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Surface Rheology Investigation of the 2-D Phase Transition in n-Dodecanol Monolayers at the Water-Air Interface Libero Liggieri,* Francesca Ravera, and Michele Ferrari CNR, Istituto per l’Energetica e le Interfasi, Sezione di Genova via De Marini 6, 16149 Genova, Italy Received August 7, 2003. In Final Form: September 16, 2003 n-Dodecanol is the last even linear chain alkanol soluble in water. Owing to this feature, previous investigations have shown in n-dodecanol monolayers at the water-air interface the existence of 2-D phase transitions, resulting in the formations of large molecular aggregates beyond a critical surface pressure. Here the rheological properties of this adsorption layer have been investigated in relation with the appearance of these phase transitions. To this aim, the surface dilational viscoelasticity has been measured as a function of the surface pressure by a specific oscillating drop procedure implemented on a pendant drop tensiometer. Different oscillation frequencies were investigated. The results show that the module and phase of the dilational viscoelasticity exhibit specific features that can be correlated to the different stages of the formation of a continuous liquidlike layer at an interface initially void from surfactant. In the regions of the gaslike 2-D phase the results agree with the prediction of the generalized Volmer model, which was already applied to interpret the equilibrium properties. In the region of coexistence between gaseous and liquid phases, just beyond the critical surface pressure, a model accounting for the presence of the aggregation process, previously developed by the authors, has suitably been applied to interpret the data. The comparison of the measured data with the model prediction allows the rate constants and other features of the aggregation process to be accessed.
1. Introduction The rheological properties of a liquid surface play an important role in many industrial and biophysical processes such as extension and contraction of lung alveolus, inkjet printing, liquid-liquid extraction, emulsification, foaming, and so on. Sodium dodecyl sulfate (SDS) is one of the most known surfactants, largely utilized for model studies and in practical applications, mainly foaming.1 n-Dodecanol is one of the most important impurities present in SDS, and it has been shown to play an important effect in increasing the surface activity of SDS solutions.2 Moreover, ndodecanol, like other alkanols, is often used in surfactant blends, playing a synergetic role in enhancing the performance of both ionic and nonionic surfactants. The lateral interaction with other surfactants is in fact favored by the small dimension and simple structure of the molecule, allowing a tighter packing in the adsorption layers3-5 to be achieved. n-Dodecanol is regarded as the last soluble of the even linear chained alcohol series; however, due to its tendency to undergo a phase transition in the adsorption layer, n-dodecanol monolayers show a behavior at the border between soluble and insoluble. Its investigation is thus of great practical and fundamental interest. In fact, such dual behavior has been recently observed for different kinds of soluble surfactants,6-8 which makes * Corresponding author. Fax:
[email protected].
the distinction between Gibbs and Langmuir monolayers somewhat fuzzy, opening a bridge between these two investigation branches, traditionally split by the utilization of different experimental techniques and theoretical approaches. The adsorption properties of medium chain alkanols9 have been explained assuming the formation of small n-mers at the interface. Longer chains show the occurrence of a first-order phase transition in alkanols Gibbs monolayers at the water-air interface, resulting in the formation of 2-D aggregates.10 The existence of these transitions has been evidenced by Langmuir trough investigations and direct visualization techniques,10,11 like Brewster angle microscopy (BAM) and ellipsometry. BAM results clearly show the formation, beyond a critical surface pressure, of a condensed 2-D phase in n-dodecanol monolayers. The condensed phase initially consists of separated dendritic-like domains, which, progressing the surface coverage, eventually merge and form a continuous layer. Similar transitions were also observed for mixed SDS-dodecanol monolayers.12 The present work deals with the rheological properties of these monolayers. These properties from one side are very relevant for practical applications and from the other side can be used to infer information on the status of the monolayer and to characterize the aggregation process. Recent results11 from interfacial stress/relaxation experiments show a dramatic decrease of the rheological
+039 0106475700. E-mail:
(1) Monin, D.; Espert, A.; Colin, A. Langmuir 2000, 16, 3873. (2) Fang, J. P.; Joos, P. Colloids Surf. 1992, 65, 113. (3) Kralchevsky, P. A.; Danov, K. D.; Kolev, V. L.; Broze, G.; Mehreteab, A. Langmuir 2003, 19, 5004. (4) Islam, M. N.; Okano, T.; Kato, T. Langmuir 2002, 18, 10068. (5) Penfold, J.; Staples, E. J.; Tucker, I.; Thomas, R. K. Colloids Surf., A 1999, 155, 11. (6) Motschmann, H.; Lunkenheimer, K. J. Colloid Interface Sci. 2002, 248, 462.
(7) Vollhardt, D.; Melzer, V. J. Phys. Chem. B 1997, 101, 3370. (8) Melzer, V.; Vollhardt, D. Phys. Rev. Lett. 1996, 76, 3770. (9) Lin, S.-Y.; Hwang, W.-B.; Lu, T.-L. Colloids Surf., A 1996, 114, 143. (10) Vollhardt, D.; Fainerman, V. B.; Emrich, G. J. Phys. Chem. B 2000, 104, 8536. (11) Ferrari, M.; Ravera, F.; Liggieri, L.; Motschmann, H.; Yi, Z.; Kraegel, J.; Miller, R.; Submitted to J. Colloid Interface Sci. (12) Vollhardt, D.; Brezesinski, G.; Siegel, S.; Emrich, G. J. Phys. Chem. B 2001, 105, 12061.
10.1021/la035442d CCC: $25.00 © 2003 American Chemical Society Published on Web 10/24/2003
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response in correspondence with the aggregation transition. The most relevant rheological parameter characterizing the properties of an adsorbed monolayer is the surface dilational viscoelasticity,13 , expressing the surface tension response to dilational stresses. The investigation of the surface tension response to low amplitude sinusoidal perturbations of the interfacial area can be suitably utilized to measure the surface dilational viscoelasticity. In this case, in fact, is given by14
)
dγ d ln A
(1)
where A is the interfacial area and γ is the surface tension. Adsorption dynamics involves time dependent processes, like the surfactant exchange with the bulk phases and reorganizations internal to the monolayer. For these reasons, is in general a complex quantity, its module and phase being dependent, further than on the coverage of the interface, on the frequency of the perturbation. Here, the dilational viscoelasticity of n-dodecanol monolayers is investigated as a function of the surface coverage, by using a specific pendant drop technique. The results provide access to the features of 2-D phases appearing during the adsorption progression. Quantitative information is obtained by comparison with models recently developed15,16 for the description of the equilibrium properties and of the rheological response of monolayers in the presence of surfactant aggregation.
Figure 1. Dynamic surface tension during the adsorption of n-dodecanol at the water/air fresh interface at 10 °C and for different aqueous concentrations.
2. Materials and Methods Ultrapure water from a Millipore-MilliQ system fed with distilled water was used for the preparation of the surfactant solutions. The surface tension of pure water was checked to be 72.5 mN/m at 20 °C and 74.0 mN/m at 10 °C. n-Dodecanol purified by double distillation has been provided by Dr. D. Vollhardt (MaxPlanck Institut fu¨r Kolloid und Grenzfla¨chenforschung). An n-dodecanol stock aqueous solution, with a concentration of 2 × 10-5 M, was prepared by stirring in an ultrasonic bath for 30 min at 40 °C. The investigated samples were prepared by dilution from the stock solution and processed within 1 day. In fact, after a longer time, the samples show a reduced surface activity, possibly due to the separation and consequent evaporation of some n-dodecanol. For the success of the study, it is mandatory to avoid any contamination of the samples. To this aim, only glass, Teflon, and stainless steel items are used in contact with the samples, during both their preparation and measurements. The experimental cell, the glassware, and any other item entering in contact with the samples are cleaned with sulfuric acid and then rinsed with distilled water and, finally, with ultrapure water. Surface tension is measured by a drop shape tensiometer ASTRA (Automatic Surface Tension Real-time Acquisition), developed in our laboratory,17 equipped with the software PAT-1 (Sintech, Germany).18 The tensiometer allows us to measure dynamic surface tension (DST) while controlled perturbations of the bubble interfacial area are applied by means of a microsyringe pump (Hamilton, Switzerland). DST is measured with an accuracy of the order of 0.1 mN/m and a maximum sampling
rate of the order of 3 Hz. Temperature can be controlled within 0.1 °C. To investigate the aging of freshly formed interfaces, a series of emerging bubbles are quickly formed and discarded at a nozzle tip, and then, by stopping the flow, a bubble of about 20 mm3 is kept. By this way an interface practically free from surfactant is obtained, as shown by the initial values of the DST (see Figures 1 and 2), which correspond to those of pure water. The apparatus has also been used to measure surface viscoelasticity as a function of the surface pressure. In this case, after forming the fresh interface according to the previous procedure, oscillations of the droplet area are applied during all the ST relaxation. The oscillation amplitudes, ∆A, were about 4% of the initial area, and with a maximum period of 20 s: much smaller than the characteristic time for the main ST relaxation process. Under these conditions, oscillations occur with a good approximation around a quasi-equilibrium state. Due to these two circumstances, a nearly sinusoidal response of the DST, with the same frequency, is expected and the module of the elasticity can be calculated as
(13) Lucassen, J. Faraday Discuss. Chem. Soc. 1975, 59, 76. (14) Noskov, B. A.; Loglio, G. Colloid Surf., A 1998, 143, 167. (15) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (16) Palazzolo, R.; Ravera, F.; Ferrari, M.; Liggieri, L. Langmuir 2002, 18, 3592. (17) Liggieri, L.; Passerone, A. High Temp. Technol. 1989, 7, 80. (18) Loglio, G.; Pandolfini, P.; Miller, R.; Makieski, A. V.; Ravera, F.; Ferrari, M.; Liggieri, L. In Novel Methods to Study Interfacial Layers; Moebius, D., Miller, R., Eds.; Studies in Interface Science Series; Elsevier-Amsterdam: 2001; Vol. 11, pp 439-483.
where A0 is the average surface area and ∆γ are the amplitudes of the DST oscillations. The phase of the elasticity is the phase shift between the DST response and the area oscillation. FFT based algorithms are utilized to calculate amplitudes and phases of the recorded DST response and surface area signals at the imposed frequency. Figure 3 shows the measured DST as a function of time and the corresponding calculated module and phase of the surface viscoelasticity, for different frequencies of the sinusoidal perturbation.
Figure 2. Dynamic surface tension during the adsorption of n-dodecanol at the water/air fresh interface with aqueous concentration c ) 12 µM and for different temperatures.
∆γ || ) A0 ∆A
(2)
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Figure 3. Dynamic surface tension of n-dodecanol at the water/air interface (T ) 10 °C, c ) 12 µM) during the aging/area oscillation investigation, at different frequencies, and corresponding calculated module (2) and phase (b) of the dilation viscoelasticity. In part a the different stages of the adsorption process are evidenced.
3. Results and Discussion Figure 1 shows the dynamic surface tensions of ndodecanol solutions with concentrations of 8, 12, and 17 µM during the aging of freshly formed interfaces. After an induction period, ST decreases till a break point, followed by the appearance of a characteristic plateau. Then, surface tension decreases again, relaxing toward
an equilibrium value. This behavior is indicative of the appearance of a surface transition at the break point. In fact, as shown in Figure 2, the critical ST decreases with the temperature. Moreover, at a given temperature, a high reproducibility of the critical ST has been observed. These results are in agreement with the previous observations obtained by Langmuir trough and BAM
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imaging, confirming the coexistence of gaseous and liquidlike 2-D phases beyond a critical surface coverage.10 It was not possible to achieve a reproducible induction period. This could be due to a very small, even if undetermined, initial surfactant load of the interface, with the surface activity of monomeric dodecanol being rather limited. The surface ST decrease after the induction period could be related to the formation of small n-mers, in coexistence with monomers as argued by Vollhardt et al.10 The investigation of the surface dilational viscoelasticity as a function of the surface coverage can be used to get more information about the aggregation process, about the phases formed in the adsorption layer, and about their structure. The plots of Figure 3 show the results of the oscillating drop experiments described above for the frequencies investigated in this study: ν ) 0.05, 0.0625, 0.1, and 0.2 Hz. Five different regions are evident during the process both in the DST and the viscoelasticity module plots. These regions can be correlated with different stages of the adsorption/aggregation process: (a) induction; (b) preaggregation; (c) aggregation; (d) aggregate network formation; (e) aggregate network/continuous monolayer. (a and b) Induction and Preaggregation Stages. Past the induction period during which the elasticity module is nearly zero, the latter increases with the surface pressure. The values of the measured module are those typical for a soluble adsorption layer.19,20 (c) Aggregation Stage. Beyond the critical surface coverage, corresponding to a surface pressure Πc, the aggregation transition is triggered. During this period the surface pressure is nearly constant, as expected during a phase transition. The elasticity modulus is dramatically decreased, since variations of the adsorption are compensated by the exchange of molecules between the condensed and the gaseous states. Even if it is difficult to measure, due to the small amplitude of the DST oscillation, during this stage, the phase shift seems to show a peak. Once a sufficient number of aggregates are formed, || starts increasing. The aggregates are initially relatively rare and small, which makes them poorly interacting, justifying the small values of ||. With progressing adsorption, the aggregates increase their size, so that their interaction becomes stronger. This stage is clearly shown by the increasing values of the elasticity module. (d) Aggregate Network Formation Stage. This stage is characterized by DST oscillations which are definitely upper limited by the critical pressure value, while clearly decreasing during the compression phase. In some signals a second plateau in the viscoelasticity module and phase is observed. This picture is compatible with the formation of a network of aggregates. At the end of the previous stage, the aggregate size is large enough to make the branches eventually merge and form a surface network. During the oscillation, the increasing of the surface area causes the destruction of the network separating the aggregates, which explains the upper bounded values of the DST. (e) Aggregate Network/Continuous Monolayer Stage. With progressing adsorption, the aggregates become even larger and more packed. The area oscillations are no longer able to separate the aggregates. Thus, the ST value during oscillations remains smaller than the critical value. (19) Wantke, K. D.; Fruehner, H.; Fang, J.; Lunkenheimer, K. J. Colloid Interface Sci. 1998, 208, 34. (20) Liggieri, L.; Attolini, V.; Ferrari, M. R.; Ravera, F. J. Colloid Interface Sci. 2002, 255, 225.
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Figure 4. Dilational viscoelasticity of an n-dodecanol adsorption layer (T ) 10 °C, c ) 12 µM) as a function of the surface pressure, at different frequencies.
The values of the elasticity modulus in this stage are comparable with those typically found for insoluble monolayers. In fact, with the aggregate molecules being in practice insoluble, the network opposes a strong reaction to area dilation/compression. The inflection point in || can be a signature of a modification of the network morphology, which may pass from a dendritic-like to a more ordinate structure, for example, a continuous layer with holes. At the end of the stage, the elasticity reaches a further plateau, which may call for the saturation of the layer by the formation of a continuos 2-D liquid layer. In this case, any further increase of the surface pressure could result in monolayer collapse, which exposes a new interface from the sublayer and makes the process restart from the critical state. This latter circumstance can be observed in Figure 4. In the experiment reported in this figure, adsorption was slowly progressing and one can even distinguish the transition points between the different stages. The descriptions of all the above stages correlate well with the different states and aggregate morphology that have been evidenced10 by direct BAM visualization of the monolayer. Moreover, the above picture is also coherent with the information available from ellipsometric measurements reported previously.11 These latter show that in the early stages of adsorption (stages a and b), compared to a limited decrease of surface tension, the thickness of the adsorbed layer increases. This might be related to a first adsorption/ accumulation process at the interface with an equilibrium of coexistence switching from monomers to small n-mers. With progressing adsorption, eventually a critical surface concentration is reached, triggering the formation of a liquid phase of large aggregates. This point was clearly evidenced by a discontinuity of the slope of the thickness function.
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For a quantitative analysis of the results, it is more useful to analyze the dilational viscoelasticity as a function of the surface pressure. This is obtained from the data shown in Figure 3, by plotting the module and phase of as a function of the reference surface pressure, Π ) γ0 - γref, during a single DST oscillation. The value of γref corresponding to a cycle is calculated by adding the calculated ST amplitude, ∆γ, to the minimum ST during the cycle. The procedure allows the plots of Figure 4 to be obtained, which also reflects the five stages discussed above. As it can be seen, there is some variability in the onset of the transition, with the critical pressure being variable between 9 and 12 mN/m. This is very likely a consequence of the dynamic conditions of the experiment, which make the transition zone a bit fuzzy, when compared with the case of static investigations, like the adsorption kinetics reported in Figure 1 or the Langmuir trough experiments presented elsewhere.10 As it appears from the plots in Figure 4, the dependence on the frequency of both the module and phase of cannot be appreciated, within the experimental error, due to the narrow frequency range investigated. The dependence of the viscoelasticity on Π in the preaggregation stages and in the coexistence region can instead be quantitatively interpreted, by using the models currently available for the description of adsorption layers showing aggregation transitions. Model for Preaggregation Stage. Coherently with the ellipsometric measurements and with the presence of the induction stage, for low surface coverage, one can adopt a model where single dodecanol molecules adsorb at the interface and form small n-mers once at the interface. In this case, the properties of a spread dodecanol monolayer below the critical surface pressure are then described15 by a generalized Volmer model, with a surface equation of state
Π + Πcoh )
RTΓ n(1 - ωmΓ)
(3)
where n is the aggregation number, ωm is the molecular area of the monomers, and Πcoh is the so-called coherence pressure (or Volmer constant). This approach was already applied to interpret equilibrium measurements,10 obtaining values of the parameters of the model which agree with those predicted by molecular simulations.21 This description is typical for a spread monolayer. Indeed, the small aggregates are most likely already insoluble. Moreover, for dodecanol, the characteristic time for the formation of small n-mers is negligible, while the characteristic time for diffusive adsorption of monomers is rather large. Thus, the formation of small n-mers is instantaneous when compared with the diffusion process. The above circumstances make the adsorption from the bulk kinetically irreversible. The characteristic time for diffusive adsorption can be evaluated by the inverse of the corresponding characteristic frequency
νD )
( )
D dcs 2π dΓ
Figure 5. Dynamic surface tension signal showing an example of the rupture of the continuous liquid-phase monolayer, evidenced by the abrupt increase of the dynamic surface tension to the critical value for the 2-D gas-liquid transition.
Figure 6. Comparison between the measured (symbols) viscoelasticity of the n-dodecanol monolayer (c ) 12 µM, T ) 10 °C, ν ) 0.0625) and the prediction of the generalized Volmer model (line), in the region of subcritical surface pressure. The values of the model parameters are those reported in Table 1.
τD ) 1/νD of the order of 105 s. Although obtained by a surface model not consistent with the Volmer one, the estimated magnitude of τD is sufficiently large, in comparison to the oscillation period, to warrant the assumption of a nearly vanishing diffusion flux during single oscillations. Under these conditions, the measured dynamic elasticity corresponds to the Gibbs elasticity, 0 ) dΠ/d ln Γ. This is also confirmed by the vanishing values of the phase of the viscoelasticity. In the framework of the generalized Volmer model, the Gibbs elasticity is
0 )
(4)
where D is the surfactant diffusion coefficient, c is the sublayer concentration, and Γ is the adsorption. To estimate νD, one can assume10 a Langmuir-like model for the adsorption of single dodecanol molecules, with the Langmuir-Szyszkovski constant aL ) 3.6 × 102 m3/mol and D ) 3.6 × 10-10 m2/s. With the value of ωm reported in Table 1, at the critical surface pressure, eq 4 provides
(5)
By solving the equation of state eq 3 in Γ, it is possible to write 0 as a function of Π
2
0
ΓRT n(1 - ωmΓ)2
0 )
RT(Π + Πcoh) + nωm(Π + Πcoh)2 RT
(6)
This dependence of the Gibbs elasticity, predicted by the generalized Volmer model, is in agreement with the module of the viscoelasticity measured in the preaggregation stage for all the investigated frequencies. An example is shown in Figure 6. The best fit values of ωm, n, and Πcoh are reported in Table 1 for the different
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Table 1. Parameters of the Generalized Volmer Model and of the Corresponding Aggregation Process for n-dodecanol Adsorbed at Water-Air (c ) 12 µM, T ) 10 °C) 10-5ωm (m2/mol)
0.05 0.0625 0.1 0.2 refs 10 and 21
ν (Hz)
n
Πcoh (mN/m)
d
kaggr ) 2πνaggr (s-1)
1.3 1.3 1.3 1.1
2 2.3 1.8 1.7
9 8 7 10
0.2 0.2 0.2 0.2
15 12 12 15
1.1
2.37
9
0.18
(7)
where
(
gΠ ) exp -
)
Π - Πc ωmδ RT
dΓ/m ) kaggr(Γm - Γc gΠ) dt
(9)
where kaggr is the aggregation rate constant. The analysis of the response to a small perturbation of a reference state provides the following general expression for the dilational elasticity above the critical surface pressure, as a function of the perturbation frequency, ν
frequencies: they are always in very good agreement with those reported previously.10,21 For Π < 1 mN/m a significant deviation from the model prediction in the subcritical region is observed, corresponding to the induction stage. At these low surface pressures a drastic decrease of viscoelasticity is clearly evident. This may correspond to the transition from adsorbed n-mers to monomers. In fact, these latter, being soluble, provide a vanishing elasticity while decreasing the surface pressure. Due to an amplitude response comparable with the accuracy of the ST measurement, a quantitative description is impossible in this surface pressure range. However, using a Langmuir model with an average molar area between ωm and 5ωm, to describe different situations of coexistence in the adsorption layer of a soluble monomer and small aggregates with n < 5, it does not provide even a qualitative agreement with the elasticity values observed for Π < 1 mN/m. In fact, in this region, the calculated values of 0 remain always much smaller than the measured ones. Thus, in conclusion, while the surface elasticity values evidence the aggregation in small n-mers in the region of Π between 1 mN/m and Πc, the physical picture during the induction period, characterized by very low surface pressure and surface elasticity, remains still unclear. This is however a zone of little practical relevance. Model for the Aggregation Stage. In correspondence with the transition, the phase of shows a peak, while a sudden decrease is observed in the module. These are typical features of a model for the surface viscoelasticity in the presence of coexistence of a gaseous phase and a condensed surface phase previously developed by the authors.16 According to this model, adsorbed molecules are distributed between the gaseous and condensed phases, according to the adsorptions Γm and Γ/m, respectively. The model also assumes a packing of the molecules in the condensed phase, which is suggested by the experimental observation that the surface pressure in the coexistence region is slightly decreasing. To account for that, the molar area of the molecules in the aggregates is written as ω/m ) (1 - δ)ωm, where δ is the packing factor (δ < 1). The surfactant packing shifts the distribution of surfactant between the two phases toward the aggregated phase. As a consequence, while increasing the surface pressure, the adsorption Γm decreases according to
Γm ) Γc gΠ
The aggregation process is described by the kinetic equation
q - ξ - j(λΨ(1 + q) - ξ) + (jν) ) 0m q(1 + ξ - λξ - j(λ + λξ + ξ)) 1 + ξ - j(λ(1 - Ψ)(1 + q) + ξ) (10) /0m (1 + ξ - λξ - j(λ + λξ + ξ)) where j ) x-1 is the imaginary unit. λ and ξ are the dimensionless frequency ξ ) (νD/2ν)1/2 and λ ) νaggr/ν, where νaggr ) kaggr/2π is the characteristic frequency of the aggregation process and νD is the characteristic frequency of the diffusion-controlled surfactant exchange, defined by eq 4. The ratio q ) Γm0/Γ/m0 and the quantities 0m ) Γm0(∂Π/∂Γm)0, /0m ) Γ/m0(∂Π/∂Γ/m)0, and Ψ ) (dΓm/dΓ)0 are specific to the reference state, indicated by the suffix ‘0’. These latter can be derived by the equation of state and, together with νD, specify (jν) for a given surface model. Moreover, owing to eq 8, it is
RTq/0m Ψ)RT + ωmδ0m
The frequency conditions which hold for the experimental study presented here are νD , ν. Under these conditions, by the limit ξ f 0 in eq 10, the surface dilational viscoelasticity reads
(jν) )
(21) Vysotsky, Y. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D.; Miller, R. Colloids Surf., A 2002, 209, 1.
(
1+q Ψ + /01(1 - jλ(1 + q)(1 - Ψ)) q 1 - jλ (12)
01 1 - jλ
)
which can be conveniently utilized to interpret the elasticity data during the first stage after the critical pressure, when non-interacting aggregates coexist with n-mers. According to the picture that one can derive from BAM observations10 and from the discussion about Figure 3a, the model should describe the data in the trans-critical region up to about Π ) 15 mN/m. Despite the limited range of applicability, eq 12 can be utilized to interpret the viscoelasticity data, c, just beyond the critical pressure to get useful information about the aggregation process. The perturbative approach leading to eq 10 is completely general and holds for any surface model provided that the conditions stated by eqs 7-9 are valid. Thus, in particular, it can be specified in the framework of the generalized Volmer model utilized to describe the range of coexistence of n-mers and large aggregates.15 Above Πc, the model leads to the surface equation of state
(8)
and Γc is the critical adsorption.
(11)
Π + Πcoh )
βΓm RT n 1 - ωm[(1 - δ)Γ + δβΓm]
(13)
with
β ) 1 - ωm(1 - δ)(Γ - Γm)
(14)
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where Γ ) Γm + Γ/m is the total adsorption and Γm is given by eqs 7 and 8. In the framework of this model, one calculates
0m )
RTΓm0 [1 - ωm(1 - δ)Γ0][β + ωm(1 - δ)Γm0] n [1 - ω (1 - δ)Γ - ω βδΓ ]2 m
0
m
m0
(15) /0m ) RT(Γ0 - Γm0) ωm(1 - δ)Γm0[β - 1 + ωm(1 - δ)Γ0] n [1 - ω (1 - δ)Γ - ω βδΓ ]2 m
0
m
m0
(16) At the critical surface pressure, being Γ/m ) 0, one can find the critical value of the dilational elasticity, c(jν), from eq 12, by expressing all quantities in terms of q and by the limit q f ∞ RTΓc n 2 1 - h - λ2(2h2Ψc - Ψc - h2) + jλ(2h2Ψc - Ψc - 2h2 + 1)
c(jν) )
[
(1 + λ2)(1 - ωmΓc)2
]
(17)
where
h ) ωm(1 - δ)Γc
(18)
and
Ψc ) lim Ψ ) qf∞
ωmδΓch2 n(1 - ωmΓc) + ωmδΓc(1 - h2)
(19)
As shown in Figure 5, by using values of n, ωm, and Πcoh similar to those utilized to describe the data before the transition, the prediction of eqs 6, 12, and 17 agrees with the measured elasticity before and just after the transition. The values of the model parameters obtained for the different excitation frequencies are reported in Table 1. Due to the vanishing amplitude of the ST response, the values of viscoelasticity phase are too undefined in the coexistence region to be useful for a quantitative analysis. However, in some signals (see Figure 1) a sudden increase of the phase can be observed close to the transition, which is another feature predicted by the adopted model. For the values of the parameters reported in Table 1, the model only provides a qualitative description of the experimental observation beyond the discontinuity corresponding to the transition. In this region, indeed, the module increases much faster than predicted by eq 12, which shows a stronger resistance of the interface to dilation/compression already before the aggregate branches come in contact. A possible justification can be found considering that a network of narrow surface “channels” is formed between the aggregates already before their branches start touching. Offering a large resistance to surfactant interfacial diffusion, these channels oppose the ST relaxation during oscillations. In fact, the surface diffusion coefficients of n-dodecanol molecules have been estimated22 to be of the order of 10-9 m2/s. Assuming the characteristic time of the area perturbations (10 s), the resulting diffusion length (22) Valkovska, D. S.; Danov, K. D. J. Colloid Interface Sci. 2000, 223, 314.
Figure 7. Comparison between the measured (symbols) viscoelasticty of the n-dodecanol monolayer (c ) 12 µM, T ) 10 °C) as a function of the surface pressure for different frequencies and the prediction (solid lines) of the rheological model assuming a surface aggregation process. The values of the model parameters are those reported in Table 1. The dashed line at ν ) 0.0625 Hz is discussed in the text.
is of the order of 100 µm: smaller than the characteristic length (some mm) of the “channels”. These retardation effects due to surface diffusion have not been considered in the approach described above, since they are quite difficult to be quantitatively modeled. In fact, from one side the channel net is ill defined and variable with the surface coverage, and from the other side the surface diffusion coefficients can only be roughly estimated. The adopted model is instead valid when aggregates are small, making suitable the estimation of δ and kaggr from the data just after the transition, and providing the data reported in Table 1. From the reported values of Kaggr, the characteristic time for the aggregation process can be estimated to be of the order of 0.5 s. The value of δ in the data interpretation has been limited to 0.2 to match its physical meaning, while one can observe (see Figure 7, ν ) 0.0625, dashed curve) that a more adequate description of the measured || in the coexistence region can be obtained by values of δ larger than 0.5. Although
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physically unsound with respect to the definition given here, these large values of the packing factor could call for a physical picture where further reductions of the occupation area of the n-dodecanol molecule in the aggregate are triggered by the surface area compression/ dilation process, for example, by forming thicker adsorption layers where the hydrophilic heads lies in alternate planes: a configuration which can be favored by the lateral interaction of the alkyl chains. 4. Summary The aim of the present work was the investigation of the rheological properties of n-dodecanol monolayers at water/air interfaces, to correlate these properties with the observed features of these monolayers, showing the coexistence of gaseous and liquid 2-D phases. The surface dilational viscoelasticity has been measured as a function of the surface pressure during the aging of fresh interfaces. To this aim harmonic perturbations were imposed to a freshly formed droplet interface in a pendant drop tensiometer, measuring the module and phase of the viscoelasticity from the harmonic response of dynamic surface tension superposed to the main aging process. The results have shown a punctual correlation between the rheological response and the features of the monolayer. In particular, the rheological response is largely damped during the aggregation transition. The viscoelasticity shows values typical for Gibbs monolayers (some 10 mN/m) in the region of low surface pressures. Beyond the
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critical pressure, the measured values rapidly increase, achieving those typical of Langmuir monolayers. The available models accurately describe the observed viscoelasticities up to values of the surface pressure just beyond the transition. The results confirm the validity of the generalized Volmer model for the description of the equilibrium features. This latter model, in conjunction with an approach developed to describe the dilational viscoelasticity in the presence of reorganization processes internal to the interface, also describes the results obtained in the region of coexistence of 2-D gaseous and liquid phases, just beyond the critical pressure, providing, in particular, access to the values of the rate constants for the aggregation process. A quantitative analysis of the observed viscoelasticity at larger surface pressures requires the development of specific models accounting for the complex interactions occurring in the monolayer between the large aggregates and for the related viscoelastic processes. Further investigation of the n-dodecanol monolayer morphology and structure, performed in dynamic conditions, by direct imaging techniques (BAM, ellipsometry, GIXD, neutron reflection, etc.) could be very helpful to address this task. Acknowledgment. This work was partially supported by the European Space Agency under the MAP project “Fundamental and Applied Studies in Emulsion Stabilitys FASES” (AO-99-052). LA035442D
