Micelle and oligomer kinetics in aqueous solutions of n-octylammonium chloride: Monomer exchange, protrusion and chain isomerization

Article abstract of DOI:10.1016/j.molliq.2010.09.001

Based on measurements of ultrasonic spectra in the frequency range between 0.1 and 2000 MHz the ultrasonic relaxation properties of solutions of the short-chain cationic surfactant n-octylammonium chloride in water are reported at solute concentrations be

Full text of DOI:10.1016/j.molliq.2010.09.001

Journal of Molecular Liquids 157 (2010) 125–132
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Journal of Molecular Liquids
journal homepage: www.elsevier.com/locate/molliq
Micelle and oligomer kinetics in aqueous solutions of n-octylammonium chloride:
Monomer exchange, protrusion and chain isomerization
⁎
Rüdiger Polacek, Udo Kaatze
Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 31 May 2010
Accepted 10 September 2010
Available online 18 September 2010
Based on measurements of ultrasonic spectra in the frequency range between 0.1 and 2000 MHz the
ultrasonic relaxation properties of solutions of the short-chain cationic surfactant n-octylammonium chloride
in water are reported at solute concentrations between 0.18∙10−3 mol cm−3 and 1.6∙10−3 mol cm−3. Two
relaxation terms are evaluated in the light of the fast monomer exchange of the Aniansson–Wall–Teubner–
Kahlweit model of micelle kinetics. Significant effects of the incomplete dissociation of counter ions as well as
of the large oligomer content at surfactant concentrations near the critical micelle concentration are shown
and effects from the formation of non-globular micelles at higher surfactant concentrations are discussed.
Two high-frequency relaxation terms are briefly considered with respect to monomer exchange of oligomers,
fluctuating protrusion, and structural isomerisation of surfactant alkyl chains.
Keywords:
Micelles
Oligomers
Kinetics
Surfactants
Ultrasonic spectrometry
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Herein, kri is the reverse rate constant of the i-th step in the
isodesmic reaction scheme
The delicate interplay between hydrophobic and hydrophilic
interactions, leading to self-aggregation of amphiphilic molecules in
water, is still today incompletely understood on a molecular basis.
Self-association is, therefore, a topic of a lively scientific debate in
liquid state physics and in colloid science [1–5]. Interest in the study of
colloidal systems arises also from their relevance to life science
because such systems provide a matrix to guide and control various
biochemical processes. Further stimulation follows from the diverse
applications of amphiphiles as cleaning agents and as surface-active
additives to stabilize dispersions, such as foams, emulsions, and
suspensions.
Ni−1 + N1⇔Ni;
i = 2; 3; …
ð2Þ
of micelle formation. Analogously, kfi denotes the forward rate
ꢀ
ꢁ
constant of i-th step. In Eq. (1), Ni is the equilibrium concentration
of species Ni, kr is the reverse rate constant for micelle sizes around
the mean ði = mÞ, and σ 2 is the variance of the Gaussian size
ꢀ
ꢁ
distribution. It is supposed that the monomer concentration N1 is
substantially larger than the concentration [Ni], i=2, 3, 4,…, of any
aggregate, so that direct association of aggregates Ni +Nj ⇔Ni + j
,
Fairly well understood are solutions of “proper micelles”. This term
is used to identify globular structures with mean aggregation number
i, j=2, 3, 4,…, can be neglected. Just a very small concentration of
oligomeric species exists with proper micelle systems. This attribute
leads to two suggestive modes of relaxation by which, after a small
disturbance, the system approaches equilibrium. For the sinusoidally
varying disturbances within acoustic fields Teubner–Kahlweit theory
[9,10] predicts a fast relaxation term with discrete relaxation time τf in
the range of nanoseconds to microseconds. It is due to the monomer
exchange between micelles and the suspending phase, a process that
changes the mean aggregation number m but keeps the concentration
of micellar species almost constant. In parallel to the fast monomer
exchange, a slow process, with relaxation time τs on the order of
milliseconds or seconds, leads to final equilibrium. The slow process
involves a redistribution of the concentrations of aggregates. In this
paper we report on the fast relaxation process which can be
favourably studied using broad-band ultrasonic spectroscopy [11].
Within the framework of the Teubner–Kahlweit model, the
P
m on the order of 50 or larger, corresponding to systems with critical
micelle concentration cmc smaller than 10−5 mol cm−3. The kinetics
of proper micelle systems is effectively represented by the Aniansson–
Wall model [6–8] which proceeds from a Gaussian distribution of
micelles:
ꢂ
ꢀ
ꢁ
ꢀ
ꢁꢃ ꢀ
ꢁ
kri + 1 Ni + 1 −kri N1
Ni = −ði−mÞkr = σ 2:
ð1Þ
=
⁎
Corresponding author. Tel.: +49 551 39 7715; fax: +49 551 39 7720.
ꢀ
ꢁ
E-mail address: uka@physik3.gwdg.de (U. Kaatze).
monomer concentration N1 is assumed to be independent of the
0167-7322/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.molliq.2010.09.001

126
R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
surfactant concentration c and to agree with the cmc. Defining the
scaled concentration Δc=c−cmc, the amplitude Af and the relaxation
rate τ−f 1 of the fast term follow as
the experimental findings. Hence the behaviour of the relaxation rate
near the cmc calls for further attention.
Another feature of micelle solutions is their tendency to form large
non-globular micelles at high surfactant concentrations. A second
monomer exchange term has been identified recently in the
ultrasonic attenuation spectra of alkyl monoglycoside solutions [22].
It has been assigned to the exchange of monomers at micelle sites
with packing of surfactants closer than characteristic for proper
globular micelles. Presently, it is not clear whether, at high surfactant
concentrations, large non-globular micelles are formed in addition to
or by growth and deformation of existing micelles.
We thus found it interesting to perform a broad band ultrasonic
spectrometry study of solutions of n-octylammonium chloride in
water. The critical micelle concentration of that system is large
enough to allow for reliable measurements at surfactant concentra-
tions below and close to the cmc. It is, on the other hand, small enough
to enable investigations into the formation of non-globular micelles at
solute concentrations distinctly larger than the cmc.
!
−1
2
πðΔVÞ σ 2
σ 2
m⋅cmc
Af =
Δc 1 +
Δc
ð3Þ
κ∞S RT m⋅cmc
and
!
kr
σ 2
σ2
m⋅cmc
τ−f 1
=
1 +
Δc
;
ð4Þ
respectively. In these relations, ΔV is the reaction volume, assumed to
be identical for all steps of the isodesmic scheme (Eq. (2)), κ∞S is the
isentropic compressibility, extrapolated to frequencies well above the
relaxation region, and R is the universal gas constant. For proper
micelle systems Eqs. (3) and (4) have been verified by experiments.
As an example, Fig. 1 reveals the linear dependence of the relaxation
rate τ−f 1 upon Δc if solute concentrations are sufficiently distant from
the cmc. Unlike non-ionic surfactant solutions, due to incomplete
2. Experimental
2.1. Surfactant solutions
ꢀ
ꢁ
dissociation of counterions, the monomer concentration N1 of ionic
surfactant systems depends noticeable on c. If this feature is
appropriately taken into account, experimental data for ionic micelle
systems [16] are also well represented by theory [17].
Aqueous solutions of n-octylammonium chloride (OACl) have
been prepared by adding appropriate amounts of hydrochloric acid
(≥99.9%, Merck, Darmstadt, Germany) to n-octylamine (≥99%,
Merck). The water used to adjust the surfactant concentration was
de-ionized and doubly distilled. The pH of the solutions has been
measured potentiometricly (pH-meter CG 707, Glass electrode H65,
Schott, Mainz, Germany). A survey of the OACl solutions is given in
Table 1 where also the density and shear viscosity data are listed. The
density ρ and the shear viscosity ηs have been determined with a
pycnometer (20 ml, Δρ/ρ≤10−3, Schott, Mainz) and a falling ball
viscometer (B/BH, Haake, Karlsruhe, Germany), respectively. Both
instruments had been calibrated against doubly distilled, degassed,
and filtered water. The viscosity of some solutions has been checked
by measurements with Ubbelohde capillary viscometers (Schott).
Temperature fluctuations during density and shear viscosity mea-
surements were smaller than 0.02 K.
At concentrations close to the cmc deviations from the predictions
of the theoretical models have been found for non-ionic [13,18] as
well as for ionic surfactants [14,16,19–21]. The inset of Fig. 1 shows
clearly that, near the cmc, the relaxation rate τ−f 1 may first decrease
with Δc to follow the theoretically predicted linear relation (Eq. (4))
at higher surfactant concentrations. This characteristic of relaxation
rate data has been suggested to reflect the high oligomer content at
solute concentrations near the cmc. In order to better apply to
surfactant solutions with high oligomer content, the Teubner–
Kahlweit model has been extended by suitable modification of the
size distribution of aggregates [15]. The extended model indeed
reveals an increase in the relaxation rate when the surfactant content
decreases to approach the cmc (Fig. 1). The effect from the formation
of oligomeric species, however, is too small to satisfactorily apply to
2.2. Critical micelle concentration
The cmc of OACl in water is tabulated [23,24] as 0.175 ∙10−3 mol cm−3
,
as originally reported by Klevens [25]. However, the substantially
larger value 0.27∙10−3 mol cm− 3 has been also reported [26].
Because of this discrepancy we have re-determined the cmc with
the aid of high-precision density measurements. A vibrating-tube
Table 1
Solute concentration c, concentration difference c−cmc, molality m, and mass fraction
Y, as well as density ρ and shear viscosity ηs for solutions of OACl in water at 25 °C.
c 10−3
c−cmc
m
Y
ρ
ηs
mol cm−3
10−3 mol cm−3
mol kg−1
g cm−3
10−3 Pa s
0.18
0.22
0.24
0.26
0.33
0.36
0.43
0.50
0.60
0.70
0.825
0.95
1.30
1.60
−0.09
−0.05
−0.03
−0.01
0.06
0.09
0.16
0.23
0.33
0.43
0.555
0.68
1.03
1.33
0.186
0.229
0.251
0.272
0.350
0.384
0.465
0.548
0.671
0.798
0.965
1.139
1.683
2.222
0.0297
0.0363
0.0396
0.0429
0.0545
0.0595
0.0711
0.0828
0.0995
0.1161
0.1371
0.1580
0.2170
0.2679
0.9969
0.9970
0.9969
0.9970
0.9962
0.9961
0.9952
0.9945
0.9934
0.9926
0.9910
0.9899
0.9864
0.9835
0.983
1.012
1.021
1.030
1.084
1.109
1.171
1.234
1.359
1.488
1.679
1.918
2.867
4.176
Fig. 1. Relaxation rates τ−f 1 of the fast monomer exchange process versus surfactant
concentration c exceeding the cmc for aqueous solutions of n-decyl-α-D-maltopyranoside
(● [12], cmc assumed to equal 2∙10−6 mol cm−3 as for n-decyl-β-D-maltopyranoside),
as well as of triethylene glycol monohexyl ether (■ [13], cmc=10−4 mol cm−3) and
n-heptylammonium chloride (▲ [14], cmc=4.5∙10−4 mol cm−3). The τf−1 values have
been derived from ultrasonic attenuation spectra. Full lines indicate the trend in the data.
Dashed line shows the results from an extended Teubner–Kahlweit model [15] of the
monomer exchange kinetics.

R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
127
densitometer with built-in reference oscillator and Peltier temper-
ature control (Δρ/ρ≤5∙10−6, Physica DMA 5000, Anton Paar, Graz,
Austria) has been used for this purpose. The density data of a series
of OACl solutions are displayed in Fig. 2, where also the shear
frequency range, two different methods and altogether nine different
specimen cells, each one matched to a particular frequency band,
were employed.
At frequencies below 23 MHz, where B′ν2 is small, a cavity
resonator method was applied in which the path-length of interaction
of the acoustical field with the liquid is virtually increased by multiple
reflections [29]. Five circular cylindrical cavity cells were used. All
cavities were completely filled with the liquid sample. The faces of
four cells were made of piezoelectric quartz discs. They served as
reflectors and, at the same time, as transducers for the connection of
the resonator cells to the electronic measurement circuit. One
biconcave resonator was provided with concavely shaped passive
glass reflectors. Separated piezoelectric devices existed for the
electrical coupling of the cell [30]. Two biplanar cells were used,
mainly differing from one another by their fundamental frequency of
thickness vibrations (νT =4 MHz, 0.8≤ν≤12 MHz; νT =8 MHz,
0.4≤ν≤23 MHz). In addition, one plano-concave (νT =1 MHz,
0.1≤ν≤2.7 MHz) and two biconcave resonators (νT =1.1 MHz,
0.1≤ν≤1.9 MHz; νT =7 MHz, 0.4≤ν≤13 MHz) were available. A
concave reflector shape reduced diffraction effects in the low-
frequency range of operation and entailed also a mode spectrum
different from biplanar cavities. In order to appropriately consider the
effect from spurious higher order modes the complex resonator
transfer function was always measured around each principal
resonance peak using computer controlled network analysers
(Hewlett Packard 8751A and 4195A). The relative level and phase of
the transfer function were analyzed to obtain the resonance frequency
viscosities from Table
1 and electrical dc conductivities are
presented. The data feature two well-known properties of surfac-
tant solutions. (i) No sharp critical concentration follows from the
concentration dependencies of the parameters. Rather there exists a
smooth transition in the concentration range around the cmc. (ii)
The transition regions of different parameters do not agree with one
another. The cmc values following from an extrapolation from
below and above to a point of intersection are 0.20∙10− 3 mol cm− 3
(σ), 0.25∙10−3 mol cm−3 (ηs), and 0.27∙10−3 mol cm−3 (ρ). We
shall use the last value in the following, in agreement with the
aforementioned value from counter ion nuclear magnetic resonance
chemical shifts [26].
2.3. Ultrasonic attenuation spectroscopy
The acoustical attenuation coefficient α of the surfactant solution
has been measured as a function of frequency ν between 100 kHz and
2 GHz. In addition to the excess contribution αexc in which we are
interested here, the total attenuation coefficient
αðνÞ = B′ν2 + αexcðνÞ
ð5Þ
contains a background contribution proportional with ν2 (B′=const.
[27,28]). For this reason, the broad frequency range of measurements
implies a considerable range of α-values for each sample. In order to
guarantee a high α-data accuracy in the complete measuring
νrn and half-power bandwidth δνn (n=1, 2, 3,…) of the undisturbed
principal resonance. Based on energy arguments, intrinsic cell losses
were considered by the widely used quality factor relation
−1
−1
−1
Q
= Q
−Q
:
ð6Þ
liquid
total
cell
−1
Here, Q
=αλ/π, where λ=cs/ν is the wavelength of the sonic
liquid
field within the liquid and cs is the sound velocity. Furthermore,
Q total =νn /δνn, and Q cell represents the cell losses. Q cell was obtained
from calibration measurements, using reference liquids with well-
known attenuation coefficient and with sound velocity as close as
possible to the sound velocity cs of the sample.
In the upper part of the measuring frequency range
(3 MHz≤ν≤2 GHz) absolute α measurements were performed by
transmitting pulse-modulated sonic waves through a sample of
varying thickness [29]. Four cells were used to cover the frequency
range. The cells differed from one another by the type of piezoelectric
transmitter and reflector and by the required sample volume V:
3≤ν≤63 MHz, X-cut quartz discs, V=130 cm3; 30≤ν≤530 MHz, Y-
cut lithium niobate discs attached to the back face of delay rods from
fused quartz, V=10 cm3; 500≤ν≤2000 MHz, lithium niobate rods
operated in
a
surface excitation mode [31], V = 3 cm3;
1100≤ν≤2000 MHz, zinc oxide films sputtered onto the back face
of delay lines made of sapphire, V=0.5 cm3. At each frequency of
measurement the transfer function of the sample cell was determined
at 400 transducer spacings. The electronic apparatus was routinely
calibrated by replacing the specimen cell for a suitable high-precision
cut-off piston attenuator. At ν≤30 MHz, diffraction effects due to the
small transducer diameter (as compared to the wavelength) have
been taken into account by a semi-empirical correction term, based on
calibration measurements with a suitable reference liquid.
2.4. Experimental accuracy
The temperature T of the cells was controlled to within 0.03 K by
circulating thermostate fluid through a thermostatic coat surrounding
the sample and also through suitable channels in the cell walls. Each
cell was additionally protected by a thermostatic box. Temperature
gradients within a specimen cell as well as temperature differences
Fig. 2. Density ρ, shear viscosity ηs, and specific electrical conductivity σ of OACl
solutions at 25 °C, displayed as function of surfactant concentration c. The dashed lines
show the extrapolation from below and above to a point of intersection.

128
R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
between different cells were smaller than 0.05 K, corresponding to an
uncertainty Δα/α less than 0.0005 in the attenuation coefficient data.
Fluctuations Δν/ν in the frequency of measurement were smaller than
10−4 throughout. The uncertainty in the frequency can thus be
neglected here. The main sources of possible experimental errors in
the resonator measurements were small disturbances in the cell
geometry and cell adjustment that might result from cleaning and
refilling procedures. Effects from an imperfect wetting of the
piezoelectric transducer surfaces, covered with gold layers to provide
the electrical contacts, may also contribute to the uncertainty in the
attenuation coefficient data. Repeated runs, using the same cells after
refilling and likewise using different cells, yielded an uncertainty Δα/
αb0.07 in the attenuation coefficients from resonator measurements.
This figure includes the small uncertainty in the attenuation
coefficients of the reference liquids (e. g. water, 25 °C: α=(21.24
0.1)·10−15 s2 m−1). The attenuation coefficients from the pulse-
modulated wave transmission measurements may be affected by
imperfect parallelism of the transmitter and receiver transducer and,
at low frequencies (ν≤30 MHz), also by an insufficient correction for
diffraction. Typical uncertainties, as estimated from repeated mea-
surements and from the agreement of data in frequency regions
where the measurement ranges of different cells overlap, are given in
Table 2. Due to the use of different methods, electronic set-ups, and
specimen cells and due to regular measurements of reference liquids
which were routinely run, systematic errors exceeding the above
values are unlikely to have remained unnoticed.
Fig. 3. Ultrasonic attenuation spectra in the frequency normalized format for
solutions of OACl in water at 25 °C (■, 1.8∙10− 4 mol cm− 3; ●, 2.2∙10−4 mol cm− 3
;
▲, 2.6∙10−4 mol cm−3; diamond, 9.5∙10−4 mol cm−3). For the 2.2∙10−4 mol cm−3
solution the dashed line indicates the limiting high-frequency contribution B′ to the
spectrum.
where ω=2πν. The relaxation spectral function R(ν) has been fitted
to the experimental spectra by a nonlinear least-squares regression
analysis that minimizes the variance
ꢄ
ꢅ
2
M
1
ðαλÞm−Rðνm; PiÞ
ðΔαλÞm
χ2
=
∑
:
ð9Þ
M−I−1 m=1
In this equation, νm, m=1, …, M, are the frequencies of measure-
ments, (αλ)m and (Δαλ)m are the data and their experimental
uncertainties, respectively, at these frequencies. The Pi, i=1, …, I, are
the parameters of R. The values of these parameters, as resulting from
the fitting procedures, are given in Table 3.
Inspection of the complete set of experimental spectra indicated in
fact a maximum of three relaxation terms per spectrum. Grouped
according to the relaxation times, however, these terms reflect four
relaxation regimes. In Table 3 the parameter values are therefore
ordered pursuant to these regimes, using subscript “0” to mark the
additional relaxation region.
3. Results and analytical description of spectra
In Fig. 3 spectra at four OACl concentrations are displayed in the
frequency normalized format α/ν2 vs. ν. In this representation of data
the background contribution (Eq. (5)), due to the Stokes–Kirchhoff
damping [32,33] and to potential molecular relaxation processes with
relaxation frequencies well above the frequency range of measure-
ments, is independent of frequency. At surfactant concentration
c=0.18∙10−3 mol cm−3 bcmc, the α/ν2 data are indeed independent
of ν and thus define parameter B′ of the background term. At
OACl concentrations close to or larger than the cmc the low-
4. Discussion
ꢂ
ꢃ
frequency value lim α= ν2 of the spectra exceeds the B′ value of
ν→0
4.1. Monomer exchange
the 0.18∙10−3 mol cm−3 solution and the α/ν2 data decrease with ν
ꢂ
ꢃ
to reach a constant B′ = lim α= ν2 at high frequencies. Hence at
Similarity of the concentration dependence in the parameters A1
and τ1 to such of aqueous solutions of other ionic [14,16,18,35] and, in
some respects, of non-ionic [12,13,22] surfactant systems suggest that
term “1” in the ultrasonic attenuation spectra reveals the fast
monomer exchange in the Teubner–Kahlweit–Aniansson–Wall
model of micelle kinetics. As shown by Fig. 5, however, the relaxation
rates τ−1 1 of term “1” do not follow a linear dependence upon c−cmc,
as predicted by Eq. (4). In correspondence with the τ−1 1 data for
ν→0
c≈cmc and cNcmc the spectra clearly reveal relaxation characteristics
within the frequency range of measurements. Obviously, the spectrum
of the solution with OACl concentration c=0.95∙10−3 mol cm−3
displays at least two relaxation regions (d(α/ν2)/dνb0), with relaxation
frequencies νr at around 1 MHz and 20 MHz, respectively. A closer look,
however, indicates up to three relaxations at cNcmc. As an example,
Fig. 4 shows the spectrum of a OACl solution (c=1.3∙10−3 mol cm−3
in the excess-attenuation-per-wavelength format:
)
αecxλ = ðαλÞexc = αλ−Bν;
ð7Þ
where B=B′cs. For the above reasons we represented the αλ spectra
by a sum of three Debye-type relaxation terms [34] with discrete
relaxation times τi:
3
Aiωτi
RðνÞ = ∑
+ Bν;
ð8Þ
1 + ω2τ2
i=1
i
Table 2
Uncertainties in the experimental attenuation coefficient data.
Fig. 4. Ultrasonic excess attenuation spectrum for an OACl solution in water
(c=1.3∙10− 3 mol cm− 3) at 25 °C. Dashed lines show the subdivision of the spectrum
into three Debye-type relaxation terms. The full line is the graph of the sum of these
terms with parameter values as given in Table 3.
ν, MHz 0.1–0.8
Δα/α 0.07
0.8–12
0.04
12–60
0.02
60–400
0.01
400–530
0.015
530–2000
0.01

R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
129
Table 3
Parameters of the relaxation spectral function used to represent the experimental data for OACl aqueous solutions at 25 °C.
c 10−3 mol cm−3
A0 10−3
A1 10−3
A2 10−3
A3 10−3
τ0 ns
τ1 ns
τ2 ns
τ3 ns
B ps
0.18
0.22
0.24
0.26
0.33
0.36
0.43
0.50
0.60
0.70
0.825
0.95
1.30
1.60
0.09
0.39
0.62
0.96
3.2
4.9
5.0
5.5
4.4
4.9
3.9
4.2
6.8
0.7
0.7
0.6
0.8
0.9
1.0
1.4
1.4
1.9
1.9
2.2
4.5
7.4
4.0
8.0
12.5
8.0
7.0
5.0
5.0
0.14
0.23
0.38
0.31
0.40
0.80
0.50
0.58
0.44
0.79
32.1
32.6
32.8
32.8
32.2
33.9
34.1
35.1
35.8
37.5
42.1
39.5
43.2
46.8
0.32
2.37
12.0
25.0
26.8
27.1
25.9
25.9
24.1
23.8
23.4
17.1
12.0
67.7
117
150
79.7
65.4
38.3
27.3
17.9
14.1
9.9
0.85
2.3
6.2
134
108
75.6
41.4
2.4
7.0
4.3
3.3
0.62
0.61
0.60
10.6
10.1
aqueous solutions of sodium dodecylsulfate (SDS, Fig. 5), the
relaxation rates decrease with surfactant concentration at c smaller
than and close to the cmc and they increase stronger than linearly at
cNcmc. Also the relaxation amplitudes A1 do not comply with the
prediction of the theoretical model (Eq. (3)) to increase with c but to
asymptotically approach the limiting value π(ΔV)2 /(κS∞RT). Rather the
A1 data reveal a relative maximum (Fig. 6) though it appears to be less
pronounced than that in the amplitudes of the SDS system.
It will be shown below that the special features in the parameters
of relaxation term “1” are due to three effects. Lessner, Teubner, and
Kahlweit have already pointed to consequences from the ionic nature
of surfactants [17]. Incomplete dissociation of the surfactant mole-
cules leads to a concentration dependence in the monomer concen-
tration [N1], as mentioned in the Introduction. For that reason the
reduced concentration
relates the concentration ½ ꢀof free counter ions to the concentra-
¯
Nc
tions ½ ꢀ and ½ ꢀ of free surfactant monomers and of class i micellar
¯
N1
¯
Ni
species, respectively. In Eq. (11) bi is the equilibrium constant of step i
and Ji =i(1−αi) denotes the number of non-dissociated monomers
per aggregate of class i. Here αi is an effective degree of counter ion
dissociation, with proper micelle systems assumed to be independent
¯
¯
¯
of i, thus αi = αm. If m−1≈ m is presumed the implicit relation
!
"
!#
ꢄ
ꢅ
ꢄ
ꢅ
1−αm¯
2−αm¯
1
c
¯
N1
c
¯
mð2−αm¯
Þ
½
ꢀ = Aγ
−1
1 + αm¯
ꢀ
ꢁ −1
ð12Þ
¯
N1
½
ꢀ
N1
ꢀ
ꢁ
¯
between the monomer concentration N1 and the total surfactant
concentration c follows. In this relation
ꢆ
ꢇ
1
2−α
pffiffiffiffiffiffi
m
¯
ð
mÞ
¯
¯
Aγ
=
2πσ mbm¯
:
ð13Þ
ꢂ
ꢀ
ꢁꢃ
ꢀ
ꢁ
¯
¯
X = c− N1ðcÞ
=
N1ðcÞ
ð10Þ
has to be used in Eqs. (3) and (4) instead of (c−cmc)/cmc. The law of
mass action
i−1Þ
ꢀ½ ꢀ½ ꢀðJ −J
ð11Þ
i
½
ꢀ = b ½
¯
Ni
¯
¯
¯
Ni−1 N1 Nc
i
Fig. 6. Normalized relaxation amplitudes A/cmc versus concentration difference c−cmc
for aqueous solutions of OACl (●, A=A1; ○, A=A0) and of sodium dodecylsulfate (▲,
A=A1 [16]) at 25 °C.
Fig. 5. Relaxation rates τ−1 versus concentration difference c−cmc for aqueous
solutions of OACl (●,τ−1 1; ○,τ0−1) and of sodium dodecylsulfate (▲,τ−1 1 [16]) at 25 °C.

130
R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
ꢀ
ꢁ
Table 4
In Fig. 7 the monomer concentration N1 following from the
Parameters of the monomer exchange derived from the ultrasonic attenuation spectra
experimental spectra, when Eqs. (10) and (11) are used in the
evaluation of the relaxation amplitudes A1, are shown on the
assumption of three different degrees of counter ion dissociation.
Only at αm =1, corresponding with the Teubner–Kahlweit model for
non-ionic surfactant systems, the monomer concentration at cNcmc
agrees fairly with the critical micelle concentration. At a smaller
degree of counter ion dissociation the monomer concentration adopts
a relative maximum at the cmc and decreases toward higher
surfactant concentrations. This feature of ionic surfactant systems is
well established now (see, e.g. Fig. 4.11 in ref. [4] and Fig. 4 in ref.
at cmcbcb0.8∙10−3 mol cm−3 on the assumption of
a concentration dependent
monomer concentration and a degree of counter ion dissociation α m = 0:33 (HACl:
n-heptyl ammonium chloride; SDS: sodium dodecylsulfate).
Surfactant
cmc 10−3
m
σ
kf 1012
kr 107 s−1
ΔV cm3
−1
mol cm−3
(s mol cm−3
)
mol−1
HACl13
OACl
0.45
0.27
0.0082
20
25
64
12.8
18.3
4.5
3.0
2.1
0.18
189
82
6.6
6.5
SDS16
0.4
11.0
[14]). Using the reasonable value αm =0.33 [36] and taking m=25
¯
Introduction, a relaxation term with relaxation frequency smaller
than that of the monomer exchange term has been found recently in
the ultrasonic spectra of solutions of alkyl monoglycosides in water
[22]. It has been assigned to the monomer exchange of micelles with
closer packing of surfactant molecules.
¯
[37], in passable agreement with m=20 [38], 25 [39], 27 [20,37], and
29 [36] for sodium octylsulfate in water, the Lessner–Teubner–
Kahlweit model yields the forward and reverse rate constants kf and
kr, respectively, for micelle sizes around the mean, as well as the
variance σ in the size distribution, and the reaction volume ΔV as
listed in Table 4. For reasons of comparison, literature data for the
homologues surfactant heptyl ammonium chloride (HACl [14]) and
for sodium dodecylsulfate (SDS [16]) are also listed in the table. The
spectra for the former system, which originally had been treated in
terms of a concentration independent monomer concentration
At increasing surfactant concentration the available number of
water molecule per molecule of solute decreases. Therefore the
surfactant molecules attempt to form structures larger than spherical
micelles so that less hydration water is required. Above a certain
concentration micelles may grow continuously to form oblate or
prolate ellipsoids or even discs or rods. Alternatively, an amount of
micelles may remain globularly shaped and additional non-spherical
micelles may be formed at increasing surfactant concentration. The
formation of this second type of aggregates may proceed continuously
or may occur discontinuously at a second critical aggregation
concentration [38–42]. The concentration ch of hydration water and
the number Zh =ch/c of hydration water molecules per OACl, as
derived from dielectric spectra [43], reveals a significant variation of
the parameters at a concentration ct (Fig. 8). Obviously at ct the
number of hydration water molecules Zh decreases due to closer
packing of the cationic surfactant within the micelles.
ꢀ
ꢁ
( N1 =cmc), have been re-evaluated to take the incomplete
dissociation of counter ions into account. It was found that for the
HACl solutions marginal effects in those parameters result rather from
the modification of the behaviour of the monomer concentration.
According to our expectations the forward and reverse rate
constants decrease when going from HACl to OACl with slightly
larger alkyl chain (Table 4). For SDS with significantly longer
hydrophobic chain both parameters are substantially smaller than
for the ammonium chlorides. Also in conformity with our idea of
micelle structures the volume change associated with the monomer
exchange of SDS is noticeably larger than with the exchange of the
short-chain surfactants, because the former involves a greater amount
of hydrophobic hydration water. The small variance in the size
distribution of the SDS system points at the formation of proper
micelles, whereas the large σ values for the short-chain amphiphiles
indicate a broad size distribution of micelles at surfactant concentra-
tions rather close to the cmc.
At cN0.7∙10−3 mol cm−3 the additional low-frequency relaxation
term (“0”) adopts detectable amplitudes (Table 3, Fig. 6). The
amplitude A0 of the term increases monotonously with c and likewise
increases the relaxation rate τ−0 1 (Fig. 5). As briefly mentioned in the
ꢀ
ꢁ
Fig. 7. Normalized monomer concentration N1 = cmc, as resulting from the relaxation
amplitudes A1 of OACl solutions when Eq. (12) is used, versus normalized surfactant
concentration c/cmc. Calculating these data a mean aggregation number m=27 has
been assumed. Dashed and dashed-dotted lines show the results for two degrees of
Fig. 8. Concentration of hydration water ch as derived from dielectric spectra of
solutions of OACl in water at 25 °C (● [43]) and number of Zh =ch/c of hydration water
molecules per surfactant (▲) versus solute concentration c. In addition to the cmc a
concentration ct is marked at which obviously a transition from globular micelles to
non-globular aggregates with closer packing of surfactants occurs.
ꢀ
ꢁ
counter ion dissociation. The full line represents N1 on the assumption of completely
dissociated counter ions (αm =1), the dashed line indicates the cmc.

R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
131
In both alternative models of micelle solutions at elevated
surfactant concentration c, the mean aggregation number m depends
upon c and, at least in the mixed solutions of spherical and non-
spherical micelles, it is intrinsically necessary to consider two size
distributions, each one for each type of micelles. In the calculations of
the rate constants and reaction volume for the monomer exchange
process (Table 4) we have, therefore, included data for a reduced
solute concentration range (cmcbcb0.8∙10−3 mol cm−3) only. The
relaxation rates τ−0 1 for term “0” of the more concentrated OACl
solutions do not differ by more than a factor of 1/20 from the
corresponding τ−1 1 values. It seems thus to be justified to consider
relaxation term “1” to be due to the monomer exchange of globular
micelles and also to the exchange of monomers at spherically shaped
parts of non-spherical micelles, like the end-caps of rod shaped
aggregates.
Slightly below the cmc there exist already non-vanishing relaxa-
tion amplitudes in the ultrasonic absorption spectra. At cbcmc the
relaxation rates τ−1 1 decrease with solute content as previously
reported for various surfactant systems [13,14,16,18–21]. A computer
simulation study [15] of the coupled scheme of reactions (Eq. (2)) had
been performed in which, in order to better apply to solutions of
short-chain surfactants, some approximations of the analytical
formulation were avoided. The shape of the size distribution had
not been fixed. Rather it had been calculated from reasonable
assumptions on the rate constants kfi and kir of the isodesmic reaction
scheme (Eq. (2)). In doing so, the rate constants have been modelled
to approach the Gaussian size distribution as a limiting form for
proper micelle systems. Reducing the surfactant concentration of
originally proper micelle systems in the computer simulation yields a
less pronounced minimum in the oligomer region of the size
distribution at cNcmc and it leads to even complete absence of the
relative minimum at the cmc. It also reveals a noticeable concentration
of oligomeric species at solute concentrations below the cmc [15].
Absence of the relative minimum in the size distribution results in
a degeneration of the two modes of relaxation by which, after a small
disturbance, equilibrium is reached in a proper micelle system. As
illustrated by the inset of Fig. 1, in principle the computer simulations
show also the initial decrease of the relaxation rates with c. In the
computer simulation data, however, the effect is much smaller than in
the experimental data, indicating that the formalism of the kinetics of
micelle formation yields only one contribution to the relaxation time.
The reaction scheme (Eq. (2)) predicts a probability for a collision
between a monomer and a complex of Ni molecules at the assumption
of uniformly distributed monomers and aggregates. Below and near
the cmc, however, fluctuations in the local concentrations may also act
a significant effect on the monomer exchange and may offer a parallel
pathway to the system, in addition to the micelle formation/decay
processes, to reach equilibrium after a small disturbance.
Fig. 9. Relaxation amplitude A2 and relaxation time τ2 of the ultrasonic spectra of OACl
solutions at 25 °C plotted against surfactant concentration c. Arrows mark the cmc and
the transition concentration ct as taken from the Zh data in Fig. 7. The shaded area
indicates the concentration range in which both high-frequency relaxation terms (“2”,
“3”) have been individually identified in the spectra.
are strong. Towards higher solute concentration τ2 decreases whereas
the relaxation amplitude A2 is almost constant at cmcbcbct.
The above-mentioned computer simulations of the extended
model of micelle kinetics used to explain the concentration depen-
dence of the relaxation rates τ−1 1 below and close to the cmc revealed
also two ultrahigh frequency relaxation terms [15]. These terms have
been assigned to the monomer exchange of small oligomeric species
in surfactant solutions with less definite minimum in the oligomer
region of the size distribution. Assumption of oligomer relaxations is
in conformity with non-vanishing relaxation amplitudes A2 and A3 at
surfactant concentrations below the cmc.
The relaxation times τ2, however, are also on the order of
relaxation times for the limited radial diffusion of monomers within
micelles, resulting in the well-known protrusion of molecules. Since
the amplitude of the relaxation term displays a stepwise increase from
A2 =1∙10− 3 to 3.2∙10− 3 fluctuating protrusion likely may also
contribute to relaxation term “2”. From theory [44] the mean diffusion
time of the protrusion process follows as
!
l2CC
D
〈tD〉 =
z2ðexpðzÞ−z−1Þ;
ð14Þ
where lCC =0.125 nm [45] denotes the length of a C–C bond in the
hydrocarbon chain axis, D is the diffusion coefficient of the radial motions
of surfactant molecules within a micelle, and z∙lCC is the mean radial
diffusion length. Using the diffusion coefficient D=2.3∙10−9 m2 s−1 as
for n-octane at 25 °C [45], z=4.3 follows as a rough estimate when
τ2=8 ns is taken the mean diffusion time. This reasonable value
indicates that, at least at cNcmc, in addition to an oligomer relaxation
process, radial diffusion of monomers within the micelles may be also
reflected by relaxation term “2”.
Relaxation time τ3 (0.14 ns≤τ3 ≤0.8 ns) almost agrees with
relaxation times of other micelle solutions which have been found
to be due to structural isomerisation of surfactant alkane chains
within the micellar cores. Examples of relevant relaxation times for
anionic, cationic, and non-ionic micelle systems are shown in Table 5.
As these relaxation times do not reveal a noticeable dependence upon
the cmc or upon the nature of the surfactant head group they are likely
due to processes occurring in the hydrocarbon core of micelles. The τ3
values for the n-octylammonium chloride system are indeed
somewhat larger than those for most other micelle solutions. As
4.2. Chain isomerisation, oligomer formation, protrusion
At surfactant concentrations up to 0.7∙10−3 mol cm−3 two further
relaxation terms with discrete relaxation times τ2 and τ3 have been
extracted from the experimental spectra (Table 3). At higher solute
content, where the additional low-frequency term due to the
monomer exchange from non-globular micelles exists, identification
of both high-frequency relaxation terms was not possible. Only one
high-frequency relaxation term has thus been considered in the
evaluation of spectra. This term seems to combine contributions from
the molecular processes reflected by both high-frequency terms at
lower solute concentration (Table 3). As a reasonable discussion of
such degenerate relaxation mode is hardly possible, the focus will be
on the spectra at c≤0.7∙10−3 mol cm−3. In that concentration range,
τ2 increases with c to adopt a relative maximum slightly above the
cmc (Fig. 9), where fluctuations in the local concentrations of species

132
R. Polacek, U. Kaatze / Journal of Molecular Liquids 157 (2010) 125–132
Table 5
been found in simulations of the extended version of the Aniansson–
Wall–Teubner–Kahlweit model. However, fluctuating protrusion of
monomers within micelles seems to also contribute to term “2”,
whereas structural isomerizations of alkyl chains within the micelle
cores likely add to term “3”. It is, therefore, impossible to evaluate the
parameters of these high-frequency relaxation terms with regard to
relevant relaxation models.
Relaxation time τ3 of the hypersonic relaxation term for micelle solutions at 25 °C.
Surfactant
cmc
c
τ3
10−3 mol cm−3 10−3 mol cm−3 ns
n-Heptylammonium chloride [14]
n-Octylammonium chloride Table 3
n-Octylammonium chloride Table 3
n-Octyltrimethylammonium
bromide [46]
0.45
0.27
0.27
0.14
0.80
0.33
0.70
0.50
0.10
0.40
0.79
0.31
Acknowledgements
n-Decyltrimethylammonium
bromide [46]
n-Dodecyltrimethylammonium
bromide [46]
n-Tetradecyltrimethylammonium
bromide [46]
n-Hexadecyltrimethylammonium
bromide [46]
Sodium decyl sulphate [46]
Sodium dodecyl sulphate [16]
n-Hexadecylsulfopropyl betaine [46] 0.00006
Pentaethylene glycol monooctyl
ether [46]
Pentaethylene glycol
monododecylether [46]
0.06
0.50
0.50
0.50
0.40
0.22
0.15
0.26
0.26
0.014
0.003
0.001
We thank Mrs. Gisa Kirschmann-Schröder for help with the
drawings. Financial support by the Deutsche Forschungsgemeinschaft
(Bonn, Germany) is gratefully acknowledged.
References
0.033
0.0083
0.50
0.40
0.10
0.18
0.26
0.17
0.29
0.25
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5. Conclusions
Broadband ultrasonic spectra of aqueous solutions of the short-
chain surfactant n-octylammonium chloride exhibit two low-fre-
quency relaxation terms with relaxation times τ0 (41 ns≤τ0 ≤134 ns,
25 °C) and τ1 (3.3 ns≤τ1 ≤150 ns, 25 °C), respectively. The latter term
is due to the fast monomer exchange in the Aniansson–Wall–
Teubner–Kahlweit isodesmic reaction model of micelle formation/
decay. Above the cmc the relaxation parameters of this term reveal a
noticeable effect from the incomplete dissociation of the chloride
counter ions, resulting in a significant reduction of the monomer
concentration with increasing total surfactant concentration c. This
variation of the monomer concentration leads to characteristics in the
relaxation parameters substantially different from non-ionic surfac-
tant systems, for which at cNcmc the monomer concentration almost
agrees with the critical micelle concentration.
Another feature of the short-chain surfactant system is the
concentration dependence in the relaxation rates of the monomer
exchange term below and close to the cmc. The relaxation rates first
decrease to increase at higher surfactant concentration as predicted
by the Aniansson–Wall–Teubner–Kahlweit model of micelle forma-
tion/decay. According to an extended model of the micelle kinetics,
this special behaviour results, at least in parts, from the comparatively
high content of oligomers in the short-chain surfactant system.
At higher surfactant content the other aforementioned low-
frequency relaxation term (“0”) indicates the formation of large
non-globular micelles, in addition to or by growth and deformation of
existing micelles. Likely this second low-frequency term is due to
different rate constants for the monomer exchange from micelle sites
with closer packing of monomers.
In the lower concentration range (cb0.8∙10−3 mol cm−3) two
high-frequency relaxation terms with relaxation times τ2 (2.2 ns≤
τ2 ≤12.5 ns, 25 °C) and τ3 (0.14 ns≤τ3 ≤0.8 ns, 25 °C) have been
assigned to the formation/decay kinetics of oligomeric species, as had
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