Mc from L-M Theory
From a classic G' & G'' curve of polymer melt viscosity over a large frequency range via WLF it is straightfoward to find the critical entanglement MWt, Mc, via the Likhtman-McLeish theory.
Mc from Likhtman-McLeish
The critical entanglement molecular weight, Mc, is intrinsic to many key features of polymers such as solution viscosity, melt viscosity (linear with MWt up to Mc then to the power of 3.4 after) and adhesion. So you would think that we know Mc for most polymers. In fact, the opposite is the case - we know Mc for just a few pure polymers and even for real-world versions of the well-studied ones. The list here has been kindly provided by my colleague Dr Hiroshi Yamamoto from a list of Me values, converted with the default value of 2 to Mc. It is sorted from low to high and you readily get a feel for going from relatively tough polymers to relatively weak ones.
Mc from rheology
Fortunately, with a rheometer capable of measuring the melt viscosity over a wide frequency range, using the standard WLF approach to combine temperature and frequency sweeps into one broad frequency plot, it is possible to measure (or at least estimate) Mc via the famous Likhtman-McLeish theory1.
Actually, the calculations behind the theory are near-impossible for most of us. Fortunately (again), the calculations have been pre-performed for us and packaged into a dataset based on the number of entanglements in the measured polymer, Z = MWt/Me (where Me is the MWt between entanglements and is traditionally assumed to be Mc/2)and the characteristic entanglement time, τe. A third parameter, cν, is a specialist parameter recommended to be 1 in the original paper and 0.1 in subsequent work.
In this app, you load your rheology data, G' and G'', then change Z and τe till you get the best visual fit. Although I could have automated this, I find it better to let human judgement arrive at the best conclusion for any specific polymer.
Once you have the best combination of Z and τe (and, maybe, cν), from your input MWt and the temperature, T, that describes the WLF fit (i.e. the WLF reference temperature) and a density ρ assumed (because the other errors are large) to be 1000 kg/m³, we can get all the values we need:
- Me = MWt/Z
- Ge = ρRT/Me [the entanglement modulus]
- τR, the Rouse time = τeZ²
In fact, it is common to add a "Ge adjustment factor" to refine the fit. Set it at first to 1 then adjust it to generally raise the curves if necessary. For the PC dataset, the value should be something like 1.8.
The L-M model is provably excellent for mondisperse linear polymers. But most of us don't live in the world of such polymers. So how reliable is L-M when applied to, say, a polydisperse linear polymer? My view is that functionally it gives us a good idea of an "effective Mc" which is what we need for our formulation work. RepTate itself contains BoB and RTP-LVE models which, I am told, do a better job of handling more complex polymers. But they are too hard for me to understand and I guess that most of use won't be using them any time soon. So this "effective Mc" seems to be at the very least a good starting point for those of us who need to formulate smarter.
The two default datasets are my hand digitized versions of a Polycaprolactone dataset from Dr McIlroy2 (T=60, MWt=100K) and a Polycarbonate dataset (T=180, Mwt=35K) taken from a review paper from Keunings and Bailly3 discussing the various methods of deriving Mc values. The digitizations and, therefore, the fits are imperfect, but they are a start. Experts will note that the PCL dataset lacks a definitive "upturn" portion to really pin down τe. A larger range of datasets will become available as soon as possible.
Into the future
It is a scandal that most of us, most of the time, formulate with almost no knowledge of Mc values in our polymer systems. It should not be a formulator's job to do complex rheology to get an Mc value - it should be the supplier's job. Perhaps suppliers know their Mc values and simply don't tell us. More likely, they simply don't know their Mc values. My hope is that it will gradually become unacceptable for any supplier to provide a polymer without an adequate Mc value as part of their technical datasheet. Suppliers should be smart enough to have RepTate and to know how to use it. But at the very least this app provides them with a "good enough" starting point.
1Alexei E. Likhtman and Tom C. B. McLeish, Quantitative Theory for Linear Dynamics of Linear Entangled Polymers, Macromolecules 2002, 35, 6332-6343
2C. McIlroy, R.S. Graham, Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts, Additive Manufacturing 24 (2018) 323–340
3Chenyang Liu, Jiasong He, Evelyne van Ruymbeke, Roland Keunings, Christian Bailly, Evaluation of different methods for the determination of the plateau modulus and the entanglement molecular weight, Polymer 47 (2006) 4461–4479