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Asphaltene-laden interfaces form soft glassy layers in contraction experiments: a mechanism for coalescence blocking Vincent O. Pauchard, Jayant P Rane, and Sanjoy Banerjee Langmuir, Just Accepted Manuscript • DOI: 10.1021/la5028042 • Publication Date (Web): 23 Sep 2014 Downloaded from http://pubs.acs.org on October 16, 2014
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Langmuir
Asphaltene-laden interfaces form soft glassy layers in contraction experiments: a mechanism for coalescence blocking Vincent Pauchard3,4, Jayant P. Rane1,2,4 and Sanjoy Banerjee1,2,4 1 Energy 2
Institute, City College of New York, New York, NY, 10031, USA Department of Chemical Engineering, City College of New York, New York, NY, 10031, USA
3 Flow
Technology group, Department of Process Technology, SINTEF Materials and Chemistry, Trondheim, Norway the Multiphase Flow Assurance Innovation Center, Norway. KEYWORDS: Asphaltenes, equation of state, IFT, interfacial coverage, Langmuir trough, jamming, soft glass rheology.
4FACE,
ABSTRACT: In previous studies, the adsorption kinetics of asphaltenes at water oil interface were interpreted utilizing a Langmuir Equation of State (EOS) based on droplet expansion experiments1-3. Long-term adsorption kinetics followed Random Sequential Adsorption (RSA) theory predictions, asymptotically reaching ~85% limiting surface coverage, which is similar to limiting random 2D close packing of disks. To extend this work beyond this slow adsorption process, we performed rapid contractions and contraction-expansions of asphaltene-laden interfaces using the pendant drop experiment to emulate a Langmuir trough. This simulates the rapid increase in interfacial asphaltene concentration that occurs during coalescence events. For the contraction of droplets aged in asphaltene solutions, deviation from the EOS consistently occurs at a surface pressure value ~21 mN/m corresponding to a surface coverage ~80%. At this point droplets lose the shape required for validity of the Laplace-Young equation, indicating solid-like surface behavior. On further contraction wrinkles appear, which disappear when the droplet is held at constant volume. Surface pressure also decreases down to an equilibrium value near that measured for slow adsorption experiments. This behavior appears to be due to a transition to a glassy interface on contraction past the packing limit, followed by relaxation towards equilibrium by desorption at constant volume. This hypothesis is supported by cycling experiments around the close-packed limit where the transition to and from a solid-like state appears to be both fast and reversible, with little hysteresis. Also, the soft glass rheology model of Sollich4 is shown to capture previously reported shear and elasticity behavior during adsorption. The results suggest that the mechanism by which asphaltenes stabilize water in oil emulsions is by blocking coalescence due to rapid formation of a glassy interface, in turn caused by interfacial asphaltenes rapidly increasing in concentration beyond the glass transition point.
1. Introduction: Water in oil emulsion stabilization by asphaltenes (indigenous hydrocarbons composed of polyaromatic cores with side alkyl chains3) is of practical importance for crude oil transport (pipe flow) and processing (dehydration). The most prevalent explanation is the formation of a crosslinked material at the water/oil interface5-9` which blocks coalescence in the emulsions. Historically this largely arose from the observation of wrinkles upon contraction of asphaltenes covered droplets10. From that starting point, confirmation of cross-linking and its relationship with emulsion stability was deduced form various experiments aimed at studying the problem. In particular, shear rheology revealed the transition from a fluid like behavior (with the adsorbed layer viscous modulus being greater than the elastic modulus, i.e., G’’>G’) to a solid like behavior (G’>G’’) after sufficiently long aging times11, 12. Furthermore shear moduli in the solid regime appeared to follow a particular form of frequency dependence 11, 12already observed for chemically cross-linked polymers at the gel point13, i.e.,
G ∝ ω , G ∝ ω = =
(1)
where G’ is interfacial elastic modulus, G” is interfacial viscous modulus, ω is frequency of oscillation, φ is phase angle and n is power exponent. As well, the condition for emulsion stability (high asphaltene concentration and poor solubility) appeared to be consistent with those causing critical gel type rheology10. This line of reasoning however contains some inconsistencies. First and most notably, stable emulsions can be obtained with short mixing times (down to a few minutes 14), which is much shorter than the time necessary for onset of critical gel rheology (which is typically hours11, 12). Second, in the bulk (3D), critical gel rheology is usually limited to the specific gel point15, 16 (before and after the gel point, the phase angle indicated in Eq. 1 shows frequency dependence) whereas for asphaltenes it seems to be valid over an extended range of conditions11. Furthermore the power exponent n is consistently reported to be higher than 0.4 (except for some values between 0.2 and 0.4 for very long pre-polymers already displaying entanglement before reaction),15-18 whereas for asphaltenes it is reported to continuously decrease with time down to values as low as 0.1511, 12. Finally, the nature of cross-linking was never really addressed. Neither the length of alkyl side chains nor
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the presence of few polar groups could justify such a mechanism. A mechanism similar to bulk nanoaggregation is also improbable since steric hindrance due to alkyl side chains limits the size of nanoaggregates in solution19. Furthermore, two independent studies20, 21 showed that when an emulsion is left at rest or under gentle stirring, droplet coalescence proceeds until surface coverage reaches a critical value ~3.2 to 3.5 mg/m2. This behavior is independent of asphaltenes concentration, solvent nature, and time of mixing. By using a reasonable estimate of the molar mass of ~750 g/mol, this critical mass coverage can be converted into a critical molecular coverage of 2.6 to 2.8 molecules/nm2. This value corresponds to ~80%, close to the asymptotic surface excess coverage obtained from droplet expansion experiments2. In these experiments surface pressure isotherms proved independent of adsorption history and could be fitted by a Langmuir Equation of State (EoS), with a surface excess coverage value of 3.3 molecules/nm2. This value corresponds both to the average core size of asphaltenes (6 to 8 rings) and to their interfacial configuration (core parallel to interface and side chains perpendicular)22-24. As remarked in18, a critical relative coverage of 80-85% is remarkably consistent with the value reported to cause coalescence arrest in Pickering emulsions and interpreted as a close packing (jamming) limit. It was therefore proposed that droplet coalescence induces an increase in asphaltenes surface coverage until the interface jams, though the mechanism by which this prevents coalescence was not discussed. This interpretation was supported by a recent analysis of adsorption kinetics of asphaltenes at the interface between water and an aliphatic mixture (which was a poor solvent for asphaltenes3). The adsorption kinetics indicated a transition from an initial diffusion limited regime to an adsorption barrier limited regime. For long times, the adsorption barrier limited regime proved to follow kinetics predicted from Random Sequential Adsorption (RSA) theory mediated by surface diffusion, with an asymptotic limit of ~85% relative coverage, which was conjectured to be due to jamming. As mentioned earlier, this asymptotic limit is remarkably consistent with theoretical simulations of 2D random close packing (RCP) or maximally random jammed (MRJ) state of size-distributed disks, which might represent asphaltenes with various core sizes. With different simulation techniques and for a large range of size distributions (type, width), RCP or MRJ occur at relative coverage values of 85±1%25-30. The conditions at which surface coverage increased slowly in these experiments are very different from the rapid increases obtained in coalescence. Further, why conditions close to jamming should block coalescence was not clear. The explanation that the formation of a cross-linked film could block coalescence did not appear to require conditions close to jamming and there were emulsion-coarsening results inconsistent with the cross-linked film mechanism as discussed earlier. The main purpose of the work presented here, then, is to investigate coalescence blocking mechanisms with a series of controlled contraction experiments on asphaltene-laden interfaces that capture the rapid increase in interfacial
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asphaltene concentration (due to reduction in specific interfacial area) associated with coalescence. A potential mechanism that may address the inconsistencies in our current understanding is offered by simulations indicating that on contracting a “dilute” interface, a glass transition occurs below RCP for a coverage value around 80%30. Informed by these findings, the present work investigates whether the transition to solid-like asphaltene laden interfaces could be primarily governed by steric interactions and hence surface coverage, in contrast to the mechanisms previously proposed. To this end, water droplets aged in asphaltenes solutions with different concentration have been contracted (and then allowed to recover at constant volume or reexpanded). These pendant drop experiments are meant to emulate Langmuir trough experiments with some advantages and disadvantages. For example, one advantage of using pendant drop experiments is that the asphaltenes are allowed to adsorb onto the drop surface in circumstances closer to those that occur in practice. In other words the forced adsorption of all types of asphaltenes structures (molecules, nanoaggregates and clusters) that occurs in Langmuir trough experiments due to evaporation of solvent is eliminated. A disadvantage is that the adsorbed amount is not known a priori. This drawback can be circumvented following a method developed for surfactants31, 32 and discussed further in the experimental section of this paper. Briefly, selection of a reference surface pressure and the determination of the surface coverage (based on the Langmuir EoS) enables comparison of surface pressure isotherms obtained with different adsorption histories.. Another advantage of the pendant droplet (used as a Langmuir trough) experiment is that it can directly provide an indicator of the solid-like nature of the interface. The droplet shape is analyzed to calculate interfacial tension by means of the Young-Laplace equation. The droplet contour is first fitted with a Laplacian shape (i.e. the shape that a droplet should assume with a fluid interface). If a significant error to the fit is observed this can be interpreted as a sign of a solid-like interface. By these means the behavior of dense asphaltene layers (in the vicinity of close packing) can be investigated. It will be shown that a deviation from the previously identified Langmuir EoS is observed starting from ~80%, in parallel with a transition to a solid-like behavior. This transition is reversible upon static recovery or reexpansion. Whether the observations can be interpreted in terms of transition to a soft glassy interface will also be investigated. 2.1. Chemicals Chemicals used in this study are essentially the same as reported in our earlier studies1, 2, 33.The aqueous phase is composed of de-ionized Milli-Q® grade water pre-mixed with 43 g/l of NaCl and 7 g/l of CaCl2. Final pH is adjusted to 7 with 0.1M NaOH. The asphaltenes have been extracted by precipitation with 20 volumes of n-heptane to 1 volume
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of crude oil from the Norwegian continental shelf and rinsing with n-heptane. A stock solution of 1000 ppm asphaltenes in toluene was prepared by sonication for 5 minutes and then stored in a sealed glass vial wrapped in aluminum foil to prevent exposure to light. Before use, the stock solution is sonicated again for 5 minutes. Then the appropriate volume of stock solution is withdrawn and further diluted with toluene before mixing with a synthetic poly-alpha olefin (2002 Nexbase® oil from Neste oil, Finland). The obtained asphaltenes solution is then sonicated for 1 minute. 2.2. Pendant droplet apparatus Experiments have been performed with a Theta Tensiometer (Biolin Scientific, Finland). An inverted 16-gauge needle is submerged in the aqueous phase such that the tip is visible in the frame of capture. A gas tight syringe (Hamilton Company, USA) is mounted in a micro-actuator (Harvard Apparatus, USA) to ensure the instantaneous creation of a droplet of a preset volume. Edge detection is used to identify the droplet shape, with the interfacial tension determined using the Young-Laplace equation 34. As discussed earlier information regarding transition to solid-like interfaces can also be derived from these experiments. Interfacial tension calculations from drop shape analysis rely on the Young Laplace equation and hence on the implicit assumption that the interface remains fluidlike. Should the interface be solid like then the drop would assume a non-Laplacian shape. This can be quantified by the residual non-convergence of fit to Young Laplace equation (r) defined as:
r=
ersum m
(2)
Here ersum is the sum of squared orthogonal distances (in pixels) between detected contour points and the YoungLaplace profile generated; m is the number of detected contour points. 2.2. Experimental procedure As mentioned above the pendant droplet apparatus is used to emulate a Langmuir trough for single contraction or expansion experiments, contraction-recovery experiments and large amplitude relatively slow cycling experiments. After a specified aging time, the droplet area is rapidly changed. The IFT is measured as a function of interfacial area. During rapid contraction or expansion, it is reasonable to assume that no asphaltenes are exchanged between the bulk and the interface (this will be further discussed later). Therefore, the mass of asphaltenes on the droplet surface Γ(t)•A(t) remains constant, where Γ(t) is the interfacial coverage of asphaltenes at time t and A(t) is the surface area of the droplet at the same time. If a reference area is chosen, one can plot IFT vs. relative coverage [Γ(t)/Γ(Aref) = Aref/A(t)] for each test condition. To compare different test conditions (different adsorption times,
different asphaltenes mass fraction), a common reference IFT, γref is chosen that defines the choice of the reference area for each test Aref=A(γref). If IFT is a unique function of surface coverage, then Γ(Aref) is unique throughout the dataset. In turn, all curves IFT vs. relative coverage [Γ(t)/Γ(Aref) = Aref/A(t)] will overlap. On the contrary, if adsorbed species undergo relaxation/reorganization over time, the curves will not overlap, intersecting at Γ(t)/Γ(Aref) = 1 instead. As reported in our earlier study1, 2, the expansion curves obtained with variable asphaltenes concentrations and adsorption time do overlap revealing a unique dependency of IFT on relative surface coverage. The master curve is well fitted by a Langmuir Equation of State:
π = − kT Γ ∞ ln(1 −
Γ ) Γ∞
(3)
With Γ∞~3.3 molecule/nm2. This EoS enabled prediction of the unique relationship observed between instantaneous dilatational elasticity and surface pressure. With the objective of studying how dense layers might deviate from the previously identified EoS, the experiments presented later have been analyzed in a somewhat different manner. Surface coverage at the reference time is calculated from initial surface pressure by recasting the equation of state as below. Then surface coverage during contraction is directly calculated from: Γ (t) = Γ(t )
( ! ) ( )
(4)
In this way the evolution of surface pressure as a function of surface coverage can still be compared between different experiments but also directly with the EoS With regard to contraction-relaxation experiments the situation is more complex since desorption can occur over long times. The procedure described above cannot therefore be directly applied without additional assumptions. These will be discussed in the following section based on observations made for contraction experiments. 3. Experimental results Two contraction experiments and one expansion experiment are shown in Figure 1(a) and (b) in terms of time evolution of area and interfacial tension respectively. Initial conditions were different. Contractions started after 5 minutes adsorption in a 50 ppm solution (IFT~32 mN/m), and after 3 hours adsorption in a 100 ppm (IFT~21 mN/m) solution, respectively. Expansion started after 6 hours aging in a 50 ppm solution (IFT~20 mN/m). Adsorption times of 3 and 6 hours largely cover the time range for cross-linking in aliphatic mixtures as reported, for example in 10. Nevertheless Figure 1(c) shows that when surface pressure is plotted versus surface coverage (following the procedure above described), the three curves superpose
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Despite the superposition of surface pressure vs. coverage curves, let us first try to interpret the transition to a non Laplacian shape with the previously suggested behavior in terms of cross-linked interfaces. This requires hypothesizing that cross-linking generates a Laplacian shape during static adsorption (until contraction begins). Then the nonLaplacian shape at large deformation could be due to the non-uniformity of strain (higher at the neck of the droplet than at the apex). Non-Laplacian shapes have however never been observed for expanded droplets, even at large expansion ratio. This means that upon expansion, aged interfaces are fluid like. Admittedly this could be due to a destruction of the network upon expansion. Whether this is true can be investigated by large amplitude cycling experiments (Figure 3(a)). Slight hysteresis can be seen in the relationship between area and surface pressure but the process is essentially reversible (Figure 3(b)). Let us for the time being assume that hysteresis is due to adsorption/desorption during cycling. The width of the hysteresis at mid-height would only correspond to variation of ±5% coverage. With respect to the magnitude of the deviation from the EOS observed for single contraction or expansion experiments, we shall then consider the same type of analysis on cycling experiments by treating each expansion or contraction leg as a separate experiment (Figure 3(c)). Contraction legs almost superpose to single contraction experiments, with a deviation from the EoS starting around 21 mN/m and a flattening above 30 mN/m. On the other hand expansion legs only slightly deviate from the EoS down to 26 mN/m, then superpose to it from 26 mN/m downward. This can be compared to the evolution of the residual error to the Laplacian fit. Upon contraction, the error starts increasing at 21 mN/m then plateaus and finally diverges around30 mN/m. We note at this point that the Langmuir EoS can safely be used below 21 mN/m, a finding we shall use in a later section. Upon expansion, the error decreases almost down to 0 at 26 mN/m. Again some hysteresis can be observed but the transition to a solid state is fully reversible and almost immediate even after repeated large expansions. All this suggests that except if cross-linking is reversible and immediate, the observations are not compatible with the hypothesis of a cross-linked network.
70 Contraction 50 ppm : 5 min 100 ppm: 3 hours Expansion 50 ppm: 6 hours
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well in the regions where surface coverage values overlap. This is confirmation of the absence of a time dependent gelling process: surface pressure only depends upon surface coverage. This is also verification of the assumption of the absence of exchange between interface and bulk during rapid area change: the effect of adsorption during expansion and desorption during contraction should shift curves in opposite directions along the vertical axis. Furthermore contraction curves follow the Langmuir EoS for surface pressure values up to 21 mN/m (Figure 1(c)). Between 21 and 30mN/m, contraction curves progressively level off compared to EOS predictions but still depends upon surface coverage only (and neither on adsorption history, nor on deformation history). Above 30mN/m, surface pressure flattens out. In the same range wrinkles appear and the droplet does not maintain its Laplacian shape (Figure 2 (a)): the interface is not fluid-like anymore but solid-like.
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356 11 30
Contraction: 50 ppm: 5 min 200 ppm: 1 min 100 ppm: 3 hours Expansion: 50 ppm: 6 hours Langmuir EOS for Asphaltenes Tilting model
25 16 20 21 26 15 31 10
5 36
(c)
0 41 0.000 0
0.465 1
2 0.930
3 1.395
4 1.860 2
Interfacial coverage (molecules/nm )
Figure 1: single contraction/expansion experiments. (a) Area vs. time (b) IFT vs. time (c) IFT vs. interfacial coverage from equation 4.Comparison with Langmuir EOS and tilting models (equation 3 and 6 respectively).
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Area(mm2)
Figure 2: Typical image of asphaltene stabilized droplet when high contraction ratio is induced to increase surface pressure of 35 mN/m. Wrinkles appear and the Laplacian shape is not maintained. 80
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Figure 3: Cycling experiments. Droplet aged 3 hours in a 100 ppm solution. (a) Time variation of area of the droplet and corresponding surface pressure during cycling experiment. (b) Surface pressure versus Area during same experiment. (c) Surface pressure versus interfacial coverage (d) Residual non-convergence of the fit against surface pressure during cycling experiment. To proceed with the interpretation of the data, observe that in terms of relative coverage, deviation from the EOS during contraction starts ~80%, which is slightly lower than the jamming limit (~85%) estimated from asymptotic adsorption data presented previously3. Kinetically the two types of tests are however very different. During static adsorption jamming limit is approached over hours due to the adsorption process, which is slow, whereas contraction only lasts a few seconds. During contraction the compaction of the interface could then undergo a kinetic arrest (i.e. a glass transition) that would not be apparent during static adsorption. This would be consistent with numerical simulations showing that upon compression an ensemble of size-distributed disks undergo a glass transition at 80% (whereas RCP is close to 85%). In other words the wrinkles reported to be a sign of a gelling process, which would require some time, could occur after very short times, provided the contraction ratio is high enough. Figure 4 actually shows that contracting by a factor 10 the volume
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306
Surface pressure (mN/m)
25 12 20 18 24 15 10 30
Sudden contraction 36 5
Figure 4: Droplet contracted by a factor 10 in volume. 45s adsorption, 100 ppm in 85-15% heptane-toluene.
80 ppm 100 ppm
0 42 0
2
4
6
8
12 14x103
10
Time (s)
Wrinkles observed with asphaltenes are however not stable and disappear after a few minutes. This reveals a mechanism lowering lateral interactions, which could be slow, desorption. To investigate this further, several contracted droplets were left at rest (i.e. at constant volume) for a few thousand seconds to monitor interfacial tension recovery (Figure 5). Surface pressure decreased monotonically (with a kink around 22-23 mN/m) down to ~19.7 mN/m at 14000 s (adsorption time + contraction time + relaxation time). These surface pressure values following ‘relaxation’ are significantly higher (by 2 to 3 mN/m) than initial values before contraction but are very close to the maximum values reached during static adsorption at the same bulk asphaltene concentrations (~19.5 mN/m after 30000 s at 80 ppm, see figure 1 in 3). This could mean that ‘relaxed’ surface pressure values correspond to an equilibrium being established between adsorption and desorption. Such a mechanism requires further examination with regard to the hypothesis made for analyzing adsorption kinetics in3, viz. that asphaltenes are “almost insoluble”, with desorption being neglected as surface coverage increases up to values ~80%. However, at equilibrium coverage itself (~85% for slow adsorption processes) desorption must equal adsorption. We will show that these phenomena can be quantitatively modeled by a simple desorption model where the rate of decrease in the surface coverage due to desorption is proportional to the surface coverage. This leads to a rapid (exponential) increase in the desorbed amount near the ‘jamming’ limit for slow adsorption. To proceed consider a quantitative analysis towards the end of ‘recovery’ (i.e., below 21 mN/m, which as shown in figure 3(c ) allows use of the Langmuir EoS to determine surface coverage from surface pressure). If recovery kinetics is controlled by desorption rate (for which the simple "#$ model alluded to earlier is = &'( Γwith kd the desorp"% tion rate constant) then Γ (t) = Γ(t )e*+,(%*
!)
(5)
With Γ(tref) being the surface coverage value at a chosen reference time, e.g. the time when 21 mN/m is reached, for which conditions we can use the Langmuir EoS to determine surface coverage.
Figure 5: Recovery curves of contraction experiments. Variable concentrations in Nexbase 2002. Variable contraction ratio. Figure 6 presents in a semi-logarithmic plot the end of the recovery tests at with tref corresponding to a surface pressure of 21 mN/m. The curves obtained are quite linear. Furthermore despite different adsorption and contraction histories, the curves obtained at 80 and 100 ppm are close to each other.. Equating the slope of the curves to the desorption rate constant enables calculation of the desorption rate at any surface coverage and comparison with the adsorption rates measured during slow (static) adsorption experiments (from 3). Figure 7 shows that adsorption and desorption curves intercept at 0.77; relative coverage corresponding to 19.6 mN/m surface pressure. This confirms that recovery is governed by desorption to equilibrium. 42.405
42.400 y = -1.08E-05.x+42.4
42.395 ln (gamma)
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2
R = 0.97 42.390
42.385 y = -1.21E-05.x+42.4 2
R = 0.966
42.380
42.375 0
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ref
1500
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(s)
Figure 6: natural logarithm of surface coverage during recovery versus rescaled time (t=0 at 21 mN/m). Same data as Figure 5.
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Figure 7: Comparison between desorption rates during recovery after contraction (black line, data calculated from slope of Figure 6) and adsorption rates during static adsorption at 100 ppm (symbols and blue line, experimental data and RSA model from3). 4. Discussion From the analysis presented above, it becomes clear that the transition between a fluid and a solid like interface is fast and reversible upon area changes. Upon contraction it occurs in a well-defined range of surface coverage around 80%. Upon re-expanding the droplet, transition to fluid like behavior occurs at a slightly higher coverage. This hysteresis is accompanied by a slight hysteresis in the surface pressure isotherm. On the other hand when a droplet is contracted sufficiently so that transition to solidlike behavior occurs and is left at rest, fluid-like behavior is recovered by desorption. A critical coverage of 80% is also consistent with glass transition during contraction as obtained by numerical simulations30. From this starting point it can be inferred that solid-like behavior can only be observed during slow (static) adsorption if equilibrium surface coverage comes close enough to 80%. Depending upon the nature of the solvent, this may require differing asphaltenes concentrations in the solution. However reaching such critical coverage in static adsorption takes very long times due to steric hindrance. This explains both the delayed rise of shear elasticity and the effect of the nature of the solvent as reported earlier11, 12. On the other hand specific interfacial area reduction due to coalescence in coarsening emulsions would allow critical coverage to be reached much quicker (in a manner similar to the contraction experiments), which would explain the time scale discrepancy between emulsion stabilization and rise of shear elasticity20, 21. An important point requires further consideration, i.e., that the transition to solid like behavior appears to be quite smooth with a relatively slow increase of the error in convergence of the pendant drop shape to a Laplacian fit which is an indicator of transition from fluid-like behavior of the interface (see figure 3 (c)). This error increases from 80 to more than 120% coverage (where wrinkling of the surface is seen). Different mechanisms could be considered to explain this relatively smooth transition. The first could
be a thermodynamic coexistence zone between a liquid phase and an ordered solid phase. As previously found, birefringence is observed in interfacial material extracted from asphaltenic emulsions35 or collected at the surface of a contracted pendant droplet36. Furthermore hysteresis of surface pressure isotherms with asphaltenes is reported to be associated with heterogeneous domains as observed by Brewster Angle Microscopy on Langmuir trough layers37. Similar observations have made with particles with order increasing with increasing compression and persisting upon expansion and also leading to heterogeneous structures38. However the rapidity of the contractions in our experiments do not favor the nucleation and growth of ordered domains from very polydisperse molecules (whereas the experimental procedures reported in 28, 29may have taken considerable time, which would allow such domains to grow). Furthermore such a transition cannot extend beyond maximum packing39. While dynamic ordering could perhaps explain the observed hysteresis during cycling, its contribution to the smoothness of the transition is likely to be small. Another possibility could be desorption of excess molecules. We have little quantitative information about the dynamics of desorption beyond the glass transition but the observation of wrinkles which exist at least for a few minutes suggests that the time-scale of desorption is much larger than the timescale of the contraction experiments (which are typically a few seconds). Furthermore as the cycling hysteresis loop is almost closed, desorption during contraction beyond jamming is could be balanced by readsorption during expansion. Experimental adsorption rates from3 indicate that re-adsorption during expansion could only account for 1/10 of the observed hysteresis. All this indicates that while not to be completely excluded, the contribution from desorption to the smoothness of glass transition is probably limited. Another hypothesis is that asphaltenes molecules have an additional degree of freedom compared to theoretical hard disks. A way to mediate the rise of lateral interactions would be for asphaltenes to progressively tilt their polyaromatic cores at the surface of water. Such behavior has been predicted by molecular dynamic simulations of small PAH molecules at the water/air interface or ice/air interface. PAH molecules mainly lay parallel to the water or ice surface but retain a certain degree of angular freedom4042. As coverage and lateral interactions increase, the prevalence of tilted configurations increases. Some reflection spectroscopy measurements on asphaltenes monolayers at air/water interface also provide indirect evidences of such behavior43. When the spreading solution concentration was low, the reflected intensity remained largely constant during compression (for mass coverage values of a few mg/nm2). This can be interpreted as a constant relative coverage polyaromatic core). Such scenarios can be tested by the data presented in Figure 1. Up to the divergence from the EoS (which occurs at ~80% coverage) asphaltenes may be assumed to lie with their cores parallel to the water surface. Beyond that point, asphaltenes, in line with the discussion above, may be considered to adopt a tilted
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configuration in order to keep relative coverage constant. This implies a surface excess coverage proportional to absolute coverage (Γ/Γ∞~0.8). In turn surface pressure would increase linearly with surface coverage: Π = &kT
#
0.2
ln(1 & 0.8)
(6)
Even if the slight increase in error in the Laplacian fit probably introduces some error in the determination of surface pressure in this range, Figure 1(c) shows that this simple model reasonably accounts for the evolution of surface pressure up to wrinkling (~30mN/m). The surface excess coverage value at the onset of wrinkling is 4.7 molecules/nm2, which corresponds to an average of 45 degrees tilt angle. The progressive tilting of asphaltenes could act similarly to droplet or bubble deformability in emulsions or foams. Such comparisons might be helpful in explaining rheological features previously ascribed to a critical gel (see equation 1).The behavior of dense emulsions and foams could perhaps be described by a soft glass rheology model proposed by Sollich and coworkers4, 44, 45. This model was based upon the description of structural disorder and metastable configurations in glassy systems from Bouchaud 46. To elaborate, in a finite disordered system, the conformational energy landscape is rough with many local minima surrounded by high energy barriers. These minima can be considered as metastable configurations (traps) from which the system hops out by thermal activation (at a rate exp(-E/x) with E the depth of the trap and x an effective noise temperature that can be seen as the out of equilibrium temperature due to slow equilibration dynamics). Based upon statistical considerations and definition of glass transition (with no more hopping at the glass transition temperature), Bouchaud concluded that the density distribution of traps ρ(E) has an exponential tail end. Sollich extended this model by assuming that plasticity or flow can be described by local hopping events activated by stress and temperature. To derive a constitutive rheological model, it is however necessary to assume that hopping involves mesoscopic domains (i.e. small groups of individual elements for which the configurations are correlated) large enough for their deformation to be described by an elastic strain variable l (measured from the initial trap position) and small enough for the macroscopic behavior to be an average of mesoscopic domains contributions. Under shear, starting from an equilibrium situation (with all mesoscopic domains located in traps), l first increases proportionally with the macroscopic shear rateand some elastic energy ½kl2(with k being the elastic modulus of the mesoscopic domain) is stored. Above a certain local yield strain ly, neighboring mesoscopic domains pass each other and hop to a new trap where elastic energy is zero again. Elastic energy at yield strain ½kly2 then defines the height of the initial trap. For an imposed local strain l below yield ly, the energy barrier to be overcome by thermal activation is then ½k(ly2-l2).
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By assuming a Bouchaud type distribution of trap heights and averaging individual domains contributions, Sollich and coworkers then derived a set of equations relating shear rate, time and shear stress, i.e. a constitutive rheological model. The model response depends upon the value of noise temperature x relative to the glass transition. Let us focus now on the model response above the glass transition (at glass transition mobility is zero, i.e. glassy state can be approached but not reached during static adsorption or is not a stable state during compression-recovery experiments). For x above 1 and below 2 the linear oscillation regime is described by: G ∝ ω8*9 , G ∝ ω8*9 = :;
