How to estimate the mobility constant and the surface energy parameter in the Cahn-Hilliard equation?

As a mathematician, I want to simulate phase separations with the Cahn-Hilliard equation
$$
\frac{\partial c}{\partial t} = M \Delta \bigg(\frac{\partial \mu}{\partial c} - \kappa \Delta c \bigg),
$$
where $c$ is the molar fraction, $\mu$ is the chemical potential (given in the unit $\text{energy}/\text{length^2}$), $M$ is the mobility constant, $\kappa$ is the surface energy parameter and $\Delta$ is the Laplacian. (Since my simulations are in 2D, the units are correspondingly chosen with the unit length of $1\mu\text{m}.$)


**Question 1:** How do we get reasonable (good) estimates for $M$ and $\kappa,$ let's say for a certain mixture such as $\text{Na}\_2\text{O-SiO}\_2$? Are there standard references?


**Literature and Problem:** In the paper of [Kim and Sanders (2020)](https://www.sciencedirect.com/science/article/abs/pii/S0022309319304624) the surface energy parameter is estimated by
$$
\kappa = \frac{\mu - \text{common tangent}}{(\partial c / \partial x)^2}
$$
while the mobility is expressed as
$$
M = \frac{D \cdot N\_A}{\partial^2 \mu / \partial c^2}
$$
where $D$ is the diffusion coefficient (given in $\text{length}^2/\text{time}$) and $N\_A$ is the number of particles per unit area (i.e. per $\text{length}^2$). According to these definitions, the corresponding unit for the mobility is
$$
\frac{\text{length}^2}{\text{energy} \cdot\text{time}}.
$$
(length is e.g. $\mu\text{m},$ energy is e.g. $\text{kJ},$ time is e.g. $\text{min}$). For the unit of the surface energy parameter we have just  

$$
\text{energy}. 
$$
The surface energy parameter estimate seems to have the correct unit due to
$$
\bigg(\frac{\text{energy}}{\text{length}^2} - \text{energy}\frac{1}{\text{length}^2}\bigg).
$$
However, the estimate of Kim and Sanders for the mobility constant seems to be incorrect, because
$$
\frac{1}{\text{time}} \not = \underbrace{\frac{\text{length}^2}{\text{energy} \cdot\text{time}}}\_{\widehat = M} \cdot \underbrace{\frac{1}{\text{length}^2}}\_{\widehat = \Delta} \cdot \underbrace{\frac{\text{energy}}{\text{length}^2}}\_{\widehat = (\frac{\partial \mu}{\partial c} - \kappa \Delta c)}.
$$


This [former question](https://chemistry.stackexchange.com/questions/165406/understanding-cahn-hilliard-equation-in-terms-of-units) was already concerned with the units of the Cahn-Hilliard equation where the mobility $M$ was determined to have the unit
$$
\frac{\text{length}^4}{\text{energy} \cdot \text{time}}, 
$$
which in fact would lead to the correct unit.


**Question 2:** Do I misunderstand something?


