A practical method for estimating specific refractive index increments for flexible non-electrolyte polymers and copolymers in pure and mixed solvents using the Gladstone-Dale and Lorentz-Lorenz equations in conjunction with molar refraction structural constants, and solvent physical property databases

Abstract

Equations for calculating the specific refractive index increment, ∂n/∂c, for polymer/solvent systems have been developed starting with either the Gladstone-Dale equation or the Lorentz-Lorenz equation for the refractive index of a dielectric medium. The resulting equations can be used to estimate ∂n/∂c as a function of polymer concentration, temperature, solvent composition, and, for copolymers, monomer composition. Simplifying assumptions have been made to generalize these equations and make them more usable: 1) molar refractions are treated as constants, 2) the density of the polymer solution, in the limit of low polymer concentration, can be approximated via the additivity of volumes approach; the densities of polymer-free solvents and solvent mixtures are not necessarily subject to this approximation, 3) the mass density of solvents and solvent mixtures can be estimated by any one of several models developed specifically for this purpose; we will consider the Rackett equation, the COSTALD equation, and the standard assumption of mass and volume additivity, and 4) the mass density of a copolymer can be obtained from a weight fraction average of the mass densities of the homopolymers from which it is comprised. Comparisons to experimental data show that (∂n/∂c)GD and (∂n/∂c)LL both yield average absolute deviations of approximately 0.009 ml/gm.

Full text

Materials Today Communications 25 (2020) 101644
Available online 21 September 2020
2352-4928/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
A practical method for estimating specic refractive index increments for
exible non-electrolyte polymers and copolymers in pure and mixed
solvents using the Gladstone-Dale and Lorentz-Lorenz equations in
conjunction with molar refraction structural constants, and solvent physical
property databases
Brian F. Hanley
Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA, 70803, United States
ARTICLE INFO
Keywords:
Refractive index increment
Molar refraction
Lorentz-Lorenz equation
Gladstone-Dale equation
ABSTRACT
Equations for calculating the specic refractive index increment,
∂
n/
∂
c, for polymer/solvent systems have been
developed starting with either the Gladstone-Dale equation or the Lorentz-Lorenz equation for the refractive
index of a dielectric medium. The resulting equations can be used to estimate
∂
n/
∂
c as a function of polymer
concentration, temperature, solvent composition, and, for copolymers, monomer composition. Simplifying as-
sumptions have been made to generalize these equations and make them more usable: 1) molar refractions are
treated as constants, 2) the density of the polymer solution, in the limit of low polymer concentration, can be
approximated via the additivity of volumes approach; the densities of polymer-free solvents and solvent mixtures
are not necessarily subject to this approximation, 3) the mass density of solvents and solvent mixtures can be
estimated by any one of several models developed specically for this purpose; we will consider the Rackett
equation, the COSTALD equation, and the standard assumption of mass and volume additivity, and 4) the mass
density of a copolymer can be obtained from a weight fraction average of the mass densities of the homopolymers
from which it is comprised. Comparisons to experimental data show that (
∂
n/
∂
c)
GD
and (
∂
n/
∂
c)
LL
both yield
average absolute deviations of approximately 0.009 ml/gm.
1. Introduction
The differential refractive index increment,
∂
n/
∂
c, plays an extraor-
dinarily important role in polymer characterization and, of late, in
polymer dynamics
1
. For example,
∂
n/
∂
c enters into the calculation of
the absolute weight average molar mass, M
w
, for bulk polymer samples
characterized by light scattering; it also enters into the determination of
a polymer’s absolute molar mass distribution when light scattering is
used as a detector for size exclusion chromatography [1,2]. In addition,
∂
n/
∂
c has become important in refractive index matched polymer/-
polymer/solvent tracer diffusion experiments and mixed solvent size
exclusion chromatography where the low molecular weight solvent and
the temperature are chosen so that
∂
n/
∂
c
B
≈0 for the background pol-
ymer/solvent pair [3–5].
At some point the properties of a homopolymer generally become
independent of its average molar mass and molar mass distribution.
Thus, for a given solvent and temperature
∂
n/
∂
c can be approximated as
constant over a wide range of molar masses. It is also possible to
correlate
∂
n/
∂
c to the molar mass of lower-molecular weight oligomers
via experimentation with these specic oligomers and thus include the
dependence of
∂
n/
∂
c on molar mass. The situation is substantially more
complex for copolymers. It is still possible to measure the average value
of the refractive index increment - 〈
∂
n/
∂
c〉- for any bulk copolymer
sample, but now,
∂
n/
∂
c varies with the composition fractions of the
copolymer for a given temperature and solvent. Because synthetic co-
polymers generally exhibit polydispersity in both molar mass and
composition, it becomes much more complicated to use light scattering
as an absolute molar mass detector in size exclusion chromatography. In
this paper we will only consider the limiting case where
∂
n/
∂
c is inde-
pendent of molar mass.
It is oftentimes desirable to have some way to estimate of
∂
n/
∂
c.
E-mail address: brianhanley@lsu.edu.
1
∂
n/
∂
c as used in this paper will generally refer to (
∂
n/
∂
c)|
c=0
. We will use the following notation to refer to nite concentration values of the refractive index
increment - (
∂
n/
∂
c)|
c
.
Contents lists available at ScienceDirect
Materials Today Communications
journal homepage: www.elsevier.com/locate/mtcomm
https://doi.org/10.1016/j.mtcomm.2020.101644
Received 25 August 2020; Received in revised form 4 September 2020; Accepted 7 September 2020

Materials Today Communications 25 (2020) 101644
2
Several models for approximating
∂
n/
∂
c from properties of the pure
polymer and solvent coupled with a theory of polarization have been
developed. Lorimer presents a review of several such models [6]. This
paper is an attempt to develop usable techniques for estimating
∂
n/
∂
c
based on the Gladstone-Dale and Lorentz-Lorenz equations. The method
described here improves upon the general formalism in a number of
ways. With the advent of modern computers, property simulation soft-
ware for solvents and solvent mixtures has become widely available.
First, then, we have incorporated one such properties estimation engine
into the calculation of
∂
n/
∂
c - the Aspen Properties add-in available for
Excel. Second, we suggest some techniques for estimating the molar
refraction of the polymer’s repeat unit from experimental estimates of
the molar refractions for the monomer(s) making up the repeat unit or
from subunits making up the repeat unit which would typically exist as
liquids at room temperature. The repeat unit’s molar refraction is then
estimated from the monomer units or subunits by making small cor-
rections based upon tabulated structural, group and atomic contribu-
tions to the molar refraction.
2. Model development
A number of empirical/semi-empirical models have been proposed
that relate the refractive index of a dielectric medium to its density.
Often these models are separable in such a way that a term related only
to the atomic polarizability properties of the substance appear. The
atomic (or molar) refraction can often be taken as independent of tem-
perature and pressure (although it remains a function of wavelength)
[7]. In addition, mixing rules have been proposed in order to extend the
applicability of these formulae to multicomponent systems [8–10]. It
appears, however, that these modelling efforts have not been effectively
extended to the calculation of the refractive index increment,
∂
n/
∂
c. In
this section, we will develop expressions for
∂
n/
∂
c based on the
Gladstone-Dale and Lorentz-Lorenz equations. We will rst consider the
Gladstone-Dale equation and then move to the Lorentz-Lorenz equation.
The Gladstone-Dale has been a common starting point for estimating
∂
n/
∂
c [6,11].
(n−1)M
ρ
=RGD (1)
R
GD
is the so-called “molar refraction energy” or “molar refraction”;
it has been found to be approximately constant for small to moderate
perturbations of the temperature for pure components that remain in the
same physical state [12]. In addition, R
GD
can be estimated with good
accuracy from tabulated atomic, group, and bond contributions [13].
With appropriate assumptions the GladstoneDale equation leads to [14].
∂
n
∂
c(T,xM
→,xS
→) = (nP(T,xM
→) − nS(T,xS
→))vP(2)
where T is the temperature, xM
→is the vector of mole fractions for the
monomers units making up a copolymer, xS
→is the vector of mole frac-
tions for pure liquids making up the solvent, and vP is the partial specic
volume of the polymer in the solvent. If additivity of volumes is assumed
for the dilute polymer solution then
∂
n
∂
c(T,xC
→,xS
→) = nP(T,xM
→) − nS(T,xS
→)
ρ
P(T,xC
→)(3)
where
ρ
P
can usually be taken to be the polymer’s mass density at the
appropriate temperature. However, some care must be taken when
applying Eq. (3). Fig. 1 is comparison the temperature dependence of the
mass density for samples of an isotactic and atactic polypropylene [15].
The molar mass distributions of polypropylenes are often determined via
GPC with 1,2,4-trichlorobenzene or 1-chloronaphthalene as the eluent.
Experiments are usually performed at temperatures between
130◦C–160◦C [16]. In this temperature range the density of a typical
isotactic polypropylene is noticeably larger than that of an atactic
sample. One might therefore expect to see non-negligible differences in
the values of
∂
n/
∂
c between isotactic and atactic polypropylene. How-
ever, reported values for
∂
n/
∂
c in these solvents do not appear to be
dependent on the tacticity of the sample (see Table 1).
Semicrystalline polymers must normally be dissolved at high tem-
peratures [17]. Polypropylene, for example, is often dissolved in one of
the aforementioned solvents at approximately 180◦C whereas GPC
Nomenclature
Symbols
c Polymer concentration (gm/mL)
∂
n/
∂
c Refractive index increment at c=0 (ml/gm)
(
∂
n/
∂
c)|
c
Refractive index increment at concentration c (ml/gm)
M Molar mass (gm/mol), 〈MP〉: average polymer
(copolymer) molar mass
〈N〉Average number of monomer repeat units
n Refractive index
ρ
Mass density (gm/mL)
R Molar refraction (ml/mol); R
M
for the monomer repeat
unit; R
S
for the solvent
R
GD
Gladstone-Dale molar refraction (ml/gm); R
GD,M
for the
monomer repeat unit; R
GD,S
for the solvent
R
LL
Lorentz-Lorenz molar refraction (ml/gm); R
LL,M
for the
monomer repeat unit; R
LL,S
for the solvent
vP Partial specic volume of the polymer in the solvent
ω
Weight fraction of polymer
x Mole fraction of polymer or copolymer in solution
xM
→Vector of monomer mole fractions in a copolymer
xS
→Vector of mole fractions for components making up the
solvent
Subscripts
P Polymer
S Solvent
M Polymer repeat unit
Fig. 1. Comparison of the mass densities of an atactic and isotactic poly-
propylene versus temperature (0 MPa). Data are reproduced from ref. [15].
B.F. Hanley

Materials Today Communications 25 (2020) 101644
3
experiments are usually performed between 130◦C–160◦C. We see, from
Fig. 1, that isotactic polypropylene at 180◦C is slightly above its melting
point; the density versus temperature curves for the isotactic and atactic
samples superpose for temperatures greater than 180◦C. For amorphous
polymers above their glass transition temperatures and for semi-
crystalline polymers above their melting points, individual chains in the
melt are expected to adopt their random ight congurations. Chains in
dilute solution from either amorphous or semicrystalline polymers are
also expected to adopt congurations close to those of a random ight
chain. Excluded volume interactions in good solvents certainly modify a
chain’s conguration; but for semicrystalline polymers chain di-
mensions above the melting point and in dilute solution are closer to one
another than are the dimensions of chains below the melting point
(where they will be partially contained in crystallites) and those in dilute
solution. In addition to polypropylene,
∂
n/
∂
c has been found to be in-
dependent of tacticity for other polymers which can be produced in
atactic and stereoregular congurations [18,19,20].
We suggest the following approximation be made in conjunction
with the additivity of polymer and solvent volumes approximation
vP≅1
ρ
P,atactic
(4)
Here
ρ
P,atactic
represents the density of the atactic stereoisomer of the
polymer regardless of the actual tacticity of the sample.
Where sufcient information is available,
∂
n/
∂
c values calculated
with Eq. (3) are in approximate agreement with experiment. However,
the utility of this expression is restricted by the limited availability of the
data underlying the calculation. Eq. (3) requires that the polymer’s
density and the refractive indexes of both the solvent and of the amor-
phous polymer be available at the temperature of interest. PVT data for
many polymers can be found in the literature [15,25]. However, nding
reliable refractive index information is much more difcult, especially at
multiple temperatures. In order to overcome this limitation we will
recast Eq. (3) by eliminating the refractive indexes in favor of the molar
refractions appearing in Eq. (1). The approximate additivity of molar
refraction contributions for atoms and structural elements within a
molecule is the crux of the method described here.
The effective molar mass for a solution of polymer or copolymer in a
solvent is given by Eq. (5)
2
M=x〈MP〉 + (1−x)MS=x〈N〉MM+ (1−x)MS(5)
where “x” is the mole fraction of polymer in the solution, 〈MP〉is the
polymer’s average molar mass, M
M
is the molecular weight of the
polymer’s repeat unit, 〈N〉is the polymer’s average degree of polymer-
ization, and M
S
is the effective molecular weight of the solvent. The
effective molar refraction for the solution can be calculated from Eq. (6)
RGD =xRGD,P+ (1−x)RGD,S=x〈N〉RGD,M+ (1−x)RGD,S(6)
where we have used the approximations of Eqs. (7) and (8) for long
chain lengths
RGD,P= 〈N〉RGD,M(7)
〈MP〉 = 〈N〉MM(8)
R
GD,S
is the molar refraction of the solvent and R
GD,M
is the molar
refraction of the polymer repeat unit. Further, the mass density of the
solution can be estimated by conjecturing that the volumes of solvent
and polymer are approximately additive (see Eq. (9))
1
ρ
=
ω
ρ
P
+1−
ω
ρ
S
(9)
This assumption generally has little effect on the uncertainty of the
∂
n/
∂
c approximation [26]. Eq. (10) and Eq. (11) are denitions relating
the mole fraction, weight fraction, and mass concentration of the poly-
mer in solution to one another.
x=
ω
〈MP〉
ω
〈MP〉+1−
ω
MS
=
ω
〈N〉MM
ω
〈N〉MM+1−
ω
MS
(10)
c=
ω ρ
(11)
Substituting these into Eq. (9) we arrive at
ρ
=(
ρ
P−
ρ
S)c+
ρ
P
ρ
S
ρ
P
(12)
And nally, substitution of the equations above into Eq. (1) gives
n=
ρ
PMSRGD,M−
ρ
SMMRGD,Sc+
ρ
PMM(MS+
ρ
SRGD,S)
ρ
PMMMS
(13)
Differentiating Eq. (13) with respect to the polymer’s concentration,
c, then yields

∂
n
∂
cGD
=
ρ
PMSRGD,M−
ρ
SMMRGD,S
ρ
PMMMS
(14)
Eq. (14) is more useful for calculations since the refractive indices of
the polymer and solvent have been eliminated in favor of the molar
refractions of these species. Recall that molar refractions are approxi-
mately constant.
The Lorentz-Lorenz relationship can also be used as a starting point
for developing a practical model for
∂
n/
∂
c.
3
n2−1
n2+2M
ρ
=RLL (15)
The quantity R
LL
is also referred to as a molar refraction. As with R
GD
,
R
LL
is approximately constant. R
LL
can also be estimated from additive
elemental and structural contributions for the compound in question
[13,27,28]. Table 2 lists some reported atomic and structural contri-
bution to the molar refraction, R
LL
. Because the molar refraction is
almost independent of temperature we can estimate the refractive index
of any polymer, solvent, or solution so long as we can estimate the
appropriate densities at the temperature of interest. The molar re-
fractions listed in Table 2 are best estimates for general use. However,
Table 1
Reported
∂
n/
∂
c values for isotactic and atactic polypropylene for the solvents 1-
chloronapthalene and 1,2,4-trichlorobenzene.
Solvent λ
(nm)
T
(◦C)
∂
n/
∂
c (cm
3
/
gm)
Reference
1-chloronaphthalene 546 140 isotactic −0.188 [21,22]
1-chloronaphthalene 546 135 isotactic −0.184 [23]
1-chloronaphthalene 546 135 isotactic −0.184 [24]
1-chloronaphthalene 546 135 atactic −0.188 [23]
1-chloronaphthalene 546 135 atactic −0.188 [24]
1,2,4-
trichlorobenzene
546 135 isotactic −0.121 [23]
1,2,4-
trichlorobenzene
546 135 atactic −0.111 [24]
1,2,4-
trichlorobenzene
546 135 atactic −0.123 [23]
2
In the developments that follow we will only consider mixed solvents
composed of two pure components and copolymers containing two distinct
repeat units. The extension of the methods developed here to multiple pure
solvents and multiple repeat units is straightforward.
3
The refractive index is known to be a function of the wavelength, λ. Most of
the available data used in the development of the models in this paper were
derived from n
D
values (λ
D
≈589 nm). Therefore, the formulae for
∂
n/
∂
c
developed here represent
∂
n
D
/
∂
c most closely. Throughout the rest of this paper
we will drop the “D” line subscript and assume that it is understood.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
4
they do not account for anomalous effects like optical exaltation.
Whenever possible, we will determine the molar refraction of a sub-
stance (solvent or monomer) directly from density and refractive index
data. If that substance is a monomer that appears as a repeat unit in a
polymer chain, then we will adjust the molar refraction for the repeat
unit using the information in Table 2 for R
LL
or from a similar compi-
lation for R
GD
[13].
Rearrangement of Eq. (15) to isolate n
2
gives
n2=M+2
ρ
RLL
M−
ρ
RLL
(16)
Eqs. (5) through (12) can then be substituted into Eq. (16), and, after
some manipulation
n2=2
ρ
PMSRLL,M−
ρ
SMMRLL,Sc+
ρ
PMM(MS+2
ρ
SRLL,S)

ρ
SMMRLL,S−
ρ
PMSRLL,Mc+
ρ
PMM(MS−
ρ
SRLL,S)(17)
Eq. (17) can be differentiated to yield (
∂
n/
∂
c)|
T,P,c
Finally, substituting c =0 into Eq. (18) we arrive at the nal
expression for the zero concentration value for the refractive index
increment,
∂
n/
∂
c
∂
n
∂
c


T,P,c=0
=
∂
n
∂
cLL
=3MS(
ρ
PMSRLL,M−
ρ
SMMRLL,S)
2
ρ
PMMMS−
ρ
SRLL,S2
MS+2
ρ
SRLL,S
MS−
ρ
SRLL,S
(19)
Eq. (19) can also be rewritten in terms of the refractive indices of the
solvent and of the polymer with the aid of Eqs. (20) and (21)
nS2−1
nS2+2MS
ρ
S=RLL,S(20)
nP2−1
nP2+2MM
ρ
P=RLL,M(21)
∂
n
∂
c


T,P,c=0
=
∂
n
∂
cLL
=(nP2−nS2)(nS2+2)
2
ρ
PnS(nP2+2)(22)
However, it is Eq. (19) which will be the most useful for approxi-
mating
∂
n/
∂
c.
Eq. (22) can be rearranged to yield an approximate relationship
between the values of
∂
n/
∂
c calculated via the Gladstone-Dale and
Lorentz-Lorenz equations:

∂
n
∂
cLL
=
∂
n
∂
cGD(nP+nS)(n2
S+2)
2nS(n2
P+2)(23)
Eq. (23) is approximate since it is a result of manipulating the
Lorentz-Lorenz equation without any reference to the Gladstone-Dale
equation. Fig. 2 is a contour plot of the ratio (
∂
n/
∂
c)
LL
/(
∂
n/
∂
c)
GD
versus n
P
and n
S
. For typical values of n
P
and n
S
, the maximum variation
in the ratio of the (
∂
n/
∂
c)
LL
/(
∂
n/
∂
c)
GD
is approximately ±10 %. Usually,
the observed variance between the two
∂
n/
∂
c estimates is smaller than
that given by Eq. (23).
3. Estimation of pure and mixed solvent mass densities
Eq. (19) requires the density of the solvent, be it a single or a mixed
Table 2
Molar refraction group and structural contributions for use in the Lorentz-Lorenz
equation. Table 5.20 taken from Lange’s Handbook of Chemistry, 15th edition, J.
Dean editor, page 5.136. Used with permission from McGraw-Hill Education.
TABLE 5.20 Atomic and Group Refractions
Group MrD Group MrD
H 1.100 N (primary aliphatic amine) 2.322
C 2.418 N (sec-aliphatic amine) 2.499
Double bond (C
–
–
C) 1.733 N (tert-aliphatic amine) 2.840
Triple bond (C
–
–
–
C) 2.398 N (primary aromatic amine) 3.21
Phenyl (C
6
H
5
) 25.463 N (sec-aromatic amine) 3.59
Naphthyl (C
10
H
7
) 43.00 N (tert-aromatic amine) 4.36
O (carbonyl) (C
–
–
O) 2.211 N (primary amide) 2.65
O (hydroxyl) (O
–
H) 1.525 N (sec amide) 2.27
O (ether, ester) (C
–
O
–
) 1.643 N (tert amide) 2.71
F (one fluoride) 0.95 N (imidine) 3.776
(polyfluorides) 1.1 N (oximido) 3.901
Cl 5.967 N (carbimido) 4.10
Br 8.865 N (hydrazone) 3.46
I 13.900 N (hydroxylamine) 2.48
S (thiocarbonyl)(C
–
–
S) 7.97 N (hydrazine) 2.47
S (thiol) (S
–
H) 7.69 N (aliphatic cyanide) (C
–
–
–
N) 3.05
S (dithia) (
–
S
–
S
–
) 8.11 N (aromatic cyanide) 3.79
Se (alkyl selenides) 11.17 N (aliphatic oxime) 3.93
3-membered ring 0.71 NO (nitroso) 5.91
4-membered ring 0.48 NO (nitrosoamine) 5.37
NO
2
(alkyl nitrate) 7.59
(alkyl nitrite) 7.44
(aliphatic nitro) 6.72
(aromatic nitro) 7.30
(nitramine) 7.51
Fig. 2. Factor relating (
∂
n/
∂
c)
LL
to (
∂
n/
∂
c)
GD
from Eq. (23) for solvent and
polymer refractive indexes ranging from 1.30 to 1.60. The gure demonstrates
that the difference between the two
∂
n/
∂
c estimates can be up to 10 % for
typical values of the indices of refraction for the solvent and the polymer.
∂
n
∂
c


T,P,c
=3
ρ
PMMMS(
ρ
PMSRLL,M−
ρ
SMMRLL,S)
2
ρ
SMMRLL,S−
ρ
PMSRLL,Mc+
ρ
PMMMS−
ρ
SRLL,S2
2(
ρ
PMSRLL,M−
ρ
SMMRLL,S)c+
ρ
PMM(MS+2
ρ
SRLL,S)
(
ρ
SMMRLL,S−
ρ
PMSRLL,M)c+
ρ
PMM(MS−
ρ
SRLL,S)
(18)
B.F. Hanley

Materials Today Communications 25 (2020) 101644
5
solvent. At least for single solvents this information can be readily
approximated with standalone software for physical properties estima-
tion or with Microsoft Excel add-ins which do the same.
4
The calcula-
tions below have all been performed with the Aspen Properties Excel
add-in (v9) because of its ease of use coupled with its large selection
of components and methods for both pure and mixed solvents.
Each Aspen Properties method has associated with it a number of
sub-methods. These sub-methods supplement the top method’s vapor/
liquid equilibrium calculations with liquid enthalpy, vapor molar vol-
ume, and liquid molar volume calculations, to name just three. Here we
are interested in the available liquid molar volume sub-methods and
their associated mixing rules. For demonstration purposes in this paper
we will focus on two of these sub-methods. The default molar volume
submethod for many method selections is the Rackett equation with its
associated mixing rules [29,30]. Generally the Rackett equation works
quite well for pure liquids and is often quite good for liquid mixtures.
Rackett’s original equation is given by
V=RTc
PcZ1+(1−Tr)2/7
c
where T
c
is the component’s critical temperature, P
c
it’s critical pres-
sure, and Z
c
it’s critical compressibility. Spencer and Danner modied
Rackett’s original equation by replacing Z
c
with a tted parameter,
usually called Z
RA
[31]. Both versions of the Rackett equation have been
extended to liquid mixtures through a set of mixing applied to the
various critical properties and the Z
RA
parameters.
Occasionally, a better molar volume sub-method with mixing rules is
available. In this paper we will only consider the COSTALD sub-method
of Hankinson and Thomson as an alternative to the Rackett sub-method
unless otherwise noted [32]. The COSTALD density method is a
Fig. 3. a) experimental densities of aqueous methanol solutions at 20
◦C [33] compared to predictions from volume additivity and from the Rackett and COSTALD
equations from Aspen Properties (v9), b) experimental densities of aqueous ethanol solutions at 20◦C [33] compared to predictions from volume additivity and from
the Rackett and COSTALD equations from Aspen Properties (v9), c) experimental densities of npentane/nheptane mixtures at 20◦C [34] compared to predictions
using the Rackett and COSTALD equations from Aspen Properties (v9), d) experimental densities of aqueous glycerol solutions at 20◦C [33] compared to predictions
from volume additivity and from the Rackett and COSTALD equations from Aspen Properties (v9). The master method in all cases was NRTL.
4
See, for example: https://www.aiche.org/dippr/eventsproducts/80
1database, https://www.aspentech.com/en/applications/engineering/physica
lpropertyestimation, https://www.prode.com/en/ppp.htm, http://gpengin
eeringsoft.com/pages/chemical_physical_properties.html.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
6
corresponding states correlation of the form
V
V∗=V(o)
R1−
ω
SRK V(δ)
R
Where V is the molar volume of the liquid, V* is a characteristic molar
volume analogous to the critical volume, V
R
(o)
is the so-called spherical
molecule function, and V
R
(δ)
is a function which, when multiplied by V
R
(o)
results in the deviation function, and
ω
SRK
is the acentric factor based on
the Soave-Redlich-Kwong equation. V
R
(o)
and V
R
(δ)
are both functions of
reduced temperature only:
V(o)
R=1+a(1−TR)1/3+b(1−TR)2/3+c(1−TR) + d(1−TR)4/3
V(δ)
R=e+fTR+gT2
R+hT3
R
TR−1.00001
a, b, c, d, e, f, g, and h are xed constants. V* is regressed by tting these
equations to experimental density data using known values of T
c
and
ω
SRK
. Some rather complex mixing rules allow for the calculation of the
molar volumes of liquid mixtures. The interested reader is referred to the
reference above.
Mixed solvents are also important in the characterization of homo-
and co- polymers. The assumption that the volumes of the individual
solvents in the mixture are additive is often unfounded. Consider Fig. 3
in which the performance of the volume additivity assumption, the
Rackett equation and that of the COSTALD molar volume sub-method
are contrasted for four liquid mixtures. For the Aspen Properties calcu-
lations the NRTL property method was chosen and then either the
Rackett or COSTALD molar volume sub-method selected for solution
density predictions. Fig. 3a is a comparison of experimentally meaured
solution densities for methanol/water mixtures at 20 ◦C, 1 atm pressure
versus methanol mole fraction [33] with density predictions via volume
additivity and from the Aspen Properties add-in for Excel using the NRTL
property method in conjunction with either the Rackett or COSTALD
molar volume sub-method. Clearly the COSTALD molar volume
sub-method comes closer to matching the data than does the Rackett
sub-method or volume additivity. Fig. 3b is a similar comparison for the
system ethanol/water at 20◦C, 1 atm. [33] In this case, none of the three
models matches the data well over the entire composition range; how-
ever, it is the Rackett molar volume sub-method which comes closer to
matching the data overall. Fig. 3c is a plot of the mass density of
n-pentane/nheptane mixtures versus npentane mole fraction at 20◦C, 1
atm [34]. Because these compounds are so similar structurally, the
assumption of volume additivity would be expected to work well. In
addition, we see that both the Rackett and COSTALD molar volume
sub-methods perform equally well here. Finally, mass density data for
glycerol/water mixtures at 20◦C, 1 atm pressure are presented in Fig. 3d
[33]. Neither the Rackett nor the COSTALD molar volume sub-methods
reproduce the experimental data paricularly well; however, the volume
additivity assumption comes close to matching the data over the entire
composition range. Other method and molar volume sub-method
choices are available; in addition, though not discussed in this paper,
adjustment of the model parameters associated with the Rackett and
COSTALD density models is also a possible remedy to improve their
predictive capabilites.
4. Estimation of polymer and copolymer mass densities
Eq. (19) also requires the mass density of the polymer sample for
which
∂
n/
∂
c is sought. The PVT behavior of polymers and copolymers is
much less well-understood and therefore less amenable to mathematical
analysis. Different polymer or copolymer samples with essentially the
same monomeric composition proles but which have been produced by
different kinetic mechanisms can have very different mass densities. The
same uncertainty in PVT behavior can also arise if the samples have
experienced different thermomechanical histories during processing.
The degree of sample crystallinity has a pronounced effect its mass
density, for example. Crystallinity in turn can be affected by the tacticity
of the polymer and tacticity differences generally arise from specic
kinetic mechanisms in the sample’s polymerization. Sample cooling rate
Fig. 4. Typical PVT-behavior of: a) an amorphous and b) a semi-crystalline polymer. Data are reproduced from ref. [15].
B.F. Hanley

Materials Today Communications 25 (2020) 101644
7
is also known to have a pronounced effect on polymer/copolymer PVT
behavior via its effect on crystallization near the glass transition. Fig. 4
illustrates some of the complex behavior described above for samples of
an amorphous and semi-crystalline polypropylene. There are two major
implications that we can draw from these data. First, in addition to
temperature and pressure, other parameters – cooling rate, deformation
history, etc. – need to be considered in order to characterize a material.
Second, and most importantly for this paper,
∂
n/
∂
c will not necessarily
be a unique function of temperature and, for copolymers, composition,
in a given solvent. Instead, the
∂
n/
∂
c function will also contain de-
pendencies on localized sample geometrical features as well as pro-
cessing history. We will therefore treat the temperature dependence of
the polymer’s mass density as a function that has been regressed from
experimental data. Recall that the appropriate density function for the
calculation of
∂
n/
∂
c is that of the amorphous polymer.
5. Calculation of the molar refraction for solvents and polymers
In the remainder of this paper, we will demonstrate some methods
for calculating
∂
n/
∂
c from molar refraction information for the mixture.
The approach below will be based upon Eq. (19) and Table 2 for the
Lorentz-Lorenz equation. Extension to the Gladstone-Dale expression of
Eq. (14) is straightforward so long as one has available atomic/struc-
tural/group contribution values to the Gladstone-Dale molar refraction
(see, for example, ref. [13]).
5.1. R
LL,S
– the molar refraction of the solvent
The most accurate value of the molar refraction for the solvent, R
LL,S
,
can be obtained from refractive index/mass density information avail-
able in the literature. If the refractive index and mass density are known
at the same temperature, then R
LL,S
can be calculated directly from Eq.
(15). For mixed solvents, the average molecular weight and the average
molar refraction can be calculated from the composition of the polymer-
free solvent
MS=x1,SM1,S+ (1−x1,S)M2,S(24)
RLL,S=x1,SRLL,1S+ (1−x1,S)RLL,2S(25)
x
1,S
is the mole fraction of solvent 1, M
1,S
is the molecular weight of
solvent 1, M
2,S
is the molecular weight of solvent 2, R
LL,1S
is the molar
refraction of solvent 1, and R
LL,2S
is the molar refraction of solvent 2.
5.2. R
LL,M
– the molar refraction of the polymer’s effective repeat unit
Eq. (19) should not only apply to homopolymers in single solvents; it
should also be valid for homopolymers or copolymers in pure or mixed
solvents. Additional information about the copolymer composition
would also be required in these cases.
MM=x1,PM1,M+ (1−x1,P)M2,M(26)
RLL,M=x1,PRLL,1M+ (1−x1,P)RLL,2M(27)
x
1,P
is the mole fraction of repeat unit 1 in the polymer, M
1,M
is the
molecular weight of repeat unit 1, M
2,M
is the molecular weight of repeat
unit 2, R
LL,1M
is the molar refraction of repeat unit 1, and R
LL,2M
is the
molar refraction of repeat unit 2.
The molar refraction for the effective monomer repeat unit in the
polymer can be calculated in a number of ways. As in the case of the
solvent, if the refractive index and mass density of the polymer are
known at the same temperature, then R
LL,M
can be calculated directly
from Eq. (15) using the molecular weight of the repeat unit. Oftentimes,
however, reliable information for one or both of these quantities is
lacking.
In order to arrive at an unambiguous estimate for R
LL,M
we suggest
here two potential methods. In the rst R
LL,M
can be approximated from
the group/structural contributions found in Table 2. This is relatively
straightforward. The second method for estimate for R
LL,M
is based on
the fact that refractive index/density data are generally readily available
for liquids, at least at one temperature.
Let us rst consider homopolymers produced either by chain growth
or step growth polymerization. If the necessary information to calculate
R
LL,S
for the monomer(s) is available, then R
LL,M
can be estimated by
removing from R
LL,S
the appropriate group contributions from Table 2.
For example, consider the chain growth polymerization of styrene
monomer to produce polystyrene (Table 3).
Sometimes the monomer(s) appearing in the repeat unit of the
polymer can be solids or powders under nominal conditions. It is much
more difcult to obtain accurate density and refractive index informa-
tion for these types of monomers. Consider, for example, the production
of poly(ethylene terephthalate) from terephthalic acid and ethylene
glycol:
nC8H6O4+nC2H6O2→(C10H8O4)n+2nH2O
Theoretically, the best way to get the molar refraction for the PETE
repeat unit would be to sum the molar refractions of terephthalic acid
and ethylene glycol and then subtract twice the molar refraction of
water from that total. In turn, the molar refractions for each compound
would be best estimated from refractive index and density information
for that compound. Molar refraction information for ethylene glycol and
water are immediately available since it is easy to nd the necessary
information for both of them. Terephthalic acid, on the other hand, is a
solid that is most often available as crystals or as a powder. Its bulk
density and refractive index are therefore difcult to get hold of. The
molar refraction for terephthalic acid could be estimated via the atomic
and group contributions to the molar refraction found in Table 2, but we
suggest here an alternate method which uses the Table 2 information to
a much lesser degree.
We note that the repeat unit in PETE can be broken up into three
separate molecules that exist in the liquid phase at standard conditions:
formic acid, benzene, and ethyl formate (see Table 4). Information about
their refractive indices and densities are much more available; hence so
are their molar refractions. An improved estimate for the molar refrac-
tion of the PETE repeat unit can be found by summing the molar re-
fractions for formic acid, benzene, and ethyl formate, and then adjusting
this total slightly by using Table 2 to subtract out the six excess hydro-
gens. This method of breaking up the repeat unit of a polymer into
molecules that exist in the liquid phase and for which density and
refractive index information are available can be applied more widely.
Further examples are given for Kevlar, Nylon 66, and bisphenol A
Table 3
Molar refraction calculations for the styrene repeat unit in polystyrene based on
the molar refraction of styrene monomer adjusted by the reported C
–
–
C bond
contribution from Table 2. Note the difference between the values of R
LL,M
estimated from group/structural/bond contributions and directly from the sty-
rene monomer itself with the small adjustment for the C
–
–
C bond.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
8
polycarbonate (Tables 5–7). 6. Comparison to experiment
Refractive index increment data are available in a number of hand-
books; [23,35] in addition, they are often reported in papers dealing
Table 4
Molar refraction calculations for the (ethylene terephthalate) repeat unit in PETE based on the molar refractions of formic acid, benzene, and ethyl formate adjusted by
the reported atomic contribution of hydrogen from Table 2. Note the rather close agreement between the values of R
LL,M
calculated from group/structural/bond
contributions and directly from formic acid, benzene and ethyl acetate, less the contributions of six hydrogens.
Table 5
Molar refraction calculations for the Kevlar repeat unit based on the molar refractions of aniline, formamide, and benzoic acid adjusted by the reported group/
structural/atomic contributions to the molar refraction found in Table 2.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
9
with light scattering for average molar mass determination or size
exclusion chromatography using a refractive index detector for the
quantication of a polymer sample’s molar mass distribution [36,37,
38].
Fig. 5 is a parity plot comparing measured values of
∂
n/
∂
c with
predictions made with the model developed here based on the Lorentz-
Lorenz equation (Eq. (19)) and with the Gladstone – Dale equation (Eq.
(14)). The dataset included measurements for mixed solvents and for
Table 6
Molar refraction calculations for the Nylon 66 repeat unit based on the molar refractions of propionic acid and propylamine adjusted by the reported group/structural/
atomic contributions to the molar refraction found in Table 2.
Table 7
Molar refraction calculations for the bisphenol A polycarbonate repeat unit based on the molar refractions of water, benzene, cumene, and formic acid adjusted by the
reported group/structural/atomic contributions to the molar refraction found in Table 2.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
10
copolymers. Pure and mixed solvent densities were calculated using the
Aspen Properties (v9) add-in for Excel. Generally, the default Rackett
equation sub-method was retained for molar volume estimates though
the COSTALD sub-method would be chosen if it gave better overall
agreement. Nominal polymer densities were obtained from data re-
ported in a number of sources [15,23,39–42]. Copolymer densities were
taken to be the weight fraction averaged densities of the two homo-
polymers, regardless of the polymer’s composition distribution. The
average absolute deviations for both (
∂
n/
∂
c)
GD
and (
∂
n/
∂
c)
LL
were found
to be approximately equal with a nominal value of 0.009 ml/gm. No
account was taken of the wavelength dependence of
∂
n/
∂
c in these
comparisons.
Fig. 6 is a comparison of experimental
∂
n/
∂
c values for poly(styrene-
co-MMA) copolymers in THF at 25◦C versus styrene mole fraction with
predictions made with Eq. (14) and with Eq. (19) [43]. Eq. (19) slightly
outperforms the Eq. (14) in this case. Fig. 7 is a comparison of experi-
mental
∂
n/
∂
c values for polystyrene in benzene/methanol mixed sol-
vents at 25◦C versus mole fraction of benzene in the solvent mixture
with (
∂
n/
∂
c)
LL
estimates for the Rackett and COSTALD molar volume
models as well as with results from the volume additivity assumption
[44]. Included in the gure are solution density data for benzene/me-
thanol mixtures at 25◦C [45]. There is virtually no change in the total
solution volume on mixing for this system. In addition, for the range of
mixed solvent compositions examined here, no effect of preferential
solvent adsorption is apparent in either the reported or calculated
∂
n/
∂
c
values or their trend with mixed solvent composition. Fig. 8 illustrates
the importance of the choice of molar volume sub-method [5] and the
mixing rules associated with that sub-method when making
∂
n/
∂
c cal-
culations with mixed solvents. On the left are data and calculations for
the system poly(methyl methacrylate) (PMMA) in the mixed solvent
bromoform/i-propanol at 25◦C. It is clear from the solvent density
Fig. 5. Parity plot comparing measured values of
∂
n/
∂
c with predictions from
the model based on the Lorentz-Lorenz equation developed in this paper (Eq.
(19)) and from the Gladstone – Dale equation (Eq. (3)). The majority of the data
were collected from refs. [23] and [35]. Data and sources are summarized at the
end of this paper.
Fig. 6. Comparison of experimental
∂
n/
∂
c values for poly(styrene-co-MMA)
copolymers in THF at 25◦C versus styrene mole fraction with predictions
made with Eq. (14) (GladstoneDale) and with Eq. (19) (Lorentz-Lorenz). Note
that the experimental data were taken with a 685 nm light source while the
calculated
∂
n/
∂
c are for a 589.6 nm light source.
Fig. 7. (top) Comparison of experimental
∂
n/
∂
c values for polystyrene in
benzene/methanol mixed solvents at 25◦C versus volume fraction of benzene in
the solvent mixture.
∂
n/
∂
c estimate using Eq. (19) with the assumption of
volume additivity for the solvent is shown in red. Also shown are
∂
n/
∂
c esti-
mates made with Eq. (19) for the Rackett and COSTALD molar volume models
available in Aspen Properties. (bottom) Comparison of solution density esti-
mates from the Rackett, COSTALD, and volume additivity models with exper-
imental data.
B.F. Hanley

Materials Today Communications 25 (2020) 101644
11
calculations for this mixed solvent combination that volume additivity
for the solvents is a good assumption. On the right are data and calcu-
lations for PMMA in the mixed solvent dioxane/water at 25◦C. The
solvent density calculations for this mixed solvent system are markedly
dependent upon the choice of solvent volume model. We see from the
∂
n/
∂
c results that the Rackett molar volume model yields values for
(
∂
n/
∂
c)
LL
that are in good agreement with experiment.
7. Conclusions
Models for the estimation of the refractive index increments, (
∂
n/
∂
c)
GD
and (
∂
n/
∂
c)
LL
, for polymers and copolymers in pure as well as
mixed solvents have been developed by differentiating the Gladstone-
Dale and the Lorentz-Lorenz equations with respect to polymer/copol-
ymer concentration. Volume additivity between polymer and solvent
was assumed in the derivation. The models make use of the fact that the
molar refraction for a pure substance is nearly constant and can be
estimated from atomic, group, and structural contributions, nominal
values of which have been tabulated. In addition, implementation of the
models for this paper included the use of the Aspen Properties add-in for
Excel. The add-in was used in this study to estimate the density of pure
solvents and solvent mixtures. The add-in gives the user exibility in
choosing the model and mixing rules for these calculations. Here, we
focused on the Rackett and COSTALD molar volume sub-methods for
density calculations. Extensive comparison of the models developed
here with reported
∂
n/
∂
c values demonstrate their applicability over a
wide range of conditions. The calculations in this paper were performed
with nominal values for the densities of the various polymer types. In
practice, the experimenter should measure the polymer sample’s den-
sity. In addition, the experimenter should have solvent density data
available for mixed solvent systems so that an appropriate molar volume
model can be chosen.
Declaration of Competing Interest
The authors report no declarations of interest.
Fig. 8. (leftmost gures) Comparison of experimental
∂
n/
∂
c values for PMMA in bromoform/i-propanol mixed solvents at 25◦C versus volume fraction of bromoform
in the solvent mixture.
∂
n/
∂
c estimates made with Eq. (14) (Gladstone-Dale) and Eq. (19) (Lorentz-Lorenz) for the Rackett molar volume model are shown. Also
shown in red is
∂
n/
∂
c estimated using Eq. (19) with the assumption of volume additivity for the solvent. (rightmost gures) Comparison of experimental
∂
n/
∂
c values
for PMMA in dioxane/water mixed solvents at 25◦C versus volume fraction of dioxane in the solvent mixture.
∂
n/
∂
c estimates made with Eq. (14) (Gladstone-Dale)
and Eq. (19) (Lorentz-Lorenz) for the Rackett molar volume model are shown. Also shown in red is
∂
n/
∂
c estimated using Eq. (19) with the assumption of volume
additivity for the solvent. Note that the predicted values for
∂
n/
∂
c fall within the range of experimental uncertainty. Data taken from ref. [23].
B.F. Hanley

Materials Today Communications 25 (2020) 101644
12
Appendix A. Derivation of the Gladstone–Dale Equation
∂
n/
∂
c
(n−1)M
ρ
=RGD
n=
ρ
PMSRGD,M−
ρ
SMMRGD,Sc+
ρ
PMMMS+
ρ
SRGD,S
ρ
PMMMS
∂
n
∂
c=
ρ
PMSRGD,M−
ρ
SMMRGD,S
ρ
PMMMS
nS−1MS
ρ
S=RGD,S
nP−1MM
ρ
P=RGD,M
∂
n
∂
c=nP−nS
ρ
P
Appendix B. Parity Plot Data
Polymer Solvent T (◦C) λ (nm) (
∂
n/
∂
c)
exp
(
∂
n/
∂
c)
LL
(
∂
n/
∂
c)
GD
Source
a
bisphenol A polycarbonate THF 23 0.177 0.1635 0.1721 http://www.ampolymer.com/dn-dc.
html
bisphenol A polycarbonate CCl4 23 0.1445 0.1371 0.1416 http://www.ampolymer.com/dn-dc.
html
nylon 12 m-cresol 25 546 −0.02 −0.0142 −0.0384 698
nylon 12 HFIP 30 632.8 0.2212 0.2359 0.2327 SEC, Mori Appendix IV
nylon 6 dichloroacetic acid 25 546 0.098 0.0883 0.0806 309
nylon 6 dichloroacetic acid 50 546 0.099 0.0929 0.0853 309
nylon 6 dichloroacetic acid 80 546 0.104 0.0988 0.0914 309
nylon 6 2,2,2 triuoroethanol 25 436 0.236 0.2250 0.2362 870
nylon 66 HFIP 25 0.241 0.1981 0.2176 http://www.ampolymer.com/dn-dc.
html
nylon 66 2,2,2 triuoroethanol 25 436 0.228 0.1878 0.2056 870
polyacrylonitrile DMSO/acrylonitrile(73/27 v/v) 25 546 0.055 0.0567 0.0583 726
polyacrylonitrile DMSO/acrylonitrile(65/35 v/v) 25 546 0.061 0.0622 0.0640 726
polyacrylonitrile DMSO/acrylonitrile(91/9 v/v) 25 546 0.04 0.0450 0.0461 726
poly(1,2 butadiene) cyclohexane 25 546 0.087 0.1056 0.1096 64
poly(1,2 butadiene) n-heptane 20? 546 0.141 0.1443 0.1516 61
poly(1,2 butadiene) THF 23? 546 0.132 0.1245 0.1301 64
poly(1,2 butadiene) dioxane 25 546 0.11 0.1096 0.1140 RIID 14
poly(1,2 butadiene) toluene 0.04 0.0297 0.0301 RIID 14
poly(tert-butyl acrylate) MEK 25 546 0.0818 0.0885 0.0917 90
poly(cis-isoprene) THF 25 546 0.148 0.1487 0.1480 655
poly(cis-isoprene) CCl4 25 632.8 0.082 0.0960 0.0906 964
poly(cis-isoprene) CCl4 25 546 0.095 0.1108 0.1065 68
poly(cis-isoprene) n-hexane 25 546 0.191 0.1808 0.1838 655
poly(cyclohexyl methacrylate) cyclohexane 25 546 0.0845 0.0872 0.0867 115
poly(cyclohexyl methacrylate) toluene 25 436 0.019 0.0275 0.0239 911
poly(epsilon caprolactone) MEK 25 488 0.108 0.1086 0.1112 RIID 144
poly(epsilon caprolactone) THF 25 488 0.079 0.0851 0.0859 RIID 144
poly(epsilon caprolactone) ethyl acetate 25 488 0.11 0.1133 0.1164 RIID 144
poly(ethyl acrylate) acetone 25 546 0.1107 0.0887 0.0949 92
poly(ethyl acrylate) CCl4 25 546 0.0363 0.0148 0.0178 RIID 14
poly(ethyl acrylate) DMF 25 546 0.032 0.0275 0.0307 RIID 14
poly(ethyl acrylate) ethyl acetate 25 546 0.0916 0.0771 0.0825 RIID 14
poly(ethyl acrylate) MEK 30 488 0.088 0.0733 0.0784 RIID 14
poly(ethyl acrylate) MEK 20 644 0.0852 0.0711 0.0762 RIID 14
poly(ethyl acrylate) water 25 0.131 0.1084 0.1163 RIID 14
poly(methyl acrylate) acetonitrile 25 546 0.118 0.1033 0.1110 96
poly(methyl acrylate) ethyl acetate 25 546 0.0967 0.0810 0.0871 96
poly(methyl methacrylate) acetone 20 436 0.132 0.1254 0.1289 89
poly(methyl methacrylate) acetone 27 436 0.107 0.1280 0.1315 131
poly(methyl methacrylate) benzene 20 436 0.001 0.0160 0.0123 6
poly(methyl methacrylate) benzene 20 546 0.007 0.0160 0.0123 6
poly(methyl methacrylate) n-butyl chloride 20 546 0.0934 0.0937 0.0940 143
(continued on next page)
B.F. Hanley

Materials Today Communications 25 (2020) 101644
13
(continued)
Polymer Solvent T (◦C) λ (nm) (
∂
n/
∂
c)
exp
(
∂
n/
∂
c)
LL
(
∂
n/
∂
c)
GD
Source
a
poly(methyl methacrylate) n-butyl chloride 25 546 0.0875 0.0956 0.0958 150
poly(methyl methacrylate) n-butyl acetate/CCl4 (20/80 v/v) 25 0.063 0.0684 0.0666 802
poly(methyl methacrylate) n-butyl acetate/CCl4 (40/60 v/v) 25 0.07 0.0748 0.0736 802
poly(methyl methacrylate) n-butyl acetate/CCl4 (80/20 v/v) 25 0.083 0.0896 0.0896 802
poly(methyl methacrylate) n-butyl acetate/CCl4 (90/10 v/v) 25 0.094 0.0938 0.0941 802
poly(methyl methacrylate) chlorobenzene 25 −0.01 −0.0016 −0.0057 681???
poly(methyl methacrylate) CCl4 25 0.059 0.0620 0.0597 802
poly(methyl methacrylate) cyclohexanone 30 0.046 0.0598 0.0572 729??
poly(methyl methacrylate) DMF 20 0.0617 0.0716 0.0702 5,6
poly(methyl methacrylate) DMF 20 436 0.06 0.0716 0.0702 897
poly(methyl methacrylate) MEK RT 546 0.111 0.1107 0.1134 154
poly(methyl methacrylate) MEK 20 546 0.1112 0.1107 0.1126 5,5,88
poly(methyl methacrylate) THF 25 546 0.098 0.0910 0.0908 637
poly(methyl methacrylate) THF 25 546 0.086 0.0910 0.0908 689
poly(methyl methacrylate) toluene 20 436 0.001 0.0194 0.0158 834
poly(methyl methacrylate) toluene 25 436 0.01 0.0215 0.0176 146
poly(methyl methacrylate) MEK 25 632 0.113 0.1126 0.1145 RIID 158
poly(methyl methacrylate) dioxane 25 546 0.0707 0.0798 0.0787 135,139
poly(methyl methacrylate) 1,2 dichloroethane 30 546 0.05 0.0652 0.0629 826
poly(n-butyl acrylate) acetone 20 436 0.112 0.1043 0.1088 RIID 14
poly(n-butyl acrylate) benzene 30 546 −0.0292 −0.0228 −0.0229 RIID 14
poly(n-butyl acrylate) cyclohexanone 30 488 0.021 0.0244 0.0245 RIID 14
poly(n-butyl acrylate) dioxane 30 488 0.043 0.0505 0.0514 RIID 14
poly(n-butyl acrylate) chlorobenzene 30 546 −0.0525 −0.0473 −0.0467 RIID 14
poly(n-butyl acrylate) n-hexane 30 546 0.0885 0.0937 0.0971 RIID 14
poly(n-butyl acrylate) THF 30 546 0.0651 0.0642 0.0658 RIID 14
poly(n-butyl methacrylate) acetone 25 436 0.1249 0.1158 0.1194 RIID 14
poly(n-vinyl carbazole) benzene 25 546 0.195 0.1904 0.2201 303
poly(n-vinyl carbazole) CCl4 25 546 0.232 0.2280 0.2662 304
poly(n-vinyl carbazole) THF 25 546 0.262 0.2530 0.2978 304
poly(phenyl methacrylate) MEK 25 546 0.182 0.1724 0.1787 115
poly(styrene co-acrylonitrile)
(0.715/0.285 m/m)
THF 23 632.8 0.1546 0.1691 0.1802 SEC, Mori Appendix IV
poly(styrene co-acrylonitrile)
(1/1 m/m)
MEK 25 546 0.198 0.1781 0.1912 753
poly(vinyl acetate) acetone 20 546 0.095 0.0957 0.1011 238
poly(vinyl acetate) MEK 25 546 0.08 0.0812 0.0855 291
poly(vinyl chloride) THF 23 632.8 0.105 0.0916 0.0919 RIID 64
poly(vinyl chloride) THF 16.5 546 0.1017 0.0896 0.0899 SEC, Mori Appendix IV
poly(vinyl chloride) cyclohexanone 25 546 0.077 0.0647 0.0625 SEC, Mori Appendix IV
poly(vinyl pyrrolidone) water 20 546 0.176 0.1804 0.1973 792
polyacrylonitrile DMA 25 546 0.0769 0.0713 0.0739 278
polyacrylonitrile DMF 20 546 0.082 0.0758 0.0789 238
polychloroprene n-butyl acetate 25 436 0.146 0.1415 0.1449 64
polychloroprene CCl4 25 436 0.0976 0.0980 0.0958 64
polychloroprene MEK 25 546 0.156 0.1546 0.1598 64
polychloroprene THF 25 632.8 0.138 0.1348 0.1371 66
polychloroprene CCl4 25 436 0.0976 0.0980 0.0958 64
polydimethyl siloxane Mw =
4000
toluene 25 546 −0.0938 −0.0946 −0.0938 978
polyethylene 1-chloronapthalene 135 546 −0.19 −0.1748 −0.1721 various
polyethylene o-dichlorobenzene 135 632.8 −0.056 −0.0661 −0.0680 RIID 148
polyethylene o-dichlorobenzene 135 633 −0.078 −0.0661 −0.0680 RIID 14
polyethylene tetralin 135 633 −0.074 −0.0653 −0.0653 RIID 14
polyethylene 1-chloronaphthalene 145 632.8 −0.177 −0.1726 −0.1709 SEC, Mori Appendix IV
polyethylene 1,2,4 TCB 135 632.8 −0.104 −0.0950 −0.0958 RIID 147
polyethylene 1-chloronaphthalene 135 632.8 −0.177 −0.1748 −0.1721 RIID 148
polyethylene n-decane 130 546 0.0937 0.0974 0.1024 23
polyethylene n-decane 149 546 0.0995 0.1031 0.1083 23
polyethylene bromobenzene 135 633 −0.083 −0.0749 −0.0769 RIID 14
polyethyleneimine methanol 35 633 0.225 0.2171 0.2098 969
polyethyleneimine water 35 633 0.21 0.2107 0.2025
polyisobutylene cyclohexane 25 546 0.105 0.1001 0.0927 32
polyisobutylene n-heptane 25 0.143 0.1389 0.1347 http://www.ampolymer.com/dn-dc.
html
polyisobutylene n-hexane 25 436 0.155 0.1518 0.1490 38
polyisobutylene THF 30 488 0.125 0.1190 0.1131 RIID 142
poly(1-butene) nonane 35 436 0.092 0.0765 0.0747 13
poly(1-butene) nonane 80 436 0.108 0.0869 0.0851 13
poly(cis-isoprene) CCl4 25 546 0.1 0.1108 0.1065 655
polypropylene 1,2,4 TCB 135 633 −0.102 −0.0958 −0.1054 RIID 14
polypropylene 1,2,4 TCB 145 633 −0.093 −0.0938 −0.1039 867
polypropylene 1-chloronapthalene 135 633 −0.177 −0.1757 −0.1817 867
polypropylene (atactic) 1,2,4 TCB 145 632.8 −0.093 −0.0938 −0.1039 RIID 149
polypropylene glycol Mw =4000 n-hexane 25 546 0.0775 0.0793 0.0980 343
(continued on next page)
B.F. Hanley

Materials Today Communications 25 (2020) 101644
14
(continued)
Polymer Solvent T (◦C) λ (nm) (
∂
n/
∂
c)
exp
(
∂
n/
∂
c)
LL
(
∂
n/
∂
c)
GD
Source
a
polypropylene glycol Mw =4000 n-hexane 57 546 0.101 0.0969 0.1149 343
polypropylene glycol Mw =4000 chlorobenzene 25 546 −0.0638 −0.0701 −0.0521 343
polystyrene acetone/cyclohexane (26.6/73.4 v/v) 25 546 0.192 0.1812 0.1933 783
polystyrene acetone/cyclohexane (38.2/61.8 v/v) 25 546 0.202 0.1874 0.2005 783
polystyrene acetone/cyclohexane (49.1/50.9 v/v) 25 546 0.207 0.1936 0.2076 783
polystyrene acetone/cyclohexane (59.1/40.9 v/v) 25 546 0.214 0.1995 0.2144 783
polystyrene acetone/cyclohexane (68.4/31.6 v/v) 25 546 0.22 0.2051 0.2210 783
polystyrene acetone/cyclohexane (77.2/22.8 v/v) 25 546 0.225 0.2105 0.2274 783
polystyrene benzene 20 546 0.1034 0.1032 0.1059 5
polystyrene benzene 20 589 0.1028 0.1032 0.1059 5
polystyrene benzene/cyclohexane (82.9/17.1 m/m) 20 436 0.137 0.1168 0.1205
polystyrene benzene/cyclohexane (65.3/34.7 m/m) 20 436 0.155 0.1292 0.1340
polystyrene benzene/cyclohexane (44.8/55.2 m/m) 20 436 0.17 0.1423 0.1487
polystyrene benzene/cyclohexane (23.6/76.4 m/m) 20 436 0.183 0.1548 0.1629 208
polystyrene benzene/cyclohexane (10/90 m/m) 20 436 0.189 0.1622 0.1714 208
polystyrene benzene/cyclohexane (4.8/95.2 m/m) 20 436 0.191 0.1649 0.1746 208
polystyrene benzene/cyclohexane (90/10 v/v) 20 546 0.113 0.1103 0.1134 708
polystyrene benzene/cyclohexane (80/20 v/v) 20 546 0.12 0.1168 0.1204 708
polystyrene benzene/cyclohexane (70/30 v/v) 20 546 0.129 0.1232 0.1275 708
polystyrene benzene/cyclohexane (60/40 v/v) 20 546 0.135 0.1297 0.1345 708
polystyrene benzene/cyclohexane (50/50 v/v) 20 546 0.142 0.1360 0.1416 708
polystyrene benzene/cyclohexane (20/80 v/v) 20 546 0.158 0.1549 0.1630 708
polystyrene benzene/cyclohexane (82.9/17.1 m/m) 70 436 0.155 0.1441 0.1468 208
polystyrene benzene/cyclohexane (65.3/34.7 m/m) 70 436 0.172 0.1554 0.1597 208
polystyrene benzene/cyclohexane (44.8/55.2 m/m) 70 436 0.186 0.1674 0.1736 208
polystyrene benzene/cyclohexane (23.6/76.4 m/m) 70 436 0.197 0.1788 0.1871 208
polystyrene benzene/cyclohexane (10/90 m/m) 70 436 0.202 0.1855 0.1952 208
polystyrene benzene/cyclohexanol 81.8/18.2 m/m) 20 436 0.121 0.1104 0.1138 208
polystyrene benzene/cyclohexanol 90.9/9.1 m/m) 20 436 0.119 0.1072 0.1102 208
polystyrene benzene/cyclohexanol 99.628/0.372 m/
m)
20 436 0.124 0.1039 0.1066 208
polystyrene benzene/n-dodecane (96.2/3.8 m/m) 20 436 0.123 0.1168 0.1194 210
polystyrene benzene/n-dodecane (92.7/7.3 m/m) 20 436 0.131 0.1265 0.1294 210
polystyrene benzene/n-dodecane (86.4/13.6 m/m) 20 436 0.14 0.1400 0.1436 210
polystyrene benzene/n-dodecane (96.2/3.8 m/m) 70 436 0.139 0.1436 0.1454 210
polystyrene benzene/n-dodecane (92.7/7.3 m/m) 70 436 0.145 0.1521 0.1544 210
polystyrene benzene/n-dodecane (86.4/13.6 m/m) 70 436 0.149 0.1638 0.1673 210
polystyrene benzene/ethanol (70.4/29.6 v/v) 19.6 436 0.156 0.1484 0.1535 611
polystyrene benzene/ethanol (94.3/5.7 v/v) 20 436 0.115 0.1127 0.1155 210
polystyrene benzene/ethanol (89.1/10.9 v/v) 20 436 0.119 0.1214 0.1245 210
polystyrene benzene/ethanol (62.1/37.9 v/v) 20 436 0.162 0.1589 0.1650 210
polystyrene benzene/n-hexane (88/12 m/m) 20 436 0.141 0.1233 0.1270 210
polystyrene benzene/n-hexane (78.6/21.4 m/m) 20 436 0.16 0.1367 0.1415 210
polystyrene benzene/n-hexane (63.2/36.8 m/m) 20 436 0.188 0.1557 0.1626 210
polystyrene benzene/n-hexane (55.1/44.9 m/m) 20 436 0.197 0.1645 0.1726 210
polystyrene benzene/MEK (25.1/74.9 m/m) 20 436 0.21 0.1828 0.1942 203
polystyrene benzene/MEK (50.2/49.8 m/m) 20 436 0.178 0.1581 0.1654 203
polystyrene benzene/MEK (75.1/24.9 m/m) 20 436 0.145 0.1319 0.1363 203
polystyrene benzene/methanol (70.9/29.1 m/m) 20 436 0.154 0.1303 0.1339 208
polystyrene benzene/methanol (83.9/16.1 m/m) 20 436 0.124 0.1172 0.1202 208
polystyrene benzene/methanol (92.4/7.6 m/m) 20 436 0.114 0.1091 0.1120 208
polystyrene CCl4 20 436 0.156 0.1388 0.1451 102
polystyrene CCl4 30 436 0.1518 0.1537 0.1586 102
polystyrene CCl4 20 436 0.165 0.1510 0.1589 102
polystyrene cyclohexane 20 546 0.1682 0.1675 0.1776 218
polystyrene cyclohexane 50 436 0.188 0.1855 0.1953 223
polystyrene DMF 20 436 0.1739 0.1639 0.1735 5,6
polystyrene ethyl acetate 20 546 0.218 0.2115 0.2287 709
polystyrene MEK 70 546 0.254 0.2296 0.2476 203
polystyrene MEK 20 546 0.2167 0.2066 0.2229 219
polystyrene toluene 20 546 0.109 0.1069 0.1099 238
polystyrene toluene 70 546 0.125 0.1328 0.1345 699
polystyrene cyclohexane/ethyl acetate (10/90 v/v) 20 546 0.216 0.2066 0.2229 709
polystyrene cyclohexane/ethyl acetate (20/80 v/v) 20 546 0.213 0.2022 0.2177 709
polystyrene cyclohexane/ethyl acetate (30/70 v/v) 20 546 0.21 0.1978 0.2126 709
polystyrene cyclohexane/ethyl acetate (40/60 v/v) 20 546 0.206 0.1935 0.2074 709
polystyrene cyclohexane/ethyl acetate (50/50 v/v) 20 546 0.202 0.1891 0.2023 709
polystyrene cyclohexane/ethyl acetate (60/40 v/v) 20 546 0.197 0.1847 0.1973 709
polystyrene cyclohexane/ethyl acetate (70/30 v/v) 20 546 0.191 0.1803 0.1922 709
polystyrene cyclohexane/ethyl acetate (80/20 v/v) 20 546 0.184 0.1760 0.1873 709
polystyrene cyclohexane/ethyl acetate (90/10 v/v) 20 546 0.176 0.1717 0.1823 709
polystyrene CCl4 35 0.145 0.1671 0.1739 RIID 200
polystyrene cyclohexane 40 514.5 0.1787 0.1832 0.1926 RIID 164
polystyrene decalin 25 436 0.12 0.1281 0.1331 RIID 14
polystyrene decalin 20 633 0.12 0.1268 0.1318 RIID 14
(continued on next page)
B.F. Hanley

Materials Today Communications 25 (2020) 101644
15
(continued)
Polymer Solvent T (◦C) λ (nm) (
∂
n/
∂
c)
exp
(
∂
n/
∂
c)
LL
(
∂
n/
∂
c)
GD
Source
a
polystyrene bromobenzene 25 436 0.043 0.0524 0.0515 RIID 14
poly(2-vinylpyridine) DMF 25? 546 0.157 0.1543 0.1642 907
poly(2-vinylpyridine) cyclohexanone 25 436 0.144 0.1391 0.1469 907
poly(2-vinylpyridine) benzene 25 436 0.107 0.0988 0.1021 907
poly(2-vinylpyridine) nitromethane/CCl4 (50/50 m/m) 25 546 0.1705 0.1452 0.1546 967
poly(2-vinylpyridine) nitromethane/CCl4 (25/75 m/m) 25 546 0.1525 0.1355 0.1434 967
poly(2-vinylpyridine) nitromethane/CCl4 (75/25 m/m) 25 546 0.1889 0.1630 0.1747 967
a
Numbers alone refer to references from the Polymer Handbook [23]; RIID – Refractive Index Increment Databook [35]; SEC, Mori – Size Exclusion
Chromatography [46].
References
[1] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953.
[2] B. Trathnigg, Size-exclusion chromatography of polymers, Encycl. Anal. Chem.
(2006). September 15.
[3] L.M. Wheeler, T.P. Lodge, B. Hanley, M. Tirrell, Translational diffusion of linear
polystyrenes in dilute and semidilute solutions of poly(viny1 methyl ether),
Macromolecules 20 (5) (1987) 1120–1129.
[4] E.T. Hanson, R. Borsali, R. Pecora, Dynamic light scattering and small-angle
neutron scattering studies of ternary rod/coil/solvent systems, Macromolecules 34
(7) (2001) 2208–2219.
[5] A.M. Striegel, P. Sinha, Absolute molar mass determination in mixed solvents. 1.
Solving for the SEC/MALS/DRI “trivial” case, Anal. Chim. Acta 1053 (2019)
186–195.
[6] J.W. Lorimer, Refractive index increments of polymers in solution: 1. General
theory, Polymer (Guildf.) 13 (2) (1972) 46–51.
[7] A.M. Striegel, Specic refractive index increment (
∂
n/
∂
c) of polymers at 660 nm
and 690 nm, Chromatographia 80 (6) (2017) 989–996.
[8] R. Mehra, Application of refractive index mixing rules in binary systems of
hexadecane and heptadecane with N-alkanols at different temperatures, J. Chem.
Sci. 115 (2) (2020) 147–154.
[9] Y. Shindo, K. Kusano, Densities and refractive indices of aqueous mixtures of
alkoxy alcohols, J. Chem. Eng. Data 24 (2) (1979) 106–110.
[10] R. Francesconi, S. Ottani, Correlation of density and refraction index for liquid
binary mixtures containing polyglycols. Use of the group contributions in the
Lorentz–Lorenz, Gladstone–Dale and Vogel equations to evaluate the density of
mixtures, J. Mol. Liq. 133 (1) (2007) 125–133.
[11] A. Bello, G.M. Guzman, Specic refractive index increments of polymers and
copolymers in several solvents, Eur. Polym. J. 2 (1) (1966) 85–91.
[12] H. Kragh, The Lorenz-Lorentz formula: origin and early history, Substantia 2
(2018) (2 SE-Feature Articles).
[13] D.W. van Krevelen, K. te Nijenhuis, Properties of Polymers, 4th ed., Elsevier,
Amsterdam, 2009.
[14] P. Outer, C.I. Carr, B.H. Zimm, Light scattering investigation of the structure of
polystyrene, J. Chem. Phys. 18 (6) (1950) 830–839.
[15] P. Zoller, D.J. Walsh, Standard Pressure-Volume-Temperature Data for Polymers,
Technomic Publishing, Lancaster, 1995.
[16] J.W. Mays, A.D. Puckett, Solution characterization of polyolens, in: C. Vasile
(Ed.), Handbook of Polyolens, Marcel-Dekker, New York, 2000, pp. 359–378.
[17] M.G. Neira-Vel´
azquez, M.T. Rodríguez-Hern´
andez, E. Hern´
andez-Hern´
andez, A.R.
Y. Ruiz-Martínez, Polymer molecular weight measurement. Handbook of Polymer
Synthesis, Characterization, and Processing, 2013, pp. 355–366. February 25.
[18] R.A. Wessling, J.E. Mark, E. Hamori, R.E. Hughes, Stereoregularity in poly
(isopropyl acrylate). II. Light-scattering results and intrinsic viscosity-molecular
weight relationships at high chain extension1a, J. Phys. Chem. 70 (6) (1966)
1903–1908.
[19] V.G.V. Schulz, W. Wunderlich, R. Kirste, Flexibilit¨
at Und Thermodynamisches
Verhalten von Isotaktischem Und Syndiotaktischem Polymethylmethacrylat, Die
Makromol. Chemie 75 (1) (1964) 22–34.
[20] C. Loucheux, Z. Czlonkowska, Comparaison Des Propri´
et´
es de Solutions de Poly-2-
Vinylpyridines Atactiques et Isotactiques, J. Polym. Sci. Part C Polym. Symp. 16 (7)
(1967) 4001–4010.
[21] R. Chiang, Light scattering studies on dilute solutions of polypropylene, J. Polym.
Sci. 28 (116) (1958) 235–238.
[22] L. Westerman, The molecular weight distribution in polypropylene, J. Polym. Sci.
Part A Gen. Pap. 1 (1) (1963) 411–423.
[23] J. Brandrup, E.H. Immergut (Eds.), Polymer Handbook, 3rd ed., John Wiley &
Sons, New York, 1989.
[24] J. Horsk´
a, J. Stejskal, P. Kratochvíl, Refractive index increments of polyolens,
J. Appl. Polym. Sci. 28 (12) (1983) 3873–3874.
[25] P. Zoller, Pressure–volume–temperature relationships of solid and molten
polypropylene and poly(butene-1), J. Appl. Polym. Sci. 23 (4) (1979) 1057–1061.
[26] Preparing Precise Polymer Solutions www.brookhaveninstruments.com/pdf
/BI-MwA/SolutionPreparationVersion1.pdf.
[27] Bauer, Fajans, Lewin, in: A. Weissberger (Ed.), Physical Methods of Organic
Chemistry, Vol. 1, Wiley-Interscience, New York, 1960.
[28] N. Lange, in: J.A. Dean (Ed.), Lange’s Handbook of Chemistry, 15th ed., McGraw-
Hill, New York, 1999.
[29] H.G. Rackett, Equation of state for saturated liquids, J. Chem. Eng. Data 15 (4)
(1970) 514–517.
[30] B.E. Pohling, J.M. Prausnitz, J.P. O’Connell, The Properties of Gases and Liquid,
5th ed., McGraw-Hill, New York, 2000.
[31] C.F. Spencer, R.P. Danner, Improved equation for prediction of saturated liquid
density, J. Chem. Eng. Data 17 (2) (1972) 236–241.
[32] R.W. Hankinson, G.H. Thomson, A new correlation for saturated densities of
liquids and their mixtures, AIChE J. 25 (4) (1979) 653–663.
[33] R.H. Perry, D.W. Green, James O. Maloney (Eds.), Perry’s Chemical Engineers’
Handbook, 7th ed., McGraw-Hill, New York, 1997.
[34] M. Ramos-Estrada, G.A. Iglesias-Silva, K.R. Hall, Experimental measurements and
prediction of liquid densities for N-alkane mixtures, J. Chem. Thermodyn. 38 (3)
(2006) 337–347.
[35] A. Theisen, C. Johann, M.P. Deacon, S.E. Harding, Refractive Increment Data-Book
for Polymer and Biomolecular Scientists, Nottingham University Press,
Nottingham, 2000.
[36] G. Allen, C. Booth, M.N. Jones, Polypropylene oxide I: an intrinsic viscosity/
molecular weight relationship, Polymer (Guildf.) 5 (1964) 195–199.
[37] E. Elbing, A.G. Parts, Interference refractometry of polymer solutions, Die
Makromol. Chemie 82 (1) (1965) 270–276.
[38] J. Veliˇ
ckovi´
c, J. Vukajlovi´
c-lipovi´
c, Refractive index increments of Di-n-
alkylitaconate polymers and copolymers, Die Angew. Makromol. Chemie 13 (1)
(1970) 79–88.
[39] K. Matsumura, T. Shimomura, T. Matsuda, M. Hirami, Melting of binary mixtures
of nylon 6 and nylon 66. A study by pressure–volume–temperature measurements,
Polym. J. 31 (10) (1999) 836–839.
[40] T. Ougizawa, G.T. Dee, D.J. Walsh, PVT properties and equations of state of
polystyrene: molecular weight dependence of the characteristic parameters in
equation-of-State theories, Polymer (Guildf.) 30 (9) (1989) 1675–1679.
[41] B. Ellis, R. Smith, Polymers A Property Database, Second, CRC Press, Boca Raton,
2009.
[42] J.E. Mark (Ed.), Polymer Data Handbook, Oxford University Press, New York,
1999.
[43] I.A.H. Ahmad, Dilute Solution Properties of Copolymers, Florida State University,
2008.
[44] M. Hert, C. Strazielle, Determination de l’increment d’indice de Refraction Dn/Dc
de Polymeres En Solution Dans Des Melanges de Solvants. Relation Entre Dn/Dc et
Le Volume Specique Partiel Du Polymere, Eur. Polym. J. 9 (6) (1973) 543–557.
[45] G.C. Williams, S. Rosenberg, H.A. Rothenberg, Physical properties of benzene-
methanol mixtures, Ind. Eng. Chem. 40 (7) (1948) 1273–1276.
[46] S. Mori, H.G. Barth, Size Exclusion Chromatography, Springer-Verlag, Berlin, 2010.
B.F. Hanley