Thermodynamic and Mechanical Focusing: Non-Ideality and Non-Linearity at

Thermodynamic and Mechanical Focusing: Non-Ideality and Non-Linearity at

Thermodynamic and Mechanical Focusing: Non-Ideality and Non-Linearity at Interfaces
Luka Pocivavsek , Kathleen D. Cao , Steven Danauskas , Enrique Cerda ,
1
4
2
2
Ka Yee C. Lee , Jaroslaw Majewski , Mati Meron , Binhua Lin
1

1

1

3

James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL 60637 USA
2
James Franck Institute and CARS, University of Chicago, IL 60637, USA
3
Departamento de Fsica and CIMAT, Universidad de Santiago, Av. Ecuador 3493, Santiago, Chile
4
Manuel Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM 87545 USA
1

Introduction
Interfaces are ubiquitous in biology: from the earliest points in
development where a sphere of cells undergoes geometric
transitions to form the first germ layers to the inside of our blood
vessels, airways, and lungs [1]. One of the key components of
biological interfaces are lipid membranes [2]. Our work has
centered on elucidating the mechanical response of model cell
membranes, lipid monolayers at liquid interfaces, under
compression. We developed a generalized continuum mechanics
model that captures both the linear and non-linear response of
supported membranes from microns thick to nanometers thin [3].
When membrane thickness is reduced to a point that the bending
stiffness is O(10-100kbT), the case with lipids, coupling between
the molecules in the membrane and underlying fluid subphase
effects the mechanical response of the membrane. This coupling
can be probed by studying lipid monolayers on different binary
subphase solutions like water and glycerol mixtures. Glycerol is the
simplest poly-alcohol. The motivation for studying glycerol is that
the extracellular environment of all tissues is made up of simple
polyols like sugars or larger polymer versions like glycopolymers
[4]. It is beginning to be appreciated that the local mechanical
environment of a cell plays a key role in its biology [4]. Here we
present a detailed physical study of the interesting and complex
mechanical and thermodynamic behavior of the lipid/glycerol/
water interface using x-ray and neutron scattering.

Our Model System
A lipid monolayer composed of a 7:3 mixture of DPPC (1,2dipalmitoyl-sn-glycero-3-phosphocholine) and POPG (1-palmitoyl-2oleoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (sodium salt)) on an
aqueous subphase is used as a model for lung surfactant.
DPPC

The Wrinkle to Fold Transition

Non-ideal interfacial mixing
probed by X-ray liquid reflectivity

~ 2cmcm

A1

Specular XR is limited to probing samples that remain stable on the order
of hours. To probe the thermal stability of the lipid/glycerol/water
interface we used the Grazing Incidence X-Ray Off-Specular (GIXOS)
technique.

Ideal mixing of binary fluids like water and poly- alcohols, e.g.
glycerol, is well established in bulk solution. However this
ideal behavior breaks down in the interfacial solvation layer
next to the lipid monolayer. Using x-ray reflectivity, we probe
the compositional degree and spatial extent of non-ideal mixing
between water and glycerol next to a model lung surfactant
layer.

A0
10 m thick polyester sheet on water

The figure on the left shows the compression of a ten micron thick polyester membrane (8cm 15cm) on
water (though any Newtonian fluid, e.g. glycerol, works). At low compressions the membrane responds
through continuous sinusoidal low amplitude wrinkles. The wrinkles cause elastic stresses to be evenly
distributed throughout the membrane. Upon further compression (middle panel) the symmetry begins to be
broken and wrinkles in the middle grow larger while others decay. The final state of the system is a fold (last
panel), where all elastic stresses are focused into a very localized region of the membrane. These transitions are
geometrically similar to the folding transitions observed in lipid monolayers (figure on right).

Specular X-ray Reflectivity Spectra
ChemMatCARS, ID15, APS, ANL

Liquid Surface
Reflection Geometry - q is
wavevector transfer [6]

Scaling Relations
U = Energy
B = Bending Stiffness
K = Substrate Stiffness
L = Sheet Length
A = Amplitude
l = Arc Length of Fold
= Displacement
= Angle of the Tangent
to the Horizontal
= Wavelength

inextensibility

Wrinkles - Linear Case

Fold - Nonlinear Case

POPG

Transition Energetically Favored
dimensionless
energy

QuickTime and a
TIFF (LZW) decompressor
are needed to see this picture.

7:3 DPPC:POPG, 25C on water,
~ 71 mN/m
Scale:

Monolayers like DPPC:POPG 7:3
form folds into the subphase at large
compressions. The folding
instability is linked to the general
wrinkle-to-fold model [3].
Interestingly the amplitude of the
folds is very sensitive to subphase
composition. On water folds are
O(1-10m) and on binary mixtures
of water/glycerol an order of
magnitude larger O(100m).

wrinkle

fold

d 0.3 c /3

For model-independent fitting [5], we perform an electron density
profile (EDP) search. An initial EDP is generated based on the
average scattering length density (SLD) of the sample, and the
estimated sample thickness. The EDP is generated by selecting a
number of boxes, typically on the order of ~0.5-2 per box, a
smoothing parameter (), and a for each box (), and a for each box (D. and E.). The
contribution of each box to the EDP is then smeared by the
Gaussian error function (erf). Each point in the generated EDP is
treated as a layer, and the reflectivity is then calculated by iterating
through each of the points of the EDP by the Parratt recursion
method.

A. XR spectra for DPPC:POPG
7:3 compressed to =30mN/m
at 25oC on different subphase
solutions: H2O (blue), 20:80
H2O:glycerol (magenta), 40:60
(black), and 64:36 (red).
Reflectivity was taken in
specular condition ( = ) with
Oxford point detector. The data
is multiplied by Fresnel (qz/qc)1/4
and plotted against qz/qc, qc =
0.0218 -1 (H2O), 0.0221 -1
(20:80), 0.0227 -1 (40:60),
0.0233 -1 (64:36).
B. and C. show that the position
of the low and high q fringes
shifts to the left as glycerol is
added to the subphase. The
fringes are points of maximal
destructive interference between
reflected waves at the interfaces.
The first minimum is
proportional to the lower bound
on the total interface thickness:
lmin 2/qz. The shift to lower q
indicates that the interface on
glycerol becomes thicker.

GIXOS was performed in an off-specular geometry ( = 0.3o/5.2mrad)
using a linear detector at two angle conditions: ==0.09o/1.5mrad and =
0.09o/1.5mrad while =4o/69mrad. With these angles, the entire reflectivity
was collected in a few minutes. To compare the specular reflectivity and
GIXOS reflectivity, the diffuse components must be divided out:

capillary wave contribution dominated by

Master Formula Relating Structure Factor to EDP

StochFit EDPs

The normalized GIXOS structure factor at 25oC both on water and 64:36
H2O:glycerol subphases is in agreement with the specular data (A and B).
This confirms that the technique is sensitive to the surface structure factor.
At 37oC, GIXOS shows that the interface on 64% glycerol remains stable
with the spectrum at 37oC overlaying almost identically that at 25oC (B).
On water however the interface shows thinning and a shift of the surface
structure factor at 37oC to the left (A).

E.

dimensionless displacement (d = /)

A wrinkled surface should be stable against further compression by a third of its
initial wavelength. Beyond this, the surface geometry becomes unstable towards the
new localized folded state, folding is thermodynamically favored at high
compression.

Interfacial Glycerol Enrichment probed by Neutron Reflectivity
DPPC d64

Stochastic Model Independent XR Fitting

D.

Stability of Enriched
Interfacial Glycerol Layer

DPPC d72

D2O

20:80

64:36

(SPEAR, Lujan Center, LANSCE, Los Alamos National Lab)

Neutrons interact with nuclei unlike X-rays which
scatter from atomic electrons. One power of
neutrons is the strong dependence of scattering on
the type of isotope. In particular deuterons and
protons are radically different scatterers. This
study probed DPPC at two different deuteration
conditions DPPC d64 with tails deuterated and
DPPC d72 with the entire lipid deuterated. The
subphase was a solution of D2O (SLD = 6.3x10-6
-2) and hydrogenated glycerol (SLD = 0.6x10-6
-2). The strong contrast between glycerol and
D2O allowed us to determine the extent of
glycerol enrichement at the interface. The profiles
show Chi maps of the best fit lipid parameters
with varying amounts of glycerol enrichment in a
10 layer underneath the headgroups. The solid
black (spectrum) and red (profile) lines are the
best fits. The dotted red line in the profile is the
fit with bulk solvent in the interfacial layer.

Glycerol enriched layer
seen at interface!

Fast Surface X-ray
Diffraction
Grazing Incidence X-ray diffraction was
performed in a special pinhole geometry using
the two-dimensional Pilatus detector.
H2O
64% glycerol

qxy

qxy

The red curves are on 25oC and the blue on 37oC.
Note that in-plane crystalline structure is lost in the
case of a water subphase but remains in the case of
glycerol at the higher temperature. This is in
agreement with the GIXOS data showing thinning
of the interface on water at higher temperature.

Conclusions
- surface structural studies
using surface sensitive X-ray
and neutron techniques have
shown that though ideally
mixed in the bulk simple
fluid solutions like water and
glycerol phase separate (demix) near the lipid interface,
with an enrichment of the
glycerol.
The work has allowed us to connect our general
model for interfacial compaction with interfacial
thermodynamics. Continuum mechanics parameters
like bending stiffness for thin systems like lipid
monolayers are intricately linked to the underlying
interfacial structure which as we show includes the
solvation shell. This has deep implications about the
role of water and simply hydrogen bonding fluids on
lipid encapsulated structures like biological cells.

References
[1] Biological Physics of the Developing Embryo, G. Forgacs and S.A.
Newman, 2005, Cambridge University Press.
[2] Mechanics of the Cell, D. Boal, 2002, Cambridge University Press.
[3] L. Pocivavsek, R. Dellsy, A. Kern, S. Johnson, B. Lin, K. Y. C. Lee, E.
Cerda, Science, 320, 912 (2008).
[4] Cell Mechanics, Y.L. Wang and D.E. Discher ed., 2007, Elsevier.
[5] Danauskas et al., StochFit, Journal of Applied Crystallography, 41, 6,
(2008) in press. StochFit is available through open source at

stochfit.sourceforge.net
[6] O.G. Shpyrko, Experimental X-ray Studies of Liquid Surfaces, Ph.D. Thesis,
Harvard University, 2004.

Further Reading
W. Lu, et al., Phys. Rev. Lett. 89, 146107 (2002).
D. G. Schultz, et al., J. Phys. Chem. B 110, 24522 (2006).
K.Y.C. Lee, Annu. Rev. Phys. Chem., 59, 771 (2008).
T. A. Witten, Rev. Mod. Phys., 79, 643 (2007).
E. Cerda, L. Mahadevan, Phys. Rev. Lett.,90, 074302 (2003)
L. Bourdieu, J. Daillant, et al., Phys. Rev. Lett. 72, 1502 (1992).
A. Gopal et al., J. Phys. Chem. B 110, 10220 (2006).

Funding
The NSF Inter-American Materials Collaboration: Chicago-Chile,
University of Chicago Materials Research Science and Engineering Center
program of the NSF, US-Israel Binational Foundation, University of
Chicago MSTP, March of Dimes, NSF/DOE for ChemMatCARS.

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