Effects of pressure, temperature and volume on the electrical conductivity of polymer electrolytes

Electrochimica Acta 57 (2011) 160–164

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Effects of pressure, temperature and volume on the electrical conductivity of polymer electrolytes J.T. Bendler a , J.J. Fontanella b,∗ , M.F. Shlesinger b,c , M.C. Wintersgill b a b c

BSC, Inc., 3046 Player Drive, Rapid City, SD 57702, USA Physics Department, U.S. Naval Academy, Annapolis, MD 21402, USA Office of Naval Research, ONR Code 30, 875 N. Randolph St., Arlington VA 22203, USA

a r t i c l e

i n f o

Article history: Received 9 December 2010 Received in revised form 27 March 2011 Accepted 4 July 2011 Available online 13 July 2011 Keywords: Polymer electrolytes Electrical conductivity High pressure Scaling Defect diffusion model

a b s t r a c t Three features of the electrical conductivity of ion conducting polymers are described. First, it is shown how the ratio of the apparent isochoric activation energy to the isobaric activation enthalpy, EV∗ /H ∗ , can be used to separate the contributions of volume and temperature to the electrical conductivity. Next, the effect of pressure on the conductivity is considered. Finally, the concept of scaling is discussed and is applied to poly(propylene glycol) containing LiCF3 SO3 . All results are interpreted in terms of the defect diffusion model (DDM). First, it is shown how EV∗ /H ∗ can be calculated using the DDM. Next, the effect of pressure on the conductivity is given a physical interpretation. Finally, it is pointed out that in the DDM, scaling arises because the critical temperature can be represented approximately by a power law. Consequently, in the DDM scaling is always approximate. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The effects of pressure, temperature and volume on the electrical relaxation time associated with the glass transition for glass-forming liquids have received a great deal of attention recently [1–5]. One aspect of this work that has been considered in some detail is separation of pure temperature effects from volume effects via the ratio of the apparent isochoric activation energy, EV or EV∗ , to the isobaric activation enthalpy, EP or H* . A second aspect of the recent work that is of interest is the concept of “thermodynamic” scaling [5–7]. In recent papers, the authors have applied the defect diffusion model (DDM) of the relaxation time to both of these problems [8–10]. In the present paper, both the ratio (EV∗ /H ∗ ) and scaling are evaluated for the electrical conductivity of two common polymer electrolytes. These results along with a separate consideration of the effect of pressure on the conductivity are discussed in terms of the DDM.

∗ Corresponding author. Tel.: +1 410 293 5507; fax: +1 410 293 5508. E-mail addresses: [email protected] (J.T. Bendler), [email protected], [email protected] (J.J. Fontanella), [email protected] (M.F. Shlesinger), [email protected], [email protected] (M.C. Wintersgill). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.07.010

2. EV∗ /H ∗ The apparent isochoric activation energy is defined by

 EV =

EV∗

=R

∂ ln  ∂(1/T )

 (1) V

where  is the electrical conductivity. Next, the isobaric activation enthalpy is given by

 ∗

EP = H = R

∂ ln  ∂(1/T )

 (2) P

so that the ratio can be written EV∗

H∗

=

((∂ ln )/(∂T ))V ((∂ ln )/(∂T ))V = ((∂ ln )/(∂T ))P ((∂ ln )/(∂T ))P

(3)

where  is the electrical resistivity ( = 1/). While the quantity given by Eq. (2) can be calculated directly from experimental data for the variation of  or  with temperature (assuming a constant pressure experiment), the evaluation of the constant volume temperature variation (Eq. (1)) requires the variation with pressure for its evaluation. Specifically,



∂ ln  ∂T



 = V

∂ ln  ∂T

 + P

˛P T



∂ ln  ∂P

 (4) T

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161

In Eq. (7) c is the total concentration of defects (i.e., the fraction of lattice sites occupied by a defect),  o is the shortest time for the jump of a defect, and TC is the critical temperature at which all defects would be clustered if the glass transition did not intervene. The critical temperature, TC , depends on pressure, i.e. TC = TC (P). Eq. (7) ignores the fact that jumps of the defects are thermally activated [8,9,15]. (1 − ı) = V(T,P)/Vo where V(T,P) is the volume of the liquid at pressure P, and absolute temperature T, and Vo is a reference volume at P = 0 at a given reference temperature.  is an integer (1, 2 or 3) describing the dimensionality of the correlation volume [16]. Finally, B∗ = −

Fig. 1. Experimental results (points) and DDM predictions (curves) for EV∗ /H ∗ vs. T − Tg for various materials.

where T is the isothermal compressibility given by 1 T = − V



∂V ∂P



(5) T

˛P =

1 V

∂V ∂T



.

(6)

P

Results for the pressure and temperature variation of the electrical conductivity for poly(propylene glycol) (PPG) of average molecular weight 1025 containing LiCF3 SO3 in the ratio of 20:1 repeat units per lithium are given elsewhere along with pressure–volume–temperature (PVT) data [11,12]. Those results were used to calculate the ratio given by Eq. (3) and the results are shown by the solid circles in Fig. 1. Next, results for the pressure and temperature dependence of the electrical conductivity of poly(ethylene glycol mono-methyl ether) (hydroxyl group at one end and a methyl ether group at the other end of each chain) of average molecular weight 350 (PEGMME 350 will be referred to in this paper simply as PEG) containing NaCF3 SO3 also in the ratio 20:1 are given elsewhere [13]. However, PVT data for that material do not seem to be available. Consequently, the PVT data for PEG:NaCF3 SO3 were approximated by the values for pure PEGMME 350 [14]. Those results were also used to calculate the ratio given by Eq. (3) and the results are shown by the open circles in Fig. 1. The experimental values for the ratio EV∗ /H ∗ are relatively high for these two polymers, being on the order of 0.7–0.75. This shows that the dynamics of ionic conductivity is controlled mostly by temperature. In fact, it places these materials among the most temperature sensitive polymers [5]. There is, of course, extra uncertainty in the results for PEG:NaCF3 SO3 because of the lack of good PVT data. The ratio given by Eq. (3) can also be calculated using the DDM because an equation for the electrical conductivity can be developed. Specifically, it has been shown that the DDM gives rise to the following equation for the electrical relaxation time [8,9,15]:



DDM ≈ c

−1/ˇKWW

o exp

(8)

do3 ˇKWW

where do is the average distance between neighboring sites at P = 0 and at a given reference temperature. The three indices on the “direct” correlation lengths Lj allow for anisotropic defect–defect interactions. If the interactions are isotropic, then L1 = L2 = L3 = L. ˇKWW is the stretched exponential parameter [17,18]. In the DDM for the electrical relaxation time, then, all of the adjustable parameters have a microscopic interpretation. The connection of Eq. (7) with the electrical conductivity is made via the diffusion constant, D, given by D=

l2 6

(9)

where l is the jump distance and the Nernst–Einstein equation

and ˛P is the isobaric volume thermal expansion coefficient given by



L1 L2 L3 ln(1 − c)

0.5

B∗ TC

(T − TC )0.5 (1 − ı)

=

q2 nl2 q2 nD = kB T 6kB T

(10)

where q is the charge associated with the charge carrier, n is the number of charge carriers per unit volume and kB is Boltzmann’s constant. These equations result in

DDM

q2 nl2 c 1/ˇKWW = exp 6kB To

(7)



0.5

−B∗ TC

(T − TC )0.5 (1 − ı)

.

(11)

In order to apply Eq. (11) to data, the following approximations were used. First, it was assumed that n, c, ˇKWW and  o are constant. In fact, each has some temperature dependence. For example, it is known that n has a small temperature dependence. For example, n often decreases by about a factor of 2 over the temperature range of typical experiments [19] and the value is thermally activated. In addition, studies have shown that n increases with pressure [20] by about a factor of 2 over the pressure range of typical experiments. However, these changes are small compared with the factor of 1010 change of the conductivity exhibited by the main subject of the present work, PPG:LiCF3 SO3 . The constancy of c is an approximation that is usually made when applying the DDM, though that is not a necessary feature of the model. Next, it has been shown that, to within experimental uncertainty, ˇKWW is constant for PPG-based materials [21,22]. However, for other materials, such as glycerol, ˇKWW does not appear to be constant [23]. Finally, as has been pointed out, it is expected that  o is also thermally activated [8,9,15]. The next assumptions are that l and n vary with the dimensions of the lattice. Specifically, it is assumed that l = lo (1 − ı)1/3 and n = no /(1 − ı). Consequently, the DDM equation for the electrical conductivity of polymer electrolytes is written as



.



 DDM =

A T (1 − ı)

1/3

exp

0.5

−B∗ TC

(T − TC )0.5 (1 − ı)

 (12)

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Eq. (12) can then be used to evaluate the ratio given by Eq. (3). Switching to the resistivity representation, the constant pressure temperature derivative is



∂ ln DDM ∂T



0.5

= P

B∗ TC [˛P + ((0.5)/(T − TC ))] 1 ˛P + − T 3 (T − TC )0.5 (1 − ı)

(13)

Next, the constant volume temperature variation can be written:



∂ ln DDM ∂T



V 0.5

0.5B∗ TC 1 = + T





(1 − ı) (T/TC )(˛P /T )((∂ ln TC )/(∂P))T − 1 (T − TC )0.5

(14)

The quantities 1/T and ˛P /3 are small compared with the last term in both Eqs. (13) and (14). Consequently, the DDM prediction for the ratio is



EV∗ H∗



= DDM

1 − (˛P /T )(T/TC )(∂TC /∂P)T . 1 + ((˛P (T − TC ))/(0.5))

(15)

All quantities in Eq. (15) except ˛P and T for PEG:NaCF3 SO3 are available. The values of TC and its pressure derivative and the PVT data are taken from Ref. [11] for PPG:LiCF3 SO3 . The values of TC and its pressure derivative for PEG:NaCF3 SO3 are taken from Ref. [13]. Again, the PVT data for PEG:NaCF3 SO3 were approximated by the values for pure PEGMME 350 [14]. The DDM results for EV∗ /H ∗ vs. temperature for PPG:LiCF3 SO3 and PEG:NaCF3 SO3 calculated using Eq. (15) are given by the curves in Fig. 1. There is good agreement between the experimental and DDM values at the lowest temperatures for both PPG:LiCF3 SO3 and PEG:NaCF3 SO3 . The agreement between the experimental and DDM values is not as satisfactory at the highest temperatures. This is expected because, in general, a change in dynamics occurs at about 1.3 Tg . (That temperature is known as TB [24] or TLL [25].) This has been shown to be the case for PPG:LiCF3 SO3 [16]. Specifically, in Ref. [16], it was shown that a liquid–liquid transition occurs at about 290 K or 1.33 Tg in that material. Because Tg = 218 K for PPG:LiCF3 SO3 the deviation should begin at about T − Tg = 72 K as is observed. A similar analysis has not been carried out for PEG:NaCF3 SO3 because the data for that material are much less complete and are complicated because of crystallinity [13]. Consequently, different forms of the DDM are necessary above and below about 1.3 Tg . Interestingly, these values of EV∗ /H ∗ are somewhat larger than the values of 0.67 and 0.55 quoted for poly(propylene glycol) and poly(propylene oxide), respectively, determined using the dielectric relaxation time for the ˛ relaxation [5]. It is tempting to attribute that difference to a difference in materials because the dielectric studies were carried out on materials that do not contain salt. However, the difference is probably more fundamental than that. For example, it has been shown that the Vogel parameters, themselves, are different for dielectric relaxation time, viscosity and conductivity for experiments carried out on the same material [15,24]. This has a natural explanation via the DDM as it is expected that different physical phenomena require different defects. Also included in Fig. 1 for comparison are the DDM predictions for EV∗ /H ∗ vs. temperature for poly(vinyl acetate) (PVAc) and polycarbonate determined using dielectric relaxation data for the ˛ relaxation [8]. Clearly, those common materials are much less temperature sensitive than are PPG:LiCF3 SO3 or PEG:NaCF3 SO3 . In fact, the effect of temperature on dynamics for PPG:LiCF3 SO3 and PEG:NaCF3 SO3 , as measured via conductivity, places them among the most temperature sensitive of materials. A physical explanation for this behavior will be given in the next section.

3. Pressure variation of the conductivity The pressure dependence of the electrical conductivity for polymer electrolytes has been addressed previously [16,26]. Both groups show that the change of conductivity with pressure is about twice as large for PPG- than for PEG-based materials. In addition, both show an approximately linear dependence between the pressure and temperature derivatives. Though the representations are different, they are mathematically equivalent. Specifically, Ingram et al. treat the data in terms of an “instantaneous” activation energy and volume given by



EA = −R and

∂ ln  ∂T −1



VA = −RT



(16) P

∂ ln  ∂P

 .

(17)

T

On the other hand, Bendler et al. plot and discuss only (∂ln/∂lnT)P and (∂ln/∂P)T [16]. However, Eq. (16) can be rewritten as



EA = RT 2

∂ ln  ∂T





= RT P

∂ ln  ∂ ln T



.

(18)

P

Consequently, (∂ln/∂lnT)P and (∂ln/∂P)T differ from EA and VA only by a factor of RT. Bendler et al. [16] discuss these results in terms of the DDM. Specifically, they show that



∂ ln DDM ∂P



≈− T

∂ ln TC ∂P



∂ ln DDM ∂ ln T





+1

(19)

P

This explained the linear dependence of a plot of (∂ln/∂lnT)P vs. (∂ln/∂P)T . It also showed why the change of conductivity with pressure is about twice as large for PPG- than for PEG-based materials because it has been found that (∂ln/∂P)T = 0.68 GPa−1 for PPG:LiCF3 SO3 [11] and 0.35 GPa−1 for PEG:NaCF3 SO3 [13]. (These values were also used in the calculations of the ratio EV∗ /H ∗ via Eq. (15).) Recent considerations make it possible to gain physical insight into the reason for the difference between these materials [8,9]. That is obtained by identifying TC with the critical demixing temperature associated with a simple Bragg–Williams (defect) phase separation transition (for nearest–neighbor pair interactions with equal occupancy of A and B sites). The governing equation is [27]



TC =



z h

4kB



=



z( ε + Pv ) 4kB

(20)

where z is the lattice coordination number, kB is Boltzmann’s constant, and h = ε + Pv is the decrease in enthalpy resulting from the formation of a defect pair: defect + defect ↔ (defect)2

(21)

where ε is the decrease in pair energy and v is the decrease in volume. From Eq. (20) it follows that



∂TC ∂P





≈ T



z v

(22)

4kB

so that the controlling molecular parameter is the net (negative) volume change, v, caused by defect pairing. Consequently, Eq. (19) can be rewritten as



∂ ln DDM ∂P



 

v

∂ ln DDM

+1

h

∂ ln T P

≈ −

T

(23)

J.T. Bendler et al. / Electrochimica Acta 57 (2011) 160–164

163

In the DDM, then, the pressure derivative of the electrical conductivity for PPG:LiCF3 SO3 and PEG:NaCF3 SO3 is governed by the volume change when defects cluster because TC (and hence h) is about the same (149 K vs. 140.5 K [11,13]) for the two materials. The conclusion is, then, that the volume change upon clustering is greater for the PPG- than for PEG-based materials. This makes physical sense because PPG contains flexible methyl side groups while PEG does not. The methyl groups generate backbone disorder, push the chains apart and bend under stress. This flexible structure allows for the large volume changes that must occur upon clustering for the PPG-based materials. Using this approach, additional insight can also be obtained into the ratio EV∗ /H ∗ . Substituting Eqs. (20) and (22) into Eq. (15) gives



EV∗

H∗





= DDM



1 − T (˛P /T ) (v/h)

1 + ((˛P (T − TC ))/(0.5))

(24)

.

At first sight, EV∗ /H ∗ should be much larger for PEG than for PPG because v is much smaller. However, as was shown previously in this note, EV∗ /H ∗ is about the same for both materials. The reason is the ratio ˛P /T . It is shown below that this ratio is much larger for PEG than for PPG effectively canceling the difference in v/h for the two materials. The physical reason for this is as follows. The ratio ˛P /T describes the dilatation of the lattice and hence the distance between defects. For example, if the temperature increases, the defect–defect distance increases. As a consequence, there would be less interaction between the defects and consequently there would be less tendency for the clustered defects to break apart i.e. the effect of temperature would be less. This would be reflected in smaller values of EV∗ /H ∗ . The opposite is true for the compressibility. Further, it is not surprising that the ratio ˛P /T is larger for PEG. It does not contain flexible side groups so it is expected that the inter-chain interactions should be larger than those for PPG. Consequently, it is not surprising that ˛P is greater for PEG (at about 295 K, ˛P is about 7.9 × 10−4 K−1 vs. 6.5 × 10−4 K−1 for PPG:LiCF3 SO3 ) and that T is smaller (at about 295 K, T is about 0.4 GPa−1 vs. 0.5 GPa−1 for PPG:LiCF3 SO3 ). 4. Scaling

Fig. 2. Plot of log() vs. 1000/(TV2.65 ) at various temperatures and pressures for PPG:LiCF3 SO3 . The symbols are the same as those for the data in Ref. [11]: the open hexagons represent the zero pressure data at a variety of temperatures. The remaining data are isothermal at various pressures and the symbols are: open circles – 268 K; solid circles – 273 K; open triangles – 285 K; solid triangles – 295 K; open diamonds – 308 K; solid diamonds – 323 K; open squares – 343 K; solid squares – 363 K.

behavior must be caused by the denominator of the last term in Eq. (25). In fact, in the DDM, approximate scaling occurs if TC (P) is given by TC (P) ≈ k

 V −ϕ

(26)

Vo

where k is a constant [9]. In that case, Eq. (25) becomes log10 (DDM ) ≈ log10 A − log10 T − log10

Another experimental observation treated in the present work is known as scaling. Specifically, it has been found that for many glass-forming liquids ln() scales at least approximately with 1/(T [V/Vo ] ) where  is the electrical relaxation time associated with the ˛ relaxation [5–7]. The assumption is that the constant (scaling parameter)  can be adjusted so that the data form a single curve, hence the term scaling. A review of the work until 2005 is given in Ref. [5] for electrical relaxation times. In order to evaluate the concept of scaling for a polymer electrolyte, the data for PPG:LiCF3 SO3 were re-analyzed. The conductivity data shown in Fig. 2 of Ref. [11] were re-plotted vs. 1/(T [V/Vo ] ) and the value of  was manually adjusted until a visual best-fit was achieved. The results are shown in Fig. 2 of the present paper. It is seen there that a reasonable degree of scaling was achieved. Because the DDM does an excellent job of reproducing the data [11], it also must account for the observed scaling. It is of interest to investigate the reason. Toward this end, Eq. (12) is rewritten as follows. log10 (DDM ) ≈ log10 A − log10 T − log10 −



V

0.434B∗ ((T/TC (P)) − 1)(V/Vo )

Vo



(25)

where, for simplicity, Vogel behavior ( = 2) has been assumed. Because at least approximate scaling exists in the DDM, the

 +

V

0.434B∗ k T (V/Vo )

ϕ+1

− k(V/Vo )

Vo



(27)

so that approximate scaling arises and it would not be surprising if  ≈ + 1. To show that Eq. (26) is approximately valid, TC (P) for PPG:LiCF3 SO3 was plotted vs. V/Vo and the results are shown in Fig. 3 for three temperatures. Also shown are the best-fit power law curves (Eq. (26)). Clearly, TC (P) can be reasonably well represented by Eq. (26). Also, the best-fit values of k and ϕ are 143 K, 150 K and 157 K and 1.74, 1.52 and 1.26 at temperatures of 268 K, 308 K and 363 K, respectively. This is about the same temperature range as the data. Interestingly, on average ϕ ≈ 1.5 so that ϕ + 1 ≈ 2.5. This is reasonably close to  ≈ 2.65. Of course, Eq. (27) contains T and V/Vo in several places so that the functional dependence is not predicted to be exactly (T(V/Vo ) +1 ). In addition, ϕ is not a constant. Thus, in the DDM scaling is approximate only. The same conclusion was reached in evaluating scaling in the context of dielectric relaxation [9]. Finally, it is noted that in the DDM, the most important contribution to the scaling parameter is how TC depends on volume. This suggests that scaling should be related to the ratio EV∗ /H ∗ because this ratio is a measure of the contribution of volume to conductivity. The relationship between scaling and volume is treated in detail elsewhere in the context of dielectric relaxation [9].

164

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Department of Defense-Army Research Office (Grant no. DAAD1901-1-0482). MFS would like to thank the Kinnear Chair of Physics for support. References

Fig. 3. Plot of TC vs. V/Vo for PPG:LiCF3 SO3 at 268 K, 308 K and 363 K. Also included are the best-fit power law curves (Eq. (26)).

5. Conclusion Three features of the electrical conductivity of ion conducting polymers have been discussed. First, it is shown that the ratio, EV∗ /H ∗ , is relatively large for PPG and PEG and that this implies that pure temperature effects on the electrical conductivity are dominant. Next, it is pointed out that the effect of pressure on the electrical conductivity of PPG-based materials is about twice as large as for PEG-based materials. Physical arguments explaining both EV∗ /H ∗ and the pressure effects are developed using the defect diffusion model. Finally, the concept of scaling is discussed and is applied to poly(propylene glycol) containing LiCF3 SO3 . It is found that, for this material, the DDM shows that scaling arises because the critical temperature can be represented approximately by a power law and that scaling, itself, is approximate. Acknowledgments This work was supported in part by the U.S. Office of Naval Research. JTB gratefully acknowledges financial support by the

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