Polymer Electrolyte Nanocomposites

Encyclopedia of Nanoscience and Nanotechnology

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Polymer Electrolyte Nanocomposites Mikrajuddin Abdullah1 , Wuled Lenggoro, Kikuo Okuyama Hiroshima University, Hiroshima, Japan

CONTENTS 1. Introduction 2. Conductivity Enhancement in Polymer Electrolytes 3. Development of Polymer Electrolyte Nanocomposites 4. Preparation Methods 5. Important Parameters 6. Charge Transport Characterizations 7. Spectroscopic Characterizations 8. Microscopic Analysis 9. Thermal Characterizations 10. Density Method 11. Electrical Properties 12. Mechanical Properties 13. Thermal Properties 14. Luminescent Composites 15. Conclusion Glossary References

1. INTRODUCTION Rechargeable cells are key components in mobile technologies, such as portable consumer electronics and electric vehicles [1]. A search for batteries that provide high energy density and multiple rechargeability has been a subject of considerable attentions. Even though battery technology developed one hundred years ago, progress and improvements in technology have been slow, particularly when compared to the growth of computer technology [2]. A Li-based battery provides a high density and flexibility of design. Today’s lithium battery has a high specific energy (>130 W h kg−1 ), a high energy density (>300 W h L−1 ), 1 Permanent address: Department of Physics, Bandung Institute of Technology, Jalan Ganeca 10 Bandung 40132, Indonesia.

ISBN: 1-58883-064-0/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

high cell voltage (3.5 V), as well as a long cycle life (500–1000) charge/discharge. Worldwide production of such devices exceeded 200 million in 1997 and it will be approximately three times that number during 2001 [1]. Since lithium produces an explosion reaction with water-based electrolytes, a search for nonaqueous electrolytes is critically important to the production of the next-generation lithium battery, using electrolytes in an solid phase in an effort to develop more environmentally friendly materials. Polymer electrolytes are potential candidates for replacing the conventional aqueous electrolytes in lithium batteries. Polymers containing esters, ethers, or mixtures thereof which have the ability to dissolve salts are the base materials for polymer electrolytes. Polymer electrolytes are generally prepared by mixing high molecular weight polymers (HMWPs) with a salt solution. The polymer serves as solid solvent, thus permitting the salt to dissociate into anions and cations. Since the mass of a cation is much smaller than that of an anion, the electrical conductivity is dominated by cation transfer. Lithium salts are usually used for this purpose since they are the most electropositive of materials (−304 V relative to the standard hydrogen electrodes) as well as the lightest metal (atomic mass 6.94 g/mol, and density 0.53 g/cm3 ) and thus facilitate the design of storage systems with high energy density (Watt hour/kg) [1]. Table 1 shows a comparison of the electrochemical properties of several metals. Until presently, however, no polymer electrolyte-based lithium batteries are commercially available in the market. Therefore, worldwide research is being focused on the development of high power and high energy density polymer electrolytes with a major attention to safety, performance, and reliability. A battery contains two electrodes: positive and negative (both sources of chemical reactions), separated by an electrolyte that contains dissociated salts through which ion carriers flow. Once these electrodes are connected to external circuits, chemical reaction appears at both electrodes to result in a deliverance of electrons to the external circuits. The properties of a battery thus strongly depend on the electrolyte, anode, and cathode. With the use of polymer electrolytes in lithium batteries, high specific energy and specific power, safe operation, flexibility in packaging, and low cost in fabrication as well as low internal voltage drop at relatively large current withdraw is expected [3]. Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 8: Pages (731–762)

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Polymer Electrolyte Nanocomposites

Table 1. Electrochemical properties of several metals that have potential applications for use in batteries. Metal Li Na Mg Zn Cd Pd

Atomic weight [g/mol]

Valence charge

Specific charge [A h kg−1 ]

Electrode potential [V]

694 2299 2431 6538 11241 20720

1 1 2 2 2 2

3862 1166 2205 820 477 250

305 271 238 076 040 013

Realization of commercial polymer electrolyte batteries is actively investigated in many companies worldwide. A major effort to develop advanced polymer batteries for electric vehicles began in the early 1990s by 3M and HydroQuebec [4]. The battery contains a lithium metal anode, a polyethylene oxide (PEO)-based polymer electrolyte, and a vanadium oxide (VOx ) cathode. The reversibility of lithium intercalation and deintercalation in the VOx is quite good but the average discharge of the cell is low. PolyPlus Battery company in the United States is developing polymer electrolyte-based lithium battery which would operate at room temperature with specific energy as high as 500 W h kg−1 [3]. In a prototype cell, using cathode made of lithium intercalated disulfide polymer, a specific energy as high as 100 W h kg−1 and charge and discharge cycles almost reproducible for over 350 cycles were observed at 90  C [5]. Moltec company reported a specific density of 180 W h kg−1 for an AA-sized battery based on organosulfur cathode [6]. Ultrafine Battery company reported a room-temperature solid polymer battery based on intercalation type electrode with a specific energy 125 W h kg−1 and charge/discharge cycling time of 500 [3]. This performance is still below the consumer expectation threshold. In 1995, Turrentine and Kurani in the United States did a survey on demand for alternative fuel cell for vehicles and found that consumers agreed to buy electric vehicles which would run for at least 200 km per battery [7].

2. CONDUCTIVITY ENHANCEMENT IN POLYMER ELECTROLYTES It is believed that in the polymer electrolytes, the cations are coiled by polymer segment leaving the anions to occupy separate positions [8]. Battery performance is limited by the speed of cation diffusion. The transport of cations takes place if there is a relaxation of the polymer segments so that cations are released from a segment and then occupy another segment. Segmental relaxation requires the presence of free volume in the polymer matrix, a condition that can be attained if the polymer is in an amorphous state. Unfortunately, most HMWPs crystallize at ambient temperatures. Ions are transported with difficulty in a crystalline matrix since no chain relaxation occurs and, as a result, the conductivity of polymer electrolytes in this phase (at ambient temperature) is depressed. The transport of ions in this state is dominated by the jumping of cations to the nearest location, which depends on the blocking potential (activation energy). This is similar to the jump of charge carriers in

crystalline solids. The characteristic time for jumping is proportional to the exponential of the blocking potential. This results in a conductivity of the order of 10−8 S/cm, a value that is far below the desired value of about 10−4 S/cm [9]. When it enters the amorphous state, that is, at temperatures above the melting point, a high conductivity appears. For a commonly used polymer, that is, polyethylene oxide, the melting temperature is 65  C. This is, of course, impractical since the operating temperature for most electronic devices is room temperature. In addition, at temperatures above the melting point, the polymer becomes soft, causing the solidstate properties to degrade. Initiated by the work of Wright and Armand [10–12], several kinds of polymer electrolytes have been intensively investigated around the world. Table 2 displays examples of polymer electrolytes and their measured conductivities at 20  C [13]. Improvements in the electrical conductivity of polymer electrolytes at ambient temperature is therefore of critical importance for technological applications. Several approaches have been explored to realize this aim. Some frequently used methods will be explained briefly here.

2.1. Preparing Low Degree of Crystallinity Polymers By considering that the presence of amorphous state is strictly important for improving the conductivity, the main strategy is to enhance the amorphous state at low temperatures. The first approach is to prepare low degree of crystallinity polymer from initial. It includes cross-linking of two polymers [13, 14], synthesis of new polymer, crosslinking high molecular weight polymer through -irradiation [15, 16], addition of plasticizers in polymer electrolytes, addition of fillers, and bending of two polymers [17, 18]. Another strategy is to prepare an amorphous polymer so as to obtain a polymer that is composed of four to five monomeric units. For this system, the chains must be sufficiently long to effectively complex cations but too short to crystallize at low temperatures. Thus the matrix would still be in the amorphous state even at low temperatures. The polymer host serves as a solvent and does not include any organic liquids.

2.2. Addition of Side Chains An alternative way to decrease the crystallinity of polymer matrix is to introduce side chain to the polymer main chain. Theoretically, chain ends and branch can be thought of as impurities, which depress the melting point of the polymer. Simple mathematical formulation can be used to explain the melting point lowering by the presence of chain ends and branch. If Xi is the mole fraction of impurities (chain ends, side chains, and branch), then the melting point of polymer, Tm , decreases according to [19] 1 1 R − o = ln1 − Xi  Tm Tm Hu

(1)

where Tmo = melting point of polymer containing only polymer chain with infinite chain length, R = gas constant, and Hu = enthalpy of fusion per mole of repeat unit. Chung and Sohn showed that the XRD intensity of polymer

733

Polymer Electrolyte Nanocomposites Table 2. Examples of polymer electrolytes with their corresponding electrical conductivities at 20  C. Polymer host

10−8

— CH2O — n

POM:LiClO4

10−8

— (CH3)CH2CH2O — n

(PPO)8 LiClO4

10−8

— CH2CH2O — n

Poly(oxymethylene), POM Poly(propylene oxide), PPO Poly(oxymethyleneoligo-ethylene), POO Poly(dimethyl siloxane), DMS

Conductivity (S/cm) at 20  C

(PEO)8 :LiClO4

Poly(ethylene oxide), PEO

— (CH2O)(CH2CH2O) — n — (CH3)2SiO — n — HC=CH(CH2)4O(CH2CH2O)n(CH2)4 — x — CH2CHO — n

(POO)25 :LiCF3 SO3

3×10−5

DMS:LiClO4

10−4

UP:LiClO4 (EO:Li+ = 32  1) (PMEGE)8 :LiClO4

10−5 10−5

—

Unsaturated ethylene Oxide segmented, UP Poly[(2-methoxy)ethyl glycidyl ether], PMEGE

Example polymer electrolyte

Repeat unit

CH2(OCH2CH2)2OCH3 PMG22 :LiCF3 SO3

CH3

—

3×10−5

— CH2C — n — — — —

Poly[(methoxy) poly(ethylene glycol)] methacrylate, PMGn (EO:Ll+ = 181)

C

O

O — (CH2CH2O)xCH3 —

—

—

—

—

—

—

CH3 CH3

—

(PEO-PPO-PEO)-SC SC = siloxane crosslinked

PEO–(CH2)3–Si–O–Si–(CH2)3–PEO O

(PEO-PPO-PEO)-SC: LiClO4 (4:1 molar)

1–3×10−5

O

PEO–(CH2)3–Si–O–Si–(CH2)3–PEO CH3 CH3

PEO grafted polysiloxane, PGPS

PGPS:LiClO4

—

CH3

10−4

—

—SiO— n CH2CH2PEO

OCH2CH2OCH2CH2OCH3

—

Poly[bis-2-(2-methoxyethoxy) ethoxy]phosphazene,MEEP

—

— P=N — n

(MEEP)4 :LiBF4 (MEEP)4 :LiN(CF3 SO2 )4 (MEEP)4 :LiC(CF3 SO2 )4

2×10−5 5×10−5 10−4

OCH2CH2OCH2CH2OCH3

decreases with the increase in the length of chain of combshaped polymer [20]. Despite depressing the melting point of polymer, the presence of side chain also promotes the solvating of a salt as reported by Ikeda and co-workers [21, 22]. The side chain has shorter relaxation time compared to the main chain. The coupling of the side chain with the ion carrier, therefore, results in an increase in the conductivity. Watanabe et al. designed comb-shaped polyether host with short polyether side chain [23]. However, the mechanical properties decreased even as the conductivity increased. High conductivity with good mechanical properties was obtained by designing a polymer of high molecular weight with trioxyethylene side chain as reported also by Ikeda et al. [24]. With 18 mol.% of side chain, the conductivity was measured to be 1.5 × 10−4 S/cm at 40  C and raised to 1.4 × 10−3 S/cm at 80  C. Composite of polymer with room-temperature molten slat is also an interesting approach to improve the conductivity of polymer electrolytes. Watanabe et al. reported the composite

consisting of chloroaluminate molten salt that possesses a conductivity of 2 × 10−3 S/cm at 303 K [25, 26]. However, the disadvantage of chloroaluminate is its hygroscopic properties such that it is impractical in application. The use of non-chloroaluminate molten salt, therefore, is required to avoid the hygroscopic problem. Tsuda et al. reported a conductivity of 2.3 × 10−2 S/cm in composite of polymer and room-temperature molten fluorohydrogenates [27].

2.3. Addition of Plasticizers Another approach to improve the conductivity is by addition of additional material into the host polymer. This approach appears to be the simplest since a pre-produced polymer can be used to make the polymer electrolytes. Previously, low molecular weight polymers were usually used to reduce the operation temperature of polymer electrolytes. The low molecular weight polymers which were added to the matrix of HMWP to reduce the crystallinity at low temperatures are frequently known as liquid plasticizers.

734

Polymer Electrolyte Nanocomposites

1 W W = 1 + 2 Tg Tg1 Tg2

(2)

10–3

10–4

10

–5

0

25

50

220

75

100

wt.% tetraglyme

Figure 2. Effect of plasticizer weight fraction on the conductivity at 25  C of a PEO-co-PPO (3:1):LiClO4 using plasticizer tetraglyme (tetraethylene glycol dimethyl ether). Data points were extracted from [29], D. R. MacFarlene et al., Electrochim. Acta 40, 2131 (1995).

particularly when the fraction of plasticizers is too high. For example, the modulus of elasticity and elastic strength significantly decreases by addition of plasticizers. This is because the plasticizers are usually low molecular weight polymer having low mechanical strength. Therefore, addition of plasticizers decreases the mechanical strength of the host polymer. Figure 3 shows the effect of plasticizer troglyme content on the elastic modulus and tensile strength of PEO-co-PPO (3:1):LiClO4 [29]. The use of moderate or large quantities of plasticizer results in the production of

2.0

Elastic modulus

Modulus [Mpa]

where W1 and W2 denote the weight fractions of component 1 and component 2, respectively, and Tg1 and Tg2 are their corresponding glass transitions. This equation tells that the glass temperature of the composite locates between the glass temperature of the components. This relation is also applicable for copolymer where Tg1 and Tg2 denote the glass temperature of polymers forming the copolymer. Reduction in the glass temperature means the enhancement in the amorphous state at low temperature, and therefore improves the conductivity at low temperatures. Figure 2 shows the enhancement of conductivity by the addition of plasticizer tetraglyme on the system of PEO-co-PPO (3:1):LiClO4 , measured at 25  C [29]. The decrease in the glass transition results in the improvement in the fraction of amorphous state at room temperature, therefore improving the conductivity. However, an improvement in conductivity is adversely accompanied by a degradation in solid-state configuration and a loss of compatibility with the lithium electrode [9],

10–2

σ [S/cm]

Feullade and Perche demonstrated the idea of plasticizing the polymer with an aprotic solution containing alkali metal salt in which the organic solution of the alkali metal salt remained trapped within the matrix of solid polymer matrix [28]. Such mixing results in formation of gels with ionic conductivity close to the liquid electrolytes. Less evaporating solvents such as ethylene carbonate (EC), propylene carbonate (PC), dimethyl formamide (DMF), diethyl phthalate (DEP), diethyl carbonate (DEC), methyl ethyl carbonate (MEC), dimethyl carbonate (DMC), -butyrolactone (BL), glycol sulfide (GS), and alkyl phthalates have been commonly investigated as plasticizers for the gel electrolytes. Figure 1 shows the effect of plasticizer content tetraglyme (tetraethylene glycol dimethyl ether) on the glass temperature of a system of PEO-co-PPO (3:1):LiClO4 [29]. The decrease in the glass temperature can be simply explained using a Fox equation:

Tg[K]

1.0

200

Tensile Strength

180

0

25

50

75

100

wt.% tetraglyme

Figure 1. Effect of plasticizer weight fraction on the glass temperature of a PEO-co-PPO (3:1):LiClO4 using plasticizer tetraglyme (tetraethylene glycol dimethyl ether). Data points were derived from [29], D. R. MacFarlene et al., Electrochim. Acta 40, 2131 (1995).

0.0

0

20

40

60

wt.% tetraglyme

Figure 3. Effect of plasticizer weight fraction on the modulus of a PEOco-PPO (3:1):LiClO4 using plasticizer tetraglyme (tetraethylene glycol dimethyl ether). Data points were extracted from [29], D. R. MacFarlene et al., Electrochim. Acta 40, 2131 (1995).

735

Polymer Electrolyte Nanocomposites

gel electrolyte. The presence of some plasticizer may also give rise to problems caused by its reaction with the lithium anode. The poor mechanical stability was accounted to be mainly due to the solubility of the polymer matrix in the plasticizer [30]. Cross-linking of the polymer with ultraviolet radiation [31], thermally [32], by photopolymerization [33], or electron beam radiation polymerization [34] was found to reduce the solubility of polymer in the solvent and also helped to trap liquid electrolytes within the polymer matrix.

3. DEVELOPMENT OF POLYMER ELECTROLYTE NANOCOMPOSITES Currently, one popular approach to improve the conductivity involves dispersing ceramic fillers (solid plasticizers) in the polymer matrix, producing what is currently known as composite polymer electrolytes. This approach was first introduced by Weston and Steele [35]. Ceramic filler was used to reduce the glass transition temperature and crystallinity of the polymer and thus allow the amorphous polymer to maintain the liquid-like characteristic at the microscopic level. Ceramic fillers that are frequently used have particle sizes in the range of about several ten nanometers up to several micrometers. Fortunately, such filler materials are commercially available in various sizes at low prices. Figure 4 shows the effect of filler content on the conductivity of polymer electrolytes PEO:LiClO4 [9]. Table 3 displays examples of polymer electrolyte nanocomposites and their conductivities at around room temperature. The inorganic filler also acts as a support matrix for the polymer, so that even at high temperature, the composite remains solid. However, at the microscopic level, it maintains a liquid-like structure, which is important for sufficient conductivity. The filler particles, due to high surface area, prevent the recrystallization of polymer when annealed above the melting point. The acid-base interaction between the filler surface group and the oxygen of the PEO leads to a Lewis acid characteristic of the inorganic filler and favors –2

Log σ [S/cm]

–4 VTF

–6

Arrhenius

–8

2.4

2.6

2.8

3.0

3.2

3.4

3.6

1000/T [1/K]

Figure 4. Arrhenius plot of electrical conductivities of: (solid) ceramicfree PEO:LiClO4 , (triangle) PEO:LiClO4 containing 10 wt.% Al2 O3 (5.8 nm), and (square) PEO:LiClO4 containing 10 wt.% TiO2 (13 nm). Data points were extracted from [9], F. Croce et al., Nature 394, 456 (1998).

Table 3. Examples of polymer electrolyte nanocomposites with their corresponding electrical conductivities at around room temperature.

Polymer electrolytes PEO:LiBF4

Fillers

nano-sized-Al2 O3 micro-sized-Al2 O3 PEO:LiCIO4 SiC EO-co-PO:LiCF3 SO3 Li13 Al03 Ti17 (PO4)3 Brached-poly(ethylene silica (12 nm size) imine):H3 PO4 PEO:LiClO4 -Al2 O3 PEO:LiClO4 AlCl3 NNPAAM PEO:LiClO4 PEO-PEG:LiI Al2 O3 PEO-PMMA:EC:LiI Al2 O3 PEG:LiCF3 O4 SiO2 C12 H25 OSO3 Li PEG:LiCF3 O4 coated-SiO2 EC:PC:PAN:LiAsF6 porous zeolite PEO:LiClO4 AlN, BaTiO3 , Bi2 O3 B4 C, BN, CaSiO3 CeO2 , Fe2 O3 , MoS2 , PbTiO3 , Si3 N4 , carbon black PEO:LiClO4 PEO:Li[(SO2 CF3 )2 N] -LiAlO2 PEO:PMMA:EC:LiI MgO PEO:AgSCN Al2 O3 PEO:AgSCN Fe2 O3 PEO:AgSCN SO2 Na2 SiO3 PEO:NaClO4 PEO:LiCF3 SO3 mineral clay PbS PEO:NH4 I CdS PEO:NH4 I Pbx Cd1−x S PEO:NH4 I

Conductivity at around rt (S/cm)

Ref.

∼10−4 ∼10−5 ∼10−7 ∼2×10−4 (40  C) ∼10−7

[36] [37] [38] [39]

∼10−5 ∼10−5 >∼10−5 ∼10−6 ∼10−8 ∼10−5 ∼5 × 10−5

[40] [40] [40] [41] [41] [42] [43]

∼10−3 ∼10−7 –10−6

[44]

∼4 × 10−6 ∼10−7 ∼88 × 10−4 ∼11 × 10−5 ∼3 × 10−6 ∼2 × 10−6 ∼10−3 ∼099 × 10−6 ∼096 × 10−6 ∼063–084 × 10−6

[46] [47] [48] [49] [50] [51] [52] [53] [54] [54] [54]

the formation of complexes with PEO. The filler then acts as cross-linking center for the PEO, reducing the tension of the polymer for self-organization and promoting stiffness. On the other hand, the acid-base interaction between the polar surface group of the filler and electrolyte ions probably favors the dissolution of the salt. Another potential application of polymer electrolyte nanocomposites is for making solar cells [55]. Dye-sensitized solar cells have attracted great scientific and technological interest as potential alternatives to classical photovoltaic devices. The cell operation mechanism involves absorption of visible light by the chemisorbed dye, followed by the electron injection from the excited synthesizer into the semiconductor conduction band. The selection of liquid electrolytes, usually containing organic solvent such as acetonitrile and propylene carbonate, assures the perfect regeneration of the dye by direct interaction of the dye oxidized state and I− /I− 3 redox couple and leads to impressively high solar-to-electrical conversion efficiencies (7–11%) [56, 57]. However, the stability and long-term operation of the cell are affected by solvent evaporation or leakage. Thus commercial exploitation of these devices needs the replacement of the liquid electrolyte by a solid charge transport medium, which not only offers hermetic sealing and stability but also reduces design restriction and endows the cell

736 with shape choices and flexibility. Katsaros et al. investigated solid-state dye-sensitized solar cells using composite polymer electrolytes using PEO and TiO2 in the presence of I− /I− 3 redox couple [55]. Initially, dye:Ru(dcbpy)2 (NCS)2 (dcbpy is 4,4’-dicarboxylic acid-2-2 -bipyridine) was attached on the surface of TiO2 nanoparticles by immersion of the TiO2 thin-film electrode overnight in ethanolic solution of the complex, followed by drying. The functionalized TiO2 nanoparticles, I− /I− 3 , and PEO were put in acetonitrile, followed by heating and drying to evaporate the solvent. Maximum incident photon to current efficiencies as high as 40% were obtained at 520 nm, only two times lower that than obtained using liquid electrolytes [58]. The overall conversion efficiency was 0.96%. For all-solid-state devices, such efficiency can be considered to be sufficiently high [59].

4. PREPARATION METHODS Now we will briefly explain several methods of preparation of polymer electrolyte nanocomposites that are commonly used. Which method should be used, of course, depends on the materials and the form of sample to be formed. One method can only produce sample in the form of thick film, and another one can produce a sample in the form of film of submicrometer thickness.

4.1. Casting Method This method is frequently used due to its simplicity. It can produce polymer film from several micrometers up to several millimeters thickness. Generally, this method includes the following steps: (a) dispersion of ceramic fillers in a salt solution, (b) addition of a specified amount of polymer to the mixture, (c) mixing by means of stirrer or ultrasonic equipment to disperse the particles homogeneously in the polymer matrix, (d) casting the mixture on a substrate, (e) finally drying in vacuum or in an atmosphere of argon. All these steps are usually performed in a glove box filled with argon gas and excluding oxygen and water to levels below 20 parts per million (ppm), to avoid the possible occurrence of a “dangerous reaction” between water and lithium. The solvent must be water-free and should be common solvent for both the salt and the polymer. Since the melting point of several polymers is as high as 65  C, the solvent must also easily evaporate so that drying can be performed at temperatures of around 65  C. Organic solvents such as acetonitrile, cyclopentanone, and propylene carbonate, plus inorganic solvents such as thionyl chloride (SOCl2 ), are typically used. Sometimes, the insertion of salt is performed after casting the film. For example, Ardel et al. prepared PVDF 2801 (Kynar)-based polymer electrolyte composites according to the following steps [60]. First, Kynar was dissolved into cyclopentanone. Nanoparticles of silica were added and the mixture was mixed for 24 h at room temperature to get homogeneous slurry. After complete dissolution, the slurry was cast on the Teflon support and spread with the use of

Polymer Electrolyte Nanocomposites

doctor blade technique. To prevent surface irregularities, the film was then covered with a box pierced with holes that allowed a slow evaporation of the cyclopentanone. After complete evaporation of the cyclopentanone, the polymer membrane was soaked in a lithium ion solution for 48 h. Several fresh lithium solutions for each soaking can be used to ensure a complete impregnation of lithium ion into the membrane.

4.2. Spin Coating The spin-coating method is very similar to the casting method. Instead of casting the film on a substrate, in this method, the mixture is dropped on a substrate and placed in a spin coater that can be rotated at adjustable rotation speed. The film thickness can be controlled easily by adjusting the viscosity (concentration) of the mixture and the speed of rotation. However, this method is only available if the viscosity of the mixture is not too high. For a gel mixture, the spin coater rotation is not enough to spread the mixture droplet to form thin film.

4.3. Hot Press Hot press technique equipment is illustrated in Figure 5. The equipment consists of: (A) weighing cylinder, (B) heating chamber, (C) basement, and (T) temperature controller. Proper amounts of polymer, salt, and filler are mixed in a mortar for about several minutes. The powder mixture is then sandwiched between two sheets of Mylar or other materials, and positioned inside the heating chamber that is controlled at temperatures lightly above the melting point of the polymer. If PEO is used as polymer matrix, temperature of 80  C is suitable [61]. The sample is then pressed overnight with a pressure that can be controlled by weighing cylinder. After heating and pressing, the sample is then slowly cooled to room temperature. The sample is then separated from the Mylar sheet and placed in a glove box.

A

B

C

T

Sample

Figure 5. Illustration of hot press equipment: (A) weighing cylinder, (B) heater, (C) base, and (T) temperature controller.

737

Polymer Electrolyte Nanocomposites

4.4. In-situ Preparation In-situ preparation explained here is the preparation of nanoparticles in the polymer matrix. Mikrajuddin et al. produced polymer electrolytes of polyethylene glycol with lithium ion by in-situ production of ZnO nanoparticles in the polymer matrix [62, 63]. The preparation methods will be briefly described as follows. Zinc acetate dihydrate, (CH3 COO)2 Zn · 2H2 O 0.1 M in 100 mL ethanol 99.5, was heated with stirring in distillation equipment at temperature of 80  C to produce about 60 mL condensate and 40 mL of hygroscopic solution. Lithium hydroxide monohydrate, LiOH · H2 O, of various concentrations was suspended in 40 mL ethanol and stirred until all the granular material dissolved. Polyethylene glycol (PEG) (Mn = 2,000,000) was suspended into each LiOH solution and then stirred with heating at around 60  C until homogeneous gel-type mixtures were obtained. The mixture temperatures were then left to go down several minutes, after which 10 mL of hygroscopic CH3 COO2 Zn · 2H2 O solution was added into each mixture. The new mixtures were then homogeneously mixed and then dried in an oven that was kept at temperature of 40  C during three days. The schematic of sample preparations is displayed in Figure 6. There are many differences between the present method and the commonly used ones. In the present approach: (a) Nanoparticles are grown in-situ in polymer matrix. (b) The size of dispersed particles is controllable. (c) Ion carriers are inserted in-situ in the polymer matrix during the growing process. (d) Finally, since the grown nanoparticles are luminescent, we obtain a new class of polymer electrolytes, namely luminescent polymer electrolytes with nanoparticles as luminescence centers. Based on the TEM picture, we found that the size of ZnO nanoparticles was 5 nm.

Ethanol

ZnAc2•2H2O

LiOH•H2O

Distilled at 80°C

Mixed around 10 min

Ethanol

Condensate (60%) Mixed at 60°C unused

PEG

Cooled at 0°C

Left cooling

Hygroscopic solution (40%) Mixed several minutes

Dried at 40°C, 3 days

ZnO nanoparticles are formed in the polymer matrix

Characterizations

Figure 6. Diagram of in-situ preparation of PEG:Li containing nanoparticles of ZnO. Adapted with permission from [62], Mikrajuddin et al., J. Electrochem. Soc. 149, H107 (2002). © 2002, Elsevier.

Chandra et al. produced polymer electrolytes PEO:NH4 I containing nanometer-sized semiconductor particles PbS, CdS, Pbx Cd1−x S [54]. Methanolic solution of PEO and NH4 I was first stirred roughly at 40  C for 8–10 h, which resulted in viscous solution of the ion conducting complexes of PEO/NH4 I. To this solution, a solution Pb(CH3 COO)2 , Cd(CH3 COO)2 , or Pd(CH3 COO)2 + Cd(CH3 COO)2 in a desired fraction was added. The stirring was continued until the viscosity was back to the value it was before adding the acetate compounds. Subsequently, H2 S was bubbled through it giving PbS, CdS, or Pbx Cd1−x S. The final viscous solution was poured in a petri dish for obtaining solution-cast film. Then the film was dried in vacuum.

5. IMPORTANT PARAMETERS To bring polymer electrolytes as well as polymer electrolyte composites, these materials should provide enough values of several properties as follows.

5.1. Electrical Conductivity Conductivity defines the density of current that can be transported in the material by applying a certain electric field. If electric field E is applied in the material, the current density will be proportional to the applied electric field, where the proportional constant is the conductivity, or, J = E

(3)

with J the current density (A/m2 ) and  (S/m or S/cm) the electrical conductivity. It is clear that high conductivity material will produce high current density upon applying a certain electric field. The value of conductivity is determined by the density of mobile ions (ion carriers) in the material (n), the scattering time of the ion (), the ion charge (q), as well as the mass of ion carrier (m), according to a relation =

nq 2  m

(4)

This equation gives the reason why most polymer electrolytes use lithium ions as ion carriers. The mass of lithium is the smallest among all metals, so it produces the highest conductivity. For industrial application, the conductivity of polymer electrolytes must be as high as 10−2 S/cm. However, until presently, this conductivity can only be achieved at high temperatures in which the polymer is present in the soft phase, or even liquid phase. The conductivity at room temperature of most reported polymers is still below 10−4 S/cm.

5.2. Transference Number Since the electrochemical process in lithium batteries involves the intercalation and de-intercalation of lithium cations throughout host compound lattice, solid polymer electrolytes with cation transference number (t + ) approaching unity are desirable for avoiding a concentration gradient during repeated charge-discharge cycles. The reported t + value for dried polymer electrolytes range from 0.06 to 0.2 [64]. For a gel polymer system, t + value of 0.4–0.5 has

738

Polymer Electrolyte Nanocomposites

been found for poly(bis-methoxy ethoxy)phosphazene [65], and 0.56 in a system of UV-cured gel polymer electrolytes based on polyethylene glycol diacrylate/polyvinylidene fluoride [66]. Transference number of a particle is defined as the ratio of the conductivity due to it and the total conductivity. Assume the total conductivity  is due to ionic, ion , and to electronic, e , then  = ion + e

(5)

The ionic and electronic transference numbers are then ti =

ion 

(6)

te =

e 

(7)

and

For pure ionic, ti = 1, and for pure electronic, te = 1. For polymer electrolyte composites, a general condition satisfied is 0 < ti  te < 1.

5.3. Crystallinity Crystallinity plays an important role in determining the conductivity of polymer electrolytes. At crystalline phase, the transport of ion carriers is very difficult so that the conductivity is very low. At amorphous phase, there is a segmental motion of polymer chain that also assists the displacement of ions. As a result, the transport of ions is relatively easy. Thus, high conductivity will result. One major route to improve the conductivity of polymer electrolytes is by increasing the fraction of amorphous states. Addition of ceramic fillers, addition of plasticizer, and production of branch polymer are efforts to improve the amorphous state in the polymer.

5.4. Mechanical Strength One objective of the use of polymer electrolyte is to make a battery or fuel cell with a strength comparable to that of liquid electrolytes. Therefore, it is expected that the improvement of conductivity is not accompanied by a decrease in the mechanical strength. It is why the addition of ceramic filler has received more attention, since the conductivity and the mechanical strength can be improved simultaneously. In contrast, the use of liquid plasticizer, although it can enhance the conductivity much higher than the addition of ceramic filler, involves such a degradation in the mechanical strength as to make this approach less interesting.

6. CHARGE TRANSPORT CHARACTERIZATIONS Electrical conductivity is the critical parameter for polymer electrolyte composites. One target of the present research in this field is to produce polymer electrolyte nanocomposites that exhibit a high electrical conductivity, especially at room temperature. Conductivity at around 10−2 –10−3 S/cm is required to bring this material into industry. The electrical conductivity relates to the value of current that can be produced by the battery. The potential produced by the battery depends on the reaction of the battery with the electrode. Even though the electrode reaction can produce high electrical potential, the use of low conductive electrolytes can produce only small amount of electric current. And since the power can be calculated simply by the relation Power = Voltage × Current, the use of low conductive materials will produce a low specific energy battery.

6.1. d.c. Conductivity Ideally, the d.c. conductivity should be measured in order to be sure that the values pertain to long-range ion movement instead of dielectric losses such as would be associated with limited or localized rattling of ions within cages. However, the difficulty in making a d.c. measurement is in finding an electrode material that is compatible with the electrolyte composites. For example, if stainless steel electrodes are attached to an electrolyte composite, as displayed in Figure 7a, and small voltage is applied across the electrodes, Li+ ions migrate preferentially toward the cathode, but pile up without being discharged at the stainless/electrolyte interface. A Li+ ion deficient layer forms at the electrolyte/stainless steel interface. The cell therefore behaves like a capacitor. There is an accumulation of ions at interface region of electrode and composite. A large instantaneous current Io presents when the cell is switched on, whose magnitude is related to the applied voltage and the resistance of the electrolytes but then falls exponentially with time, as illustrated in Figure 7b. The characteristic time of current decreasing is relatively fast so that it is difficult to make an accurate measurement. Therefore, the a.c. method is commonly used in the present to make measurement over a wide range of frequencies. The d.c. value can be extracted from the a.c. data. Many a.c. measurements are performed with blocking I

(a) +

(b)

Io

CPE

5.5. Storage Time Battery or fuel cell made from polymer electrolytes should have to operate several weeks or several months. Thus, the properties of polymer electrolytes should not change too much during this time. For example, the conductivity should not depend so much on the storage time. Ideally, the properties should be time independent. However, in reality, the properties tend to degrade with storage time.

+

time Figure 7. (a) Polymer electrolyte composites sandwiched between two blocking electrodes. (b) The decay of current when a constant d.c. voltage is applied between two electrodes.

739

Polymer Electrolyte Nanocomposites

Z  = Z    − iZ   

(8)

where is the frequency, Z    is the real part of impedance, contributed by resistive part, Z    is the imaginary√part of impedance, contributed by capacitive part, and i = −1, the imaginary number. As an illustration, Figure 8 shows examples of simple RC circuits and the corresponding plot of impedance (Nyquist plot). For a serial arrangement of a resistor and a capacitor, as displayed in Figure 8b left, the impedance can be written as i Z =R− C

(9)

or Z = R Z  =

1 C

(10a) (10b)

It is clear that the real part of impedance is constant, independent of the frequency, while the imaginary part depends on the frequency. For very small frequency, the imaginary part is very large and this value decreases inversely with frequency. For frequency approaches to infinity, the imaginary part of impedance closes to zero and the impedance

(a)

Z′′ [Ω]

R

Z′ [Ω]

R

C

R •

Z′′ [Ω]

(b)

R Z′ [Ω] R Z′′ [Ω]

(c)

C

(d)

ωmCR = 1 •

•

Z′ [Ω]

R

R C2

Z′′ [Ω]

electrode such that no discharge or reaction occurs at the electrode/electrolyte interface. Because the current will flow back and forth, no ions pile up on electrode surface, especially when using a high a.c. frequency. This is why the a.c. resistance (impedance) tends to decrease with increase in the frequency. The electrodes that are commonly used are platinum, stainless steel, gold, and indium tin oxide (ITO) glass. The complex impedance method is widely used to determine the resistance of the sample. The principle of the method is based on measurements of cell impedance, which are taken over a wide range of frequency and then analyzed in the complex impedance plane which is useful for determining the appropriate equivalent circuits for a system and for estimating the values of the circuit parameters. Impedance is nothing but the a.c. resistance of the cell. The value in general contains the real and the imaginary part. An electrochemical cell, in general, exhibits resistive, capacitive, as well as inductive properties. The resistive property contributes to the real part of the impedance, while the capacitive and the inductive properties contribute to the imaginary part of the impedance. Therefore, an electrochemical cell can be considered as a network of resistor, capacitor, as well as conductor. Which arrangement for which cell is usually determined after performing a measurement, by analyzing the form of impedance curve. A capacitor that presents as an open circuit in a d.c. network and an inductor that appears as a straight conductor wire in a d.c. circuit, both appear as imaginary resistors in the a.c. circuit. Until presently, the inductive properties of the electrochemical cell are ignored so that the polymer electrolyte composite is considered only as a network of resistor and capacitor. The complex impedance can be written in a general form as

ωmC1R = 1 • R

C1

•

Z′ [Ω]

Figure 8. Examples of simple RC circuits and the corresponding impedance (Nyquist) plots.

value at this very high frequency equals to resistance. Thus the Nyquist plot for this arrangement appears as a vertical straight line, starting from a lower frequency value at the upper part downwards when the frequency increases, as shown in Figure 8b right. The intersection of this line with horizontal axis (the real value of impedance) corresponds to the resistance. For a parallel arrangement of resistor R and capacitance C, as appears in Figure 8c left, the real and imaginary parts of the impedance are given by Z =

R 1 +  RC2

(11a)

and Z  = R

RC 1 +  RC2

(11b)

and the corresponding Nyquist plot appears in Figure 8c right. The Nyquist plot appears as an arc. The intersection of this arc with the vertical axis at a low frequency (right arc) corresponds to the resistance. The frequency at the peak of the arc, m , satisfies the relation m RC

=1

(12)

From the intersection point at the low frequency region and the position of the arc peak, the resistance and the capacitance of the system can be determined.

740

Polymer Electrolyte Nanocomposites

More complex arrangements, have a more complex expression for the impedance. For example, a combination of serial and parallel circuit as appears in Figure 8d left has the impedance as −1
1 1 + (13) + i C1 Z= R i C2 with the corresponding Nyquist plot appearing in Figure 8d right. It contains a vertical line that intersects the horizontal axis at Z  = R, and an arc with the peak satisfies m RC1 = 1. Again, from these two values, one can determine R and C1 . The value of C2 is determined by measuring the vertical component of impedance at a certain frequency, say ∗ . If  the vertical component of impedance at this point is Z ∗ , the value of C2 satisfies Z

 ∗

=

1 ∗C 2

(14)

Sometimes, the form of curve is not as simple as that described here. However, in principle, we can find some circuit arrangement such that the theoretical Nyquist plot is in agreement with the measured data. Some computer software is commercially available for extracting the equivalent circuit for measured data. As an example, the impedance measurement of a system of PEO:LiCF3 SO3 containing Li14 Al04 Ge16 (PO4 3 fillers is displayed in Figure 9a [67]. The corresponding a.c. circuit that can produce this impedance data appears in Figure 9b, with Rb = bulk resistance, Rpb = phase boundary resistance, Rint = interfacial resistance, Cpb = phase boundary capacity, Cint = interfacial capacity, Zd = diffusive impedance. The corresponding parameter values that can properly fit the measured data are

Z′′ [kΩ]

=

1 $ Re A

(15)

where Re = resistance of polymer electrolyte, $ = material thickness (electrode spacing), and A = material cross section. The common procedure for measuring the temperature dependence of conductivity is as follows. (a) Heat the sample at a required temperature. Sometimes, it needs a half hour or more to equilibrate the sample temperature. (b) Measure the impedance at all range of frequency. Sometimes, it can take from tens of mHz up to several MHz. The computerized measurement is usually performed since a great number of data should be collected for each setting temperature. (c) Change the setting temperature, and again collect the impedance data in all frequency regions. (d) Analyze the collected data and find the equivalent circuit. (e) Determine the resistance of the electrolytes based on the impedance plot and the equivalent circuit at each setting temperature. (f) Calculate the conductivity at each setting temperature.

6.2. a.c. Conductivity

0.6

Despite the d.c. conductivity, the a.c. conductivity sometimes gives important information such as the dielectric properties of the composites. The frequency dependence of a.c. conductivity in polymer electrolytes can be written as [38]

0.4

ac   = dc + A

0.2

0

Rb = 593 ", Rpb = 1637 ", Cpb = 31 nF, and Cint = 19 #F [67]. From the measured resistance of polymer electrolytes, the electrical conductivity can be calculated using a simple equation

0

0.5

1.0

1.5

2.0

2.5

3.0

Z′ [kΩ] Rpb

Rint

Rb

Zd

Cpb

(16)

where dc = d.c. conductivity, A and n are the material parameters, 0 < n < 1, and is an angular frequency. The curve might consist of three regions, a spike at low frequency, followed by a plateau at medium frequency, and another spike at high frequency. The high frequency part corresponds to bulk relaxation phenomena, while the plateau region is connected to d.c. part of conductivity. The lower spike is connected to electrode/electrolyte phenomena. Fitting the curve with Eq. (16), one can determine the parameters A and n, and from those parameters, the hopping frequency [68],

Cint

Figure 9. (Top) Impedance plot PEO:LiCF3 SO3 containing Li14 Al04 Ge16 (PO4 3 obtained from experiment. Data points were extracted from [67], C. J. Leo et al., Solid State Ionics 148, 159 (2002). (Bottom) The suggested RC circuit for data in (a). See text for the explanation of symbols.

n

p

=

  1/n dc

A

(17)

By fitting the experimental data of ac − , one can determine the dc and p at each temperature. Using this approach, Siekierski et al. [69] found in a system of

741

Polymer Electrolyte Nanocomposites p

–7.2

(18)

where C is either dc or p , and Co is the corresponding prefactor. Furthermore, the temperature-dependent dielectric constant can also be obtained from the ac data. The real part of the dielectric constant, , , can be expressed as   , = ac ,o

  , s 2 ds P 2 2 0 s −

2   s, s ds P 2 2 0 s −

(20)

(21)

6.3. Diffusion Coefficient Electrical conductivity can also be determined by measuring the diffusion coefficient. From the temperature-dependent diffusion coefficient, the temperature dependence of electrical conductivity can be determined using Nerst–Einstein equation =

ne2 D kT

- 2 Ds t + A1 L2

–7.8

0.0

0.5

1.0

(23)

where 1 = measured cell potential, Ds = salt diffusion coefficient, L = electrolyte thickness, t = time, and A1 = a constant. It appears that Ds is proportional to the slope of curve ln 1 with respect to t. The dependence of salt diffusion constant on the salt concentration is displayed in Figure 10 [71] for system of PEO:NaCF3 SO3 at 83  C. The Ds decreases

1.5

2.0

2.5

Salt concentration [mol/L]

Figure 10. Effect of salt concentration on the diffusion coefficient for PEO:NaCF3 SO3 system at 83  C. Data points were extracted from [71], Y. Ma et al., J. Electrochem. Soc. 142, 1859 (1995).

as the salt concentration increases from about 8 ×10−8 cm s for dilute solution. Diffusion coefficient can also be determined from the Nyquist plot as discussed by Strauss et al. [72]. The medium frequency arc is attributed to the solid/electrolyte interface. At lower frequencies, the impedance is affected by concentration gradient (diffusion) and ionic aggregates. The diffusion impedance of symmetric electrolyte with no-blocking electrode, such as Li/CPE/Li, can be written as D=

RT L n2 F 2 Cb ZDC

(24)

where R = the gas constant, n = ratio of EO/cations, F = Faraday number, Cb = bulk concentration of cation, T = temperature, and L = electrolyte thickness. Lorimer also introduced another formula for calculating the diffusion constant, that is [73],

(22)

where n = charge carrier concentration e = electron charge, and D = diffusion coefficient. Salt diffusion coefficient can be obtained by galvanostatically polarizing a symmetric cell containing no-blocking electrode for a short period of time. For example, assume a cell containing Li-based polymer electrolytes and lithium electrodes at both sides. When the current is turned off, the induced concentration profile is allowed to relax. At long time after the current interrupt, the following equation is applicable [71]: ln 1 =

–7.6

–8.0

where P denotes the principal part of the integral [70]. On the other hand, if the imaginary part has been known, the real part can be determined using a relation ,   =

–7.4

(19)

 is the imaginary part of the a.c. conducttivity, where ac and ,o is the permitivity of vacuum. The complex dielectric constant can be written as ,  = ,   + i,  , and the imaginary part can be obtained from the real part using a Kramer–Kronig relation

,   = −

–7.0

satisfy the

Log Ds [cm2/s]

PEO3 :LiClO4 + -Al2 O3 , that both dc and Arrhenius expression   E C = Co exp − kT

D=

mL

2

254

(25)

where m = the frequency at the maxima of low frequency arc, and L = electrolyte thickness. The values predicted by Eq. (25), however, are around 6–10 times as large as that predicted by Eq. (14). The error can be contributed by the shift of m due to the formation of ion pairs [72].

6.4. Transference Number Transference number can be calculated by analyzing the arc impedance spectrum of symmetrical cell with no blocking electrode. The transference number can be calculated by comparing the width of the skew low frequency semicircle, Zd , with the value of the bulk resistance, that is [74], t+ =

1 1 + Zd /Rb

(26)

Transference number can also be determined by measurement of the electrochemical potential of the cell as illustrated in Figure 11 [75]. Suppose the polymer composite is

742

Polymer Electrolyte Nanocomposites Electrode 1

Table 4. Transference number of several composites.

Electrode 2

Transference number Temperature Ref.

Composites

µ1

CPE

µ2

Figure 11. A simple experiment for determining the transference number to electrodes with differential chemical potentials.

sandwiched between two electrodes with different chemical potential #1 and #2 . The electrochemical potential across this cell is given by 1  #2 1 E= t d# = (27) t # − #1  z F #1 i z F i 2 where z = absolute value of the valence of the mobile ion in the electrolyte; and F = Faraday number. For pure ionic composite, ti = 1 so that Epure = #2 − #1  z −1 F −1

(28)

E = ti Epure

(29)

Thus By measuring E and calculating Epure , we can obtain ti . Another method based on a combination of d.c. polarization and a.c. impedance has been introduced by Evans et al. This method involves measuring the resistance and current across a symmetrical Li/electrolyte/Li cell polarized by a d.c. voltage [76]. The t + is given by t+ =

IS V − Io Ro  Io V − IS RS 

(30)

where V = d.c. voltage applied to the cell, Ro = initial resistance of the passivating layer, RS = steady-state resistance of the passivating layer, Io = initial current, and IS = steadystate current. The d.c. polarization potential usually used is several tens of millivolts. This equation is applicable for ideal, dilute solutions. However, Doyle and Newman state that although this equation is not strictly applicable in concentrated electrolytes, the ratio of steady-state to initial current provides useful information on the contribution by organic additives to the ionic conductivity of polymer electrolytes [77]. The simplification of Eq. (30) was also used, that is, t + = ISS /IO . However, significant errors resulted from neglect of kinetic resistances at the electrode/electrolyte interface [78]. Transference numbers of some composites appear in Table 4.

7. SPECTROSCOPIC CHARACTERIZATIONS 7.1. NMR Spectroscopy A moving ion would substantially modify the interaction of electromagnetic waves with matter. Investigating this interaction gives a better understanding of ion dynamics on

PEO:LiCF3 SO3 + -LiAlO2 (4 #m) PEO:LiBF4 + -LiAlO2 (4 #m) PEO:LiClO4 + TiO2 (13 nm) PEO:LiClO4 + Al2 O3 (6 nm) (PEO)30 LiClO4 (PEO)8 LiClO4 (PEO)8 LiClO4 + SiO2

0.29 0.26 0.5–0.6 0.31–0.33 0.18–0.19 0.19–0.20 0.22–0.23

90 90 90 90 100 90 100



C C  C  C  C  C  C 

[79] [79] [79] [80] [80] [80] [80]

a microscopic scale. An example of method for studying the ion dynamics is nuclear magnetic resonance (NMR) spectroscopy. This method probes the spin of ion using an electromagnetic wave in radio frequency. In amorphous single-phase polymer electrolytes, there is usually found a straight relationship between polymer segmental motion and ionic mobility by observing a strong correlation between the onset of NMR line-narrowing and the glass transition [81]. NMR has contributed significantly to the understanding of the physical properties of the composite polymer electrolytes mainly because it offers the possibility to selectively study the ionic and polymer chain dynamics. For example, measurement of the temperature dependence of 7 Li lineshapes and spin-lattice relaxation allows the determination of the activation energy and the correlation time of the cation motion. Gang et al. described the 7 Li line-narrowing in the composite of PEO:LiBF4 + -LiAlO2 (10–30 wt%) in the temperature range of 270–270 K [82]. Dai et al. reported wide line and high resolution solid-state 7 Li NMR [83]. In material, each spin interacts with other spins, giving rise to spin-spin interaction or relaxation time T2 . Furthermore, a new thermal equilibrium distribution of the spin, which has to be mediated through lattice, is forced by the magnetic field. The characterization time required for the excess energy to be given to the lattice or for attainment of new thermal equilibrium is expressed in terms of spinlattice or thermal relaxation time T1 . Under simultaneous application of static and radio frequency magnetic field in perpendicular direction, Hz = Ho

(31a)

Hx = 2H1 cos t

(31b)

The interaction of this magnetic field with the nuclear spin results in the Bloch susceptibility [84] 9 =

1 9 2 o

o T2

9  =

1 9 2 o

o T2

1 + T23 

T2  o −  2 2 2 o −  +  H 1 T 1 T2

(32a)

1 + T22 

1 − 2 +  2 H12 T1 T2 o

(32b)

where o = Ho , and  is the gyromagnetic ratio of the spin. For low RF field H1 ,  2 H12 T1 T2 1, then Eqs. (32a) and (32b) give the familiar absorption curve with half-width  =

o

−

 1/T2

(33)

However, the exact lineshape and linewidth can be determined using Van Vleck method of moment. This method

743

Polymer Electrolyte Nanocomposites

allows the connection of the absorption line with the motional behavior of the nuclei. The second moment, M2 is given by   − o 2 f   d M2 = −

=

 1 − 3 cos