Jeong_Douglas_Alkane_SI_Sep08_Correction

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Supplementary Information: Mass Dependence of the Activation
Enthalpy and Entropy of Unentangled Linear Alkane Chains
Cheol Jeong and Jack F. Douglas
Materials Science and Engineering Division, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899, USA
(September 8, 2015)
I. Free-Volume Models of the Molecular Friction Coefficient, ζ
A number of “free volume” models [S1-S6] have been introduced to rationalize the
T dependence of relaxation and diffusion in glass-forming liquids and free-volume
models have been adapted to model polymer self-diffusion D and the shear viscosity  of
unentangled and entangled polymer melts, based on the Rouse and Reptation mean field
models, respectively. [S5-S6] Leaving aside evidence described in the main text
indicating that the Rouse model does not provide a quantitative description of our
simulated unentangled alkane melts, we analyse the assumptions underlying free volume
models that have been applied in modelling unentangled alkane melts.
In the free volume approach of von Meerwall et al. [S7], the self-diffusion coefficient
D of alkanes with molecular mass M was modelled as the product of three factors,
D  A exp[ Ea* / RT ] M 1 exp[ Bd / f (T , M )]
(S1)
where the first term is an Arrhenius factor borrowed from Eyring’s transition state theory
(TST) [S8], the second M 1 factor is associated with the assumption that the chain
segmental dynamics can be described by the Rouse model, and the last factor models the
effect of ‘free-volume’. Although the factorization D, and the ‘friction coefficient’ ζ of
the Rouse model, into Arrhenius and free-volume factors follows earlier reasoning by
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Macedo and coworkers [S9], this assumption is not uniformly followed in modelling of
polymer melts based on free-volume theory. In Macedo and coworkers, the free-volume
contribution to D is modelled following a specific argument made by Bueche [S10] that
attributes free volume to the presence of chain ‘free ends’. [S5, S10] (Variants of the free
chain end argument have also been developed by Fox and Flory [S5] and Ferry and
coworkers. [S6]) Based on the Bueche free-volume model, Macedo and coworkers [S9]
estimate the fractional free-volume f (T , M ) in terms of the alkane melt density,
 (T , M ) ,
f (T , M )  f (T , )  2Ve (T )  (T , M ) / M
(S2)
where f (T ,  ) is the free-volume fraction in the infinite mass limit and Ve (T ) is the free
-volume contribution of the chain ‘free ends’. Formally, f (T , M ) vanishes at low
temperatures and for n = 100 this extrapolated ‘critical condition’ arises for a T near T =
120 K. This finding is consistent with the relaxation time divergence temperature T0,
indicated in Fig. 8 of the main article.
Although Eq. (S1) involves adjustable parameters, von Meerwall and coworkers
found that it provides a good fit to unentangled alkane melt D data where Ea* in Eq. (S1)
[not to be confused with the activation energy of transition state theory given the
incorporation of the free-volume factor in (S1)] is assumed to be M independent and the
Rouse and the free chain end models are assumed to be applicable. We next consider the
implications of this model for understanding the mass scaling of D in alkanes.
Since the Arrhenius term is taken to be independent of M in this fitting scheme, the
change in the mass scaling exponent for D derives from the free-volume factor,
exp[ Bd / f (T , M )] . If we adopt the expression for f (T , M ) of von Meerwall and
coworkers and further assume a power law scaling, D ~ M   , then we find that the
resulting effective mass scaling exponent  depends rather strongly on M; this behaviour
is illustrated in Fig. S1 for representative T values. Evidently, a stable power law scaling
of D with M does not exist in this free-volume model for unentangled polymers.
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-1.0

-1.5
303 K
343 K
383 K
423 K
-2.0
-2.5
-3.0
1.0
1.5
2.0
2.5
log n
FIG. S1. Effective mass scaling exponent  from the free-volume model of von
Meerwall and coworkers using Eqs. (4), (5a), (5b), and (6) of Ref. [S7]. Note that 
reproduces the estimates of van Meerwall and coworkers in the limit of unentangled
chains [S7], i.e., β = 2.7, 2.4, 2.2, and 1.9 for 303 K, 343 K, 383 K, and 423 K,
respectively. The red line and arrow indicate the mass range investigated by van
Meerwall et al. [S7].
We note by comparison that the free-volume model of Pearson and coworkers [S11]
does not include the Arrhenius factor A exp[ Ea / RT ] of von Meerwall and coworkers
[S7] and that these authors focus rather on the effective activation energy of the
relaxation time rather than . The variability of the effective activation energy in the
model of Pearson and coworkers [S11], a quantity related to  in the modelling of van
Meerwall and coworkers, derives from M dependence of the characteristic temperature T0
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and the thermal expansion coefficient of the polymer melt. Although von Meerwall and
coworkers [S7] and Pearson and coworkers [S11] both assume a common Rouse factor
M 1 in their expression for D, these models are otherwise built on different
developments of free-volume theory.
At first, we tried to adopt these conventional free-volume analyses, but ultimately we
decided against them for the following reasons:
(i)
Our simulations of unentangled alkane chains indicate that the chains are highly
non-Gaussian, but the Rouse theory assumes the chains are Gaussian. There is
then a problem of consistency.
(ii)
Even within the realm of the free-volume model, we cannot rationalize the ad
hoc combination [S7] of the Arrhenius and free volume factors on which the
analysis of von Meerwall and coworkers is based.
(iii)
The application of the free volume model by Pearson et al. relies on the
application of free-volume theory at temperatures above the onset temperature
TA where D and  of the alkane exhibit an Arrhenius temperature dependence.
The applicability of this free-volume model at such elevated temperatures is
unclear.
(iv)
The fitting of the D and  data involves multiple fit parameters whose physical
significance is uncertain.
(v)
A consistent friction coefficient  cannot be obtained when multiple transport
properties are considered. [S12] In our view, this finding provides further
evidence that the use of the Rouse model is unsuitable for describing
unentangled polymer melts.
(vi)
Recent simulations have shown that the chain free end argument for the density
fails dramatically for star polymers. [S13]
Despite these technical problems, however, the fitting of the alkane D data by von
Meerwall et al. [S.7] does provide insight into the transition state theory activation energy,
determined in the high temperature regime where this theory should be applicable. To
appreciate this point, we note that the free-volume factor, exp[ Bd / f (T , M )] , originally
introduced empirically by Doolittle [S2], reduces to a simpler power law in the liquid
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density in the high temperature liquid regime. [S3] In particular, Batchinski’s original
free-volume theory [S1] (on which Doolittle’s free-volume model was latter based)
predicts D to have the high temperature scaling,
D ~ (vs  v0 ) / v0
(S3)
where vs is the specific volume and v0 is the limiting volume at which fluidity
extrapolates to zero. Eq. (S3), a high temperature variant of free-volume theory, holds
remarkably well for a large number of liquids [S1] in a T range in which Arrhenius
diffusion is applicable, i.e., T > TA. If we further adopt a simple two-state model of the
liquid specific volume vs [S14] in which vs can be approximated by a low temperature
constant, plus an Arrhenius term describing T dependence of vs with the excitation of
‘free-volume’ in the liquid upon heating. The difference (vs  v0 ) / v0 in Eq. (S3) can then
be expected to have an Arrhenius T dependence, as often observed in practice. We thus
obtain a formal link between the Arrhenius relation for D described in the main text, and
an expression derived from free-volume modelling. Based on this formal correspondence,
we can infer that the analysis of von Meerwall and coworkers indicates that the alkane
activation energy has two contributions - a constant intramolecular term associated with
relaxation of the monomer units and a second term related to chain packing interactions
(i.e., density), an intermolecular interaction contribution. This is exactly the interpretation
of the TST activation energy of polymers advocated by Tabor [S15] and Bershtein et al.
[S16], where the first term is identified with the barrier height of the bond rotational
potential and the second intermolecular term is proportional to the cohesive interaction
strength. This formal correspondence also explains long-standing observations linking the
measurements of density and pressure to viscosity [S17] which ultimately led to a
realization that the activation energy ΔHa should be related to the heat of vaporization.
[S8] This alternative view of free-volume theory in relation to activated transport should
provide an interesting starting point for further research into the molecular significance of
the activation energy for D in polymer melts.
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II. Non-Gaussian Conformations of Alkanes in the Melt
The ratio  R 2  /6  S 2  is a basic measure of the extent to which polymer chains
follow Gaussian chain statistics since this quantity equals 1 for Gaussian random coils
and other values quantify the extent of deviation from the flexible chain model. For the
alkane melt data described in the main article, this ratio becomes significantly larger than
1 and exhibits a maximum for rod-like values of this ratio where the chains then buckle,
n  nbuck . Kratky and Porod. [S17] introduced the worm-like chain (WLC) model to
describe the model the relative persistent conformations of short alkane chains.
Yamakawa and coworkers [S18] derived analytic WLC model expressions for the
average mean-squared end-to-end distance  R 2  and the radius of gyration  S 2  ,
R 2  2l p L  2l p 2 (1  e
 L/lp
(S4)
)
and
S2 
lp L
3
 2l p 2  2
l p3
L
2
lp4
2
L
(1  e
 L/lp
)
(S5)
where l p is the persistence length and L is the contour length of a linear chain.
These expressions recover the asymptotic random coil and rod limit limits,
lim
L / l p 
lim
L / l p 0
R 2  2l p L  N l p2  n 


2


l
L
N
l
p
S2  p 
n 

3
6

random coils
(S6)
rods.
(S7)
and
lim
L / lP  0
lim
L / l p 0
R 2  L2  n 2 


2

L
2
2
S 
n 
12


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FIG. S1. Average  R 2  as a function of the number of bonds n 1 in alkanes. The
dashed line indicates the fitting results using the wormlike chain model. [S19]
For alkanes, L corresponds to the end-to-end distance of a fully extended chain so that
L  (n  1)b sin( / 2) where n is the number of monomers, b is covalent bond distance
between backbone carbons, and θ is the angle between two neighbouring bonds.
Assuming the molecular parameters, b = 1.53 Å and θ = 110̊, we take l p as a free
parameter in Eq. (S4) to model the n of  R 2  and  S 2  . While the WLC describes
 R 2  well for all n, resulting in l p = 7.9 Å, it fails to describe  S 2  quantitatively. Eq.
(S7) underestimates  S 2  because WLC model ignores local chain structure, but by
incorporating excluded volume interactions into this semi-flexible chain model [S20] can
an improved theoretical description can be obtained of the L dependence of  S 2  in
comparison to simulation estimates.
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Supplementary Information References
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