Relationship between segmental anisotropy in polarizability and stationary
ultrasonically induced birefringence in polymer solutions
Hiroyasu Nomuraa, Satoru Andob, Tatsuro Matsuokab and Shinobu Kodab
a
Laboratory of Chemistry, Department of Natural Science, School of Science and Technology,
Tokyo Denki University, Hatoyama，Hiki-Gun，Saitama, 350-0394，Japan.
b
Department of Molecular Design and Engineering, Graduate School of Engineering, Nagoya
University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan.
Stationary ultrasonically induced birefringence in various polymer solutions was
measured in order to investigate its relation to the anisotropy in polarizability of polymer
chain. From the concentration dependence of stationary ultrasonically induced birefringence
in polystyrene(PS)-toluene, polycarbonate(PC)-chloroform, polybutadiene(PBD)-toluene
solutions, the intrinsic values of the birefringence were obtained. Linear relationship
between the intrinsic value of the birefringence per a segment and the segmental anisotropy in
polarizability was obtained.
In aqueous solutions of polyelectrolyte, sodium
polystyrenesulfonate (NaPSS) and tetramethylammonium polystyrenesulfonate (TMAPSS),
the birefringence decreased with ionic strength by addition of the salts. The stationary
birefringence per ultrasonic intensity for all polymer solutions investigated here decreased
with increasing frequency.
1. INTRODUCTION
Ultrasonically induced birefringence has been observed in various liquids, colloidal
and polymer solutions [1-12]. For the small anisotropic molecules such as liquid crystals,
the orientational relaxation time is smaller or of the same order of magnitude so that the
velocity gradient caused by ultrasound can directly induce the sinusoidal orientation. This
causes the sinusoidal birefringence and it is proportional to the ultrasonic amplitude A, that is,
the square root of the ultrasonic intensity W U , since the velocity gradient is proportional to
A [1-5,10]. On the other hand, for large anisotropic particles such as colloidal particles, the
orientational relaxation time is much larger than the period of the ultrasonic wave so that the
orientational motion cannot follow the sinusoidal velocity gradient. However, the radiation
1
pressure, which is one of the typical quadratic acoustic effects, produces the stationary torque
on the particle that induces the uniform and stationary orientation of the particles in the
solutions [1,6-9]. In this case, the stationary birefringence is induced. The induced
birefringence is proportional to the square of the ultrasonic amplitude A2, that is, the
ultrasonic intensity WU .
From the point of view of detection technique, the both “non-biased” [1-5,7,10,11] and
“biased” [6,8,9,11,12] techniques have been used in the experiment of the ultrasonically
induced birefringence. The latter can detect only the stationary birefringence since the
sinusoidal birefringence is averaged out in the temporal and spatial manner. The former
detects the phase retardation by the root mean square way so that it can detect both the
sinusoidal and stationary birefringence. If the both sinusoidal and stationary birefringence
co-exits, the predominant birefringence will be observed.
The mechanism of the ultrasonic induced birefringence in polymer solutions is of
strong interest since a polymer molecule is a long chain consisting of the small monomer unit.
Jerrard observed the root mean square of the ultrasonically induced birefringence, nrms in
polystyrene (PS)-toluene and polyisobutylene-cyclohexane solutions using the “non-biased”
technique [1, 12] and found that it was proportional to the square root of the ultrasonic
intensity, W U [1]. Recently we observed that not only the root mean square of the
birefringence nrms by the “non-biased” technique but also the stationary birefringence nst by
“biased” technique in polystyrene(PS)-toluene solutions [11]. The former was proportional
to the square root of the ultrasonic intensity, W U but the latter was proportional to WU. In
our previous study, it was suggested that the stationary birefringence, nst, observed by the
biased detection technique was to be related to the local motion of the polymer segment not to
the Rouse-Zimm mode [12]. However, the detailed mechanism was not clarified yet.
For polymer solutions, it is very important to know the interactions between the
segments in a chain. For the reason, it is better to measure the birefringence in lower
concentration ranges. In this study, we focused on the experimental studies about the
stationary birefringence because the sign of the birefringence is obtained and the high signal
to noise ratio is realized [11]. We paid our attention to the relationship between the
stationary birefringence and the segmental anisotropy in polarizability. First we obtained the
intrinsic values of the stationary birefringence for three polymer solutions and related those to
the segmental anisotropy in polarizability. Second we measured the ultrasonically induced
birefringence in aqueous solutions of the polyelectrolyte with addition of salt. Addition of
the salt to the polyelectrolyte solutions changes drastically the segment size and the
distribution of polymer chains.
We also measured the frequency dependence of the stationary birefringence to give a
2
clue to the clarification of the mechanism of the ultrasonically induced birefringence in
polymer solutions.
2. EXPERIMENTAL
2.1 Sample
Monodisperse polystyrene samples (PS) with molecular weight from 7.6×102 to
1.0×106 were purchased from the Polymer Source Inc. The index of polydispersity, Mw/Mn
is lower than 1.1. Polycarbonate (PC) sample was obtained from Aldrich chemical and its
weight-average molecular weight, Mw, was 6.4×104. Polybutadiene (PBD) sample was
purchased from the Scientific Polymer Products Inc. and its Mw was 200,000. The cis-form
content was 98wt%. To remove antioxidant in the PBD sample, it was dissolved in benzene
at 5wt% and precipitated by ethanol. The precipitate was dissolved in toluene again and
filtered and freeze-dried. Solvent toluene and chloroform were of spectral quality grade
purchased from Wako Pure Chemical Co. Ltd. without further purification.
Sodium polystyrenesulphonate (NaPSS) was purchased from Scientific Polymer Products
Inc. and its Mw was 7×104. The sample NaPSS was dialyzed with distilled water and
precipitated by acetone. The precipitate was dissolved in distilled water again and filtered
and freeze-dried. Polystyrenesulphonic acid (HPSS) was obtained by ion-exchange
operation. Tetrametylammonium polystyrenesulphonate (TMAPSS) were prepared by
neutralization of the HPSS with addition of an aqueous solution of tetramethylammonium
hydroxide (TMAOH), which was purchased from Nakalai Tesque, Inc. All solutions were
prepared with water deionized after distillation. Sodium chloride (NaCl) was purchased
from Wako Pure Chemical Co. Ltd. and tetrametylammonium chloride (TMACl) was
provided from Nacalai Tesque, Inc.
In this paper, the concentration is represented by the mole number of the monomer unit
in the 1 dm3 solutions.
2.2 Apparatus
The stationary birefringence nst was measured by the “biased” detection technique
previously described [8,11] from the variation of the light intensity transmitted through the
sample cell as,
nst 
 I   I 
4 d
I0
(1)
where  is the wavelength of the laser light (632.8nm), d is the optical path length (16mm)
3
and I0 is the transmitted light intensity with the analyzer being parallel with the polarizer.
The angle of the analyzer was set at a small offset angle , the variations of the transmitted
light intensity are described as I+ and I-, respectively. In the biased detection, the linear
birefringence is averaged both spatially and temporary and only the stationary component can
be detected and the sign of the birefringence can be determined. Ultrasonic intensity, WU,
was obtained from the light intensity diffracted by the sound wave [1-6,8-13]. Details of the
experimental setup were given elsewhere [11]. Frequency dependence studies were carried
out in the frequency range from 15 to 65 MHz. All experiments were performed at
temperature of 25C.
3. RESULTS AND DISCUSSION
3.1 Intrinsic values of the stationary ultrasonically induced birefringence and its relation
to the segmental anisotropy in polarizability
1
Figure 1 shows the concentration dependence of nst  WU C for PS-toluene,
1
PC-chloroform and PBD-toluene solutions. Data of nst  WU for the PS and PC solutions
cited from the literature [12]. Sign of birefringence of the PBD solution is negative likely to
the PC solution. A PBD chain has no side group and its sign of the anisotropy in the
 WU
1
  lim n
C 0
st
 WU
1
C
(2)
-7
st
-2
-3
-1
n
-1
2
-1
3
nst·WU · C / 10 cm ·W ·dm ·mol
-1
polarizability is the same as that for PC molecules. The result for the PBD solution supports
that the main chain of the polymer perpendicularly aligns to the ultrasonic field as
investigated previously [12].
Extrapolated
values
of
1
nst  WU C take finite ones, we can
1
therefore define the intrinsic value of the
PS
0
stationary ultrasonic birefringence as
PBD
follows,
-1
In our previous study, we indicated that the
nst in polymer solution was related to the
local segmental motion [12]. Therefore it
is worth to relate the intrinsic value of the
stationary birefringence to the segmental
anisotropy in polarizability,  s .
-4
PC
-5
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
-3
C / mol·dm
Figure 1. Concentration dependence of
ultrasonically induced birefringence per
concentration in polymer solutions.
4
Table 1. Segmental anisotropy in polarizability and intrinsic value of the stationary
ultrasonically induced birefringence. Details for the calculation are described in the text.
Sample
Ms
ns
/-
/-
/-
104
254
54
850
490
200
8.2
1.9
3.7
s /
10-25cm3
[nst·Wu-1 ] ns 107/
cm2 W-1 dm3 mol-1
[nst·Wu-1 ] 107 /
cm2·W-1·dm3·mol-1
-135
147
27.9
0.33
-3.2
-0.27
2.7
-7.8
-1.7
6
from the molecular weight of monomer unit,
M 0 and the molecular weight of a segment,
M s , which was determined by limiting
-7
-1
birefringence with the anisotropy in
polarizability on the basis of the segment unit,
the intrinsic values of the birefringence
multiplied by the number of monomer unit in
1
a segment, ns , [nst  WU ]  ns was
calculated. The value of ns was estimated
4
2
PS
0
PBD
2
-1
3
The segmental anisotropy in polarizability
obtained from the flow birefringence [14-16]
are listed in the Table 1. To compare the
[nst·WU ]·ns / 10 cm ·W ·dm ·mol
-1
PS
PC
PBD
M0
-2
-4
-6
PC
-8
-10
-12
-200 -150 -100
-50
0
50
100
150
200
s / 10 cm
-25
3
Figure 2. Plots of the intrinsic value of the
stationary
ultrasonically
induced
birefringence per a segment against
segmental anisotropy in polarizability.
rigidity of the Rouse mode obtained by the
viscoelastic
measurements
at
high
frequencies [17,18]. Figure 2 shows the plots the intrinsic values of the stationary
birefringence per a segment against the segmental anisotropy in polarizability. The linear
relationship is obtained. This means that the stationary ultrasonically induced birefringence
of polymer solutions is related to the segmental anisotropy in polarizability of polymer chains.
3.2 Effect of the addition of the salt for the stationary birefringence in polyelectrolyte
solutions
The Kuhn statistical segmental anisotropy in polarizability,  s , is related to the
monomer anisotropy in polarizability,  m , as,
 s  Z m
(3)
where Z is the parameter which reflect to the stiffness of the polymer chain, the restricted
rotation and the steric effects between monomer units [19].
In aqueous solutions of
5
polyelectrolyte, we can change the stiffness
of polymer chain drastically with addition of
the salt. If the nst is related to the
anisotropy of the segment of polymer chains
as mentioned in the previous section, the
change in chain stiffness should influence
the nst values. Figure 3 shows the ionic
1
strength dependence of nst  WU for NaPSS
and TMAPSS solutions. The ionic strength
3.5
NaPSS:0.5mol/l (NaCl)
TMAPSS:0.5mol/l (TMACl)
2.5
2.0
1.0
0.5
PS
 / mPa·s
1.5
-1
-8
2
nst·WU / 10 cm ·W
-1
3.0
20
10
0
0.4
0.8
1.2
1.6
2.0
Ionic Strength
0.0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Ionic Strength
was controlled with the addition of salt.
The NaCl was added into the NaPSS
solutions, while the TMACl was added to
Figure 3. Ionic strength dependence of
ultrasonically induced birefringence in
aqueous solutions of NaPSS and TMAPSS.
Ionic strength dependence of the viscosity is
shown in the inset.
the TMAPSS solutions.
As shown in
1
Figure 3, the nst  WU value increased
with decreasing ionic strength.
This
tendency is related to the ionic strength
dependence of the viscosity as shown in the inset of Figure 3. It is well known that the
decrease of the persistence length with addition of the salts decreases the viscosity of aqueous
solutions of polyelectrolyte. Since the persistence length is directly correlated to the chain
stiffness, the segmental anisotropy in polarizability decreases with the addition of salt taking
Eq.(3) into consideration.
Therefore the decrease of the nst values with addition of the
1
salt is caused by the decease in the stiffness. The dashed line indicates that the nst  WU
value of PS-toluene solution at the same molarity. This result is quite reasonable because the
polyelectrolyte behaves flexible polymer in the presence of excess salt and the fact also
supports our assumption that the stationary birefringence of polymer solutions is directly
related to the segmental anisotropy in polarizability of polymer chains.
1
The nst  WU values of the NaPSS solution are smaller than those of TMAPSS
solutions at high concentration range. Sodium ions are bound to PSS ions through so called
the “ion-binding” effect [20-22], while TMA ions are not. This may cause the small
1
viscosity and also nst  WU values for the NaPSS solutions.
3.3 Frequency dependence of the stationary birefringence in polymer solutions
1
Frequency dependence of the nst  WU for PS-toluene, PC-chloroform, PBD-toluene
and TMAPSS-water solutions are shown in the Figures 4 (a) to (d), respectively. For all the
1
solutions, the nst  WU value decreases with increasing frequency likely to “relaxation
phenomena”. For PS-toluene solutions, the effect of the molecular weight was investigated.
6
10
-1
-1
8
-1
-8
12
35
MW=4,350
MW=6,400
MW=9,000
MW=250,000
6
4
2
0
10
20
30
40
30
(b)
-3
2
14
PS
-3
1.3mol·dm
PC 0.4mol·dm
25
-8
(a)
- nst·WU / 10 cm ·W
16
2
nst·WU / 10 cm ·W
-1
18
20
15
10
5
0
10
50 60 70
20
Frequency / MHz
30
40
50 60 70
Frequency / MHz
-1
(c)
-3
PBD 0.95mol·dm
2
5
(d)
-3
TMAPSS 0.5mol·dm
4
-8
4
3
-1
-1
-8
nst·WU / 10 cm ·W
5
2
- nst·WU / 10 cm ·W
-1
6
2
1
0
10
20
30
40
3
2
1
0
10
50 60 70
Frequency / MHz
20
30
40
50 60 70
Frequency / MHz
Figure 4. Frequency dependence of the stationary birefringence per ultrasonic intensity
for (a) PS-toluene, (b) PBD-toluene, (c) PC-chloroform and (d) TMAPSS-water solutions.
1
The difference in nst  WU values between at lower and higher frequency region increased
with increasing the molecular weight.
For the root mean square of birefringence, n rms , in polymer solution, Jerrard
1
observed that it was proportional to WU and the value of nrms  WU
increased with
increasing frequency in the frequency range from 1 to 5MHz [1]. It can then be concluded
that the sinusoidal birefringence, i.e. the linear birefringence, in polymer solutions is caused
by the sinusoidal velocity gradient generated by the ultrasound.
For colloid solutions, the stationary birefringence increased with increasing frequency
of the ultrasound [1,6,8]. The radiation pressure caused by the velocity difference between
the fluid and the particles produces the torque and it produces the stationary orientation of the
disk- or rod-like particle to the preferential direction. The stationary birefringence is then
observed. Since the velocity difference between the fluid and particles increases with an
1
increase in frequency in a non-ideal fluid, the nst  WU values increased with frequency.
In polymer solutions, the normal stress difference can be observed and it is well
known the normal stress difference is a typical non-linear effect that comes from the
7
non-linearity of the Finger strain tensor [23,24]. We put forward a theory of the non-linear
ultrasonically induced birefringence taking into account of the non-linearity of the Finger
strain tensor [25]. However, the estimated value stationary birefringence for the PS-toluene
solution 105 times was smaller than the experimental value [12]. In addition, the predicted
frequency dependence was not consistent with the experimental results in this study.
Frequency dependence of the stationary birefringence in polymer solution observed in
this study cannot be explained by the mechanism of for the ultrasonically birefringence
discussed so far. Ultrasonic intensity used in the birefringence measurement ranged from
0.001 to 1 W·cm-1. The corresponding pressure amplitude of ultrasound estimated as from
0.054 to 1.7 atm using values of the density and sound velocity of 103 kg·m-3 and 1500 m·s-1,
respectively. Since the pressure amplitude is not so small, the linearization of the fluid
dynamics equation is not adequate for our situation. This suggests strongly that quadratic
acoustic effect including the streaming will occur in solutions and it causes the birefringence
of polymer chains in solutions. Furthermore, in polymer solutions, it is considered that
many nonlinear effects, arising from the coupling between local segmental motion and the
solvent flow and so on, exist. To clarify the mechanism the stationary birefringence, the
further theoretical treatment including the non-linearity of fluid dynamics and the polymer
dynamics should be required.
4.CONCLUSION
In this study, we investigated the relation of the stationary birefringence to the
segmental anisotropy in polarizability. From the concentration dependence of stationary
ultrasonically induced birefringence in PS-toluene, PC-chloroform, PBD-toluene solutions,
the intrinsic values of the birefringence were obtained. Linear relationship was obtained
between the intrinsic value of the stationary birefringence per a segment and the segmental
anisotropy in polarizability. This suggests that the stationary ultrasonically induced
birefringence of polymer solutions is related to the segmental anisotropy of polymer chains.
In aqueous solutions of polyelectrolyte, the stationary birefringence decreased with ionic
strength with addition of the salts since the segmental anisotropy in polarizability decreased
due to decease of the chain stiffness with addition of the salt.
1
For all polymer solutions investigated here, the nst  WU value decreased with
increasing frequency. This tendency is not consistent with the present treatment for the
ultrasonically induced birefringence. In order to solve this, it is necessary to measure the
ultrasonically induced birefringence with wide ranges of frequency by both biased and
non-biased technique in more details. Also a new theory is required including the
non-linearity of fluid dynamics and the polymer dynamics for the ultrasonically induced
8
birefringence in polymer solutions.
ACKNOWLEDGMENTS
This work is partly supported by Grant-in-Aid for Scientific Research (No. 12650881)
from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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9