Clarification on polar chain length equation

[ImagineBaggins]

Hello again,

I have a question about the equation for the average chain length term used for polar calculations, specifically the way it is limited to a maximum value of 2.

In your code (if I understand correctly; I am not familiar with Rust notation), you appear to be limiting the pure-component m_i/m_j to 2:

impl MeanSegmentNumbers {
    ...
    for i in 0..npoles {
        let mi = parameters.m[comp[i]].min(2.0);
        for j in i..npoles {
            let mj = parameters.m[comp[j]].min(2.0);
            let mij = (mi * mj).sqrt();
            ...
        }
    }
    ...
}

(the above starting @ line 94 in polar.rs)

In the original paper by Gross & Vrabec (2006) on dipoles, it is instead the post-average quantity that is limited to 2:

(the QQ (2005) and DQ (2008) papers do not mention this limit at all)

However, these two forms are not mathematically equivalent; $\min (\sqrt{x \cdot y},2 ) \neq \sqrt{\min(x,2) \cdot \min(y,2)}$. So I was wondering if you know if either of these is incorrect or misrepresented (or if I am misunderstanding). It seems like the maximum error (for m_ij) ranges between ~27% and ~42% (depending on how you define the error) for {m_i,m_j} ≥ 1, so I would like to use the correct form for my own implementation of this model.

Thank you again for all your work on FeOs; it's been very helpful for my research!

[ImagineBaggins]

I also have a similar question regarding the "moment squared" terms (for DD/QQ/DQ). I apologize for the fact that this may be moreso a question about the original papers by G&V rather than FeOs, but seeing as FeOs has implemented these models, I figure you must have at least an interpretation of the models (if not the actual answer to my question, since many of you have worked under/with Prof. Gross).

I'm having trouble following the FeOs code, but the relevant part for DD interactions can be found between lines 167 - 237 in polar.rs.

In Gross & Vrabec, 2006 (dipoles), the A₂^DD part of the Padé approximant is given as:

with

It's not immediately clear to me if the ε_ii in the definition of μ_i*² is divided by k_B (as it is in nearly every other occurrence within SAFT models), or by k_BT, or not divided by anything (which would require multiplying by k_B since the SAFT ε parameter is usually provided as ε/k_B).

In addition, the σ_ii, ε_ii terms in the same denominator appear to cancel with those in the summation expression (I think you have done this in FeOs' code, but not sure).

I haven't found any future publications dealing with PCP-SAFT to offer a different presentation of these equations, but one: the thesis of A.J. de Villiers (2011), which presents A₂^DD like this:

which, if correct, seems to confirm the view that the ε_ii in the denominator of the expression for μ_i*² is not scaled by anything.

But I don't know for a fact that this interpretation is correct since this is just one other author, so I figured I would append this question to my previous one. How exactly does FeOs handle the εs in these summation expressions? I would appreciate any guidance on the matter!

[g-bauer]

The definition of $\mu_i^{*2}$ as denoted above (with $\epsilon$ in energy units) is correct. The squared dipole moment $\mu_i^2$ in Debye is made dimensionless by dividing by units of energy times cubic length to yield the dimensionless squared dipole moment. We calculate this when parsing input parameters and store it as mu2 in the parameters object (see here) so when you see parameters.mu2 it is already the dimensionless squared dipole moment (same for parameters.q2 and $q^2$).

Both equations you posted for $A_2^{dd}$ are the same from a quick glance. We implement eq 8 as written above (using $\epsilon_k$). The term you are looking for is used here to build an array containing $\frac{\epsilon_{k,ii} \sigma_{ii}^3 \mu_i^{*2}}{T}$. Note that we actually calculate $\beta A^{DD}$ (not divided by $N$) and thus use the partial densities in the double sum and not the mole fractions (see here) and then multiply by the volume.

[ImagineBaggins]

Thank you! I appreciate the detailed help and clarification.

[g-bauer]

Concerning the limitation of m values to 2: You read our implementation correctly - we limit the pure substance values and then combine them. Why we implemented it that way I'd have to check (it's been a while since we implemented this contribution) and come back to you. I'll edit this post.