Fractionation

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FRACTIONATION

Introduction

Synthetic polymers are heterogeneous materials. Measuring this heterogeneity
is the key to understand structure-property relationships, polymerization mech-
anisms and kinetics, and polymer reaction engineering. Polymer fractionation
techniques attempt to fractionate polymers according to speciﬁc characteristics of
their microstructures as deﬁned by their distributions of molecular weight, chem-
ical composition, comonomer sequence length, tacticity, and long-chain branching.

All synthetic and some natural polymers have a distribution of molecular

weights caused by the statistical nature of the polymerization mechanism. Poly-
mers produced via living polymerization have the narrowest molecular weight dis-
tributions, while long-chain branched polymers or polymers made with multiple-
site catalysts such as heterogeneous Ziegler–Natta catalysts have the broadest
molecular weight distributions. Nonetheless, regardless of the polymerization
mechanism involved in their production, all polymers have a molecular weight
distribution that must be quantiﬁed to unequivocally deﬁne their microstructures
and their physical properties. This explains why the vast majority of polymer frac-
tionation techniques were developed to fractionate polymers according to molec-
ular weight.

For instance, the instantaneous chain-length distribution of many polymers

is given by Flory–Schultz’s most probable distribution (1),

exp

−

(1)

where f

is the number distribution of chains of length n, and N

is the number-

average chain-length. The molecular weight distribution can be obtained from the
chain-length distribution by multiplying the latter by the molecular weight of the
repeating unit.

The equivalent weight chain-length distribution, w

, is given by

exp

−

(2)

Polymers whose chain-length distributions follow equation 1 have a poly-

dispersity index N

of 2, where N

is the weight-average chain length of the

polymer. Figure 1 shows the number and weight chain-length distributions of a
polymer with N

= 1000. Notice that, even for the relatively narrow polydispersity

of 2, a signiﬁcant range of chain lengths is encompassed by the distribution.

Nowadays the most common technique for determining the molecular weight

distribution of polymers is size-exclusion chromatography (SEC), also known as
gel-permeation chromatography (GPC). Other methods for measuring molecular
weight distributions exist, but the ease of use and universality of SEC makes it
the technique of choice for this type of fractionation.

Copolymerization adds another level of complexity to polymer microstruc-

ture. Superimposed on the molecular weight distribution, we now have to

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Fig. 1.

Flory–Schultz’s most probable chain length distributions for N

= 1000. f

(—–)

Number distribution; w

(– – –) weight distribution.

account for the distributions of chemical composition and comonomer sequence
length. The instantaneous bivariate weight distribution of chain length and
chemical composition, w

, of many linear binary copolymers was derived by

Stockmayer (2):

exp

−

πF

− F

)

exp

−

− F

)

(3)

where y is the deviation from the average fraction of monomer 1 in the copolymer
F

. The parameter

κ is deﬁned as a function of F

and the reactivity ratios r

and

κ =

− 4F

− F

)(1

− r

)

(4)

The average fraction of comonomer in the copolymer can be estimated as

usual using the Mayo–Lewis equation:

− 1)f

+ f

+ r

− 2)f

+ 2(1 − r

+ r

(5)

where f

is the molar fraction of monomer 1 in the reactor.

Notice that equation 3 reduces to Flory–Schultz’s most probable distribution

upon integration over all copolymer compositions, ie integrating y from

− ∞ to

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FRACTIONATION

Fig. 2.

Stockmayer’s distribution for chains of different length of a model random copoly-

mer (r

= 1.0) with F

= 0.5 and N

= 1000.

= 250;

= 500;

2000.

+ ∞. Consequently, linear binary copolymers described by Stockmayer’s bivariate
distribution also obey Flory–Schultz’s distribution for chain length.

Figure 2 compares the chemical composition distribution of polymers of dif-

ferent chain lengths for a random binary copolymer (r

= 1.0). Notice how the

distribution gets broader with decreasing chain-lengths. It is easy to understand
this behaviour considering that, as chain length tends to inﬁnity, the composition
of all copolymer chains should be the same for a given probability of polymeriza-
tion of monomer 1 and monomer 2. On the other hand, chains with short lengths
are more likely to deviate from the average copolymer composition on purely sta-
tistical grounds, in the same way as it is more likely to get three consecutive heads
by tossing a coin three times than it is to get one hundred consecutive heads by
tossing a coin one hundred times.

Figure 3 shows how the chemical composition distribution of copolymer

chains of the same length varies when the comonomer incorporation varies from
alternating (r

→ 0) to random (r

= 1) to block (r

→ ∞). As expected,

the distribution gets broader with increasing tendency to form comonomer blocks:
all chains of a perfectly alternating binary copolymer have the same composition
(ie F

= F

= 0.5) but the tendency to form long comonomer sequences or blocks

→ ∞) may broaden the distribution considerably.
There are no general analytical solutions to describe the chemical composi-

tion distribution of ternary or higher copolymers, although an analytical solution
for the particular case of random multicomponent copolymers has been derived
(3). For nonrandom polymerizations, the chemical composition distribution of mul-
ticomponent copolymers can be calculated using techniques such as Monte Carlo
simulation or population balances (4).

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Fig. 3.

Stockmayer’s distribution for chains with F

= 0.5 and N

= 1000 as a function

of reactivity ratio product. Alternating: r

= 0.01; random: r

= 1; and block: r

10.

= 0.01;

= 1.0;

= 10.

Asymmetrical monomers such as propylene and styrene form polymers with

different tacticity. Isotactic and syndiotactic chains have regular molecular struc-
tures and are generally semicrystalline, while atactic polymers are amorphous
because of the random placement of the monomer units in the chain. The tacticity
distribution may be modeled as a comonomer distribution, where the different
monomer orientations during chain insertion (meso or racemic) are treated as
pseudo-comonomer types. Therefore, at least in principle, Stockmayer’s distribu-
tion can be used to approximate the tacticity distribution of homopolymers.

The most common techniques to measure the chemical composition distri-

bution of semicrystalline copolymers are temperature rising elution fractionation
(TREF) and crystallization analysis fractionation (Crystaf). These two techniques
are based on the differences among the crystallizabilities of copolymer chains
with distinct comonomer contents. Because variations in tacticity affects polymer
crystallinity, Crystaf and TREF have also been used to measure the tacticity dis-
tribution of polymers. No direct methods are available for measuring the complete
chemical composition distribution of amorphous copolymers, although some ﬁeld
ﬂow fractionation, ultracentrifugation, and chromatography methods are sensi-
tive to differences in comonomer content. The use of matrix-assisted laser desorp-
tion ionization time-of-ﬂight (MALDI-TOF) mass spectrometry as a detector for
SEC has also been proposed to determine the bivariate distribution of molecular
weight and chemical composition of some copolymers, either semicrystalline or
amorphous.

The last and most complex polymer microstructural distribution is the one

for long-chain branching. Long-chain branches can be formed according to very

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FRACTIONATION

different mechanisms, and analytical solutions for long-chain branch distributions
are only known for a few particular cases. For the case of terminal branching
with coordination catalysts, the instantaneous joint distribution of chain-length,
chemical composition, and long-chain branching, w

,n,y

, is given by the equation

(5,6)

,n,y

(2i

+ 1)!

+ 1

+ 2

exp

−

πF

− F

)

exp

−

− F

)

(6)

where i is the number of long-chain branches per chain. Notice that Stockmayer’s
distribution, equation 3, is a particular solution of equation 6 for linear chains, ie
for i

= 0.

Figure 4 and Figure 5 illustrate how the distributions of chain length and

chemical composition vary as a function of the number of long-chain branches per
chain for a model copolymer. Notice how chains with more long-chain branches
are longer and have narrower distributions of chain length and chemical com-
position. The increase in average chain length with increasing branching is self-
explanatory; the narrowing of the distributions is caused by the averaging effect of
randomly combining chains of different lengths and compositions via long-chain
branch-formation reactions.

Unfortunately, there are no techniques that can be used to fractionate poly-

mers exclusively according to long-chain branching. A measure of average long-
chain branching as a function of molecular weight can be obtained when SEC
is combined with a viscometer or a light-scattering detector, but the results are

Fig. 4.

Chain-length distribution as a function of long-chain branching frequency for a

model polymer with N

= 1000, F

= 0.5, and r

= 1.0. The curves are obtained by

integrating equation 6 over the entire composition range,

− ∞ < y < + ∞. For details see

Ref. 6.

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Fig. 5.

Chemical composition distribution as a function of long-chain branching frequency

for a model polymer with N

= 1000, F

= 0.5, and r

= 1.0.— -- —Linear;

LCB;

3 LCB;

10 LCB. The curves are obtained by integrating equation 6 over

the entire chain-length range, 0

< n < ∞. For details see Ref. 6.

semiquantitative at the best for most polymers because of the complex fractiona-
tion process that takes place in nonuniformly branched polymers. Considering the
importance that long-chain branching has on altering rheological and mechanical
properties of polymers, this poses a signiﬁcant limitation on the understanding of
a very important class of polymers.

These three theoretical distributions describe only a very small portion of

the diversity of polymer microstructures that are produced every day in academia
and industry. Even for the polymerization systems they describe, they are only
strictly valid as instantaneous distributions. If conditions in the polymerization
reactor ﬂuctuate as a function of time or spatial location, the distributions for
the polymer product may be considerably more complex. In this case, it is very
difﬁcult to ﬁnd a mathematical model precise enough to describe the complete
polymer microstructure, and we must rely solely on experimental fractionation
for its determination. In fact, the comparison of experimentally-measured mi-
crostructural distributions with the ones predicted by theory is a powerful tool to
investigate polymerization mechanisms and understand polymer reactor nonide-
alities. Nonetheless, these distributions are essential to realize the complexity of
polymer microstructure and the interdependency of the distributions of molecular
weight, chemical composition (or tacticity), and long-chain branching. This inter-
dependency should always be kept in mind when interpreting the fractionation
data from any experimental technique.

In this article we will review some relevant fractionation techniques that can

be used to elucidate the main aspects of the polymer microstructures delineated
above.

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FRACTIONATION

Batch Fractionation

It has been claimed that polymer batch fractionation is almost as old as polymer
science itself (7). Batch fractionation relies on the separation of polymer chains,
initially present in a dilute solution, in two phases (liquid–liquid or liquid–solid).
Fractionation occurs by altering the goodness of the solvent, either by changing
the ratio of solvent to nonsolvent or by varying the temperature of the polymer
solution. The nonsolvent, most often, must be miscible with the solvent, but should
not be able to dissolve the polymer.

Batch fractionation was developed to determine the molecular weight distri-

bution of polymers. Even though molecular weight fractionations nowadays are
more efﬁciently achieved with chromatographic techniques such as size-exclusion
fractionation, batch fractionation is still important when the objective is to obtain
large fractions of polymers with narrow molecular weight distributions that may
be required, for instance, to establish structure-property relationships or to eluci-
date polymerization mechanisms. In this way, batch fractionation is still relevant
today, albeit to serve a different purpose from which it was originally developed.

Semicrystalline polymers can be fractionated according to their molecular

weights by the solvent/nonsolvent approach at temperatures above their melting
point, or by crystallizability by the temperature variation approach. Amorphous
polymers are fractionated only according to their molecular weights. We will de-
scribe the fractionation of amorphous polymers ﬁrst, followed by a discussion on
the fractionation of semicrystalline polymers later.

A comprehensive compendium of laboratory procedures for batch fraction-

ation of polymers was written by Francuskiewicz (8). The readers are directed
to this reference for more speciﬁc details on the several experimental techniques
described below.

Fractionation of Amorphous and Semicrystalline Polymers by

Molecular Weight.

Flory–Huggin’s theory for dilute polymer solutions is still

convenient to describe the liquid–liquid batch fractionation of amorphous poly-
mers between a polymer-rich phase and a polymer-lean phase. Even though Flory–
Huggin’s theory for dilute solution is based on several simplifying assumptions,
it correctly describes the trends during the batch fractionation of polymers and
provides a useful theoretical framework to rationalize its qualitative features. The
discussion in this section will follow closely the elegant development adopted by
Flory in his remarkable book on polymer chemistry (9).

Semicrystalline polymers can be fractionated by either molecular weight

or crystallizability. All the considerations that will be made in this section
for the fractionation of amorphous polymers by chain length are equally valid
for semicrystalline polymers, provided that the polymer is kept above its
melting point throughout the fractionation to eliminate chain crystallization
effects.

For a liquid–liquid system, the volume fractions of polymer of chain length n

in the polymer-rich phase,

∗

, and in the polymer-lean phase,

, are conveniently

described by the simple exponential equation:

∗

= e

σ n

(7)

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The parameter

σ can be estimated with the expression

σ = υ

−

− υ

∗

−

∗

+ χ

− υ

)

−

− υ

∗

(8)

where N

is the number average chain length,

χ is the Flory–Huggin’s interaction

parameter between polymer and solvent, and

∗

and

are the volume fractions

of solvent in the polymer-rich and polymer-lean phases, respectively. In the fol-
lowing discussion, the exact value of the parameter

σ is immaterial; it sufﬁces to

know that

σ depends on the properties of the polymer and solvent(s) used in the

fractionation.

Calling V and V

∗

the volume fractions of the polymer-lean and polymer-rich

phases, respectively, and deﬁning R

∗

, the fraction of polymer chains with chain

length n in the polymer-lean phase, f

, is given by

+ V

∗

+ R

∗

+ Re

σn

(9)

Similarly, the fraction of polymer with chain length n in the polymer-rich

phase, f

∗

, is given by the expression

∗

= 1 − f

σn

+ Re

σn

(10)

Equation 7 shows that, for positive values of

σ , the partition ratio

∗

always greater than 1. Therefore, the polymer chains, independently of their chain
lengths, will always have a higher volume fraction in the polymer-rich phase. Since
the partition ratio increases exponentially with chain length, longer chains will be
more selective towards the polymer-rich phase than shorter chains. This simply
reﬂects the fact that shorter chains have fewer units to interact with the poor
solvent in the polymer-lean phase, and therefore are less discriminating than the
longer chains. Batch fractionation of amorphous polymers relies solely on this
small selectivity difference among chains of different lengths.

Precipitation Fractionation.

In this method, fractions of decreasing molec-

ular weights are obtained in each successive fractionation step. Because of the
decreasing molecular weight of the fractions, this technique is also called down-
ward precipitation fractionation, to distinguish it from another variant named
upward precipitation fractionation that will be described later in this section. In
either case, it is instructive to notice that precipitation fractionation is a mis-
nomer, since the polymer does not in fact precipitate, but rather forms a separate
polymer-rich liquid phase.

The precipitation fractionation method starts with a dilute polymer solution.

Liquid–liquid separation, with the formation of a polymer-rich and a polymer-lean
phase, is induced by either decreasing the temperature or by adding a speciﬁed
amount of a nonsolvent. A third, less common technique, involves the evaporation
of solvent from a solvent/nonsolvent mixture. All these procedures increase the
value of

χ, until it eventually exceeds its critical value, thus leading to phase

separation.

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FRACTIONATION

The Flory–Huggin’s interaction parameter,

χ, is deﬁned by the equation

χ =

(

− δ

)

(11)

where V

is the molar volume of the solvent, T is the temperature, R is the gas

constant, and

and

are the solubility parameters for the solvent and polymer,

respectively. Therefore, a decrease in temperature will increase the value of

χ,

and consequently the value of

σ as well (cf eq. 8). An exception to this rule may

happen for systems that have a lower critical solution temperature (LCST). In the
latter case, an increase in temperature will instead increase the value of

χ.

Similarly, if the solubility parameter for the solvent/nonsolvent mixture is

calculated by the simple additive rule

= υ

solvent

+ υ

nonsolvent

(12)

then adding a nonsolvent to the solution will increase the term (

)

in equation

11 and accordingly increase the value of

χ, leading to the formation of the polymer-

rich phase.

If the initial polymer solution is very dilute, the volume of the polymer-rich

phase, V

∗

, will be much smaller than the volume of the polymer-lean phase, V;

ie R

1. Since the partition ratio for longer chains is higher than for shorter

chains, the concentration of longer chains will be higher than the concentration of
the shorter chains in the polymer-rich phase. Because of this selective partition,
it is advantageous to assure that the volume of the polymer-lean phase is much
larger than the volume of the polymer-rich phase. In this case, the majority of the
less selective shorter chains will stay in the polymer-lean phase, simply because
its volume is much larger than the volume of the polymer-rich phase. This effect
is illustrated, in a rather exaggerated manner, in Figure 6, where a plot of f

versus n for several R values is shown when the value of the parameter

σ was

selected (arbitrarily) so that f

2000

1
2

. Notice how the separation efﬁciency gets

increasingly better as R tends to zero. In this theoretical limit, an almost per-
fect partition between the phases would be possible, but evidently this operation
condition cannot be reached experimentally.

The polymer-rich phase is then separated from the polymer-lean phase, and a

new fraction of lower molecular weight can be collected by either further lowering
the temperature or by adding more nonsolvent. The polymer-rich phase is once
more separated from the polymer-lean phase, and the process is repeated as many
times as required.

Since chains of all molecular weights will be partitioned between the two

phases, precipitation fractionation is not capable, even at very high dilutions, to
obtain polymer fractions of very sharp chain-length distributions. This effect is
illustrated in Figure 7 for a model polymer that follows the most probable chain-
length distribution given by equation 2. Notice that even at high dilutions (small
R), there is signiﬁcant overlapping of the distributions of each of the fractions.

This interesting phenomenon is further illustrated in Figure 8 for the sim-

ulated fractionation of a model polymer into ﬁve fractions. The dotted lines in-
dicate the chain-length distributions of the polymer in the polymer-lean phase,

FRACTIONATION

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Fig. 6.

Molar fraction of polymer in the polymer-lean phase (f

) as a function of chain

length (n) and the ratio of the volume of polymer-rich to polymer-lean phases (R).
R

= 1.0 E-1;— —R = 1.0 E-2;

= 1.0 E-5;

= 1.0 E-20. The value of the

parameter

σ was selected so that f

2000

1
2

Fig. 7.

Chain-length distributions in the polymer-lean fraction as a function of the ratio of

volumes of polymer-rich to polymer-lean phases (R).

Parent distribution;

= 1.0 E-1;— —R = 1.0 E-2;

= 1.0 E-3.

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FRACTIONATION

Fig. 8.

Simulated chain-length distributions for precipitation fractionation of a model

polymer sample that follows Flory–Schultz’s distribution with N

= 1000. The following

parameters were used to simulate the fractionation: R

= 1 × 10

− 3

and

σ = 3 × 10

− 3

4.5

× 10

− 3

, 6.75

× 10

− 3

, 1.01

× 10

− 2

, and 1.52

× 10

− 2

and the solid lines the chain-length distributions in the polymer-rich phase. The
chain-length distribution of the original model polymer follows the most proba-
ble distribution with N

= 1000. The parameter σ is arbitrarily increased from

fractions 1 to 5 to simulate either a decrease in temperature or the addition
of a nonsolvent. The distribution of the ﬁrst polymer-rich solution is given by
w

− f

(1)

], where w

is the most probable chain-length distribution of the parent

resin. Similarly, the distribution of the ﬁrst polymer-lean phase is simply w

(1)

After separation of the polymer-rich phase, the second fraction is obtained by de-
creasing the temperature of (or adding nonsolvent to) the polymer-lean phase.
The distribution of the second polymer-rich phase is w

(1)

− f

(2)

], and that

of the polymer-lean phase is w

(1)

(2)

. It is easy to see that this process can be

repeated for k fractions, leading to the following equations for the chain-length
distributions of polymer in the two phases:

(k)
n

= w

= 1

(i)

, polymer-lean phase

(13)

∗(k)

= w

− 1

= 1

(i)

− f

(k)

= w

− 1)

− f

(k)

, polymer-rich phase

(14)

where w

(k)

is the chain-length distribution of the kth fraction and the asterisk

superscript indicates the polymer-rich phase. Notice that even though the chain-
length distributions of the fractions is narrower than that of the parent sample,

FRACTIONATION

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signiﬁcant overlapping takes place in all fractions because of the lower values of
the partition ratios for the shorter chains. The reader is directed to the review by
Kamide (7) for extensive and more rigorous simulations of batch fractionations of
polymer solutions.

Rigorously speaking, for the case of solvent/nonsolvent precipitation frac-

tionation the general mathematical treatment described above must be modiﬁed
to treat the ternary solvent/nonsolvent/polymer system, but the trends are the
same. It has been proposed that solvent/nonsolvent fractionation can be more
effective for chain-length fractionation than temperature reduction, and in fact
solvent/nonsolvent fractionation is generally a more commonly used technique to
fractionate polymer chains according to molecular weight.

Upward Precipitation Fractionation.

In this method, fractions of increas-

ing molecular weights are obtained in each successive fractionation step. A large
amount of nonsolvent is added to the original polymer solution, driving most of
the polymer to the polymer-rich phase. Contrarily to the downward precipitation
fractionation discussed above, in this method most of the polymer will be concen-
trated in the polymer-rich phase. The polymer-lean phase is recovered as the ﬁrst
fraction with the lowest average molecular weight. The remaining polymer-rich
phase is completely redissolved and more nonsolvent is added, preferentially driv-
ing the high molecular weight chains into the polymer-rich phase. The polymer-
lean phase is recovered as the second fraction with higher molecular weight and
the cycle of redissolutions and solvent/nonsolvent additions is repeated. A similar
result can be obtained by altering the temperature of the solution. The mathe-
matical description adopted above for the downward precipitation fractionation
method is also adequate for the upward precipitation fractionation method.

Notice that the high molecular weight fractions are only obtained after sev-

eral fractions of lower molecular weights have been recovered. Considering the
lower phase selectivity of the low molecular weight chains, the upward precipita-
tion fractionation method can remove most of the shorter chains before isolating
the high molecular weight fractions. As a consequence, the high molecular weight
fractions isolated with the upward precipitation fractionation method will have
narrower molecular weight distributions than the ones obtained with the down-
ward precipitation fractionation technique.

It should be clear that both upward and downward precipitation fractiona-

tions require the use of very large amounts of solvent to ensure the low polymer
concentration necessary for efﬁcient fractionation. To minimize this inconve-
nience, several refractionation mechanisms have been suggested. The refractiona-
tion schemes involve the fractionation of polymer-rich solutions into two fractions
that must subsequently be refractionated several times until the required degree
of separation is achieved (10).

An informative summary of the several steps required to fractionate poly-

mers by precipitation fractionation is given in Figure 9.

Extraction or Solution Fractionation.

In extraction fractionation, polymer

chains of increasing molecular weights are extracted from the polymer-rich phase
by gradually increasing the goodness of the solvent, either by increasing the tem-
perature or the ratio of solvent to nonsolvent. Many of the considerations made
above for the precipitation fractionation methods are also valid for the extraction
fractionation techniques.

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FRACTIONATION

Fig. 9.

Steps required for precipitation fractionation. From Table 5.2 in Ref. 8. Copyright

(1994) Springer-Verlag.

A large contact area between the polymer-rich and polymer-lean phases is

a very important requirement in extraction fractionation to reduce mass-transfer
limitations between the phases during extraction. In the simplest extraction tech-
nique, direct extraction, this is achieved by using ﬁnely divided polymer particles or
powder. Direct extraction is commonly performed in Sohxlet extractors. Diffusion
limitations, polymer swelling, and equipment plugging make direct extraction a
technique of limited use.

The contact area between the phases can be enlarged in ﬁlm extraction and

column extraction. The discussion of column extraction techniques will be post-
poned to the section on TREF. In ﬁlm extraction, a thin polymer ﬁlm is deposited
onto a support material with a large surface area. This technique can also be
performed inside Sohxlet extractors, but it is also prone to several experimental
difﬁculties (11).

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In coacervate extraction the polymer is initially present as a dilute solution. A

large amount of nonsolvent is added, causing most of the polymer to separate into
the polymer-rich phase. The polymer-lean phase, containing the low molecular
weight chains, is recovered as the ﬁrst fraction. Then more solvent is added to the
polymer-rich phase. The liquid–liquid mixture is vigorously stirred to ensure good
contact between the phases. Upon equilibration, the newly formed lean phase is
recovered as the second fraction and the process is repeated as many times as
required. A similar fractionation can be obtained by increasing the temperature
of the solution.

As apparent from the last paragraph, coacervate extraction is similar to up-

ward precipitation fractionation. However, differently from the last technique,
the polymer-rich phase is not redissolved during coacervate extraction. As a con-
sequence, smaller volumes of solvent and nonsolvent are required, but care must
be taken to avoid occlusion of chains in the polymer-rich phase and mass-transfer
limitations during fractionation. Similarly to upward precipitation fractionation,
coacervate extraction is more adequate to isolate high molecular weight fractions
with narrower molecular weight distributions.

Fractionation Equipment.

Until recently, no fully automated instrument

existed to perform batch fractionation of polymers. Most of the older experimen-
tal apparatuses consisted of simple glassware equipment that required consid-
erable operator time and the manipulation of sizeable amounts of solvent. Some
of these experimental fractionation systems have been thoroughly described by
Francuskiewicz (8).

A fully automated batch fractionation instrument called mc2-PREP is now

available. Figure 10 shows a schematic of mc2-PREP. The two stirred frac-
tionation vessels placed inside a temperature-programmable oven can fraction-
ate polymers into eight fractions by either solvent/nonsolvent or temperature
variation techniques. Operator intervention is only required to precipitate and
ﬁlter the polymer fractions after they are isolated from the parent polymer
solution.

The coacervate extraction method is recommended in the molecular weight

fractionation mode of mc2-PREP (the “m” in mc2-PREP), although the precipita-
tion method (both upward and downward) is also possible in principle. As usual,
the initial polymer solution is precipitated by the addition of a large amount
of nonsolvent or reducing the temperature. The polymer-lean phase is trans-
ferred from the fractionation vessel to a fraction collection vial between each
solvent/nonsolvent injection or temperature-increase steps through the transfer
line located inside the fractionation vessel. The two fractionation vessels can be
operated in parallel, with either the same or different samples. At the end of the
fractionation, the fractions collected in the fraction collection vials are precipi-
tated and ﬁltered. mc2-PREP can also be used to fractionate polymers according
to crystallizability, as described in the next section.

Fractionation of Semicrystalline Polymers by Crystallizability.

The

temperature variation method is the most common approach to fractionate
semicrystalline polymer chains according to crystallizability. In this case, since
chain crystallization is the main fractionation mechanism, the temperature
should be lowered below the melting point of the polymer in solution during the
fractionation.

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FRACTIONATION

Fig. 10.

Diagram of mc2-PREP. Courtesy of Polymer Char.

Chain crystallizability is strongly inﬂuenced by the degree of regularity of

the chains. In the case of binary copolymers, generally the comonomer present in
the smallest fraction does not crystallize and acts as a chain defect. The melting
temperature, therefore, decreases with increasing amounts of this comonomer.
Typical examples are copolymers of ethylene and

α-oleﬁns, where the α-oleﬁn

molecules are generally too large to be included in the crystalline lattice and lead
to the formation of smaller polymer crystallites with lower melting temperatures.
For the case when the two comonomers crystallize, such as for ethylene/propylene
copolymers, the comonomer present in less amount acts as a defect and the melting
temperature passes through a minimum located between the melting points of the
two semicrystalline homopolymers. A similar phenomenon takes place with stere-
oregular homopolymers such as isotactic polypropylene. In this case, propylene
molecules inserted in a racemic conﬁguration will decrease the melting tempera-
ture of the isotactic polymer. Similarly, regioirregular insertions will also decrease
the melting point of the polymer (12).

The melting point depression of a random binary AB copolymer, where the

comonomer B does not crystallize, is described by the equation 15

−

= −

lnx

(15)

where T

is the melting point of an inﬁnitely long-chain of homopolymer A, T

the melting temperature of the random copolymer AB,

is enthalpy of melting

FRACTIONATION

Vol. 10

of the repeating unit A, and x

is the molar fraction of the crystallizable unit A. In

principle, this equation can also be used to describe the melting point depression
of stereoregular homopolymers due to stereoirregular insertions. For instance,
in the case of isotactic polypropylene, x

could represent the molar fraction of

meso insertions and (1

− x

) would then represent the molar fraction of racemic

insertions.

Fractionation by crystallizability takes place by either decreasing or increas-

ing the temperature, similarly to what is done in the methods of fractional pre-
cipitation or coacervate extraction for molecular weight fractionation. However,
differently from those, solid–liquid phase separation occurs during fractionation
by crystallizability. We will call Crystaf-mode the fractionation that takes place
upon cooling the polymer solution, and TREF-mode the fractionation that takes
place by dissolving previously precipitated polymer crystallites. This nomencla-
ture is consistent with the continuous analysis techniques of Crystaf and TREF,
which will be described later in this article, and is more adequate for fractionation
by crystallizability.

Irrespective of the fractionation mode, one starts with a dilute polymer solu-

tion at high temperature. In Crystaf-mode, fractions of decreasing crystallinity (in-
creasing noncrystallizable comonomer content) are obtained as the temperature
is reduced. The more crystalline chains are precipitated at higher temperatures
and separated from the remaining solution before the temperature is lowered to
obtain the next polymer fraction. In TREF-mode, all or most of the polymer is pre-
cipitated by reducing the temperature to its lowest value at the beginning of the
fractionation. The ﬁrst fraction is collected at the lowest fractionation temperature
and it is, therefore, the one that has the highest fraction of the non-crystallizable
comonomer. Fractions with increasing crystallizabilities and lower comonomer
contents are collected by increasing the temperature at predetermined intervals.
A slow cooling rate is a requirement for good fractionation even in the TREF-mode,
since it permits the growth of well-formed crystals and enhances the resolution of
the fractionation.

Fractionation Equipment.

Similar to molecular weight fractionation, most

of the conventional equipment for crystallization fraction is rather simple and
requires signiﬁcant operator time. Fully automated crystallization fractionations
are possible using mc2-PREP, in both Crystaf-mode and TREF-mode (the “c2” in
mc2-PREP stands for these two fractionation modes).

In TREF-mode the polymer solution is initially placed inside a crystallization

vessel at high temperature. The temperature is then slowly reduced, precipitating
most of the polymer. The polymer that remains in solution is transferred via the
in-line ﬁlter to a fraction collection vial as the ﬁrst, least crystalline fraction. More
solvent is transferred to the fractionation vessel and the temperature is increased
by a predetermined amount, dissolving the next, more crystalline fraction. The
process is then repeated with the recovery of polymer solutions at increasing dis-
solution temperatures. Two polymer samples can be fractionated simultaneously
in the TREF-mode, one in each fractionation vessel.

The two fractionation vessels are required to fractionate one sample in

Crystaf-mode. The polymer solution is initially placed in one of the fractiona-
tion vessels at high temperature (vessel 1). The temperature is then decreased,
causing part of the polymer to crystallize out of solution. The polymer solution

Vol. 10

FRACTIONATION

is transferred from vessel 1 to vessel 2 via an in-line ﬁlter to avoid sampling
the polymer crystallites that constitute the ﬁrst, high crystallinity fraction. The
crystallized polymer remaining in vessel 1 is then redissolved and the resulting so-
lution is transferred to a fraction collection vial. The fractionation process is then
repeated by decreasing the temperature in vessel 2, transferring the remaining
solution to vessel 1, and recovering the polymer precipitated in vessel 2 as the
second fraction. These steps can be repeated several times to obtain the number
of polymer fractions of decreasing crystallinity required by the fractionation.

Several of the issues related to the fractionation efﬁciency of these two tech-

niques have been investigated with their continuous counterparts, Crystaf and
TREF, and will be discussed in the next section.

Crystallization Analysis Fractionation

Crystallization analysis fractionation (Crystaf) fractionates polymer chains ac-
cording to differences in crystallizability. Crystaf can be used to fractionate poly-
mers due to differences in chemical composition, comonomer sequence length, and
tacticity. It may also respond to long-chain branching, provided that the polymer
is branched enough to affect its crystallinity. The fractionation principle operative
in Crystaf was discussed in the section on batch fractionation for the case of slowly
cooling (or warming) solutions of semicrystalline polymers.

In Crystaf, a dilute solution of a semicrystalline polymer is slowly cooled

from a high temperature to room or subambient temperatures, while the poly-
mer concentration is monitored as a function of crystallization temperature by
an on-line mass detector to obtain its cumulative distribution. The differential
distribution is calculated by taking the ﬁrst derivative of the cumulative distribu-
tion with respect to the crystallization temperature. Both cumulative and differ-
ential Crystaf curves for a bimodal blend of two ethylene/

α-oleﬁn copolymers are

shown in Figure 11. A calibration curve must be used to translate the crystalliza-
tion temperature distribution into a polymer microstructural distribution, as will
be explained later in this section.

The Crystaf apparatus shown in Figure 12 has ﬁve crystallization vessels

that can be operated in parallel. Sample injection, dissolution, analysis, and dis-
posal are completely automated. Small aliquots of the solution are taken through
an in-line ﬁlter to avoid sampling polymer crystals with the polymer solution and
sent to the on-line mass detector. The mass detector is commonly an infrared cell
that is less sensitive to temperature ﬂuctuations of the polymer solution. Addi-
tional detectors can also be installed on the sampling line to measure complemen-
tary properties, such as viscosity and copolymer composition.

Usually, Crystaf is operated at a cooling rate of 0.1

◦

C/min. It will be shown

later that cooling rates can signiﬁcantly inﬂuence the results of Crystaf analysis.
The concentration of the polymer solution is not considered to affect the fraction-
ation results, provided it is kept in the range of 0.2–1.0 mg/mL (14). The type of
solvent only affects the crystallization temperature: polymers crystallize at lower
temperatures in the presence of better solvents, but the effect on fractionation
efﬁciency is negligible (15).

FRACTIONATION

Vol. 10

Fig. 11.

Cumulative and differential Crystaf curves of a blend of polyoleﬁns.

The raw Crystaf proﬁle provides information about the distribution of crys-

tallizabilities of the polymer sample that can be correlated to molecular structure
with the use of a proper calibration curve. Contrary to SEC, there is no universal
calibration curve for Crystaf: for quantitative analysis, calibration curves should
be prepared with narrow standards that have microstructures as similar as pos-
sible to the sample being analyzed.

Calibration standards for Crystaf are not commercially available and have

to be prepared by either preparative fractionation of a broad sample with similar
characteristics (generally obtained by using the batch fractionation techniques de-
scribed previously) or by synthesis of samples with narrow distributions. Figure 13
shows the Crystaf proﬁles of a set of calibration standards for ethylene/1-hexene
random copolymers, and Figure 14 is the resulting calibration curve (16). The cal-
ibration curves for Crystaf for random (or near random) copolymers are generally
linear, in good agreement with the prediction of equation 15 for copolymers with
small amounts of comonomer. In this case, the following approximations apply:

−

− T

∼

− T

(16)

lnx

= − (1 − x

)

= − x

(17)

Substituting these expressions in equation 15,

∼

= T

−

(18)

Vol. 10

FRACTIONATION

Fig. 12.

Schematic of Crystaf. Courtesy of Polymer Char.

which shows that the melting temperature depression is a linear function of the
fraction of non-crystallizable comonomer in the copolymer.

One of the main difﬁculties for the quantiﬁcation of Crystaf is the nonuniver-

sality of its calibration curves. Even for a series of ethylene/

α-oleﬁn copolymers,

the calibration curves will vary as a function of comonomer type, as illustrated
in Figure 15. The general rule of thumb for these copolymers (from propene to
1-octene) is, the longer the

α-oleﬁn, the lower the crystallization temperature for

a given

α-oleﬁn molar fraction. This has been explained by several authors on

the basis of the difference in the degree of inclusion of the

α-oleﬁn in the crys-

talline lattice: shorter

α-oleﬁns are more likely to cocrystallize with ethylene and

therefore depress the crystallization temperature to a lesser extent.

Sequence length distribution is another factor that may affect calibration

curves in Crystaf. This is easy to visualize if we compare the extreme cases of
block and random ethylene/

α-oleﬁn copolymers with the same α-oleﬁn content. We

start by assuming that the

α-oleﬁn units cannot crystallize and the crystallization

temperature is a function only of the average length of the ethylene sequences in
the chain. Since the average sequence of crystallizable ethylene units is much
longer in block copolymers than in random copolymers with the same comonomer
content, block copolymers will have much higher crystallization temperatures
than random copolymers. This limitation is easily overcome by using calibration
standards with microstructure similar to the polymers being analyzed.

FRACTIONATION

Vol. 10

Fig. 13.

Crystaf proﬁles for a set of random ethylene/1-hexene copolymer calibration stan-

dards. —

䊉

— 0r (0 mL); —

䊊

— 5d (0.5 mL); —

— 10e (1.0 mL); —— 23b (2.3 mL);

41a (4.1 mL); . . . 41e (4.1 mL); — —65b (6.5 mL); —

䊏

— 100f (10.0 mL); —

ⵧ

— 150a

(15.0 mL); —

䉫

— 200a (20.0 mL). From Ref. 16.

Molecular weight effects are practically negligible in determining the Crystaf

peak temperatures for polyethylene chains with molecular weights higher than
5000, and even these effects can be corrected by taking into account the terminal
methyl groups of the chains. Likewise, small amounts of long-chain branching (in
the range of 1 to 4 LCB per 1000 C atoms) do not inﬂuence Crystaf proﬁles appre-
ciably. On the other hand, the shape of Crystaf proﬁles is affected by molecular

Fig. 14.

Crystaf calibration curve for random ethylene/1-hexene copolymers.

ASTM;

– – – % incorporation

= −0.1263 · Tpeak + 10.8065;

䉫

dePooter;

% incorporation

−0.1317 · Tpeak + 11.2467. From Ref. 16.

Vol. 10

FRACTIONATION

Fig. 15.

Crystaf calibration curves for ethylene/1-butene (EB-01) and ethylene/1-octene

(EO-01) copolymers.

䊉

EB-01;

䊊

EO-01. From Ref. 17. Copyright (2000) Wiley-VCH.

weight: distributions with a low molecular weight tail are characterized by a low
crystallinity tail in Crystaf, perhaps because the low molecular weight chains
crystallize at lower temperatures (18,19).

Cooling rate is the most important operating variable in Crystaf. It signif-

icantly affects Crystaf peak temperatures (Fig. 16) and the cocrystallization of

Fig. 16.

Crystaf peak temperatures as a function of cooling rate for an ethylene/1-hexene

copolymer sample. —

䊉

— 0.0033 CPM; —

䊊

— 0.01 CPM; —

— 0.02 CPM; —— 0.05 CPM;

—

䊏

— 0.1 CPM; —

ⵧ

— 0.2 CPM; —

䉬

— 0.5 CPM; —

䉫

— 1 CPM; —

䉱

— 2 CPM. CPM

◦

per minute. From Ref. 14.

FRACTIONATION

Vol. 10

chains with different average comonomer contents. This is particularly important
to keep in mind when analyzing polymers made with multiple-site catalysts, such
as heterogeneous Ziegler–Natta systems, where chain populations with different
comonomer contents are present (17). Good resolution among these populations
can only be achieved under very slow cooling rates, often slower than the stan-
dard Crystaf rates of 0.1

◦

C/min. Figure 17 illustrates this situation for ternary

blends of ethylene/1-hexene copolymers made with a metallocene catalyst. The
dotted lines represent the Crystaf proﬁle estimated for the blend if no cocrystal-
lization takes place. Cocrystallization becomes more signiﬁcant with increasing
cooling rate and proximity of the crystallization peaks of the different polymer
populations. Interestingly, the type of

α-oleﬁn does not seem to play a role in the

cocrystallization phenomena (20).

Temperature Rising Elution Fractionation

Temperature rising elution fractionation (TREF) is based on the same principles of
Crystaf, but involves two consecutive steps: precipitation (or crystallization) and
elution. In the precipitation step, polymer is crystallized from a dilute solution by
slowly decreasing the temperature inside a column packed with an inert support.
Alternatively, the precipitation step can be done in a stirred vessel in the pres-
ence of the support and the polymer-coated support is subsequently transferred to
the TREF column for the elution step. In the elution step, the polymer deposited
onto the support is eluted from the column with a continuous ﬂow of solvent at in-
creasing temperatures. An on-line detector measures the concentration of the poly-
mer solution exiting the column as a function of elution temperature. Figure 18
shows a schematic of a TREF apparatus and a typical TREF curve for linear low
density polyethylene.

Polymer fractionation takes place during the precipitation step according

to crystallizability. The precipitation step in TREF is essentially analogous to the
crystallization that takes place in Crystaf. As for Crystaf, a slow cooling rate is the
most important requirement to minimize cocrystallization of polymer populations
with different crystallizabilities and enhance peak resolution. Since there is no
monitoring of the polymer solution concentration during the precipitation step,
the elution step is required to measure the concentration of polymer recovered as
a function of elution temperature.

Given that the fractionation mechanism in Crystaf and TREF are very simi-

lar, they both provide analogous information about polymer microstructure, with
the TREF proﬁles shifted to higher temperatures due to the supercooling effect
in Crystaf, as exempliﬁed in Figure 19. Crystaf analysis times tend to be shorter
because of the extra time required by the additional elution step in TREF.

Table 1 compares the analysis times for Crystaf and TREF at different cool-

ing and heating rates (14). Similar to Crystaf, TREF calibration curves are not
universal and depend on the microstructure of the polymer being analyzed (21).

It has long been considered that the elution step in TREF was mainly re-

quired to recover and monitor the concentration of the polymer already fraction-
ated during the precipitation step. However, recent studies with ethylene/

α-oleﬁn

copolymer blends indicate that for the same cooling rate, TREF can better resolve

Vol. 10

FRACTIONATION

Fig. 17.

Cocrystallization of ethylene/1-hexene copolymers in Crystaf as a function of cool-

ing rate. Dotted lines are the proﬁles calculated from the Crystaf analyses of the individual
components in the blend. . . . Calculation;

experiment. From Ref. 14.

FRACTIONATION

Vol. 10

Fig. 18.

TREF schematic and typical TREF proﬁles of a Ziegler–Natta linear low density

polyethylene.

Fig. 19.

Comparison of Crystaf and TREF proﬁles for an ethylene/1-butene Ziegler–Natta

copolymer.

EB-01 Crystaf; - - - - - EB-01 TREF. From Ref. 17. Copyright (2000)

Wiley-VCH.

Table 1. Comparison of Crystaf and TREF Analysis Times

Tref analysis time for several heating rates

Cooling rate,

Crystaf analysis

◦

C/min

time, min

0.5

◦

C/min

0.2

◦

C/min

0.1

◦

C/min

0.5

235

425

620

945

0.2

430

620

815

1140

0.1

755

945

1140

1465

0.05

1405

1595

1790

2115

0.02

3355

3545

3740

4065

0.01

6605

6795

6990

7315

From Ref. 14.

Vol. 10

FRACTIONATION

Fig. 20.

Comparison of TREF and Crystaf blend peak resolution for ethylene/1-hexene

copolymers as a function of cooling rate.

Tref; . . . Crystaf. From Ref. 14.

the peaks of the individual populations, as indicated in Figure 20. The slow heating
during the elution step may lead to recrystallization of the crystals formed during
the precipitation step, enhancing the peak resolution of TREF as compared to
Crystaf. This is an attractive TREF feature when analyzing complex blends or
polymers made with multiple-site catalysts (14).

The performance of TREF is relatively insensitive to polymer concentration

in the range of 1–4 mg/mL. When compared to Crystaf, TREF peak positions
are less dependent on the cooling rate, maybe because of recrystallization during
the elution step. Solvent ﬂow rate and heating rate can also affect peak position
and breadth, but it has been demonstrated that these variations can be elimi-
nated by operating TREF at constant ratio of heating rate to solvent ﬂow rate,
as illustrated in Figure 21. These conditions ensure that each volume element
exiting the column contains polymer chains that were eluted during the same
time/temperature interval (14).

Solvent and support type have only a minor effect on TREF fractionation.

Better solvents lead to lower elution temperatures. The effect of different support
types has not been systematically investigated, although some authors claim that
it may lead to peak broadening in TREF (15,21).

TREF can also be operated in preparative mode. In this case, a larger column

is used and samples are collected at predetermined time intervals for posterior
analysis by other techniques. The efﬁciency of TREF preparative fractionation is
essentially the same as for batch fractionation by temperature.

An interesting variation of TREF uses supercritical ﬂuids as solvents and

columns containing a high surface stainless steel mesh packing. The main advan-
tage of supercritical TREF (CITREF: critical, isobaric TREF) over conventional
TREF is that supercritical solvents such as propane can be easily separated from
the polymer by pressure reduction after the column, recompressed, and recycled
back to the column in a closed loop. Because of this, CITREF can be used as a

100

FRACTIONATION

Vol. 10

Fig. 21.

Comparison of the TREF proﬁles for an ethylene/1-hexene sample measured at

a ratio of 1:1:1 for cooling rate: heating rate: solvent ﬂow rate. —

䊉

— Rate

= 0.1

◦

C/min;

—

䊊

— rate

= 0.2

◦

C/min; —

— rate = 0.5

◦

C/min. From Ref. 14.

preparative method to fractionate much larger polymer samples than can be prac-
tically accomplished with conventional TREF using organic chlorinated solvents
(22–24). Interestingly, because the density of supercritical ﬂuids can be changed
over a wide range of values by varying the pressure, CITREF can also be operated
at constant temperature and varying pressure to fractionate polymers according
to their molecular weights (25), provided they are kept above their melting points
for the case of semicrystalline polymers. In this way, the term (

− δ

)

in equa-

tion 11 can be regulated by varying the pressure of the supercritical ﬂuid without
the addition of a nonsolvent.

Mathematical Models for Crystaf and TREF.

Existing mathematical

models for Crystaf and TREF suffer from several limitations because of the com-
plex nature of the fractionation process that takes place in these techniques. For
binary linear copolymers made with single site catalysts, Stockmayer’s distribu-
tion (eq. 3) can be used to model the chemical composition distribution, but its
translation into actual Crystaf and TREF curves is a much harder task.

A very simple model relating Crystaf or TREF curves to Stockmayer’s distri-

bution has been proposed (26), but this approach was later found to be unsatisfac-
tory to describe quantitatively the chemical composition distribution of polyoleﬁns
made with single site catalysts, as illustrated in Figure 22 (27). Notice, however,
that even though the Crystaf proﬁles and Stockmayer’s distribution do not agree
quantitatively, Stockmayer’s distribution predicts the correct trends, particularly
the broadening of the chemical composition distribution with decreasing molecular
weight of the samples. A new model was proposed on the basis of the crystallization
of the longest crystallizable monomer sequence per chain (28). In this approach,
the longest ethylene sequence per chain, and not the overall comonomer distri-
bution, determines the temperature at which the chain crystallizes in Crystaf or

Vol. 10

FRACTIONATION

101

Fig. 22.

Comparison of Crystaf proﬁles and Stockmayer’s distributions for a series of

ethylene/1-octene copolymers. Experimental Crystaf curves are indicated with symbols
and the solid curves are the respective Stockmayer’s distributions.

䊉

Sample A;

䉬

sample

sample H. Molecular weights of samples A: M

= 29 600; D: M

= 38 000; H: M

19 800. From Ref. 27. Copyright (1998) Wiley-VCH.

elutes in TREF. Therefore, the distribution of the longest ethylene sequences, and
not the chemical composition distribution, is used to model the fractionation with
Crystaf and TREF. This approach led to modest success for some ethylene/1-octene
copolymers, but was later found not to be generally applicable to a broader range
of polymer samples. It is certain that crystallization kinetics play a very important
role in both Crystaf and TREF fractionations and that models that rely simply on
thermodynamic equilibrium assumptions are not capable of describing these tech-
niques quantitatively (18). Much more research is still required to obtain reliable
phenomenological models for these two fractionation techniques.

Size-Exclusion Chromatography

Size-exclusion chromatography (SEC), also known as gel-permeation chromatog-
raphy (GPC), is currently the standard technique to fractionate polymers accord-
ing to molecular weight due to its short analysis time and good reproducibility
(see C

HROMATOGRAPHY

, S

IZE

XCLUSION

). There exist many excellent reviews and

book chapters covering different aspects of SEC fractionation (29–35). SEC is a
standard liquid chromatographic technique and, as such, involves the transport
of a polymer sample dissolved in a mobile phase through a packed column. Even
though SEC can be run in preparative mode using large columns to fractionate
sizeable polymer samples, it is much more commonly operated in analytical mode
with small columns and only a few milligrams of polymer.

102

FRACTIONATION

Vol. 10

Fig. 23.

SEC schematic.

SEC is commonly referred as room-temperature SEC or high temperature

SEC, but the separation mechanism for both techniques is the same. As the name
indicates, room-temperature SEC is used for polymers that are soluble at room
temperature in the mobile phase; tetrahydrofuran is a commonly used solvent in
this case, but many other solvents are equally adequate. High temperature SEC
is mostly used for polyoleﬁns and their copolymers since they are only soluble at
temperatures from 120 to 150

◦

C in solvents such as trichlorobenzene. In general,

solvent type does not play a very important role in SEC, provided that it does not
interfere with the column pore structure (when cross-linked polymer beads are
used) and with the on-line detectors at the exit of the columns. However, care must
be taken to avoid non size-exclusion effects, particularly for room-temperature
SEC.

Figure 23 shows a schematic of SEC. The principle of SEC operation can be

succinctly described as follows: the polymer sample, present as a dilute solution
in a vial, is injected into the mobile phase as a narrow pulse and continuously
ﬂows through a series of columns containing packing with variable pore sizes.
The elution time of the polymer chains depends on their size in solution or hydro-
dynamic radius: chains with higher volumes penetrate into fewer pores and elute
ﬁrst, while chains with smaller volumes penetrate into more pores (or penetrate
deeper into the pores) and elute later. An on-line detector at the exit of the columns,
most commonly a refractive index detector, is used to monitor the concentration
of polymer chains eluting from the last column in the series, and a calibration
curve is applied to relate elution volumes to molecular weights. Retention time is
the time taken by a polymer fraction to elute from the columns. Likewise, elution
volume is the volume of mobile phase required to elute a given polymer fraction
from the columns.

In reality, even if all the pores were of the same diameter, fractionation by

size would still take place since, because of solute-to-wall exclusion effects inside
the pores, the available pore volume is higher for smaller chains than for larger
ones, as illustrated in Figure 24. In practice, columns with a range of pore sizes
are always used.

Vol. 10

FRACTIONATION

103

Fig. 24.

Mechanism of SEC separation. Dotted lines indicate solution-to-wall exclusion

volumes.

The relation between elution volume in the column and chain size in solution

can be described with the simple equation:

= V

+ KV

(19)

where V

is the total elution volume, V

is the interstitial volume between the

packing particles, V

is the pore volume of the packing particles, and K is a par-

tition coefﬁcient for the polymer chains between the stationary and the mobile
phases. The partition coefﬁcient K depends on the sizes of pores and polymer
chains. It is equal to zero (K

= 0) for chains that are too large to penetrate into

any of the pores (total exclusion limit), and equal to one (K

= 1) for chains that are

small enough to diffuse into all pores (total permeation or penetration limit). These
two limits, indicated in Figure 25, set the boundaries for polymer size resolution
in SEC. Chains whose sizes are below the total permeation limit or above the total
exclusion limit cannot be fractionated by SEC. Some mathematical models have
been proposed to relate the value of K to pore sizes and radius of gyration of the
polymer chains (32,36).

104

FRACTIONATION

Vol. 10

Fig. 25.

An SEC calibration curve, showing the total permeation and total exclusion

limits.

The partition coefﬁcient can also be understood from a thermodynamic point

of view with equation (37):

= exp

−

= exp

−

(20)

Ideally, enthalpic effects are completely absent and SEC fractionation takes

place on a size-exclusion basis only, ie

K ∼

= exp

(21)

Therefore, the value of K is proportional to the decrease in entropy experi-

enced as the polymer chains diffuse into the pores of the packing and 0

≤ K ≤ 1.

The size of polymer chains in solution is a function of their chain length and

topology, as well as solvent type and temperature. For linear chains, the size in so-
lution is directly proportional to chain length, but for chains containing long-chain
branches this relation depends on the type, number, and topology of the branches,
making the quantiﬁcation of SEC results signiﬁcantly more complex (33).

In principle, SEC can be calibrated with standards of narrow molecular

weight distribution of the polymer that one wishes to analyze. In this way, a
calibration curve relating molecular weight to elution volume is obtained, in a
fashion similar to what is done for the calibration of TREF or crystallization anal-
ysis fractionation. This approach, however, suffers from the limitation that the
calibration curve is only applicable to one speciﬁc polymer. Additionally, narrow
standards are not available for many different types of polymers. In principle, the
batch fractionation techniques discussed above could be used to obtain such nar-
row standards, but these processes are time-consuming and, as explained, do not
permit the isolation of samples with very narrow molecular weight distributions.

One of the great advantages of SEC is that it can be calibrated using a

universal calibration curve that is independent of polymer type, provided that the
chains behave like random coils in solution. This remarkably elegant solution to
a rather complex problem is often taken for granted, but it is perhaps the most
important reason why SEC is so widely applicable to different polymer types. The

Vol. 10

FRACTIONATION

105

universal calibration curve relies on the principle that polymer chains with the
same volume in solution will elute at the same time from an SEC column. In other
words, chains with the same hydrodynamic volume have the same retention time
and elution volume.

The product of the molecular weight M and the intrinsic viscosity [

η] is called

the hydrodynamic volume and is proportional to the size of polymer chains in
solution:

[

η]M =

2
0

(22)

In equation 22, ¯

2
0

is the root-mean-square end-to-end distance of the polymer

chain, and

α and are constants that depend on the type of solvent and polymer

(30). When the logarithm of the hydrodynamic volume is plotted as a function
of elution volume, an elegant relationship arises that is independent of polymer
type, as illustrated in Figure 26 (31).

According to equation 22, polymers with the same root-mean-square end-

to-end distances have the same hydrodynamic volume. Polystyrene standards
spanning a wide range of molecular weight averages with very narrow molec-
ular weight distributions (with polydispersity indices typically below 1.1) can be
produced via living anionic polymerization and are, therefore, ideal calibration
standards for SEC. The hydrodynamic volume of an unknown polymer, x, can be
compared to that of a polystyrene standard, ps, with the equation:

[

η]

= M

[

η]

(23)

The choice of a polystyrene standard is one of mere convenience; in

principle, any other polymer standard with narrow molecular weight dis-
tribution could be used to generate the universal calibration curve. Equa-
tion 23 can be made more useful by substituting the Mark–Houwink
relationship,

[

η] = KM

(24)

and solving for M

to obtain

logM

+ a

log

+ a

logM

(25)

Calibration curves for polystyrene narrow standards are very often linear

functions in the general form,

= a − b log M

(26)

where a and b are positive constants determined empirically. Higher degree
polynomials are used when a simple linear regression is not adequate. The
experimentally-measured SEC elution proﬁle for polymer x is converted into
the molecular weight distribution as follows: for each retention volume V

, the

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Fig. 26.

Universal calibration curve for SEC.

䊉

Polystyrene (linear);

䊊

polystyrene

(“comb”);

+ polystyrene (“star”); polystyrene–poly(methylmethacrylate) copolymer

(heterograft);

× poly(methylmethacrylate) (linear); poly(vinylchloride); polystyrene–

poly(methylmethacrylate) copolymer (graft-comb);

poly(phenyl siloxane);

polystyrene–

poly(methylmethacrylate) copolymer (statistical-linear);

polybutadiene. From Ref. 31.

polystyrene-equivalent molecular weight, M

, is calculated using the universal

calibration curve (eq. 26). This value is then substituted into equation 25 and
the molecular weight of the sample, M

, is calculated provided that the Mark–

Houwink constants are known for polystyrene and the polymer sample at the

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107

required analysis temperature and solvent. Calibration curves using broad stan-
dards may also be used, but are less common than the ones that use narrow
standards (30).

Once the molecular weight distribution is established, molecular weight av-

erages are calculated according to the following expressions:

w(M)

(27)

Mw(M) dM

(28)

w(M) dM

(29)

w(M) dM

(30)

where w(M) is the differential molar mass distribution measured by the mass
on-line detector:

w(M)

d(logM)

(31)

The term

is the height of the chromatogram and

d(log M)

is the slope of

the calibration curve, both measured at the elution volume V.

Many modern SEC instruments are provided with on-line viscometers to

measure the intrinsic viscosity of the polymer sample during the analysis. In this
case, the Mark–Houwink constants are estimated during the SEC analysis and
do not need to be known a priori.

Many round-robins have been performed to test the intra- and inter-

laboratory reproducibility of SEC. D’Agnillo and co-workers (38) reviewed the
previous literature in the area and provided the most recent round-robin for high
temperature SEC.

Fractionation Equipment.

The main requirements for a commercial SEC

unit are as follows: (1) reliable solvent pumps to keep a constant ﬂow of mobile
phase; (2) fractionation columns in the right range of porosity to fractionate a
wide range of molecular weights; (3) an injection system capable of injecting the
polymer sample without signiﬁcantly disturbing the ﬂow of the mobile phase; (4)
an efﬁcient mass detector and, optionally, chain-length or composition-sensitive
detectors; and (5) an automated data acquisition system. All of the commercially
available SEC units have incorporated these required features.

A reliable high pressure liquid pumping system (generally reciprocating-

dual piston design to avoid pressure ﬂuctuations) is one of the most important
parts of SEC. Since SEC calibration is based on the retention time in the columns
and the logarithm of the molecular weight varies very often as a linear function

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of the elution volume or retention time, any ﬂuctuation in ﬂow rate will lead to
signiﬁcantly inaccurate molecular weight distributions. Pump pressures generally
vary from 1000 to 4000 psi. These high pressures are required to overcome the
pressure loss in the columns due to the small particle sizes of the packing. The
ﬂow rate of the mobile phase is kept in the range of 1–3 mL/min.

The injection system should be able to inject a small sample aliquot with

precision, with injection volumes usually varying from 10 to 200

µL. The polymer

sample solutions are commonly prepared with concentrations varying from 0.05 to
0.1% wt/vol. For high temperature SEC, polymer solutions are generally prepared
directly in the injection compartment under spinning and ﬁltering, if required. Off-
line sample preparation setups are also available. The sample injection systems
are provided with a sample carousel to permit the sequential injection of several
samples.

SEC columns are commonly packed with microporous glass beads or cross-

linked styrene–divinylbenzene gels, the latter being more common especially for
applications involving chains soluble in organic solvents. Several other, less com-
monly used packings, are also available (33,39). The column packing can have
pores of constant size or a distribution of pore sizes; the latter are called linear
columns since they generally lead to a linear calibration curve. For polystyrene–
divinylbenzene columns, the pore sizes depend on the degree of cross-linking since
they are formed by swelling the packing with solvent. Pore sizes in the range
of 50 to 10

˚A correspond to the effective size range of most polymer molecules

in solution. The size of the particles in the packing is also important: gel parti-
cle sizes in the range of 5–10

µm allow good packing and minimum channeling;

µm particles with increased pore volume packing, providing enhanced reso-

lution at low molecular weights, are also available. SEC columns are generally
300–600-mm long and 7.5 mm in diameter, but shorter mini-bore columns 250-mm
long and 4.6 mm in diameter can also be used to decrease solvent usage. Generally
two or three columns are used in series, preceded by a shorter guard column to
ﬁlter polymer gels and other impurities that may plug or damage the columns.
All columns will degrade with usage, particularly polystyrene/divinylbenzene
columns used at high temperatures. Column degradation alters their porosity
and, consequently, polymer retention time. Because of this, SEC columns must
be recalibrated at regular intervals to ensure that the results are reliable. It is
common practice to monitor the state of the columns by injecting broad and nar-
row standards during the analysis of other unknown samples. Eventually, column
degradation will lead to peak broadening and loss of resolution.

Differential refractive index detectors are the standard SEC mass detectors.

They have the advantage of being able to respond to all polymer types, provided
that the choice of solvent is adequate. To maximize detector response, the refrac-
tive indices of the polymer and solvent should differ signiﬁcantly. Infrared and
ultraviolet detectors have also been used successfully as mass detectors. Viscome-
ters can be added to allow the calculation of the intrinsic viscosity of unknown
polymer samples. They are also essential for the analysis of polymers with long-
chain branches. Light-scattering detectors permit the direct measurement of the
weight-average molecular weight of the polymer fraction present in the detector
cell and can, therefore, be used to determine molecular weight distributions as an
absolute technique dispensing the use of calibration curves. The combination of

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109

SEC with refractive index, viscosity, and light-scattering detectors is referred to as
triple detector system. This powerful conﬁguration has found many applications in
industry and academia, but it is also more complex to operate than single-detector
SEC systems.

As any other liquid chromatographic technique, SEC analysis can be in-

ﬂuenced by peak broadening due to axial dispersion and ﬂow irregularities. A
description of this phenomenon is beyond the scope of this article. For more infor-
mation, see the brief summary presented by Styring and Hamielec and the related
references cited by these authors (33).

SEC of Complex Polymers.

For the case of linear homopolymers and lin-

ear and chemically homogeneous copolymers, there exists a direct relationship be-
tween the size of the polymer chains in solution and molecular weight. In this case,
the separation mechanism operative in SEC will lead to fractionation by molecular
weight only. On the other hand, complex polymers may have long-chain branches
and, in the case of copolymers, are heterogeneous in chemical composition. Since
the radius of gyration (or size in solution) is affected by both long-chain branch-
ing and chemical composition, chains with different molecular weights may have
the same retention volume, and consequently the species present in the detector
cell may have a distribution of molecular weights. In this case, other microstruc-
tural factors besides molecular weight affect SEC fractionation. Interaction
chromatography (IC) is sometimes used to resolve these polymers. IC makes use
of the enthalpic term of equation 20 to achieve the fractionation. Since the en-
thalpic term is inﬂuenced by the chemical nature of the polymer, chains with
distinct chemical compositions can be fractionated. Chang recently published a
comprehensive review on this subject (37).

Field Flow Fractionation

The power of ﬁeld ﬂow fractionation (FFF) lies in the simplicity of the design of its
main component, the FFF separation channel. In FFF, a pulse of a polymer solu-
tion is injected into a carrier liquid and ﬂows through a rectangular thin channel.
An external physical ﬁeld is applied perpendicularly to the axial ﬂow, acting as a
driving force for the fractionation of the polymer molecules according to molecular
weight and/or other microstructural properties. Figure 27 shows a schematic of
a FFF channel. Contrarily to chromatographic techniques such as SEC, the FFF
channels contain no packing and can, therefore, be used to analyze polymer chains
with a broader range of hydrodynamic radii from 1 nm to 100

µm. Several books

and reviews have been recently published on this topic (40–45). The reader is re-
ferred to the review by C¨olfen and Antonietti (43) for a comprehensive and up-to-
date overview of principles, techniques, and applications of several FFF methods.

The nature of the external ﬁeld determines the type of fractionation achieved

in FFF. Even though many different types of ﬁelds have been examined, the most
common are thermal, ﬂow, and gravitational/centrifugal. Figure 28 summarizes
some types of FFF methods with their respective dates of invention.

The fractionation mechanism in FFF is as simple as it is elegant (Fig. 29).

Since the axial ﬂow is laminar, a parabolic velocity proﬁle is established in the
rectangular channel. Consider ﬁrst the case of ﬂow-FFF (Fl-FFF) of a linear

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Fig. 27.

Schematic of a FFF channel. From Ref. 40.

homopolymer: the carrier liquid ﬂows perpendicularly through the porous walls
of the channel, establishing an external ﬂow ﬁeld that is normal to the axial
ﬂow (cross ﬂow). A narrow pulse of polymer solution is injected at one end of the
column and the axial ﬂow is stopped for a short period of time. The cross ﬂow
forces the polymer molecules to accumulate near the opposite wall (accumulation
wall), generating a transversal polymer concentration gradient. According to
Fick’s law, this concentration gradient will act as the driving force for molecular
diffusion in the opposite direction of the external ﬁeld. After a short time, a

Fig. 28.

Some types of FFF with their times of invention. From Ref. 43. Copyright (2000)

Springer-Verlag.

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111

Fig. 29.

FFF ﬂow proﬁles and mechanism of fractionation. The molecular weight of the

particles increases in the order A

> B > C. From Ref. 40.

steady state is reached in a process called relaxation with the establishment of
a polymer layer of a given thickness near the accumulation wall. Chains with
smaller hydrodynamic radii (smaller molecular weights) have higher diffusion
coefﬁcients and will move farther away from the accumulation wall than chains
with higher hydrodynamic radii. When the axial ﬂow is restarted, generating a
parabolic velocity proﬁle inside the channel, smaller chains will, on average, be
transported with higher axial velocities than larger chains and will consequently
elute from the channel at shorter residence times (or shorter elution volumes).
This mode of operation, where the intensity of the driving force is constant
throughout the channel, is called the classic or normal model of operation. The
intensity of the driving forces can be varied as a function of analysis time to permit
the fractionation of polymers with broad molecular weight distributions. This
mode of operation is called ﬁeld intensity programming. Some other FFF modes
of operation include steric-FFF (46), hyperlayer-FFF (47), and focusing-FFF (48).
Hydrodynamic chromatography can also be seen as a special case of FFF where
the external driving forces are absent (43). More details about these modes of
operation can be found in the cited references.

Among all FFF techniques, Fl-FFF is the most universal, since the separation

relies only on the difference between the diffusion coefﬁcients of the solute species.
It can be applied to fractionate polymers (down to molecular weights of 10

g/mol),

colloids, and particles with hydrodynamic radii from 1 nm to 50

µm in either

water or organic solvents, although it has been used much more frequently for
water-soluble polymers. The transversal ﬂow driving force has the same effect on
molecules of all chain lengths, and thus forces them towards the accumulation wall
regardless of their sizes. However, since Fick’s diffusivity is inversely proportional
to molecular weight, the shorter chains will diffuse away from the accumulation
wall faster than the longer ones. As a consequence, in Fl-FFF the fractionation
mechanism is regulated mainly by molecular weight. In symmetric Fl-FFF, the
two walls are permeable to the transversal ﬂow, while in asymmetric Fl-FFF only
one of the walls is permeable. Figure 30 shows an example of Fl-FFF used in the
fractionation of proteins.

In thermal-FFF (Th-FFF), a temperature gradient is imposed on the FFF

channel (generally 10–100 K), causing the chains to diffuse towards the cold
wall by a mechanism called thermal diffusion (Soret effect). Th-FFF is commonly
used to fractionate synthetic polymers soluble in organic solvents with molecu-
lar weights from 10

to 10

g/mol. In principle, it can also be used to fractionate

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Fig. 30.

Fl-FFF fractionation of three proteins. From Ref. 40.

hydrophilic polymers, but in practice Fl-FFF is the predominant technique for
these polymers. Th-FFF is very useful for shear-sensitive chains with very high
molecular weights or macromolecular aggregates. Unfortunately, the fractionation
of shorter polymer chains requires the use of very high temperature gradients that
may cause several operational problems.

The ratio of the thermal diffusivity D

, to the molecular diffusivity D is the

main fractionation parameter in Th-FFF. D

depends on polymer and solvent com-

position, but it is virtually independent of chain length and long-chain branching.
For random copolymers, D

varies linearly with monomer composition, but more

complex behavior is observed for block copolymers. In principle, this dependency
on chemical composition can be used to fractionate copolymers according to their
composition, as is beautifully illustrated in Figure 31 (49).

Gravitational or centrifugal forces are used to promote fractionation in sedi-

mentation FFF (S-FFF). S-FFF is applicable for polymers with molecular weights
higher than 10

g/mol and colloids or particles with diameters larger than 30 nm,

both in water and organic solvents. The fractionation is regulated basically by the
differences in density and the intensity of the centrifugal forces. It is mostly used
to fractionate large biopolymers, polymer latexes, inorganic particles, emulsions,
biological cells, etc. Figure 32 shows the schematic of an S-FFF channel coiled
around a centrifugal rotor. An example of fractionation of polystyrene latex beads
by S-FFF is illustrated in Figure 33.

Theoretical Considerations.

The laminar ﬂow patterns inside the FFF

channel are well understood from basic ﬂuid dynamic models and permit a very

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113

Fig. 31.

Th-FFF according to chemical composition. From Ref. 49. Copyright (1989)

Springer-Verlag.

elegant theoretical treatment for different FFF modes of operation, as opposed
to chromatographic techniques where the internal packing generates ﬂow pat-
terns that are difﬁcult to describe rigorously with simple mathematical models.
Because of this precise mathematical treatment, FFF can be used not only as a
fractionation technique, but also to determine fundamental physicochemical prop-
erties of the sample under analysis. This also allows a new concept for a universal

Fig. 32.

S-FFF channel. From Ref. 40.

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Fig. 33.

S-FFF of polystyrene beads of different diameters. From Ref. 40.

calibration curve for fractionation by molecular weight. Basic mathematical rela-
tionships for Th-FFF, Fl-FFF, and S-FFF will be developed in this section, but the
readers should be aware that signiﬁcant deviations from this ideal behavior may
occur during actual fractionation experiments. For a more elaborate mathemati-
cal treatment of these techniques, the reader is referred to the review by C¨olfen
and Antonietti (43).

Very concise expressions can be developed for FFF if the following assump-

tions are made: (1) the FFF channel can be modeled as the space between two
parallel inﬁnite plates, (2) the ﬂow proﬁle is parabolic, (3) a steady state concen-
tration proﬁle is established in the channel, and (4) the external ﬁeld is uniform.
With these assumptions, a simple mass balance across the cross section of the
channel leads to the expression for the transversal ﬂux J

= u

c(x)

− D

dc(x)

(32)

where u

is the drift velocity of the chains toward the accumulation wall caused by

the external ﬁeld, c(x) is the concentration proﬁle of solute molecules as a function
of the distance x from the accumulation wall, and D is their diffusion coefﬁcient.
The diffusion coefﬁcient of polymers is a function of their molecular weight and
other microstructural characteristics.

Solving this equation at the steady state, ie J

= 0, leads to an exponential

solution for c(x),

c(x)

= c

exp

−

(33)

where c

is the polymer concentration at the accumulation wall. Therefore, the

normal mode of operation generates an exponential concentration proﬁle with
thickness inversely proportional to the molecular weight of the polymer solute. (In
the case of focusing FFF, the external forces are varied in such a way as to balance

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115

Fig. 34.

Concentration gradients in classical and focused FFF. Reprinted from Ref. 45 by

courtesy of Marcel Dekker, Inc.

the opposing diffusional forces, creating a focused band at a deﬁned position above
the accumulation wall where the external force is exactly counterbalanced, as
illustrated in Figure 34.)

The two most important parameters in FFF are the dimensionless retention

parameter

λ and the ratio of the retention time of an unretained solution to the

retention time of a retained solution, R

. An unretained solution is one whose

retention time is not affected by the external ﬁeld (the carrier liquid, for instance).
The retention parameter is deﬁned by the equation

λ =

(34)

where l is the thickness of the solute separation layer, w is the width of the FFF
channel, k is Boltzmann’s constant, T is the temperature, and F is the transversal
force that acts on the solute molecules. The parameter

λ is, therefore, the fraction

of the channel width occupied by the solute molecules.

The parameter

λ cannot be measured directly, but must be related to R via

the equation

= 6λ

coth

− 2λ

(35)

where V

is the volume of the separation channel and V

is the retention volume

for the retained solution.

Since

λ 1 for most cases, the following approximation applies:

R ∼

= 6(λ − 2λ

) ∼

= 6λ

(36)

Once

λ is estimated from an experimentally-measured value of R, it can be

used to calculate the values of several physicochemical parameters for the polymer
chains. For Fl-FFF, the following expression holds:

λ =

(37)

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where ˙

is the transversal ﬂow rate. The equivalent expressions for Th-FFF and

S-FFF are as follows:

λ =

(dT

/dx)w

(38)

λ =

−

(39)

where D

is the thermal diffusion coefﬁcient, dT/dx is the temperature gradient

across the channel,

r is the centrifugal acceleration, M is the molecular mass

of the solute, R is the gas constant, and

ρ and ρ

are the solvent and solute

densities, respectively. Equations 37 to 39 are very helpful to understand the basic
physical parameters that affect separation inside FFF channels. Notice that the
layer thickness, as estimated by the parameter

λ, is always inversely proportional

to the ﬁeld strength, ie, ˙

, dT/dx, or

r. This permits the fractionation of a

very broad range of molecular sizes by simply decreasing the intensity of the
external ﬁeld, as done in the mode of operation called ﬁeld intensity programming.
For instance, the analysis of a sample with broad molecular weight distribution
can be started at high ﬁeld intensity. If this intensity was kept throughout the
fractionation, very large elution times would be required to elute the highest
molecular weight species. Decreasing the ﬁeld intensity in a programmed way
leads to a reduction of the retention time of larger chains, thus shortening the
analysis time. This procedure, however, increases the complexity of the analytical
procedure and may also lead to undesirable secondary relaxation phenomena:
solute–solute and solute-accumulation wall interactions at high ﬁeld strengths.

Use of Fl-FFF and Th-FFF for Molecular Weight Fractionation.

Th-

FFF and SEC have been often compared since both techniques are generally used
to obtain the molecular weight distribution of synthetic polymers in organic sol-
vents. Nonetheless, SEC is much more widely used than FFF, despite the fact
that these techniques can be seen as complementary. One of the advantages of
SEC is the existence of a universal calibration curve applicable to all polymer
types. Even though universal calibration curves can also be obtained for FFF,
their concept is radically different from that of the universal calibration curve
for SEC. For SEC, the universal calibration curve can be used for all polymers
in the same SEC instrument and column set, but will differ from instrument to
instrument. Even in the same instrument, the calibration curves will “drift” as the
columns age and change their pore structure, particularly for high temperature
SEC. Contrarily, the universal calibration curve for FFF is applicable, in principle,
to a given polymer–solvent combination in all FFF setups, but will be different
for each polymer–solvent system. Another practical advantage of SEC that may
explain its predominance over FFF is a more straightforward mode of operation.
Many of the operation modes of FFF, while very efﬁcient, require considerably
more expertise from the analyst.

In general, SEC is capable of analyzing lower molecular weight chains (using

packing with small pore radii) but has limitations for very high molecular weight
chains that can suffer chain scission because of the high shear stresses inside the

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117

SEC columns and/or generate very high viscosities and pressure drops. On the
other hand, Th-FFF is not very effective for fractionating low molecular weight
chains because a very high temperature gradient is required (D is large for small
chains, cf eq. 38), but becomes more attractive for higher molecular weight chains.
Additionally, a series of SEC columns can be used to fractionate samples with
very broad molecular weight distributions (up to ﬁve decades or more) since the
elution volume in SEC is a linear (or quasi-linear) function of the logarithm of the
molecular weight. Calibration curves for FFF, however, show a now-linear relation
between elution volume and the logarithm of the molecular weight, as indicated
in equation 40 below. This implies that in the normal mode of operation, FFF
fractionation is limited to samples with narrow molecular weight distributions.
As we have seen, this limitation can be reduced by operating FFF in the ﬁeld
programming mode, but at the price of a more elaborate analysis procedure (43).

FFF molecular weight calibration curves can be represented by the relation

log V

= a + S

logM

(40)

where V

is the retention volume, M is the molar mass of the polymer, and a and

are constants for a given polymer–solvent system. The power of this approach

is that the calibration constants can be related to the characteristics of the FFF
equipment used to obtain the calibration curve and then extended to any other FFF
setup. In other words, after equation 40 is determined for a given polymer-solution
system in one FFF setup, in principle it can be used in any other FFF setup, even
if some of the operation conditions and channel dimensions are different. This
procedure will be outlined in the next paragraphs.

The following expressions apply for linear or uniformly branched polymers:

= ϕ

− n

(41)

= AM

− b

(42)

where the parameters

, n, A, and b are universal constants for a given polymer–

solvent system. Substituting equation 42 in equation 41, the following expression
results:

− b

= ϕ

− n

(43)

Since it is know that D

is nearly independent of molecular weight, then

≈ n provided that the measurements are made at the same temperature.

Equations 41 and 42 can be substituted into their respective deﬁnitions of

λ,

equations 38 and 37, to obtain

λ =

− n

(dT

/dx)w

− n

(

T/w)w

− n

(44)

λ =

− b

(45)

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The parameter

λ can be calculated from the experimentally-measured value

of R using equation 35. Using the approximation R

= 6λ (accurate within 2% when

< 0.06), it is possible to obtain closed-form expressions for the dependency of

the retention volume on molecular weight:

for Th-FFF

(46)

6Aw

for Fl-FFF

(47)

Comparing equations 46 and 47 with the universal calibration curve, equa-

tion 40, the following expressions for the calibration constants are obtained:

= log

and S

= n for Th-FFF

(48)

= log

6Aw

and S

= b for Fl-FFF

(49)

The beauty of these equations is that they are related only to the dimensions

of the FFF channel, w and V

; the strength of the external ﬁelds,

T or ˙V

; and the

polymer–solvent speciﬁc constants,

, A, n and b. (For more rigorous calibrations,

some corrections of these equations may be required.) They can, therefore, be
applied to analyze a known polymer–solvent system in different FFF instruments
under a wide range of experimental conditions (50).

Fractionation Equipment.

The basic setup of FFF is essentially the same

required for chromatographic techniques, with the chromatographic columns re-
placed by the FFF channel. The same requirements for precise sample injection,
stable ﬂow of the carrier liquid, and adequate sample detection described for SEC
also apply to FFF.

FFF channels are usually 50- to 500-

µm thick, 10- to 100-cm long, and have

an aspect ratio of 20 to 200 (width/thickness). Particular details such as materi-
als of construction, geometry of the channels, and speciﬁc arrangements of ﬂow
inlets and outlets can vary among different setups and operation modes. Details
for several equipment conﬁgurations can be found in the references cited in this
section.

Differential refractive index detectors are the most common choice for FFF

mass detectors. As also the case for SEC, the use of multiple detectors is becoming
increasingly important with FFF. Low angle laser light scattering (LALLS) and
multiangle laser light scattering (MALLS) detectors are used for absolute deter-
mination of molecular weights. Many other detectors (UV, ICP-MS, differential
viscometer, IR, electrospray mass spectroscopy) have also been used with FFF.

FFF is manly an analytical technique, but some effort has be made to run it

in preparative or at least micro-preparative mode. The use of modiﬁed continuous
FFF techniques or SPLITT channels (where the incoming polydisperse sample
is split into two outlet ﬂows with narrower distributions) is brieﬂy described by
C¨olfen and Antonietti (43).

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119

Continuous Polymer Fractionation

Continuous polymer fractionation (CPF) is a liquid–liquid extraction technique
developed to fractionate large amounts of polymer (51). It was originally devel-
oped to fractionate homopolymers by molecular weight and has been applied for
polyvinyl chloride (52), polyisobutylene (53,54), polyethylene (55), and polycarbon-
ate (56). Recently, attempts to extend CPF to copolymers have been made (57) and
it was shown that, under certain operational conditions, fractionation by chemical
composition could also be achieved (58).

Being continuous, the advantage of CPF over the other techniques previously

discussed lies in the fact that larger polymer samples can be fractionated in rela-
tively shorter times. This is particularly important if large fractions with narrow
MWD or CCD are required for the establishment of structure-property relation-
ships or to remove, on a technical scale, chain populations that are undesirable
for a given application of specialty polymers.

In a CPF liquid–liquid extraction column, the feed stream containing the

polymer solution (polymer rich phase) ﬂows countercurrently to the eluent stream
containing the extracting agent. Chains with low molecular weight present in the
polymer-rich phase (gel phase) are extracted by the eluent stream (sol phase), and
the two phases exit from the bottom and top of the column because of their density
differences. Mixed (solvent/nonsolvent) or single solvents can be used with CPF,
but mixed solvents provide more ﬂexible operational conditions. The same princi-
ples discussed before for batch fractionation can be extended to CPF. Weinmann
and co-workers provide detailed theoretical treatment of CPF (56). Naturally, a
single pass through a CPF column only permits the fractionation into two low–
high molecular weight fractions. If more fractions are required, it is necessary to
use columns in series, in which case successive extractions of the low molecular
weight components can be achieved by changing the solvent/nonsolvent ratio or
column temperature in the series.

A recent article describes two conﬁgurations for CPF columns (57): both were

glass columns with high diameter ratios of 2.4 m/3.5 cm and 1.9 m/1.5 cm. The
larger column was ﬁlled with glass beads of 8- and 10-mm diameter in a 1:1 ratio,
while the smaller one with glass beads of 6 and 8 mm in a 3:1 ratio. The feed inlet
was positioned at 3/4 of the total column high, since theoretical and experimental
results indicate that this placement increases the separation efﬁciency (56). In
this case, the temperature of the upper

1
4

of the column can be decreased, creating

a reﬂux effect that is beneﬁcial to the fractionation. Two to three theoretical plates
per meter and a fractionation capacity of 10–100 kg of polymer per hour and square
meter of cross section have been reported for these columns (51).

A related technique for the fractionation of large amounts of polymer, con-

tinuous spin fractionation (CSF), has also been recently disclosed (59).

Mass Spectrometry

Traditional mass spectrometry techniques involving the study of ions in the gas
phase are not adequate for the analysis of polymers because of their very low vapor
pressures and their tendency to suffer chain scission during ionization. On the

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other hand, recent developments on soft ionization techniques such as electrospray
ionization (ESI) and matrix-assisted laser desorption ionization (MALDI) mass
spectrometry have found many applications in the polymer ﬁeld (60–63). Since
ESI is more commonly used with biopolymers (63) and most common applications
of mass spectrometry (qv) for synthetic polymers use MALDI, we will restrict our
discussion in this section to the later technique.

Mass spectrometry experiments consist of two equally important steps: ion-

ization/vaporization and detection. Although traditional techniques such as elec-
tron impact ionization are too harsh and lead to the fragmentation of polymer
molecules, the softer ionization technique of MALDI can be successfully used to
fractionate polymers of narrow molecular weight distribution. The ionized ana-
lyte is accelerated through a magnetic ﬁeld with voltage V, acquiring a velocity
that is proportional to its mass-to-charge ratio (m/z). Time-of-ﬂight (TOF) analyz-
ers have been found to be the most adequate for polymer analysis. In the TOF
spectrometer, the ions are separated according to their velocities and their m/z
is evaluated on the basis of the time it takes for the ionized molecules to travel a
ﬂight path of length L to the detector (64):

2zeV

(50)

Commercial MALDI-TOF instruments in the range up to 10,000 g/mol can

have a resolution of up to 4 g/mol, which permits a remarkable resolution of
molecular weight distributions (65). This impressive resolution is illustrated in
Figure 35 for the molecular weight distribution of a polymethylmethacrylate
sample (66).

The original promise of obtaining the complete molecular weight distribu-

tion of polymers with MALDI-TOF did not hold true except for polymers with very
narrow distributions (polydispersity indices less than 1.2), in which case the anal-
ysis can be very accurate and reproducible (67). The reasons for this are complex
and mainly caused by the fact that both the ionization process and the detector
are mass-sensitive and discriminate towards the shorter chains in the population.
The detector effect can be easily understood from inspection of Figure 1, where the
number and weight chain length distributions of a polymer that follows Flory–
Schultz’s most probable distribution are compared. It is clear that the number of
shorter chains greatly exceeds the number of longer chains, but the longer chains
account for a much higher mass fraction of the sample. The signal of refractive
index detectors, commonly used with SEC and FFF, is proportional to the mass
of the polymer in the detection cell and is therefore very sensitive to the high
molecular weight end of the distribution. On the other hand, the TOF detector is
sensitive to the number, not the weight, of chains, and therefore the signal for the
longer chains is quickly lost in the background noise. This is elegantly illustrated
in Figure 36, where the molecular weight distributions of three polydimethylsilox-
ane samples measured by MALDI-TOF are much narrower and shifted to the low
molecular weight end of the distribution than the ones measured by SEC.

In order to circumvent this limitation but still preserve the detailed

molecular weight information generated by this powerful detection technique,
MALDI-TOF has been used as an absolute detector, both on-line and off-line, for

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121

Fig. 35.

MALDI analysis of a polymethylmethacrylate resin.

Raw data;

Theoretical isotope distribution. From Ref. 66.

narrow molecular weight fractions obtained by SEC (65,69,70). When applied
to molecular weight measurements of homopolymers alone, MALDI-TOF brings
little advantages to the already sophisticated SEC triple detector systems
discussed above. However, when applied to copolymers, this approach permits
the simultaneous estimation of the bivariate distribution of molecular weight
and chemical composition, as explained by Suddaby and co-workers (65) and
illustrated in Figure 37. In brief, this methodology consists in matching the
molecular weights of the MALDI peaks for each SEC fraction to combinations
of the molar masses of the comonomers using a polymerization kinetics model.
Therefore, SEC/MALDI-TOF may have the potential to surpass any other known
technique for the elucidation of the microstructure of synthetic copolymers.

Moreover, even though the complete molecular weight distribution for broad

polymers may not be accurately recovered by MALDI-TOF alone, the very detailed
molecular weight information provided for the lower molecular weight species is
ideally suitable for the study of polymerization reaction mechanisms, chain end
types, dendrimer analyses, and polymer identiﬁcation (60–63,68).

Mass spectrometry experiments in general, and MALDI-TOF in particular,

require the sample to be in the gas phase and ionized. Ionization is commonly
achieved by the complexation of a cation with the polymer molecule. Polymers
containing heteroatoms such as polyethers, polyamides, polyesters, polymethyl-
metacrylates, polysiloxanes, and polycarbonates are reasonably simple to be
cationized by the addition of sodium or potassium salts. Apolar polymers contain-
ing double bonds such as polystyrene, polybutadiene, and polyisoprene can also

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Fig. 36.

Overlay of SEC and MALDI-TOF proﬁles for three poly(dimethylsiloxane) sam-

be cationized with silver or copper salts. Until recently, apolar polymers with no
functional groups, such as polyethylene and polypropylene, could not be analyzed
by MALDI [with the exception of low molecular weight waxes and long alkanes
(67)], but some alternative routes involving the functionalization of their terminal
double bonds have been developed and seem to lead to sound analytical results
(71–74). This is particularly attractive for the analysis of polyethylene made with
metallocene catalysts that generally contains a large fraction of vinyl-terminated
chains. Unfortunately, polymer–cation interactions are still the subject of much
debate, even for polymer containing heteroatoms. For instance, varying the cation
type in the analysis of polymethylmethacrylate samples can cause changes of
20–35% in their estimated molecular weight distributions (60).

The selection of the matrix for MALDI analysis is also far from triv-

ial. In his comprehensive review, Nielen listed a table with more than
200 polymer/matrix/solvents entries for MALDI-TOF analysis. Homogeneous

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123

Fig. 37.

Bivariate distribution of degree of polymerization (DP) and chemical composi-

tion for a copolymer of methylmethacrylate and n-butylmethacrylate determined by SEC–
MALDI. Reprinted with permission from Ref. 65. Copyright (1996) American Chemical
Society.

cocrystallization of matrix and polymer sample is essential for accurate analy-
sis with MALDI. Generally, it is recommended that the polarities of matrix and
polymer be close, but often a trial-and-error process is required. Several meth-
ods also exist for transferring the mixture of matrix, salt, and sample solutions
onto the MALDI target, with electrospray deposition being considered the best
approach (60).

Interaction Chromatography and Other Polymer
Fractionation Techniques

The fractionation techniques described above are the ones that have proved to be
applicable to the widest class of synthetic polymers over the last decades. Ev-
idently, many more fractionation techniques have been used with many poly-
mer types. Some of these methods will be described very brieﬂy in the next
paragraphs.

Ultracentrifugation has been used mainly for the characterization of biopoly-

mers and gels in aqueous solutions. Some recent equipment developments have

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stirred renewed interest in this fractionation method (75,76). Ultracentrifugation
can be used in three main modes of operations: sedimentation velocity, equilib-
rium, and density gradient. In the sedimentation-velocity method, the centrifuge
is operated at high rotational speed (above 40,000 rpm), generating a concentra-
tion proﬁle in the fractionation cell that can be correlated to the molecular weight
distribution of the polymer. In the equilibrium method, the centrifuge is operated
at lower rotational speeds and long analysis times are required to establish the
equilibrium between sedimentation and diffusion. The density-gradient method is
mainly used to estimate the chemical composition or tacticity distributions of poly-
mers and involves the use of two solvents of different densities. More details on
centrifugation methods, including the description of new equipment apparatuses
can be found in Borchard and co-workers (75).

Turbidimetric methods have also found some applications for the detection of

polymer fractionation. In this case, polymer precipitation is promoted via the addi-
tion of a nonsolvent or temperature variation, according to the methods described
for batch fractionation, and the turbidity created by the precipitation of the poly-
mer chains is measured with a photometer. Both molecular weight and chemical
composition fractionations can be achieved, based on the precipitation method uti-
lized. However, only qualitative information about the distributions of molecular
weight or chemical composition of the samples can be achieved by this method,
because it is very difﬁcult to relate the detector response to the actual microstruc-
tural distributions. Since this method of detection does not require the recovery
of the individual polymer fractions, it may be very attractive for samples that
form very viscous solutions, such as high molecular weight polymers, or samples
containing inorganic ﬁllers that would otherwise clog continuous fractionation
equipment such as TREF, Crystaf, or SEC. Cloud-point titration is a variation of
this method, used for the detection of incipient turbidity in a polymer solution.
It is especially useful for the determination of the theta composition of polymer
solutions. The principles of turbidimetric methods have been reviewed by Elias
(77).

Many variations of interaction or adsorption chromatography have also been

used to fractionate polymers. Although powerful, they are very often polymer-
speciﬁc and beyond the scope of the present review. The reader is referred to the
chapter of fractionation of polymers in the Polymer Handbook for an overview of
some of these techniques (78) and to the review by Chang (37) for more details.

The power of interaction chromatography techniques will be brieﬂy illus-

trated with a well-designed example for the fractionation by molecular weight of
the six polystyrene samples shown in Figure 38 (79). The samples were analyzed
in a high performance liquid chromatography (HPLC) system equipped with C18
bonded silica columns with various pores sizes using a solvent mixture containing
a good solvent, CH

, and a poor solvent, CH

CN, for polystyrene. At the high-

est concentration of the good solvent (65% CH

by volume) the entropic effects

dominate (recall eq. 20) and size exclusion is the main mechanism of separation.
Thus, the chains with higher molecular weight have lower retention times. As
the fraction of the good solvent CH

decreases to 57%, the fractionation res-

olution is totally lost and all samples elute at the same retention time. At this
operation condition, called the chromatographic critical point or the adsorption-
size exclusion transition point, enthalpic and entropic effects cancel each other and

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125

no separation takes place. For higher concentrations of nonsolvent, solute-support
interaction effects start to dominate, increasing the enthalpic term in equation 20.
At these conditions, longer chains having more repeating units interact strongly
with the support and are retained longer in the columns, reversing the sequence
of retention times observed during the size-exclusion mode of fractionation. A
similar behavior is observed when the solvent mixture composition remains the
same (CH

/CH

CN: 57/43, the chromatographic critical point at 30.5

◦

C) and

the temperature is allowed to vary. According to equation 20, it is expected that
the enthalpic effects will decrease as the temperature increases, and therefore
fractionation by size exclusion will predominate at higher fractionation temper-
atures. This is indeed observed in the series of fractionations varying from 20 to
50

◦

C, as indicated in Figure 38.

Interaction chromatography can also be used to fractionate model polymers

according to long-chain branching. An application for star-shaped polystyrenes
was presented by Chang and co-workers (79).

Fig. 38.

Solvent composition and temperature effects for the fractionation of six

polystyrene samples of different molecular weights M

: (1) 2.5

× 10

, (2) 12

× 10

, (3) 29

, (4) 165

× 10

, (5) 502

× 10

, and (6) 1800

× 10

. The solvent mixture is CH

/CH

CN.

is a good solvent and CH

CN is a poor solvent for polystyrene. The chromatograms

on the left were measured at a constant temperature of 30.5

◦

C and the ones on the right

were measured at a constant ratio CH

/CH

CN of 57/43 (vol/vol). From Ref. 79. Copy-

right (1996) Wiley-VCH.

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FRACTIONATION

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Cross-Fractionation Chromatography and Related Techniques

Cross fractionation, also called orthogonal fractionation, is a term that can be ap-
plied to any technique that permits the fractionation of polymers according to two
or more microstructural characteristics. For instance, polymer chains can be ﬁrst
separated into several fractions with narrow chemical composition distributions
by TREF and each fraction can then be analyzed by SEC to recover their molec-
ular weight distributions. When done off-line, these TREF–SEC fractionations
are very time-consuming but can generate a wealth of information on polymer
microstructure (80,81).

The term cross-fractionation chromatography (CFC) has been lately applied

to a more speciﬁc method that combines TREF and SEC fractionation in an auto-
mated cross-fractionation instrument. The TREF fractions are collected in step-
wise mode at predetermined temperature intervals and directly injected into the
SEC columns (82). This system is very attractive because it permits the direct esti-
mation of the bivariate distribution of molecular weight and chemical composition
distribution of semicrystalline polymers. Evidently, all the considerations made
above for the fractionation of complex polymers with TREF and SEC must be kept
in mind when interpreting the results of these fractionations, since the separation
mechanisms are not truly based on molecular weight and chemical composition
but rather on hydrodynamic volume and crystallizability. CFC has been used in a
large number of investigations, especially to study the polymerization mechanism
of ethylene, propylene, and higher

α-oleﬁns with single- and multiple-site cata-

lysts (83–89). Figure 39 shows the bivariate distribution of an ethylene/propylene
copolymer measured by CFC and illustrates well the potential of this
technique.

Interaction chromatographic separation methods that are sensitive to chem-

ical composition have also been coupled with SEC to achieve similar cross-
fractionation results for other polymer types (90–94), and fully automated in-
struments have been developed for HPLC–SEC orthogonal fractionations (95). It
is generally preferred that the samples be ﬁrst fractionated according to chemical
composition by HPLC and then separated according to molecular weight by SEC.

Fig. 39.

Cross-fractionation chromatograph of an ethylene–propylene sample. Reprinted

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FRACTIONATION

127

Fig. 40.

Molecular weight distribution and average comonomer content for an ethylene/1-

butene sample made with a Ziegler–Natta catalyst. Reprinted from Ref. 99. Copyright
(2001), with permission from Elsevier.

The advantage of this sequence lies in the fact that SEC of complex copolymers
responds to both molecular weight and chemical composition, while the inﬂuence
of molecular weight on HPLC fractionation can be minimized by the selection of
adequate analysis conditions. The importance of polymer cross-fractionation tech-
niques is increasing because of the rapidly growing use of polymers with complex
microstructures and would require a separate chapter for complete discussion.
The interested reader is referred to the article by Pasch for a review on recent
advances in the ﬁeld of cross fractionation of polymers (96).

A simpler approach to CFC consists in measuring the average chemical com-

position of polymer eluting from SEC columns using an on-line Fourier transform
infrared (FTIR) detector. Even though liquid ﬂow cells can be used, signal-to-noise
ratio problems may arise because of the low concentration of polymer species in
the detector cell. Alternatively, the SEC efﬂuent can be sent to a solvent evapo-
rative interface where the polymer is deposited as a thin layer onto the surface
of a germanium disk. After SEC fractionation, the disk is attached to an infrared
spectrometer and the polymer microstructure is analyzed as a function of elution
time (97,98). Unfortunately, this technique is not free of experimental difﬁculties
either because it is very often difﬁcult to obtain a uniform ﬁlm deposited onto
the disk. The type of information that can be obtained with SEC–FTIR is illus-
trated in Figure 40 for an ethylene/1-butene copolymer made with a heterogeneous
Ziegler–Natta catalyst.

The SEC–FTIR combination, although attractive due to its simplicity, is not

able to recover the bivariate distribution of molecular weight and chemical com-
position, but can only measure average composition as a function of molecular
weight. Consequently, it is not a true cross-fractionation technique. A method
to recover the chemical composition component of the distribution was proposed
for the case of polyoleﬁns made with single and multiple-site catalysts using a

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polymerization kinetics model and molecular weight distribution deconvolution
techniques (99,100).

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Vol. 10

FRACTIONATION

131

98. T. Provder, M. Whited, D. Huddleston, and C. Y. Kuo, Prog. Org. Coat. 32, 155 (1997).
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(1996).

OAO

B. P. S

OARES

University of Waterloo

FRACTOGRAPHY.

See Volume 2.

FRACTURE.

See Volume 2.

FREE RADICAL POLYMERIZATION.

See R

ADICAL

OLYMERIZATION

FRP.

See R

ADICAL

OLYMERIZATION

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