# Advances and obstacles to unify the concept of “printability” in Ceramics Direct Ink Writing

Mathilde Maillard, Sylvain Fournier, Esther García-Tuñón, Edwin-Joffrey Courtial

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## To cite this version:

Mathilde Maillard, Sylvain Fournier, Esther García-Tuñón, Edwin-Joffrey Courtial. Advances and obstacles to unify the concept of “printability” in Ceramics Direct Ink Writing. Open Ceramics, 2025, 23, pp.100808. 10.1016/j.oceram.2025.100808. hal-05134710

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Review

Advances and obstacles to unify the concept of “printability” in Ceramics
Direct Ink Writing

Mathilde Maillard<sup>aID,*</sup>, Sylvain Fournier<sup>a</sup>, Esther García-Tuñón<sup>b</sup>, Edwin-Jeffrey Courtial<sup>c</sup>

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ARTICLE INFO

Keywords:

Printability

Extrusion

Rheology

A B S T R A C T

In Additive Manufacturing, “printability” describes the ability to shape a functional 3D object from a starting material. This concept is not easy to define because it depends on several factors including material considerations (microstructure and rheological properties), as well as process parameters that cannot be evaluated by a single metric. Some printability criteria have been proposed to classify and map the rheological properties of DIW inks. However, these criteria fail to capture all the relevant rheological aspects, and to consider DIW system technical specifications, processing parameters, and final part properties, while rheological characterisation methods remain inconsistent and lack standardisation. This review tackles this gap in the field by providing an overview of literature investigating the links between rheological properties and printability for ceramics DIW. Based on recent findings, we present an overview of specific rheological tests and parameters that correlate with printing behaviour to achieve high-quality outcomes in ceramic 3D printing.

1. Introduction

Unlike other materials, ceramics need to undergo a sequence of
processes for shaping and consolidation respectively. A challenge in
Additive Manufacturing (AM) of ceramics is to successfully fabricate
defect-free and functional ceramics parts with an accurate shape with-
out requiring additional refinement steps such as polishing. Ceramics
AM encompasses a wide range of diverse printing methods, such as
stereolithography (SLA), digital light processing (DLP), fused deposi-
tion modelling (FDM) and direct ink writing (DIW), while extrusion-
based techniques dominate in other domains such as bio-printing [1].
Each AM technique involves different pre- and post-shaping steps, and
as such the concept of “printability” and associated metrics will differ
amongst them. Prior work in DIW [2,3], state that “good printability” is
achieved through the careful selection of ink formulation and printing
parameters.

What is common to all AM techniques is that the success of a
print or 'printability' is typically assessed based on two performance
indicators: (1) the shape fidelity of the printed part compared to the
model, and (2) its final performance. The latter refers to the 3D object's
ability to satisfy the intended requirements depending on its use (e.g.,
mechanical strength for structural ceramics, or its electromechanical

coupling factor for piezoelectrics). Thus, printability can be described
as the ability to achieve a successful print that satisfies both criteria (3D
shaping and function). Ink formulation, rheological properties, system
specifications and process/printing parameters are implicit in the first
"3D shaping" criterion (Fig. 1). Quantitative and measurable metrics
linked to these criteria have been used to construct "printability maps",
which are graphs of specific rheological indicators used to compare
inks (similarly as Ashby-type charts for material's properties), with
delimited "printable" regions [4], or trends that correlate with printing
resolution [5]. Maps of rheological properties are an essential tool in
DIW, however, a current limitation is that they fail to account for all
the constraints associated with "printability". These maps cannot (yet)
capture the diversity of requirements and considerations including:
(a) the chosen DIW setup, including the various DIW machines and
parameters variants, (b) materials constraints, and (c) post-shaping
processes needed to obtain a ceramic part. In total, there are at least
six aspects that need to be considered (Fig. 1), which can be broken
down into three levels: ink constraints (1 and 2), technique/process
considerations (3 and 4), and ceramics manufacturing considerations
(6) (Fig. 1).

<footnote><sup>a</sup> Univ. Lyon, INSA Lyon, Universite Lyon 1, CNRS, MATEIS, UMR5510, 69621 Villeurbanne, France</footnote>

<footnote><sup>b</sup> School of Engineering and Materials Innovation Factory, University of Liverpool, UK</footnote>

<footnote><sup>c</sup> 3d.FAB, CNRS UMR 5246, ICBMS (Institute of Molecular and Supramolecular Chemistry and Biochemistry), Université Lyon 1, 69622, Villeurbanne, Auvergne-Rhône-Alpes, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: mathilde.maillard@univ-lille.fr (M. Maillard), Esther.GTunon@liverpool.ac.uk (E. García-Tuñón).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Scheme illustrating relevant aspects (numbered from 1 to 6) that should be considered to assess the “printability” of ceramic inks for DIW. 3D shape fidelity (5), is determined by interrelated factors (1 to 4), while final properties (6), or performance is also considered to assess “printability” in ceramics DIW. A rigorous and systematic characterisation at each stage in the process enables linking rheology, shape fidelity and quantitative measures of final performance, for example through structural and mechanical characterisation of printed parts.</figcaption></figure>

**Level 1: Technique considerations** (3 and 4, Fig. 1). The DIW equipment will determine the technical specifications and working limits, which should be taken into account as they may affect the forces applied to the ink. For example, the extrusion can be driven by syringe pumps with or without a screw, and controlled by an air-pressurised system or by a plunger displacement connected to a servo actuator. Apart from the diversity of extrusion systems, DIW variants include embedded DIW, or in situ consolidation methods (e.g., UV curing). Designing inks with tailored flow behaviours for each DIW variant is essential to meet the requirements of the chosen method. DIW requires the design of yield stress fluids (YSF) able to flow through the chosen printing nozzle tip, exhibiting neither microstructure disruption nor shape distortion during extrusion to ensure good dimensional accuracy [6,7]. In the case of DIW systems with in situ consolidation methods (e.g., UV curing), a wider range of complex fluids that evolve over time could be considered as “printable”. The process parameters for each DIW variant can be adjusted within specific ranges depending on the hardware and software available, which places an additional barrier to standardisation. Printing parameters (4, Fig. 1) such as volumetric flow, nozzle’s geometry and dimensions, printing height from the substrate, and so on, also affect shape fidelity.

**Level 2: Ink constraints** (1 and 2, Fig. 1). Ceramics DIW relies on the shaping of an ink comprising ceramic particles and organic compounds, therefore thermal treatments (for debinding and sintering) of the printed part are essential for organics removal, particles bonding and to increase part density. While a low amount of organics promotes a rapid thermal processing, high density and low grain growth, it also increases the risk of extrusion inaccuracy or failure during debinding. Achieving a compromise between ceramic and organics fraction is the most common ink optimisation usually performed to improve “printability”. The choice of organics and solvents determines the microstructure and therefore the ink’s rheological properties. A detailed discussion of the different microstructures that can be found in ceramics DIW is beyond the scope of this manuscript; however, this has been covered by previous reviews in the field that will be useful to the reader [4,8]. Any DIW ink, regardless of its microstructure, must meet specific rheological requirements for successful printing. An ink is deemed as “printable” when: (1) it can be extruded through narrow nozzle tips (the microstructure has the ability to yield and flow, *flowability*) [5]; (2) the flow of material is continuous and steady (the microstructure is not compromised, e.g., no phase separation or filter

press effects take place) [9]; (3) the ink’s microstructure is able to rebuild (or “recover”) and achieve enough strength to retain the filament shape with accuracy, and the 3D object is built without slumping or collapsing. Alternative terms for *flowability* currently used within ceramics DIW are: *pumpability*, *extrudability*, or *buildability* [10–12].

**Level 3: Ceramics Manufacturing considerations** (6, Fig. 1). Ceramics DIW involves a multi-step manufacturing process from shaping to consolidation. The forming step converts a ceramic slurry or paste with high particle loading, into a solid “green part”, which must undergo subsequent post-shaping processes like debinding and sintering [13,14]. These post-shaping steps lead to substantial changes in the composition and properties of the material, for example, through burning of the organic part of the ink, densification of particles, and grain growth. This means that a DIW ink must also be designed to ensure that consolidation can be achieved and that the mechanical and functional performance is not compromised. It is therefore critical to find a compromise between shaping and post-shaping requirements, thus adding complexity to the criteria definition to quantify “printability” in ceramics DIW (Fig. 1). The relationship between the rheological behaviour of DIW inks, the shape fidelity of the printed parts and the mechanical properties of final parts, is currently not clearly established [15,16]. In this article, we present a workflow of rheological assessments that can help evaluating different properties linked to “printability”. These metrics could then be systematically compared with shape fidelity metrics, and measures of final performance (e.g., mechanical strength). With this roadmap, we aim to advance towards the standardisation of rheological characterisation and enhance the reliability of ceramics DIW manufacturing processes.

## 2. Complex fluids for ceramics DIW: general considerations

Considering that DIW inks are subjected to various mechanical stresses throughout the printing process, their rheological study can be a useful tool to predict their behaviour and adapt printing parameters to ensure print quality. The composition of DIW inks is not only constrained by the forming step (extrusion in this case) but also by consolidation processes to ensure that high final density is achieved. DIW inks often contain a high amount (higher than 30 vol.%) of ceramic particles with sizes ranging from nm to µm, and various organic compounds aiming to improve particles dispersion and suspension
stability, to reduce foaming, or to specifically modify the ink rheology. The rheological properties of such concentrated suspensions are strongly impacted by particle solid loading, as depicted by the Krieger-Dougherty model which shows that the viscosity follows an exponential growth for increasing solid loading [17,18]. Particle properties (surface chemistry and surface charge, particle size and size distribution, surface area and wettability) do play a critical role in the ink rheological behaviour at rest and while flowing [19–21]. For example, particle size and agglomeration are limited by the nozzle diameter to avoid nozzle clogging [19–22]. The dimension of agglomerates at rest and under shear is itself dependant of particle surface properties, that can be modified by organic compounds such as dispersing agents [23]. Finally, the strong influence of particles characteristics (from their shape, to their concentration and interactions) on the structure and rheology of DIW inks, discussed in detail by Wei et al. [24], highlights the need to understand the material structure at rest and under shear, to complement the use of printability criteria for ink optimisation.

The rheological response of fluids with a high concentration of ceramic particles depends both on the timescale, the force and type of deformation imposed, and this response often varies in time or can be dependent of shear history (thixotropic) [25]. These complex materials are best described as thixotropic elasto-visco-plastic (or TEVP) materials, which poses many challenges in both experimentally measuring their rheological characteristics as well as attempts to model each isolated aspect of the constitutive response [25]. Each of these features or contributions needs to be studied by carefully choosing the experimental methodology that is relevant to each step in the DIW process. For example, a flow sweep (steady shear measurement of the ink response (stress, τ, Pa) over a range of increasing or decreasing shear rates (γ, s⁻¹)) may capture the ink behaviour when it is flowing (“liquid-like” state), but cannot describe the solid-like behaviour of the material before or after extrusion. A Dynamic Strain Sweep or DSS (measurement of the stress response (σ, Pa) to an oscillatory strain input of increasing or decreasing amplitude (γ₀, %)) can be used to quantify the viscoelastic properties of the ink before extrusion in the linear viscoelastic regime (LVER) and to compare nonlinear behaviours associated with different yielding phenomena [5], but it cannot capture thixotropic effects, or the re-structuring extent and timescales. Decoupling plastic and viscous dissipation during yielding and flow is even more challenging and it would require more advanced experiments and analyses using recovery rheology [26,27], (which should not be confused with the concept of “recovery” commonly used in DIW). The term “recovery” in DIW refers to the ability of the material to re-structure after exiting the nozzle and retain the filament shape without spreading. The elasto-viscoplastic KDR model [28] can capture the features of YSF for DIW, for example for F127 [29], a common formulation base used in ceramics DIW.

### 3. Linking rheological characterisation with the different processing conditions in ceramics DIW

Conducting comprehensive rheological characterisation requires linking ink formulation to its rheology and printing performance [30]. The DIW process can be divided in three distinct shearing conditions: (1) flow initiation and (in principle) steady state flow, (2) transient shear conditions upon exiting the nozzle tip, and (3) no shearing forces, except the settling of next upper layers, acting on the printed object. Hence, the following terms are commonly used: “flowability”, “extrudability” or “pumpability” (for stage 1), “recoverability” (stage 2) and “material strength” (stage 3) [5].

**Flow.** DIW inks are non-Newtonian materials with a YSF behaviour, meaning that they behave as solids at stresses below their yield stress (σ<sub>y</sub>) and flow above this threshold. However, the yielding phenomenon can be more or less sudden depending on the ink exhibiting brittle or ductile behaviours [28,31,32]. Yielding is understood as a process, which can be highly heterogeneous [33] thus questioning the definition of a singular “yield point”. In fact, yielding can result from different

modifications in the ink's structure, such as the disentanglement of polymer chains [34] or the de-clustering of ceramic particles [35]. Yielding is influenced by the type of stress applied, its duration, as well as shear-history for thixotropic fluids. When subjected to continuous shear, DIW inks are mostly shear thinning, meaning that their viscosity η decreases with increasing shear rate (γ̇). However, high particle loading, insufficient particle dispersion or stability, can be responsible of shear-thickening effects at high shear, which may disrupt ink flow during printing. The ability to flow (referred to as “flowability” or “extrudability”) is essential in DIW, which is often assessed using flow sweeps (in continuous shear) and DSS (in oscillatory shear). The latter probes the response of the material under a wide range of deformations from small to large amplitude.

**“Recovery”**. Upon exiting the nozzle tip, the formulation's microstructure needs to rebuild to maintain the filament shape and enable 3D shaping. In DIW, the term “recovery” refers to the ability of the ink structure to reform with enough strength to retain the shape of the printed object right after extrusion and to prevent collapse. Thixotropic (sensitive to shear history) inks should be carefully characterised. For example, start-up experiments followed by shear cessation can assess the thixotropic behaviour of an ink [36]. These measurements investigate how the shear history affects the thixotropic restructuring time (which corresponds to the time after shear cessation τ<sub>ss</sub><sup>-</sup> to reach a plateau or steady state τ<sub>ss</sub><sup>-</sup>, Fig. 2, Table 1). Indeed, the strength of the ink at rest and after extrusion can be very different. Therefore, it is the strength after extrusion (in the printing timescale) that should be used as a reference rheological indicator. However, continuous shear tests for DIW inks can lead to flow instabilities and high uncertainty even at small shear rates [33,37]. Alternatively, “recovery” tests using a sequence of oscillatory shear experiments (e.g., from small to large to small amplitude SAOS-LAOS-SAOS, also referred to as 3ITT) are frequently performed with less uncertainty. 3ITT tests are used to determine the ability of a formulation to “recover”, even if it has been argued that they are less directly representative of the DIW printing process [38]. In practice, neither continuous or oscillatory shear experiments provide realistic processing conditions during extrusion [33]. Despite these limitations, shear rheology provides versatile and accessible experimental tools to probe the constitutive relations of YSF.

**Material Strength.** Once deposited on the substrate, although the material is not subjected to any shearing stresses, its strength is critical to ensure that the printed part is a good replica of the pre-designed structure, without collapsing or deforming as additional layers are added on top. Measurements of material strength (storage modulus G', and stress values, τ<sub>y</sub><sup>dyn</sup> or σ<sub>f</sub>, Table 1) are widely used in DIW literature [4,5]. Since the ink's rheological response depends on the timescale and the imposed force, the conditions of the DIW process play a significant role on the final printability. Flow rate, contractions in the flow (e.g., syringe to nozzle), nozzle tip diameter and length, print velocity and height will determine the shearing conditions [39,40]. Therefore, printing parameters should be carefully chosen depending on the materials' rheological response [41,42].

There are additional ink's features that also play a role in DIW, for example the ability to elongate or “stretch”, which are described as “drawability”. The extensional properties of materials for DIW are rarely quantified in the ceramic field, however, approaches in the broader YSF field are emerging. For example, using an extensional gravity rheometer based on the analysis of extruded filaments [43], or measuring the strain to break using filament stretching extensional rheometry (FiSER) to calculate the draw ratio, DR [44]. Elongational properties should also be considered in the future development and refinement of printability criteria.
# 4. The need for consistency in rheological analysis to assess “print-ability”

## 4.1. Practical considerations

Considering the large number of phenomena that affect rheological measurements and the importance of rheological properties for successful 3D printing, rigorous rheological measurements and correct data collection are essential [45]. Researchers in the DIW ceramics field can benefit from adopting consistent rheology protocols and reporting the same metrics to enable systematic comparison and to provide a robust database for our community. To avoid confusion about characteristic values (e.g., between the yield stress and the yield point), we provide a definition for each metric here described (Table 1). The following tips and signposted resources might be useful when setting up a rheological experiment.

The measuring geometry should be carefully chosen considering the sample's properties, to correctly quantify them. For highly concentrated suspensions, it is advised to use parallel-plate geometries instead of cone-plate, to prevent the particles jamming the truncation gap [46]. If/when using a cone-plate for suspensions, its truncation gap should be at least ten times the size of the particles in the formulation to minimise the risk of jamming [46]. Solvent traps should be used to prevent drying, and textured or roughened geometries to prevent wall slip [47]. Sample conditioning, also called pre-shear procedure, should not be overlooked since it can damage the structure of the ink, and therefore the rheological results as explained in Section 4.5. Experiment conditions (e.g., the controlled variables, shear rate ($\dot{\gamma}$) or controlled stress ($\tau$) in continuous shear, or frequency ($\omega$) and strain or stress input ($\gamma(t)$), or $\sigma(t)$) range in oscillatory shear) must be set up to suit the material of interest, and within the technical limits specified by the manufacturer. The obtained data should be analysed to identify instabilities or flawed results, like a sudden drop in a flow curve which can suggest wall slip, or that the sample was expelled out of the rheometer geometry, leading to bad data [45,48]. Equipment technical specifications and working limits will differ depending on the rheometer setup. In any case, the rheometer torque limits impose a measurable range. Inertial effects at high shear rates and frequencies also result in unreal data.

## 4.2. Shear rheology: continuous shear tests

In continuous shear, the geometry rotates in the same direction, following an imposed shear rate ($\dot{\gamma}$) or stress ($\tau$), that can be set as a constant step (hold) or time dependant ($\dot{\gamma}(t)$). Continuous shear tests (including flow sweeps, start-up, and shear-cessation tests) can provide useful metrics for DIW (Table 1).

**Continuous shear: Start-up experiments** consist of applying a stepped shear rate (constant $\dot{\gamma}$) and measuring the stress (Fig. 2). Often, shear cessation ($\dot{\gamma} \to 0$) is performed following the start-up step, aiming to quantify the steady state stress reached after shear cessation, called thixotropic stress ($\tau_{ss}^{-}$), and the corresponding time at which it is reached, called thixotropic time ($\tau_{ss}^{+}$). This measurement is often repeated several times to investigate the effect of the shear history on the thixotropic behaviour.

**Continuous shear: Flow sweeps tests** consist of increasing the shear rate ($\dot{\gamma}$) imposed to the sample. The shear rate can be increased by discrete logarithmic steps (Fig. 3a). At each step the stress ($\tau$) is measured (reported on Fig. 3b for each value of $\dot{\gamma}$) when steady-state is reached or at a certain length of time defined by the user. The flow $\tau$ vs. $\dot{\gamma}$ curve shows a plateau region at low $\dot{\gamma}$ that can be interpreted as a yield stress as stated by the Herschel-Bulkley (HB) model: $\tau = \tau_0 + K\dot{\gamma}^n$, where $\tau_0$ is the yield stress, K the consistency index and n the flow index ($n = 1$ for Newtonian, $n < 1$ shear thinning, and $n > 1$ for shear thickening fluids). A detailed discussion on yield measurements is provided in Section 4.4. Fitting the flow sweep results to a simple

Generalised Newtonian Fluid model (GNF, e.g., power law or HB) that can provide a good representation of the liquid-like state when the ink is flowing, the degree of shear-thinning of an ink ($n$), and could highlight the occurrence of shear-thickening events at high $\dot{\gamma}$ values.

However, when working with highly concentrated ceramic inks, collecting data with good reliability and reproducibility in continuous shear can be challenging [33,37]. “Flow curves” are popular in DIW printing [49], however, these measurements can be affected by flow instabilities including transient effects and shear banding at low $\dot{\gamma}$ values [33], while slip and fracture can take place at high $\dot{\gamma}$ [37]. These instabilities are associated with sudden changes of stress and/or viscosity with increasing $\dot{\gamma}$, and gap defects (e.g., material being expelled out, or detached from the plate). Also, during a flow sweep a large total strain could be accumulated, damaging the ink’s structure, which is likely for high solid loading inks that may behave as pastes as defined by Coussot et al. [32]. Transient effects at low shear can be avoided by performing the experiment in reverse, from high to low $\dot{\gamma}$, while slip and fracture can be minimised by working at relatively small shear rates up to $10 \text{ s}^{-1}$.

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Startup test followed by Shear cessation. The imposed shear rate ($\dot{\gamma}$) is represented in black while the measured stress ($\tau$) is shown in blue. The incomplete and delayed recovery highlights the thixotropic behaviour of the material within the timescales of the experiment that is relevant for DIW.</figcaption></figure>

## 4.3. Shear rheology: oscillatory tests

**Shear rheology: oscillatory tests** are frequently used in DIW, owing to the versatile and reliable experiments that can be performed and combined in different ways, including: amplitude sweeps (fixed frequency, $\omega$, and varying strain amplitude, $\gamma_0$) known as dynamic strain sweep (DSS) or large amplitude oscillatory shear test (LAOS); time sweeps (fixed $\omega$ and $\gamma_0$); frequency sweeps (varying $\omega$ and fixed $\gamma_0$ in the linear viscoelastic regime, LVER); and structure evolution tests combining a series of amplitude, time and frequency sweeps.

A dynamic strain sweep (DSS, Fig. 4) imposes a sinusoidal signal on the material with varying (generally increasing) strain amplitudes ($\gamma_0$) and fixed frequency ($\omega$). The response of the material is also a sinusoidal signal, which is in phase with the input strain in the linear viscoelastic region (LVER). The LVER corresponds to the region where the relationship of stress ($\sigma_0$) and strain amplitude ($\gamma_0$) is linear, and $G'$ is independent of the strain imposed, and is a measure of the stored energy under small deformations or at “rest”, in the solid-like state. As the strain increases beyond the LVER, the sinusoidal stress response becomes distorted due to nonlinear behaviours, which can be visualised in Lissajous-Bowditch plots and quantified through different mathematical frameworks (e.g., Fourier Transform Rheology [50] and the Sequence of Physical Processes [51]). When performing an increasing strain amplitude DSS, LAOS analyses enable the quantification of distinctive behaviours (onset ($\sigma_{nl}$, $\gamma_{nl}$) and relative intensities of higher harmonics, $\frac{I_n}{I_l}$) during the yielding processes that correlate with printing resolution in DIW [5,52]. LAOS tests can provide a comprehensive
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a) Imposed shear rate $\dot{\gamma}$ and recorded stress $\tau^+$ for a flow sweep test performed for increasing $\dot{\gamma}$. The values of $\tau$ for each point displayed on the $\tau$ vs $\dot{\gamma}$ plots are obtained from stable flow data (in the red quadrants). (b) $\tau$ vs $\dot{\gamma}$ flow curve for increasing $\dot{\gamma}$ in black and decreasing in grey. Herschel-Bulkley model fit allow to determine $\tau_{\gamma}^{\text{stat}}$ for the first one and $\tau_{\gamma}^{\text{dyn}}$ for the latter. (c) Corresponding viscosity $\eta$ vs $\dot{\gamma}$ plot displaying a shear-thinning behaviour and the shear history dependency.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. (a) Strain sinusoidal sequence imposed in an increasing amplitude DSS. (b) Characteristic LAOS (commonly called DSS) fingerprint for a yield stress fluid used in DIW showing type III strain softening behaviour with increasing strain amplitude [50]. (c) Characteristic L-B plots are included for three representative points in the sweep. (i) LVER showing almost a purely elastic behaviour represented by a straight line. (ii) In the flow transition just before the crossover, the distorted ellipsoidal shape shows that the energy dissipation has increased and the behaviour is non-linear. And (iii) in the LAOS region the rectangular shape gets close to an idealised perfect plastic response [53].</figcaption></figure>

set of metrics (Table 1, Section 5) to characterise the solid-to-liquid transition. The extent of the LVER (and onset of nonlinearities) can be accurately determined using Fourier Transform rheology, while Lissajous-Bowditch (L-B) plots provide qualitative and visual information that capture non-linear behaviours, and their evolution in the flow transition (Fig. 4c). L-B plots are a visual tool that can be very helpful to the DIW community, however these are rarely reported. These plots provide evidence of linear and nonlinear behaviours and help distinguishing ideal behaviours [50]. A purely elastic behaviour corresponds to a straight line ($\sigma/\sigma_0$ vs $\gamma/\gamma_0$, Fig. 4c-i), a circle corresponds to purely viscous, and rectangular shapes are associated to perfect plastic (Fig. 4c-iii). The dissipation ratio, $\phi$, is a scalar metric that can quantify the deviation from these ideal behaviours [53].

A sequence of SAOS (small amplitude oscillatory shear)-LAOS-SAOS or three interval thixotropic test 3ITT, can be used to determine the ability of the structure to recover after being subjected to an intermediate interval at large deformations, thus mimicking the

transitions during processing conditions. This is done at a large strain amplitude ($\gamma_0$, within the liquid-like regime, $G' < G''$) to account for the worst-case scenario. From this test, it is possible to quantify recovery timescales: the mutation times ($\lambda_I, \lambda_{II}, \lambda_{III}$). Mutation times quantify the kinetics of structural evolution (Table 1), and are determined from the gradients in the curve scaled by the recovered storage modulus [5, 37]. The mutation times are calculated using Eq. (1),

$$ \lambda_{mut} = \left[ \frac{1}{G'_{rec}} \left( \frac{dG'(t)}{dt} \right)_{rebuild} \right]^{-1}, \quad (1) $$

where $\lambda_{mut}$ is the mutation time that can be determined for different regions in the recovery interval, $\frac{dG'(t)}{dt}$ is the gradient of the storage modulus, and $G'_{rec}$ is the average storage modulus when it reaches a plateau in the third (SAOS) interval. The extent of the recovery is calculated as $\%G' = \frac{G'_{rec}}{G'_{LVER}} \times 100$, which provides a measure of the ability of the structure to rebuild or reform its initial configuration (see Fig. 5).
<figure><img src="image_8.png" /><figcaption>Fig. 5. In a 3ITT, the material is subjected to a sequence of oscillatory intervals from small to large, to small, to replicate the states of the ink through the extrusion process from solid (before extrusion) to liquid (when flowing) and back to solid-like (upon exiting the nozzle). The extent and kinetics of the recovery can be quantified from the *G'* evolution in the third interval (Eq. (1)).</figcaption></figure>

## 4.4. Yield stress determination in DIW

One of the issues in DIW and more broadly in rheology, is the inconsistency in the yield stress determination [32]. This is partly due to the lack of consensus on the existence and universal determination of a “yield point” as stated in the introduction and in the practical considerations section. Despite these inconsistencies, the literature agrees that this measure of *Material Strength* (Table 1), is directly linked to shape fidelity and the capacity to stack layers without slumping or collapsing, also affecting the final properties of printed parts [4]. For example, the yield stress is sometimes defined as the moduli crossover ($G' = G''$), in the solid-to-liquid transition in a DSS test, and at the start of flow in continuous shear (viscosity, $\eta$ vs. shear rate, $\dot{\gamma}$) [54,55]. Flow sweeps 4.2 can be used to determine the static yield stress ($\tau_y^{\text{stat}}$) [56] from a ramp-up flow curve, and the dynamic yield stress ($\tau_y^{\text{dyn}}$) from the ramp-down curve (Fig. 3). The static yield stress ($\tau_y^{\text{stat}}$) is the stress required to initiate flow (ramp up), while the dynamic yield stress ($\tau_y^{\text{dyn}}$, Table 1) represents the minimum value of stress needed to maintain flow (ramp-down) [57,58]. However, yield stress determination using DSS, or continuous shear, is not settled in the field. For example, according to Dinkgreve et al. oscillatory measurements do not allow to accurately determine “*the apparent yield stress, or apparent yield strain*” of thixotropic samples [59], because values obtained from oscillatory are often higher compared to other measurements (performed by creep tests for example). Other works choose DSS over continuous shear experiments for yield stress determination based on the likelihood of flow instabilities during the latter [33], which could be responsible for the disparity between oscillatory and continuous shear results.

To avoid confusion on the terminology, we compile all the rheological indicators (Table 1) using self-explicative labels, chosen to represent specific points in the frame of reference. For example, the onset of nonlinearities is indicated by the subscript *nl* (which marks the end of the linear viscoelastic region, LVER) and subscript *f* for “flow” at the moduli crossover ($G' = G''$).

## 4.5. Pre-shear

The shear history of thixotropic complex fluids (such as ceramic inks) can have a significant impact on its stress–strain relationship [60]. Types of ageing can be split into two phenomena, the shear induced ageing and the long term ageing. Sample preparation should not be overlooked because it can change the microstructure and as a consequence the rheological results. This phenomenon called “memory effect” must be taken into account, ensuring that the ink’s rheological properties are measured in the same solid-like state as the ink is right before printing. For instance, a ceramic paste manipulated with a

<figure><img src="image_9.png" /><figcaption>Fig. 6. Increasing $\gamma_0$ ramp DSS results for two different inks subjected to two types of shear “history”: right after syringe extrusion and geometry positioning for the first one, and after the first DSS for the second one. The ink labelled *low memory* effect displays almost the same $G'$ values in the LVER for both DSS experiments. However, after loading the sample using a syringe prior measurement, the *high memory* effect ink displays lower $G'$ values with a nonlinear trend at small strain amplitudes ($\gamma_0$).</figcaption></figure>

syringe shows a non-linear trend in $G'$ values at small strain amplitudes (Fig. 6). Loading the same sample with a spoon will minimise the damage to the microstructure and lead to a different result. In short, loading the sample in the geometry (with normal forces acting on the sample depending on the closure profile chosen), and setting pre-shear (or sample conditioning) before testing, will affect the ink’s structure and properties [61]. This “memory effect” is a characteristic of each ink formulation. Some inks will be sensitive to shear history but there are cases where the impact is not significant, as illustrated in Fig. 6. If the structure breaks-down upon extrusion, these “memory” effects can also affect printing performance and the ink’s “printability” might be compromised. When choosing pre-shear conditions, it is important to investigate: (1) that the set conditions are representing the processing conditions during DIW, and (2) the potential damage of the microstructure.

## 5. Printability characteristic metrics for ceramics DIW

A flowchart of rheological measurements linked to “printability” indicators has been proposed by Rau et al. [62] with the aim to facilitate ink design *step by step*. However, the proposed roadmap does not fully capture all the rheological behaviours that affect “printability”, for example, the quantitative measures of “material strength”
<table><caption>**Table 1**
Rheological metrics for ceramics DIW obtained in continuous and oscillatory shear rheology.</caption><thead><tr><th colspan="5">Continuous Shear</th></tr><tr><th>Test</th><th colspan="2">Rheological metrics</th><th>Corresponding printability property</th></tr></thead><tbody><tr><td rowspan="4">Flow sweep<br>(up and down)</td><td>τ<sub>y</sub><sup>stat</sup></td><td>Static yield stress</td><td>(Pa)</td><td>Material Strength</td></tr><tr><td>τ<sub>y</sub><sup>dyn</sup></td><td>Dynamic yield stress</td><td>(Pa)</td><td>Material Strength</td></tr><tr><td>n</td><td>Flow index</td><td>(-)</td><td>Flowability</td></tr><tr><td>η<sub>0</sub></td><td>Zero-shear viscosity</td><td>(Pa.s)</td><td>Material Strength</td></tr><tr><td rowspan="3">Start-up</td><td>τ<sub>SS</sub><sup>+</sup></td><td>Steady state stress at the printing shear rate</td><td>(Pa)</td><td>Material Strength</td></tr><tr><td>t<sub>SS</sub><sup>+</sup></td><td>Time to reach steady state</td><td>(s)</td><td>Flowability</td></tr><tr><td>τ<sub>max</sub></td><td>Maximum stress at the printing shear rate</td><td>(Pa)</td><td>Flowability and Material Strength</td></tr><tr><td rowspan="2">Shear cessation</td><td>τ<sub>SS</sub><sup>-</sup></td><td>Steady State stress after shear</td><td>(Pa)</td><td>Recoverability</td></tr><tr><td>t<sub>SS</sub><sup>-</sup></td><td>Time to reach steady state</td><td>(s)</td><td>Recoverability</td></tr><tr><th colspan="5">Oscillatory shear</th></tr><tr><th>Test</th><th colspan="2">Rheological metrics</th><th>Corresponding printability property</th></tr><tr><td rowspan="8">DSS</td><td>G'<sub>LVER</sub></td><td>Storage modulus in the LVER</td><td>(Pa)</td><td>Material Strength</td></tr><tr><td>γ<sub>nl</sub></td><td>Strain at the nonlinear onset</td><td>(-)</td><td>Flowability</td></tr><tr><td>σ<sub>nl</sub></td><td>Stress at the nonlinear onset</td><td>(Pa)</td><td>Flowability</td></tr><tr><td>I<sub>3</sub>/I<sub>1</sub></td><td>Relative third harmonic intensity</td><td>(-)</td><td>Flowability</td></tr><tr><td>φ</td><td>Dissipation ratio</td><td>(-)</td><td>Flowability</td></tr><tr><td>σ<sub>max</sub></td><td>Stress overshoot before the crossover</td><td>(Pa)</td><td>Flowability</td></tr><tr><td>γ<sub>f</sub></td><td>Strain at moduli crossover (G' = G'')</td><td>(-)</td><td>Flowability</td></tr><tr><td>σ<sub>f</sub></td><td>Stress at moduli crossover (G' = G'')</td><td>(Pa)</td><td>Flowability & Material Strength</td></tr><tr><td rowspan="3">3ITT</td><td>G'<sub>rec</sub></td><td>Storage modulus in the “recovery plateau”</td><td>(Pa)</td><td>Recoverability</td></tr><tr><td>%G'</td><td>Extent of recovery</td><td>(-)</td><td>Recoverability</td></tr><tr><td>λ<sub>I</sub></td><td>Mutation time</td><td>(s)</td><td>Recoverability</td></tr><tr><td>Time sweep</td><td>G'</td><td>Storage modulus evolution in time</td><td>(Pa)</td><td>Recoverability & Ageing (Material Strength)</td></tr></tbody></table>

and “recoverability” that are compiled in Table 1. Here we provide a summary of rheological metrics (that describe the ink behaviour above and below the yield stress, from solid-like to liquid-like) linked to “printability”, aiming to adopt a consistent terminology in the field and work towards the establishment of more comprehensive printability criteria and rheological maps to enhance the reliability of ceramics’ DIW manufacturing processes. A barrier we have encountered is that *flowability* and *recoverability* have not been consistently measured and defined in DIW literature. This table and review purposely do not provide specific values or intervals that metrics should reach for an ink to be printable, owing to the diversity of inks, printers, and printing parameters that also affect “printability” and shape fidelity.

## 5.1. Flowability metrics

The printability metrics often combine data extracted from simple rheological tests. Feilden et al. [54] defined a figure of merit ($\frac{G'_{eq}}{\tau_y}$) as a printability criterion. Noting that in this case $\tau_y$ is determined as the stress at the moduli crossover ($G' = G''$), and to avoid confusion, we refer to it as $\sigma_f$, using the subscript f for flow (Table 1), since this value is taken at the solid-to-liquid transition. Corker et al. [37] proposed a combination of oscillatory and continuous shear tests to quantify structure evolution and recovery timescales (mutation times), as well as the flow transition index (FTI) as an additional metric. Agrawal and García-Tuñón et al. [5] recently proposed the use of three Ashby-type maps using a comprehensive set of metrics from LAOS analyses to quantify the flowability, recoverability and material strength of DIW inks. The flowability map is constructed plotting a dimensionless energy dissipation ratio ($\frac{\phi_{SAOS}}{\phi_{LAOS}}$) vs the $FTI^{-1}$, which is comprised between 0 and 1, enabling to compare the yielding process of different inks.

However, even when reporting all these metrics, they can be misleading if they are used isolated. We recommend performing a careful analysis of all the data collected along with any metric reported. For example, plotting oscillation stress amplitude ($\sigma_0$) vs strain amplitude ($\gamma_0$) results from a DSS test, or plotting the storage and loss modulus ($G'$ and $G''$) vs stress amplitude, $\sigma_0$ (instead of strain) can reveal considerable stress overshoots in the flow transition region, which are associated with microstructure disruption [33,52].

## 5.2. Recoverability metrics

The extent of the recovery can be assessed by comparing the storage modulus in the LVER ($G'_{LVER}$), or first step in the 3ITT test, with the storage modulus in the third plateau ($G'_{rec}$), while the mutation time, $\lambda_I$ (Eq. (1)) quantifies the timescales of the recovery. Agrawal and García-Tuñón proposed a recoverability map that correlates the timescale and extent of recovery plotting $\lambda_I$ vs (%$G'_{rec}$) [5], and shows that better resolution is achieved (as intuitively expected) when the extent of recovery is maximum in the shortest timescales. Metrics obtained from shear cessation experiments ($\tau_{ss}^{-}$, and $t_{ss}^{-}$, Table 1) can also be used to quantify the stress and time to reach steady state within the experiment time frame. Other approaches to assess thixotropy have been proposed, for example, Wang and Ewoldt provide a framework for time-dependent dynamics that can be directly applied to quantify thixotropy in DIW [63].

## 5.3. Material strength metrics

Material strength has also been widely used as a printability indicator in many other studies in DIW [4,54] plotting the storage modulus ($G'_{LVER}$) vs the “yield” stress (note that we refer to it as $\sigma_f$, Table 1). However, this measure of strength in the linear regime (Linear Visco Elastic Region, LVER) is only appropriate in the absence of thixotropy. Therefore, it is recommended to quantify the material strength from a recovery test ($G'_{rec}$) [5]. Alternatively, applying successive DSS experiments (with increasing amplitude, decreasing amplitude and increasing again) enables the quantification of the impact of shear history (Section 4.5) on the sample.

## 5.4. Where to start with rheological characterisation?

Quantifying the printability of ceramic inks requires to effectively utilise rheological characterisation, and to continue to adapt and develop protocols to better mimic the printing process and its subsequent stages to ensure a comprehensive understanding of the entire process. Based on the state-of-the-art, we propose to perform rheological analyses that enable to quantify the yielding process (flowability, using LAOS) and the extent and timescales of structural recovery (Table 1).
<figure><img src="image_10.png" /><figcaption>Fig. 7. Rheology guide to determine rheological indicators associated with “printability” of ceramic feedstocks for DIW: Proposed flowchart to perform rheological analysis following (1) practical considerations for rheological analysis discussed in Section 4, (2) performing preliminary tests on materials already known in literature (e.g., Pluronic F-127, 25 wt%) and compare the results to validate your protocol and data collection, (3) apply the 5 tests (flow sweep, start-up, DSS, 3ITT and time sweep) to obtain the rheological metrics compiled in Table 1. Plot the results in context with literature and report them along with printing parameters and printer specifications.</figcaption></figure>

From these experiments, material strength can also be determined. A simple flowchart shows how to perform rheological analysis when users prepare a (new) ink formulation, considering the quantitative metrics presented in Table 1, and the summarised guidelines in Fig. 7.

## 6. Linking rheology and performance of printed parts: a bespoke printability roadmap for ceramics DIW

Unifying a quantitative “printability” definition and rheological criteria requires a collective effort to: (1) create a comprehensive database, including rheological indicators, printing parameters, printer specification and relevant final properties, and (2) reconsider the meaning of existing printability criteria, what they represent, and their potential limitations (Table 2). A common barrier is that optimal values of the rheological metrics listed in Table 1 simultaneously depend on ink formulation and printing conditions as reported by Inês dos Santos et al. [64] or del Mazo et al. [8]. Here, we compile a summary of the existing criteria, and provide a critical discussion of the behaviours they describe (e.g., flowability, recoverability, or strength) and their limitations.

M'Barki et al. [56] proposed a dimensionless printability criterion $\Xi$, which takes values > 1 when the dynamic yield stress ($\tau_y^{dyn}$) of an ink overcomes gravitational and surface tension forces. If this criterion is met, the printed object's distortions should be minimised. This “printability” number accounts for ink density, nozzle diameter, and it can thus provide a maximum object height. This criterion can therefore be associated to a measure of strength of the ink formulation, and shape fidelity of the printed object (Table 2). Feilden et al. [54] introduced a figure of merit $\Phi$ (note the distinction between this parameter, and the dissipation ratio introduced in Table 1), $\frac{G'}{\sigma_f} > 20$, which is similar to Chan et al.'s [65] $K$ parameter. Both of these parameters are determined using the equilibrium modulus ($G'_{eq}$) and the yield stress ($\tau_y^{dyn}$, or $\sigma_f$), and are therefore associated with Material Strength. The Flow Transition Index (FTI) [5,37], a dimensionless parameter, is calculated from DSS tests, as the ratio between the stress at the nonlinear onset ($\sigma_{nl}$ or end of the LVER) and the flow stress ($\sigma_f$, stress at the moduli crossover, $G' = G''$). FTI$^{-1} = \frac{\sigma_{nl}}{\sigma_f}$ can vary between ≈ 0 and 1. When FTI$^{-1} \to 0$, $\sigma_f \gg \sigma_{nl}$ which means that the transition from solid-like to liquid-like, also called flow transition, is extensive. While, FTI$^{-1} \to 1$ when $\sigma_f \approx \sigma_{nl}$ means that the transition is abrupt. This criterion combined with the trends of the dissipation ratio is indicative of the type of yielding, or flowability.

Inês dos Santos et al. proposed a recovery rate parameter $\bar{R}$ based on the 3ITT test slope analysis (Table 2). This criterion evaluates the

recovery rate, while 3ITT test is also used to evaluate the percentage of recovery of the storage modulus. This recoverability criterion is similar to the determination of the mutation times, $\lambda_{mut}$, proposed by Corker et al. [37] and the map proposed by Agrawal and García-Tuñón [5] (Table 2). These criteria are both associated with the ability to recover, however they cannot account for flowability or material strength.

From this critical discussion, the conclusions that can be drawn are: (1) the need to use complementary metrics to describe the interplay between these aspects [5], and (2) that a single criterion cannot be used to predict or quantify printability. Among the criteria that account for material strength, M'Barki's printability number [56] is the only criterion that accounts for the object's dimension and the printing parameters (nozzle diameter and extrusion velocity). However, systematic studies of the impact of printing conditions on this criterion are yet to be reported. Although other studies report shape fidelity measures depending on printing velocity and nozzle tip dimensions [66], this remains rare for ceramic materials. Ashby-type maps using some of the criteria here discussed are currently being developed in the field [4,5]. However, a common criticism on the value of these maps is that they cannot capture the impact of non-rheological factors, such as printer technical specifications, and printing parameters (including flow rate, printing velocity, nozzle type, geometry and dimensions, and height from the substrate). Despite this limitation, there is still value on pursuing the establishment of these maps and to further develop them to account for printing settings. For example, by using dimensionless numbers as proposed by Wei et al. [24].

**Process considerations:** In the absence of thixotropy, Geffrault et al. showed that different patterns emerge depending on flow conditions ($V_e$, $V_m$) and height away from the substrate (H), providing a theoretical base to identify suitable printing parameters [9]. Correlating processing parameters to the resulting printing patterns, could also serve to improve ‘in-situ’ monitoring strategies. Recent work combining rheology and ‘in situ’ monitoring during the printing process enables adjusting printing parameters on the go [67]. Hossain et al. evidence the links between observable haptic feeling and rheological quantities that could lead us to better understand the link between extrusion quality and rheological properties [68].

**Final performance considerations:** In bioprinting applications, the term printability also accounts for cell survival and bioactivity of the printed object [69–71]. A recent definition of bioprintability given by Mosnier et al. claims that final performance (as shape fidelity and mechanical strength) can be improved by fine-tuning of the ink recovery and other rheological properties [72]. Following a similar concept, we propose that the final performance of 3D-printed ceramic
<table><caption>**Table 2**
Printability criteria and their rheological interpretations.</caption><thead><tr><th>Printability criterion</th><th>Physical meaning</th><th>Dependency on printing parameters</th><th>Level of printability</th><th>Limitations</th></tr></thead><tbody><tr><td>Yield-Forces ratio Ξ [56]</td><td>Minimising object distortion</td><td>Minimal</td><td>Material Strength</td><td>Surface tension can not be easily determined for yield stress fluids. Cannot capture yielding type, or recovery timescales</td></tr><tr><td>Modulus-yield product K [65]</td><td>Avoid slumping</td><td>Medium</td><td>Material Strength</td><td>Cannot capture yielding type, or recovery timescales</td></tr><tr><td>Modulus-yield ratio Φ [54]</td><td>Low deformation & extrudability</td><td>High</td><td>Material Strength</td><td>Cannot capture yielding type, or recovery timescales</td></tr><tr><td>Flow Transition Index FTI [37,52] Energy dissipation ratio Φ<sub>LAOS</sub>/Φ<sub>LAOS</sub> [5]</td><td>Yielding type (associated with printing resolution)</td><td>Minimal</td><td>Flowability</td><td>High dependency on materials. Cannot capture microstructure disruption</td></tr><tr><td>Recovery rate R̄ [64]</td><td>Extent of elastic recovery</td><td>High</td><td>Recovery</td><td>Dependency on the recovery rate measurement method</td></tr><tr><td>Extent of recovery %G' Mutation time λ<sub>1</sub> G'<sub>rec</sub>, and σ<sub>f</sub> [5]</td><td>Re-structuring extent and timescales (associated with printing resolution)</td><td>Minimal</td><td>Recovery & Material Strength</td><td>Cannot capture structural disruption in the solid-to-liquid transition</td></tr></tbody></table>

objects should be used as an additional printability indicator (Figs. 1 and 7) as proposed by prior work [73]. Indeed, material strength of ink formulations is not only important as a “printability” criterion, it is also linked to the final properties of printed parts (linking back to Fig. 1). Tuning the recovery rate of the recovered “strength” enables to manipulate the surface finish and to control the final properties of printed parts. The control of surface topography prevents the formation of cracks or high stress concentration due to surface defects that increase the failure of the printed part.

What is common to any research area within DIW, is the agreement on the importance of the design and understanding of YSF for DIW. This 3D printing technology is a promising solution for rapid fabrication of high-added value materials, with reduced raw material usage and energy consumption. Improving our understanding and control of rheological behaviours has the potential to enhance the mechanical and functional properties of printed structures. Progress towards this goal requires the systematic investigation of DIW ceramic inks from an inter-disciplinary perspective, including rheology, fluid mechanics and ceramics manufacturing. Based on these reflections, the schematic in Fig. 8 illustrates the broader context and considerations that determine the printability of ceramic inks in DIW. Rheology plays a central role at the base of inter-related materials and process considerations: from ink formulation to final performance of 3D printed object in ceramics DIW. The proposed rheological framework (sections 1 to 5) is combined with a process-focused approach (Fig. 8). For example, aspects to consider from a process-focused perspective include: the adjustment of printing parameters, the technical limits of the machine itself to widen the range of printable material, or even environmental conditions (e.g., temperature, humidity). For example, nozzle direction has been optimised to improve final performance of 3D printed objects [74–76]. The combined materials and process approach requires a deeper fundamental understanding in non-Newtonian fluid mechanics (Fig. 8) and needs multi-disciplinary collaboration across different scientific communities to advance the field [77].

## 7. Data science and machine learning in printability for DIW ceramics

The digital era opens up new opportunities in this context. Data-driven formulation optimisation offers new avenues for the development of predictive strategies in ink formulation and the quantification of “printability” thus accelerating the field by minimising trial-and-error approaches. Using rheological metrics as input features to predict the final performance of printed ceramic parts presents both a promising opportunity to advance direct ink writing (DIW) towards ideal printability and a significant challenge in applying machine learning (ML) methods to rheological data [24,38]. As discussed in this review, printability maps can be challenging to achieve due to the diversity of materials, printers and printing parameters. Moreover, the heterogeneous experimental approaches in rheological characterisation for DIW could lead to large uncertainty of results for printability criteria. To overcome this, ML approaches could help accelerating the development of bespoke printability maps based on a specific ink formulation to optimise the entire process, from formulation to final printed parts [78]. However, ML approaches for this experimental research could be challenging. Firstly, because ML approaches require a degree of simplification which is difficult to achieve due to the large number of parameters and variables in 3D printing. Secondly, training these systems require robust inputs from experimental data (as schematically represented in Fig. 9), which can be time consuming and resource demanding in the DIW field. To overcome this challenge, random forest regression models or multiple linear regression could be explored if sufficient data were available especially for complex fluid modelling, as inks used in DIW [79]. Rheological metrics appear as relevant inputs to quantify print success with measurable scores as illustrated in Fig. 8. However, as discussed here, these rheological experiments must be performed with care, while the curation and analysis of results must be rigorous and comprehensive to provide useful metrics. This concern has highlighted the need to bring communities together, to invest time and resources on the same subject [80].
RHEOLOGY AND PRINTABILITY IN CERAMICS DIW

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Broader context on the concept of printability of ceramic inks for DIW inspired by Mosnier et al. [72]. Rheology is central to the DIW process, and sits at the base of inter-related materials and process considerations, which include from ink formulation to final performance of 3D printed object in ceramics DIW.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Summarised machine learning approaches in ceramic DIW for printability scores or probabilities in recent literature.</figcaption></figure>

The existing literature on ceramics DIW seems to focus mainly on achieving a fundamental understanding of material properties with less emphasis on process-related parameters. Whereas, in bioprinting literature, more effort seems to be directed towards printability through trial-and-error optimisation of different parameters (ink concentrations, viscosities, and printer settings) [81,82]. Recently, researchers have combined efficient data-driven approaches and rheology to discuss the best model to improve the accuracy of AI models, thanks to physics-informed machine learning (PIML). This approach allows the integration of fundamental physic law models and showing complex material behaviour [83]. Open-source mathematical modelling solutions [84] open up new opportunities to analyse rheological data and fitting to different models, even for complex materials such as the YSFs discussed in this short review. A non-rheology focused analysis combined with ‘in-situ’ monitoring to quantify printability as a measurable score exemplifies a different approach [27,85].

8. Conclusion

This review provides a detailed overview of established rheological experiments to determine specific rheological indicators associated with three characteristic features that a DIW ink should exhibit: *flowability*, *recoverability* and *material strength*.

Existing rheological criteria of ceramic inks for DIW have been critically discussed, highlighting advances in the field as well as current limitations. Based on the emerging reflections, we proposed a flowchart for rheological characterisation and a broader contextual framework in DIW of ceramics, combining material-focused and process-focused approaches.

Building bridges across research communities in ceramics manufacturing, rheology and non-Newtonian fluid mechanics can accelerate progress in the DIW field by improving our fundamental understanding, and by paving the way to standardisation of 3D printing DIW processes.

The success of data driven approaches requires improved and consistent rheological characterisation, data extraction, and analysis, with
the aim of training machine learning models for ceramics DIW. Rheologists, ceramists, and computational engineers are working together towards this goal.

## CRediT authorship contribution statement

**Mathilde Maillard:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Conceptualization.
**Sylvain Fournier:** Writing – review & editing, Methodology, Conceptualization.
**Esther García-Tuñón:** Writing – review & editing, Validation.
**Edwin-Joffrey Courtial:** Writing – review & editing, Supervision, Funding acquisition.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was supported by a UKRI Future Leaders Fellowship MR/V021117/1.

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