Nonlinear oscillation of a dielectric elastomer balloon

Transcription

1 Research Artice Received: 28 May 2009 Revised: 31 Juy 2009 Accepted: 15 October 2009 Pubished onine in Wiey Interscience: 1 February 2010 ( DOI /pi.2767 Noninear osciation of a dieectric eastomer baoon Jian Zhu, Shengqiang Cai and Zhigang Suo Abstract Much of the existing iterature on dieectric eastomers has focused on quasi-static deformation. However, in some potentia appications, the eastomer deforms at high frequencies and undergoes noninear osciation. Whie noninear osciation has been studied in many areas of science and engineering, we are unaware of any theoretica anaysis of noninear osciation of dieectric eastomers. This paper reports a theoretica study of the dynamic behavior of a dieectric eastomer baoon subject to a combination of pressure and votage. When the pressure and votage are static, the baoon may reach a state of equiibrium. We determine the stabiity of the state of equiibrium, and cacuate the natura frequency of the sma-ampitude osciation around the state of equiibrium. We focus on the parametric responses of the dieectric eastomer baoon. When the votage is sinusoida, the baoon resonates at mutipe frequencies of excitation, giving rise to superharmonic, harmonic and subharmonic responses. When the frequency of excitation varies continuousy, the osciating ampitude of the baoon may jump, exhibiting hysteresis. c 2010 Society of Chemica Industry Keywords: dieectric eastomer actuator; arge deformation; dynamics; noninear osciation 378 INTRODUCTION Subject to a votage through its thickness, a thin membrane of a dieectric eastomer reduces in thickness and expands in area. The strain of the membrane induced by the votage can be arge, readiy exceeding 100%. This and reated phenomena are being deveoped for appications in eectromechanica transducers. 1 7 Much of the existing iterature on dieectric eastomers has focused on quasi-static deformation, where the effect of inertia is negigibe. In some potentia appications, however, eastomers deform at high frequencies, up to 50 khz, and function as vibration sources, 8 high-speed pumps 9 12 and acoustic equipment. 13,14 In a arge-strain, high-frequency appication, the eastomer may undergo noninear osciation. Whie noninear osciation has been studied in many areas of science and engineering, 15,16 we are unaware of any theoretica anaysis of noninear osciation of dieectric eastomers. To expore the subject, this paper reports a study of an ideaized system: a dieectric eastomer baoon of a spherica shape, subject to a combination of pressure and votage. Dieectric eastomer baoons have been studied as pumps 12 and oudspeakers, 13 as we as eements of she-ike actuators. 17,18 The pan of this paper is as foows. The foowing section derives the equation of motion using the method of virtua work. The section after that describes the state of equiibrium when the pressure and votage are static. We then discuss the smaampitude osciation around the state of equiibrium, and the stabiity of the state of equiibrium against sma perturbation. The section after that studies parametric excitation, where the pressure is static but the votage is sinusoida. Finay, we show that, when the frequency of excitation is varied continuousy, the osciating ampitude of the baoon can jump, exhibiting hysteresis. EQUATION OF MOTION Figure 1 shows a spherica baoon of radius R and thickness H in the undeformed state. The membrane of the baoon is a dieectric eastomer, taken to be incompressibe, of density ρ.the membrane is coated on both faces with compiant eectrodes. We wi negect any stress in the eectrodes, but may add the mass of the eectrodes to that of the membrane. When the pressure inside the baoon exceeds the pressure outside by p and the two eectrodes are subject to a votage,the baoon deforms to radius r and the two eectrodes gain charges +Q and Q.Letλ be the stretch of the membrane, namey λ = r R Let D be the eectric dispacement in the membrane, namey D = (1) Q 4πr 2 (2) The baoon is taken to deform under an isotherma condition, and the fixed temperature wi not be considered expicity. Consequenty, the baoon is a thermodynamic system of two independent variabes, λ and D. We next formuate a mode to evove this system in time t. The thermodynamics of the dieectric eastomer is characterized by the density of the Hemhotz free energy as a function of the Correspondence to: Zhigang Suo, Schoo of Engineering and Appied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mai: suo@seas.harvard.edu Schoo of Engineering and Appied Sciences, Harvard University, Cambridge, MA 02138, USA Poym Int 2010; 59: c 2010 Society of Chemica Industry

2 pr mh Noninear osciation of a dieectric eastomer baoon 1.6 R p r F 1.2 ef 2 = 0 mh H h 0.2 Reference State Current State Figure 1. Dieectric eastomer baoon deforms under a pressure and avotage. two independent variabes, W(λ, D). We wi adopt the mode of an idea dieectric eastomer. 19 The mode assumes that the eastomer is a network of ong and fexibe poymers with a ow density of crossinks, so that the crossinks amost do not constrain the process of poarization. Once the effect of crossinks on poarization is negected, the dieectric behavior of the eastomer is iquidike, unaffected by deformation. Consequenty, the free-energy function of the dieectric eastomer is written as a sum of two parts: W(λ, D) = µ 2 (2λ2 + λ 4 3) + D2 (3) 2ε The first part is the eastic energy, where µ is the shear moduus. For simpicity, we use the neo-hookean mode to describe the easticity of the network; other modes of easticity may be used. The second part in Eqn (3) is the dieectric energy, where ε is the permittivity. For an idea dieectric eastomer, the permittivity is taken to be a constant independent of deformation. The mode of idea dieectric eastomers has been used amost excusivey in the iterature to anayze devices. However, recent experiments on VHB (an acryic-based eastomer made by 3M), the most studied dieectric eastomer, have shown that the permittivity may vary by a factor of two when the eastomer undergoes arge deformation. 18 See Zhao and Suo 20 for a review of thermodynamic modes for dieectric eastomers. Whie the genera considerations in this paper are vaid for arbitrary function W(λ, D), we wi use the function (3) in numerica cacuations. When the charge on the eectrodes varies by δq, the appied votage does work δq. When the radius of the baoon varies by δr, the pressure does work 4πr 2 pδr, and the inertia force does work 4πR 2 H ρ(d 2 r/dt 2 )δr. We negect any viscous force. Thermodynamics dictates that, for arbitrary variation of the system, the variation of the free energy of the membrane shoud equa the work done by the votage, the pressure and the inertia, namey 4πR 2 HδW = δq + 4πr 2 pδr 4πR 2 Hρ d2 r δr (4) 2 dt Inserting Eqns (1) and (2) into Eqn (4), and recaing that the baoon is a system of two independent variabes, λ and D, we obtain that W(λ, D) = 2 Dλ + pr λ H H λ2 R 2 ρ d2 λ dt 2 (5) W(λ, D) = D H λ2 (6) Equation (5) baances momentum and Eqn (6) enforces eectrostatic equiibrium. For an idea dieectric eastomer, Eqn (6) recovers Figure 2. Subject to a static pressure and votage, the baoon may reach a state of equiibrium. The pressure is potted as a function of the equiibrium stretch at severa vaues of the votage. the iquid-ike dieectric behavior, D = εe, where E is the eectric fied. Inserting Eqn (3) into Eqns (5) and (6), and eiminating D, we obtain that d 2 λ + g(λ, p, ) = 0 (7) 2 dt with g(λ, p, ) = 2λ 2λ 5 pr µh λ2 2 ε 2 µh 2 λ3 (8) where T is the dimensioness time (T = t/r ρ/µ). Equation (7) is the equation of motion that evoves the stretch of the baoon as a function of time, λ(t). This equation is consistent with that derived in Mockensturm and Goubourne. 12 In writing Eqns (7) and (8), we use the dimensioness pressure (pr/µh) and votage ( ε/µ /H). BALLOON IN A STATE OF EQUILIBRIUM UNDER STATIC PRESSURE AND VOLTAGE When the two oading parameters, p and, are both static, the baoon may reach a state of equiibrium, of stretch λ eq.inthestate of equiibrium, the equation of motion, Eqn (7), reduces to g(λ eq, p, ) = 0 (9) This noninear agebraic equation determines λ eq for given vaues of p and. Figure 2 pots Eqn (9) as pressure stretch curves at severa vaues of votage. When = 0, the probem reduces to that of a pressurized baoon, a we-known probem in the iterature of noninear easticity. 21 As the baoon expands, the pressure first increases, reaches a peak and then decreases. The peak pressure corresponds to a critica state. When the appied pressure is above the peak, the baoon cannot reach a state of equiibrium. When the appied pressure is beow the peak, corresponding to each vaue of the pressure are two vaues of the stretch. The vaue of the stretch on the rising part of the curve corresponds to a stabe state of equiibrium, and the vaue of the stretch on the descending part of the curve corresponds to a state of unstabe equiibrium. When 0, charges of opposite signs are induced on the two eectrodes. The attraction between the eectrodes causes the membrane of the baoon to reduce in thickness and increase in 379 Poym Int 2010; 59: c 2010 Society of Chemica Industry

3 pr mh J Zhu, S Cai, Z Suo 0.3 pr mh = ef 2 mh ω o R r / m = Figure 3. Votage is potted as a function of the equiibrium stretch at severa vaues of the pressure. area. As shown in Fig. 2, the votage owers the critica pressure, and increases the stretch of any stabe state of equiibrium. Figure 3 pots the condition of equiibrium (9) as votage stretch curves at severa vaues of pressure. When p = 0, as the baoon expands, the votage first increases, reaches a peak 12,22 26 and then decreases. This behavior is understood as foows. The votage induces a positive charge on one eectrode, and negative charge on the other eectrode. The oppositey charged eectrodes attract each other, so that the dieectric membrane reduces its thickness. For a prescribed votage, the reduction in the thickness of the membrane increases the eectric fied. This positive feedback eads to eectromechanica instabiity, or pu-in instabiity. The peak of the curve corresponds to the critica state. The critica votage is reduced in the presence of the pressure. SMALL-AMPLITUDE OSCILLATION AROUND A STATE OF EQUILIBRIUM Consider a state of equiibrium, λ eq. When the baoon is perturbed from this state of equiibrium, we write λ(t) = λ eq + (T) (10) where (T) is the ampitude of perturbation, and is taken to be sma. We then expand the function g(λ, p, ) as a power series in around the equiibrium stretch λ eq. Consequenty, to the eading order in, the equation of motion (Eqn (7)) becomes d 2 dt 2 where the partia derivative g(λ, p, ) λ g + λ (λ eq, p, ) = 0 (11) = λ 6 2 pr µh λ 6 ε 2 µh 2 λ2 (12) is evauated at λ = λ eq, and is a constant independent of time. We may ca this partia derivative the stiffness of the baoon. When the stiffness is negative, the perturbation (T) wi grow exponentiay in time, and the state of equiibrium is unstabe. When the stiffness is positive, the baoon wi osciate around the state of equiibrium, and the state of equiibrium is stabe. Inspecting Eqn (11), we see that the natura frequency of the ef 2 mh 2 Figure 4. Potted on the pane of (p, ) are curves of constant natura frequency. The curve abeed by ω 0 = 0 corresponds to the critica conditions. Above this curve, the baoon cannot reach a state of equiibrium. Beow the curve, the baoon can reach a stabe state of equiibrium. In this region, each point (p, ) corresponds to a state of stabe equiibrium, and the baoon can osciate around the state at the natura frequency ω 0. sma-ampitude osciation around the stabe state of equiibrium is determined by ω 2 0 = g λ (λ eq, p, ) (13) where ω 0 = ω 0 R ρ/µis the dimensioness natura frequency and ω 0 is the natura frequency. When p and are prescribed at static vaues, the equiibrium stretch λ eq is determined by Eqn (9), and the dimensioness natura frequency ω 0 is determined by Eqn (13). To avoid soving the noninear agebraic Eqn (9), we rearrange Eqns (9) and (13) to express p and in terms of λ eq and ω 0,namey ε 2 µh 2 = λ 2 eq + 7λ 8 eq 1 2 λ 2 eq ω2 0 (14) pr µh = 4λ 1 eq 16λ 7 eq + λ 1 eq ω2 0 (15) Figure 4 pots the curves of constant natura frequency on the pane of (p, ), using the foowing procedure. In Eqns (14) and (15), we fix ω 0. For an arbitrary vaue of λ eq, we obtain a point (p, ). By varying λ eq, we can pot the curve for a constant vaue of ω 0. Each point on the curve abeed by ω 0 = 0 corresponds to a critica state of the baoon. The coordinates of the point gives the critica vaues of p and. When the pressure and votage fa above this curve, the baoon cannot reach a state of equiibrium. When the pressure and votage fa beow the curve, the baoon can reach a stabe state of equiibrium, and the natura frequency of the sma-ampitude osciation around the state of equiibrium can be read from the diagram. As indicated in Fig. 4, the frequency is normaized by a group of parameters, R 1 µ/ρ. These parameters are fixed for a given baoon. However, a change in the static pressure or the static votage wi tune the natura frequency. PARAMETRIC EXCITATION When the pressure or the votage varies with time, the dynamic behavior of the baoon can be very compex. To iustrate the c 2010 Society of Chemica Industry Poym Int 2010; 59:

4 Noninear osciation of a dieectric eastomer baoon compexity, we prescribe a static pressure p and a sinusoida votage: (a): F ac / F dc = 0.1 (t) = dc + ac sin( t) (16) where dc is the DC votage, ac is the ampitude of the AC votage, and is the frequency of excitation. The equation of motion, Eqn (7), becomes Ampitude d 2 λ dt 2 + 2λ 2λ 5 pr ( µh λ2 2 ε 2 dc µh ) 2 ac sin( t) λ 3 = 0 dc (17) where = R ρ/µ is the dimensioness excitation frequency. The osciatory votage is a source of energy, and appears in a coefficient of the ordinary differentia Eqn (17). Phenomena of this type are known as parametric excitation. 15,16 The time-dependent stretch λ(t) can be obtained by soving Eqn (17) for any initia conditions. We set pr/µh = 0.1 and ε 2 dc /µh2 = 0.1. Were these vaues of pressure and votage static, the baoon coud attain a state of equiibrium λ eq = 1.029, or osciate around the state of equiibrium at the natura frequency ω 0 R ρ/µ = We use this state of equiibrium as the initia conditions in our numerica simuations. We then appy the osciatory votage with specific vaues of ac and. Once the numerica soution of λ(t) attains a steady state of osciation, we define the ampitude of osciation as haf of the difference between the maxima and minima vaues of the stretch. Figure 5 pots the ampitude of osciation as a function of the frequency of excitation. The baoon resonates most strongy when the frequency of excitation is around the natura frequency, ω 0. The baoon aso resonates when the frequency of excitation is severa times the natura frequency, for exampe, 2ω 0, a response known as subharmonic resonance. 16 In addition, the baoon resonates when the frequency of excitation is a fraction of the natura frequency, for exampe, ω 0 /2, a response known as superharmonic resonance. Resonance at mutipe vaues of the frequency of excitation is common for parametric excitation. 27,28 As expected, the peak ampitudes at subharmonic, harmonic and superharmonic resonances increase with the AC votage. The peak ampitudes of the fundamenta frequency, as shown in Figs 5(b) and (c), are even unbounded when ac / dc increases to 0.3 and. The peak ampitudes aso increase with the DC votage and the pressure. Figure 6 shows the numerica resuts of λ(t) for three vaues of the frequency of excitation, ω 0 /2, ω 0,2ω 0. Independent of the frequency of excitation, the baoon aways osciates near the natura frequency ω 0, as defined by the sma-ampitude osciation around a state of equiibrium. Figure 6 shows a phase shift between the votage input and the stretch output. Such a phase shift is often found in damped and undamped noninear osciators; for exampe, the undamped Duffing osciator is discussed in Jordan and Smith. 15 In genera, not ony the viscous effect, but aso the noninearity can ead to a phase shift between the excitation and the osciation. Experimenta data reported in the iterature show that the frequency of the osciation is doube, or is cose to, that of the AC votage. 10,11 For exampe, in Fig. 70 of Fox, 11 ω 0 /2, whie in Fig. 72 in Fox, 11 ω 0. Our numerica resuts not ony show superharmonic and harmonic resonance, but aso show subharmonic resonance with 2ω 0. This discrepancy between theory and experiment needs to be resoved in future studies. Ampitude Ampitude Frequency of Excitation, (b): F ac / F dc = 0.3 (c): F ac / F dc = W R r / m Figure 5. Excited by a sinusoida votage, the baoon resonates at severa vaues of the frequency of excitation. The osciating ampitude of the baoon is potted as a function of the frequency of excitation for pr/µh = 0.1andε 2 dc /µh2 = 0.1, at seected vaues of ac / dc. JUMP IN OSCILLATING AMPLITUDE WHEN THE FREQUENCY OF EXCITATION VARIES To obtain other essentia characteristics of the noninear osciation, we study Eqn (17) using the method of harmonic baance. 15,16 The time-dependent stretch is approximated as λ(t) = λ eq + a(t)cosωt + b(t)sinωt (18) where λ eq is the stretch in the state of equiibrium, a and b are time-dependent ampitudes and ω is the dimensionessfrequency of osciation. We assume that the ampitudes vary sowy with time. We use the truncated Fourier series, and negect terms of high frequencies, 2ω, 3ω, etc. We are interested in the harmonic osciation in a steady state, where a and b are constants, and the baoon osciates at a frequency equa to the frequency of excitation, ω =. With the method of harmonic baance, we substitute Eqn (18) into Eqn (17), set the coefficients of the constant cos T and sin T to be zero, and negect terms of higher frequencies. Then we obtain three noninear equations for λ eq, a and b. Wesovethese noninear equations for a and b using the Newton Raphson method. Figure 7 pots the osciating ampitude of the baoon, 381 Poym Int 2010; 59: c 2010 Society of Chemica Industry

5 J Zhu, S Cai, Z Suo 0.6 (a) W R r / m = 1.55 C F ac / F dc Ampitude 0.3 B 0.2 E F ac / F dc (b) W R r / m = D F O A O Frequency of Excitation, W R r / m Figure 7. Steady-state soutions when pr/µh = 0.1, ε 2 dc /µh2 = 0.1and ac / dc = 0.1. (c) W R r / m = 6.19 F ac / F dc Figure 6. Superharmonic, harmonic and subharmonic responses (pr/µh = 0.1, ε 2 dc /µh2 = 0.1and ac / dc = 0.1). 382 a 2 + b 2, as a function of the frequency of excitation. Upon increasing the frequency of excitation, the steady-state soution wi start from point O, to A, to D, then jump to point E, then to F, and to O. However, upon decreasing the frequency of excitation, the steady state wi start from point O,toF,toEand C. Simiar phenomena of hysteresis have been reported in many parametricay excited osciators. 16,29 The above interpretations are better understood when we study how the ampitudes vary with time, a(t)andb(t). Since a and b are taken to vary with time sowy, d 2 a/dt 2 and d 2 b/dt 2 are negected. Taking second derivative of Eqn (18) with respect to T, we obtain that d 2 ( λ dt 2 = ω 2 a + 2ω db ) ( cos ωt + 2ω da ) dt dt ω2 b sin ωt (19) Substituting Eqn (19) into Eqn (17), setting = ω and equating the coefficients of cos( T) and sin( T) to zero, we obtain that da db = F(a, b); = G(a, b) (20) dt dt The two functions F(a, b)andg(a, b) are engthy and are not isted here. Given initia vaues a(0) and b(0), we can evove Eqn (20) to obtain a(t) andb(t). Figure 8. Phase pane of a and b when R ρ/µ = 2.4, pr/µh = 0.1, ε 2 dc /µh2 = 0.1and ac / dc = 0.1. Figure 8 shows the phase pane of (a, b). Steady-state soution 1, reated to point A in Fig. 7, is a center point. Steady-state soution 2, reated to point B in Fig. 7, is a sadde point. Steady-state soution 3, reated to point C in Fig. 7, is a center point. Figure 8 shows that the parametric response depends on the initia condition. For exampe, if the initia condition is cose to the center point 1, a(0) = 0andb(0) = 0.1, the fina path wi cyce around this center point. If the initia condition is cose to the sadde point 2, however, the fina path wi not stay near the sadde point, but wi foow either a arger cyce, say path (1), or a sma cyce, say path (2). The stabiity anaysis of steady-state soution is as foows. With the infinitesima perturbation for the steady-state soution, et a(t) = a 0 + α(t); b(t) = b 0 + β(t) (21) where a 0 and b 0 are the steady-state soution and α(t) andβ(t) are infinitesima perturbation. Substituting Eqn (21) into Eqn (20), keeping the inear term for α and β, we obtain [ dα ] [ F dt dβ = a G dt a F ] [ ] b α G β b (22) c 2010 Society of Chemica Industry Poym Int 2010; 59:

6 Noninear osciation of a dieectric eastomer baoon The partia derivatives of the functions F(a, b) andg(a, b) are evauated at the steady-state soution (a 0, b 0 ). The steady-state soution is stabe when the eigenvaues of the matrix in Eqn (22) have negative rea parts. This anaysis produces resuts indicated on the phase pane. CONCLUDING REMARKS We describe a dieectric eastomer baoon with an equation of motion of one degree of freedom. When the pressure and the votage are static, the baoon may reach a state of equiibrium. We study the stabiity of the state of equiibrium against sma perturbation, and cacuate the natura frequency of the smaampitude osciation around the state of equiibrium. The natura frequency is tunabe by varying the pressure or the votage. When the pressure is static but the votage is sinusoida, the baoon resonates at mutipe vaues of the frequency of excitation, giving rise to superharmonic, harmonic and subharmonic responses. Furthermore, when the frequency of excitation is changed continuousy, the osciating ampitude of the baoon may jump at certain vaues of the frequency of excitation, exhibiting hysteresis. We hope to further anayze the noninear dynamic behavior by using modes of many degrees of freedom, incuding dissipation duetoviscosityandeakage.weasohopethatfutureexperiments wi confirm the theoretica predictions. ACKNOWLEDGEMENTS This work was supported by the Nationa Science Foundation through a project on Large Deformation and Instabiity in Soft Active Materias. JZ acknowedges the support of NSERC postdoctora feowship of Canada. REFERENCES 1 Perine R, Kornbuh R, Pei QB and Joseph J, Science 287:836 (2000). 2 Pante JS and Dubowsky S, Int J Soids Struct 43:7727 (2006). 3 Wisser M and Mazza E, Sens Actuators A 134:494 (2007). 4 Kofod G, Wirges W, Paajanen M and Bauer S, App Phys Lett 90: (2007). 5 Patrick L, Gabor K and Sivain M, Sens Actuators A 135:748 (2007). 6 Carpi F, De Rossi D, Kornbuh R, Perine R and Sommer-Larsen P, Dieectric Eastomers as Eectromechanica Transducers. Esevier, Oxford (2008). 7 Carpi F, Frediani G and De Rossi D, IEEE/ASME Trans Mechatronics (in press). 8 Bonwit N, Heim J, Rosentha M, Duncheon C and Beavers A, Proc SPIE 6168:39 (2006). 9 Fox JW and Goubourne NC, J Mech Phys Soids 56:2669 (2008). 10 Fox JWand Goubourne NC, J Mech Phys Soids 57:1417 (2009). 11 Fox JW, Eectromechanica characterization of the static and dynamic response of dieectric eastomer membranes. Master thesis, Virginia Poytechnic Institute and State University (2007). 12 Mockensturm EM and Goubourne NC, Int J Non-Linear Mech 41:388 (2006). 13 ChibaS, Waki M, KornbuhR and PerineR,Proc SPIE 6524: (2007). 14 Heydt R, Kornbuh R, Eckere J and Perine R, Proc SPIE 6168:61681M (2006). 15 Jordan DW and Smith P, Non-inear Ordinary Differentia Equations. Carendon Press, Oxford (1987). 16 Nayfeh AH and Mook DT, Non-inear Osciation. Wiey, New York (1979). 17 Lochmatter P, Deveopment of a she-ike eectroactive poymer (EAP) actuator. Dissertation, ETH, Zurich (2007). 18 Wisser M and Mazza E, Sens Actuators A 138:384 (2007). 19 Zhao X, Hong W and Suo Z, Phys Rev B 76: (2007). 20 Zhao X and Suo Z, J App Phys 104: (2008). 21 Hozapfe GA, Non-inear Soid Mechanics.JohnWiey&Sons,NewYork (2000). 22 Stark KH and Garton CG, Nature 176:1225 (1955). 23 Zhao X and Suo Z, App Phys Lett 91: (2007). 24 Norris AN, App Phys Lett 92: (2008). 25 Liu YJ, Liu LW, Zhang Z, Shi L and Leng JS, App Phys Lett 93: (2008). 26 Diaz-Caeja R, Riande E and Sanchis MJ, App Phys Lett 93: (2008). 27 Turner KL, Mier SA, Hartwe PG, MacDonad NC, Strogatz SH and Adams SG, Nature 396:149 (1998). 28 De SK and Auru NR, Phys Rev Lett 94: (2005). 29 Lifshitz R and Cross MC, Phys Rev B 67: (2003). 383 Poym Int 2010; 59: c 2010 Society of Chemica Industry