Ionics

, Volume 18, Issue 6, pp 599–606

Dielectric behaviour of cellulose acetate-based polymer electrolytes

Authors

  • N. I. Harun
    • Faculty of Applied SciencesUniversiti Teknologi MARA
  • R. M. Ali
    • Faculty of Applied SciencesUniversiti Teknologi MARA
  • A. M. M. Ali
    • Faculty of Applied SciencesUniversiti Teknologi MARA
    • Ionic Materials & Devices (iMADE) Research Laboratory, Institute of ScienceUniversiti Teknologi MARA
    • Faculty of Applied SciencesUniversiti Teknologi MARA
    • Ionic Materials & Devices (iMADE) Research Laboratory, Institute of ScienceUniversiti Teknologi MARA
Original Paper

DOI: 10.1007/s11581-011-0653-0

Cite this article as:
Harun, N.I., Ali, R.M., Ali, A.M.M. et al. Ionics (2012) 18: 599. doi:10.1007/s11581-011-0653-0

Abstract

The present work deals with the findings on the dielectric behaviour of cellulose acetate (CA) and its complexes consisting of ammonium tetrafluoroborate (NH4BF4) and polyethylene glycol with a molecular weight of 600 g/mol (PEG600) that were prepared using the solution casting method. The highest σ obtained for CA-NH4BF4 film was 2.18 × 10−7 S cm−1 and enhanced to 1.41 × 10−5 S cm−1 with the addition of 30 wt.% PEG600. The dielectric behaviours of the selected samples were analyzed using complex impedance Z*, complex admittance A*, complex permittivity ɛ*, and complex electric modulus M*-based frequency and temperature dependence in the range of 10 Hz–1 MHz and 303–363 K, respectively. The variation in dielectric permittivity (εr and εi) as a function of frequency at different temperatures exhibits a dispersive behaviour at low frequencies and decays at higher frequencies. The variation in dielectric permittivity as a function of temperature at different frequencies is typical of polar dielectrics in which the orientation of dipoles is facilitated with the rising temperature, and thereby the permittivity is increased. Modulus analysis was also performed to understand the mechanism of electrical transport process, whereas relaxation time was determined from the variation in loss tangent with temperature at different frequencies.

Keywords

Cellulose acetateDielectric permittivityElectrical modulusLoss tangent

Introduction

Solid polymer electrolytes (SPEs) have been proved to be prospective candidates for advanced electrochemical device applications because of their characteristics such as viscoelasticity and flexibility, as well as high ionic conductivity. Interest in these materials grew mainly because of the pioneering measurements of the ionic conductivity of polymer salt complexes reported by Wright et al. and the development of ionic conductivity of polymer salt complexes by Armand et al. [1]. Aside from poly(ethylene oxide) [2]—the most frequently used polyether—a few types of biopolymers such as chitosan [35] and cellulose with its derivatives [610] are also used as polymer matrixes in developing various SPEs. Both chitosan and cellulose films are homogenous with high mechanical strength [3]. However, most biopolymer films have very low electrical conductivity at ambient temperature in their actual state. The salt-solvating power and the sufficient mobility of ions necessary for ionic conduction are imparted by incorporating plasticizer [11].

Although such systems have evolved a great deal, understanding the ion transport behaviour of these polymer electrolytes is important. Dielectric analysis is an informative technique used to determine the molecular motions and structural relaxations present in polymeric materials possessing permanent dipole moments [12]. In dielectric measurements, the material is exposed to an alternating electric field generated by applying a sinusoidal voltage. This process causes alignment of dipoles in the material, resulting in polarization. The capacitance and the conductance of the material are measured over a range of temperature and frequency and are related to the dielectric constant (εr) and dielectric loss (εi), respectively. εr represents the amount of dipole alignment (both induced and permanent), and εi measures the energy required to align dipoles or move ions. A further analysis of the dielectric behaviour would be more successfully achieved using electric modulus formalism, which is used to suppress the signal intensity associated with electrode polarization. Thus, the electric modulus spectra provide an opportunity to investigate conductivity and its associated relaxation in ionic conductors and polymers.

In this work, SPEs based on cellulose acetate (CA) as host polymer complexes with ammonium tetrafluoroborate (NH4BF4) as doping salt and polyethylene glycol with a molecular weight of 600 g/mol (PEG600) as plasticizer were prepared. Their conductive performances were evaluated using an electrical impedance spectroscopy (IS) instrument.

Experimental

Materials

CA with an acetyl content of 39.8 wt.% (Aldrich), NH4BF4 (Fluka), PEG600 (Fluka), and acetone (Aldrich) were used in this study.

Sample preparation

The polymer electrolytes comprising CA as a host polymer, NH4BF4 as a doping salt, and PEG600 as a plasticizer were prepared by the solution casting technique using acetone as solvent. CA (1 g) was dissolved in 30 mL acetone for several hours to obtain the homogenous solution. Subsequently, 5–50 wt.% NH4BF4 and 5–40 wt.% PEG600 were added into the solution. After complete dissolution of the complexes, the solutions were casted in Petri dishes and left to dry at room temperature (~ 30 °C) to form thin films of (a) CA-NH4BF4 and (b) CA-NH4BF4-PEG600. The films were then placed in dry cabinet for further drying before they were used.

Material characterization

IS was measured using a HIOKI 3532-50 LCR Hi-tester interfaced into a computer with frequency ranging from 10 Hz to 1 MHz in the temperature range of 303–363 K. The measurements were performed by inserting the film between two stainless steel electrodes that act as blocking electrodes in a temperature-controlled chamber (ESPEC SH-221). The complex impedance data [Z(ω)] can be represented by its real (Zr) and imaginary (Zi) parts by the following equation:
$$ Z\left( \omega \right) = {Z_{\text{r}}} + {Z_{\text{i}}} $$
(1)
The bulk and the surface phenomena can be separated by presenting the impedance data as a complex impedance plot (Zi versus Zr). Other related functions besides complex impedance used in IS are complex admittance A(ω), complex permittivity ε(ω), and complex electrical modulus M(ω). The equations for the dielectric constant εr, the dielectric loss εi, the real electrical modulus Mr, and the imaginary electrical modulus Mi are as follows:
$$ {\varepsilon_{\text{r}}} = \frac{{{Z_{\text{i}}}}}{{\omega {C_{\text{o}}}\left( {{Z_{\text{r}}}^2 + {Z_{\text{i}}}^2} \right)}} $$
(2)
$$ {\varepsilon_{\text{i}}} = \frac{{{Z_{\text{r}}}}}{{\omega {C_{\text{o}}}\left( {{Z_{\text{r}}}^2 + {Z_{\text{i}}}^2} \right)}} $$
(3)
$$ {M_{\text{r}}} = \frac{{{\varepsilon_{\text{r}}}}}{{\left( {{\varepsilon_{\text{r}}}^2 + {\varepsilon_{\text{i}}}^2} \right)}} $$
(4)
$$ {M_{\text{i}}} = \frac{{{\varepsilon_{\text{i}}}}}{{\left( {{\varepsilon_{\text{i}}}^2 + {\varepsilon_{\text{r}}}^2} \right)}} $$
(5)
where Co is equal to εoA/t, and εo is the permittivity of the free space; A is the electrolyte–electrode contact area; t is the thickness of the sample; and ω is equal to 2πf, with f being the frequency in hertz.
The dielectric relaxation peak can be studied from the measurement of either εr (or εi) or tan δ as a frequency where
$$ \tan \delta = \frac{{{\varepsilon_{\text{i}}}}}{{{\varepsilon_{\text{r}}}}} $$
(6)

Results and discussion

Ionic conductivity at 303 K

The conductivity of the CA-NH4BF4 and CA-NH4BF4-PEG600 systems at 303 K is illustrated in Fig. 1. The variation in conductivity for both plots was divided into two regions, i.e increase and decrease in conductivity. From Fig. 1a, it can be observed that conductivity increases up to 2.18 × 10−7 S cm−1 when 25 wt.% of NH4BF4 (A6) is added which can be attributed to the increasing number of mobile ions and then decreases with increasing salt concentration due to the formation of ion aggregates [13]. The existence of ion aggregates at higher salt concentrations will clog the ionic migration, which in turn decreases the conductivity. Figure 1b shows the effect of PEG600 on the conductivity of the highest conducting CA-NH4BF4 sample. The addition of PEG600 improves the conductivity to a maximum value of 1.41 × 10−5 S cm−1 at 30 wt.% of PEG600 (B6) due to the ability of plasticizer to dissociate more salt into ions. Their low viscosity facilitates ion diffusion, and their hydroxyl group reduces the ion association in the polymer–salt complexes [1416]. The conductivity tends to decline with further addition of PEG600 due to ion association dominated in the bulk.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig1_HTML.gif
Fig. 1

Conductivity of a CA-NH4BF4 and b CA-NH4BF4-PEG600 at 303 K

Frequency dependence of dielectric behaviour

Dielectric permittivity, εr, and εi

The dielectric constant (εr) as a function of frequency at different temperatures for the highest conducting unplasticized sample—CA-25 wt.% NH4BF4 (A6)—and the highest conducting plasticized sample—CA-25 wt.%–30 wt.% PEG600 (B6)—is given in Fig. 3 and its inset, respectively. The value of εr for B6 is relatively higher than that of the unplasticized sample A6. This observation is also true for the conductivity. The addition of plasticizer is expected to increase the degree of salt dissociation, which alternatively increases the ionic mobility by reducing the potential barrier to ionic motion, resulting in decreased anion–cation coordination of the salt and also a more favourable chain flexibility motion of the polymer host.

Figure 2 shows the increase in dielectric constant with temperature in the range of 303–363 K. Thermal energy assists in the dissolution of salt into ions, thereby increasing the number of free ions leading to an increase in conductivity with temperature. According to Nithya et al. [1], the dielectric constant increases with the increase in temperature because of the total polarization that arises from the dipole orientation and the trapped charge carriers. The dielectric dispersion rises drastically toward low frequencies and decays at higher frequencies. The dielectric profile plot is higher at low frequencies possibly because of the different types of polarization effects. These effects may be caused by one or more of the contribution polarization factors, that is, electronic, atomic, ionic, interfacial, and so on. Higher dielectric constants at low frequencies depend on ionic vibration or movement, ion–ion orientation, and space charge effects. Thus, the higher value of εr at the low-frequency region is most probably due to electrode polarization and space charge effects confirming the non-Debye dependence [1, 17, 18] or familiarly known as ω(n-1) variation. This dispersion reflects the existence of space charge polarization where enough time is provided for the charges to build up at the interface before the applied field changes direction, hence giving a large value of εr [5].
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig2_HTML.gif
Fig. 2

Dielectric constant with frequency at a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K for A6 and B6 (inset)

The decay of εr at higher frequencies suggests that less ionic polarization occurred in the bulk. At high-frequency regimes, less excess ion accumulating at the electrode–electrolyte interface than that in the bulk is found, resulting in low dielectric constant. This result again reflects on the space charge and electrode polarization effect. At such frequencies, the periodic reversal of the electric field occurs so fast that no excess ion diffusion in the direction of the field is found. Hence, εr decreases with the increase in frequency [1, 19].

Dielectric loss (εi) is the direct measure of dissipated energy and generally contributes in the ionic transport and the polarization of the charge or the dipole. Figure 3 and its inset show the plot of dielectric loss (εi versus log f) for the highest unplasticized and plasticized room temperature conducting samples (A6 and B6) at various temperatures. From the plots, the εi value of plasticized CA–salt complexes is observed to be one order higher than that of the unplasticized CA–salt complex system. Dielectric dispersion rises sharply at low frequencies and then slowly decays at higher frequencies. The higher value of dielectric loss at lower frequencies is caused by the “free” charge motion within the material. These values do not correspond to the bulk dielectric of the material because of the “free” charge build up at the electrode–electrolyte interface.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig3_HTML.gif
Fig. 3

Dielectric loss with frequency at a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K for A6 and B6 (inset)

At low frequencies (see Fig. 3), εi has great value because of the space charge polarization where there is enough time for the charges to build up at the interface before the applied field changes direction contributing to the large apparent value of εi. In contrast, no time is provided at high frequencies due to increasing rate of electric field to change direction. Consequently the decrease of the space charge polarization leads to decrease in εi value [1, 3].

Modulus formalism, Mr and Mi

A further analysis of the dielectric behaviour would be more successfully achieved using dielectric moduli, which suppress the effect of electrode polarization. The variation in the real part of modulus formalism (Mr) at different temperatures for A6 and B6 is depicted in Fig. 4 and its inset. The characteristic plot of B6 is similar to that of the unplasticized sample A6 at various temperatures.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig4_HTML.gif
Fig. 4

Frequency dependence of the real part of electrical modulus at a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K for A6 and B6 (inset)

The value of Mr is very low and approaching zero in the low-frequency region. As frequency increases (see Fig. 4), the value of Mr increases and reaches a maximum constant value of M = 1/ε at higher frequencies for all temperatures. These observations may be related to a lack of restoring force governing the mobility of charge carriers under the action of an induced electric field. This type of behaviour supports the conduction phenomena because of the long-range mobility of charge carriers [20].

Figure 5 and its inset show the plot of imaginary part of modulus formalism (Mi) versus log f for the A6 and B6 samples at various temperatures. The appearance of the long tail at lower frequencies is associated with the long range ionic motion in which ions can perform successful-hopping from one site to the neighbouring site, resulting in DC conductivity. A small Mi value associated with large capacitance, and its higher capacitance makes a negligible contribution to electric modulus. This observation further confirms the non-Debye behaviour in the sample.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig5_HTML.gif
Fig. 5

Frequency dependence of the imaginary part of electrical modulus at a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K for A6 and B6 (inset)

Mi peaks are not found in the plasticized system, whereas they appear in the unplasticized system (Fig. 5). The peak positions in A6 are shifted toward higher frequencies with increasing temperature. The possible presence of peaks in the modulus formalism at higher frequencies for all the polymer system and temperature indicates that the polymer electrolyte films are ionic conductors [3]. The constancy of the height of the modulus plot suggests the invariance of the dielectric constant and the distribution of relaxation times with temperature. Beyond 333 K, no appearance of relaxation peak is found. The relaxation peaks are expected to be displaced toward higher frequencies and so are not observed in this plot because of the frequencies being in the range exceeding that permitted by the instrument used in the present study.

Dissipation factor or loss tangent (tan δ) is the ratio of loss factor (εi) to relative permittivity (εr), and it measures the ratio of the electric energy lost to the energy stored in a periodic field. The values of tan δ as a function of frequency for A6 and B6 at different temperatures in the range of 303–363 K are given in Fig. 6 and its inset, respectively. Similar to the unplasticized system, the plasticized system also exhibits tan δ peaks of dielectric relaxation but is obtained only for the lower temperature region in the range between 303 and 323 K, as shown in the Fig. 6. Beyond 323 K, the relaxation frequency is higher than that of the frequency limit of the instrument used in the measurement. These peaks shift toward sides with higher frequency as temperature increases. These features suggest the presence of dielectric relaxation phenomenon in the material because of the flipping or fluctuations of the boundaries of polar clusters [18].
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig6_HTML.gif
Fig. 6

Temperature variation of a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K in loss tangent (tan δ) at different frequencies for A6 and B6 (inset)

The values of relaxation frequency (fr) and relaxation time (τ) with the equation (τ = 1/2πfr is given in Table 1 for A6 and B6. The relaxation frequency depends on temperatures. At higher temperatures, more thermal energy will be available; hence, the dipolar orientation will be faster, which means relaxation frequency could increase.
Table 1

Temperature dependence of relaxation frequency (fr) and relaxation time (τ) of A6 and B6

T (K)

fr (Hz)

τ (s)

(Tan δ) max

A6

B6

A6

B6

A6

B6

303

4.57 × 103

9.12 × 104

3.48 × 10−5

1.75 × 10−6

4.09

6.33

313

1.20 × 104

1.82 × 105

1.33 × 10−5

8.74 × 10−7

4.73

6.66

323

2.40 × 104

4.37 × 105

6.63 × 10−6

3.64 × 10−7

5.28

6.85

333

3.31 × 104

a

4.81 × 10−6

a

5.55

a

343

3.39 × 104

a

4.69 × 10−6

a

5.59

a

353

4.27 × 104

a

3.73 × 10−6

a

5.64

a

363

7.94 × 104

a

2.00 × 10−6

a

5.66

a

aThe relaxation frequency is beyond 1 MHz, which was the limit of the measurement

Temperature dependence of dielectric behaviour

Dielectric permittivity, εr, and εi

Figure 7 and its inset illustrate the variation in εr with temperature at different selected frequencies for A6 and B6. The values of εr is observed to increase gradually with an increase in temperature for both the unplasticized (A6) and the plasticized system (B6). However, for the system that contains plasticizer—PEG600 in this work—εr increases at a decreasing rate and almost constant at temperatures above 350 K, implying that the dissociation of the salt has reached its maximum. Similar results were obtained in the study of Buraidah et al. [13]. Furthermore, the variation in εr with temperature is different for nonpolar and polar polymers. εr is independent of temperature for nonpolar polymers, whereas it increases with the increase in temperature for polar polymers [19].
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig7_HTML.gif
Fig. 7

Temperature dependence of dielectric constant at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for A6 and B6 (inset)

The behaviour of εr with temperature can be explained as follows: at relatively low temperature, the charge carriers on most cases cannot orient themselves with respect to the direction of the applied field. Therefore, they possess a weak contribution to the polarization and εr. As temperature increases, the bound charge carriers obtain enough excitation thermal energy to be able to obey the change in the external field more easily. This effect in turn enhances their contribution to the polarization, leading to an increase in εr of the sample [21].

Figure 8 and its inset show the variation in dielectric loss (εiεi) with temperature at different frequencies for A6 and B6 samples. From these graphs, the trend of variations in εi with respect to temperature is similar to that of εr with respect to temperature. The dielectric loss (εi) increases with decreasing applied frequency. This behaviour could be discussed by the process of dielectric polarization mechanism similar to the conduction process.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig8_HTML.gif
Fig. 8

Temperature dependence of dielectric loss at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for A6 and B6 (inset)

Modulus formalism, Mr and Mi

Figures 9 and 10 with their insets show the temperature dependence of Mr and Mi at selected frequencies for A6 and B6 (inset). Both Mr and Mi decrease with increasing temperature investigated in this work because of the increase in ionic conductivity in the sample provided by mobile ions. The complex impedance plots can be used as a method to detect the presence of ions and their temperature dependence within the present sample. This capability is due to the powerful ability of IS to investigate molecular mobility, phase transitions, conductivity mechanisms, and interfacial effects in polymers and complex systems.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig9_HTML.gif
Fig. 9

Temperature dependence of real modulus formalism at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for A6 and B6 (inset)

https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig10_HTML.gif
Fig. 10

Temperature dependence of imaginary modulus formalism at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for A6 and B6 (inset)

Figure 11 shows the complex impedance plots (Zi vs. Zr) for A6 and B6 at different temperatures, supporting the fact that the increase in temperature causes both Mr and Mi to decrease. Therefore, bulk resistance decreases with the increase in temperature from 303 to 363 K, indicating the dominance of ammonium ions, that is, the system is almost an ionic conductor. Consequently, both Mr and Mi decrease with increasing temperature. The complex impedance plots are commonly used to separate the bulk material (depressed semicircle) and the electrode surface polarization phenomena (tilted spike). The electrode polarization phenomena (tilted spike) occur because of the formation of an electric double layer capacitance through the free charge build up at the interface between the electrolyte and the electrode surfaces in plane geometry.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig11_HTML.gif
Fig. 11

Complex impedance plots at various temperatures: a 303 K, b 313 K, c 323 K, d 333 K, e 343 K, f 353 K, and g 363 K for A6 and B6 (inset)

Figure 12 shows the variation in loss tangent (tan δ) with temperature at different frequencies of the A6 sample. tan δ exhibits two different trends at low-frequency and high-frequency regimes. The value of tan δ initially increases with increasing temperature between 100 kHz and 1 MHz (high frequencies) and to a maximum value at 333 K. The presence of tan δ peaks in the pattern suggests the existence of relaxation phenomenon in the material, which may be attributed to β-relaxation because of the orientation of the polar groups present in the side group of the polymer. This result is in good agreement with those of previous studies [18, 19]. At lower frequencies, which are between 100 Hz and 1 kHz, tan δ at first shows the decrease in the low-temperature region, which lies in the range of 303–333 K. Thereafter, tan δ shows an almost constant value at high temperatures beyond 333 K.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig12_HTML.gif
Fig. 12

Temperature dependence of loss tangent at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for A6

Figure 13 illustrates the variation in loss tangent with temperature at selected frequencies for the B6 sample. At low frequencies (from 100 Hz to 10 kHz), no peaks appeared. However, at frequencies between 100 kHz and 1 MHz, the relaxation spectra exhibit the peaks, and the peaks are observed to shift toward the higher temperature region as the frequency increases. The presence of tan δ peaks in the spectra suggests the existence of relaxation phenomenon of material [18]. At low frequencies, the relaxation spectra of tan δ show an almost constant value within the temperature range investigated.
https://static-content.springer.com/image/art%3A10.1007%2Fs11581-011-0653-0/MediaObjects/11581_2011_653_Fig13_HTML.gif
Fig. 13

Temperature dependence of loss tangent at different frequencies: a 100 Hz, b 1 kHz, c 10 kHz, d 100 kHz, and e 1 MHz for B6

Conclusion

The frequency and temperature dependence of dielectric behaviour for the highest conducting unplasticized sample—CA-25 wt.% NH4BF4 (A6)—and the highest conducting plasticized sample—CA-25 wt.%–30 wt.% PEG600 (B6)—were studied. The frequency-dependent εr and εi show the presence of electrode polarization phenomena at lower frequencies and decay at higher frequencies. The modulus formalism approaching zero at low frequency and reaching maximum constant value at high frequency supports the conduction phenomena because of the long-range mobility of charge carriers. The behaviour of εr and εi with temperature is typical of polar dielectrics in which the orientation of dipoles is facilitated with the rising temperature, and thereby the permittivity is increased. The complex impedance plots (Zi vs. Zr) support the fact that the increase in temperature causes both Mr and Mi to decrease. The nature of variation in tan δ as a function of frequency and temperature indicates the presence of dielectric relaxation in the material.

Acknowledgment

N.I. Harun would like to thank the Universiti Teknologi MARA for the scholarship (NSF) awarded.

Copyright information

© Springer-Verlag 2011