Heat capacity anomaly in a self-aggregating system

Heat capacity anomaly in a self-aggregating system: Triblock copolymer 17R4 in
water
Lorenzo V. Dumancas, David E. Simpson, and D. T. Jacobs
Citation: The Journal of Chemical Physics 142, 174902 (2015); doi: 10.1063/1.4919633
View online: http://dx.doi.org/10.1063/1.4919633
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THE JOURNAL OF CHEMICAL PHYSICS 142, 174902 (2015)
Heat capacity anomaly in a self-aggregating system: Triblock copolymer
17R4 in water
Lorenzo V. Dumancas, David E. Simpson, and D. T. Jacobsa)
Department of Physics, The College of Wooster, Wooster, Ohio 44691, USA
(Received 9 March 2015; accepted 9 April 2015; published online 4 May 2015)
The reverse Pluronic, triblock copolymer 17R4 is formed from poly(propylene oxide) (PPO) and
poly(ethylene oxide) (PEO): PPO14 − PEO24 − PPO14, where the number of monomers in each block
is denoted by the subscripts. In water, 17R4 has a micellization line marking the transition from
a unimer network to self-aggregated spherical micelles which is quite near a cloud point curve
above which the system separates into copolymer-rich and copolymer-poor liquid phases. The phase
separation has an Ising-like, lower consolute critical point with a well-determined critical temperature
and composition. We have measured the heat capacity as a function of temperature using an adiabatic
calorimeter for three compositions: (1) the critical composition where the anomaly at the critical point
is analyzed, (2) a composition much less than the critical composition with a much smaller spike when
the cloud point curve is crossed, and (3) a composition near where the micellization line intersects
the cloud point curve that only shows micellization. For the critical composition, the heat capacity
anomaly very near the critical point is observed for the first time in a Pluronic/water system and
is described well as a second-order phase transition resulting from the copolymer-water interaction.
For all compositions, the onset of micellization is clear, but the formation of micelles occurs over a
broad range of temperatures and never becomes complete because micelles form differently in each
phase above the cloud point curve. The integrated heat capacity gives an enthalpy that is smaller than
the standard state enthalpy of micellization given by a van’t Hoff plot, a typical result for Pluronic
systems. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4919633]
I. INTRODUCTION
We investigate the reverse Pluronic, triblock copolymer
17R4 that has a structure PPO14 − PEO24 − PPO14, with the
subscripts denoting the number of monomers in each block.
In H2O, 17R4 is an entropically driven system that shows
both micellization at lower temperatures and separation into
coexisting liquid phases at higher temperatures.1 We present
here new data for the heat capacity due to micellization and to
phase separation of 17R4 in H2O as functions of temperature
and composition that include a composition that passes through
the lower critical solution point.2 This system is particularly
interesting because the micellization line is close to the cloud
point curve.1
A. Self-aggregation into micelles
Micellar systems interest scientists because they are a
beautiful example of self-assembly, the molecules spontaneously forming higher order structures. Micelles can form from
triblock copolymers when one block of the polymer is attracted
to the solvent and one block of the copolymer is repelled by
the solvent. These amphiphilic copolymers, or “unimers,” can
self-assemble into spherical aggregates, or “micelles,” that
are dynamic entities, nanometers in diameter, in chemical
equilibrium with the free copolymers, and polydisperse in
a)Author to whom correspondence should be addressed. Electronic mail:
djacobs@wooster.edu
0021-9606/2015/142(17)/174902/6/$30.00
size and aggregation number (number of copolymers in the
micelle).
When micelles form, they all do not contain the same
number of copolymers and all do not have the same size.
The nature of the distribution of sizes reflects all the
factors that control the micelle formation. Micelles, like
synthetic polymers, are always polydisperse. In fact, they
are polydisperse at several levels: the relative block lengths
and total molecular masses of the copolymers are not exactly
the same, the aggregation numbers of the resulting micelles
are polydisperse, and the resulting sizes of the micelles are
polydisperse.3 When a second liquid phase is present, the
micelles can distribute themselves between the phases and can
have different distributions in the coexisting phases, reflecting
differences in molecular interactions in the two phases.
Micellizations can be entropically driven (∆Smic > 0,
∆Hmic > 0, typical when water is the solvent) or enthalpically
driven (∆Smic < 0, ∆Hmic < 0, for organic solvents). Micellization occurs when ∆Gmic = ∆Hmic − T∆Smic < 0, which is
only above a temperature Tfloor in the first case, or only below
a Tceiling in the second case. The copolymer concentration
at which micellization occurs at constant temperature is
called the critical micelle concentration or CMC while the
temperature at which micellization occurs at constant copolymer concentration (Tfloor or Tceiling) is the critical micelle
temperature or CMT. The CMT and CMC are two ways of
looking at the same micellization line that separates a region
of unimers from a region with micelles.
142, 174902-1
© 2015 AIP Publishing LLC
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Dumancas, Simpson, and Jacobs
J. Chem. Phys. 142, 174902 (2015)
The temperature dependence of the micellization line
can be related to thermodynamic quantities using the van’t
Hoff plot of ln (CMC) plotted versus 1/T, where the slope
0
determines the standard state enthalpy of micellization ∆Hmic
while the intercept gives the standard state entropy of
0
micellization ∆Smic
,
ln (CMC) =
0
∆S 0
∆G0mic ∆Hmic
=
− mic ,
RT
RT
R
(1)
which has been derived by Tanford4 for ideal solutions
of monodisperse polymers when the micelle aggregation
number is large. The assumptions leading to Eq. (1) have
been discussed in our previous paper5 where we found the
micellization line for 17R4/H2O to be described by Eq. (1) with
0
0
∆Hmic
= 125 ± 5 kJ/mol, ∆Smic
= 461 ± 17 J/(mol K), and
0
thus ∆Gmic = −20 ± 11 kJ/mol even though the aggregation
number is small, only about 10 in this system.1
Thus, the extent of micellization can be roughly calculated
from heat capacity measurements.
C. Phase boundaries and critical point
In addition to the micellization, the reverse Pluronic
17R4 in aqueous solution exhibits a macroscopic separation
into two liquid phases.1,5 Because of the polydispersity
of the copolymer, the cloud point curve is not the same
as the coexistence curve in this system.5,13 This differs
from a monodisperse system such as a polymer of narrow
polydispersity in a solvent,14 or methanol and cyclohexane,15
in which the opalescence due to the onset of phase separation—
the “cloud point”—coincides with the coexistence curve. This
distinction is discussed in some detail by Huff et al.,5 who
measured the micellization line, coexistence curves, and the
cloud point curve in 17R4/H2O as shown in Fig. 1.
D. Structure
B. Heat capacity and enthalpy
The heat capacity is a good measure of any energy loss or
gain in the sample that could arise from a background specific
heat of the solvent, a structural change in the unimers forming
micelles with the associated expulsion of water molecules,
micelles aggregating, or the formation of correlated regions of
unimers in the solvent as occurs near a critical point. While the
0
only captures
standard state enthalpy of micellization ∆Hmic
the formation of micelles, the integral of the heat capacity
captures the enthalpies associated with the many processes
0
just listed. The difference between ∆Hmic
and the enthalpy
∆Hmic from integrating the heat capacity has been recently
detailed.6 Experiments on Pluronic (PEO-PPO-PEO) systems
0
,
have found that ∆Hmic is considerably smaller than ∆Hmic
an effect that is attributed to the copolymer polydispersity
where the longer PEO blocks determine the C MC but the
calorimetric enthalpy relates to the whole sample.7 Published
calorimetry data on aqueous solutions of Pluronics resulted
0
to be 0.94 for L44,8 0.86 for
in the ratio of ∆Hmic/∆Hmic
P237,9 and 0.65 for P3339 and for P94.7 The broad shape of
the micellization’s heat capacity as a function of temperature
indicates that while the onset of micellization is fairly sharp,
micelles continue to form as temperature increases, typically
until an equilibrium saturation occurs when the heat capacity
returns to a background value.
Experiments almost always measure the heat capacity at
constant pressure Cp , which has been shown by Artyomov and
Freed10 to be considerably less than incompressible model
predictions for the heat capacity C (C ≈ 1.4Cp ) by extending a Flory-Huggins type model to compressible systems
to accommodate the volume change on polymerization or
micellization.11 They note that the approximate relation of
Wheeler et al.12 for the extent of polymerization (equivalent
to micellization) Φ predicted from an incompressible model
would be larger by the factor 1.4 when measuring Cp in a
compressible system,
1.4
Φ(T) ≈
0
∆Hmic

Cp dT.
(2)
The copolymer structure of 17R4/D2O has recently been
measured16 using dynamic light scattering and small-angle
neutron scattering in the three regions that roughly correspond
to the phase diagram in Fig. 1. Region I is cloudy due to
large clusters of hydrophobic PPO rich “knots” bridged by
dissolved PEO chains that form a loose network that scatters
light readily. Region I is sharply separated from region II where
flower micelles reversibly form with PPO blocks at the center
and PEO blocks looping into the solvent while the network of
“knots” disappears so the sample is clear. Two phases coexist
in region III: one phase has a high concentration of 17R4
while the other has a much lower concentration5 with micelles
in both phases.16 The unimers in region II for 17R4/D2O have
a characteristic hydrodynamic radius of 0.28 nm at 35 ◦C and
0.49 nm at 40 ◦C while the flower micelles are about 3.0 nm.
Zhou and Chu1 used light scattering on 17R4/H2O to determine
the unimer size as 0.14 nm and the micelle size as 4.0 nm at
FIG. 1. Phase diagram of 17R4 in water (closed symbols from Ref. 5; open
symbols from Ref. 1). Region I is unimers linked in a loose network and
separated by the micellization line (dotted-dashed curve) from region II that
has unimers and spherical micelles. Region II is separated from region III by
the cloud point curve (solid line). Region III has two phases with the lower
phase rich in 17R4. CP marks the lower critical solution point. The vertical
dashed lines show the compositions measured in this experiment with the red
triangles marking the observed CMT and cloud points.
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Dumancas, Simpson, and Jacobs
J. Chem. Phys. 142, 174902 (2015)
growth of critical fluctuations that occurs in addition to the
micellization process. The heat capacity due to the critical
point appears as a lambda-like spike that is largest for the
critical composition (volume fraction φc = 0.22 ± 0.01 for
17R4/H2O)5 but can be observed as a smaller spike at other
compositions when crossing the cloud point curve.
While the measurements reported here are the first to
probe the heat capacity for a Pluronic in aqueous solution at
the critical composition, others have reported a small spike in
the heat capacity when crossing the cloud point curve in the
Pluronic/water systems L64,1,23,24 25R425–27 and F68, F88,
F98, and F108.28
FIG. 2. The turbidity in the three regions shown in Fig. 1 for a 0.163 volume
fraction of 17R4 in water. In region III, the inverted triangles are the lower
phase (rich in 17R4) and the upright triangles are for the upper phase.
II. MATERIALS AND METHODS
A. Sample preparation
40 ◦C. A larger micelle size is expected when the solvent is
H2O rather than D2O.16
E. Turbidity
The cloudy and clear regions can be measured by the
turbidity τ, which is an extinction coefficient for a transmitted
beam of light that enters the sample of length L with an intensity I0 and exits with an intensity It : τ = −(1/L) ln(It /I0).14
Using a measurement technique described previously,14 we can
illustrate the cloudiness for one composition of 17R4 and water
in Fig. 2. The large turbidity values in region I indicate cloudiness while the small values in region II show how clear the sample becomes. Region III has different turbidities in each phase
reflecting the different concentration and aggregation number
of micelles in each phase. These turbidity results support the
phase diagram previously reported5 and shown in Fig. 1.
F. Critical phenomena
The 17R4/H2O phase diagram shows a lower critical
solution point (critical point (CP) in Fig. 1).5 Phenomena near
liquid-liquid critical points can be described by power laws in
the reduced temperature (t = |Tc − T |/Tc ) with universal Ising
exponents.2,17,18 The shape of the heat capacity Cp near the
critical point is given by
Cp = C0 +
A± −α
t ,
α
(3)
where C0 is the background heat capacity, A± are the
amplitudes in the one-phase (+) and two-phase (−) regions
of the sample, and α = 0.11 is the universal, Ising exponent
value.18–20 The location of the critical point in temperature
and composition (Tc and φc ) is not universal and depends
on the system. Theory and experiments in liquid-liquid and
polymer-solvent systems give the universal relationship19,21
A+/A− = 0.54, while A+ is related to the amplitude of
the correlation length amplitude ξ0 by A+ξ03/k B = 0.019.22
When a system is near its CP, the size of the composition
fluctuations is measured by the correlation length that diverges as the power-law ξ = ξ0t −0.63 as the critical point is
approached (t → 0). The heat capacity can determine the
The reverse Pluronic 17R4 sample, PPO14 − PEO24
− PPO14, was a gift from BASF chemical company. BASF
reported that the sample had an average molecular mass of
2650 g/mol, a density of 1.048 g/ml, and a 0.09 wt. % water
content. The 17R4 came from the same source and batch as
our previous work5 and the more recent structure paper16 and
its purity has been described. The H2O used was from the
Barnstead NANOpure system, with resistivity greater than
18 MΩ-cm.
Each copolymer solution was prepared by massing the
desired amount of copolymer into a glass vial and adding
nano-pure water until the total desired composition was
achieved. All samples were vigorously shaken to ensure
complete mixing and then the heat capacity cell was filled
with approximately 17 ml before sealing. Mass fractions were
converted to volume fractions by using the densities of H2O
and 17R4 and by assuming no change of volume on mixing.
Three compositions were prepared and measured: φ = 0.226,
0.126, and 0.071 volume fraction 17R4 in water with an
uncertainty of ±0.001 for each.
B. Adiabatic calorimeter
The heat capacity of each sample was measured using an
adiabatic calorimeter, which we have already described.21,29
The fluid sample was sealed in a cylindrical, gold plated,
copper “cell” with a Kalrez (DuPont perfluoroelastomer)
o-ring. The cell is within a set of nested cylinders that are
temperature controlled using a LabVIEW program that measures and controls the temperature of each stage to follow the
temperature of the cell (measured to 0.1 mK). The fluids are
sloshed while heating to achieve thermal equilibrium, which
is especially important in the two-phase region.30 The cell
is heated for a fixed period of time by applying a constant
current to a manganin wire wrapped in grooves on the outside
of the cell body. After the heater is turned off and the sloshing
stops, the thermostat reaches a constant plateau temperature
before the next heating period. This step process ensures the
adiabatic condition and has been described previously.21,29
This corresponds to a very slow scan rate of 1 mK/min
versus the much faster scan rates of 500–1000 mK/min for
differential scanning calorimeters. A very slow scan rate is
particularly important near a critical point. The specific heat
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174902-4
Dumancas, Simpson, and Jacobs
FIG. 3. Heat capacity data are shown for three volume fraction concentrations of 17R4 in water: 0.226 (∆, red), 0.126 ( ⃝, black), and 0.071 (∇, green).
The uncertainty in the heat capacity is 1% which is about the size of the
symbols. A broad hump shows micellization while the sharp peak in the two
larger compositions indicates crossing the cloud point where the system phase
separates. The solid vertical lines correspond to the CMT values while the
dashed lines are the cloud point values, both shown as red symbols in Fig. 1.
of the empty copper cell was determined from a calibration
using nanopure water and was taken into account in reporting
the heat capacities reported here.
III. RESULTS AND DISCUSSION
We consider the heat capacity behavior of the three
copolymer concentrations of 17R4 and water in three regions:
Region I, where a cloudy, knotted network of unimers makes
the mixture opaque; region II, where micelles coexist with
free copolymers and the mixture is transparent with some
opalescence near the micellization line or near the cloud
point, and region III where two liquid phases coexist. Fig. 3
shows the heat capacity data of the fluid mixtures normalized
by the sample volume. Each composition starts with a nearly
constant heat capacity that increases at the micellization
temperature (CMT) as micelles form; the number of micelles
and perhaps their aggregation number continue to change
as the temperature increases, which is typical micellization
behavior in Pluronic systems.7 Our system is different than
others because the cloud point is close to the micellization
line as shown in Fig. 1. Thus for the two higher concentrations, the fluids separate into two phases at a temperature
before the micellization process is complete. The two phase
(cloud point) boundary is indicated by the sharp peak in
the heat capacity for those concentrations. It is particularly
pronounced for the critical composition sample and is much
smaller at the lower composition (0.126) due to being further
from the critical point. These two larger compositions show
a steep decline in Cp after crossing the cloud point curve
as micelles form differently in each phase due to the strong
temperature dependence of their 17R4 concentrations.5
The lowest composition (0.071) is near the intersection
of the micellization line and cloud point curve around 37.5
± 1 ◦C where micelles form in each phase. While the minimum of the cloud point curve is near this composition and
could be mistaken for the critical point, it is actually quite far
from the true critical point because of the polydispersity in
J. Chem. Phys. 142, 174902 (2015)
this system.5 Thus, there is no discernable spike in the heat
capacity due to the growth of critical fluctuations. In fact, the
17R4-rich lower phase would be very small in volume, giving
little contribution to the heat capacity averaged over the
whole volume. Thus, the heat capacity for the 0.071 composition measures both the changing number of unimers in the
upper phase5 as well as the micellization formed from them.
The onsets of micellization (CMT) are shown as the
vertical solid lines in Fig. 3 and are roughly determined
(1σ uncertainty of ±1 ◦C) from the gradual increase in the
heat capacity above a fairly constant value in region I. These
CMT values (37.5 ◦C for φ = 0.071, 35 ◦C for φ = 0.126,
and 31 ◦C for φ = 0.226) are consistent though systematically
smaller than those observed visually5 as a cloudy (region I)
to clear (region II) transition. The heat capacity indicates
that some micelles are beginning to form from the knotted
network16 of unimers in region I before the system becomes
totally clear with flower micelles in region II. This supports
the recent theoretical prediction for the onset of micelle
formation below the micellization line.31
The cloud point temperatures are at the spikes in the
heat capacity for the two larger compositions (shown as
vertical dashed lines in Fig. 3) and are better determined than
the CMT. The two cloud point temperatures (41.9 ± 0.1 ◦C
for φ = 0.126 and 44.60 ± 0.05 ◦C for φ = 0.226) are quite
consistent with those observed visually5 for comparable
concentrations (see Fig. 1).
A. Critical point
17R4/H2O has a lower critical solution point with a
coexistence curve that is well described as a simple powerlaw with an exponent consistent with that predicted by the
Ising model in critical phenomena.5 Our heat capacity data
for the critical composition φ = 0.226 can also be fitted to
the functional form predicted for critical phenomena as given
by Eq. (3), using the Ising exponent value for α = 0.11 and
the universal amplitude ratio A+/A− = 0.54 when within the
immediate vicinity of the critical point (within ±1 ◦C of Tc ).
The fitted curve is shown as the solid line in Fig. 4 where
the other parameters in Eq. (3) are C0 = 4.36 ± 0.05 (J/ml)/K
FIG. 4. For the critical composition sample (0.226), the sharp spike in
the heat capacity as the temperature goes through the critical point is well
described by an Ising-like critical anomaly as given by Eq. (3), (black solid
line).
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174902-5
Dumancas, Simpson, and Jacobs
and A+ = 0.0198 ± 0.0036 (J/ml)/K (uncertainties are quoted
at the 3σ level) with a reduced chi-square less than one. This
confirms the location of the critical point found previously5
and that this system falls within the Ising universality class
of critical phenomena. The value of A+ allows an estimate
for the correlation length amplitude as ξ0 = 0.35 ± 0.01 nm
which is consistent with the hydrodynamic radius for a 17R4
unimer (0.49 nm) found at 40 ◦C for 17R4/D2O.16 The unimer
(and not the micelle) size is the relevant comparison to
the correlation length since the critical point is due to the
interaction of water with just the copolymer.
B. Enthalpy and extent of micellization
The enthalpy can be determined from numerically integrating the heat capacity using a trapezoidal method once
the background heat capacity is subtracted and the units are
converted to (J/mol)/K. The resulting enthalpy values vary
from 50% to 70% of the standard state enthalpy of micel0
determined by a van’t Hoff plot.5 As described
lization ∆Hmic
earlier, the integrated heat capacity gives the enthalpy due to
several processes and not just to the formation of micelles
from a fixed number of unimers which the standard state
enthalpy represents.7 In addition, the micellization process
is incomplete since each composition is interrupted by the
cloud point causing different copolymer concentrations in
each phase that vary strongly with temperature.5 Even when
micellization was completed in other Pluronic systems, the
integrated heat capacity was typically much less than the
standard state enthalpy.9,32
The extent of micellization Φ can be calculated from
Eq. (2), for the three prepared compositions using the
measured heat capacities at constant pressure Cp and the stan0
dard state value5 of the enthalpy ∆Hmic
= 125 ± 5 kJ/mole.
The heat capacity associated with the critical point for
the critical composition 0.226 was subtracted before it was
integrated so the enthalpy better represented the micellization
process. The incomplete micellization seen in Fig. 5 is
consistent with the heat capacities not returning to their
J. Chem. Phys. 142, 174902 (2015)
baseline (region I) values. The uncertainties in the final value
of Φ results from two sources: the subtraction of the baseline
value of Cp and the addition in quadrature of the uncertainties
during the integration. This prediction of the extent of
micellization for this system could be tested in the future.
IV. CONCLUSIONS
We determined the average heat capacity at constant
pressure in three distinct regions as we varied temperature
and composition of the amphiphilic triblock copolymer
17R4/H2O system:33 Region I with no micelles but a knotted
network of unimers, region II where unimers and flower
micelles coexist, and region III where micelles form in each
of two phases.16 The heat capacity indicates the temperatures
of micellization that are consistent with, although slightly
smaller than, previous observations in this system. This
indicates some micellar formation below the micellization
line determined visually,5 as predicted recently.31
The heat capacity associated with the phase transition
as the system goes from region II to region III is observed
as a spike that is quite large at the critical composition
and smaller at a composition roughly half of that. For the
first time, the heat capacity in the immediate vicinity of the
critical point for a Pluronic/water system was observed and
analyzed to show that the phase transition is second order
with a critical exponent that is the universal Ising model value
seen in many diverse, near-critical systems. Using amplitude
relations predicted and verified in other systems, we can
determine a correlation length amplitude that is consistent
with the hydrodynamic radius of unimers in region II found in
the similar system 17R4/D2O.16 This confirms that the phase
transition is due to the interaction of the copolymer and water
and not due to the micelles.
The enthalpy calculated by integrating the heat capacity
is significantly smaller than the standard state value, which
has also been observed in many Pluronic/water systems.32
Finally, we make a calculation of the extent of micellization
using the simple relation of Wheeler and co-workers.12
ACKNOWLEDGMENTS
We thank Sandra Greer for helpful conversations and
Kelly Patton for her assistance. This research was funded by
Research Corporation and by the NSF-REU Grant No. DMR0649112.
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2A.
FIG. 5. The extent of micellization can be calculated using the Wheeler
relation given by Eq. (2). The colors and symbols are the same as in Fig. 3:
volume fraction of 17R4 in water of 0.226 (∆, red), 0.126 ( ⃝, black), and
0.071 (∇, green). The uncertainties are smaller than the symbol size at low
temperatures but gradually accumulate from the numerical integration with
the error bars shown only for the final value for clarity.
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174902-6
11R.
Dumancas, Simpson, and Jacobs
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