+
-+ +
Electrolytic Solution
The surrounding ionsDebye-Hückel Model
give rise to:
Excess charge density - ρr
Reference Ion Point Charge Medium
dielectric constant - ε The solvent molecules
provide a:
Fig. (2.1.3) The Debye-Hückel model upon selecting one ion as a point charge reference ion, replacing the solvent molecules the solvent molecules by a continuous medium of dielectric constant ε and the remaining ions by an excess charge density [5]
2.1.3.
ebye-Hückel Theory – Review and Limitations
he success of the Debye-Hückel model for ion-ion interaction of solute in solvents is so remarkable and the implications so wide that ebye-Hückel approach is regarded as one of the most significant pieces of theory in part of electrochemistry. It even rates
mong the leading pieces of physical chemistry of the first half of the 20th century. However ne must not overstress the achievement of the Debye-Hückel limiting law as seen in Equ.
.1.1]:
D
T
D the ionics a
o [2
( )
12log f± =−A z+z− I Equ. [2.1.1]
This expression indicates that the activity coefficient must decrease linearly with the square root of the ionic strength or, in
f±
the case of 1:1 valent electrolytes withc12 s c
f±
. Models are always simplifications of reality. They never treat all its complexities and thu an never be a perfect fit between experiments and prediction based on a model. The inadequacies of the Debye-Hückel limiting law are exposed. If one examines the experimental log versus I this is a curve Fig. (2.1.4) [5] not just in extreme regions, but also at higher concentrations, it turns out that the simple Debye-Hückel limiting law fails. The plot log f±versus I is not a straight line as promised by Equ. [2.1.1]. Furthermore, the curves depend not only on valence type (1:1 or 2:2) but also on the particular electrolyte (NaCl or KCl). The agreement of the
versus
Debye-Hückel limiting law with experiment improved with decreasing concentration and became excellent for the limiting tangent to the log f± I as seen in Fig. (2.1.4). With increasing concentration, however the experimental data deviated more and more from theoretical values as the concentrations continued to proceed above 1 N, there was an increase in with an increase in concentration, whereas theory indicated a continued decrease.
Because the Debye-Hückel theory attempts to represent all these various aspects of real situations within solutions, the experimentally calibrated ion size parameter varies with concentration. The Debye-Hückel theory is an outstanding piece of work in itself; but at the same time, it is unfortunately inadequate in explaining and modelling some aspects of the behaviours of ion-polymer interactions. Such questions were tackled at the beginning of the
f±
second half of the 20th century by Flory and Huggins [7 - 8] and modified aspects of Debye-m at around a siDebye-milar tiDebye-me.
Hückel theory where also intergraded by Guggenhei
0.2 0.4 0.6
0.2 1.0 1 2. 1.4 1.6
- 0.6 - 0.4
- 0.8 - 0.2 0.0
I f
±log
Limiting Law Equation 3
Experimental Curve
Fig. (2.1.4) The experimental log f+/- . vs. I1/2 curve is a straight line only at extremely low concentrations [5]
2.1.4.
The Conductivity Mechanism:
To this point, we have only discussed the relationship of ions within dilute liquid systems and we have only vaguely referred to solid-polymeric electrolyte systems. The mechanism; to how charge is transported ch systems has largely remained overlooked. The model most commonly encountered for conductivity in polymer electrolytes is the free volume theory. The conductivity of s is often expressed by the Vogel-Tamman-Fulcher (VTF), which will be discussed later in section {2.3}. The experimental results for poly(ethylene oxide) (PEO) ne
polymeric investigations a linear plot for the conductivity versus (T-T0)-1 [9]. The temperature dependence of conductivity of (PEO) systems many be attributed to its excellent solvating character which implies that the X- anion has a lesser tendency to ion-paring with complexed M+ cations. As was previously mentioned in Section {1} will be developed in later sections; a comprehensive theoretical approach to explaining the formation of ion-paring and the mechanism of conductivity in polymer electrolytes has not been definitively established and is still in many scientific circles hotly debated.
In the instance of (PEO) systems at low salt concentrations, at a constant temperature a linear relationship of conductivity and salt concentration is observed. Nevertheless, this simple behaviour is not observed above the “critical” salt concentration. This phenomenon has been explained in the past through the proposal that salt molecules are associated in quantities in which dipoles (ion pairs), ion triplets; free ions, and large ionic aggregates are assumed to form at high salt concentrations. It was suggested by Torell [10], that the molecules tend to associate forming “multiplets of higher order”. The formation of larger aggregates such as the indicated triple ion species, are one of a number of aspects of these systems as one increases the salt concentration successively. In parallel to the increase in salt concentration, there is a corresponding drop in conductivity and rise in the viscosity of the polymeric test material, whereby this situation is related to the presence of “transient” or ionic cross-linking and an associated increase in Tg values [11].
within su such system
tworks containing for example sodium tetraphenylborate exhibited in early
The evidence of formation of such higher ionic aggregates can be determined with of transport numbers1. The modern tools in etermining the existence of such pairs through the interpretation of transport numbers are made possible by the use of NMR and steady-state current methods (see section {2.2.4}). The
ntrib
e Hittorf ell [14 - 16] the transfer of anions, which would normally occur between cathode and anode ompar
various spectroscopic methods and in the form d
co ution of the species to the overall conductivity of system is still hotly debated. For example, the author Angell [12] maintains in (PEO) systems, that the cations are mobile charge carrying species of the system and other authors argue the contra giving the transport number of the cation in such systems as zero. The usual method for the acquisition of transference numbers2 in polymer electrolytes is by means of pulsed gradient NMR and the steady-state current methods. The later method only functions well enough when the system under investigation is completely dissociated and other conditions are fulfilled. Therefore the method is very limited and system specific. It is difficult to determine the nature of the information produced by the steady-state method. Since quantitive, analysis of the non-equilibrium species is extremely complex and difficult to attain. The classical example proposed by Bruce & Vincent [13] which will be discussed in the next section {2.2} is the method for cation transport number in a fully dissociated salt, which can be calculated from the ratio of the steady state current, where a uniform concentration gradient extends right across the cell from cathode to anode. In three compartments known by design as th
c
c tments is prevented by diffusion down the concentration gradient. It appears that most transport numbers, which are widely quoted in the field of polymer electrolytes are not classical quantities according to Cameron & Ingram [17]. They refer to the net migration of charged species caused by the application of an electric field. The steady-state current method is more informative for many practical purposes since it simulates accurately the conditions in a electrochemical cell where the concentration gradient is generated.
1 If potential is applied to an electrolyte and the current measured, the transport number t of any charge species is the proportion of the overall electrolyte conductivity due to the species. The sum of the transport numbers for all species present is equal to unity.
2 If potential is applied to an electrolyte and the current measured, a transference number T refers to the proportion of current carried by a constituent of the salt present.