Self-Consistent. Field(SCF) theory for grafted polymer layers are compared in detail with the recent. Monte. Carlo simulations using the bond- fluctuation model.
Phys.
J.
II
France
2
(1992)
547-560
1992,
MARCH
PAGE
547
Classification
Physics
Abstracts
36.20
61A0K
Carlo Monte of a polymer Pik-Yin
and E. B.
Physik,
fir
self-consi§tent
theory
field
brush
Lai
Institut
of the
test
Zhul1~la
Universit£t-Mainz,
Johannes-Gutenberg
Postfach
3980,
Mainz,
D-6500
Germany
(RecHved
August 1991, accepted
29
in
final
form 8
1991)
November
Field(SCF) theory for grafted analytic predictions from the Self-Consistent detail with Monte Carlo compared in the simulations using the bondrecent are Quantities describing the equilibrium fluctuation model. of the brush derived structure are Carlo data with no free In most from the SCF theory and compared with the Monte parameter. predictions. Causes for discrepancies the results in agreement with the SCF also cases are are The
Abstract
polymer layers
discussed.
Introduction.
1
understandBecause of their wide applications in polymer technology and their importance in ing the fundamental problems of polymeric materials, grafted polymer layers 11, 2] have been subject of recent interest both theoretically [3 24] and experirnentauy [25 33]. Theoretia cally, analytic Field (SCF) theory [7,10 11,14] gives a very detailed solution Self-Consistent properties of the polymer brush while experiments usually can of the equilibrium structural only provide more global information like the thickness, concentration profile and the force profile between two interacting brushes. On the other hand, detailed information like the polybe chain fluctuations conveniently obtained in simulation trajectory, etc. computer can a mer and compared with the SCF theory. Furthermore, in a simulation the system is precisely chartheories
simulation
results
analytic SCF predictions
the
chains be
used in the
Carlo(MC)
Monte
recent
with
approximations
and
acterised
the
[35
38]
was
used
in the
in
simulations.
by the MC simulations are listed stretching and binary interactions
tested
chain
detail.
(*) Academy
Alexander of
von
Science,
Humboldt
Fellow.
Leningrad
199004,
or
In
this
paper,
In the derived
shall
we
[34] of the polymer brush in bond-fluctuation The model of
between
Permanent U-S-S-R-
avoided.
are
a
address:
The Institute
excluded
volume
ofmacromolecular
solvent
macromolecular
next section, all the predictions using analytic SCF theory with monomers.
compare
good
that
will
Gaussian
parameter Compounds,
JOURNAL
548
the
for
MC
the
observed
idea
The
results
deviations
Analytic
2.
model
bond-fluctuation
Then
ies.
calculated
using
with
SCF
the
N°3
results
independent studpossible causes for the
other
from
predictions
and
SCF.
from
interactions
monomer
are
theory
SCF
of
then
II
discussed.
are
results
is
compared
PHYSIQUE
DE
polymer brushes represented by
for
be
can
the assumption that dependent potential U(z)
based
is a
monomer-
on
position potential
where
z
win volume determine the grafting surface. This monomer fraction #(z) which in turn gives rise to U(z). The equations determining #(z) thus have to stretched observation [12] that grafted chains are be solved self-consistently- One exploits the reduced which partition in the and the number of polymer chain configurations is greatly case minimizes the total free energy. Flucfunction is dominated by the chain configuration that and the is reduced chain be neglected problem path can tuations about this minimal to a one ill- The mechanics problem and U(z) is found to be parabolic [7,10 classical dimensional summarized below, the details of this theory can be analytic results of the SCF theory are
the
is
distance
found
in
from
references
the
[7] and
ill].
Monodispersed polymer chains surface with grafting density I/S of the free per
unit
volume),
then
has
one
are
in
(R()
where
is
the
average
binary
coefficient
of
interaction
square
2
end-to
hence
monomer,
coverage
concentration
the then a
=
the
a~/S,
volume
is
flat
impenetrable the density of
monomer
distance
iii]
of =
~~~
unperturbed,
an
vi~
where
v
is
Gaussian the
chain.
second
virial
obtains
one
profile
2
=
)U(z)
(2)
parabolic [7,10 polymer units, #(z)
also
is
of
=
ii]. If a~ is the volume of a a~$(z). Introducing surface
obtain
we
(R() /(Na~)
a
~ 8iiiN
~°~~~~~~
monomers,
fraction
~(~) where p = dimensional
"
end
I(z) and
to
f[$(z)] is per chain). If $(z) (number concentration
layer with
~~~~
"
between
interactions,
monomer
a
grafted
end
one
(S-area
ill]
~fjj( )j &I(( Assuming
with
monomers
considered
interactions
volume
energy
of N
is the
bond-fluctuation
stiffness model
~
~~~
~2/3(~~~~) 8pv 2 parameter
of
calculated
are
((~ chain.
a
in the
(~ )2j
(~)
h
The
of p and v for the three h is the brush height and
values
appendix,
given by h
The
chain,
"trajectory" is given by
of
the
=
grafted chain,
(~)l"ii'3N
which
is
position
the
(4) of
the
ith
monomer
along
the
[7] ~~
'
~~
~~
2N
~~~
N°3
TEST
MC
probability
The
OF
distribution
SCF
THE
of the free
THEORY
These the
major SCF SCF
above
Using (3), thickness,
predictions
results trivial
it is
will
obtain
to
first ~
j~j z-component
of the
of the
square
iRizi Using (5), (6) and
I,
N »
The
orientation
of the
ith
where
(£~)Q~ I, after
is the
bond
bond
is
some
+i
probability
Finally, the
Z
"
~
,2
between
j
relative
mean
~ ~
ith
one
constant
chains
of the
brush
is
defined
as
181
~~~
square
)j~P~~)2/3p/2
jgj
,2
by (cos R;) which
~~)i/~llz;I
defined
is
as
lz;-i)1
=
along
monomers
p;(z)
monomer,
(1°1
Using (5)
chain.
a
)
also be
can
32
@)PN(ZN)dZ~ displacement of
(h the
ith
sin
jj3 monomer
izii =
and
obtained
(h
can
sin
be
~~
)2
~~
measured
by
j~~j
_~
iz;i~
(11)
gets
)($)~ =
a
,2
~
CDS
)l'~~ is
5
=
iz;i~
which
~
successive
iAzii Using (5) and (6),
of the
~j~
ix ~~ z
use
(~)
~j(jij~l«~vp«)~'~
°;I
~
~"
measure
a
now
simulations.
obtains
of the
distribution
gyration
of
ljl'B l'~jjl
one
ICOS
The
is
We
MC
x2
8
characterised
length
algebra,
which
data.
in the
ij fiz; ( L z;i~i
+
~
ICOS R;)
N »
simulation measured
~(~l~v~jl/3p/
~
radius
j~2N jj~
gz
MC are
gets,
one
jj~2
iii (6)
#(z)
of
moment
8
The
549
$@~
other
the
BRUSH
"
compared with our quantities that
be
the
derive
to
POLYMER
given by [7,10
end is
PN(~)
OF A
independent
of
I, N and
a.
i
=
o.1528
1i4l
JOURNAL
550
3.
Monte
The
MC
model
data
from
in
N
taken
Here
data.
and
monomers
two
no
and
is
there
of
0.02.
2.71~
ci
allow
To
volume
excluded
the
coverage
N°3
take
a
2.
=
a
precise
parameter
v
details from
Thus
when
other
ranges For the
stiffness
a
than
a
@
that
SCF
the the
a
relevant
cubic results
good
in
lattice with solvent
bond length characterised by the in the simulations,
concentration p for
work can chain length
is
MC in
fluctuation
The
and is
to
and
simple
avoidance.
comparison parameter
in
comparing the polymer brush
of
range
0.2
model
sites
self-
from 2 to
bond simulation
to
of the
lattice
consider
[34]. The of the 0.025
features of 8
quantitative and
reference
some
cube
We
interaction
in The
ranged
a
site.
common
a
along a chain monomers the bond length, (£~)~/~
of
root-mean-square
studies
simulations.
briefly mention occupies a monomer
monomer-monomer
no
successive
between
(£~)~/~
one
MC
surface
share should
can
simulation
the these
will
we
Each
predictions,
SCF
in
[34]. The
reference
analysing the the
from
used
was
10 to 80.
=
II
results.
were
38]
[35
found
be
Carlo
PHYSIQUE
DE
theory, the 3-dimensional
values bond-
independently using the results from are can v 3-dimensional bond-fluctuation model as given in the appendix. other studies [37, 38] of the We get v ci 11.55 and p ci 2.7 ~ 0.2 and these values will be used in the subsequent analysis from the results of the present study by a of the data. The product up can also be obtained linear least fit of (z) as a function of Na~/~ as implied in equation (7). As shown in square figure I (upper part), the data do lie nicely on a straight line as predicted in (7). Linear least with the independent fit (dashed line) gives up ci 33 ~ 2 which is in good agreement square Appendix). estimate, ci 31.2 (see model
fluctuation
essential.
be
and p
obtained
50
~4
-
40
,'
,' ,'
30 ,,'
~, ,, 20
~
Q
~ ~
~gA~'
10
f, ~q~
~4
,
0
Fig.
First
I.
dashed
line is
Symbols:
The
Na~/~
a
=
brush in
set of data) (z) data. The solid line is the least square fit for a 0.025(o), o.05(n), o.075(6), o.lo(o), o.15(V)> o.20(*).
moment
of the
density profile (z) (upper the
thickness
figure
I
can
be
(lower part).
characterised The
data
SCF
by (R(z)Q~ whose MC fall
nicely
on
a
(R(z)~'~
and
straight
result
results line
Na~/~. vs.
from
The
equation (8).
plotted against verifying the scaling are
MC
N°3
behavior. line is
Smau calculated
TEST
systematic from
the
2
1. o
B m ,
cq
~
6
h4
~ 2
0.
0
~
OF
THE
deviations
SCF
results
SCF
THEORY
for low in
(9)
OF A
values which
of
show
POLYMER
Na~'~ almost
occur
BRUSH
as
perfect
expected. agreement.
551
The
solid
JOURNAL
552
be
can
h
estimated
from
(2.8~ 0.2)Na~/~
ci
theoretically
[2] to
decay
#(z) This is
#(h)
from
decay
a
#(h)
0 in
=
which is
figure
c
(40, o,05) (*), (30,o.1) (+), (30,o.05) (~). Fig. in
=
1. 5
f
I-D
>.
~
b
fi Q3
$ °
5
0. 0
2
.6
4
il(N
B
1.0
1.2
1)
projection (cos 9;) of the local orientation of bond connecting vector normalized position of the ith bond along the chain. Solid is the curve SCF result from equation (11). Symbols: (N, a) (80, 0.1) (o), (60, 0.2) (n), (60, o-1) (/£), (40,o.1) (o), (40,0,05) (V)> (30,0.2) (*), (30,0.1) (+), (30, 0.075) (~), (30, 0.05) (~). Fig.
7.
monomer
Scaring plot
I
-1
and I
of
vs,
the
the
=
556
JOURNAL
PHYSIQUE II
DE
N°3
25
»
20 cq
fi
~
~
h4 ~ -
*
~
V
~
O
15
-
~
c~pw
~
V
i
~
a
~
a
6
-
°
a
lo
o
a
~
o
o
05 0
2
4
.6
B
1.0
1.2
ilN
a)
25
o
20 m
~
~
f
-
~
o
~
~
15
Q
~__
'4
»
~
j
~
~
~
g
j
6
©
~
~
$
10
05 2
0
Fig. 8. a) Plot (14). Symbols: a chain
length
at
4.
Discussions
The
plausible
a
of =
=
.6
4
ilN
B
0
((bz;)~)/(z;)~ os, ilN with N 30. Solid curve is the 0.2(o), 0.15(n), o.10(/£), 0.075(o), o.05(V), 0.025(*). b) 20(o), 30(a), 40(/£), 60(o), 80(T7). Symbols: N o,05. =
1.2
b) SCF Same
result as
a)
from but
for
equation various
=
conclusion.
and
discrepancies
between the SCF theory and the MC results are (I) (it) correction to the Gaussian elastic flee energy due to the finite of the cllain and higher order terms in the equation of state [10,16,17]. (iii) fluctuations of a chain about the minimal path are important and (iv) corrections to the mean-field description due to strong spatial fluctuations in concentration. Since of our data showed the predicted scaling quite well and in many most the cases finite
chain
causes
length in extensibility
for
the
the
simulations
MC
N°3
TEST
discrepancies do discrepancies found smoothening of the
observed these be the
for
behavior
N
THE
here
of the
too
used
N
profiles).
Our
coverage
considered
of
range
to
length,
chain
short
BRUSH
POLYMER
OF A
increasing
with due
not
are
cutoff
THEORY
SCF
decrease
not
the
20 for
ci
OF
already
data
show
of
most
simulation(except
This
here.
that
believe
we
the
in
557
may
predicted scaling rapid approach to the the
previously in noted model has also been bond-fluctuation scaling regime for the other studies [37, 38]. Gaussian elastic free corrections references [10] and [16], As has been pointed out in to the a~/~ observable deviations h/(Na) Indeed there important is small. when not are energy ci are pN(z) much sharper, #(z) flatter and (cos 9;) shows chains for the a > 0,I (a~/~ > 0.46) data uniformly stretched) from the SCF theory. This correction to Gaussian stretching more are asymptotic
be
can
the
examined
in
nth-moments
#
using
detail
more
of
that
to
MC
our
Dn,
of pN,
data.
defined
It
been
has
shown
the
that
[10]
as
iz"i ~
depends only SCF
which
the
on
form
assumed
of the
elastic
elastic
Gaussian
free free
ji~j
izii
= "
but
energy
equation
the
not
gives Di
energy
of
ratio
"
2/x
of
state
and D2
0.6366
=
used. "
Analytic
1/2.
The
for Di and D2 are shown in figure 9 as a function as a a. increases, Di and D2 decrease slowly and deviations from the Gaussian stretching assumption become significant when a > 0.I as expected. The decrease of Dn with increasing a is expected I/(n + I) from a simple calculation. since at a I, one gets Dn interactions At larger the brush at the time is denser and higher order are coverage, same concentration important. The self-consistent potential is no longer proportional to the local that and the full form of the equation of state should be used [10,16,17]. However note we though the shape of the distribution functions differ appreciably from the SCF theory at higher coverage, their peak values are still quite close to the SCF predictions. As a result, first like (z) and the average trajectory (z;) are well described by the SCF results even moments
data
MC
of
=
to
up
a
mizes
indeed
is
very
in
most
that
=
analytic
the
is
indicate
data
0.2.
=
the
In
The
SCF
'action'.
the low
case.
and
cases
treatment, is
The
exception
valid
the
when
is of
one
chains
deviation
from
the
dominating stretched.
are
at
course
conformations
chain
numerous
strongest
the
assumes
one
This
the
end
to
prediction
where
the
partition
the is
that
mini-
part of the brush,
most
brush
of the
contributes SCF
configuration
chain For
observed
And
sum. near
this
concentration the
indeed
brush
end.
where discrepancies arising from this should disappear in the numerical SCF calculations considered. all possible chain configurations are Finally there is a correction to the mean-field picture which the interacassumes monomer mean-field tions can be represented by a background potential U(z). It is known [39] that such dilute where spatial density are model would fail for the very fluctuations of case monomer in most part of the brush is quite high and the interactions large. The concentration monomer represented slow-varying background is near be safely by potential. Again the exception can a The
the
brush
end
where
surprising
that
the
brush
end
since
concentration
agreement
(iii)
both
and
is low
and
analytic
with
the
(iv)
contribute
fluctuation
SCF
effects
prediction
corrections.
strong.
are
is
However
least we
Thus
satisfactory should point
it is near
out
not
the that
major picture of the SCF predictions. In view of the fact that no free parameter is used in fitting the results to the theory, and all the MC results be can described quite well by the theory (not only the scaling dependences but also the prefactors), find the analytic SCF theory gives a good description of the system. we found that the SCF theory gives an prediction for the detailed To conclude, we accurate of the polymer brush. The prediction is particular good over the entire structure coverage
all
these
deviations
do
not
alter
the
JOURNAL
558
PHYSIQUE
DE
II
N°3
a
4 Q
2
0.
0
cr
Fig.
Di (upper data) and D2 (lower data)
9.
dashed
fines
range
considered
observed
the
are
SCF
higher
for N
a
20(o), 30(n), 40(/£), 60(o), 80(T7).
=
free
elastic
The
energy.
deviations More prominent thickness and the average chain trajectory. fluctuations about end are probably due to strong chain configuration behavior. Other observable discrepancies and correction to the mean-field correction due to higher order in the equation of state and to terms are extension of the chain. free energy for the finite
coverages elastic
Gaussian
the
os,
Gaussian
brush
the most-likely path at
the
for the
the
near
using
results
Appendix. In this appendix, independent studies
lations
of
where
lI is the
neighbor The
hard
coefficients
50(
v(2)
=
well
by
model
and
B;'s
core
(good
and
pressure
data).
MC
the
calculate
are
I
solvent
exclusion,
Thus
using ci
the
11.55.
case)
fraction
is the
the
virial
parameters
of the
and p
v
using the model
values
their
value
82
=
13.504
"
of
occupied
the
MC
from simu-
3-dimensional
up
to
I
simple 7
=
can
3)l~131~ [40],
we
dual
cubic be
[38] for N shown that it it is 3, >
reference
lattice (a~i = 8i = # lattice gas with thirdreference [40]. obtained in
sites in the
of the
enumerated
~13) + IN
from
for
results
and
is [38]_
coefficients
v(N) have been exactly v(3)=73.9229. And for N ~lN)
to be
value
the
Bond-fluctuation equation of state [38] of the bulk polymer system [37]. The equation of state
concentrated
bond-fluctuation
in
we
of the
=
1,2,3 can
be
27, to give v(1) approximated quite =
lA2)
~12)1
obtain
the
second
virial
coefficient
v
N°3
TEST
MC
The
(with
estimation N up
THE
parameter
stiffness
of
THEORY
SCF
in
POLYMER
BRUSH
simulation results MC data for the most dense system taken. concentrated For such a
The
expect
data,
MC
these
estimated
we
p
ci
lRil
#
559
extensive
the
on
solution.
reference
lR~l From
based
p is
polymer
concentrated
OF A
[3'7] (corresponding to # = 0.5) were that the square of the end-to-end distances of chains, can one close to those in melt state in which Gaussian behavior is expected, I-e-
investigated system, very
of the
200) [37]
to
OF
(R~),
already
are
PNa~
=
lA3)
2.7 + 0.2.
Acknowledgements. This
grant
research no.
reading
the
One of
Foundation.
boldt
Bundesministerium fur Forschung und Technolgie (BMFT) (E.B.Z.) acknowledges Alexander Humsupport from the us von grateful to A. Halperin for discussions and K. Binder for a critical
supported by
is
03M4040.
We
are
manuscript.
of the
References
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[11] Zhulina
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Pavlushkov
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21
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E.
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V.,
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[12] Semenov A. N., Soy. Phys. JEPT 61 (1985) 733. Macromolecules M, and Ho S. 22 (1989) 965. [13] Muthukumar Macromolecules 22 (1989) 853; [14] Miner S., Witten T. and Cates M., Macromolecules Miner S., Wang Z.-G. and Witten T., 22 (1989) 489. [15] Birshtein T. M., Liatskaya Yu V, and Zhufina E. B.,Polymer 31 (1990) 2185. [16] Shim D. F. K. and Cates M. E., J. Phvs. France 50 (1989) 3535. [17] Lai P.-Y, and Halperin A. Macromolecules, in press. submitted. [18] Marko J. F. and Witten T. A., Phvs. Rev. Lett. 66 (1991) 1541; Macromolecules, Macromolecules Alexander S., Europhvs. Lett. 6 (1988) 329; 22 (1989) 2403 [19] Halperin A. and 23 (1991). [20] Klushin L. I. and Skvortscv A. M., Macromolecules Macromolecules 22 (1989) 4054. [21] Murat M. and Grest G. S., [22] Murat M. and Grest G. S., Phvs. Rev. Lett., 63 (1989) 1074. Macromolecules A. and Toral R., 23 (1990) 2016. [23] Chakrabarti
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560
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II
N°3
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(Adam Hilger, Bristol) to be published. Macromolecules I, and Kremer K., 21 (1988) 2819; J. Phvs. H.-P, and Binder K., J. Chem. Phvs. 94 (1991) 2294.
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J.
Chem.
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Rochester
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p.
G.,
Alder
B. J,
and
Ree F.
H., J.
Chem.
Phvs.
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(1954)
3528.
Press,
Ithaca,
NY
Revue
livres
de
physique
Hydrodynamique GUYON,
E.
HULIN,
J.-P.
Collection
Savoirs
L.
PETIT
(InterEditions,
actuels
Editions
1991).
CNRS,
du
et
Fluides J.
dcoulement,
en
mdthodes
modkles
et
PADET
Enseignement
Collection Les
programmes
Commenqons (es
auteurs
par le affichent
de
Physique
la
propos£s par premier (GHP volont6
une
suite).
la
s'adresser
de
livres
deux
ces
dans
1990) 368 p.,
(Masson,
h des
prime
de
sont
Tout
en
abord
r£cusant
ne
physiciens,
180 F.
pas
plutbt
compl£mentaires.
l'influence
des
explicitement
faisant
m6caniciens,
r£f£rence
h
I'£cole
anglaise et h P. G. de Gennes qui, h ma l'honneur connaissance, mis h France le terme a en «Hydrodynamique Physique» dans collage de France milieu des ann£es70. II son cours au au logiquement par un chapitre sur la physique des fluides, d£crivant de faqon simple leurs commence propridt£s fondamentales, des propri£t6s macroscopiques, diffusion effets de surface, h et transport, leur microscopique. GHP abordent ensuite notion viscosit£, origine microscopique la de structure son (gaz et liquides), et les diff£rents r£gimes d'£coulement derri~re cylindre, prdtexte h la pr£sentation un du modble de Landau d'une bifurcation dl£mentaire puis, dans un trop bref paragraphe, de la transition h
turbulence.
la
A
part
une
allusion
autre
et
courte
une
annexe
sur
le
chaos,
ce
sera
tout
sur
la
large pan de l'hydrodynamique, bien pour lequel il faut avouer que les physiciens pourraient apporter plus dans le futur, aurait sans doute m£rit£ mieux. De toute faqon, dbs le chapitre 3, GHL rejoignent le g£n£rat des hydrodynamiciens, d£crivant le des fluides, courant mouvement cin£matique (C3), dynamique (C4), lois de conservation (C5), £coulements potentiels (C6), vorticit£ (C7). Les £coulements (foible nombre de Reynolds) et les couches limites laminaires font rampants « ensuite l'objet chacun d'un chapitre, le livre se terminant Etude sommaire des instabilit£s sur une hydrodynamiques et une Evocation des propri£t£s de l'h£lium superfluide. Le plan adopt£ dans le second livre (JP dans la suite) est int£ressant. d£matte classiquement Le livre l'hydrodynamique et leur les Equations de ddrivation (Cl), mais passe tout de suite h la notion de sur similitude (C2) et situe d'emblde le problbme majeur pos£ par la turbulence (C3) avant d'examiner les diff£rents d'£coulement int£ressent qui l'ing£nieur (typiquement, au voisinage d'une types extemes plaque), laminaires (C4) puis turbulents (C5), intemes (typiquement, dans conduite) niveau une au th£orique (C6) puis pratique (C7). A l'intersection des deux naturellement les Equations de l'hydrodynamique, trouve programmes, on turbulence.
Ce
»
mais sous
aussi un
les
dimensions
sans
situations
GHP II
est
comme
plus
de
Press, ici.
sun
la
A
de
une
dans
le
les
traitement deux
de la
livres
couche
limite
laminaire.
La
similitude
est
vue
GHP
l'inverse
(cf. le
Physical
Dynamics
(2nde ddition, » franqais » du livre de l'auteur anglais, les auteurs calcul analytique par ont du mat h remplacer un Naturellement, appr6cie explications (dynamique qh et16 les semi-qualitatives on solitaires...) mais mdme, l'argument rejoint trop vite l'approche de tout chapitre sur les £coulements potentiels). Enfin, si je me r£fbre h mes du notes
comparer
physique. suspensions, ondes
conventionnelle
et
pr£sente simplement (es diff£rents nombres d'application h des rapports de flux, tandis que JP donne plus de d6tails en vue L'analyse math£matique de la couche limite suit des lignes trds paralldles mais signification physique des approximations permettant de comprendre la structure
1988). On
analyse
des
similitude
vitesse.
tentant
Clarendon examin£
de
industrielles.
insiste
profil
du
notions
angle plutbt physique
GHP
constate
au
livre
alors
de
Tritton,
rapidement
le
«
caractdre
Fluid
finalement
trds
«
JOURNAL
562
de
de
cours
dean
Tout
h
Palais
de
r£f£rences illustrant
la
livre
de
dans
s6rie
Tritton,
de
films,
D£couverte
h
diff£rentes
certains
et
«
de
An
Par
leur
diverses de
contre, livre
elles
National of
plus
en
Committee
les
encore
auteurs
restent
JP, si
dans l'on
»
Fluid £dit6
Enfln «
Pour
de
tr~s
en
M.
Van
appr£ciera
de
nombreuses
Experiments in comp16ment d'une
Illustrated
«
Mechanics
par
on
excepte
explicafives,
figures
des
notamment
for
Motion
sources.
Walker pour
lh
que
tirdes
sont
Fluid
autres
J.
trouve
N° 3
II
ouvrage.
comporte,
Ici,
Album
rubriques
passages.
de
GHP
pddagogique. produit par le
»
1n6dit), je
rest£
titre
le
vocation
Mechanics
remarquable
inscrite
le
comme
illustrations Fluid
(malheureusement
Gennes
l'ambition
de
PHYSIQUE
DE
en
Dyke,
aussi
que
des GHP
documents
du
donnent
les
d£crivant exp6riences des humoristiques particulibrechapitres, les symboles « trdfle, carreau, sous-paragraphes, les signaux «virage la
les
Science
»
citations
(Achille Talon !) et des du en exergue pique», soit certaines Equations soit des marquant stationnement dangereux » pour les points ddlicats ou « interdit » pour les sauter passages que l'on peut premidre lecture, l'ouvrage m'a laiss£ impression d'aust£rit£ art£nu6 n'a la pr£sence, h la en une que pas r£sum£ forme d'encadr£ off sont regroup£es les formules fin de chaque chapitre, d'un essentielles. sous (On notera que, bien qu'ils soient issus de cours donn6s h dirt£rents niveaux, livres ne des deux aucun d'approfondir les sujets abordds.) comporte de s£ries de probldmes permettant deux diff£rentes, et malgr£ les r£serves Pour conclure, m'apparaissent, raisons ces ouvrages pour des contributions exprimdes, des majeures, en langue franqaise, au domaine de l'hydrodynamique. Its sont susceptibles d'aider bien dvidemment l'enseignant et I'£tudiant de cette matidre, mais aussi d'un point chercheur l'ing6nieur de vue plus g£n£rat, le qui a besoin de comprendre la physique des dcoulements ou auxquels il peut dtre confront£. ment
cceur,
bienvenues
P.
MANNEV~LE
(Saclay).
