Challenges in Rheological Characterization of Highly Concentrated Suspensions — A Case Study for Screen-printing Silver Pastes

The viscosity is a key parameter in fluid processing and for multiphase fluids its dependence on shear rate is often determined using parallel-plate rotational rheometry. For highly concentrated suspensions this is neither a straight forward nor a trivial task and the definition of a suitable measuring protocol can be challenging. Here it is demonstrated how highly concentrated silver pastes can be rheologically characterized combining rotational rheometry and video recordings. A robust experimental protocol for determination of the steady shear viscosity is established and the accessible shear rate range is determined. Figure 1 represents an overview of apparent viscosity and apparent shear stress vs. the applied apparent shear rate for paste B. The measurement is done with a plate roughness of R_q = 2 – 4 µm. Cutouts from video recordings permit the division of the obtained flow curve into three sections. In section one wall slip dominates. The upper plate glides without paste deformation. In this section the shear stress is constant. Paste deformation sets in at _min, app = 0.07 s^-1 marking the onset of section two. At the same time the shear stress begins to increase. The deformation of the paste and the stress increase monotonically until section three is reached. At a critical shear rate or angular speed the paste creeps out of the gap and at the same time apparent viscosity and shear stress drop strongly due to the sample spill. Accordingly, the viscosity and shear stress curves exhibit a characteristic kink which occurs at about
_max, app = 2.5 s^-1. This _{max, app} marks the onset of sample spill. The higher the shear rate the faster the paste is ejected. The viscosity of the paste is accessible only in the shear rate range at _{min, app} < _app < _{max, app}. However, since the deformation inside the gap is not known and a priori must not be the same as observed at the rim the viscosity data even in that nominal shear rate range has to be treated as the apparent values. The shear stress τ_app at the rim of the plate r_max is calculated from the applied torque T in the following way
τ_app = T (2π r_max³)^-1 [3 + d(ln T) / d(ln _app)]. The apparent shear rate _app at the edge of the plate is calculated from the angular velocity Ω of the plate and the gap height h according to _app = Ω (r_max / h)³⁴. Since the true deformation and stresses inside the gap are not known these calculated stress and shear rate values have to be treated as apparent or nominal values.

In soft matter often a critical stress, the so-called apparent yield stress τ_{y, app}, is found at which a transition from an elastic reversible deformation to irreversible flow is observed. This yield stress is a key factor in paste formulation regarding classical screen-printing as well as emerging additive manufacturing techniques. A high yield stress is desirable to ensure shape accuracy after printing. Generally, the yield stress is determined from the kink in the deformation vs. shear stress curve using the tangent intersection method as exemplarily shown in Figure 2. Often this is done using a so-called vane geometry guaranteeing reliable and significant results without the effect of slip^35,36. Measuring the yield stress with the parallel-plate geometry is another option which has to be carefully validated. Wall slip or shear banding phenomena often observed in highly filled suspensions may interfere with the yield stress evaluation. Therefore, the effect of plate roughness on τ_{y, app} determination is investigated. Results for the yield stress values obtained for paste A and B in stress sweep experiments at different plate roughness are shown in Figure 3. Increasing the plate roughness causes an increase in calculated yield stress, whereas the variation in gap height h does not affect determination of this quantity. Figure 4 displays cutouts of the video taken for paste A at a plate roughness of R_q = 1.15 µm and a gap height of h = 1 mm. Soot particles were placed on the sample rim as a marker and endoscopic video imaging was used to characterize the deformation at the sample rim. The paste glides on the bottom plate at stresses up to τ_app = 600 Pa while it sticks to the upper plate. A plug flow is formed in the measuring gap, i.e. the sample is not deformed and the determination of a yield stress or viscosity is pointless even though a kink in the corresponding apparent deformation vs. shear stress curve seems to imply a transition from elastic deformation to viscous flow. Similar behavior is obtained for other gap heights for paste A as well as paste B. So a plate roughness of R_q = 1.15 µm is not appropriate for the determination of yield stress or viscosity of such highly filled silver pastes. In contrast, for a plate roughness R_q = 2 – 4 µm (as declared by the manufacturer) video imaging confirms the formation of a shear deformation profile at the rim (Figure 5) as necessary for a reliable and well-defined rheological measurement. Plug flow is avoided and for paste A uniform flow sets in at τ_app = 1,360 Pa. Similar flow behavior was observed for paste B. So this choice of plate roughness allows for a reliable yield stress measurement. Choosing a higher plate roughness R_q = 9 µm results in higher yield stress values than obtained for plate roughness R_q = 1.15 µm and R_q = 2 – 4 µm. This effect is much more pronounced for paste A than for paste B. The video recordings show that no shear profile is formed with paste A during this measurement (Figure 6). At a stress τ_app = 1,880 Pa the upper plate starts moving without paste deformation. A stress of τ_app = 2,605 Pa causes gliding of the paste on the bottom plate still without paste deformation. The critical stress corresponding to the kink in the deformation vs. stress curve does not mark the transition from elastic to viscous deformation, i.e. it is not the apparent yield stress. Instead it marks the onset of slip and plug flow and has to be considered as critical slip stress τ_slip. In contrast, no plug flow was observed for paste B using the R_q = 9 µm plate (Figure 7). The deformation of the paste starts at τ_app = 1,430 Pa and is fully developed at τ_app = 1,597 Pa. At higher shear stress (τ_app = 1,880 Pa) shear banding occurs, i.e. only an intermediate narrow layer of the sample is sheared. The yield stress obtained from the deformation vs. stress data with the R_q = 9 µm plate is close to that obtained with R_q = 2 – 4 µm in case of paste B, but it is pointless to use this rough plate for τ_{y, app} determination of paste A. To double-check the parallel-plate R_q = 2 – 4 µm results, the yield stress was also measured with the vane geometry. This geometry is inherently not affected by wall slip effects and the onset of rapid vane rotation at a certain applied stress is unequivocally related to the structural breakdown within the paste in the cylindrical plane defined by the diameter of the vane^35,36. Figure 8 shows that the results obtained using the vane geometry agree very well with those obtained from parallel-plate rheometry with R_q = 2 – 4 µm. Based on the findings presented above, all further experiments were performed using a plate with roughness R_q = 2 - 4 µm except for the wall slip velocity measurements. The plates with roughness R_q = 1.15 µm and R_q = 9 µm cannot be recommended for yield stress and viscosity determination of silver pastes or other highly filled suspensions similar to those investigated here. Finally, it is stated that the yield stress of paste A is higher than that of paste B.

Wall slip is another important parameter for successful printing. The higher the wall slip, the better the paste flows through the screen mesh openings³². The wall slip velocity, i.e. the relative velocity of the moving plate and the adjacent paste layer, can be evaluated directly from video recordings irrespective of plug flow or shear deformation prevailing in the gap. A smooth upper plate and a rough bottom plate have to be used when performing these experiments^25,27,28,30. If the sample within the gap is at rest, the slip velocity is directly given by the speed of the upper plate. Figure 9 displays the wall slip velocity vs. shear stress as determined under this latter conditions using a plate with R_q = 1.15 µm. Slip clearly occurs at stresses far below the yield stress similar as observed for concentrated emulsions and pastes of soft microgel particles^28,29,30. For paste A, higher wall slip velocities are obtained than for paste B irrespective of applied stress. In both cases, slip velocity increases linearly with applied stress. However, the obtained slope m_A = 0.33 µm (Pa s)^-1 for paste A is nearly three times higher than the slope m_B = 0.12 µm (Pa s)^-1 obtained for paste B. Similar as observed previously^28,29,30, a characteristic slip velocity V* at about a stress level corresponding to the yield stress is found and above τ_y slip is hardly measurable. For paste A and B, V_A* = 0.37 mm s^-1 and V_B* = 0.11 mm s^-1, respectively.

The sample spillover is attributed to the strong centrifugal force acting on the high density micron-sized particles and should therefore be controlled by the angular or rotational speed n_crit at which the centrifugal force dominates over viscous friction. To test this, the measuring gap height h was increased from 0.2 mm to 2 mm. The intensity of sample spillover increases with gap height h and shear rate. The wider the gap height the earlier sample spill sets in, i.e. _crit is lower (Figure 10). Figure 11 demonstrates that sample spill sets in at a critical angular speed n_crit irrespective of sample height h between 0.5 mm and 2 mm. For paste A, the critical rotational speed is n_{crit, A} ≈ 0.6 min^-1 and for paste B it is n_crit, B ≈ 1.7 min^-1. The finding n_{crit, A} < n_{crit, B} might be due to different vehicle viscosity or due to varying silver particle size. However, both pastes exhibit much higher n_crit values for a measuring gap height h = 0.2 mm. Thus, decreasing the gap height allows for a wider shear rate range in which viscosity determination is possible. The reason for the high n_crit values found for h = 0.2 mm is not yet clear. This might be due to a stronger contribution of surface tension at the sample rim or due to the formation of aggregates clogging the narrow gap. Further investigations are necessary to clarify that. Figure 10 further confirms that for _app = 0.07 s^-1 – 2.5 s^-1 the apparent viscosity data obtained at different gap heights do not vary systematically, i.e. wall slip is negligible under these experimental conditions. Varying the shear rate from high to low or from low to high values yields the same viscosity data as long as n_crit is not exceeded, i.e. no spillage takes place, i.e. there is no evidence of irreversible structural change within the sample.

A capillary rheometer is used to determine the paste viscosity especially at process-oriented high shear rates. The Weissenberg-Rabinowitsch correction for non-parabolic velocity profile is done here to get the true shear rate in the case of non-Newtonian fluids³⁴. The entrance pressure loss is negligible because of the high L/d ratio >> 1, but the occurrence of wall slip has not been investigated in this case and therefore data have to be treated as apparent viscosity values. Figure 12 displays the apparent viscosity for both pastes A and B determined with parallel-plate rotational rheometry and capillary rheometry. Remarkably, the data obtained from both experimental techniques seem to agree very well for both pastes suggesting that wall slip is of minor relevance in the capillary rheometry measurements performed here. Finally, paste A and B exhibit similar apparent viscosity at low shear rates but viscosity of paste A is higher than that of paste B in the high shear regime.

Steady shear measurements using parallel-plate rotational rheometers have to be performed carefully and may be disturbed by wall slip, shear banding or sample spillage as extensively discussed above. Therefore, using oscillatory shear experiments have been suggested to characterize structural breakdown and recovery of silver pastes during the screen-printing process. This is done in a three stage oscillatory test as suggested by Hoornstra, Zhou and Thibert^10,15,21. First, an amplitude sweep has to be performed to determine the linear and non-linear response regime at a pre-selected frequency (Figure 13). The linear viscoelastic regime distinguishes itself by constant, -independent modulus values and G' > G''. The decay of the storage modulus G' at large is chosen as a criterion to identify the onset of the non-linear response regime. The characteristic deformation amplitude _c marking the transition from linear to non-linear response is defined as the amplitude at which G' has decreased to 80% of its average initial value G'₀ in the linear regime: G'(_c) = 0.8 G'₀. In stage I and III of the test, a small oscillation amplitude within the linear viscoelastic response regime, i.e. < _c is selected to characterize the rest structure of the paste (stage I) as well as the time dependence and degree of recovery in stage III of the test after the destruction of the initial structure due to a high deformation amplitude applied in stage II. Figure 14 shows corresponding results for paste B. In stage I, the paste is deformed at = 0.025% and G' is higher than G'', i.e. the elastic behavior of the paste is predominant. When deformation is increased in stage II, G'' is higher than G' ensuring the structural breakdown within the paste during this period of large deformation. Stage III simulates the resting of the finger lines on the substrate after printing. In this stage G' is higher than G'' again, but G' and also G'' are both lower than the respective initial G' and G'' values before the paste structure was destroyed. The video recordings confirm that this is not related to the effects like wall slip, plug flow or sample spill. The paste deforms uniformly during oscillatory shear, sticks to the plate, and indicates neither wall slip nor sample spill. Therefore, it can be concluded that the incomplete recovery of the shear moduli indicates an irreversible structural change within the sample due to the applied large amplitude shear in stage II. The data shown in Figure 15 (a) reveal that the degree of irreversible structural change does not depend on the duration of the large deformation amplitude oscillatory shear applied in stage II. The results in stage III do not vary for different periods of shearing time t_II. However, the value of the selected deformation amplitude in stage II has a strong impact on the degree of structural recovery. This is obvious from Figure 15 (b) showing the difference between the storage modulus values determined in stage III and I ΔG' normalized by the initial modulus value G'_I. For > 20%, i.e. at deformations corresponding to the crossover of G' and G'' (see Figure 13) paste B recovers only 30% of its initial value and paste A only 10%. This is considered a paste property crucial for various coating operations and its dependence on paste composition will be addressed in future work. Note, the structural recovery immediately after cessation of large amplitude oscillatory shear in stage II would be of utmost relevance especially for the screen-printing process but the commercial rheometer used here does not allow for determination of reliable data within the first second after changing .

Filament stretching experiments have been performed to simulate the snap-off during screen-printing. The snap-off belongs to the final step of screen-printing. Figure 16 demonstrates that the filament lengths at break increases with increasing stretching velocity for both pastes. Breakage always occurs at lower stretch ratio (x_br - x₀) / x₀ = Δx_br / x₀ for paste B than for paste A, but this difference seems to decrease slightly with increasing stretching velocity. Since the filament breakage of paste B happens at a lower stretch ratio this paste may have better snap-off properties.

The results obtained with the capillary breakup elongational rheometer imaging setup are confirmed by the tensile tester experiments. Corresponding results are shown in Figure 17. Again the stretch ratio at which the filaments break increase with increasing stretching velocity (Figure 17 (a)) and paste B breaks at lower Δx_br / x₀ values than paste A. However, the absolute values Δx_br / x₀ obtained with the capillary breakup elongational rheometer are always higher than corresponding values obtained with the tensile tester. This is attributed to the different plate diameters d = 6 mm and d = 5 mm, i.e. different initial sample volumes for the capillary breakup elongational rheometer and tensile tester. Finally the tensile tester also provides a second characteristic parameter the maximum force F_max acting on the filament during stretching. This quantity also increases linearly with increasing separation speed but in this case the values obtained for paste B are larger than those for paste A. Further investigations will be needed to disclose the relevance of F_max for the screen-printing process or other coating operations.

Figure 1. Apparent viscosity in controlled shear rate parallel-plate rotational rheometry. Overview of the resulting apparent viscosity and shear stress measured in controlled shear rate mode with a parallel-plate rheometer (short: PP), plate roughness R_q = 2 – 4 µm and gap height h = 1 mm. Classification of apparent shear rate in three sections based on video recordings during measurement. Wall slip, paste deformation and sample spill are highlighted in the video recording cutouts. Paste B is chosen exemplarily here, but similar results were obtained for paste A. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 2. Applying the tangent intersection point method for yield stress determination. The deformation γ, i.e. the displacement of the upper plate divided by the gap height, is plotted versus the applied nominal shear stress τ_app. The shear stress controlled measurement is done with a plate roughness R_q = 2 – 4 µm and a gap height h = 1 mm. Paste B is chosen exemplarily here, but similar results were obtained for paste A. Please click here to view a larger version of this figure.

Figure 3. Effect of plate roughness on apparent yield stress. The resulting apparent yield stress for different plate roughness R_q = 1.15 µm, R_q = 2 – 4 µm and R_q = 9 µm vs. gap height h. Results depend on the plate roughness R_q and is independent of varying the measuring gap height (left: paste A, right: paste B). The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 4. Cutouts from video recordings at varying shear stress. Here using the example of paste A measured with plate roughness R_q = 1.15 µm. Please click here to view a larger version of this figure.

Figure 5. Cutouts from video recordings at varying shear stress. Here using the example of paste A measured with plate roughness R_q = 2 - 4 µm. Please click here to view a larger version of this figure.

Figure 6. Cutouts from video recordings at varying shear stress. Here using the example of paste A measured with plate roughness R_q = 9 µm. Please click here to view a larger version of this figure.

Figure 7. Cutouts from video recordings at varying shear stress. Here using the example of paste B measured with plate roughness R_q = 9 µm. Please click here to view a larger version of this figure.

Figure 8. Resulting yield stress. A comparison of the yield stress for paste A and B determined with the vane geometry and the R_q = 2 – 4 µm parallel-plate geometry. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 9. Dependence of wall slip on shear stress. Wall slip velocity v_slip vs. shear stress τ for paste A and B determined with R_q = 1.15 µm parallel-plate geometry at gap height h = 1 mm. The characteristic wall slip velocity obtained at the yield stress of the material is indicated as V*. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 10. Effect of gap height on sample spill. Apparent viscosity at varying measuring gap height h in controlled shear rate mode for paste B and plate roughness R_q = 2 – 4 µm. The gap height is varied between h = 0.2 mm and h = 2.0 mm. Downward kink of viscosity curve sets in at higher shear rates when a lower gap height is chosen. The gap height is varied between h = 0.2 mm and h = 2.0 mm. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 11. The critical rotational speed at which sample spill sets in. Rotational speed n_crit at which the downward kink of the viscosity curve sets in vs. gap height h using parallel-plate geometry with R_q = 2 – 4 µm (left: paste A, right: paste B). Sample spillover sets in at this characteristic rotational speed as confirmed by video recordings. For paste A sample spill sets in at n_{crit, A} = 0.6 min^-1 and for paste B at n_{crit, B} = 1.7 min^-1 for gap height h ≥ 0.5 mm. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 12. Viscosity in a wide shear rate range. Apparent viscosity of paste A and B determined in a wide shear rate range using parallel-plate rotational rheometry (gap height h = 0.2 mm and h = 1 mm; R_q = 2 – 4 µm) and capillary rheometry (d = 0.5 mm and L = 40 mm). The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 13. Determination of the linear and non-linear response regime in oscillatory shear. Amplitude sweep test for paste B: G', G'' vs. deformation amplitude at fixed frequency f = 1 Hz. Test performed using a rotational rheometer equipped with parallel-plate geometry (R_q = 2 - 4 µm, gap height h = 1 mm). The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 14. Three stage structural recovery test. Three stage structural recovery test for paste B performed at constant frequency f = 1 Hz with a parallel-plate rotational rheometer (plate roughness R_q = 2 – 4 µm). The applied deformation amplitude in stage I is 0.025%, in stage II = 80% and in stage III = 0.025%. Video recordings confirm homogeneous sample deformation throughout the gap, no wall slip, shear banding, plug flow or sample spillage occurs. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 15. Structural recovery tests. (a) Effect of shearing time on structural recovery. Structural recovery of paste B for _II = 80% and different duration of stage II, t_II = 50 s, 150 s and 600 s. (b) Effect of deformation amplitude on structural recovery. Relative irreversible structural change
(G'(t → ∞) – G'(t = 0)) / G' (t = 0) = ΔG' / G'_I as a function of deformation amplitude applied in stage II of the three stage structural recovery test determined for paste A and B at constant t_II = 150 s. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Figure 16. Optical determination of filament breakage in elongational deformation. Critical tensile stretch ratio at which filament breakage occurs for pastes A and B at different stretching velocities as obtained using the capillary breakup elongational rheometer (initial gap height h = 1 mm). Please click here to view a larger version of this figure.

Figure 17. Tensile testing — axial force during elongational deformation. Resulting Δx_br / x₀ (a) and resulting force F_max (b) vs. stretching velocity obtained for paste A and B with the tensile tester. The initial gap height is h = 1 mm and the piston diameter d = 5 mm. The insert displays raw force F vs. stretch ratio data for paste B obtained at v = 30 mm s^-1 to demonstrate the determination of F_max and Δx_br. The error bars are calculated as the standard deviation obtained from at least three independent measurements. Please click here to view a larger version of this figure.

Table 1. Overview of applied methods, corresponding characteristic information and differences between specific flow properties of paste A and B. Please click here to view a larger version of this table.