Toward Establishing Uniqueness of Experimentally Determined Transference Numbers

The passage of current through a battery results in the development of concentration gradients in the electrolytic phase. For a fully characterized binary electrolyte, where the conductivity, salt diffusion coefficient, cation transference number, and the thermodynamic factor are known, concentration and potential gradients in the electrolytic phase can be modeled using Newman’s concentrated solution theory. We report two methods for measuring the transference number: the standard method based on electrochemical measurements ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">echem</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ) and electrophoretic NMR ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">eNMR</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ). The electrochemical approach requires combining measurements from multiple experiments; the equations used to determine the cation transference number and the thermodynamic factor are coupled, nonlinear algebraic equations. In the electrophoretic-NMR-based approach, however, the equations used to determine the cation transference number and the thermodynamic factor are decoupled. We find for a liquid electrolyte comprised of a lithium salt dissolved in tetraglyme, the values of the transference numbers obtained by these two methods are distinct. For example, at 30 °C, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">echem</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> = −1.02 ± 1.11 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">eNMR</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> = 0.25 ± 0.04. The corresponding thermodynamic factors are also different. While the magnitude of the predicted concentration gradients based on the two sets of parameters are different, the predicted current-voltage relationships are similar.