. The higher molecular weight derivatives are not amenable to analysis by gas chromatography, and hence we used liquid chromatography instead. This work was an essential part of a comprehensive study of degradation of the ether derivatives, the results of which are to be reported separately. [START_REF] Hörsing | Virtual Computational Chemistry Laboratory[END_REF]

Experimental

Chemicals and equipment

This study targeted four ether derivatives of TBBPA: TBBPA (97%, cas 79-94-7), TBBPAdihydroxyethylether (TBBPA-DHEE, > 95%, cas 4162-45-2), TBBPA-diallylether (TBBPA-DAE, > 95%, cas 25327-89-3), TBBPA-dibromopropylether (TBBPA-DBPE, > 90% cas 21850-44-2), and TBBPA-dimethyl ether (TBBPA-DMe, 98%, cas 37853-61-5). The chemical structures and molecular weights of those derivatives are given in Table 1.

TBBPA was purchased from Sigma-Aldrich (Sweden) and three of the TBBPA derivatives were from Tokyo Chemical Industry (Labora Chemicon, Sweden). Dimethylated TBBPA was synthesized in our laboratory (see below). Several solvents representing a wide range of polarities and dipole moments were used (see Table 2 where the solvents are listed in order of increasing polarity): pentane, hexane, toluene, dichloromethane, 1-propanol, tetrahydrofurane (THF), ethyl acetate, acetone, methanol, and acetonitrile. The methanol and 1-propanol were of HPLC grade, and all other chemicals were of pro analysis quality (VWR International, Sweden). The solvents were chosen not only on the basis of 3 polarity, but also with regard to elementary structure and functional groups, considering them as aromatic (toluene), hydroxylated (n-propanol, methanol), ether (THF), halogenated (dichloromethane), ketone (acetone), and ester (ethyl acetate) eluents.

In the SPE analyses, we used Isolute ENV+ and OASIS HLB cartridges (200 mg; purchased from Sorbent and Waters, respectively), and both types were packed with PS-DVB-based polymers that had been produced as general sorbents suitable for compounds with properties ranging from hydrophilic to lipophilic. According to the manufacturer, this is feasible because ENV+ and OASIS HLB contain, respectively, hydroxybenzene and vinylpyrrolidone units in the PS-DVB backbone.

Synthesis of TBBPA-dimethyl ether

The non-commercial TBBPA-DMe (i.e. the methyl ether derivative of TBBPA) was synthesized by the following procedure. First, sodium was added to methanol to generate methoxide, and then TBBPA was added and transformed to its deprotonized form as it dissolved in the strong alkaline solution. Next, methyl iodide was added, and the solution was stirred for 15 min and then refluxed at 40 °C. The target compound, sodium, and methyl iodide were added in molar proportions of 5:12:24. The target was gradually methylated, and small aliquots of the solution were subsequently analysed by full-scan GC-MS to confirm the identity of the reaction product, and then the solution was diluted in water and extracted with three portions of dichloromethane. The collected organic phase was purified from unreacted reagents by extraction with two portions of water. Thereafter, the cleaned extract was dried using sodium sulphate and then evaporated to complete dryness to give the TBBPA-DMe (98%) as white crystals. A chromatographic system comprising a Hewlett-Packard 6890 GC and 5973 MS was used to establish the identity and purity of the product. Separation was done on a capillary column (5% phenyl VF-5MS, dimensions 15 m, 0.25 mm, and 0.25 µm i.d.; Scantec Lab, Sweden). The extract was silylated with TMS (trimethylsilyl) reagent prior analysis. The TMS reagent was prepared from bis(trimethylsilyl)-trifluoracetamide (BSTFA), chlorotrimethylsilane (TMCS) in pyridine (10:1:10). Detection was performed by electron ionization (70 eV) in the full-scan mode (m/z 50-700), and the mass spectrum obtained is shown in Fig. 1.

Instrumentation and chromatographic conditions

The targets were analysed using a Waters HPLC instrument (pump 510) equipped with a UV detector (Waters 481) at a wavelength of 280 nm, which had proven to be relevant for all the targets. Retention of the targets was evaluated using traditional silica-based columns: C8 and C18 (4.6 × 150 mm, 5 µm particle size; Waters, Sweden), and a reversed-phase Synergi column (4 u Fusion-RP 80A; dimensions 4.4 × 100 mm) comprising a dual phase with a polar non-nitrogen-containing ligand together with C18 on a silica surface purchased from Phenomenex (Genetec AB, Sweden). Unfortunately, it was not possible to obtain any additional information on the chemical structure of the polar-embedded C18 Synergi column.

The mobile phase consisted of selected concentrations of methanol (Table 3) and 3% acetic acid to ensure complete protonization of the phenolic targets. The analyses were performed under isocratic conditions at a flow rate of 1 mL min -1 , and the extract (20 µL) was injected into the system (Waters 717 plus autosampler). Chromeleon software (version 6, Dionex) was used to evaluate the chromatograms.

Extraction procedures

Each test was performed in duplicate, and the areas of the chromatograms were compared with the area recorded by direct analysis of the target. Calibration curves including five different concentrations (range 0-100 mg l -1 ) of each target showed R 2 = 0.99-1.00. Prior to use, all glassware was washed and heated as previously described [START_REF] Jonsson | [END_REF] in order to avoid contamination. Glass bottles (120 mL) were used as sample vessels in the silica sorption tests. The target (20 µg) was spread on the glass surface by drop-wise application of acetone (1.00 cm 3 ). Thereafter, the acetone was allowed to evaporate at ambient temperature for 20 h, and the target was subsequently extracted using two portions of solvent (10 cm 3 each). The bottles were capped with Teflon-coated rubber stoppers. To increase the extraction capacity, the bottles were heated at 60 ºC for 5 min in a water bath and then vigorously shaken by hand for 10 min. The extraction was continued for 20 min on a rotary table at 180 rpm, after which the extract was transferred to a conical glass tube and evaporated to dryness under a gentle stream of nitrogen gas. The residues were diluted in methanol (200 µL) before analysis. A test showed that there had been no losses during the evaporation process.

Liquid extraction of the solid silica phase

In some experiments, we also silanized the inside of the glass sample bottle with dimethyldichlorosilane (Sigma-Aldrich, Sweden; 3.4% in toluene) in an attempt to reduce adsorption of the targets to the silica surface. After the silanization, the bottles were dried for approximately 6 h at ambient temperature before performing the drop-wise addition of target solutions. The extraction was performed as described above.

Different extraction protocols (including varied volumes and portions of solvent) and procedures (variation in time, temperature, and performance) were tested to ensure a robust and reliable method for accurate comparisons of the solvents that were investigated.

Solid phase extraction of the water phase

The SPE cartridges were conditioned and dried according to the protocol provided by the manufacturers: conditioning with methanol (3 cm 3 ); equilibration with the same matrix as the sample (3 cm 3 ); application of the sample; vacuum drying (30 min); elution with the chosen solvent.

K ow calculations

A log K ow value was available for TBBPA (4.5; WHO, 1995) but not for the other investigated derivatives. To allow comparison of the results, log K ow values for all derivatives were calculated with ALOGPS 2.1 software, 21 which uses several calculation programs and reports average values and standard deviations (Table 1).

Results and discussion

Desorption from a silica surface

A more comprehensive test protocol including a wide range of solvents with different polarities and dipole moments (Table 2) was employed for the most hydrophobic derivative (TBBPA-DBPE), since we had initially experienced difficulties with desorption of that target. The use of two hydrophobic solvents, hexane (representing n-alkane solvents) and toluene (representing aromatic solvents), which presumably appealed to the hydrophobic and aromatic properties of the TBBPA-backbone structure, showed that toluene was not favoured over hexane. In fact, the best recovery was achieved using hexane (44% compared to 21% for toluene), which has both a lower polarity index and a smaller dipole moment than toluene. Acetonitrile was the most powerful water-soluble extraction medium for the DBPE derivative (yield 73%), probably because that solvent has a high polarity index in combination with a large dipole moment (Table 2). Methanol was a poor solvent (extraction yield 39%), perhaps because it lacks hydrophobic properties due to the presence of only one carbon atom in the molecular structure. Acetone, the chosen solvent for the stock solutions, was completely unable to desorb the DBPE derivative (below detection limit). Consequently, the ability of the solvents to extract the TBBPA ether derivatives from the silica surface of the experimental vessel could not have depended solely on the solubility properties of the derivatives, but must have also been determined by the degree to which sorption sites were exposed on the solid surface. In other words, a polar surface attracts hydrophobic compounds better than a liquid with the same polarity. 24 When a solution of the target is spread on a glass surface and the solvent is then allowed to evaporate, as in our study, the silica sites participating in the adsorption mechanism will be maximized. The best overall recoveries in our investigation were obtained using 1-propanol (Table 3).

Adsorption of hydrophobic compounds on silica (glass) surfaces is a well-known problem for analytical chemists trying to evaluate their protocols. Also, since considerable amounts of the target compounds can escape analysis due to adsorption on the glassware that is used, it is important to conduct experiments in a manner that will avoid that phenomenon. For example, the glass surface can be wetted with a hydrophobic solvent such as hexane before the targets (e.g. phthalates) are introduced into the sample vessel. 14 Another well accepted procedure is to treat silica surfaces with silylating agents in order to reduce polar interactions (e.g. with organosulphur compounds 25 ). In our study, silanization of the glass vessels increased the extraction yield of the DBPE derivative from 53% to 86% when 1-propanol was used as solvent. In general, solvent extractions of the silica surface resulted in relative standard deviations (RSDs) of up to 12% (n = 3). and the highest value was valid for TBBPA-DMe. In fact, we found that the smaller the yield of the target, the larger the RSD value.

Desorption from PS-DVB based polymers

All targets were completely adsorbed on both ISOLUTE ENV+ and OASIS HLB in tests using SPE columns, as confirmed by analysis of the matrix solution that passed through the cartridges. In runs using the ENV+ cartridge, only methanol was to some degree capable of extracting all targets, with the exception of the two most hydrophobic compounds TBBPA-DAE and TBBPA-DBPE (data not shown). It is very likely that the aromatic structure of the SPE cartridge in combination with the hydroxyl moiety of the hydroxylated polymer appealed to the inherent characteristics of the most hydrophobic derivatives to such an extent that neither the hydrophobic nor the polar interactions were diminished by any of the solvents. In an earlier study performed in our laboratory, 25 it was found that a highly hydrophobic compound (diethylhexyl phthalate) was adsorbed to PS-DVB polymers via hydrophobic as well as polar interactions. The results of that investigation showed that polar attractions were indeed involved and that it was necessary to use a polar eluent to achieve better recovery. Obviously, none of the characteristics of the solvents used in the present study possessed the desorption properties needed to extract the two most hydrophobic compounds (highest calculated log K ow ) from ISOLUTE ENV+.

OASIS HLB exhibited the same behaviour when methanol was applied as eluent. Nonetheless, in tests using that polymer, ethyl acetate proved to be the best eluting solvent for all targets, including the troublesome TBBPA derivatives DAE and DBPE (Table 3). Regardless of the solvent employed, the lowest recovery was seen for TBPPA-DMe (18% and 45% for methanol and ethyl acetate, respectively).

To avoid difficulties associated with chromatographic separation of compounds co-eluted from an SPE column, it is advisable to carry out a washing step prior to the elution. In our study, no targets were observed in the eluted extracts when we used up to 70% methanol as washing solution. However, it should be kept in mind that complex sample matrices may lower the adsorption capacity of the SPE cartridge, and therefore each matrix should be tested to ensure a safe washing solution. 26,27 In general, elution of the OASIS HLB catridge resulted in RSD values of up to 7.5% (n = 3), and as for the sorption study with silica, the highest value was valid for TBBPA-DMe, i.e. as for the desorption from the silica surface, the smaller the yield, the larger the RSD.

The solid phase extraction sorbents were chosen to appeal to the targets with both hydrophobic and hydrophilic properties. However, the strong hydrophobic properties of the TBBPA derivatives with highest calculated log K ow values (DAE and DBPE) should probably benefit from a sorbent with less pronounced hydrophobic attraction, e.g. silica based, in order to reduce the 1) dual sorption power of the sorbent and 2) subsequent dual elution capacity of the eluting solvent.

Chromatographic retardation

The TBBPA derivatives with the highest log K ow values were too strongly retarded on the HPLC columns when lower concentrations of methanol were used as mobile phase, and hence retention times longer than 60 min were excluded from the test protocol. Furthermore, to eliminate the influence of various column dimensions on the retention times, we used the dimensionless capacity factor (k) presented in Table 4, which were calculated according to the following equation:

tM tM tR k - = (eq. 1)
where tR and tM refer to retention times for the target and the mobile phase, respectively. It is usually recommended to have peaks with capacity factors that are greater than 1 but less than 10-15,and the latter limit is considered appropriate due to problems associated with resolution and detection. [START_REF] Yost | Practical liquid chromatography -an introduction[END_REF] We found that when using weaker mobile phases (i.e. lower methanol concentrations), retardation was less pronounced (lower capacity factors) on the dual-phase column than on the C8 and C18 columns (Table 4). The weakest overall retardation of the targets occurred on the C8 column, and even though the dual-phase column comprised polar units, the influence of its non-polar C18 units predominated, as indicated by the higher capacity factor values for the dual-phase than for the C8 column. These observations suggest that a C8 column is just as suitable as the studied Synergi dual-phase column for analysis of targets with a broad range of hydrophobicities. Increasing the fraction of the polar units among the C18 chains would probably enhance the characteristic identity of the Synergi column as compared to the traditional alkyl columns, and thereby strengthen the manufacturer's assertion that dual columns are more suitable for compounds with widely varying hydrophobicities. A literature survey shows that, the relatively new Synergi Fusion-RP column employed in the present study has been in use since 2006 but has been utilized in only a few scientific studies. Two of those investigations 29,30 indicated that the Synergi column offered chromatographic benefits, and one 31 suggested that it has drawbacks compared to other columns, and the remaining four studies merely mentioned that the Synergi column had been used, without any further comments.

Physico-chemical characteristics of the TBBPAs

To fully understand the extent of partitioning processes, it is necessary to consider the importance of uneven distribution of electrons on the target molecules. The compounds of interest here consist of ether or hydroxyl functional groups together with bromine atoms attached to the carbon skeleton, thus they are inherently monopolar (ether and bromine) and bipolar (hydroxyl). Additionally, the aromatic ring structure also gives the molecules polarizing effects. Such physico-chemical properties have a substantial impact on adsorption to surfaces and the ability to desorb into and dissolve in the surrounding medium. 24 The highest calculated log K ow values (Table 1) were noted for TBBPA-DBPE and TPPBA-DAE, which have the same number of carbon and hydrogen atoms but differ in that the former has two bromine atoms on each ether group, and the latter contains a double bond (allyl). The allyl groups in the TBBPA-DAE molecule add exposed delocalized electrons that contribute more to the polar characteristics than is the case for the electronegative bromines in TBBPA-DBPE. This is indicated by the lower log K ow (Table 1) and capacity factor values of the DAE derivative compared to TBBPA-DBPE (Table 4), although that conclusion was neither confirmed nor refuted by the findings of the SPE experiments (Table 3).

Comparing the two hydroxylated targets, i.e. TBBPA and TBPPA-DHEE, the three dimensional positions of the hydroxyl groups suggest that on TBBPA the hydroxyls are hidden to a greater extent than the bromine atoms are and hence cannot exert a polar effect that is as powerful as that of the more exposed hydroxyl groups on the DHEE derivative (cf. molecular structures in Table 1). In other words, the hydroxyl groups on the DHEE derivative should have a greater influence on the polarity of the molecule than the hydroxyl groups on TBBPA do. This is indeed indicated by the shorter retention times of TBBPA-DHEE, noted as lower capacity factors in HPLC analysis (Table 4) and a lower calculated log K ow value (Table 1).

The polarity of hydroxyl groups is effectively concealed by methylation. 24 In our study, this is shown by the observation that TBBPA had a lower log K ow (Table 1) and lower capacity factor values compared to its methoxylated derivative TBBPA-DMe (Table 4). The fact that the lowest recoveries of TBBPA-DMe were seen upon extraction from silica and desorption from the SPE cartridge indicates that polar and non-polar sorption strengths are equal for this compound, and thus both must be complemented in the extraction/eluting solvent. Consequently it will be necessary to find a more powerful eluting medium for this compound before a satisfactory analytical protocol can be achieved.

In our experiments, compared to TBBPA-DMe, extraction yields were greater for targets with lower calculated log K ow as well as those with higher such values, which suggests that those compounds will be easier to extract from environmental matrices than TBBPA-DMe.

It is highly likely that the targets exposing free hydroxyl groups, i.e. TBBPA and TBBPA-DHEE, are influenced by the pH of the surrounding medium. A study has indicated pK a values of 7.5 and 8.5 for TBBPA 1 , which shows that this compound is at least partly deprotonated under normal environmental conditions. The molecular structure of the DHEE derivative suggests that its pK a should be at least one or probably several units higher than the corresponding value for TBBPA, since the alkoxides are not stabilized as extensively by the aromatic backbone and its bromine atoms as is the case for the phenolate on the TBBPA molecule. By comparison, 2-phenoxyethanol has a pK a of 15.1. 32 Consequently, we suggest that DHEE is a very strong base and therefore exist as a neutral compound in nature.

Conclusions

The observed results from the partitioning tests show that the inherent physico-chemical properties of the studied ether derivatives of TBBPA differ to such extent, and therefore cover a wide range of sorption properties, that it is impossible to cover the targets with sufficient yields in one single analytical protocol. In common for all compounds were i) the inherent properties cover both polar and nonpolar characteristics, which are not consistent with their calculated log K ow values and ii) the polarization property of the silica hydroxyl groups on glassware surfaces, which exert strong sorption qualities on the targets, suggests that glassware used in the analyses should be treated prior usage in order to reduce this sorption effect.

For the individual compounds, it was definitely shown that the methoxy derivative of TBBPA, i.e. TBBPA-DMe, showed strongest sorption properties in all studied extraction perspectives, since this compound presented the lowest recoveries irrespectively of studied adsorbent and eluting solvent. These findings would definitely obstruct environmental trace analyses of this compound.

Regarding the two hydroxylated targets TBBPA and its DHEE derivative, pH should be taken into consideration for TBBPA during analysis. The extraction pH of TBBPA should be 5.5 or less, which is two pH units below the lowest pK a for that compound. In contrary, the DHEE derivative most probably exists only as a neutral compound in aqueous environments and is therefore not affected by pH changes during analytical extraction procedures.

Fig. 1 .

 1 Fig. 1. Mass spectrum of the synthesized TBBPA-DMe. The isotope ratios (derived from the four bromine atoms on the TBBPA skeleton) of the mass fragments in the two clusters were 18:72:100:65:17 (litt. 17:68:100:65:16). Other investigators 7 have used m/z 555, 557, 559, and 572 as diagnostic fragments for this compound.

Table 1 .

 1 Chemical structures, molecular weights, and octanol-water partitioning coefficients (log K ow ) of the target compounds listed in order of increasing log K ow values ) Log K ow was calculated using ALOGPS 2.1 software 21 .

	Ether derivative			Chemical structure	Mw	Log K ow	a)
				Br		Br
		O H			CH 3		OH
	TBBPA-DHEE			O			O	632	6.02 (± 0.68)
				Br	CH 3	Br
				Br		Br
					CH 3
	TBBPA	H	O			OH	544	6.70 (± 0.62)
				Br	CH 3	Br
				Br		Br
					CH 3
	TBBPA-DMe	C H 3	O			O CH 3	572	7.36 (± 0.79)
				Br	CH 3	Br
	TBBPA-DAE TBBPA-DBPE	CH 3 CH 3 CH 3 r Br O F Br Br o O Br	Br Br Br	O O	Br	624 944	8.63 (± 1.03) 9.99 (± 1.56)
		Br		Br	P CH 3	Br	Br
					e
					e r
							R
							e
								v i e
								w
								2

). Other investigators

[START_REF] Watanabe | [END_REF]

have used m/z 555, 557, 559, and 572 as diagnostic fragments for this compound. a

Table 2 .

 2 Polarity index and dipole moment of each of the solvents used in this study (Tables Polarity Index and Dipole Moment 22 ), listed in order of increasing polarity index, and extraction yields of TBBPA-DBPE from silica For comparison, water has a polarity index of 10 and dipole moment of 1.9

		Polarity	Dipole	Yield of DBPE
		Index*	Moment*	from silica (%)
	Pentane	0.0	0.0
	Hexane	0.1	0.1	44
	Toluene	2.4	0.3	21
	Dichloromethane 3.1	1.1
	THF	4.0	1.8
	1-Propanol	4.0	3.1	64
	Ethyl acetate	4.4	1.9
	Acetone	5.1	2.7	b.d
	Methanol	5.1	2.9	39
	Acetonitrile	5.8	3.4	73

*

Table 3 .

 3 Yields of TBBPA and ether derivatives of that compound obtained using different solvents in extractions from a silica surface (glass) or a solid-phase cartridge consisting of vinylpyrrolidone units on a PS-DVB polymer backbone; OASIS HLB.

	Sorbent	Silica surface	OASIS HLB
	Desorbing solvent	1-propanol	Methanol	Ethyl acetate
	TBBPA-DHEE	67	96	99
	TBBPA	65	52	61
	TBBPA-DMe	23	18	41
	TBBPA-DAE	69	b.d.	47
	TBBPA-DBPE	72	b.d.	72
	b.d. = below detection limit

Table 4 .

 4 Capacity factors of the TBBPA derivatives listed according to their calculated log K ow values (Table1) at different methanol concentrations (70-98%) when eluted from three different HPLC columns. Capacity factors corresponding to retention times > 60 min.

	Column	C18				C8			Dual-phase
	Methanol conc. 98% 90% 80% 70%	98% 90% 80% 70% 98% 90% 80% 70%
	TBBPA-DHEE	0.3	0.9	2.5	8.3	0.4 0.8 2.1 7.1	0.1	1.1	1.9	5.6
	TBBPA	0.5	1.2	3.7	13	0.5 0.9 2.8 9.7	0.2	1.4	2.7	8.5
	TBBPA-DMe	2.2	5.8	26	*	0.7 2.5 9.4 41	1.6	1.7	12	56
	TBBPA-DAE	2.0	12	*	*	1.0 3.2 23	*	2.0	4.5	24	*
	TBBPA-DBPE	3.6	26	*	*	1.3 5.8	*	*	3.9	11	*

*

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Acknowledgements

We are grateful to Daniel Nästorp, MSc, and Fredrik Björnehäll, MSc, for assistance in the laboratory. This study was funded by the Department of Water and Environmental Studies, Linköping University, Linköping, Sweden.