Quantification of thermodynamics of aqueous solutions of poly(ethylene glycols):
Role of calorimetry
Lalaso V. Mohite, Vinay A. Juvekar
Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
a b s t r a c t
Keywords:
Polyethylene glycols
Calorimetry
Enthalpy of mixing
Activity
Binodal curve
A correlation based on generalized Flory-Huggins model, which relates the activity data of aqueous
poly(ethylene glycol) (PEG) solutions at low temperatures to the binodal curve in the LCST region, has
been developed. The temperature dependent parameters of the model are estimated by regression of the
experimental data of the enthalpy of mixing of aqueous PEG solutions, obtained using isothermal titration calorimetry. The temperature independent parameters are estimated from the data of the activity of
PEG solutions at a single temperature. An attempt to develop a similar correlation based on the hydrogen
bond model has been unsuccessful. It is also shown that the correlation based on the activity data alone,
does not have the ability to predict the binodal curve and the use of the enthalpy data is essential for this
purpose.
1. Introduction
Poly(ethylene glycol) (PEG) is an industrially important nonionic
polymer. Its hydrophilic ethereal oxygen atoms interact strongly
and favorably with water through hydrogen bonds. As a result,
it is completely miscible with water at moderate temperatures.
The interaction between PEG and water progressively weakens
with increasing temperature due to the reduction in the enthalpy
of hydrogen bond and finally, above the lower critical solution
temperature (LCST), phase separation occurs. This phase behavior is utilized in diverse applications including the separation of
biomolecules [1,2] and metal ions [2,3], bioconversion [4], and
organic synthesis [4]. To understand and optimize these applications, thermodynamic properties of aqueous solutions of PEG
need to be quantified. This requires both the accurate experimental data and a good model which allows prediction of properties of
the system in the range of temperature and composition in which
experimental data are not available.
A large number of experimental studies have been reported
in the literature on the thermodynamic properties of aqueous
PEG solutions. These include activity of water in the solution and
phase separation behavior. The data obtained from these studies are well documented by Wohlfarth [5]. The most widely used
methods for activity measurements are vapor pressure osmome-
try [6], laser-light scattering [7], isopiestic method [7,8], dew point
method [9], and sedimentation technique [10]. Different methods
of measurements differ in terms of their accuracy as well as the
range of temperature and composition in which each is applicable. Every technique has a relatively narrow range of temperature
and polymer compositions over which it is accurate. For example,
the sedimentation technique is accurate only at lower temperatures (<313 K) [10], whereas vapor pressure osmometry is valid only
above the temperature of about 313 K [6].
Regarding the phase separation studies, the coexistence curves
for aqueous PEG solutions are obtained from the cloud-point data.
The cloud point is measured by using either thermo-optical analysis
method [11,12] or through visual observations [13–15].
Several models have been used for predicting the behavior
of aqueous PEG solutions viz. those based on the osmotic virial
expansion [7], those based on equations of state [16,17], the group
contribution schemes [18], and those based on the lattice mean
field theory [11,14–16,19–24]. Among these models, those based on
the mean field theory are most widely used. Two types of mean
field models are reported. The first is the modified form of the
Flory-Huggins theory [11,14–16,19] and the second accounts for
the thermodynamics of hydrogen bond formation [20–24]. A good
thermodynamic model should be able to relate the activity of PEG
solutions at low temperatures with its phase behavior at high temperatures. Unfortunately, models described above use two separate
sets of parameters, one to correlate the low-temperature activity
data for PEG solutions and the other to correlate the phase separation data. The parameters obtained from the activity data in the
42
low-temperature range (278–343 K) are not suitable to predict LCST
and the coexistence curve of PEG. This failure stems from inaccuracies in either the models or the experimental data. A very high
accuracy of the low-temperature activity data is needed since the
data need to be extrapolated over a wide interval of the temperature
beyond the range of the measurement and small inaccuracies in
the parameter estimates are magnified. Same can be said about the
inaccuracies in the model. The task is made difficult by the fact that
each activity measurement technique has a relatively narrow range
of temperature over which it is accurate. The data obtained using
two or more technique needs to be combined in order to extend the
range of temperature. This procedure is also a source of error.
One possible way to overcome this difficulty is to supplement the activity data with the measurements of the partial
molar enthalpy of water (or PEG) in the solution. Partial molar
enthalpy is related to the derivative of the chemical potential
with respect to temperature, through Gibbs-Helmholtz equation.
This derivative dependence on temperature results in a better
temperature-extrapolation ability of the partial molar enthalpy
than that of the chemical potential (or the activity coefficient)
itself. A calorimeter could be used for the measurement of the
enthalpy of mixing of PEG and water, from which the partial molar
enthalpies of the constituents could be derived. These could then
be used to estimate the model parameters which predict temperature dependence of the activity coefficient. Another advantage of
the calorimetric technique is that it is accurate over a wider range
of temperature (278–348 K).
Calorimetry therefore appears suitable for correlating lowtemperature activity data of PEG to the phase behavior data. The
main objective of the present work is to verify this expectation. If
such a correlation is found, it can be used for predicting the thermodynamic data in the range of temperature and concentration,
where experimental measurements cannot be performed by the
available techniques. It can also be used as a benchmark for comparing various methods of measurement of activity of PEG solution
and also for discriminating different models for thermodynamics
of PEG–water system.
In the present work, the enthalpy of mixing of the aqueous
solutions of PEG has been measured using isothermal titration
calorimeter. The range of temperature covered is 288.15–348.15 K
and that of the concentration is 0–53% (w/w). We have used
generalized Flory-Huggins theory and hydrogen bond model of
Dormidontova [23] to correlate both the activity as well as the
calorimetry data. It is important to note that the calorimetric data
alone are insufficient for computing the activity because, when the
expression for the activity is substituted in the Gibbs-Helmholtz
equation, temperature independent parameters in the expression
are eliminated during the differentiation and thus cannot be estimated using the calorimetry data. Hence, the calorimetry data
needs to be supplemented with the activity data in order to determine the temperature independent coefficients.
Using the correlations based on the two models viz. generalized
Flory-Huggins theory and hydrogen bond model of Dormidontova
[23], and using the activity data obtained from different techniques,
an attempt has been made to predict the coexistence curve for
PEG–water system in the region of LCST. The work is presented as
follows. We first describe the models for thermodynamic of PEG
solution. We then present the methodology to estimate the coefficients of these models from the data on the enthalpy of mixing
and the activity of water in the PEG solutions. Next, we present the
results of the experiment on measurements of the enthalpy of mixing of aqueous solutions of PEG. This is followed by the analysis of
the results to obtain the model parameters. The different models
are then discriminated based on the accuracy, and the best model
is arrived at. Finally, the selected model has been used to grade the
quality of the reported solution activity data.
2. Models for solution thermodynamics
2.1. Generalized Flory-Huggins model
According to the generalized Flory-Huggins model, the total free
energy of polymer solution, F, is given by [25]
0p
np
F
0
=
ln(p ) + nw ln(1 − p ) + gnw p + np
+ nw w
RT
rp
RT
RT
(1)
In the above equation, np and nw represent the moles of polymer
and water in the aqueous PEG solution, respectively. The number
of Kuhn segments in PEG chain is represented by rp . 0p and 0w are
the chemical potential of pure polymer and pure water, respectively.
The term g is the generalized Flory-Huggins parameter and should
be considered as the function of the volume fraction of the polymer,
p , and temperature, T. We have considered g to be independent of
rp . In the range of chain lengths of PEG investigated in this work,
the entropic term in the Flory-Huggins model (the first term on the
right of Eq. (1)) is adequate to account for the effect of the chain
length. The dependence of g on rp needs to be considered only for
short chain polymers where the end-group effect is significant.
The expressions for the chemical potential of water and polymer
segment in the polymer solution can be obtained from Eq. (1) as
follows:
w − 0w
=
RT
∂(F/RT )
∂nw
np
= ln(1 − p ) + p
vw
1−
rp vp
+ gp2 − (1 − p )p2
∂g
∂p
(2)
p − 0p
RT
=
∂(F/RT )
∂np
+g
=
nw
1
ln(p ) + (1 − p )
rp
vp
vp
2
2
(1 − p ) + (1 − p ) p
vw
vw
vp
1
−
vw
rp
∂g
(3)
∂p
where vp and vw are the partial molar volumes of polymer segment and water, respectively. The ratio of vp to vw is assumed to be
independent of the polymer concentration and temperature.
The activity of water in the solution can be obtained using Eq.
(2) as
ln aw =
w − 0w
RT
= ln(1 − p ) + p
1−
vw
rp vp
+ gp2 − (1 − p )p2
∂g
∂p
(4)
We define the chain length independent activity of water, ˇw ,
as
ln ˇw = ln aw +
vw
rp vp
p = ln(1 − p ) + p + gp2 − (1 − p )p2
∂g
∂p
(5)
The advantage of using the chain length independent activity is
that the relation between ˇw and p for the solutions of different
chain lengths of PEG molecules could be represented by a single
curve.
The partial molar enthalpies of species (polymer segment and
water) are related to the respective chemical potentials by the
Gibbs-Helmholtz equation:
hi − h0i = −RT 2
∂(((i − 0i )/RT ))
∂T
(i = w or p)
(6)
43
where h0i are the partial molar enthalpy of pure component,
i.
The expression of the partial molar enthalpy of water and
polymer is obtained in terms of the generalized Flory-Huggins
parameter using Eqs. (2), (3) and (6):
∂g(T, p )
∂2 g(T, p )
hw − h0w
+ (1 − p )p2
= −p2
2
∂T
∂p ∂T
RT
hp − h0p
Substitution of the second and third derivative of the free energy
of polymer solution (Eq. (1)) in the above equations yield
− pc (1 − pc )
The enthalpy of mixing of n1 moles of polymer solution-1 with
n2 moles of polymer solution-2 can be expressed in terms of partial
molar enthalpies of polymer and water as follows:
Hmix = (n1 + n2 )[xp (hp − h0p ) + xw (hw − h0w )] − n1 [xp1 (hp1 − h0p )
+ xw1 (hw1 − h0w )] − n2 [xp2 (hp2 − h0p ) + xw2 (hw2 − h0w )]
(9)
Here, xi1 , xi2 and xi (i = p or w) are the mole fractions of polymer
(p) or water (w) in solution-1, solution-2 and in the mixture, respectively. The terms hi1 , hi2 and hi (i = p or w) are the corresponding
partial molar enthalpies.
Substitution of the expressions for partial molar enthalpies from
Eqs. (7) and (8) into Eq. (9) allows us to express the enthalpy of
mixing in terms of the temperature derivative of the generalized
Flory-Huggins parameter.
The conditions for the phase equilibrium between two separated
phases (˛-phase and ˇ-phase) of a binary aqueous polymer solution
are given by
ˇ
˛
w = w
(10)
ˇ
˛
p = p
(11)
ln
1 − p˛
ˇ
+ (p˛ − p )
ˇ
1 − p
1−
vw
rp vp
−
2
pc
− 3(1 − 2pc )
ˇ
ˇ
p
vp
1
−
vw
rp
p
np
F
=
ln
RT
rp
rp e
p˛
(13)
=0
+ nw
1 − p
ln
e
+ nw p
p vw
1−p−x
1 − p vp
(14)
p vw
1 − p vp
ln
Fp
RT
Fw
RT
p vw
ln 1 − p − x
1 − p vp
2(1 − p )
e
(17)
where is the Flory-Huggins type interaction parameter, which
is a function of temperature alone. The terms Fp and Fw are
the free energies of PEG–water and water–water hydrogen bonds,
respectively. The quantity x represents the fraction of the total
number of the proton acceptors on PEG chain, which is hydrogen
bonded to water and p represents the fraction of the total number of proton acceptors on water, which is hydrogen bonded to
water.
Expression for x and p are obtained by minimizing the
free energy (expressed by Eq. (17)) with respect to x and p.
Equating the partial derivatives of the free energy with respect
to x and p independently to zero, we obtain the following
equations:
ˇ
The values of p˛ and p at specific temperature T is obtained by
solving Eqs. (12) and (13), simultaneously.
The critical point is given by the following condition:
∂p3
+ 2np x ln x + (1 − x) ln(1 − x) − x
(12)
p
=
ˇ
ˇ 2
ˇ 2 ˇ ∂g(T, p ) − g(T, p )(1 − p ) + (1 − p ) p
=0
∂p ˇ
∂p2
(16)
The total free energy of aqueous solution of PEG, on the basis
of the recently developed hydrogen bond model by Dormidontova
[23] is expressed as
− 2nw p + x
∂g(T, p ) 2
2
g(T, p˛ )(1 − p˛ ) + (1 − p˛ ) p˛
∂p ∂3 (F/RT )
=0
2.2. Hydrogen bond model
+ 2nw
∂2 (F/RT )
Tc ,pc
vp
+
vw
Tc ,pc
p
+ (p − p˛ )
∂p2
The critical temperature, Tc , and critical polymer volume
fraction, pc , are obtained by solving the above two equations
simultaneously.
p
∂2 g(T, p ) + 2nw p ln p + (1 − p) ln(1 − p) − p
ˇ
ˇ2
ˇ
ˇ2 ∂g(T, p ) − g(T, p )p − (1 − p )p
=0
∂p ˇ
p˛
(15)
∂3 g(T, p ) − pc (1 − pc )
∂p3
∂g(T,
)
p
+ g(T, p˛ )p˛2 − (1 − p˛ )p˛2
∂p ˛
=0
Tc ,pc
∂p2
1
ln
rp
∂2 g(T, p ) Tc ,pc
∂g(T, p ) vw
1
+
− 6
2
rp vp
∂p (1 − pc )
Tc ,pc
1
Substituting Eqs. (2) and (3) into Eqs. (10) and (11), one can
obtain following equations, which govern the phase equilibria:
(7)
vp
vp
∂2 g(T, p )
2 ∂g(T, p )
2
=−
(1 − p )
−
(1 − p ) p
(8)
vw
vw
∂T
∂p ∂T
RT 2
∂g(T, p ) 1
1 vw
+
− 2g(Tc , pc ) − 2(1 − 2pc )
pc rp vp
1 − p
∂p 2np
Fp
ln x − ln(1 − x) −
RT
×
ln 1 − p − x
p vw
1 − p vp
− 2nw
p vw
1 − p vp
+ ln[2(1 − p )]
=0
(18)
44
Fw
RT
2nw ln p − ln(1 − p) −
×
p vw
1 − p vp
ln 1 − p − x
where p0 represent the value of p in pure water and is obtained
from Eq. (22) by substituting p = 0. Thus
− 2nw
+ ln[2(1 − p )]
=0
(19)
The mole of polymer and water in the solution can be relate to
the volume fractions of polymer through the following equations:
np =
p VT
vp
(1 − p )VT
nw =
;
(20)
vw
p0 = 2 exp
F w
RT
x = 2 exp
p = 2 exp
Fp
RT
(1 − x)(1 − p )
F w
RT
(1 − p)(1 − p )
p vw
1−p−x
1 − p vp
hw − h0w
∂
= −p2
−
∂T
RT 2
p vw
1−p−x
1 − p vp
−
hp − h0p
=−
RT 2
Ei
Si
Fi
=
−
RT
RT
R
(i = w or p)
(23)
The entropic loss of hydrogen bond formation is related to the
characteristic space bond angle for the hydrogen bond to remain
stable, as follows [26]:
Si
= − ln
R
1 − cos i
(24)
2
where i is the critical angle for the PEG–water (i = p) and
water–water (i = w) hydrogen bond formation.
The expressions for the chemical potentials of water and polymer, obtained by differentiating the expression for free energy (Eq.
(17)) with respect to nw and np respectively, are
w − 0w
vw
= ln(1 − p ) + p 1 −
RT
rp vp
p vw
+ 2 ln 1 − p − x
1 − p vp
+ p2
+ 2p(1 − p ) − 4 ln(1 − p0 ) − 2p0
p − 0p
RT
=
1
ln
rp
p
rp
+ (1 − p )
+ 2xp + 2 ln(1 − x) + 2p
vp
1
−
vw
rp
(25)
+
vp
2
(1 − p )
vw
vp
(1 − p )
vw
(26)
The chain length independent activity of water, ˇw (see Eq. (5))
is obtained from Eq. (25) as
ln ˇw = ln aw +
vw
p
rp vp
= ln(1 − p ) + p + p2 + 2 ln 1 − p − x
+ 2x
+
p vw
1 − p vp
2p (1 − p )p(1 − p)(1 − p + xvw /vp )
Ep
RT 2
(1 − p2 )(1 − p ) − x2 p vw /vp
+ 2(p − p0 )
(29)
2p (1 − p )x(1 − x)(1 − p + xvw /vp )
(1 − p2 )(1 − p ) − x2 p vw /vp
Ep
− 2x
2
2(1 − p ) p(1 − p)(1 − p + xvw /vp )vp /vw
(1 − p2 )(1 − p ) − x2 p (vw /vp )
RT 2
Ew
RT 2
(30)
Substitution of the expressions for partial molar enthalpies from
Eqs. (29) and (30) into Eq. (9) allows us to express the enthalpy of
mixing in terms of the temperature derivative of the hydrogen bond
model parameter.
The conditions for the phase equilibrium between two separated
phases (˛-phase and ˇ-phase) are obtained from substituting Eqs.
(25) and (26) into Eqs. (10) and (11):
ln
1 − p˛
ˇ
1 − p
+ 2 ln
+2
vw
+ 2x
+ 2 ln(1 − p)
vp p
2p2 x(1 − x)(1 − p + xvw /vp )vw /vp
vp
2 ∂
(1 − p )
vw
∂T
+
The free energies of PEG–water and water–water hydrogen
bonds, Fp and Fw , can be expressed in terms of entropic loss
S and energetic gain E of hydrogen bond as
Ew
RT 2
(21)
(22)
(28)
(1 − p2 )(1 − p ) − x2 p vw /vp
×
2
The expression for the partial molar enthalpy of water and polymer is obtained through the Gibbs-Helmholtz equation (refer Eq.
(6)), as shown below:
where VT is the total volume of the polymer solution.
Substituting np and nw from Eq. (20) into Eqs. (18) and (19), and
subsequent rearrange yields the following expressions for x and p:
(1 − p0 )
+ (p˛
ˇ
− p )
vw
ˇ2
1−
+ (p˛2 − p )
rp vp
1 − p˛ − x˛ (p˛ /(1 − p˛ ))(vw /vp )
ˇ
ˇ
1 − pˇ − xˇ (p /(1 − p ))(vw /vp )
vw ˛ ˛
ˇ
(x p − xˇ p ) + 2 ln
vp
1 − p˛
1 − pˇ
ˇ
+ 2[p˛ (1 − p˛ ) − pˇ (1 − p )] = 0
1
ln
rp
p˛
ˇ
p
ˇ
+ (p
− p˛ )
vp
1
−
rp
vw
(31)
vp
1 − x˛
ˇ 2
ˇ
2
[(1 − p˛ ) − (1 − p ) ] + 2(x˛ p˛ − xˇ p ) + 2 ln
vw
1 − xˇ
vp ˛
ˇ
+2
[p (1 − p˛ ) − pˇ (1 − p )] = 0
(32)
vw
+
where x˛ and p˛ are the values of x and p in phase ˛. These fractions
can be estimated by simultaneously solving Eqs. (21) and (22) in
phase ˛. Similarly, xˇ and pˇ are obtained in phase ˇ. The values of
ˇ
p˛ and p at a given temperature T is obtained by solving Eqs. (31)
and (32), simultaneously.
The critical point is obtained by taking the second and third
derivative of free energy of polymer solution with respect to p
(refer Eq. (14)) and leads to the following equations:
vw
+ 2 ln(1 − p) + 2p(1 − p ) − 4 ln(1 − p0 ) − 2p0
vp p
(27)
2
2(1 − pc + xc vw /vp )
1
vw 1
−
− 2c +
=0
rp vp pc
1 − pc
(1 − p2c )(1 − pc ) − xc2 pc vw /vp
(33)
45
−
2
2(1 − pc + xc vw /vp )
vw 1
1
−
+
2
2
3
rp vp pc
(1 − pc )
[(1 − p2c )(1 − p ) − xc2 p vw /vp ]
×
1 − p2c + xc2
vw
vp
(1 − p2c )(1 − pc ) − xc2 p
vw
vp
vw
vw
− 2xc (1 − xc )(1 − pc )
(1 + pc )(1 − pc ) + xc
vp
vp
− 2pc (1 − pc )
vw
xc2 pc
vp
vw
− 1 − pc − pc xc
vp
(1 − pc )
=0
(34)
where c , xc and pc are the respective values of , x and p at the
critical point. The critical temperature, Tc , and critical polymer volume fraction, pc , are obtained by solving Eqs. (21), (22), (33) and
(34) simultaneously.
3. Procedure for regression of the model parameters
The polymer–water interaction parameter g(T, p ) in the FloryHuggins theory decides the thermodynamic behavior of polymer
solutions. In the literature, a variety of the forms of correlations
for this parameter have been reported [11,14,25,27]. In the present
analysis, we have tried the following empirical form of g(T, p ):
n
⎡
ndata
||R|| = ⎣
j=1
bi (T )pi
(35)
where the temperature dependent coefficients bi (T) are expressed
as [11,27]
bi (T ) = bi˛ +
T
+ bi ln T
i = 0, 1, . . . , n
(36)
bi˛ , biˇ and bi are constants.
This form requires 3(n + 1) empirical constants to be determined
using the experimental data. For this form of g(T, p ), we have
∂g(T, p ) ∂bi i
=
p =
∂T
∂T
n
n
i=0
−
i=0
and
∂2 g(T, p ) ∂bi i−1 i
=
i
=
∂p ∂T
∂T p
n
i=1
n
i=0
biˇ
T2
exp
rj
− rjmodel
j
2 ⎤1/2
⎦
(39)
exp
where rj
and rjmodel (j = 1, . . ., ndata) respectively represent the
experimental value and the corresponding model prediction of the
quantity to be fitted (Hmix or ˇw ), and j is the standard deviation.
There are five unknown parameters of hydrogen bond model,
namely, the Flory-Huggins type interaction parameter (), energetic gain per water–water hydrogen bond (Ew ) and PEG–water
hydrogen bond (Ep ), critical hydrogen bond angle between
water–water (w ) and PEG–water (p ). The parameters w , p ,
Ew and Ep are temperature independent, whereas the interaction parameter () is temperature dependent. The standard form of
temperature dependence of is [23]
i=0
biˇ
3.2. Hydrogen bond model
3.1. Generalized Flory-Huggins model
g(T, p ) =
constants. In Method-3, we combine the data on the enthalpy of
mixing of PEG solutions (calorimetry data) and the activity of water
(all methods), and use them for simultaneous regression of all the
constants.
Three different values of n, viz. n = 1, 2 and 3 are used in order
to test the effect of n (degree of polynomial in p in Eq (35)) on
the quality of the estimates. The nonlinear least-square method of
Levenberg-Marquardt is used for the regression in all the cases. The
l2 -norm of the residual (||R||) is used for judging the quality of the
regression. It is defined as
+
−
bi
T
biˇ
T2
+
pi
bi
T
(37)
pi−1
(38)
We see from the above equations that coefficients bi˛ are
eliminated during partial differentiation of g with respect to temperature. Since the expressions for partial molar enthalpies (Eqs.
(7) and (8)), involve only ∂g(T, p )∂T and ∂2 g(T, p )/∂p ∂T, bi˛ will
also be absent in the expressions for the partial molar enthalpy. The
same is true for the enthalpy of mixing (Eq. (9)) in which the only
thermodynamic terms are the partial molar enthalpies. Thus the
regression of the data on the enthalpy of mixing will allow us to
estimate biˇ and bi , but not bi˛ .
Three different methods are used for estimating the constants.
In Method-1, we use the data on the enthalpy of mixing of PEG
solutions. As mentioned above, these data are not sufficient to
determine all the constants required for fully expressing g. We need
additional data to determine bi˛ . To obtain these constants (bi˛ ), we
use the activity data of water in PEG solution at a suitable single
temperature, which we call the base temperature. In Method-2, we
use only the data on the activity of water in PEG solutions over a
range of temperatures and compositions, for regression of all the
= ah +
bh
T
(40)
where ah and bh are constants.
The values of the energy gain parameters used by Dormidontova
[23] are: Ew /R = 1800 K and Ep /R = 2100 K. We retain these values. As a result, only four unknown constants are left, viz. ah , bh . w
and p . For estimation of these, we have used three different methods. In Method-1, bh , w and p are estimated from the regression
of the calorimetric data. The parameter ah , is estimated using the
data on the activity of the solution as a function of composition at
one temperature. Method-2, is based on the regression of the activity data alone. In Method-3, all unknown constants are estimated
from the regression of the combined data (the calorimetric data,
and the activity data from all methods). Again, the quality of estimate is judged by the same procedure as described in connection
with the generalized Flory-Huggins model.
4. Measurement of enthalpy of mixing
PEG having the molecular weights 4600 (PEG4600), 8000
(PEG8000), 20,000 (PEG20000), and 35,000 (PEG35000) were purchased from Sigma–Aldrich, Germany and used without further
purification. The enthalpy of mixing of the aqueous solutions of
PEG was measured by isothermal titration calorimeter CSC 4200
(Calorimetry Science Corporation, USA). The sensitivity of calorimeter is ±0.5 ␮J. An aqueous solution PEG of known concentration was
injected (6–10 injections per run) from a gastight syringe, through
a stainless steel cannula (Hamilton 1725LT), to 1.3 mL of aqueous
PEG solution contained in a 1.3 mL cylindrical stainless steel cell.
A computer-controlled syringe pump was used for the injection.
A turbine agitator, provided with the cell, was set at 300 rpm to
give uniform stirring to the solution in the cell. All the solutions
were prepared by mixing the polymer with deionized water. Concentrations of PEG in the syringe and in the cell were selected in
such way that the heat evolved did not cross the upper limit of
46
Table 1
Enthalpy of mixing of aqueous PEG solution obtained from Fig. 1.
injection
injection
Peak numbera
mcell
p (g)
mcell
w (g)
mp
(mg)
mw
(mg)
Hmix b (mJ)
1
2
3
4
5
6
7
8
9
10
0.4038
0.4028
0.4018
0.4008
0.3998
0.3988
0.3978
0.3968
0.3959
0.3949
0.9597
0.9606
0.9614
0.9622
0.9631
0.9639
0.9647
0.9655
0.9664
0.9672
2.0811
2.0811
2.0811
2.0811
2.0811
2.0811
2.0811
2.0811
2.0811
2.0811
8.2359
8.2359
8.2359
8.2359
8.2359
8.2359
8.2359
8.2359
8.2359
8.2359
−12.5746
−12.4841
−12.2782
−12.0861
−11.8662
−11.7122
−11.4777
−11.2611
−11.0976
−10.9185
injection
injection
cell
mcell
/mw
: mass of polyp /mw : mass of polymer/water in the cell; mp
mer/water added per injection.
a
Peak numbers in the first column correspond those in Fig. 1, counted from the
left to the right.
b
Enthalpy of mixing is equal to the area under the peak in Fig. 1.
Fig. 1. Typical output of calorimeter during titration of aqueous PEG solution. Temperature = 298.15 K, Mp = 8000, volume of the solution in the cell = 1.30 mL, volume of
the titrant/injection = 10 ␮L, initial weight fraction of PEG in the cell = 0.2962 (w/w),
weight fraction of PEG in the titrant = 0.2017 (w/w).
the measurement. The calorimeter was pre-calibrated using a precise electrical input. The measurements of the enthalpy of mixing
of aqueous PEG8000 solutions were carried out at seven different temperatures in the range of 288.15–348.15 K. The calorimetric
measurements on PEG of other molecular weights were performed
only at one temperature (298.15 K) and concentration, in order to
verify the non-dependence of the partial molar enthalpy on the
molecular weight. For each titration, 10 injections were used. Two
titrations were performed for a fixed concentration of the solution in the cell, one using 5 ␮L per injection and the other using
10 ␮L per injection. Weight fraction of PEG in the cell was varied in the range of 0.1–0.53 and that in the syringe from 0 to
0.45.
Fig. 1 shows the typical output from the isothermal titration
calorimeter and Table 1 shows the values of the enthalpy of mixing
obtained from the output. It is seen from the table that the heat
evolved per injection is of the order of 10 mJ. This is much larger
than the least count of the instrument, which is 0.5 ␮J. These data
also show a high sensitivity of the enthalpy of mixing to composition.
The mixing enthalpy is seen to be negative (mixing is exothermic) and it gradually decreases with increase in the amount of the
polymer added (Table 1). In this experiment, a dilute polymer solution in the syringe is injected into a concentrated solution in the cell.
There are two heat effects involved during the mixing process. The
polymer, which in the syringe, was surrounded by more molecules
of water, enters the cell in which the concentration of water is lower.
As a result the polymer loses a fraction of hydrogen bonds which it
had formed when it was in the syringe. This loss of hydrogen bonds
should be accompanied by absorption of heat (endothermic step).
On the other hand, water molecules form more hydrogen bonds in
the concentrated polymer solution in the cell compared to the rela-
tively dilute solution in the syringe. This is an exothermic step. The
net effect is exothermic since the number of water molecules in the
injected solution is much larger than the total number of hydrogen bond sites on the injected polymer. As more and more amount
of the polymer is injected, the solution in the cell becomes more
dilute and the difference in the concentration between the solution in cell and the syringe reduces and hence the exothermicity of
mixing progressively reduces.
5. Results and discussion
5.1. Generalized Flory-Huggins theory
We first discuss the results of the regression analysis of the data
using the three proposed methods.
Method-1. Here, the data of the enthalpy of mixing are used
to estimate the set of constants biˇ , bi . The additional parameters required for solving these equations are the number of Kuhn
segments (rp ) in polymer chain and the ratio of partial molar volume of the PEG segment, to that of the water molecule (vp /vw ).
The number of Kuhn segments is estimated using the relation [25]
rp = 0.0141Mp , where Mp is the molecular weight of PEG. The partial
molar volumes are obtained using the density data of aqueous PEG
solution [7,28,29]. The solution density is practically independent of
Mp . The dependence of vp /vw on the temperature and polymer concentration is found to be very small. Therefore, The value of vp /vw
is taken as constant equal to 3.3 for all polymer concentrations and
temperatures.
Table 2 lists the regression estimates of the constants biˇ , bi
(i = 0, 1, . . ., n), for n = 1, 2, and 3. The table also list the l2 -norm
of the residuals, ||R||, for the best fit values of the constants. It is
seen that the value of ||R|| for n = 1 is significantly larger than those
for n = 2 and 3. Hence the linear form (n = 1) is not used in the further analysis. The values of ||R|| for n = 2 and 3 are not significantly
different from each other and hence both are accepted.
Table 2
The least square estimates of constants biˇ and bi , obtained from enthalpy of mixing.
n
i=0
bi (T )pi
biˇ (i = 0, 1, . . ., n)
bi (i = 0, 1, . . ., n)
||R|| × 102
n=1
b0ˇ = −1.4900 × 10 , b1ˇ = −1.5067 × 10
b0 = −3.0933, b1 = −3.3231
1.6987
n=2
b0ˇ = −1.7168 × 103 , b1ˇ = −1.5470 × 103 ,
b2ˇ = −6.9445 × 102
b0ˇ = −3.1670 × 103 , b1ˇ = −3.4347 × 103 ,
b2ˇ = 6.5417 × 102 , b3ˇ = −6.7884 × 103
b0 = −3.5519, b1 = −3.4314, b2 = −1.2222
0.6697
b0 = −8.0678, b1 = −9.3077, b2 = 2.9763, b3 = −21.098
0.6170
n=3
3
3
47
Table 3a
Estimates of the constants bi˛ (acceptable set of constants are indicated by asterisk; the best set is denoted by double asterisk).
n
i=0
bi (T )pi
Method
Dew point method [9]
Laser light scattering and isopiestic
method [7]
Sedimentation [10]
n=2
Vapor pressure osmometry [6]
Isopiestic method [8]
Dew point method [9]
Laser light scattering and isopiestic
method [7]
Sedimentation [10]
n=3
Vapor pressure osmometry [6]
Isopiestic method [8]
Base temperature (K)
bi˛ (i = 0, 1, . . ., n)
i=0
i=1
298.15
298.15
313.15
303.15(**)
313.15(*)
318.15
328.15
293.15
333.15
26.263
26.462
26.466
26.310
26.299
26.187
26.479
26.340
26.435
24.709
24.724
24.743
24.657
24.645
24.640
24.753
24.734
24.698
298.15
298.15
313.15
303.15(*)
313.15(*)
318.15
328.15
293.15
333.15
56.806
57.117
57.049
56.890
56.865
57.933
56.678
56.650
57.078
64.443
64.636
64.544
64.445
64.408
65.716
64.129
64.050
64.700
The remaining temperature independent constants, i.e. bi˛ (i = 0,
1, . . ., n) are estimated from the reported data on the activity of
water in PEG solutions [6–10]. The data for a single temperature
(base temperature) is used. Since activity data have been obtained
using different measurement techniques, it is necessary to make a
choice. The following three criteria are used for the selection of the
best data. First, the quality of regression is checked on the basis of
l2 -norm. The second criterion is the accuracy of prediction of the
critical constants, i.e. Tc (LCST) and pc (the critical volume fraction
of PEG) using the estimated parameters. These critical constants
are obtained by simultaneously solving Eqs. (15) and (16). The third
criterion is the accuracy of predication of the binodal curve. The
binodal curve is estimated by solving Eqs. (12) and (13) simultaneously.
The values of the estimated parameters are listed in Table 3a,
and the correlations are compared in Table 3b. These estimates are
with reference to PEG (Mp = 15,000). We compare the predicted Tc
and pc with the reported experimental values for Mp = 15,000. Bae
et al. [11] have reported Tc = 390.40 K and pc = 0.09160; and Fischer
and Borchard [15] reported Tc = 388.47 K, and pc = 0.1099.
In some cases, Eqs. (15) and (16) are incompatible and do not
yield a real solution. We can therefore neither obtain the critical
i=2
8.8597
9.5741
9.4822
9.0804
9.0759
8.7491
9.4670
9.1122
9.4461
−19.513
−18.971
−18.958
−19.334
−19.310
−19.327
−19.084
−18.765
−19.356
i=3
–
–
–
–
–
–
–
–
–
142.68
143.27
142.89
142.86
142.77
146.20
141.80
141.28
143.50
temperature nor the critical polymer composition using these correlations. Blanks appearing in the last two columns of Table 3b
represent these cases.
The four selected correlations are indicated by asterisks in the
third column of Table 3. Two are for n = 2 and the rest two for n = 3.
Among the four selected correlations, the one based on n = 2 and
the base temperature of 303.15 K yields the most accurate value of
the critical temperature and is shown by double asterisk.
It is important to note that all four selected correlations are
based on sedimentation method for the activity data. A possible reason is that the sedimentation velocity can be accurately measured
over a wider range of polymer concentrations. Other methods do
not possess such a wider composition range. For example, isopiestic method is not accurate in dilute solutions, whereas laser
light scattering is not accurate in concentrated solutions. Vapor
pressure osmometry and dew point techniques have applicability over a limited range of concentrations [30]. Small errors these
methods introduce, at the boundaries of their limited range, are
possibly magnified during extrapolation to high temperatures and
concentrations required for the prediction of LCST and the critical composition. Sedimentation method seems to be relatively free
from these errors.
Table 3b
Comparison of the estimates of the parameters listed in Tables 2 and 3a (acceptable set of constants are indicated by asterisk; the best set is denoted by double asterisk).
n
i=0
n=2
bi (T )pi
Method used for activity measurement
Selected base temperature (K)
||R|| × 103
Tc (K)
pc
Dew point method [9]
Laser light scattering and Isopiestic
method [7]
298.15
298.15
313.15
303.15(**)
313.15(*)
345.15
355.15
293.15
333.15
3.2518
0.4115
2.4831
0.2237
0.2853
0.4011
0.1208
7.9588
3.4903
380.35
–
–
388.15
391.48
389.04
–
399.25
351.94
0.2280
–
–
0.1394
0.1286
0.2052
–
0.2714
0.0298
298.15
298.15
313.15
303.15(*)
313.15(*)
345.15
355.15
293.15
333.15
3.2463
0.3482
2.4830
0.2133
0.2536
0.2338
0.0561
7.3564
3.4801
381.97
–
–
393.44
389.40
–
342.74
–
405.82
0.2551
–
–
0.0488
0.0404
–
0.0221
–
0.2990
Sedimentation [10]
Vapor pressure osmometry [6]
Isopiestic method [8]
Dew point method [9]
Laser light scattering and Isopiestic
method [7]
n=3
Sedimentation [10]
Vapor pressure osmometry [6]
Isopiestic method [8]
Critical constants
48
Fig. 2. Comparison of the predicted binodal curve with the experimental data (n = 2).
Lines are predictions based on the two selected correlations from Table 3 for n = 2.
The solid line (—) corresponds to base temperature of 303.15 K and dashed line (- - -)
corresponds to 313.15 K. Reported experimental data are indicated by points: ()
Bae et al. [Mp = 15,000] [11]; () Fisher et al. [Mp = 15,481] [14].
Fig. 3. Comparison of the predicted binodal curve with the experimental data (n = 3).
Lines are predictions based on the two selected correlations from Table 3 for n = 3.
The solid line (—) corresponds to base temperature of 303.15 K and dashed line (- -) to 313.15 K. Reported experimental data are indicated by points: () Bae et al.
[Mp = 15,000] [11]; () Fisher et al. [Mp = 15,481] [14].
We use each of these four selected sets of parameters to predict
the LCST part of the binodal curve. Fig. 2 compares the predicted
binodal curves, for n = 2, with experimental data for aqueous PEG
(Mp = 15,000) solution. It is seen that the predicted binodal curve
matches very well with the experiments for the parameter set
obtained from the activity data at the base temperature of 303.15 K.
The curve, obtained using the base temperature of 313.15 K, deviates
significantly from the experiment at higher PEG concentrations.
A similar comparison for selected correlations for n = 3 is made
in Fig. 3. The predicted binodal curves show two minima, one at a
low-polymer concentration and the other at a higher concentration.
This implies that the correlations based on n = 3 (cubic form) predict
two values of LCST. This is not realistic. Hence these correlations are
not accepted.
Method-2. This method is based on the activity data alone. In
the literature, there are only two works which measure the activity
of water in aqueous PEG solution over a sufficiently wide range of
temperatures and PEG concentrations. The activity of water using
sedimentation [10] technique is available in the temperature range
of 293–313 K, and that based on vapor pressure osmometry [6] is
available in the range of 308–338 K. The other data do not cover sufficiently wide range of temperature and/or concentrations. Hence,
Table 4a
The least square estimates of the constants bi˛ (i = 0, 1, . . ., n) using Method-2.
n
i=0
bi (T )pi
Method
Sedimentation [10]
bi˛
n=1
biˇ
bi
bi˛
n=2
biˇ
bi
bi˛
n=3
biˇ
bi
Vapor pressure osmometry [6]
Sedimentation [10] and Vapor pressure osmometry [6]
b0˛ × 10−2
b1˛ × 10−2
b0ˇ × 10−4
b1ˇ × 10−4
b0 × 10−1
b1 × 10−1
1.4544
2.4326
−0.6975
−1.1453
−2.1355
−3.5973
4.5413
8.0758
−2.2170
−3.8997
−6.6625
−11.886
1.2866
2.5205
−0.6232
−1.1847
−1.8847
−3.7284
b0˛ × 10−2
b1˛ × 10−2
b2˛ × 10−2
b0ˇ × 10−4
b1ˇ × 10−4
b2ˇ × 10−4
b0 × 10−1
b1 × 10−1
b2 × 10−1
2.8381
1.6266
5.1783
−1.3373
−0.7726
−2.3944
−4.1886
−2.4014
−7.6834
6.6645
10.072
3.8916
−3.2876
−4.8911
−1.9835
−9.7646
−14.810
−5.6746
3.3200
2.2679
6.8806
−1.5492
−1.0581
−3.1423
−4.9096
−3.3589
−10.231
b0˛ × 10−2
b1˛ × 10−2
b2˛ × 10−2
b3˛ × 10−2
b0ˇ × 10−4
b1ˇ × 10−4
b2ˇ × 10−4
b3ˇ × 10−4
b0 × 10−1
b1 × 10−1
b2 × 10−1
b3 × 10−1
4.6954
5.4513
−0.2900
12.785
−2.1884
−2.5253
0.1114
−5.8588
−6.9479
−8.0837
0.4407
−18.994
−53.876
−50.347
−46.175
−122.00
26.387
24.774
22.295
60.235
79.123
73.873
67.972
178.91
5.1073
5.4441
3.0796
10.775
−2.3729
−2.5310
−1.3645
−4.9932
−7.5622
−8.0676
−4.6042
−15.976
49
Table 4b
Comparison of the estimates of the parameters using Method-2.
n
i=0
bi (T )pi
Method
Sedimentation
[10]
Vapor pressure
osmometry [6]
Combination of Sedimentation [10]
and Vapor pressure osmometry [6]
n=1
||R|| × 103
Tc (K)
pc
6.2900
335.89
0.0297
1.0779
344.25
0.0364
7.2309
330.57
0.03096
n=2
||R|| × 103
Tc (K)
pc
2.3863
–a
–a
1.0379
342.89
0.0339
3.2707
358.25
0.1552
n=3
||R|| × 103
Tc (K)
pc
2.1273
352.24
0.0187
0.9716
339.36
0.0207
3.0682
368.88
0.1509
a
In these cases, the equations for the critical constants have no solution.
we consider only these two set for data for the estimation of the
constants. The data are used both individually and together. Table 4
presents the relevant data for PEG (Mp = 15,000) (4(a) lists the estimates of the parameters, and 4(b) the comparison of the estimates).
From Table 4b, it is seen that even though the values of ||R|| are
satisfactory in all cases, predicted critical temperature and critical
concentration of PEG deviate considerably from the experimental
values. Hence Method-2 is inferior to Method-1 and is therefore not
given any further consideration.
Method-3. In this method, the parameters of the model (see Eqs.
(35) and (36)) are obtained by regression of the combined data on
the mixing enthalpy and the solution activity. In the solution activity data, we include all the methods (vapor pressure osmometry [6],
laser-light scattering [7], isopiestic method [7,8], dew point method
[9], and sedimentation technique [10]). Table 5 presents the results
of Method-3 for PEG of Mp = 15,000. It is seen that although the
values of ||R|| are satisfactory in all cases, the predictions of the
model are not satisfactory. Among various values of n, only n = 1
gives a good estimate of the critical temperature. But in this case,
the prediction of the critical composition is poor. Rest of the values
of n neither predict correct critical temperature nor correct critical
composition. Hence Method-3 is not acceptable.
Based on the aforementioned analysis, we select the correlation
denoted by double asterisk in Table 3. The expression for the generalized Flory-Huggins parameter, based on this correlation, is given
below (T is expressed in Kelvin):
g(T, p ) = b0 (T ) + b1 (T )p + b2 (T )p2
(41)
where
b0 (T ) = 26.310 −
1.7168 × 103
− 3.5519 ln T
T
(42)
b1 (T ) = 24.657 −
1.5470 × 103
− 3.4314 ln T
T
(43)
and
b2 (T ) = 9.0804 −
6.9445 × 102
− 1.2222 ln T
T
(44)
The accuracy of the correlation is illustrated through parity plots
of the heat of mixing of solutions of PEG in Fig. 4(a) and (b). Abscissa
represents experimental values and the ordinate represents the values predicted using the correlation. In Fig. 4(a), we have used the
data from our experiment for different molecular weights of PEG
and in Fig. 4(b), we have presented the experimental data of Grossman et al. [8] for two molecular weights of PEG. It is seen that except
for some points in Fig. 4(b), rest of the points in both the figures lie
close to the parity line. The disparate points correspond to very high
concentrations of PEG (55–95%, w/w) and they lie above the parity
line. It is known that PEG forms aggregate in water at temperatures below 340 K and concentrations above 50% (w/w) [31,32]. As
a result, PEG is only partially hydrated in this region and we expect
lower values of enthalpy of mixing than that predicted using complete hydration. This explains why predicted values in Fig. 4(b) lie
above the parity line in the high-concentration region. However,
since these concentrations are outside the binodal region, these
deviations do not affect the predictions of the binodal curve.
Further validation of the correlation is sought by comparing it
with the reported experimental binodal curves for the molecular
weights of PEG in the range of 8000–35,000. Comparison is shown
in Fig. 5 and indicates a reasonably good match between the experiments and the predictions.
Attempt was made to extend the present correlation to predict
the closed loop binodal curve for aqueous PEG15000 as shown in
Fig. 6. It is seen that the binodal curve predicted by the model does
not close around the upper critical solution temperature (UCST). A
possible reason for this deviation is the failure of our assumption
that vp /vw is independent of temperature, pressure and the solution
composition. The assumption of constant vp /vw is most likely to be
valid for temperatures up to and around LCST (it has been shown
that LCST is independent of pressure over a wide range [33], indi-
Table 5
Analysis of Method-3.
n
i=0
bi (T )pi
bi˛ (i = 0, 1, . . ., n)
biˇ × 10−3 (i = 0, 1, . . ., n)
n=1
b0˛ = 24.543, b1˛ = 26.456
b0ˇ = −1.5741, b1ˇ = −1.6449
n=2
b0˛ = 33.088, b1˛ = 29.133,
b2˛ = 24.166
b0ˇ = −2.0396,
b1ˇ = −1.7622, b2ˇ = −1.4011
n=3
b0˛ = 99.452, b1˛ = 116.19,
b2˛ = −40.887, b3˛ = 314.57
b0ˇ = −5.1531,
b1ˇ = −5.8465, b2ˇ = 1.6542,
b3ˇ = −14.766
bi (i = 0, 1, . . ., n)
b0 = −3.3110,
b1 = −3.6730
b0 = −4.5326,
b1 = −4.0860,
b2 = −3.3951
b0 = −14.344,
b1 = −16.936, b2 = 6.1623,
b3 = −46.441
||R|| × 102
Critical constants
Tc (K)
pc
3.9574
387.44
0.2818
3.0753
350.83
0.0303
2.9568
414.50
0.0292
50
Fig. 5. Comparison of the predicted binodal curves for aqueous solutions of PEG
with the experimental data. Predicted binodal curves are based on the correlation
presented in Eqs. (41)–(44). They are represented by lines: (—) for PEG8000, (- - -)
for PEG15000, and (· · ·) for PEG35000. Experimental data are indicated by points:
() Bae et al. [Mp = 8000] [11], (䊉) Saraiva et al. [Mp = 8420] [12], () Bae et al.
[Mp = 15,000] [11], () Fisher et al. [Mp = 12,000] [14], () Fisher et al. [Mp = 15,481]
[14], ( ) Fisher et al. [Mp = 33,500] [14].
Fig. 4. (a) Parity plot of enthalpy of mixing of PEG solutions. Predicted values of the
heat of mixing are based on Eq. (9) and the correlation presented by Eqs. (41)–(44).
The experimental data are from the present work: () PEG4600, () PEG8000, ()
PEG20000, () PEG35000. (b) Parity plot for enthalpy of mixing of PEG solutions.
Predicted values of the heat of mixing are based on Eq. (9) and the correlation presented by Eqs. (41)–(44). The experimental data are from Grossmann et al. [8]: ()
PEG6230, T = 298.15 K; () PEG39000, T = 298.15 K; () PEG6230, T = 333.15 K; and
(䊉) PEG39000, T = 333.15 K.
of the generalized Flory-Huggins model. In Method-2, all constants
of hydrogen bond model, i.e. ah , bh , w and p are estimated using
the reported data of the activity of water in PEG solutions, measured
using sedimentation technique [10] and vapor pressure osmometry
[6]. In Method-3, the combined data on the enthalpy of mixing and
activity of water (measured using vapor pressure osmometry [6],
laser-light scattering [7], isopiestic method [7,8], dew point method
[9], sedimentation technique [10]) are used for estimation of all
hydrogen bond constants. Table 6 lists the estimated values of the
constants obtained by the three methods along with l2 -norm of the
residuals ||R||. For comparison, we have also listed the values of the
parameters reported by Dormidontova [23]. These are obtained by
fitting the coexistence curve. We find a large difference among the
four sets of estimates.
cating that vp /vw is independent of pressure). However, UCST lies
near the critical point of water (647.1 K) (The actual critical point
is expected to decrease further with addition of PEG and likely to
be close to UCST). Since large changes in density of the solution
are expected near UCST, the ratio vp /vw is expected be much more
sensitive to both the pressure and the temperature in this region.
Unfortunately, the density data for this system near UCST are not
available, and hence our conjuncture cannot be verified.
5.2. Hydrogen bond model
The results of the regression analysis based on the hydrogen
bond model are done using Method-1, -2 and -3. In Method-1, the
enthalpy of mixing is used to estimate the parameters bh , w and
p . The temperature independent parameter ah is estimated from
the reported data on the activity of water in PEG solution. We use the
activity data obtained by the sedimentation technique at 303.15 K.
This choice is based on our experience from the regression analysis
Fig. 6. Prediction of closed loop binodal curve for PEG15000. The solid line represents the prediction. The reported experimental data are: () Bae et al. [Mp = 15,000]
[11]; () Fisher et al. [Mp = 15,481] [14].
51
Table 6
Estimates of the parameters of the hydrogen bond model.
Data used
||R|| × 102
Model parameters
ah
bh
w
p
Enthalpy of mixing
Sedimentation [10], Temperature: 303.15 K
–
0.6792
−145.46
–
/7.3874
–
/5.7172
–
6.7088
1.0084
Method-2
Sedimentation [10]
Vapor pressure osmometry [6]
1.0131
−0.2519
−284.82
341.31
/9.5431
/1.3387
/8.5286
/1.3030
11.386
2.6070
Method-3
Enthalpy of mixing and reported activity data
0.4889
−155.08
/7.4461
/9.3738
9.3040
Reported model parameters [23]
–
−0.211
93.5
/4.75
/8.35
Method-1
Fig. 7. (a) Parity plot of activity of water in PEG solutions. Predicted values of the activity of water are based on Eq. (27) and the parameters of hydrogen bond model presented
in Table 6 (Method-1). The experimental data are from sedimentation [10] method at 303.15 K () and isopiestic [8] technique at 333.15 K (). (b) Parity plot of enthalpy of
mixing of aqueous PEG solutions. Predicted values of the enthalpy of mixing are based on hydrogen bond model [23] (Eqs. (9), (29) and (30)) and the parameters of hydrogen
bond model (Method-1) presented in Table 6. The experimental data are from the present study represented by points: () T = 288.15, () T = 298.15, () T = 308.15, ()
T = 318.15, (♦) T = 328.15, (夽) T = 338.15, and ( ) T = 348.15 K.
The sets of the parameters obtained using Method-1, -2 and
-3 were used for estimation of the lower solution critical temperatures by solving the relevant equations (Eqs. (21), (22), (33)
and (34)). However, all three methods failed since Eqs. (33) and
(34) did not yield a solution. To diagnose the reason for this failure, we have drawn the parity plots of activity of water [8,10]
(Fig. 7(a)) and enthalpy of mixing (Fig. 7(b)). Abscissa represents
the experimental values and the ordinate represents the values
predicted by the hydrogen bond model (Method-1). It is seen
from Fig. 7(a) that the parity is excellent at 303.15 K, but poor at
333.15 K. This indicates that the hydrogen bond model of Dormidontova [23] does not depict the effect of temperature on the
activity correctly. This fact is more clearly evident from the parity
plot of enthalpy of mixing (Fig. 7(b)), which show a considerable disparity of the model predictions from the experimental
results.
Based on this analysis, our previous choice of the correlation, given by Eqs. (41)–(44), in conjunction with Eq. (1), is still
the best one. We therefore accept it as the final choice. The
normalized sensitivity coefficients (NSC) of the constants of the correlation with respect to the critical temperature (LCST) are listed in
Table 7:
NSC =
bij
Tc
dTc
dbij
(i = 0, 1, 2; j = ˛, ˇ, )
(45)
From Table 7, it is seen that the critical temperature is
very sensitive to the parameters: b0˛ , b1˛ , b0 and b1 . Since
b0˛ , b1˛ are estimated from the activity data and b0 and b1
from the data on enthalpy of mixing, the accuracy of both
these measurements are important for the accuracy of the final
correlation.
Table 7
Normalized sensitivity coefficients (NSC) of the constants in the accepted correlation.
˛
j⇒
i
ˇ
bi˛
NSCa
biˇ
NSCa
bi
NSCa
26.310
24.657
9.0804
−69.641
37.945
7.2532
−1.7168 × 103
−1.5470 × 103
−6.9445 × 102
11.707
−6.1330
−1.4290
−3.5519
−3.4314
−1.2222
56.047
−31.480
−5.8199
⇓
0
1
2
a
NSC =
bij
Tc
dTc dbij
(i = 0, 1, 2; j = ˛, ˇ, ).
52
Fig. 8. Activity of water in PEG solutions by sedimentation technique [10]: comparison with the correlation. Experimental data (Mp = 8000) are represented by points:
() T = 283.15, (䊉) T = 293.15, () T = 303.15, and () T = 313.15 K. The lines correspond
to prediction based on the correlation (—) T = 283.15, (- - -) T = 293.15, (- · -) T = 303.15,
and (· · ·) T = 313.15 K.
We use our correlation to grade the reported experimental data
on the activity of water in aqueous PEG solutions. The values of
the molecular weight independent activity (ˇw ) are compared with
the correlation in Figs. 8–12. The standard deviations are listed in
Table 8. We see from these figures and Table 8 that barring for
the small deviations, almost all the reported experimental data
show a closed match with the correlation. Among these, sedimentation (T > 298.15 K) and vapor pressure osmometry are the best
methods, followed by dew point, light scattering and isopiestic
techniques.
Although, the activity data are close agreement with the correlation, when used alone they are not adequate to provide
us a correlation which can predict the critical constants and
Fig. 9. Activity of water in PEG solutions by dew point method [9]: comparison
with the correlation. Experimental data (T = 298.15 K) are represented by points: ()
Mp = 6000, (䊉) Mp = 8000, () Mp = 10,000, and () Mp = 20,000. Prediction is indicted
by (—) solid line.
Fig. 10. Activity of water in PEG solutions by isopiestic method [8]: comparison with the correlation. Experimental data are represented by points: (䊉)
T = 293.15 K, Mp = 6230, () T = 293.15 K, Mp = 39,000, () T = 333.15 K, Mp = 6230, and
() T = 333.15 K, Mp = 39,000. Predictions are indicted by lines: (—) T = 293.15, (- - -)
T = 333.15.
the binodal curve. The reason is the high sensitivity of the
critical constants to the errors in the measurement of the activity. Activity data with much higher accuracy and over wider
temperature range are needed for this purpose and the existing measurement techniques are not capable of achieving the
needed accuracy and the range of temperatures. The use of
calorimetry is therefore essential in order achieve an accurate correlation.
Fig. 11. Activity of water in PEG solutions by combination of isopiestic and laser
light scattering [7] methods: comparison with the correlation. Experimental data
are represented by points: (䊉) T = 298.15 K, Mp = 6700; () T = 298.15 K, Mp = 20,000;
() T = 298.15 K, Mp = 34,400; () T = 313.15 K, Mp = 6700; and () T = 313.15 K,
Mp = 34,400. Predictions are represented by lines: (—) T = 298.15, (- - -) T = 313.15.
53
parameters of the correlation, calorimetric data needs to be supplemented with activity data, it has been shown that correlation
based on the activity data alone is far from accurate. This brings
out the key role of the calorimetric measurements. The correlation has been used to grade the techniques for measurement of
activity of PEG solutions. It is also shown that the hydrogen bond
model of Dormidontova [23] inaccurate in predicting the effect of
temperature on the activity.
Acknowledgements
The authors would like to thank (i) Unilever Industries Private
Limited for providing the funding for the research and (ii) Professor
V.M. Naik from Indian Institute of Technology Bombay for valuable
suggestions.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.fluid.2009.01.003.
Fig. 12. Activity of water in PEG solutions by vapor pressure osmometry [6]: comparison with the correlation. Experimental data are represented by points: ()
T = 308.15 K, (䊉) T = 318.15 K, () T = 328.15 K, and () T = 338.15 K. Predictions are
represented by lines: (—) T = 308.15 K, (- - -) T = 318.15 K, (· · ·) T = 328.15 K and (- · -)
T = 338.15 K.
Table 8
The standard deviation between predicted and reported activity of water.
a × 103
Method
Base temperature (K)
Sedimentation [10]
283.15
293.15
303.15
308.15
313.15
3.0061
0.5374
0.0707
0.5551
1.1424
Dew point method [9]
298.15
2.6035
Laser light scattering and Isopiestic
method [7]
293.15
333.15
1.8712
17.855
Isopiestic method [8]
298.15
313.15
2.5304
8.3329
Vapor pressure osmometry [6]
308.15
318.15
328.15
338.15
0.1112
0.5561
0.2291
0.0771
a
=
ndata 1/2
rjexp −rjmodel 2
(N−1)
.
j=1
6. Conclusions
We have demonstrated the use of the data on the enthalpy of
mixing of PEG solutions, to obtain a correlation based on the generalized Flory-Huggins theory, which is capable of predicting, at one
end, the low-temperature activity of water in PEG solutions, and,
at the other end, the phase separation behavior of PEG–water system. This is possible due to the ability of calorimetry to correctly
estimate the dependence of the activity on temperature, through
Gibbs-Helmholtz equation. Although, in order to estimate all the
References
[1] R. Hatti-Kaul, Mol. Biotechnol. 19 (2001) 269–278.
[2] J.G. Huddleston, H.D. Willauer, S.T. Griffin, R.D. Rogers, Ind. Eng. Chem. Res. 38
(1999) 2523–2539.
[3] R.D. Rogers, A.H. Bond, C.B. Bauer, J. Zhang, S.T. Griffin, J. Chromatogr. B 680
(1996) 221–229.
[4] J. Chen, S.K. Spear, J.G. Huddleston, R.D. Rogers, Green Chem. 7 (2005) 64–82.
[5] C. Wohlfarth, Thermodynamic Data of Aqueous Polymer Solutions, CRC Press,
New York, 2003.
[6] A. Eliassi, H. Modarress, G.A. Mansoori, J. Chem. Eng. Data 44 (1999) 52–55.
[7] H. Hasse, H.P. Kany, R. Tintinger, G. Maurer, Macromolecules 28 (1995)
3540–3552.
[8] C. Grossmann, R. Tintinger, J. Zhu, G. Maurer, Fluid Phase Equilib. 106 (1995)
111–138.
[9] L. Ninni, M.S. Camargo, A.J.A. Meirelles, Thermochim. Acta 328 (1999) 169–176.
[10] C.B. Stanley, H.H. Strey, Macromolecules 36 (2003) 6888–6893.
[11] Y.C. Bae, J.J. Shim, D.S. Soane, J.M. Prausnitz, J. Appl. Polym. Sci. 47 (1993)
1193–1206.
[12] A. Saraiva, O. Persson, A. Fredenslund, Fluid Phase Equilib. 91 (1993) 291–311.
[13] W. Borchard, S. Frahn, V. Fischer, Macromol. Chem. Phys. 195 (1994)
3311–3324.
[14] V. Fischer, W. Borchard, M. Karas, J. Phys. Chem. 100 (1996) 15992–15999.
[15] V. Fischer, W. Borchard, J. Phys. Chem. B 104 (2000) 4463–4470.
[16] M. Yu, H. Nishiumi, J.D.S. Arons, Fluid Phase Equilib. 83 (1993) 357–364.
[17] W. Li, Z.Q. Zhu, M. Li, Chem. Eng. J. 78 (2000) 179–185.
[18] E.A. Tritopoulou, G.D. Pappa, E.C. Voutsas, I.G. Economou, D.P. Tassios, Ind. Eng.
Chem. Res. 42 (2003) 5399–5408.
[19] Y. Hu, S.M. Lambert, D.S. Soane, J.M. Prausnitz, Macromolecules 24 (1991)
4356–4363.
[20] R. Kjellander, E. Florin, J. Chem. Soc. Faraday Trans. I 77 (1981) 2053–2077.
[21] G.A. Karlstrom, J. Phys. Chem. 89 (1985) 4962–4964.
[22] A. Matsuyama, F. Tanaka, Phys. Rev. Lett. 65 (1990) 341–344.
[23] E.E. Dormidontova, Macromolecules 35 (2002) 987–1001.
[24] E.E. Dormidontova, Macromolecules 37 (2004) 7747–7761.
[25] S.K. Pattanayek, V.A. Juvekar, Macromolecules 35 (2002) 9574–9585.
[26] E.E. Dormidontova, G.T. Brinke, J. Chem. Phys. 113 (2000) 4814–4826.
[27] C. Qian, S.J. Mumby, B.E. Eichinger, Macromolecules 24 (1991) 1655–1661.
[28] R.D.C. Cruz, R.J. Martins, M.J.E.D. Cardoso, O.E. Barcia, J. Appl. Polym. Sci. 91
(2004) 2685–2689.
[29] L. Ninni, H. Burd, W.H. Fung, A.J.A. Meirelles, J. Chem. Eng. Data 48 (2003)
324–329.
[30] C. Wohlfarth, in: G. Wypych (Ed.), Handbook of Solvents, ChemTec Publishing,
Toronto, 2001, pp. 146–242.
[31] S. Bekiranov, R. Bruinsma, P. Pincus, Phys. Rev. E 55 (1997) 577–585.
[32] B. Hammouda, D.L. Ho, S. Kline, Macromolecules 37 (2004) 6932–6937.
[33] S. Saeki, N. Kuwahara, M. Nakata, M. Kaneko, Polymer 17 (1976) 685–689.