J. Phys. Chem. B 2009, 113, 12283–12292
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Monomer Exchange and Rotational Isomerization of Alkyl Monoglycosides in Water Julian Haller† and Udo Kaatze* Drittes Physikalisches Institut, Georg-August-UniVersita¨t, Friedrich-Hund-Platz 1, 37077 Go¨ttingen, Germany ReceiVed: June 12, 2009; ReVised Manuscript ReceiVed: July 22, 2009
Ultrasonic attenuation spectra between 100 kHz and 400 MHz are reported for solutions of hexyl-, heptyl, octyl-, and nonyl-β-D-glucopyranoside in water. Results for a mixture of an octyl glucopyranoside with an octyl galactopyranoside are also presented. The analytical descriptions of the spectra in terms of relaxation spectral functions feature up to four relaxation regions. The high-frequency relaxation processes, with relaxation times at around 2 and 10 ns, respectively, correspond with molecular processes also existing in saccharide solutions. They are assigned to the exocyclic hydroxymethyl group isomerization as well as to the rotation around glycosidic bond angles. The low frequency terms, with relaxation times in the range from 30 ns to 3 µs, are characteristic of micelle solutions. They are discussed in terms of two modes of monomer exchange. One mode reflects the well-known exchange kinetics of systems with Gaussian size distribution of globular micelles. The other mode is considered to be due to the monomer exchange at micelle sites with closer packing of surfactants, as is characteristic of large micelles with nonglobular shape. TABLE 1: Survey of the Surfactants Investigated in this Study (MW Denotes the Molar Weight)
1. Introduction Alkyl glycosides attract interest due to their wide range of application, their availability from renewable resources, their biodegradability, and their marginal irritation. The favorable features of these nonionic detergents, first synthesized by Fischer in 1893,1 have only recently been utilized on a grand scale, after technical processes for the industrial production of alkyl glycosides had been developed.2 Present applications are to be found in biochemistry and medicine, in pharmaceutical and cosmetic preparations, as well as in cleaning and large-scale processes.3,4 Especially n-alkyl-β-D-glucopyranosides are used in biochemistry for the nondenaturing extraction and purification of membrane proteins,5,6 for the solubilization of liposomes,7 and for the permeability enhancement of membranes.8 Alkyl glycosides also improve specific drug delivery to organs.9,10 An enhancement of the acceptance of insulin via nose11 and eye12 drops by alkyl glycosides has been reported. Another example of the appropriateness of alkyl glycosides is their use in industrial separation processes, such as the flotation of metals. So far, most papers on alkyl glycosides are related to the optimization of the synthesis of these detergents, but phase diagrams of various alkyl glycoside/water systems have also been determined using various experimental techniques.13-17 Also, the structure of different phases has been intensively investigated. Evidence for the existence of rod-shaped octylβ-D-glucopyranoside micelles in water, for example, has been obtained from dielectric,18 1H NMR,15 self-diffusion,19 shear viscosity,18 as well as specific heat20 measurements. Little is known, however, about the molecular dynamics of alkyl glycoside solutions. In this paper, we report on broadband ultrasonic attenuation measurements of aqueous solutions of alkyl monoglycosides, predominantly of glucopyranosides. The ultrasonic spectra are analyzed to yield the relaxation term due to the exchange of monomers between the micelles and the suspending phase. Two * Corresponding author. E-mail:
[email protected]. † Present address: PTB Braunschweig, Bundesallee 100, 38116 Braunschweig, Germany.
surfactant n-hexyl-β-D-glucopyranoside n-heptyl-β-D-glucopyranoside n-octyl-β-D-glucopyranoside n-nonyl-β-D-glucopyranoside n-octyl-β-D-galactopyranoside
shorthand MW/g · logogram mol-1 C 6 G1 C 7 G1 C 8 G1 C 9 G1 C8Gal
264.3 278.3 292.4 306.4 292.4
cmc/10-3mol · cm-3 0.2540 0.07040 0.018-0.02641 0.006540 0.029540
further terms are found to be related to conformational isomerizations of the surfactant molecules. The monomer exchange term is compared to the micelle formation/decay kinetics of nonionic poly(ethylene glycol) monoalkyl ether surfactant systems,21-23 and it is also discussed in the light of the Teubner-Kahlweit-Aniansson-Wall24-28 model for solutions of proper micelles. As short-chain surfactant systems have shown characteristic deviations from the theoretical predictions of this model,29 attention is particularly directed to the appropriateness of the theory for the description of the alkyl glycoside systems. The additional relaxation terms are evaluated with regard to the conformational variety of carbohydrates,30-34 which is believed to play a key role in the molecular language of life.35-39 2. Experimental Section 2.1. Alkyl Glycoside Solutions. The alkyl glycosides used in this study, their shorthand logograms and molar weights, as well as the literature values of their critical micelle concentration (cmc) in aqueous solutions are presented in Table 1. According to our expectations, the cmc of the series of alkyl glucopyranosides decreases substantially with the length of the hydrophobic alkyl group. In contrast, the cmc varies only insignificantly when the hydrophilic glucopyranoside headgroup is replaced by the galactopyranoside ring. The structures of a glucopyranoside and a galactopyranoside are shown in Figure 1. All alkyl glycosides have been purchased from Anatrace Inc., Maumee, Ohio, and have been used as delivered. Their purity
10.1021/jp905523p CCC: $40.75 2009 American Chemical Society Published on Web 08/19/2009
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Haller and Kaatze
Figure 1. Structure of n-octyl-β-D-glucopyranoside (top) and n-octylβ-D-galactopyranoside (bottom).
was 98% or better. For all alkyl glycosides, a highly concentrated stock solution was first prepared by weighing appropriate amounts of surfactants into volumetric flasks and adding deionized, bidistilled, and UV-sterilized water up to the line measure. To avoid uptake of water from the air, the powdery surfactants have always been treated in a dry nitrogen atmosphere. Measurement solutions have been prepared by dilution of parts of the stock solution. When not used in measurements, the solutions have been stored at 6 °C in a fridge. For each concentration series of measurements, a small part of the stock solution was retained and was finally diluted to the lowest concentration used in the series. A comparison of the density F of the measurement solution with that of the control sample did not reveal any differences. Densities have been measured to within ∆F/F ) (5 × 10-6, using a high-precision vibrating tube densitometer with built-in reference oscillator and Peltier temperature control (Physica DMA 5000, Anton Paar, Graz, Austria). The instrument has been calibrated against doubly distilled, degassed, and filtered water. Temperature fluctuations during the density measurements were smaller than 0.01 K. 2.2. Broadband Ultrasonic Spectrometry. Measurements of the ultrasonic attenuation coefficient R as a function of frequency ν were performed in the frequency range between 0.1 and 400 MHz. In that frequency range, R varies by several orders of magnitude, due to its asymptotic high-frequency background part which is proportional to the squared frequency
R(ν) ) Rexc(ν) + B'ν2
(1)
Here, Rexc(ν) is the part exceeding the background part B′ν2 in the total attenuation coefficient. Although we are interested in the excess attenuation here, the frequency dependence in the background part requires different methods of measurements. At low frequencies, where B′ν2 is small, a cavity resonator technique was applied in which multiple reflections enhance the sensitivity of measurements by virtually increasing the pathway of interactions between the sample and the sonic field. At higher frequencies, where the background part predominates, a method was appropriate in which pulse-modulated ultrasonic waves are transmitted through a specimen cell of varying sample length. In a resonator method suitable at frequencies below 12 MHz, the liquid was contained in a circular cavity cell with concavely shaped focusing faces. Two cells were used,42 the one for measurements at frequencies below 2 MHz was provided with passive, concavely shaped reflector faces made of glass. Quartz discs, used as piezoelectric transducers to couple the cell to the electronic circuit, were attached to the reflectors. The faces of the other cell, operated in the frequency range between 0.3 and 12 MHz, were formed by thin quartz discs and were mounted
in such a way that they were slightly bent. In this way, the discs served as both transducers and concave reflectors. The complex transfer functions of the cells were measured around the desired principal resonance peaks using a computercontrolled network analyzer. This mode of operation allowed us to carefully consider the effects from higher-order modes by fitting the transfer function data to an appropriate sum of resonance terms. The resonance frequency νr and half-power bandwidth δν of a principal resonance were used to calculate the quality factor Qtot ) νr/δν which was corrected for the intrinsic cell losses using the common relation
1/Qtot ) 1/Q + 1/Qcell
(2)
which is based on energy arguments. In this equation, Q ) π/Rλ is the quality factor due to the liquid losses and Qcell represents the intrinsic cell losses. Here, Rλ ) Rλ means the attenuation per unit wavelength, with λ ) cs/ν and with cs denoting the sound velocity. For the determination of the intrinsic cell losses, water with well-known sound attenuation coefficient43 was used as a reference liquid. At frequencies between 3 and 440 MHz, absolute R measurements were performed by transmitting pulse-modulated ultrasonic waves through cells of alterable sample length l. Two cells were employed, differing from one another mainly in their dimensions and in the piezoelectric transducers that were utilized as transmitter and receiver, respectively: 3 e ν e 60 MHz, X-cut quartz discs,44 νT ) 1 MHz, 2rT ) 40 mm; 30 e ν e 440 MHz, Y-cut lithium niobate discs,45 νT ) 10.8 MHz, 2rT ) 12 mm; νT, fundamental frequency of transducer thickness vibrations; rT, tranducer radius. At each frequency of measurement, the transfer function T(l) of the sample cell was determined at 400 l values. The characteristic line of the electronic apparatus was routinely recorded. In these calibration measurements, the cell was replaced by a calculable high-precision below-cutoff piston attenuator.46 Diffraction effects in the specimen cell were considered by a semiempirical correction term42 with its parameters obtained from calibration measurements with suitable reference liquids. By circulating thermostat fluid through suitable channels of the specimen cells and by additionally providing the cell with a thermostatic coat, temperature fluctuations in the sample did not exceed ( 0.03 K. Uncertainties in the attenuation coefficient data from these temperature fluctuations were negligibly small. The temperature was measured with an uncertainty of less than 0.02 K. Uncertainties in the frequency ν of measurements (∆ν/ν < 10-4) were also small. The experimental uncertainties in the R data, as following from repeated measurements, including cell cleaning and refilling procedures between different runs, are those given below: ∆R/R ) 0.1, 0.1-3 MHz; ∆R/R ) 0.02, 3-25 MHz; ∆R/R ) 0.01, 25-300 MHz; ∆R/R ) 0.02, 300-400 MHz. Due to the use of different cells and different principles of measurements, systematic errors are unlikely to have remained unnoticed. 2.3. Sound Velocity. The sound velocity cs of the samples, which is required for the calculation of the attenuation-per-unitwavelength data Rλ () Rλ ) Rcs/ν), was obtained as a byproduct of the ultrasonic attenuation coefficient measurements. In the frequency range of resonator measurements (ν < 12 MHz), cs was derived from the frequency distance of successive resonance peaks of the cavity resonator cells. The uncertainty in the sound velocity data was ∆cs/cs ) 0.005. The cs values have been verified by high-frequency variable path-length measurements in which, at small cell length, the periodicity resulting from a
Alkyl Monoglycosides in Water
J. Phys. Chem. B, Vol. 113, No. 36, 2009 12285 excess attenuation spectra measured in this study. We thus used the function 3
R(ν) )
A ωτ
∑ 1 +n(ωτn )2 + Bν
n)1
(3)
n
to analytically represent the experimental Rλ data. The spectra of several solutions, however, require a further relaxation term at low frequencies. These spectra have been represented by the extended function
Figure 2. Ultrasonic excess attenuation spectra for two solutions of n-hexyl-β-D-glucopyranoside in water at 25 °C (O, 0.15 × 10-3 mol · cm-3; b, 0.2 × 10-3 mol · cm-3). Dotted lines show the subdivision of the former spectrum into two Debye relaxation terms and dashed lines the subdivision of the latter in three such terms. Full lines indicate the sum of these terms, respectively.
R*(ν) ) R(ν) +
A0ωτ0
(4)
1 + (ωτ0)2
The relaxation amplitudes An and relaxation times τn, n ) 0, ..., 3, as well as the B values of the measured spectra were found by fitting the relevant spectral function to the measured attenuation per wavelength data using a regression analysis that minimizes the variance
superposition of a standing wave pattern and the exponentially decaying wave can be used to determine λ and thus cs. 3. Results and Analytical Description of Spectra Figure 2 shows excess attenuation-per-wavelength spectra (eq 1) for two C6G1 solutions, both with a surfactant concentration below the cmc. The spectrum at concentration c () 0.15 mol/ L), significantly smaller than the cmc () 0.25 × 10-3mol · cm-3), reveals only one excess attenuation region ((Rλ)exc ) Rexcλ > 0). A close inspection of the experimental data shows inappropriateness of a single Debye-type relaxation term with discrete relaxation time. In fact, a superposition of two Debye terms, as indicated by the dotted lines, applies to that spectrum. The spectrum for the solution with concentration c () 0.20 × 10-3 mol · cm-3) closer to the cmc exhibits an additional relaxation term with the relaxation frequency at about 1 MHz which will be assigned to the monomer exchange between micelles and the suspending phase. Hence, this spectrum has to be represented by a sum of three Debye terms. It turns out that a three-Debye-term model is well suited for almost all
χ2 )
(
M (Rλ)m - R˜(νm, Pj) 1 M - J - 1 m)1 ∆(Rλ)m
∑
)
2
(5)
In this equation νm, m ) 1, ..., M, are the frequencies of measurements; (Rλ)m and ∆(Rλ)m are the data and their experimental uncertainties, respectively, at these frequencies; and R˜(νm) ) R(νm) or R*(νm), depending on the appropriateness of the model functions. The Pj, j ) 1, ..., J, are the parameters of R˜(νm). The values of these parameters resulting from the fitting procedure are given in Tables 2-4. 4. Discussion 4.1. Splitting of the Low-Frequency Relaxation Regime. As briefly mentioned before, finding that the low-frequency relaxation regime exists close to or above the cmc only is an
TABLE 2: Surfactant Concentration c, Density G, and Sound Velocity cs, as well as Parameters of the Relaxation Spectral Functions R(ν) and R*(ν), Respectively, for Aqueous Solutions of Alkyl Monoglucosides at 25 °Ca F
c surfactant C6 G 1 C7G1
C8G1
C8G1-C8Gal C9 G 1
a#
10 mol · cm 3
0.15 0.20 0.05 0.06 0.07 0.075 0.10 0.15 0.20 0.025 0.03 0.05 0.075 0.10 0.20 0.05# 0.067# 0.075 0.10 0.15 0.20
-3
g · cm
cs -3
1.004854 1.007506 0.999720 1.000003 1.000540 1.000643 1.002000 1.004143 1.006200 0.998192 0.998425 0.999142 0.999918 1.001147 1.005238 1.001300 1.001355 0.999642 1.000538 1.002341 1.004130
m·s
A1 -1
1519 1527 1506 1506 1508 1511 1511 1510 1509 1502 1502 1502 1503 1500 1500 1503 1503 1499 1499 1499 1499
10
-3
τ1
A2 -3
τ2
10
τ3
B
-3
ns
ps
1.8 1.9 2.9 2.2 1.0 1.0 1.0 0.7 1.8 2.0 0.8 1.8 1.0 1.2 2.2 2.2 1.0 0.8 1.0 1.5
33.9 35.1 32.7 32.8 32.8 33.0 33.6 36.0 38.7 32.8 32.7 33.6 35.7 36.6 42.9 36.7 36.7 38.2 38.1 43.0 47.1
ns
10
0.31
114
0.01 0.13
9.0 18.0
0.31 3.5 11.0 14.4 1.0 1.9 5.8 6.6 6.5 5.6 7.0 7.3 2.1 2.1 2.0 1.9
1013 268 174 128 3066 1737 839 582 358 159 378 355 887 647 402 302
0.10 0.12 0.18 0.46 0.54
3.3 4.6 4.2 5.2 14.0
0.45 0.63 0.13 0.13 0.10 0.11 0.23 0.40 0.85
0.09 0.02 0.37 0.88 0.22 0.17 0.37 0.44 0.71 0.86
12.6 16.0 6.7 7.7 18.4 12.6 5.4 6.0 5.9 7.0
0.10 0.28 0.57 0.39 1.06 0.41 0.48 0.01 0.71 0.60 0.87
indicates the concentration of C8Gal; the total glucoside concentration is 0.1 mol/L.
ns
A3
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TABLE 3: Parameters of Term “0” in the Relaxation Spectral Function R*(ν) of Solutions of Alkyl Glucosides in Water at 25 °C c surfactant
-3
A0
10 mol · cm
C 7 G1
-3
10
0.10 0.15 0.20 0.20 0.075 0.10 0.15 0.20
C8G1 C9G1
-3
4.2 3.6 1.2 2.3 0.10 0.12 0.18 0.20
τ0
RH(ν) ) AH
ns
AH(ωτH)mH [1 + (ωτH)2sH]mH+nH/2sH
with the normalization relation
∫-∞∞ GH(ln(τ/τH))d ln(τ/τh) ) 0
(6)
reflecting a continuous relaxation time distribution function GH(ln(τ/τH)). In eq 6, AH is an amplitude, and mH, nH, sH ∈]0,1] are parameters controlling the shape and width of GH(ln(τ/τH)). The principal relaxation time τH, according to -1
τH ) (2πνmax) (mH /nH)
1/(2sH)
∫-∞∞ GH(ln(τ/τH))ωτ(1 + (ωτ)2)-1d ln(τ/τH) (8)
1200 763 538 1131 53 45 36 56
indication of an underlying monomer exchange between the micelles and the suspending phase. The broadening of this relaxation regime in the experimental spectra is an unexpected result. The Teubner-Kahlweit-Aniansson-Wall model24-28 of the formation/decay kinetics of proper micelle systems predicts this spectral range to reflect one of two pathways along which micelle solutions, after a small disturbance, will reach their equilibrium size distribution of aggregates. This theoretical model implies a deep relative minimum in the oligomer region of the distribution. The minimum separates the monomers from the micelles, and fundamentally, it leads to the two modes of relaxation indicated above. The system first responds to a disturbance by a fast monomer exchange, with relaxation times in the nanosecond to microsecond range. This exchange causes a change in the mean aggregation number of micelles. In parallel, a slow redistribution of the system (with relaxation times on the order of milliseconds or seconds) occurs. It involves a change in the number density of micelles. Both relaxations are characterized by a discrete relaxation time. A broadening of the monomer exchange term (the lowfrequency relaxation region in the present spectra) has already been found in the spectra for solutions of other nonionic21 and also ionic47 micelles and has been empirically considered by a Hill-type relaxation term48-50
RH(ν) )
is characteristic of a Debye term. The Hill relaxation time distribution is defined by
(7)
is related to the frequency νmax at which RH(ν) adopts its maximum. A Hill term can indeed be used to represent the lowfrequency relaxation regime of the present spectra within the limits of uncertainty. Judging from the variance χ2 (eq 5), such a term applies almost as well to the experimental findings as a sum of two Debye terms. The trends in the width of the relaxation time distribution function, however, disagree with the previous results. As illustrated by the relaxation time distribution function in the Hill relaxation term of some n-heptylammonium chloride aqueous solutions in Figure 3, this function is particularly broad at surfactant concentrations near the cmc. Above the cmc, the distribution narrows with increasing surfactant concentration to finally approach the δ-function that
(9)
The functions shown in Figure 3 have been calculated by analytical continuation51 from the corresponding spectral functions RH(ν). The appearance of a Hill relaxation term instead of a Debye term in the spectra of previously studied surfactant solutions21,47 has been assigned to the fact that the minimum in the size distribution of aggregates decreases to finally vanish completely when the cmc is approached.29 Simultaneously, the relaxation times of both processes converge. As a result, near the cmc the relaxation terms do not appear well separated from one another but rather display one broadened relaxation region in the ultrasonic spectra. By contrast, the low-frequency relaxation regimes in the present spectra of alkyl monoglucoside solutions become broad at surfactant concentrations well above the cmc: c g 1.4 · cmc, C7G1; c g 8 · cmc, C8G1; c g 11 · cmc, C9G1. Hence, the concentration characteristics in the low-frequency relaxation regime of the alkyl monoglucoside solution spectra do not suggest broadening due to the special features in the size distribution of aggregates near the cmc.29 In fact, the broadening in the alkyl glycoside solution spectra points to the existence of two types of micelles or of micelles with nonspherical shape at high surfactant content. We thus proceed from the assumption of two monomer exchange relaxation terms. 4.2. Dominating Monomer Exchange Term, Rate Constants. Amplitudes A1 and relaxation rates 1/τ1 of the dominating monomer exchange term in the spectra are shown as a function of reduced concentration
X ) (c - cmc)/cmc
(10)
in Figure 4. The relaxation rates of the C8G1 and C9G1 solutions increase linearly with X, as predicted by the Teubner-KahlweitAniansson-Wall theory
(
1 1 X ) kb 2 + τ1 m j σ
)
(11)
with m j and σ2 denoting the mean aggregation number as well as the variance in the Gaussian size distribution b (ki+1 [Sji+1] - kib[Sji])/[Sji] ) -(i - m j )kb /σ2
(12)
of the micelles (i ) 2, 3, ...). Here, kbi is the backward rate constant of the ith step in the underlying isodesmic reaction scheme of micelle formation, and kb is the backward rate constant at micelle sizes around the mean (i ≈ m j ). The relaxation rates of the C7G1 solutions with rather high cmc display a small curvature, which is presumably a consequence of the fact that the formation of small micellar aggregates already starts at concentrations slightly below the cmc.
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J. Phys. Chem. B, Vol. 113, No. 36, 2009 12287
TABLE 4: Surfactant Concentration c and Temperature T, Density G, and Sound Velocity cs, as well as Parameters of the Relaxation Spectral Function R(ν) for Aqueous Solutions of C8G1 c
T
F
cs
A1
τ1
A2
τ2
A3
τ3
B
10-3mol · cm-3
°C
g · cm-3
m · s-1
10-3
ns
10-3
ns
10-3
ns
ps
0.025
15 25 30 15 25 30 10 15 20 25 30
1.000343 0.998192 0.996737 1.001385 0.999142 0.997671 1.002961 1.002208 1.001186 0.999918 0.998424
1473 1502 1514 1473 1502 1514 1458 1475 1487 1503 1513
1.0 1.0 7.8 5.8 5.6 9.8 7.7 7.0 6.6 5.8
3066 1489 1686 839 662 1288 869 692 582 442
2.1 0.8 1.6 1.0 1.0 2.0 1.8 2.1
43.9 32.8 28.9 46.8 33.6 30.7 56.7 48.4 41.3 35.7 31.7
0.05 0.075
A linear regression analysis of the relaxation rate data in terms j values (Table 5). Using estimates of eq 11 yields kb/σ2 and kb/m of mean aggregation numbers m j from the literature, kb, σ, and kf have also been derived (Table 5). The forward rate constant kf ) kfmj at the maximum of the size distribution of micelles has been simply calculated according to kf ) kb/cmc.24,25 The inset to Figure 5 shows the m j (τ1-1kb-1 - σ-2) data of the three series of alkyl glycoside solutions, for which measurements have been performed above the cmc. The data are proportional to X, thus confirming the applicability of eq 11. In Figure 5, a log-log plot of the backward rate constant kb ) kbmj versus critical micelle concentration cmc is presented. Also included are data for other nonionic, cationic, as well as anionic
Figure 3. Relaxation time distribution function (eqs 8 and 9) of the Hill relaxation term (eq 6) in the spectra of some n-heptylammonium chloride aqueous solutions at 25 °C.47
Figure 4. Relaxation rates τ-1 1 and amplitudes A1 of term “1” in the spectral function (eqs 3 and 4) for aqueous solutions of alkyl monoglucosides, at 25 °C, displayed as a function of reduced concentration (1, C6G1; b, C7G1; 9, C8G1; 2, C9G1).
0.04 0.09
59.0 12.6
0.30 0.15 0.10 0.02
3.9 6.8 3.0 16.0
0.31 0.28 0.40 0.28 0.34 0.50 0.57 0.54
TABLE 5: Parameters in the Relaxation Rate Relation (Equation 11) for the Monomer Exchange at 25 °Ca kb/m j 6 -1
surfactant 10 s C 7G 1 C8G1 C9G1
23.3 0.85 0.12
kb/σ2
kb
106s-1 mj 2.9 0.3 0.15
63 80 95
kf
106s-1 109cm3 · mol-1 · s-1 205 68 12
2936 2709 1802
σ 8.4 15.4 9
a Mean aggregation numbers have been taken from the literature41 and have been also obtained by interpolation. For C8G1 solutions, the m j value and cmc ) 0.025 mol/L have been selected from a broad variety of data.
Figure 5. Bilogarithmic plot of the backward rate constant kb at micellar sizes around the mean m j versus critical micelle concentration cmc for aqueous solutions of alkyl monoglucosides (2, CiG1; i denotes the number of methyl groups per surfactant chain), alkyl maltopyranosides (b, CiG2; ref 52), poly(ethylene glycol) monoalkyl ethers (0, CiE3; ), CiE4; ∆, CiE5; refs 21, 22, 53, and 54), alkyl ammonium chlorides (solid triangle pointing left, CiACl; refs 47, 55, and 56), alkyl trimethylammonium bromides (solid triangle pointing right, CiTABr; ref 57), and sodium alkyl sulfates (O, CiSO4Na, refs 58 and 59). The shaded area is given to accentuate the trend in the data. For alkyl monoglucoside solutions (b, C7G1; 9, C8G1; 2, C9G1), the inset shows -2 -1 the quantity a ) m j (τ-1 1 kb - σ ) as a function of reduced concentration X. The line with slope 1 represents eq 11.
micelle solutions. Our data fit the tendency in the kb values to increase with cmc. This tendency is a reasonable result, as both the probability for a molecule to escape from a proper micelle and the probability for a surfactant to be solved as a monomer will increase with decreasing length of its hydrophobic group. The data in Figure 5 also indicate an effect of the headgroup. At a given length of the surfactant alkyl group, more hydrophilic ionic surfactants reveal larger kb and cmc values than less hydrophilic nonionic surfactants. 4.3. Dominating Monomer Exchange Term, Reaction Volume. The reaction volume ∆V associated with the monomer exchange is related to the relaxation amplitude A1 in the ultrasonic spectra according to24,25
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A1 )
j )X (σ2 /m π(∆V)2 cmc κ∞S RT 1 + (σ2 /m j )X
Haller and Kaatze
(13)
with κS∞denoting the adiabatic compressibility extrapolated to frequencies well above the relaxation region and with the universal gas constant R. Using eq 13 it is assumed that the reaction volume is identical for all steps (∆V ) ∆Vi, i ) 2, 3, ...) in the isodesmic reaction scheme, which is the basis of the Teubner-Kahlweit model. The ∆V values obtained from the amplitudes, using the (rather uncertain) σ and m j values from Table 5 and using also κS∞ ≈ -1 -2 F cs , are displayed as a function of the number of methyl groups per alkyl chain of surfactant in Figure 6, where also results for solutions of poly(ethylene glycol) monoalkyl ethers in water are given. The data for the latter define a straight line that can be extrapolated back to pass through the origin. This finding has been taken as an indication of the reaction volume to be completely controlled by the hydrophobic chain of the surfactant. It has also been concluded that obviously the hydration properties of all methyl groups change in a similar way when a surfactant monomer is exchanged between a micelle and the suspending phase. This feature of nonionic surfactants is different from ionic detergents for which it is normally assumed that the first two methyl groups near the surface of micelles are in contact with water.60 From the slope in the ∆Vversus-i relation in Figure 6, it follows that ∆V(CH2) ) 1.6 cm3 · mol-1 for the volume change per methyl group of poly(ethylene glycol) monoalkyl ether surfactants. The ∆V value from the C7G1 series almost fits the data of the CiEj solutions (Figure 6), whereas the reaction volumes from the C8G1 and C9G1 series noticeably deviate from the trend of the CiEj data. Since the alkyl glucosides feature indications of a second monomer exchange term in their ultrasonic spectra (Table 3), the comparatively small ∆V values of those surfactants likely result from the use of the total reduced concentration X in eq 13. In fact, however, only a part of the molecules is essentially involved in the dominating monomer exchange because the other part has a share in the second exchange process. This view is supported by the dependence of the reaction volume per methyl group, ∆V(CH2), upon the reduced concentration X, as shown in the inset of Figure 6. Assuming, in conformity with the CiEj solutions, the reaction volume to reflect the change in the hydration of the alkyl chain when a monomer enters a micelle and assuming, in addition, all i methyl groups to cause an identical volume change, ∆V(CH2) has simply been calculated as ∆V/i. The reaction volumes per methyl group display two opposing effects (Figure 6). At surfactant concentrations slightly above the cmc, the ∆V/i values are small and increase with X. Obviously, in this concentration region, a significant amount of surfactant forms rather open oligomeric structures in which not just the hydrophilic head groups but also some adjacent methyl groups are in contact with water. At increasing concentration, larger micelles with more compact surface are formed, and the wetting of methyl groups is thus reduced. At even higher surfactant content, the second exchange process appears, leading to a reduction of surfactants participating in the dominating monomer exchange. 4.4. Second Monomer Exchange Term. The additional relaxation term “0” in some spectra of the alkyl monoglucoside solutions appears at the low frequency tail of the measurement range. For the 0.1 × 10-3 mol · cm-3 C7G1 and the 0.2 × 10-3 mol · cm-3 C8G1 solutions, for example, the relaxation frequency
Figure 6. Reaction volume ∆V as a function of the number i of methyl groups per surfactant alkyl chain for aqueous solutions of alkyl monoglucosides (b, CiG1; only the largest ∆V values of a concentration series are presented) and poly(ethylene glycol) monoalkyl ethers (O, CH3(CH2)i-1OCH2CH2)jOH; CiEj; refs 21-23). For the alkyl monoglucosides, the inset shows the reaction volume per methyl group of the alkyl chain, ∆V/i, as a function of reduced concentration X (b, C7G1; 9, C8G1; 2, C9G1).
(2πτ0)-1 is as small as 130 kHz. For this reason, a meaningful evaluation of the relaxation times in terms of rate constants is impossible. We thus attempt to just assign the low-frequency relaxation term to a molecular process in the surfactant solutions. Limited radial diffusion of monomers within the micelles, leading to protrusion but not to a complete exchange of molecules, is evidently characterized by relaxation times smaller than those of the monomer exchange process. For sodium dodecylsulfate micelles in water, protrusion relaxation times on the order of 2 ns have been found.59 In addition, protrusion is a zero-order reaction which should be characterized by a relaxation time independent of concentration, in contrast to the experimental relaxation times of the C7G1 system (Table 3). For this reason, we conclude that the relaxation term “0” reflects a complete monomer exchange. Two alternative models account for the existence of a second monomer exchange term. One proceeds from the existence of globular micelles at and slightly above the cmc. At increasing surfactant concentration, the micelles are assumed to continuously increase in size. Such growth is only possible if the micelles change their shape to become prolate or oblate ellipsoids or even rods or discs. Within the framework of this model, the mean aggregation number m j strictly depends upon the concentration, and furthermore, the size distribution of micelles at high surfactant content is no longer Gaussian. The appearance of two relaxation processes for the monomer exchange is assigned to different molecular packing at different sites of the micelles. For rod-shaped micelles, for instance, the exchange of monomers at the end-caps will largely correspond to that of globular micelles, whereas the rate constants for the access and emission of monomers at the middle section of micelles may noticeably differ. The other model likewise implies globular micelles at the cmc. They are, however, assumed to also exist at higher surfactant concentration. The system responds to increasing surfactant concentration by additionally forming nonspherical micelles. The formation of such a second type of aggregate may occur continuously or, alternatively, discontinuously, defining a second critical micelle concentration.61-63 In a strict sense, it is again incorrect to consider, within the framework of this model, only one size distribution of micelles in the evaluation of data. Rather, both types of micelles should display their own distribution of sizes. Let us note, however, that this model also accounts for the existence of a second monomer exchange term in the ultrasonic spectra.
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Figure 7. Ultrasonic excess attenuation spectra for 0.2 × 10-3 mol · cm-3 solutions of C6G1 (b), D-(+)-maltose (∆; ref 33), and D-glucose (O; ref 33) in water at 25 °C. The dashed line indicates relaxation term “2” in the spectrum of the alkyl monoglucoside solution. The dotted line shows the corresponding relaxation term in the maltose solution spectrum. Full lines are graphs of the spectral functions describing the experimental spectra.
Given the result that nonspherical micelles exist in the solutions of alkyl monoglycosides, the question arises why the backward rate constants, though based on the TeubnerKahlweit-Aniansson-Wall model of spherically shaped proper micelles, so nicely fit the previous results for poly(ethylene glycol) monoalkyl ether solutions (Figure 5). According to eq 11, the backward rate constant
kb ) m j ∆(τ-1 1 )/∆X
(14)
is given by the slope in the linear dependence of the relaxation rate upon reduced concentration. From this relation, it follows that a supposable variation in m j does not change kb beyond the limits of scatter of the data in Figure 5. For the same reason, it was possible to calculate reasonable rate constants using the rather uncertain m j and σ data from the literature. 4.5. Rotation Around Glycosidic Angles. Figure 7 shows ultrasonic excess attenuation spectra for solutions of C6G1, of D-(+)-maltose and D-glucose in water. In all cases, the solute concentration is 0.2 × 10-3 mol · cm-3. Besides the lowfrequency relaxation term due to the monomer exchange in the surfactant solution and the high-frequency term that is common to all solutions, the spectra of the alkyl monoglucoside and the maltose solutions reveal an intermediate relaxation term with a relaxation frequency around 10 MHz. As such a term has never been found with a solution of monosaccharides31 but is characteristic of disaccharide solutions,32,33 it is assigned to a rotation of a saccharide ring around the glycosidic angles Φ and Ψ. For disaccharides, several local minima exist in the Φ, Ψ potential energy surface,64,65 and it is assumed that the ultrasonic relaxation term reflects the transitions between such minima.32 On the basis of a thermal activation model Eyring relation66
( )
ln
pτ-1 2 ∆Hq ) const kBT RT
(15)
has yielded an activation enthalpy ∆Hq ) (24 ( 8) kJ · mol-1 for D-(+)-maltose32 if equipartition of two disaccharide conformational isomers is assumed. This ∆Hq value corresponds to that of the hydrogen network fluctuations of water at room temperature.67 We assume analogous local minima also in the Φ, Ψ potential energy surface of the alkyl monoglucosides and thus assign relaxation term “2” to the saccharide ring rotation around the direction of the bond to the alkyl chain of the surfactant. The relaxation rate of such unimolecular (zero-order) reaction should
Figure 8. Relaxation rates τ-1 2 (closed symbols) and amplitudes A2 of the term assigned to the rotation around glucosidic bond angles, displayed against solute concentration c for solutions of C9G1 (b,O) and D-(+)-maltose (2, ∆; ref 33) in water at 25 °C. Dashed lines are drawn just to indicate the trends in the data.
be independent of solute concentration c, and the relaxation amplitude should linearly increase with c.68 As indicated by the data in Figure 8, the relaxation rates τ2-1 of the C9G1 solutions are almost constant but are significantly larger than those for the D-(+)-maltose solutions. Obviously, the rotation around the bond between the glucose headgroup and the alkyl chain is somewhat faster than that around the bond between two glucose rings. This is likely an effect of a different activation enthalpy. The slight decrease of the τ2-1 values with c may result from an increasing viscosity. A viscosity effect had also been found in the relaxation rates of disaccharide solutions. Here, the formation of large micelles with closer packed surfactant head groups may increase the viscous friction at increasing solute concentration. The relaxation amplitudes A2 of both series of solutions in Figure 8 increase with c. At the same solute concentration, however, the values for the surfactant system are much larger than for the maltose solutions. If for lack of further information only insignificant differences in the distribution of Φ, Ψ-conformers of both types of solutes are assumed, the plausible result follows that the rotation of the glucose headgroup in a micelle is associated with a distinctly larger volume change than the rotation of both glucose rings relative to one another, as observed with aqueous solution of maltose. A quantitative discussion of term “2” in the ultrasonic spectra of alkyl monoglycoside solutions is impossible since, on the one hand, the parameter values of this intermediate relaxation term are subject to considerable experimental uncertainties and because, on the other hand, the solutions containing monomers, globular micelles, and nonspherical micelles are too complicated to enable clear conclusions from the data. 4.6. Exocyclic Hydroxymethyl Group Isomerization. All alkyl monoglycosides exhibit a high-frequency relaxation term with a relaxation time on the order of 1 ns (Table 2). Such a term is common to the ultrasonic spectra of saccharide solutions containing exocyclic hydroxymethyl groups30-32,69 and has thus been assigned to the rotation of the exocyclic -CH2OH group relative to the saccharide ring. The relaxation rate τ3-1 for the alkylglycoside solutions is somewhat larger than that for the solutions of D-glucose and D-(+)-fructose (Figure 9), indicating that the change of the exocyclic group between two potential energy minima is facilitated at the micelle surfaces or that the distribution of isomers is different. This is likely an effect of the activation enthalpy barrier which will be reduced when the pyranose ring is not fully surrounded by water and thus hydrogen bonding with water molecules is minimized. The assumption of exocyclic -CH2OH isomerization70 is supported by NMR studies71-73 and is in conformity with energy
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∆VS ) ∆V -
Figure 9. Relaxation rates τ-1 3 (top) and amplitudes A3 (bottom) of the term due to the exocyclic hydroxymethyl group isomerization plotted versus solute concentration c for solutions of C6G1 (1), C7G1 (b), C8G1 (9), C9G1 (2), D-glucose (O), and D-(+)-fructose (∆) at 25 °C. The dashed line indicates the trend in the monosaccharide data.
differences from computer models.74 NMR measurements revealed an equilibrium71 kqf
gg 798 gt
(16)
Rp ∆H0 FCp
(20)
is the isentropic volume change, with ∆V and ∆H0 denoting the isothermal reaction volume and reaction enthalpy, respectively; F is the density; and Rp and Cp are the thermal expansion coefficient and heat capacity, respectively, at constant pressure. In principle, the increase in the amplitudes, when going from D-glucose to alkyl monoglucosides, may be due to an increase in Γ or ∆VS. Since, however, the distribution of conformers in D-glucose aqueous solutions already leads to a stoichiometric factor Γ ) ([0.53]-1 + [0.45]-1)-1c ) 0.24c close to the maximum possible value 0.25c, we conclude that the conformational isomerization of the exocyclic -CH2OH groups in a micelle needs more space than in water. Assuming the free energy change associated with the isomerization to be given by the enthalpy difference (∆G0 ≈ ∆H0, ∆S0 ≈ 0), ∆H0 < 1 kJ · mol-1 follows for D-glucose solutions from the van’t Hoff equation, evidencing that the enthalpy term in eq 20 is negligibly small (
