Liquid Bridge Model

The Liquid Bridge Model introduces a cohesive force between two particles (or between one particle and a Geometry) when the particles are wet.

The current model is based on the publications by Mikami et al and Schmelzle et al, where the forces and the amount of liquid involved in the bridge are estimated for the contact between two spheres or a sphere with a wall.

You can use this model for other types of particles in EDEM, such as Multi-Spheres or Sphero-Cylinders.

Note: Consider using this model when the material is slightly wet or partially saturated, but not when it is fully saturated.

Bridge formation

The liquid bridge is formed when physical contact occurs between two particles or one particle with a Geometry. When a gap is opened between the two entities in contact, the force is still active and changes with the size of the gap, and will only work if a Contact Radius is provided.

Note: The contact may exist before that due to the existence of a Contact Radius, however, the force is only activated when the contact is physical. A Contact Radius larger than the physical radius and large enough to include the gap at which the bridge breaks is mandatory for this model to work properly. For more information, see Rupture Distance.

The quantity of liquid attached to the particle is represented by the Custom Properties 'Liquid Mass Ratio' (mass of liquid attached to the particle over particle mass) and 'Volume Added' (volume of liquid attached to the particle). You can use both the Custom Properties simultaneously, but 'Volume Added' may be considered for deprecation in the next releases. The liquid represented by both Custom Properties is added and considered by the model as the available liquid.

Calculating Particle-Particle Force value

For a given gap ℎ, the bridge force between two particles is always aligned with the normal direction at the contact point and its value is:

$F = π r_{p} γ \hat{F}$

Here, γ is the surface tension of the liquid and $\hat{F}$ is computed as follows:

$\hat{F} = \exp (A \hat{h} + B) + C$

where

$\hat{h}$ is a dimensionless distance defined as $\hat{h} = h / r_{p}$ , and r_pis the harmonic mean of the radii of the particles in contact and parameters A, B, and C are computed as follows:

$A = - 1.1 {\hat{V}}^{- 0.53}$

$B = (- 0.34 \ln \hat{V} - 0.96) θ^{2} - 0.019 \ln \hat{V} + 0.48$

$C = 0.0042 \ln \hat{V} + 0.078$

The value θ is the wetting angle between the liquid and the material of the particle (in radians). In case the materials of the particles in contact is different, the average of both wetting angles is used.

$\hat{V}$ is the dimensionless volume of liquid in the bridge, defined as:

$\hat{V} = V / r_{p}^{3}$

Rupture Distance

The condition for the liquid bridge to break is defined by the following value of the dimensionless distance:

${\hat{h}}_{r u p t u r e} = (0.62 θ + 0.99) {\hat{V}}^{0.34}$

Calculating Particle-Geometry Force value

For a given gap h, the bridge force between two particles is always aligned with the normal direction at the contact point and its value is:

$F = π r_{p} γ \hat{F}$

γ is the surface tension of the liquid and $\hat{F}$ is computed as follows:

where $\hat{h}$ is a dimensionless distance defined as $\hat{h} = h / r_{p}$ , and r_p is the radius of the particle and parameters A, B, and C are computed as follows:

$A = - 1.9 {\hat{V}}^{0.51}$

$B = (- 0.016 \ln \hat{V} - 0.76) θ^{2} - 0.12 \ln \hat{V} + 1.2$

$C = 0.013 \ln \hat{V} + 0.18$

θ is the wetting angle between the liquid and the material of the particle (in radians). In case the materials of the particles in contact is different, the average of both wetting angles is used. $\hat{V}$ is the dimensionless volume of liquid in the bridge, defined as:

$\hat{V} = V / r_{p}^{3}$

Rupture Distance

The condition for the liquid bridge to break is defined by the following value of the dimensionless distance:

${\hat{h}}_{r u p t u r e} = (0.22 θ + 0.95) {\hat{V}}^{0.32}$

Liquid Volume and Liquid Exchange

The amount of liquid forming a bridge is provided by the particles in contact.

The available volume of fluid surrounding the particles and their radii define how much liquid is provided by each particle in Shi, D., & McCarthy, J. J. (2008). Numerical simulation of liquid transfer between particles. Powder Technology, 184(1), 64-75.

$V_{i} = \frac{L_{i}}{2} (1 - \sqrt{1 - \frac{r_{j}^{2}}{{(r_{i} + r_{j})}^{2}}})$

where i and j stand for each of both particles and can be swapped in the equation. In the case of a particle contacting a wall, all the liquid is provided by the particle and equals $V_{i} = \frac{L_{i}}{2}$ , as a particular case of the previous equation. Once the liquid bridge reaches its rupture distance, the liquid must be redistributed between both particles in contact. For this, the Schmelzle and Nirschl approach has been adopted. The fraction of the liquid volume recovered by particle i is:

$T F_{i} = E + (1 - E) {(1 - \frac{1}{1 + V})}^{D}$

where

$E = - 0.073 θ_{j}^{2} + 0.44 θ + 0.53$

$D = 0.076 θ_{j}^{2} - 0.186 θ + 0.3$

and V is the total volume of liquid present in the bridge. In order to ensure the symmetry of this equation, the model introduces a correction to the value of TF_i:

$\hat{T F_{i}} = \frac{T F_{i}}{T F_{i} + T F_{j}}$

In the case of Particle-Geometry, all the liquid goes back to the particle. In the case of Particle-Particle, the liquid is split and sent to both particles. The amounts sent to each particle respond to the fractions defined by TF_i and affect equally affect the portions of liquid brought by the custom property 'Volume Added' and 'Liquid Mass Ratio'.

Mass and inertia of the liquid attached to a particle

The mass and the inertia of the liquid content of a particle is considered through a particle body force model.

Using the Liquid Bridge Model

To use the Liquid Bridge model on a CPU or GPU:

Add the model to the Physics of a given EDEM simulation.
In the Creator Tree, select Physics.
Select Particle to Particle (mandatory) and/or Particle to Geometry (optional) and/or Particle Body Force (optional) from the Interaction dropdown list.
Click Edit Contact Chain at the lower section of the Physics panel.
Under Plug-in Models, select the LiquidBridgeAndExchange checkbox.
Select the plug-in and click the icon in the lower-right section of the Physics panel to configure it.

In the Liquid Bridge Model Parameter Editor dialog box, specify values for the following:


For	Specify
Particle-Particle
General parameters	This section must not be left blank as the properties of the bulk materials and the liquid can be only defined in this section. Model Activation Time: Units: s. Range: [0.0, ∞). Before this time, no liquid bridges between particles are formed and therefore, no forces are introduced by this model. Distance cutoff (relative to radius): Units: None. Range: [0.0, ∞). This dimensionless distance is multiplied by the radius of the particle in order to establish a gap between particles below which the bridge force can no longer increase its value. When this distance is different between two particle particles of different size, the largest distance is used.
Liquid properties	The following liquid properties will be applied to both Particle-Particle liquid bridges and Particle-Geometry liquid bridges: Density: Units: kg/m³ Range: (0.0, ∞). Density of the liquid. Surface tension: Units: N/m. Range: [0.0, ∞). Indicates the surface tension of the fluid in the current environment (usually air).
Properties associated with bulk materials	Specify the list of materials if you want them to form liquid bridges with other particles or with Geometries. Even if one bulk material is not in the list, no liquid bridge will be formed by any particle of this bulk material. Wetting angle: Units: angle. Range: [0.0, ∞). Indicates the wetting angle between the liquid and the material of the particle.
Particle-Geometry (optional)
General Parameters	Model Activation Time: Units: s. Range: [0.0, ∞). Before this time, no liquid bridges between particles and Geometries are formed and therefore, no forces are introduced by this model.
Properties associated with equipment materials	Specify the list of equipment materials if you want them to form liquid bridges with particles. Even if one equipment material is not in the list, no liquid bridge will be formed by any particle with this equipment material. If one bulk material is added to this list, it will be ignored and a warning message will be displayed. Wetting angle: Units: angle. Range: [0.0, ∞). Indicates the wetting angle between the liquid and the equipment material.
Particle-Body Force (optional)
General Parameters	This section is optional and can be used to add the weight and the inertia forces added by the attached fluid to the particles. Liquid Density: Units: s. (0.0, ∞). Indicates the density of the liquid. This field is only used if part of the liquid attached to the particle is carried by the Custom Property '`Volume Added`'. The field will be ignored if all the fluid is carried by the Custom Property '`Liquid Mass Ratio`', but you must add the model to the Particle Body Force section.