Thermodynamic properties and swelling behavior of glycolipid monolayers at interfaces

Article abstract of DOI:10.1021/jp0028103

Synthetic glycolipids with lactose headgroups (N = 1, 2, and 3) were synthesized, and their thermodynamic properties were systematically studied by Langmuir isotherms using a film balance. The molar transition entropy and the molar latent heat were calculated by applying the Clausius-Clapeyron equation. It has been demonstrated that the phase behavior of the glycolipid monolayers is comparable to that of ordinary phospholipids, despite the lower degree of cooperativity between the larger headgroups. The glycolipid monolayer was transferred onto a solid surface by Langmuir-Blodgett deposition, and the swelling behavior was investigated by ellipsometry. The surface grafting density was precisely controlled, and the water disjoining pressure inside the lactose layer was quantitatively measured. The measured swelling curves were analyzed in terms of the theoretical descriptions for the grafted polymer brushes . For the lipids with lactose units of N = 2 and 3, the disjoining pressure-thickness relation could fit very well to these theoretical approaches, even though the statistical limit N a?? 1 is hardly fulfilled. The results suggest entropic effects of the headgroups on the interaction between the neighboring molecules. On the other hand, the theoretical description of the swelling behavior of the lipids with one lactose unit failed due to the rodlike structure of lactose. The unique properties of these glycolipids at interfaces, such as (i) the phase behavior comparable to that of ordinary phospholipids and (ii) the polymer-like swelling behavior, play very important roles in biological systems. Mimicking the complex interactions between oligosaccharide headgroups in the plasma membranes, the synthetic glycolipids designed in this study are quite realistic models for the glycocalix.

Full text of DOI:10.1021/jp0028103

5178
J. Phys. Chem. B 2001, 105, 5178-5185
Thermodynamic Properties and Swelling Behavior of Glycolipid Monolayers at Interfaces
Matthias F. Schneider, Gerald Mathe, and Motomu Tanaka*
Lehrstuhl fu¨r Biophysik E22, Technische UniVersita¨t Mu¨nchen, James-Franck-Strasse,
D-85748 Garching, Germany
Christian Gege and Richard R. Schmidt
Fachbereich Chemie, UniVersita¨t Konstanz, Fach M-725, D-78457 Konstanz, Germany
ReceiVed: August 2, 2000; In Final Form: January 22, 2001
Synthetic glycolipids with lactose headgroups (N ) 1, 2, and 3) were synthesized, and their thermodynamic
properties were systematically studied by Langmuir isotherms using a film balance. The molar transition
entropy and the molar latent heat were calculated by applying the Clausius-Clapeyron equation. It has been
demonstrated that the phase behavior of the glycolipid monolayers is comparable to that of ordinary
phospholipids, despite the lower degree of cooperativity between the larger headgroups. The glycolipid
monolayer was transferred onto a solid surface by Langmuir-Blodgett deposition, and the swelling behavior
was investigated by ellipsometry. The surface grafting density was precisely controlled, and the water disjoining
pressure inside the lactose layer was quantitatively measured. The measured swelling curves were analyzed
in terms of the theoretical descriptions for the grafted polymer “brushes”. For the lipids with lactose units of
N ) 2 and 3, the disjoining pressure-thickness relation could fit very well to these theoretical approaches,
even though the statistical limit N . 1 is hardly fulfilled. The results suggest entropic effects of the headgroups
on the interaction between the neighboring molecules. On the other hand, the theoretical description of the
swelling behavior of the lipids with one lactose unit failed due to the “rodlike” structure of lactose. The
unique properties of these glycolipids at interfaces, such as (i) the phase behavior comparable to that of
ordinary phospholipids and (ii) the “polymer-like” swelling behavior, play very important roles in biological
systems. Mimicking the complex interactions between oligosaccharide headgroups in the plasma membranes,
the synthetic glycolipids designed in this study are quite realistic models for the glycocalix.
1. Introduction
the steric repulsion by the PEG headgroups.17-20 A similar
approach has been chosen to stabilize the colloidal dispersions
using the soft, hydrated chains.21 Moreover, it has been reported
that the PEG brushes grafted onto solid supports could passivate
the surface against the nonspecific adsorption of the proteins.22-25
There have been many reports on the effects of the ethylene
glycol chain on their morphologies and interfacial properties;26-33
however, quantitative studies of the thermodynamic and elastic
properties of glycolipids themselves in the well-defined artificial
model systems are still missing.
In the present study, lipids covalently attached to lactose
oligomers (the number of lactose units, N ) 1, 2, and 3) were
synthesized, and their thermodynamic properties and swelling
behavior were studied. Since each lactose unit takes a linear
conformation, they are expected to be rather simple glycocalix
models. The design of such well-defined biocompatible inter-
faces includes a variety of scientific applications toward the
generation of cell and tissue surface models (e.g., basal laminas
of blood vessels) to investigate the basic principles of cell
adhesion and cell growth control.34 Indeed, the glycolipid with
a lactose spacer between sialyl-Lewis-X headgroup and a
ceramide moiety exhibited similar efficiency in cell-rolling
experiments.13,35 In the first part of this study, Langmuir
pressure-area isotherms were measured at several temperatures
in order to estimate thermodynamic and structural parameters
of the monolayers at the air/water interface. In the second part,
glycolipid monolayers were transferred onto the hydrophilic
Glycolipids, whose polar headgroups are attached to oligo-
saccharides, play fundamental and essential roles in cell-cell
and cell-matrix interaction by (i) specific recognition by
counterpart lectins, (ii) modulation of adhesion receptors, and
(iii) glycolipid-glycolipid interaction.1-7 For example, blood
group- and tumor-associated antigens such as sialyl-Lewis-X
and sialyl-Lewis-A interact specifically with selectins.8,9 Al-
though such phenomena have been widely studied, little is
known regarding the interaction mechanism on a molecular
level. Indeed, most of the experiments were concerned with the
measurement of rather macroscopic quantities such as cell
rolling assays to determine the rolling velocity and shear force
resistance.10-13
It should be noted that such oligosaccharides only not serve
as specific recognition sites but also provide a soft “cushion”
between tissue-forming cells because of their unique swelling
behavior.14,15 To study the complex interplay of weak generic
forces and strong specific forces, several artificial models have
been proposed. Among various glycocalix models, phospholipids
with poly(ethylene glycol) chains (PEG lipids) have been widely
applied in numerous fields.16 For example, vesicles doped with
PEG lipids showed a remarkable increase in the blood circula-
tion times compared with ordinary vesicles, corresponding to
* Corresponding author. Tel: ++49-89-28912539. Fax: ++49-89-
28912469. E-mail: mtanaka@ph.tum.de.
10.1021/jp0028103 CCC: $20.00 © 2001 American Chemical Society
Published on Web 05/10/2001

Glycolipid Monolayers at Interfaces
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5179
Figure 1. Synthesis of the investigated derivatives Lac N (N ) 1, 2, and 3) via repetitive glycosylation.
silicon wafers. Equilibrium thickness of the saccharide layer
was measured as a function of disjoining pressure by using
ellipsometry. Relative humidity (i.e., partial water vapor pres-
sure) in the measuring chamber was precisely controlled between
20% and 98%. The obtained results were discussed in terms of
the scaling theory of de Gennes and Alexander36-39 and a self-
consistent field (SCF) approach of Milner.40,41
connected di-lactose intermediate 4 (J1,2 ) 7.9 Hz for 1c-H;
shift of the 4b-H signal from 3.85 to 5.29 ppm after acetyla-
tion). Again, de-O-isopropylidenation furnished the following
acceptor 5, and de-O-benzoylation furnished the desired Lac 2
derivative under the conditions described above. Repetitive
coupling of 5 with donor 1 yielded the elongated protected
glycolipid 6, which was de-O-isopropylidenated in the same
manner. Because of the low solubility of Lac 3, the de-O-
benzoylation and purification were performed in dimethyl
sulfoxide (DMSO) mixtures.
2. Materials and Methods
2.1. Synthesis of the Glycolipids. 2.1.1. General. Solvents
were purified according to the standard procedures. Flash
chromatography was performed on J. T. Baker silica gel 60 (40-
63 µm) or J. T. Baker RP-18 silica gel (40 µm) at a pressure of
approximately 0.4 bar. Thin-layer chromatography was per-
formed on Merck silica gel plastic plates 60F254 (TLC), glass
plates 60F254 (HPTLC), or glass plates RP-18 60F254S (RP-18
TLC); compounds were visualized by UV light and charring
with 15% aqueous H2SO4. Optical rotations were measured on
a Perkin-Elmer polarimeter 241 in a 1 dm cell at 293 K. NMR
measurement were recorded on a Bruker AC250 Cryospec or
Bruker DRX spectrometer (using the deuterated solvent as
internal standard, unless otherwise stated). The carbohydrate
monomers were assigned in alphabetical order, beginning from
the aglycon, based in part on carbon-proton shift-correlation
heteronuclear multiple quantum coherence (HMQC); only the
anomeric signals are shown. The water content of the glycolipids
was calculated according elemental analysis.
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-3,4-
di-O-isopropylidene-â-D-galactopyranosyl)-(1f4)-2,3,6-tri-O-
benzoyl-â-D-glucopyranoside (2). A solution of (2,6-di-O-
benzoyl-3,4-di-O-isopropylidene-â-D-galactopyranosyl)-(1 f 4)-
2,3,6-tri-O-benzoyl-â-D-glucopyranosyl trichloroacetimidate 142
(3.81 g, 3.64 mmol) and 1,2-di-O-hexadecyl-sn-glycerol35,43
(1.97 g, 3.64 mmol) in dry CH2Cl2 (25 mL) was treated at room
temperature with BF3‚Et2O (138 µL, 1.09 mmol). After stirring
for 15 min, the mixture was neutralized with NEt3 and
concentrated in vacuo. Flash chromatography on silica with a
mixture of toluene/ethyl acetate (12/1) as eluent afforded 2 (4.88
g, 94%) as a colorless solid. TLC (10/1 toluene/ethyl acetate)
Rf ) 0.39. mp 377 K. [R]D ) 28.4 (c ) 1.0 in CHCl3).1H NMR
(250 MHz, CDCl3): δ 4.72 (d, J1,2 ) 7.9 Hz, 1H; 1a-H).
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-â-D-
galactopyranosyl)-(1f4)-2,3,6-tri-O-benzoyl-â-D-glucopyrano-
side (3). Compound 2 (410 mg, 287 µmol) was dissolved in
CH2Cl2 (5 mL), and 50% aqueous CF3COOH (5 mL) was added.
After stirring vigorously for 6 h, the mixture was diluted with
CH2Cl2 (50 mL). The organic phase was washed with a saturated
aqueous NaHCO3 solution, dried (Na2SO4), and concentrated.
The residue was purified by flash chromatography (eluent: 4/1
toluene/acetone) to give 3 (392 mg, 98%) as colorless foam.
TLC (4/1 toluene/acetone) Rf ) 0.32. [R]D ) 27.2 (c ) 1.0 in
CHCl3).
2.1.2. Synthesis. In the following section, the investigated
glycolipids were named Lac N, corresponding to the number
of lactose units, N ) 1, 2, and 3. The synthetic route to these
glycolipids is shown in Figure 1: glycosidation of known lactose
trichloroacetimidate 142 with 1,2-di-O-hexadecyl-sn-glycerol35,43
as lipid anchor was performed in the presence of BF3‚Et2O as
catalyst to afford the lactose intermediate 2 in 94% yield. The
â-configuration of the newly formed glycosidic bond was
1
confirmed in the H NMR spectrum (J1,2 ) 7.9 Hz). Acid-
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(â-D-galactopyranosyl)-
(1f4-â-D-glucopyranoside (Lac 1). A solution of sodium
methoxide (1 M) in CH3OH (1.7 mL) was added to a solution
of 3 (600 mg, 433 µmol) in dry CH3OH and dry CH2Cl2 (40
mL each). After stirring overnight, the solution was neutralized
with Amberlite IR120 (H+), filtered, and absorbed on RP-18
silica. Flash chromatography on RP-18 silica with a mixture of
0/4/1 f 0/1/0 f 65/25/4 CHCl3/CH3OH/H2O as eluent afforded
catalyzed cleavage of the isopropylidene group furnished 3,
which served as an acceptor for the next glycosylation. Removal
of the O-benzoyl protective groups with sodium methoxide in
CH2Cl2/CH3OH afforded Lac 1 in almost quantitative yield after
purification on reversed phase (RP-18) silica gel.
Elongation of 3 with donor 1 in the presence of BF3‚Et2O as
catalyst at 253 K afforded exclusively the expected â(1-3)-

5180 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Schneider et al.
Lac 1 (419 mg, 97%) as a colorless powder after lyophilization
from H2O/dioxane (3/1). HPTLC (80/20/2 CHCl3/CH3OH/H2O)
Rf ) 0.58. [R]D ) - 12.3 (c ) 1.0 in 65/25/4 CHCl3/CH3OH/
H2O). 1H NMR (600 MHz, TMS, 65/25/4 CDCl3/CD3OD/
D2O): δ 4.29 (d, J1,2 ) 7.8 Hz, 1H; 1a-H), 4.33 (d, J1,2 ) 7.0
Hz, 1H; 1b-H). C47H92O13‚H2O (883.3).
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-3,4-
di-O-isopropylidene-â-D-galactopyranosyl)-(1f4)-(2,3,6-tri-O-
benzoyl-â-D-glucopyranosyl)-(1f3)-(2,6-di-O-benzoyl-â-D-
galactopyranosyl)-(1f4)-2,3,6-tri-O-benzoyl-â-D-gluco-
pyranoside (4). A solution of 1 (160 mg, 153 µmol)42 and 3
(193 mg, 139 µmol) in dry CH2Cl2 (2 mL) was treated at 253
K with BF3‚Et2O (5.3 µL, 42 µmol). After stirring for 15 min,
the mixture was neutralized with NEt3 and concentrated in
vacuo. Flash chromatography (eluent: 10/1 toluene/acetone)
afforded 4 (275 mg, 87%) as a colorless foam. TLC (9/1 toluene/
acetone) Rf ) 0.36. [R]D ) 42.1 (c ) 1.0 in CHCl3). 1H NMR
(600 MHz, CDCl3): δ 4.45 (d, J1,2 ) 7.9 Hz, 1H; 1d-H), 4.57
(d, J1,2 ) 7.6 Hz, 1H; 1b-H), 4.62 (d, J1,2 ) 7.9 Hz, 1H; 1c-
H), 4.68 (d, J1,2 ) 7.9 Hz, 1H; 1a-H).
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-â-D-
galactopyranosyl)-(1f4)-(2,3,6-tri-O-benzoyl-â-D-glucopyrano-
syl)-(1f3)-(2,6-di-O-benzoyl-â-D-galactopyranosyl)-(1f4)-
(2,3,6-tri-O-benzoyl-â-D-glucopyranosyl)-(1f3)-(2,6-di-O-
benzoyl-â-D-galactopyranosyl)-(1f4)-2,3,6-tri-O-benzoyl-â-D-
glucopyranoside (7). Compound 6 (1.11 g, 356 µmol) was
treated as described for 3 to give after flash chromatography
(eluent: 3/1 toluene/acetone) derivative 7 (958 mg, 87%) as
colorless foam. TLC (3/1 toluene/acetone) Rf ) 0.34. [R]D
48.9 (c ) 1.0 in CHCl3).
)
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(â-D-galactopyranosyl)-
(1f4)-(â-D-glucopyranosyl)-(1f3)-(â-D-galactopyranosyl)-
(1f4)-(â-D-glucopyranosyl)-(1f3)-(â-D-galactopyranosyl)-
(1f4)-â-D-glucopyranoside (Lac 3). A solution of sodium
methoxide (1 M) in CH3OH (3.3 mL) was added to a solution
of 7 (402 mg, 131 µmol) in dry CH3OH and dry DMSO (10
mL each). The solution was stirred for 3 days, diluted with some
water, and absorbed on RP-18 silica. Flash chromatography on
RP-18 silica with a mixture of 0/1 f 2/1 f 1/1 f 1/2 DMSO/
H2O as eluent afforded Lac 3 (136 mg, 67%) as a colorless
powder after lyophilization from H2O. RP-18 TLC (1/2 DMSO/
CH3OH) Rf ) 0.14.
2.2. Sample Preparation. Thermally oxidized silicon wafers
were given as a gift from Wacker Chemtronic (Burghausen,
Germany). Freshly distilled ultrapure water (Millipore, Molsheim,
France) was used as a subphase throughout this study. All other
chemicals were purchased from Fluka (Neu-Ulm, Germany) and
used without further purification. The spreading solutions were
prepared by dissolving each glycolipid into the mixture of
chloroform/methanol (70/30 by volume) at a concentration of
1 × 10-3 M.
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-â-D-
galactopyranosyl)-(1f4)-(2,3,6-tri-O-benzoy-â-D-glucopyrano-
syl)-(1f3)-(2,6-di-O-benzoyl-â-D-galactopyranosyl)-(1f4)-
2,3,6-tri-O-benzoyl-â-D-glucopyranoside (5). Compound 4 (2.85
g, 1.25 mmol) was treated as described for 3 to give after flash
chromatography (eluent: 5/1 toluene/acetone) derivative 5 (2.63
g, 94%) as colorless foam. TLC (4/1 toluene/acetone) Rf ) 0.30.
[R]D ) 45.2 (c ) 1.0 in CHCl3).
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(â-D-galactopyranosyl)-
(1f4)-(â-D-glucopyranosyl)-(1f3)-(â-D-galactopyranosyl)-
(1f4)-â-D-glucopyranoside (Lac 2). A solution of sodium
methoxide (1 M) in CH3OH (1.8 mL) was added to a solution
of 5 (199 mg, 88.5 µmol) in dry CH3OH (40 mL). After stirring
overnight, the solution (with precipitate) was evaporated and
dissolved in CHCl3/CH3OH/H2O (65/25/4). After neutralization
with Amberlite IR120 (H+), the solvent was filtered and
absorbed on RP-18 silica. Flash chromatography on RP-18 silica
with a mixture of 0/3/1 f 0/1/0 f 65/25/4 CHCl3/CH3OH/
H2O as eluent afforded Lac 2 (85.5 mg, 80%) as a colorless
powder after lyophilization from H2O/dioxane (3/1). HPTLC
(70/30/4 CHCl3/CH3OH/H2O) Rf ) 0.49. [R]D ) 0.6 (c ) 1.0
2.3. Film Balance Experiments. Langmuir pressure-area
isotherms were measured with a film balance designed in our
laboratory (trough area: 982.5 cm2). After the lipids were spread
onto the subphase, the surface pressure was monitored with a
Wilhelmy plate in the course of compression at a rate of 1.3
cm2/min. The whole trough was temperature-controlled with a
commercially available heating/cooling system. Typical errors
in temperature control were in the range of (0.5 K.
2.4. Static Swelling Measurements. Equilibrium thickness
of the glycolipid monolayer deposited onto hydrophilic Si/SiO2
substrates by the Langmuir-Blodgett method was measured as
a function of relative humidity using a conventional point
ellipsometer (Plasmos GmbH Prozesstechnik, Mu¨nchen, Ger-
many). The ellipsometer was attached to a film balance equipped
with a closed chamber, which enables precise control of the
relative humidity of the atmosphere between 4% and 98%
(Figure 2). Throughout the experiment, a constant flow (5
L/min) of the adjusted humid air was applied. During the
spreading and compression of the monolayer, a vertically
mounted Si/SiO2 substrate remained dipped in the subphase.
After the surface pressure reached to a well-defined pressure
value of p ) 25 mN/m, the substrate was pulled out from the
subphase at a rate of ∼0.1 mm/s. The total thickness of the
deposited monolayer, which consists of the lactose layer and
the layer of the alkyl chains, was measured at a constant
wavelength (λ ) 632.8 nm) and a fixed angle of incidence (70°).
Thickness of the lactose layer was calculated by assuming the
alkyl layer thickness of 1.8 nm and the refractive index of n )
1.5.31-33 By measuring the two ellipsometric angles, Ψ and ∆,
we can derive the ratio between the complex reflection coef-
ficients, Rp/Rs,
1
in 65/25/4 65/25/4CHCl3/CH3OH/H2O). H NMR (600 MHz,
TMS, 65/25/4 CDCl3/CD3OD/D2O): δ 4.30 (d, J1,2 ) 8.2 Hz,
1H; 1a-H), 4.37 (t, 1H; 1d-H), 4.44 (d, J1,2 ) 7.7 Hz, 1H;
1b-H), 4.57 (d, J1,2 ) 7.7 Hz, 1H; 1c-H). C59H112O23‚H2O
(1207.5).
(1,2-Di-O-hexadecyl-sn-3-glyceryl)-O-(2,6-di-O-benzoyl-3,4-
di-O-isopropylidene-â-D-galactopyranosyl)-(1f4)-(2,3,6-tri-O-
benzoyl-â-D-glucopyranosyl)-(1f3)-(2,6-di-O-benzoyl-â-D-ga-
lactopyranosyl)-(1f4)-(2,3,6-tri-O-benzoyl-â-D-glucopyranosyl)-
(1f3)-(2,6-di-O-benzoyl-â-D-galactopyranosyl)-(1f4)-2,3,6-tri-
O-benzoyl-â-D-glucopyranoside (6). A solution of 1 (540 mg,
513 µmol)42 and 5 (958 mg, 429 µmol) in dry CH2Cl2 (8 mL)
was treated at 253 K with BF3‚Et2O (16 µL, 129 µmol), and
then the ice bad was removed. After stirring for 30 min, the
mixture was neutralized with NEt3 and concentrated in vacuo.
Flash chromatography (eluent: petroleum ether/ethyl acetate 2/1
3
f /2) afforded 6 (1.15 g, 86%) as a colorless foam. TLC
(petroleum ether/ethyl acetate 2/1) Rf ) 0.20. [R]D ) 56.8 (c )
1.0 in CHCl3). 1H NMR (600 MHz, CDCl3): δ 4.40 (d, J1,2
)
8.0 Hz, 1H; 1b-H), 4.43 (d, J1,2 ) 8.0 Hz, 1H; 1d-H), 4.56-
4.58 [m, 2H. HMQC: 4.56 (1c-H), 4.58 (1f-H)], 4.60 (d, J1,2
) 7.9 Hz, 1H; 1a-H), 4.67 (d, J1,2 ) 7.8 Hz, 1H; 1e-H).
Rp/Rs ) tan Ψ exp(-i∆)
(1)

Glycolipid Monolayers at Interfaces
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5181
of the film, which is a function of the thickness d and the water
volume fraction within the film.51,52 The disjoining pressure can
be expressed by use of van’t Hoffs law
RT
p ) -
ln(X)
(3)
( )
Vm
where Vm is the molar volume of water, T is the temperature. R
stands for the gas constant, and X is the relative humidity inside
the chamber. To analyze the force-distance relationship within
this interlayer, we plotted the determined layer thickness as a
function of the relative humidity (i.e., disjoining pressure),
yielding the disjoining pressure-thickness curve.
3. Theoretical Concepts
The measured swelling curves were analyzed by applying
two different physical concepts: (i) the scaling theory by
Alexander and de Gennes36-39 and (ii) the mean field approach
by Milner.40,41 Although our experimental systems do not fulfill
a symmetrically planar situation with the polymers confined
between two parallel plates, all the equations discussed in the
following chapter were applicable to the experimental data by
multiplying all theoretical expressions by the order of unity.
3.1. Scaling Theory. A general expression for the free energy
F of grafted polymers on the surface is given by de Gennes,
Alexander, and Daoud36-39 by
Figure 2. Schematic drawing of the ellipsometer coupled to a covered
film balance and a humidity chamber. The top of the chamber was
sealed with an o-ring, and bottom was connected to the covered film
balance. The glycolipid monolayer could be transferred onto the
substrate by the Langmuir-Blodgett method. By mixing dry air and
water-saturated air, the relative humidity inside the chamber was
controlled between 4% and 98%. The flow rate was 5 L/min, and the
chamber volume was about 3 L.
Rp and Rs are given as the ratio between the incident and
5/3
5/4
a
D
Na
D
Na3
σD
D
evanescent electric fields; Rp ) Ep,evan/Ep,incid, and Rs ) Es,evan
/
F ≈ kTN
- δ
+ δ
+
( )
( )
(
)
E
s,incid. The initial thickness of the monolayer (i.e., thickness
2
-1/3
under low humidity conditions) was determined between 4%
and 20% of relative humidity. The thickness change in this
regime was in the range of the resolution limit of the ellipsom-
eter used (1-2 Å). Between two subsequent equilibrium
thickness measurements, the film was allowed to equilibrate
for about 10 min. Furthermore, the thickness was measured at
decreasing humidity as well as at increasing humidity to ensure
the equilibrium between the deposited monolayer and the humid
air. A refractive index of n ) 1.56 for the bulk polysaccharide
was assumed32,44-46 to evaluate the initial thickness of the lactose
layer by analyzing Rp/Rs in terms of the Fresnel equations.47
For the thickness evaluation of the swollen lactose film
containing increasing amounts of water, the refractive index was
calculated by applying the Garnet equation,48,49 which relates
the refractive index of a homogeneous solution, nF, to the
volume fraction of the solute, Φ
ê
a
kT
N
-1 + kT ln σ (4)
( ) ( )
a
N is the number of monomer segments with length a, and D is
the thickness of the polymer layer. δ represents the surface
adsorption energy per monomer in units of kT, while σ and ê
the mean area per polymer and the blob diameter, respectively.38
The first term describes the energy necessary to confine a
polymer molecule behaving as an ideal chain inside a blob. The
second term stands for the adsorption energy of a chain on the
surface, while the third term for the repulsion between over-
lapped polymer chains. The fourth term describes the so-called
“brush” regime. The last term represents the translational entropy
of the polymers, which can be neglected by assuming that the
alkyl chains are immobilized on the surface.
In case the distance between the grafting points dp is less
than the Flory radius, dp < RF, the polymer chain takes a “brush”
conformation with the blob diameter of ê. The equilibrium
3Φ
st
brush
st
2
thickness D
can be described as D
≈ aNd-p 2/3. The
nF ) nM
1 +
(2)
n02 + 2nM
(
brush
st
brush
interaction potential of the polymer V
is given by
- Φ
n02 - n2M
)
x
st
5/4
D
7/4
kT
dp3
4
5
brush
4
7
D
st
brush
Vbrush
≈
D
+
(5)
(
)
n0 and nM are the refractive indices of the pure solute and the
pure solvent, respectively. The volume fraction Φ is simply
related to the layer thickness d by Φ ) d0/d, where d0 is the
initial film thickness. The equilibrium film thickness at certain
humidity can be calculated self-consistently, starting from the
refractive index of the initial layer and successively applying
the Garnet formula.
st
[
D
]
(
)
Dbrush
yielding the resulting pressure of
st
9/4
D
3/4
kT
brush
D
st
st
brush
P
≈
-
(6)
(
)
3
[
D
]
(
)
dp
Dbrush
The dominating forces between these two interfaces separated
by a thin lactose layer can be described in the context of the
disjoining pressure model.50 This external pressure, p, can be
defined as the negative derivative of Gibbs free energy G, with
respect to the film thickness d, p ) -∂G/∂d. G is the free energy
3.2. Self-Consistent-Field Approach. The self-consistent-
field (SCF) approach of Milner et al.40,41 is based on the
terminally fixed linear chains exhibiting a high grafting density.

5182 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Schneider et al.
In contrast to the scaling approach, the quality of the solvent is
not so crucial in this treatment. When the film is compressed
scf
from the equilibrium thickness D , the interaction energy
brush
per unit area can be described as
scf
brush
π2
1
2
D
1
10
D
D
2
5
scf
brush
V
≈ -
N(σw)2/3
+
-
scf
scf
( )
12
[
2D
]
(
)
(
)
Dbrush
Dbrush
(7)
(8)
and the resulting interfacial pressure is given by
scf
brush
2
π2
1
1
2
D
scf
D
scf
brush
P
≈
N(σw)2/3
-
+
-
(
)
scf
( )
12
[
D
D
D
brush
brush
4
1
2
D
scf
]
(
)
Dbrush
σ ) d-p 2 is the grafting density, and w stands for the “excluded
volume”.
4. Results and Discussion
4.1. Langmuir Isotherms. For each glycolipid (the number
of lactose units, N ) 1, 2, 3), the pressure-area isotherms were
measured at several different temperature conditions between
283 and 308 K. To eliminate hysteresis effects, we monitored
the isotherms during the expansion as well as during the
compression.
The Langmuir isotherms of the Lac 1 lipid are shown in
Figure 3a. At T e 298 K, the isotherms exhibited no liquid
expanded phase and were dominated by the condensation of
the dihexadecyl chains from a gas phase to a liquid condensed
phase. Such behavior can be explained by the stiffness of the
short, fully hydrated, and stretched “rodlike” lactose moieties.
At T ) 303 K, an onset of a plateaulike regime was observed,
corresponding to a first-order phase transition from the liquid
expanded to the liquid condensed state. Further rise in temper-
ature leads to an increase in the transition pressure and a
decrease in the coexistence region. Such a systematic tendency
coincides with the approach to a critical point, which is well-
known from the previous studies on ordinary phospholipid
monolayers.53,54
As presented in Figure 3b,c, qualitatively similar isotherms
were observed for the monolayers of Lac 2 and Lac 3. In
accordance with the increase in the lactose units, a systematic
increase in the transition pressure, pK, at a given temperature
was observed (Figure 4a). The obtained results suggested that
the steric interactions between neighboring lipid molecules were
dominated by the strong repulsion between the headgroups.
However, the qualitative shape of the coexistence region was
still dominated by the lateral packing density of the alkyl chains
and not by the “polymer-like” effects of the headgroups. Indeed,
a similar tendency was observed in the previous study for the
monolayers of PEG lipids with shorter chains.33
It should be also noticed that the slope of the isotherms in
the coexistence region was increased with the increase in the
size of the lactose headgroups. Such a slope in the Langmuir
isotherms can be generally explained by the stabilization of
domains due to (i) small amounts of impurities (g0.2 mol
%),55,56 (ii) “intermediate” states of the alkyl chains,57 or (iii)
the strong interaction between the headgroups.58 The first two
approaches are based on nonequilibrium effects, which do not
follow the Gibbs phase rule; however, the third interpretation
explains this slope by the continuous compression of the
Figure 3. Langmuir isotherms of the monolayers of (a) Lac 1 at T )
283, 293, 298, and 303 K, (b) Lac 2 at T ) 288, 293, 298, and 303 K,
and (c) Lac 3 at T ) 288, 293, 298, and 303 K. The fluid-gas
coexistence line was fitted by polynomial of 4th order (broken line).
At high temperature and pressure conditions, the coexistence line can
be approximated as parabola, whose vertex coincides with the critical
point. The points of intersection of the linear extrapolated lines in panel
a were taken to define ALE and ALC. To estimate the deviation in A, we
took several points.
headgroups. From a thermodynamic point of view, any slope
can be described by a decrease in cooperativity of the transition,
while a classical first-order transition assumes infinite cooper-
ativity. Therefore, it is plausible that the larger headgroups result
in a less effective interaction between the alkyl chains.

Glycolipid Monolayers at Interfaces
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5183
Figure 4. (a) Transition pressure pK and (b) molar latent heat of the
phase transition q plotted as a function of temperature T for Lac 1
(open circle), Lac 2 (closed square), and Lac 3 (open square).
By applying the Clausius-Clapeyron equation
dpK
∆s
∆q
)
)
(9)
dT
(ALE - ALC) T(ALE - ALC)
we can estimate thermodynamic quantities such as the molar
latent heat, q, or the molar transition entropy, s ) q/T, from
the variation of the transition pressure with the absolute
Figure 5. Absolute disjoining pressure vs thickness of the lactose layers
for (a), Lac 1, (b) Lac 2, and (c) Lac 3. The measured values (open
circles) were compared with the theoretical predictions based on the
self-consistent-field approach (solid lines) and the scaling theory (broken
lines). The swelling behavior of Lac 2 and Lac 3 could be well
explained by both of the “brush” models.
temperature, dpK/dT. It should be noted that the values ALE
-
A
LC and ∆q can differ up to 50%, depending on how the onset
(ALE) and the end point (ALC) of the transition were defined.
As shown in Figure 3a, we defined the onset and the end point
according to the manner previously reported.59,60 The increase
in pK as well as the decrease in ALE - ALC with temperature
observed in this study had been also reported for phospholipid
monolayers, and they can be understood in terms of a two-
dimensional van der Waals gas model. The phase coexistence
region can be fitted by polynomial of 4th order (Figure 3 a-c).
At high surface pressure and high temperatures, the phase
coexistence line can be approximated well as a parabola, whose
vertex coincides with the critical point. This justifies, at least
as a rough estimation, the application of a two-dimensional van
der Waals equation in the vicinity of this critical point.
Expanding of the van der Waals equation near the critical point
derives
the linear assumption between pK and T (Figure 4a), the critical
pressure can be derived from the critical temperature, TC,
calculated according to eq 10. This leads to a critical pressure
pC ) 9-16 mN/m and a critical temperature of TC ) 313-316
K, respectively. The extracted values were in good agreement
with the critical pressure derived from the parabola fit, which
also supports the validity of our theoretical model. Moreover,
they are also comparable to those of phospholipids with
dihexadecyl chains.54 Figure 4b shows the temperature depen-
dence of the latent heat ∆q. When the lactose moiety was longer,
the corresponding latent heat was less than that of the shorter
one, which can be attributed to the lower degree of cooperativity
due to the larger headgroups.
4.2. Swelling Behavior of Glycolipid Monolayers. The
glycolipid monolayers were transferred onto the substrate at T
) 293 K and at a lateral pressure of p ) 25 mN/m. The areas
per molecule were 37 Å2 (Lac 1), 37 Å2 (Lac 2), and 40 Å2
(Lac 3), respectively. In this regime, the glycolipids take the
1/2
A - A
AC
TC - T
C ) 2
(10)
(
)
TC
where AC is the area at the maximum of the parabola.61 From

5184 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Schneider et al.
The difference between the swelling behavior of Lac 1 from
those of Lac 2 and Lac 3 might be explained by the influence
of the alkyl chains on the phase transition of the glycolipid
monolayers. Actually, the Langmuir isotherms of Lac 1 were
dominated by the condensation of the alkyl chains at T e 298
K (Figure 3a). In accordance with the increase in the lactose
units, the headgroups gained conformational entropy. The steric
interaction between the neighboring lipids is strongly influenced
by the repulsion between headgroups. Thus, the swelling curves
could be well explained by the “brush like” behavior of the
lactose groups.
5. Conclusions
Interfacial properties of the synthetic glycolipids with lactose
headgroups (Lac N; N ) 1, 2, and 3) were systematically studied
by applying two experimental techniques. Thermodynamic and
structural properties were obtained from film balance experi-
ments. The Langmuir isotherms clearly show an increase in the
transition pressure and in the slope of the transition regime,
according to the increasing number of lactose units. This reflects
the increase in the repulsion between headgroups, which
impedes upon the compression of alkyl chains as well as an
increase in the solubility, which in turn causes a decrease in
cooperativity. From the phase coexistence regions in the
isotherms, the molar transition entropy, s, and the molar latent
heat, q, were estimated by applying the Clausius-Clapeyron
equation. Furthermore, by applying a two-dimensional van der
Waals gas model, the critical temperature and the critical
pressure were estimated, TC ) 313-316 K and pC ) 9-16
mN/m, respectively. It is worth noting that the phase behavior
of the glycolipid monolayers is comparable to that of ordinary
phospholipids, despite the lower degree of cooperativity between
the larger headgroups.
Figure 6. Absolute disjoining pressure as a function of relative swelling
ratio of Lac 1, d/d0, normalized to the thickness of “dry” layer. At
high disjoining pressures from 2 × 108 to 7 × 107 Pa, a power law p
(d/d0)n was fitted to the disjoining pressure curves, yielding an
exponent of n ≈ -9.
liquid condensed phase, where the alkyl chains orient nearly
perpendicular to the surface. The relative humidity was varied
between 30% and 98%, corresponding to a change of the
disjoining pressure between 1.69 × 109 and 2.83 × 106 Pa.
In Figure 5a-c, the disjoining pressure is plotted versus the
absolute thickness of the swollen lactose layer. In each plot,
both the experimental data (open squares) as well as two
theoretical fits are presented based on the scaling approach and
the SCF model. To expose possible power law dependence
between the disjoining pressure against the thickness, we
presented all results as log-log plot. The swelling behavior of
Lac 1 could hardly be interpreted as “brushes” neither by the
scaling approach nor by the SCF theory, even though the
swelling ratio of ∼2.0 in the low-pressure regime (∼107 Pa) is
still in a plausible range from the corresponding ratios of dextran
(∼2.0) and hyaluronic acid (∼2.7).46 This observation suggests
that the very short headgroups behave like “rigid-rod” but not
like “polymer chains”, similar to what had been observed for
the PEG lipid monolayers with shorter chains.33 This is also in
a good agreement with the Langmuir isotherms of Lac 1,
showing the qualitatively similar characteristics to phospholipid
monolayers. At high disjoining pressures from 2 × 108 to 7 ×
The swelling behaviors of these lipids were investigated by
ellipsometry coupled to a film balance and a humidity chamber,
which enables the precise control of the surface grafting density
and the quantitative measurements of the water disjoining
pressure inside the lactose layer. Theoretical interpretation was
attempted by applying the scaling theory and the SCF approach
for the grafted polymer “brushes”. Both of the theoretical
treatments failed to explain the disjoining pressure-thickness
relation of the Lac 1 monolayer. This might be due to the
“rodlike” structure of lactose. For Lac 2 and Lac 3, the swelling
curves could be fitted very well by these theoretical approaches,
suggesting entropic effects of the headgroups on the interaction
between the neighboring molecules. The random conformations
of the lactose groups resulted in the “polymer-like” swelling
behavior, driven by the water uptake of the saccharide layer.
107 Pa, a power law p
(d/d0)n was fitted to the disjoining
pressure curves, yielding an exponent of n ≈ -9 (Figure 6). In
this high disjoining pressure regime, typical intermolecular
distances, r, are comparable to Bohr radius (∼0.5 Å), and the
swelling should be mainly dominated by short-range repulsive
interactions caused by the overlapping of molecular orbitals.
The hard core repulsion of the Lennard-Jones potential scales
as r-12. This exponent corresponds to a scaling law of p (d/
-9 50
0
As described above, the unique properties of these glycolipids
at the interface can be specified in terms of (i) the dominating
effects of alkyl chains on the phase behavior, which is
comparable to that of ordinary phospholipids, and (ii) the
specific swelling behavior due to the hydration, which can be
interpreted as that of “polymer brushes”. The first characteristics
is very important in biological systems to keep the cell
membrane structure stable, while the second plays a key role
to form a soft “cushion”, preventing the nonspecific adhesion
between cells and tissues. For the study of the complex interplay
of various physical forces in the cell-cell recognition processes,
the synthetic glycolipids designed here are quite unique and
suitable glycocalix models. The quantitative study of the various
generic forces at biocompatible interfaces includes a variety of
scientific applications toward the generation of models of cell
d )
that agrees with the power law obtained from our
experiments. Similar power law dependencies in the high
disjoining pressure regime could be also observed for Lac 2
and Lac 3.
In Figure 5b,c, the swelling behaviors of Lac 2 and Lac 3
were compared with theoretical predictions for “polymer
brushes” based on the scaling approach and the SCF theory.
Indeed, both of the “brush” models fit obviously better to the
measured disjoining pressure curves than the “mushroom”
models, which is quite evident from the conditions of prepara-
tions (i.e., high transfer pressure, relatively short headgroups).
Also, it should be noted that the thickness of the lactose layers
(e3 nm) is still far away from the basic statistical condition, N
. 1. These results also showed a good agreement with results
for PEG lipids with longer chains.33

Glycolipid Monolayers at Interfaces
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5185
(27) Kenworthy, A. K.; Simon, S. A.; McIntosh, T. J. Biophys. J. 1995,
68, 1903.
(28) Kenworthy, A. K.; Hristova, K.; Needham, D.; McIntosh, T. J.
Biophys. J. 1995, 68, 1921.
and tissue surfaces to investigate the basic principles of the
interactions between oligosaccharide headgroups in plasma
membranes.
(29) Kuhl, T. L.; Majewski, J.; Wong, J. Y.; Steinberg, S.; Leckband,
D. E.; Israelachvili, J. N.; Smith, G. S. Biophys. J. 1998, 75, 2352.
(30) Naumann, C. A.; Brooks, C. F.; Fuller, G. G.; Knoll, W.; Frank,
C. W. Langmuir 1999, 15, 7752.
(31) Baekmark, T. R.; Elender, G.; Lasic, D. D.; Sackmann, E. Langmuir
1995, 11, 3975.
(32) Elender, G. Ph.D. Thesis, Technische Universita¨t Mu¨nchen,
Mu¨nchen, Germany, 1996.
(33) Mathe, G.; Gege, C.; Neumaier, K. R.; Schmidt, R. R.; Sackmann,
E. Langmuir 2000, 16, 3835.
Acknowledgment. The authors express the deepest gratitude
to Prof. Erich Sackmann for his constant support and helpful
suggestions throughout this work. We are also grateful to Dr.
Armin Geyer for his help in the structural assignments by NMR
experiments. We appreciate Wacker Chemtronic (Burghausen,
Germany) for providing thermally oxidized silicon wafers as a
gift. This work was supported by the Deutsche Forschungsge-
meinschaft (DFG Sa 246/27-2, Schm 213/36-2) and the Fonds
der Chemischen Industrie. One of the authors (M. T.) is an
Alexander von Humboldt Fellow.
(34) Sackmann, E.; Tanaka, M. Trends Biotechnol. 2000, 18, 58.
(35) Gege, C.; Bendas, G.; Vogel, J.; Rothe, U.; Schmidt, R. R. Chem.
Eur. J. 2000, 6, 111.
(36) De Gennes, P. G. J. Phys. 1976, 37, 1445.
(37) Alexander, S. J. Phys. II France 1977, 38, 983-987.
(38) De Gennes, P. G. Macromolecules 1980, 13, 1069-1075.
(39) Daoud, M.; De Gennes, P. G. J. Phys. II 1977, 38, 85-93.
(40) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988,
21, 2610-2619.
(41) Milner, S. T.; Witten, T. A.; Cates, M. E. Europhys. Lett. 1988, 5,
413-418.
(42) Lay, L.; Windmu¨ller, R.; Reinhardt, S.; Schmidt, R. R. Carbohydr.
Res. 1997, 303, 39.
(43) Abdelmageed, O. H.; Duclos, R. I., Jr.; Griffin, R. G.; Siminovitch,
D. J.; Ruocco, M. J.; Makriyannis, A. Chem. Phys. Lipids 1989, 50, 163.
(44) Elender, G.; Ku¨hner, M.; Sackmann, E. Biosens. Bioelectron. 1996,
11, 565-577.
(45) In Handbook of Chemistry and Physics; Lide, D. R., Ed.; CRC
Press Inc.: New York, 1997.
(46) Mathe, G.; Albersdo¨rfer, A.; Neumaier, K. R.; Sackmann, E.
Langmuir 1999, 15, 8726.
(47) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light;
North Holland: Amsterdam, 1977.
References and Notes
(1) Curatolo, W. Biochim. Biophys. Acta 1987, 906, 137.
(2) Hakomori, S. Pure Appl. Chem. 1991, 63, 473.
(3) Hakomori, S.; Igarashi, Y. J. Biochem. 1995, 118, 1091.
(4) Springer, T. A. Annu. ReV. Physiol. 1995, 57, 827.
(5) Gabius, H. J.; Gabius, S. Glycoscience; Chapmann & Hall:
Weinheim, 1997.
(6) Geyer, A.; Gege, C.; Schmidt, R. R. Angew. Chem., Int. Ed. Engl.
1999, 38, 1466.
(7) Geyer, A.; Gege, C.; Schmidt, R. R. Angew. Chem., Int. Ed. Engl.
2000, 39, 3246.
(8) Feizi, T. Curr. Opin. Struct. Biol. 1993, 3, 701.
(9) Varki, A. Proc. Natl. Acad. Sci. U.S.A. 1994, 91, 7390.
(10) Lawrence, M. B.; Springer, T. A. Cell 1991, 65, 859.
(11) Goetz, D. J.; Brandley, B. K.; Hammer, D. A. Ann. Biomed. Eng.
1996, 24, 87-98.
(12) Brunk, D. K.; Goetz, D. J.; Hammer, D. A. Biophys. J. 1996, 71,
2902-2907.
(13) Vogel, J.; Bendas, G.; Bakowsky, M.; Hummel, G.; Schmidt, R.
R.; Kettmann, U.; Rothe, U. Biochim. Biophys. Acta 1998, 1372, 205.
(14) Andrade, S. Surface and Interfacial Aspects of Biomedical Poly-
mers; Plenum Press: New York, 1985.
(15) Israelachvili, J. N.; Wennerstro¨m, H. Nature 1996, 379, 219.
(16) Harris, J. M. Poly(ethyleneglycol) Chemistry; Plenum Press: New
York, 1992.
(48) Garnet, M. Philos. Trans. R. Soc. London, Ser. A 1904, 203, 385-
420.
(49) Garnet, M. Philos. Trans. R. Soc. London, Ser. A 1906, 205, 237-
288.
(50) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.;
Academic Press: London, 1992.
(17) Blume, G.; Cevc, G. Biochim. Biophys. Acta 1990, 1029, 91.
(18) Kilbanov, A. L.; Maruyama, K.; Torchilin, V. P.; Huang, L. FEBS
Lett. 1990, 268, 235.
(19) Papahadjopoulos, D.; Allen, T. M.; Gabizon, A.; Mayhew, E.;
Matthay, K.; Huang, S. K.; Lee, K.-D.; Woodle, M. C.; Lasic, D. D.;
Redemann, C.; Martin, F. J. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 11460.
(20) Wu, N. Z.; Da, D.; Rudoll, T. L.; Needham, D.; Whorton, A. R.;
Dewhirst, M. W. Cancer Res. 1993, 53, 3765.
(21) Napper, D. H. Polymeric Stability of Colloidal Dispersions;
Academic Press: New York, 1983.
(22) Lee, J. H.; Kopecek, J.; Andrade, S. J. Biomed. Mater. Res. 1989,
23, 351.
(23) Bridgett, M. J.; Davies, M. C.; Dencer, S. P. Biomaterials 1992,
13, 411.
(24) Desai, N. P.; Hossainy, S. F. A.; Hubbell, J. A. Biomaterials 1992,
13, 417.
(25) Yang, Z.; Galloway, J. A.; Yu, H. Langmuir 1999, 15, 8405.
(26) Kuhl, T. L.; Leckband, D. E.; Lasic, D. D.; Israelachvili, J. N.
Biophys. J. 1994, 66, 1479.
(51) Rand, R. P. Annu. ReV. Biophys. Bioeng. 1981, 10, 277-314.
(52) Landau, L. D.; Lifschitz, E. M. Lehrbuch der theoretischen Physik,
Band V, Statistische Physik Teil1, 7th Auflage ed.; Akademie-Verlag:
Berlin, 1987.
(53) Albrecht, O.; Gruler, H.; Sackmann, E. J. Phys. 1978, 39, 301-
313.
(54) Mo¨hwald, H. In Structure and Dynamics of Membranes; Lipowsky,
R., Sackmann, E. H., Ed.; Elsevier: Amsterdam, 1995.
(55) Pallas, N. R.; Pethica, B. A. Langmuir 1985, 1, 509.
(56) Miller, A.; Mo¨hwald, H. J. Chem. Phys. 1987, 86, 4258.
(57) Mouritsen, O. G. Biochim. Biophys. Acta 1983, 731, 217.
(58) Scott, H. L., Jr. Biochim. Biophys. Acta 1975, 406, 329.
(59) Ko¨berl, M.; Hinz, H.-J. R.; G. Chem. Phys. Lipids 1997, 85, 23-
43.
(60) Johann, C.; Garidel, P.; Mennicke, L.; Blume, A. Biophys. J. 1996,
72, 3215-3228.
(61) Goldenfeld, N. Lectures on Phase Transitions and the Renormal-
ization Group; Addiso-Wesley Publishing: Reading, MA, 1992.