friction

Frictional pair force classes apply a force and torque on every particle in the simulation state. The following general expression for Markovian tangential friction forces is implemented for interactions between two spherical particles and is discussed in detail in Hofmann et al. 2025.

For two particles $i$ and $j$ with radii $R_{i,j}$, center positions $\mathbf{r}_{i,j}$, angular velocities $\mathbf{\omega}_{i,j}$, and translational velocities $\mathbf{v}_{i,j}$, their surface velocities at the contact point are given by

$$\begin{align*} \mathbf{u}_i &= \mathbf{v}_i+\mathbf{\omega}_i \times \mathbf{\hat{r}}_{ij}R_i, \\ \mathbf{u}_j &= \mathbf{v}_j-\mathbf{\omega}_j \times \mathbf{\hat{r}}_{ij}R_j \, , \end{align*}$$

where $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/r_{ij}$. With these expressions, we calculate the relative tangential velocity $\mathbf{u}^\perp_{i,j}$ at the contact point

$$\mathbf{u}^\perp_{i,j} = \mathbf{P}(\mathbf{\hat{r}}_{ij})(\mathbf{v}_j -\mathbf{v}_i) -(\mathbf{\omega}_iR_i+\mathbf{\omega}_jR_j) \times \mathbf{\hat{r}}_{ij},$$

with the projection operator

$$\mathbf{P}(\mathbf{\hat{r}}_{ij})=1-\mathbf{\hat{r}}_{ij}\mathbf{\hat{r}}_{ij}.$$

We model the tangential friction force at the contact point very generally as

$$\mathbf{F}^\mathrm{f,contact}_i = -\mathbf{F}^\mathrm{f,contact}_j = f(u^\perp_{i,j} ,r_{i,j})\mathbf{\hat{u}}^\perp_{i,j}$$

where $\mathbf{\hat{u}}^\perp_{i,j}=\mathbf{u}^\perp_{i,j}/u^\perp_{i,j}$, and $f(u^\perp_{i,j},r_{i,j})$ is an arbitrary scalar function. The functional form of $f(u^\perp_{i,j},r_{i,j})$ specifies the frictional model.

In addition, a stochastic force satisfying the fluctuation-dissipation relation can be included. It has the form

$$\mathbf{F}^\mathrm{R,contact}_{i} = -\mathbf{F}^\mathrm{R,contact}_j = \sqrt{D(u^\perp_{ij},r_{ij})}\Big[\mathbf{P} (\mathbf{\hat{r}}_{ij})\mathbf{\xi}_{ij} - \mathbf{\hat{r}}_{ij} \times \mathbf{N} _{ij}\Big]$$

where $\mathbf{\xi}_{ij}$ and $\mathbf{N}_{ij}$ are Gaussian white noise vectors with correlations

$$\begin{align*} \langle \mathbf{\xi}_{ij} \mathbf{\xi}_{kl} \rangle &= \mathbf{1}kT (\delta_{ik}\delta_{jl}-\delta_{il} \delta_{jk})/\delta t \\ \langle \mathbf{N}_{ij} \mathbf{N}_{kl} \rangle &= \mathbf{1}kT(\delta_{ik} \delta_{jl}+\delta_{il}\delta_{jk})/ \delta t \, . \end{align*}$$

The function $D(u,r)$ is calculated as

$$D(u,r) = \frac{1}{kT\nu}\int_u^\infty \mathrm{d}u'f(u',r)\mathrm{exp}(-\frac{u'^2 -u^2}{2kT\nu})$$

with $\nu=(1/m_i+1/m_j)+(R^2_i/I_i+R_j^2/I_j)$.

The suface force $\mathbf{F}_i^\mathrm{contact}=\mathbf{F}^\mathrm{f,contact}_i +\mathbf{F}^\mathrm{R,contact}_i$ generates a center-of-mass force and a torque acting on particle $i$,

$$\mathbf{F}_{ij} = \mathbf{F}_i^\mathrm{contact},\quad \mathbf{\tau}_{ij}=R_i\hat{ \mathbf{r}}_{ij}\times\mathbf{F}^\mathrm{contact}_i,$$

which is the pair frictional contact force and torque resulting from the friction with particle $j$.

FrictionalPair does not support any shifting modes.

Classes

• FrictionLJCoulomb
• FrictionLJCoulombNewton
• FrictionLJLinear
• FrictionalPair