Phoretic motion of soft vesicles and droplets: an XFEM/particle-based numerical solution

Abstract

When immersed in solution, surface-active particles interact with solute molecules and migrate along gradients of solute concentration. Depending on the conditions, this phenomenon could arise from either diffusiophoresis or the Marangoni effect, both of which involve strong interactions between the fluid and the particle surface. We introduce here a numerical approach that can accurately capture these interactions, and thus provide an efficient tool to understand and characterize the phoresis of soft particles. The model is based on a combination of the extended finite element—that enable the consideration of various discontinuities across the particle surface—and the particle-based moving interface method—that is used to measure and update the interface deformation in time. In addition to validating the approach with analytical solutions, the model is used to study the motion of deformable vesicles in solutions with spatial variations in both solute concentration and temperature.

Appendices

Appendix 1: Expressions and derivatives of shape functions

In the numerical study, XFEM is implemented in the element shape functions to account for the discontinuities. For split elements, the extended interpolation matrices ${\mathbf {N}}^4(x)$ and ${\mathbf {N}}^9(x)$ in Eq. (35) and (36) have the form:

$$\begin{aligned} {\mathbf {N}}^4(x)= & {} \left[ \begin{array}{cccccc}N_1^4&...&N_4^4&\varPsi _1N_1^4&...&\varPsi _4N_4^4\end{array}\right] \nonumber \\ {\mathbf {N}}^9({\mathbf {x}})= & {} \left[ \begin{array}{cccccccccc}N_1^9&{}0&{}...&{}N_9^9&{}0&{}\varPsi _1N_1^9&{}0&{}...&{}\varPsi _9N_9^9&{}0\\ 0&{}N_1^9&{}0&{}...&{}N_9^9&{}0&{}\varPsi _1N_1^9&{}0&{}...&{}\varPsi _9N_9^9\end{array}\right] \nonumber \\ \end{aligned}$$

(52)

where $\varPsi _I=H-H_I$ is the Heaviside enrichment function applying on the Ith node. For normal elements that are not splitted, the Heaviside enrichment functions vanish to zero and only the normal shape functions are retain. On the interface, a simple $1-D$ shape function is used to interpolate the quantities on the interface :

$$\begin{aligned} \mathbf {\overline{N}}=[\overline{N}_1 \overline{N}_2] \end{aligned}$$

(53)

To compute the deformation rate of the fluid, the rate of deformation of fluid and the divergence of fluid are written as:

$$\begin{aligned} {\mathbf {D}}=(\mathbf {\nabla }{\mathbf {v}})^s={\mathbf {B}}\cdot {\mathbf {v}}^e\;\;\; \text {and}\;\;\; \mathbf {\nabla }\cdot {\mathbf {v}}={{\hat{\mathbf {B}}}}\cdot {\mathbf {v}}^e \end{aligned}$$

we write the rate ${\mathbf {B}}$ and ${\hat{\mathbf {B}}}$ matrices that are related to the nodal velocities to deformation rate and velocity divergence in the cylindrical coordinates as

$$\begin{aligned} \begin{aligned}&{\mathbf {B}}=\left[ \begin{array}{cccc}{\mathbf {B}}_1&...&{\mathbf {B}}_{n9+m9}\end{array}\right] ,\quad \hat{{\mathbf {B}}}=\left[ \begin{array}{ccc}\hat{{\mathbf {B}}}_1&...&\hat{{\mathbf {B}}}_{n9+m9}\end{array}\right] \\&{ with} \\&{\mathbf {B}}_I=\left[ \begin{array}{cc}\frac{\partial N_I^9({\mathbf {x}})}{\partial \rho }&{}0\\ 0&{} \frac{\partial N_I^9({\mathbf {x}})}{\partial z}\\ \frac{N_I^9({\mathbf {x}})}{\rho }&{}\frac{N_I^9({\mathbf {x}})}{\rho }\\ \frac{\partial N_I^9({\mathbf {x}})}{\partial z}&{}\frac{\partial N_I^9({\mathbf {x}})}{\partial \rho }\end{array}\right] , \\&\hat{\mathbf {B}}_{I}=\left[ \begin{array}{cc}\frac{\partial N_I^9({\mathbf {x}})}{\partial \rho }+\frac{N^9_I({\mathbf {x}})}{r}&\frac{\partial N_I^9({\mathbf {x}})}{\partial z}\end{array}\right] \end{aligned} \end{aligned}$$

(54)

where the subscript n corresponds to the normal element nodes and m is corresponded to the enriched degrees of freedom. Similarly, the matrix that relates to the spatial rate of change in concentration reads:

$$\begin{aligned} \begin{aligned}&{\mathbf {B}}^4=\left[ \begin{array}{ccc}{\mathbf {B}}^4_1&...&{\mathbf {B}}^4_{n4+m4}\end{array}\right] \\&with\\&{\mathbf {B}}^4_I=\left[ \begin{array}{cc}\frac{\partial N_I^4({\mathbf {x}})}{\partial \rho }&\frac{\partial N_I^9({\mathbf {x}})}{\partial z} \end{array}\right] ^T, \end{aligned} \end{aligned}$$

(55)

Appendix 2: Discretized weak form

With the matrix forms of the fields given, plug in the matrix form for ${\mathbf {v}}$, p in Eq. (30) and c in Eq. (31), the discretized weak form for fluid is then read:

$$\begin{aligned}&\sum _{e=1}^{nel}\{\varvec{\omega }_v^{eT}\}\left[ \int _{\varOmega ^e}(r{\mathbf {B}}^T\cdot {\mathbf {B}}\cdot {\mathbf {v}}^e-{\mathbf {B}}^T\cdot {\mathbf {N}}^4\cdot {\mathbf {p}}^e)rd\varOmega ^e\right. \nonumber \\&\quad +\int _{\varGamma ^e}\left( {{\mathbf {N}}^9}^T\cdot {\mathbf {a}}^T\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\cdot \mathbf {\lambda }^{e+}rd\varGamma ^e \right. \nonumber \\&\quad \left. -\int _{\varGamma ^e}{{\mathbf {N}}^9}^T\cdot {\mathbf {a}}^T\cdot \frac{r^-}{l^-}\mathbf {\overline{N}}\mathbf {\lambda }^{e-}\right) rd{\varGamma ^e}+\left. \int _{\varGamma ^e}{{\mathbf {N}}^9}^T\cdot \overline{{\mathbf {f}}}rd\varGamma ^e\right] \nonumber \\&\quad +\sum _{e=1}^{nel}\{\varvec{\omega }_p\}^T\left[ \int _{\varOmega ^e}{{\mathbf {N}}^4}^T\cdot \hat{{\mathbf {B}}}\cdot {\mathbf {v}}^erd\varOmega ^e\right. \nonumber \\&\quad +\left. \int _{\varGamma ^e}{{\mathbf {N}}^4_{[]}}^T\cdot \mathbf {\overline{N}}\cdot \mathbf {\lambda }^prd\varGamma ^e\right] \nonumber \\&\quad +\sum _{e=1}^{nel}\{\varvec{\omega }_{\lambda _p}\}^T\left[ \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot {\mathbf {N}}^4_{[]}\cdot {\mathbf {p}}^erd\varGamma ^e-\int _{\varGamma ^e}\overline{{\mathbf {N}}}^T\cdot \overline{{\mathbf {f}}} r d\varGamma ^e\right] \nonumber \\&\quad +\sum _{e=1}^{nel}\{\varvec{\omega }_{\lambda ^+}^e\}^T\left[ \int _{\varGamma ^e}\mathbf {\overline{N}}\cdot \left( r^+{\mathbf {a}}^T\cdot {\mathbf {P}}^\perp \cdot \mathbf {B_{+}}\right. \right. \nonumber \\&\quad +\left. \left. \frac{r^+}{l^+}{\mathbf {a}}^T\cdot {\mathbf {N}}_{+}\right) \cdot {\mathbf {v}}^erd\varGamma ^e\right. \nonumber \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\cdot {\mathbf {v}}_ard\varGamma ^e-\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\cdot \mathbf {\overline{v}}^\parallel rd\varGamma ^e\right] \nonumber \\&\quad +\sum _{e=1}^{nel}\{\omega _{\lambda -}\}^T\left[ \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot (r{\mathbf {a}}^T\cdot {\mathbf {P}}^\perp \cdot {\mathbf {B}}_-\right. \nonumber \\&\quad -\left. \frac{r^-}{l^-}{\mathbf {a}}\cdot {\mathbf {N}}_v)\cdot {\mathbf {v}}^erd\varGamma ^e\right. +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^-}{l^-}\mathbf {\overline{N}}\cdot \mathbf {\overline{v}}^{\parallel e}rd\varGamma ^e\right] \nonumber \\&\quad +\sum _{e=1}^{nel}\{\varvec{\omega }_{\overline{v}^\parallel }\}^T\left[ \int _{\varGamma ^e}\mathbf {\overline{N}}\cdot \left( -\frac{r^+}{l^+}{\mathbf {a}}\cdot {\mathbf {N}}_++\frac{r^-}{l^-}{\mathbf {a}}\cdot {\mathbf {N}}_- \right) \cdot {\mathbf {v}}^erd\varGamma ^e\right. \nonumber \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\cdot \mathbf {\lambda }^+rd\varGamma ^e+\int _{\varGamma ^e}\mathbf {\overline{N}}\cdot \left( -\frac{r^-}{l^-}\mathbf {\overline{N}}\cdot \mathbf {\lambda }^-\right) rd\varGamma ^e\right. \nonumber \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \left( \frac{r^+}{l^+}-\frac{r^-}{l^-}\right) \mathbf {\overline{N}}\cdot \mathbf {\overline{v}}^\parallel rd\varGamma ^e\right. \nonumber \\&\quad -\left. \int _{\varGamma ^e}\overline{{\mathbf {N}}}^T\cdot \left( \frac{r^+}{l^+}{\mathbf {v}}_a \right) rd\varGamma ^e\right] =0 \end{aligned}$$

(56)

where the $+$ and − sign on the ${\mathbf {N}}$ and ${\mathbf {B}}$ are associated with the interpolation on the $+$ or − side of the interface respectively. The matrix ${\mathbf {N}}^4_{[]}=\left[ \begin{array}{cccccc}0&...&0&N_1^4&...&N_4^4\end{array}\right]$ is associated with the pressure jump at the interface.

In a similar way, the discretized weak form for transportation of solute reads:

$$\begin{aligned} \begin{aligned}&\sum _{e=1}^{nel}\{\varvec{\omega }_c\}^T\left[ \int _{\varOmega ^e}{{\mathbf {N}}^4}^T\cdot {\mathbf {N}}^4\cdot {{\dot{\mathbf {c}}}^e}rd\varOmega ^e\right. \\&\quad +\left. \int _{\varOmega ^e}{{\mathbf {N}}^4}^T\cdot {\mathbf {v}}^{eT}\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}^erd\varOmega ^e\right. \\&\quad +\left. \int _{\varOmega ^e}{{\mathbf {B}}^4}^T\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}^erd\varGamma ^e\right. \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot {\mathbf {N}}_{[]}^4\cdot {\mathbf {c}}^erd\varGamma ^e-\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot [-{\mathbf {n}}\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}_t\right. \\&\quad +\left. \frac{1}{K}({\mathbf {N}}^+-{\mathbf {N}}^-)\cdot {\mathbf {c}}_t+K{\mathbf {P}}^\parallel \cdot {\mathbf {B}}^4\cdot {\mathbf {q}}^\parallel ]rd\varGamma ^e\right. \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot [-{\mathbf {n}}\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}_t+\frac{1}{K}({\mathbf {N}}^+-{\mathbf {N}}^-)\cdot {\mathbf {c}}_t\right. \\&\quad +\left. K{\mathbf {P}}^\parallel \cdot {\mathbf {B}}^4\cdot {\mathbf {q}}^\parallel ]rd\varGamma ^e\right] \\&\quad +\sum _{e=1}^{nel}\{\varvec{\omega }_{\lambda c}\}^T\left[ \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot {\mathbf {N}}_{[]}^4\cdot {\mathbf {c}}^erd\varGamma ^e\right. \\&\quad +\left. \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot [-{\mathbf {n}}\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}_t+\frac{1}{K}({\mathbf {N}}^+-{\mathbf {N}}^-)\cdot {\mathbf {c}}_t\right. \\&\quad +\left. K{\mathbf {P}}^\parallel \cdot {\mathbf {B}}^4\cdot {\mathbf {q}}^\parallel ]rd\varGamma ^e\right] =0 \end{aligned} \end{aligned}$$

where ${\mathbf {v}}^e$ is the calculated fluid velocity from the stokes equation, ${\mathbf {c}}_t$ is the nodal value of solute concentration obtained from previous timestep, $\dot{{\mathbf {c}}}^e$ is the rate of change of solute concentration. The tangential flux ${\mathbf {q}}^\parallel$ is computed according to Eq. (11) for solute concentration or (18) for temperature using the calculated nodal value ${\mathbf {c}}_t$.

Appendix 3: Matrix form

For Stokes flow problem (Eq. 33), in each element, the vector ${\mathbf {d}}(t)$ is composed of the fluid degrees of freedom ${\mathbf {d}}(t)=\left[ {\mathbf {v}}(t),{\mathbf {p}}(t)\right] ^T$ and $\mathbf {\overline{d}}(t)$ is composed of the interface degrees of freedom $\mathbf {\overline{d}}(t)=\left[ \overline{v}^\parallel ,\lambda _p,\lambda ^+,\lambda ^-\right] ^T$. The submatrix ${\mathbf {K}}^T$ and $\mathbf {\overline{K}}^T$ are corresponding to fluid domain $\varOmega$ and interface $\varGamma$ respectively, while ${\mathbf {I}}^t_1$ and ${\mathbf {I}}^T_1$ are associated to fluid-interface interactions. The global force vector ${\mathbf {F}}^t_f$ is corresponding to the external force as well as the force produced by the interface, and $\mathbf {\overline{F}}^T_f$ is associated with the jump conditions on the interface including pressure and tangential velocity. The detailed expressions of for the submatrices in (33) are:

$$\begin{aligned} {\mathbf {K}}^t_e= & {} \left[ \begin{array}{cc}{\mathbf {K}}_{vv}^e&{}{\mathbf {K}}_{vp}^e\\ {\mathbf {K}}_{pv}^e&{}0\end{array}\right] \quad \mathbf {\overline{K}}^t_e=\left[ \begin{array}{cccc}{\mathbf {K}}^e_{\overline{v}^\parallel \overline{v}^\parallel }&{}0&{}{\mathbf {K}}^e_{\overline{v}^\parallel \lambda ^+}&{}{\mathbf {K}}^e_{\overline{v}^\parallel \lambda ^-}\\ 0&{}0&{}0&{}0\\ {\mathbf {K}}^e_{\lambda ^+\overline{v}^\parallel }&{}0&{}0&{}0\\ {\mathbf {K}}^e_{\lambda ^-\overline{v}^\parallel }&{}0&{}0&{}0\end{array}\right] \\ {\mathbf {I}}^t_{e1}= & {} \left[ \begin{array}{cc}{\mathbf {K}}^e_{\overline{v}^\parallel v}&{}0 \\ 0&{}{\mathbf {K}}^e_{\lambda _p p}\\ {\mathbf {K}}^e_{\lambda ^+v}&{}0\\ {\mathbf {K}}^e_{\lambda ^-v}&{}0\end{array}\right] \quad {\mathbf {I}}^t_{e2}=\left[ \begin{array}{cccc}0&{}0&{}{\mathbf {K}}^e_{v\lambda ^+}&{}{\mathbf {K}}^e_{v\lambda ^-}\\ 0&{}{\mathbf {K}}^e_{p\lambda _p}&{}0&{}0\end{array}\right] \\ {\mathbf {F}}^t_{fe}= & {} \left[ \begin{array}{cc}{\mathbf {F}}^e_v&{\mathbf {F}}^e_p\end{array}\right] ^T\quad \quad \mathbf {\overline{F}}^t_{fe}=\left[ \begin{array}{cccc}{\mathbf {F}}^e_{\overline{v}^\parallel }&{\mathbf {F}}^e_{\lambda _p}&{\mathbf {F}}^e_{\lambda +}&0\end{array}\right] ^T \end{aligned}$$

with the element matrices forms:

$$\begin{aligned}&{\mathbf {K}}_{vv}^e=\int _{\varOmega ^e}r{\mathbf {B}}^T\cdot {\mathbf {B}}\;rd\varOmega ^e\\&{\mathbf {K}}_{vp}^e=\int _{\varOmega ^e}-{\mathbf {B}}^T\cdot {\mathbf {N}}^4\;rd\varOmega ^e \\&{\mathbf {K}}_{v\lambda ^+}^e=\int _{\varGamma ^e}{\mathbf {N}}_+^T\cdot {\mathbf {a}}\frac{r^+}{l^+}\mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{v\lambda ^-}^e=\int _{\varGamma ^e}-{\mathbf {N}}_+^T\cdot {\mathbf {a}}\frac{r^-}{l^-}\mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{pv}^e=\int _{\varOmega ^e}-{{\mathbf {N}}^4}^T\cdot {\mathbf {B}}\;rd\varOmega ^e\\&{\mathbf {K}}_{p\lambda _p}=\int _{\varGamma ^e}{{\mathbf {N}}_{[]}^4}^T\cdot \mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{\overline{v}^\parallel v}=\int _{\varGamma ^e}\mathbf {\overline{N}}^T \cdot \left( -\frac{r}{l^+}{\mathbf {a}}\cdot {\mathbf {N}}^9_+-\frac{r}{l^-} {\mathbf {a}}\cdot {\mathbf {N}}^9_-\right) \;rd\varGamma ^e\\&{\mathbf {K}}_{\overline{v}^\parallel \overline{v}^\parallel }=\int _{\varGamma ^e} \mathbf {\overline{N}}^T\left( \frac{r^+}{l^+}+\frac{r^-}{l^-}\right) \overline{N}\;rd\varGamma ^e\\&{\mathbf {K}}_{\overline{v}^\parallel \lambda ^+}=\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{\overline{v}^\parallel \lambda ^-}=\int _{\varGamma ^e}-\mathbf {\overline{N}}^T\cdot \frac{r^-}{l^-}\mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{\lambda _p p}=\int _{\varGamma ^e}-\mathbf {\overline{N}}^T\cdot {\mathbf {N}}_{[]}^4\;rd\varGamma ^e\\&{\mathbf {K}}_{\lambda ^+v}=\int _{\varGamma ^e}\mathbf {\overline{N}}^T \cdot \left( r^+{\mathbf {a}}^T\cdot P^\perp \cdot \mathbf {B_+}+\frac{r^+}{l^+}{\mathbf {N}}_+\cdot {\mathbf {a}}\right) \;rd\varGamma ^e\\&{\mathbf {K}}_{\lambda ^+\overline{v}^\parallel }=\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^+}{l^+}\mathbf {\overline{N}}\;rd\varGamma ^e\\&{\mathbf {K}}_{\lambda ^-v}=\int _{\varGamma ^e} \mathbf {\overline{N}}^T\cdot \left( r^-{\mathbf {a}}^T\cdot {\mathbf {P}}^\perp \cdot {\mathbf {B}}_--\frac{r^-}{l^-}\mathbf {N_-\cdot a}\right) \;rd\varGamma ^e\\&{\mathbf {K}}_{\lambda ^-\overline{v}^\parallel }=\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \frac{r^-}{l^-}\mathbf {\overline{N}}\;rd\varGamma ^e \end{aligned}$$

with the element force vector:

$$\begin{aligned}&{\mathbf {F}}^e_v=\int _{\varOmega ^e}-{\mathbf {N}}^T\cdot \mathbf {\overline{f}}\;rd\varOmega ^e\\&{\mathbf {F}}^e_{\overline{v}^\parallel }=\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \left( \frac{r}{l^+}{\mathbf {v}}_a\right) \; rd\varGamma ^e\\&{\mathbf {F}}^e_{\lambda _p}=\int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot \mathbf {\overline{f}}\; rd\varGamma ^e\\&{\mathbf {F}}^e_{\lambda ^+}=\int _{\varGamma ^e}-\mathbf {\overline{N}}^T\cdot \frac{r}{l^+}{\mathbf {v}}_a\;rd\varGamma ^e \end{aligned}$$

Regarding the transport problem (34), the vector $[\dot{{\mathbf {d}}}_c]=[\dot{{\mathbf {c}}},0]^T$ is composed of the rate of change of the solute concentration and the submatrix ${\mathbf {A}}^T$ is corresponding to the fluid domain. The vector $[{\mathbf {d}}_c]=[{\mathbf {c}}, \lambda _c]^T$, is composed of the solute concentration degrees of freedom, with the submatrix ${\mathbf {K}}_c$ is associated with the convection and diffusion of solute molecules. The global force vector ${\mathbf {F}}_c$ accounts for the jump of concentration on the interface, as well as the emission or adsorption of solute on the interface. The detailed expressions of submatrices for Eq. (34) are:

$$\begin{aligned} {\mathbf {A}}^t_e= & {} \left[ \begin{array}{cc}{\mathbf {K}}^e_{cc}&{}0 \\ 0&{}0\end{array}\right] \quad \quad {{\mathbf {K}}_c^t}_e=\left[ \begin{array}{cc}{\mathbf {K}}^e_{cv}&{}{\mathbf {K}}^e_{c\lambda }\\ {\mathbf {K}}^e_{\lambda c}&{}0\end{array}\right] \nonumber \\ {\mathbf {F}}_c^e= & {} \left[ \begin{array}{cc}{\mathbf {F}}_c&{\mathbf {F}}_{\lambda c}\end{array}\right] \end{aligned}$$

(57)

and the element matrices written as:

$$\begin{aligned} {\mathbf {K}}^e_{cc}= & {} \int _{\varOmega ^e}{{\mathbf {N}}^4}^T\cdot {\mathbf {N}}^4\;rd\varOmega ^e\\ {\mathbf {K}}^e_{cv}= & {} \int _{\varOmega ^e}({{\mathbf {N}}^4}^T\cdot {\mathbf {N}}\cdot {\mathbf {v}}_s^T\cdot {\mathbf {B}}^4+{{\mathbf {B}}^4}^T\cdot {\mathbf {B}}^4)\;rd\varOmega ^e\\ {\mathbf {K}}^e_{c\lambda }= & {} \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot {\mathbf {N}}_{[]}^4\;rd\varGamma ^e\\ {\mathbf {K}}^e_{\lambda c}= & {} \int _{\varGamma ^e}{\mathbf {N}}_{[]}^4\cdot \mathbf {\overline{N}}\;rd\varGamma ^e \end{aligned}$$

with the force vector:

$$\begin{aligned} {\mathbf {F}}_c^e= & {} \int _{\varGamma ^e}-\mathbf {\overline{N}}^T\cdot [-{\mathbf {n}}^T\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}_t \\&+\frac{1}{K}({\mathbf {N}}^4_+-{\mathbf {N}}^4_-)\cdot {\mathbf {c}}_t+K{\mathbf {P}}^\parallel \cdot {\mathbf {B}}^4\cdot {\mathbf {q}}^\parallel ]\;rd\varGamma ^e. \\ {\mathbf {F}}_{\lambda c}^e= & {} \int _{\varGamma ^e}\mathbf {\overline{N}}^T\cdot [-{\mathbf {n}}^T\cdot {\mathbf {B}}^4\cdot {\mathbf {c}}_t \\&+\frac{1}{K}({\mathbf {N}}^4_+-{\mathbf {N}}^4_-)\cdot {\mathbf {c}}_t+K{\mathbf {P}}^\parallel \cdot {\mathbf {B}}^4\cdot {\mathbf {q}}^\parallel ]\;rd\varGamma ^e. \end{aligned}$$