Multi-parameter Improvement Method for ( Micro-) Structural Properties of High Performance Ceramics

Many pH-measurement electrodes rely on porous diaphragms to create a liquid electrolyte junction between reference-electrolyte and the fluid to be measured. In field applications, the diaphragm is required to meet partly contradictory improvement criteria. To minimize measurement errors and to ensure durability of the measurement device, the diaphragm is supposed to maximize electrolyte conductivity and reference-electrolyte outflow velocity, while simultaneously minimizing reference electrolyte flow rate. The task of optimizing the overall performance of this small piece of ceramics has lead to the development of a novel multi-parameter improvement scheme for its (micro-) structural design. The method encompasses the consideration of microscopic material design parameters, such as porosity, poretortuosity and constrictivity, macroscopic material parameters such as diaphragm diameter and length, as well as process parameters like internal electrode pressure or the electrolyte viscosity and specific resistivity. Comprising sets of design parameters to dimensionless groups, concrete design guidelines as well as the introduction of a three-dimensional improvement space concept are proposed. The novel design space concept allows the improvement of each possible diaphragm-based measurement set-up, by considering the simultaneous, dimensionless interaction of all relevant design parameters.


INTRODUCTION
Many pH-probes require a liquid junction between a reference electrolyte and the substrate solution.Its main task is to provide a conductive bridge between the two liquids, while simultaneously preventing the reference electrolyte to be polluted by substrate ions [4].In the case of this study, the device in charge of maintaining such a junction is a small, cylindrical, porous piece of ceramics, Ddia= 1 mm in width, Ldia= 2 mm in length, referred to as diaphragm [5]- [7].A schematic illustration of the type of pH-probes in question is shown in Figure 1 [1].
The overall task within this research is to conduct knowledge-based improvement of existing macro-and micro-structural diaphragm design options.Thereby diaphragmimprovement shall be defined as an increase of probe durability and a simultaneous decrease of measurement errors, which co-occur with too little junction-conductivity or ionic pollution of the reference electrolyte [8], [9].

Functionality of Porous Diaphragms
The pH-measurement set-up under consideration, features a pressurized (p ≈ 2.0bar) reference electrolyte (3M KCl + additives), which is contained within a reference glass electrode, as seen in Figure 1, left [1], [10].The diaphragm both separates and connects the internal reference electrolyte and the external measurement solution.On the one hand the diaphragm's pores constitute a passage for electrons [e-], thus providing the necessary conductivity Sohm for the measurement procedure.On the other hand, anions [A+] or cations [X-], dissolved in the external measurement solution tend to diffuse through the pores to the inside and thus pollute the reference electrolyte [5]-[7|.While the first effect is strongly desired, the latter is to be averted as well as possible, since measurement accuracy decreases with increasing electrolyte pollution.Thus a flux of reference electrolyte, induced by the pressure gradient over the length of the diaphragm, is used to flush out substrate-ions diffusing upstream.This means that gradual reference electrolyte is accepted for the sake of measurement accuracy [10].
The bottom line at this point is, that the "ideal" porous diaphragm has to fulfill several, partly contradicting tasks: Provide much open cross sectional area Adia, in order to achieve highest possible conductivity Sohm.

(ii)
Flush out most external ions by inducing highest possible maximum outflow velocities vmax within the pores.(iii) Ensure maximum probe durability, namely minimum electrolyte depletion, by minimizing electrolyte outflow rate V .

Characterization of Diaphragm Microstructure
Obviously, the task of improving diaphragm functionality, as described in chapter 1.1, goes beyond the adaption of macrostructure parameters, such as diaphragm length Ldia or width Ddia.It has to entail a thorough study and controlled adaption of related microstructure parameters in relation to porosity, pore-shapes and sizes.As proposed in [1] and [11]- [21], three main micro-structural parameters are chosen to characterize a diaphragm's pore collective: (i) diaphragm porosity, (ii) pore constrictivity and (iii) pore tortuosity.They shall shortly be described in the following.

Diaphragm Porosity εdia(-)
The dimensionless porosity of the diaphragm, defined in Equ.1 and sketched in Figure 2 (top section), relates the total pore volume Vp to the total volume of the diaphragm Vdia.
A sensible variation of this quantity is the open pore porosity εdia,eff, which is defined by Equ.2 and only considers open pore space, effectively filled with electrolyte Vp,eff.

Pore Constrictivity βp(-)
The dimensionless constrictivity, defined by Equ.3 and sketched in Figure 2 (middle section), is a measure for the occurrence of pore-bottlenecks and relates minimum pore diameters Dp,min to maximum pore diameters Dp,max.

Pore Tortuosity τp(-)
The dimensionless pore tortuosity, shown in Equ.3 and sketched in Figure 2 (bottom section), is the ratio between the effective pore length Lp and the minimum possible pore length, which is the actual diaphragm length Ldia.

How to adapt Structural Parameters
Microstructure related parameters can be modified by adjusting production related parameters [2], [3].One such example is shown in Figure 3.There porosity, tortuosity and constrictivity of three diaphragm prototypes, made out of one and the same material matrix, are varied due to gradual increase of the sintering temperature.

Problem Definition
Since the given tasks in regard to diaphragm functionality involve flow-, conductivity-, process-and microstructure-related aspects, a thorough problem definition is necessary.This relates to laying out a clear picture of all possible degrees of freedom as well as an unmistakable statement regarding the aspects we wish to improve.

Design Parameters / Degrees of Freedom
Table 1 gives an overview of all degrees of freedom (=design parameters), their respective symbols, units and related, dimensionless design factors.The design parameters concern mainly ceramic macro-or microstructure, but also relate to the measurement process and electrolyte properties.Based on the main objectives to increase sensor durability while simultaneously decreasing its measurement error rate, three distinct improvement criteria can be defined: the velocity criterion, the flow-rate criterion and the conductivity criterion.

Velocity Criterion
The velocity criterion, according Eqn.5, states that the maximum occurring outflow velocity vmax within the diaphragm's pores must be maximized in order to flush out as many diffusing ions as possible.The overall rate of ions diffusing upstream does not necessarily relate to average outflow velocities, but rather to the maximum encountered outflow velocity.The latter is likely to occur at bottleneck formations along the pores.

Flow-rate Criterion
The flow-rate criterion, according Eqn.6, states that the outgoing flow-rate of reference electrolyte must be minimized in order to increase sensor lifetime.Obviously this criterion is contradictory to the velocity criterion, when applied to a given pore geometry.

Conductivity Criterion
The conductivity criterion, according Eqn.7, states that the electric conductivity Sohm of the entire diaphragm must be maximized in order to let electrons pass as easily as possible, leading to a minimization of measurement errors at varying PH levels.Since e.g.wider, shorter pores will yield higher conductivity, this criterion, at first inspection, stands in contradiction to the flow-rate criterion but goes in-line with the velocity criterion. max.

Semi-Analytical Model of Simplified Physics
A simplified, semi-analytical model has been created to derive generalized designguidelines for material-and process parameters.The model has been validated by CFD calculations and is used to depict the full path from improvement criteria towards possible design options.

Connecting Improvement Criteria with Design Parameters
The basic model equations are shown in Table 2 and provide the context between design parameters and improvement criteria.

The model encompasses: (i)
A standard formulation for creeping pipe flow [23], [24] extended by the calculation of maximum outflow velocities at the bottlenecks and thus a relation to pore constrictivity β. (ii) A formulation of the material's overall permeability K(m 2 ), according Eqn.8, [1] and [22], which relates microstructure aspects to this macrostructure property.
In Eqn.8, K0(m2) is the intrinsic permeability in relation to the ceramic surface V  roughness, w(-) is a dimensionless proportionality constant (hereby chosen to be equal to 1 and thus unused) and M (-) is the dimensionless M-factor, or permeability factor, according Eqn.9.The latter includes the link to microstructure properties of the material.
( ) Based on [1], [12] and [17], the coefficients x, y and z in Eqn.9 are chosen as: 1.15, 0.37 and 4.39 respectively.In Eqn.9, p (m 3  Dia/mp) is a proportionality factor which relates the porosity to the cross sectional area of an average pore.Thus p is defined by p= n -1 *LDia*p', where n (1/m 2 Dia) is the number of pores per cross-sectional area of the diaphragm and p'(m 2 p/m 3 p) is a coefficient related to the porosity's isotropy, which is hereby assumed as 1.
(iii) Ohm's law of electric conductivity through electrolyte-filled pores.
(iv) The assumption that overall electric conductivity and permeability behave in 1:1 analogy to intrinsic material properties and the microstructure, [11], [12] and [21].Thus it can be concluded that the overall electric conductivity of the electrolyte-filled diaphragm depends on the intrinsic electrolyte resistivity ρohm (Ωm) and on the microstructure-related M-factor (Eqn.9).(v) A formulation for the overall diaphragm-performance, or performance factor P, which shall be defined as an assembly of all process variables to be improved Φi and their respective improvement coefficients γi, according Eqn.10.Thereby γi is +1 in case of desired maximization and γi is -1 in case of desired minimization of the process variable.
Table 2: Overview of process variables to be improved and their context to the design parameters according to the semi-analytic model.The simple case of viscid, single phase, creeping flow, driven by a given pressure gradient of ≈0.5bar/mm, was modelled by using the OpenFoam®-based ico Foam solver, converging to a steady state (Figure 4, bottom).

Context to Design Parameters Φi = f(Ψi
In order to get a reasonable sense of the quality of our model-based permeability predictions the following steps were conducted: Three different types of material matrices were analyzed, digitalized and evaluated by FIB-tomography according [1], [25]- [30]; The presence of pore-former additive was simulated by adding five different amounts of spherical pores per reconstructed matrix-type; (iii) The electrolyte flow through those digital material samples was then simulated by OpenFoam® and predicted by our model; The results of this validation procedure are depicted in Figure 5, where the relative change of outflow-rate, compared to a reference material sample is printed out against the relative amount of added pore former.Based on these results, it can be shown that deviations between well known and widely validated CFD simulations and our model predictions are lower than 12%.This result is valid within a relatively wide range of microstructure properties and a pressure gradient of ≈0.5bar/mm.It can be seen that the semi-analytical model shows a slight systematic overestimation of flow rate deviations.

Comparing Design Options by Introduction of Dimensionless Design Factors
In order to evaluate the impact of design parameter modifications on the process variables to be improved, a dimensionless approach is imperative.To achieve this, dimensionless design factors Fi,j are introduced.They relate any design parameter Ψi,j of type i and design option j to its adapted counterpart within a new design option j+1, according Eqn.11.
, 1 , , While Table 1 (right column) denotes all design factors to their respective design parameters, Table 3 shows how a change of design factors impacts the relation of essential process variables Φi,j, according to Eqn. 12. , 1 , ,

( )
Table 3: Ratios of process variables Φi,j+1/ Φi,j of type i, belonging to consecutive design options j+1 and j as well as their context to the dimensionless design factors Fi,j according to the semi-analytical model.

Measures to Improve Individual Improvement Criteria
The relations shown in Table 3 already contain generalized information on how to improve individual process variables, according to the improvement criteria defined in chapter 2.1.2.Table 4 provides an overview of the measures, derived from Table 3.
Table 4: Overview of measures to independently modify process variables such that their respective improvement criterion is matched more closely, by design parameter increase ("+") or decrease ("-").Non-dependencies are marked by "x".  4 clearly demonstrates that, adapting one single design parameter to improve performance with respect to one individual improvement criterion might cause problems with respect to other criteria.Thus, a system-based approach is needed, which considers all design parameter-criteria-interactions as well as respective degrees of proportionality.A first step in this context is the depiction of a dependency diagram, as seen in Figure 6.Red coloration means that the respective criterion will "suffer" according the denoted proportionality, as the design parameter is increased.

Parameter Type
Green coloration means that the respective criterion will "benefit" according the denoted proportionality, as the design parameter is increased.Note that the coefficients are x=1.15,y=0.37 and z=4.39 [1], that indices "1" and "0" are used for design options j+1 and j respectively and that p is a constant proportionality factor relating the porosity to the cross-sectional area of an individual pore, according chapter 2.2.1.

Introducing the Switching Factors α, β and Ω
A mere overview of all design-parameter-criteria-interactions does not suffice to allow for a controlled "switching of focus" between the improvement criteria from one design option to the next.Thus, the switching factors α, β and Ω have been introduced.They are ratios between design options and their focus in regard to selected improvement criteria.

Switching Factor α
According to Eqn.13, α relates the development of the velocity criterion to the development of the flow rate criterion as design option j shifts to design option j+1.If α>1 then the focus obviously shifts from flow rate criterion to velocity criterion.This means that, if design option j+1 is chosen, starting from option j and α is >1, then the relative increase of outflow velocity will outweigh the relative decrease of outflow rate.Properties of High Performance Ceramics Eqn.14 shows how α relates to the individual design factors Fi,j which connect the design parameters Ψi,j and Ψi,j+1.
Switching Factor β According to Eqn.15, β relates the development of the flow rate criterion to the development of the conductivity criterion as design option j shifts to design option j+1.If β>1 then the focus obviously shifts from conductivity criterion to flow rate criterion.This means that, if design option j+1 is chosen, starting from option j and β is >1, then the relative decrease of outflow rate will outweigh the relative increase of conductivity.Eqn.16 shows how β relates to the individual design factors Fi,j which connect the design parameters Ψi,j and Ψi,j+1.

Switching Factor or Performance Factor Ω
The switching factor Ω is hereby also called the performance factor.This is because, according to Eqn.17, Ω relates all improvement criteria as design option j shifts to design option j+1.If Ω>1 then the relative change in process variables improving the overall performance, outweighs the relative change in process variables decreasing overall performance.
Eqn.18 shows how Ω relates to the individual design factors Fi,j which connects the design parameters Ψi,j and Ψi,j+1.
These definitions and dependencies can be re-formulated such that the development of the process variables to be improved Φi, can be shown in relation to a certain set of chosen switching factors as seen in Eqn.19.

(
) The necessary equations are summarized in Table 5.Note that, in order for this scheme to make physical sense, all switching factors must be larger than zero.
Table 5: Overview of process variable ratios between design options j+1 and j and their context to the dimensionless switching factors α, β and Ω according to Eqn.13 to Eqn.19.All switching factors must be larger than zero.

Context to Switching Factors
Φi,j+1/ Φi,j = f(α,β,Ω ) On the basis of the model assumptions, definitions and according derivations within chapter 2, some concrete measures of how to improve diaphragm performance can now be presented in chapter 3.1.Furthermore a method to find design options to simultaneously improve all process variables in terms of improvement criteria is shown in chapter 3.2.The improvement space concept introduced there, can also be used to shift the focus between various improvement criteria in a very controlled way.

Increasing the Overall Performance by Adjustment of Ω
The performance factor Ω relates all process variables in terms of improvement criteria.
Thus an increase of Ω can be counted as an increase of the overall performance of the diaphragm.
Note that an increase of Ω does not necessarily mean that design option j+1 is better with respect to all individual improvement criteria, but rather that relative improvements outweigh the setbacks.Having that in mind, an inspection of Eqn.18 provides valuable insights on how to improve the overall performance of a diaphragm by adjusting the related design factors.Thus several essential design guidelines can be extracted.They are summarized in Table 6.
Table 6: Overview of impact of design parameter changes on performance factor and related design guidelines.Favorable design parameter increase is denoted as "+", favorable design parameter decrease is marked as "-".Non-dependencies are marked with "x".

Introducing the Improvement Space Concept
Practical application has shown that it is not always enough to have design guidelines to increase overall performance as defined by the relation of process variables and improvement criteria.Quite frequently material samples exist, which excel in one or two out of three improvement criteria already, but lack behind expectations regarding the others.Thus, a tool is required to shift the focus between improvement criteria, while simultaneously increasing or at least keeping overall performance of the medium.This requirement has led to the introduction of the improvement space concept.
The improvement space features three dimensions: the switching factors α, β and Ω.Thus movement within that space always means a change from reference design option j (α=1, β=1, Ω=1) to any advanced design option j+1 (0<α<∞, 0<β<∞, Ω≥1).More specifically, movement in each one of the three directions means the following: Direction α>1 New design option j+1 will feature higher outflow velocity (good) but also higher outflow rate (bad).Direction β>1 New design option j+1 will feature lower outflow rate (good) but also lower conductivity (bad).Direction Ω>1 New design option j+1 will feature higher overall performance (good) and thus more options to perform better in regard to anyone of the improvement criteria.
Let Ω be constant and ≥1, then the improvement space turns into an improvement plane, which is illustrated in Figure 10, along with typical regional features.Assuming the concrete example, of diaphragm length reduction of 13% (b=0.87),Ω becomes 1.15.Now a range of design options emerges which allow for a certain freedom of movement within α-(shifting between velocity and flow rate) and β-(shifting between flow rate and conductivity) direction, while diaphragm performance with respect to every single improvement criterion can still be achieved.Figure 11 illustrates design options with higher conductivity (bottom, right, green), with lower flow rate (bottom, left, green), with higher outflow velocity (top, right, green) and design options which perform better with respect to all three criteria (top, left, green).Note that the green triangle within the top, left subplot of Figure 11 represents the intersection of the three individual improvement zones.It is found by intersecting the three critical functions where Φi,j+1/Φi,j=1.These critical borderlines can be derived from Table 5 For vmax,j+1/vmax,j=1: For Vj+1/Vj=1: Also, note that the corners of the improvement triangle represent design options where one particular improvement criterion is emphasized, while the others remain unchanged, as compared to the reference case.Thus, the top-left corner (intersection between conductivity-borderline and velocity-borderline) corresponds to a minimization of flow rate, the top-right corner (intersection between conductivity-borderline and flow rate borderline) corresponds to a maximization of flow velocity and the bottom-center corner (intersection between velocity-borderline and flow rate borderline) corresponds to a maximization of conductivity.
The full improvement space becomes accessible, when Ω≥1 can be varied as well.As the performance factor Ω increases, the overall diaphragm improvement triangle within the αβ plane gets larger and larger, as seen in Figure 12.
Since the improvement space is a completely non-dimensional concept, it is universally applicable as long as the model assumptions (chapter 2) hold.The generalized procedure of sensibly applying the improvement space concept can be summarized as follows: (i) Choose reference sample j, analyze its design parameters (according Table 1) and process variables (maximum outflow velocity, conductivity and outflow rate) in relation to any target values.

(ii)
Choose any one of the design-guidelines proposed in Table 6, to obtain a performance factor Ω>1. (iii) Calculate the new performance factor for the relation between new design option j+1 and reference case j, according Eqn.18.(iv) Draw the critical borderlines according Eqn.20 to Eqn.22 in α-β space to obtain an improvement triangle, as seen in Figure 12, with Ω being the parameter.(v) Choose suitable α or β values to obtain any desired process variables of design option j+1 in relation to reference case j. (vi) Use α, β and Ω values, as well as process variables of reference design option j to verify process variables of new design option j+1, according Table 5. (vii) Use relations from Table 3, as well as chosen restrictions and design parameters of reference case j to calculate design factors and corresponding design parameters of new design option j+1.

CONCLUSION
In total, a method was presented which allows the controlled modification of a ceramic diaphragm, applied as liquid junction in pH-probes.This multi-parameter improvement method is designed to increase, shift and adapt the diaphragm's performance in regard to three distinct improvement criteria: (i) the maximization of outflow velocity through the pores, (ii) the minimization of electrolyte outflow rate and (iii) the maximization of electric conductivity.
In this context, a semi-analytical model was developed to relate diaphragm design parameters as well as their dimensionless design factors to process variables and their respective improvement criteria.The model was validated by comparison to wellestablished OpenFoam®-based CFD simulations, conducted within digitally reconstructed and modified material sample meshes.The dependencies found within the semi-analytic model, were then evaluated and used to introduce the switching factors α and β, as well as the performance factor Ω. The latter was found to represent one way to define and thus increase overall diaphragm performance.
On this basis it is hereby proposed that a new, improved diaphragm design option should feature high porosity, low constrictivity and low tortuosity, thus it should yield many, straight pores with defined bottlenecks.Furthermore the sample should be as short as possible and the electrolyte should feature low intrinsic resistivity.Design parameters such as diaphragm cross section, electrolyte viscosity or internal probe over-pressure do not generally lead to an increase or decrease of the overall performance, but can rather be used to balance out two improvement criteria at a time.

Figure 1 :
Figure 1: Schematic illustration of the pH-probe (1a) and of the liquid junction (diaphragm) inside the probe [1].

2 .
Validating the Model by Comparison to CFD Those aspects of the model concerning flow through the pores (velocity and flow rate calculation), have been validated by conducting OpenFoam®-based CFD calculations.As shown in Figure 4, material samples were analysed by FIB-tomography, then their microstructure was digitalised (top-left, top-middle) [25]-[30] and transferred into a finite volume mesh (top-right).

Figure 6 :
Figure 6: Dependency diagram depicting all design parameter-criteriainteractions.Red coloration means that the respective criterion will "suffer" according the denoted proportionality, as the design parameter is increased.Green coloration means that the respective criterion will "benefit" according the denoted proportionality, as the design parameter is increased.Note that the coefficients are x=1.15,y=0.37 and z=4.39[1], that indices "1" and "0" are used for design options j+1 and j respectively and that p is a constant proportionality factor relating the porosity to the cross-sectional area of an individual pore, according chapter 2.2.1.

Figure 7 :
Figure 7: Dimensionless performance factor Ω against dimensionless tortuosityrelated design factor g. Increasing tortuosity means lower overall diaphragm performance.

Figure 8 :
Figure 8: Dimensionless performance factor Ω against dimensionless porosityrelated design factor e. increasing porosity means higher overall diaphragm performance.

Figure 9 :
Figure 9: Dimensionless performance factor Ω against dimensionless constrictivity-related design factor f. Graphs are parameterized by reference sample constrictivity β0.Increasing constrictivity means lower overall diaphragm performance.If reference constrictivity is low, the effect is more pronounced.

Figure 10 :
Figure 10: Conceptual illustration of the improvement plane for constant Ω≥1.

Figure 11 :
Figure 11: Illustration of improvement zones (green) for conductivity criterion (bottom, right), flow rate criterion (bottom, left), velocity criterion (top, right) and the overall diaphragm improvement triangle (top left) for Ω=1.15.Red zones symbolize design options where at least one of the three (velocity, flow rate, conductivity) improvement criteria gets worse.

Figure 12 :
Figure12: Diaphragm improvement triangle (green tones) where each improvement criterion gets "better", as it gets larger with increasing performance factor Ω. Red zone symbolizes design options where at least one of the three (velocity, flow rate, conductivity) improvement criteria gets "worse".

Table 1 :
Overview of design parameters and dimensionless design factors.*For definition of p, see chapter 2.2.1.

Parameter Velocity Criterion:
ohm S → 2.2.5.From Individual Parameters to System Approach: The Dependency Diagram Table