Journal of Molecular Structure 485–486 (1999) 569–584
Diagnosing long-chain branching in polyethylenes
J. Janzen a,*, R.H. Colby b
b
a
Phillips Petroleum Company Research Center, Bartlesville, OK 74004, USA
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Received 15 January 1999; accepted 4 February 1999
Abstract
We propose a novel method for assessing sparse long-chain branching in synthetic polymers such as high-density polyethylene at levels far below the limits of detectability by the usual methods of solution viscometry, size-exclusion chromatography, and NMR spectrometry on solutions. The new method exploits the extreme sensitivity of melt Newtonian viscosity to
random branching architecture, along with the systematic phenomenological description thereof developed recently in fundamental studies by Lusignan et al. The method satisfies the only validation criterion presently available to us: it finds long-branch
contents in quantitative agreement with stoichiometric yields calculated for several series of linear precursor polyethylenes
treated with very low levels of peroxide. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: Long-chain branching; Polyethylene; Entangled randomly branched polymers
One of the most vexing (and long-standing)
unsolved [1] problems in polymer physics is that of
characterizing ‘‘long-chain branching’’ (LCB) in
polyethylenes synthesized by various methods.
This problem is important because flow behavior
(‘‘rheology’’) of these products is enormously sensitive to LCB concentrations far too low to be detectable by spectroscopic (NMR, IR) or chromatographic
(SEC) techniques. Thus polyethylene manufacturers
are often faced with ‘‘processability’’ issues that
depend directly upon polymer properties that are not
explainable with spectroscopic or chromatographic
characterization data. Rheological characterization
becomes the method of last resort, but when the
rheological test results are in hand, we often still
wonder what molecular structures gave rise to those
results.
In this report the authors suggest that for progress to
be made in this area, it will be necessary to regard
rheometers themselves as probes of molecular structure, because they are the only instruments that
measure properties sufficiently sensitive to subtle
molecular-structure features to provide information
that can help answer the structural questions.
The approach that we suggest is admittedly yet
crude and approximate, but we believe that it is a
necessary first step toward a satisfactory resolution
of what we are tempted to call, ruefully, ‘‘the dreaded
polyethylene long-chain branching problem.’’ We
would, furthermore, emphasize that this step becomes
possible only after recognizing it necessary to relegate
the classical analyses 1 based on the seminal paper of
* Corresponding author. Tel.: 1 1-918-661-7756; fax: 1 1-918662-2870.
E-mail address: jyj@ppco.com (J. Janzen)
1
At least two reviews [2,3] of these techniques are useful,
although there are numerous misprints in the latter one.
1. Introduction
0022-2860/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S0022-286 0(99)00097-6
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J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Zimm and Stockmayer [4] to a role that is secondary
to some more modern ideas of polymer physics which
are based on scaling and percolation arguments.
2. Background
It has been 30 years since Porter et al. [5] declared,
‘‘The flow of branched polyethylene melts appears
distinct and as yet unexplained despite a hundred
published studies.’’ Thus they began one of innumerable papers attributing ‘‘anomalous’’ flow behavior
observed in polyethylenes to the presence of longchain branches, which they considered to be chain
segments longer than about 285 carbons. Among the
properties that they discussed as requiring distinct
explanation were: (a) zero-shear melt viscosities
departing markedly from the 3.4-power dependence
on molecular weight expected for linear polyethylene
and (b) larger-than-expected sensitivity of viscosities
to temperature, expressed as flow-activation energies.
Earlier, Schreiber and Bagley [6] proposed interpreting the ratio (h 0)lin/(h 0)br simply as an index
covariant with amount of long-chain branching,
where (h 0)br denotes the Newtonian limiting melt
viscosity observed at a given temperature for a sample
of interest and (h 0)lin is the corresponding viscosity
for a linear polymer of equal molecular weight, after
both have been corrected for small effects due to short
branches. Later, it has also been suggested [7,8] that
similar interpretation be placed on values of flow-activation energies. These approaches have not led to
methods that can be said to have achieved widespread
acceptance, for reasons that will become clear in later
discussion.
The extensive mainstream literature on long-branch
characterization in synthetic polymers is preponderantly comprised of variations on the theme of g-ratio
analysis, where the essential idea is to relate, quantitatively, experimental measurements sensitive to the
sizes of polymer molecules in dilute solution, such as
light scattering or intrinsic viscosities, to sizes
predicted theoretically for structures having specific
(regular or random) branching arrangements. The
pioneering publication giving theoretical g ratios is
that of Zimm and Stockmayer [4], and an early experimental application to polyethylenes was reported by
Billmeyer [9]. Subsequent developments up to 1975
were reviewed by Small [2], and Rudin has twice
provided more recent tutorials [10,11]. Publications
discussing related methods continue to appear, up to
the present [12,13].
There are three lingering difficulties in making the
g-ratio methods quantitative. The first is in knowing
which of several available theoretical expressions for
g most closely relates to the polymer structure of
interest when the latter has been synthesized by
methods that introduce branching by incompletely
understood mechanisms. The second is in knowing
how the theoretical g ratio, defined [14] as
ks2 l0 †br =…ks2 l0 †lin ; relates in detail to the experimental
measurement results when the physics of the latter is
imperfectly understood, as is the case with intrinsic
viscosity measurements. This difficulty can be highlighted merely by noting that the quantitative aspects
are still being debated [2,13] some 40 years after
Zimm and Kilb [15] argued that k < 1/2 in the relation
‰hŠbr
ˆ gk
‰hŠlin
1†
(where the left-hand side is a ratio of u -state intrinsic
viscosities), which differed from what Zimm and
Stockmayer [4] had proposed ten years earlier,
namely that k < 3/2. The third difficulty is that very
large rheological anomalies are observed in many
instances in which chromatographic and solution
viscometric experimental sensitivity is insufficient to
establish that the left-hand side of (1) differs significantly from unity, hence the analysis fails entirely—
rheologically significant long branches are below the
detection limit in dilute solution measurements [16].
The remaining traditional approach to longbranching characterization is through spectroscopy,
principally counting branch points by means of 13C
NMR in solution [17]. This approach has a sensitivity
limitation similar to that just mentioned for chromatographic and solution viscometric techniques: Because
of limited solubility and the rather low natural abundance of 13C nuclei, it is not feasible to count branchpoint carbons in concentrations below about 1 in 10 4,
which is still considerably higher than concentrations
we are faced with in many practical polyethylenes. In
addition, because the chemical shifts measured by
NMR are inherently insensitive to features of the
environment of a nucleus that lie beyond distances
corresponding to about 6 C–C steps, physics prevents
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
571
Fig. 1. Zero-shear viscosities obtained by fitting the Carreau–Yasuda model to complex viscosity data, uh*…v†u; for high-density polyethylenes
synthesized with various catalysts and reaction conditions. Continuous line is the linear-polymer benchmark.
this spectroscopy from perceiving distances between
branch points that even approach characteristic
lengths for chain entanglement, and we are compelled
to turn to melt rheometric measurements as the lone
remaining alternative source of large-scale structural
information. Let us say this again in no uncertain
terms: solution 13C NMR techniques are helpless in
discerning branch-vertex concentrations below about
10 24, and they are similarly helpless in discerning
branch lengths beyond about C6, while the rheologically relevant branch lengths are those long enough to
entangle, that is, at least C150, and such branches are
rheologically highly important at concentrations far
below 10 24.
For the purpose of providing an illustrative data set
to further motivate the exposition, we plot in Fig. 1
viscosity data for an extensive set of polyethylene
samples collected recently by Max P. McDaniel as
part of ongoing efforts to study long-branching characteristics produced by variations in the chemical
preparation and operating conditions of ethylene polymerization catalysts of both titanium-halide and chromium-oxide types. The specific details of these
variations in catalyst and polymerization chemistry
need not concern us in the present report; it will
suffice for our purposes that a wide (and, lacking
further insight, confusing) range of results was
produced. It is at first glance additionally confusing
that points lying well below the line in the figure are
routinely reported in the literature for LDPEs [18,19].
In Fig. 1, the continuous line is the 3.41-power
correlation (for 1908C) reported by Arnett and
Thomas [20] for strictly linear model polyethylenes
made by hydrogenating a series of anionically polymerized narrow-distribution polybutadienes and
extrapolating to zero ethyl-branch content. This line
agrees very closely with what (at the same temperature) was reported for a series of linear polyethylene
fractions (covering a slightly less wide range of Mw)
by Mendelson, Bowles, and Finger (MBF) [18] and
there are additional results in the literature in good
agreement with this, as well. Hence we have considerable confidence in this relation as a linear-polyethylene benchmark. The discrete data points in Fig. 1
were determined on samples furnished by Dr. McDaniel using methods described elsewhere [21,22] for
both h 0 and Mw. Thus the Mw values used in this figure
are SEC results uncorrected for branching effects, and
we denote that they are indicated results from SEC
using universal calibration with linear standards,
rather than absolute molecular weights, by adding a
subscript, viz. (Mw)sec.
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J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Fig. 2. Relation between zero-shear viscosities and molecular weights for the series of polyesters, studied by Lusignan et al., that were made to
have essentially constant average molecular mass between random branch points.
Attempts have been made in the past, at least by one
of us (J.J.) and some other colleagues, to discern a
systematic interpretation of results such as those
plotted in Fig. 1. We have looked for such features
as isopleths of constant flow-activation energy at
more-or-less constant displacements above the linear
benchmark, but without success. The scheme to be
proposed below is based on a distinctly different
point of view born of recent fundamental work [23–
25] on rheological properties of entangled randomly
branched polymers, and it makes clear the reasons
why the earlier attempts have failed.
3. Method
Lusignan [24] has remarked on the lack of a
proper theory for dealing with observations of
the sort we are interested in here, but he and
coworkers [23–25] have nonetheless proposed a
generalized phenomenological description which is
easily recast into a condensed form suitable for our
purposes.
The needed primary result from Lusignan, Mourey,
Wilson, and Colby (LMWC) [23–25], restated here
without rederivation, 2 summarizes the dependence of
zero-shear melt viscosity on molecular weight and an
average chain length adjacent to random branch
points, when present, as follows:
8
AMw for Mb , Mc
>
>
>
"
2:4 #
>
>
M
>
w
> AM 1 1
>
for Mc , Mw , Mb
w
>
<
Mc
h0 ˆ
"
2:4 #
>
>
M
Mw s=g
>
b
>
AM
1
1
>
b
>
Mc
Mb
>
>
>
:
for Mc , Mb , Mw
2†
Here Mc is a critical molecular mass for entanglement
of random branches, Mw is the mass-average molecular mass [14], and Mb is an average molecular
mass between a branch point and its adjacent vertexes,
either chain ends or other branch points. The numerical prefactor A carries the dimensions of viscosity,
2
In this report, the symbols s and g (not otherwise defined) are to
be interpreted having the same meaning as in Refs. [23–25]. Additional polymer-specific notation and nomenclature will be found in
Ref. [14].
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
573
Table 1
Parameters for randomly branched polyesters, used to calculate model curve in Fig. 2
Parameter
Value
Source
A, (Pa s)/(g/mol)
B
2.35(10 25)
3.89
M1, g/mol
Mc, g/mol
MKuhn, g/mol
Mb, g/mol
s/g
72/5 ˆ 14.4
2100
10.5M1 ˆ 151.2
40 000
6.22
Fit 3.4-power line to first three points
In Eq. (3), makes s/g ˆ 6.22 at Mb =Mc ˆ 40
000/2, 100 ˆ 19
Ref. [24], p. 177
Ref. [24], p. 172
From assuming C∞ ˆ 7
Observed break in h 0 vs. M
Fit last five points with power law
and is specific for a chosen polymer system at a given
temperature.
The second line on the right-hand side of (2)
obviously gives the usual [26,27] 3.4-power dependence of h 0 on Mw for linear polymers with Mw q
Mc. The general behavior observed in branched polymers, namely viscosities either greater or less [28]
than those of linear counterparts with the same Mw,
is captured in the third line of (2) through a dependence of the exponent s/g on Mb:
s
3
9
Mb
ˆ max 1; 1 B ln
3†
g
90MKuhn
2
8
Eqs. (2) and (3) comprise the general descriptive
model to be employed below. If A, B, Mc, and MKuhn
are known constants, then given measured values of
h 0 and Mw, it is possible to solve (2) (numerically) for
Mb, in cases 3 where Mc , Mb , Mw.
4. Model predictions
Fig. 2 compares the model comprised of Eqs. (2)
and (3) with measured data for a series of polyesters
studied by LMWC [25]. This series was synthesized
to produce materials varying in molecular weight but
having a nearly constant ration Mb/Mc and hence a
3
Such cases appear to be usual in the three main types of polyethylenes that we wish to consider here, namely: (a) ‘‘LDPEs’’,
low-density products made by free-radical polymerizations at high
pressures; (b) ‘‘HDPEs’’, high-density products of high molecular
weight from catalysts based on chromium oxides or titanium
halides; and (c) ‘‘LLDPEs’’, lower-density variants of the
HDPEs, made by copolymerization with increased levels of a olefins.
constant viscosity exponent s/g . The discrete
experimental data points in the figure are from
Table 2 of Ref. [25], and the continuous trace is
calculated using the parameter values listed in
Table 1.
Fig. 3 displays the general behavior of the model
after re-parameterizing for polyethylene at 1908C,
that is, using the parameter values collected in
Table 2.
Fig. 3 can be regarded as a sort of nomograph
which could be used to look up an estimate of
Mb/Mc (and hence Mb itself) from a pair of coordinates, (Mw, h 0). However, in the region between
the 3.4-power (linear-polymer) line and the dashed
curve representing an upper bound, this gives an
ambiguous result, because, as may be inferred by
inspection of the figure, every point in this region
is at an intersection of two different isopleths of
constant Mb/Mc. From another perspective, if one
takes a vertical slice, varying Mb/Mc while holding
Mw fixed, it is found that h 0 passes through a
maximum (a point on the upper-bound locus) and
then declines. Thus, in the enclosed region on and
above the 3.4-power line the model (for a given h 0)
has two solutions for Mb/Mc, and only one in the
region just below.
Interestingly, in addition to representing linear
polymers, the 3.4-power line is also expected to
represent h 0(Mw), regardless of molecular weight,
for branched polymers that happen to have a longbranch density such that Mb/Mc equals what we
might call a ‘‘quasilinear’’ ratio (obtained by solving
Eq. (3) for s/g ˆ 3.4):
Mb
90MKuhn
3:4 2 1:5
ˆ
exp
4†
Mc ql
Mc
9=8†B
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J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Table 2
Parameters for calculating model curves in Fig. 3, for polyethylenes at 1908C
Parameter
Value
Source
A, (Pa s)/(g/mol)
B
M1, g/mol
Me, g/mol
Mc, g/mol
MKuhn, g/mol
90MKuhn/Mc
Mb/Mc
s/g
5.22(10 26)
6
14.027
840
2100
10.4M1
6.252
Varied independently
Varied
Ref. [20]
This work
–CH2 – molecular weight
Ref. [29]
2.5Me
Makes a C∞ ˆ 6.93 and kr 2l0/M ˆ 1.20 Å 2mol/g
This table
Chosen to span 2 , Mb/Mc , 10 6/2100
Eq. (3) (function of Mb/Mc)
Assuming 10.4 C–C bonds per Kuhn segment is intermediate between matching C∞ ˆ 6.7 from Flory [30] and kr 2l0/M ˆ 1.25 from Fetters,
Lohse, and Colby [29].
a
For this ratio, the parameter values in Table 2 imply
Mb
< 8:3
5†
Mc ql
[28] that analogous crossovers occurred at values of 3
and 4 in regular 3-arm and 4-arm star polybutadienes,
respectively.
and from the values in Table 1 we obtain
Mb
< 10
Mc ql
5. Application
6†
Values of this magnitude (around 9) for randomly
branched polymers seem to us to be entirely plausible
by comparison with the findings of Kraus and Gruver
Practical application of the principal results, Eqs.
(2) and (3), which would consist of solving numerically for Mb/Mc given measured values for Mw and h 0,
is not quite as straightforward as it sounds, because of
Fig. 3. Selected viscosity isopleths given by the model (Eqs. (2) and (3)), with parameter values (Table 2) estimated for polyethylene.
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
575
Fig. 4. Cartoon representation of the topology assumed for randomly long-branched polyethylenes.
some considerations that we have alluded to but not
yet taken into account. One is that in practice the
measurements we have access to are more often
linear-equivalent molecular weights, (Mw)sec, than
the needed absolute values, Mw. Another is that we
need a way to disambiguate the dual solutions for Mb/
Mc at a single point (Mw, h 0). A third is that we have
not specified a topological model that will allow a
concrete structural interpretation of Mb/Mc in familiar
terms. In this section we indicate in principle how to
deal with these questions, and point out specific
assumptions we make to allow us to proceed in practice to obtain results that appear to be at least semiquantitatively plausible.
least one terminal vertex an arm. Thus a linear
polymer has one edge that is an arm; it also has two
terminal vertexes, no long-branch vertexes, and no
interior edges. A star polymer is a Cayley tree having
f arms and one long-branch vertex of functionality f,
but no interior edges.
Fig. 4 shows a cartoon example [31] of a Cayley
tree (with f ˆ 3, v1 ˆ 12, and v3 ˆ 10), annotated to
illustrate some additional nomenclature that will
allow translation from Mb/Mc, the primary result
obtained by solving Eq. (2), into some more familiar
terms traditionally used in discussions of LCB. Graph
theory dictates 5 that b, the total number of edges, is
related to the total number of vertexes through
b 1 1 ˆ v1 1 v3
5.1. Assumed topological structure and some
nomenclature
We assume, as is customary [1,31], that we are
dealing generally with polymer molecules adequately
describable topologically as Cayley trees having, on
average, a number vf of f-functional long-branch
vertexes 4 and v1 terminal vertexes (chain ends) per
mass-average molecule (where f is an integer greater
than 2). We call any chain segment between two adjacent vertexes and edge [32]. (Adjacency of two
vertexes means that there is a connection between
the two that does not pass through any other vertex.)
We call a segment between two adjacent long-branch
vertexes an interior edge; and a segment having at
4
We use ‘‘long-branch vertex’’ as another term for what Zimm
and Stockmayer [4] called a ‘‘branch point.’’
7†
and also that
v1 ˆ v 3 1 2
8†
Together, Eqs. (7) and (8) imply
b ˆ 2v3 1 1
9†
and this is easily verified in the example of Fig. 4 by
counting the edges and finding that there are 21 of
them; this happens to equal 2(10) 1 1.
A statistic commonly used to quantify long-chain
branch content is a , the fraction of the total carbons
5
Bugada and Rudin [33] reported, for a lot of LDPEs, edge
lengths that do not satisfy the graph-theoretic constraints laid out
here, and Usami and coworkers [34] followed suit. The discrepancy
has been pointed out by Martin [35], whose reasoning was correct
although his arguments were not couched in formal graph-theoretic
language.
576
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Fig. 5. Predicted effects on viscosity of one long-chain branch point per mass-average molecule, in two polyethylenes of different molecular
weights.
that are long-branch vertexes. This is defined as
v3
Mw =M1
aˆ
10†
Now since Mb must also be simply the total molecular
mass divided among b edges,
Mb ˆ
Mw
b
11†
Then substituting for b from (9) and eliminating v3
between (10) and (11) leads to
M1 21
Mb 2 Mw21 †
2
aˆ
12†
Two other statistics used [2,33–36] as LCB descriptors are Cb, the number of carbons per edge, and l , the
number of long-branch vertexes per unit of molecular
weight. These are easily obtained from the foregoing
as
Cb ˆ
Mb
M1
13†
and
lˆ
a
v
ˆ f
M1
Mw
14†
5.2. Assumed corrections for hydrodynamic volume
effects in SEC
If (Mw)sec and Mw are related through some function
that depends on LCB, we can formally incorporate the
necessary correction into the model by simply
carrying the relation along with (2) and (3) as another
equation to be satisfied simultaneously. As mentioned
above, it is not known what specific choice might be
most accurate, so to illustrate the argument we simply
select commonplace forms often invoked in the literature. Eq. (48) of Zimm and Stockmayer, [4] written in
present notation, is
‰hŠbr
ˆ g22a
‰hŠlin
15†
where a is the exponent in the Mark-HouwinkSakurada [37,38] equation
‰hŠlin ˆ KMva
16†
Our numerical computations will be done using the
value a ˆ 0.725; this is taken from Eq. (8) of
Wagner’s critical review [37] as applicable to the
solvent that we use for SEC.
To get a simple explicit molecular weight correction [39], we combine Eqs. (2) and (6) from Rudin’s
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
577
Fig. 6. Results obtained by inverting the model described in the text, using the data shown in Fig. 1, plotted to show that the average number of
carbons between long-branch vertexes is slightly less that the average number of carbons per mass-average molecule.
[10] chapter 6 to obtain
M
ˆ
Msec
‰hŠbr
‰hŠlin
21=…11a†
where
17†
and then combine this with (15), leading to
M
a22
ˆg
Msec
11a
21†
18†
Finally, we shall use (18) as an approximation to
the desired average correction ratio [39], in the
form
Mw
a22
ˆg
Mw †sec
11a
19†
and we take g as given by Eq. (44) of Zimm and
Stockmayer [4]; that is
g ˆ kg3 lw
kg3 lw ˆ
#
(
!1=2 "
)
6 1 2 1 v3
2 1 v3 †1=2 1 …v3 †1=2
ln
21
v3
v3 2
2 1 v3 †1=2 2 …v3 †1=2
20†
6
Erratum: This chapter has Eq. (44) of Zimm and Stockmayer
printed incorrectly, as Eq. (10).
Fig. 5, calculated from the model that now includes
Eqs. (10), (12), (19), (20) and (21), is provided as an
illustration of the combined effects of LCB and
corrections according to Eq. (19) for two instances
having v3 ˆ 1, that is, one long-branch vertex per
mass-average molecule in, respectively, molecules
of 5000 and 10 000 carbons. Horizontal error bars
are drawn to represent experimental uncertainty of
9% in …Mw †sec values. Hydrodynamic size corrections
are seen to be not greater than this experimental
uncertainty at the a levels shown, namely 1 to
2(10 24), while the melt viscosity increase relative to
linear polymer of the same molecular weight is a
factor of 1500 in the larger molecule with the lower
a , but only a factor of about 9 in the smaller molecule,
even though the a is larger.
The relative insensitivity of calculated g ratios to
lower amounts of LCB provides the key to knowing
578
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Fig. 7. Results obtained by inverting the model described in the text, using the data shown in Fig. 1, expressed as the fraction of total carbons
that are long-branch vertexes.
experimentally which of the two solutions for Mb =Mc
to accept. With on-line viscometry [11], we can distinguish whether or not the correction given by Eq. (17)
is appreciably greater than unity. If it is, we take the
solution corresponding to the larger value of v3; if not,
the smaller value is admitted. Thus we use an
experimental indication of the intrinsic-viscositybased g ratio to disambiguate dual numerical solutions
to Eq. (2), without relying on accurate values in both
sides of Eq. (15) to effect the entire analysis. It also
appears that a large value of flow-activation energy
can be another qualitative pointer to the lower of two
Fig. 8. Results obtained by inverting the model described in the text, using the data shown in Fig. 1, plotted as viscosity enhancement ratios
versus long-branch vertex concentrations. Molecular weights are not constant for a given a .
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
579
Fig. 9. Comparison between independent estimates of long-chain branch contents in several series of polyethylenes treated with known very
low levels of peroxide. The abscissas (stoichiometric predictions) were calculated assuming a yield of one trifunctional long-branch vertex per
oxygen atom added as peroxide. The ordinates were obtained from the model using molecular weights measured via SEC and zero-shear
viscosities obtained, as in Fig. 1, by fitting complex viscosity data.
solutions for Mb =Mc (higher LCB density), even
though it does not appear feasible to use such values
directly as monotonic measures of LCB.
6. Results and discussion
6.1. Case studies
In Figs. 6–8 we display, in three different ways,
LCB statistics obtained using the data of Fig. 1 as
input for the analysis outlined above.
Fig. 6 plots Cb vs. Mw =M1 , that is, the number of
carbons per edge compared with the total number of
carbons per mass-average molecule. At first glance
this might appear to be a miraculous ordering of the
data from Fig. 1, but that impression is spurious. All
that is evidenced is that the branch concentrations are
very low, that is, the average edge molecular mass is
just slightly less than that of the total molecule. Fig. 6
is included only to underscore the magnitude of the
edge lengths at which rheological effects of very few
long-branch vertexes are still significant and easily
detected.
Fig. 7 plots the long-branch vertex fraction vs.
molecular weight. This illustrates that there is not a
simple relation between the two, but that various longbranch densities can be made at a given Mw ;
depending on the chemistry of the catalyst preparation
and polymerization conditions. It is beyond the scope
of the present report to explore these dependencies in
detail, but preliminary indications are that the a s do
bear some systematic relations to conditions of
polymer genesis, and it is hoped that the proposed
technique will prove to be of utility in furthering
investigations into these relations.
Fig. 8 plots the viscosity enhancement ratio
h0 †br =…h0 †lin against LCB concentration, as a demonstration of the now-expected circumstance 7 that,
again, there is no simple monotonic relation between
the two, because of the large confounding effect of
length of entangling edges, which the present analysis
scheme attempts to account for.
It is apparent from Figs. 7 and 8 that the present
method can routinely indicate a values much less than
10 24. These are, as was emphasized in the background
discussion, well below detection limits of customary
spectroscopic techniques. Hence there are not many
7
See again Fig. 5.
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J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Fig. 10. Binary blend study confirming expected mass-wise averaging of long-branch vertex concentration estimates, as well as of massaverage molecular weights.
choices when it comes to validating the proposed
technique in this range. Of course we regard the extensive studies by LMWC, which laid the essential
groundwork, as providing substantial validating
evidence in advance. Additional data for direct
comparison, however, come from applying the
analysis to samples into which stoichiometrically
calculable amounts of long-branch vertexes have
been introduced by treatment with known very small
amounts of peroxides. Fig. 9 displays such comparisons for several series of polyethylenes so treated. The
starting materials varied in Mw ; and we customarily
observe that the rise in h0 with a given peroxide dose
steepens with starting Mw (or h0 ), as our descriptive
scheme would predict. Experiments of this sort also
appear to be fairly reproducible, since we obtain
initial slopes of h0 vs. added peroxide concentration
that are in good quantitative agreement with published
results of Hughes [7] and of Bersted [40], when we
use starting materials with h0 values similar to theirs.
The assumption made in the stoichiometric calculation of the x-axis values for Fig. 9 is that one trifunctional long-branch vertex is yielded by each oxygen
atom introduced as peroxide. This is the usual starting
assumption, according to the review by Lazár and
coworkers, [41] who also mention the free-radical
reaction mechanism by which this may be supposed
to occur. The striking thing observed in Fig. 9, then, is
that the stoichiometrically predicted a values and
those obtained by the proposed analysis of measured
h0 and …Mw †sec data are in excellent quantitative
agreement. That is, the slopes for all the sample sets
are satisfactorily close to unity, including that for the
one set which had appreciable LCB indicated present
before peroxide addition.
Another way of checking plausibility of our results
that has occurred to us is to see if they follow the
expected mass-average blending rule in a series of
binary blends of components with different indicated
a values. Fig. 10 presents results for such a series of
blends. Two polyethylenes differing both in Mw and a
were extruder blended in varying proportions, with
the results shown: Both Mw s and a s determined
according to the proposed technique interpolate linearly with mass-fractional composition, as is required
of mass-intensive properties. This alone by no means
proves that our results are correct, but it does tend to
obviate one possible argument by which they could be
called into question.
As a final example of practical applications, we
give results obtained by our analysis for the several
high-pressure low-density polyethylene fractions
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
581
Fig. 11. Zero-shear viscosity and molecular weight data from Mendelson et al. for two series of low-density polyethylene fractions, each from a
single parent.
studied by MBF [18]. These are included for the
reason that they represent the upper end of the range
of a values that are of interest as far as commercial
polyethylenes are concerned. The requisite input
coordinates …Mw †sec and h0 were taken as given by
Tables II and V of Ref. [18]; these are plotted in
Fig. 11. For the analysis by our method, the prefactor
Table 3
LCB statistics inferred for LDPE fractions of Mendelson, Bowles,
and Finger [18]
Sample
(Mw)sec, kg/mol
h 0, Pa s
10 4a
Cb
E-F2
E-F3
E-F4
E-F5
E-F6
E-F7
E-F8
P-F1
P-F2
P-F3
P-F4
P-F5
P-F6
P-F7
12.3
20.0
33.6
56.2
110
305
1240
5.0
12.4
26.2
57.4
81.5
134
238
3.0
15.0
94
555
5.9(10 3)
7.3(10 4)
8(10 5)
0.325
4.35
67
1.85(10 3)
1.80(10 4)
2(10 5)
2(10 6)
4.05
2.52
3.15
3.77
4.28
4.88
5.35
3.35
2.55
2.32
3.27
3.29
3.62
3.99
528
855
995
1046
1066
1010
934
292
623
1034
1169
1266
1264
1212
A (Table 2) was increased to 1.07(10 25), to allow for
the fact that MBF determined h0 at 150 instead of
1908C. This shift is calculated using the flow-activation energy value of 29.29 kJ/mol determined by
Arnett and Thomas [20]. Table 3 lists the resulting
LCB statistics for this set of materials, and Fig. 12
shows, as filled symbols, the LDPE Cb results from
Table 3 in comparison with HDPE results plotted
previously in Fig. 6.
The noteworthy aspect of these statistics is that for
Mw =M1 $1000 the indicated edge lengths are roughly
constant, and of the order of 10 3 carbons. This is
appreciably larger than values reported in the literature [35] for LDPEs based on NMR spectroscopy,
namely Cb , 200. We interpret this as entirely plausible: In as much as NMR spectroscopy almost
certainly counts many branches much too short to
entangle, while our method should only sense
branches that are truly ‘‘long’’, that is, rheologically
significant, we would expect to find an appreciable
difference between the LCB concentrations indicated
by the two methods, and this difference is straightforwardly attributable to branches that are ‘‘short’’
(C,150) by rheological standards but ‘‘long’’ (C.6)
on the scale discernible by NMR. To our knowledge,
582
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
Fig. 12. Results obtained by inverting the model described in the text, using the data shown in Fig. 11, plotted in comparison with the HDPE
results shown previously in Fig. 6.
ability to assess this difference experimentally, in
terms even approaching semiquantitative, is without
precedent.
6.2. Uncertainties and limitations
The fourth paragraph of the introduction acknowledged in advance that the scheme of analysis outlined
above is not to be considered refined or complete. In
order to present the basic ideas expeditiously, it has
been necessary to gloss over several fine points that
will probably need to be revisited as additional information becomes available.
Specifically, first there is some uncertainty in the
values of the two important parameters: B and the
constellation 90MKuhn =Mc : LMWC [25] estimated
B < 2.1, although reanalysis of their data, as in Fig.
2, indicates B < 3.9. This, however, is still too small
to account for observations of the sort plotted in Fig.
1. We have therefore taken B ˆ 6 because this gives a
least upper bound (dashed curve in Fig. 3) for the data
of Fig. 1. This may turn out to be too conservative, and
a somewhat larger value may be indicated by future
observations. We note that in the range of interest, s
values predicted by the theory of Rubinstein, Zurek,
McLeish, and Ball [42] (dashed line in Fig. 11 of Ref.
[25]) are well approximated with B < 7.4, and that the
polyethylene cases reported in Ref. [25] are very close
to agreement with this theoretical result, rather than
with B , 4. This raises the question of whether or not
B should be regarded as a universal constant or as a
parameter varying somewhat among different specific
polymer systems (and possibly temperature dependent, also). It would not be surprising to find B
affected by different values of f, and of course we
cannot be certain that we have no contributions
from vertexes with f ˆ 4 in our experimental materials, although our calculations are done assuming f ˆ
3 in the model. Further consideration of these questions is beyond the scope of the present report, as is
any more nearly rigorous accounting for possible
effects of distributions of molecular weights, edge
lengths, and f.
In this first presentation we have also neglected
correction for the presence of any short-chain
branches. This, however, will be rather simple to
deal with: Short-chain branches introduced by a olefin comonomer incorporation can be accounted
for just by making what will usually be a rather
small adjustment to the value of M1, in the manner
employed by Arnett and Stacy [43]. Suitable corrections can be estimated straightforwardly from NMR
J. Janzen, R.H. Colby / Journal of Molecular Structure 485–486 (1999) 569–584
583
information on total branch content, if available, or
from solid-state density, once this has been related to
short-branch content. At the present stage of development, we have not felt it worthwhile to attempt to
include such corrections, because of the possibility
that considerably greater inaccuracy could be introduced by the assumptions we made in arriving at Eq.
(19). Interestingly, by the way, the effect of whatever
inaccuracy may be present in Eq. (19) will be greater
at higher LCB concentrations; that is, we expect that
the lowest a values ( p 10 24) indicated by our
analysis will generally be the most accurate, as long
as the input viscosity and molecular weight data are
free of error. This behavior of error due to bias in the
model is opposite what one usually expects.
us by Dr. Ashish M. Sukhadia. The binary blends (Fig.
10) were furnished by Dr. Rex L. Bobsein.
7. Conclusions
[9]
[10]
This report has outlined a novel scheme for diagnosing long-chain branching in polyethylenes, using
data from SEC and melt viscometry. This scheme
accounts for the strong dependence of viscosity on
very low concentrations of non-linear chain molecules
with distances between branch points that are very
much longer than entanglement spacings. It also
accounts for the eventual reduction of melt viscosity,
relative to that of linear chains, as distances between
long-branch vertexes decrease to small multiples of
entanglement spacings. Thus the method proposed is
applicable to polyethylenes spanning the range of
types from HDPEs, with very sparse LCB, to highpressure LDPEs, customarily regarded as having the
greatest degrees of LCB.
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Acknowledgements
We acknowledge indispensible contributions by
several individuals to various facets of the work
reported here. Dr. David C. Rohlfing and Michael J.
Hicks have provided zero-shear viscosity and flowactivation energy data on numerous materials, and
SEC data were obtained by Dr. Timothy W. Johnson
and Delores J. Henson. Dr. Max P. McDaniel
provided the assortment of sparsely long-chain
branched samples described in the body of the report,
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