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GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L07304, doi:10.1029/2007GL032860, 2008
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Role of Korteweg stresses in geodynamics
Gabriele Morra1,2 and David A. Yuen3
Received 21 December 2007; revised 26 February 2008; accepted 29 February 2008; published 3 April 2008.
[1] It has come to our attention that the constitutive
relationship used in the modeling of geodynamical flow
problems with strongly variable physical properties,
should have additional terms in the stress tensor, known
in the literature as Korteweg stresses (K-stresses). These
stresses arising at diffuse interfaces, which can best be
explained in terms of density gradients, have already been
mentioned in the literature for more than one hundred
years, but have not received attention recently until a
combination of experimental and numerical evidence have
confirmed their existence. We will discuss the important
potential role these new terms have for geophysics. It has
many ramifications in geodynamics, ranging from mantle
convection to earthquakes and magma fragmentation.
Citation: Morra, G., and D. A. Yuen (2008), Role of
Korteweg stresses in geodynamics, Geophys. Res. Lett., 35,
L07304, doi:10.1029/2007GL032860.
1. Introduction
[2] Stresses generated in geodynamics are strongly influenced by density gradients along interfaces, such as phase
transitions and compositional gradients. The nature of the
interface between two different miscible fluids has been the
topic of intense study for more than 150 years in the fields of
physics and chemistry (e.g. Gibbs, 1876). Most geoscientists
have not scrutinized in detail the precise nature of the
interface and the stresses produced there by the density
gradients. Korteweg [1901] has proposed a rheological
relationship for a capillary type of stresses, based on the
density and its spatial gradients. They have to be known as
Korteweg stresses (sK), or K-stresses, and are illustrated in
Figure 1a. In brief, they represent additional terms in the
constitutive relationship, which may be important in geodynamics. Joseph [1990, 1996], Mungall [1994], Petitjeans
and Maxworthy [1996], Chen and Meiburg [1996, 2002],
Anderson et al. [1998], Brenner [2005], Pojman et al.
[2006], Chen et al. [2006], and others have called attention
to the role played by K-stresses but, up to now, its significance has not been properly appreciated among earth scientists. In this letter we will draw attention to this kind of
capillary stresses associated with diffuse interfaces and
evaluate its potentially important role for geophysics, since
it has great impact on many areas.
1
Department of Geological Sciences, University Roma Tre, Rome, Italy.
Also at Geophysics Department, ETH Zurich, Zurich, Switzerland.
3
Department of Geology and Geophysics and Minnesota Supercomputing
Institute, University of Minnesota, Minneapolis, Minnesota, USA.
2
Copyright 2008 by the American Geophysical Union.
0094-8276/08/2007GL032860$05.00
2. Experimental Indications
[3] Two seminal experiments for understanding the dynamics of light and low viscous diapirs were carried out
more than twenty years ago by Olson and Singer [1985]
(Figure 1b) and Griffiths [1986], who found similar results
for rising cavity plumes. A particular detail, well documented in both works, is the unexplained departure from the
Stokes terminal velocity of the plumes. Olson and Singer
[1985] found systematically a lower velocity of a cavity
plume for all parameter changes tested the role of the wall
effects but they found it to be negligible and quantified a
scaling difference from an expected vexp/t2/3, where v is the
ascent rate, to and this was observed to be vobs/t2/5.
Griffiths [1986] also observed the same lower rising velocity and assessed this to be 22%. He sought an explanation
from the influences of the boundary conditions, although he
also stated somewhat paradoxically that the rising velocity
of the plume remains constant for a distance longer than two
diameters of the plume head (r) and a large lateral size of the
container (l) with r/l = 0.07.
[4] Joseph [1990] proposed a new interpretation of
the experimental results by putting forward the role of
K-stresses at the miscible interface zone between plume and
surrounding fluid. Although he could not quantify the force
involved, he displayed a long list of laboratory experiments,
carried out between 1870 and 1930, in which the ‘‘membrane’’ character of the miscible interface was investigated
and demonstrated experimentally. Most strikingly, he recovered an empirical expression for the stresses that one would
expect to be generated at a smooth infinitesimally thin
interface, originally proposed by Korteweg [1901], which
depends on density and density gradient terms, formally
analogous to an elastic membrane. However, although he
showed beyond any doubt the similarity between miscible
interface and interface membrane, he could not definitively
demonstrate the precise role of K-stresses in the above
experiments, because quantification of the K-stresses required the physical determination of unknown constants in
front of the expression to be evaluated.
[5] Since that time several works have attempted to find
evidence of the role of this force in analogue laboratory
experiments, analyzed in concert with numerical models
[e.g., Petitjeanes and Maxworthy, 1996; Chen and Meiburg,
1996]. These works confirmed unequivocally a departure
from the usual Stokes prediction of a diapir tip velocity for
capillary systems, where the surface tension is expected to
have a major role. More recently [Chen and Meiburg, 2002]
developed a numerical code that solved both the Stokes
equation and the K-stresses and explicitly showed how the
Korteweg terms in a capillary geometry would exert a
significant effect on the speed of the tip of a rising diapir
(Figure 1c). Finally, Pojman et al. [2006] found the first
direct experimental evidence and explicit quantification of
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an effective interfacial tension (EIT) between two miscible
fluids.
[6] Independently, a set of experimental works with
mixing fluids close to critical point in a Hele-Shaw domain
[Vailati and Giglio, 1997; Cicuta et al., 2000, 2001]
intentionally aimed at understanding the arising of giant
fluctuations in the long term during the mixing of two
miscible fluids, have shown how the capillary stresses are
expected to grow in the beginning of a free diffusion phase
and disappear when the coherence of the interface is
destroyed by nonequilibrium fluctuations. Here the Kstresses were measured indirectly through a relationship
between the scaling length of the interface and the associated stress [Cicuta et al., 2000; Vailati and Giglio, 1997].
Following this approach, we estimated bounds on the Kstresses in the Earth’s mantle (section 4).
[7] These many experiments strongly encourage further
investigations into the role played by the K-stresses on the
evolution of a miscible interface. Nevertheless we want to
point out that it is still very difficult to directly measure them
in the laboratory experiments above mentioned, therefore
other unknown causes associated to K-stresses might lead to
the observed existent discrepancies [Ribe et al., 2007].
3. Mathematical Formulation of the Korteweg
Stresses
[8] When inertial forces can be neglected as in the mantle
dynamics and magma chambers, the fluid dynamics is
described by the equilibrium between external forces (buoyancy Drg in geodynamics) and the internal response of the
system described in general by the divergence of a tensor of
the deviatoric stresses (rt). Most stress tensor models
depend from velocity gradient (strain rate) and/or displacement gradient (strain). The traditional Newtonian rheology,
where a linear viscosity term m expresses the ratio between
shear deformation rate and stresses:
sSij ¼ m @i uj þ @j mi 2=3ldij @i ui :
ð1Þ
While much effort has been expended in proposing a nonlinear viscosity based on strain-rate dependent formulations
of the stress tensor, no attention at all has been paid toward
formulating a theory that involves density and its gradients
terms. Such a formulation is not new and has been already
put forth by Korteweg [1901] and echoed by Joseph [1990]
and Brenner [2005] in a mathematical form analogous to the
elastic stresses at the boundary between two immiscible
fluids:
sKij ¼ d ij ar2 r þ brrrr þ dð@i rÞ @j r þ g@i @j r
ð2Þ
where the constants a, b, d, and g need to be determined
from theory or observation. The term between parentheses
on the left is the bulk expression of the K-stresses, while the
terms on the right express their shear component. Therefore
the Stokes equation can be recast in a new format where
now the stress term depends separately on strain rates and
density gradients:
0 ¼ rgi þ @j sSij þ sKij
ð3Þ
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Although their nature is fundamentally different, K-stresses
can be represented analogously to the surface tension that
appears at the boundary between two immiscible fluids.
This analogy has been exploited by Pojman et al. [2006]
and Zoltowski et al. [2007] for extracting K-stresses from
laboratory experiments and more recently by Chen et al.
[2006], who numerically explore the entire range of
viscosity and density parameters by means of a nondimensional formulation, showing the scale-free nature of
Korteweg stresses.
4. Dynamical Consequences From Forces
due to Density Gradients
[9] The diagram displayed in Figure 2 illustrates an
overall view of the scales in which K-stresses are involved.
Global geodynamics triggers K-stresses producing effects at
all length scales, depending on the physics dictating each
scale. We consider three major lengths (micro, meso and
macro scales) in an attempt to offer a first estimate of their
contribution.
[10] In the lower mantle compositional fluctuations are
broader and less steep, strain-rates are low except for
ascending plumes and descending slabs. At the macro-scale,
the downwelling of cold slabs and the upwelling of hot
plumes in the mantle environment naturally generate steep
temperature and compositional gradients. Also phase
changes, being possibly very sharp (O(10 km)), produce
estimated steep density gradient due to density steps of 1 –
10%. Thus stresses associated with small density gradients
may exert significant effects only over larger distances.
Therefore, we believe that an effort should be engendered
from the mineral physics community to assess the probable
values of K-stresses under deep Earth conditions, especially
in view of the post-perovskite phase transition [Murakami et
al., 2004] and the high-spin to low-spin transition of Fe in
the mid-mantle [Crowhurst et al., 2008].
[11] At an intermediate scale between grain size up to
km, deep mantle rocks are expected to display layering,
mixing, eventually melting. K-stresses appear at each density transition; therefore layering will be also characterized
by an oscillatory presence of Korteweg stresses. The shear
K-stresses are proportional to the curvature of the boundary,
therefore they will influence the size of fingering and grain
growth dynamics, as shown numerically and experimentally
by Chen et al. [2005] and Pojman et al. [2006]. Nondimensional quantities have been put forward and can be
re-employed also for estimating the role of K-stresses in
geodynamics (Bond number Bo = rgl2/t and Capillary
number Ca = mu/t, where t are K-stresses, instead of
surface tension).
[12] At the large end of the spectrum shown in Figure 2,
the effects of K-stresses are expected to behave similarly to
the experiments of Olson and Singer [1985] and Griffiths
[1986]. In this case a Rayleigh-Taylor instability trigger a
rising mantle anomaly delimited by a diffuse interface that
has an asymptotic thickness, which depends on the diffusion
coefficient and the velocity of the process.
[13] We try here to assess the magnitude of the K-stresses
in geodynamics, following the analytical work of Vailati
and Giglio [1997] and the experimental results of Cicuta et
al. [2000, 2001]. As noted above, there is a physical
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Figure 1. (a) When two miscible fluids are mixed dynamically, for example, through a rising plume (bottom right) a
diffused boundary across the shell with a density (r) gradient forms, analogous to a membrane structure. Korteweg stresses
(K-stresses) appear at the central part of the ‘‘membrane’’, where the gradient reaches its peak. The bulk component is a
function of density, density gradient and the Laplacian of the density, while the shear contribution depends on the
quadractic density gradient. (b) Snapshot of the rising of a miscible cavity diapir performed in a laboratory with two
miscible fluids [from Olson and Singer, 1985]. The velocity of the diapir in function of a dimensionless time for three sets
of experiments with three diapirs with different sets of discharge and three buoyancies, respectively (1.8 102, 3.8
104) mL/s and (291, 196, 80) cm/s2. Dashed and continuous lines represent the expected Stokes rising
103, 3.3
velocity without and with wall correction, respectively. All points show a systematic consistent negative departure from the
expected trends, following a different power law (t5/3 instead of t7/5). (c) Numerical results (pattern and streamlines) of the
growth of a miscible diapir in a capillary tube [from Chen and Meiburg, 2002], performed solving Stokes law (examples a
and b) and adding the K-stresses (examples c and d) with d = 104. The last column (example e) shows the difference
between the two models: the main difference is at the tip of the plume. The plot on the right shows that the sudden decrease
of the peak velocity appears for values of delta close to d = 104.
relationship between non-equilibrium density fluctuations
and the K-stresses: when the thickness of the layer grows,
interface excitations also grow, increasing the role of
capillary forces, and it is only when the non-equilibrium
fluctuations enter into play that K-stresses decay. This is not
only clearly illustrated from a theoretical point of view but
has been also demonstrated in laboratory experiments, as
shown by Cicuta et al. [2000] and its implications in global
geodynamics should be further studied.
[14] First, we calculate major length scales associated to
the mantle density. When two fluids characterized by
different densities are allowed to mix together, density
anomalies go through a time interval in which non-equilibrium fluctuations appear and survive up to a length-scale
Figure 2. When large-scale geodynamics induces density gradients in some regions of the mantle, they provoke
K-stresses that will modify the momentum equation at all scales. At each scale a different physical consequences will
produce local density fluctuations. The associated K-stresses will contribute to the momentum equation and potentially
trigger a positive feedback mechanism.
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called the roll-off that can be determined from the wave
vector qRO [Vailati and Giglio, 1997]:
qRO ¼
g@z r 1=4
mD
ð4Þ
where the associated wave-length is then lRO = 2p/qRO. For
lower mantle conditions for example, taking (g = 10 m/s2,
D = 106 m2/s, m = 1022 Pas), lRO oscillates between
200km for a sharp plume head boundary (Dr = 100 Kg/m3
in Dz = 10 km) and 2000 km for a very mild density
fluctuation (Dr = 1 Kg/m3 in Dx = 1000 km).
[15] A quantity analogous to the roll-off is available for
capillary stresses [Cicuta et al., 2000] that links the density
variation, gravity and surface tension with the capillary
wave vector:
qcap ¼
gDr 1=2
t
ð5Þ
where t is the integral of the longitudinal stress along the
membrane. Although we don’t have a handle on the surface
stress expected for Earth conditions, we can invert (5) in
order to calculate upper bounds, following the approach of
Cicuta et al. [2000]. The diffusion between two miscible
fluids is defined by a transition from an initial phase in
which capillary stresses are dominating to a second stage in
which the non-equilibrium fluctuations overwhelm the
capillary effects. The transition from the first to the second
stage occurs after a cross-over time [Cicuta et al., 2001]
tco ¼
1
Dq2
1þ
1
qRO
q
4
ð6Þ
which is extremely large under mantle conditions O(1015s)
until q > qRO, i.e. l < lRO. Beyond this threshold tco decays
extremely fast (fourth power of q). We can therefore use the
relationships (4), (5) and (6) for estimating a lower bound
for qcap, being qmincap = qRO = 2p/(200 km). An upper
bound for qcap can be set more simply considering the
location of the steepest macroscopic gradient at the geodynamic scale, that we assume to be the boundary over the
head of a plume Dz = 10 km, therefore qmaxcap = 2p/(10 km).
[16] Employing our estimate for qcap, we can invert
(5) obtain an estimate for the lower and upper bound of
the forces at the boundary (t = gDr/q2cap). Assuming Dr =
2109 N/m and t max
100 Kg/m3, one finds t min
11
810 N/m. For a the peak case of the surface over the
head of a plume with compositional thickness of 10 km,
and assuming a uniform shear stress through its all
200 KPa and speakmax
thickness, we get speakmin
80 MPa, while for the very smooth background compositional gradient related to the 200km large fluctuations, one
obtains sbgmin 10 KPa and sbgmax 4 MPa. The order
of magnitude represents an important contribution that the
K-stresses are expected to apply in the mantle.
5. Discussions and Ramifications
[17] In this work we have shown from several lines of
experimental evidence demonstrating the inherent inconsis-
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tencies between Stokes law predictions and laboratory
observations. These would argue for the neglect of a
fundamental type of force in geodynamics, called Korteweg
or capillary, which may be important. Although it is today
difficult to evaluate the exact magnitude of K-stresses under
lower mantle conditions, we put forward a very first
estimate of the stresses created by density gradients. For a
more precise assessment, the constants associated with
K-stresses may be evaluated from first-principles calculations, taking into account effects of grain-boundary which
will allow for cross-scaling from micro to mesocales.
[18] Although they are formally similar, Korteweg and
elastic stresses are physically distinct, therefore they contribute an additional stress to the Earth, which might play a
role in creating the conditions for earthquakes release for
example, inside a subducting plate, where temperature
gradient are maximum, and at phase transitions, where they
might represent an important contribution to the local stress,
for example at the 660 km transition [Morris, 1992].
Because K-stresses will be proportional to the local curvature of the phase transition, their estimates in mantle
convection models require accurate assessment of the phase
transition near a mantle phase boundary [Richter, 1973].
Although the high-low spin transition is estimated to be
very smooth and through several hundred kilometers, its
global contribution integrated through the Earth sphericity
should be assessed in order to rule out its contribution. All
fluctuations of chemical, phase and thermal layering also
induce alternating density gradients producing fluctuations
in the stresses. Besides the giant fluctuations phenomena,
the equivalent of a spinodal decomposition immiscible
fluid, observed in the laboratory by Vailati and Giglio
[1997], can make it possible for background density anomalies with a length-scale O (100 to 500 km) to exist in the
mantle. They must be sought out by accurate analysis of
seismic data or by seismic imaging.
[19] It is almost impossible to assess the influence of
K-stresses without quantifying the smallest scale effects.
Evaluation of the consequences of one scale to another will
require accurate numerical modelling, using different techniques. They might also act and influence phenomena
beyond geodynamics, such as core dynamics, where there
is evidence for density stratification at the base of the outer
core [Souriau and Poupinet, 1991].
[20] It has also been experimentally observed the presence
of K-stresses in silicates melts [Mungall, 1994]. K-stresses
might help to explain magma fracturing [Papale, 1999]. A
recent review [Zhang et al., 2007] has shown that the most
important parameter for triggering magma fragmentation
seems to be critical differential pressure between the bubbles
immersed in the magma and the ambient pressure, which
controls the dynamics of the breakup of bubbles with dire
environmental consequences. Experimental data [Spieler et
al., 2004] can also fit with the interpretation that the critical
tensile stress at the outer wall of the melt shell, combined to
vesicularity, would explain magma fragmentation. We want
to stress here that, because local fluctuations of density will
naturally create local K-stresses changes, combined with the
critical stresses in proximity to the bubbles, this might
enhance conditions for dramatic plastic behavior [Alidibirov
and Dingwell, 1996].
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[21] Acknowledgments. We thank ‘‘Disco’’ Dan Joseph for invaluable discussions in 1990 and 2007. We are grateful for continuous
encouragement from Klaus Regenauer-Lieb. We are grateful to A. Vailati
and M. Giglio for sharing their knowledge, to Artem Oganov for interesting
discussions, to Hans Jurgen Hermann for his insights on relating surface
tension to a free energy formulation, and to Melanie Ross for technical
assistance. This work was also supported by the ITR program of National
Science Foundation. As part of the Eurohorcs/ESF - European Young
Investigators Awards Scheme, it was also supported by funds from the
National Research Council of Italy and other National Funding Agencies
participating in the 3rd Memorandum of Understanding, as well as from the
EC Sixth Framework Programme.
References
Alidibirov, M., and D. B. Dingwell (1996), Magma fragmentation by rapid
decompression, Nature, 380, 146 – 149.
Anderson, D. M., G. B. McFadden, and A. A. Wheeler (1998), Diffuseinterface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30,
139 – 165.
Brenner, H. (2005), Navier-Stokes revisited, Physica A, 349, 60 – 132.
Chen, C. Y., and E. Meiburg (1996), Miscible displacements in a capillary
tube. Part 2. Numerical simulations, J. Fluid Mech., 326, 57 – 90.
Chen, C. Y., and E. Meiburg (2002), Miscible displacements in capillary
tubes: Influence of Korteweg stresses and divergence effects, Phys.
Fluids, 14, 2052 – 2058.
Chen, C. Y., C. H. Chen, and J. A. Miranda (2005), Numerical study of
miscible fingering in a time-dependent gap Hele-Shaw cell, Phys. Rev. E,
71, 056304, doi:10.1103/PhysRevE.71.056304.
Chen, C. Y., C. H. Chen, and J. A. Miranda (2006), Numerical study of
pattern formation in miscible rotating Hele-Shaw flows, Phys. Rev. E, 73,
046306, doi: 10.1103/PhysRevE.73.046306.
Cicuta, P., A. Vailati, and M. Giglio (2000), Equilibrium and nonequilibrium fluctuations at the interface between two fluid phases, Phys. Rev. E,
62, 4920 – 4926.
Cicuta, P., A. Vailati, and M. Giglio (2001), Capillary-to-bulk crossover of
nonequilibrium fluctuations in the free diffusion of a near-critical binary
liquid mixture, Appl. Opt., 40, 4140 – 4145.
Crowhurst, J. C., J. M. Brown, A. F. Goncharov, and S. D. Jacobsen (2008),
Elasticity of (Mg, Fe) O through the spin transition of iron in the lower
mantle, Science, 319, 451 – 453.
Griffiths, R. W. (1986), Thermals in extremely viscous fluids, including
the effects of temperature-dependent viscosity, J. Fluid Mech., 166,
115 – 138.
Joseph, D. D. (1990), Fluid dynamics of two miscible liquids with slow
diffusion and Korteweg stresses, Eur. J. Mech. B, 9, 565 – 596.
Joseph, D. D. (1996), Non-solenoidal velocity effects and Korteweg stresses
in simple mixtures of incompressible liquids, Physica D, 97, 104 – 125.
L07304
Korteweg, D. J. (1901), Sur la forme que prennent les equations du mouvements des fluides si l’on tient compte des forces capillaires causees par
des variations de densite considerables mais continues et sur la theorie de
la capillarite dans l’hypothese d’une variation continue de la densite,
Arch. Neerl. Sci. Exactes Nat., Ser. II, 6, 1 – 24.
Morris, S. (1992), Stress relief during solid-state transformations in minerals, Proc. R. Soc. London, Ser. A, 436, 203 – 216.
Mungall, J. E. (1994), Interfacial tension in miscible two-fluid systems with
linear viscoelastic rheology, Phys. Rev. Lett., 73, 288 – 291.
Murakami, M., K. Hirose, K. Kawamura, N. Sata, and Y. Ohishi (2004),
Post-perovskite phase transition in MgSiO3, Science, 304, 855 – 858,
doi:10.1126/science.1095932.
Olson, P., and H. Singer (1985), Creeping plumes, J. Fluid Mech., 158,
511 – 531.
Papale, P. (1999), Strain-induced magma fragmentation in explosive eruptions, Nature, 397, 425 – 428.
Petitjeans, P., and T. Maxworthy (1996), Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech., 326, 37 – 56.
Pojman, J. A., C. Whitmore, M. L. TurcoLiveri, R. Lombardo, J. Marszalek,
R. Parker, and B. Zoltowski (2006), Evidence for the existence of an
effective interfacial tension between miscible fluids: Isobutyric acidwater and 1 – butanol-water in a spinning-drop tensiometer, Langmuir
2006, 22, 2569 – 2577.
Ribe, N., A. Davaille, U. Christensen, (2007), Fluid dynamics of mantle
plumes, in Mantle Plumes: A Multidisciplinary Approach, chap. 1,
pp. 1 – 48, Springer, New York.
Richter, F. M. (1973), Finite amplitude convection through a phase boundary, Geophys. J. R. Astron. Soc., 35, 265 – 276.
Souriau, A., and G. Poupinet (1991), A study of the outermost liquid core
using differential travel times of the SKS, SKKS and S3KS phases, Phys.
Earth Planet. Inter., 68, 183 – 199.
Spieler, O., B. Kennedy, U. Kueppers, D. B. Dingwell, B. Scheu, and
J. Taddeucci (2004), The fragmentation threshold of pyroclastic rocks,
Earth Planet. Sci. Lett., 226, 139 – 148.
Vailati, A., and M. Giglio (1997), Giant fluctuations in a free diffusion
process, Nature, 390, 262 – 265.
Zhang, Y., Z. Xu, M. Zhu, and H. Wang (2007), Silicate melt properties
and volcanic eruptions, Rev. Geophys., 45, RG4004, doi:10.1029/
2006RG000216.
Zoltowski, B., Y. Chekanov, J. Masere, J. A. Pojman, and V. Volpert (2007),
Evidence for the existence of an effective interfacial tension between
miscible fluids. 2. Dodecyl acrylate-poly (dodecyl acrylate) in a spinning
drop tensiometer, Langmuir 2007, 23, 5522 – 5531.
G. Morra, Department of Geological Sciences, University Roma Tre,
I-00146 Roma, Italy. (gabriele.morra@erdw.ethz.ch)
D. A. Yuen, Department of Geology and Geophysics, University of
Minnesota, Minneapolis, MN 55455 – 0219, USA.
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