Factors Affecting the Use of Impedance Spectroscopy in the Characterisation of the Freezing Stage of the Lyophilisation Process: the Impact of Liquid Fill Height in Relation to Electrode Geometry

Abstract

This study aims to investigate the application of impedance spectroscopy using fixed electrode geometries on a standard glass vial in the characterisation of the freezing process of solutions at different fill liquid volumes. Impedance spectra (between 10 and 10⁶ Hz) were recorded every 3 min, during the freezing cycle on a solution of 30 mg/mL sucrose contained within 10 mL glass vials having an electrode system (two thin copper foils: w, 18 mm; h, 5 mm) affixed to the external surface of the vial. A fill factor (Φ) was defined in terms of the relative height of the solution volume to the height of the electrodes from the base of the vial. Solution volumes of 1.5 to 5 mL (corresponding to Φ= 0.5–1.6) were investigated to establish the applicability of having a fixed electrode geometry for a range of solution volumes. A linear relationship between the time duration of the ice formation/solidification phase and the fill factor suggests that fixed electrode geometries may be used to investigate a range of fill volumes. The benefit of this approach is that it does not invade the solution and hence records the freezing process without providing additional nucleation sites and in a manner which is representative of the entire fill volume.

Appendices

Appendices

Appendix 1: Estimation of the Peak Frequencies of Liquid and Frozen Sucrose Solution (30 mg/mL) Within Tubing Vial of Different Vial Geometries

In a simple approximation, the impedance of the object under test can be described as a combination of resistor and capacitor; the resistance of which is defined by following equation

$$ R={K}_1d/{A}_{\mathrm{CS}}\sigma $$

(1)

where K ₁ is a geometrical coefficient, d is the internal diameter of the vial, A _CS is an area of effective vertical cross section of the sample (i.e. the solution within the vial) and σ is specific conductivity of the sample. The capacitance can be defined by the following equation.

$$ C={\varepsilon}_0\varepsilon \kern0.5em A/l $$

(2)

Where ε ₀ is the permittivity of a vacuum, ε is the dielectric constant of glass, A is the area of the electrodes and l is thickness of the glass wall. For the purpose of these calculations, the wall thickness is assumed to be constant for all sizes of vial.

Multiplying R by C, we obtain

$$ \tau = RC={K}_1{\varepsilon}_0\varepsilon dA/\sigma l{A}_{\mathrm{CS}} $$

(3)

where τ is the known as the time constant of the serial RC circuit. It is this time constant which defines the position of the interfacial relaxation peak in the experimental frequency window (where f _peak = 1/2πτ).

In the first approximation, A _CS can be presented as A _CS = AK ₂ where K ₂ is a constant coefficient (associated with the fixed cylindrical shape of the sample volume). Then Eq. (3) can be rewritten as

$$ \tau = RC={K}_1{\varepsilon}_0\varepsilon dA/\sigma lA{K}_2 $$

(4)

A in the numerator and denominator can be cancelled thus giving

$$ \tau = RC={K}_1{\varepsilon}_0\varepsilon d/\sigma l\ {K}_2 $$

(5)

For simplicity, let us denote

$$ {K}_p={K}_1{\varepsilon}_0\varepsilon / l\sigma\ {K}_2 $$

(6)

As all members in the right side of Eq. (6) are constants then K _p (the proportionality coefficient) is also a constant, and expression (5) can be simplified to

$$ \tau ={K}_pd $$

(7)

and respectively

$$ {f}_{\mathrm{peak}}=1/2\pi \tau =1/2\pi {K}_pd $$

(8)

Expression (8) shows that the frequency position of the peak has an approximately inverse dependence on the internal diameter of the vial.

Having measured the experimental value of f _peak for 10 mL (f _peak10 ml), it is then straight forward to estimate f _peak(xml) for different sized vials from the ratio of the diameters, according to the formula below.

$$ {f}_{\mathrm{peak}\mathrm{x}}={f}_{\mathrm{peak}\left(10\ \mathrm{ml}\right)}\times {d}_{10\ \mathrm{ml}}/{d}_{xml} $$

(9)

Table I gives theoretical estimates for the peak frequency for different sized vials, for both the liquid state and the frozen state.

Table I Estimated Position of Pseudo-relaxation Peak for a 30 mg/mL Solution of Sucrose in Distilled Water Within Glass Tubing Vials of Varying Diameter (but Constant Wall Thickness, 2 mm)

Appendix 2: Thermal Mass Contributions from the External Electrodes

The basis for these calculations is to first determine the position of the top of the guard electrode from the base of the vial (dimension A; Fig. 7) for a fill volume that provides a constant ratio of the liquid cross sectional area to the liquid fill height, for all sizes of vial and which is equal to that for a 3-mL fill volume in a 10-mL vial.

Fig. 7

Schematic of the electrode assembly. A electrode height, B length of guard electrode, C height of stimulating/sensing electrode, D width of stimulating/sensing electrode, E width of guard electrode, F spacing between guard electrode and sensing electrode, and G is the height of the side segment of the guard electrode

Once the position of the top of the guard electrode is defined, then the next step is to estimate the length of the sensing/stimulating electrode (dimension D; Fig. 7). This dimension is defined by the ratio of the electrode length to the circumference of the vial, which is fixed at 0.4 for all vials. All other dimensions, i.e. the separation/gap between the guard and the sensing/stimulating electrodes and the width of the guard electrode are fixed at 1 and 1.5, respectively. Knowing the dimensions A, D, E and F permits the calculation of all other dimensions, from which the total area of the electrode assembly can then be calculated (Table II). The mass of the electrode assembly is then determined from the specific weight of the electrode material (0.4 mg/mm² for the copper foil used on the 10 mL vial) and the %increase in vial weight from attaching the copper foil is then determined from the weight of the vial. Table III shows the dimensions of the electrode assemblies and the results of these calculations of %increase in mass.

Table II Calculation of Liquid Cross Sectional Area (a) and Liquid Fill Height (h) Corresponding to a Fixed Ratio of a/h = 45 mm

Table III Theoretical Calculations of the %Increase in Thermal Mass on Fixing an Electrode Pair (with Guard Electrodes) to Different Volumes of Tubing Vial