In our extensive research on sand casting foundry coatings, we have dedicated significant effort to understanding and quantifying the rheological behavior that governs their technological performance. The ability to predict and control properties such as suspensibility, brushability, sagging resistance, and leveling is crucial for producing high-quality castings. Through a combination of theoretical analysis and experimental verification, we have established that the Casson model provides an exceptionally accurate mathematical description of the flow behavior of sand casting foundry coatings. This model serves as the foundation for a computer-aided analysis system that enables us to evaluate and optimize coating formulations systematically.
Rheological Behavior and Mathematical Model
The coatings used in sand casting foundry are typically pseudoplastic fluids with a yield value and thixotropic characteristics. Their internal structure, composed of high-molecular-weight compounds (binders, thickeners), bentonite (as binder and thixotropic agent), and refractory fillers, aligns well with the assumptions made by Casson in 1959 for similar suspensions. According to the Casson hypothesis, the molecules within the fluid form chain-like aggregates due to attractive forces; the size of these aggregates determines the viscosity. When the suspension flows, the aggregates are broken down by shear stress, and their equilibrium size depends on the shear rate. Consequently, both the aggregate size and viscosity vary with shear rate. This leads to the Casson model, expressed as:
$$ \eta^n = \eta_\infty^n + \tau_0^n \dot{\gamma}^{-n} $$
where:
- \(\eta\) is the viscosity (Pa·s)
- \(\dot{\gamma}\) is the shear rate (s⁻¹)
- \(\eta_\infty\) is the viscosity at infinite shear rate (high-shear viscosity, Pa·s)
- \(\tau_0\) is the yield stress (minimum shear stress required to initiate flow, in 10⁻⁵ N/cm²)
- \(n\) is an exponent in the range 0 to 1
From the above equation, we can also write:
$$ \eta^n = \eta_\infty^n \left(1 + \frac{\tau_0^n}{\eta_\infty^n \dot{\gamma}^{-n}}\right) $$
Defining a characteristic shear rate \(\dot{\gamma}_m\) as:
$$ \dot{\gamma}_m^n = \frac{\tau_0^n}{\eta_\infty^n} $$
Then:
$$ \eta^n = \eta_\infty^n \left(1 + \dot{\gamma}_m^n \dot{\gamma}^{-n}\right) $$
The parameters \(\eta_\infty\), \(\tau_0\), and \(\dot{\gamma}_m\) are fundamental physical constants in the Casson model, where \(\dot{\gamma}_m\) reflects the relative magnitude of structural properties and internal friction.
To validate the applicability of the Casson model for sand casting foundry coatings, we conducted a series of tests on various commercial and experimental coatings. The measurements were performed using a rotational viscometer (HAAKE RV-2) with a maximum shear rate of 1694 s⁻¹, following the ASTM D2196-68 standard for rheological testing of non-Newtonian materials. The results, summarized in the table below, show the regression correlation coefficients and the optimal Casson exponent \(n\) for each coating.
| Coating ID | Type | Density (g/cm³) | Max Correlation Coefficient | \(n\) value at max R |
|---|---|---|---|---|
| SH1 | Water-based zircon flour | 1.89 | 0.9999 | 0.65 |
| SH2 | Water-based corundum flour | 1.84 | 0.9850 | 0.80 |
| SH3 | Water-based bauxite | 1.68 | 0.9999 | 0.60 |
| SC1 | Alcohol-based zircon quick-dry | 1.72 | 0.9999 | 0.70 |
| SC2 | Quick-dry coating | 1.78 | 0.9423 | 0.85 |
| SC3 | Water-based coating | 1.88 | 0.9986 | 0.70 |
| WH1 | Water-based high suspension coating | 1.86 | 0.9999 | 0.65 |
| WH2 | Water-based chromite flour | 1.70 | 0.9070 | 0.80 |
| WH3 | Mixed-base quick-dry coating | 1.68 | 0.9753 | 0.75 |
| Holcote | Water-based zircon (imported) | 1.90 | 0.9997 | 0.65 |
| Foseco | Water-based zircon (imported) | 1.91 | 0.9939 | 0.65 |
As the table demonstrates, all tested coatings exhibited extremely high correlation coefficients (above 0.90) with the Casson model, and the exponent \(n\) consistently fell within the range of 0.5 to 0.85, with a concentration between 2/3 and 1/2. This confirms that the Casson model is indeed a superior mathematical representation for the rheological behavior of sand casting foundry coatings.
Quantitative Description of Rheological Parameters
To facilitate computer-aided analysis, we need to define a set of rheological parameters that are directly linked to the technological properties of sand casting foundry coatings. Through our study, we identified four key parameters:
- Yield stress \(\tau_0\) – describes low-shear viscosity and static structural strength.
- High-shear viscosity \(\eta_\infty\) – characterizes brushability.
- Casson B (structural thixotropy coefficient) – quantifies shear-thinning behavior.
- Casson M (time-dependent thixotropy coefficient) – quantifies recovery behavior after shear cessation.
Low-Shear Viscosity and Yield Stress
At very low shear rates (e.g., \(\dot{\gamma} \to 0.1\) to 1 s⁻¹), the contribution of \(\eta_\infty\) becomes negligible because \(\tau_0/\dot{\gamma}\) is typically 400–1000 times larger than \(\eta_\infty\). Thus, we can approximate the viscosity in this regime as:
$$ \eta \approx \frac{\tau_0}{\dot{\gamma}} \quad \text{(for low shear rates)} $$
Therefore, \(\tau_0\) serves as a reliable indicator of low-shear viscosity and the coating’s ability to remain suspended without sedimentation.
High-Shear Viscosity for Brushability
At high shear rates (around 10⁴ s⁻¹ during brushing), the viscosity approaches \(\eta_\infty\) because the term \(\tau_0^n \dot{\gamma}^{-n}\) becomes negligible. Hence:
$$ \eta^n \approx \eta_\infty^n \quad \text{(for high shear rates)} $$
Coating formulations with low \(\eta_\infty\) require less force during brushing, which is desirable for manual application.
Structural Thixotropy (Shear-Thinning) – Casson B
We define the structural thixotropy coefficient Casson B as the ratio of viscosity at a reference low shear rate \(\dot{\gamma}_0\) to the high-shear viscosity \(\eta_\infty\). This ratio quantifies how much the coating thins when sheared. From the Casson model:
$$ \frac{\eta_0}{\eta_\infty} = \left[1 + \left(\frac{\dot{\gamma}_m}{\dot{\gamma}_0}\right)^n\right]^{1/n} $$
Thus, we set:
$$ \text{Casson B} = \frac{\eta_0}{\eta_\infty} = \left[1 + \left(\frac{\dot{\gamma}_m}{\dot{\gamma}_0}\right)^n\right]^{1/n} $$
A larger Casson B indicates stronger shear-thinning behavior, which is beneficial for easy brushing combined with good sag resistance after application.
Time-Dependent Thixotropy – Casson M
The time-dependent thixotropy (recovery after shear) can be evaluated using the hysteresis loop method. By measuring the ascending and descending flow curves with a rotational viscometer, we convert them into Casson coordinates (linearized plot of \(\sqrt{\eta}\) vs. \(1/\sqrt{\dot{\gamma}}\) when \(n=0.5\), or more generally using the exponent \(n\)). The ascending curve yields a higher yield stress \(\tau_0\) (shear-induced breakdown), while the descending curve yields a lower yield stress \(\tau_\infty\) (recovery not yet complete). We define the time-dependent thixotropy coefficient Casson M as:
$$ \text{Casson M} = \frac{\tau_0 – \tau_\infty}{\tau_0} $$
This dimensionless parameter reflects the relative change in yield stress due to shear history and recovery time. A lower Casson M means faster recovery (good sag resistance but may cause brush marks); a moderate value allows proper leveling before the viscosity rises too high.
Relationships Between Rheological Parameters and Technological Properties
Based on systematic experiments and production experience in sand casting foundry, we have established empirical relationships linking the four rheological parameters to key coating properties. These are summarized in the following table.
| Technological property | Primary parameter | Secondary parameter | Desired range for optimal performance |
|---|---|---|---|
| Suspensibility | \(\tau_0\) | – | \(\tau_0 > 60\) N/m² (for zircon coatings) |
| Brushability | \(\eta_\infty\) | Casson B | \(\eta_\infty < 10\)–15 Pa·s; Casson B > 100–150 s⁻¹ |
| Sag resistance | \(\tau_0\), Casson M | – | \(\tau_0 > 60\) N/m²; Casson M < 40% |
| Leveling | \(\tau_0\), Casson M | – | \(\tau_0 > 100\) N/m²; Casson M between 20% and 40% |
Furthermore, we have identified how common coating ingredients affect these rheological parameters, as shown in Table 3.
| Ingredient | Effect on \(\tau_0\) | Effect on \(\eta_\infty\) | Effect on Casson B | Effect on Casson M |
|---|---|---|---|---|
| Bentonite | Medium | Medium | Small | Medium |
| CMC or alginate | Small | Small | Medium | Small |
| Bentonite + polymer compound | Large | Small | Medium | Large |
Computer-Aided Analysis System
To implement a practical computer-aided analysis for sand casting foundry coatings, we use a rotational viscometer (such as the NXS-11 type) that provides shear stress vs. shear rate data over a limited range. The system collects at least 10 data points and performs linear regression in the transformed Casson coordinates to determine the exponent \(n\) and the parameters \(\tau_0\), \(\eta_\infty\), and \(\dot{\gamma}_m\). Then, using the definitions above, we calculate Casson B and Casson M from the hysteresis loop data.
The software workflow can be outlined as follows:
- Input raw data from the viscometer (shear rate and corresponding shear stress).
- Perform regression analysis to find the optimal Casson exponent \(n\) and establish the rheological model.
- Compute the four rheological parameters \(\tau_0\), \(\eta_\infty\), Casson B, and Casson M.
- Evaluate the coating’s technological properties using the relationships in Table 2.
- If the properties do not meet requirements, the system suggests modifications to the formulation based on the ingredient effect table (Table 3).
- Output the results and graphical representations (flow curves, viscosity curves).
This approach allows us to iteratively optimize sand casting foundry coating formulations with minimal experimental effort, saving both time and materials. The combination of the Casson model and the defined rheological parameters provides a scientifically sound foundation for computer-aided design of coatings.
Conclusions
Through extensive theoretical and experimental investigations, we have reached the following conclusions:
- The Casson model \(\eta^n = \eta_\infty^n + \tau_0^n \dot{\gamma}^{-n}\) fits the rheological behavior of sand casting foundry coatings with extremely high accuracy. The exponent \(n\) is typically in the range of 0.5 to 0.85, with the majority of coatings exhibiting \(n\) between 2/3 and 1/2.
- Four rheological parameters are sufficient to quantitatively describe the key aspects of coating flow: \(\tau_0\) for low-shear static properties, \(\eta_\infty\) for high-shear brushability, Casson B for shear-thinning (structural thixotropy), and Casson M for time-dependent recovery (temporal thixotropy).
- Empirical relationships between these parameters and technological properties (suspensibility, brushability, sag resistance, leveling) have been established, along with the influence of common coating ingredients.
- A computer-aided analysis system integrating the Casson model, the defined parameters, and the empirical knowledge base enables efficient formulation design and evaluation for sand casting foundry coatings.
Our work provides a practical tool for foundry engineers and coating developers to achieve consistent, high-performance coatings through quantitative rheological control.