Phoretic velocityΒΆ
According to Hu et al., Phys. Rev. Lett. (123), 2019, the phoretic velocity is given by
\[\vec{U}
=
-
\vec{e}_{\vt}
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\vat{
\frac{1}{\vr}
\pder{}{c}{\vt}
}{\vr = 1}
d \vt
=
M
\left[
-
\vec{e}_x
\Re \left( C_1^s \right)
+
\vec{e}_y
\Im \left( C_1^s \right)
\right],\]
which is derived here.
Using
\[ \begin{align}\begin{aligned}\vec{e}_{\vt}
&
=
-
\vec{e}_x \sin \vt
+
\vec{e}_y \cos \vt\\&
=
-
\vec{e}_x \frac{I}{2} \left[
\expp{I \vt}
-
\expp{- I \vt}
\right]
+
\vec{e}_y \frac{1}{2} \left[
\expp{I \vt}
+
\expp{- I \vt}
\right]\end{aligned}\end{align} \]
and
\[\vat{\pder{}{c}{\vt}}{\vr = 1}
=
\sum_k I k C_k^s \expp{I k \vt},\]
we see that
\[ \begin{align}\begin{aligned}-
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\frac{1}{\vr}
\pder{}{c}{\vt}
\vec{e}_{\vt}
d \vt
=
&
-
\vec{e}_x
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\sum_k I k C_k^s \expp{I k \vt}
\frac{I}{2} \left[
\expp{I \vt}
-
\expp{- I \vt}
\right]
d \vt\\&
-
\vec{e}_y
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\sum_k I k C_k^s \expp{I k \vt}
\frac{1}{2} \left[
\expp{I \vt}
+
\expp{- I \vt}
\right]
d \vt\\=
&
+
\vec{e}_x
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\sum_k
\frac{k C_k^s}{2} \left[
\expp{I \left( k + 1 \right) \vt}
-
\expp{I \left( k - 1 \right) \vt}
\right]
d \vt\\&
-
I
\vec{e}_y
\frac{1}{2 \pi}
\int_0^{2 \pi}
M
\sum_k
\frac{k C_k^s}{2} \left[
\expp{I \left( k + 1 \right) \vt}
+
\expp{I \left( k - 1 \right) \vt}
\right]
d \vt.\end{aligned}\end{align} \]
It is readily apparent that the integral of the exponential functions in the range of \(\left[ 0, 2 \pi \right]\) is zero. Exceptions are when the exponent is zero:
\[\int_0^{2 \pi}
1
d \vt
=
2 \pi.\]
Thus the phoretic velocity leads to
\[\vec{U}
=
M
\left[
\vec{e}_x
\left(
-
\frac{C_1^s}{2}
-
\frac{C_{-1}^s}{2}
\right)
+
I
\vec{e}_y
\left(
-
\frac{C_1^s}{2}
+
\frac{C_{-1}^s}{2}
\right)
\right].\]
Because of \(c \in \mathbb{R}\), \(C_k^s\) fulfills the complex-conjugate property:
\[ \begin{align}\begin{aligned}\Re \left( C_{-1}^s \right)
&
=
\Re \left( C_1^s \right),\\\Im \left( C_{-1}^s \right)
&
=
-
\Im \left( C_1^s \right).\end{aligned}\end{align} \]
Assigning these relations yields
\[\vec{U}
=
M
\left[
-
\vec{e}_x
\Re \left( C_1^s \right)
+
\vec{e}_y
\Im \left( C_1^s \right)
\right],\]
and thus we conclude
\[ \begin{align}\begin{aligned}U_x
=
-
M
\Re \left( C_1^s \right),\\U_y
=
+
M
\Im \left( C_1^s \right).\end{aligned}\end{align} \]